---------------------------------------- -------------------------------- ,transcrib]'s ,note3 ,?is v]sion ( ! text _c repres5t c]ta9 typographical effects4 ,italics are delimit$ ) ! #8' "* z italic4 ,?is is a v visual "w1 ) num]\s diagrams & meticul\s discussion ( !m4 ,! ,,html or e,book v]sions w\ld be m suitable if ^? aspects ( ! "w are ( int]e/4 ,!re is a s+le footnote foll[+ ! ,translator's note4 ,if an article spans sev]al pages1 m>g9al p>agraph notes 7 repeat$4 ,^! r$undant notes h be5 remov$4 ,m9or errors1 attributable to ! pr9t]1 h be5 correct$4 ,please see ! transcrib]'s note at ! end ( ?is text = details reg>d+ ! h&l+ ( any textual issues enc\nt]$ dur+ its prep>ation4 ,! ,engli% ,works ( ,?omas ,hobbes3 ,volume #a ,illu/ration3 ,,?omas ,,hobbes4 ,w ,humphrys sc4 ,london4 ,publi%$ ,m>* #a/ #ahci1 by ,john ,bohn4 #ag1 ,h5rietta ,/reet1 ,cov5t ,g>d54 ,,! ,,engli% ,,works ,,( ,,?omas ,,hobbes ,,( ,,malmesbury2 ,,n[ ,,fir/ ,,collect$ ,,& ,,edit$ ,,by ,,sir ,,william ,,moleswor?1 ,,b>t4 --- ,,vol4 ,i4 --- ,,london3 ,,john ,,bohn1 ,,h5rietta ,,/reet1 ,,cov5t ,,g>d54 --- ,,mdcccxxxix4 ,,london3 ,c4 ,,ri*>ds1 ,,pr9t]1 ,,/4 ,,m>t9',s ,,lane4 ,,to ,,george ,,grote1 ,,esq4 ,m4,p4 ,,= ,,! ,,city ,,( ,,london4 -------------- ,,de> ,,grote1 ,i d$icate to y ?is edition ( ! ,works ( ,hobbes2 fir/1 because ,i "k y w be well pleas$ to see a complete collection ( all ! writ+s ( an ,au?or = :om y h s high an admiration4 ,secondly1 because ,i am indebt$ to y = my fir/ acqua9tance ) ! speculations ( "o ( ! greate/ & mo/ orig9al ?9k]s 9 ! ,engli% language1 ^: works1 ,i h (t5 he>d y regret1 7 s sc>ce1 & s mu* less read & /udi$ ?an !y des]v$ to be4 ,x n[1 !re=e1 gives me great satisfaction to be able to gratify a wi%1 y h frequ5tly express$1 t "s p]son1 :o _h "t & due rev]5ce = t illu/ri\s man1 w\ld und]take to edite 8 works1 & br+ 8 views aga9 be=e 8 c\ntrym51 :o h s long & s unju/ly neglect$ him4 ,& likewise1 ,i am desir\s1 9 "s way1 to express ! s9c]e reg>d & respect t ,i feel = y1 & ! gratitude t ,i owe y = ! valuable in/ruction1 t ,i h obta9$ f y\r society1 & f ! fri5d%ip ) :i* y h hon\r$ me1 dur+ ! _m ye>s we h be5 companions 9 political life4 ,y\rs1 truly1 ,,william ,,moleswor?4 ,febru>y #be?1 #ahci4 #gi1 ,eaton ,squ>e1 ,london4 ---------------------------------------- -------------------------------- ,,elem5ts ,,( ,,philosophy4 ,,! ,,fir/ ,,section1 ,,conc]n+ ,,body1 ,,writt5 ,,9 ,,lat9 ,,by ,,?omas ,,hobbes ,,( ,,malmesbury ,,& ,,translat$ ,,into ,,engli%4 ,,! ,,translator ,,to ,,! ,,read]4 ,if1 :5 ,i _h f9i%$ my translation ( ?is fir/ section ( ! ,elem5ts ( ,philosophy1 ,i _h pres5tly committ$ ! same to ! press1 x might h come to y\r h&s soon] ?an n[ x do?4 ,b z ,i und]took x ) mu* diffid5ce ( my own ability to p]=m x well2 s ,i ?\ght fit1 be=e ,i publi%$ x1 to pray ,mr4 ,hobbes to view1 correct1 & ord] x accord+ to 8 own m9d & pleasure4 ,:]e=e1 ?\gh y f9d "s places enl>g$1 o!rs alt]$1 & two *apt]s1 ,,xviii & ,,xx1 almo/ :olly *ang$1 y may nev]!less rema9 assur$1 t z n[ ,i pres5t x to y1 x do? n at all v>y f ! au?or's own s5se & mean+4 ,z = ! ,six ,lessons to ! ,savilian ,pr(essors at ,ox=d1 !y are n ( my translation1 b 7 writt51 z "h y h !m 9 ,engli%1 by ,mr4 ,hobbes himself2 & are jo9$ to ?is book1 because !y are *iefly 9 def5ce ( ! same4,a ,footnote ,a3 ,!y w be publi%$ 9 a sep>ate volume1 ) o!r works ( a simil> description4 ,w4 ,m4 ,,! ,,au?or',s ,,epi/le ,,d$icatory1 ,,to ,,! ,,"r ,,honorable1 ,,my ,,mo/ ,,honor$ ,,"l1 ,,william1 ,,e>l ,,( ,,devon%ire4 ------- ,?is fir/ section ( ! ,elem5ts ( ,philosophy1 ! monum5t ( my s]vice & y\r ,lord%ip's b\nty1 ?\gh1 aft] ! ,?ird ,section publi%$1 long def]r$1 yet at la/ f9i%$1 ,i n[ pres5t1 my mo/ excell5t ,"l1 & d$icate to y\r ,lord%ip4 ,a little book1 b full2 & great en\gh1 if m5 c\nt well = great2 & to an att5tive read] v]s$ 9 ! demon/rations ( ma!maticians1 t is1 to y\r ,lord%ip1 cle> & easy to und]/&1 & almo/ new ?r\gh\t1 )\t any (f5sive novelty4 ,i "k t t "p ( philosophy1 :]e9 are consid]$ l9es & figures1 has be5 deliv]$ to u notably improv$ by ! anci5ts2 & )al a mo/ p]fect patt]n ( ! logic by :i* !y 7 enabl$ to f9d out & demon/rate su* excell5t !orems z !y h done4 ,i "k also t ! hypo!sis ( ! e>?'s diurnal motion 0 ! inv5tion ( ! anci5ts2 b t bo? x1 & a/ronomy1 t is1 cele/ial physics1 spr++ up toge!r ) x1 7 by succe$+ philosoph]s /rangl$ ) ! sn>es ( words4 ,& !re=e ! beg9n+ ( a/ronomy1 except obs]vations1 ,i ?9k is n to be d]iv$ f f>!r "t ?an f ,nicolaus ,cop]nicus2 :o 9 ! age next prec$+ ! pres5t reviv$ ! op9ion ( ,py?agoras1 ,ari/>*us1 & ,philolaus4 ,aft] him1 ! doctr9e ( ! motion ( ! e>? be+ n[ receiv$1 & a difficult "q !reupon aris+ conc]n+ ! desc5t ( heavy bodies1 ,galileus 9 our "t1 /riv+ ) t difficulty1 0 ! fir/ t op5$ to u ! gate ( natural philosophy univ]sal1 :i* is ! k ( ! nature ( motion4 ,s t nei!r c ! age ( natural philosophy be reckon$ high] ?an to him4 ,la/ly1 ! sci5ce ( man's body1 ! mo/ pr(itable "p ( natural sci5ce1 0 fir/ discov]$ ) admirable sagacity by our c\ntryman ,doctor ,h>vey1 pr9cipal ,physician to ,k+ ,james & ,k+ ,*>les1 9 8 books ( ! ,motion ( ! ,blood1 & ( ! ,g5]ation ( ,liv+ ,creatures2 :o is ! only man ,i "k1 t conqu]+ envy1 ha? e/abli%$ a new doctr9e 9 8 life-"t4 ,be=e ^!1 !re 0 no?+ c]ta9 9 natural philosophy b e man's exp]im5ts to himself1 & ! natural hi/ories1 if !y may be call$ c]ta91 t are no c]ta9] ?an civil hi/ories4 ,b s9ce ^!1 a/ronomy & natural philosophy 9 g5]al h1 = s little "t1 be5 extraord9>ily advanc$ by ,joannes ,kepl]us1 ,petrus ,gass5dus1 & ,m>9us ,m]s5nus2 & ! sci5ce ( human bodies 9 special by ! wit & indu/ry ( physicians1 ! only true natural philosoph]s1 especially ( our mo/ le>n$ m5 ( ! ,college ( ,physicians 9 ,london4 ,natural ,philosophy is !re=e b "y2 b ,civil ,philosophy yet mu* y\ng]1 z be+ no old] 7,i say x provok$1 & t my detractors may "k h[ little !y h wr\ght ^u me7 ?an my own book ,de ,cive4 ,b :at8 7 !re no philosoph]s natural nor civil among ! anci5t ,greeks8 ,!re 7 m5 s call$2 witness ,lucian1 by :om !y are d]id$2 witness div]s cities1 f :i* !y h be5 (t5 by public edicts bani%$4 ,b x foll[s n t !re 0 philosophy4 ,!re walk$ 9 old ,greece a c]ta9 phantasm1 = sup]ficial gravity1 ?\gh full )9 ( fraud & fil?1 a little l philosophy2 :i* unw>y m51 ?9k+ to be x1 adh]$ to ! pr(essors ( x1 "s to "o1 "s to ano!r1 ?\gh !y disagre$ among !mselves1 & ) great sal>y put _! *ildr5 to !m to be taught1 in/ead ( wisdom1 no?+ b to dispute1 &1 neglect+ ! laws1 to det]m9e e "q accord+ to _! own fancies4 ,! fir/ doctors ( ! ,*ur*1 next ! ,apo/les1 born 9 ^? times1 :il/ !y endeav\r$ to def5d ! ,*ri/ian fai? aga9/ ! ,g5tiles by natural reason1 began also to make use ( philosophy1 & ) ! decrees ( ,holy ,scripture to m+le ! s5t5ces ( hea!n philosoph]s2 & fir/ "s h>mless ones ( ,plato1 b aft]w>ds also _m fooli% & false ones out ( ! physics & metaphysics ( ,ari/otle2 & br++ 9 ! enemies1 betray$ unto !m ! citadel ( ,*ri/ianity4 ,f t "t1 in/ead ( ! wor%ip ( ,god1 !re ent]$ a ?+ call$ s*ool div9ity1 walk+ on "o foot firmly1 :i* is ! ,holy ,scripture1 b halt$ on ! o!r rott5 foot1 :i* ! ,apo/le ,paul call$ va91 & might h call$ p]nici\s philosophy2 = x ha? rais$ an inf9ite numb] ( controv]sies 9 ! ,*ri/ian _w conc]n+ religion1 & f ^? controv]sies1 w>s4 ,x is l t ,empusa 9 ! ,a!nian comic poet1 :i* 0 tak5 9 ,a!ns = a gho/ t *ang$ %apes1 hav+ "o braz5 leg1 b ! o!r 0 ! leg ( an ass1 & 0 s5t1 z 0 believ$1 by ,hecate1 z a sign ( "s approa*+ evil =tune4 ,aga9/ ?is ,empusa ,i ?9k !re _c be inv5t$ a bett] exorcism1 ?an to di/+ui% betwe5 ! rules ( religion1 t is1 ! rules ( hon\r+ ,god1 :i* we h f ! laws1 & ! rules ( philosophy1 t is1 ! op9ions ( private m52 & to yield :at is due to religion to ! ,holy ,scripture1 & :at is due to philosophy to natural reason4 ,& ?is ,i %all d1 if ,i b h&le ! ,elem5ts ( ,philosophy truly & cle>ly1 z ,i endeav\r to d4 ,!re=e hav+ 9 ! ,?ird ,section1 :i* ,i h publi%$ & d$icat$ to y\r ,lord%ip1 long s9ce r$uc$ all p[] ecclesia/ical & civil by /rong argum5ts ( reason1 )\t repugnance to ,god's ^w1 to "o & ! same sov]eign au?ority2 ,i int5d n[1 by putt+ into a cle> me?od ! true f\ndations ( natural philosophy1 to fright & drive away ?is metaphysical ,empusa2 n by skirmi%1 b by lett+ 9 ! light ^u h]4 ,= ,i am confid5t1 if any confid5ce ( a writ+ c proce$ f ! writ]'s fe>1 circumspection1 & diffid5ce1 t 9 ! ?ree =m] p>ts ( ?is book all t ,i h said is suffici5tly demon/rat$ f def9itions2 & all 9 ! f\r? "p f suppositions n absurd4 ,b if !re appe> to y\r ,lord%ip any?+ less fully demon/rat$ ?an to satisfy e read]1 ! cause 0 ?is1 t ,i pr(ess$ to write n all to all1 b "s ?+s to geometricians only4 ,b t y\r ,lord%ip w be satisfi$1 ,i _c d\bt4 ,!re rema9s ! second section1 :i* is conc]n+ ,man4 ,t "p !re(1 :]e ,i h&le ! ,optics1 conta9+ six *apt]s1 toge!r ) ! tables ( ! figures belong+ to !m1 ,i h already writt5 & engrav5 ly+ by me above ^! six ye>s4 ,! re/ %all1 z soon z ,i c1 be add$ to x2 ?\gh by ! contumelies & petty injuries ( "s unskilful m51 ,i "k already1 by exp]i5ce1 h[ mu* great] ?anks w be due ?an paid me1 = tell+ m5 ! tru? ( :at m5 are4 ,b ! bur!n ,i h tak5 on me ,i mean to c>ry ?r\gh2 n /riv+ to appease1 b r to rev5ge myself ( envy1 by encreas+ x4 ,= x cont5ts me t ,i h y\r ,lord%ip's fav\r1 :i*1 be+ all y require1 ,i ackn[l$ge2 & = :i*1 ) my pray]s to ,almighty ,god = y\r ,lord%ip's safety1 ,i %all1 to my p[]1 be always ?ankful4 ,y\r ,lord%ip's mo/ humble s]vant1 ,,?omas ,,hobbes4 ,london1 ,april #bc1 #afee4 ,,! ,,au?or',s ,,epi/le ,,to ,,! ,,read]4 ,?9k n1 ,c\rte\s ,read]1 t ! philosophy1 ! elem5ts :]e( ,i am go+ to set 9 ord]1 is t :i* makes philosoph]s' /ones1 nor t :i* is f\nd 9 ! metaphysic codes2 b t x is ! natural reason ( man1 busily fly+ up & d[n among ! creatures1 & br++ back a true report ( _! ord]1 causes & effects4 ,philosophy1 !re=e1 ! *ild ( ! _w & y\r own m9d1 is )9 y\rself2 p]haps n fa%ion$ yet1 b l ! _w its fa!r1 z x 0 9 ! beg9n+1 a ?+ confus$4 ,d1 !re=e1 z ! /atu>ies d1 :o1 by hew+ (f t :i* is sup]flu\s1 d n make b f9d ! image4 ,or imitate ! creation3 if y w be a philosoph] 9 good e>ne/1 let y\r reason move ^u ! deep ( y\r own cogitations & exp]i5ce2 ^? ?+s t lie 9 confusion mu/ be set asund]1 di/+ui%$1 & e "o /amp$ ) its own "n set 9 ord]2 t is to say1 y\r me?od mu/ resemble t ( ! creation4 ,! ord] ( ! creation 01 light1 di/9ction ( day & night1 ! firmam5t1 ! lum9>ies1 s5sible creatures1 man2 &1 aft] ! creation1 ! comm&m5t4 ,!re=e ! ord] ( contemplation w be1 reason1 def9ition1 space1 ! />s1 s5sible quality1 man2 & aft] man is gr[n up1 subjection to comm&4 ,9 ! fir/ "p ( ?is section1 :i* is entitl$ ,logic1 ,i set up ! light ( reason4 ,9 ! second1 :i* ha? = title ! ,gr\nds ( ,philosophy1 ,i di/+ui% ! mo/ common notions by accurate def9ition1 = ! avoid+ ( confusion & obscurity4 ,! ?ird "p conc]ns ! expansion ( space1 t is ,geometry4 ,! f\r? conta9s ! ,motion ( ! ,/>s1 toge!r ) ! doctr9e ( s5sible qualities4 ,9 ! second section1 if x please ,god1 %all be h&l$ ,man4 ,9 ! ?ird section1 ! doctr9e ( ,subjection is h&l$ already4 ,?is is ! me?od ,i foll[$2 & if x l y1 y may use ! same2 = ,i d b prop\nd1 n comm5d to y any?+ ( m9e4 ,b :atsoev] %all be ! me?od y w l1 ,i w\ld v fa9 comm5d philosophy to y1 t is to say1 ! /udy ( wisdom1 = want ( :i* we h all suff]$ mu* damage lately4 ,= ev5 !y1 t /udy weal?1 d x out ( love to wisdom2 = _! treasures s]ve !m b = a look+-glass1 :]e9 to behold & contemplate _! own wisdom4 ,nor d !y1 t love to be employ$ 9 public bus9ess1 aim at any?+ b place :]e9 to %[ _! wisdom4 ,nei!r d voluptu\s m5 neglect philosophy1 b only because !y "k n h[ great a pleasure x is to ! m9d ( man to be ravi%$ 9 ! vigor\s & p]petual embraces ( ! mo/ beaute\s _w4 ,la/ly1 ?\gh = no?+ else1 yet because ! m9d ( man is no less impati5t ( empty "t ?an nature is ( empty place1 to ! end y be n =c$ = want ( :at to d1 to be tr\blesome to m5 t h bus9ess1 or take hurt by fall+ into idle company1 b h some:at ( y\r own :]e) to fill up y\r "t1 ,i recomm5d unto y to /udy philosophy4 ,f>ewell4 ,t4 ,h4 ,,titles ,,( ,,! ,,*apt]s4 ,,"p ,,fir/1 ,,or ,,logic4 ,,*ap4 ,,page4 #a4 ,( ,philosophy #a #b4 ,( ,names #ac #c4 ,( ,proposition #bi #d4 ,( ,syllogism #dd #e4 ,( ,err+1 ,falsity1 & ,captions #ee #f4 ,( ,me?od #fe ,,"p ,,second1 ,,or ,,! ,,fir/ ,,gr\nds ,,( ,,philosophy4 #g4 ,( ,place & ,"t #ia #h4 ,( ,body & ,accid5t #aja #i4 ,( ,cause & ,effect #abj #aj4 ,( ,p[] & ,act #abg #aa4 ,( ,id5tity & ,diff]5ce #acb #ab4 ,( ,quantity #ach #ac4 ,( ,analogism1 or ! ,same ,proportion #add #ad4 ,( ,/rait & ,crook$1 ,angle & ,figure #agf ,,"p ,,?ird1 ,,( ,,! ,,proportions ,,( ,,motions ,,& ,,magnitudes4 #ae4 ,( ! ,nature1 ,prop]ties1 & div]s #bjc ,consid]ations ( ,motion & ,endeav\r #af4 ,( ,motion ,accel]at$ & ,uni=m1 & ( ,motion #bah by ,conc\rse #ag4 ,( ,figures ,defici5t #bdf #ah4 ,( ! ,equation ( ,/rait ,l9es ) ! ,crook$ #bfh ,l9es ( ,p>abolas1 & o!r ,figures made 9 imitation ( ,p>abolas #ai4 ,( ,angles ( ,incid5ce & ,reflection1 equal by #bgc ,supposition #bj4 ,( ! ,dim5sion ( a ,circle1 & ! ,division ( #bhg ,angles or ,ar*es #ba4 ,( ,circul> ,motion #cag #bb4 ,( o!r ,v>iety ( ,motions #ccc #bc4 ,( ! ,c5tre ( ,equipond]ation ( ,bodies #cej press+ d[nw>ds 9 ,/rait ,p>allel ,l9es #bd4 ,( ,refraction & ,reflection #cgd ,,"p ,,f\r?1 ,,( ,,physics1 ,,or ,,! ,,ph5om5a ,,( ,,nature4 #be4 ,( ,s5se & ,animal ,motion #chg #bf4 ,( ! ,_w & ( ! ,/>s #daj #bg4 ,( ,light1 ,heat1 & ( ,col\rs #dde #bh4 ,( ,cold1 ,w9d1 ,h>d1 ,ice1 ,re/itution ( ,bodies #dff b5t1 ,diaphan\s1 ,lightn+ & ,?und]1 & ( ! ,heads ( ,riv]s #bi4 ,( ,s\nd1 ,od\r1 ,sav\r1 & ,t\* #dhe #cj4 ,( ,gravity #ejh ---------------------------------------- -------------------------------- ,,computation ,,or ,,logic4 ------- ,,*apt] ,i4 ,,( ,,philosophy4 #a4 ,! ,introduction4--#b4 ,! ,def9ition ( ,philosophy expla9$4--#c4 ,ratioc9ation ( ! ,m9d4--#d4 ,prop]ties1 :at !y are4--#e4 ,h[ ,prop]ties are kn[n by ,g5]ation1 & contr>ily4--#f4 ,! ,scope ( ,philosophy4--#g4 ,! ,utility ( x4--#h4 ,! ,subject4--#i4 ,! ,p>ts ( x4--#aj4 ,! ,epilogue4 ,sid5ote3 ,introduction4 ,philosophy seems to me to be among/ m5 n[1 9 ! same mann] z corn & w9e are said to h be5 9 ! _w 9 anci5t "t4 ,= f ! beg9n+ !re 7 v9es & e>s ( corn gr[+ "h & !re 9 ! fields2 b no c>e 0 tak5 = ! plant+ & s[+ ( !m4 ,m5 liv$ !re=e ^u acorns2 or if any 7 s bold z to v5ture ^u ! eat+ ( ^? unkn[n & d\btful fruits1 !y did x ) dang] ( _! heal?4 ,9 l mann]1 e man br\ght ,philosophy1 t is1 ,natural ,reason1 into ! _w ) him2 = all m5 c reason to "s degree1 & conc]n+ "s ?+s3 b :]e !re is ne$ ( a long s]ies ( reasons1 !re mo/ m5 w&] out ( ! way1 & fall into error = want ( me?od1 z x 7 = want ( s[+ & plant+1 t is1 ( improv+ _! reason4 ,& f h5ce x comes to pass1 t !y :o cont5t !mselves ) daily exp]i5ce1 :i* may be lik5$ to fe$+ ^u acorns1 & ei!r reject1 or n mu* reg>d philosophy1 are commonly e/eem$1 & are1 inde$1 m5 ( s\nd] judgm5t ?an ^? :o1 f op9ions1 ?\gh n vulg>1 yet full ( unc]ta9ty1 & c>elessly receiv$1 d no?+ b dispute & wrangle1 l m5 t are n well 9 _! wits4 ,i confess1 inde$1 t t "p ( philosophy by :i* magnitudes & figures are comput$1 is highly improv$4 ,b because ,i h n obs]v$ ! l advancem5t 9 ! o!r p>ts ( x1 my purpose is1 z f> =? z ,i am able1 to lay op5 ! few & fir/ ,elem5ts ( ,philosophy 9 g5]al1 z s _m se$s f :i* pure & true ,philosophy may h]eaft] spr+ up by little & little4 ,i am n ignorant h[ h>d a ?+ x is to we$ out ( m5's m9ds su* invet]ate op9ions z h tak5 root !re1 & be5 confirm$ 9 !m by ! au?ority ( mo/ eloqu5t writ]s2 especially see+ true 7t is1 accurate7 ,philosophy pr(ess$ly rejects n only ! pa9t & false col\rs ( language1 b ev5 ! v ornam5ts & graces ( ! same2 & ! fir/ gr\nds ( all sci5ce are n only n beautiful1 b poor1 arid1 &1 9 appe>ance1 de=m$4 ,nev]!less1 !re be+ c]ta9ly "s m51 ?\gh b few1 :o are delight$ ) tru? & /r5g? ( reason 9 all ?+s1 ,i ?\ght ,i might d well to take ?is pa9s = ! sake ev5 ( ^? few4 ,i proce$ !re=e to ! matt]1 & take my beg9n+ f ! v def9ition ( philosophy1 :i* is ?is4 ,sid5ote3 ,def9ition ( ,philosophy expla9$4 #b4 ,,philosophy is su* k ( effects or appe>ances1 z we acquire by true ratioc9ation f ! k we h fir/ ( _! causes or g5]ation2 ,& aga91 ( su* causes or g5]ations z may be f kn[+ fir/ _! effects4 ,= ! bett] und]/&+ ( :i* def9ition1 we mu/ consid]1 fir/1 t al?\gh ,s5se & ,memory ( ?+s1 :i* are common to man & all liv+ creatures1 be k1 yet because !y are giv5 u imm$iately by nature1 & n gott5 by ratioc9ation1 !y are n philosophy4 ,secondly1 see+ ,exp]i5ce is no?+ b memory2 & ,prud5ce1 or prospect into ! future "t1 no?+ b expectation ( su* ?+s z we h already _h exp]i5ce (1 ,prud5ce also is n to be e/eem$ philosophy4 ,by ,,ratioc9ation1 ,i mean computation4 ,n[ to compute1 is ei!r to collect ! sum ( _m ?+s t are add$ toge!r1 or to "k :at rema9s :5 "o ?+ is tak5 out ( ano!r4 ,ratioc9ation1 !re=e1 is ! same ) addition & sub/raction2 & if any man add multiplication & division1 ,i w n be aga9/ x1 see+ multiplication is no?+ b addition ( equals "o to ano!r1 & division no?+ b a sub/raction ( equals "o f ano!r1 z (t5 z is possible4 ,s t all ratioc9ation is compreh5d$ 9 ^! two op]ations ( ! m9d1 addition & sub/raction4 ,sid5ote3 ,ratioc9ation ( ! ,m9d4 #c4 ,b h[ by ! ratioc9ation ( our m9d1 we add & sub/ract 9 our sil5t ?\ghts1 )\t ! use ( words1 x w be necess>y = me to make intelligible by an example or two4 ,if !re=e a man see some?+ af> (f & obscurely1 al?\gh no appellation _h yet be5 giv5 to any?+1 he w1 not)/&+1 h ! same idea ( t ?+ = :i* n[1 by impos+ a "n on x1 we call x body4 ,aga91 :51 by com+ ne>]1 he sees ! same ?+ ?us & ?us1 n[ 9 "o place & n[ 9 ano!r1 he w h a new idea !re(1 namely1 t = :i* we n[ call su* a ?+ animat$4 ,?irdly1 :5 /&+ ne>]1 he p]ceives ! figure1 he>s ! voice1 & sees o!r ?+s :i* are signs ( a rational m9d1 he has a ?ird idea1 ?\gh x h yet no appellation1 namely1 t = :i* we n[ call any?+ rational4 ,la/ly1 :51 by look+ fully & di/9ctly ^u x1 he conceives all t he has se5 z "o ?+1 ! idea he has n[ is comp\nd$ ( 8 =m] ideas1 :i* are put toge!r 9 ! m9d 9 ! same ord] 9 :i* ^! ?ree s+le names1 body1 animat$1 rational1 are 9 spee* comp\nd$ into ?is "o "n1 body-animat$-rational1 or man4 ,9 l mann]1 ( ! sev]al conceptions ( f\r sides1 equality ( sides1 & "r angles1 is comp\nd$ ! conception ( a squ>e4 ,= ! m9d may conceive a figure ( f\r sides )\t any conception ( _! equality1 & ( t equality )\t conceiv+ a "r angle2 & may jo9 toge!r all ^! s+le conceptions into "o conception or "o idea ( a squ>e4 ,& ?us we see h[ ! conceptions ( ! m9d are comp\nd$4 ,aga91 :osoev] sees a man /&+ ne> him1 conceives ! :ole idea ( t man2 & if1 z he goes away1 he foll[ him ) 8 eyes only1 he w lose ! idea ( ^? ?+s :i* 7 signs ( 8 be+ rational1 :il/1 nev]!less1 ! idea ( a body-animat$ rema9s /ill be=e 8 eyes1 s t ! idea ( rational is sub/ract$ f ! :ole idea ( man1 t is to say1 ( body-animat$-rational1 & !re rema9s t ( body-animat$2 & a :ile aft]1 at a great] di/ance1 ! idea ( animat$ w be lo/1 & t ( body only w rema92 s t at la/1 :5 no?+ at all c be se51 ! :ole idea w vani% out ( sight4 ,by :i* examples1 ,i ?9k1 x is manife/ en\gh :at is ! int]nal ratioc9ation ( ! m9d )\t words4 ,we mu/ n !re=e ?9k t computation1 t is1 ratioc9ation1 has place only 9 numb]s1 z if man 7 di/+ui%$ f o!r liv+ creatures 7:i* is said to h be5 ! op9ion ( ,py?agoras7 by no?+ b ! faculty ( numb]+2 = magnitude1 body1 motion1 "t1 degrees ( quality1 action1 conception1 proportion1 spee* & names 79 :i* all ! k9ds ( philosophy consi/7 are capable ( addition & sub/raction4 ,n[ su* ?+s z we add or sub/ract1 t is1 :i* we put into an acc\nt1 we are said to consid]1 9 ,greek 1 9 :i* language also signifies to compute1 reason1 or reckon4 ,sid5ote3 ,prop]ties1 :at !y are4 #d4 ,b effects & ! appe>ances ( ?+s to s5se1 are faculties or p[]s ( bodies1 :i* make u di/+ui% !m f "o ano!r2 t is to say1 conceive "o body to be equal or unequal1 l or unlike to ano!r body2 z 9 ! example above1 :5 by com+ ne> en\gh to any body1 we p]ceive ! motion & go+ ( ! same1 we di/+ui% x !reby f a tree1 a column1 & o!r fix$ bodies2 & s t motion or go+ is ! prop]ty !re(1 z be+ prop] to liv+ creatures1 & a faculty by :i* !y make u di/+ui% !m f o!r bodies4 ,sid5ote3 ,h[ ,prop]ties are kn[n by ,g5]ation1 & contr>ily4 #e4 ,h[ ! k ( any effect may be gott5 f ! k ( ! g5]ation !re(1 may easily be und]/ood by ! example ( a circle3 = if !re be set be=e u a pla9 figure1 hav+1 z ne> z may be1 ! figure ( a circle1 we _c possibly p]ceive by s5se :e!r x be a true circle or no2 ?an :i*1 nev]!less1 no?+ is m easy to be kn[n to him t kn[s fir/ ! g5]ation ( ! prop\nd$ figure4 ,= let x be kn[n t ! figure 0 made by ! circumduction ( a body :]e( "o end rema9$ unmov$1 & we may reason ?us2 a body c>ri$ ab\t1 reta9+ always ! same l5g?1 applies itself fir/ to "o radius1 !n to ano!r1 to a ?ird1 a f\r?1 & successively to all2 &1 !re=e1 ! same l5g?1 f ! same po9t1 t\*e? ! circumf]5ce 9 e "p !re(1 :i* is z mu* z to say1 z all ! radii are equal4 ,we "k1 !re=e1 t f su* g5]ation proce$s a figure1 f ^: "o middle po9t all ! extreme po9ts are rea*$ unto by equal radii4 ,& 9 l mann]1 by kn[+ fir/ :at figure is set be=e u1 we may come by ratioc9ation to "s g5]ation ( ! same1 ?\gh p]haps n t by :i* x 0 made1 yet t by :i* x might h be5 made2 = he t kn[s t a circle has ! prop]ty above decl>$1 w easily "k :e!r a body c>ri$ ab\t1 z is said1 w g5]ate a circle or no4 ,sid5ote3 ,scope ( ,philosophy4 #f4 ,! end or scope ( philosophy is1 t we may make use to our b5efit ( effects =m]ly se52 or t1 by application ( bodies to "o ano!r1 we may produce ! l effects ( ^? we conceive 9 our m9d1 z f> =? z matt]1 /r5g?1 & indu/ry1 w p]mit1 = ! commodity ( human life4 ,= ! inw>d glory & triumph ( m9d t a man may h = ! ma/]+ ( "s difficult & d\btful matt]1 or = ! discov]y ( "s hidd5 tru?1 is n wor? s mu* pa9s z ! /udy ( ,philosophy requires2 nor ne$ any man c>e mu* to tea* ano!r :at he kn[s himself1 if he ?9k t w be ! only b5efit ( 8 lab\r4 ,! end ( k is p[]2 & ! use ( !orems 7:i*1 among geometricians1 s]ve = ! f9d+ out ( prop]ties7 is = ! con/ruction ( problems2 &1 la/ly1 ! scope ( all speculation is ! p]=m+ ( "s action1 or ?+ to be done4 ,sid5ote3 ,utility ( ,philosophy4 #g4 ,b :at ! utility ( philosophy is1 especially ( natural philosophy & geometry1 w be be/ und]/ood by reckon+ up ! *ief commodities ( :i* mank9d is capable1 & by comp>+ ! mann] ( life ( su* z enjoy !m1 ) t ( o!rs :i* want ! same4 ,n[1 ! greate/ commodities ( mank9d are ! arts2 namely1 ( measur+ matt] & motion2 ( mov+ pond]\s bodies2 ( ar*itecture2 ( navigation2 ( mak+ in/rum5ts = all uses2 ( calculat+ ! cele/ial motions1 ! aspects ( ! />s1 & ! p>ts ( "t2 ( geography1 c4 ,by :i* sci5ces1 h[ great b5efits m5 receive is m easily und]/ood ?an express$4 ,^! b5efits are enjoy$ by almo/ all ! p ( ,europe1 by mo/ ( ^? ( ,asia1 & by "s ( ,africa3 b ! ,am]icans1 & !y t live ne> ! ,poles1 d totally want !m4 ,b :y8 ,h !y %>p] wits ?an ^!8 ,h n all m5 "o k9d ( s\l1 & ! same faculties ( m9d8 ,:at1 !n1 makes ?is diff]5ce1 except philosophy8 ,philosophy1 !re=e1 is ! cause ( all ^! b5efits4 ,b ! utility ( moral & civil philosophy is to be e/imat$1 n s mu* by ! commodities we h by kn[+ ^! sci5ces1 z by ! calamities we receive f n kn[+ !m4 ,n[1 all su* calamities z may be avoid$ by human indu/ry1 arise f w>1 b *iefly f civil w>2 = f ?is proce$ slaught]1 solitude1 & ! want ( all ?+s4 ,b ! cause ( w> is n t m5 are will+ to h x2 = ! w has no?+ = object b good1 at lea/ t :i* seeme? good4 ,nor is x f ?is1 t m5 "k n t ! effects ( w> are evil2 = :o is !re t ?9ks n pov]ty & loss ( life to be great evils8 ,! cause1 !re=e1 ( civil w> is1 t m5 "k n ! causes nei!r ( w> nor peace1 !re be+ b few 9 ! _w t h le>n$ ^? duties :i* unite & keep m5 9 peace1 t is to say1 t h le>n$ ! rules ( civil life suffici5tly4 ,n[1 ! k ( ^! rules is moral philosophy4 ,b :y h !y n le>n$ !m1 unless = ?is reason1 t none hi!rto h taught !m 9 a cle> & exact me?od8 ,= :at %all we say8 ,c\ld ! anci5t ma/]s ( ,greece1 ,egypt1 ,rome1 & o!rs1 p]suade ! unskilful multitude to _! innum]able op9ions conc]n+ ! nature ( _! gods1 :i* !y !mselves knew n :e!r !y 7 true or false1 & :i* 7 inde$ manife/ly false & absurd2 & c\ld !y n p]suade ! same multitude to civil duty1 if !y !mselves _h und]/ood x8 ,or %all ^? few writ+s ( geometricians :i* are extant1 be ?\ght suffici5t = ! tak+ away ( all controv]sy 9 ! matt]s !y treat (1 & %all ^? innum]able & huge volumes ( e?ics be ?\ght unsuffici5t1 if :at !y tea* _h be5 c]ta9 & well demon/rat$8 ,:at1 !n1 c be imag9$ to be ! cause t ! writ+s ( ^? m5 h increas$ sci5ce1 & ! writ+s ( ^! h increas$ no?+ b words1 sav+ t ! =m] 7 writt5 by m5 t knew1 & ! latt] by su* z knew n1 ! doctr9e !y taught only = o/5tation ( _! wit & eloqu5ce8 ,nev]!less1 ,i d5y n b ! read+ ( "s su* books is v delightful2 = !y are mo/ eloqu5tly writt51 & conta9 _m cle>1 :olesome & *oice s5t5ces1 :i* yet are n univ]sally true1 ?\gh by !m univ]sally pron\nc$4 ,f :5ce x comes to pass1 t ! circum/ances ( times1 places1 & p]sons be+ *ang$1 !y are no less frequ5tly made use ( to confirm wick$ m5 9 _! purposes1 ?an to make !m und]/& ! precepts ( civil duties4 ,n[ t :i* is *iefly want+ 9 !m1 is a true & c]ta9 rule ( our actions1 by :i* we might "k :e!r t we und]take be j or unju/4 ,= x is to no purpose to be bidd5 9 e ?+ to d "r1 be=e !re be a c]ta9 rule & measure ( "r e/abli%$1 :i* no man hi!rto ha? e/abli%$4 ,see+1 !re=e1 f ! n kn[+ ( civil duties1 t is1 f ! want ( moral sci5ce1 proce$ civil w>s1 & ! greate/ calamities ( mank9d1 we may v well attribute to su* sci5ce ! production ( ! contr>y commodities4 ,& ?us mu* is suffici5t1 to say no?+ ( ! praises & o!r cont5tm5t proce$+ f philosophy1 to let y see ! utility ( ! same 9 e k9d !re(4 ,sid5ote3 ,subject ( ,philosophy4 #h4 ,! subject ( ,philosophy1 or ! matt] x treats (1 is e body ( :i* we c conceive any g5]ation1 & :i* we may1 by any consid]ation !re(1 comp>e ) o!r bodies1 or :i* is capable ( composition & resolution2 t is to say1 e body ( ^: g5]ation or prop]ties we c h any k4 ,& ?is may be d$uc$ f ! def9ition ( philosophy1 ^: pr(ession x is to se>* out ! prop]ties ( bodies f _! g5]ation1 or _! g5]ation f _! prop]ties2 &1 !re=e1 :]e !re is no g5]ation or prop]ty1 !re is no philosophy4 ,!re=e x excludes ,!ology1 ,i mean ! doctr9e ( ,god1 et]nal1 +5]able1 incompreh5sible1 & 9 :om !re is no?+ nei!r to divide nor comp\nd1 nor any g5]ation to be conceiv$4 ,x excludes ! doctr9e ( angels1 & all su* ?+s z are ?\ght to be nei!r bodies nor prop]ties ( bodies2 !re be+ 9 !m no place nei!r = composition nor division1 nor any capacity ( m & less1 t is to say1 no place = ratioc9ation4 ,x excludes hi/ory1 z well natural z political1 ?\gh mo/ useful 7nay necess>y7 to philosophy2 because su* k is b exp]i5ce1 or au?ority1 & n ratioc9ation4 ,x excludes all su* k z is acquir$ by ,div9e inspiration1 or revelation1 z n d]iv$ to u by reason1 b by ,div9e grace 9 an in/ant1 &1 z x 71 by "s s5se sup]natural4 ,x excludes n only all doctr9es :i* are false1 b su* also z are n well-gr\nd$2 = :atsoev] we "k by "r ratioc9ation1 c nei!r be false nor d\btful2 &1 !re=e1 a/rology1 z x is n[ held =?1 & all su* div9ations r ?an sci5ces1 are exclud$4 ,la/ly1 ! doctr9e ( ,god's wor%ip is exclud$ f philosophy1 z be+ n to be kn[n by natural reason1 b by ! au?ority ( ! ,*ur*2 & z be+ ! object ( fai?1 & n ( k4 ,sid5ote3 ,p>ts ( ,philosophy4 #i4 ,! pr9cipal p>ts ( philosophy are two4 ,= two *ief k9ds ( bodies1 & v diff]5t f "o ano!r1 (f] !mselves to su* z se>* aft] _! g5]ation & prop]ties2 "o :]e( be+ ! "w ( nature1 is call$ a natural body1 ! o!r is call$ a commonweal?1 & is made by ! wills & agreem5t ( m54 ,& f ^! spr+ ! two p>ts ( philosophy1 call$ natural & civil4 ,b see+ t1 = ! k ( ! prop]ties ( a commonweal?1 x is necess>y fir/ to "k ! dispositions1 affections1 & mann]s ( m51 civil philosophy is aga9 commonly divid$ into two p>ts1 :]e( "o1 :i* treats ( m5's dispositions & mann]s1 is call$ e?ics2 & ! o!r1 :i* takes cognizance ( _! civil duties1 is call$ politics1 or simply civil philosophy4 ,9 ! fir/ place1 !re=e 7aft] ,i h set d[n su* premises z app]ta9 to ! nature ( philosophy 9 g5]al71 ,i w disc\rse ( bodies natural2 9 ! second1 ( ! dispositions & mann]s ( m52 & 9 ! ?ird1 ( ! civil duties ( subjects4 ,sid5ote3 ,epilogue4 #aj4 ,to conclude2 see+ !re may be _m :o w n l ?is my def9ition ( philosophy1 & w say1 t1 f ! lib]ty :i* a man may take ( s def9+ z seems be/ to himself1 he may conclude any ?+ f any ?+ 7?\gh ,i ?9k x no h>d matt] to demon/rate t ?is def9ition ( m9e agrees ) ! s5se ( all m572 yet1 le/ 9 ?is po9t !re %\ld be any cause ( dispute betwixt me & !m1 ,i "h und]take no m ?an to deliv] ! elem5ts ( t sci5ce by :i* ! effects ( any?+ may be f\nd out f ! kn[n g5]ation ( ! same1 or contr>ily1 ! g5]ation f ! effects2 to ! end t !y :o se>* aft] o!r philosophy1 may be admoni%$ to seek x f o!r pr9ciples4 ------- ,,*apt] ,,ii4 ,,( ,,names4 #a4 ,! necessity ( s5sible ,monim5ts or ,m>ks = ! help ( ,memory3 a ,m>k def9$4--#b4 ,! necessity ( ,m>ks = ! signification ( ! conceptions ( ! ,m9d4--#c4 ,names supply bo? ^? necessities4--#d4 ,! ,def9ition ( a ,"n4--#e4 ,names are ,signs n ( ,?+s1 b ( our ,cogitations4--#f4 ,:at x is we give ,names to4--#g4 ,names ,positive & ,negative4--#h4 ,contradictory ,names4--#i4 ,a ,common ,"n4--#aj4 ,names ( ! ,fir/ & ,second ,int5tion4--#aa4 ,univ]sal1 ,p>ticul>1 ,individual1 & ,indef9ite ,names4--#ab4 ,names ,univocal & ,equivocal4--#ac4 ,absolute & ,relative ,names4--#ad4 ,simple & ,comp\nd$ ,names4--#ae4 ,a ,pr$icam5t describ$4--#af4 ,"s ?+s to be not$ conc]n+ ,pr$icam5ts4 ,sid5ote3 ,necessity ( s5sible ,monim5ts or ,m>ks = ! help ( ,memory4 #a4 ,h[ uncon/ant & fad+ m5's ?\ghts are1 & h[ mu* ! recov]y ( !m dep5ds ^u *ance1 !re is none b kn[s by infallible exp]i5ce 9 himself4 ,= no man is able to rememb] quantities )\t s5sible & pres5t measures1 nor col\rs )\t s5sible & pres5t patt]ns1 nor numb] )\t ! names ( numb]s dispos$ 9 ord] & le>n$ by he>t4 ,s t :atsoev] a man has put toge!r 9 8 m9d by ratioc9ation )\t su* helps1 w pres5tly slip f him1 & n be revocable b by beg9n+ 8 ratioc9ation anew4 ,f :i* x foll[s1 t1 = ! acquir+ ( philosophy1 "s s5sible monim5ts are necess>y1 by :i* our pa/ ?\ghts may be n only r$uc$1 b also regi/]$ e "o 9 its own ord]4,sid5ote3 ,a ,m>k def9$ ,^! monim5ts ,i call ,,m>ks1 namely1 s5sible ?+s tak5 at pleasure1 t1 by ! s5se ( !m1 su* ?\ghts may be recall$ to our m9d z are l ^? ?\ghts = :i* we took !m4 ,sid5ote3 ,necessity ( ,m>ks = ! signification ( ! conceptions ( ! ,m9d4 #b4 ,aga91 ?\gh "s "o man1 ( h[ excell5t a wit soev]1 %\ld sp5d all 8 "t p>tly 9 reason+1 & p>tly 9 inv5t+ m>ks = ! help ( 8 memory1 & advanc+ himself 9 le>n+2 :o sees n t ! b5efit he reaps to himself w n be mu*1 & to o!rs none at all8 ,= unless he communicate 8 notes ) o!rs1 8 sci5ce w p]i% ) him4 ,b if ! same notes be made common to _m1 & s "o man's inv5tions be taught to o!rs1 sci5ces w !reby be increas$ to ! g5]al good ( mank9d4 ,x is !re=e necess>y1 = ! acquir+ ( philosophy1 t !re be c]ta9 signs1 by :i* :at "o man f9ds out may be manife/$ & made kn[n to o!rs4 ,n[1 ^? ?+s we call ,,signs are ! antec$5ts ( _! consequ5ts1 & ! consequ5ts ( _! antec$5ts1 z (t5 z we obs]ve !m to g be=e or foll[ aft] 9 ! same mann]4 ,= example1 a ?ick cl\d is a sign ( ra9 to foll[1 & ra9 a sign t a cl\d has gone be=e1 = ?is reason only1 t we seldom see cl\ds )\t ! consequ5ce ( ra91 nor ra9 at any "t b :5 a cl\d has gone be=e4 ,& ( signs1 "s are natural1 :]e( ,i h already giv5 an example1 o!rs are arbitr>y1 namely1 ^? we make *oice ( at our own pleasure1 z a bu% hung up1 signifies t w9e is to be sold !re2 a /one set 9 ! gr\nd signifies ! b\nd ( a field2 & words s & s connect$1 signify ! cogitations & motions ( our m9d4 ,! diff]5ce1 !re=e1 betwixt m>ks & signs is ?is1 t we make ^? = our own use1 b ^! = ! use ( o!rs4 ,sid5ote3 ,names supply bo? ^? necessities4 #c4 ,words s connect$ z t !y become signs ( our ?\ghts1 are call$ ,,spee*1 ( :i* e "p is a "n4 ,b see+ 7z is said7 bo? m>ks & signs are necess>y = ! acquir+ ( philosophy1 7m>ks by :i* we may rememb] our own ?\ghts1 & signs by :i* we may make our ?\ghts kn[n to o!rs71 names d bo? ^! (fices2 b !y s]ve = m>ks be=e !y be us$ z signs4 ,= ?\gh a man 7 alone 9 ! _w1 !y w\ld be useful to him 9 help+ him to rememb]2 b to tea* o!rs1 7unless !re 7 "s o!rs to be taught7 ( no use at all4 ,aga91 names1 ?\gh /&+ s+ly by !mselves1 are m>ks1 because !y s]ve to recal our own ?\ghts to m9d2 b !y _c be signs1 o!rwise ?an by be+ dispos$ & ord]$ 9 spee* z p>ts ( ! same4 ,= example1 a man may beg9 ) a ^w1 :]eby ! he>] may frame an idea ( some?+ 9 8 m9d1 :i*1 nev]!less1 he _c conceive to be ! idea :i* 0 9 ! m9d ( him t spake1 b t he w\ld say some?+ :i* began ) t ^w1 ?\gh p]haps n z by itself1 b z "p ( ano!r ^w4 ,s t ! nature ( a "n consi/s pr9cipally 9 ?is1 t x is a m>k tak5 = memory's sake2 b x s]ves also by accid5t to signify & make kn[n to o!rs :at we rememb] ourselves1 &1 !re=e1 ,i w def9e x ?us3 ,sid5ote3 ,def9ition ( a ,"n4 #d4 ,a ,,"n is a ^w tak5 at pleasure to s]ve = a m>k1 :i* may raise 9 our m9d a ?\ght l to "s ?\ght we _h be=e1 & :i* be+ pron\nc$ to o!rs1 may be to !m a sign ( :at ?\ght ! speak] _h1 or _h n be=e 9 8 m9d4 ,& x is = brevity's sake t ,i suppose ! orig9al ( names to be arbitr>y1 judg+ x a ?+ t may be assum$ z unque/ionable4 ,= consid]+ t new names are daily made1 & old ones laid aside2 t div]se nations use diff]5t names1 & h[ impossible x is ei!r to obs]ve similitude1 or make any comp>ison betwixt a "n & a ?+1 h[ c any man imag9e t ! names ( ?+s 7 impos$ f _! natures8 ,= ?\gh "s names ( liv+ creatures & o!r ?+s1 :i* our fir/ p>5ts us$1 7 taught by ,god himself2 yet !y 7 by him arbitr>ily impos$1 & aft]w>ds1 bo? at ! ,t[] ( ,babel1 & s9ce1 9 process ( "t1 gr[+ ev]y:]e out ( use1 are q =gott51 & 9 _! room h succe$$ o!rs1 inv5t$ & receiv$ by m5 at pleasure4 ,moreov]1 :atsoev] ! common use ( words be1 yet philosoph]s1 :o 7 to tea* _! k to o!rs1 _h always ! lib]ty1 & sometimes !y bo? _h & w h a necessity1 ( tak+ to !mselves su* names z !y please = ! signify+ ( _! mean+1 if !y w\ld h x und]/ood4 ,nor _h ma!maticians ne$ to ask leave ( any b !mselves to "n ! figures !y inv5t$1 p>abolas1 hyp]boles1 cissoeides1 quadratices1 c4 or to call "o magnitude ,a1 ano!r ,b4 ,sid5ote3 ,names are signs n ( ?+s1 b ( our cogitations4 #e4 ,b see+ names ord]$ 9 spee* 7z is def9$7 are signs ( our conceptions1 x is manife/ !y are n signs ( ! ?+s !mselves2 = t ! s\nd ( ?is ^w /one %\ld be ! sign ( a /one1 _c be und]/ood 9 any s5se b ?is1 t he t he>s x collects t he t pron\nces x ?9ks ( a /one4 ,&1 !re=e1 t disputation1 :e!r names signify ! matt] or =m1 or some?+ comp\nd$ ( bo?1 & o!r l subtleties ( ! metaphysics1 is kept up by err+ m51 & su* z und]/& n ! words !y dispute ab\t4 ,sid5ote3 ,:at x is we give names to4 #f4 ,nor1 inde$1 is x at all necess>y t e "n %\ld be ! "n ( some?+4 ,= z ^!1 a man1 a tree1 a /one1 are ! names ( ! ?+s !mselves1 s ! images ( a man1 ( a tree1 & ( a /one1 :i* are repres5t$ to m5 sleep+1 h _! names also1 ?\gh !y be n ?+s1 b only fictions & phantasms ( ?+s4 ,= we c rememb] ^!2 &1 !re=e1 x is no less necess>y t !y h names to m>k & signify !m1 ?an ! ?+s !mselves4 ,also ?is ^w future is a "n1 b no future ?+ has yet any be+1 nor d we "k :e!r t :i* we call future1 %all ev] h a be+ or no4 ,nev]!less1 see+ we use 9 our m9d to knit toge!r ?+s pa/ ) ^? t are pres5t1 ! "n future s]ves to signify su* knitt+ toge!r4 ,moreov]1 t :i* nei!r is1 nor has be51 nor ev] %all1 or ev] c be1 has a "n1 namely1 t :i* nei!r is nor has be51 c42 or m briefly ?is1 impossible4 ,to conclude2 ?is ^w no?+ is a "n1 :i* yet _c be ! "n ( any ?+3 = :51 = example1 we sub/ract #b & #c f #e1 & s no?+ rema9+1 we w\ld call t sub/raction to m9d1 ?is spee* no?+ rema9s1 & 9 x ! ^w no?+ is n unuseful4 ,& = ! same reason we say truly1 less ?an no?+ rema9s1 :5 we sub/ract m f less2 = ! m9d feigns su* rema9s z ^! = doctr9e's sake1 & desires1 z (t5 z is necess>y1 to call ! same to memory4 ,b see+ e "n has "s relation to t :i* is nam$1 ?\gh t :i* we "n be n always a ?+ t has a be+ 9 nature1 yet x is lawful = doctr9e's sake to apply ! ^w ?+ to :atsoev] we "n2 z if x 7 all "o :e!r t ?+ be truly exi/5t1 or be only feign$4 ,sid5ote3 ,names ,positive & ,negative4 #g4 ,! fir/ di/9ction ( names is1 t "s are positive1 or affirmative1 o!rs negative1 :i* are also call$ privative & indef9ite4 ,positive are su* z we impose = ! lik5ess1 equality1 or id5tity ( ! ?+s we consid]2 negative1 = ! div]sity1 unlik5ess1 or inequality ( ! same4 ,examples ( ! =m] are1 a man1 a philosoph]2 = a man d5otes any "o ( a multitude ( m51 & a philosoph]1 any "o ( _m philosoph]s1 by reason ( _! similitude2 also1 ,socrates is a positive "n1 because x signifies always "o & ! same man4 ,examples ( negatives are su* positives z h ! negative p>ticle n add$ to !m1 z n-man1 n-philosoph]4 ,b positives 7 be=e negatives2 = o!rwise !re c\ld h be5 no use at all ( ^!4 ,= :5 ! "n ( :ite 0 impos$ ^u c]ta9 ?+s1 & aft]w>ds ^u o!r ?+s1 ! names ( black1 blue1 transp>5t1 c4 ! inf9ite dissimilitudes ( ^! ) :ite c\ld n be compreh5d$ 9 any "o "n1 save t :i* _h 9 x ! negation ( :ite1 t is to say1 ! "n n-:ite1 or "s o!r equival5t to x1 9 :i* ! ^w :ite is repeat$1 su* z unlike to :ite1 c4 ,& by ^! negative names1 we take notice ourselves1 & signify to o!rs :at we h n ?\ght (4 ,sid5ote3 ,contradictory names4 #h4 ,positive & negative names are contradictory to "o ano!r1 s t !y _c bo? be ! "n ( ! same ?+4 ,besides1 ( contradictory names1 "o is ! "n ( any?+ :atsoev]2 = :atsoev] is1 is ei!r man1 or n-man1 :ite or n-:ite1 & s ( ! re/4 ,& ?is is s manife/1 t x ne$s no f>!r pro( or explication2 = !y t say ! same ?+ _c bo? be1 & n be1 speak obscurely2 b !y t say1 :atsoev] is1 ei!r is1 or is n1 speak also absurdly & ridicul\sly4 ,! c]ta9ty ( ?is axiom1 viz4 ( two contradictory names1 "o is ! "n ( any?+ :atsoev]1 ! o!r n1 is ! orig9al & f\ndation ( all ratioc9ation1 t is1 ( all philosophy2 & !re=e x ought to be s exactly prop\nd$1 t x may be ( itself cle> & p]spicu\s to all m52 z inde$ x is1 sav+ to su*1 z read+ ! long disc\rses made ^u ?is subject by ! writ]s ( metaphysics 7:i* !y believe to be "s egregi\s le>n+7 ?9k !y und]/& n1 :5 !y d4 ,sid5ote3 ,a common "n4 #i4 ,secondly1 ( names1 "s are common to _m ?+s1 z a man1 a tree2 o!rs prop] to "o ?+1 z he t writ ! ,iliad1 ,hom]1 ?is man1 t man4 ,& a common "n1 be+ ! "n ( _m ?+s sev]ally tak51 b n collectively ( all toge!r 7z man is n ! "n ( all mank9d1 b ( e "o1 z ( ,pet]1 ,john1 & ! re/ sev]ally7 is !re=e call$ an univ]sal "n2 & !re=e ?is ^w univ]sal is nev] ! "n ( any ?+ exi/5t 9 nature1 nor ( any idea or phantasm =m$ 9 ! m9d1 b always ! "n ( "s ^w or "n2 s t :5 a liv+ creature1 a /one1 a _s1 or any o!r ?+1 is said to be univ]sal1 x is n to be und]/ood1 t any man1 /one1 c4 ev] 0 or c be univ]sal1 b only t ^! words1 liv+ creature1 /one1 c4 are univ]sal names1 t is1 names common to _m ?+s2 & ! conceptions answ]+ !m 9 our m9d1 are ! images & phantasms ( sev]al liv+ creatures1 or o!r ?+s4 ,& !re=e1 = ! und]/&+ ( ! ext5t ( an univ]sal "n1 we ne$ no o!r faculty b t ( our imag9ation1 by :i* we rememb] t su* names br+ sometimes "o ?+1 sometimes ano!r1 into our m9d4 ,also ( common names1 "s are m1 "s less common4 ,m common1 is t :i* is ! "n ( m ?+s2 less common1 ! "n ( few] ?+s2 z liv+ creature is m common ?an man1 or horse1 or lion1 because x compreh5ds !m all3 & !re=e a m common "n1 9 respect ( a less common1 is call$ ! g5us1 or a g5]al "n2 & ?is 9 respect ( t1 ! species1 or a special "n4 ,sid5ote3 ,names ( ! fir/ & second int5tion4 #aj4 ,& f h5ce proce$s ! ?ird di/9ction ( names1 :i* is1 t "s are call$ names ( ! fir/1 o!rs ( ! second int5tion4 ,( ! fir/ int5tion are ! names ( ?+s1 a man1 /one1 c43 ( ! second are ! names ( names & spee*es1 z univ]sal1 p>ticul>1 g5us1 species1 syllogism1 & ! l4 ,b x is h>d to say :y ^? are call$ names ( ! fir/1 & ^! ( ! second int5tion1 unless p]haps x 0 fir/ int5d$ by u to give names to ^? ?+s :i* are ( daily use 9 ?is life1 & aft]w>ds to su* ?+s z app]ta9 to sci5ce1 t is1 t our second int5tion 0 to give names to names4 ,b :atsoev] ! cause h]e( may be1 yet ?is is manife/1 t g5us1 species1 def9ition1 c4 are names ( words & names only2 & !re=e to put g5us & species = ?+s1 & def9ition = ! nature ( any ?+1 z ! writ]s ( metaphysics h done1 is n "r1 see+ !y be only significations ( :at we ?9k ( ! nature ( ?+s4 ,sid5ote3 ,univ]sal1 p>ticul>1 individual1 & indef9ite names4 #aa4 ,f\r?ly1 "s names are ( c]ta9 & det]m9$1 o!rs ( unc]ta9 & undet]m9$ signification4 ,( det]m9$ & c]ta9 signification is1 fir/1 t "n :i* is giv5 to any "o ?+ by itself1 & is call$ an individual "n2 z ,hom]1 ?is tree1 t liv+ creature1 c4 ,secondly t :i* has any ( ^! words1 all1 e1 bo?1 ei!r1 or ! l add$ to x2 & x is !re=e call$ an univ]sal "n1 because x signifies e "o ( ^? ?+s to :i* x is common2 & ( c]ta9 signification = ?is reason1 t he :i* he>s1 conceives 9 8 m9d ! same ?+ t he :i* speaks w\ld h him conceive4 ,( indef9ite signification is1 fir/1 t "n :i* has ! ^w "s1 or ! l add$ to x1 & is call$ a p>ticul> "n2 secondly1 a common "n set by itself )\t any note ei!r ( univ]sality or p>ticul>ity1 z man1 /one1 & is call$ an indef9ite "n2 b bo? p>ticul> & indef9ite names are ( unc]ta9 signification1 because ! he>] kn[s n :at ?+ x is ! speak] w\ld h him conceive2 & !re=e 9 spee*1 p>ticul> & indef9ite names are to be e/eem$ equival5t to "o ano!r4 ,b ^! words1 all1 e1 "s1 c4 :i* d5ote univ]sality & p>ticul>ity1 are n names1 b p>ts only ( names2 s t e man1 & t man :i* ! he>] conceives 9 8 m9d1 are all "o2 & "s man1 & t man :i* ! speak] ?\ght (1 signify ! same4 ,f :5ce x is evid5t1 t ! use ( signs ( ?is k9d1 is n = a man's own sake1 or = 8 gett+ ( k by 8 own private m$itation 7= e man has 8 own ?\ghts suffici5tly det]m9$ )\t su* helps z ^!7 b = ! sake ( o!rs2 t is1 = ! tea*+ & signify+ ( our conceptions to o!rs2 nor 7 !y inv5t$ only to make u rememb]1 b to make u able to disc\rse ) o!rs4 ,sid5ote3 ,names univocal & equivocal4 #ab4 ,fif?ly1 names are usually di/+ui%$ into univocal & equivocal4 ,univocal are ^? :i* 9 ! same tra9 ( disc\rse signify always ! same ?+2 b equivocal ^? :i* mean sometimes "o ?+ & sometimes ano!r4 ,?us1 ! "n triangle is said to be univocal1 because x is always tak5 9 ! same s5se2 & p>abola to be equivocal1 = ! signification x has sometimes ( allegory or similitude1 & sometimes ( a c]ta9 geometrical figure4 ,also e metaphor is by pr(ession equivocal4 ,b ?is di/9ction belongs n s mu* to names1 z to ^? t use names1 = "s use !m prop]ly & accurately = ! f9d+ out ( tru?2 o!rs draw !m f _! prop] s5se1 = ornam5t or deceit4 ,sid5ote3 ,absolute & relative names4 #ac4 ,six?ly1 ( names1 "s are absolute1 o!rs relative4 ,relative are su* z are impos$ = "s comp>ison1 z fa!r1 son1 cause1 effect1 l1 unlike1 equal1 unequal1 ma/]1 s]vant1 c4 ,& ^? t signify no comp>ison at all are absolute names4 ,b1 z x 0 not$ above1 t univ]sality is to be attribut$ to words & names only1 & n to ?+s1 s ! same is to be said ( o!r di/9ctions ( names2 = no ?+s are ei!r univocal or equivocal1 or relative or absolute4 ,!re is also ano!r di/9ction ( names into concrete & ab/ract2 b because ab/ract names proce$ f proposition1 & c h no place :]e !re is no affirmation1 ,i %all speak ( !m h]eaft]4 ,sid5ote3 ,simple & comp\nd$ names4 #ad4 ,la/ly1 !re are simple & comp\nd$ names4 ,b "h x is to be not$1 t a "n is n tak5 9 philosophy1 z 9 gramm>1 = "o s+le ^w1 b = any numb] ( words put toge!r to signify "o ?+2 = among philosoph]s s5ti5t animat$ body passes b = "o "n1 be+ ! "n ( e liv+ creature1 :i* yet1 among gramm>ians1 is acc\nt$ ?ree names4 ,also a simple "n is n "h di/+ui%$ f a comp\nd$ "n by a preposition1 z 9 gramm>4 ,b ,i call a simple "n1 t :i* 9 e k9d is ! mo/ common or mo/ univ]sal2 & t a comp\nd$ "n1 :i* by ! jo9+ ( ano!r "n to x1 is made less univ]sal1 & signifies t m conceptions ?an "o 7 9 ! m9d1 = :i* t latt] "n 0 add$4 ,= example1 9 ! conception ( man 7z is %[n 9 ! =m] *apt]47 ,fir/1 he is conceiv$ to be some?+ t has ext5sion1 :i* is m>k$ by ! ^w body4 ,body1 !re=e1 is a simple "n1 be+ put = t fir/ s+le conception2 aft]w>ds1 ^u ! sight ( su* & su* motion1 ano!r conception arises1 = :i* he is call$ an animat$ body2 & ?is ,i "h call a comp\nd$ "n1 z ,i d also ! "n animal1 :i* is equival5t to an animat$ body4 ,&1 9 ! same mann]1 an animat$ rational body1 z also a man1 :i* is equival5t to x1 is a m comp\nd$ "n4 ,& by ?is we see h[ ! composition ( conceptions 9 ! m9d is answ]able to ! composition ( names2 =1 z 9 ! m9d "o idea or phantasm succe$s to ano!r1 & to ?is a ?ird2 s to "o "n is add$ ano!r & ano!r successively1 & ( !m all is made "o comp\nd$ "n4 ,nev]!less we mu/ n ?9k bodies :i* are )\t ! m9d1 are comp\nd$ 9 ! same mann]1 namely1 t !re is 9 nature a body1 or any o!r imag9able ?+ exi/5t1 :i* at ! fir/ has no magnitude1 & !n1 by ! addition ( magnitude1 comes to h quantity1 & by m or less quantity to h d5sity or r>ity2 & aga91 by ! addition ( figure1 to be figurate1 & aft] ?is1 by ! injection ( light or col\r1 to become lucid or col\r$2 ?\gh su* has be5 ! philosophy ( _m4 ,sid5ote3 ,a pr$icam5t describ$4 #ae4 ,! writ]s ( logic h endeav\r$ to dige/ ! names ( all ! k9ds ( ?+s into c]ta9 scales or degrees1 by ! cont9ual subord9ation ( names less common1 to names m common4 ,9 ! scale ( bodies !y put 9 ! fir/ & highe/ place body simply1 & 9 ! next place "u x less common names1 by :i* x may be m limit$ & det]m9$1 namely animat$ & inanimat$1 & s on till !y come to individuals4 ,9 l mann]1 9 ! scale ( quantities1 !y assign ! fir/ place to quantity1 & ! next to l9e1 sup]ficies1 & solid1 :i* are names ( less latitude2 & ^! ord]s or scales ( names !y usually call pr$icam5ts & categories4 ,& ( ?is ord9ation n only positive1 b negative names also are capable2 :i* may be exemplifi$ by su* =ms ( ! pr$icam5ts z foll[3 ,,! ,,=m ,,( ,,! ,,pr$icam5t ,,( ,,body4 ,n-,body1 or ,accid5t4 ,n animat$4 ,body ,n liv+ ,creature4 ,animat$ ,liv+ ,n ,man4 ,creature ,n ,pet]4 ,man ,pet]4 ,quantity1 or s mu*4 ,bo? ,accid5t & ,body ,absolutely1 z are consid]$ or ,quality1 or su*4 ,comp>atively1 :i* is call$ _! ,relation4 ,,! ,,=m ,,( ,,! ,,pr$icam5t ,,( ,,quantity4 ,n cont9ual1 z ,numb]4 ,l9e4 ,quantity4 ,( itself1 z ,sup]ficies4 ,solid4 ,cont9ual ,"t1 by ,l9e4 ,motion1 by ,l9e & ,by accid5t1 z ,"t4 ,=ce1 by ,motion & ,solid4 ,:]e1 x is to be not$1 t l9e1 sup]ficies1 & solid1 may be said to be ( su* & su* quantity1 t is1 to be orig9ally & ( _! own nature capable ( equality & inequality2 b we _c say !re is ei!r majority or m9ority1 or equality1 or inde$ any quantity at all1 9 "t1 )\t ! help ( l9e & motion2 nor 9 motion1 )\t l9e & "t2 nor 9 =ce1 o!rwise ?an by motion & solid4 ,,! ,,=m ,,( ,,! ,,pr$icam5t ,,( ,,quality4 ,see+4 ,he>+4 ,prim>y ,smell+4 ,ta/+4 ,p]ception ,t\*+4 by ,s5se ,imag9ation4 ,second>y ,affection pleasant4 ,quality unpleasant4 ,by ,see+1 z ,light & ,col\r4 ,by ,he>+1 z ,s\nd4 ,s5sible ,by ,smell+1 z ,od\rs4 ,quality ,by ,ta/+1 z ,sav\rs4 ,by ,t\*+1 z ,h>dness1 ,heat1 ,cold1 c4 ,,! ,,=m ,,( ,,! ,,pr$icam5t ,,( ,,relation4 ,magnitudes1 z ,equality & ,inequality4 ,qualities1 z ,lik5ess & ,unlik5ess4 ,relation ( ,toge!r ,9 ,place4 ,9 ,"t4 ,ord] ,9 ,place ,=m]4 ,n toge!r ,lat] ,9 ,"t ,=m]4 ,lat] ,sid5ote3 ,"s ?+s to be not$ conc]n+ pr$icam5ts4 #af4 ,conc]n+ :i* pr$icam5ts x is to be not$1 9 ! fir/ place1 t z ! division is made 9 ! fir/ pr$icam5t into contradictory names1 s x might h be5 done 9 ! re/4 ,=1 z !re1 body is divid$ into animat$ & n-animat$1 s1 9 ! second pr$icam5t1 cont9ual quantity may be divid$ into l9e & n-l9e1 & aga91 n-l9e into sup]ficies & n-sup]ficies1 & s 9 ! re/2 b x 0 n necess>y4 ,secondly1 x is to be obs]v$1 t ( positive names ! =m] compreh5ds ! latt]2 b ( negatives ! =m] is compreh5d$ by ! latt]4 ,= example1 liv+-creature is ! "n ( e man1 & !re=e x compreh5ds ! "n man2 b1 on ! contr>y1 n-man is ! "n ( ev]y?+ :i* is n-liv+-creature1 & !re=e ! "n n-liv+-creature1 :i* is put fir/1 is compreh5d$ by ! latt] "n1 n-man4 ,?irdly1 we mu/ take he$ t we d n ?9k1 t z names1 s ! div]sities ( ?+s !mselves may be se>*$ out & det]m9$ by su* di/9ctions z ^!2 or t argum5ts may be tak5 f h5ce 7z "s h done ridicul\sly7 to prove t ! k9ds ( ?+s are n inf9ite4 ,f\r?ly1 ,i w\ld n h any man ?9k ,i deliv] ! =ms above = a true & exact ord9ation ( names2 = ?is _c be p]=m$ z long z philosophy rema9s imp]fect2 nor t by plac+ 7= example7 light 9 ! pr$icam5t ( qualities1 :ile ano!r places ! same 9 ! pr$icam5t ( bodies1 ,i pret5d t ei!r ( u ought = ?is to be drawn f 8 op9ion2 = ?is is to be done only by argum5ts & ratioc9ation1 & n by dispos+ ( words into classes4 ,la/ly1 ,i confess ,i h n yet se5 any great use ( ! pr$icam5ts 9 philosophy4 ,i believe ,ari/otle :5 he saw he c\ld n dige/ ! ?+s !mselves into su* ord]s1 might nev]!less desire out ( 8 own au?ority to r$uce words to su* =ms1 z ,i h done2 b ,i d x only = ?is end1 t x may be und]/ood :at ?is ord9ation ( words is1 & n to h x receiv$ = true1 till x be demon/rat$ by good reason to be s4 ------- ,,*apt] ,,iii4 ,,( ,,proposition4 #a4 ,div]s k9ds ( spee*4--#b4 ,proposition def9$4--#c4 ,subject1 pr$icate1 & copula1 :at !y are2 & ab/ract & concrete :at4 #d4 ,! use & abuse ( names ab/ract4--#e4 ,proposition1 univ]sal & p>ticul>4--#f4 ,affirmative & negative4--#g4 ,true & false4--#h4 ,true & false belongs to spee*1 & n to ?+s4--#i4 ,proposition1 prim>y1 n prim>y1 def9ition1 axiom1 petition4--#aj4 ,proposition1 necess>y & cont+5t4--#aa4 ,categorical & hypo!tical4--#ab4 ,! same proposition div]sely pron\nc$4--#ac4 ,propositions t may be r$uc$ to ! same categorical proposition1 are equipoll5t4--#ad4 ,univ]sal propositions conv]t$ by contradictory names1 are equipoll5t4--#ae4 ,negative propositions are ! same1 :e!r negation be be=e or aft] ! copula4--#af4 ,p>ticul> propositions simply conv]t$1 are equipoll5t4--#ag4 ,:at are subalt]n1 contr>y1 subcontr>y1 & contradictory propositions4--#ah4 ,consequ5ce1 :at x is4--#ai4 ,falsity _c foll[ f tru?4--#bj4 ,h[ "o proposition is ! cause ( ano!r4 ,sid5ote3 ,div]s k9ds ( spee*4 #a4 ,f ! connexion or contexture ( names arise div]s k9ds ( spee*1 :]e( "s signify ! desires & affections ( m52 su* are1 fir/1 int]rogations1 :i* d5ote ! desire ( kn[+3 z1 ,:o is a good man8 ,9 :i* spee* !re is "o "n express$1 & ano!r desir$ & expect$ f him ( :om we ask ! same4 ,!n pray]s1 :i* signify ! desire ( hav+ some?+2 promises1 ?reats1 wi%es1 comm&s1 compla9ts1 & o!r significations ( o!r affections4 ,spee* may also be absurd & insignificant2 z :5 !re is a succession ( words1 to :i* !re c be no succession ( ?\ghts 9 m9d to answ] !m2 & ?is happ5s (t5 to su*1 z1 und]/&+ no?+ 9 "s subtle matt]1 d1 nev]!less1 to make o!rs believe !y und]/&1 speak ( ! same incoh]5tly2 = ! connection ( incoh]5t words1 ?\gh x want ! end ( spee* 7:i* is signification7 yet x is spee*2 & is us$ by writ]s ( metaphysics almo/ z frequ5tly z spee* significative4 ,9 philosophy1 !re is b "o k9d ( spee* useful1 :i* "s call 9 ,lat9 dictum1 o!rs enuntiatum et pronunciatum2 b mo/ m5 call x proposition1 & is ! spee* ( ^? t affirm or d5y1 & expresse? tru? or falsity4 ,sid5ote3 ,proposition def9$4 #b4 ,a ,,proposition is a spee* consi/+ ( two names copulat$1 by :i* he t speake? signifies he conceives ! latt] "n to be ! "n ( ! same ?+ :]e( ! =m] is ! "n2 or 7:i* is all "o7 t ! =m] "n is compreh5d$ by ! latt]4 ,= example1 ?is spee*1 man is a liv+ creature1 9 :i* two names are copulat$ by ! v]b is1 is a proposition1 = ?is reason1 t he t speaks x conceives bo? liv+ creature & man to be names ( ! same ?+1 or t ! =m] "n1 man1 is compreh5d$ by ! latt] "n1 liv+ creature4 ,n[ ! =m] "n is commonly call$ ! subject1 or antec$5t1 or ! conta9$ "n1 & ! latt] pr$icate1 consequ5t1 or conta9+ "n4 ,! sign ( connection among/ mo/ nations is ei!r "s ^w1 z ! ^w is 9 ! proposition man is a liv+ creature1 or "s case or t]m9ation ( a ^w1 z 9 ?is proposition1 man walke? 7:i* is equival5t to ?is1 man is walk+72 ! t]m9ation by :i* x is said he walke?1 r ?an he is walk+1 signifie? t ^? two are und]/ood to be copulat$1 or to be names ( ! same ?+4 ,b !re are1 or c]ta9ly may be1 "s nations t h no ^w :i* answ]s to our v]b is1 :o nev]!less =m propositions by ! position only ( "o "n aft] ano!r1 z if in/ead ( man is a liv+ creature1 x %\ld be said man a liv+ creature2 = ! v ord] ( ! names may suffici5tly %[ _! connection2 & !y are z apt & useful 9 philosophy1 z if !y 7 copulat$ by ! v]b is4 ,sid5ote3 ,subject1 pr$icate1 & copula1 :at !y are1 & ab/ract & concrete :at4 #c4 ,:]e=e1 9 e proposition ?ree ?+s are to be consid]$1 viz4 ! two names1 :i* are ! subject1 & ! pr$icate1 & _! copulation2 bo? :i* names raise 9 our m9d ! ?\ght ( "o & ! same ?+2 b ! copulation makes u ?9k ( ! cause = :i* ^? names 7 impos$ on t ?+4 ,z1 = example1 :5 we say a body is moveable1 ?\gh we conceive ! same ?+ to be design$ by bo? ^? names1 yet our m9d re/s n !re1 b se>*es f>!r :at x is to be a body1 or to be moveable1 t is1 :]e9 consi/s ! diff]5ce betwixt ^! & o!r ?+s1 = :i* ^! are s call$1 o!rs are n s call$4 ,!y1 !re=e1 t seek :at x is to be any ?+1 z to be moveable1 to be hot1 c4 seek 9 ?+s ! causes ( _! names4 ,& f h5ce arises t di/9ction ( names 7t\*$ 9 ! la/ *apt]7 into concrete & ab/ract4 ,= concrete is ! "n ( any ?+ :i* we suppose to h a be+1 & is !re=e call$ ! subject1 9 ,lat9 suppositum1 & 9 ,greek 2 z body1 moveable1 mov$1 figurate1 a cubit high1 hot1 cold1 l1 equal1 ,appius1 ,l5tulus1 & ! l2 &1 ab/ract is t :i* 9 any subject d5otes ! cause ( ! concrete "n1 z to be a body1 to be moveable1 to be mov$1 to be figurate1 to be ( su* quantity1 to be hot1 to be cold1 to be l1 to be equal1 to be ,appius1 to be ,l5tulus1 c4 ,or names equival5t to ^!1 :i* are mo/ commonly call$ ab/ract names1 z corporiety1 mobility1 motion1 figure1 quantity1 heat1 cold1 lik5ess1 equality1 & 7z ,cic]o has x7 ,appiety & ,l5tulity4 ,( ! same k9d also are inf9itives2 = to live & to move are ! same ) life & motion1 or to be liv+ & to be mov$4 ,b ab/ract names d5ote only ! causes ( concrete names1 & n ! ?+s !mselves4 ,= example1 :5 we see any ?+1 or conceive 9 our m9d any visible ?+1 t ?+ appe>s to u1 or is conceiv$ by u1 n 9 "o po9t1 b z hav+ p>ts di/ant f "o ano!r1 t is1 z be+ ext5d$ & fill+ "s space4 ,see+ !re=e we call ! ?+ s conceiv$ body1 ! cause ( t "n is1 t t ?+ is ext5d$1 or ! ext5sion or corporiety ( x4 ,s :5 we see a ?+ appe> sometimes "h1 sometimes !re1 & call x mov$ or remov$1 ! cause ( t "n is t x is mov$ or ! motion ( ! same4 ,& ^! causes ( names are ! same ) ! causes ( our conceptions1 namely1 "s p[] ( action1 or affection ( ! ?+ conceiv$1 :i* "s call ! mann] by :i* any ?+ works ^u our s5ses1 b by mo/ m5 !y are call$ accid5ts2 ,i say accid5ts1 n 9 t s5se 9 :i* accid5t is oppos$ to necess>y2 b s1 z be+ nei!r ! ?+s !mselves1 nor p>ts !re(1 d nev]!less accompany ! ?+s 9 su* mann]1 t 7sav+ ext5sion7 !y may all p]i%1 & be de/roy$1 b c nev] be ab/ract$4 ,sid5ote3 ,! use & abuse ( names ab/ract4 #d4 ,!re is also ?is diff]5ce betwixt concrete & ab/ract names1 t ^? 7 inv5t$ be=e propositions1 b ^! aft]2 = ^! c\ld h no be+ till !re 7 propositions1 f ^: copula !y proce$4 ,n[ 9 all matt]s t conc]n ?is life1 b *iefly 9 philosophy1 !re is bo? great use & great abuse ( ab/ract names2 & ! use consi/s 9 ?is1 t )\t !m we _c1 = ! mo/ "p1 ei!r reason1 or compute ! prop]ties ( bodies2 = :5 we w\ld multiply1 divide1 add1 or sub/ract heat1 light1 or motion1 if we %\ld d\ble or add !m toge!r by concrete names1 say+ 7= example7 hot is d\ble to hot1 light d\ble to light1 or mov$ d\ble to mov$1 we %\ld n d\ble ! prop]ties1 b ! bodies !mselves t are hot1 light1 mov$1 c4 :i* we w\ld n d4 ,b ! abuse proce$s f ?is1 t "s m5 see+ !y c consid]1 t is 7z ,i said be=e7 br+ into acc\nt ! increas+s & decreas+s ( quantity1 heat & o!r accid5ts1 )\t consid]+ _! bodies or subjects 7:i* !y call ab/ract+1 or mak+ to exi/ ap>t by !mselves7 !y speak ( accid5ts1 z if !y might be sep>at$ f all bodies4 ,& f h5ce proce$ ! gross errors ( writ]s ( metaphysics2 =1 because !y c consid] ?\ght )\t ! consid]ation ( body1 !y inf] !re is no ne$ ( a ?9k+-body2 & because quantity may be consid]$ )\t consid]+ body1 !y ?9k also t quantity may be )\t body1 & body )\t quantity2 & t a body has quantity by ! addition ( quantity to x4 ,f ! same f\nta9 spr+ ^? insignificant words1 ab/ract sub/ance1 sep>at$ ess5ce1 & ! l2 z also t confusion ( words d]iv$ f ! ,lat9 v]b e/1 z ess5ce1 ess5tiality1 entity1 entitative2 besides reality1 aliquiddity1 quiddity1 c4 :i* c\ld nev] h be5 he>d ( among su* nations z d n copulate _! names by ! v]b is1 b by adjective v]bs1 z runne?1 reade?1 c4 or by ! m]e plac+ ( "o "n aft] ano!r2 & yet see+ su* nations compute & reason1 x is evid5t t philosophy has no ne$ ( ^? words ess5ce1 entity1 & o!r ! l b>b>\s t]ms4 ,sid5ote3 ,proposition1 univ]sal & p>ticul>4 #e4 ,!re are _m di/9ctions ( propositions1 :]e( ! fir/ is1 t "s are univ]sal1 o!rs p>ticul>1 o!rs indef9ite1 & o!rs s+ul>2 & ?is is commonly call$ ! di/9ction ( quantity4 ,an univ]sal proposition is t ^: subject is affect$ ) ! sign ( an univ]sal "n1 z e man is a liv+ creature4 ,p>ticul>1 t ^: subject is affect$ ) ! sign ( a p>ticul> "n1 z "s man is le>n$4 ,an indef9ite proposition has = its subject a common "n1 & put )\t any sign1 z man is a liv+ creature1 man is le>n$4 ,& a s+ul> proposition is t ^: subject is a s+ul> "n1 z ,socrates is a philosoph]1 ?is man is black4 ,sid5ote3 ,affirmative & negative4 #f4 ,! second di/9ction is into affirmative & negative1 & is call$ ! di/9ction ( quality4 ,an affirmative proposition is t ^: pr$icate is a positive "n1 z man is a liv+ creature4 ,negative1 t ^: pr$icate is a negative "n1 z man is n a /one4 ,sid5ote3 ,true false4 #g4 ,! ?ird di/9ction is1 t "o is true1 ano!r false4 ,a true proposition is t1 ^: pr$icate conta9s1 or compreh5ds its subject1 or ^: pr$icate is ! "n ( e ?+1 ( :i* ! subject is ! "n2 z man is a liv+ creature is !re=e a true proposition1 because :atsoev] is call$ man1 ! same is also call$ liv+ creature2 & "s man is sick1 is true1 because sick is ! "n ( "s man4 ,t :i* is n true1 or t ^: pr$icate does n conta9 its subject1 is call$ a false proposition1 z man is a /one4 ,n[ ^! words true1 tru?1 & true proposition1 are equival5t to "o ano!r2 = tru? consi/s 9 spee*1 & n 9 ! ?+s spok5 (2 & ?\gh true be sometimes oppos$ to app>5t or feign$1 yet x is always to be ref]r$ to ! tru? ( proposition2 = ! image ( a man 9 a glass1 or a gho/1 is !re=e d5i$ to be a v man1 because ?is proposition1 a gho/ is a man1 is n true2 = x _c be d5i$ b t a gho/ is a v gho/4 ,& !re=e tru? or v]ity is n any affection ( ! ?+1 b ( ! proposition conc]n+ x4 ,z = t :i* ! writ]s ( metaphysics say1 t a ?+1 "o ?+1 & a v ?+1 are equival5t to "o ano!r1 x is b trifl+ & *ildi%2 = :o does n "k1 t a man1 "o man1 & a v man1 signify ! same4 ,sid5ote3 ,true false belongs to spee*1 & n to ?+s4 #h4 ,& f h5ce x is evid5t1 t tru? & falsity h no place b among/ su* liv+ creatures z use spee*4 ,= ?\gh "s brute creatures1 look+ ^u ! image ( a man 9 a glass1 may be affect$ ) x1 z if x 7 ! man himself1 & = ?is reason fe> x or fawn ^u x 9 va92 yet !y d n appreh5d x z true or false1 b only z l2 & 9 ?is !y are n deceiv$4 ,:]e=e1 z m5 owe all _! true ratioc9ation to ! "r und]/&+ ( spee*2 s also !y owe _! errors to ! misund]/&+ ( ! same2 & z all ! ornam5ts ( philosophy proce$ only f man1 s f man also is d]iv$ ! ugly absurdity ( false op9ions4 ,= spee* has some?+ 9 x l to a spid]'s web1 7z x 0 said ( old ( ,solon's laws7 = by contexture ( words t5d] & delicate wits are ensn>$ & /opp$2 b /rong wits break easily ?r\gh !m4 ,f h5ce also ?is may be d$uc$1 t ! fir/ tru?s 7 arbitr>ily made by ^? t fir/ ( all impos$ names ^u ?+s1 or receiv$ !m f ! imposition ( o!rs4 ,= x is true 7= example7 t man is a liv+ creature1 b x is = ?is reason1 t x pleas$ m5 to impose bo? ^? names on ! same ?+4 ,sid5ote3 ,proposition1 prim>y1 n prim>y1 def9ition1 axiom1 petition4 #i4 ,f\r?ly1 propositions are di/+ui%$ into prim>y & n prim>y4 ,prim>y is t :]e9 ! subject is explicat$ by a pr$icate ( _m names1 z man is a body1 animat$1 rational2 = t :i* is compreh5d$ 9 ! "n man1 is m l>gely express$ 9 ! names body1 animat$1 & rational1 jo9$ toge!r2 & x is call$ prim>y1 because x is fir/ 9 ratioc9ation2 = no?+ c be prov$1 )\t und]/&+ fir/ ! "n ( ! ?+ 9 "q4 ,n[ prim>y propositions are no?+ b def9itions1 or p>ts ( def9itions1 & ^! only are ! pr9ciples ( demon/ration1 be+ tru?s con/itut$ arbitr>ily by ! inv5tors ( spee*1 & !re=e n to be demon/rat$4 ,to ^! propositions1 "s h add$ o!rs1 :i* !y call prim>y & pr9ciples1 namely1 axioms1 & common notions2 :i*1 7?\gh !y be s evid5t t !y ne$ no pro(7 yet1 because !y may be prov$1 are n truly pr9ciples2 & ! less to be receiv$ = su*1 9 reg>d propositions n intelligible1 & sometimes manife/ly false1 are ?ru/ on u "u ! "n ( pr9ciples by ! clam\r ( m51 :o obtrude = evid5t to o!rs1 all t !y !mselves ?9k true4 ,also c]ta9 petitions are commonly receiv$ into ! numb] ( pr9ciples2 z1 = example1 t a /raight l9e may be drawn betwe5 two po9ts1 & o!r petitions ( ! writ]s ( geometry2 & ^! are inde$ ! pr9ciples ( art or con/ruction1 b n ( sci5ce & demon/ration4 ,sid5ote3 ,proposition necess>y cont+5t4 #aj4 ,fif?ly1 propositions are di/+ui%$ into necess>y1 t is1 necess>ily true2 & true1 b n necess>ily1 :i* !y call cont+5t4 ,a necess>y proposition is :5 no?+ c at any "t be conceiv$ or feign$1 :]e( ! subject is ! "n1 b ! pr$icate also is ! "n ( ! same ?+2 z man is a liv+ creature is a necess>y proposition1 because at :at "t soev] we suppose ! "n man agrees ) any ?+1 at t "t ! "n liv+-creature also agrees ) ! same4 ,b a cont+5t proposition is t1 :i* at "o "t may be true1 at ano!r "t false2 z e cr[ is black2 :i* may p]haps be true n[1 b false h]eaft]4 ,aga91 9 e necess>y proposition1 ! pr$icate is ei!r equival5t to ! subject1 z 9 ?is1 man is a rational liv+ creature2 or "p ( an equival5t "n1 z 9 ?is1 man is a liv+ creature1 = ! "n rational-liv+-creature1 or man1 is comp\nd$ ( ^! two1 rational & liv+-creature4 ,b 9 a cont+5t proposition ?is _c be2 = ?\gh ?is 7 true1 e man is a li>1 yet because ! ^w li> is no "p ( a comp\nd$ "n equival5t to ! "n man1 t proposition is n to be call$ necess>y1 b cont+5t1 ?\gh x %\ld happ5 to be true always4 ,& !re=e ^? propositions only are necess>y1 :i* are ( sempit]nal tru?1 t is1 true at all times4 ,f h5ce also x is manife/1 t tru? adh]es n to ?+s1 b to spee* only1 = "s tru?s are et]nal2 = x w be et]nally true1 if man1 !n liv+-creature2 b t any man1 or liv+-creature1 %\ld exi/ et]nally1 is n necess>y4 ,sid5ote3 ,categorical hypo!tical4 #aa4 ,a six? di/9ction ( propositions is into categorical & hypo!tical4 ,a categorical proposition is t :i* is simply or absolutely pron\nc$1 z e man is a liv+-creature1 no man is a tree2 & hypo!tical is t :i* is pron\nc$ conditionally1 z1 if any ?+ be a man1 ! same is also a liv+-creature1 if any?+ be a man1 ! same is also n-a-/one4 ,a categorical proposition1 & an hypo!tical answ]+ x1 d bo? signify ! same1 if ! propositions be necess>y2 b n if !y be cont+5t4 ,= example1 if ?is1 e man is a liv+-creature1 be true1 ?is also w be true1 if any ?+ be a man1 ! same is also a liv+-creature2 b 9 cont+5t propositions1 ?\gh ?is be true1 e cr[ is black1 yet ?is1 if any ?+ be a cr[1 ! same is black1 is false4 ,b an hypo!tical proposition is !n rightly said to be true1 :5 ! consequ5ce is true1 z e man is a liv+-creature1 is rightly said to be a true proposition1 because ( :atsoev] x is truly said t is a man1 x _c b be truly said also1 ! same is a liv+ creature4 ,& !re=e :5soev] an hypo!tical proposition is true1 ! categorical answ]+ x1 is n only true1 b also necess>y2 :i* ,i ?\ght wor? ! not+1 z an argum5t1 t philosoph]s may 9 mo/ ?+s reason m solidly by hypo!tical ?an categorical propositions4 ,sid5ote3 ,! same proposition div]sely pron\nc$4 #ab4 ,b see+ e proposition may be1 & uses to be1 pron\nc$ & writt5 9 _m =ms1 & we are oblig$ to speak 9 ! same mann] z mo/ m5 speak1 yet !y t le>n philosophy f ma/]s1 _h ne$ to take he$ !y be n deceiv$ by ! v>iety ( expressions4 ,& !re=e1 :5soev] !y meet ) any obscure proposition1 !y ought to r$uce x to its mo/ simple & categorical =m2 9 :i* ! copulative ^w is mu/ be express$ by itself1 & n m+l$ 9 any mann] ei!r ) ! subject or pr$icate1 bo? :i* mu/ be sep>at$ & cle>ly di/+ui%$ "o f ano!r4 ,= example1 if ?is proposition1 man c n s91 be comp>$ ) ?is1 man _c s91 _! diff]5ce w easily appe> if !y be r$uc$ to ^!1 man is able n to s91 &1 man is n able to s91 :]e ! pr$icates are manife/ly diff]5t4 ,b !y ought to d ?is sil5tly by !mselves1 or betwixt !m & _! ma/]s only2 = x w be ?\ght bo? ridicul\s & absurd1 = a man to use su* language publicly4 ,be+ !re=e to speak ( equipoll5t propositions1 ,i put 9 ! fir/ place all ^? = equipoll5t1 t may be r$uc$ purely to "o & ! same categorical proposition4 ,sid5ote3 ,propositions t may be r$uc$ to ! same categorical proposition1 are equipoll5t4 #ac4 ,secondly1 t :i* is categorical & necess>y1 is equipoll5t to its hypo!tical proposition2 z ?is categorical1 a "r-l9$ triangle has its ?ree angles equal to two "r angles1 to ?is hypo!tical1 if any figure be a "r-l9$ triangle1 ! ?ree angles ( x are equal to two "r angles4 ,sid5ote3 ,univ]sal propositions conv]t$ by contradictory names1 are equipoll5t4 #ad4 ,also1 any two univ]sal propositions1 ( :i* ! t]ms ( ! "o 7t is1 ! subject & pr$icate7 are to ! t]ms ( ! o!r1 & _! ord] inv]t$1 z ^!1 e man is a liv+ creature1 & e ?+ t is n a liv+-creature is n a man1 are equipoll5t4 ,= see+ e man is a liv+ creature is a true proposition1 ! "n liv+ creature conta9s ! "n man2 b !y are bo? positive names1 & !re=e 7by ! la/ article ( ! prec$5t *apt]7 ! negative "n n man1 conta9s ! negative "n n liv+-creature2 :]e=e e ?+ t is n a liv+-creature1 is n a man1 is a true proposition4 ,likewise ^!1 no man is a tree1 no tree is a man1 are equipoll5t4 ,= if x be true t tree is n ! "n ( any man1 !n no "o ?+ c be signifi$ by ! two names man & tree1 :]e=e no tree is a man is a true proposition4 ,also to ?is1 :atsoev] is n a liv+-creature is n a man1 :]e bo? ! t]ms are negative1 ?is o!r proposition is equipoll5t1 only a liv+ creature is a man4 ,sid5ote3 ,negative propositions are ! same1 :e!r ! negation be be=e or aft] ! copula4 #ae4 ,f\r?ly1 negative propositions1 :e!r ! p>ticle ( negation be set aft] ! copula z "s nations d1 or be=e x1 z x is 9 ,lat9 & ,greek1 if ! t]ms be ! same1 are equipoll5t3 z1 = example1 man is n a tree1 &1 man is n-a-tree1 are equipoll5t1 ?\gh ,ari/otle d5y x4 ,also ^!1 e man is n a tree1 & no man is a tree1 are equipoll5t1 & t s manife/ly1 z x ne$s n be demon/rat$4 ,sid5ote3 ,p>ticul> propositions simply conv]t$1 are equipoll5t4 #af4 ,la/ly1 all p>ticul> propositions t h _! t]ms inv]t$1 z ^!1 "s man is bl9d1 "s bl9d ?+ is a man1 are equipoll5t2 = ei!r ( ! two names1 is ! "n ( "s "o & ! same man2 & !re=e 9 :i* soev] ( ! two ord]s !y be connect$1 !y signify ! same tru?4 ,sid5ote3 ,:at are subalt]n1 contr>y1 subcontr>y1 & contradictory propositions4 #ag4 ,( propositions t h ! same t]ms1 & are plac$ 9 ! same ord]1 b v>i$ ei!r by quantity or quality1 "s are call$ subalt]n1 o!rs contr>y1 o!rs subcontr>y1 & o!rs contradictory4 ,subalt]n1 are univ]sal & p>ticul> propositions ( ! same quality2 z1 e man is a liv+ creature1 "s man is a liv+ creature2 or1 no man is wise1 "s man is n wise4 ,( ^!1 if ! univ]sal be true1 ! p>ticul> w be true also4 ,contr>y1 are univ]sal propositions ( diff]5t quality2 z1 e man is happy1 no man is happy4 ,& ( ^!1 if "o be true1 ! o!r is false3 also1 !y may bo? be false1 z 9 ! example giv54 ,subcontr>y1 are p>ticul> propositions ( diff]5t quality2 z1 "s man is le>n$1 "s man is n le>n$2 :i* _c be bo? false1 b !y may be bo? true4 ,contradictory are ^? t diff] bo? 9 quantity & quality2 z1 e man is a liv+ creature1 "s man is n a liv+-creature2 :i* c nei!r be bo? true1 nor bo? false4 ,sid5ote3 ,consequ5ce1 :at x is4 #ah4 ,a proposition is said to foll[ f two o!r propositions1 :5 ^! be+ grant$ to be true1 x _c be d5i$ b ! o!r is true also4 ,= example1 let ^! two propositions1 e man is a liv+ creature1 &1 e liv+ creature is a body1 be suppos$ true1 t is1 t body is ! "n ( e liv+ creature1 & liv+ creature ! "n ( e man4 ,see+ !re=e1 if ^! be und]/ood to be true1 x _c be und]/ood t body is n ! "n ( e man1 t is1 t e man is a body is false1 ?is proposition w be said to foll[ f ^? two1 or to be necess>ily inf]r$ f !m4 ,sid5ote3 ,falsity _c foll[ f tru?4 #ai4 ,t a true proposition may foll[ f false propositions1 may happ5 sometimes2 b false f true1 nev]4 ,= if ^!1 e man is a /one1 & e /one is a liv+ creature1 7:i* are bo? false7 be grant$ to be true1 x is grant$ also t liv+ creature is ! "n ( e /one1 & /one ( e man1 t is1 t liv+ creature is ! "n ( e man2 t is to say1 ?is proposition e man is a liv+ creature1 is true1 z x is inde$ true4 ,:]e=e a true proposition may sometimes foll[ f false4 ,b if any two propositions be true1 a false "o c nev] foll[ f !m4 ,= if true foll[ f false1 = ?is reason only1 t ! false are grant$ to be true1 !n tru? f two tru?s grant$ w foll[ 9 ! same mann]4 ,sid5ote3 ,h[ "o proposition is ! cause ( ano!r4 #bj4 ,n[1 see+ none b a true proposition w foll[ f true1 & t ! und]/&+ ( two propositions to be true1 is ! cause ( und]/&+ t also to be true :i* is d$uc$ f !m2 ! two antec$5t propositions are commonly call$ ! causes ( ! inf]r$ proposition1 or conclusion4 ,& f h5ce x is t logicians say1 ! premises are causes ( ! conclusion2 :i* may pass1 ?\gh x be n prop]ly spok52 = ?\gh und]/&+ be ! cause ( und]/&+1 yet spee* is n ! cause ( spee*4 ,b :5 !y say1 ! cause ( ! prop]ties ( any ?+1 is ! ?+ itself1 !y speak absurdly4 ,= example1 if a figure be prop\nd$ :i* is triangul>2 see+ e triangle has all its angles toge!r equal to two "r angles1 f :5ce x foll[s t all ! angles ( t figure are equal to two "r angles1 !y say1 = ?is reason1 t t figure is ! cause ( t equality4 ,b see+ ! figure does n itself make its angles1 & !re=e _c be said to be ! effici5t-cause1 !y call x ! =mal-cause2 :]eas inde$ x is no cause at all2 nor does ! prop]ty ( any figure foll[ ! figure1 b has its be+ at ! same "t ) x2 only ! k ( ! figure goes be=e ! k ( ! prop]ties2 & "o k is truly ! cause ( ano!r k1 namely ! effici5t cause4 ,& ?us mu* conc]n+ proposition2 :i* 9 ! progress ( philosophy is ! fir/ /ep1 l ! mov+ t[>ds ( "o foot4 ,by ! due addition ( ano!r /ep ,i %all proce$ to syllogism1 & make a complete pace4 ,( :i* 9 ! next *apt]4 ------- ,,*apt] ,,iv4 ,,( ,,syllogism4 #a4 ,! def9ition ( syllogism4--#b4 ,9 a syllogism !re are b ?ree t]ms4--#c4 ,major1 m9or1 & middle t]m2 also major & m9or proposition1 :at !y are4--#d4 ,! middle t]m 9 e syllogism ought to be det]m9$ 9 bo? ! propositions to "o & ! same ?+4--#e4 ,f two p>ticul> propositions no?+ c be conclud$4--#f4 ,a syllogism is ! collection ( two propositions into "o sum4--#g4 ,! figure ( a syllogism1 :at x is4--#h4 ,:at is 9 ! m9d answ]+ to a syllogism4--#i4 ,! fir/ indirect figure1 h[ x is made4--#aj4 ,! second indirect figure1 h[ made4--#aa4 ,h[ ! ?ird indirect figure is made4--#ab4 ,!re are _m moods 9 e figure1 b mo/ ( !m useless 9 philosophy4--#ac4 ,an hypo!tical syllogism :5 equipoll5t to a categorical4 ,sid5ote3 ,def9ition ( syllogism4 #a4 ,a ,,spee*1 consi/+ ( ?ree propositions1 f two ( :i* ! ?ird foll[s1 is call$ a ,,syllogism2 & t :i* foll[s is call$ ! conclusion2 ! o!r two premises4 ,= example1 ?is spee*1 e man is a liv+ creature1 e liv+ creature is a body1 !re=e1 e man is a body1 is a syllogism1 because ! ?ird proposition foll[s f ! two fir/2 t is1 if ^? be grant$ to be true1 ?is mu/ also be grant$ to be true4 ,sid5ote3 ,9 a syllogism !re are b ?ree t]ms4 #b4 ,f two propositions :i* h n "o t]m common1 no conclusion c foll[2 & !re=e no syllogism c be made ( !m4 ,= let any two premises1 a man is a liv+ creature1 a tree is a plant1 be bo? ( !m true1 yet because x _c be collect$ f !m t plant is ! "n ( a man1 or man ! "n ( a plant1 x is n necess>y t ?is conclusion1 a man is a plant1 %\ld be true4 ,coroll>y3 !re=e1 9 ! premises ( a syllogism !re c be b ?ree t]ms4 ,besides1 !re c be no t]m 9 ! conclusion1 :i* 0 n 9 ! premises4 ,= let any two premises be1 a man is a liv+ creature1 a liv+ creature is a body1 yet if any o!r t]m be put 9 ! conclusion1 z man is two-foot$2 ?\gh x be true1 x _c foll[ f ! premises1 because f !m x _c be collect$1 t ! "n two-foot$ belongs to a man2 & !re=e1 aga91 9 e syllogism !re c be b ?ree t]ms4 ,sid5ote3 ,major1 m9or & middle t]m2 also major & m9or proposition1 :at !y are4 #c4 ,( ^! t]ms1 t :i* is ! pr$icate 9 ! conclusion1 is commonly call$ ! major2 t :i* is ! subject 9 ! conclusion1 ! m9or1 & ! o!r is ! middle t]m2 z 9 ?is syllogism1 a man is a liv+ creature1 a liv+ creature is a body1 !re=e1 a man is a body1 body is ! major1 man ! m9or1 & liv+ creature ! middle t]m4 ,also ( ! premises1 t 9 :i* ! major t]m is f\nd1 is call$ ! major proposition1 & t :i* has ! m9or t]m1 ! m9or proposition4 ,sid5ote3 ,! middle t]m 9 e syllogism to be det]m9$ 9 bo? propositions to "o & ! same ?+4 #d4 ,if ! middle t]m be n 9 bo? ! premises det]m9$ to "o & ! same s+ul> ?+1 no conclusion w foll[1 nor syllogism be made4 ,= let ! m9or t]m be man1 ! middle t]m liv+ creature1 & ! major t]m lion2 & let ! premises be1 man is a liv+ creature1 "s liv+ creature is a lion1 yet x w n foll[ t e or any man is a lion4 ,by :i* x is manife/1 t 9 e syllogism1 t proposition :i* has ! middle t]m = its subject1 ought to be ei!r univ]sal or s+ul>1 b n p>ticul> nor indef9ite4 ,= example1 ?is syllogism1 e man is a liv+ creature1 "s liv+ creature is f\r-foot$1 !re=e "s man is f\r-foot$1 is !re=e faulty1 because ! middle t]m1 liv+ creature1 is 9 ! fir/ ( ! premises det]m9$ only to man1 = !re ! "n ( liv+ creature is giv5 to man only1 b 9 ! latt] premise x may be und]/ood ( "s o!r liv+ creature besides man4 ,b if ! latt] premise _h be5 univ]sal1 z "h1 e man is a liv+ creature1 e liv+ creature is a body1 !re=e e man is a body1 ! syllogism _h be5 true2 = x w\ld h foll[$ t body _h be5 ! "n ( e liv+ creature1 t is ( man2 t is to say1 ! conclusion e man is a body _h be5 true4 ,likewise1 :5 ! middle t]m is a s+ul> "n1 a syllogism may be made1 ,i say a true syllogism1 ?\gh useless 9 philosophy1 z ?is1 "s man is ,socrates1 ,socrates is a philosoph]1 !re=e1 "s man is a philosoph]2 = ! premises be+ grant$1 ! conclusion _c be d5i$4 ,sid5ote3 ,f two p>ticul> propositions no?+ c be conclud$4 #e4 ,& !re=e ( two premises1 9 bo? :i* ! middle t]m is p>ticul>1 a syllogism _c be made2 = :e!r ! middle t]m be ! subject 9 bo? ! premises1 or ! pr$icate 9 bo?1 or ! subject 9 "o1 & ! pr$icate 9 ! o!r1 x w n be necess>ily det]m9$ to ! same ?+4 ,= let ! premises be1 ,"s man is bl9d1 ,9 bo? :i* ! middle ,"s man is le>n$1 t]m is ! subject1 x w n foll[ t bl9d is ! "n ( any le>n$ man1 or le>n$ ! "n ( any bl9d man1 see+ ! "n le>n$ does n conta9 ! "n bl9d1 nor ?is t2 & !re=e x is n necess>y t bo? %\ld be names ( ! same man4 ,s f ^! premises1 ,e man is a liv+-creature1 ,9 bo? :i* ! middle ,e horse is a liv+-creature1 t]m is ! pr$icate1 no?+ w foll[4 ,= see+ liv+ creature is 9 bo? ( !m indef9ite1 :i* is equival5t to p>ticul>1 & t man may be "o k9d ( liv+ creature1 & horse ano!r k9d1 x is n necess>y t man %\ld be ! "n ( horse1 or horse ( man4 ,or if ! premises be1 ,e man is a ,9 "o ( :i* ! liv+-creature1 middle-t]m ,"s liv+ creature is f\r-foot$1 is ! subject1 & 9 ! o!r ! pr$icate1 ! conclusion w n foll[1 because ! "n liv+ creature be+ n det]m9$1 x may 9 "o ( !m be und]/ood ( man1 9 ! o!r ( n-man4 ,sid5ote3 ,a syllogism is ! collection ( two propositions into "o sum4 #f4 ,n[ x is manife/ f :at has be5 said1 t a syllogism is no?+ b a collection ( ! sum ( two propositions1 jo9$ toge!r by a common t]m1 :i* is call$ ! middle t]m4 ,& z proposition is ! addition ( two names1 s syllogism is ! add+ toge!r ( ?ree4 ,sid5ote3 ,! figure ( a syllogism :at x is4 #g4 ,syllogisms are usually di/+ui%$ accord+ to _! div]sity ( figures1 t is1 by ! div]se position ( ! middle t]m4 ,& aga9 9 figure !re is a di/9ction ( c]ta9 moods1 :i* consi/ ( ! diff]5ces ( propositions 9 quantity & quality4 ,! fir/ figure is t1 9 :i* ! t]ms are plac$ "o aft] ano!r accord+ to _! latitude ( signification2 9 :i* ord] ! m9or t]m is fir/1 ! middle t]m next1 & ! major la/2 z1 if ! m9or t]m be man1 ! middle t]m1 liv+ creature1 & ! major t]m1 body1 !n1 man is a liv+-creature1 is a body1 w be a syllogism 9 ! fir/ figure3 9 :i*1 man is a liv+ creature is ! m9or proposition2 ! major1 liv+ creature is a body1 & ! conclusion1 or sum ( bo?1 man is a body4 ,n[ ?is figure is call$ direct1 because ! t]ms /& 9 direct ord]2 & x is v>i$ by quantity & quality into f\r moods3 ( :i* ! fir/ is t :]e9 all ! t]ms are positive1 & ! m9or t]m univ]sal1 z e man is a liv+ creature1 e liv+ creature is a body3 9 :i* all ! propositions are affirmative1 & univ]sal4 ,b if ! major t]m be a negative "n1 & ! m9or an univ]sal "n1 ! figure w be 9 ! second mood1 z1 e man is a liv+ creature1 e liv+ creature is n a tree1 9 :i* ! major proposition & conclusion are bo? univ]sal & negative4 ,to ^! two1 are commonly add$ two m1 by mak+ ! m9or t]m p>ticul>4 ,also x may happ5 t bo? ! major & middle t]ms are negative t]ms1 & !n !re arises ano!r mood1 9 :i* all ! propositions are negative1 & yet ! syllogism w be good2 z1 if ! m9or t]m be man1 ! middle t]m n a /one1 & ! major t]m n a fl9t1 ?is syllogism1 no man is a /one1 :atsoev] is n a /one is n a fl9t1 !re=e1 no man is a fl9t1 is true1 ?\gh x consi/ ( ?ree negatives4 ,b 9 philosophy1 ! pr(ession :]e( is to e/abli% univ]sal rules conc]n+ ! prop]ties ( ?+s1 see+ ! diff]5ce betwixt negatives & affirmatives is only ?is1 t 9 ! =m] ! subject is affirm$ by a negative "n1 & by a positive 9 ! latt]1 x is sup]flu\s to consid] any o!r mood 9 direct figure1 besides t1 9 :i* all ! propositions are bo? univ]sal & affirmative4 ,sid5ote3 ,:at is 9 ! m9d answ]+ to a syllogism4 #h4 ,! ?\ghts 9 ! m9d answ]+ to a direct syllogism1 proce$ 9 ?is mann]2 fir/1 !re is conceiv$ a phantasm ( ! ?+ nam$1 ) t accid5t or quality !re(1 = :i* x is 9 ! m9or proposition call$ by t "n :i* is ! subject2 next1 ! m9d has a phantasm ( ! same ?+ ) t accid5t1 or quality1 = :i* x ha? ! "n1 t 9 ! same proposition is ! pr$icate2 ?irdly1 ! ?\ght returns ( ! same ?+ z hav+ t accid5t 9 x1 = :i* x is call$ by ! "n1 t is ! pr$icate ( ! major proposition2 & la/ly1 rememb]+ t all ^? are ! accid5ts ( "o & ! same ?+1 x concludes t ^? ?ree names are also names ( "o & ! same ?+2 t is to say1 ! conclusion is true4 ,= example1 :5 ?is syllogism is made1 man is a liv+ creature1 a liv+ creature is a body1 !re=e1 man is a body1 ! m9d conceives fir/ an image ( a man speak+ or disc\rs+1 & rememb]s t t1 :i* s appe>s1 is call$ man2 !n x has ! image ( ! same man mov+1 & rememb]s t t1 :i* appe>s s1 is call$ liv+ creature2 ?irdly1 x conceives an image ( ! same man1 z fill+ "s place or space1 & rememb]s t :at appe>s s is call$ body2 & la/ly1 :5 x rememb]s t t ?+1 :i* 0 ext5d$1 & mov$ & spake1 0 "o & ! same ?+1 x concludes t ! ?ree names1 man1 liv+ creature1 & body1 are names ( ! same ?+1 & t !re=e man is a liv+ creature is a true proposition4 ,f :5ce x is manife/1 t liv+ creatures t h n ! use ( spee*1 h no conception or ?\ght 9 ! m9d1 answ]+ to a syllogism made ( univ]sal propositions2 see+ x is necess>y to ?9k n only ( ! ?+1 b also by turns to rememb] ! div]s names1 :i* = div]s consid]ations !re( are appli$ to ! same4 ,sid5ote3 ,! fir/ indirect figure h[ made4 #i4 ,! re/ ( ! figures arise ei!r f ! inflexion1 or inv]sion ( ! fir/ or direct figure2 :i* is done by *ang+ ! major1 or m9or1 or bo? ! propositions1 into conv]t$ propositions equipoll5t to !m4 ,f :5ce foll[ ?ree o!r figures2 ( :i*1 two are inflect$1 & ! ?ird inv]t$4 ,! fir/ ( ^! ?ree is made by ! conv]sion ( ! major proposition4 ,= let ! m9or1 middle1 & major t]ms /& 9 direct ord]1 ?us1 man is a liv+ creature1 is n a /one1 :i* is ! fir/ or direct figure2 ! inflection w be by conv]t+ ! major proposition 9 ?is mann]1 man is a liv+ creature1 a /one is n a liv+ creature2 & ?is is ! second figure1 or ! fir/ ( ! indirect figures2 9 :i* ! conclusion w be1 man is n a /one4 ,= 7hav+ %[n 9 ! la/ *apt]1 art4 #ad1 t univ]sal propositions1 conv]t$ by contradiction ( ! t]ms1 are equipoll5t7 bo? ^? syllogisms conclude alike2 s t if ! major be read 7l ,hebrew7 backw>ds1 ?us1 a liv+ creature is n a /one1 x w be direct aga91 z x 0 be=e4 ,9 l mann] ?is direct syllogism1 man is n a tree1 is n a pe>-tree1 w be made indirect by conv]t+ ! major proposition 7by contradiction ( ! t]ms7 into ano!r equipoll5t to x1 ?us1 man is n a tree1 a pe>-tree is a tree2 = ! same conclusion w foll[1 man is n a pe>-tree4 ,b = ! conv]sion ( ! direct figure into ! fir/ indirect figure1 ! major t]m 9 ! direct figure ought to be negative4 ,= ?\gh ?is direct1 man is a liv+ creature1 is a body1 be made indirect1 by conv]t+ ! major proposition1 ?us1 ,man is a liv+ creature1 ,n a body is n a liv+ creature1 ,!re=e1 ,e man is a body2 ,yet ?is conv]sion appe>s s obscure1 t ?is mood is ( no use at all4 ,by ! conv]sion ( ! major proposition1 x is manife/1 t 9 ?is figure1 ! middle t]m is always ! pr$icate 9 bo? ! premises4 ,sid5ote3 ,second indirect figure h[ made4 #aj4 ,! second indirect figure is made by conv]t+ ! m9or proposition1 s z t ! middle t]m is ! subject 9 bo?4 ,b ?is nev] concludes univ]sally1 & !re=e is ( no use 9 philosophy4 ,nev]!less ,i w set d[n an example ( x2 by :i* ?is direct ,e man is a liv+ creature1 ,e liv+ creature is a body1 by conv]sion ( ! m9or proposition1 w /& ?us1 ,"s liv+ creature is a man1 ,e liv+ creature is a body1 ,!re=e1 ,"s man is a body4 ,= e man is a liv+ creature _c be conv]t$ into ?is1 e liv+ creature is a man3 & !re=e if ?is syllogism be re/or$ to its direct =m1 ! m9or proposition w be "s man is a liv+ creature1 & consequ5tly ! conclusion w be "s man is a body1 see+ ! m9or t]m man1 :i* is ! subject 9 ! conclusion1 is a p>ticul> "n4 ,sid5ote3 ,h[ ! ?ird indirect figure is made4 #aa4 ,! ?ird indirect or inv]t$ figure1 is made by ! conv]sion ( bo? ! premises4 ,= example1 ?is direct syllogism1 ,e man is a liv+ creature1 ,e liv+ creature is n a /one1 ,!re=e1 ,e man is n a /one1 be+ inv]t$1 w /& ?us1 ,e /one is n a liv+ creature1 ,:atsoev] is n a liv+ creature1 is n a man1 ,!re=e1 ,e /one is n a man2 :i* conclusion is ! conv]se ( ! direct conclusion1 & equipoll5t to ! same4 ,! figures1 !re=e1 ( syllogisms1 if !y be numb]$ by ! div]se situation ( ! middle t]m only1 are b ?ree2 9 ! fir/ :]e(1 ! middle t]m has ! middle place2 9 ! second1 ! la/2 & 9 ! ?ird1 ! fir/ place4 ,b if !y be numb]$ accord+ to ! situation ( ! t]ms simply1 !y are f\r2 = ! fir/ may be di/+ui%$ aga9 into two1 namely1 into direct & inv]t$4 ,f :5ce x is evid5t1 t ! controv]sy among logicians conc]n+ ! f\r? figure1 is a m]e 1 or cont5tion ab\t ! "n !re(2 =1 z = ! ?+ itself1 x is pla9 t ! situation ( ! t]ms 7n consid]+ ! quantity or quality by :i* ! moods are di/+ui%$7 makes f\r diff]5ces ( syllogisms1 :i* may be call$ figures1 or h any o!r "n at pleasure4 ,sid5ote3 ,!re are _m moods 9 e figure1 b mo/ ( !m useless 9 philosophy4 #ab4 ,9 e "o ( ^! figures !re are _m moods1 :i* are made by v>y+ ! premises accord+ to all ! diff]5ces !y are capable (1 by quantity & quality2 z namely1 9 ! direct figure !re are six moods2 9 ! fir/ indirect figure1 f\r2 9 ! second1 f\rte52 & 9 ! ?ird1 eighte54 ,b because f ! direct figure ,i reject$ z sup]flu\s all moods besides t :i* consi/s ( univ]sal propositions1 & ^: m9or proposition is affirmative1 ,i d1 toge!r ) x1 reject ! moods ( ! re/ ( ! figures :i* are made by conv]sion ( ! premises 9 ! direct figure4 ,sid5ote3 ,an hypo!tical syllogism :5 equipoll5t to a categorical4 #ac4 ,z x 0 %[$ be=e1 t 9 necess>y propositions a categorical & hypo!tical proposition are equipoll5t2 s likewise x is manife/ t a categorical & hypo!tical syllogism are equival5t4 ,= e categorical syllogism1 z ?is1 ,e man is a liv+ creature1 ,e liv+ creature is a body1 ,!re=e1 ,e man is a body1 is ( equal =ce ) ?is hypo!tical syllogism3 ,if any ?+ be a man1 ! same is also a liv+ creature1 ,if any ?+ be a liv+ creature1 ! same is a body1 ,!re=e1 ,if any ?+ be a man1 ! same is a body4 ,9 l mann]1 ?is categorical syllogism 9 an indirect figure1 ,no /one is a liv+ creature1 ,e man is a liv+ creature1 ,!re=e1 ,no man is a /one1 ,or1 ,no /one is a man1 is equival5t to ?is hypo!tical syllogism3 ,if any ?+ be a man1 ! same is a liv+ creature1 ,if any ?+ be a /one1 ! same is n a liv+ creature1 ,!re=e1 ,if any ?+ be a /one1 ! same is n a man1 ,or1 ,if any ?+ be a man1 ! same is n a /one4 ,& ?us mu* seems suffici5t = ! nature ( syllogisms2 7= ! doctr9e ( moods & figures is cle>ly deliv]$ by o!rs t h writt5 l>gely & pr(itably ( ! same74 ,nor are precepts s necess>y z practice = ! atta9+ ( true ratioc9ation2 & !y t /udy ! demon/rations ( ma!maticians1 w soon] le>n true logic1 ?an !y t sp5d "t 9 read+ ! rules ( syllogiz+ :i* logicians h made2 no o!rwise ?an little *ildr5 le>n to g1 n by precepts1 b by ex]cis+ _! feet4 ,?is1 !re=e1 may s]ve = ! fir/ pace 9 ! way to ,philosophy4 ,9 ! next place ,i %all speak ( ! faults & errors into :i* m5 t reason unw>ily are apt to fall2 & ( _! k9ds & causes4 ------- ,,*apt] ,v4 ,,( ,,err+1 ,,falsity1 ,,& ,,captions4 #a4 ,err+ & falsity h[ !y diff]4 ,error ( ! m9d by itself )\t ! use ( words1 h[ x happ5s4--#b4 ,a sev5fold incoh]5cy ( names1 e "o ( :i* makes always a false proposition4--#c4 ,examples ( ! fir/ mann] ( incoh]5cy4--#d4 ,( ! second4--#e4 ,( ! ?ird4--#f4 ,( ! f\r?4--#g4 ,( ! fif?4--#h4 ,( ! six?4--#i4 ,( ! sev5?4--#aj4 ,falsity ( propositions detect$ by resolv+ ! t]ms ) def9itions cont9u$ till !y come to simple names1 or names t are ! mo/ g5]al ( _! k9d4--#aa4 ,( ! fault ( a syllogism consi/+ 9 ! implication ( ! t]ms ) ! copula4--#ab4 ,( ! fault :i* consi/s 9 equivocation4--#ac4 ,sophi/ical captions are (t5] faulty 9 ! matt] ?an 9 ! =m ( syllogisms4 ,sid5ote3 ,err+ falsity h[ !y diff]4 ,error ( ! m9d by itself1 )\t ! use ( words1 h[ x happ5s4 #a4 ,m5 are subject to err n only 9 affirm+ & d5y+1 b also 9 p]ception1 & 9 sil5t cogitation4 ,9 affirm+ & d5y+1 :5 !y call any ?+ by a "n1 :i* is n ! "n !re(2 z if f see+ ! sun fir/ by reflection 9 wat]1 & aft]w>ds aga9 directly 9 ! firmam5t1 we %\ld to bo? ^? appe>ances give ! "n ( sun1 & say !re are two suns2 :i* none b m5 c d1 = no o!r liv+ creatures h ! use ( names4 ,?is k9d ( error only des]ves ! "n ( falsity1 z aris+1 n f s5se1 nor f ! ?+s !mselves1 b f pron\nc+ ra%ly2 = names h _! con/itution1 n f ! species ( ?+s1 b f ! w & cons5t ( m54 ,& h5ce x comes to pass1 t m5 pron\nce falsely1 by _! own neglig5ce1 9 dep>t+ f su* appellations ( ?+s z are agre$ ^u1 & are n deceiv$ nei!r by ! ?+s1 nor by ! s5se2 = !y d n p]ceive t ! ?+ !y see is call$ sun1 b !y give x t "n f _! own w & agreem5t4 ,tacit errors1 or ! errors ( s5se & cogitation1 are made1 by pass+ f "o imag9ation to ! imag9ation ( ano!r diff]5t ?+2 or by feign+ t to be pa/1 or future1 :i* nev] 01 nor ev] %all be2 z :51 by see+ ! image ( ! sun 9 wat]1 we imag9e ! sun itself to be !re2 or by see+ swords1 t !re has be5 or %all be fight+1 because x uses to be s = ! mo/ "p2 or :5 f promises we feign ! m9d ( ! promis] to be su* & su*2 or la/ly1 :5 f any sign we va9ly imag9e some?+ to be signifi$1 :i* is n4 ,& errors ( ?is sort are common to all ?+s t h s5se2 & yet ! deception proce$s nei!r f our s5ses1 nor f ! ?+s we p]ceive2 b f ourselves :ile we feign su* ?+s z are b m]e images to be some?+ m ?an images4 ,b nei!r ?+s1 nor imag9ations ( ?+s1 c be said to be false1 see+ !y are truly :at !y are2 nor d !y1 z signs1 promise any ?+ :i* !y d n p]=m2 = !y inde$ d n promise at all1 b we f !m2 nor d ! cl\ds1 b we1 f see+ ! cl\ds1 say x %all ra94 ,! be/ way1 !re=e1 to free ourselves f su* errors z arise f natural signs1 is fir/ ( all1 be=e we beg9 to reason conc]n+ su* conjectural ?+s1 to suppose ourselves ignorant1 & !n to make use ( our ratioc9ation2 = ^! errors proce$ f ! want ( ratioc9ation2 :]eas1 errors :i* consi/ 9 affirmation & negation1 7t is1 ! falsity ( propositions7 proce$ only f reason+ amiss4 ,( ^!1 !re=e1 z repugnant to philosophy1 ,i w speak pr9cipally4 ,sid5ote3 ,a sev5fold incoh]5cy ( names1 all ( :i* make always a false proposition4 #b4 ,errors :i* happ5 9 reason+1 t is1 9 syllogiz+1 consi/ ei!r 9 ! falsity ( ! premises1 or ( ! inf]5ce4 ,9 ! fir/ ( ^! cases1 a syllogism is said to be faulty 9 ! matt] ( x2 & 9 ! second case1 9 ! =m4 ,i w fir/ consid] ! matt]1 namely1 h[ _m ways a proposition may be false2 & next ! =m1 & h[ x comes to pass1 t :5 ! premises are true1 ! inf]5ce is1 not)/&+1 false4 ,see+1 !re=e1 t proposition only is true1 7*ap1 ,,iii1 art4 #g7 9 :i* are copulat$ two names ( "o & ! same ?+2 & t always false1 9 :i* names ( diff]5t ?+s are copulat$1 look h[ _m ways names ( diff]5t ?+s may be copulat$1 & s _m ways a false proposition may be made4 ,n[1 all ?+s to :i* we give names1 may be r$uc$ to ^! f\r k9ds1 namely1 bodies1 accid5ts1 phantasms1 & names !mselves2 & !re=e1 9 e true proposition1 x is necess>y t ! names copulat$1 be bo? ( !m names ( bodies1 or bo? names ( accid5ts1 or bo? names ( phantasms1 or bo? names ( names4 ,= names o!rwise copulat$ are incoh]5t1 & con/itute a false proposition4 ,x may happ51 also1 t ! "n ( a body1 ( an accid5t1 or ( a phantasm1 may be copulat$ ) ! "n ( a spee*4 ,s t copulat$ names may be incoh]5t sev5 mann] ( ways4 #a4 ,if ! "n ( a ,body ! "n ( an ,accid5t4 #b4 ,if ! "n ( a ,body ! "n ( a ,phantasm4 #c4 ,if ! "n ( a ,body be ! "n ( a ,"n4 #d4 ,if ! "n ( an ,accid5t copulat$ ! "n ( a ,phantasm4 #e4 ,if ! "n ( an ,accid5t ) ! "n ( a ,"n4 #f4 ,if ! "n ( a ,phantasm ! "n ( a ,"n4 #g4 ,if ! "n ( a ,body1 ! "n ( a ,spee*4 ,accid5t1 or ,phantasm ,( all :i* ,i w give "s examples4 ,sid5ote3 ,examples ( ! fir/ mann] ( incoh]5cy4 #c4 ,aft] ! fir/ ( ^! ways propositions are false1 :5 ab/ract names are copulat$ ) concrete names2 z 79 ,lat9 & ,greek7 esse e/ ens1 ess5tia e/ ens1 7i472 quidditas e/ ens1 & _m ! l1 :i* are f\nd 9 ,ari/otle's ,metaphysics4 ,also1 ! und]/&+ worke?1 ! und]/&+ und]/&e?1 ! sight see?2 a body is magnitude1 a body is quantity1 a body is ext5sion2 to be a man is a man1 :it5ess is a :ite ?+1 c42 :i* is z if "o %\ld say1 ! runn] is ! runn+1 or ! walk walke?4 ,moreov]1 ess5ce is sep>at$1 sub/ance is ab/ract$3 & o!rs l ^!1 or d]iv$ f ^!1 7) :i* common philosophy ab\nds47 ,= see+ no subject ( an accid5t 7t is1 no body7 is an accid5t3 no "n ( an accid5t ought to be giv5 to a body1 nor ( a body to an accid5t4 ,sid5ote3 ,! second4 #d4 ,false1 9 ! second mann]1 are su* propositions z ^!2 a gho/ is a body1 or a _s1 t is1 a ?9 body2 s5sible species fly up & d[n 9 ! air1 or are mov$ hi!r & ?i!r1 :i* is prop] to bodies2 also1 a %ad[ is mov$1 or is a body2 light is mov$1 or is a body2 col\r is ! object ( sight1 s\nd ( he>+2 space or place is ext5d$2 & innum]able o!rs ( ?is k9d4 ,= see+ gho/s1 s5sible species1 a %ad[1 light1 col\r1 s\nd1 space1 c4 appe> to u no less sleep+ ?an wak+1 !y _c be ?+s )\t u1 b only phantasms ( ! m9d t imag9es !m2 & !re=e ! names ( ^!1 copulat$ ) ! names ( bodies1 _c con/itute a true proposition4 ,sid5ote3 ,! ?ird4 #e4 ,false propositions ( ! ?ird k9d1 are su* z ^!2 g5us e/ ens1 univ]sale e/ ens1 ens de ente prdicatur4 ,= g5us1 & univ]sale1 & pr$ic>e1 are names ( names1 & n ( ?+s4 ,also1 numb] is inf9ite1 is a false proposition2 = no numb] c be inf9ite1 b only ! ^w numb] is !n call$ an indef9ite "n :5 !re is no det]m9$ numb] answ]+ to x 9 ! m9d4 ,sid5ote3 ,! f\r?4 #f4 ,to ! f\r? k9d belong su* false propositions z ^!1 an object is ( su* magnitude or figure z appe>s to ! behold]s2 col\r1 light1 s\nd1 are 9 ! object2 & ! l4 ,= ! same object appe>s sometimes great]1 sometimes less]1 sometimes squ>e1 sometimes r\nd1 accord+ to ! div]sity ( ! di/ance & m$ium2 b ! true magnitude & figure ( ! ?+ se5 is always "o & ! same2 s t ! magnitude & figure :i* appe>s1 is n ! true magnitude & figure ( ! object1 nor any?+ b phantasm2 & !re=e1 9 su* propositions z ^!1 ! names ( accid5ts are copulat$ ) ! names ( phantasms4 ,sid5ote3 ,! fif?4 #g4 ,propositions are false 9 ! fif? mann]1 :5 x is said t ! def9ition is ! ess5ce ( a ?+2 :it5ess1 or "s o!r accid5t1 is ! g5us1 or univ]sal4 ,= def9ition is n ! ess5ce ( any ?+1 b a spee* signify+ :at we conceive ( ! ess5ce !re(2 & s also n :it5ess itself1 b ! ^w :it5ess1 is a g5us1 or an univ]sal "n4 ,sid5ote3 ,! six?4 #h4 ,9 ! six? mann] !y err1 t say ! idea ( any?+ is univ]sal2 z if !re c\ld be 9 ! m9d an image ( a man1 :i* 7 n ! image ( "s "o man1 b a man simply1 :i* is impossible2 = e idea is "o1 & ( "o ?+2 b !y are deceiv$ 9 ?is1 t !y put ! "n ( ! ?+ = ! idea !re(4 ,sid5ote3 ,! sev5?4 #i4 ,!y err 9 ! sev5? mann]1 t make ?is di/9ction betwe5 ?+s t h be+1 t "s ( !m exi/ by !mselves1 o!rs by accid5t2 namely1 because ,socrates is a man is a necess>y proposition1 & ,socrates is a musician a cont+5t proposition1 !re=e !y say "s ?+s exi/ necess>ily or by !mselves1 o!rs cont+5tly or by accid5t2 :]eby1 see+ necess>y1 cont+5t1 by itself1 by accid5t1 are n names ( ?+s1 b ( propositions1 !y t say any ?+ t has be+1 exi/s by accid5t1 copulate ! "n ( a proposition ) ! "n ( a ?+4 ,9 ! same mann] also1 !y err1 :i* place "s ideas 9 ! und]/&+1 o!rs 9 ! fancy2 z if f ! und]/&+ ( ?is proposition1 man is a liv+ creature1 we _h "o idea or image ( a man d]iv$ f s5se to ! memory1 & ano!r to ! und]/&+2 :]e9 t :i* deceives !m is ?is1 t !y ?9k "o idea %\ld be answ]able to a "n1 ano!r to a proposition1 :i* is false2 = proposition signifies only ! ord] ( ^? ?+s "o aft] ano!r1 :i* we obs]ve 9 ! same idea ( man2 s t ?is proposition1 man is a liv+ creature raises b "o idea 9 u1 ?\gh 9 t idea we consid] t fir/1 = :i* he is call$ man1 & next t1 = :i* he is call$ liv+ creature4 ,! falsities ( propositions 9 all ^! sev]al mann]s1 is to be discov]$ by ! def9itions ( ! copulat$ names4 ,sid5ote3 ,falsity ( propositions detect$ by resolv+ ! t]ms ) def9itions4 #aj4 ,b :5 names ( bodies are copulat$ ) names ( bodies1 names ( accid5ts ) names ( accid5ts1 names ( names ) names ( names1 & names ( phantasms1 ) names ( phantasms1 if we1 nev]!less1 rema9 /ill d\btful :e!r su* propositions are true1 we ought !n 9 ! fir/ place to f9d out ! def9ition ( bo? ^? names1 & aga9 ! def9itions ( su* names z are 9 ! =m] def9ition1 & s proce$ by a cont9ual resolution till we come to a simple "n1 t is1 to ! mo/ g5]al or mo/ univ]sal "n ( t k9d2 & if aft] all ?is1 ! tru? or falsity !re( be n evid5t1 we mu/ se>* x out by philosophy1 & ratioc9ation1 beg9n+ f def9itions4 ,= e proposition1 univ]sally true1 is ei!r a def9ition1 or "p ( a def9ition1 or ! evid5ce ( x dep5ds ^u def9itions4 ,sid5ote3 ,( ! fault ( a syllogism consi/+ 9 implication ( ! t]ms ) ! copula4 #aa4 ,t fault ( a syllogism :i* lies hid 9 ! =m !re(1 w always be f\nd ei!r 9 ! implication ( ! copula ) "o ( ! t]ms1 or 9 ! equivocation ( "s ^w2 & 9 ei!r ( ^! ways !re w be f\r t]ms1 :i* 7z ,i h %ewn7 _c /& 9 a true syllogism4 ,n[ ! implication ( ! copula ) ei!r t]m1 is easily detect$ by r$uc+ ! propositions to pla9 & cle> pr$ication2 z 7= example7 if any man %\ld argue ?us1 ,! h& t\*e? ! p51 ,! p5 t\*e? ! pap]1 ,!re=e1 ,! h& t\*e? ! pap]2 ! fallacy w easily appe> by r$uc+ x1 ?us3 ,! h&1 is1 t\*+ ! p51 ,! p51 is1 t\*+ ! pap]1 ,!re=e1 ,! h&1 is1 t\*+ ! pap]2 :]e !re are manife/ly ^! f\r t]ms1 ! h&1 t\*+ ! p51 ! p51 & t\*+ ! ! pap]4 ,b ! dang] ( be+ deceiv$ by sophisms ( ?is k9d1 does n seem to be s great1 z t ,i ne$ insi/ long] ^u !m4 ,sid5ote3 ,( ! fault :i* consi/s 9 equivocation4 #ab4 ,& ?\gh !re may be fallacy 9 equivocal t]ms1 yet 9 ^? t be manife/ly su*1 !re is none at all2 nor 9 metaphors1 = !y pr(ess ! transf]r+ ( names f "o ?+ to ano!r4 ,nev]!less1 sometimes equivocals 7& ^? n v obscure7 may deceive2 z 9 ?is argum5tation3--,x belongs to metaphysics to treat ( pr9ciples2 b ! fir/ pr9ciple ( all1 is1 t ! same ?+ _c bo? exi/ & n exi/ at ! same "t2 & !re=e x belongs to metaphysics to treat :e!r ! same ?+ may bo? exi/ & n exi/ at ! same "t2 :]e ! fallacy lies 9 ! equivocation ( ! ^w pr9ciple2 = :]eas ,ari/otle 9 ! beg9n+ ( 8 ,metaphysics1 says1 t ! treat+ ( pr9ciples belongs to prim>y sci5ce1 he und]/&s by pr9ciples1 causes ( ?+s1 & c]ta9 exi/5ces :i* he calls prim>y2 b :]e he says a prim>y proposition is a pr9ciple1 by pr9ciple1 !re1 he means ! beg9n+ & cause ( k1 t is1 ! und]/&+ ( words1 :i*1 if any man want1 he is incapable ( le>n+4 ,sid5ote3 ,sophi/ical captions are (t5] faulty 9 ! matt] ?an 9 ! =m ( syllogisms4 #ac4 ,b ! captions ( sophi/s & sceptics1 by :i* !y 7 wont1 ( old1 to d]ide & oppose tru?1 7 faulty = ! mo/ "p1 n 9 ! =m1 b 9 ! matt] ( syllogism2 & !y deceiv$ n o!rs (t5] ?an !y 7 !mselves deceiv$4 ,= ! =ce ( t fam\s argum5t ( ,z5o aga9/ motion1 consi/$ 9 ?is proposition1 :atsoev] may be divid$ into p>ts1 inf9ite 9 numb]1 ! same is inf9ite2 :i* he1 )\t d\bt1 ?\ght to be true1 yet nev]!less is false4 ,= to be divid$ into inf9ite p>ts1 is no?+ else b to be divid$ into z _m p>ts z any man w4 ,b x is n necess>y t a l9e %\ld h p>ts inf9ite 9 numb]1 or be inf9ite1 because ,i c divide & subdivide x z (t5 z ,i please2 = h[ _m p>ts soev] ,i make1 yet _! numb] is f9ite2 b because he t says p>ts1 simply1 )\t add+ h[ _m1 does n limit any numb]1 b leaves x to ! det]m9ation ( ! he>]1 !re=e we say commonly1 a l9e may be divid$ inf9itely2 :i* _c be true 9 any o!r s5se4 ,sid5ote3 ,conclusion4 ,& ?us mu* may suffice conc]n+ syllogism1 :i* is1 z x 71 ! fir/ pace t[>ds philosophy2 9 :i* ,i h said z mu* z is necess>y to tea* any man f :5ce all true argum5tation has its =ce4 ,& to enl>ge ?is treatise ) all t may be heap$ toge!r1 w\ld be z sup]flu\s1 z if "o %\ld 7z ,i said be=e7 give a "y *ild precepts = ! tea*+ ( him to g2 = ! art ( reason+ is n s well le>n$ by precepts z by practice1 & by ! read+ ( ^? books 9 :i* ! conclusions are all made by sev]e demon/ration4 ,& s ,i pass on to ! way ( philosophy1 t is1 to ! me?od ( /udy4 ------- ,,*apt] ,,vi4 ,,( ,,me?od4 #a4 ,me?od & sci5ce def9$4--#b4 ,x is m easily kn[n conc]n+ s+ul>1 ?an univ]sal ?+s1 t !y are2 & contr>ily1 x is m easily kn[n conc]n+ univ]sal1 ?an s+ul> ?+s1 :y !y are1 or :at are _! causes4--#c4 ,:at x is philosoph]s seek to "k4--#d4 ,! fir/ "p1 by :i* pr9ciples are f\nd out1 is purely analytical4--#e4 ,! highe/ causes1 & mo/ univ]sal 9 e k9d1 are kn[n by !mselves4--#f4 ,me?od f pr9ciples f\nd out1 t5d+ to sci5ce simply1 :at x is4--#g4 ,t me?od ( civil & natural sci5ce1 :i* proce$s f s5se to pr9ciples1 is analytical2 & aga91 t1 :i* beg9s at pr9ciples1 is syn!tical4--#h4 ,! me?od ( se>*+ out1 :e!r any ?+ prop\nd$ be matt] or accid5t4--#i4 ,! me?od ( seek+ :e!r any accid5t be 9 ?is1 or 9 t subject4--#aj4 ,! me?od ( se>*+ aft] ! cause ( any effect prop\nd$4--#aa4 ,words s]ve to inv5tion1 z m>ks2 to demon/ration1 z signs4--#ab4 ,! me?od ( demon/ration is syn!tical4--#ac4 ,def9itions only are prim>y & univ]sal propositions4--#ad4 ,! nature & def9ition ( a def9ition4--#ae4 ,! prop]ties ( a def9ition4--#af4 ,! nature ( a demon/ration4--#ag4 ,! prop]ties ( a demon/ration1 & ord] ( ?+s to be demon/rat$4--#ah4 ,! faults ( a demon/ration4--#ai4 ,:y ! analytical me?od ( geometricians _c be treat$ ( 9 ?is place4 ,sid5ote3 ,me?od & sci5ce def9$4 #a4 ,= ! und]/&+ ( me?od1 x w be necess>y = me to repeat ! def9ition ( philosophy1 deliv]$ above 7,*ap4 ,i1 art4 #b47 9 ?is mann]1 ,philosophy is ! k we acquire1 by true ratioc9ation1 ( appe>ances1 or app>5t effects1 f ! k we h ( "s possible production or g5]ation ( ! same2 & ( su* production1 z has be5 or may be1 f ! k we h ( ! effects4 ,,me?od1 !re=e1 9 ! /udy ( philosophy1 is ! %orte/ way ( f9d+ out effects by _! kn[n causes1 or ( causes by _! kn[n effects4 ,b we are !n said to "k any effect1 :5 we "k t !re be causes ( ! same1 & 9 :at subject ^? causes are1 & 9 :at subject !y produce t effect1 & 9 :at mann] !y "w ! same4 ,& ?is is ! sci5ce ( causes1 or1 z !y call x1 ( ! 4 ,all o!r sci5ce1 :i* is call$ ! 1 is ei!r p]ception by s5se1 or ! imag9ation1 or memory rema9+ aft] su* p]ception4 ,! fir/ beg9n+s1 !re=e1 ( k1 are ! phantasms ( s5se & imag9ation2 & t !re be su* phantasms we "k well en\gh by nature2 b to "k :y !y be1 or f :at causes !y proce$1 is ! "w ( ratioc9ation2 :i* consi/s 7z is said above1 9 ! #a/ ,*apt]1 ,art4 #b7 9 composition1 & division or resolution4 ,!re is !re=e no me?od1 by :i* we f9d out ! causes ( ?+s1 b is ei!r compositive or resolutive1 or p>tly compositive1 & p>tly resolutive4 ,& ! resolutive is commonly call$ analytical me?od1 z ! compositive is call$ syn!tical4 ,sid5ote3 ,x is easi] kn[n conc]n+ s+ul> ?an univ]sal ?+s1 t !y are2 & contr>ily x is easi] kn[n conc]n+ univ]sal ?an s+ul> ?+s1 :y !y are1 or :at are _! causes4 #b4 ,x is common to all sorts ( me?od1 to proce$ f kn[n ?+s to unkn[n2 & ?is is manife/ f ! cit$ def9ition ( philosophy4 ,b 9 k by s5se1 ! :ole object is m kn[n1 ?an any "p !re(2 z :5 we see a man1 ! conception or :ole idea ( t man is fir/ or m kn[n1 ?an ! p>ticul> ideas ( 8 be+ figurate1 animate1 & rational2 t is1 we fir/ see ! :ole man1 & take notice ( 8 be+1 be=e we obs]ve 9 him ^? o!r p>ticul>s4 ,& !re=e 9 any k ( ! 1 or t any ?+ is1 ! beg9n+ ( our se>* is f ! :ole idea2 & contr>ily1 9 our k ( ! 1 or ( ! causes ( any ?+1 t is1 9 ! sci5ces1 we h m k ( ! causes ( ! p>ts ?an ( ! :ole4 ,= ! cause ( ! :ole is comp\nd$ ( ! causes ( ! p>ts2 b x is necess>y t we "k ! ?+s t are to be comp\nd$1 be=e we c "k ! :ole comp\nd4 ,n[1 by p>ts1 ,i d n "h mean p>ts ( ! ?+ itself1 b p>ts ( its nature2 z1 by ! p>ts ( man1 ,i d n und]/& 8 head1 8 %\ld]s1 8 arms1 c4 b 8 figure1 quantity1 motion1 s5se1 reason1 & ! l2 :i* accid5ts be+ comp\nd$ or put toge!r1 con/itute ! :ole nature ( man1 b n ! man himself4 ,& ?is is ! mean+ ( t common say+1 namely1 t "s ?+s are m kn[n to u1 o!rs m kn[n to nature2 = ,i d n ?9k t !y1 :i* s di/+ui%1 mean t some?+ is kn[n to nature1 :i* is kn[n to no man2 & !re=e1 by ^? ?+s1 t are m kn[n to u1 we are to und]/& ?+s we take notice ( by our s5ses1 &1 by m kn[n to nature1 ^? we acquire ! k ( by reason2 = 9 ?is s5se x is1 t ! :ole1 t is1 ^? ?+s t h univ]sal names1 7:i*1 = brevity's sake1 ,i call univ]sal7 are m kn[n to u ?an ! p>ts1 t is1 su* ?+s z h names less univ]sal1 7:i* ,i !re=e call s+ul>72 & ! causes ( ! p>ts are m kn[n to nature ?an ! cause ( ! :ole2 t is1 univ]sals ?an s+ul>s4 ,sid5ote3 ,:at x is philosoph]s seek to "k4 #c4 ,9 ! /udy ( philosophy1 m5 se>* aft] sci5ce ei!r simply or indef9itely2 t is1 to "k z mu* z !y c1 )\t prop\nd+ to !mselves any limit$ "q2 or !y enquire into ! cause ( "s det]m9$ appe>ance1 or endeav\r to f9d out ! c]ta9ty ( some?+ 9 "q1 z :at is ! cause ( light1 ( heat1 ( gravity1 ( a figure prop\nd$1 & ! l2 or 9 :at subject any prop\nd$ accid5t is inh]5t2 or :at may conduce mo/ to ! g5]ation ( "s prop\nd$ effect f _m accid5ts2 or 9 :at mann] p>ticul> causes ought to be comp\nd$ = ! production ( "s c]ta9 effect4 ,n[1 accord+ to ?is v>iety ( ?+s 9 "q1 sometimes ! analytical me?od is to be us$1 & sometimes ! syn!tical4 ,sid5ote3 ,! fir/ "p1 by :i* pr9ciples are discov]$1 is purely analytical4 #d4 ,b to ^? t se>* aft] sci5ce indef9itely1 :i* consi/s 9 ! k ( ! causes ( all ?+s1 z f> =? z x may be atta9$1 7& ! causes ( s+ul> ?+s are comp\nd$ ( ! causes ( univ]sal or simple ?+s7 x is necess>y t !y "k ! causes ( univ]sal ?+s1 or ( su* accid5ts z are common to all bodies1 t is1 to all matt]1 be=e !y c "k ! causes ( s+ul> ?+s1 t is1 ( ^? accid5ts by :i* "o ?+ is di/+ui%$ f ano!r4 ,&1 aga91 !y mu/ "k :at ^? univ]sal ?+s are1 be=e !y c "k _! causes4 ,moreov]1 see+ univ]sal ?+s are conta9$ 9 ! nature ( s+ul> ?+s1 ! k ( !m is to be acquir$ by reason1 t is1 by resolution4 ,= example1 if !re be prop\nd$ a conception or idea ( "s s+ul> ?+1 z ( a squ>e1 ?is squ>e is to be resolv$ into a pla91 t]m9at$ ) a c]ta9 numb] ( equal & /raight l9es & "r angles4 ,= by ?is resolution we h ^! ?+s univ]sal or agreeable to all matt]1 namely1 l9e1 pla91 7:i* conta9s sup]ficies7 t]m9at$1 angle1 /raightness1 rectitude1 & equality2 & if we c f9d out ! causes ( ^!1 we may comp\nd !m altoge!r into ! cause ( a squ>e4 ,aga91 if any man prop\nd to himself ! conception ( gold1 he may1 by resolv+1 come to ! ideas ( solid1 visible1 heavy1 7t is1 t5d+ to ! c5tre ( ! e>?1 or d[nw>ds7 & _m o!r m univ]sal ?an gold itself2 & ^! he may resolve aga91 till he come to su* ?+s z are mo/ univ]sal4 ,& 9 ?is mann]1 by resolv+ cont9ually1 we may come to "k :at ^? ?+s are1 ^: causes be+ fir/ kn[n sev]ally1 & aft]w>ds comp\nd$1 br+ u to ! k ( s+ul> ?+s4 ,i conclude1 !re=e1 t ! me?od ( atta9+ to ! univ]sal k ( ?+s1 is purely analytical4 ,sid5ote3 ,! highe/ causes1 & mo/ univ]sal 9 e k9d1 are kn[n by !mselves4 #e4 ,b ! causes ( univ]sal ?+s 7( ^?1 at lea/1 t h any cause7 are manife/ ( !mselves1 or 7z !y say commonly7 kn[n to nature2 s t !y ne$ no me?od at all2 = !y h all b "o univ]sal cause1 :i* is motion4 ,= ! v>iety ( all figures arises out ( ! v>iety ( ^? motions by :i* !y are made2 & motion _c be und]/ood to h any o!r cause besides motion2 nor has ! v>iety ( ^? ?+s we p]ceive by s5se1 z ( col\rs1 s\nds1 sav\rs1 c4 any o!r cause ?an motion1 resid+ p>tly 9 ! objects t "w ^u our s5ses1 & p>tly 9 ourselves1 9 su* mann]1 z t x is manife/ly "s k9d ( motion1 ?\gh we _c1 )\t ratioc9ation1 come to "k :at k9d4 ,= ?\gh _m _c und]/& till x be 9 "s sort demon/rat$ to !m1 t all mutation consi/s 9 motion2 yet ?is happ5s n f any obscurity 9 ! ?+ itself1 7= x is n intelligible t any?+ c dep>t ei!r f re/1 or f ! motion x has1 except by motion71 b ei!r by hav+ _! natural disc\rse corrupt$ ) =m] op9ions receiv$ f _! ma/]s1 or else = ?is1 t !y d n at all b5d _! m9d to ! enquir+ out ( tru?4 ,sid5ote3 ,me?od f pr9ciples f\nd out1 t5d+ to sci5ce simply1 :at x is4 #f4 ,by ! k !re=e ( univ]sals1 & ( _! causes 7:i* are ! fir/ pr9ciples by :i* we "k ! ( ?+s7 we h 9 ! fir/ place _! def9itions1 7:i* are no?+ b ! explication ( our simple conceptions47 ,= example1 he t has a true conception ( place1 _c be ignorant ( ?is def9ition1 place is t space :i* is possess$ or fill$ adequately by "s body2 & s1 he t conceives motion aright1 _c b "k t motion is ! privation ( "o place1 & ! acquisition ( ano!r4 ,9 ! next place1 we h _! g5]ations or descriptions2 z 7= example7 t a l9e is made by ! motion ( a po9t1 sup]ficies by ! motion ( a l9e1 & "o motion by ano!r motion1 c4 ,x rema9s1 t we enquire :at motion begets su* & su* effects2 z1 :at motion makes a /raight l9e1 & :at a circul>2 :at motion ?ru/s1 :at draws1 & by :at way2 :at makes a ?+ :i* is se5 or he>d1 to be se5 or he>d sometimes 9 "o mann]1 sometimes 9 ano!r4 ,n[ ! me?od ( ?is k9d ( enquiry1 is compositive4 ,= fir/ we are to obs]ve :at effect a body mov$ produce?1 :5 we consid] no?+ 9 x besides its motion2 & we see pres5tly t ?is makes a l9e1 or l5g?2 next1 :at ! motion ( a long body produces1 :i* we f9d to be sup]ficies2 & s =w>ds1 till we see :at ! effects ( simple motion are2 & !n1 9 l mann]1 we are to obs]ve :at proce$s f ! addition1 multiplication1 subtraction1 & division1 ( ^! motions1 & :at effects1 :at figures1 & :at prop]ties1 !y produce2 f :i* k9d ( contemplation sprung t "p ( philosophy :i* is call$ geometry4 ,f ?is consid]ation ( :at is produc$ by simple motion1 we are to pass to ! consid]ation ( :at effects "o body mov$ worke? ^u ano!r2 & because !re may be motion 9 all ! sev]al p>ts ( a body1 yet s z t ! :ole body rema9 /ill 9 ! same place1 we mu/ enquire fir/1 :at motion cause? su* & su* motion 9 ! :ole1 t is1 :5 "o body invades ano!r body :i* is ei!r at re/ or 9 motion1 :at way1 & ) :at swiftness1 ! invad$ body %all move2 &1 aga91 :at motion ?is second body w g5]ate 9 a ?ird1 & s =w>ds4 ,f :i* contemplation %all be drawn t "p ( philosophy :i* treats ( motion4 ,9 ! ?ird place we mu/ proce$ to ! enquiry ( su* effects z are made by ! motion ( ! p>ts ( any body1 z1 h[ x comes to pass1 t ?+s :5 !y are ! same1 yet seem n to be ! same1 b *ang$4 ,& "h ! ?+s we se>* aft] are s5sible qualities1 su* z light1 col\r1 transp>5cy1 opacity1 s\nd1 od\r1 sav\r1 heat1 cold1 & ! l2 :i* because !y _c be kn[n till we "k ! causes ( s5se itself1 !re=e ! consid]ation ( ! causes ( see+1 he>+1 smell+1 ta/+1 & t\*+1 belongs to ?is ?ird place2 & all ^? qualities & *anges1 above m5tion$1 are to be ref]r$ to ! f\r? place2 :i* two consid]ations compreh5d t "p ( philosophy :i* is call$ physics4 ,& 9 ^! f\r p>ts is conta9$ :atsoev] 9 natural philosophy may be explicat$ by demon/ration1 prop]ly s call$4 ,= if a cause 7 to be r5d]$ ( natural appe>ances 9 special1 z1 :at are ! motions & influ5ces ( ! heav5ly bodies1 & ( _! p>ts1 ! reason h]e( mu/ ei!r be drawn f ! p>ts ( ! sci5ces above m5tion$1 or no reason at all w be giv51 b all left to unc]ta9 conjecture4 ,aft] physics we mu/ come to moral philosophy2 9 :i* we are to consid] ! motions ( ! m9d1 namely1 appetite1 av]sion1 love1 b5evol5ce1 hope1 fe>1 ang]1 emulation1 envy1 c42 :at causes !y h1 & ( :at !y be causes4 ,& ! reason :y ^! are to be consid]$ aft] physics is1 t !y h _! causes 9 s5se & imag9ation1 :i* are ! subject ( physical contemplation4 ,also ! reason1 :y all ^! ?+s are to be se>*$ aft] 9 ! ord] above-said1 is1 t physics _c be und]/ood1 except we "k fir/ :at motions are 9 ! smalle/ p>ts ( bodies2 nor su* motion ( p>ts1 till we "k :at x is t makes ano!r body move2 nor ?is1 till we "k :at simple motion w effect4 ,& because all appe>ance ( ?+s to s5se is det]m9$1 & made to be ( su* & su* quality & quantity by comp\nd$ motions1 e "o ( :i* has a c]ta9 degree ( velocity1 & a c]ta9 & det]m9$ way2 !re=e1 9 ! fir/ place1 we are to se>* out ! ways ( motion simply 79 :i* geometry consi/s72 next ! ways ( su* g5]at$ motions z are manife/2 &1 la/ly1 ! ways ( int]nal & invisible motions 7:i* is ! enquiry ( natural philosoph]s74 ,&1 !re=e1 !y t /udy natural philosophy1 /udy 9 va91 except !y beg9 at geometry2 & su* writ]s or disput]s !re(1 z are ignorant ( geometry1 d b make _! read]s & he>]s lose _! "t4 ,sid5ote3 ,t me?od ( civil & natural sci5ce1 proce$+ f s5se to pr9ciples1 is analytical2 & aga91 t :i* beg9s at pr9ciples is syn!tical4 #g4 ,civil & moral philosophy d n s adh]e to "o ano!r1 b t !y may be sev]$4 ,= ! causes ( ! motions ( ! m9d are kn[n1 n only by ratioc9ation1 b also by ! exp]i5ce ( e man t takes ! pa9s to obs]ve ^? motions )9 himself4 ,&1 !re=e1 n only !y t h atta9$ ! k ( ! passions & p]turbations ( ! m9d1 by ! syn!tical me?od1 & f ! v fir/ pr9ciples ( philosophy1 may by proce$+ 9 ! same way1 come to ! causes & necessity ( con/itut+ commonweal?s1 & to get ! k ( :at is natural "r1 & :at are civil duties2 &1 9 e k9d ( gov]nm5t1 :at are ! rights ( ! commonweal?1 & all o!r k app]ta9+ to civil philosophy2 = ?is reason1 t ! pr9ciples ( ! politics consi/ 9 ! k ( ! motions ( ! m9d1 & ! k ( ^! motions f ! k ( s5se & imag9ation2 b ev5 !y also t h n le>n$ ! fir/ "p ( philosophy1 namely1 geometry & physics1 may1 not)/&+1 atta9 ! pr9ciples ( civil philosophy1 by ! analytical me?od4 ,= if a "q be prop\nd$1 z1 :e!r su* an action be j or unju/2 if t unju/ be resolv$ into fact aga9/ law1 & t notion law into ! comm& ( him or !m t h co]cive p[]2 & t p[] be d]iv$ f ! wills ( m5 t con/itute su* p[]1 to ! end !y may live 9 peace1 !y may at la/ come to ?is1 t ! appetites ( m5 & ! passions ( _! m9ds are su*1 t1 unless !y be re/ra9$ by "s p[]1 !y w always be mak+ w> ^u "o ano!r2 :i* may be kn[n to be s by any man's exp]i5ce1 t w b exam9e 8 own m9d4 ,&1 !re=e1 f h5ce he may proce$1 by comp\nd+1 to ! det]m9ation ( ! ju/ice or inju/ice ( any prop\nd$ action4 ,s t x is manife/1 by :at has be5 said1 t ! me?od ( philosophy1 to su* z seek sci5ce simply1 )\t prop\nd+ to !mselves ! solution ( any p>ticul> "q1 is p>tly analytical1 & p>tly syn!tical2 namely1 t :i* proce$s f s5se to ! inv5tion ( pr9ciples1 analytical2 & ! re/ syn!tical4 ,sid5ote3 ,! me?od ( se>*+ out1 :e!r any?+ prop\nd$ be matt] or accid5t4 #h4 ,to ^? t seek ! cause ( "s c]ta9 & prop\nd$ appe>ance or effect1 x happ5s1 sometimes1 t !y "k n :e!r ! ?+1 ^: cause is s\ght aft]1 be matt] or body1 or "s accid5t ( a body4 ,= ?\gh 9 geometry1 :5 ! cause is s\ght ( magnitude1 or proportion1 or figure1 x be c]ta9ly kn[n t ^! ?+s1 namely magnitude1 proportion1 & figure1 are accid5ts2 yet 9 natural philosophy1 :]e all que/ions are conc]n+ ! causes ( ! phantasms ( s5sible ?+s1 x is n s easy to disc]n betwe5 ! ?+s !mselves1 f :i* ^? phantasms proce$1 & ! appe>ances ( ^? ?+s to ! s5se2 :i* h deceiv$ _m1 especially :5 ! phantasms h be5 made by light4 ,= example1 a man t looks ^u ! sun1 has a c]ta9 %9+ idea ( ! magnitude ( ab\t a foot ov]1 & ?is he calls ! sun1 ?\gh he "k ! sun to be truly a great deal bigg]2 &1 9 l mann]1 ! phantasm ( ! same ?+ appe>s sometimes r\nd1 by be+ se5 af> (f1 & sometimes squ>e1 by be+ ne>]4 ,:]eupon x may well be d\bt$1 :e!r t phantasm be matt]1 or "s body natural1 or only "s accid5t ( a body2 9 ! exam9ation ( :i* d\bt we may use ?is me?od4 ,! prop]ties ( matt] & accid5ts already f\nd out by u1 by ! syn!tical me?od1 f _! def9itions1 are to be comp>$ ) ! idea we h be=e u2 & if x agree ) ! prop]ties ( matt] or body1 !n x is a body2 o!rwise x is an accid5t4 ,see+1 !re=e1 matt] _c by any endeav\r ( ours be ei!r made or de/roy$1 or increas$1 or dim9i%$1 or mov$ out ( its place1 :]eas t idea appe>s1 vani%es1 is increas$ & dim9i%$1 & mov$ hi!r & ?i!r at pleasure2 we may c]ta9ly conclude t x is n a body1 b an accid5t only4 ,& ?is me?od is syn!tical4 ,sid5ote3 ,! me?od ( seek+ :e!r any accid5t be 9 ?is or 9 t subject4 #i4 ,b if !re be a d\bt made conc]n+ ! subject ( any kn[n accid5t 7= ?is may be d\bt$ sometimes1 z 9 ! prec$5t example1 d\bt may be made 9 :at subject t spl5d\r & app>5t magnitude ( ! sun is71 !n our enquiry mu/ proce$ 9 ?is mann]4 ,fir/1 matt] 9 g5]al mu/ be divid$ into p>ts1 z1 into object1 m$ium1 & ! s5ti5t itself1 or su* o!r p>ts z seem mo/ con=mable to ! ?+ prop\nd$4 ,next1 ^! p>ts are sev]ally to be exam9$ h[ !y agree ) ! def9ition ( ! subject2 & su* ( !m z are n capable ( t accid5t are to be reject$4 ,= example1 if by any true ratioc9ation ! sun be f\nd to be great] ?an its app>5t magnitude1 !n t magnitude is n 9 ! sun2 if ! sun be 9 "o det]m9$ /raight l9e1 & "o det]m9$ di/ance1 & ! magnitude & spl5d\r be se5 9 m l9es & di/ances ?an "o1 z x is 9 reflection or refraction1 !n nei!r t spl5d\r nor app>5t magnitude are 9 ! sun itself1 &1 !re=e1 ! body ( ! sun _c be ! subject ( t spl5d\r & magnitude4 ,& = ! same reasons ! air & o!r p>ts w be reject$1 till at la/ no?+ rema9 :i* c be ! subject ( t spl5d\r & magnitude b ! s5ti5t itself4 ,& ?is me?od1 9 reg>d ! subject is divid$ into p>ts1 is analytical2 & 9 reg>d ! prop]ties1 bo? ( ! subject & accid5t1 are comp>$ ) ! accid5t conc]n+ ^: subject ! enquiry is made1 x is syn!tical4 ,sid5ote3 ,me?od ( se>*+ = ! cause ( any effect1 prop\nd$4 #aj4 ,b :5 we seek aft] ! cause ( any prop\nd$ effect1 we mu/ 9 ! fir/ place get into our m9d an exact notion or idea ( t :i* we call cause1 namely1 t a cause is ! sum or aggregate ( all su* accid5ts1 bo? 9 ! ag5ts & ! pati5t1 z concur to ! produc+ ( ! effect prop\nd$2 all :i* exi/+ toge!r1 x _c be und]/ood b t ! effect exi/e? ) !m2 or t x c possibly exi/ if any "o ( !m be abs5t4 ,?is be+ kn[n1 9 ! next place we mu/ exam9e s+ly e accid5t t accompanies or prec$es ! effect1 z f> =? z x seems to conduce 9 any mann] to ! production ( ! same1 & see :e!r ! prop\nd$ effect may be conceiv$ to exi/1 )\t ! exi/5ce ( any ( ^? accid5ts2 & by ?is means sep>ate su* accid5ts1 z d n concur1 f su* z concur to produce ! said effect2 :i* be+ done1 we are to put toge!r ! concurr+ accid5ts1 & consid] :e!r we c possibly conceive1 t :5 ^! are all pres5t1 ! effect prop\nd$ w n foll[2 & if x be evid5t t ! effect w foll[1 !n t aggregate ( accid5ts is ! entire cause1 o!rwise n2 b we mu/ /ill se>* out & put toge!r o!r accid5ts4 ,= example1 if ! cause ( light be prop\nd$ to be s\ght out2 fir/1 we exam9e ?+s )\t u1 & f9d t :5soev] light appe>s1 !re is "s pr9cipal object1 z x 7 ! f\nta9 ( light1 )\t :i* we _c h any p]ception ( light2 &1 !re=e1 ! concurr5ce ( t object is necess>y to ! g5]ation ( light4 ,next we consid] ! m$ium1 & f9d1 t unless x be dispos$ 9 a c]ta9 mann]1 namely1 t x be transp>5t1 ?\gh ! object rema9 ! same1 yet ! effect w n foll[2 &1 !re=e1 ! concurr5ce ( transp>5cy is also necess>y to ! g5]ation ( light4 ,?irdly1 we obs]ve our own body1 & f9d t by ! indisposition ( ! eyes1 ! bra91 ! n]ves1 & ! he>t1 t is1 by ob/ructions1 /upidity1 & debility1 we are depriv$ ( light1 s t a fitt+ disposition ( ! organs to receive impressions f )\t is likewise a necess>y "p ( ! cause ( light4 ,aga91 ( all ! accid5ts inh]5t 9 ! object1 !re is none t c conduce to ! effect+ ( light1 b only action 7or a c]ta9 motion71 :i* _c be conceiv$ to be want+1 :5soev] ! effect is pres5t2 =1 t any?+ may %9e1 x is n requisite t x be ( su* or su* magnitude or figure1 or t ! :ole body ( x be mov$ out ( ! place x is 9 7unless x may p]haps be said1 t 9 ! sun1 or o!r body1 t :i* causes light is ! light x ha? 9 itself2 :i* yet is b a trifl+ exception1 see+ no?+ is meant !reby b ! cause ( light2 z if any man %\ld say t ! cause ( light is t 9 ! sun :i* produce? x72 x rema9s1 !re=e1 t ! action1 by :i* light is g5]at$1 is motion only 9 ! p>ts ( ! object4 ,:i* be+ und]/ood1 we may easily conceive :at x is ! m$ium contributes1 namely1 ! cont9uation ( t motion to ! eye2 &1 la/ly1 :at ! eye & ! re/ ( ! organs ( ! s5ti5t contribute1 namely1 ! cont9uation ( ! same motion to ! la/ organ ( s5se1 ! he>t4 ,& 9 ?is mann] ! cause ( light may be made up ( motion cont9u$ f ! orig9al ( ! same motion1 to ! orig9al ( vital motion1 light be+ no?+ b ! alt]ation ( vital motion1 made by ! impression ^u x ( motion cont9u$ f ! object4 ,b ,i give ?is only = an example1 = ,i %all speak m at l>ge ( light1 & ! g5]ation ( x1 9 its prop] place4 ,9 ! mean "t x is manife/1 t 9 ! se>*+ out ( causes1 !re is ne$ p>tly ( ! analytical1 & p>tly ( ! syn!tical me?od2 ( ! analytical1 to conceive h[ circum/ances conduce sev]ally to ! production ( effects2 & ( ! syn!tical1 = ! add+ toge!r & comp\nd+ ( :at !y c effect s+ly by !mselves4 ,& ?us mu* may s]ve = ! me?od ( inv5tion4 ,x rema9s t ,i speak ( ! me?od ( tea*+1 t is1 ( demon/ration1 & ( ! means by :i* we demon/rate4 ,sid5ote3 ,words s]ve to inv5tion z m>ks2 to demon/ration z signs4 #aa4 ,9 ! me?od ( inv5tion1 ! use ( words consi/s 9 ?is1 t !y may s]ve = m>ks1 by :i*1 :atsoev] we h f\nd out may be recall$ to memory2 = )\t ?is all our inv5tions p]i%1 nor w x be possible = u to g on f pr9ciples beyond a syllogism or two1 by reason ( ! weakness ( memory4 ,= example1 if any man1 by consid]+ a triangle set be=e him1 %\ld f9d t all its angles toge!r tak5 are equal to two "r angles1 & t by ?9k+ ( ! same tacitly1 )\t any use ( words ei!r und]/ood or express$2 & x %\ld happ5 aft]w>ds t ano!r triangle1 unlike ! =m]1 or ! same 9 diff]5t situation1 %\ld be (f]$ to 8 consid]ation1 he w\ld n "k readily :e!r ! same prop]ty 7 9 ?is la/ or no1 b w\ld be =c$1 z (t5 z a diff]5t triangle 7 br\ght be=e him 7& ! diff]5ce ( triangles is inf9ite7 to beg9 8 contemplation anew2 :i* he w\ld h no ne$ to d if he _h ! use ( names1 = e univ]sal "n d5otes ! conceptions we h ( inf9ite s+ul> ?+s4 ,nev]!less1 z ,i said above1 !y s]ve z m>ks = ! help ( our memory1 :]eby we regi/] to ourselves our own inv5tions2 b n z signs by :i* we decl>e ! same to o!rs2 s t a man may be a philosoph] alone by himself1 )\t any ma/]2 ,adam _h ?is capacity4 ,b to tea*1 t is1 to demon/rate1 supposes two at ! lea/1 & syllogi/ical spee*4 ,sid5ote3 ,! me?od ( demon/ration is syn!tical4 #ab4 ,& see+ tea*+ is no?+ b lead+ ! m9d ( him we tea*1 to ! k ( our inv5tions1 9 t track by :i* we atta9$ ! same ) our own m9d2 !re=e1 ! same me?od t s]v$ = our inv5tion1 w s]ve also = demon/ration to o!rs1 sav+ t we omit ! fir/ "p ( me?od :i* proce$$ f ! s5se ( ?+s to univ]sal pr9ciples1 :i*1 because !y are pr9ciples1 _c be demon/rat$2 & see+ !y are kn[n by nature1 7z 0 said above 9 ! #e? article7 !y ne$ no demon/ration1 ?\gh !y ne$ explication4 ,! :ole me?od1 !re=e1 ( demon/ration1 is syn!tical1 consi/+ ( t ord] ( spee* :i* beg9s f prim>y or mo/ univ]sal propositions1 :i* are manife/ ( !mselves1 & proce$s by a p]petual composition ( propositions into syllogisms1 till at la/ ! le>n] und]/& ! tru? ( ! conclusion s\ght aft]4 ,sid5ote3 ,def9itions only are prim>y1 univ]sal propositions4 #ac4 ,n[1 su* pr9ciples are no?+ b def9itions1 :]e( !re are two sorts2 "o ( names1 t signify su* ?+s z h "s conceivable cause1 & ano!r ( su* names z signify ?+s ( :i* we c conceive no cause at all4 ,names ( ! =m] k9d are1 body1 or matt]1 quantity1 or ext5sion1 motion1 & :atsoev] is common to all matt]4 ,( ! second k9d1 are su* a body1 su* & s great motion1 s great magnitude1 su* figure1 & :atsoev] we c di/+ui% "o body f ano!r by4 ,& names ( ! =m] k9d are well en\gh def9$1 :51 by spee* z %ort z may be1 we raise 9 ! m9d ( ! he>] p]fect & cle> ideas or conceptions ( ! ?+s nam$1 z :5 we def9e motion to be ! leav+ ( "o place1 & ! acquir+ ( ano!r cont9ually2 = ?\gh no ?+ mov$1 nor any cause ( motion be 9 t def9ition1 yet1 at ! he>+ ( t spee*1 !re w come into ! m9d ( ! he>] an idea ( motion cle> en\gh4 ,b def9itions ( ?+s1 :i* may be und]/ood to h "s cause1 mu/ consi/ ( su* names z express ! cause or mann] ( _! g5]ation1 z :5 we def9e a circle to be a figure made by ! circumduction ( a /raight l9e 9 a plane1 c4 ,besides def9itions1 !re is no o!r proposition t ought to be call$ prim>y1 or 7accord+ to sev]e tru?7 be receiv$ into ! numb] ( pr9ciples4 ,= ^? axioms ( ,euclid1 see+ !y may be demon/rat$1 are no pr9ciples ( demon/ration1 ?\gh !y h by ! cons5t ( all m5 gott5 ! au?ority ( pr9ciples1 because !y ne$ n be demon/rat$4 ,also1 ^? petitions1 or po/ulata1 7z !y call !m7 ?\gh !y be pr9ciples1 yet !y are n pr9ciples ( demon/ration1 b ( con/ruction only2 t is1 n ( sci5ce1 b ( p[]2 or 7:i* is all "o7 n ( !orems1 :i* are speculations1 b ( problems1 :i* belong to practice1 or ! do+ ( some?+4 ,b z = ^? common receiv$ op9ions1 ,nature abhors vacuity1 ,nature do? no?+ 9 va91 & ! l1 :i* are nei!r evid5t 9 !mselves1 nor at all to be demon/rat$1 & :i* are (t5] false ?an true1 !y are mu* less to be ackn[l$g$ = pr9ciples4 ,to return1 !re=e1 to def9itions2 ! reason :y ,i say t ! cause & g5]ation ( su* ?+s1 z h any cause or g5]ation1 ought to ent] into _! def9itions1 is ?is4 ,! end ( sci5ce is ! demon/ration ( ! causes & g5]ations ( ?+s2 :i* if !y be n 9 ! def9itions1 !y _c be f\nd 9 ! conclusion ( ! fir/ syllogism1 t is made f ^? def9itions2 & if !y be n 9 ! fir/ conclusion1 !y w n be f\nd 9 any fur!r conclusion d$uc$ f t2 &1 !re=e1 by proce$+ 9 ?is mann]1 we %all nev] come to sci5ce2 :i* is aga9/ ! scope & int5tion ( demon/ration4 ,sid5ote3 ,! nature def9ition ( a def9ition4 #ad4 ,n[1 see+ def9itions 7z ,i h said7 are pr9ciples1 or prim>y propositions1 !y are !re=e spee*es2 & see+ !y are us$ = ! rais+ ( an idea ( "s ?+ 9 ! m9d ( ! le>n]1 :5soev] t ?+ has a "n1 ! def9ition ( x c be no?+ b ! explication ( t "n by spee*2 & if t "n be giv5 x = "s comp\nd$ conception1 ! def9ition is no?+ b a resolution ( t "n into its mo/ univ]sal p>ts4 ,z :5 we def9e man1 say+ man is a body animat$1 s5ti5t1 rational1 ^? names1 body animat$1 c4 are p>ts ( t :ole "n man2 s t def9itions ( ?is k9d always consi/ ( g5us & diff]5ce2 ! =m] names be+ all1 till ! la/1 g5]al2 & ! la/ ( all1 diff]5ce4 ,b if any "n be ! mo/ univ]sal 9 its k9d1 !n ! def9ition ( x _c consi/ ( g5us & diff]5ce1 b is to be made by su* circumlocution1 z be/ explicate? ! =ce ( t "n4 ,aga91 x is possible1 & happ5s (t51 t ! g5us & diff]5ce are put toge!r1 & yet make no def9ition2 z ^! words1 a /raight l9e1 conta9 bo? ! g5us & diff]5ce2 b are n a def9ition1 unless we %\ld ?9k a /raight l9e may be ?us def9$1 a /raight l9e is a /raight l9e3 & yet if !re 7 add$ ano!r "n1 consi/+ ( diff]5t words1 b signify+ ! same ?+ :i* ^! signify1 !n ^! might be ! def9ition ( t "n4 ,f :at has be5 said1 x may be und]/ood h[ a def9ition ought to be def9$1 namely1 t x is a proposition1 ^: pr$icate resolves ! subject1 :5 x may2 & :5 x may n1 x exemplifies ! same4 ,sid5ote3 ,prop]ties ( a def9ition4 #ae4 ,! prop]ties ( a def9ition are3 ,fir/1 t x takes away equivocation1 z also all t multitude ( di/9ctions1 :i* are us$ by su* z ?9k !y may le>n philosophy by disputation4 ,= ! nature ( a def9ition is to def9e1 t is1 to det]m9e ! signification ( ! def9$ "n1 & to p>e f x all o!r signification besides :at is conta9$ 9 ! def9ition itself2 & !re=e "o def9ition does z mu*1 z all ! di/9ctions 7h[ _m soev]7 t c be us$ ab\t ! "n def9$4 ,secondly1 t x gives an univ]sal notion ( ! ?+ def9$1 repres5t+ a c]ta9 univ]sal picture !re(1 n to ! eye1 b to ! m9d4 ,= z :5 "o pa9ts a man1 he pa9ts ! image ( "s man2 s he1 t def9es ! "n man1 makes a repres5tation ( "s man to ! m9d4 ,?irdly1 t x is n necess>y to dispute :e!r def9itions are to be admitt$ or no4 ,= :5 a ma/] is in/ruct+ 8 s*ol>1 if ! s*ol> und]/& all ! p>ts ( ! ?+ def9$1 :i* are resolv$ 9 ! def9ition1 & yet w n admit ( ! def9ition1 !re ne$s no fur!r controv]sy betwixt !m1 x be+ all "o z if he refus$ to be taught4 ,b if he und]/& no?+1 !n c]ta9ly ! def9ition is faulty2 = ! nature ( a def9ition consi/s 9 ?is1 t x exhibit a cle> idea ( ! ?+ def9$2 & pr9ciples are ei!r kn[n by !mselves1 or else !y are n pr9ciples4 ,f\r?ly1 t1 9 philosophy1 def9itions are be=e def9$ names4 ,= 9 tea*+ philosophy1 ! fir/ beg9n+ is f def9itions2 & all progression 9 ! same1 till we come to ! k ( ! ?+ comp\nd$1 is compositive4 ,see+1 !re=e1 def9ition is ! explication ( a comp\nd$ "n by resolution1 & ! progression is f ! p>ts to ! comp\nd1 def9itions mu/ be und]/ood be=e comp\nd$ names2 nay1 :5 ! names ( ! p>ts ( any spee* be explicat$1 x is n necess>y t ! def9ition %\ld be a "n comp\nd$ ( !m4 ,= example1 :5 ^! names1 equilat]al1 quadrilat]al1 "r-angl$1 are suffici5tly und]/ood1 x is n necess>y 9 geometry t !re %\ld be at all su* a "n z squ>e2 = def9$ names are receiv$ 9 philosophy = brevity's sake only4 ,fif?ly1 t comp\nd$ names1 :i* are def9$ "o way 9 "s "o "p ( philosophy1 may 9 ano!r "p ( ! same be o!rwise def9$2 z a p>abola & an hyp]bole h "o def9ition 9 geometry1 & ano!r 9 rhetoric2 = def9itions are in/itut$ & s]ve = ! und]/&+ ( ! doctr9e :i* is treat$ (4 ,&1 !re=e1 z 9 "o "p ( philosophy1 a def9ition may h 9 x "s "o fit "n = ! m brief explanation ( "s proposition 9 geometry2 s x may h ! same lib]ty 9 o!r p>ts ( philosophy2 = ! use ( names is p>ticul> 7ev5 :]e _m agree to ! settl+ ( !m7 & arbitr>y4 ,six?ly1 t no "n c be def9$ by any "o ^w2 because no "o ^w is suffici5t = ! resolv+ ( "o or m words4 ,sev5?ly1 t a def9$ "n ought n to be repeat$ 9 ! def9ition4 ,= a def9$ "n is ! :ole comp\nd1 & a def9ition is ! resolution ( t comp\nd into p>ts2 b no total c be "p ( itself4 ,sid5ote3 ,nature ( a demon/ration4 #af4 ,any two def9itions1 t may be comp\nd$ into a syllogism1 produce a conclusion2 :i*1 because x is d]iv$ f pr9ciples1 t is1 f def9itions1 is said to be demon/rat$2 & ! d]ivation or composition itself is call$ a demon/ration4 ,9 l mann]1 if a syllogism be made ( two propositions1 :]e( "o is a def9ition1 ! o!r a demon/rat$ conclusion1 or nei!r ( !m is a def9ition1 b bo? =m]ly demon/rat$1 t syllogism is also call$ a demon/ration1 & s successively4 ,! def9ition !re=e ( a demon/ration is ?is1 a demon/ration is a syllogism1 or s]ies ( syllogisms1 d]iv$ & cont9u$1 f ! def9itions ( names1 to ! la/ conclusion4 ,& f h5ce x may be und]/ood1 t all true ratioc9ation1 :i* take? its beg9n+ f true pr9ciples1 produce? sci5ce1 & is true demon/ration4 ,= z = ! orig9al ( ! "n1 al?\gh t1 :i* ! ,greeks call$ 1 & ! ,lat9s demon/ratio1 0 und]/ood by !m = t sort only ( ratioc9ation1 9 :i*1 by ! describ+ ( c]ta9 l9es & figures1 !y plac$ ! ?+ !y 7 to prove1 z x 7 be=e m5's eyes1 :i* is prop]ly 1 or to %ew by ! figure2 yet !y seem to h done x = ?is reason1 t unless x 7 9 geometry1 79 :i* only !re is place = su* figures7 !re 0 no ratioc9ation c]ta91 & end+ 9 sci5ce1 _! doctr9es conc]n+ all o!r ?+s be+ no?+ b controv]sy & clam\r2 :i*1 nev]!less1 happ5$1 n because ! tru? to :i* !y pret5d$ c\ld n be made evid5t )\t figures1 b because !y want$ true pr9ciples1 f :i* !y might d]ive _! ratioc9ation2 &1 !re=e1 !re is no reason b t if true def9itions 7 premis$ 9 all sorts ( doctr9es1 ! demon/rations also w\ld be true4 ,sid5ote3 ,prop]ties ( a demon/ration1 & ord] ( ?+s to be demon/rat$4 #ag4 ,x4 is prop] to me?odical demon/ration1 ,fir/1 t !re be a true succession ( "o reason to ano!r1 accord+ to ! rules ( syllogiz+ deliv]$ above4 ,secondly1 t ! premises ( all syllogisms be demon/rat$ f ! fir/ def9itions4 ,?irdly1 t aft] def9itions1 he t tea*es or demon/rates any ?+1 proce$ 9 ! same me?od by :i* he f\nd x out2 namely1 t 9 ! fir/ place ^? ?+s be demon/rat$1 :i* imm$iately succe$ to univ]sal def9itions 79 :i* is conta9$ t "p ( philosophy :i* is call$ philosophia prima74 ,next1 ^? ?+s :i* may be demon/rat$ by simple motion 79 :i* geometry consi/s74 ,aft] geometry1 su* ?+s z may be taught or %ew$ by manife/ action1 t is1 by ?ru/+ f1 or pull+ t[>ds4 ,& aft] ^!1 ! motion or mutation ( ! invisible p>ts ( ?+s1 & ! doctr9e ( s5se & imag9ations1 & ( ! int]nal passions1 especially ^? ( m51 9 :i* are compreh5d$ ! gr\nds ( civil duties1 or civil philosophy2 :i* takes up ! la/ place4 ,& t ?is me?od ought to be kept 9 all sorts ( philosophy1 is evid5t f h5ce1 t su* ?+s z ,i h said are to be taught la/1 _c be demon/rat$1 till su* z are prop\nd$ to be fir/ treat$ (1 be fully und]/ood4 ,( :i* me?od no o!r example c be giv51 b t treatise ( ! elem5ts ( philosophy1 :i* ,i %all beg9 9 ! next *apt]1 & cont9ue to ! end ( ! "w4 ,sid5ote3 ,faults ( a demon/ration4 #ah4 ,besides ^? p>alogisms1 ^: fault lies ei!r 9 ! falsity ( ! premises1 or ! want ( true composition1 ( :i* ,i h spok5 9 ! prec$5t *apt]1 !re are two m1 :i* are frequ5t 9 demon/ration2 "o :]e( is commonly call$ petitio pr9cipii2 ! o!r is ! suppos+ ( a false cause2 & ^! d n only deceive unskilful le>n]s1 b sometimes ma/]s !mselves1 by mak+ !m take t = well demon/rat$1 :i* is n demon/rat$ at all4 ,petitio pr9cipii is1 :5 ! conclusion to be prov$ is disguis$ 9 o!r words1 & put = ! def9ition or pr9ciple f :5ce x is to be demon/rat$2 & ?us1 by putt+ = ! cause ( ! ?+ s\ght1 ei!r ! ?+ itself or "s effect ( x1 !y make a circle 9 _! demon/ration4 ,z = example1 he t w\ld demon/rate t ! e>? /&s /ill 9 ! c5tre ( ! _w1 & %\ld suppose ! e>?'s gravity to be ! cause !re(1 & def9e gravity to be a quality by :i* e heavy body t5ds t[>ds ! c5tre ( ! _w1 w\ld lose 8 lab\r2 = ! "q is1 :at is ! cause ( t quality 9 ! e>?8 &1 !re=e1 he t supposes gravity to be ! cause1 puts ! ?+ itself = its own cause4 ,( a false cause ,i f9d ?is example 9 a c]ta9 treatise :]e ! ?+ to be demon/rat$ is ! motion ( ! e>?4 ,he beg9s1 !re=e1 ) ?is1 t see+ ! e>? & ! sun are n always 9 ! same situation1 x mu/ ne$s be t "o ( !m be locally mov$1 :i* is true2 next1 he affirms t ! vap\rs1 :i* ! sun raises f ! e>? & sea1 are1 by reason ( ?is motion1 necess>ily mov$1 :i* also is true2 f :5ce he inf]s ! w9ds are made1 & ?is may pass = grant$2 & by ^! w9ds he says1 ! wat]s ( ! sea are mov$1 & by _! motion ! bottom ( ! sea1 z if x 7 beat5 =w>ds1 moves r\nd2 & let ?is also be grant$2 :]e=e1 he concludes1 ! e>? is mov$2 :i* is1 nev]!less1 a p>alogism4 ,=1 if t w9d 7 ! cause :y ! e>? 01 f ! beg9n+1 mov$ r\nd1 & ! motion ei!r ( ! sun or ! e>? 7 ! cause ( t w9d1 !n ! motion ( ! sun or ! e>? 0 be=e ! w9d itself2 & if ! e>? 7 mov$1 be=e ! w9d 0 made1 !n ! w9d c\ld n be ! cause ( ! e>?'s revolution2 b1 if ! sun 7 mov$1 & ! e>? /& /ill1 !n x is manife/ ! e>? might rema9 unmov$1 not)/&+ t w9d2 & !re=e t motion 0 n made by ! cause :i* he allege?4 ,b p>alogisms ( ?is k9d are v frequ5t among ! writ]s ( physics1 ?\gh none c be m elaborate ?an ?is 9 ! example giv54 ,sid5ote3 ,:y ! analytical me?od ( geometricians _c be treat$ ( 9 ?is place4 #ai4 ,x may to "s m5 seem p]t95t to treat 9 ?is place ( t art ( ! geometricians1 :i* !y call logi/ica1 t is1 ! art1 by :i*1 f suppos+ ! ?+ 9 "q to be true1 !y proce$ by ratioc9ation1 till ei!r !y come to some?+ kn[n1 by :i* !y may demon/rate ! tru? ( ! ?+ s\ght =2 or to some?+ :i* is impossible1 f :5ce !y collect t to be false1 :i* !y suppos$ true4 ,b ?is art _c be explicat$ "h1 = ?is reason1 t ! me?od ( x c nei!r be practis$1 nor und]/ood1 unless by su* z are well v]s$ 9 geometry2 & among geometricians !mselves1 !y1 t h mo/ !orems 9 read9ess1 are ! mo/ ready 9 ! use ( ?is logi/ica2 s t1 inde$1 x is n a di/9ct ?+ f geometry itself2 = !re are1 9 ! me?od ( x1 ?ree p>ts2 ! fir/ :]e( consi/s 9 ! f9d+ out ( equality betwixt kn[n & unkn[n ?+s1 :i* !y call equation2 & ?is equation _c be f\nd out1 b by su* z "k p]fectly ! nature1 prop]ties1 & transpositions ( proportion1 z also ! addition1 subtraction1 multiplication1 & division ( l9es & sup]ficies1 & ! extraction ( roots2 :i* are ! p>ts ( no mean geometrician4 ,! second is1 :5 an equation is f\nd1 to be able to judge :e!r ! tru? or falsity ( ! "q may be d$uc$ f x1 or no2 :i* yet requires great] k4 ,& ! ?ird is1 :5 su* an equation is f\nd1 z is fit = ! solution ( ! "q1 to "k h[ to resolve ! same 9 su* mann]1 t ! tru? or falsity may !reby manife/ly appe>2 :i*1 9 h>d que/ions1 _c be done )\t ! k ( ! nature ( crook$-l9$ figures2 b he t und]/&s readily ! nature & prop]ties ( ^!1 is a complete geometrician4 ,x happ5s besides1 t = ! f9d+ out ( equations1 !re is no c]ta9 me?od1 b he is be/ able to d x1 t has ! be/ natural wit4 ,,"p ,,ii4 ,,! ,,fir/ ,,gr\nds ,,( ,,philosophy4 ------- ,,*apt] ,,vii4 ,,( ,,place ,,& ,,"t4 #a4 ,?+s t h no exi/5ce1 may nev]!less be und]/ood & comput$4--#b4 ,:at is ,space4--#c4 ,"t4--#d4 ,"p4--#e4 ,division4--#f4 ,"o4--#g4 ,numb]4--#h4 ,composition4--#i4 ,! :ole4--#aj4 ,spaces & times contigu\s1 & cont9ual4--#aa4 ,beg9n+1 end1 way1 f9ite1 inf9ite4--#ab4 ,:at is inf9ite 9 p[]4 ,no?+ inf9ite c be truly said to be ei!r :ole1 or "o2 nor inf9ite spaces or times1 _m4--#ac4 ,division proce$s n to ! lea/4 ,sid5ote3 ,?+s t h no exi/5ce1 may nev]!less be und]/ood & comput$4 #a4 ,9 ! tea*+ ( natural philosophy1 ,i _c beg9 bett] 7z ,i h already %ewn7 ?an f privation2 t is1 f feign+ ! _w to be annihilat$4 ,b1 if su* annihilation ( all ?+s be suppos$1 x may p]haps be ask$1 :at w\ld rema9 = any man 7:om only ,i except f ?is univ]sal annihilation ( ?+s7 to consid] z ! subject ( philosophy1 or at all to reason ^u2 or :at to give names unto = ratioc9ation's sake4 ,i say1 !re=e1 !re w\ld rema9 to t man ideas ( ! _w1 & ( all su* bodies z he _h1 be=e _! annihilation1 se5 ) 8 eyes1 or p]ceiv$ by any o!r s5se2 t is to say1 ! memory & imag9ation ( magnitudes1 motions1 s\nds1 col\rs1 c4 z also ( _! ord] & p>ts4 ,all :i* ?+s1 ?\gh !y be no?+ b ideas & phantasms1 happ5+ int]nally to him t imag9e?2 yet !y w appe> z if !y 7 ext]nal1 & n at all dep5d+ ^u any p[] ( ! m9d4 ,& ^! are ! ?+s to :i* he w\ld give names1 & subtract !m f1 & comp\nd !m ) "o ano!r4 ,= see+1 t aft] ! de/ruction ( all o!r ?+s1 ,i suppose man /ill rema9+1 & namely t he ?9ks1 imag9es1 & rememb]s1 !re c be no?+ = him to ?9k ( b :at is pa/2 nay1 if we d b obs]ve dilig5tly :at x is we d :5 we consid] & reason1 we %all f9d1 t ?\gh all ?+s be /ill rema9+ 9 ! _w1 yet we compute no?+ b our own phantasms4 ,= :5 we calculate ! magnitude & motions ( heav5 or e>?1 we d n asc5d into heav5 t we may divide x into p>ts1 or measure ! motions !re(1 b we d x sitt+ /ill 9 our closets or 9 ! d>k4 ,n[ ?+s may be consid]$1 t is1 be br\ght into acc\nt1 ei!r z int]nal accid5ts ( our m9d1 9 :i* mann] we consid] !m :5 ! "q is ab\t "s faculty ( ! m9d2 or z species ( ext]nal ?+s1 n z really exi/+1 b appe>+ only to exi/1 or to h a be+ )\t u4 ,& 9 ?is mann] we are n[ to consid] !m4 ,sid5ote3 ,:at is ,space4 #b4 ,if !re=e we rememb]1 or h a phantasm ( any ?+ t 0 9 ! _w be=e ! suppos$ annihilation ( ! same2 & consid]1 n t ! ?+ 0 su* or su*1 b only t x _h a be+ )\t ! m9d1 we h pres5tly a conception ( t we call space3 an imag9>y space inde$1 because a m]e phantasm1 yet t v ?+ :i* all m5 call s4 ,= no man calls x space = be+ already fill$1 b because x may be fill$2 nor does any man ?9k bodies c>ry _! places away ) !m1 b t ! same space conta9s sometimes "o1 sometimes ano!r body2 :i* c\ld n be if space %\ld always accompany ! body :i* is once 9 x4 ,& ?is is ( itself s manife/1 t ,i %\ld n ?9k x ne$$ any expla9+ at all1 b t ,i f9d space to be falsely def9$ by c]ta9 philosoph]s1 :o inf] f !nce1 "o1 t ! _w is inf9ite 7= tak+ space to be ! ext5sion ( bodies1 & ?9k+ ext5sion may encrease cont9ually1 he inf]s t bodies may be inf9itely ext5d$72 &1 ano!r1 f ! same def9ition1 concludes ra%ly1 t x is impossible ev5 to ,god himself to create m worlds ?an "o2 =1 if ano!r _w 7 to be creat$1 he says1 t see+ !re is no?+ )\t ?is _w1 & !re=e 7accord+ to 8 def9ition7 no space1 t new _w mu/ be plac$ 9 no?+2 b 9 no?+ no?+ c be plac$2 :i* he affirms only1 )\t %[+ any reason = ! same2 :]eas ! contr>y is ! tru?3 = m _c be put into a place already fill$1 s mu* is empty space fitt] ?an t1 :i* is full1 = ! receiv+ ( new bodies4 ,hav+ !re=e spok5 ?us mu* = ^! m5's sakes1 & = !irs t ass5t to !m1 ,i return to my purpose1 & def9e space ?us3 ,,space is ! phantasm ( a ?+ exi/+ )\t ! m9d simply2 t is to say1 t phantasm1 9 :i* we consid] no o!r accid5t1 b only t x appe>s )\t u4 ,sid5ote3 ,"t4 #c4 ,z a body leaves a phantasm ( its magnitude 9 ! m9d1 s also a mov$ body leaves a phantasm ( its motion1 namely1 an idea ( t body pass+ out ( "o space into ano!r by cont9ual succession4 ,& ?is idea1 or phantasm1 is t1 :i* 7)\t rec$+ mu* f ! common op9ion1 or f ,ari/otle's def9ition7 ,i call ,"t4 ,= see+ all m5 confess a ye> to be "t1 & yet d n ?9k a ye> to be ! accid5t or affection ( any body1 !y mu/ ne$s confess x to be1 n 9 ! ?+s )\t u1 b only 9 ! ?\ght ( ! m9d4 ,s :5 !y speak ( ! times ( _! pr$ecessors1 !y d n ?9k aft] _! pr$ecessors are gone1 t _! times c be any :]e else ?an 9 ! memory ( ^? t rememb] !m4 ,& z = ^? t say1 days1 ye>s1 & mon?s are ! motions ( ! sun & moon1 see+ x is all "o to say1 motion pa/ & motion de/roy$1 & t future motion is ! same ) motion :i* is n yet begun1 !y say t1 :i* !y d n mean1 t !re nei!r is1 nor has be51 nor %all be any "t3 = ( :atsoev] x may be said1 x has be5 or x %all be1 ( ! same also x might h be5 said h]et(ore1 or may be said h]eaft]1 x is4 ,:at !n c days1 mon?s1 & ye>s1 be1 b ! names ( su* computations made 9 our m9d8 ,"t !re=e is a phantasm1 b a phantasm ( motion1 = if we w\ld "k by :at mom5ts "t passes away1 we make use ( "s motion or o!r1 z ( ! sun1 ( a clock1 ( ! s& 9 an h\r-glass1 or we m>k "s l9e ^u :i* we imag9e some?+ to be mov$1 !re be+ no o!r means by :i* we c take notice ( any "t at all4 ,& yet1 :5 ,i say "t is a phantasm ( motion1 ,i d n say ?is is suffici5t to def9e x by2 = ?is ^w "t compreh5ds ! notion ( =m] & latt]1 or ( succession 9 ! motion ( a body1 9 z mu* z x is fir/ "h !n !re4 ,:]e=e a complete def9ition ( "t is su* z ?is1 ,,"t is ! phantasm ( be=e & aft] 9 motion2 :i* agrees ) ?is def9ition ( ,ari/otle1 "t is ! numb] ( motion accord+ to =m] & latt]2 = t numb]+ is an act ( ! m9d2 & !re=e x is all "o to say1 "t is ! numb] ( motion accord+ to =m] & latt]2 & "t is a phantasm ( motion numb]$4 ,b t o!r def9ition1 "t is ! measure ( motion1 is n s exact1 = we measure "t by motion & n motion by "t4 ,sid5ote3 ,"p4 #d4 ,"o space is call$ "p ( ano!r space1 & "o "t "p ( ano!r "t1 :5 ?is conta9s t & some?+ besides4 ,f :5ce x may be collect$1 t no?+ c rightly be call$ a ,,"p1 b t :i* is comp>$ ) some?+ t conta9s x4 ,sid5ote3 ,division4 #e4 ,& !re=e to make p>ts1 or to "p or ,,divide space or "t1 is no?+ else b to consid] "o & ano!r )9 ! same2 s t if any man divide space or "t1 ! div]se conceptions he has are m1 by "o1 ?an ! p>ts he makes2 = 8 fir/ conception is ( t :i* is to be divid$1 !n ( "s "p ( x1 & aga9 ( "s o!r "p ( x1 & s =w>ds z long z he goes on 9 divid+4 ,b x is to be not$1 t "h1 by division1 ,i d n mean ! sev]+ or pull+ asund] ( "o space or "t f ano!r 7= does any man ?9k t "o hemisph]e may be sep>at$ f ! o!r hemisph]e1 or ! fir/ h\r f ! second87 b div]sity ( consid]ation2 s t division is n made by ! op]ation ( ! h&s b ( ! m9d4 ,sid5ote3 ,"o4 #f4 ,:5 space or "t is consid]$ among o!r spaces or times1 x is said to be ,,"o1 namely "o ( !m2 = except "o space might be add$ to ano!r1 & subtract$ f ano!r space1 & s ( "t1 x w\ld be suffici5t to say space or "t simply1 & sup]flu\s to say "o space or "o "t1 if x c\ld n be conceiv$ t !re 7 ano!r4 ,! common def9ition ( "o1 namely1 t "o is t :i* is undivid$1 is obnoxi\s to an absurd consequ5ce2 = x may !nce be inf]r$1 t :atsoev] is divid$ is _m ?+s1 t is1 t e divid$ ?+1 is divid$ ?+s1 :i* is insignificant4 ,sid5ote3 ,numb]4 #g4 ,,numb] is "o & "o1 or "o "o & "o1 & s =w>ds2 namely1 "o & "o make ! numb] two1 & "o "o & "o ! numb] ?ree2 s are all o!r numb]s made2 :i* is all "o z if we %\ld say1 numb] is unities4 ,sid5ote3 ,composition4 #h4 ,to ,,comp\nd space ( spaces1 or "t ( times1 is fir/ to consid] !m "o aft] ano!r1 & !n altoge!r z "o2 z if "o %\ld reckon fir/ ! head1 ! feet1 ! arms1 & ! body1 sev]ally1 & !n = ! acc\nt ( !m all toge!r put man4 ,& t :i* is s put = all ! sev]als ( :i* x consi/s1 is call$ ! ,,:ole2 & ^? sev]als1 :5 by ! division ( ! :ole !y come aga9 to be consid]$ s+ly1 are p>ts !re(2 & !re=e ! :ole & all ! p>ts tak5 toge!r are ! same ?+4 ,& z ,i not$ above1 t 9 division x is n necess>y to pull ! p>ts asund]2 s 9 composition1 x is to be und]/ood1 t = ! mak+ up ( a :ole !re is no ne$ ( putt+ ! p>ts toge!r1 s z to make !m t\* "o ano!r1 b only ( collect+ !m into "o sum 9 ! m9d4 ,= ?us all m51 be+ consid]$ toge!r1 make up ! :ole ( mank9d1 ?\gh nev] s mu* disp]s$ by "t & place2 & twelve h\rs1 ?\gh ! h\rs ( sev]al days1 may be comp\nd$ into "o numb] ( twelve4 ,sid5ote3 ,! :ole4 #i4 ,?is be+ well und]/ood1 x is manife/1 t no?+ c rightly be call$ a :ole1 t is n conceiv$ to be comp\nd$ ( p>ts1 & t x may be divid$ into p>ts2 s t if we d5y t a ?+ has p>ts1 we d5y ! same to be a :ole4 ,= example1 if we say ! s\l c h no p>ts1 we affirm t no s\l c be a :ole s\l4 ,also x is manife/1 t no?+ has p>ts till x be divid$2 & :5 a ?+ is divid$1 ! p>ts are only s _m z ! division makes !m4 ,aga91 t a "p ( a "p is a "p ( ! :ole2 & ?us any "p ( ! numb] f\r1 z two1 is a "p ( ! numb] eight2 = f\r is made ( two & two2 b eight is comp\nd$ ( two1 two1 & f\r1 & !re=e two1 :i* is a "p ( ! "p f\r1 is also a "p ( ! :ole eight4 ,sid5ote3 ,spaces & times contigu\s & cont9ual4 #aj4 ,two spaces are said to be ,,contigu\s1 :5 !re is no o!r space betwixt !m4 ,b two times1 betwixt :i* !re is no o!r "t1 are call$ imm$iate1 z ,a ,b1 ,b ,c4,sid5ote3 ,a ,b ,c ,& any two spaces1 z well z times1 are said to be ,,cont9ual1 :5 !y h "o common "p1 z ,a ,c1 ,b ,d1 :]e ! "p ,b ,c is common2 ,sid5ote3 ,a ,b ,c ,d & m spaces & times are cont9ual1 :5 e two :i* are next "o ano!r are cont9ual4 ,a ,b ,c ,d ------------- ,a ,b ,c ------------ ,sid5ote3 ,beg9n+1 end1 way1 f9ite1 inf9ite4 #aa4 ,t "p :i* is betwe5 two o!r p>ts1 is call$ a ,,mean2 & t :i* is n betwe5 two o!r p>ts1 an ,,extreme4 ,& ( extremes1 t :i* is fir/ reckon$ is ! ,,beg9n+1 & t :i* la/1 ! ,,end2 & all ! means toge!r tak5 are ! ,,way4 ,also1 extreme p>ts & limits are ! same ?+4 ,& f h5ce x is manife/1 t beg9n+ & end dep5d ^u ! ord] 9 :i* we numb] !m2 & t to t]m9ate or limit space & "t1 is ! same ?+ ) imag9+ _! beg9n+ & end2 z also t e ?+ is ,,f9ite or ,,inf9ite1 accord+ z we imag9e or n imag9e x limit$ or t]m9at$ e way2 & t ! limits ( any numb] are unities1 & ( ^!1 t :i* is ! fir/ 9 our numb]+ is ! beg9n+1 & t :i* we numb] la/1 is ! end4 ,:5 we say numb] is inf9ite1 we mean only t no numb] is express$2 = :5 we speak ( ! numb]s two1 ?ree1 a ?\s&1 c4 !y are always f9ite4 ,b :5 no m is said b ?is1 numb] is inf9ite1 x is to be und]/ood z if x 7 said1 ?is "n numb] is an indef9ite "n4 ,sid5ote3 ,:at is inf9ite 9 p[]4 ,no?+ inf9ite c be truly said to be ei!r :ole or "o2 nor inf9ite spaces or times1 _m4 #ab4 ,space or "t is said to f9ite 9 p[]1 or t]m9able1 :5 !re may be assign$ a numb] ( f9ite spaces or times1 z ( paces or h\rs1 ?an :i* !re c be no great] numb] ( ! same measure 9 t space or "t2 & inf9ite 9 p[] is t space or "t1 9 :i* a great] numb] ( ! said paces or h\rs may be assign$1 ?an any numb] t c be giv54 ,b we mu/ note1 t1 al?\gh 9 t space or "t :i* is inf9ite 9 p[]1 !re may be numb]$ m paces or h\rs ?an any numb] t c be assign$1 yet _! numb] w always be f9ite2 = e numb] is f9ite4 ,& !re=e 8 ratioc9ation 0 n good1 t und]tak+ to prove ! _w to be f9ite1 reason$ ?us2 ,if ! _w be inf9ite1 !n !re may be tak5 9 x "s "p :i* is di/ant f u an inf9ite numb] ( paces3 b no su* "p c be tak52 :]e=e ! _w is n inf9ite2 because t consequ5ce ( ! major proposition is false2 = 9 an inf9ite space1 :atsoev] we take or design 9 our m9d1 ! di/ance ( ! same f u is a f9ite space2 = 9 ! v design+ ( ! place !re(1 we put an end to t space1 ( :i* we ourselves are ! beg9n+2 & :atsoev] any man ) 8 m9d cuts (f bo? ways f inf9ite1 he det]m9es ! same1 t is1 he makes x f9ite4 ,( inf9ite space or "t1 x _c be said t x is a :ole or "o3 n a :ole1 because n comp\nd$ ( p>ts2 = see+ p>ts1 h[ _m soev] !y be1 are sev]ally f9ite1 !y w also1 :5 !y are all put toge!r1 make a :ole f9ite3 nor "o1 because no?+ c be said to be "o1 except !re be ano!r to comp>e x )2 b x _c be conceiv$ t !re are two spaces1 or two times1 inf9ite4 ,la/ly1 :5 we make "q :e!r ! _w be f9ite or inf9ite1 we h no?+ 9 our m9d answ]+ to ! "n _w2 = :atsoev] we imag9e1 is !re=e f9ite1 ?\gh our computation rea* ! fix$ />s1 or ! n9? or t5?1 nay1 ! ?\s&? sph]e4 ,! mean+ ( ! "q is ?is only1 :e!r ,god has actually made s great an addition ( body to body1 z we are able to make ( space to space4 ,sid5ote3 ,division proce$s n to ! lea/4 #ac4 ,&1 !re=e1 t :i* is commonly said1 t space & "t may be divid$ inf9itely1 is n to be s und]/ood1 z if !re might be any inf9ite or et]nal division2 b r to be tak5 9 ?is s5se1 :atsoev] is divid$1 is divid$ into su* p>ts z may aga9 be divid$2 or ?us1 ! lea/ divisible ?+ is n to be giv52 or1 z geometricians h x1 no quantity is s small1 b a less may be tak52 :i* may easily be demon/rat$ 9 ?is mann]4 ,let any space or "t1 t :i* 0 ?\ght to be ! lea/ divisible1 be divid$ into two equal p>ts1 ,a & ,b4 ,i say ei!r ( !m1 z ,a1 may be divid$ aga94 ,= suppose ! "p ,a to be contigu\s to ! "p ,b ( "o side1 & ( ! o!r side to "s o!r space equal to ,b4 ,?is :ole space1 !re=e1 be+ great] ?an ! space giv51 is divisible4 ,:]e=e1 if x be divid$ into two equal p>ts1 ! "p 9 ! middle1 :i* is ,a1 w be also divid$ into two equal p>ts2 & !re=e ,a 0 divisible4 ------- ,,*apt] ,,viii4 ,,( ,,body ,,& ,,accid5t4 #a4 ,body def9$4--#b4 ,accid5t def9$4--#c4 ,h[ an accid5t may be und]/ood to be 9 its subject4--#d4 ,magnitude1 :at x is4--#e4 ,place1 :at x is1 & t x is immovable4--#f4 ,:at is full & empty4--#g4 ,"h1 !re1 some:]e1 :at !y signify4--#h4 ,_m bodies _c be 9 "o place1 nor "o body 9 _m places4--#i4 ,contigu\s & cont9ual1 :at !y are4--#aj4 ,! def9ition ( motion4 ,no motion intelligible b ) "t4--#aa4 ,:at x is to be at re/1 to h be5 mov$1 & to be mov$4 ,no motion to be conceiv$1 )\t ! conception ( pa/ & future4--#ab4 ,a po9t1 a l9e1 sup]ficies & solid1 :at !y are4--#ac4 ,equal1 great]1 & less 9 bodies & magnitudes1 :at !y are4--#ad4 ,"o & ! same body has always "o & ! same magnitude4--#ae4 ,velocity1 :at x is4--#af4 ,equal1 great]1 & less 9 times1 :at !y are4--#ag4 ,equal1 great]1 & less1 9 velocity1 :at4--#ah4 ,equal1 great]1 & less1 9 motion1 :at4--#ai4 ,t :i* is at re/1 w always be at re/1 except x be mov$ by "s ext]nal ?+2 & t :i* is mov$1 w always be mov$1 unless x be h9d]$ by "s ext]nal ?+4--#bj4 ,accid5ts are g5]at$ & de/roy$1 b bodies n s4--#ba4 ,an accid5t _c dep>t f its subject4--#bb4 ,nor be mov$4--#bc4 ,ess5ce1 =m1 & matt]1 :at !y are4--#bd4 ,fir/ matt]1 :at4--#be4 ,t ! :ole is great] ?an any "p !re(1 :y demon/rat$4 ,sid5ote3 ,body def9$4 #a4 ,hav+ und]/ood :at imag9>y space is1 9 :i* we suppos$ no?+ rema9+ )\t u1 b all ^? ?+s to be de/roy$1 t1 by exi/+ h]et(ore1 left images ( !mselves 9 our m9ds2 let u n[ suppose "s "o ( ^? ?+s to be plac$ aga9 9 ! _w1 or creat$ anew4 ,x is necess>y1 !re=e1 t ?is new-creat$ or replac$ ?+ d n only fill "s "p ( ! space above m5tion$1 or be co9cid5t & coext5d$ ) x1 b also t x h no dep5dance ^u our ?\ght4 ,& ?is is t :i*1 = ! ext5sion ( x1 we commonly call body2 & because x dep5ds n ^u our ?\ght1 we say is a ?+ subsi/+ ( itself2 z also exi/+1 because )\t u2 &1 la/ly1 x is call$ ! subject1 because x is s plac$ 9 & subject$ to imag9>y space1 t x may be und]/ood by reason1 z well z p]ceiv$ by s5se4 ,! def9ition1 !re=e1 ( body may be ?is1 a body is t1 :i* hav+ no dep5dance ^u our ?\ght1 is co9cid5t or coext5d$ ) "s "p ( space4 ,sid5ote3 ,accid5t def9$4 #b4 ,b :at an accid5t is _c s easily be expla9$ by any def9ition1 z by examples4 ,let u imag9e1 !re=e1 t a body fills any space1 or is coext5d$ ) x2 t coext5sion is n ! coext5d$ body3 &1 9 l mann]1 let u imag9e t ! same body is remov$ out ( its place2 t remov+ is n ! remov$ body3 or let u ?9k ! same n remov$2 t n remov+ or re/ is n ! re/+ body4 ,:at1 !n1 are ^! ?+s8 ,!y are accid5ts ( t body4 ,b ! ?+ 9 "q is1 :at is an accid5t8 :i* is an enquiry aft] t :i* we "k already1 & n t :i* we %\ld enquire aft]4 ,= :o does n always & 9 ! same mann] und]/& him t says any ?+ is ext5d$1 or mov$1 or n mov$8 ,b mo/ m5 w h x be said t an accid5t is some?+1 namely1 "s "p ( a natural ?+1 :51 inde$1 x is no "p ( ! same4 ,to satisfy ^! m51 z well z may be1 !y answ] be/ t def9e an accid5t to be ! mann] by :i* any body is conceiv$2 :i* is all "o z if !y %\ld say1 an accid5t is t faculty ( any body1 by :i* x works 9 u a conception ( itself4 ,:i* def9ition1 ?\gh x be n an answ] to ! "q prop\nd$1 yet x is an answ] to t "q :i* %\ld h be5 prop\nd$1 namely1 :5ce does x happ5 t "o "p ( any body appe>s "h1 ano!r !re8 ,= ?is is well answ]$ ?us3 x happ5s f ! ext5sion ( t body4 ,or1 h[ comes x to pass t ! :ole body1 by succession1 is se5 n[ "h1 n[ !re8 & ! answ] w be1 by reason ( its motion4 ,or1 la/ly1 :5ce is x t any body possesse? ! same space = sometime8 & ! answ] w be1 because x is n mov$4 ,= if conc]n+ ! "n ( a body1 t is1 conc]n+ a concrete "n1 x be ask$1 :at is x8 ! answ] mu/ be made by def9ition2 = ! "q is conc]n+ ! signification ( ! "n4 ,b if x be ask$ conc]n+ an ab/ract "n1 :at is x8 ! cause is dem&$ :y a ?+ appe>s s or s4 ,z if x be ask$1 :at is h>d8 ,! answ] w be1 h>d is t1 :]e( no "p gives place1 b :5 ! :ole gives place4 ,b if x be dem&$1 :at is h>dness8 a cause mu/ be %ewn :y a "p does n give place1 except ! :ole give place4 ,:]e=e1 ,i def9e an accid5t to be ! mann] ( our conception ( body4 ,sid5ote3 ,h[ an accid5t may be und]/ood to be 9 its subject4 #c4 ,:5 an accid5t is said to be 9 a body1 x is n s to be und]/ood1 z if any ?+ 7 conta9$ 9 t body2 z if1 = example1 r$ness 7 9 blood1 9 ! same mann]1 z blood is 9 a bloody clo?1 t is1 z a "p 9 ! :ole2 = s1 an accid5t w\ld be a body also4 ,b1 z magnitude1 or re/1 or motion1 is 9 t :i* is great1 or :i* re/e?1 or :i* is mov$1 7:i*1 h[ x is to be und]/ood1 e man und]/&s7 s also1 x is to be und]/ood1 t e o!r accid5t is 9 its subject4 ,& ?is1 also1 is explicat$ by ,ari/otle no o!rwise ?an negatively1 namely1 t an accid5t is 9 its subject1 n z any "p !re(1 b s z t x may be away1 ! subject /ill rema9+2 :i* is "r1 sav+ t !re are c]ta9 accid5ts :i* c nev] p]i% except ! body p]i% also2 = no body c be conceiv$ to be )\t ext5sion1 or )\t figure4 ,all o!r accid5ts1 :i* are n common to all bodies1 b peculi> to "s only1 z to be at re/1 to be mov$1 col\r1 h>dness1 & ! l1 d p]i% cont9ually1 & are succe$$ by o!rs2 yet s1 z t ! body nev] p]i%e?4 ,& z = ! op9ion t "s may h1 t all o!r accid5ts are n 9 _! bodies 9 ! same mann] t ext5sion1 motion1 re/1 or figure1 are 9 ! same2 = example1 t col\r1 heat1 od\r1 virtue1 vice1 & ! l1 are o!rwise 9 !m1 &1 z !y say1 inh]5t2 ,i desire !y w\ld susp5d _! judgm5t = ! pres5t1 & expect a little1 till x be f\nd out by ratioc9ation1 :e!r ^! v accid5ts are n also c]ta9 motions ei!r ( ! m9d ( ! p]ceiv]1 or ( ! bodies !mselves :i* are p]ceiv$2 = 9 ! se>* ( ?is1 a great "p ( natural philosophy consi/s4 ,sid5ote3 ,magnitude1 :at x is4 #d4 ,! ext5sion ( a body1 is ! same ?+ ) ! magnitude ( x1 or t :i* "s call real space4 ,b ?is magnitude does n dep5d ^u our cogitation1 z imag9>y space do?2 = ?is is an effect ( our imag9ation1 b magnitude is ! cause ( x2 ?is is an accid5t ( ! m9d1 t ( a body exi/+ out ( ! m9d4 ,sid5ote3 ,place1 :at x is1 & t x is immovable4 #e4 ,t space1 by :i* ^w ,i "h und]/& imag9>y space1 :i* is co9cid5t ) ! magnitude ( any body1 is call$ ! place ( t body2 & ! body itself is t :i* we call ! ?+ plac$4 ,n[ place1 & ! magnitude ( ! ?+ plac$1 diff]4 ,fir/ 9 ?is1 t a body keeps always ! same magnitude1 bo? :5 x is at re/1 & :5 x is mov$2 b :5 x is mov$1 x does n keep ! same place4 ,secondly 9 ?is1 t place is a phantasm ( any body ( su* & su* quantity & figure2 b magnitude is ! peculi> accid5t ( e body2 = "o body may at sev]al times h sev]al places1 b has always "o & ! same magnitude4 ,?irdly 9 ?is1 t place is no?+ out ( ! m9d1 nor magnitude any ?+ )9 x4 ,& la/ly1 place is feign$ ext5sion1 b magnitude true ext5sion2 & a plac$ body is n ext5sion1 b a ?+ ext5d$4 ,besides1 place is immovable2 =1 see+ t :i* is mov$1 is und]/ood to be c>ri$ f place to place1 if place 7 mov$1 x w\ld also be c>ri$ f place to place1 s t "o place mu/ h ano!r place1 & t place ano!r place1 & s on inf9itely1 :i* is ridicul\s4 ,& z = ^?1 t1 by mak+ place to be ( ! same nature ) real space1 w\ld f !nce ma9ta9 x to be immovable1 !y also make place1 ?\gh !y d n p]ceive !y make x s1 to be a m]e phantasm4 ,= :il/ "o affirms t place is !re=e said to be immovable1 because space 9 g5]al is consid]$ !re2 if he _h rememb]$ t no?+ is g5]al or univ]sal besides names or signs1 he w\ld easily h se5 t t space1 :i* he says is consid]$ 9 g5]al1 is no?+ b a phantasm1 9 ! m9d or ! memory1 ( a body ( su* magnitude & su* figure4 ,& :il/ ano!r says3 real space is made immovable by ! und]/&+2 z :51 "u ! sup]ficies ( runn+ wat]1 we imag9e o!r & o!r wat] to come by cont9ual succession1 t sup]ficies fix$ !re by ! und]/&+1 is ! immovable place ( ! riv]3 :at else does he make x to be b a phantasm1 ?\gh he d x obscurely & 9 p]plex$ words8 ,la/ly1 ! nature ( place does n consi/ 9 ! sup]ficies ( ! ambi5t1 b 9 solid space2 = ! :ole plac$ body is coext5d$ ) its :ole place1 & e "p ( x ) e answ]+ "p ( ! same place2 b see+ e plac$ body is a solid ?+1 x _c be und]/ood to be coext5d$ ) sup]ficies4 ,besides1 h[ c any :ole body be mov$1 unless all its p>ts be mov$ toge!r ) x8 ,or h[ c ! int]nal p>ts ( x be mov$1 b by leav+ _! place8 ,b ! int]nal p>ts ( a body _c leave ! sup]ficies ( an ext]nal "p contigu\s to x2 &1 !re=e1 x foll[s1 t if place be ! sup]ficies ( ! ambi5t1 !n ! p>ts ( a body mov$1 t is1 bodies mov$1 are n mov$4 ,sid5ote3 ,:at is full & empty4 #f4 ,space1 or place1 t is possess$ by a body1 is call$ full1 & t :i* is n s possess$1 is call$ empty4 ,sid5ote3 ,"h1 !re1 some:]e1 :at !y signify4 #g4 ,"h1 !re1 9 ! c\ntry1 9 ! city1 & o!r ! l names1 by :i* answ] is made to ! "q :]e is x8 are n prop]ly names ( place1 nor d !y ( !mselves br+ into ! m9d ! place t is s\ght2 = "h & !re signify no?+1 unless ! ?+ be %ewn at ! same "t ) ! f+] or some?+ else2 b :5 ! eye ( him t seeks1 is1 by po9t+ or "s o!r sign1 direct$ to ! ?+ s\ght1 ! place ( x is n h]eby def9$ by him t answ]s1 b f\nd out by him t asks ! "q4 ,n[ su* %ew+s z are made by words only1 z :5 we say1 9 ! c\ntry1 or 9 ! city1 are "s ( great] latitude ?an o!rs1 z :5 we say1 9 ! c\ntry1 9 ! city1 9 su* a /reet1 9 a h\se1 9 ! *amb]1 9 b$1 c4 ,= ^! d1 by little & little1 direct ! seek] ne>] to ! prop] place2 & yet !y d n det]m9e ! same1 b only re/ra9 x to a less] space1 & signify no m1 ?an t ! place ( ! ?+ is )9 a c]ta9 space design$ by ^? words1 z a "p is 9 ! :ole4 ,& all su* names1 by :i* answ] is made to ! "q :]e8 h1 = _! highe/ g5us1 ! "n some:]e4 ,f :5ce x may be und]/ood1 t :atsoev] is some:]e1 is 9 "s place prop]ly s call$1 :i* place is "p ( t great] space t is signifi$ by "s ( ^! names1 9 ! c\ntry1 9 ! city1 or ! l4 ,sid5ote3 ,_m bodies _c be 9 "o place1 nor "o body 9 _m places4 #h4 ,a body1 & ! magnitude1 & ! place !re(1 are divid$ by "o & ! same act ( ! m9d2 =1 to divide an ext5d$ body1 & ! ext5sion !re(1 & ! idea ( t ext5sion1 :i* is place1 is ! same ) divid+ any "o ( !m2 because !y are co9cid5t1 & x _c be done b by ! m9d1 t is by ! division ( space4 ,f :5ce x is manife/1 t nei!r two bodies c be toge!r 9 ! same place1 nor "o body be 9 two places at ! same "t4 ,n two bodies 9 ! same place2 because :5 a body t fills its :ole place is divid$ into two1 ! place itself is divid$ into two also1 s t !re w be two places4 ,n "o body 9 two places2 = ! place t a body fills be+ divid$ into two1 ! plac$ body w be also divid$ into two2 =1 z ,i said1 a place & ! body t fills t place1 are divid$ bo? toge!r2 & s !re w be two bodies4 ,sid5ote3 ,contigu\s & cont9ual1 :at !y are4 #i4 ,two bodies are said to be contigu\s to "o ano!r1 & cont9ual1 9 ! same mann] z spaces are2 namely1 ^? are contigu\s1 betwe5 :i* !re is no space4 ,n[1 by space ,i und]/&1 "h z =m]ly1 an idea or phantasm ( a body4 ,:]e=e1 ?\gh betwe5 two bodies !re be put no o!r body1 & consequ5tly no magnitude1 or1 z !y call x1 real space1 yet if ano!r body may be put betwe5 !m1 t is1 if !re int]c$e any imag9$ space :i* may receive ano!r body1 !n ^? bodies are n contigu\s4 ,& ?is is s easy to be und]/ood1 t ,i %\ld wond] at "s m51 :o be+ o!rwise skilful en\gh 9 philosophy1 are ( a diff]5t op9ion1 b t ,i f9d t mo/ ( ^? t affect metaphysical subtleties w&] f tru?1 z if !y 7 l$ out ( _! way by an ignis fatuus4 ,= c any man t has 8 natural s5ses1 ?9k t two bodies mu/ !re=e necess>ily t\* "o ano!r1 because no o!r body is betwe5 !m8 ,or t !re c be no vacuum1 because vacuum is no?+1 or z !y call x1 non ens8 ,:i* is z *ildi%1 z if "o %\ld reason ?us2 no man c fa/1 because to fa/ is to eat no?+2 b no?+ _c be eat54 ,cont9ual1 are any two bodies t h a common "p2 & m ?an two are cont9ual1 :5 e two1 t are next to "o ano!r1 are cont9ual4 ,sid5ote3 ,! def9ition ( motion4 ,no motion intelligible b ) "t4 #aj4 ,,motion is a cont9ual rel9qui%+ ( "o place1 & acquir+ ( ano!r2 & t place :i* is rel9qui%$ is commonly call$ ! t]m9us a quo1 z t :i* is acquir$ is call$ ! t]m9us ad quem2 ,i say a cont9ual rel9qui%+1 because no body1 h[ little soev]1 c totally & at once g out ( its =m] place into ano!r1 s1 b t "s "p ( x w be 9 a "p ( a place :i* is common to bo?1 namely1 to ! rel9qui%$ & ! acquir$ places4 ,sid5ote3 ,a ,g ,b ,i ,e ----------- 3 3 3 3 ----------- ,c ,h ,d ,k ,f ,= example1 let any body be 9 ! place ,a ,c ,b ,d2 ! same body _c come into ! place ,b ,d ,e ,f1 b x mu/ fir/ be 9 ,g ,h ,i ,k1 ^: "p ,g ,h ,b ,d is common to bo? ! places ,a ,c ,b ,d1 & ,g ,h ,i ,k1 & ^: "p ,b ,d ,i ,k1 is common to bo? ! places ,g ,h ,i ,k1 & ,b ,d ,e ,f4 ,n[ x _c be conceiv$ t any ?+ c be mov$ )\t "t2 = "t is1 by ! def9ition ( x1 a phantasm1 t is1 a conception ( motion2 &1 !re=e1 to conceive t any ?+ may be mov$ )\t "t1 7 to conceive motion )\t motion1 :i* is impossible4 ,sid5ote3 ,:at x is to be at re/1 to h be5 mov$1 & to be mov$4 ,no motion to be conceiv$ )\t ! conception ( pa/ & future4 #aa4 ,t is said to be at re/1 :i*1 dur+ any "t1 is 9 "o place2 & t to be mov$1 or to h be5 mov$1 :i*1 :e!r x be n[ at re/ or mov$1 0 =m]ly 9 ano!r place ?an t :i* x is n[ 94 ,f :i* def9itions x may be inf]r$1 fir/1 t :atsoev] is mov$1 has be5 mov$2 = if x be /ill 9 ! same place 9 :i* x 0 =m]ly1 x is at re/1 t is1 x is n mov$1 by ! def9ition ( re/2 b if x be 9 ano!r place1 x has be5 mov$1 by ! def9ition ( mov$4 ,secondly1 t :at is mov$1 w yet be mov$2 = t :i* is mov$1 leave? ! place :]e x is1 & !re=e w be 9 ano!r place1 & consequ5tly w be mov$ /ill4 ,?irdly1 t :atsoev] is mov$1 is n 9 "o place dur+ any "t1 h[ little soev] t "t be2 = by ! def9ition ( re/1 t :i* is 9 "o place dur+ any "t1 is at re/4 ,!re is a c]ta9 sophism aga9/ motion1 :i* seems to spr+ f ! n und]/&+ ( ?is la/ proposition4 ,= !y say1 t1 if any body be mov$1 x is mov$ ei!r 9 ! place :]e x is1 or 9 ! place :]e x is n2 bo? :i* are false2 & !re=e no?+ is mov$4 ,b ! falsity lies 9 ! major proposition2 = t :i* is mov$1 is nei!r mov$ 9 ! place :]e x is1 nor 9 ! place :]e is n2 b f ! place :]e x is1 to ! place :]e x is n4 ,inde$ x _c be d5i$ b t :atsoev] is mov$1 is mov$ some:]e1 t is1 )9 "s space2 b !n ! place ( t body is n t :ole space1 b a "p ( x1 z is said above 9 ! sev5? article4 ,f :at is above demon/rat$1 namely1 t :atsoev] is mov$1 has also be5 mov$1 & w be mov$1 ?is also may be collect$1 t !re c be no conception ( motion1 )\t conceiv+ pa/ & future "t4 ,sid5ote3 ,a po9t1 a l9e1 sup]ficies1 & solid1 :at !y are4 #ab4 ,?\gh !re be no body :i* has n "s magnitude1 yet if1 :5 any body is mov$1 ! magnitude ( x be n at all consid]$1 ! way x makes is call$ a l9e1 or "o s+le dim5sion2 & ! space1 ?r\gh :i* x passe?1 is call$ l5g?2 & ! body itself1 a po9t2 9 :i* s5se ! e>? is call$ a po9t1 & ! way ( its ye>ly revolution1 ! ecliptic l9e4 ,b if a body1 :i* is mov$1 be consid]$ z long1 & be suppos$ to be s mov$1 z t all ! sev]al p>ts ( x be und]/ood to make sev]al l9es1 !n ! way ( e "p ( t body is call$ bread?1 & ! space :i* is made is call$ sup]ficies1 consi/+ ( two dim5sions1 "o :]e( to e sev]al "p ( ! o!r is appli$ :ole4 ,aga91 if a body be consid]$ z hav+ sup]ficies1 & be und]/ood to be s mov$1 t all ! sev]al p>ts ( x describe sev]al l9es1 !n ! way ( e "p ( t body is call$ ?ickness or dep?1 & ! space :i* is made is call$ solid1 consi/+ ( ?ree dim5sions1 any two :]e( are appli$ :ole to e sev]al "p ( ! ?ird4 ,b if a body be consid]$ z solid1 !n x is n possible t all ! sev]al p>ts ( x %\ld describe sev]al l9es2 = :at way soev] x be mov$1 ! way ( ! foll[+ "p w fall into ! way ( ! "p be=e x1 s t ! same solid w /ill be made :i* ! =emo/ sup]ficies w\ld h made by itself4 ,& !re=e !re c be no o!r dim5sion 9 any body1 z x is a body1 ?an ! ?ree :i* ,i h n[ describ$2 ?\gh1 z x %all be %ew$ h]eaft]1 velocity1 :i* is motion accord+ to l5g?1 may1 by be+ appli$ to all ! p>ts ( a solid1 make a magnitude ( motion1 consi/+ ( f\r dim5sions2 z ! goodness ( gold1 comput$ 9 all ! p>ts ( x1 makes ! price & value !re(4 ,sid5ote3 ,equal1 great1 great]1 & less1 9 bodies & magnitudes1 :at !y are4 #ac4 ,bodies1 h[ _m soev] !y be1 t c fill e "o ! place ( e "o1 are said to be equal e "o to e o!r4 ,n[1 "o body may fill ! same place :i* ano!r body fille?1 ?\gh x be n ( ! same figure ) t o!r body1 if s be t x may be und]/ood to be r$ucible to ! same figure1 ei!r by flexion or transposition ( ! p>ts4 ,& "o body is great] ?an ano!r body1 :5 a "p ( t is equal to all ?is2 & less1 :5 all t is equal to a "p ( ?is4 ,also1 magnitudes are equal1 or great]1 or less]1 ?an "o ano!r1 = ! same consid]ation1 namely1 :5 ! bodies1 ( :i* !y are ! magnitudes1 are ei!r equal1 or great]1 or less1 c4 ,sid5ote3 ,"o & ! same body has always "o & ! same magnitude4 #ad4 ,"o & ! same body is always ( "o & ! same magnitude4 ,= see+ a body & ! magnitude & place !re( _c be compreh5d$ 9 ! m9d o!rwise ?an z !y are co9cid5t1 if any body be und]/ood to be at re/1 t is1 to rema9 9 ! same place dur+ "s "t1 & ! magnitude !re( be 9 "o "p ( t "t great]1 & 9 ano!r "p less1 t body's place1 :i* is "o & ! same1 w be co9cid5t sometimes ) great]1 sometimes ) less magnitude1 t is1 ! same place w be great] & less ?an itself1 :i* is impossible4 ,b !re w\ld be no ne$ at all ( demon/rat+ a ?+ t is 9 itself s manife/1 if !re 7 n "s1 ^: op9ion conc]n+ bodies & _! magnitudes is1 t a body may exi/ sep>at$ f its magnitude1 & h great] or less magnitude be/[$ ^u x1 mak+ use ( ?is pr9ciple = ! explication ( ! nature ( r>um & d5sum4 ,sid5ote3 ,velocity1 :at x is4 #ae4 ,motion1 9 z mu* z a c]ta9 l5g? may 9 a c]ta9 "t be transmitt$ by x1 is call$ ,,velocity or swiftness3 c4 ,= ?\gh swift be v (t5 und]/ood ) relation to sl[] or less swift1 z great is 9 respect ( less1 yet nev]!less1 z magnitude is by philosoph]s tak5 absolutely = ext5sion1 s also velocity or swiftness may be put absolutely = motion accord+ to l5g?4 ,sid5ote3 ,equal1 great]1 & less1 9 times1 :at !y are4 #af4 ,_m motions are said to be made 9 equal times1 :5 e "o ( !m beg9s & ends toge!r ) "s o!r motion1 or if x _h begun toge!r1 w\ld also h end$ toge!r ) ! same4 ,= "t1 :i* is a phantasm ( motion1 _c be reckon$ b by "s expos$ motion2 z 9 dials by ! motion ( ! sun or ( ! h&2 & if two or m motions beg9 & end ) ?is motion1 !y are said to be made 9 equal times2 f :5ce also x is easy to und]/& :at x is to be mov$ 9 great] or long] "t1 & 9 less "t or n s long2 namely1 t t is long] mov$1 :i* beg9n+ ) ano!r1 ends lat]2 or end+ toge!r1 began soon]4 ,sid5ote3 ,equal1 great]1 & less1 9 velocity1 :at4 #ag4 ,motions are said to be equally swift1 :5 equal l5g?s are transmitt$ 9 equal times2 & great] swiftness is t1 :]e9 great] l5g? is pass$ 9 equal "t1 or equal l5g? 9 less "t4 ,also t swiftness by :i* equal l5g?s are pass$ 9 equal p>ts ( "t1 is call$ uni=m swiftness or motion2 & ( motions n uni=m1 su* z become swift] or sl[] by equal increas+s or decreas+s 9 equal p>ts ( "t1 are said to be accel]at$ or ret>d$ uni=mly4 ,sid5ote3 ,equal1 great]1 & less1 9 motion1 :at4 #ah4 ,b motion is said to be great]1 less1 & equal1 n only 9 reg>d ( ! l5g? :i* is transmitt$ 9 a c]ta9 "t1 t is1 9 reg>d ( swiftness only1 b ( swiftness appli$ to e smalle/ p>ticle ( magnitude2 = :5 any body is mov$1 e "p ( x is also mov$2 & suppos+ ! p>ts to be halves1 ! motions ( ^? halves h _! swiftness equal to "o ano!r1 & sev]ally equal to t ( ! :ole2 b ! motion ( ! :ole is equal to ^? two motions1 ei!r ( :i* is ( equal swiftness ) x2 & !re=e x is "o ?+ = two motions to be equal to "o ano!r1 & ano!r ?+ = !m to be equally swift4 ,& ?is is manife/ 9 two horses t draw abrea/1 :]e ! motion ( bo? ! horses toge!r is ( equal swiftness ) ! motion ( ei!r ( !m s+ly2 b ! motion ( bo? is great] ?an ! motion ( "o ( !m1 namely1 d\ble4 ,:]e=e motions are said to be simply equal to "o ano!r1 :5 ! swiftness ( "o1 comput$ 9 e "p ( its magnitude1 is equal to ! swiftness ( ! o!r comput$ also 9 e "p ( its magnitude3 & great] ?an "o ano!r1 :5 ! swiftness ( "o comput$ z above1 is great] ?an ! swiftness ( ! o!r s comput$2 & less1 :5 less4 ,besides1 ! magnitude ( motion comput$ 9 ?is mann] is t :i* is commonly call$ ,,=ce4 ,sid5ote3 ,t :i* is at re/ w always be at re/1 except x be mov$ by "s ext]nal ?+4 #ai4 ,:atsoev] is at re/1 w always be at re/1 unless !re be "s o!r body besides x1 :i*1 by endeav\r+ to get into its place by motion1 suff]s x no long] to rema9 at re/4 ,= suppose t "s f9ite body exi/ & be at re/1 & t all space besides be empty2 if n[ ?is body beg9 to be mov$1 x w c]ta9ly be mov$ "s way2 see+ !re=e !re 0 no?+ 9 t body :i* did n dispose x to re/1 ! reason :y x is mov$ ?is way is 9 some?+ out ( x2 & 9 l mann]1 if x _h be5 mov$ any o!r way1 ! reason ( motion t way _h also be5 9 some?+ out ( x2 b see+ x 0 suppos$ t no?+ is out ( x1 ! reason ( its motion "o way w\ld be ! same ) ! reason ( its motion e o!r way1 :]e=e x w\ld be mov$ alike all ways at once2 :i* is impossible4 ,sid5ote3 ,t :i* is mov$ w always be mov$1 unless x be h9d]$ by "s ext]nal ?+4 ,9 l mann]1 :atsoev] is mov$1 w always be mov$1 except !re be "s o!r body besides x1 :i* cause? x to re/4 ,= if we suppose no?+ to be )\t x1 !re w be no reason :y x %\ld re/ n[1 r ?an at ano!r "t2 :]e=e its motion w\ld cease 9 e p>ticle ( "t alike2 :i* is n intelligible4 ,sid5ote3 ,accid5ts are g5]at$ & de/roy$1 b bodies n s4 #bj4 ,:5 we say a liv+ creature1 a tree1 or any o!r specifi$ body is g5]at$ or de/roy$1 x is n to be s und]/ood z if !re 7 made a body ( t :i* is n-body1 or n a body ( a body1 b ( a liv+ creature n a liv+ creature1 ( a tree n a tree1 c4 t is1 t ^? accid5ts = :i* we call "o ?+ a liv+ creature1 ano!r ?+ a tree1 & ano!r by "s o!r "n1 are g5]at$ & de/roy$2 & t !re=e ! same names are n to be giv5 to !m n[1 :i* 7 giv5 !m be=e4 ,b t magnitude = :i* we give to any ?+ ! "n ( body is nei!r g5]at$ nor de/roy$4 ,= ?\gh we may feign 9 our m9d t a po9t may swell to a huge bulk1 & t ?is may aga9 contract itself to a po9t2 t is1 ?\gh we may imag9e some?+ to arise :]e be=e 0 no?+1 & no?+ to be !re :]e be=e 0 some?+1 yet we _c compreh5d 9 our m9d h[ ?is may possibly be done 9 nature4 ,& !re=e philosoph]s1 :o tie !mselves to natural reason1 suppose t a body c nei!r be g5]at$ nor de/roy$1 b only t x may appe> o!rwise ?an x did to u1 t is1 "u diff]5t species1 & consequ5tly be call$ by o!r & o!r names2 s t t :i* is n[ call$ man1 may at ano!r "t h ! "n ( n-man2 b t :i* is once call$ body1 c nev] be call$ n-body4 ,b x is manife/1 t all o!r accid5ts besides magnitude or ext5sion may be g5]at$ & de/roy$2 z :5 a :ite ?+ is made black1 ! :it5ess t 0 9 x p]i%e?1 & ! blackness t 0 n 9 x is n[ g5]at$2 & !re=e bodies1 & ! accid5ts "u :i* !y appe> div]sely1 h ?is diff]5ce1 t bodies are ?+s1 & n g5]at$2 accid5ts are g5]at$1 & n ?+s4 ,sid5ote3 ,an accid5t _c dep>t f its subject #ba4 ,& !re=e1 :5 any ?+ appe>s o!rwise ?an x did by reason ( o!r & o!r accid5ts1 x is n to be ?\ght t an accid5t goes out ( "o subject into ano!r1 7= !y are n1 z ,i said above1 9 _! subjects z a "p 9 ! :ole1 or z a conta9$ ?+ 9 t :i* conta9s x1 or z a ma/] ( a family 9 8 h\se17 b t "o accid5t p]i%e?1 & ano!r is g5]at$4 ,= example1 :5 ! h&1 be+ mov$1 moves ! p51 motion does n g out ( ! h& into ! p52 = s ! writ+ might be cont9u$ ?\gh ! h& /ood /ill2 b a new motion is g5]at$ 9 ! p51 & is ! p5's motion4 ,sid5ote3 ,nor be mov$4 #bb4 ,& !re=e also x is improp] to say1 an accid5t is mov$2 z :51 in/ead ( say+1 figure is an accid5t ( a body c>ri$ away1 we say1 a body c>ries away its figure4 ,sid5ote3 ,ess5ce1 =m1 & matt]1 :at !y are4 #bc4 ,n[ t accid5t = :i* we give a c]ta9 "n to any body1 or ! accid5t :i* d5om9ates its subject1 is commonly call$ ! ,,ess5ce !re(2 z rationality is ! ess5ce ( a man2 :it5ess1 ( any :ite ?+1 & ext5sion ! ess5ce ( a body4 ,& ! same ess5ce1 9 z mu* z x is g5]at$1 is call$ ! ,,=m4 ,aga91 a body1 9 respect ( any accid5t1 is call$ ! ,,subject1 & 9 respect ( ! =m x is call$ ! ,,matt]4 ,also1 ! production or p]i%+ ( any accid5t makes its subject be said to be *ang$2 only ! production or p]i%+ ( =m makes x be said x is g5]at$ or de/roy$2 b 9 all g5]ation & mutation1 ! "n ( matt] /ill rema9s4 ,= a table made ( wood is n only wood51 b wood2 & a /atue ( brass is brass z well z braz52 ?\gh ,ari/otle1 9 8 ,metaphysics1 says1 t :atsoev] is made ( any ?+ ought n to be call$ 1 b 2 z t :i* is made ( wood1 n 1 b 1 t is1 n wood1 b wood54 ,sid5ote3 ,fir/ matt]1 :at4 #bd4 ,& z = t matt] :i* is common to all ?+s1 & :i* philosoph]s1 foll[+ ,ari/otle1 usually call mat]ia prima1 t is1 fir/ matt]1 x is n any body di/9ct f all o!r bodies1 nor is x "o ( !m4 ,:at !n is x8 ,a m]e "n2 yet a "n :i* is n ( va9 use2 = x signifies a conception ( body )\t ! consid]ation ( any =m or o!r accid5t except only magnitude or ext5sion1 & aptness to receive =m & o!r accid5t4 ,s t :5soev] we h use ( ! "n body 9 g5]al1 if we use t ( mat]ia prima1 we d well4 ,= z :5 a man n kn[+ :i* 0 fir/1 wat] or ice1 w\ld f9d out :i* ( ! two 7 ! matt] ( bo?1 he w\ld be fa9 to suppose "s ?ird matt] :i* 7 nei!r ( ^! two2 s he t w\ld f9d out :at is ! matt] ( all ?+s1 ought to suppose su* z is n ! matt] ( any?+ t exi/s4 ,:]e=e mat]ia prima is no?+2 & !re=e !y d n attribute to x ei!r =m or any o!r accid5t besides quantity2 :]eas all s+ul> ?+s h _! =ms & accid5ts c]ta94 ,mat]ia prima1 !re=e1 is body 9 g5]al1 t is1 body consid]$ univ]sally1 n z hav+ nei!r =m nor any accid5t1 b 9 :i* no =m nor any o!r accid5t b quantity are at all consid]$1 t is1 !y are n drawn into argum5tation4 ,sid5ote3 ,t ! :ole is great] ?an any "p !re(1 :y demon/rat$4 #be4 ,f :at has be5 said1 ^? axioms may be demon/rat$1 :i* are assum$ by ,euclid 9 ! beg9n+ ( 8 fir/ elem5t1 ab\t ! equality & inequality ( magnitudes2 ( :i*1 omitt+ ! re/1 ,i w "h demon/rate only ?is "o1 ! :ole is great] ?an any "p !re(2 to ! end t ! read] may "k t ^? axioms are n indemon/rable1 & !re=e n pr9ciples ( demon/ration2 & f h5ce le>n to be w>y h[ he admits any ?+ = a pr9ciple1 :i* is n at lea/ z evid5t z ^! are4 ,great] is def9$ to be t1 ^: "p is equal to ! :ole ( ano!r4 ,n[ if we suppose any :ole to be ,a1 & a "p ( x to be ,b2 see+ ! :ole ,b is equal to itself1 & ! same ,b is a "p ( ,a2 !re=e a "p ( ,a w be equal to ! :ole ,b4 ,:]e=e1 by ! def9ition above1 ,a is great] ?an ,b2 :i* 0 to be prov$4 ------- ,,*apt] ,,ix4 ,,( ,,cause ,,& ,,effect4 #a4 ,action & passion1 :at !y are4--#b4 ,action & passion m$iate & imm$iate4--#c4 ,cause simply tak54 ,cause )\t :i* no effect foll[s1 or cause necess>y by supposition4--#d4 ,cause effici5t & mat]ial4--#e4 ,an entire cause is always suffici5t to produce its effect4 ,at ! same in/ant t ! cause is entire1 ! effect is produc$4 ,e effect has a necess>y cause4--#f4 ,! g5]ation ( effects is cont9ual4 ,:at is ! beg9n+ 9 causation4--#g4 ,no cause ( motion b 9 a body contigu\s & mov$4--#h4 ,! same ag5ts & pati5ts1 if alike dispos$1 produce l effects ?\gh at diff]5t times4--#i4 ,all mutation is motion4--#aj4 ,cont+5t accid5ts1 :at !y are4 ,sid5ote3 ,action & passion1 :at !y are4 #a4 ,a body is said to "w ^u or act1 t is to say1 d some?+ to ano!r body1 :5 x ei!r g5]ates or de/roys "s accid5t 9 x3 & ! body 9 :i* an accid5t is g5]at$ or de/roy$ is said to suff]1 t is1 to h some?+ done to x by ano!r body2 z :5 "o body by putt+ =w>ds ano!r body g5]ates motion 9 x1 x is call$ ! ,,ag5t2 & ! body 9 :i* motion is s g5]at$1 is call$ ! ,,pati5t2 s fire t w>ms ! h& is ! ag5t1 & ! h&1 :i* is w>m$1 is ! pati5t4 ,t accid5t1 :i* is g5]at$ 9 ! pati5t1 is call$ ! ,,effect4 ,sid5ote3 ,action & passion1 m$iate1 & imm$iate4 #b4 ,:5 an ag5t & pati5t are contigu\s to "o ano!r1 _! action & passion are !n said to be imm$iate1 o!rwise1 m$iate2 & :5 ano!r body1 ly+ betwixt ! ag5t & pati5t1 is contigu\s to !m bo?1 x is !n itself bo? an ag5t & a pati5t2 an ag5t 9 respect ( ! body next aft] x1 ^u :i* x works1 & a pati5t 9 respect ( ! body next be=e x1 f :i* x suff]s4 ,also1 if _m bodies be s ord]$ t e two :i* are next to "o ano!r be contigu\s1 !n all ^? t are betwixt ! fir/ & ! la/ are bo? ag5ts & pati5ts1 & ! fir/ is an ag5t only1 & ! la/ a pati5t only4 ,sid5ote3 ,cause simply tak54 #c4 ,an ag5t is und]/ood to produce its det]m9$ or c]ta9 effect 9 ! pati5t1 accord+ to "s c]ta9 accid5t or accid5ts1 ) :i* bo? x & ! pati5t are affect$2 t is to say1 ! ag5t ha? its effect precisely su*1 n because x is a body1 b because su* a body1 or s mov$4 ,= o!rwise all ag5ts1 see+ !y are all bodies alike1 w\ld produce l effects 9 all pati5ts4 ,& !re=e ! fire1 = example1 does n w>m1 because x is a body1 b because x is hot2 nor does "o body put =w>d ano!r body because x is a body1 b because x is mov$ into ! place ( t o!r body4 ,sid5ote3 ,cause )\t :i* no effect foll[s1 or cause necess>y by supposition4,! cause1 !re=e1 ( all effects consi/s 9 c]ta9 accid5ts bo? 9 ! ag5ts & 9 ! pati5ts2 :i* :5 !y are all pres5t1 ! effect is produc$2 b if any "o ( !m be want+1 x is n produc$2 & t accid5t ei!r ( ! ag5t or pati5t1 )\t :i* ! effect _c be produc$1 is call$ causa s9e qua non1 or cause necess>y by supposition1 z also ! cause requisite = ! production ( ! effect4 ,b a ,,cause simply1 or an entire cause1 is ! aggregate ( all ! accid5ts bo? ( ! ag5ts h[ _m soev] !y be1 & ( ! pati5t1 put toge!r2 :i* :5 !y are all suppos$ to be pres5t1 x _c be und]/ood b t ! effect is produc$ at ! same in/ant2 & if any "o ( !m be want+1 x _c be und]/ood b t ! effect is n produc$4 ,sid5ote3 ,cause effici5t1 & mat]ial4 #d4 ,! aggregate ( accid5ts 9 ! ag5t or ag5ts1 requisite = ! production ( ! effect1 ! effect be+ produc$1 is call$ ! effici5t cause !re(2 & ! aggregate ( accid5ts 9 ! pati5t1 ! effect be+ produc$1 is usually call$ ! mat]ial cause2 ,i say ! effect be+ produc$2 = :]e !re is no effect1 !re c be no cause2 = no?+ c be call$ a cause1 :]e !re is no?+ t c be call$ an effect4 ,b ! effici5t & mat]ial causes are bo? b p>tial causes1 or p>ts ( t cause1 :i* 9 ! next prec$5t article ,i call$ an entire cause4 ,& f h5ce x is manife/1 t ! effect we expect1 ?\gh ! ag5ts be n defective on _! "p1 may nev]!less be fru/rat$ by a defect 9 ! pati5t2 & :5 ! pati5t is suffici5t1 by a defect 9 ! ag5ts4 ,sid5ote3 ,an entire cause is always suffici5t to produce its effect4 #e4 ,an entire cause is always suffici5t = ! production ( its effect1 if ! effect be at all possible4 ,= let any effect :atsoev] be prop\nd$ to be produc$2 if ! same be produc$1 x is manife/ t ! cause :i* produc$ x 0 a suffici5t cause2 b if x be n produc$1 & yet be possible1 x is evid5t t some?+ 0 want+ ei!r 9 "s ag5t1 or 9 ! pati5t1 )\t :i* x c\ld n be produc$2 t is1 t "s accid5t 0 want+ :i* 0 requisite = its production2 & !re=e1 t cause 0 n entire1 :i* is contr>y to :at 0 suppos$4 ,sid5ote3 ,at ! same in/ant t ! cause is entire1 ! effect is produc$4 ,x foll[s also f h5ce1 t 9 :atsoev] in/ant ! cause is entire1 9 ! same in/ant ! effect is produc$4 ,= if x be n produc$1 some?+ is /ill want+1 :i* is requisite = ! production ( x2 & !re=e ! cause 0 n entire1 z 0 suppos$4 ,sid5ote3 ,e effect has a necess>y cause4 ,& see+ a necess>y cause is def9$ to be t1 :i* be+ suppos$1 ! effect _c b foll[2 ?is also may be collect$1 t :atsoev] effect is produc$ at any "t1 ! same is produc$ by a necess>y cause4 ,= :atsoev] is produc$1 9 z mu* z x is produc$1 _h an entire cause1 t is1 _h all ^? ?+s1 :i* be+ suppos$1 x _c be und]/ood b t ! effect foll[s2 t is1 x _h a necess>y cause4 ,& 9 ! same mann] x may be %ewn1 t :atsoev] effects are h]eaft] to be produc$1 %all h a necess>y cause2 s t all ! effects t h be51 or %all be produc$1 h _! necessity 9 ?+s antec$5t4 ,sid5ote3 ,! g5]ation ( effects is cont9ual4 ,:at is ! beg9n+ 9 causation4 #f4 ,& f ?is1 t :5soev] ! cause is entire1 ! effect is produc$ 9 ! same in/ant1 x is manife/ t causation & ! production ( effects consi/ 9 a c]ta9 cont9ual progress2 s t z !re is a cont9ual mutation 9 ! ag5t or ag5ts1 by ! work+ ( o!r ag5ts ^u !m1 s also ! pati5t1 ^u :i* !y "w1 is cont9ually alt]$ & *ang$4 ,= example3 z ! heat ( ! fire increases m & m1 s also ! effects !re(1 namely1 ! heat ( su* bodies z are next to x1 & aga91 ( su* o!r bodies z are next to !m1 increase m & m accord+ly2 :i* is already no little argum5t t all mutation consi/s 9 motion only2 ! tru? :]e( %all be fur!r demon/rat$ 9 ! n9? article4 ,b 9 ?is progress ( causation1 t is1 ( action & passion1 if any man compreh5d 9 8 imag9ation a "p !re(1 & divide ! same into p>ts1 ! fir/ "p or beg9n+ ( x _c be consid]$ o!rwise ?an z action or cause2 =1 if x %\ld be consid]$ z effect or passion1 !n x w\ld be necess>y to consid] some?+ be=e x1 = its cause or action2 :i* _c be1 = no?+ c be be=e ! beg9n+4 ,& 9 l mann]1 ! la/ "p is consid]$ only z effect2 = x _c be call$ cause1 if no?+ foll[ x2 b aft] ! la/1 no?+ foll[s4 ,& f h5ce x is1 t 9 all action ! beg9n+ & cause are tak5 = ! same ?+4 ,b e "o ( ! int]m$iate p>ts are bo? action & passion1 & cause & effect1 accord+ z !y are comp>$ ) ! antec$5t or subsequ5t "p4 ,sid5ote3 ,no cause ( motion b 9 a body contigu\s & mov$4 #g4 ,!re c be no cause ( motion1 except 9 a body contigu\s & mov$4 ,= let !re be any two bodies :i* are n contigu\s1 & betwixt :i* ! int]m$iate space is empty1 or1 if fill$1 fill$ ) ano!r body :i* is at re/2 & let "o ( ! prop\nd$ bodies be suppos$ to be at re/2 ,i say x %all always be at re/4 ,= if x %all be mov$1 ! cause ( t motion1 by ! #h? *apt]1 article #ai1 w be "s ext]nal body2 &1 !re=e1 if betwe5 x & t ext]nal body !re be no?+ b empty space1 !n :atsoev] ! disposition be ( t ext]nal body or ( ! pati5t itself1 yet if x be suppos$ to be n[ at re/1 we may conceive x w cont9ue s till x be t\*$ by "s o!r body4 ,b see+ cause1 by ! def9ition1 is ! aggregate ( all su* accid5ts1 :i* be+ suppos$ to be pres5t1 x _c be conceiv$ b t ! effect w foll[1 ^? accid5ts1 :i* are ei!r 9 ext]nal bodies1 or 9 ! pati5t itself1 _c be ! cause ( future motion4 ,& 9 l mann]1 see+ we may conceive t :atsoev] is at re/ w /ill be at re/1 ?\gh x be t\*$ by "s o!r body1 except t o!r body be mov$2 !re=e 9 a contigu\s body1 :i* is at re/1 !re c be no cause ( motion4 ,:]e=e !re is no cause ( motion 9 any body1 except x be contigu\s & mov$4 ,! same reason may s]ve to prove t :atsoev] is mov$1 w always be mov$ on 9 ! same way & ) ! same velocity1 except x be h9d]$ by "s o!r contigu\s & mov$ body2 & consequ5tly t no bodies1 ei!r :5 !y are at re/1 or :5 !re is an int]position ( vacuum1 c g5]ate or ext+ui% or less5 motion 9 o!r bodies4 ,!re is "o t has writt5 t ?+s mov$ are m resi/$ by ?+s at re/1 ?an by ?+s contr>ily mov$2 = ?is reason1 t he conceiv$ motion n to be s contr>y to motion z re/4 ,t :i* deceiv$ him 01 t ! words re/ & motion are b contradictory names2 :]eas motion1 inde$1 is n resi/$ by re/1 b by contr>y motion4 ,sid5ote3 ,! same ag5ts & pati5ts1 if alike dispos$1 produce l effects1 ?\gh at diff]5t times4 #h4 ,b if a body "w ^u ano!r body at "o "t1 & aft]w>ds ! same body "w ^u ! same body at ano!r "t1 s t bo? ! ag5t & pati5t1 & all _! p>ts1 be 9 all ?+s z !y 72 & !re be no diff]5ce1 except only 9 "t1 t is1 t "o action be =m]1 ! o!r lat] 9 "t2 x is manife/ ( itself1 t ! effects w be equal & l1 z n diff]+ 9 any?+ besides "t4 ,& z effects !mselves proce$ f _! causes1 s ! div]sity ( !m dep5ds ^u ! div]sity ( _! causes also4 ,sid5ote3 ,all mutation is motion4 #i4 ,?is be+ true1 x is necess>y t mutation c be no?+ else b motion ( ! p>ts ( t body :i* is *ang$4 ,= fir/1 we d n say any?+ is *ang$1 b t :i* appe>s to our s5ses o!rwise ?an x appe>$ =m]ly4 ,secondly1 bo? ^? appe>ances are effects produc$ 9 ! s5ti5t2 &1 !re=e1 if !y be diff]5t1 x is necess>y1 by ! prec$+ article1 t ei!r "s "p ( ! ag5t1 :i* 0 =m]ly at re/1 is n[ mov$1 & s ! mutation consi/s 9 ?is motion2 or "s "p1 :i* 0 =m]ly mov$1 is n[ o!rwise mov$1 & s also ! mutation consi/s 9 ?is new motion2 or :i*1 be+ =m]ly mov$1 is n[ at re/1 :i*1 z ,i h %ewn above1 _c come to pass )\t motion2 & s aga91 mutation is motion2 or la/ly1 x happ5s 9 "s ( ^! mann]s to ! pati5t1 or "s ( its p>ts2 s t mutation1 h[soev] x be made1 w consi/ 9 ! motion ( ! p>ts1 ei!r ( ! body :i* is p]ceiv$1 or ( ! s5ti5t body1 or ( bo?4 ,mutation !re=e is motion1 namely1 ( ! p>ts ei!r ( ! ag5t or ( ! pati5t2 :i* 0 to be demon/rat$4 ,& to ?is x is consequ5t1 t re/ _c be ! cause ( any?+1 nor c any action proce$ f x2 see+ nei!r motion nor mutation c be caus$ by x4 ,sid5ote3 ,cont+5t accid5ts4 #aj4 ,accid5ts1 9 respect ( o!r accid5ts :i* prec$e !m1 or are be=e !m 9 "t1 & ^u :i* !y d n dep5d z ^u _! causes1 are call$ cont+5t accid5ts2 ,i say1 9 respect ( ^? accid5ts by :i* !y are n g5]at$2 =1 9 respect ( _! causes1 all ?+s come to pass ) equal necessity2 = o!rwise !y w\ld h no causes at all2 :i*1 ( ?+s g5]at$1 is n intelligible4 ------- ,,*apt] ,x4 ,,( ,,p[] ,,& ,,act4 #a4 ,p[] & cause are ! same ?+4--#b4 ,an act is produc$ at ! same in/ant 9 :i* ! p[] is pl5>y4--#c4 ,active & passive p[] are p>ts only ( pl5>y p[]4--#d4 ,an act1 :5 said to be possible4--#e4 ,an act necess>y & cont+5t1 :at4--#f4 ,active p[] consi/s 9 motion4--#g4 ,cause1 =mal & f9al1 :at !y are4 ,sid5ote3 ,p[] & cause are ! same ?+4 #a4 ,correspond5t to cause & effect1 are ,,p[] & ,,act2 nay1 ^? & ^! are ! same ?+s2 ?\gh1 = div]s consid]ations1 !y h div]s names4 ,= :5soev] any ag5t has all ^? accid5ts :i* are necess>ily requisite = ! production ( "s effect 9 ! pati5t1 !n we say t ag5t has p[] to produce t effect1 if x be appli$ to a pati5t4 ,b1 z ,i h %ewn 9 ! prec$5t *apt]1 ^? accid5ts con/itute ! effici5t cause2 & !re=e ! same accid5ts1 :i* con/itute ! effici5t cause1 con/itute also ! p[] ( ! ag5t4 ,:]e=e ! p[] ( ! ag5t & ! effici5t cause are ! same ?+4 ,b !y are consid]$ ) ?is diff]5ce1 t cause is s call$ 9 respect ( ! effect already produc$1 & p[] 9 respect ( ! same effect to be produc$ h]eaft]2 s t cause respects ! pa/1 p[] ! future "t4 ,also1 ! p[] ( ! ag5t is t :i* is commonly call$ active p[]4 ,9 l mann]1 :5soev] any pati5t has all ^? accid5ts :i* x is requisite x %\ld h1 = ! production ( "s effect 9 x1 we say x is 9 ! p[] ( t pati5t to produce t effect1 if x be appli$ to a fitt+ ag5t4 ,b ^? accid5ts1 z is def9$ 9 ! prec$5t *apt]1 con/itute ! mat]ial cause2 & !re=e ! p[] ( ! pati5t1 commonly call$ passive p[]1 & mat]ial cause1 are ! same ?+2 b ) ?is diff]5t consid]ation1 t 9 cause ! pa/ "t1 & 9 p[] ! future1 is respect$4 ,:]e=e ! p[] ( ! ag5t & pati5t toge!r1 :i* may be call$ entire or pl5>y p[]1 is ! same ?+ ) entire cause2 = !y bo? consi/ 9 ! sum or aggregate ( all ! accid5ts1 z well 9 ! ag5t z 9 ! pati5t1 :i* are requisite = ! production ( ! effect4 ,la/ly1 z ! accid5t produc$ is1 9 respect ( ! cause1 call$ an effect1 s 9 respect ( ! p[]1 x is call$ an act4 ,sid5ote3 ,an act is produc$ at ! same in/ant 9 :i* ! p[] is pl5>y4 #b4 ,z !re=e ! effect is produc$ 9 ! same in/ant 9 :i* ! cause is entire1 s also e act t may be produc$1 is produc$ 9 ! same in/ant 9 :i* ! p[] is pl5>y4 ,& z !re c be no effect b f a suffici5t & necess>y cause1 s also no act c be produc$ b by suffici5t p[]1 or t p[] by :i* x c\ld n b be produc$4 ,sid5ote3 ,active & passive p[] are p>ts only ( pl5>y p[]4 #c4 ,& z x is manife/1 z ,i h %ewn1 t ! effici5t & mat]ial causes are sev]ally & by !mselves p>ts only ( an entire cause1 & _c produce any effect b by be+ jo9$ toge!r1 s also p[]1 active & passive1 are p>ts only ( pl5>y & entire p[]2 nor1 except !y be jo9$1 c any act proce$ f !m2 & !re=e ^! p[]s1 z ,i said 9 ! fir/ article1 are b conditional1 namely1 ! ag5t has p[]1 if x be appli$ to a pati5t2 & ! pati5t has p[]1 if x be appli$ to an ag5t2 o!rwise nei!r ( !m h p[]1 nor c ! accid5ts1 :i* are 9 !m sev]ally1 be prop]ly call$ p[]s2 nor any action be said to be possible = ! p[] ( ! ag5t alone or ( ! pati5t alone4 ,sid5ote3 ,an act1 :5 said to be possible4 #d4 ,= t is an impossible act1 = ! production ( :i* !re is no p[] pl5>y4 ,= see+ pl5>y p[] is t 9 :i* all ?+s concur1 :i* are requisite = ! production ( an act1 if ! p[] %all nev] be pl5>y1 !re w always be want+ "s ( ^? ?+s1 )\t :i* ! act _c be produc$2 :]e=e t act %all nev] be produc$2 t is1 t act is ,,impossible3 & e act1 :i* is n impossible1 is ,,possible4 ,e act1 !re=e1 :i* is possible1 %all at "s "t be produc$2 = if x %all nev] be produc$1 !n ^? ?+s %all nev] concur :i* are requisite = ! production ( x2 :]e=e t act is impossible1 by ! def9ition2 :i* is contr>y to :at 0 suppos$4 ,sid5ote3 ,an act necess>y & cont+5t1 :at4 #e4 ,a necess>y act is t1 ! production :]e( x is impossible to h9d]2 & !re=e e act1 t %all be produc$1 %all necess>ily be produc$2 =1 t x %all n be produc$1 is impossible2 because1 z is already demon/rat$1 e possible act %all at "s "t be produc$2 nay1 ?is proposition1 :at %all be1 %all be1 is z necess>y a proposition z ?is1 a man is a man4 ,b "h1 p]haps1 "s man may ask :e!r ^? future ?+s1 :i* are commonly call$ cont+5ts1 are necess>y4 ,i say1 !re=e1 t g5]ally all cont+5ts h _! necess>y causes1 z is %ewn 9 ! prec$+ *apt]2 b are call$ cont+5ts 9 respect ( o!r ev5ts1 ^u :i* !y d n dep5d2 z ! ra91 :i* %all be tomorr[1 %all be necess>y1 t is1 f necess>y causes2 b we ?9k & say x happ5s by *ance1 because we d n yet p]ceive ! causes !re(1 ?\gh !y exi/ n[2 = m5 commonly call t casual or cont+5t1 :]e( !y d n p]ceive ! necess>y cause2 & 9 ! same mann] !y us$ to speak ( ?+s pa/1 :5 n kn[+ :e!r a ?+ be done or no1 !y say x is possible x nev] 0 done4 ,:]e=e1 all propositions conc]n+ future ?+s1 cont+5t or n cont+5t1 z ?is1 x w ra9 tomorr[1 or ?is1 tomorr[ ! sun w rise1 are ei!r necess>ily true1 or necess>ily false2 b we call !m cont+5t1 because we d n yet "k :e!r !y be true or false2 :]eas _! v]ity dep5ds n ^u our k1 b ^u ! =ego+ ( _! causes4 ,b !re are "s1 :o ?\gh !y confess ?is :ole proposition1 tomorr[ x w ei!r ra91 or n ra91 to be true1 yet !y w n ackn[l$ge ! p>ts ( x1 z1 tomorr[ x w ra91 or1 tomorr[ x w n ra91 to be ei!r ( !m true by itself2 because !y say nei!r ?is nor t is true det]m9ately4 ,b :at is ?is det]m9ately true1 b true ^u our k1 or evid5tly true8 ,& !re=e !y say no m b t x is n yet kn[n :e!r x be true or no2 b !y say x m obscurely1 & d>k5 ! evid5ce ( ! tru? ) ! same words1 ) :i* !y endeav\r to hide _! own ignorance4 ,sid5ote3 ,active p[] consi/s 9 motion4 #f4 ,9 ! #i? article ( ! prec$+ *apt]1 ,i h %ewn t ! effici5t cause ( all motion & mutation consi/s 9 ! motion ( ! ag5t1 or ag5ts2 & 9 ! fir/ article ( ?is *apt]1 t ! p[] ( ! ag5t is ! same ?+ ) ! effici5t cause4 ,f :5ce x may be und]/ood1 t all active p[] consi/s 9 motion also2 & t p[] is n a c]ta9 accid5t1 :i* diff]s f all acts1 b is1 inde$1 an act1 namely1 motion1 :i* is !re=e call$ p[]1 because ano!r act %all be produc$ by x aft]w>ds4 ,= example1 if ( ?ree bodies ! fir/ put =w>d ! second1 & ?is ! ?ird1 ! motion ( ! second1 9 respect ( ! fir/ :i* produce? x1 is ! act ( ! second body2 b1 9 respect ( ! ?ird1 x is ! active p[] ( ! same second body4 ,sid5ote3 ,cause1 =mal & f9al1 :at !y are4 #g4 ,! writ]s ( metaphysics reckon up two o!r causes besides ! effici5t & mat]ial1 namely1 ! ,,ess5ce1 :i* "s call ! =mal cause1 & ! ,,end1 or f9al cause2 bo? :i* are nev]!less effici5t causes4 ,= :5 x is said ! ess5ce ( a ?+ is ! cause !re(1 z to be rational is ! cause ( man1 x is n intelligible2 = x is all "o1 z if x 7 said1 to be a man is ! cause ( man2 :i* is n well said4 ,& yet ! k ( ! ess5ce ( any?+1 is ! cause ( ! k ( ! ?+ itself2 =1 if ,i fir/ "k t a ?+ is rational1 ,i "k f !nce1 t ! same is man2 b ?is is no o!r ?an an effici5t cause4 ,a f9al cause has no place b 9 su* ?+s z h s5se & w2 & ?is also ,i %all prove h]eaft] to be an effici5t cause4 ------- ,,*apt] ,,xi4 ,,( ,,id5tity ,,& ,,diff]5ce4 #a4 ,:at x is = "o ?+ to diff] f ano!r4--#b4 ,to diff] 9 numb]1 magnitude1 species1 & g5us1 :at4--#c4 ,:at is relation1 proportion1 & relatives4--#d4 ,proportionals1 :at4--#e4 ,! proportion ( magnitudes to "o ano!r1 :]e9 x consi/s4--#f4 ,relation is no new accid5t1 b "o ( ^? t 7 9 ! relative be=e ! relation or comp>ison 0 made4 ,also ! causes ( accid5ts 9 ! correlatives1 are ! cause ( relation4--#g4 ,( ! beg9n+ ( individuation4 ,sid5ote3 ,:at x is = "o ?+ to diff] f ano!r4 #a4 ,hi!rto ,i h spok5 ( body simply1 & accid5ts common to all bodies1 z magnitude1 motion1 re/1 action1 passion1 p[]1 possible1 c42 & ,i %\ld n[ desc5d to ^? accid5ts by :i* "o body is di/+ui%$ f ano!r1 b t x is fir/ to be decl>$ :at x is to be di/9ct & n di/9ct1 namely1 :at are ! ,,same & ,,diff]5t2 = ?is also is common to all bodies1 t !y may be di/+ui%$ & diff]5c$ f "o ano!r4 ,n[1 two bodies are said to diff] f "o ano!r1 :5 some?+ may be said ( "o ( !m1 :i* _c be said ( ! o!r at ! same "t4 ,sid5ote3 ,to diff] 9 numb]1 magnitude1 species1 & g5us1 :at4 #b4 ,&1 fir/ ( all1 x is manife/ t no two bodies are ! same2 = see+ !y are two1 !y are 9 two places at ! same "t2 z t1 :i* is ! same1 is at ! same "t 9 "o & ! same place4 ,all bodies !re=e diff] f "o ano!r 9 numb]1 namely1 z "o & ano!r2 s t ! same & diff]5t 9 numb]1 are names oppos$ to "o ano!r by contradiction4 ,9 magnitude bodies diff] :5 "o is great] ?an ano!r1 z a cubit long1 & two cubits long1 ( two p\nd weight1 & ( ?ree p\nd weight4 ,& to ^!1 equals are oppos$4 ,bodies1 :i* diff] m ?an 9 magnitude1 are call$ unlike2 & ^?1 :i* diff] only 9 magnitude1 l4 ,also1 ( unlike bodies1 "s are said to diff] 9 ! species1 o!rs 9 ! g5us2 9 ! species1 :5 _! diff]5ce is p]ceiv$ by "o & ! same s5se1 z :ite & black2 & 9 ! g5us1 :5 _! diff]5ce is n p]ceiv$ b by div]s s5ses1 z :ite & hot4 ,sid5ote3 ,:at is relation1 proportion1 & relatives4 #c4 ,& ! lik5ess1 or unlik5ess1 equality1 or inequality ( "o body to ano!r1 is call$ _! ,,relation2 & ! bodies !mselves relatives or correlatives2 ,ari/otle calls !m 2 ! fir/ :]e( is usually nam$ ! antec$5t1 & ! second ! consequ5t2 & ! relation ( ! antec$5t to ! consequ5t1 accord+ to magnitude1 namely1 ! equality1 ! excess or defect !re(1 is call$ ! ,,proportion ( ! antec$5t to ! consequ5t2 s t proportion is no?+ b ! equality or inequality ( ! magnitude ( ! antec$5t comp>$ to ! magnitude ( ! consequ5t by _! diff]5ce only1 or comp>$ also ) _! diff]5ce4 ,= example1 ! proportion ( ?ree to two consi/s only 9 ?is1 t ?ree exce$s two by unity2 & ! proportion ( two to five 9 ?is1 t two1 comp>$ ) five1 is defici5t ( x by ?ree1 ei!r simply1 or comp>$ ) ! numb]s diff]5t2 & !re=e 9 ! proportion ( unequals1 ! proportion ( ! less to ! great]1 is call$ ,,defect2 & t ( ! great] to ! less1 ,,excess4 ,sid5ote3 ,proportionals1 :at4 #d4 ,besides1 ( unequals1 "s are m1 "s less1 & "s equally unequal2 s t !re is proportion ( proportions1 z well z ( magnitudes2 namely1 :]e two unequals h relation to two o!r unequals2 z1 :5 ! inequality :i* is betwe5 #b & #c1 is comp>$ ) ! inequality :i* is betwe5 #d & #e4 ,9 :i* comp>ison !re are always f\r magnitudes2 or1 :i* is all "o1 if !re be b ?ree1 ! middlemo/ is twice numb]$2 & if ! proportion ( ! fir/ to ! second1 be equal to ! proportion ( ! ?ird to ! f\r?1 !n ! f\r are said to be proportionals2 o!rwise !y are n proportionals4 ,sid5ote3 ,! proportion ( magnitudes to "o ano!r1 :]e9 x consi/s4 #e4 ,! proportion ( ! antec$5t to ! consequ5t consi/s 9 _! diff]5ce1 n only simply tak51 b also z comp>$ ) "o ( ! relatives2 t is1 ei!r 9 t "p ( ! great]1 by :i* x exce$s ! less1 or 9 ! rema9d]1 aft] ! less is tak5 out ( ! great]2 z ! proportion ( two to five consi/s 9 ! ?ree by :i* five exce$s two1 n 9 ?ree simply only1 b also z comp>$ ) five or two4 ,= ?\gh !re be ! same diff]5ce betwe5 two & five1 :i* is betwe5 n9e & twelve1 namely ?ree1 yet !re is n ! same inequality2 & !re=e ! proportion ( two to five is n 9 all relation ! same ) t ( n9e to twelve1 b only 9 t :i* is call$ ari?metical4 ,sid5ote3 ,relation is no new accid5t1 b "o ( ^? t 7 9 ! relative1 be=e ! relation or comp>ison 0 made4 ,also ! causes ( accid5ts 9 correlatives are ! cause ( relation4 #f4 ,b we mu/ n s ?9k ( relation1 z if x 7 an accid5t diff]+ f all ! o!r accid5ts ( ! relative2 b "o ( !m1 namely1 t by :i* ! comp>ison is made4 ,= example1 ! lik5ess ( "o :ite to ano!r :ite1 or its unlik5ess to black1 is ! same accid5t ) its :it5ess2 & equality & inequality1 ! same accid5t ) ! magnitude ( ! ?+ comp>$1 ?\gh "u ano!r "n3 = t :i* is call$ :ite or great1 :5 x is n comp>$ ) some?+ else1 ! same :5 x is comp>$1 is call$ l or unlike1 equal or unequal4 ,& f ?is x foll[s t ! causes ( ! accid5ts1 :i* are 9 relatives1 are ! causes also ( lik5ess1 unlik5ess1 equality & inequality2 namely1 t he1 t makes two unequal bodies1 makes also _! inequality2 & he1 t makes a rule & an action1 makes also1 if ! action be congru\s to ! rule1 _! congruity2 if incongru\s1 _! incongruity4 ,& ?us mu* conc]n+ comp>ison ( "o body ) ano!r4 ,sid5ote3 ,( ! beg9n+ ( individuation4 #g4 ,b ! same body may at diff]5t times be comp>$ ) itself4 ,& f h5ce spr+s a great controv]sy among philosoph]s ab\t ! beg9n+ ( individuation1 namely1 9 :at s5se x may be conceiv$ t a body is at "o "t ! same1 at ano!r "t n ! same x 0 =m]ly4 ,= example1 :e!r a man gr[n old be ! same man he 0 :il/ he 0 "y1 or ano!r man2 or :e!r a city be 9 diff]5t ages ! same1 or ano!r city4 ,"s place individuity 9 ! unity ( matt]2 o!rs1 9 ! unity ( =m2 & "o says x consi/s 9 ! unity ( ! aggregate ( all ! accid5ts toge!r4 ,= matt]1 x is plead$ t a lump ( wax1 :e!r x be sph]ical or cubical1 is ! same wax1 because ! same matt]4 ,= =m1 t :5 a man is gr[n f an infant to be an old man1 ?\gh 8 matt] be *ang$1 yet he is /ill ! same num]ical man2 = t id5tity :i* _c be attribut$ to ! matt]1 ought probably to be ascrib$ to ! =m4 ,= ! aggregate ( accid5ts1 no in/ance c be made2 b because1 :5 any new accid5t is g5]at$1 a new "n is commonly impos$ on ! ?+1 !re=e he1 t assign$ ?is cause ( individuity1 ?\ght ! ?+ itself also 0 become ano!r ?+4 ,accord+ to ! fir/ op9ion1 he t s9s1 & he t is puni%$1 %\ld n be ! same man1 by reason ( ! p]petual flux & *ange ( man's body2 nor %\ld ! city1 :i* makes laws 9 "o age & abrogates !m 9 ano!r1 be ! same city2 :i* 7 to conf\nd all civil rights4 ,accord+ to ! second op9ion1 two bodies exi/+ bo? at once1 w\ld be "o & ! same num]ical body4 ,= if1 = example1 t %ip ( ,!seus1 conc]n+ ! diff]5ce :]e( made by cont9ual rep>ation 9 tak+ out ! old planks & putt+ 9 new1 ! sophi/]s ( ,a!ns 7 wont to dispute1 71 aft] all ! planks 7 *ang$1 ! same num]ical %ip x 0 at ! beg9n+2 & if "s man _h kept ! old planks z !y 7 tak5 out1 & by putt+ !m aft]w>ds toge!r 9 ! same ord]1 _h aga9 made a %ip ( !m1 ?is1 )\t d\bt1 _h also be5 ! same num]ical %ip ) t :i* 0 at ! beg9n+2 & s !re w\ld h be5 two %ips num]ically ! same1 :i* is absurd4 ,b1 accord+ to ! ?ird op9ion1 no?+ w\ld be ! same x 02 s t a man /&+ w\ld n be ! same he 0 sitt+2 nor ! wat]1 :i* is 9 ! vessel1 ! same ) t :i* is p\r$ out ( x4 ,:]e=e ! beg9n+ ( individuation is n always to be tak5 ei!r f matt] alone1 or f =m alone4 ,b we mu/ consid] by :at "n any?+ is call$1 :5 we inquire conc]n+ ! id5tity ( x4 ,= x is "o ?+ to ask conc]n+ ,socrates1 :e!r he be ! same man1 & ano!r to ask :e!r he be ! same body2 = 8 body1 :5 he is old1 _c be ! same x 0 :5 he 0 an infant1 by reason ( ! diff]5ce ( magnitude2 = "o body has always "o & ! same magnitude2 yet1 nev]!less1 he may be ! same man4 ,& !re=e1 :5soev] ! "n1 by :i* x is ask$ :e!r a ?+ be ! same x 01 is giv5 x = ! matt] only1 !n1 if ! matt] be ! same1 ! ?+ also is individually ! same2 z ! wat]1 :i* 0 9 ! sea1 is ! same :i* is aft]w>ds 9 ! cl\d2 & any body is ! same1 :e!r ! p>ts ( x be put toge!r1 or disp]s$2 or :e!r x be congeal$1 or dissolv$4 ,also1 if ! "n be giv5 = su* =m z is ! beg9n+ ( motion1 !n1 z long z t motion rema9s1 x w be ! same individual ?+2 z t man w be always ! same1 ^: actions & ?\ghts proce$ all f ! same beg9n+ ( motion1 namely1 t :i* 0 9 8 g5]ation2 & t w be ! same riv] :i* fl[s f "o & ! same f\nta91 :e!r ! same wat]1 or o!r wat]1 or some?+ else ?an wat]1 fl[ f !nce2 & t ! same city1 ^: acts proce$ cont9ually f ! same in/itution1 :e!r ! m5 be ! same or no4 ,la/ly1 if ! "n be giv5 = "s accid5t1 !n ! id5tity ( ! ?+ w dep5d ^u ! matt]2 =1 by ! tak+ away & supply+ ( matt]1 ! accid5ts t 71 are de/roy$1 & o!r new ones are g5]at$1 :i* _c be ! same num]ically2 s t a %ip1 :i* signifies matt] s figur$1 w be ! same z long z ! matt] rema9s ! same2 b if no "p ( ! matt] be ! same1 !n x is num]ically ano!r %ip2 & if "p ( ! matt] rema9 & "p be *ang$1 !n ! %ip w be p>tly ! same1 & p>tly n ! same4 ------- ,,*apt] ,,xii4 ,,( ,,quantity4 #a4 ,! def9ition ( quantity4--#b4 ,! exposition ( quantity1 :at x is4--#c4 ,h[ l9e1 sup]ficies1 & solid1 are expos$4--#d4 ,h[ "t is expos$4--#e4 ,h[ numb] is expos$4--#f4 ,h[ velocity is expos$4--#g4 ,h[ weight is expos$4--#h4 ,h[ ! proportion ( magnitudes is expos$4--#i4 ,h[ ! proportion ( times & velocities is expos$4 ,sid5ote3 ,def9ition ( quantity4 #a4 ,:at & h[ manifold dim5sion is1 has be5 said 9 ! #h? *apt]1 namely1 t !re are ?ree dim5sions1 l9e or l5g?1 sup]ficies1 & solid2 e "o ( :i*1 if x be det]m9$1 t is1 if ! limits ( x be made kn[n1 is commonly call$ quantity2 = by quantity all m5 und]/& t :i* is signifi$ by t ^w1 by :i* answ] is made to ! "q1 ,h[ mu* is x8 ,:5soev]1 !re=e1 x is ask$1 = example1 ,h[ long is ! j\rney8 x is n answ]$ indef9itely1 l5g?2 nor1 :5 x is ask$1 ,h[ big is ! field8 is x answ]$ indef9itely1 sup]ficies2 nor1 if a man ask1 ,h[ great is ! bulk8 indef9itely1 solid3 b x is answ]$ det]m9ately1 ! j\rney is a hundr$ miles2 ! field is a hundr$ acres2 ! bulk is a hundr$ cubical feet2 or at lea/ 9 "s su* mann]1 t ! magnitude ( ! ?+ enquir$ aft] may by c]ta9 limits be compreh5d$ 9 ! m9d4 ,,quantity1 !re=e1 _c o!rwise be def9$1 ?an to be a dim5sion det]m9$1 or a dim5sion1 ^: limits are set out1 ei!r by _! place1 or by "s comp>ison4 ,sid5ote3 ,! exposition ( quantity1 :at x is4 #b4 ,& quantity is det]m9$ two ways2 "o1 by ! s5se1 :5 "s s5sible object is set be=e x2 z :5 a l9e1 a sup]ficies or solid1 ( a foot or cubit1 m>k$ out 9 "s matt]1 is object$ to ! eyes2 :i* way ( det]m9+1 is call$ exposition1 & ! quantity s kn[n is call$ expos$ quantity2 ! o!r by memory1 t is1 by comp>ison ) "s expos$ quantity4 ,9 ! fir/ mann]1 :5 x is ask$ ( :at quantity a ?+ is1 x is answ]$1 ( su* quantity z y see expos$4 ,9 ! second mann]1 answ] _c be made b by comp>ison ) "s expos$ quantity2 = if x be ask$1 h[ long is ! way8 ! answ] is1 s _m ?\s& paces2 t is1 by comp>+ ! way ) a pace1 or "s o!r measure1 det]m9$ & kn[n by exposition2 or ! quantity ( x is to "s o!r quantity kn[n by exposition1 z ! diamet] ( a squ>e is to ! side ( ! same1 or by "s o!r ! l means4 ,b x is to be und]/ood1 t ! quantity expos$ mu/ be "s /&+ or p]man5t ?+1 su* z is m>k$ out 9 consi/5t or durable matt]2 or at lea/ some?+ :i* is revocable to s5se2 = o!rwise no comp>ison c be made by x4 ,see+1 !re=e1 by :at has be5 said 9 ! next prec$+ *apt]1 comp>ison ( "o magnitude ) ano!r is ! same ?+ ) proportion2 x is manife/1 t quantity det]m9$ 9 ! second mann] is no?+ else b ! proportion ( a dim5sion n expos$ to ano!r :i* is expos$2 t is1 ! comp>ison ( ! equality or inequality !re( ) an expos$ quantity4 ,sid5ote3 ,h[ l9e1 sup]ficies1 & solids1 are expos$4 #c4 ,l9es1 sup]ficies1 & solids1 are expos$1 fir/1 by motion1 9 su* mann] z 9 ! #h? *apt] ,i h said !y are g5]at$2 b s z t ! m>ks ( su* motion be p]man5t2 z :5 !y are design$ ^u "s matt]1 z a l9e ^u pap]2 or grav5 9 "s durable matt]4 ,secondly1 by apposition2 z :5 "o l9e or l5g? is appli$ to ano!r l9e or l5g?1 "o bread? to ano!r bread?1 & "o ?ickness to ano!r ?ickness2 :i* is z mu* z to describe a l9e by po9ts1 a sup]ficies by l9es1 & a solid by sup]ficies2 sav+ t by po9ts 9 ?is place are to be und]/ood v %ort l9es2 &1 by sup]ficies1 v ?9 solids4 ,?irdly1 l9es & sup]ficies may be expos$ by section1 namely1 a l9e may be made by cutt+ an expos$ sup]ficies2 & a sup]ficies1 by ! cutt+ ( an expos$ solid4 ,sid5ote3 ,h[ "t is expos$4 #d4 ,"t is expos$1 n only by ! exposition ( a l9e1 b also ( "s moveable ?+1 :i* is mov$ uni=mly ^u t l9e1 or at lea/ is suppos$ s to be mov$4 ,=1 see+ "t is an idea ( motion1 9 :i* we consid] =m] & latt]1 t is succession1 x is n suffici5t = ! exposition ( "t t a l9e be describ$2 b we mu/ also h 9 our m9d an imag9ation ( "s moveable ?+ pass+ ov] t l9e2 & ! motion ( x mu/ be uni=m1 t "t may be divid$ & comp\nd$ z (t5 z !re %all be ne$4 ,&1 !re=e1 :5 philosoph]s1 9 _! demon/rations1 draw a l9e1 & say1 ,let t l9e be "t1 x is to be und]/ood z if !y said1 ,let ! conception ( uni=m motion ^u t l9e1 be "t4 ,= ?\gh ! circles 9 dials be l9es1 yet !y are n ( !mselves suffici5t to note "t by1 except also !re be1 or be suppos$ to be1 a motion ( ! %ad[ or ! h&4 ,sid5ote3 ,h[ numb] is expos$4 #e4 ,numb] is expos$1 ei!r by ! exposition ( po9ts1 or ( ! names ( numb]1 "o1 two1 ?ree1 c42 & ^? po9ts mu/ n be contigu\s1 s z t !y _c be di/+ui%$ by notes1 b !y mu/ be s plac$ t !y may be disc]n$ "o f ano!r2 =1 f ?is x is1 t numb] is call$ discreet quantity1 :]eas all quantity1 :i* is design$ by motion1 is call$ cont9ual quantity4 ,b t numb] may be expos$ by ! names ( numb]1 x is necess>y t !y be recit$ by he>t & 9 ord]1 z "o1 two1 ?ree1 c42 = by say+ "o1 "o1 "o1 & s =w>d1 we "k n :at numb] we are at beyond two or ?ree2 :i* also appe> to u 9 ?is mann]1 n z numb]1 b z figure4 ,sid5ote3 ,h[ velocity is expos$4 #f4 ,= ! exposition ( velocity1 :i*1 by ! def9ition !re(1 is a motion :i*1 9 a c]ta9 "t1 passe? ov] a c]ta9 space1 x is requisite1 n only t "t be expos$1 b t !re be also expos$ t space :i* is transmitt$ by ! body1 ^: velocity we w\ld det]m9e2 & t a body be und]/ood to be mov$ 9 t space also2 s t !re mu/ be expos$ two l9es1 ^u "o ( :i* uni=m motion mu/ be und]/ood to be made1 t ! "t may be det]m9$2 &1 ^u ! o!r1 ! velocity is to be comput$4 ,z if we w\ld expose ! velocity ( ! we w\ld expose ! velocity ( ! body ,a1 we draw two l9es ,a ,b & ,c ,d1 & place a body 9 ,c also2 :i* done1 we say ! velocity ( ! body ,a is s great1 t x passe? ov] ! l9e ,a ,b 9 ! same "t 9 :i* ! body ,c passe? ov] ! l9e ,c ,d ) uni=m motion4 ,a ,b ------ ,c ,d ------ ,sid5ote3 ,h[ weight is expos$4 #g4 ,weight is expos$ by any heavy body1 ( :at matt] soev]1 s x be always alike heavy4 ,sid5ote3 ,h[ ! proportion ( magnitudes is expos$4 #h4 ,! proportion ( two magnitudes is !n expos$1 :5 ! magnitudes !mselves are expos$1 namely1 ! proportion ( equality1 :5 ! magnitudes are equal2 & ( inequality1 :5 !y are unequal4 ,= see+1 by ! #e? article ( ! prec$+ *apt]1 ! proportion ( two unequal magnitudes consi/s 9 _! diff]5ce1 comp>$ ) ei!r ( !m2 & :5 two unequal magnitudes are expos$1 _! diff]5ce is also expos$2 x foll[s1 t :5 magnitudes1 :i* h proportion to "o ano!r1 are expos$1 _! proportion also is expos$ ) !m2 &1 9 l mann]1 ! proportion ( equals1 :i* consi/s 9 ?is1 t !re is no diff]5ce ( magnitude betwixt !m1 is expos$ at ! same "t :5 ! equal magnitudes !mselves are expos$4 ,= example1 if ! expos$ l9es ,a ,b & ,c ,d be equal1 ! proportion ( equality is expos$ 9 !m2 & if ! expos$ l9es1 ,e ,f & ,e ,g be unequal1 ! proportion :i* ,e ,f has to ,e ,g1 & t :i* ,e ,g has to ,e ,f are also expos$ 9 !m2 = n only ! l9es !mselves1 b also _! diff]5ce1 ,g ,f1 is expos$4 ,! proportion ( unequals is quantity2 = ! diff]5ce1 ,g ,f1 9 :i* x consi/s1 is quantity4 ,b ! proportion ( equality is n quantity2 because1 betwe5 equals1 !re is no diff]5ce2 nor is "o equality great] ?an ano!r1 z "o inequality is great] ?an ano!r inequality4 ,a ,b ------- ,c ,d ------- ,e ,f ,f ------- ,sid5ote3 ,h[ ! proportion ( times & velocities is expos$4 ,! proportion ( two times1 or ( two uni=m velocities1 is !n expos$1 :5 two l9es are expos$ by :i* two bodies are und]/ood to be mov$ uni=mly2 & !re=e ! same two l9es s]ve to exhibit bo? _! own proportion1 & t ( ! times & velocities1 accord+ z !y are consid]$ to be expos$ = ! magnitudes !mselves1 or = ! times or velocities4 ,= let ! two l9es ,a & ,b be expos$2 _! proportion !re=e 7by ! la/ =ego+ article7 is expos$2 & if !y be consid]$ z drawn ) equal & uni=m velocity1 !n1 see+ _! times are great]1 or equal1 or less1 accord+ z ! same spaces are transmitt$ 9 great]1 or equal1 or less "t1 ! l9es ,a & ,b w exhibit ! equality or inequality1 t is1 ! proportion ( ! times4 ,to conclude1 if ! same l9es1 ,a & ,b1 be consid]$ z drawn 9 ! same "t1 !n1 see+ _! velocities are great]1 or equal1 or less1 accord+ z !y pass ov] 9 ! same "t long]1 or equal1 or %ort] l9es1 ! same l9es1 ,a & ,b1 w exhibit ! equality1 or inequality1 t is1 ! proportion ( _! velocities4 ,a --- ,b ------- ,,*apt] ,,xiii4 ,,( ,,analogism1 ,,or ,,! ,,same ,,proportion4 #a1 #b1 #c1 #d4 ,! nature & def9ition ( proportion1 ari?metical & geometrical4--#e4 ,! def9ition1 & "s prop]ties ( ! same ari?metical proportion4--#f1 #g4 ,! def9ition & transmutations ( analogism1 or ! same geometrical proportion4--#h1 #i44 ,! def9itions ( hyp]logism & hypologism1 t is1 ( great] & less proportion1 & _! transmutations4--#aj1 #aa1 #ab4 ,comp>ison ( analogical quantities1 accord+ to magnitude4--#ac1 #ad1 #ae4 ,composition ( proportions4--#af1 #ag1 #ah1 #ai1 #bj1 #ba1 #bb1 #bc1 #bd1 #be4 ,! def9ition & prop]ties ( cont9ual proportion4--#bf1 #bg1 #bh1 #bi4 ,comp>ison ( ari?metical & geometrical proportions4 ,note1 t 9 ?is *apt] ! sign signifies t ! quantities betwixt :i* x is put1 are add$ toge!r2 & ?is sign - ! rema9d] aft] ! latt] quantity is tak5 out ( ! =m]4 ,s t ,a ,b is equal to bo? ,a & ,b toge!r2 & :]e y see ,a - ,b1 !re ,a is ! :ole1 ,b ! "p tak5 out ( x1 & ,a - ,b ! rema9d]4 ,also1 two lett]s1 set toge!r )\t any sign1 signify1 unless !y belong to a figure1 t "o ( ! quantities is multipli$ by ! o!r2 z ,a ,b signifies ! product ( ,a multipli$ by ,b4 ,sid5ote3 ,! nature & def9ition ( proportion1 ari?metical geometrical4 #a4 ,great & little are n intelligible1 b by comp>ison4 ,n[ t1 to :i* !y are comp>$1 is some?+ expos$2 t is1 "s magnitude ei!r p]ceiv$ by s5se1 or s def9$ by words1 t x may be compreh5d$ by ! m9d4 ,also t1 to :i* any magnitude is comp>$1 is ei!r great] or less1 or equal to x4 ,& !re=e proportion 7:i*1 z ,i h %ewn1 is ! e/imation or compreh5sion ( magnitudes by comp>ison17 is ?reefold1 namely1 proportion ( equality1 t is1 ( equal to equal2 or ( excess1 :i* is ( ! great] to ! less2 or ( defect1 :i* is ! proportion ( ! less to ! great]4 ,aga91 e "o ( ^! proportions is two-fold2 = if x be ask$ conc]n+ any magnitude giv51 h[ great x is1 ! answ] may be made by comp>+ x two ways2 fir/1 by say+ x is great] or less ?an ano!r magnitude1 by s mu*2 z sev5 is less ?an t51 by ?ree unities2 & ?is is call$ ari?metical proportion4 ,secondly1 by say+ x is great] or less ?an ano!r magnitude1 by su* a "p or p>ts !re(2 z sev5 is less ?an t51 by ?ree t5? p>ts ( ! same t54 ,& ?\gh ?is proportion be n always explicable by numb]1 yet x is a det]m9ate proportion1 & ( a diff]5t k9d f ! =m]1 & call$ geometrical proportion1 & mo/ commonly proportion simply4 #b4 ,proportion1 :e!r x be ari?metical or geometrical1 _c be expos$ b 9 two magnitudes1 7( :i* ! =m] is commonly call$ ! antec$5t1 & ! latt] ! consequ5t ( ! proportion7 z ,i h %ewn 9 ! #h? article ( ! prec$+ *apt]4 ,&1 !re=e1 if two proportions be to be comp>$1 !re mu/ be f\r magnitudes expos$1 namely1 two antec$5ts & two consequ5ts2 = ?\gh x happ5 sometimes t ! consequ5t ( ! =m] proportion be ! same ) ! antec$5t ( ! latt]1 yet 9 t d\ble comp>ison x mu/ ( necessity be twice numb]$2 s t !re w be always f\r t]ms4 #c4 ,( two proportions1 :e!r !y be ari?metical or geometrical1 :5 ! magnitudes comp>$ 9 bo? 7:i* ,euclid1 9 ! fif? def9ition ( 8 six? book1 calls ! quantities ( proportions17 are equal1 !n "o ( ! proportions _c be ei!r great] or less ?an ! o!r2 = "o equality is nei!r great] nor less ?an ano!r equality4 ,b ( two proportions ( inequality1 :e!r !y be proportions ( excess or ( defect1 "o ( !m may be ei!r great] or less ?an ! o!r1 or !y may bo? be equal2 = ?\gh !re be prop\nd$ two magnitudes t are unequal to "o ano!r1 yet !re may be o!r two m1 unequal1 & o!r two equally unequal1 & o!r two less unequal ?an ! two :i* 7 prop\nd$4 ,& f h5ce x may be und]/ood1 t ! proportions ( excess & defect are quantity1 be+ capable ( m & less2 b ! proportion ( equality is n quantity1 because n capable nei!r ( m1 nor ( less4 ,& !re=e proportions ( inequality may be add$ toge!r1 or subtract$ f "o ano!r1 or be multipli$ or divid$ by "o ano!r1 or by numb]2 b proportions ( equality n s4 #d4 ,two equal proportions are commonly call$ ! same proportion2 &1 x is said1 t ! proportion ( ! fir/ antec$5t to ! fir/ consequ5t is ! same ) t ( ! second antec$5t to ! second consequ5t4 ,& :5 f\r magnitudes are ?us to "o ano!r 9 geometrical proportion1 !y are call$ proportionals2 & by "s1 m briefly1 analogism4 ,& great] proportion is ! proportion ( a great] antec$5t to ! same consequ5t1 or ( ! same antec$5t to a less consequ5t2 & :5 ! proportion ( ! fir/ antec$5t to ! fir/ consequ5t is great] ?an t ( ! second antec$5t to ! second consequ5t1 ! f\r magnitudes1 :i* are s to "o ano!r1 may be call$ hyp]logism4 ,less proportion is ! proportion ( a less antec$5t to ! same consequ5t1 or ( ! same antec$5t to a great] consequ5t2 & :5 ! proportion ( ! fir/ antec$5t to ! fir/ consequ5t is less ?an t ( ! second to ! second1 ! f\r magnitudes may be call$ hypologism4 ,sid5ote3 ,! def9ition & "s prop]ties ( ! same ari?metical proportion4 #e4 ,"o ari?metical proportion is ! same ) ano!r ari?metical proportion1 :5 "o ( ! antec$5ts exce$s its consequ5t1 or is exce$$ by x1 z mu* z ! o!r antec$5t exce$s its consequ5t1 or is exce$$ by x4 ,& !re=e1 9 f\r magnitudes1 ari?metically proportional1 ! sum ( ! extremes is equal to ! sum ( ! means4 ,= if ,a4 ,b 33 ,c4 ,d be ari?metically proportional1 & ! diff]5ce on bo? sides be ! same excess1 or ! same defect1 ,e1 !n ,b ,c 7if ,a be great] ?an ,b7 w be equal to ,a - ,e ,c2 & ,a ,d w be equal to ,a ,c - ,e2 b ,a - ,e ,c & ,a ,c - ,e are equal4 ,or if ,a be less ?an ,b1 !n ,b ,c w be equal to ,a ,e ,c2 & ,a ,d w be equal to ,a ,c ,e2 b ,a ,e ,c & ,a ,c ,e are equal4 ,also1 if !re be nev] s _m magnitudes1 ari?metically proportional1 ! sum ( !m all w be equal to ! product ( half ! numb] ( ! t]ms multipli$ by ! sum ( ! extremes4 ,= if ,a4 ,b 33 ,c4 ,d 33 ,e4 ,f be ari?metically proportional1 ! c\ples ,a ,f1 ,b ,e1 ,c ,d w be equal to "o ano!r2 & _! sum w be equal to ,a ,f1 multipli$ by ! numb] ( _! comb9ations1 t is1 by half ! numb] ( ! t]ms4 ,if1 ( f\r unequal magnitudes1 any two1 toge!r tak51 be equal to ! o!r two toge!r tak51 !n ! greate/ & ! lea/ ( !m w be 9 ! same comb9ation4 ,let ! unequal magnitudes be ,a1 ,b1 ,c1 ,d2 & let ,a ,b be equal to ,c ,d2 & let ,a be ! greate/ ( !m all2 ,i say ,b w be ! lea/4 ,=1 if x may be1 let any ( ! re/1 z ,d1 be ! lea/4 ,see+ !re=e ,a is great] ?an ,c1 & ,b ?an ,d1 ,a,b w be great] ?an ,c ,d2 :i* is contr>y to :at 0 suppos$4 ,if !re be any f\r magnitudes1 ! sum ( ! greate/ & lea/1 ! sum ( ! means1 ! diff]5ce ( ! two greate/1 & ! diff]5ce ( ! two lea/1 w be ari?metically proportional4 ,=1 let !re be f\r magnitudes1 :]e( ,a is ! greate/1 ,d ! lea/1 & ,b & ,c ! means2 ,i say ,a ,d4 ,b ,c 33 ,a - ,b4 ,c - ,d are ari?metically proportional4 ,= ! diff]5ce betwe5 ! fir/ antec$5t & its consequ5t is ?is1 ,a ,d - ,b - ,c2 & ! diff]5ce betwe5 ! second antec$5t & its consequ5t ?is1 ,a - ,b - ,c ,d2 b ^! two diff]5ces are equal2 & !re=e1 by ?is #e? article1 ,a ,d4 ,b ,c 33 ,a - ,b4 ,c - ,d are ari?metically proportional4 ,if1 ( f\r magnitudes1 two be equal to ! o!r two1 !y w be 9 reciprocal ari?metical proportion4 ,= let ,a ,b be equal to ,c ,d1 ,i say ,a4 ,c 33 ,d4 ,b are ari?metically proportional4 ,= if !y be n1 let ,a4 ,c 33 ,d4 ,e 7suppos+ ,e to be great] or less ?an ,b7 be ari?metically proportional1 & !n ,a ,e w be equal to ,c ,d2 :]e=e ,a ,b & ,c ,d are n equal2 :i* is contr>y to :at 0 suppos$4 ,sid5ote3 ,! def9ition & transmutations ( analogism1 or ! same geometrical proportion4 #f4 ,"o geometrical proportion is ! same ) ano!r geometrical proportion2 :5 ! same cause1 produc+ equal effects 9 equal times1 det]m9es bo? ! proportions4 ,if a po9t uni=mly mov$ describe two l9es1 ei!r ) ! same1 or diff]5t velocity1 all ! p>ts ( !m :i* are contempor>y1 t is1 :i* are describ$ 9 ! same "t1 w be two to two1 9 geometrical proportion1 :e!r ! antec$5ts be tak5 9 ! same l9e1 or n4 ,=1 f ! po9t ,a 79 ! #aj? figure at ! end ( ! #ad? *apt]7 let ! two l9es1 ,a ,d1 ,a ,g1 be describ$ ) uni=m motion2 & let !re be tak5 9 !m two p>ts ,a ,b1 ,a ,e1 & aga91 two o!r p>ts1 ,a ,c1 ,a ,f2 9 su* mann]1 t ,a ,b1 ,a ,e1 be contempor>y1 & likewise ,a ,c1 ,a ,f contempor>y4 ,i say fir/ 7tak+ ! antec$5ts ,a ,b1 ,a ,c 9 ! l9e ,a ,d1 & ! conqu5ts ,a ,e1 ,a ,f 9 ! l9e ,a ,g7 t ,a ,b4 ,a ,c 33 ,a ,e4 ,a ,f are proportionals4 ,= see+ 7by ! #h? *ap1 & ! #ae? art47 velocity is motion consid]$ z det]m9$ by a c]ta9 l5g? or l9e1 9 a c]ta9 "t transmitt$ by x1 ! quantity ( ! l9e ,a ,b w be det]m9$ by ! velocity & "t by :i* ! same ,a ,b is describ$2 & = ! same reason1 ! quantity ( ! l9e ,a ,c w be det]m9$ by ! velocity & "t1 by :i* ! same ,a ,c is describ$2 & !re=e ! proportion ( ,a ,b to ,a ,c1 :e!r x be proportion ( equality1 or ( excess or defect1 is det]m9$ by ! velocities & times by :i* ,a ,b1 ,a ,c are describ$2 b see+ ! motion ( ! po9t ,a ^u ,a ,b & ,a ,c is uni=m1 !y are bo? describ$ ) equal velocity2 & !re=e :e!r "o ( !m h to ! o!r ! proportion ( majority or ( m9ority1 ! sole cause ( t proportion is ! diff]5ce ( _! times2 & by ! same reason x is evid5t1 t ! proportion ( ,a ,e to ,a ,f is det]m9$ by ! diff]5ce ( _! times only4 ,see+ !re=e ,a ,b1 ,a ,e1 z also ,a ,c1 ,a ,f are contempor>y1 ! diff]5ce ( ! times 9 :i* ,a ,b & ,a ,c are describ$1 is ! same ) t 9 :i* ,a ,e & ,a ,f are describ$4 ,:]e=e ! proportion ( ,a ,b to ,a ,c1 & ! proportion ( ,a ,e to ,a ,f are bo? det]m9$ by ! same cause4 ,b ! cause1 :i* s det]m9es ! proportion ( bo?1 works equally 9 equal times1 = x is uni=m motion2 & !re=e 7by ! la/ prec$5t def9ition7 ! proportion ( ,a ,b to ,a ,c is ! same ) t ( ,a ,e to ,a ,f2 & consequ5tly ,a ,b4 ,a ,c 33 ,a ,e4 ,a ,f are proportionals2 :i* is ! fir/4 ,secondly1 7tak+ ! antec$5ts 9 diff]5t l9es7 ,i say1 ,a ,b4 ,a ,e 33 ,a ,c4 ,a ,f are proportionals2 = see+ ,a ,b1 ,a ,e are describ$ 9 ! same "t1 ! diff]5ce ( ! velocities 9 :i* !y are describ$ is ! sole cause ( ! proportion !y h to "o ano!r4 ,& ! same may be said ( ! proportion ( ,a ,c to ,a ,f4 ,b see+ bo? ! l9es ,a ,d & ,a ,g are pass$ ov] by uni=m motion1 ! diff]5ce ( ! velocities 9 :i* ,a ,b1 ,a ,e are describ$1 w be ! same ) ! diff]5ce ( ! velocities1 9 :i* ,a ,c1 ,a ,f are describ$4 ,:]e=e ! cause :i* det]m9es ! proportion ( ,a ,b to ,a ,e1 is ! same ) t :i* det]m9es ! proportion ( ,a ,c to ,a ,f2 & !re=e ,a ,b4 ,a ,e 33 ,a ,c4 ,a ,f1 are proportionals2 :i* rema9$ to be prov$4 ,coroll4 ,i4 ,if f\r magnitudes be 9 geometrical proportion1 !y w also be proportionals by p]mutation1 t is1 by transpos+ ! middle t]ms4 ,= ,i h %[n1 t n only ,a ,b4 ,a ,c 33 ,a ,e4 ,a ,f1 b also t1 by p]mutation1 ,a ,b4 ,a ,e 33 ,a ,c4 ,a ,f are proportionals4 ,coroll4 ,,ii4 ,if !re be f\r proportionals1 !y w also be proportionals by inv]sion or conv]sion1 t is1 by turn+ ! antec$5ts into consequ5ts4 ,= if 9 ! la/ analogism1 ,i _h = ,a ,b1 ,a ,c1 put by inv]sion ,a ,c1 ,a ,b1 & 9 l mann] conv]t$ ,a ,e1 ,a ,f into ,a ,f1 ,a ,e1 yet ! same demon/ration _h s]v$4 ,= z well ,a ,c1 ,a ,b1 z ,a ,b1 ,a ,c are ( equal velocity2 & ,a ,c1 ,a ,f1 z well z ,a ,f1 ,a ,c are contempor>y4 ,coroll4 ,,iii4 ,if proportionals be add$ to proportionals1 or tak5 f !m1 ! aggregates1 or rema9d]s1 w be proportionals4 ,= contempor>ies1 :e!r !y be add$ to contempor>ies1 or tak5 f !m1 make ! aggregates or rema9d]s contempor>y1 ?\gh ! addition or subtraction be ( all ! t]ms1 or ( ! antec$5ts alone1 or ( ! consequ5ts alone4 ,coroll4 ,,iv4 ,if bo? ! antec$5ts ( f\r proportionals1 or bo? ! consequ5ts1 or all ! t]ms1 be multipli$ or divid$ by ! same numb] or quantity1 ! products or quoti5ts w be proportionals4 ,= ! multiplication & division ( proportionals1 is ! same ) ! addition & subtraction ( !m4 ,coroll4 ,v4 ,if !re be f\r proportionals1 !y w also be proportionals by composition1 t is1 by comp\nd+ an antec$5t ( ! antec$5t & consequ5t put toge!r1 & by tak+ = consequ5t ei!r ! consequ5t s+ly1 or ! antec$5t s+ly4 ,= ?is composition is no?+ b addition ( proportionals1 namely1 ( consequ5ts to _! own antec$5ts1 :i* by supposition are proportionals4 ,coroll4 ,,vi4 ,9 l mann]1 if ! antec$5t s+ly1 or consequ5t s+ly1 be put = antec$5t1 & ! consequ5t be made ( bo? put toge!r1 ^! also w be proportionals4 ,= x is ! inv]sion ( proportion by composition4 ,coroll4 ,,vii4 ,if !re be f\r proportionals1 !y w also be proportionals by division1 t is1 by tak+ ! rema9d] aft] ! consequ5t is subtract$ f ! antec$5t1 or ! diff]5ce betwe5 ! antec$5t & consequ5t = antec$5t1 & ei!r ! :ole or ! subtract$ = consequ5t2 z if ,a4 ,b 33 ,c4 ,d be proportionals1 !y w by division be ,a - ,b4 ,b 33 ,c - ,d4 ,d1 & ,a - ,b4 ,a 33 ,c - ,d4 ,c2 & :5 ! consequ5t is great] ?an ! antec$5t1 ,b - ,a4 ,a 33 ,d - ,c4 ,c1 & ,b - ,a4 ,b 33 ,d - ,c4 ,d4 ,= 9 all ^! divisions1 proportionals are1 by ! v supposition ( ! analogism ,a4 ,b 33 ,c4 ,d1 tak5 f ,a & ,b1 & f ,c & ,d4 ,coroll4 ,,viii4 ,if !re be f\r proportionals1 !y w also be proportionals by ! conv]sion ( proportion1 t is1 by inv]t+ ! divid$ proportion1 or by tak+ ! :ole = antec$5t1 & ! diff]5ce or rema9d] = consequ5t4 ,z1 if ,a4 ,b 33 ,c4 ,d be proportionals1 !n ,a4 ,a - ,b 33 ,c4 ,c - ,d1 z also ,b4 ,a - ,b 33 ,d4 ,c - ,d w be proportionals4 ,= see+ ^! inv]t$ be proportionals1 !y are also !mselves proportionals4 ,coroll4 ,,ix4 ,if !re be two analogisms :i* h _! quantities equal1 ! second to ! second1 & ! f\r? to ! f\r?1 !n ei!r ! sum or diff]5ce ( ! fir/ quantities w be to ! second1 z ! sum or diff]5ce ( ! ?ird quantities is to ! f\r?4 ,let ,a4 ,b 33 ,c4 ,d & ,e4 ,b 33 ,f4 ,d be analogisms2 ,i say ,a ,e4 ,b 33 ,c ,f4 ,d are proportionals4 ,= ! said analogisms w by p]mutation be ,a4 ,c 33 ,b4 ,d1 & ,e4 ,f 33 ,b4 ,d2 & !re=e ,a4 ,c 33 ,e4 ,f w be proportionals1 = !y h bo? ! proportion ( ,b to ,d common4 ,:]e=e1 if 9 ! p]mutation ( ! fir/ analogism1 !re be add$ ,e & ,f to ,a & ,c1 :i* ,e & ,f are proportional to ,a & ,c1 !n 7by ! ?ird coroll47 ,a ,e4 ,b 33 ,c ,f4 ,d w be proportionals2 :i* 0 to be prov$4 ,also 9 ! same mann] x may be %[n1 t ,a - ,e4 ,b 33 ,c - ,f4 ,d are proportionals4 #g4 ,if !re be two analogisms1 :]e f\r antec$5ts make an analogism1 _! consequ5ts also %all make an analogism2 z also ! sums ( _! antec$5ts w be proportional to ! sums ( _! consequ5ts4 ,= if ,a4 ,b 33 ,c4 ,d & ,e4 ,f 33 ,g4 ,h be two analogisms1 & ,a4 ,e 33 ,c4 ,g be proportionals1 !n by p]mutation ,a4 ,c 33 ,e4 ,g1 & ,e4 ,g 33 ,f4 ,h1 & ,a4 ,c 33 ,b4 ,d w be proportionals2 :]e=e ,b4 ,d 33 ,e4 ,g1 t is1 ,b4 ,d 33 ,f4 ,h1 & by p]mutation ,b4 ,f 33 ,d4 ,h are proportionals2 :i* is ! fir/4 ,secondly1 ,i say ,a ,e4 ,b ,f 33 ,c ,g4 ,d ,h w be proportionals4 ,= see+ ,a4 ,e 33 ,c4 ,g are proportionals1 ,a ,e4 ,e 33 ,c ,g4 ,g w also by composition be proportionals1 & by p]mutation ,a ,e4 ,c ,g 33 ,e4 ,g w be proportionals2 :]e=e1 also ,a ,e4 ,c ,g 33 ,b ,f4 ,d ,h w be proportionals4 ,aga91 see+1 z is %[n above1 ,b4 ,f 33 ,d4 ,h are proportionals1 ,b ,f4 ,f 33 ,d ,h4 ,h w also by composition be proportionals2 & by p]mutation ,b ,f4 ,d ,h 33 ,f4 ,h w also be proportionals2 :]e=e ,a ,e4 ,c ,g 33 ,b ,f4 ,d ,h are proportionals2 :i* rema9$ to be prov$4 ,coroll4 ,by ! same reason1 if !re be nev] s _m analogisms1 & ! antec$5ts be proportional to ! antec$5ts1 x may be demon/rat$ also t ! consequ5ts w be proportional to ! consequ5ts1 z also ! sum ( ! antec$5ts to ! sum ( ! consequ5ts4 ,sid5ote3 ,! def9ition ( hyp]logism & hypologism1 t is1 ( great] & less proportion1 & _! transmutations4 #h4 ,9 an hyp]logism1 t is1 :]e ! proportion ( ! fir/ antec$5t to its consequ5t is great] ?an ! proportion ( ! second antec$5t to its consequ5t1 ! p]mutation ( ! proportionals1 & ! addition ( proportionals to proportionals1 & sub/raction ( !m f "o ano!r1 z also _! composition & division1 & _! multiplication & division by ! same numb]1 produce always an hyp]logism4 ,= suppose ,a4 ,b 33 ,c 4 ,d & ,a4 ,c 33 ,e4 ,f be analogisms1 ,a ,e4 ,b 33 ,c ,f 4 ,d w also be an analogism2 b ,a ,e4 ,b 33 ,c4 ,d w be an hyp]logism2 :]e=e by p]mutation1 ,a ,e4 ,c 33 ,b4 ,d is an hyp]logism1 because ,a4 ,b 33 ,c4 ,d is an analogism4 ,secondly1 if to ! hyp]logism ,a ,e4 ,b 33 ,c4 ,d ! proportionals ,g & ,h be add$1 ,a ,e ,g4 ,b 33 ,c ,h4 ,d w be an hyp]logism1 by reason ,a ,e ,g4 ,b 33 ,c ,f ,h4 ,d is an analogism4 ,also1 if ,g & ,h be tak5 away1 ,a ,e - ,g4 ,b 33 ,c - ,h4 ,d w be an hyp]logism2 = ,a ,e - ,g4 ,b 33 ,c ,f - ,h4 ,d is an analogism4 ,?irdly1 by composition ,a ,e ,b4 ,b 33 ,c ,d4 ,d w be an hyp]logism1 because ,a ,e ,b4 ,b 33 ,c ,f ,d4 ,d is ah analogism1 & s x w be 9 all ! v>ieties ( composition4 ,f\r?ly1 by division1 ,a ,e - ,b4 ,b 33 ,c - ,d4 ,d w by an hyp]logism1 by reason ,a ,e - ,b4 ,b 33 ,c ,f - ,d4 ,d is an analogism4 ,also ,a ,e - ,b4 ,a ,e 33 ,c - ,d4 ,c is an hyp]logism2 = ,a ,e - ,b4 ,a ,e 33 ,c ,f - ,d4 ,c is an analogism4 ,fif?ly1 by multiplication #d ,a #d ,e4 ,b 33 #d ,c4 ,d is an hyp]logism1 because #d ,a4 ,b 33 #d ,c4 ,d is an analogism2 & by division ,a ,e4 ,b 33 ,c4,d is an hyp]logism1 because ,a4 ,b 33 ,c4 ,d is an analogism4 #i4 ,b if ,a ,e4 ,b 33 ,c4 ,d be an hyp]logism1 !n by inv]sion ,b4 ,a ,e 33 ,d4 ,c w be an hypologism1 because ,b4 ,a 33 ,d4 ,c be+ an analogism1 ! fir/ consequ5t w be too great4 ,also1 by conv]sion ( proportion1 ,a ,e4 ,a ,e - ,b 33 ,c4 ,c - ,d is an hypologism1 because ! inv]sion ( x1 namely ,a ,e - ,b4 ,a ,e 33 ,c - ,d4 ,c is an hyp]logism1 z ,i h %[n b n[4 ,s also ,b4 ,a ,e - ,b 33 ,d4 ,c - ,d is an hypologism1 because1 z ,i h newly %[n1 ! inv]sion ( x1 namely ,a ,e - ,b4 ,b 33 ,c - ,d4 ,d is an hyp]logism4 ,note t ?is hypologism ,a ,e4 ,a ,e - ,b 33 ,c4 ,c - ,d is commonly ?us express$2 if ! proportion ( ! :ole1 7,a ,e7 to t :i* is tak5 out ( x 7,b71 be great] ?an ! proportion ( ! :ole 7,c7 to t :i* is tak5 out ( x 7,d71 !n ! proportion ( ! :ole 7,a ,e7 to ! rema9d] 7,a ,e - ,b7 w be less ?an ! proportion ( ! :ole 7,c7 to ! rema9d] 7,c - ,d74 ,sid5ote3 ,comp>ison ( analogical quantities1 accord+ to magnitude4 #aj4 ,if !re be f\r proportionals1 ! diff]5ce ( ! two fir/1 to ! diff]5ce ( ! two la/1 w be z ! fir/ antec$5t is to ! second antec$5t1 or z ! fir/ consequ5t to ! second consequ5t4 ,= if ,a4 ,b 33 ,c4 ,d be proportionals1 !n by division ,a - ,b4 ,b 33 ,c - ,d4 ,d w be proportionals2 & by p]mutation ,a - ,b4 ,c - ,d 33 ,b4 ,d2 t is1 ! diff]5ces are proportional to ! consequ5ts1 & !re=e !y are s also to ! antec$5ts4 #aa4 ,( f\r proportionals1 if ! fir/ be great] ?an ! second1 ! ?ird also %all be great] ?an ! f\r?4 ,= see+ ! fir/ is great] ?an ! second1 ! proportion ( ! fir/ to ! second is ! proportion ( excess2 b ! proportion ( ! ?ird to ! f\r? is ! same ) t ( ! fir/ to ! second2 & !re=e also ! proportion ( ! ?ird to ! f\r? is ! proportion ( excess2 :]e=e ! ?ird is great] ?an ! f\r?4 ,9 ! same mann] x may be prov$1 t :5soev] ! fir/ is less ?an ! second1 ! ?ird also is less ?an ! f\r?2 & :5 ^? are equal1 t ^! also are equal4 #ab4 ,if !re be f\r proportionals :atsoev]1 ,a4 ,b 33 ,c4,d1 & ! fir/ & ?ird be multipli$ by any "o numb]1 z by #b2 & aga9 ! second & f\r? be multipli$ by any "o numb]1 z by #c2 & ! product ( ! fir/ #b ,a1 be great] ?an ! product ( ! second #c ,b2 ! product also ( ! ?ird #b ,c1 w be great] ?an ! product ( ! f\r? #c ,d4 ,b if ! product ( ! fir/ be less ?an ! product ( ! second1 !n ! product ( ! ?ird w be less ?an t ( ! f\r?4 ,& la/ly1 if ! products ( ! fir/ & second be equal1 ! products ( ! ?ird & f\r? %all also be equal4 ,n[ ?is !orem is all "o ) ,euclid's def9ition ( ! same proportion2 & x may be demon/rat$ ?us4 ,see+ ,a4 ,b 33 ,c4 ,d are proportionals1 by p]mutation also 7art4 #f1 coroll4 ,i47 ,a4 ,c 33 ,b 4 ,d w be proportionals2 :]e=e 7by coroll4 ,,iv4 art4 #f7 #b ,a4 #b ,c 33 #c ,b4 #c ,d w be proportionals2 & aga91 by p]mutation1 #b ,a4 #c ,b 33 #b ,c4 #c ,d w be proportionals2 & !re=e1 by ! la/ article1 if #b ,a be great] ?an #c ,b1 !n #b ,c w be great] ?an #c ,d2 if less1 less2 & if equal1 equal2 :i* 0 to be demon/rat$4 ,sid5ote3 ,composition ( proportions4 ,if any ?ree magnitudes be prop\nd$1 or ?ree ?+s :atsoev] t h any proportion "o to ano!r1 z ?ree numb]s1 ?ree times1 ?ree degrees1 c42 ! proportions ( ! fir/ to ! second1 & ( ! second to ! ?ird1 toge!r tak51 are equal to ! proportion ( ! fir/ to ! ?ird4 ,let !re be ?ree l9es1 = any proportion may be r$uc$ to ! proportion ( l9es1 ,a ,b1 ,a ,c1 ,a ,d2 & 9 ! fir/ place1 let ! proportion z well ( ! fir/ ,a ,b to ! second ,a ,c1 z ( ! second ,a ,c to ! ?ird ,,ad1 be ! proportion ( defect1 or ( less to great]2 ,i say ! proportions toge!r tak5 ( ,a ,b to ,a ,c1 & ( ,a ,c to ,a ,d1 are equal to ! proportion ( ,a ,b to ,a ,d4 ,suppose ! po9t ,a to be mov$ ov] ! :ole l9e ,a ,d ) uni=m motion2 !n ! proportions z well ( ,a ,b to ,a ,c1 z ( ,a ,c to ,a ,d1 are det]m9$ by ! diff]5ce ( ! times 9 :i* !y are describ$2 t is1 ,a ,b has to ,a ,c su* proportion z is det]m9$ by ! diff]5t times ( _! description2 & ,a ,c to ,,ad su* proportion z is det]m9$ by _! times4 ,b ! proportion ( ,a ,b to ,a ,d is su* z is det]m9$ by ! diff]5ce ( ! times 9 :i* ,a ,b & ,a ,d are describ$2 & ! diff]5ce ( ! times 9 :i* ,a ,b & ,a ,c are describ$1 toge!r ) ! diff]5ce ( ! times 9 :i* ,a ,c & ,a ,d are describ$1 is ! same ) ! diff]5ce ( ! times 9 :i* ,a ,b & ,a ,d are describ$4 ,& !re=e1 ! same cause :i* det]m9es ! two proportions ( ,a ,b to ,a ,c & ( ,a ,c to ,a ,d1 det]m9es also ! proportion ( ,a ,b to ,a ,d4 ,:]e=e1 by ! def9ition ( ! same proportion1 deliv]$ above 9 ! #f? article1 ! proportion ( ,a ,b to ,a ,c toge!r ) ! proportion ( ,a ,c to ,a ,d1 is ! same ) ! proportion ( ,a ,b to ,a ,d4 ,a ,b ,c ,d ------------- ,9 ! second place1 let ,a ,d be ! fir/1 ,a ,c ! second1 & ,a ,b ! ?ird1 & let _! proportion be ! proportion ( excess1 or ! great] to less2 !n1 z be=e1 ! proportions ( ,a ,d to ,a ,c1 & ( ,a ,c to ,a ,b1 & ( ,a ,d to ,a ,b1 w be det]m9$ by ! diff]5ce ( _! times2 :i* 9 ! description ( ,a ,d & ,a ,c1 & ( ,a ,c & ,a ,b toge!r tak51 is ! same ) ! diff]5ce ( ! times 9 ! description ( ,a ,d & ,a ,b4 ,:]e=e ! proportion ( ,a ,d to ,a ,b is equal to ! two proportions ( ,a ,d to ,a ,c & ( ,a ,c to ,a ,b4 ,9 ! la/ place4 ,if "o ( ! proportions1 namely ( ,a ,d to ,a ,b1 be ! proportion ( excess1 & ano!r ( !m1 z ( ,a ,b to ,a ,c be ! proportion ( defect1 ?us also ! proportion ( ,a ,d to ,a ,c w be equal to ! two proportions toge!r tak5 ( ,a ,d to ,a ,b1 & ( ,a ,b to ,a ,c4 ,= ! diff]5ce ( ! times 9 :i* ,,ad & ,,ab are describ$1 is excess ( "t2 = !re goes m "t to ! description ( ,a ,d ?an ( ,a ,b2 & ! diff]5ce ( ! times 9 :i* ,a ,b & ,a ,c are describ$1 is defect ( "t1 = less "t goes to ! description ( ,a ,b ?an ( ,a ,c2 b ?is excess & defect be+ add$ toge!r1 make ,d ,b - ,b ,c1 :i* is equal to ,d ,c1 by :i* ! fir/ ,a ,d exce$s ! ?ird ,a ,c2 & !re=e ! proportions ( ! fir/ ,a ,d to ! second ,a ,b1 & ( ! second ,a ,b to ! ?ird ,a ,c1 are det]m9$ by ! same cause :i* det]m9es ! proportion ( ! fir/ ,a ,d to ! ?ird ,a ,c4 ,:]e=e1 if any ?ree magnitudes1 c4 ,coroll4 ,i4 ,if !re be nev] s _m magnitudes hav+ proportion to "o ano!r1 ! proportion ( ! fir/ to ! la/ is comp\nd$ ( ! proportions ( ! fir/ to ! second1 ( ! second to ! ?ird1 & s on till y come to ! la/2 or1 ! proportion ( ! fir/ to ! la/ is ! same ) ! sum ( all ! int]m$iate proportions4 ,= any numb] ( magnitudes hav+ proportion to "o ano!r1 z ,a1 ,b1 ,c1 ,d1 ,e be+ prop\nd$1 ! proportion ( ,a to ,e1 z is newly %[n1 is comp\nd$ ( ! proportions ( ,a to ,d & ( ,d to ,e2 & aga91 ! proportion ( ,a to ,d1 ( ! proportions ( ,a to ,c1 & ( ,c to ,d2 & la/ly1 ! proportion ( ,a to ,c1 ( ! proportions ( ,a to ,b1 & ( ,b to ,c4 ,coroll4 ,,ii4 ,f h5ce x may be und]/ood h[ any two proportions may be comp\nd$4 ,= if ! proportions ( ,a to ,b1 & ( ,c to ,d1 be prop\nd$ to be add$ toge!r1 let ,b h to some?+ else1 z to ,e1 ! same proportion :i* ,c has to ,d1 & let !m be set 9 ?is ord]1 ,a1 ,b1 ,e2 = s ! proportion ( ,a to ,e w evid5tly be ! sum ( ! two proportions ( ,a to ,b1 & ( ,b to ,e1 t is1 ( ,c to ,d4 ,or let x be z ,d to ,c1 s ,a to some?+ else1 z to ,e1 & let !m be ord]$ ?us1 ,e1 ,a1 ,b2 = ! proportion ( ,e to ,b w be comp\nd$ ( ! proportions ( ,e to ,a1 t is1 ( ,c to ,d1 & ( ,a to ,b4 ,also1 x may be und]/ood h[ "o proportion may be tak5 out ( ano!r4 ,= if ! proportion ( ,c to ,d be to be subtract$ out ( ! proportion ( ,a to ,b1 let x be z ,c to ,d1 s ,a to some?+ else1 z ,e1 & sett+ !m 9 ?is ord]1 ,a1 ,e1 ,b1 & tak+ away ! proportion ( ,a to ,e1 t is1 ( ,c to ,d1 !re w rema9 ! proportion ( ,e to ,b4 ,coroll4 ,,iii4 ,if !re be two ord]s ( magnitudes :i* h proportion to "o ano!r1 & ! sev]al proportions ( ! fir/ ord] be ! same & equal 9 numb] ) ! proportions ( ! second ord]2 !n1 :e!r ! proportions 9 bo? ord]s be successively answ]able to "o ano!r1 :i* is call$ ord9ate proportion1 or n successively answ]able1 :i* is call$ p]turb$ proportion1 ! fir/ & ! la/ 9 bo? w be proportionals4 ,= ! proportion ( ! fir/ to ! la/ is equal to all ! int]m$iate proportions2 :i* be+ 9 bo? ord]s ! same1 & equal 9 numb]1 ! aggregates ( ^? proportions w also be equal to "o ano!r2 b to _! aggregates1 ! proportions ( ! fir/ to ! la/ are equal2 & !re=e ! proportion ( ! fir/ to ! la/ 9 "o ord]1 is ! same ) ! proportion ( ! fir/ to ! la/ 9 ! o!r ord]4 ,:]e=e ! fir/ & ! la/ 9 bo? are proportionals4 ,sid5ote3 ,composition ( proportions4 #ad4 ,if any two quantities be made ( ! mutual multiplication ( _m quantities1 :i* h proportion to "o ano!r1 & ! effici5t quantities on bo? sides be equal 9 numb]1 ! proportion ( ! products w be comp\nd$ ( ! sev]al proportions1 :i* ! effici5t quantities h to "o ano!r4 ,a ,b4 ,a4 ,c ,b4 ,c4 ,c ,d4 ,e4 ,fir/1 let ! two products be ,a ,b & ,c ,d1 :]e( "o is made ( ! multiplication ( ,a into ,b1 & ! o!r ( ! multiplication ( ,c into ,d4 ,i say ! proportion ( ,a ,b to ,c ,d is comp\nd$ ( ! proportions ( ! effici5t ,a to ! effici5t ,c1 & ( ! effici5t ,b to ! effici5t ,d4 ,= let ,a ,b1 ,c ,b & ,c ,d be set 9 ord]2 & z ,b is to ,d1 s let ,c be to ano!r quantity z ,e2 & let ,a1 ,c1 ,e be set also 9 ord]4 ,!n 7by coroll4 ,,iv4 ( ! #f? art47 x w be z ,a ,b ! fir/ quantity to ,c ,b ! second quantity 9 ! fir/ ord]1 s ,a to ,c 9 ! second ord]2 & aga91 z ,c ,b to ,c ,d 9 ! fir/ ord]1 s ,b to ,d1 t is1 by con/ruction1 s ,c to ,e 9 ! second ord]2 & !re=e 7by ! la/ coroll>y7 ,a ,b4 ,c ,d 33 ,a4 ,e w be proportionals4 ,b ! proportion ( ,a to ,e is comp\nd$ ( ! proportions ( ,a to ,c1 & ( ,b to ,d2 :]e=e also ! proportion ( ,a ,b to ,c ,d is comp\nd$ ( ! same4 ,secondly1 let ! two products be ,a ,b ,f1 & ,c ,d ,g1 ea* ( !m made ( ?ree effici5ts1 ! fir/ ( ,a1 ,b & ,f1 & ! second ( ,c1 ,d & ,g2 ,i say1 ! proportion ( ,a ,b ,f to ,c ,d ,g is comp\nd$ ( ! proportions ( ,a to ,c1 ( ,b to ,d1 & ( ,f to ,g4 ,= let !m be set 9 ord] z be=e2 & z ,b is to ,d1 s let ,c be to ano!r quantity ,e2 & aga91 z ,f is to ,g1 s let ,e be to ano!r1 ,h2 & let ! fir/ ord] /& ?us1 ,,abf1 ,,cbf1 ,,cdf & ,,cdg2 & ! second ord] ?us1 ,a1 ,c1 ,e1 ,h4 ,!n ! proportion ( ,a ,b ,f to ,c ,b ,f 9 ! fir/ ord]1 w be z ,a to ,c 9 ! second2 & ! proportion ( ,,cbf to ,,cdf 9 ! fir/ ord]1 z ,b to ,d1 t is1 z ,c to ,e 7by con/ruction7 9 ! second ord]2 & ! proportion ( ,,cdf to ,,cdg 9 ! fir/1 z ,f to ,g1 t is1 z ,e to ,h 7by con/ruction7 9 ! second ord]2 & !re=e ,a ,b ,f4 ,c ,d ,g33 ,a4 ,h w be proportionals4 ,b ! proportion ( ,a to ,h is comp\nd$ ( ! proportions ( ,a to ,c1 ,b to ,d1 & ,f to ,g4 ,:]e=e ! proportion ( ! product ,a ,b ,f to ,c ,d ,g is also comp\nd$ ( ! same4 ,& ?is op]ation s]ves1 h[ _m soev] ! effici5ts be t make ! quantities giv54 ,,abf4 ,a4 ,,cbf4 ,c4 ,,cdf4 ,e4 ,,cdg4 ,h4 ,f h5ce arise? ano!r way ( comp\nd+ _m proportions into "o1 namely1 t :i* is suppos$ 9 ! #e? def9ition ( ! #f? book ( ,euclid2 :i* is1 by multiply+ all ! antec$5ts ( ! proportions into "o ano!r1 & 9 l mann] all ! consequ5ts into "o ano!r4 ,& f h5ce also x is evid5t1 9 ! fir/ place1 t ! cause :y p>allelograms1 :i* are made by ! duction ( two /raight l9es into "o ano!r1 & all solids :i* are equal to figures s made1 h _! proportions comp\nd$ ( ! proportions ( ! effici5ts2 & 9 ! second place1 :y ! multiplication ( two or m fractions into "o ano!r is ! same ?+ ) ! composition ( ! proportions ( _! sev]al num]ators to _! sev]al d5om9ators4 ,= example1 if ^! fractions #ab1 #bc1 #cd be to be multipli$ into "o ano!r1 ! num]ators #a1 #b1 #c1 are fir/ to be multipli$ into "o ano!r1 :i* make #f2 & next ! d5om9ators #b1 #c1 #d1 :i* make #bd2 & ^! two products make ! fraction #fbd4 ,9 l mann]1 if ! proportions ( #a to #b1 ( #b to #c1 & ( #c to #d1 be to be comp\nd$1 by work+ z ,i h %[n above1 ! same proportion ( #f to #bd w be produc$4 #ae4 ,if any proportion be comp\nd$ ) itself inv]t$1 ! comp\nd w be ! proportion ( equality4 ,= let any proportion be giv51 z ( ,a to ,b1 & let ! inv]se ( x be t ( ,c to ,d2 & z ,c to ,d1 s let ,b be to ano!r quantity2 = ?us !y w be comp\nd$ 7by ! second coroll4 ( ! #ab? art47 ,n[ see+ ! proportion ( ,c to ,d is ! inv]se ( ! proportion ( ,a to ,b1 x w be z ,c to ,d1 s ,b to ,a2 & !re=e if !y be plac$ 9 ord]1 ,a1 ,b1 ,a1 ! proportion comp\nd$ ( ! proportions ( ,a to ,b1 & ( ,c to ,d1 w be ! proportion ( ,a to ,a1 t is1 ! proportion ( equality4 ,& f h5ce ! cause is evid5t :y two equal products h _! effici5ts reciprocally proportional4 ,=1 = ! mak+ ( two products equal1 ! proportions ( _! effici5ts mu/ be su*1 z be+ comp\nd$ may make ! proportion ( equality1 :i* _c be except "o be ! inv]se ( ! o!r2 = if betwixt ,a & ,a any o!r quantity1 z ,c1 be int]pos$1 _! ord] w be ,a1 ,c1 ,a1 & ! lat] proportion ( ,c to ,a w be ! inv]se ( ! =m] proportion ( ,a to ,c4 ,sid5ote3 ,! def9ition & prop]ties ( cont9ual proportion4 #af4 ,a proportion is said to be multipli$ by a numb]1 :5 x is s (t5 tak5 z !re be unities 9 t numb]2 & if ! proportion be ( ! great] to ! less1 !n %all also ! quantity ( ! proportion be increas$ by ! multiplication2 b :5 ! proportion is ( ! less to ! great]1 !n z ! numb] increase?1 ! quantity ( ! proportion dim9i%e?2 z 9 ^! ?ree numb]s1 #d1 #b1 #a1 ! proportion ( #d to #a is n only ! duplicate ( #d to #b1 b also twice z great2 b inv]t+ ! ord] ( ^? numb]s ?us1 #a1 #b1 #d1 ! proportion ( #a to #b is great] ?an t ( #a to #d2 & !re=e ?\gh ! proportion ( #a to #d be ! duplicate ( #a to #b1 yet x is n twice s great z t ( #a to #b1 b contr>ily ! half ( x4 ,9 l mann]1 a proportion is said to be divid$1 :5 betwe5 two quantities are int]pos$ "o or m means 9 cont9ual proportion1 & !n ! proportion ( ! fir/ to ! second is said to be subduplicate ( t ( ! fir/ to ! ?ird1 & subtriplicate ( t ( ! fir/ to ! f\r?1 c4 ,?is mixture ( proportions1 :]e "s are proportions ( excess1 o!rs ( defect1 z 9 a m]*ant's acc\nt ( debtor & cr$itor1 is n s easily reckon$ z "s ?9k2 b make? ! composition ( proportions sometimes to be addition1 sometimes sub/raction2 :i* s\nde? absurdly to su* z h always by composition und]/ood addition1 & by dim9ution sub/raction4 ,!re=e to make ?is acc\nt a little cle>]1 we are to consid] 7t :i* is commonly assum$1 & truly7 t if !re be nev] s _m quantities1 ! proportion ( ! fir/ to ! la/ is comp\nd$ ( ! proportions ( ! fir/ to ! second1 & ( ! second to ! ?ird1 & s on to ! la/1 )\t reg>d+ _! equality1 excess1 or defect2 s t if two proportions1 "o ( inequality1 ! o!r ( equality1 be add$ toge!r1 ! proportion is n !reby made great] nor less2 z = example1 if ! proportions ( ,a to ,b & ( ,b to ,b be comp\nd$1 ! proportion ( ! fir/ to ! second is z mu* z ! sum ( bo?1 because proportion ( equality1 be+ n quantity1 nei!r augm5te? quantity nor less5e? x4 ,b if !re be ?ree quantities1 ,a1 ,b1 ,c1 unequal1 & ! fir/ be ! greate/1 ! la/ lea/1 !n ! proportion ( ,b to ,c is an addition to t ( ,a to ,b1 & makes x great]2 & on ! contr>y1 if ,a be ! lea/1 & ,c ! greate/ quantity1 !n do? ! addition ( ! proportion ( ,b to ,c make ! comp\nd$ proportion ( ,a to ,c less ?an ! proportion ( ,a to ,b1 t is1 ! :ole less ?an ! "p4 ,! composition !re=e ( proportions is n 9 ?is case ! augm5tation ( !m1 b ! dim9ution2 = ! same quantity 7,euclid ,v4 #h7 comp>$ ) two o!r quantities1 ha? a great] proportion to ! less] ( !m ?an to ! great]4 ,likewise1 :5 ! proportions comp\nd$ are "o ( excess1 ! o!r ( defect1 if ! fir/ be ( excess1 z 9 ^! numb]s1 #h1 #f1 #i1 ! proportion comp\nd$1 namely1 ( #h to #i1 is less ?an ! proportion ( "o ( ! p>ts ( x1 namely1 ( #h to #f2 b if ! proportion ( ! fir/ to ! second be ( defect1 & t ( ! second to ! ?ird be ( excess1 z 9 ^! numb]s1 #f1 #h1 #d1 !n %all ! proportion ( ! fir/ to ! ?ird be great] ?an t ( ! fir/ to ! second1 z #f ha? a great] proportion to #d ?an to #h2 ! reason :]e( is manife/ly ?is1 t ! less any quantity is defici5t ( ano!r1 or ! m "o exce$e? ano!r1 ! proportion ( x to t o!r is ! great]4 ,suppose n[ ?ree quantities 9 cont9ual proportion1 ,a ,b #d1 ,a ,c #f1 ,a ,d #i4 ,because !re=e ,a ,d is great] ?an ,a ,c1 b n great] ?an ,a ,d1 ! proportion ( ,a ,d to ,,ac w be 7by ,euclid1 ,v4 #h7 great] ?an t ( ,a ,d to ,a ,d2 & likewise1 because ! proportions ( ,a ,d to ,a ,c1 & ( ,a ,c to ,a ,b are ! same1 ! proportions ( ,a ,d to ,a ,c & ( ,a ,c to ,a ,b1 be+ bo? proportions ( excess1 make ! :ole proportion ( ,a ,d to ,a ,b1 or ( #i to #d1 n only ! duplicate ( ,a ,d to ,a ,c1 t is1 ( #i to #f1 b also ! d\ble1 or twice s great4 ,on ! o!r side1 because ! proportion ( ,a ,d to ,a ,d1 or #i to #i1 be+ proportion ( equality1 is no quantity1 & yet great] ?an t ( ,a ,c to ,a ,d1 or #f to #i1 x w be z #j - #i to #j - #f1 s ,a ,c to ,a ,d1 & aga91 z #j - #i to #j - #f1 s #j - #f to #j - #d2 b #j - #d1 #j - #f1 #j - #i are 9 cont9ual proportion2 & because #j - #d is great] ?an #j - #f1 ! proportion ( #j - #d to #j - #f w be d\ble to ! proportion ( #j - #d to #j - #i1 d\ble ,i say1 & yet n duplicate1 b subduplicate4 ,if any be unsatisfi$ ) ?is ratioc9ation1 let him fir/ consid] t 7by ,euclid ,v4 #h7 ! proportion ( ,a ,b to ,a ,c is great] ?an t ( ,a ,b to ,a ,d1 :]esoev] ,d be plac$ 9 ! l9e ,a ,c prolong$2 & ! fur!r (f ! po9t ,d is f ,c1 s mu* ! great] is ! proportion ( ,a ,b to ,a ,c ?an t ( ,a ,b to ,a ,d4 ,!re is !re=e "s po9t 7:i* suppose be ,e7 9 su* di/ance f ,c1 z t ! proportion ( ,a ,b to ,a ,c w be twice z great z t ( ,a ,b to ,a ,e4 ,t consid]$1 let him det]m9e ! l5g? ( ! l9e ,a ,e1 & demon/rate1 if he c1 t ,a ,e is great] or less ?an ,a ,d4 ,c ,a----------,e ,by ! same me?od1 if !re be m quantities ?an ?ree1 z ,a1 ,b1 ,c1 ,d1 9 cont9ual proportion1 & ,a be ! lea/1 x may be made appe> t ! proportion ( ,a to ,b is triple magnitude1 ?\gh subtriple 9 multitude1 to ! proportion ( ,a to ,d4 #ag4 ,if !re be nev] s _m quantities1 ! numb] :]e( is odd1 & _! ord] su*1 t f ! middlemo/ quantity bo? ways !y proce$ 9 cont9ual proportion1 ! proportion ( ! two :i* are next on ei!r side to ! middlemo/ is subduplicate to ! proportion ( ! two :i* are next to ^! on bo? sides1 & subtriplicate ( ! proportion ( ! two :i* are yet "o place m remote1 c4 ,= let ! magnitudes be ,c1 ,b1 ,a1 ,d1 ,e1 & let ,a1 ,b1 ,c1 z also ,a1 ,d1 ,e be 9 cont9ual proportion2 ,i say ! proportion ( ,d to ,b is subduplicate ( ! proportion ( ,e to ,c4 ,= ! proportion ( ,d to ,b is comp\nd$ ( ! proportions ( ,d to ,a1 & ( ,a to ,b once tak52 b ! proportion ( ,e to ,c is comp\nd$ ( ! same twice tak52 & !re=e ! proportion ( ,d to ,b is subduplicate ( ! proportion ( ,e to ,c4 ,& 9 ! same mann]1 if !re 7 ?ree t]ms on ei!r side1 x might be demon/rat$ t ! proportion ( ,d to ,b w\ld be subtriplicate ( t ( ! extremes1 c4 #ah4 ,if !re be nev] s _m cont9ual proportionals1 z ! fir/1 second1 ?ird1 c4 _! diff]5ces w be proportional to !m4 ,= ! second1 ?ird1 c4 are sev]ally consequ5ts ( ! prec$+1 & antec$5ts ( ! foll[+ proportion4 ,b 7by art4 #aj7 ! diff]5ce ( ! fir/ antec$5t & consequ5t1 to diff]5ce ( ! second antec$5t & consequ5t1 is z ! fir/ antec$5t to ! second antec$5t1 t is1 z ! fir/ t]m to ! second1 or z ! second to ! ?ird1 c4 9 cont9ual proportionals4 #ai4 ,if !re be ?ree cont9ual proportionals1 ! sum ( ! extremes1 toge!r ) ! mean twice tak51 ! sum ( ! mean & ei!r ( ! extremes1 & ! same extreme1 are cont9ual proportionals4 ,= let ,a4 ,b4 ,c be cont9ual proportionals4 ,see+1 !re=e1 ,a4 ,b 33 ,b4 ,c are proportionals1 by composition also ,a ,b4 ,b 33 ,b ,c4 ,c w be proportionals2 & by p]mutation ,a ,b4 ,b ,c 33 ,b4 ,c w also be proportionals2 & aga91 by composition ,a #b;,b ,c4 ,b ,c 33 ,b ,c4 ,c2 :i* 0 to be prov$4 #bj4 ,9 f\r cont9ual proportionals1 ! greate/ & ! lea/ put toge!r is a great] quantity ?an ! o!r two put toge!r4 ,let ,a4 ,b 33 ,c4 ,d be cont9ual proportionals2 :]e( let ! greate/ be ,a1 & ! lea/ be ,d2 ,i say ,a ,d is great] ?an ,b ,c4 ,= by art4 #aj1 ,a - ,b4 ,c - ,d 33 ,a4 ,c are proportionals2 & !re=e ,a - ,b is1 by art4 #aa1 great] ?an ,c - ,d4 ,add ,b on bo? sides1 & ,a w be great] ?an ,c ,b - ,d4 ,& aga91 add ,d on bo? sides1 & ,a ,d w be great] ?an ,b ,c2 :i* 0 to be prov$4 ,sid5ote3 ,! def9ition & prop]ties ( cont9ual proportion4 #ba4 ,if !re be f\r proportionals1 ! extremes multipli$ into "o ano!r1 & ! means multipli$ into "o ano!r1 w make equal products4 ,let ,a4 ,b 33 ,c4 ,d be proportionals2 ,i say ,a ,d is equal to ,b ,c4 ,= ! proportion ( ,a ,d to ,b ,c is comp\nd$1 by art4 #ac1 ( ! proportions ( ,a to ,b1 & ,d to ,c1 t is1 its inv]se ,b to ,a2 & !re=e1 by art4 #ad1 ?is comp\nd$ proportion is ! proportion ( equality2 & !re=e also1 ! proportion ( ,a ,d to ,b ,c is ! proportion ( equality4 ,:]e=e !y are equal4 #bb4 ,if !re be f\r quantities1 & ! proportion ( ! fir/ to ! second be duplicate ( ! proportion ( ! ?ird to ! f\r?1 ! product ( ! extremes to ! product ( ! means1 w be z ! ?ird to ! f\r?4 ,let ! f\r quantities be ,a1 ,b1 ,c & ,d2 & let ! proportion ( ,a to ,b be duplicate ( ! proportion ( ,c to ,d1 ,i say ,a ,d1 t is1 ! product ( ,a into ,d is to ,b ,c1 t is1 to ! product ( ! means1 z ,c to ,d4 ,= see+ ! proportion ( ,a to ,b is duplicate ( ! proportion ( ,c to ,d1 if x be z ,c to ,d1 s ,d to ano!r1 ,e1 !n ,a4 ,b 33 ,c4 ,e w be proportionals2 = ! proportion ( ,a to ,b is by supposition duplicate ( ! proportion ( ,c to ,d2 & ,c to ,e duplicate also ( t ( ,c to ,d by ! def9ition1 art4 #ae4 ,:]e=e1 by ! la/ article1 ,a ,e or ,a into ,e is equal to ,b ,c or ,b into ,c2 b1 by coroll4 ,,iv4 art4 #f1 ,a ,d is to ,a ,e z ,d to ,e1 t is1 z ,c to ,d2 & !re=e ,a ,d is to ,b ,c1 :i* z ,i h %[n is equal to ,a ,e1 z ,c to ,d2 :i* 0 to be prov$4 ,moreov]1 if ! proportion ( ! fir/ ,a to ! second ,b be triplicate ( ! proportion ( ! ?ird ,c to ! f\r? ,d1 ! product ( ! extremes to ! product ( ! means w be duplicate ( ! proportion ( ! ?ird to ! f\r?4 ,= if x be z ,c to ,d s ,d to ,e1 & aga91 z ,d to ,e s ,e to ano!r1 ,f1 !n ! proportion ( ,c to ,f w be triplicate ( ! proportion ( ,c to ,d2 & consequ5tly1 ,a4 ,b 33 ,c4 ,f w be proportionals1 & ,a ,f equal to ,b ,c4 ,b z ,a ,d to ,a ,f1 s is ,d to ,f2 & !re=e1 also1 z ,a ,d to ,b ,c1 s ,d to ,f1 t is1 s ,c to ,e2 b ! proportion ( ,c to ,e is duplicate ( ! proportion ( ,c to ,d2 :]e=e1 also1 ! proportion ( ,a ,d to ,b ,c is duplicate ( t ( ,c to ,d1 z 0 prop\nd$4 #bc4 ,if !re be f\r proportionals1 & a mean be int]pos$ betwixt ! fir/ & second1 & ano!r betwixt ! ?ird & f\r?1 ! fir/ ( ^! means w be to ! second1 z ! fir/ ( ! proportionals is to ! ?ird1 or z ! second ( !m is to ! f\r?4 ,= let ,a4 ,b 33 ,c4 ,d be proportionals1 & let ,e be a mean betwixt ,a & ,b1 & ,f a mean betwixt ,c & ,d2 ,i say ,a4 ,c 33 ,e4 ,f are proportionals4 ,= ! proportion ( ,a to ,e is subduplicate ( ! proportion ( ,a to ,b1 or ( ,c to ,d4 ,also1 ! proportion ( ,c to ,f is subduplicate ( t ( ,c to ,d2 & !re=e ,a4 ,e 33 ,c4 ,f are proportionals2 & by p]mutation ,a4 ,c 33 ,e4 ,f are also proportionals2 :i* 0 to be prov$4 #bd4 ,any ?+ is said to be divid$ into extreme & mean proportion1 :5 ! :ole & ! p>ts are 9 cont9ual proportion4 ,z = example1 :5 ,a ,b4 ,a4 ,b are cont9ual proportionals2 or :5 ! /raight l9e ,a ,c is s divid$ 9 ,b1 t ,a ,c4 ,a ,b4 ,b ,c are 9 cont9ual proportion4 ,& if ! same l9e ,a ,c be aga9 divid$ 9 ,d1 s z t ,a ,c4 ,c ,d4 ,a ,d be cont9ual proportionals2 !n also ,a ,c4 ,a ,b4 ,a ,d w be cont9ual proportionals2 & 9 l mann]1 ?\gh 9 contr>y ord]1 ,c ,a4 ,c ,d4 ,c ,b w be cont9ual proportionals2 :i* _c happ5 9 any l9e o!rwise divid$4 ,a ,b ,c ------------ ,d #be4 ,if !re be ?ree cont9ual proportionals1 & aga91 ?ree o!r cont9ual proportions1 :i* h ! same middle t]m1 _! extremes w be 9 reciprocal proportion4 ,= let ,a4 ,b4 ,c & ,d4 ,b4 ,e be cont9ual proportionals1 ,i say ,a4 ,d 33 ,e4 ,c %all be proportionals4 ,= ! proportion ( ,a to ,d is comp\nd$ ( ! proportions ( ,a to ,b1 & ( ,b to ,d2 & ! proportion ( ,e to ,c is comp\nd$ ( ^? ( ,e to ,b1 t is1 ( ,b to ,d1 & ( ,b to ,c1 t is1 ( ,a to ,b4 ,:]e=e1 by equality1 ,a4 ,d 33 ,e4 ,c are proportionals4 ,sid5ote3 ,comp>ison ( ari?metical & geometrical proportion4 ,if any two unequal quantities be made extremes1 & !re be int]pos$ betwixt !m any numb] ( means 9 geometrical proportion1 & ! same numb] ( means 9 ari?metical proportion1 ! sev]al means 9 geometrical proportion w be less ?an ! sev]al means 9 ari?metical proportion4 ,= betwixt ,a ! less]1 & ,e ! great] extreme1 let !re be int]pos$ ?ree means1 ,b1 ,c1 ,d1 9 geometrical proportion1 & z _m m1 ,f1 ,g1 ,h1 9 ari?metical proportion2 ,i say ,b w be less ?an ,f1 ,c ?an ,g1 & ,d ?an ,h4 ,= fir/1 ! diff]5ce betwe5 ,a & ,f is ! same ) t betwe5 ,f & ,g1 & ) t betwe5 ,g & ,h1 by ! def9ition ( ari?metical proportion2 & !re=e1 ! diff]5ce ( ! proportionals :i* /& next to "o ano!r1 to ! diff]5ce ( ! extremes1 is1 :5 !re is b "o mean1 half _! diff]5ce2 :5 two1 a ?ird "p ( x2 :5 ?ree1 a qu>t]1 c42 s t 9 ?is example x is a qu>t]4 ,b ! diff]5ce betwe5 ,d & ,e1 by art4 #ag1 is m ?an a qu>t] ( ! diff]5ce betwe5 ! extremes1 because ! proportion is geometrical1 & !re=e ! diff]5ce betwe5 ,a & ,d is less ?an ?ree qu>t]s ( ! same diff]5ce ( ! extremes4 ,9 l mann]1 if ! diff]5ce betwe5 ,a & ,d be und]/ood to be divid$ into ?ree equal p>ts1 x may be prov$1 t ! diff]5ce betwe5 ,a & ,c is less ?an two qu>t]s ( ! diff]5ce ( ! extremes ,a & ,e4 ,& la/ly1 if ! diff]5ce betwe5 ,a & ,c be divid$ into two equal p>ts1 t ! diff]5ce betwe5 ,a & ,b is less ?an a qu>t] ( ! diff]5ce ( ! extremes ,a & ,e4 ,a ,a - - ,b ,f --- --- ,c ,g ----- ----- ,d ,h ------- ------- ,e ,e --------- --------- ,f ! consid]ation h]e(1 x is manife/1 t ,b1 t is ,a toge!r ) some?+ else :i* is less ?an a f\r? "p ( ! diff]5ce ( ! extremes ,a & ,e1 is less ?an ,f1 t is1 ?an ! same ,a ) some?+ else :i* is equal to ! said f\r? "p4 ,also1 t ,c1 t is ,a ) some?+ else :i* is less ?an two f\r? p>ts ( ! said diff]5ce1 is less ?an ,g1 t is1 ?an ,a toge!r ) ! said two-f\r?s4 ,& la/ly1 t ,d1 :i* exce$s ,a by less ?an ?ree-f\r?s ( ! said diff]5ce1 is less ?an ,h1 :i* exce$s ! same ,a by ?ree entire f\r?s ( ! said diff]5ce4 ,& 9 ! same mann] x w\ld be if !re 7 f\r means1 sav+ t in/ead ( f\r?s ( ! diff]5ce ( ! extremes we are to take fif? p>ts2 & s on4 #bg4 ,,lemma4 ,if a quantity be+ giv51 fir/ "o quantity be bo? add$ to x & subtract$ f x1 & !n ano!r great] or less1 ! proportion ( ! rema9d] to ! aggregate1 is great] :]e ! less quantity is add$ & sub/ract$1 ?an :]e ! great] quantity is add$ & sub/ract$4 ,let ,b be add$ to & sub/ract$ f ! quantity ,a2 s t ,a - ,b be ! rema9d]1 & ,a ,b ! aggregate2 & aga91 let ,c1 a great] quantity ?an ,b1 be add$ to & sub/ract$ f ! same ,a1 s t ,a - ,c be ! rema9d] & ,a ,c ! aggregate2 ,i say ,a - ,b4 ,a ,b 33 ,a - ,c4 ,a ,c w be an hyp]logism4 ,= ,a - ,b4 ,a 33 ,a - ,c4 ,a is an hyp]logism ( a great] antec$5t to ! same consequ5t2 & !re=e ,a - ,b4 ,a ,b 33 ,a - ,c4 ,a ,c is a mu* great] hyp]logism1 be+ made ( a great] antec$5t to a less consequ5t4 #bh4 ,if unequal p>ts be tak5 f two equal quantities1 & betwixt ! :ole & ! "p ( ea* !re be int]pos$ two means1 "o 9 geometrical1 ! o!r 9 ari?metical proportion2 ! diff]5ce betwixt ! two means w be greate/1 :]e ! diff]5ce betwixt ! :ole & its "p is greate/4 ,= let ,a ,b & ,a ,b be two equal quantities1 f :i* let two unequal p>ts be tak51 namely1 ,a ,e ! less1 & ,a ,f ! great]2 & betwixt ,a ,b & ,a ,e let ,a ,g be a mean 9 geometrical proportion1 & ,a ,h a mean 9 ari?metical proportion4 ,also betwixt ,a ,b & ,a ,f let ,a ,i be a mean 9 geometrical proportion1 & ,a ,k a mean 9 ari?metical proportion2 ,i say ,h ,g is great] ?an ,k ,i4 ,a ,e ,g ,h ,b ------------- ------------- ,a ,f ,i ,k ,b ,= 9 ! fir/ place we h ?is ,a ,b4 ,a ,g 33 ,b ,g4 ,g ,e1 by analogism article #ah4 ,!n by composition we h ?is ,a ,b ,a ,g4 ,a ,b 33 ,b ,g ,g ,e t is1 ,b ,e4 ,b ,g4 ,& by tak+ ! halves ( ! ,a ,b ,a ,g4 ,a ,b 33 ,b ,g ,g ,e1 antec$5ts ?is ?ird t is1 ,b ,h4 ,b ,g4 ,& by conv]sion a f\r? ,a ,b4 ,a ,b ,a ,g 33 ,b ,g4 ,b ,h4 ,& by division ?is fif? ,a ,b - ,a ,g4 ,a ,b ,a ,g 33 ,h ,g4 ,b ,h4 ,& by d\bl+ ! fir/ antec$5t & ! fir/ consequ5t ,a ,b - ,a ,g4 ,a ,b ,a ,g 33 ,h ,g4 ,b ,h4 ,also by ! same me?od may be f\nd out ?is analogism ,a ,b - ,a ,i4 ,a ,b ,,ai 33 ,k ,i4 ,b ,k4 ,n[ see+ ! proportion ( ,a ,b to ,a ,e is great] ?an t ( ,a ,b to ,a ,f1 ! proportion ( ,a ,b to ,a ,g1 :i* is half ! great] proportion1 is great] ?an ! proportion ( ,a ,b to ,a ,i ! half ( ! less proportion2 & !re=e ,a ,i is great] ?an ,a ,g4 ,:]e=e ! proportion ( ,a ,b - ,a ,g to ,a ,b ,a ,g1 by ! prec$5t lemma1 w be great] ?an ! proportion ( ,a ,b - ,a ,i to ,a ,b ,a ,i2 & !re=e also ! proportion ( ,h ,g to ,b ,h w be great] ?an t ( ,k ,i to ,b ,k1 & mu* great] ?an ! proportion ( ,k ,i to ,b ,h1 :i* is great] ?an ,b ,k2 = ,b ,h is ! half ( ,b ,e1 z ,b ,k is ! half ( ,b ,f1 :i*1 by supposition1 is less ?an ,b ,e4 ,:]e=e ,h ,g is great] ?an ,k ,i2 :i* 0 to be prov$4 ,coroll4 ,x is manife/ f h5ce1 t if any quantity be suppos$ to be divid$ into equal p>ts inf9ite 9 numb]1 ! diff]5ce betwe5 ! ari?metical & geometrical means w be inf9itely little1 t is1 none at all4 ,& ^u ?is f\ndation1 *iefly1 ! art ( mak+ ^? numb]s1 :i* are call$ ,log>i?ms1 seems to h be5 built4 #bi4 ,if any numb] ( quantities be prop\nd$1 :e!r !y be unequal1 or equal to "o ano!r2 & !re be ano!r quantity1 :i* multipli$ by ! numb] ( ! prop\nd$ quantities1 is equal to !m all2 t o!r quantity is a mean 9 ari?metical proportion to all ^? prop\nd$ quantities4 ------- ,,*ap4 ,,xiv4 ,,( ,,/rait ,,& ,,crook$1 ,,angle ,,& ,,figure4 #a4 ,! def9ition & prop]ties ( a /rait l9e4--#b4 ,! def9ition & prop]ties ( a plane sup]ficies4--#c4 ,sev]al sorts ( crook$ l9es4--#d4 ,! def9ition & prop]ties ( a circul> l9e4--#e4 ,! prop]ties ( a /rait l9e tak5 9 a plane4--#f44 ,! def9ition ( tang5t l9es4--#g4 ,! def9ition ( an angle1 & ! k9ds !re(4--#h4 ,9 conc5tric circles1 ar*es ( ! same angle are to "o ano!r1 z ! :ole circumf]5ces are4--#i4 ,! quantity ( an angle1 9 :at x consi/s4 --#aj4 ,! di/9ction ( angles1 simply s call$4--#aa4 ,( /rait l9es f ! c5tre ( a circle to a tang5t ( ! same4 --#ab4 ,! g5]al def9ition ( p>allels1 & ! prop]ties ( /rait p>allels4--#ac4 ,! circumf]5ces ( circles are to "o ano!r1 z _! diamet]s are4--#ad4 ,9 triangles1 /rait l9es p>allel to ! bases are to "o ano!r1 z ! p>ts ( ! sides :i* !y cut (f f ! v]tex4--#ae4 ,by :at fraction ( a /rait l9e ! circumf]5ce ( a circle is made4--#af4 ,t an angle ( cont+5ce is quantity1 b ( a diff]5t k9d f t ( an angle simply s call$2 & t x c nei!r add nor take away any ?+ f ! same4--#ag4 ,t ! incl9ation ( planes is angle simply s call$4--#ah4 ,a solid angle :at x is4--#ai4 ,:at is ! nature ( asymptotes4--#bj4 ,situation1 by :at x is det]m9$4--#ba4 ,:at is l situation2 :at is figure2 & :at are l figures4 ,sid5ote3 ,! def9ition end prop]ties ( a /rait l9e4 #a4 ,betwe5 two po9ts giv51 ! %orte/ l9e is t1 ^: extreme po9ts _c be drawn f>!r asund] )\t alt]+ ! quantity1 t is1 )\t alt]+ ! proportion ( t l9e to any o!r l9e giv54 ,= ! magnitude ( a l9e is comput$ by ! greate/ di/ance :i* may be betwe5 its extreme po9ts2 s t any "o l9e1 :e!r x be ext5d$ or b[$1 has always "o & ! same l5g?1 because x c h b "o greate/ di/ance betwe5 its extreme po9ts4 ,& see+ ! action1 by :i* a /rait l9e is made crook$1 or contr>ily a crook$ l9e is made /rait1 is no?+ b ! br++ ( its extreme po9ts ne>] to "o ano!r1 or ! sett+ ( !m fur!r asund]1 a crook$ l9e may rightly be def9$ to be t1 ^: extreme po9ts may be und]/ood to be drawn fur!r asund]2 & a /rait l9e to be t1 ^: extreme po9ts _c be drawn fur!r asund]2 & comp>atively1 a m crook$1 to be t l9e ^: extreme po9ts are ne>] to "o ano!r ?an ^? ( ! o!r1 suppos+ bo? ! l9es to be ( equal l5g?4 ,n[1 h[soev] a l9e be b[$1 x makes always a s9us or cavity1 sometimes on "o side1 sometimes on ano!r2 s t ! same crook$ l9e may ei!r h its :ole cavity on "o side only1 or x may h x "p on "o side & "p on ! o!r side4 ,:i* be+ well und]/ood1 x w be easy to und]/& ! foll[+ comp>isons ( /rait & crook$ l9es4 ,fir/1 if a /rait & a crook$ l9e h _! extreme po9ts common1 ! crook$ l9e is long] ?an ! /rait l9e4 ,= if ! extreme po9ts ( ! crook$ l9e be drawn out to _! greate/ di/ance1 x w be made a /rait l9e1 ( :i* t1 :i* 0 a /rait l9e f ! beg9n+1 w be b a "p2 & !re=e ! /rait l9e 0 %ort] ?an ! crook$ l9e1 :i* _h ! same extreme po9ts4 ,& = ! same reason1 if two crook$ l9es h _! extreme po9ts common1 & bo? ( !m h all _! cavity on "o & ! same side1 ! out]mo/ ( ! two w be ! longe/ l9e4 ,secondly1 a /rait l9e & a p]petually crook$ l9e _c be co9cid5t1 no1 n 9 ! lea/ "p4 ,= if !y %\ld1 !n n only "s /rait l9e w\ld h its extreme po9ts common ) "s crook$ l9e1 b also !y w\ld1 by reason ( _! co9cid5ce1 be equal to "o ano!r2 :i*1 z ,i h newly %[n1 _c be4 ,?irdly1 betwe5 two po9ts giv51 !re c be und]/ood b "o /rait l9e2 because !re _c be m ?an "o lea/ int]val or l5g? betwe5 ! same po9ts4 ,= if !re may be two1 !y w ei!r be co9cid5t1 & s bo? ( !m w be "o /rait l9e2 or if !y be n co9cid5t1 !n ! application ( "o to ! o!r by ext5sion w make ! ext5d$ l9e h its extreme po9ts at great] di/ance ?an ! o!r2 & consequ5tly1 x 0 crook$ f ! beg9n+4 ,f\r?ly1 f ?is la/ x foll[s1 t two /rait l9es _c include a sup]ficies4 ,= if !y h bo? _! extreme po9ts common1 !y are co9cid5t2 & if !y h b "o or nei!r ( !m common1 !n at "o or bo? ends ! extreme po9ts w be disjo9$1 & include no sup]ficies1 b leave all op5 & undet]m9$4 ,fif?ly1 e "p ( a /rait l9e is a /rait l9e4 ,= see+ e "p ( a /rait l9e is ! lea/ t c be drawn betwe5 its own extreme po9ts1 if all ! p>ts %\ld n con/itute a /rait l9e1 !y w\ld altoge!r be long] ?an ! :ole l9e4 ,sid5ote3 ,! def9ition & prop]ties ( a plane sup]ficies4 #b4 ,a plane or a plane sup]ficies1 is t :i* is describ$ by a /rait l9e s mov$1 t all ! sev]al po9ts !re( describe sev]al /rait l9es4 ,a /rait l9e1 !re=e1 is necess>ily all ( x 9 ! same plane :i* x describes4 ,also ! /rait l9es1 :i* are made by ! po9ts t describe a plane1 are all ( !m 9 ! same plane4 ,moreov]1 if any l9e :atsoev] be mov$ 9 a plane1 ! l9es1 :i* are describ$ by x1 are all ( !m 9 ! same plane4 ,all o!r sup]ficies1 :i* are n plane1 are crook$1 t is1 are ei!r concave or convex4 ,& ! same comp>isons1 :i* 7 made ( /rait & crook$ l9es1 may also be made ( plane & crook$ sup]ficies4 ,=1 fir/1 if a plane & crook$ sup]ficies be t]m9at$ ) ! same l9es1 ! crook$ sup]ficies is great] ?an ! plane sup]ficies4 ,= if ! l9es1 ( :i* ! crook$ sup]ficies consi/s1 be ext5d$1 !y w be f\nd to be long] ?an ^? ( :i* ! plane sup]ficies consi/s1 :i* _c be ext5d$1 because !y are /rait4 ,secondly1 two sup]ficies1 :]e( ! "o is plane1 & ! o!r cont9ually crook$1 _c be co9cid5t1 no1 n 9 ! lea/ "p4 ,= if !y 7 co9cid5t1 !y w\ld be equal2 nay1 ! same sup]ficies w\ld be bo? plane & crook$1 :i* is impossible4 ,?irdly1 )9 ! same t]m9at+ l9es !re c be no m ?an "o plane sup]ficies2 because !re c be b "o lea/ sup]ficies )9 ! same4 ,f\r?ly1 no numb] ( plane sup]ficies c include a solid1 unless m ?an two ( !m end 9 a common v]tex4 ,= if two planes h bo? ! same t]m9at+ l9es1 !y are co9cid5t1 t is1 !y are b "o sup]ficies2 & if _! t]m9at+ l9es be n ! same1 !y leave "o or m sides op54 ,fif?ly1 e "p ( a plane sup]ficies is a plane sup]ficies4 ,= see+ ! :ole plane sup]ficies is ! lea/ ( all ^?1 t h ! same t]m9at+ l9es2 & also e "p ( ! same sup]ficies is ! lea/ ( all ^?1 t are t]m9at$ ) ! same l9es2 if e "p %\ld n con/itute a plane sup]ficies1 all ! p>ts put toge!r w\ld n be equal to ! :ole4 ,sid5ote3 ,sev]al sorts ( crook$ l9es4 #c4 ,( /raitness1 :e!r x be 9 l9es or 9 sup]ficies1 !re is b "o k9d2 b ( crook$ness !re are _m k9ds2 = ( crook$ magnitudes1 "s are congru\s1 t is1 are co9cid5t :5 !y are appli$ to "o o!r2 o!rs are incongru\s4 ,aga91 "s are or uni=m1 t is1 h _! p>ts1 h[soev] tak51 congru\s to "o ano!r2 o!rs are or ( sev]al =ms4 ,moreov]1 ( su* z are crook$1 "s are cont9ually crook$1 o!rs h p>ts :i* are n crook$4 ,sid5ote3 ,def9ition & prop]ties ( a circul> l9e4 #d4 ,if a /rait l9e be mov$ 9 a plane1 9 su* mann]1 t :ile "o end ( x /&s /ill1 ! :ole l9e be c>ri$ r\nd ab\t till x come aga9 into ! same place f :5ce x 0 fir/ mov$1 x w describe a plane sup]ficies1 :i* w be t]m9at$ e way by t crook$ l9e1 :i* is made by t end ( ! /rait l9e :i* 0 c>ri$ r\nd4 ,n[ ?is sup]ficies is call$ a ,,circle2 & ( ?is circle1 ! unmov$ po9t is ! c5tre2 ! crook$ l9e :i* t]m9ates x1 ! p]imet]2 & e "p ( t crook$ l9e1 a circumf]5ce or ar*2 ! /rait l9e1 :i* g5]at$ ! circle1 is ! semidiamet] or radius2 & any /rait l9e1 :i* passe? ?r\gh ! c5tre & is t]m9at$ on bo? sides 9 ! circumf]5ce1 is call$ ! diamet]4 ,moreov]1 e po9t ( ! radius1 :i* describes ! circle1 describes 9 ! same "t its own p]imet]1 t]m9at+ its own circle1 :i* is said to be conc5tric to all ! o!r circles1 because ?is & all ^? h "o common c5tre4 ,:]e=e 9 e circle1 all /rait l9es f ! c5tre to ! circumf]5ce are equal4 ,= !y are all co9cid5t ) ! radius :i* g5]ates ! circle4 ,also ! diamet] divides bo? ! p]imet] & ! circle itself into two equal p>ts4 ,= if ^? two p>ts be appli$ to "o ano!r1 & ! semip]imet]s be co9cid5t1 !n1 see+ !y h "o common diamet]1 !y w be equal2 & ! semicircles w be equal also2 = ^! also w be co9cid5t4 ,b if ! semip]imet]s be n co9cid5t1 !n "s "o /rait l9e1 :i* passes ?r\gh ! c5tre1 :i* c5tre is 9 ! diamet]1 w be cut by !m 9 two po9ts4 ,:]e=e1 see+ all ! /rait l9es f ! c5tre to ! circumf]5ce are equal1 a "p ( ! same /rait l9e w be equal to ! :ole2 :i* is impossible4 ,= ! same reason ! p]imet] ( a circle w be uni=m1 t is1 any "o "p ( x w be co9cid5t ) any o!r equal "p ( ! same4 ,sid5ote3 ,! prop]ties ( a /rait l9e tak5 9 a plane4 #e4 ,f h5ce may be collect$ ?is prop]ty ( a /rait l9e1 namely1 t x is all conta9$ 9 t plane :i* conta9s bo? its extreme po9ts4 ,= see+ bo? its extreme po9ts are 9 ! plane1 t /rait l9e1 :i* describes ! plane1 w pass ?r\gh !m bo?2 & if "o ( !m be made a c5tre1 & at ! di/ance betwe5 bo? a circumf]5ce be describ$1 ^: radius is ! /rait l9e :i* describes ! plane1 t circumf]5ce w pass ?r\gh ! o!r po9t4 ,:]e=e betwe5 ! two prop\nd$ po9ts1 !re is "o /rait l9e1 by ! def9ition ( a circle1 conta9$ :olly 9 ! prop\nd$ plane2 & !re=e if ano!r /rait l9e might be drawn betwe5 ! same po9ts1 & yet n be conta9$ 9 ! same plane1 x w\ld foll[1 t betwe5 two po9ts two /rait l9es may be drawn2 :i* has be5 demon/rat$ to be impossible4 ,x may also be collect$1 t if two planes cut "o ano!r1 _! common section w be a /rait l9e4 ,= ! two extreme po9ts ( ! int]section are 9 bo? ! int]sect+ planes2 & betwe5 ^? po9ts a /rait l9e may be drawn2 b a /rait l9e betwe5 any two po9ts is 9 ! same plane1 9 :i* ! po9ts are2 & see+ ^! are 9 bo? ! planes1 ! /rait l9e :i* connects !m w also be 9 bo? ! same planes1 & !re=e x is ! common section ( bo?4 ,& e o!r l9e1 t c be drawn betwe5 ^? po9ts1 w be ei!r co9cid5t ) t l9e1 t is1 x w be ! same l9e2 or x w n be co9cid5t1 & !n x w be 9 nei!r1 or b 9 "o ( ^? planes4 ,z a /rait l9e may be und]/ood to be mov$ r\nd ab\t :il/ "o end !re( rema9s fix$1 z ! c5tre2 s 9 l mann] x is easy to und]/&1 t a plane may be circumduc$ ab\t a /rait l9e1 :il/ ! /rait l9e rema9s /ill 9 "o & ! same place1 z ! axis ( t motion4 ,n[ f h5ce x is manife/1 t any ?ree po9ts are 9 "s "o plane4 ,= z any two po9ts1 if !y be connect$ by a /rait l9e1 are und]/ood to be 9 ! same plane 9 :i* ! /rait l9e is2 s1 if t plane be circumduc$ ab\t ! same /rait l9e1 x w 9 its revolution take 9 any ?ird po9t1 h[soev] x be situate2 & !n ! ?ree po9ts w be all 9 t plane2 & consequ5tly ! ?ree /rait l9es :i* connect ^? po9ts1 w also be 9 ! same plane4 ,sid5ote3 ,def9ition ( tang5t l9es4 #f4 ,two l9es are said to t\* "o ano!r1 :i* be+ bo? drawn to "o & ! same po9t1 w n cut "o ano!r1 ?\gh !y be produc$1 produc$1 ,i say1 9 ! same mann] 9 :i* !y 7 g5]at$4 ,& !re=e if two /rait l9es t\* "o ano!r 9 any "o po9t1 !y w be contigu\s ?r\gh _! :ole l5g?4 ,also two l9es cont9ually crook$ w d ! same1 if !y be congru\s & be appli$ to "o ano!r accord+ to _! congruity2 o!rwise1 if !y be incongru\sly appli$1 !y w1 z all o!r crook$ l9es1 t\* "o ano!r1 :]e !y t\*1 b 9 "o po9t only4 ,:i* is manife/ f ?is1 t !re c be no congruity betwe5 a /rait l9e & a l9e t is cont9ually crook$2 = o!rwise ! same l9e might be bo? /rait & crook$4 ,besides1 :5 a /rait l9e t\*es a crook$ l9e1 if ! /rait l9e be nev] s little mov$ ab\t ^u ! po9t ( contact1 x w cut ! crook$ l9e2 = see+ x t\*es x b 9 "o po9t1 if x incl9e any way1 x w d m ?an t\* x2 t is1 x w ei!r be congru\s to x1 or x w cut x2 b x _c be congru\s to x2 & !re=e x w cut x4 ,sid5ote3 ,! def9ition ( an angle1 & ! k9ds !re(4 #g4 ,an angle1 accord+ to ! mo/ g5]al acceptation ( ! ^w1 may be ?us def9$2 :5 two l9es1 or _m sup]ficies1 concur 9 "o sole po9t1 & div]ge e :]e else1 ! quantity ( t div]g5ce is an ,,angle4 ,& an angle is ( two sorts2 =1 fir/1 x may be made by ! concurr5ce ( l9es1 & !n x is a sup]ficial angle2 or by ! concurr5ce ( sup]ficies1 & !n x is call$ a solid angle4 ,aga91 f ! two ways by :i* two l9es may div]ge f "o ano!r1 sup]ficial angles are divid$ into two k9ds4 ,= two /rait l9es1 :i* are appli$ to "o ano!r1 & are contigu\s 9 _! :ole l5g?1 may be sep>at$ or pull$ op5 9 su* mann]1 t _! concurr5ce 9 "o po9t w /ill rema92 & ?is sep>ation or op5+ may be ei!r by circul> motion1 ! c5tre :]e( is _! po9t ( concurr5ce1 & ! l9es w /ill reta9 _! /raitness1 ! quantity ( :i* sep>ation or div]g5ce is an angle simply s call$2 or !y may be sep>at$ by cont9ual flexion or curvation 9 e imag9able po9t2 & ! quantity ( ?is sep>ation is t1 :i* is call$ an angle ( cont+5ce4 ,besides1 ( sup]ficial angles simply s call$1 ^?1 :i* are 9 a plane sup]ficies1 are plane2 & ^?1 :i* are n plane1 are d5om9at$ f ! sup]ficies 9 :i* !y are4 ,la/ly1 ^? are /rait-l9$ angles1 :i* are made by /rait l9es2 z ^? :i* are made by crook$ l9es are crook$-l9$2 & ^? :i* are made bo? ( /rait & crook$ l9es1 are mix$ angles4 ,sid5ote3 ,9 conc5tric circles1 ar*es ( ! same angle are to "o ano!r1 z ! :ole circumf]5ces are4 #h4 ,two ar*es int]cept$ betwe5 two radii ( conc5tric circles1 h ! same proportion to "o ano!r1 :i* _! :ole p]imet]s h to "o ano!r4 ,= let ! po9t ,a 79 ! fir/ figure7 be ! c5tre ( ! two circles ,b ,c ,d & ,e ,f ,g1 9 :i* ! radii ,a ,e ,b & ,a ,f ,c int]cept ! ar*es ,b ,c & ,e ,f2 ,i say ! proportion ( ! ar* ,b ,c to ! ar* ,e ,f is ! same ) t ( ! p]imet] ,b ,c ,d to ! p]imet] ,e ,f ,g4 ,= if ! radius ,a ,f ,c be und]/ood to be mov$ ab\t ! c5tre ,a ) circul> & uni=m motion1 t is1 ) equal swiftness ev]y:]e1 ! po9t ,c w 9 a c]ta9 "t describe ! p]imet] ,b ,c ,d1 & 9 a "p ( t "t ! ar* ,b ,c2 & because ! velocities are equal by :i* bo? ! ar* & ! :ole p]imet] are describ$1 ! proportion ( ! magnitude ( ! p]imet] ,b ,c ,d to ! magnitude ( ! ar* ,,bc is det]m9$ by no?+ b ! diff]5ce ( ! times 9 :i* ! p]imet] & ! ar* are describ$4 ,b bo? ! p]imet]s are describ$ 9 "o & ! same "t1 & bo? ! ar*es 9 "o & ! same "t2 & !re=e ! proportions ( ! p]imet] ,b ,c ,d to ! ar* ,b ,c1 & ( ! p]imet] ,e ,f ,g to ! ar* ,e ,f1 are bo? det]m9$ by ! same cause4 ,:]e=e ,b ,c ,d4 ,b ,c 33 ,e ,f ,g4 ,e ,f are proportionals 7by ! #f? art4 ( ! la/ *apt]71 & by p]mutation ,b ,c ,d4 ,e ,f ,g 33 ,b ,c4 ,e ,f w also be proportionals2 :i* 0 to be demon/rat$4 ,sid5ote3 ,! quantity ( an angle1 9 :at x consi/s4 #i4 ,no?+ is contribut$ t[>ds ! quantity ( an angle1 nei!r by ! l5g?1 nor by ! equality1 nor by ! inequality ( ! l9es :i* compreh5d x4 ,= ! l9es ,a ,b & ,a ,c compreh5d ! same angle :i* is compreh5d$ by ! l9es ,a ,e & ,a ,f1 or ,a ,b & ,a ,f4 ,nor is an angle ei!r increas$ or dim9i%$ by ! absolute quantity ( ! ar*1 :i* subt5ds ! same2 = bo? ! great] ar* ,b ,c & ! less] ar* ,e ,f are subt5d$ to ! same angle4 ,b ! quantity ( an angle is e/imat$ by ! quantity ( ! subt5d+ ar* comp>$ ) ! quantity ( ! :ole p]imet]4 ,& !re=e ! quantity ( an angle simply s call$ may be ?us def9$3 ! quantity ( an angle is an ar* or circumf]5ce ( a circle1 det]m9$ by its proportion to ! :ole p]imet]4 ,s t :5 an ar* is int]cept$ betwe5 two /rait l9es drawn f ! c5tre1 look h[ great a portion t ar* is ( ! :ole p]imet]1 s great is ! angle4 ,f :5ce x may be und]/ood1 t :5 ! l9es :i* conta9 an angle are /rait l9es1 ! quantity ( t angle may be tak5 at any di/ance f ! c5tre4 ,b if "o or bo? ( ! conta9+ l9es be crook$1 !n ! quantity ( ! angle is to be tak5 9 ! lea/ di/ance f ! c5tre1 or f _! concurr5ce2 = ! lea/ di/ance is to be consid]$ z a /rait l9e1 see+ no crook$ l9e c be imag9$ s little1 b t !re may be a less /rait l9e4 ,& al?\gh ! lea/ /rait l9e _c be giv51 because ! lea/ giv5 l9e may /ill be divid$1 yet we may come to a "p s small1 z is n at all consid]able2 :i* we call a po9t4 ,& ?is po9t may be und]/ood to be 9 a /rait l9e :i* t\*es a crook$ l9e2 = an angle is g5]at$ by sep>at+1 by circul> motion1 "o /rait l9e f ano!r :i* t\*es x1 z has be5 said above 9 ! #g? article4 ,:]e=e an angle1 :i* two crook$ l9es make1 is ! same ) t :i* is made by two /rait l9es :i* t\* !m4 ,sid5ote3 ,! di/9ction ( angles1 simply s call$4 #aj4 ,f h5ce x foll[s1 t v]tical angles1 su* z are ,a ,b ,c1 ,d ,b ,f 9 ! second figure1 are equal to "o ano!r4 ,= if1 f ! two semip]imet]s ,d ,a ,c1 ,f ,d ,a1 :i* are equal to "o ano!r1 ! common ar* ,d ,a be tak5 away1 ! rema9+ ar*es ,a ,c1 ,d ,f w be equal to "o ano!r4 ,ano!r di/9ction ( angles is into "r & oblique4 ,a "r angle is t1 ^: quantity is ! f\r? "p ( ! p]imet]4 ,& ! l9es1 :i* make a "r angle1 are said to be p]p5dicul> to "o ano!r4 ,also1 ( oblique angles1 t :i* is great] ?an a "r1 is call$ an obtuse angle2 & t :i* is less1 an acute angle4 ,f :5ce x foll[s1 t all ! angles t c possibly be made at "o & ! same po9t1 toge!r tak51 are equal to f\r "r angles2 because ! quantities ( !m all put toge!r make ! :ole p]imet]4 ,also1 t all ! angles1 :i* are made on "o side ( a /rait l9e1 f any "o po9t tak5 9 ! same1 are equal to two "r angles2 = if t po9t be made ! c5tre1 t /rait l9e w be ! diamet] ( a circle1 by ^: circumf]5ce ! quantity ( an angle is det]m9$2 & t diamet] w divide ! p]imet] into two equal p>ts4 ,sid5ote3 ,( /rait l9es f ! c5tre ( a circle to a tang5t ( ! same4 #aa4 ,if a tang5t be made ! diamet] ( a circle1 ^: c5tre is ! po9t ( contact1 a /rait l9e drawn f ! c5tre ( ! =m] circle to ! c5tre ( ! latt] circle1 w make two angles ) ! tang5t1 t is1 ) ! diamet] ( ! latt] circle1 equal to two "r angles1 by ! la/ article4 ,& because1 by ! #f? article1 ! tang5t has on bo? sides equal incl9ation to ! circle1 ea* ( !m w be a "r angle2 z also ! semidiamet] w be p]p5dicul> to ! same tang5t4 ,moreov]1 ! semidiamet]1 inasmu* z x is ! semidiamet]1 is ! lea/ /rait l9e :i* c be drawn f ! c5tre to ! tang5t2 & e o!r /rait l9e1 t rea*es ! tang5t1 w pass out ( ! circle1 & w !re=e be great] ?an ! semidiamet]4 ,9 l mann]1 ( all ! /rait l9es1 :i* may be drawn f ! c5tre to ! tang5t1 t is ! greate/ :i* makes ! greate/ angle ) ! p]p5dicul>2 :i* w be manife/1 if ab\t ! same c5tre ano!r circle be describ$1 ^: semidiamet] is a /rait l9e tak5 ne>] to ! p]p5dicul>1 & !re be drawn a p]p5dicul>1 t is1 a tang5t1 to ! same4 ,f :5ce x is also manife/1 t if two /rait l9es1 :i* make equal angles on ei!r side ( ! p]p5dicul>1 be produc$ to ! tang5t1 !y w be equal4 ,sid5ote3 ,! g5]al def9ition ( p>allels2 ! prop]ties ( /rait p>allels4 #ab4 ,!re is 9 ,euclid a def9ition ( /rait-l9$ p>allels2 b ,i d n f9d t p>allels 9 g5]al are any:]e def9$2 & !re=e = an univ]sal def9ition ( !m1 ,i say t any two l9es :atsoev]1 /rait or crook$1 z also any two sup]ficies1 are ,,p>allel2 :5 two equal /rait l9es1 :]esoev] !y fall ^u !m1 make always equal angles ) ea* ( !m4 ,f :i* def9ition x foll[s2 fir/1 t any two /rait l9es1 n incl9$ opposite ways1 fall+ ^u two o!r /rait l9es1 :i* are p>allel1 & int]cept+ equal p>ts 9 bo? ( !m1 are !mselves also equal & p>allel4 ,z if ,a ,b & ,c ,d 79 ! ?ird figure71 incl9$ bo? ! same way1 fall ^u ! p>allels ,a ,c & ,b ,d1 & ,a ,c & ,b ,d be equal1 ,a ,b & ,c ,d w also be equal & p>allel4 ,= ! p]p5dicul>s ,b ,e & ,d ,f be+ drawn1 ! "r angles ,e ,b ,d & ,f ,d ,h w be equal4 ,:]e=e1 see+ ,e ,f & ,b ,d are p>allel1 ! angles ,e ,b ,a & ,f ,d ,c w be equal4 ,n[ if ,d ,c be n equal to ,b ,a1 let any o!r /rait l9e equal to ,b ,a be drawn f ! po9t ,d2 :i*1 see+ x _c fall ^u ! po9t ,c1 let x fall ^u ,g4 ,:]e=e ,a ,g w be ei!r great] or less ?an ,b ,d2 & !re=e ! angles ,e ,b ,a & ,f ,d ,c are n equal1 z 0 suppos$4 ,:]e=e ,a ,b & ,c ,d are equal2 :i* is ! fir/4 ,aga91 because !y make equal angles ) ! p]p5dicul>s ,b ,e & ,d ,f2 !re=e ! angle ,c ,d ,h w be equal to ! angle ,a ,b ,d1 &1 by ! def9ition ( p>allels1 ,a ,b & ,c ,d w be p>allel2 :i* is ! second4 ,t plane1 :i* is includ$ bo? ways )9 p>allel l9es1 is call$ a ,,p>allelogram4 ,coroll4 ,i4 ,f ?is la/ x foll[s1 t ! angles ,a ,b ,d & ,c ,d ,h are equal1 t is1 t a /rait l9e1 z ,b ,h1 fall+ ^u two p>allels1 z ,a ,b & ,c ,d1 makes ! int]nal angle ,a ,b ,d equal to ! ext]nal & opposite angle ,c ,d ,h4 ,coroll4 ,,ii4 ,& f h5ce aga9 x foll[s1 t a /rait l9e fall+ ^u two p>allels1 makes ! alt]nate angles equal1 t is1 ! angle ,a ,g ,f1 9 ! f\r? figure1 equal to ! angle ,g ,f ,d4 ,= see+ ,g ,f ,d is equal to ! ext]nal opposite angle ,e ,g ,b1 x w be also equal to its v]tical angle ,a ,g ,f1 :i* is alt]nate to ,g ,f ,d4 ,coroll4 ,,iii4 ,t ! int]nal angles on ! same side ( ! l9e ,f ,g are equal to two "r angles4 ,= ! angles at ,f1 namely1 ,g ,f ,c & ,g ,f ,d1 are equal to two "r angles4 ,b ,g ,f ,d is equal to its alt]nate angle ,a ,g ,f4 ,:]e=e bo? ! angles ,g ,f ,c & ,a ,g ,f1 :i* are int]nal on ! same side ( ! l9e ,f ,g1 are equal to two "r angles4 ,coroll4 ,,iv4 ,t ! ?ree angles ( a /rait-l9$ pla9 triangle are equal to two "r angles2 & any side be+ produc$1 ! ext]nal angle w be equal to ! two opposite int]nal angles4 ,= if !re be drawn by ! v]tex ( ! pla9 triangle ,a ,b ,c 7fig4 #e7 a p>allel to any ( ! sides1 z to ,a ,b1 ! angles ,a & ,b w be equal to _! alt]nate angles ,e & ,f1 & ! angle ,c is common4 ,b1 by ! #aj? article1 ! ?ree angles ,e1 ,c & ,f1 are equal to two "r angles2 & !re=e ! ?ree angles ( ! triangle are equal to ! same2 :i* is ! fir/4 ,aga91 ! two angles ,b & ,d are equal to two "r angles1 by ! #aj? article4 ,:]e=e tak+ away ,b1 !re w rema9 ! angles ,a & ,c1 equal to ! angle ,d2 :i* is ! second4 ,coroll4 ,v4 ,if ! angles ,a & ,b be equal1 ! sides ,a ,c & ,c ,b w also be equal1 because ,a ,b & ,e ,f are p>allel2 &1 on ! contr>y1 if ! sides ,a ,c & ,c ,b be equal1 ! angles ,a & ,b w also be equal4 ,= if !y be n equal1 let ! angles ,b & ,g be equal4 ,:]e=e1 see+ ,g ,b & ,e ,f are p>allels1 & ! angles ,g & ,b equal1 ! sides ,g ,c & ,c ,b w also be equal2 & because ,c ,b & ,a ,c are equal by supposition1 ,c ,g & ,c ,a w also be equal2 :i* _c be1 by ! #aa? article4 ,coroll4 ,,vi4 ,f h5ce x is manife/1 t if two radii ( a circle be connect$ by a /rait l9e1 ! angles !y make ) t connect+ l9e w be equal to "o ano!r2 & if !re be add$ t segm5t ( ! circle1 :i* is subt5d$ by ! same l9e :i* connects ! radii1 !n ! angles1 :i* ^? radii make ) ! circumf]5ce1 w also be equal to "o ano!r4 ,= a /rait l9e1 :i* subt5ds any ar*1 makes equal angles ) ! same2 because1 if ! ar* & ! subt5se be divid$ 9 ! middle1 ! two halves ( ! segm5t w be congru\s to "o ano!r1 by reason ( ! uni=mity bo? ( ! circumf]5ce ( ! circle1 & ( ! /rait l9e4 ,sid5ote3 ,! circumf]5ces ( circles are to "o ano!r z _! diamet]s are4 #ac4 ,p]imet]s ( circles are to "o ano!r1 z _! semidiamet]s are4 ,= let !re be any two circles1 z1 9 ! fir/ figure1 ,b ,c ,d ! great]1 & ,e ,f ,g ! less]1 hav+ _! common c5tre at ,a2 & let _! semidiamet]s be ,a ,c & ,a ,e4 ,i say1 ,a ,c has ! same proportion to ,a ,e1 :i* ! p]imet] ,b ,c ,d has to ! p]imet] ,e ,f ,g4 ,= ! magnitude ( ! semidiamet]s ,a ,c & ,a ,e is det]m9$ by ! di/ance ( ! po9ts ,c & ,e f ! c5tre ,a2 & ! same di/ances are acquir$ by ! uni=m motion ( a po9t f ,a to ,c1 9 su* mann]1 t 9 equal times ! di/ances acquir$ be equal4 ,b ! p]imet]s ,b ,c ,d & ,e ,f ,g are also det]m9$ by ! same di/ances ( ! po9ts ,c & ,e f ! c5tre ,a2 & !re=e ! p]imet]s ,b ,c ,d & ,e ,f ,g1 z well z ! semidiamet]s ,a ,c & ,a ,e1 h _! magnitudes det]m9$ by ! same cause1 :i* cause makes1 9 equal times1 equal spaces4 ,:]e=e1 by ! #ac? *apt] & #f? article1 ! p]imet]s ( circles & _! semidiamet]s are proportionals2 :i* 0 to be prov$4 ,sid5ote3 ,9 triangles /rait l9es p>allel to ! bases are to "o ano!r1 z ! p>ts ( ! sides :i* !y cut (f f ! v]tex4 #ad4 ,if two /rait l9es1 :i* con/itute an angle1 be cut by /rait-l9$ p>allels1 ! int]cept$ p>allels w be to "o ano!r1 z ! p>ts :i* !y cut (f f ! v]tex4 ,let ! /rait l9es ,a ,b & ,a ,c1 9 ! #f? figure1 make an angle at ,a1 & be cut by ! two /rait-l9$ p>allels ,b ,c & ,d ,e1 s t ! p>ts cut (f f ! v]tex 9 ei!r ( ^? l9es1 z 9 ,a ,b1 may be ,a ,b & ,a ,d4 ,i say1 ! p>allels ,b ,c & ,d ,e are to "o ano!r1 z ! p>ts ,a ,b & ,a ,d4 ,= let ,a ,b be divid$ into any numb] ( equal p>ts1 z into ,a ,f1 ,f ,d1 ,d ,b2 & by ! po9ts ,f & ,d1 let ,f ,g & ,d ,e be drawn p>allel to ! base ,b ,c1 & cut ,a ,c 9 ,g & ,e2 & aga91 by ! po9ts ,g & ,e1 let o!r /rait l9es be drawn p>allel to ,a ,b1 & cut ,b ,c 9 ,h & ,i4 ,if n[ ! po9t ,a be und]/ood to be mov$ uni=mly ov] ,a ,b1 & 9 ! same "t ,b be mov$ to ,c1 & all ! po9ts ,f1 ,d1 & ,b be mov$ uni=mly & ) equal swiftness ov] ,f ,g1 ,d ,e1 & ,b ,c2 !n %all ,b pass ov] ,b ,h1 equal to ,f ,g1 9 ! same "t t ,a passes ov] ,a ,f2 & ,a ,f & ,f ,g w be to "o ano!r1 z _! velocities are2 & :5 ,a is 9 ,f1 ,d w be 9 ,k2 :5 ,a is 9 ,d1 ,d w be 9 ,e2 & 9 :at mann] ! po9t ,a passes by ! po9ts ,f1 ,d1 & ,b1 9 ! same mann] ! po9t ,b w pass by ! po9ts ,h1 ,i1 & ,c2 & ! /rait l9es ,f ,g1 ,d ,k1 ,k ,e1 ,b ,h1 ,h ,i1 & ,i ,c1 are equal1 by reason ( _! p>allelism2 & !re=e1 z ! velocity 9 ,a ,b is to ! velocity 9 ,b ,c1 s is ,a ,d to ,d ,e2 b z ! velocity 9 ,a ,b is to ! velocity 9 ,b ,c1 s is ,a ,b to ,b ,c2 t is to say1 all ! p>allels w be sev]ally to all ! p>ts cut (f f ! v]tex1 z ,a ,f is to ,f ,g4 ,:]e=e1 ,a ,f4 ,g ,f 33 ,a ,d4 ,d ,e 33 ,a ,b4 ,b ,c are proportionals4 ,! subt5ses ( equal angles 9 diff]5t circles1 z ! /rait l9es ,b ,c & ,f ,e 79 fig4 #a71 are to "o ano!r z ! ar*es :i* !y subt5d4 ,= 7by art4 #h7 ! ar*es ( equal angles are to "o ano!r z _! p]imet]s are2 & 7by art4 #ac7 ! p]imet]s z _! semidiamet]s2 b ! subt5ses ,b ,c & ,f ,e are p>allel to "o ano!r by reason ( ! equality ( ! angles :i* !y make ) ! semidiamet]s2 & !re=e ! same subt5ses1 by ! la/ prec$5t article1 w be proportional to ! semidiamet]s1 t is1 to ! p]imet]s1 t is1 to ! ar*es :i* !y subt5d4 ,sid5ote3 ,by :at fraction ( a /rait l9e ! circumf]5ce ( a circle is made4 #ae4 ,if 9 a circle any numb] ( equal subt5ses be plac$ imm$iately aft] "o ano!r1 & /rait l9es be drawn f ! extreme po9t ( ! fir/ subt5se to ! extreme po9ts ( all ! re/1 ! fir/ subt5se be+ produc$ w make ) ! second subt5se an ext]nal angle d\ble to t1 :i* is made by ! same fir/ subt5se1 & a tang5t to ! circle t\*+ x 9 ! extreme po9ts !re(2 & if a /rait l9e :i* subt5ds two ( ^? ar*es be produc$1 x w make an ext]nal angle ) ! ?ird subt5se1 triple to ! angle :i* is made by ! tang5t ) ! fir/ subt5se2 & s cont9ually4 ,= ) ! radius ,a ,b 79 fig4 #g7 let a circle be describ$1 & 9 x let any numb] ( equal subt5ses1 ,b ,c1 ,c ,d1 & ,d ,e1 be plac$2 also let ,b ,d & ,b ,e be drawn2 & by produc+ ,b ,c1 ,b ,d & ,b ,e to any di/ance 9 ,g1 ,h & ,i1 let !m make angles ) ! subt5ses :i* succe$ "o ano!r1 namely1 ! ext]nal angles ,g ,c ,d1 & ,h ,d ,e4 ,la/ly1 let ! tang5t ,k ,b be drawn1 mak+ ) ! fir/ subt5se ! angle ,k ,b ,c4 ,i say ! angle ,g ,c ,d is d\ble to ! angle ,k ,b ,c1 & ! angle ,h ,d ,e triple to ! same angle ,k ,b ,c4 ,= if ,a ,c be drawn cutt+ ,b ,d 9 ,m1 & f ! po9t ,c !re be drawn ,l ,c p]p5dicul> to ! same ,a ,c1 !n ,c ,l & ,m ,d w be p>allel1 by reason ( ! "r angles at ,c & ,m2 & !re=e ! alt]ne angles ,l ,c ,d & ,b ,d ,c w be equal3 z also ! angles ,b ,d ,c & ,c ,b ,d w be equal1 because ( ! equality ( ! /rait l9es ,b ,c & ,c ,d4 ,:]e=e ! angle ,g ,c ,d is d\ble to ei!r ( ! angles ,c ,b ,d or ,c ,d ,b2 & !re=e also ! angle ,g ,c ,d is d\ble to ! angle ,l ,c ,d1 t is1 to ! angle ,k ,b ,c4 ,aga91 ,c ,d is p>allel to ,b ,e1 by reason ( ! equality ( ! angles ,c ,b ,e & ,d ,e ,b1 & ( ! /rait l9es ,c ,b & ,d ,e2 & !re=e ! angles ,g ,c ,d & ,g ,b ,e are equal2 & consequ5tly ,g ,b ,e1 z also ,d ,e ,b is d\ble to ! angle ,k ,b ,c4 ,b ! ext]nal angle ,h ,d ,e is equal to ! two int]nal ,d ,e ,b & ,d ,b ,e2 & !re=e ! angle ,h ,d ,e is triple to ! angle ,k ,b ,c1 c42 :i* 0 to be prov$4 ,coroll4 ,i4 ,f h5ce x is manife/1 t ! angles ,k ,b ,c & ,c ,b ,d1 z also1 t all ! angles t are compreh5d$ by two /rait l9es meet+ 9 ! circumf]5ce ( a circle & insi/+ ^u equal ar*es1 are equal to "o ano!r4 ,coroll4 ,,ii4 ,if ! tang5t ,b ,k be mov$ 9 ! circumf]5ce ) uni=m motion ab\t ! c5tre ,b1 x w 9 equal times cut (f equal ar*es2 & w pass ov] ! :ole p]imet] 9 ! same "t 9 :i* itself describes a semip]imet] ab\t ! c5tre ,b4 ,coroll4 ,,iii4 ,f h5ce also we may und]/&1 :at x is t det]m9es ! b5d+ or curvation ( a /rait l9e into ! circumf]5ce ( a circle2 namely1 t x is fraction cont9ually increas+ 9 ! same mann]1 z numb]s1 f "o upw>ds1 increase by ! cont9ual addition ( unity4 ,= ! indef9ite /rait l9e ,k ,b be+ brok5 9 ,b accord+ to any angle1 z t ( ,k ,b ,c1 & aga9 9 ,c accord+ to a d\ble angle1 & 9 ,d accord+ to an angle :i* is triple1 & 9 ,e accord+ to an angle :i* is quadruple to ! fir/ angle1 & s cont9ually1 !re w be describ$ a figure :i* w inde$ be rectil9eal1 if ! brok5 p>ts be consid]$ z hav+ magnitude2 b if !y be und]/ood to be ! lea/ t c be1 t is1 z s _m po9ts1 !n ! figure describ$ w n be rectil9eal1 b a circle1 ^: circumf]5ce w be ! brok5 l9e4 ,coroll4 ,,iv4 ,f :at has be5 said 9 ?is pres5t article1 x may also be demon/rat$1 t an angle 9 ! c5tre is d\ble to an angle 9 ! circumf]5ce ( ! same circle1 if ! int]cept$ ar*es be equal4 ,= see+ t /rait l9e1 by ^: motion an angle is det]m9$1 passes ov] equal ar*es 9 equal times1 z well f ! c5tre z f ! circumf]5ce2 & :ile t1 :i* is f ! circumf]5ce1 is pass+ ov] half its own p]imet]1 x passes 9 ! same "t ov] ! :ole p]imet] ( t :i* is f ! c5tre1 ! ar*es1 :i* x cuts (f 9 ! p]imet] ^: c5tre is ,a1 w be d\ble to ^?1 :i* x makes 9 its own semip]imet]1 ^: c5tre is ,b4 ,b 9 equal circles1 z ar*es are to "o ano!r1 s also are angles4 ,x may also be demon/rat$1 t ! ext]nal angle made by a subt5se produc$ & ! next equal subt5se is equal to an angle f ! c5tre insi/+ ^u ! same ar*2 z 9 ! la/ diagram1 ! angle ,g ,c ,d is equal to ! angle ,c ,a ,d2 = ! ext]nal angle ,g ,c ,d is d\ble to ! angle ,c ,b ,d2 & ! angle ,c ,a ,d insi/+ ^u ! same ar* ,c ,d is also d\ble to ! same angle ,c ,b ,d or ,k ,b ,c4 ,sid5ote3 ,t an angle ( cont+5ce is quantity1 b ( a diff]5t k9d f t ( an angle simply s call$2 & t x c nei!r add nor take away any?+ f ! same4 #af4 ,an angle ( cont+5ce1 if x be comp>$ ) an angle simply s call$1 h[ little soev]1 has su* proportion to x z a po9t has to a l9e2 t is1 no proportion at all1 nor any quantity4 ,= fir/1 an angle ( cont+5ce is made by cont9ual flexion2 s t 9 ! g5]ation ( x !re is no circul> motion at all1 9 :i* consi/s ! nature ( an angle simply s call$2 & !re=e x _c be comp>$ ) x accord+ to quantity4 ,secondly1 see+ ! ext]nal angle made by a subt5se produc$ & ! next subt5se is equal to an angle f ! c5tre insi/+ ^u ! same ar*1 z 9 ! la/ figure ! angle ,g ,c ,d is equal to ! angle ,c ,a ,d1 ! angle ( cont+5ce w be equal to t angle f ! c5tre1 :i* is made by ,a ,b & ! same ,a ,b2 = no "p ( a tang5t c subt5d any ar*2 b z ! po9t ( contact is to be tak5 = ! subt5se1 s ! angle ( cont+5ce is to be acc\nt$ = ! ext]nal angle1 & equal to t angle ^: ar* is ! same po9t ,b4 ,n[1 see+ an angle 9 g5]al is def9$ to be ! op5+ or div]g5ce ( two l9es1 :i* concur 9 "o sole po9t2 & see+ "o op5+ is great] ?an ano!r1 x _c be d5i$1 b t by ! v g5]ation ( x1 an angle ( cont+5ce is quantity2 = :]esoev] !re is great] & less1 !re is also quantity2 b ?is quantity consi/s 9 great] & less flexion2 = h[ mu* ! great] a circle is1 s mu* ! ne>] comes ! circumf]5ce ( x to ! nature ( a /rait l9e2 = ! circumf]5ce ( a circle be+ made by ! curvation ( a /rait l9e1 ! less t /rait l9e is1 ! great] is ! curvation2 & !re=e1 :5 "o /rait l9e is a tang5t to _m circles1 ! angle ( cont+5ce1 :i* x makes ) a less circle1 is great] ?an t :i* x makes ) a great] circle4 ,no?+ !re=e is add$ to or tak5 f an angle simply s call$1 by ! addition to x or tak+ f x ( nev] s _m angles ( cont+5ce4 ,& z an angle ( "o sort c nev] be equal to an angle ( ! o!r sort1 s !y _c be ei!r great] or less ?an "o ano!r4 ,f :5ce x foll[s1 t an angle ( a segm5t1 t is1 ! angle1 :i* any /rait l9e makes ) any ar*1 is equal to ! angle :i* is made by ! same /rait l9e1 & ano!r :i* t\*es ! circle 9 ! po9t ( _! concurr5ce2 z 9 ! la/ figure1 ! angle :i* is made betwe5 ,g ,b & ,b ,k is equal to t :i* is made betwe5 ,g ,b & ! ar* ,b ,c4 ,sid5ote3 ,t ! incl9ation ( planes is angle simply s call$4 #ag4 ,an angle1 :i* is made by two planes1 is commonly call$ ! incl9ation ( ^? planes2 & because planes h equal incl9ation 9 all _! p>ts1 in/ead ( _! incl9ation an angle is tak51 :i* is made by two /rait l9es1 "o ( :i* is 9 "o1 ! o!r 9 ! o!r ( ^? planes1 b bo? p]p5dicul> to ! common section4 ,sid5ote3 ,a solid angle :at x is4 #ah4 ,a solid angle may be conceiv$ two ways4 ,fir/1 = ! aggregate ( all ! angles1 :i* are made by ! motion ( a /rait l9e1 :ile "o extreme po9t !re( rema9+ fix$1 x is c>ri$ ab\t any pla9 figure1 9 :i* ! fix$ po9t ( ! /rait l9e is n conta9$4 ,& 9 ?is s5se1 x seems to be und]/ood by ,euclid4 ,n[ x is manife/1 t ! quantity ( a solid angle s conceiv$ is no o!r1 ?an ! aggregate ( all ! angles 9 a sup]ficies s describ$1 t is1 9 ! sup]ficies ( a pyramidal solid4 ,secondly1 :5 a pyramis or cone has its v]tex 9 ! c5tre ( a sph]e1 a solid angle may be und]/ood to be ! proportion ( a sph]ical sup]ficies subt5d+ t v]tex to ! :ole sup]ficies ( ! sph]e4 ,9 :i* s5se1 solid angles are to "o ano!r z ! sph]ical bases ( solids1 :i* h _! v]tex 9 ! c5tre ( ! same sph]e4 ,sid5ote3 ,:at is ! nature ( asymptotes4 #ai4 ,all ! ways1 by :i* two l9es respect "o ano!r1 or all ! v>iety ( _! position1 may be compreh5d$ "u f\r k9ds2 = any two l9es :atsoev] are ei!r p>allels1 or be+ produc$1 if ne$ be1 or mov$ "o ( !m to ! o!r p>allelly to itself1 !y make an angle2 or else1 by ! l production & motion1 !y t\* "o ano!r2 or la/ly1 !y are asymptotes4 ,! nature ( p>allels1 angles1 & tang5ts1 has be5 already decl>$4 ,x rema9s t ,i speak briefly ( ! nature ( asymptotes4 ,asymptosy dep5ds ^u ?is1 t quantity is inf9itely divisible4 ,& f h5ce x foll[s1 t any l9e be+ giv51 & a body suppos$ to be mov$ f "o extreme !re( t[>ds ! o!r1 x is possible1 by tak+ degrees ( velocity always less & less1 9 su* proportion z ! p>ts ( ! l9e are made less by cont9ual division1 t ! same body may be always mov$ =w>ds 9 t l9e1 & yet nev] rea* ! end ( x4 ,= x is manife/1 t if any /rait l9e1 z ,a ,f1 79 ! #h? figure7 be cut any:]e 9 ,b1 & aga9 ,b ,f be cut 9 ,c1 & ,c ,f 9 ,d1 & ,d ,f 9 ,e1 & s et]nally1 & !re be drawn f ! po9t ,f1 ! /rait l9e ,f ,f at any angle ,a ,f ,f2 & la/ly1 if ! /rait l9es ,a ,f1 ,b ,f1 ,c ,f1 ,d ,f1 ,e ,f1 c41 hav+ ! same proportion to "o ano!r ) ! segm5ts ( ! l9e ,a ,f1 be set 9 ord] & p>allel to ! same ,a ,f1 ! crook$ l9e ,a ,b ,c ,d ,e1 & ! /rait l9e ,f ,f1 w be asymptotes1 t is1 !y w always come ne>] & ne>] toge!r1 b nev] t\* "o ano!r4 ,n[1 because any l9e may be cut et]nally accord+ to ! proportions :i* ! segm5ts h to "o ano!r1 !re=e ! div]s k9ds ( asymptotes are inf9ite 9 numb]1 & n necess>y to be fur!r spok5 ( 9 ?is place4 ,9 ! nature ( asymptotes 9 g5]al !re is no m1 ?an t !y come /ill ne>] & ne>]1 b nev] t\*4 ,b 9 special 9 ! asymptosy ( hyp]bolic l9es1 x is und]/ood !y %\ld approa* to a di/ance less ?an any giv5 quantity4 ,sid5ote3 ,situation1 by :at x is det]m9$4 #bj4 ,,situation is ! relation ( "o place to ano!r2 & :]e !re are _m places1 _! situation is det]m9$ by f\r ?+s2 by _! di/ances f "o ano!r2 by sev]al di/ances f a place assign$2 by ! ord] ( /rait l9es drawn f a place assign$ to ! places ( !m all2 & by ! angles :i* are made by ! l9es s drawn4 ,= if _! di/ances1 ord]1 & angles1 be giv51 t is1 be c]ta9ly kn[n1 _! sev]al places w also be s c]ta9ly kn[n1 z t !y c be no o!r4 ,sid5ote3 ,:at is l situation2 :at is figure2 & :at are l figures4 #ba4 ,po9ts1 h[ _m soev] !y be1 h l situation ) an equal numb] ( o!r po9ts1 :5 all ! /rait l9es1 t are drawn f "s "o po9t to all ^!1 h sev]ally ! same proportion to ^?1 t are drawn 9 ! same ord] & at equal angles f "s "o po9t to all ^?4 ,= let !re be any numb] ( po9ts z ,a1 ,b1 & ,c1 79 ! #i? figure7 to :i* f "s "o po9t ,d let ! /rait l9es ,d ,a1 ,d ,b1 & ,d ,c be drawn2 & let !re be an equal numb] ( o!r po9ts1 z ,e1 ,f1 & ,g1 & f "s po9t ,h let ! /rait l9es ,h ,e1 ,h ,f1 & ,h ,g be drawn1 s t ! angles ,a ,d ,b & ,b ,d ,c be sev]ally & 9 ! same ord] equal to ! angles ,e ,h ,f & ,f ,h ,g1 & ! /rait l9es ,d ,a1 ,d ,b1 & ,d ,c proportional to ! /rait l9es ,h ,e1 ,h ,f1 & ,h ,g2 ,i say1 ! ?ree po9ts ,a1 ,b1 & ,c1 h l situation ) ! ?ree po9ts ,e1 ,f1 & ,g1 or are plac$ alike4 ,= if ,h ,e be und]/ood to be laid ^u ,d ,a1 s t ! po9t ,h be 9 ,d1 ! po9t ,f w be 9 ! /rait l9e ,d ,b1 by reason ( ! equality ( ! angles ,a ,d ,b & ,e ,h ,f2 & ! po9t ,g w be 9 ! /rait l9e ,d ,c1 by reason ( ! equality ( ! angles ,b ,d ,c & ,f ,h ,g2 & ! /rait l9es ,a ,b & ,e ,f1 z also ,b ,c & ,f ,g1 w be p>allel1 because ,a ,d4 ,e ,h 33 ,b ,d4 ,f ,h 33 ,c ,d4 ,g ,h are proportionals by con/ruction2 & !re=e ! di/ances betwe5 ! po9ts ,a & ,b1 & ! po9ts ,b & ,c1 w be proportional to ! di/ances betwe5 ! po9ts ,e & ,f1 & ! po9ts ,f & ,g4 ,:]e=e1 9 ! situation ( ! po9ts ,a1 ,b1 & ,c1 & ! situation ( ! po9ts ,e1 ,f & ,g1 ! angles 9 ! same ord] are equal2 s t _! situations diff] 9 no?+ b ! inequality ( _! di/ances f "o ano!r1 & ( _! di/ances f ! po9ts ,d & ,h4 ,n[1 9 bo? ! ord]s ( po9ts1 ^? inequalities are equal2 = ,a ,b4 ,b ,c 33 ,e ,f4 ,f ,g1 :i* are _! di/ances f "o ano!r1 z also ,d ,a4 ,d ,b4 ,d ,c 33 ,h ,e4 ,h ,f4 ,h ,g1 :i* are _! di/ances f ! assum$ po9ts ,d & ,h1 are proportionals4 ,_! diff]5ce1 !re=e1 consi/s solely 9 ! magnitude ( _! di/ances4 ,b1 by ! def9ition ( l1 7*apt] ,i4 article #b7 ^? ?+s1 :i* diff] only 9 magnitude1 are l4 ,:]e=e ! po9ts ,a1 ,b1 & ,c1 h to "o ano!r l situation ) ! po9ts ,e1 ,f1 & ,g1 or are plac$ alike2 :i* 0 to be prov$4 ,,figure is quantity1 det]m9$ by ! situation or plac+ ( all its extreme po9ts4 ,n[ ,i call ^? po9ts extreme1 :i* are contigu\s to ! place :i* is )\t ! figure4 ,9 l9es !re=e & sup]ficies1 all po9ts may be call$ extreme2 b 9 solids only ^? :i* are 9 ! sup]ficies t includes !m4 ,l figures are ^?1 ^: extreme po9ts 9 "o ( !m are all plac$ l all ! extreme po9ts 9 ! o!r2 = su* figures diff] 9 no?+ b magnitude4 ,& l figures are alike plac$1 :5 9 bo? ( !m ! homologal /rait l9es1 t is1 ! /rait l9es :i* connect ! po9ts :i* answ] "o ano!r1 are p>allel1 & h _! proportional sides incl9$ ! same way4 ,& see+ e /rait l9e is l e o!r /rait l9e1 & e plane l e o!r plane1 :5 no?+ b plan5ess is consid]$2 if ! l9es1 :i* include planes1 or ! sup]ficies1 :i* include solids1 h _! proportions kn[n1 x w n be h>d to "k :e!r any figure be l or unlike to ano!r prop\nd$ figure4 ,& ?us mu* conc]n+ ! fir/ gr\nds ( philosophy4 ,! next place belongs to geometry2 9 :i* ! quantities ( figures are s\ght out f ! proportions ( l9es & angles4 ,:]e=e x is necess>y = him1 t w\ld /udy geometry1 to "k fir/ :at is ! nature ( quantity1 proportion1 angle & figure4 ,hav+ !re=e expla9$ ^! 9 ! ?ree la/ *apt]s1 ,i ?\ght fit to add !m to ?is "p2 & s pass to ! next4 ---------------------------------------- -------------------------------- ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xiv4 ,fig4 #a-aj ---------------------------------------- -------------------------------- ,,"p ,,iii ,,proportions ,,( ,,motions ,,& ,,magnitudes4 ------- ,,*apt] ,,xv4 ,,( ,,! ,,nature1 ,,prop]ties1 ,,& ,,div]s ,,consid]ations ,,( ,,motion ,,& ,,endeav\r4 #a44 ,repetition ( "s pr9ciples ( ! doctr9e ( motion =m]ly set d[n4--#b4 ,o!r pr9ciples add$ to !m4--#c4 ,c]ta9 !orems conc]n+ ! nature ( motion4--#d4 ,div]s consid]ations ( motion4--#e4 ,! way by :i* ! fir/ endeav\r ( bodies mov$ t5de?4--#f4 ,9 motion :i* is made by conc\rse1 "o ( ! mov5ts ceas+1 ! endeav\r is made by ! way by :i* ! re/ t5d4--#g4 ,all endeav\r is propagat$ 9 inf9itum4--#h4 ,h[ mu* great] ! velocity or magnitude is ( a mov5t1 s mu* ! great] is ! efficacy !re( ^u any o!r body 9 its way4 ,sid5ote3 ,repetition ( "s pr9ciples ( ! doctr9e ( motion =m]ly set d[n4 #a4 ,! next ?+s 9 ord] to be treat$ ( are ,,motion & ,,magnitude1 :i* are ! mo/ common accid5ts ( all bodies4 ,?is place !re=e mo/ prop]ly belongs to ! elem5ts ( geometry4 ,b because ?is "p ( philosophy1 hav+ be5 improv$ by ! be/ wits ( all ages1 has af=d$ great] pl5ty ( matt] ?an c well be ?ru/ toge!r )9 ! n>r[ limits ( ?is disc\rse1 ,i ?\ght fit to admoni% ! read]1 t be=e he proce$ fur!r1 he take into 8 h&s ! works ( ,euclid1 ,ar*im$es1 ,apollonius1 & o!r z well anci5t z mod]n writ]s4 ,= to :at end is x1 to d ov] aga9 t :i* is already done8 ,! little !re=e t ,i %all say conc]n+ geometry 9 "s ( ! foll[+ *apt]s1 %all be su* only z is new1 & conduc+ to natural philosophy4 ,i h already deliv]$ "s ( ! pr9ciples ( ?is doctr9e 9 ! eigh? & n9? *apt]s2 :i* ,i %all briefly put toge!r "h1 t ! read] 9 go+ on may h _! light ne>] at h&4 ,fir/1 !re=e1 9 *ap4 ,,viii4 art4 #aj1 motion is def9$ to be ! cont9ual privation ( "o place1 & acquisition ( ano!r4 ,secondly1 x is !re %[n1 t :atsoev] is mov$ is mov$ 9 "t4 ,?irdly1 9 ! same *apt]1 art4 #aa1 ,i h def9$ re/ to be :5 a body rema9s = "s "t 9 "o place4 ,f\r?ly1 x is !re %[n1 t :atsoev] is mov$ is n 9 any det]m9$ place2 z also t ! same has be5 mov$1 is /ill mov$1 & w yet be mov$2 s t 9 e "p ( t space1 9 :i* motion is made1 we may consid] ?ree times1 namely1 ! pa/1 ! pres5t1 & future "t4 ,fif?ly1 9 art4 #ae ( ! same *apt]1 ,i h def9$ velocity or swiftness to be motion consid]$ z p[]1 namely1 t p[] by :i* a body mov$ may 9 a c]ta9 "t transmit a c]ta9 l5g?2 :i* also may m briefly be enunciat$ ?us1 velocity is ! quantity ( motion det]m9$ by "t & l9e4 ,six?ly1 9 ! same *apt]1 art4 #af1 ,i h %[n t motion is ! measure ( "t4 ,sev5?ly1 9 ! same *apt]1 art4 #ag1 ,i h def9$ motions to be equally swift1 :5 9 equal times equal l5g?s are transmitt$ by !m4 ,eigh?ly1 9 art4 #ah ( ! same *apt]1 motions are def9$ to be equal1 :5 ! swiftness ( "o mov$ body1 comput$ 9 e "p ( its magnitude1 is equal to ! swiftness ( ano!r1 comput$ also 9 e "p ( its magnitude4 ,f :5ce x is to be not$1 t motions equal to "o ano!r1 & motions equally swift1 d n signify ! same ?+2 = :5 two horses draw abrea/1 ! motion ( bo? is great] ?an ! motion ( ei!r ( !m s+ly2 b ! swiftness ( bo? toge!r is b equal to t ( ei!r4 ,n9?ly1 9 art4 #ai ( ! same *apt]1 ,i h %[n1 t :atsoev] is at re/ w always be at re/1 unless !re be "s o!r body besides x1 :i* by gett+ into its place suff]s x no long] to rema9 at re/4 ,& t :atsoev] is mov$1 w always be mov$1 unless !re be "s o!r body besides x1 :i* h9d]s its motion4 ,t5?ly1 9 *ap4 ,,ix4 art4 #g1 ,i h demon/rat$1 t :5 any body is mov$ :i* 0 =m]ly at re/1 ! imm$iate effici5t cause ( t motion is 9 "s o!r mov$ & contigu\s body4 ,elev5?ly1 ,i h %[n 9 ! same place1 t :atsoev] is mov$1 w always be mov$ 9 ! same way1 & ) ! same swiftness1 if x be n h9d]$ by "s o!r mov$ & contigu\s body4 ,sid5ote3 ,o!r pr9ciples add$ to !m4 #b4 ,to :i* pr9ciples ,i %all "h add ^? t foll[4 ,fir/1 ,i def9e ,,endeav\r to be motion made 9 less space & "t ?an c be giv52 t is1 less ?an c be det]m9$ or assign$ by exposition or numb]2 t is1 motion made ?r\gh ! l5g? ( a po9t1 & 9 an in/ant or po9t ( "t4 ,= ! expla9+ ( :i* def9ition x mu/ be rememb]$1 t by a po9t is n to be und]/ood t :i* has no quantity1 or :i* _c by any means be divid$2 = !re is no su* ?+ 9 nature2 b t1 ^: quantity is n at all consid]$1 t is1 :]e( nei!r quantity nor any "p is comput$ 9 demon/ration2 s t a po9t is n to be tak5 = an indivisible1 b = an undivid$ ?+2 z also an in/ant is to be tak5 = an undivid$1 & n = an indivisible "t4 ,9 l mann]1 endeav\r is to be conceiv$ z motion2 b s z t nei!r ! quantity ( ! "t 9 :i*1 nor ( ! l9e 9 :i* x is made1 may 9 demon/ration be at all br\ght into comp>ison ) ! quantity ( t "t1 or ( t l9e ( :i* x is a "p4 ,& yet1 z a po9t may be comp>$ ) a po9t1 s "o endeav\r may be comp>$ ) ano!r endeav\r1 & "o may be f\nd to be great] or less ?an ano!r4 ,= if ! v]tical po9ts ( two angles be comp>$1 !y w be equal or unequal 9 ! same proportion :i* ! angles !mselves h to "o ano!r4 ,or if a /rait l9e cut _m circumf]5ces ( conc5tric circles1 ! inequality ( ! po9ts ( int]section w be 9 ! same proportion :i* ! p]imet]s h to "o ano!r4 ,& 9 ! same mann]1 if two motions beg9 & end bo? toge!r1 _! endeav\rs w be equal or unequal1 accord+ to ! proportion ( _! velocities2 z we see a bullet ( lead desc5d ) great] endeav\r ?an a ball ( wool4 ,secondly1 ,i def9e ,,impetus1 or quickness ( motion1 to be ! swiftness or velocity ( ! body mov$1 b consid]$ 9 ! sev]al po9ts ( t "t 9 :i* x is mov$4 ,9 :i* s5se impetus is no?+ else b ! quantity or velocity ( endeav\r4 ,b consid]$ ) ! :ole "t1 x is ! :ole velocity ( ! body mov$ tak5 toge!r ?r\gh\t all ! "t1 & equal to ! product ( a l9e repres5t+ ! "t1 multipli$ into a l9e repres5t+ ! ari?metically mean impetus or quickness4 ,:i* ari?metical mean1 :at x is1 is def9$ 9 ! #bi? article ( *apt] ,,xiii4 ,& because 9 equal times ! ways t are pass$ are z ! velocities1 & ! impetus is ! velocity !y g )al1 reckon$ 9 all ! sev]al po9ts ( ! times1 x foll[e? t dur+ any "t :atsoev]1 h[soev] ! impetus be increas$ or decreas$1 ! l5g? ( ! way pass$ ov] %all be increas$ or decreas$ 9 ! same proportion2 & ! same l9e %all repres5t bo? ! way ( ! body mov$1 & ! sev]al impetus or degrees ( swiftness :]e) ! way is pass$ ov]4 ,& if ! body mov$ be n a po9t1 b a /rait l9e mov$ s z t e po9t !re( make a sev]al /rait l9e1 ! plane describ$ by its motion1 :e!r uni=m1 accel]at$1 or ret>d$1 %all be great] or less1 ! "t be+ ! same1 9 ! same proportion ) t ( ! impetus reckon$ 9 "o motion to ! impetus reckon$ 9 ! o!r4 ,= ! reason is ! same 9 p>allelograms & _! sides4 ,= ! same cause also1 if ! body mov$ be a plane1 ! solid describ$ %all be /ill great] or less 9 ! proportions ( ! sev]al impetus or quicknesses reckon$ ?r\gh "o l9e1 to ! sev]al impetus reckon$ ?r\gh ano!r4 ,?is und]/ood1 let ,a ,b ,c ,d1 79 figure #a1 *ap4 ,,xvii47 be a p>allelogram2 9 :i* suppose ! side ,a ,b to be mov$ p>allelly to ! opposite side ,c ,d1 decreas+ all ! way till x vani% 9 ! po9t ,c1 & s describ+ ! figure ,a ,b ,e ,f ,c2 ! po9t ,b1 z ,a ,b decrease?1 w !re=e describe ! l9e ,b ,e ,f ,c2 & suppose ! "t ( ?is motion design$ by ! l9e ,c ,d2 & 9 ! same "t ,c ,d1 suppose ! side ,a ,c to be mov$ p>allel & uni=mly to ,b ,d4 ,f ! po9t ,o tak5 at adv5ture 9 ! l9e ,c ,d1 draw ,o ,r p>allel to ,b ,d1 cutt+ ! l9e ,b ,e ,f ,c 9 ,e1 & ! side ,a ,b 9 ,r4 ,& aga91 f ! po9t ,q tak5 also at adv5ture 9 ! l9e ,c ,d1 draw ,q ,s p>allel to ,b ,d1 cutt+ ! l9e ,b ,e ,f ,c 9 ,f1 & ! side ,a ,b 9 ,s2 & draw ,e ,g & ,f ,h p>allel to ,c ,d1 cutt+ ,a ,c 9 ,g & ,h4 ,la/ly1 suppose ! same con/ruction done 9 all ! po9ts possible ( ! l9e ,b ,e ,f ,c4 ,i say1 t z ! proportions ( ! swiftness :]e) ,q ,f1 ,o ,e1 ,d ,b1 & all ! re/ suppos$ to be drawn p>allel to ,d ,b & t]m9at$ 9 ! l9e ,b ,e ,f ,c1 are to ! proportions ( _! sev]al times design$ by ! sev]al p>allels ,h ,f1 ,g ,e1 ,a ,b1 & all ! re/ suppos$ to be drawn p>allel to ! l9e ( "t ,c ,d & t]m9at$ 9 ! l9e ,b ,e ,f ,c1 ! aggregate to ! aggregate1 s is ! area or plane ,d ,b ,e ,f ,c to ! area or plane ,a ,c ,f ,e ,b4 ,= z ,a ,b decreas+ cont9ually by ! l9e ,b ,e ,f ,c vani%e? 9 ! "t ,c ,d into ! po9t ,c1 s 9 ! same "t ! l9e ,d ,c cont9ually decreas+ vani%e? by ! same l9e ,c ,f ,e ,b into ! po9t ,b2 & ! po9t ,d describe? 9 t decreas+ motion ! l9e ,d ,b equal to ! l9e ,a ,c describ$ by ! po9t ,a 9 ! decreas+ motion ( ,a ,b2 & _! swiftnesses are !re=e equal4 ,aga91 because 9 ! "t ,g ,e ! po9t ,o describe? ! l9e ,,oe1 & 9 ! same "t ! po9t ,s describe? ! l9e ,s ,e1 ! l9e ,o ,e %all be to ! l9e ,,se1 z ! swiftness :]e) ,,oe is describ$ to ! swiftness :]e) ,,se is describ$4 ,9 l mann]1 because 9 ! same "t ,h ,f ! po9t ,q describe? ! l9e ,q ,f1 & ! po9t ,r ! l9e ,r ,f1 x %all be z ! swiftness by :i* ,q ,f is describ$ to ! swiftness by :i* ,r ,f is describ$1 s ! l9e itself ,q ,f to ! l9e itself ,r ,f2 & s 9 all ! l9es t c possibly be drawn p>allel to ,b ,d 9 ! po9ts :]e !y cut ! l9e ,b ,e ,f ,c4 ,b all ! p>allels to ,b ,d1 z ,s ,e1 ,r ,f1 ,a ,c1 & ! re/ t c possibly be drawn f ! l9e ,a ,b to ! l9e ,b ,e ,f ,c1 make ! area ( ! plane ,a ,b ,e ,f ,c2 & all ! p>allels to ! same ,b ,d1 z ,q ,f1 ,o ,e1 ,d ,b & ! re/ drawn to ! po9ts :]e !y cut ! same l9e ,b ,e ,f ,c1 make ! area ( ! plane ,b ,e ,f ,c ,d4 ,z !re=e ! aggregate ( ! swiftnesses :]e) ! plane ,b ,e ,f ,c ,d is describ$1 is to ! aggregate ( ! swiftnesses :]e) ! plane ,a ,c ,f ,e ,b is describ$1 s is ! plane itself ,b ,e ,f ,c ,d to ! plane itself ,a ,c ,f ,e ,b4 ,b ! aggregate ( ! times repres5t$ by ! p>allels ,a ,b1 ,g ,e1 ,h ,f & ! re/1 make? also ! area ,a ,c ,f ,e ,b4 ,& !re=e1 z ! aggregate ( all ! l9es ,q ,f1 ,o ,e1 ,d ,b & all ! re/ ( ! l9es p>allel to ,b ,d & t]m9at$ 9 ! l9e ,b ,e ,f ,c1 is to ! aggregate ( all ! l9es ,h ,f1 ,g ,e1 ,a ,b & all ! re/ ( ! l9es p>allel to ,c ,d & t]m9at$ 9 ! same l9e ,b ,e ,f ,c2 t is1 z ! aggregate ( ! l9es ( swiftness to ! aggregate ( ! l9es ( "t1 or z ! :ole swiftness 9 ! p>allels to ,d ,b to ! :ole "t 9 ! p>allels to ,c ,d1 s is ! plane ,b ,e ,f ,c ,d to ! plane ,a ,c ,f ,e ,b4 ,& ! proportions ( ,q ,f to ,f ,h1 & ( ,o ,e to ,e ,g1 & ( ,d ,b to ,b ,a1 & s ( all ! re/ tak5 toge!r1 are ! proportions ( ! plane ,d ,b ,e ,f ,c to ! plane ,a ,b ,e ,f ,c4 ,b ! l9es ,q ,f1 ,o ,e1 ,d ,b & ! re/ are ! l9es t design ! swiftness2 & ! l9es ,h ,f1 ,g ,e1 ,a ,b & ! re/ are ! l9es t design ! times ( ! motions2 & !re=e ! proportion ( ! plane ,d ,b ,e ,f ,c to ! plane ,a ,b ,e ,f ,c is ! proportion ( all ! velocities tak5 toge!r to all ! times tak5 toge!r4 ,:]e=e1 z ! proportions ( ! swiftnesses1 c42 :i* 0 to be demon/rat$4 ,! same holds also 9 ! dim9ution ( ! circles1 :]e( ! l9es ( "t are ! semidiamet]s1 z may easily be conceiv$ by imag9+ ! :ole plane ,a ,b ,c ,d turn$ r\nd ^u ! axis ,b ,d2 = ! l9e ,b ,e ,f ,c w be ev]y:]e 9 ! sup]ficies s made1 & ! l9es ,h ,f1 ,g ,e1 ,a ,b1 :i* are "h p>allelograms1 w be !re cyl9d]s1 ! diamet]s ( ^: bases are ! l9es ,h ,f1 ,g ,e1 ,a ,b1 c4 & ! altitude a po9t1 t is to say1 a quantity less ?an any quantity t c possibly be nam$2 & ! l9es ,q ,f1 ,o ,e1 ,d ,b1 c4 small solids ^: l5g?s & bread?s are less ?an any quantity t c be nam$4 ,b ?is is to be not$1 t unless ! proportion ( ! sum ( ! swiftnesses to ! proportion ( ! sum ( ! times be det]m9$1 ! proportion ( ! figure ,d ,b ,e ,f ,c to ! figure ,a ,b ,e ,f ,c _c be det]m9$4 ,?irdly1 ,i def9e ,,resi/ance to be ! endeav\r ( "o mov$ body ei!r :olly or 9 "p contr>y to ! endeav\r ( ano!r mov$ body1 :i* t\*e? ! same4 ,i say1 :olly contr>y1 :5 ! endeav\r ( two bodies proce$s 9 ! same /rait l9e f ! opposite extremes1 & contr>y 9 "p1 :5 two bodies h _! endeav\r 9 two l9es1 :i*1 proce$+ f ! extreme po9ts ( a /rait l9e1 meet )\t ! same4 ,f\r?ly1 t ,i may def9e :at x is to ,,press1 ,i say1 t ( two mov$ bodies "o presses ! o!r1 :5 ) its endeav\r x makes ei!r all or "p ( ! o!r body to g out ( its place4 ,fif?ly1 a body1 :i* is press$ & n :olly remov$1 is said to ,,re/ore itself1 :51 ! press+ body be+ tak5 away1 ! p>ts :i* 7 mov$ d1 by reason ( ! int]nal con/itution ( ! press$ body1 return e "o into its own place4 ,& ?is we may obs]ve 9 spr+s1 9 bl[n bladd]s1 & 9 _m o!r bodies1 ^: p>ts yield m or less to ! endeav\r :i* ! press+ body makes at ! fir/ arrival2 b aft]w>ds1 :5 ! press+ body is remov$1 !y d1 by "s =ce )9 !m1 re/ore !mselves1 & give _! :ole body ! same figure x _h be=e4 ,six?ly1 ,i def9e ,,=ce to be ! impetus or quickness ( motion multipli$ ei!r into itself1 or into ! magnitude ( ! mov5t1 by means :]e( ! said mov5t works m or less ^u ! body t resi/s x4 ,sid5ote3 ,c]ta9 !orems conc]n+ ! nature ( motion4 #c4 ,hav+ premis$ ?us mu*1 ,i %all n[ demon/rate1 fir/1 t if a po9t mov$ come to t\* ano!r po9t :i* is at re/1 h[ little soev] ! impetus or quickness ( its motion be1 x %all move t o!r po9t4 ,= if by t impetus x d n at all move x out ( its place1 nei!r %all x move x ) d\ble ! same impetus4 ,= no?+ d\bl$ is /ill no?+2 & = ! same reason x %all nev] move x ) t impetus1 h[ _m times soev] x be multipli$1 because no?+1 h[ev] x be multipli$1 w = ev] be no?+4 ,:]e=e1 :5 a po9t is at re/1 if x d n yield to ! lea/ impetus1 x w yield to none2 & consequ5tly x w be impossible t t1 :i* is at re/1 %\ld ev] be mov$4 ,secondly1 t :5 a po9t mov$1 h[ little soev] ! impetus !re( be1 falls ^u a po9t ( any body at re/1 h[ h>d soev] t body be1 x w at ! fir/ t\* make x yield a little4 ,= if x d n yield to ! impetus :i* is 9 t po9t1 nei!r w x yield to ! impetus ( nev] s _m po9ts1 :i* h all _! impetus sev]ally equal to ! impetus ( t po9t4 ,= see+ all ^? po9ts toge!r "w equally1 if any "o ( !m h no effect1 ! aggregate ( !m all toge!r %all h no effect z _m times told z !re are po9ts 9 ! :ole body1 t is1 /ill no effect at all2 & by consequ5t !re w\ld be "s bodies s h>d t x w\ld be impossible to break !m2 t is1 a f9ite h>dness1 or a f9ite =ce1 w\ld n yield to t :i* is inf9ite2 :i* is absurd4 ,coroll4 ,x is !re=e manife/1 t re/ does no?+ at all1 nor is ( any efficacy2 & t no?+ b motion gives motion to su* ?+s z be at re/1 & takes x f ?+s mov$4 ,?irdly1 t cessation 9 ! mov5t does n cause cessation 9 t :i* 0 mov$ by x4 ,= 7by numb] #aa ( art4 #a ( ?is *apt]7 :atsoev] is mov$ p]sev]es 9 ! same way & ) ! same swiftness1 z long z x is n h9d]$ by some?+ t is mov$ aga9/ x4 ,n[ x is manife/1 t cessation is n contr>y motion2 & !re=e x foll[s t ! /&+ /ill ( ! mov5t does n make x necess>y t ! ?+ mov$ %\ld also /& /ill4 ,coroll4 ,!y are !re=e deceiv$1 t reckon ! tak+ away ( ! imp$im5t or resi/ance = "o ( ! causes ( motion4 ,sid5ote3 ,div]s consid]ations ( motions4 #d4 ,motion is br\ght into acc\nt = div]s respects2 fir/1 z 9 a body undivid$1 t is1 consid]$ z a po9t2 or1 z 9 a divid$ body4 ,9 an undivid$ body1 :5 we suppose ! way1 by :i* ! motion is made1 to be a l9e2 & 9 a divid$ body1 :5 we compute ! motion ( ! sev]al p>ts ( t body1 z ( p>ts4 ,secondly1 f ! div]sity ( ! regulation ( motion1 x is 9 body1 consid]$ z undivid$4 sometimes uni=m & sometimes multi=m4 ,uni=m is t by :i* equal l9es are always transmitt$ 9 equal times2 & multi=m1 :5 9 "o "t m1 9 ano!r "t less space is transmitt$4 ,aga91 ( multi=m motions1 !re are "s 9 :i* ! degrees ( accel]ation & ret>dation proce$ 9 ! same proportions1 :i* ! spaces transmitt$ h1 :e!r duplicate1 or triplicate1 or by :atsoev] numb] multipli$2 & o!rs 9 :i* x is o!rwise4 ,?irdly1 f ! numb] ( ! mov5ts2 t is1 "o motion is made by "o mov5t only1 & ano!r by ! conc\rse ( _m mov5ts4 ,f\r?ly1 f ! position ( t l9e 9 :i* a body is mov$1 9 respect ( "s o!r l9e2 & f h5ce "o motion is call$ p]p5dicul>1 ano!r oblique1 ano!r p>allel4 ,fif?ly1 f ! position ( ! mov5t 9 respect ( ! mov$ body2 f :5ce "o motion is pulsion or driv+1 ano!r traction or draw+4 ,pulsion1 :5 ! mov5t makes ! mov$ body g be=e x2 & traction1 :5 x makes x foll[4 ,aga91 !re are two sorts ( pulsion2 "o1 :5 ! motions ( ! mov5t & mov$ body beg9 bo? toge!r1 :i* may be call$ trusion or ?ru/+ & vection2 ! o!r1 :5 ! mov5t is fir/ mov$1 & aft]w>ds ! mov$ body1 :i* motion is call$ p]cussion or /roke4 ,six?ly1 motion is consid]$ sometimes f ! effect only :i* ! mov5t works 9 ! mov$ body1 :i* is usually call$ mom5t4 ,n[ mom5t is ! excess ( motion :i* ! mov5t has above ! motion or endeav\r ( ! resi/+ body4 ,sev5?ly1 x may be consid]$ f ! div]sity ( ! m$ium2 z "o motion may be made 9 vacuity or empty place2 ano!r 9 a fluid2 ano!r 9 a consi/5t m$ium1 t is1 a m$ium ^: p>ts are by "s p[] s consi/5t & coh]+1 t no "p ( ! same w yield to ! mov5t1 unless ! :ole yield also4 ,eigh?ly1 :5 a mov$ body is consid]$ z hav+ p>ts1 !re arises ano!r di/9ction ( motion into simple & comp\nd4 ,simple1 :5 all ! sev]al p>ts describe sev]al equal l9es2 comp\nd$1 :5 ! l9es describ$ are unequal4 ,sid5ote3 ,! way by :i* ! fir/ endeav\r ( bodies mov$ t5de?4 #e4 ,all endeav\r t5ds t[>ds t "p1 t is to say1 9 t way :i* is det]m9$ by ! motion ( ! mov5t1 if ! mov5t be b "o2 or1 if !re be _m mov5ts1 9 t way :i* _! conc\rse det]m9es4 ,= example1 if a mov$ body h direct motion1 its fir/ endeav\r w be 9 a /rait l9e2 if x h circul> motion1 its fir/ endeav\r w be 9 ! circumf]5ce ( a circle4 ,sid5ote3 ,9 motion1 :i* x made by conc\rse1 "o ( ! mov5ts ceas+1 ! endeav\r is made by ! way by :i* ! re/ t5d4 #f4 ,& :atsoev] ! l9e be1 9 :i* a body has its motion f ! conc\rse ( two mov5ts1 z soon z 9 any po9t !re( ! =ce ( "o ( ! mov5ts ceases1 !re imm$iately ! =m] endeav\r ( t body w be *ang$ into an endeav\r 9 ! l9e ( ! o!r mov5t4 ,:]e=e1 :5 any body is c>ri$ on by ! conc\rse ( two w9ds1 "o ( ^? w9ds ceas+1 ! endeav\r & motion ( t body w be 9 t l9e1 9 :i* x w\ld h be5 c>ri$ by t w9d alone :i* bl[s /ill4 ,& 9 ! describ+ ( a circle1 :]e t :i* is mov$ has its motion det]m9$ by a mov5t 9 a tang5t1 & by ! radius :i* keeps x 9 a c]ta9 di/ance f ! c5tre1 if ! ret5tion ( ! radius cease1 t endeav\r1 :i* 0 9 ! circumf]5ce ( ! circle1 w n[ be 9 ! tang5t1 t is1 9 a /rait l9e4 ,=1 see+ endeav\r is comput$ 9 a less "p ( ! circumf]5ce ?an c be giv51 t is1 9 a po9t1 ! way by :i* a body is mov$ 9 ! circumf]5ce is comp\nd$ ( innum]able /rait l9es1 ( :i* e "o is less ?an c be giv52 :i* are !re=e call$ po9ts4 ,:]e=e :5 any body1 :i* is mov$ 9 ! circumf]5ce ( a circle1 is fre$ f ! ret5tion ( ! radius1 x w proce$ 9 "o ( ^? /rait l9es1 t is1 9 a tang5t4 ,sid5ote3 ,all endeav\r is propagat$ 9 inf9itum4 #g4 ,all endeav\r1 :e!r /rong or weak1 is propagat$ to inf9ite di/ance2 = x is motion4 ,if !re=e ! fir/ endeav\r ( a body be made 9 space :i* is empty1 x w always proce$ ) ! same velocity2 = x _c be suppos$ t x c receive any resi/ance at all f empty space2 & !re=e1 7by art4 #g1 *ap1 ,,ix7 x w always proce$ 9 ! same way & ) ! same swiftness4 ,& if its endeav\r be 9 space :i* is fill$1 yet1 see+ endeav\r is motion1 t :i* /&s next 9 its way %all be remov$1 & endeav\r fur!r1 & aga9 remove t :i* /&s next1 & s inf9itely4 ,:]e=e ! propagation ( endeav\r1 f "o "p ( full space to ano!r1 proce$s inf9itely4 ,besides1 x rea*es 9 any in/ant to any di/ance1 h[ great soev]4 ,= 9 ! same in/ant 9 :i* ! fir/ "p ( ! full m$ium removes t :i* is next x1 ! second also removes t "p :i* is next to x2 & !re=e all endeav\r1 :e!r x be 9 empty or 9 full space1 proce$s n only to any di/ance1 h[ great soev]1 b also 9 any "t1 h[ little soev]1 t is1 9 an in/ant4 ,nor makes x any matt]1 t endeav\r1 by proce$+1 gr[s weak] & weak]1 till at la/ x c no long] be p]ceiv$ by s5se2 = motion may be ins5sible2 & ,i d n "h exam9e ?+s by s5se & exp]i5ce1 b by reason4 ,sid5ote3 ,h[ mu* great] ! velocity or magnitude is ( a mov5t1 s mu* ! great] is ! efficacy !re( ^u any o!r body 9 its way4 #h4 ,:5 two mov5ts are ( equal magnitude1 ! swift] ( !m works ) great] =ce ?an ! sl[]1 ^u a body t resi/s _! motion4 ,also1 if two mov5ts h equal velocity1 ! great] ( !m works ) m =ce ?an ! less4 ,= :]e ! magnitude is equal1 ! mov5t ( great] velocity makes ! great] impression ^u t body ^u :i* x falls2 & :]e ! velocity is equal1 ! mov5t ( great] magnitude fall+ ^u ! same po9t1 or an equal "p ( ano!r body1 loses less ( its velocity1 because ! resi/+ body works only ^u t "p ( ! mov5t :i* x t\*es1 & !re=e abates ! impetus ( t "p only2 :]eas 9 ! mean "t ! p>ts1 :i* are n t\*$1 proce$1 & reta9 _! :ole =ce1 till !y also come to be t\*$2 & _! =ce has "s effect4 ,:]e=e1 = example1 9 batt]ies a long] ?an a %ort] piece ( timb] ( ! same ?ickness & velocity1 & a ?ick] ?an a sl5d]] piece ( ! same l5g? & velocity1 "w a great] effect ^u ! wall4 ------- ,,*apt] ,,xvi4 ,,( ,,motion ,,accel]at$ ,,& ,,uni=m1 ,,& ,,( ,,motion ,,by ,,conc\rse4 #a4 ,! velocity ( any body1 9 :at "t soev] x be comput$1 is t :i* is made ( ! multiplication ( ! impetus1 or quickness ( its motion into ! "t4--#b-e4 ,9 all motion1 ! l5g?s :i* are pass$ ?r\gh are to "o ano!r1 z ! products made by ! impetus multipli$ into ! "t4--#f4 ,if two bodies be mov$ ) uni=m motion ?r\gh two l5g?s1 ! proportion ( ^? l5g?s to "o ano!r w be comp\nd$ ( ! proportions ( "t to "t1 & impetus to impetus1 directly tak54--#g4 ,if two bodies pass ?r\gh two l5g?s ) uni=m motion1 ! proportion ( _! times to "o ano!r w be comp\nd$ ( ! proportions ( l5g? to l5g?1 & impetus to impetus reciprocally tak52 also ! proportion ( _! impetus to "o ano!r w be comp\nd$ ( ! proportions ( l5g? to l5g?1 & "t to "t reciprocally tak54--#h4 ,if a body be c>ri$ on ) uni=m motion by two mov5ts toge!r1 :i* meet 9 an angle1 ! l9e by :i* x passes w be a /rait l9e1 subt5d+ ! complem5t ( t angle to two "r angles4--#i1 c4 ,if a body be c>ri$ by two mov5ts toge!r1 "o ( !m be+ mov$ ) uni=m1 ! o!r ) accel]at$ motion1 & ! proportion ( _! l5g?s to _! times be+ explicable 9 numb]s1 h[ to f9d out :at l9e t body describes4 ,sid5ote3 ,! velocity ( any body1 9 :at "t soev] x be comput$1 is t :i* is made ( ! multiplication ( ! impetus or quickness ( its motion into ! "t4 #a4 ,! velocity ( any body1 9 :atsoev] "t x be mov$1 has its quantity det]m9$ by ! sum ( all ! sev]al quicknesses or impetus1 :i* x ha? 9 ! sev]al po9ts ( ! "t ( ! body's motion4 ,= see+ velocity1 7by ! def9ition ( x1 *ap1 ,,viii1 art4 #ae7 is t p[] by :i* a body c 9 a c]ta9 "t pass ?r\gh a c]ta9 l5g?2 & quickness ( motion or impetus1 7by *ap4 ,,xv1 art4 #b1 num4 #b7 is velocity tak5 9 "o po9t ( "t only1 all ! impetus1 toge!r tak5 9 all ! po9ts ( "t1 w be ! same ?+ ) ! mean impetus multipli$ into ! :ole "t1 or :i* is all "o1 w be ! velocity ( ! :ole motion4 ,coroll4 ,if ! impetus be ! same 9 e po9t1 any /rait l9e repres5t+ x may be tak5 = ! measure ( "t3 & ! quicknesses or impetus appli$ ord9ately to any /rait l9e mak+ an angle ) x1 & repres5t+ ! way ( ! body's motion1 w design a p>allelogram :i* %all repres5t ! velocity ( ! :ole motion4 ,b if ! impetus or quickness ( motion beg9 f re/ & increase uni=mly1 t is1 9 ! same proportion cont9ually ) ! times :i* are pass$1 ! :ole velocity ( ! motion %all be repres5t$ by a triangle1 "o side :]e( is ! :ole "t1 & ! o!r ! greate/ impetus acquir$ 9 t "t2 or else by a p>allelogram1 "o ( ^: sides is ! :ole "t ( motion1 & ! o!r1 half ! greate/ impetus2 or la/ly1 by a p>allelogram hav+ = "o side a mean proportional betwe5 ! :ole "t & ! half ( t "t1 & = ! o!r side ! half ( ! greate/ impetus4 ,= bo? ^! p>allelograms are equal to "o ano!r1 & sev]ally equal to ! triangle :i* is made ( ! :ole l9e ( "t1 & ( ! greate/ acquir$ impetus2 z is demon/rat$ 9 ! elem5ts ( geometry4 ,sid5ote3 ,9 all motion1 ! l5g?s :i* are pass$ ?r\gh are to "o ano!r1 z ! products made by ! impetus multipli$ into "t4 #b4 ,9 all uni=m motions ! l5g?s :i* are transmitt$ are to "o ano!r1 z ! product ( ! mean impetus multipli$ into its "t1 to ! product ( ! mean impetus multipli$ also into its "t4 ,= let ,a ,b 79 fig4 #a7 be ! "t1 & ,a ,c ! impetus by :i* any body passes ) uni=m motion ?r\gh ! l5g? ,d ,e2 & 9 any "p ( ! "t ,a ,b1 z 9 ! "t ,a ,f1 let ano!r body be mov$ ) uni=m motion1 fir/1 ) ! same impetus ,a ,c4 ,?is body1 !re=e1 9 ! "t ,a ,f ) ! impetus ,a ,c w pass ?r\gh ! l5g? ,a ,f4 ,see+1 !re=e1 :5 bodies are mov$ 9 ! same "t1 & ) ! same velocity & impetus 9 e "p ( _! motion1 ! proportion ( "o l5g? transmitt$ to ano!r l5g? transmitt$1 is ! same ) t ( "t to "t1 x foll[e?1 t ! l5g? transmitt$ 9 ! "t ,a ,b ) ! impetus ,a ,c w be to ! l5g? transmitt$ 9 ! "t ,a ,f ) ! same impetus ,a ,c1 z ,a ,b itself is to ,a ,f1 t is1 z ! p>allelogram ,a ,i is to ! p>allelogram ,a ,h1 t is1 z ! product ( ! "t ,a ,b into ! mean impetus ,a ,c is to ! product ( ! "t ,a ,f into ! same impetus ,a ,c4 ,aga91 let x be suppos$ t a body be mov$ 9 ! "t ,a ,f1 n ) ! same b ) "s o!r uni=m impetus1 z ,a ,l4 ,see+ !re=e1 "o ( ! bodies has 9 all ! p>ts ( its motion ! impetus ,a ,c1 & ! o!r 9 l mann] ! impetus ,a ,l1 ! l5g? transmitt$ by ! body mov$ ) ! impetus ,a ,c w be to ! l5g? transmitt$ by ! body mov$ ) ! impetus ,a ,l1 z ,a ,c itself is to ,a ,l1 t is1 z ! p>allelogram ,a ,h is to ! p>allelogram ,f ,l4 ,:]e=e1 by ord9ate proportion x w be1 z ! p>allelogram ,a ,i to ! p>allelogram ,f ,l1 t is1 z ! product ( ! mean impetus into ! "t is to ! product ( ! mean impetus into ! "t1 s ! l5g? transmitt$ 9 ! "t ,a ,b ) ! impetus ,a ,c1 to ! l5g? transmitt$ 9 ! "t ,a ,f ) ! impetus ,a ,l2 :i* 0 to be demon/rat$4 ,coroll4 ,see+1 !re=e1 9 uni=m motion1 z has be5 %[n1 ! l5g?s transmitt$ are to "o ano!r z ! p>allelograms :i* are made by ! multiplication ( ! mean impetus into ! times1 t is1 by reason ( ! equality ( ! impetus all ! way1 z ! times !mselves1 x w also be1 by p]mutation1 z "t to l5g?1 s "t to l5g?2 & 9 g5]al1 to ?is place are applicable all ! prop]ties & transmutations ( analogisms1 :i* ,i h set d[n & demon/rat$ 9 *apt] ,,xiii4 #c4 ,9 motion begun f re/ & uni=mly accel]at$1 t is1 :]e ! impetus increase? cont9ually accord+ to ! proportion ( ! times1 x w also be1 z "o product made by ! mean impetus multipli$ into ! "t1 to ano!r product made likewise by ! mean impetus multipli$ into ! "t1 s ! l5g? transmitt$ 9 ! "o "t to ! l5g? transmitt$ 9 ! o!r "t4 ,= let ,a ,b 79 fig4 #a7 repres5t a "t2 9 ! beg9n+ ( :i* "t ,a1 let ! impetus be z ! po9t ,a2 b z ! "t goes on1 s let ! impetus increase uni=mly1 till 9 ! la/ po9t ( t "t ,a ,b1 namely 9 ,b1 ! impetus acquir$ be ,b ,i4 ,aga91 let ,a ,f repres5t ano!r "t1 9 ^: beg9n+ ,a1 let ! impetus be z ! po9t itself ,a2 b z ! "t proce$s1 s let ! impetus increase uni=mly1 till 9 ! la/ po9t ,f ( ! "t ,a ,f ! impetus acquir$ be ,f ,k2 & let ,d ,e be ! l5g? pass$ ?r\gh 9 ! "t ,a ,b ) impetus uni=mly increas$4 ,i say1 ! l5g? ,d ,e is to ! l5g? transmitt$ 9 ! "t ,a ,f1 z ! "t ,a ,b multipli$ into ! mean ( ! impetus increas+ ?r\gh ! "t ,a ,b1 is to ! "t ,a ,f multipli$ into ! mean ( ! impetus increas+ ?r\gh ! "t ,a ,f4 ,= see+ ! triangle ,a ,b ,i is ! :ole velocity ( ! body mov$ 9 ! "t ,a ,b1 till ! impetus acquir$ be ,b ,i2 & ! triangle ,a ,f ,k ! :ole velocity ( ! body mov$ 9 ! "t ,a ,f ) impetus increas+ till !re be acquir$ ! impetus ,f ,k2 ! l5g? ,d ,e to ! l5g? acquir$ 9 ! "t ,a ,f ) impetus increas+ f re/ 9 ,a till !re be acquir$ ! impetus ,f ,k1 w be z ! triangle ,a ,b ,i to ! triangle ,a ,f ,k1 t is1 if ! triangles ,a ,b ,i & ,a ,f ,k be l1 9 duplicate proportion ( ! "t ,a ,b to ! "t ,a ,f2 b if unlike1 9 ! proportion comp\nd$ ( ! proportions ( ,a ,b to ,a ,f & ( ,b ,i to ,f ,k4 ,:]e=e1 z ,a ,b ,i is to ,a ,f ,k1 s let ,d ,e be to ,d ,p2 = s1 ! l5g? transmitt$ 9 ! "t ,a ,b ) impetus increas+ to ,b ,i1 w be to ! l5g? transmitt$ 9 ! "t ,a ,f ) impetus increas+ to ,f ,k1 z ! triangle ,a ,b ,i is to ! triangle ,a ,f ,k2 b ! triangle ,a ,b ,i is made by ! multiplication ( ! "t ,a ,b into ! mean ( ! impetus increas+ to ,b ,i2 & ! triangle ,a ,f ,k is made by ! multiplication ( ! "t ,a ,f into ! mean ( ! impetus increas+ to ,f ,k2 & !re=e ! l5g? ,d ,e :i* is transmitt$ 9 ! "t ,a ,b ) impetus increas+ to ,b ,i1 to ! l5g? ,d ,p :i* is transmitt$ 9 ! "t ,a ,f ) impetus increas+ to ,f ,k1 is z ! product :i* is made ( ! "t ,a ,b multipli$ into its mean impetus1 to ! product ( ! "t ,a ,f multipli$ also into its mean impetus2 :i* 0 to be prov$4 ,coroll4 ,i4 ,9 motion uni=mly accel]at$1 ! proportion ( ! l5g?s transmitt$ to t ( _! times1 is comp\nd$ ( ! proportions ( _! times to _! times1 & impetus to impetus4 ,coroll4 ,,ii4 ,9 motion uni=mly accel]at$1 ! l5g?s transmitt$ 9 equal times1 tak5 9 cont9ual succession f ! beg9n+ ( motion1 are z ! diff]5ces ( squ>e numb]s beg9n+ f unity1 namely1 z #c1 #e1 #g1 c4 ,= if 9 ! fir/ "t ! l5g? transmitt$ be z #a1 9 ! fir/ & second times ! l5g? transmitt$ w be z #d1 :i* is ! squ>e ( #b1 & 9 ! ?ree fir/ times x w be z #i1 :i* is ! squ>e ( #c1 & 9 ! f\r fir/ times z #af1 & s on4 ,n[ ! diff]5ces ( ^! squ>es are #c1 #e1 #g1 c4 ,coroll4 ,,iii4 ,9 motion uni=mly accel]at$ f re/1 ! l5g? transmitt$ is to ano!r l5g? transmitt$ uni=mly 9 ! same "t1 b ) su* impetus z 0 acquir$ by ! accel]at$ motion 9 ! la/ po9t ( t "t1 z a triangle to a p>allelogram1 :i* h _! altitude & base common4 ,= see+ ! l5g? ,d ,e 79 fig4 #a7 is pass$ ?r\gh ) velocity z ! triangle ,a ,b ,i1 x is necess>y t = ! pass+ ?r\gh ( a l5g? :i* is d\ble to ,d ,e1 ! velocity be z ! p>allelogram ,a ,i2 = ! p>allelogram ,a ,i is d\ble to ! triangle ,a ,b ,i4 #d4 ,9 motion1 :i* beg9n+ f re/ is s acl]at$1 t ! impetus !re( increases cont9ually 9 proportion duplicate to ! proportion ( ! times 9 :i* x is made1 a l5g? transmitt$ 9 "o "t w be to a l5g? transmitt$ 9 ano!r "t1 z ! product made by ! mean impetus multipli$ into ! "t ( "o ( ^? motions1 to ! product ( ! mean impetus multipli$ into ! "t ( ! o!r motion4 ,= let ,a ,b 79 fig4 #b7 repres5t a "t1 9 ^: fir/ in/ant ,a let ! impetus be z ! po9t ,a2 b z ! "t proce$s1 s let ! impetus increase cont9ually 9 duplicate proportion to t ( ! times1 till 9 ! la/ po9t ( "t ,b ! impetus acquir$ be ,b ,i2 !n tak+ ! po9t ,f any:]e 9 ! "t ,a ,b1 let ! impetus ,f ,k acquir$ 9 ! "t ,a ,f be ord9ately appli$ to t po9t ,f4 ,see+ !re=e ! proportion ( ,f ,k to ,b ,i is suppos$ to be duplicate to t ( ,a ,f to ,a ,b1 ! proportion ( ,a ,f to ,a ,b w be subduplicate to t ( ,f ,k to ,b ,i2 & t ( ,a ,b to ,a ,f w be 7by *ap4 ,,xiii4 art4 #af7 duplicate to t ( ,b ,i to ,f ,k2 & consequ5tly ! po9t ,k w be 9 a p>abolical l9e1 ^: diamet] is ,a ,b & base ,b ,i2 & = ! same reason1 to :at po9t soev] ( ! "t ,a ,b ! impetus acquir$ 9 t "t be ord9ately appli$1 ! /rait l9e design+ t impetus w be 9 ! same p>abolical l9e ,a ,k ,i4 ,:]e=e ! mean impetus multipli$ into ! :ole "t ,a ,b w be ! p>abola ,a ,k ,i ,b1 equal to ! p>allelogram ,a ,m1 :i* p>allelogram has = "o side ! l9e ( "t ,a ,b & = ! o!r ! l9e ( ! impetus ,a ,l1 :i* is two-?irds ( ! impetus ,b ,i2 = e p>abola is equal to two-?irds ( t p>allelogram ) :i* x has its altitude & base common4 ,:]e=e ! :ole velocity 9 ,a ,b w be ! p>allelogram ,a ,m1 z be+ made by ! multiplication ( ! impetus ,a ,l into ! "t ,a ,b4 ,& 9 l mann]1 if ,f ,n be tak51 :i* is two-?irds ( ! impetus ,f ,k1 & ! p>allelogram ,f ,o be complet$1 ,f ,o w be ! :ole velocity 9 ! "t ,a ,f1 z be+ made by ! uni=m impetus ,a ,o or ,f ,n multipli$ into ! "t ,a ,f4 ,let n[ ! l5g? transmitt$ 9 ! "t ,a ,b & ) ! velocity ,a ,m be ! /rait l9e ,d ,e2 & la/ly1 let ! l5g? transmitt$ 9 ! "t ,a ,f ) ! velocity ,a ,n be ,d ,p2 ,i say t z ,a ,m is to ,a ,n1 or z ! p>abola ,a ,k ,i ,b to ! p>abola ,a ,k ,f1 s is ,d ,e to ,d ,p4 ,= z ,a ,m is to ,f ,l1 t is1 z ,a ,b is to ,a ,f1 s let ,d ,e be to ,d ,g4 ,n[ ! proportion ( ,a ,m to ,a ,n is comp\nd$ ( ! proportions ( ,a ,m to ,f ,l1 & ( ,f ,l to ,a ,n4 ,b z ,a ,m to ,f ,l1 s by con/ruction is ,d ,e to ,d ,g2 & z ,f ,l is to ,a ,n 7see+ ! "t 9 bo? is ! same1 namely1 ,a ,f71 s is ! l5g? ,d ,g to ! l5g? ,d ,p2 = l5g?s transmitt$ 9 ! same "t are to "o ano!r z _! velocities are4 ,:]e=e by ord9ate proportion1 z ,a ,m is to ,a ,n1 t is1 z ! mean impetus ,a ,l multipli$ into its "t ,a ,b1 is to ! mean impetus ,a ,o multipli$ into ,a ,f1 s is ,d ,e to ,d ,p2 :i* 0 to be prov$4 ,coroll4 ,i4 ,l5g?s transmitt$ ) motion s accel]at$1 t ! impetus increase cont9ually 9 duplicate proportion to t ( _! times1 if ! base repres5t ! impetus1 are 9 triplicate proportion ( _! impetus acquir$ 9 ! la/ po9t ( _! times4 ,= z ! l5g? ,d ,e is to ! l5g? ,d ,p1 s is ! p>allelogram ,a ,m to ! p>allelogram ,a ,n1 & s ! p>abola ,a ,k ,i ,b to ! p>abola ,a ,k ,f4 ,b ! proportion ( ! p>abola ,a ,k ,i ,b to ! p>abola ,a ,k ,f is triplicate to ! proportion :i* ! base ,b ,i has to ! base ,f ,k4 ,:]e=e also ! proportion ( ,d ,e to ,d ,p is triplicate to t ( ,b ,i to ,f ,k4 ,coroll4 ,,ii4 ,l5g?s transmitt$ 9 equal times succe$+ "o ano!r f ! beg9n+1 by motion s accel]at$1 t ! proportion ( ! impetus be duplicate to ! proportion ( ! times1 are to "o ano!r z ! diff]5ces ( cubic numb]s beg9n+ at unity1 t is z #g1 #ai1 #cg1 c4 ,= if 9 ! fir/ "t ! l5g? transmitt$ be z #a1 ! l5g? at ! end ( ! second "t w be z #h1 at ! end ( ! ?ird "t z #bg1 & at ! end ( ! f\r? "t z #fd1 c42 :i* are cubic numb]s1 ^: diff]5ces are #g1 #ai1 #cg1 c4 ,coroll4 ,,iii4 ,9 motion s accel]at$1 z t ! l5g? transmitt$ be always to ! l5g? transmitt$ 9 duplicate proportion to _! times1 ! l5g? uni=mly transmitt$ 9 ! :ole "t1 & ) impetus all ! way equal to t :i* is la/ acquir$1 is z a p>abola to a p>allelogram ( ! same altitude & base1 t is1 z #b to #c4 ,= ! p>abola ,a ,k ,i ,b is ! impetus increas+ 9 ! "t ,a ,b2 & ! p>allelogram ,a ,i is ! greate/ uni=m impetus multipli$ into ! same "t ,a ,b4 ,:]e=e ! l5g?s transmitt$ w be z a p>abola to a p>allelogram1 c41 t is1 z #b to #c4 #e4 ,if ,i %\ld proce$ to ! explication ( su* motions z are made by impetus increas+ 9 proportion triplicate1 quadruplicate1 qu9tuplicate1 c41 to t ( _! times1 x w\ld be a lab\r inf9ite & unnecess>y4 ,= by ! same me?od by :i* ,i h comput$ su* l5g?s1 z are transmitt$ ) impetus increas+ 9 s+le & duplicate proportion1 any man may compute su* z are transmitt$ ) impetus increas+ 9 triplicate1 quadruplicate1 or :at o!r proportion he pleases4 ,9 mak+ :i* computation he %all f9d1 t :]e ! impetus increase 9 proportion triplicate to t ( ! times1 !re ! :ole velocity w be design$ by ! fir/ p>abola/] 7( :i* see ! next *apt]72 & ! l5g?s transmitt$ w be 9 proportion quadruplicate to t ( ! times4 ,& 9 l mann]1 :]e ! impetus increase 9 quadruplicate proportion to t ( ! times1 t !re ! :ole velocity w be design$ by ! second p>abola/]1 & ! l5g?s transmitt$ w be 9 qu9tuplicate proportion to t ( ! times2 & s on cont9ually4 ,sid5ote3 ,if two bodies be mov$ ) uni=m motion ?r\gh two l5g?s1 ! proportion ( ^? l5g?s to "o ano!r1 w be comp\nd$ ( ! proportions ( "t to "t1 & impetus to impetus1 directly tak54 #f4 ,if two bodies ) uni=m motion transmit two l5g?s1 ea* ) its own impetus & "t1 ! proportion ( ! l5g?s transmitt$ w be comp\nd$ ( ! proportions ( "t to "t1 & impetus to impetus1 directly tak54 ,let two bodies be mov$ uni=mly 7z 9 fig4 #c71 "o 9 ! "t ,a ,b ) ! impetus ,a ,c1 ! o!r 9 ! "t ,a ,d ) ! impetus ,a ,e4 ,i say ! l5g?s transmitt$ h _! proportion to "o ano!r comp\nd$ ( ! proportions ( ,a ,b to ,a ,d1 & ( ,a ,c to ,a ,e4 ,= let any l5g? :atsoev]1 z ,z1 be transmitt$ by "o ( ! bodies 9 ! "t ,a ,b ) ! impetus ,a ,c2 & any o!r l5g?1 z ,x1 be transmitt$ by ! o!r body 9 ! "t ,a ,d ) ! impetus ,a ,e2 & let ! p>allelograms ,a ,f & ,a ,g be complet$4 ,see+ n[ ,z is to ,x 7by art4 #b7 z ! impetus ,a ,c multipli$ into ! "t ,a ,b is to ! impetus ,a ,e multipli$ into ! "t ,a ,d1 t is1 z ,a ,f to ,a ,g2 ! proportion ( ,z to ,x w be comp\nd$ ( ! same proportions1 ( :i* ! proportion ( ,a ,f to ,a ,g is comp\nd$2 b ! proportion ( ,a ,f to ,a ,g is comp\nd$ ( ! proportions ( ! side ,a ,b to ! side ,a ,d1 & ( ! side ,a ,c to ! side ,a ,e 7z is evid5t by ! ,elem5ts ( ,euclid71 t is1 ( ! proportions ( ! "t ,a ,b to ! "t ,a ,d1 & ( ! impetus ,a ,c to ! impetus ,a ,e4 ,:]e=e also ! proportion ( ,z to ,x is comp\nd$ ( ! same proportions ( ! "t ,a ,b to ! "t ,a ,d1 & ( ! impetus ,a ,c to ! impetus ,a ,e2 :i* 0 to be demon/rat$4 ,coroll4 ,i4 ,:5 two bodies are mov$ ) uni=m motion1 if ! times & impetus be 9 reciprocal proportion1 ! l5g?s transmitt$ %all be equal4 ,= if x 7 z ,a ,b to ,a ,d 79 ! same fig4 #c7 s reciprocally ,a ,e to ,a ,c1 ! proportion ( ,a ,f to ,a ,g w\ld be comp\nd$ ( ! proportions ( ,a ,b to ,a ,d1 & ( ,a ,c to ,a ,e1 t is1 ( ! proportions ( ,a ,b to ,a ,d1 & ( ,a ,d to ,a ,b4 ,:]e=e1 ,a ,f w\ld be to ,a ,g z ,a ,b to ,a ,b1 t is1 equal2 & s ! two products made by ! multiplication ( impetus into "t w\ld be equal2 & by consequ5t1 ,z w\ld be equal to ,x4 ,coroll4 ,,ii4 ,if two bodies be mov$ 9 ! same "t1 b ) diff]5t impetus1 ! l5g?s transmitt$ w be z impetus to impetus4 ,= if ! "t ( bo? ( !m be ,a ,d1 & _! diff]5t impetus be ,a ,e & ,a ,c1 ! proportion ( ,a ,g to ,d ,c w be comp\nd$ ( ! proportions ( ,a ,e to ,a ,c & ( ,a ,d to ,a ,d1 t is1 ( ! proportions ( ,a ,e to ,a ,c & ( ,a ,c to ,a ,c2 & s ! proportion ( ,a ,g to ,d ,c1 t is1 ! proportion ( l5g? to l5g?1 w be z ,a ,e to ,a ,c1 t is1 z t ( impetus to impetus4 ,9 l mann]1 if two bodies be mov$ uni=mly1 & bo? ( !m ) ! same impetus1 b 9 diff]5t times1 ! proportion ( ! l5g?s transmitt$ by !m w be z t ( _! times4 ,= if !y h bo? ! same impetus ,a ,c1 & _! diff]5t times be ,a ,b & ,a ,d1 ! proportion ( ,a ,f to ,d ,c w be comp\nd$ ( ! proportions ( ,a ,b to ,a ,d & ( ,a ,c to ,a ,c2 t is1 ( ! proportions ( ,a ,b to ,a ,d & ( ,a ,d to ,a ,d2 & !re=e ! proportion ( ,a ,f to ,d ,c1 t is1 ( l5g? to l5g?1 w be ! same ) t ( ,a ,b to ,a ,d1 :i* is ! proportion ( "t to "t4 ,sid5ote3 ,if two bodies pass ?r\gh two l5g?s ) uni=m motion1 ! proportion ( _! times to "o ano!r1 w be comp\nd$ ( ! proportions ( l5g? to l5g?1 & impetus to impetus reciprocally tak52 also ! proportion ( _! impetus to "o ano!r1 w be comp\nd$ ( ! proportions ( l5g? to l5g?1 & "t to "t reciprocally tak54 #g4 ,if two bodies pass ?r\gh two l5g?s ) uni=m motion1 ! proportion ( ! times 9 :i* !y are mov$ w be comp\nd$ ( ! proportions ( l5g? to l5g? & impetus to impetus reciprocally tak54 ,= let any two l5g?s be giv51 z 79 ! same fig4 #c7 ,z & ,x1 & let "o ( !m be transmitt$ ) ! impetus ,a ,c1 ! o!r ) ! impetus ,a ,e4 ,i say ! proportion ( ! times 9 :i* !y are transmitt$1 w be comp\nd$ ( ! proportions ( ,z to ,x1 & ( ,a ,e1 :i* is ! impetus ) :i* ,x is transmitt$1 to ,a ,c1 ! impetus ) :i* ,z is transmitt$4 ,= see+ ,a ,f is ! product ( ! impetus ,a ,c multipli$ into ! "t ,a ,b1 ! "t ( motion ?r\gh ,z w be a l9e1 :i* is made by ! application ( ! p>allelogram ,a ,f to ! /rait l9e ,a ,c1 :i* l9e is ,a ,b2 & !re=e ,a ,b is ! "t ( motion ?r\gh ,z4 ,9 l mann]1 see+ ,a ,g is ! product ( ! impetus ,a ,e multipli$ into ! "t ,a ,d1 ! "t ( motion ?r\gh ,x w be a l9e :i* is made by ! application ( ,a ,g to ! /rait l9e ,a ,d2 b ,a ,d is ! "t ( motion ?r\gh ,x4 ,n[ ! proportion ( ,a ,b to ,a ,d is comp\nd$ ( ! proportions ( ! p>allelogram ,a ,f to ! p>allelogram ,a ,g1 & ( ! impetus ,a ,e to ! impetus ,a ,c2 :i* may be demon/rat$ ?us4 ,put ! p>allelograms 9 ord] ,a ,f1 ,a ,g1 ,d ,c1 & x w be manife/ t ! proportion ( ,a ,f to ,d ,c is comp\nd$ ( ! proportions ( ,a ,f to ,a ,g & ( ,a ,g to ,d ,c2 b ,a ,f is to ,d ,c z ,a ,b to ,a ,d2 :]e=e also ! proportion ( ,a ,b to ,a ,d is comp\nd$ ( ! proportions ( ,a ,f to ,a ,g & ( ,a ,g to ,d ,c4 ,& because ! l5g? ,z is to ! l5g? ,x z ,a ,f is to ,a ,g1 & ! impetus ,a ,e to ! impetus ,a ,c z ,a ,g to ,d ,c1 !re=e ! proportion ( ,a ,b to ,a ,d w be comp\nd$ ( ! proportions ( ! l5g? ,z to ! l5g? ,x1 & ( ! impetus ,a ,e to ! impetus ,a ,c2 :i* 0 to be demon/rat$4 ,9 ! same mann] x may be prov$1 t 9 two uni=m motions ! proportion ( ! impetus is comp\nd$ ( ! proportions ( l5g? to l5g? & ( "t to "t reciprocally tak54 ,= if we suppose ,a ,c 79 ! same fig4 #c7 to be ! "t1 & ,a ,b ! impetus ) :i* ! l5g? ,z is pass$ ?r\gh2 & ,a ,e to be ! "t1 & ,a ,d ! impetus ) :i* ! l5g? ,x is pass$ ?r\gh1 ! demon/ration w proce$ z 9 ! la/ article4 ,sid5ote3 ,if a body be c>ri$ on ) uni=m motion by two mov5ts toge!r1 :i* meet 9 an angle1 ! l9e by :i* x passes w be a /rait l9e1 subt5d+ ! complem5t ( t angle to #b "r angles4 #h4 ,if a body be c>ri$ by two mov5ts toge!r1 :i* move ) /rait & uni=m motion1 & concur 9 any giv5 angle1 ! l9e by :i* t body passes w be a /rait l9e4 ,let ! mov5t ,a ,b 79 fig4 #d7 h /rait & uni=m motion1 & be mov$ till x come into ! place ,c ,d2 & let ano!r mov5t ,a ,c1 hav+ likewise /rait & uni=m motion1 & mak+ ) ! mov5t ,a ,b any giv5 angle ,c ,a ,b1 be und]/ood to be mov$ 9 ! same "t to ,d ,b2 & let ! body be plac$ 9 ! po9t ( _! conc\rse1 ,a4 ,i say ! l9e :i* t body describes ) its motion is a /rait l9e4 ,= let ! p>allelogram ,a ,b ,d ,c be complet$1 & its diagonal ,a ,d be drawn2 & 9 ! /rait l9e ,a ,b let any po9t ,e be tak52 & f x let ,e ,f be drawn p>allel to ! /rait l9es ,a ,c & ,b ,d1 cutt+ ,a ,d 9 ,g2 & ?r\gh ! po9t ,g let ,h ,i be drawn p>allel to ! /rait l9es ,a ,b & ,c ,d2 & la/ly1 let ! measure ( ! "t be ,a ,c4 ,see+ !re=e bo? ! motions are made 9 ! same "t1 :5 ,a ,b is 9 ,c ,d1 ! body also w be 9 ,c ,d2 & 9 l mann]1 :5 ,a ,c is 9 ,b ,d1 ! body w be 9 ,b ,d4 ,b ,a ,b is 9 ,c ,d at ! same "t :5 ,a ,c is 9 ,b ,d2 & !re=e ! body w be 9 ,c ,d & ,b ,d at ! same "t2 :]e=e x w be 9 ! common po9t ,d4 ,aga91 see+ ! motion f ,a ,c to ,b ,d is uni=m1 t is1 ! spaces transmitt$ by x are 9 proportion to "o ano!r z ! times 9 :i* !y are transmitt$1 :5 ,a ,c is 9 ,e ,f1 ! proportion ( ,a ,b to ,a ,e w be ! same ) t ( ,e ,f to ,e ,g1 t is1 ( ! "t ,a ,c to ! "t ,a ,h4 ,:]e=e ,a ,b w be 9 ,h ,i 9 ! same "t 9 :i* ,a ,c is 9 ,e ,f1 s t ! body w at ! same "t be 9 ,e ,f & ,h ,i1 & !re=e 9 _! common po9t ,g4 ,& 9 ! same mann] x w be1 :]esoev] ! po9t ,e be tak5 betwe5 ,a & ,b4 ,:]e=e ! body w always be 9 ! diagonal ,a ,d2 :i* 0 to be demon/rat$4 ,coroll4 ,f h5ce x is manife/1 t ! body w be c>ri$ ?r\gh ! same /rait l9e ,a ,d1 ?\gh ! motion be n uni=m1 provid$ x h l accel]ation2 = ! proportion ( ,a ,b to ,a ,e w always be ! same ) t ( ,a ,c to ,a ,h4 ,sid5ote3 ,if a body be c>ri$ by two mov5ts toge!r1 "o ( !m be+ mov$ ) uni=m1 ! o!r ) accel]at$ motion1 & ! proportion ( _! l5g?s to _! times be+ explicable 9 numb]s1 h[ to f9d out :at l9e t body describes4 #i4 ,if a body be c>ri$ by two mov5ts toge!r1 :i* meet 9 any giv5 angle1 & are mov$1 ! "o uni=mly1 ! o!r ) motion uni=mly accel]at$ f re/1 t is1 t ! proportion ( _! impetus be z t ( _! times1 t is1 t ! proportion ( _! l5g?s be duplicate to t ( ! l9es ( _! times1 till ! l9e ( greate/ impetus acquir$ by accel]ation be equal to t ( ! l9e ( "t ( ! uni=m motion2 ! l9e 9 :i* ! body is c>ri$ w be ! crook$ l9e ( a semip>abola1 ^: base is ! impetus la/ acquir$1 & v]tex ! po9t ( re/4 ,let ! /raight l9e ,a ,b 79 fig4 #e7 be und]/ood to be mov$ ) uni=m motion to ,c ,d2 & let ano!r mov5t 9 ! /rait l9e ,a ,c be suppos$ to be mov$ 9 ! same "t to ,b ,d1 b ) motion uni=mly accel]at$1 t is1 ) su* motion1 t ! proportion ( ! spaces :i* are transmitt$ be always duplicate to t ( ! times1 till ! impetus acquir$ be ,b ,d equal to ! /rait l9e ,a ,c2 & let ! semip>abola ,a ,g ,d ,b be describ$4 ,i say t by ! conc\rse ( ^? two mov5ts1 ! body w be c>ri$ ?r\gh ! semip>abolical crook$ l9e ,a ,g ,d4 ,= let ! p>allelogram ,a ,b ,d ,c be complet$2 & f ! po9t ,e1 tak5 any:]e 9 ! /rait l9e ,a ,b1 let ,e ,f be drawn p>allel to ,a ,c & cutt+ ! crook$ l9e 9 ,g2 & la/ly1 ?r\gh ! po9t ,g let ,h ,i be drawn p>allel to ! /rait l9es ,a ,b & ,c ,d4 ,see+ !re=e ! proportion ( ,a ,b to ,a ,e is by supposition duplicate to ! proportion ( ,e ,f to ,e ,g1 t is1 ( ! "t ,a ,c to ! "t ,a ,h1 at ! same "t :5 ,a ,c is 9 ,e ,f1 ,a ,b w be 9 ,h ,i2 & !re=e ! mov$ body w be 9 ! common po9t ,g4 ,& s x w always be1 9 :at "p soev] ( ,a ,b ! po9t ,e be tak54 ,:]e=e ! mov$ body w always be f\nd 9 ! p>abolical l9e ,a ,g ,d2 :i* 0 to be demon/rat$4 #aj4 ,if a body be c>ri$ by two mov5ts toge!r1 :i* meet 9 any giv5 angle1 & are mov$ ! "o uni=mly1 ! o!r ) impetus increas+ f re/1 till x be equal to t ( ! uni=m motion1 & ) su* accel]ation1 t ! proportion ( ! l5g?s transmitt$ be e :]e triplicate to t ( ! times 9 :i* !y are transmitt$2 ! l9e1 9 :i* t body is mov$1 w be ! crook$ l9e ( ! fir/ semip>abola/] ( two means1 ^: base is ! impetus la/ acquir$4 ,let ! /rait l9e ,a ,b 79 ! #f? figure7 be mov$ uni=mly to ,c ,d2 & let ano!r mov5t ,a ,c be mov$ at ! same "t to ,b ,d ) motion s accel]at$1 t ! proportion ( ! l5g?s transmitt$ be ev]y:]e triplicate to ! proportion ( _! times2 & let ! impetus acquir$ 9 ! end ( t motion be ,b ,d1 equal to ! /rait l9e ,a ,c2 & la/ly1 let ,a ,g ,d be ! crook$ l9e ( ! fir/ semip>abola/] ( two means4 ,i say1 t by ! conc\rse ( ! two mov5ts toge!r1 ! body w be always 9 t crook$ l9e ,a ,g ,d4 ,= let ! p>allelogram ,a ,b ,d ,c be complet$2 & f ! po9t ,e1 tak5 any:]e 9 ! /rait l9e ,a ,b1 let ,e ,f be drawn p>allel to ,a ,c1 & cutt+ ! crook$ l9e 9 ,g2 & ?r\gh ! po9t ,g let ,h ,i be drawn p>allel to ! /rait l9es ,a ,b & ,c ,d4 ,see+ !re=e ! proportion ( ,a ,b to ,a ,e is1 by supposition1 triplicate to ! proportion ( ,e ,f to ,e ,g1 t is1 ( ! "t ,a ,c to ! "t ,a ,h1 at ! same "t :5 ,a ,c is 9 ,e ,f1 ,a ,b w be 9 ,h ,i2 & !re=e ! mov$ body w be 9 ! common po9t ,g4 ,& s x w always be1 9 :at "p soev] ( ,a ,b ! po9t ,e be tak52 & by consequ5t1 ! body w always be 9 ! crook$ l9e ,a ,g ,d2 :i* 0 to be demon/rat$4 #aa4 ,by ! same me?od x may be %[n1 :at l9e x is t is made by ! motion ( a body c>ri$ by ! conc\rse ( any two mov5ts1 :i* are mov$ "o ( !m uni=mly1 ! o!r ) accel]ation1 b 9 su* proportions ( spaces & times z are explicable by numb]s1 z duplicate1 triplicate1 c41 or su* z may be design$ by any brok5 numb] :atsoev]4 ,= :i* ?is is ! rule4 ,let ! two numb]s ( ! l5g? & "t be add$ toge!r2 & let _! sum be ! d5om9ator ( a fraction1 ^: num]ator mu/ be ! numb] ( ! l5g?4 ,seek ?is fraction 9 ! table ( ! ?ird article ( ! ,x,v,i,ith *apt]2 & ! l9e s\ght w be t1 :i* d5om9ates ! ?ree-sid$ figure not$ on ! left h&2 & ! k9d ( x w be t1 :i* is numb]$ above ov] ! fraction4 ,= example1 let !re be a conc\rse ( two mov5ts1 :]e( "o is mov$ uni=mly1 ! o!r ) motion s accel]at$1 t ! spaces are to ! times z #e to #c4 ,let a fraction be made ^: d5om9ator is ! sum ( #e & #c1 & ! num]ator #e1 namely ! fraction #eh4 ,seek 9 ! table1 & y w f9d #eh to be ! ?ird 9 t r[1 :i* belongs to ! ?ree-sid$ figure ( f\r means4 ,:]e=e ! l9e ( motion made by ! conc\rse ( two su* mov5ts1 z are la/ ( all describ$1 w be ! crook$ l9e ( ! ?ird p>abola/] ( f\r means4 #ab4 ,if motion be made by ! conc\rse ( two mov5ts1 :]e( "o is mov$ uni=mly1 ! o!r beg9n+ f re/ 9 ! angle ( conc\rse ) any accel]ation :atsoev]2 ! mov5t1 :i* is mov$ uni=mly1 %all put =w>d ! mov$ body 9 ! sev]al p>allel spaces1 less ?an if bo? ! mov5ts _h uni=m motion2 & /ill less & less1 z ! motion ( ! o!r mov5t is m & m accel]at$4 ,let ! body be plac$ 9 ,a1 79 ! #g? figure7 & be mov$ by two mov5ts1 by "o ) uni=m motion f ! /rait l9e ,a ,b to ! /rait l9e ,c ,d p>allel to x2 & by ! o!r ) any accel]ation1 f ! /rait l9e ,a ,c to ! /rait l9e ,b ,d p>allel to x2 & 9 ! p>allelogram ,a ,b ,d ,c let a space be tak5 betwe5 any two p>allels ,e ,f & ,g ,h4 ,i say1 t :il/ ! mov5t ,a ,c passes ?r\gh ! latitude :i* is betwe5 ,e ,f & ,g ,h1 ! body is less mov$ =w>ds f ,a ,b t[>ds ,c ,d1 ?an x w\ld h be51 if ! motion f ,a ,c to ,b ,d _h be5 uni=m4 ,= suppose t :il/ ! body is made to desc5d to ! p>allel ,e ,f by ! p[] ( ! mov5t f ,a ,c t[>ds ,b ,d1 ! same body 9 ! same "t is mov$ =w>ds to any po9t ,f 9 ! l9e ,e ,f1 by ! p[] ( ! mov5t f ,a ,b t[>ds ,c ,d2 & let ! /rait l9e ,a ,f be drawn & produc$ indet]m9ately1 cutt+ ,g ,h 9 ,h4 ,see+ !re=e1 x is z ,a ,e to ,a ,g1 s ,e ,f to ,g ,h2 if ,a ,c %\ld desc5d t[>ds ,b ,d ) uni=m motion1 ! body 9 ! "t ,g ,h1 7= ,i make ,a ,c & its p>allels ! measure ( "t17 w\ld be f\nd 9 ! po9t ,h4 ,b because ,a ,c is suppos$ to be mov$ t[>ds ,b ,d ) motion cont9ually accel]at$1 t is1 9 great] proportion ( space to space1 ?an ( "t to "t1 9 ! "t ,g ,h ! body w be 9 "s p>allel beyond x1 z betwe5 ,g ,h & ,b ,d4 ,suppose n[ t 9 ! end ( ! "t ,g ,h x be 9 ! p>allel ,i ,k1 & 9 ,i ,k let ,i ,l be tak5 equal to ,g ,h4 ,:5 !re=e ! body is 9 ! p>allel ,i ,k1 x w be 9 ! po9t ,l4 ,:]e=e :5 x 0 9 ! p>allel ,g ,h1 x 0 9 "s po9t betwe5 ,g & ,h1 z 9 ! po9t ,m2 b if bo? ! motions _h be5 uni=m1 x _h be5 9 ! po9t ,h2 & !re=e :il/ ! mov5t ,a ,c passes ov] ! latitude :i* is betwe5 ,e ,f & ,g ,h1 ! body is less mov$ =w>ds f ,a ,b t[>ds ,c ,d1 ?an x w\ld h be51 if bo? ! motions _h be5 uni=m2 :i* 0 to be demon/rat$4 #ac4 ,any l5g? be+ giv51 :i* is pass$ ?r\gh 9 a giv5 "t ) uni=m motion1 to f9d out :at l5g? %all be pass$ ?r\gh 9 ! same "t ) motion uni=mly accel]at$1 t is1 ) su* motion t ! proportion ( ! l5g?s pass$ ?r\gh be cont9ually duplicate to t ( _! times1 & t ! l9e ( ! impetus la/ acquir$ be equal to ! l9e ( ! :ole "t ( ! motion4 ,let ,a ,b 79 ! #h? figure7 be a l5g?1 transmitt$ ) uni=m motion 9 ! "t ,a ,c2 & let x be requir$ to f9d ano!r l5g?1 :i* %all be transmitt$ 9 ! same "t ) motion uni=mly accel]at$1 s t ! l9e ( ! impetus la/ acquir$ be equal to ! /rait l9e ,a ,c4 ,let ! p>allelogram ,a ,b ,d ,c be complet$2 & let ,b ,d be divid$ 9 ! middle at ,e2 & betwe5 ,b ,e & ,b ,d let ,b ,f be a mean proportional2 & let ,a ,f be drawn & produc$ till x meet ) ,c ,d produc$ 9 ,g2 & la/ly1 let ! p>allelogram ,a ,c ,g ,h be complet$4 ,i say1 ,a ,h is ! l5g? requir$4 ,= z duplicate proportion is to s+le proportion1 s let ,a ,h be to ,a ,i1 t is1 let ,a ,i be ! half ( ,a ,h2 & let ,i ,k be drawn p>allel to ! /rait l9e ,a ,c1 & cutt+ ! diagonal ,a ,d 9 ,k1 & ! /rait l9e ,a ,g 9 ,l4 ,see+ !re=e ,a ,i is ! half ( ,a ,h1 ,i ,l w also be ! half ( ,b ,d1 t is1 equal to ,b ,e2 & ,i ,k equal to ,b ,f2 = ,b ,d1 t is1 ,g ,h1 ,b ,f1 & ,b ,e1 t is1 ,i ,l1 be+ cont9ual proportionals1 ,a ,h1 ,a ,b & ,a ,i w also be cont9ual proportionals4 ,b z ,a ,b is to ,a ,i1 t is1 z ,a ,h is to ,a ,b1 s is ,b ,d to ,i ,k1 & s also is ,g ,h1 t is1 ,b ,d to ,b ,f2 & !re=e ,b ,f & ,i ,k are equal4 ,n[ ! proportion ( ,a ,h to ,a ,i is duplicate to ! proportion ( ,a ,b to ,a ,i1 t is1 to t ( ,b ,d to ,i ,k1 or ( ,g ,h to ,i ,k4 ,:]e=e ! po9t ,k w be 9 a p>abola1 ^: diamet] is ,a ,h1 & base ,g ,h1 :i* ,g ,h is equal to ,a ,c4 ,! body !re=e proce$+ f re/ 9 ,a1 ) motion uni=mly accel]at$ 9 ! "t ,a ,c1 :5 x has pass$ ?r\gh ! l5g? ,a ,h1 w acquire ! impetus ,g ,h equal to ! "t ,a ,c1 t is1 su* impetus1 z t ) x ! body w pass ?r\gh ! l5g? ,a ,c 9 ! "t ,a ,c4 ,:]e=e any l5g? be+ giv51 c41 :i* 0 prop\nd$ to be done4 #ad4 ,any l5g? be+ giv51 :i* 9 a giv5 "t is transmitt$ ) uni=m motion1 to f9d out :at l5g? %all be transmitt$ 9 ! same "t ) motion s accel]at$1 t ! l5g?s transmitt$ be cont9ually 9 triplicate proportion to t ( _! times1 & ! l9e ( ! impetus la/ ( all acquir$ be equal to ! l9e ( "t giv54 ,let ! giv5 l5g? ,a ,b 79 ! #i? figure7 be transmitt$ ) uni=m motion 9 ! "t ,a ,c2 & let x be requir$ to f9d :at l5g? %all be transmitt$ 9 ! same "t ) motion s accel]at$1 t ! l5g?s transmitt$ be cont9ually 9 triplicate proportion to t ( _! times1 & ! impetus la/ acquir$ be equal to ! "t giv54 ,let ! p>allelogram ,a ,b ,d ,c be complet$2 & let ,b ,d be s divid$ 9 ,e1 t ,b ,e be a ?ird "p ( ! :ole ,b ,d2 & let ,b ,f be a mean proportional betwe5 ,b ,d & ,b ,e2 & let ,a ,f be drawn & produc$ till x meet ! /rait l9e ,c ,d 9 ,g2 & la/ly1 let ! p>allelogram ,a ,c ,g ,h be complet$4 ,i say1 ,a ,h is ! l5g? requir$4 ,= z triplicate proportion is to s+le proportion1 s let ,a ,h be to ano!r l9e1 ,a ,i1 t is1 make ,a ,i a ?ird "p ( ! :ole ,a ,h2 & let ,i ,k be drawn p>allel to ! /rait l9e ,a ,c1 cutt+ ! diagonal ,a ,d 9 ,k1 & ! /rait l9e ,a ,g 9 ,l2 !n1 z ,a ,b is to ,a ,i1 s let ,a ,i be to ano!r1 ,a ,n2 & f ! po9t ,n let ,n ,q be drawn p>allel to ,a ,c1 cutt+ ,a ,g1 ,a ,d1 & ,f ,k produc$ 9 ,p1 ,m1 & ,o2 & la/ ( all1 let ,f ,o & ,l ,m be drawn1 :i* w be equal & p>allel to ! /rait l9es ,b ,n & ,i ,n4 ,by ?is con/ruction1 ! l5g?s transmitt$ ,a ,h1 ,a ,b1 ,a ,i1 & ,a ,n1 w be cont9ual proportionals2 &1 9 l mann]1 ! times ,g ,h1 ,b ,f1 ,i ,l & ,n ,p1 t is1 ,n ,q1 ,n ,o1 ,n ,m & ,n ,p1 w be cont9ual proportionals1 & 9 ! same proportion ) ,a ,h1 ,a ,b1 ,a ,i & ,a ,n4 ,:]e=e ! proportion ( ,a ,h1 ,a ,b1 ,a ,i & ,a ,n4 ,:]e=e ! proportion ( ,a ,h to ,a ,n is ! same ) t ( ,b ,d1 t is1 ( ,n ,q to ,n ,p2 & ! proportion ( ,n ,q to ,n ,p triplicate to t ( ,n ,q to ,n ,o1 t is1 triplicate to t ( ,b ,d to ,i ,k2 :]e=e also ! l5g? ,a ,h is to ! l5g? ,a ,n 9 triplicate proportion to t ( ! "t ,b ,d1 to ! "t ,i ,k2 & !re=e ! crook$ l9e ( ! fir/ ?ree-sid$ figure ( two means ^: diamet] is ,a ,h1 & base ,g ,h equal to ,a ,c1 %all pass ?r\gh ! po9t ,o2 & consequ5tly1 ,a ,h %all be transmitt$ 9 ! "t ,a ,c1 & %all h its la/ acquir$ impetus ,g ,h equal to ,a ,c1 & ! proportions ( ! l5g?s acquir$ 9 any ( ! times triplicate to ! proportions ( ! times !mselves4 ,:]e=e ,a ,h is ! l5g? requir$ to be f\nd out4 ,by ! same me?od1 if a l5g? be giv5 :i* is transmitt$ ) uni=m motion 9 any giv5 "t1 ano!r l5g? may be f\nd out :i* %all be transmitt$ 9 ! same "t ) motion s accel]at$1 t ! l5g?s transmitt$ %all be to ! times 9 :i* !y are transmitt$1 9 proportion quadruplicate1 qu9tuplicate1 & s on inf9itely4 ,= if ,b ,d be divid$ 9 ,e1 s t ,b ,d be to ,b ,e z #d to #a2 & !re be tak5 betwe5 ,b ,d & ,b ,e a mean proportional ,f ,b2 & z ,a ,h is to ,a ,b1 s ,a ,b be made to a ?ird1 & aga9 s t ?ird to a f\r?1 & t f\r? to a fif?1 ,a ,n1 s t ! proportion ( ,a ,h to ,a ,n be quadruplicate to t ( ,a ,h to ,a ,b1 & ! p>allelogram ,n ,b ,f ,o be complet$1 ! crook$ l9e ( ! fir/ ?ree-sid$ figure ( ?ree means w pass ?r\gh ! po9t ,o2 & consequ5tly1 ! body mov$ w acquire ! impetus ,g ,h equal to ,a ,c 9 ! "t ,a ,c4 ,& s ( ! re/4 #ae4 ,also1 if ! proportion ( ! l5g?s transmitt$ be to t ( _! times1 z any numb] to any numb]1 ! same me?od s]ves = ! f9d+ out ( ! l5g? transmitt$ ) su* impetus1 & 9 su* "t4 ,= let ,a ,c 79 ! #aj? figure7 be ! "t 9 :i* a body is transmitt$ ) uni=m motion f ,a to ,b2 & ! p>allelogram ,a ,b ,d ,c be+ complet$1 let x be requir$ to f9d out a l5g? 9 :i* t body may be mov$ 9 ! same "t ,a ,c f ,a1 ) motion s accel]at$1 t ! proportion ( ! l5g?s transmitt$ to t ( ! times be cont9ually z #c to #b4 ,let ,b ,d be s divid$ 9 ,e1 t ,b ,d be to ,b ,e z #c to #b2 & betwe5 ,b ,d & ,b ,e let ,b ,f be a mean proportional2 & let ,a ,f be drawn & produc$ till x meet ) ,c ,d produc$ 9 ,g2 & mak+ ,a ,m a mean proportional betwe5 ,a ,h & ,a ,b1 let x be z ,a ,m to ,a ,b1 s ,a ,b to ,a ,i2 & s ! proportion ( ,a ,h to ,a ,i w be to t ( ,a ,h to ,a ,b z #c to #b2 = ( ! proportions1 ( :i* t ( ,a ,h to ,a ,m is "o1 t ( ,a ,h to ,a ,b is two1 & t ( ,a ,h to ,a ,i is ?ree2 & consequ5tly1 z #c to #b to t ( ,g ,h to ,b ,f1 & 7,f ,k be+ drawn p>allel to ,b ,i & cutt+ ,a ,d 9 ,k7 s likewise to t ( ,g ,h or ,b ,d to ,i ,k4 ,:]e=e ! proportion ( ! l5g? ,a ,h to ,a ,i is to ! proportion ( ! "t ,b ,d to ,i ,k z #c to #b2 & !re=e if 9 ! "t ,a ,c ! body be mov$ ) accel]at$ motion1 z 0 prop\nd$1 till x acquire ! impetus ,h ,g equal to ,a ,c1 ! l5g? transmitt$ 9 ! same "t w be ,a ,h4 #af4 ,b if ! proportion ( ! l5g?s to t ( ! times _h be5 z #d to #c1 !re %\ld !n h be5 tak5 two mean proportionals betwe5 ,a ,h & ,a ,b1 & _! proportion %\ld h be5 cont9u$ "o t]m fur!r1 s t ,a ,h to ,a ,b might h ?ree ( ! same proportions1 ( :i* ,a ,h to ,a ,i has f\r2 & all ?+s else %\ld h be5 done z is already %[n4 ,n[ ! way h[ to int]pose any numb] ( means betwe5 two l9es giv51 is n yet f\nd out4 ,nev]!less ?is may /& = a g5]al rule2 if !re be a "t giv51 & a l5g? be transmitt$ 9 t "t ) uni=m motion2 z = example1 if ! "t be ,a ,c1 & ! l5g? ,a ,b1 ! /rait l9e ,a ,g1 :i* det]m9es ! l5g? ,c ,g or ,a ,h1 transmitt$ 9 ! same "t ,a ,c ) any accel]at$ motion1 %all s cut ,b ,d 9 ,f1 t ,b ,f %all be a mean proportional betwe5 ,b ,d & ,b ,e1 ,b ,e be+ s tak5 9 ,b ,d1 t ! proportion ( l5g? to l5g? be ev]y:]e to ! proportion ( "t to "t1 z ! :ole ,b ,d is to its "p ,b ,e4 #ag4 ,if 9 a giv5 "t two l5g?s be transmitt$1 "o ) uni=m motion1 ! o!r ) motion accel]at$ 9 any proportion ( ! l5g?s to ! times2 & aga91 9 "p ( ! same "t1 p>ts ( ! same l5g?s be transmitt$ ) ! same motions1 ! :ole l5g? w exce$ ! o!r l5g? 9 ! same proportion 9 :i* "o "p exce$s ! o!r "p4 ,= example1 let ,a ,b 79 ! #h? figure7 be a l5g? transmitt$ 9 ! "t ,a ,c1 ) uni=m motion2 & let ,a ,h be ano!r l5g? transmitt$ 9 ! same "t ) motion uni=mly accel]at$1 s t ! impetus la/ acquir$ be ,g ,h equal to ,a ,c2 & 9 ,a ,h let any "p ,a ,i be tak51 & transmitt$ 9 "p ( ! "t ,a ,c ) uni=m motion2 & let ano!r "p ,a ,b be tak5 & transmitt$ 9 ! same "p ( ! "t ,a ,c ) motion uni=mly accel]at$2 ,i say1 t z ,a ,h is to ,a ,b1 s w ,a ,b be to ,a ,i4 ,let ,b ,d be drawn p>allel & equal to ,h ,g1 & divid$ 9 ! mid/ at ,e1 & betwe5 ,b ,d & ,b ,e let a mean proportional be tak5 z ,b ,f2 & ! /rait l9e ,a ,g1 by ! demon/ration ( art4 #ac1 %all pass ?r\gh ,f4 ,& divid+ ,a ,h 9 ! mid/ at ,i1 ,a ,b %all be a mean proportional betwe5 ,a ,h & ,a ,i4 ,aga91 because ,a ,i & ,a ,b are describ$ by ! same motions1 if ,i ,k be drawn p>allel & equal to ,b ,f or ,a ,m1 & divid$ 9 ! mid/ at ,n1 & betwe5 ,i ,k & ,i ,n be tak5 ! mean proportional ,i ,l1 ! /rait l9e ,a ,f w1 by ! demon/ration ( ! same art4 #ac1 pass ?r\gh ,l4 ,& divid+ ,a ,b 9 ! mid/ at ,o1 ! l9e ,a ,i w be a mean proportional betwe5 ,a ,b & ,a ,o4 ,:]e ,a ,b is divid$ 9 ,i & ,o1 9 l mann] z ,a ,h is divid$ 9 ,b & ,i2 & z ,a ,h to ,a ,b1 s is ,a ,b to ,a ,i4 ,:i* 0 to be prov$4 ,coroll4 ,also z ,a ,h to ,a ,b1 s is ,h ,b to ,b ,i2 & s also ,b ,i to ,i ,o4 ,& z ?is1 :]e "o ( ! motions is uni=mly accel]at$1 is prov$ out ( ! demon/ration ( art4 #ac2 s1 :5 ! accel]ations are 9 d\ble proportion to ! times1 ! same may be prov$ by ! demon/ration ( art4 #ad2 & by ! same me?od 9 all o!r accel]ations1 ^: proportions to ! times are explicable 9 numb]s4 #ah4 ,if two sides1 :i* conta9 an angle 9 any p>allelogram1 be mov$ 9 ! same "t to ! sides opposite to !m1 "o ( !m ) uni=m motion1 ! o!r ) motion uni=mly accel]at$2 t side1 :i* is mov$ uni=mly1 w affect z mu* ) its conc\rse ?r\gh ! :ole l5g? transmitt$1 z x w\ld d if ! o!r motion 7 also uni=m1 & ! l5g? transmitt$ by x 9 ! same "t 7 a mean proportional betwe5 ! :ole & ! half4 ,let ! side ,a ,b ( ! p>allelogram ,a ,b ,d ,c1 79 ! #aa? figure7 be und]/ood to be mov$ ) uni=m motion till x be co9cid5t ) ,c ,d2 & let ! "t ( t motion be ,a ,c or ,b ,d4 ,also 9 ! same "t let ! side ,a ,c be und]/ood to be mov$ ) motion uni=mly accel]at$1 till x be co9cid5t ) ,b ,d2 !n divid+ ,a ,b 9 ! middle 9 ,e1 let ,a ,f be made a mean proportional betwe5 ,a ,b & ,a ,e2 & draw+ ,f ,g p>allel to ,a ,c1 let ! side ,a ,c be und]/ood to be mov$ 9 ! same "t ,a ,c ) uni=m motion till x be co9cid5t ) ,f ,g4 ,i say1 ! :ole ,a ,b conf]s z mu* to ! velocity ( ! body plac$ 9 ,a1 :5 ! motion ( ,a ,c is uni=mly accel]at$ till x comes to ,b ,d1 z ! "p ,a ,f conf]s to ! same1 :5 ! side ,a ,c is mov$ uni=mly & 9 ! same "t to ,f ,g4 ,= see+ ,a ,f is a mean proportional betwe5 ! :ole ,a ,b & its half ,a ,e1 ,b ,d w 7by ! #ac? article7 be ! la/ impetus acquir$ by ,a ,c1 ) motion uni=mly accel]at$ till x come to ! same ,b ,d2 & consequ5tly1 ! /rait l9e ,f ,b w be ! excess1 by :i* ! l5g?1 transmitt$ by ,a ,c ) motion uni=mly accel]at$1 w exce$ ! l5g? transmitt$ by ! same ,a ,c 9 ! same "t ) uni=m motion1 & ) impetus e :]e equal to ,b ,d4 ,:]e=e1 if ! :ole ,a ,b be mov$ uni=mly to ,c ,d 9 ! same "t 9 :i* ,a ,c is mov$ uni=mly to ,f ,g1 ! "p ,f ,b1 see+ x concurs n at all ) ! motion ( ! side ,a ,c :i* is suppos$ to be mov$ only to ,f ,g1 w conf] no?+ to its motion4 ,aga91 suppos+ ! side ,a ,c to be mov$ to ,b ,d ) motion uni=mly accel]at$1 ! side ,a ,b ) its uni=m motion to ,c ,d w less put =w>ds ! body :5 x is accel]at$ 9 all ! p>allels1 ?an :5 x is n at all accel]at$2 & by h[ mu* ! great] ! accel]ation is1 by s mu* ! less x w put x =w>ds1 z is %[n 9 ! #ab? article4 ,:5 !re=e ,a ,c is 9 ,f ,g ) accel]at$ motion1 ! body w n be 9 ! side ,c ,d at ! po9t ,g1 b at ! po9t ,d2 s t ,g ,d w be ! excess1 by :i* ! l5g? transmitt$ ) accel]at$ motion to ,b ,d exce$s ! l5g? transmitt$ ) uni=m motion to ,f ,g2 s t ! body by its accel]ation avoids ! action ( ! "p ,a ,f1 & comes to ! side ,c ,d 9 ! "t ,a ,c1 & makes ! l5g? ,c ,d1 :i* is equal to ! l5g? ,a ,b4 ,:]e=e uni=m motion f ,a ,b to ,c ,d 9 ! "t ,a ,c1 works no m 9 ! :ole l5g? ,a ,b ^u ! body uni=mly accel]at$ f ,a ,c to ,b ,d1 ?an if ,a ,c 7 mov$ 9 ! same "t ) uni=m motion to ,f ,g2 ! diff]5ce consi/+ only 9 ?is1 t :5 ,a ,b works ^u ! body uni=mly mov$ f ,a ,c to ,f ,g1 t1 by :i* ! accel]at$ motion exce$s ! uni=m motion1 is altoge!r 9 ,f ,b or ,g ,d2 b :5 ! same ,a ,b works ^u ! body accel]at$1 t1 by :i* ! accel]at$ motion exce$s ! uni=m motion1 is disp]s$ ?r\gh ! :ole l5g? ,a ,b or ,c ,d1 yet1 s t if x 7 collect$ & put toge!r1 x w\ld be equal to ! same ,f ,b or ,g ,d4 ,:]e=e1 if two sides :i* conta9 an angle1 c42 :i* 0 to be demon/rat$4 #ai4 ,if two transmitt$ l5g?s h to _! times any o!r proportion explicable by numb]1 & ! side ,a ,b be s divid$ 9 ,e1 t ,a ,b be to ,a ,e 9 ! same proportion :i* ! l5g?s transmitt$ h to ! times 9 :i* !y are transmitt$1 & betwe5 ,a ,b & ,a ,e !re be tak5 a mean proportional ,a ,f2 x may be %[n by ! same me?od1 t ! side1 :i* is mov$ ) uni=m motion1 works z mu* ) its conc\rse ?r\gh ! :ole l5g? ,a ,b1 z x w\ld d if ! o!r motion 7 also uni=m1 & ! l5g? transmitt$ 9 ! same "t ,a ,c 7 t mean proportional ,a ,f4 ,& ?us mu* conc]n+ motion by conc\rse4 ---------------------------------------- -------------------------------- ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xvi4 ,fig4 #a-aa ---------------------------------------- -------------------------------- ------- ,,*ap4 ,,xvii4 ,,( ,,figures ,,defici5t4 #a4 ,def9itions ( a defici5t figure2 ( a complete figure2 ( ! complem5t ( a defici5t figure2 & ( proportions :i* are proportional & comm5surable to "o ano!r4--#b4 ,! proportion ( a defici5t figure to its complem5t4--#c4 ,! proportions ( defici5t figures to ! p>allelograms 9 :i* !y are describ$1 set =? 9 a table4--#d4 ,! description & production ( ! same figures4--#e4 ,! draw+ ( tang5ts to !m4--#f4 ,9 :at proportion ! same figures exce$ a /rait-l9$ triangle ( ! same altitude & base4--#g4 ,a table ( solid defici5t figures describ$ 9 a cyl9d]4--#h4 ,9 :at proportion ! same figures exce$ a cone ( ! same altitude & base4--#i4 ,h[ a pla9 defici5t figure may be describ$ 9 a p>allelogram1 s t x be to a triangle ( ! same base & altitude1 z ano!r defici5t figure1 pla9 or solid1 twice tak51 is to ! same defici5t figure1 toge!r ) ! complete figure 9 :i* x is describ$4--#aj4 ,! transf]r+ ( c]ta9 prop]ties ( defici5t figures describ$ 9 a p>allelogram to ! proportions ( ! spaces transmitt$ ) sev]al degrees ( velocity4--#aa4 ,( defici5t figures describ$ 9 a circle4--#ab4 ,! proposition demon/rat$ 9 art4 #b confirm$ f ! elem5ts ( philosophy4--#ac4 ,an unusual way ( reason+ conc]n+ ! equality betwe5 ! sup]ficies ( a portion ( a sph]e & a circle4--#ad4 ,h[ f ! description ( defici5t figures 9 a p>allelogram1 any numb] ( mean proportionals may be f\nd out betwe5 two giv5 /rait l9es4 ,sid5ote3 ,def9ition ( a defici5t figure4 #a4 ,i call ^? defici5t figures :i* may be und]/ood to be g5]at$ by ! uni=m motion ( "s quantity1 :i* decreases cont9ually1 till at la/ x h no magnitude at all4 ,sid5ote3 ,def9itions ( a complete figure2 ( ! complem5t ( a defici5t figure2 & ( proportions :i* are proportional comm5surable to "o ano!r4 ,& ,i call t a complete figure1 answ]+ to a defici5t figure1 :i* is g5]at$ ) ! same motion & 9 ! same "t1 by a quantity :i* reta9s always its :ole magnitude4 ,! complem5t ( a defici5t figure is t :i* be+ add$ to ! defici5t figure makes x complete4 ,f\r proportions are said to be proportional1 :5 ! fir/ ( !m is to ! second z ! ?ird is to ! f\r?4 ,= example1 if ! fir/ proportion be duplicate to ! second1 & aga91 ! ?ird be duplicate to ! f\r?1 ^? proportions are said to be proportional4 ,& comm5surable proportions are ^?1 :i* are to "o ano!r z numb] to numb]4 ,z :5 to a proportion giv51 "o proportion is duplicate1 ano!r triplicate1 ! duplicate proportion w be to ! triplicate proportion z #b to #c2 b to ! giv5 proportion x w be z #b to #a2 & !re=e ,i call ^? ?ree proportions comm5surable4 ,sid5ote3 ,! proportion ( a defici5t figure to its complem5t4 #b4 ,a defici5t figure1 :i* is made by a quantity cont9ually decreas+ to no?+ by proportions ev]y:]e proportional & comm5surable1 is to its complem5t1 z ! proportion ( ! :ole altitude to an altitude dim9i%$ 9 any "t is to ! proportion ( ! :ole quantity1 :i* describes ! figure1 to ! same quantity dim9i%$ 9 ! same "t4 ,let ! quantity ,a ,b 79 fig4 #a71 by its motion ?r\gh ! altitude ,a ,c1 describe ! complete figure ,a ,d2 & aga91 let ! same quantity1 by decreas+ cont9ually to no?+ 9 ,c1 describe ! defici5t figure ,a ,b ,e ,f ,c1 ^: complem5t w be ! figure ,b ,d ,c ,f ,e4 ,n[ let ,a ,b be suppos$ to be mov$ till x lie 9 ,g ,k1 s t ! altitude dim9i%$ be ,g ,c1 & ,a ,b dim9i%$ be ,g ,e2 & let ! proportion ( ! :ole altitude ,a ,c to ! dim9i%$ altitude ,g ,c1 be1 = example1 triplicate to ! proportion ( ! :ole quantity ,a ,b or ,g ,k to ! dim9i%$ quantity ,g ,e4 ,& 9 l mann]1 let ,h ,i be tak5 equal to ,g ,e1 & let x be dim9i%$ to ,h ,f2 & let ! proportion ( ,g ,c to ,h ,c be triplicate to t ( ,h ,i to ,h ,f2 & let ! same be done 9 z _m p>ts ( ! /rait l9e ,a ,c z is possible2 & a l9e be drawn ?r\gh ! po9ts ,b1 ,e1 ,f & ,c4 ,i say ! defici5t figure ,a ,b ,e ,f ,c is to its complem5t ,b ,d ,c ,f ,e z #c to #a1 or z ! proportion ( ,a ,c to ,g ,c is to ! proportion ( ,a ,b1 t is1 ( ,g ,k to ,g ,e4 ,= 7by art4 #b1 *apt] ,,xv47 ! proportion ( ! complem5t ,b ,e ,f ,c ,d to ! defici5t figure ,a ,b ,e ,f ,c is all ! proportions ( ,d ,b to ,b ,a1 ,o ,e to ,e ,g1 ,q ,f to ,f ,h1 & ( all ! l9es p>allel to ,d ,b t]m9at$ 9 ! l9e ,b ,e ,f ,c1 to all ! p>allels to ,a ,b t]m9at$ 9 ! same po9ts ( ! l9e ,b ,e ,f ,c4 ,& see+ ! proportions ( ,d ,b to ,o ,e1 & ( ,d ,b to ,q ,f c4 are ev]y:]e triplicate ( ! proportions ( ,a ,b to ,g ,e1 & ( ,a ,b to ,h ,f c4 ! proportions ( ,h ,f to ,a ,b1 & ( ,g ,e to ,a ,b c4 7by art4 #af1 *ap4 ,,xiii471 are triplicate ( ! proportions ( ,q ,f to ,d ,b1 & ( ,o ,e to ,d ,b c4 & !re=e ! defici5t figure ,a ,b ,e ,f ,c1 :i* is ! aggregate ( all ! l9es ,h ,f1 ,g ,e1 ,a ,b1 c4 is triple to ! complem5t ,b ,e ,f ,c ,d made ( all ! l9es ,q ,f1 ,o ,e1 ,d ,b1 c42 :i* 0 to be prov$4 ,x foll[s f h5ce1 t ! same complem5t ,b ,e ,f ,c ,d is #ad ( ! :ole p>allelogram4 ,& by ! same me?od may be calculat$ 9 all o!r defici5t figures1 g5]at$ z above decl>$1 ! proportion ( ! p>allelogram to ei!r ( its p>ts2 z t :5 ! p>allels increase f a po9t 9 ! same proportion1 ! p>allelogram w be divid$ into two equal triangles2 :5 "o increase is d\ble to ! o!r1 x w be divid$ into a semip>abola & its complem5t1 or into #b & #a4 ,! same con/ruction /&+1 ! same conclusion may o!rwise be demon/rat$ ?us4 ,let ! /rait l9e ,c ,b be drawn cutt+ ,g ,k 9 ,l1 & ?r\gh ,l let ,m ,n be drawn p>allel to ! /rait l9e ,a ,c2 :]e=e ! p>allelograms ,g ,m & ,l ,d w be equal4 ,!n let ,l ,k be divid$ into ?ree equal p>ts1 s t x may be to "o ( ^? p>ts 9 ! same proportion :i* ! proportion ( ,a ,c to ,g ,c1 or ( ,g ,k to ,g ,l1 ha? to ! proportion ( ,g ,k to ,g ,e4 ,!re=e ,l ,k w be to "o ( ^? ?ree p>ts z ! ari?metical proportion betwe5 ,g ,k & ,g ,l is to ! ari?metical proportion betwe5 ,g ,k & ! same ,g ,k want+ ! ?ird "p ( ,l ,k2 & ,k ,e w be some:at great] ?an a ?ird ( ,l ,k4 ,see+ n[ ! altitude ,a ,g or ,m ,l is1 by reason ( ! cont9ual decrease1 to be suppos$ less ?an any quantity t c be giv52 ,l ,k1 :i* is int]cept$ betwe5 ! diagonal ,b ,c & ! side ,b ,d1 w be also less ?an any quantity t c be giv52 & consequ5tly1 if ,g be put s ne> to ,a 9 g1 z t ! diff]5ce betwe5 ,c g & ,c ,a be less ?an any quantity t c be assign$1 ! diff]5ce also betwe5 ,c l 7remov+ ,l to l7 & ,c ,b1 w be less ?an any quantity t c be assign$2 & ! l9e g l be+ drawn & produc$ to ! l9e ,b ,d 9 k1 cutt+ ! crook$ l9e 9 e1 ! proportion ( ,g k to ,g l w /ill be triplicate to ! proportion ( ,g k to ,g e1 & ! diff]5ce betwe5 k & e1 ! ?ird "p ( k l1 w be less ?an any quantity t c be giv52 & !re=e ! p>allelogram e ,d w diff] f a ?ird "p ( ! p>allelogram ,a e by a less diff]5ce ?an any quantity t c be assign$4 ,aga91 let ,h ,i be drawn p>allel & equal to ,g ,e1 cutt+ ,c ,b 9 ,p1 ! crook$ l9e 9 ,f1 & ,o ,e 9 ,i1 & ! proportion ( ,c g to ,c ,h w be triplicate to ! proportion ( ,h ,f to ,h ,p1 & ,i ,f w be great] ?an ! ?ird "p ( ,p ,i4 ,b aga91 sett+ ,h 9 h s ne> to g1 z t ! diff]5ce betwe5 ,c h & ,c g may be b z a po9t1 ! po9t ,p w also 9 p be s ne> to l1 z t ! diff]5ce betwe5 ,c p & ,c l w be b z a po9t2 & draw+ h p till x meet ) ,b ,d 9 i1 cutt+ ! crook$ l9e 9 f & hav+ drawn e o p>allel to ,b ,d1 cutt+ ,d ,c 9 o1 ! p>allelogram f o w diff] less f ! ?ird "p ( ! p>allelogram g f1 ?an by any quantity t c be giv54 ,& s x w be 9 all o!r spaces g5]at$ 9 ! same mann]4 ,:]e=e ! diff]5ces ( ! ari?metical & geometrical means1 :i* are b z s _m po9ts ,b1 e1 f1 c4 7see+ ! :ole figure is made up ( s _m indivisible spaces7 w con/itute a c]ta9 l9e1 su* z is ! l9e ,b ,e ,f ,c1 :i* w divide ! complete figure ,a ,d into two p>ts1 :]e( "o1 namely1 ,a ,b ,e ,f ,c1 :i* ,i call a defici5t figure1 is triple to ! o!r1 namely1 ,b ,d ,c ,f ,e1 :i* ,i call ! complem5t !re(4 ,& :]eas ! proportion ( ! altitudes to "o ano!r is 9 ?is case ev]y:]e triplicate to t ( ! decreas+ quantities to "o ano!r2 9 ! same mann]1 if ! proportion ( ! altitudes _h be5 ev]y:]e quadruplicate to t ( ! decreas+ quantities1 x might h be5 demon/rat$ t ! defici5t figure _h be5 quadruple to its complem5t2 & s 9 any o!r proportion4 ,:]e=e1 a defici5t figure1 :i* is made1 c4 :i* 0 to be demon/rat$4 ,! same rule holde? also 9 ! dim9ution ( ! bases ( cyl9d]s1 z is demon/rat$ 9 ! second article ( *apt] ,,xv4 ,sid5ote3 ,! proportion ( defici5t figures to ! p>allelograms 9 :i* !y are describ$1 set =? 9 a table4 #c4 ,by ?is proposition1 ! magnitudes ( all defici5t figures1 :5 ! proportions by :i* _! bases decrease cont9ually are proportional to ^? by :i* _! altitudes decrease1 may be comp>$ ) ! magnitudes ( _! complem5ts2 & consequ5tly1 ) ! magnitudes ( _! complete figures4 ,& !y w be f\nd to be1 z ,i h set !m d[n 9 ! foll[+ tables2 9 :i* ,i comp>e a p>allelogram ) ?ree-sid$ figures2 & fir/1 ) a /rait-l9$ triangle1 made by ! base ( ! p>allelogram cont9ually decreas+ 9 su* mann]1 t ! altitudes be always 9 proportion to "o ano!r z ! bases are1 & s ! triangle w be equal to its complem5t2 or ! proportions ( ! altitudes & bases w be z #a to #a1 & !n ! triangle w be half ! p>allelogram4 ,secondly1 ) t ?ree-sid$ figure :i* is made by ! cont9ual decreas+ ( ! bases 9 subduplicate proportion to t ( ! altitudes2 & s ! defici5t figure w be d\ble to its complem5t1 & to ! p>allelogram z #b to #c4 ,!n1 ) t :]e ! proportion ( ! altitudes is triplicate to t ( ! bases2 & !n ! defici5t figure w be triple to its complem5t1 & to ! p>allelogram z #c to #d4 ,also ! proportion ( ! altitudes to t ( ! bases may be z #c to #b2 & !n ! defici5t figure w be to its complem5t z #c to #b1 & to ! p>allelogram z #c to #e2 & s =w>ds1 accord+ z m mean proportionals are tak51 or z ! proportions are m multipli$1 z may be se5 9 ! foll[+ table4 ,= example1 if ! bases decrease s1 t ! proportion ( ! altitudes to t ( ! bases be always z #e to #b1 & x be dem&$ :at proportion ! figure made has to ! p>allelogram1 :i* is suppos$ to be unity2 !n1 see+ t :]e ! proportion is tak5 five times1 !re mu/ be f\r means2 look 9 ! table among/ ! ?ree-sid$ figures ( f\r means1 & see+ ! proportion 0 z #e to #b1 look 9 ! upp]mo/ r[ = ! numb] #b1 & desc5d+ 9 ! second column till y meet ) t ?ree-sid$ figure1 y w f9d #eg2 :i* %[s t ! defici5t figure is to ! p>allelogram z #eg to #a1 or z #e to #g4 #a #b #c #d #e #f #g ,p>allelogram #a 3 3 3 3 3 3 ---- ,/rait-sid$ triangle #a_/b 3 3 3 3 3 3 ---- ,?ree-sid$ figure ( #a mean #b_/c 3 3 3 3 3 3 -------- ,?ree-sid$ figure ( #b means #c_/d #c_/e 3 3 3 3 3 ------------ ,?ree-sid$ figure ( #c means #d_/e #d_/f #d_/g 3 3 3 3 ---------------- ,?ree-sid$ figure ( #d means #e_/f #e_/g #e_/he_/i 3 3 3 -------------------- ,?ree-sid$ figure ( #e means #f_/g #f_/h #f_/if_/ajf_/aa 3 3 ------------------------ ,?ree-sid$ figure ( #f means #g_/h #g_/ig_/aag_/aag_/abg_/ac 3 ---------------------------- ,?ree-sid$ figure ( #g means #h_/ih_/ajh_/aah_/abh_/ach_/adh_/ae ---------------------------- ,sid5ote3 ,description production ( ! same figures4 #d4 ,n[ = ! bett] und]/&+ ( ! nature ( ^! ?ree-sid$ figures1 ,i w %[ h[ !y may be describ$ by po9ts2 & fir/1 ^? :i* are 9 ! fir/ column ( ! table4 ,any p>allelogram be+ describ$1 z ,a ,b ,c ,d 79 figure #b7 let ! diagonal ,b ,d be drawn2 & ! /rait-l9$ triangle ,b ,c ,d w be half ! p>allelogram2 !n let any numb] ( l9es1 z ,e ,f1 be drawn p>allel to ! side ,b ,c1 & cutt+ ! diagonal ,b ,d 9 ,g2 & let x be ev]y:]e1 z ,e ,f to ,e ,g1 s ,e ,g to ano!r1 ,e ,h2 & ?r\gh all ! po9ts ,h let ! l9e ,b ,h ,h ,d be drawn2 & ! figure ,b ,h ,h ,d ,c w be t :i* ,i call a ?ree-sid$ figure ( "o mean1 because 9 ?ree proportionals1 z ,e ,f1 ,e ,g & ,e ,h1 !re is b "o mean1 namely1 ,e ,g2 & ?is ?ree-sid$ figure w be #bc ( ! p>allelogram1 & is call$ a p>abola4 ,aga91 let x be z ,e ,g to ,e ,h1 s ,e ,h to ano!r1 ,e ,i1 & let ! l9e ,b ,i ,i ,d be drawn1 mak+ ! ?ree-sid$ figure ,b ,i ,i ,d ,c2 & ?is w be #cd ( ! p>allelogram1 & is by _m call$ a cubic p>abola4 ,9 l mann]1 if ! proportions be fur!r cont9u$ 9 ,e ,f1 !re w be made ! re/ ( ! ?ree-sid$ figures ( ! fir/ column2 :i* ,i ?us demon/rate4 ,let !re be drawn /rait l9es1 z ,h ,k & ,g ,l1 p>allel to ! base ,d ,c4 ,see+ !re=e ! proportion ( ,e ,f to ,e ,h is duplicate to t ( ,e ,f to ,e ,g1 or ( ,b ,c to ,b ,l1 t is1 ( ,c ,d to ,l ,g1 or ( ,k ,m 7produc+ ,k ,h to ,a ,d 9 ,m7 to ,k ,h1 ! proportion ( ,b ,c to ,b ,k w be duplicate to t ( ,k ,m to ,k ,h2 b z ,b ,c is to ,b ,k1 s is ,d ,c or ,k ,m to ,k ,n1 & !re=e ! proportion ( ,k ,m to ,k ,n is duplicate to t ( ,k ,m to ,k ,h2 & s x w be :]esoev] ! p>allel ,k ,m be plac$4 ,:]e=e ! figure ,b ,h ,h ,d ,c is d\ble to its complem5t ,b ,h ,h ,d ,a1 & consequ5tly #bc ( ! :ole p>allelogram4 ,9 ! same mann]1 if ?r\gh ,i be drawn ,o ,p ,i ,q p>allel & equal to ,c ,d1 x may be demon/rat$ t ! proportion ( ,o ,q to ,o ,p1 t is1 ( ,b ,c to ,b ,o1 is triplicate t ( ,o ,q to ,o ,i1 & !re=e t ! figure ,b ,i ,i ,d ,c is triple to its complem5t ,b ,i ,i ,d ,a1 & consequ5tly ( ! :ole p>allelogram1 c4 ,secondly1 su* ?ree-sid$ figures z are 9 any ( ! transv]se r[s1 may be ?us describ$4 ,let ,a ,b ,c ,d 79 fig4 #c7 be a p>allelogram1 ^: diagonal is ,b ,d4 ,i w\ld describe 9 x su* figures1 z 9 ! prec$+ table ,i call ?ree-sid$ figures ( ?ree means4 ,p>allel to ,d ,c1 ,i draw ,e ,f z (t5 z is necess>y1 cutt+ ,b ,d 9 ,g2 & betwe5 ,e ,f & ,e ,g1 ,i take ?ree proportionals ,e ,h1 ,e ,i & ,e ,k4 ,if n[ !re be drawn l9es ?r\gh all ! po9ts ,h1 ,i & ,k1 t ?r\gh all ! po9ts ,h w make ! figure ,b ,h ,d ,c1 :i* is ! fir/ ( ^? ?ree-sid$ figures2 & t ?r\gh all ! po9ts ,i1 w make ! figure ,b ,i ,d ,c1 :i* is ! second2 & t :i* is drawn ?r\gh all ! po9ts ,k1 w make ! figure ,b ,k ,d ,c ! ?ird ( ^? ?ree-sid$ figures4 ,! fir/ ( ^!1 see+ ! proportion ( ,e ,f to ,e ,g is quadruplicate ( t ,e ,f to ,e ,h1 w be to its complem5t z #d to #a1 & to ! p>allelogram z #d to #e4 ,! second1 see+ ! proportion ( ,e ,f to ,e ,g is to t ( ,e ,f to ,e ,i z #d to #b1 w be d\ble to its complem5t1 & #df or #bc ( ! p>allelogram4 ,! ?ird1 see+ ! proportion ( ,e ,f to ,e ,g is t ( ,e ,f to ,e ,k z #d to #c1 w be to its complem5t z #d to #c1 & to ! p>allelogram z #d to #g4 ,any ( ^! figures be+ describ$ may be produc$ at pleasure1 ?us2 let ,a ,b ,c ,d 79 fig4 #d7 be a p>allelogram1 & 9 x let ! figure ,b ,k ,d ,c be describ$1 namely1 ! ?ird ?ree-sid$ figure ( ?ree means4 ,let ,b ,d be produc$ indef9itely to ,e1 & let ,e ,f be made p>allel to ! base ,d ,c1 cutt+ ,a ,d produc$ 9 ,g1 & ,b ,c produc$ 9 ,f2 & 9 ,g ,e let ! po9t ,h be s tak51 t ! proportion ( ,f ,e to ,f ,g may be quadruplicate to t ( ,f ,e to ,f ,h1 :i* may be done by mak+ ,f ,h ! greate/ ( ?ree proportionals betwe5 ,f ,e & ,f ,g2 ! crook$ l9e ,b ,k ,d produc$1 w pass ?r\gh ! po9t ,h4 ,= if ! /rait l9e ,b ,h be drawn1 cutt+ ,c ,d 9 ,i1 & ,h ,l be drawn p>allel to ,g ,d1 & meet+ ,c ,d produc$ 9 ,l2 x w be z ,f ,e to ,f ,g1 s ,c ,l to ,c ,i1 t is1 9 quadruplicate proportion to t ( ,f ,e to ,f ,h1 or ( ,c ,d to ,c ,i4 ,:]e=e if ! l9e ,b ,k ,d be produc$ accord+ to its g5]ation1 x w fall ^u ! po9t ,h4 ,sid5ote3 ,! draw+ ( tang5ts to !m4 #e4 ,a /rait l9e may be drawn s z to t\* ! crook$ l9e ( ! said figure 9 any po9t1 9 ?is mann]4 ,let x be requir$ to draw a tang5t to ! l9e ,b ,k ,d ,h 79 fig4 #d7 9 ! po9t ,d4 ,let ! po9ts ,b & ,d be connect$1 & draw+ ,d ,a equal & p>allel to ,b ,c1 let ,b & ,a be connect$2 & because ?is figure is by con/ruction ! ?ird ( ?ree means1 let !re be tak5 9 ,a ,b ?ree po9ts1 s1 t by !m ! same ,a ,b be divid$ into f\r equal p>ts2 ( :i* take ?ree1 namely1 ,a ,m1 s t ,a ,b may be to ,a ,m1 z ! figure ,b ,k ,d ,c is to its complem5t4 ,i say1 ! /rait l9e ,m ,d w t\* ! figure 9 ! po9t giv5 ,d4 ,= let !re be drawn any:]e betwe5 ,a ,b & ,d ,c a p>allel1 z ,r ,q1 cutt+ ! /rait l9e ,b ,d1 ! crook$ l9e ,b ,k ,d1 ! /rait l9e ,m ,d1 & ! /rait l9e ,a ,d1 9 ! po9ts ,p1 ,k1 ,o & ,q4 ,r ,k w !re=e1 by con/ruction1 be ! lea/ ( ?ree means 9 geometrical proportion betwe5 ,r ,q & ,r ,p4 ,:]e=e 7by coroll4 ( art4 #bh1 *apt] ,,xiii47 ,r ,k w be less ?an ,r ,o2 & !re=e ,m ,d w fall )\t ! figure4 ,n[ if ,m ,d be produc$ to ,n1 ,f ,n w be ! greate/ ( ?ree means 9 ari?metical proportion betwe5 ,f ,e & ,f ,g2 & ,f ,h w be ! greate/ ( ?ree means 9 geometrical proportion betwe5 ! same ,f ,e & ,f ,g4 ,:]e=e 7by ! same coroll4 ( art4 #bh1 *apt] ,,xiii47 ,f ,h w be less ?an ,f ,n2 & !re=e ,d ,n w fall )\t ! figure1 & ! /rait l9e ,m ,n w t\* ! same figure only 9 ! po9t ,d4 ,sid5ote3 ,9 :at proportion ! same figures exce$ a /rait-l9$ triangle ( ! same altitude & base4 #f4 ,! proportion ( a defici5t figure to its complem5t be+ kn[n1 x may also be kn[n :at proportion a /rait-l9$ triangle has to ! excess ( ! defici5t figure above ! same triangle2 & ^! proportions ,i h set d[n 9 ! foll[+ table2 :]e if y seek1 = example1 h[ mu* ! f\r? ?ree-sid$ figure ( five means exce$s a triangle ( ! same altitude & base1 y w f9d 9 ! conc\rse ( ! f\r? column ) ! ?ree-sid$ figures ( five means #baj2 by :i* is signifi$1 t t ?ree-sid$ figure exce$s ! triangle by two-t5?s or by "o-fif? "p ( ! same triangle4 #a #b #c #d #e #f #g ,! triangle #a 3 3 3 3 3 3 --- ,a ?ree-sid$ fig4 ( #a mean #a_/c 3 3 3 3 3 3 ------- ,a ?ree-sid$ fig4 ( #b means#b_/d #a_/e 3 3 3 3 3 ----------- ,! ,a ?ree-sid$ fig4 ( #c means#c_/e #b_/f #a_/g 3 3 3 3 --------------- excess,a ?ree-sid$ fig4 ( #d means#d_/f #c_/g #b_/h #a_/i 3 3 3 ------------------- ( ,a ?ree-sid$ fig4 ( #e means#e_/g #d_/h #c_/ib_/aja_/aa 3 3 ----------------------- ,a ?ree-sid$ fig4 ( #f means#f_/h #e_/id_/ajc_/aab_/aba_/ac 3 --------------------------- ,a ?ree-sid$ fig4 ( #g means#g_/if_/aje_/aad_/abc_/acb_/ada_/ae --------------------------- ,sid5ote3 ,a table ( solid defici5t figures describ$ 9 a cyl9d]4 #g4 ,9 ! next table are set d[n ! proportion ( a cone & ! solids ( ! said ?ree-sid$ figures1 namely1 ! proportions betwe5 !m & a cyl9d]4 ,z = example1 9 ! conc\rse ( ! second column ) ! ?ree-sid$ figures ( f\r means1 y h #ei2 :i* gives y to und]/&1 t ! solid ( ! second ?ree-sid$ figure ( f\r means is to ! cyl9d] z #ei to #a1 or z #e to #i4 #a #b #c #d #e #f #g ,a cyl9d] #a 3 3 3 3 3 3 ---- 3 3 3 3 3 3 ,a cone #a_/c 3 3 3 3 3 3 -------- 3 3 3 3 3 ,a ?ree-sid$ fig4 ( #a mean #b_/d 3 3 3 3 3 -------- 3 3 3 3 3 ,! ,a ?ree-sid$ fig4 ( #b means #c_/e #c_/g 3 3 3 3 3 solids ---------------- 3 3 3 ( ,a ?ree-sid$ fig4 ( #c means #d_/f #d_/h #d_/aj 3 3 3 -------------------- 3 3 ,a ?ree-sid$ fig4 ( #d means #e_/g #e_/i #e_/aae_/ac 3 3 ------------------------ 3 ,a ?ree-sid$ fig4 ( #e means #f_/h #f_/ajf_/abf_/adf_/af 3 ---------------------------- ,a ?ree-sid$ fig4 ( #f means #g_/i #g_/aag_/acg_/aeg_/agg_/ai ---------------------------- ,a ?ree-sid$ fig4 ( #g means #h_/ajh_/abh_/adh_/afh_/ahh_/bjh_/bb ---------------------------- ,sid5ote3 ,9 :at proportion ! same figures exce$ a cone ( ! same altitude & base4 #h4 ,la/ly1 ! excess ( ! solids ( ! said ?ree-sid$ figures above a cone ( ! same altitude & base1 are set d[n 9 ! table :i* foll[s3 #a #b #c #d #e #f #g ,! ,cone #a 3 3 3 3 3 3 3 ----- ,( ! solid ( a ?ree- #f_/ab 3 3 3 3 3 3 sid$ figure ( #a mean ,! ex- ---------- cesses ,ditto ditto1 #b means #ab_/ae #f_/ba 3 3 3 3 3 ( ! --------------- solids ,ditto ditto1 #c means #ah_/ahab_/bd #f_/cj 3 3 3 3 ( ^! -------------------- ?ree- ,ditto ditto1 #d means #bd_/baah_/bgab_/cc #f_/ci 3 3 3 sid$ ------------------------- figures ,ditto ditto1 #e means #cj_/bdbd_/cjah_/cfab_/db #f_/dh 3 3 above a ------------------------------ cone4 ,ditto ditto1 #f means #cf_/bgcj_/ccbd_/ciah_/deab_/ea #f_/eg 3 ----------------------------------- ,ditto ditto1 #g means #db_/cjcf_/cfcj_/dbbd_/dhah_/edab_/fj #f_/ff ----------------------------------- ,sid5ote3 ,h[ a pla9 defici5t figure may be describ$ 9 a p>allelogram1 s t x be to a triangle ( ! same base & altitude1 z ano!r defici5t figure1 pla9 or solid1 twice tak51 is to ! same defici5t figure1 toge!r ) ! complete figure1 9 :i* x is describ$4 #i4 ,if any ( ^! defici5t figures1 ( :i* ,i h n[ spok51 z ,a ,b ,c ,d 79 ! #e? figure7 be inscrib$ )9 ! complete figure ,b ,e1 hav+ ,a ,d ,c ,e = its complem5t2 & !re be made ^u ,c ,b produc$ ! triangle ,a ,b ,i2 & ! p>allelogram ,a ,b ,i ,k be complet$2 & !re be drawn p>allel to ! /rait l9e ,c ,i1 any numb] ( l9es1 z ,m ,f1 cutt+ e "o ( !m ! crook$ l9e ( ! defici5t figure 9 ,d1 & ! /rait l9es ,a ,c1 ,a ,b & ,a ,i 9 ,h1 ,g1 & ,l2 & z ,g ,f is to ,g ,d1 s ,g ,l be made to ano!r1 ,g ,n2 & ?r\gh all ! po9ts ,n !re be drawn ! l9e ,a ,n ,i3 !re w be a defici5t figure ,a ,n ,i ,b1 ^: complem5t w be ,a ,n ,i ,k4 ,i say1 ! figure ,a ,n ,i ,b is to ! triangle ,a ,b ,i1 z ! defici5t figure ,a ,b ,c ,d twice tak5 is to ! same defici5t figure toge!r ) ! complete figure ,b ,e4 ,= z ! proportion ( ,a ,b to ,a ,g1 t is1 ( ,g ,m to ,g ,l1 is to ! proportion ( ,g ,m to ,g ,n1 s is ! magnitude ( ! figure ,a ,n ,i ,b to t ( its complem5t ,a ,n ,i ,k1 by ! second article ( ?is *apt]4 ,b1 by ! same article1 z ! proportion ( ,a ,b to ,a ,g1 t is1 ( ,g ,m to ,g ,l1 is to ! proportion ( ,g ,f to ,g ,d1 t is1 by con/ruction1 ( ,g ,l to ,g ,n1 s is ! figure ,a ,b ,c ,d to its complem5t ,a ,d ,c ,e4 ,& by composition1 z ! proportion ( ,g ,m to ,g ,l1 toge!r ) t ( ,g ,l to ,g ,n1 is to ! proportion ( ,g ,m to ,g ,l1 s is ! complete figure ,b ,e to ! defici5t figure ,a ,b ,c ,d4 ,& by conv]sion1 z ! proportion ( ,g ,m to ,g ,l is to bo? ! proportions ( ,g ,m to ,g ,l & ( ,g ,l to ,g ,n1 t is1 to ! proportion ( ,g ,m to ,g ,n1 :i* is ! proportion comp\nd$ ( bo?1 s is ! defici5t figure ,a ,b ,c ,d to ! complete figure ,b ,e4 ,b x 01 z ! proportion ( ,g ,m to ,g ,l to t ( ,g ,m to ,g ,n1 s ! figure ,a ,n ,i ,b to its complem5t ,a ,n ,i ,k4 ,& !re=e1 ,a ,b ,c ,d4 ,b ,e 33 ,a ,n ,i ,b4 ,a ,n ,i ,k are proportionals4 ,& by composition1 ,a ,b ,c ,d ,b ,e4 ,a ,b ,c ,d 33 ,b ,k4 ,a ,n ,i ,b are proportionals4 ,& by d\bl+ ! consequ5ts1 ,a ,b ,c ,d ,b ,e4 #b ,a ,b ,c ,d 33 ,b ,k4 #b ,a ,n ,i ,b are proportionals4 ,& by tak+ ! halves ( ! ?ird & ! f\r?1 ,a ,b ,c ,d ,b ,e4 #b ,a ,b ,c ,d 33 ,a ,b ,i4 ,a ,n ,i ,b are also proportionals2 :i* 0 to be prov$4 ,sid5ote3 ,! transf]r+ ( c]ta9 prop]ties ( defici5t figures describ$ 9 a p>allelogram to ! proportions ( spaces transmitt$ ) sev]al degrees ( velocity4 #aj4 ,f :at has be5 said ( defici5t figures describ$ 9 a p>allelogram1 may be f\nd out :at proportions spaces1 transmitt$ ) accel]at$ motion 9 det]m9$ times1 h to ! times !mselves1 accord+ z ! mov$ body is accel]at$ 9 ! sev]al times ) "o or m degrees ( velocity4 ,= let ! p>allelogram ,a ,b ,c ,d1 9 ! #f? figure1 & 9 x ! ?ree-sid$ figure ,d ,e ,b ,c be describ$2 & let ,f ,g be drawn any:]e p>allel to ! base1 cutt+ ! diagonal ,b ,d 9 ,h1 & ! crook$ l9e ,b ,e ,d 9 ,e2 & let ! proportion ( ,b ,c to ,b ,f be1 = example1 triplicate to t ( ,f ,g to ,f ,e2 :]eupon ! figure ,d ,e ,b ,c w be triple to its complem5t ,b ,e ,d ,a2 & 9 l mann]1 ,i ,e be+ drawn p>allel to ,b ,c1 ! ?ree-sid$ figure ,e ,k ,b ,f w be triple to its complem5t ,b ,k ,e ,i4 ,:]e=e ! p>ts ( ! defici5t figure cut (f f ! v]tex by /rait l9es p>allel to ! base1 namely1 ,d ,e ,b ,c & ,e ,k ,b ,f1 w be to "o ano!r z ! p>allelograms ,a ,c & ,i ,f2 t is1 9 proportion comp\nd$ ( ! proportions ( ! altitudes & bases4 ,see+ !re=e ! proportion ( ! altitude ,b ,c to ! altitude ,b ,f is triplicate to ! proportion ( ! base ,d ,c to ! base ,f ,e1 ! figure ,d ,e ,b ,c to ! figure ,e ,k ,b ,f w be quadruplicate to ! proportion ( ! same ,d ,c to ,f ,e4 ,& by ! same me?od1 may be f\nd out :at proportion any ( ! said ?ree-sid$ figures has to any "p ( ! same1 cut (f f ! v]tex by a /rait l9e p>allel to ! base4 ,n[ z ! said figures are und]/ood to be describ$ by ! cont9ual decreas+ ( ! base1 z ( ,c ,d1 = example1 till x end 9 a po9t1 z 9 ,b2 s also !y may be und]/ood to be describ$ by ! cont9ual increas+ ( a po9t1 z ( ,b1 till x acquire any magnitude1 z t ( ,c ,d4 ,suppose n[ ! figure ,b ,e ,d ,c to be describ$ by ! increas+ ( ! po9t ,b to ! magnitude ,c ,d4 ,see+ !re=e ! proportion ( ,b ,c to ,b ,f is triplicate to t ( ,c ,d to ,f ,e1 ! proportion ( ,f ,e to ,c ,d w1 by conv]sion1 z ,i %all pres5tly demon/rate1 be triplicate to t ,b ,f to ,b ,c4 ,:]e=e if ! /rait l9e ,b ,c be tak5 = ! measure ( ! "t 9 :i* ! po9t ,b is mov$1 ! figure ,e ,k ,b ,f w repres5t ! sum ( all ! increas+ velocities 9 ! "t ,b ,f2 & ! figure ,d ,e ,b ,c w 9 l mann] repres5t ! sum ( all ! increas+ velocities 9 ! "t ,b ,c4 ,see+ !re=e ! proportion ( ! figure ,e ,k ,b ,f to ! figure ,d ,e ,b ,c is comp\nd$ ( ! proportions ( altitude to altitude1 & base to base2 & see+ ! proportion ( ,f ,e to ,c ,d is triplicate to t ( ,b ,f to ,b ,c2 ! proportion ( ! figure ,e ,k ,b ,f to ! figure ,d ,e ,b ,c w be quadruplicate to t ( ,b ,f to ,b ,c2 t is1 ! proportion ( ! sum ( ! velocities 9 ! "t ,b ,f1 to ! sum ( ! velocities 9 ! "t ,b ,c1 w be quadruplicate to ! proportion ( ,b ,f to ,b ,c4 ,:]e=e if a body be mov$ f ,b ) velocity s increas+1 t ! velocity acquir$ 9 ! "t ,b ,f be to ! velocity acquir$ 9 ! "t ,b ,c 9 triplicate proportion to t ( ! times !mselves ,b ,f to ,b ,c1 & ! body be c>ri$ to ,f 9 ! "t ,b ,f2 ! same body 9 ! "t ,b ,c w be c>ri$ ?r\gh a l9e equal to ! fif? proportional f ,b ,f 9 ! cont9ual proportion ( ,b ,f to ,b ,c4 ,& by ! same mann] ( work+1 we may det]m9e :at spaces are transmitt$ by velocities increas+ accord+ to any o!r proportions4 ,x rema9s t ,i demon/rate ! proportion ( ,f ,e to ,c ,d to be triplicate to t ( ,b ,f to ,b ,c4 ,see+ !re=e ! proportion ( ,c ,d1 t is1 ( ,f ,g to ,f ,e is subtriplicate to t ( ,b ,c to ,b ,f2 ! proportion ( ,f ,g to ,f ,e w also be subtriplicate to t ( ,f ,g to ,f ,h4 ,:]e=e ! proportion ( ,f ,g to ,f ,h is triplicate to t ( ,f ,g1 t is1 ( ,c ,d to ,f ,e4 ,b 9 f\r cont9ual proportionals1 ( :i* ! lea/ is ! fir/1 ! proportion ( ! fir/ to ! f\r?1 7by ! #af? article ( *apt] ,,xiii471 is subtriplicate to ! proportion ( ! ?ird to ! same f\r?4 ,:]e=e ! proportion ( ,f ,h to ,g ,f is subtriplicate to t ( ,f ,e to ,c ,d2 & !re=e ! proportion ( ,f ,e to ,c ,d is triplicate to t ( ,f ,h to ,f ,g1 t is1 ( ,b ,f to ,b ,c2 :i* 0 to be prov$4 ,x may f h5ce be collect$1 t :5 ! velocity ( a body is increas$ 9 ! same proportion ) t ( ! times1 ! degrees ( velocity above "o ano!r proce$ z numb]s d 9 imm$iate succession f unity1 namely1 z #a1 #b1 #c1 #d1 c4 ,& :5 ! velocity is increas$ 9 proportion duplicate to t ( ! times1 ! degrees proce$ z numb]s f unity1 skipp+ "o1 z #a1 #c1 #e1 #g1 c4 ,la/ly1 :5 ! proportions ( ! velocities are triplicate to ^? ( ! times1 ! progression ( ! degrees is z t ( numb]s f unity1 skipp+ two 9 e place1 z #a1 #d1 #g1 #aj1 c41 & s ( o!r proportions4 ,= geometrical proportionals1 :5 !y are tak5 9 e po9t1 are ! same ) ari?metical proportionals4 ,sid5ote3 ,( defici5t figures describ$ 9 a circle4 #aa4 ,moreov]1 x is to be not$ t z 9 quantities1 :i* are made by any magnitudes decreas+1 ! proportions ( ! figures to "o ano!r are z ! proportions ( ! altitudes to ^? ( ! bases2 s also x is 9 ^?1 :i* are made ) motion decreas+1 :i* motion is no?+ else b t p[] by :i* ! describ$ figures are great] or less4 ,& !re=e 9 ! description ( ,ar*im$es' spiral1 :i* is done by ! cont9ual dim9ution ( ! semidiamet] ( a circle 9 ! same proportion 9 :i* ! circumf]5ce is dim9i%$1 ! space1 :i* is conta9$ )9 ! semidiamet] & ! spiral l9e1 is a ?ird "p ( ! :ole circle4 ,= ! semidiamet]s ( circles1 inasmu* z circles are und]/ood to be made up ( ! aggregate ( !m1 are s _m sectors2 & !re=e 9 ! description ( a spiral1 ! sector :i* describes x is dim9i%$ 9 duplicate proportions to ! dim9utions ( ! circumf]5ce ( ! circle 9 :i* x is inscrib$2 s t ! complem5t ( ! spiral1 t is1 t space 9 ! circle :i* is )\t ! spiral l9e1 is d\ble to ! space )9 ! spiral l9e4 ,9 ! same mann]1 if !re be tak5 a mean proportional ev]y:]e betwe5 ! semidiamet] ( ! circle1 :i* conta9s ! spiral1 & t "p ( ! semidiamet] :i* is )9 ! same1 !re w be made ano!r figure1 :i* w be half ! circle4 ,& to conclude1 ?is rule s]ves = all su* spaces z may be describ$ by a l9e or sup]ficies decreas+ ei!r 9 magnitude ( p[]2 s t if ! proportions1 9 :i* !y decrease1 be comm5surable to ! proportions ( ! times 9 :i* !y decrease1 ! magnitudes ( ! figures !y describe w be kn[n4 ,sid5ote3 ,! proposition demon/rat$ 9 art4 #b confirm$ f ! elem5ts ( philosophy4 #ab4 ,! tru? ( t proposition1 :i* ,i demon/rat$ 9 art4 #b1 :i* is ! f\ndation ( all t has be5 said conc]n+ defici5t figures1 may be d]iv$ f ! elem5ts ( philosophy1 z hav+ its orig9al 9 ?is2 t all equality & inequality betwe5 two effects1 t is1 all proportion1 proce$s f1 & is det]m9$ by1 ! equal & unequal causes ( ^? effects1 or f ! proportion :i* ! causes1 concurr+ to "o effect1 h to ! causes :i* concur to ! produc+ ( ! o!r effect2 & t !re=e ! proportions ( quantities are ! same ) ! proportions ( _! causes4 ,see+1 !re=e1 two defici5t figures1 ( :i* "o is ! complem5t ( ! o!r1 are made1 "o by motion decreas+ 9 a c]ta9 "t & proportion1 ! o!r by ! loss ( motion 9 ! same "t2 ! causes1 :i* make & det]m9e ! quantities ( bo? ! figures1 s t !y c be no o!r ?an !y are1 diff] only 9 ?is1 t ! proportions by :i* ! quantity :i* g5]ates ! figure proce$s 9 describ+ ( ! same1 t is1 ! proportions ( ! rema9d]s ( all ! times & altitudes1 may be o!r proportions ?an ^? by :i* ! same g5]at+ quantity decreases 9 mak+ ! complem5t ( t figure1 t is1 ! proportions ( ! quantity :i* g5]ates ! figure cont9ually dim9i%$4 ,:]e=e1 z ! proportion ( ! times 9 :i* motion is lo/1 is to t ( ! decreas+ quantities by :i* ! defici5t figure is g5]at$1 s w ! defect or complem5t be to ! figure itself :i* is g5]at$4 ,sid5ote3 ,an unusual way ( reason+ conc]n+ ! equality betwe5 ! sup]ficies ( a portion ( a sph]e & a circle4 #ac4 ,!re are also o!r quantities :i* are det]m9able f ! k ( _! causes1 namely1 f ! comp>ison ( ! motions by :i* !y are made2 & t m easily ?an f ! common elem5ts ( geometry4 ,= example1 t ! sup]ficies ( any portion ( a sph]e is equal to t circle1 ^: radius is a /rait l9e drawn f ! pole ( ! portion to ! circumf]5ce ( its base1 ,i may demon/rate 9 ?is mann]4 ,let ,b ,a ,c 79 fig4 #g7 be a portion ( a sph]e1 ^: axis is ,a ,e1 & ^: base is ,b ,c2 & let ,a ,b be ! /rait l9e drawn f ! pole ,a to ! base 9 ,b2 & let ,a ,d1 equal to ,a ,b1 t\* ! great circle ,b ,a ,c 9 ! pole ,a4 ,x is to be prov$ t ! circle made by ! radius ,a ,d is equal to ! sup]ficies ( ! portion ,b ,a ,c4 ,let ! pla9 ,a ,e ,b ,d be und]/ood to make a revolution ab\t ! axis ,a ,e2 & x is manife/ t by ! /rait l9e ,a ,d a circle w be describ$2 & by ! ar* ,a ,b ! sup]ficies ( a portion ( a sph]e2 & la/ly1 by ! subt5se ,a ,b ! sup]ficies ( a "r cone4 ,n[ see+ bo? ! /rait l9e ,a ,b & ! ar* ,a ,b make "o & ! same revolution1 & bo? ( !m h ! same extreme po9ts ,a & ,b1 ! cause :y ! sph]ical sup]ficies1 :i* is made by ! ar*1 is great] ?an ! conical sup]ficies1 :i* is made by ! subt5se1 is1 t ,a ,b ! ar* is great] ?an ,a ,b ! subt5se2 & ! cause :y x is great] consi/s 9 ?is1 t al?\gh !y be bo? drawn f ,a to ,b1 yet ! subt5se is drawn /rait1 b ! ar* angul>ly1 namely1 accord+ to t angle :i* ! ar* makes ) ! subt5se1 :i* angle is equal to ! angle ,d ,a ,b 7= an angle ( cont+5ce adds no?+ to an angle ( a segm5t1 z has be5 %[n 9 *apt] ,,xiv1 article #af47 ,:]e=e ! magnitude ( ! angle ,d ,a ,b is ! cause :y ! sup]ficies ( ! portion1 describ$ by ! ar* ,a ,b1 is great] ?an ! sup]ficies ( ! "r cone describ$ by ! subt5se ,a ,b4 ,aga91 ! cause :y ! circle describ$ by ! tang5t ,a ,d is great] ?an ! sup]ficies ( ! "r cone describ$ by ! subt5se ,a ,b 7not)/&+ t ! tang5t & ! subt5se are equal1 & bo? mov$ r\nd 9 ! same "t7 is ?is1 t ,a ,d /&s at "r angles to ! axis1 b ,a ,b obliquely2 :i* obliquity consi/s 9 ! same angle ,d ,a ,b4 ,see+ !re=e ! quantity ( ! angle ,d ,a ,b is t :i* makes ! excess bo? ( ! sup]ficies ( ! portion1 & ( ! circle made by ! radius ,a ,d1 above ! sup]ficies ( ! "r cone describ$ by ! subt5se ,a ,b2 x foll[s1 t bo? ! sup]ficies ( ! portion & t ( ! circle d equally exce$ ! sup]ficies ( ! cone4 ,:]e=e ! circle made by ,a ,d or ,a ,b1 & ! sph]ical sup]ficies made by ! ar* ,a ,b1 are equal to "o ano!r2 :i* 0 to be prov$4 ,sid5ote3 ,h[ f ! description ( defici5t figures 9 a p>allelogram1 any numb] ( mean proportionals may be f\nd out betwe5 two giv5 /rait l9es4 #ad4 ,if ^! defici5t figures1 :i* ,i h describ$ 9 a p>allelogram1 7 capable ( exact description1 !n any numb] ( mean proportionals might be f\nd out betwe5 two /rait l9es giv54 ,= example1 9 ! p>allelogram ,a ,b ,c ,d1 79 figure #h7 let ! ?ree-sid$ figure ( two means be describ$ 7:i* _m call a cubical p>abola72 & let ,r & ,s be two giv5 /rait l9es2 betwe5 :i*1 if x be requir$ to f9d two mean proportionals1 x may be done ?us4 ,let x be z ,r to ,s1 s ,b ,c to ,b ,f2 & let ,f ,e be drawn p>allel to ,b ,a1 & cut ! crook$ l9e 9 ,e2 !n ?r\gh ,e let ,g ,h be drawn p>allel & equal to ! /rait l9e ,a ,d1 & cut ! diagonal ,b ,d 9 ,i2 = ?us we h ,g ,i ! greate/ ( two means betwe5 ,g ,h & ,g ,e1 z appe>s by ! description ( ! figure 9 article #d4 ,:]e=e1 if x be z ,g ,h to ,g ,i1 s ,r to ano!r l9e1 ,t1 t ,t w be ! greate/ ( two means betwe5 ,r & ,s4 ,& !re=e if x be aga9 z ,r to ,t1 s ,t to ano!r l9e1 ,x1 t w be done :i* 0 requir$4 ,9 ! same mann]1 f\r mean proportionals may be f\nd out1 by ! description ( a ?ree-sid$ figure ( f\r means2 & s any o!r numb] ( means1 c4 ---------------------------------------- -------------------------------- ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xvii4 ,fig4 #a-h ---------------------------------------- -------------------------------- ------- ,,*apt] ,,xviii4 ,,( ,,! ,,equation ,,( ,,/rait ,,l9es ,,) ,,! ,,crook$ ,,l9es ,,( ,,p>abolas ,,& ,,o!r ,,figures ,,made ,,9 ,,imitation ,,( ,,p>abolas4 #a4 ,to f9d ! /rait l9e equal to ! crook$ l9e ( a semip>abola4--#b4 ,to f9d a /rait l9e equal to ! crook$ l9e ( ! fir/ semip>abola/]1 or to ! crook$ l9e ( any o!r ( ! defici5t figures ( ! table ( ! #c;d article ( ! prec$5t *apt]4 ,sid5ote3 ,to f9d a /rait l9e equal to ! crook$ l9e ( a semip>abola4 #a4 ,a p>abola be+ giv51 to f9d a /rait l9e equal to ! crook$ l9e ( ! semip>abola4 ,let ! p>abolical l9e giv5 be ,a ,b ,c 79 figure #a71 & ! diamet] f\nd be ,a ,d1 & ! base drawn ,d ,c2 & ! p>allelogram ,a ,d ,c ,e be+ complet$1 draw ! /rait l9e ,a ,c4 ,!n divid+ ,a ,d into two equal p>ts 9 ,f1 draw ,f ,h equal & p>allel to ,d ,c1 cutt+ ,a ,c 9 ,k1 & ! p>abolical l9e 9 ,o2 & betwe5 ,f ,h & ,f ,o take a mean proportional ,f ,p1 & draw ,a ,o1 ,a ,p & ,p ,c4 ,i say t ! two l9es ,a ,p & ,p ,c1 tak5 toge!r z "o l9e1 are equal to ! p>abolical l9e ,a ,b ,o ,c4 ,= ! l9e ,a ,b ,o ,c be+ a p>abolical l9e1 is g5]at$ by ! conc\rse ( two motions1 "o uni=m f ,a to ,e1 ! o!r 9 ! same "t uni=mly accel]at$ f re/ 9 ,a to ,d4 ,& because ! motion f ,a to ,e is uni=m1 ,a ,e may repres5t ! times ( bo? ^? motions f ! beg9n+ to ! end4 ,let !re=e ,a ,e be ! "t2 & consequ5tly ! l9es ord9ately appli$ 9 ! semip>abola w design ! p>ts ( "t :]e9 ! body1 t describe? ! l9e ,a ,b ,o ,c1 is 9 e po9t ( ! same2 s t z at ! end ( ! "t ,a ,e or ,d ,c x is 9 ,c1 s at ! end ( ! "t ,f ,o x w be 9 ,o4 ,& because ! velocity 9 ,a ,d is increas$ uni=mly1 t is1 9 ! same proportion ) ! "t1 ! same l9es ord9ately appli$ 9 ! semip>abola w design also ! cont9ual augm5tation ( ! impetus1 till x be at ! greate/1 design$ by ! base ,d ,c4 ,!re=e suppos+ uni=m motion 9 ! l9e ,a ,f1 9 ! "t ,f ,k ! body 9 ,a by ! conc\rse ( ! two uni=m motions 9 ,a ,f & ,f ,k w be mov$ uni=mly 9 ! l9e ,a ,k2 & ,k ,o w be ! increase ( ! impetus or swiftness ga9$ 9 ! "t ,f ,k2 & ! l9e ,a ,o w be uni=mly describ$ by ! conc\rse ( ! two uni=m motions 9 ,a ,f & ,f ,o 9 ! "t ,f ,o4 ,f ,o draw ,o ,l p>allel to ,e ,c1 cutt+ ,a ,c 9 ,l2 & draw ,l ,n p>allel to ,d ,c1 cutt+ ,e ,c 9 ,n1 & ! p>abolical l9e 9 ,m2 & produce x on ! o!r side to ,a ,d 9 ,i2 & ,i ,n1 ,i ,m & ,i ,l w be1 by ! con/ruction ( a p>abola1 9 cont9ual proportion1 & equal to ! ?ree l9es ,f ,h1 ,f ,p & ,f ,o2 & a /rait l9e p>allel to ,e ,c pass+ ?r\gh ,m w fall on ,p2 & !re=e ,o ,p w be ! increase ( impetus ga9$ 9 ! "t ,f ,o or ,i ,l4 ,la/ly1 produce ,p ,m to ,c ,d 9 ,q2 & ,q ,c or ,m ,n or ,p ,h w be ! increase ( impetus proportional to ! "t ,f ,p or ,i ,m or ,d ,q4 ,suppose n[ uni=m motion f ,h to ,c 9 ! "t ,p ,h4 ,see+ !re=e 9 ! "t ,f ,p ) uni=m motion & ! impetus increas$ 9 proportion to ! times1 is describ$ ! /raight l9e ,a ,p2 & 9 ! re/ ( ! "t & impetus1 namely1 ,p ,h1 is describ$ ! l9e ,c ,p uni=mly2 x foll[e? t ! :ole l9e ,a ,p ,c is describ$ ) ! :ole impetus1 & 9 ! same "t :]e) is describ$ ! p>abolical l9e ,a ,b ,c2 & !re=e ! l9e ,a ,p ,c1 made ( ! two /rait l9es ,a ,p & ,p ,c1 is equal to ! p>abolical l9e ,a ,b ,c2 :i* 0 to be prov$4 ,sid5ote3 ,to f9d a /rait l9e equal to ! crook$ l9e ( ! fir/ semip>abola/] or to ! crook$ l9e ( any o!r ( ! defici5t figures ( ! table ( art4 #c ( ! prec$+ *apt]4 #b4 ,to f9d a /rait l9e equal to ! crook$ l9e ( ! fir/ semip>abola/]4 ,sid5ote3 ,discussion ( ,figure #ah4b ,let ,a ,b ,c be ! crook$ l9e ( ! fir/ semip>abola/]2 ,a ,d ! diamet]2 ,d ,c ! base2 & let ! p>allelogram complet$ be ,a ,d ,c ,e1 ^: diagonal is ,a ,c4 ,divide ! diamet] into two equal p>ts 9 ,f1 & draw ,f ,h equal & p>allel to ,d ,c1 cutt+ ,a ,c 9 ,k1 ! crook$ l9e 9 ,o1 & ,e ,c 9 ,h4 ,!n draw ,o ,l p>allel to ,e ,c1 cutt+ ,a ,c 9 ,l2 & draw ,l ,n p>allel to ! base ,d ,c1 cutt+ ! crook$ l9e 9 ,m1 & ! /rait l9e ,e ,c 9 ,n2 & produce x on ! o!r side to ,a ,d 9 ,i4 ,la/ly1 ?r\gh ! po9t ,m draw ,p ,m ,q p>allel & equal to ,h ,c1 cutt+ ,f ,h 9 ,p2 & jo9 ,c ,p1 ,a ,p & ,a ,o4 ,i say1 ! two /rait l9es ,a ,p & ,p ,c are equal to ! crook$ l9e ,a ,b ,o ,c4 ,= ! l9e ,a ,b ,o ,c1 be+ ! crook$ l9e ( ! fir/ semip>abola/]1 is g5]at$ by ! conc\rse ( two motions1 "o uni=m f ,a to ,e1 ! o!r 9 ! same "t accel]at$ f re/ 9 ,a to ,d1 s z t ! impetus increase? 9 proportion p]petually triplicate to t ( ! increase ( ! "t1 or :i* is all "o1 ! l5g?s transmitt$ are 9 proportion triplicate to t ( ! times ( _! transmission2 = z ! impetus or quicknesses increase1 s ! l5g?s transmitt$ increase also4 ,& because ! motion f ,a to ,e is uni=m1 ! l9e ,a ,e may s]ve to repres5t ! "t1 & consequ5tly ! l9es1 ord9ately drawn 9 ! semip>abola/]1 w design ! p>ts ( "t :]e9 ! body1 beg9n+ f re/ 9 ,a1 describe? by its motion ! crook$ l9e ,a ,b ,o ,c4 ,& because ,d ,c1 :i* repres5ts ! greate/ acquir$ impetus1 is equal to ,a ,e1 ! same ord9ate l9es w repres5t ! sev]al augm5tations ( ! impetus increas+ f re/ 9 ,a4 ,!re=e1 suppos+ uni=m motion f ,a to ,f1 9 ! "t ,f ,k !re w be describ$1 by ! conc\rse ( ! two uni=m motions ,a ,f & ,f ,k1 ! l9e ,a ,k uni=mly1 & ,k ,o w be ! increase ( impetus 9 ! "t ,f ,k2 & by ! conc\rse ( ! two uni=m motions 9 ,a ,f & ,f ,o w be describ$ ! l9e ,a ,o uni=mly4 ,?r\gh ! po9t ,l draw ! /rait l9e ,l ,m ,n p>allel to ,d ,c1 cutt+ ! /rait l9e ,a ,d 9 ,i1 ! crook$ l9e ,a ,b ,c 9 ,m1 & ! /rait l9e ,e ,c 9 ,n2 & ?r\gh ! po9t ,m ! /rait l9e ,p ,m ,q p>allel & equal to ,h ,c1 cutt+ ,d ,c 9 ,q & ,f ,h 9 ,p4 ,by ! conc\rse !re=e ( ! two uni=m motions 9 ,a ,f & ,f ,p 9 ! "t ,f ,p w be uni=mly describ$ ! /rait l9e ,a ,p2 & ,l ,m or ,o ,p w be ! increase ( impetus to be add$ = ! "t ,f ,o4 ,& because ! proportion ( ,i ,n to ,i ,l is triplicate to ! proportion ( ,i ,n to ,i ,m1 ! proportion ( ,f ,h to ,f ,o w also be triplicate to ! proportion ( ,f ,h to ,f ,p2 & ! proportional impetus ga9$ 9 ! "t ,f ,p is ,p ,h4 ,s t ,f ,h be+ equal to ,d ,c1 :i* design$ ! :ole impetus acquir$ by ! accel]ation1 !re is no m increase ( impetus to be comput$4 ,n[ 9 ! "t ,p ,h suppose an uni=m motion f ,h to ,c2 & by ! two uni=m motions 9 ,c ,h & ,h ,p w be describ$ uni=mly ! /rait l9e ,p ,c4 ,see+ !re=e ! two /rait l9es ,a ,p & ,p ,c are describ$ 9 ! "t ,a ,e ) ! same increase ( impetus1 :]e) ! crook$ l9e ,a ,b ,o ,c is describ$ 9 ! same "t ,a ,e1 t is1 see+ ! l9e ,a ,p ,c & ! l9e ,a ,b ,o ,c are transmitt$ by ! same body 9 ! same "t & ) equal velocities1 ! l9es !mselves are equal2 :i* 0 to be demon/rat$4 ,by ! same me?od 7if any ( ! semip>abola/]s 9 ! table ( art4 #c ( ! prec$5t *apt] be exhibit$7 may be f\nd a /rait l9e equal to ! crook$ l9e !re(1 namely1 by divid+ ! diamet] into two equal p>ts1 & proce$+ z be=e4 ,yet no man hi!rto ha? comp>$ any crook$ ) any /rait l9e1 ?\gh _m geometricians ( e age h endeav\r$ x4 ,b ! cause1 :y !y h n done x1 may be ?is1 t !re be+ 9 ,euclid no def9ition ( equality1 nor any m>k by :i* to judge ( x besides congruity 7:i* is ! #h? axiom ( ! fir/ ,book ( 8 ,elem5ts7 a ?+ ( no use at all 9 ! comp>+ ( /rait & crook$2 & o!rs aft] ,euclid 7except ,ar*im$es & ,apollonius1 & 9 our "t ,bonav5tura7 ?9k+ ! indu/ry ( ! anci5ts _h rea*$ to all t 0 to be done 9 geometry1 ?\ght also1 t all t c\ld be prop\nd$ 0 ei!r to be d$uc$ f :at !y _h writt51 or else t x 0 n at all to be done3 x 0 !re=e disput$ by "s ( ^? anci5ts !mselves1 :e!r !re might be any equality at all betwe5 crook$ & /rait l9es2 :i* "q ,ar*im$es1 :o assum$ t "s /rait l9e 0 equal to ! circumf]5ce ( a circle1 seems to h despis$1 z he _h reason4 ,& !re is a late writ] t grante? t betwe5 a /rait & a crook$ l9e !re is equality2 b n[1 says he1 s9ce ! fall ( ,adam1 )\t ! special assi/ance ( ,div9e ,grace x is n to be f\nd4 ---------------------------------------- -------------------------------- ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xviii4 ,fig4 #a-b ---------------------------------------- -------------------------------- ------- ,,*apt] ,,xix4 ,,( ,,angles ,,( ,,incid5ce ,,& ,,reflection1 ,,equal ,,by ,,supposition4 #a4 ,if two /rait l9es fall+ ^u ano!r /rait l9e be p>allel1 ! l9es reflect$ f !m %all also be p>allel4--#b4 ,if two /rait l9es drawn f "o po9t fall ^u ano!r /rait l9e1 ! l9es reflect$ f !m1 if !y be drawn out ! o!r way1 w meet 9 an angle equal to ! angle made by ! l9es ( incid5ce4--#c4 ,if two /rait p>allel l9es1 drawn n oppositely1 b f ! same p>ts1 fall ^u ! circumf]5ce ( a circle1 ! l9es reflect$ f !m1 if produc$ !y meet )9 ! circle1 w make an angle d\ble to t :i* is made by two /rait l9es drawn f ! c5tre to ! po9ts ( incid5ce4--#d4 ,if two /rait l9es drawn f ! same po9t )\t a circle fall ^u ! circumf]5ce1 & ! l9es reflect$ f !m be+ produc$ meet )9 ! circle1 !y w make an angle equal to twice t angle1 :i* is made by two /rait l9es drawn f ! c5tre to ! po9ts ( incid5ce1 toge!r ) ! angle :i* ! incid5t l9es !mselves make4--#e4 ,if two /rait l9es drawn f "o po9t fall ^u ! concave circumf]5ce ( a circle1 & ! angle !y make be less ?an twice ! angle at ! c5tre1 ! l9es reflect$ f !m & meet+ )9 ! circle w make an angle1 :i* be+ add$ to ! angle ( ! incid5t l9es w be equal to twice ! angle at ! c5tre4--#f4 ,if ?r\gh any "o po9t two unequal *ords be drawn cutt+ "o ano!r1 & ! c5tre ( ! circle be n plac$ betwe5 !m1 & ! l9es reflect$ f !m concur :]esoev]1 !re _c ?r\gh ! po9t1 ?r\gh :i* ! two =m] l9es 7 drawn1 be drawn any o!r /rait l9e ^: reflect$ l9e %all pass ?r\gh ! common po9t ( ! two =m] l9es reflect$4--#g4 ,9 equal *ords ! same is n true4--#h4 ,two po9ts be+ giv5 9 ! circumf]5ce ( a circle1 to draw two /rait l9es to !m1 s t _! reflect$ l9es may conta9 any angle giv54--#i4 ,if a /rait l9e fall+ ^u ! circumf]5ce ( a circle be produc$ till x rea* ! semidiamet]1 & t "p ( x1 :i* is int]cept$ betwe5 ! circumf]5ce & ! semidiamet]1 be equal to t "p ( ! semidiamet] :i* is betwe5 ! po9t ( conc\rse & ! c5tre1 ! reflect$ l9e w be p>allel to ! semidiamet]4--#aj4 ,if f a po9t )9 a circle1 two /rait l9es be drawn to ! circumf]5ce1 & _! reflect$ l9es meet 9 ! circumf]5ce ( ! same circle1 ! angle made by ! reflect$ l9es w be a ?ird "p ( ! angle made by ! incid5t l9es4 ,sid5ote3 ,angles ( incid5ce & reflection4 ,:e!r a body fall+ ^u ! sup]ficies ( ano!r body & be+ reflect$ f x1 d make equal angles at t sup]ficies1 x belongs n to ?is place to dispute1 be+ a k :i* dep5ds ^u ! natural causes ( reflection2 ( :i* hi!rto no?+ has be5 said1 b %all be spok5 ( h]eaft]4 ,9 ?is place1 !re=e1 let x be suppos$ t ! angle ( incid5ce is equal to ! angle ( reflection2 t our pres5t se>* may be appli$1 n to ! f9d+ out ( ! causes1 b "s consequ5ces ( ! same4 ,i call an angle ( incid5ce1 t :i* is made betwe5 a /rait l9e & ano!r l9e1 /rait or crook$1 ^u :i* x falls1 & :i* ,i call ! l9e reflect+2 & an angle ( reflection equal to x1 t :i* is made at ! same po9t betwe5 ! /rait l9e :i* is reflect$ & ! l9e reflect+4 ,sid5ote3 ,if two /rait l9es fall+ ^u ano!r /rait l9e be p>allel1 ! l9es reflect$ f !m %all also be p>allel4 #a4 ,if two /rait l9es1 :i* fall ^u ano!r /rait l9e1 be p>allel1 _! reflect$ l9es %all be also p>allel4 ,let ! two /rait l9es ,a ,b & ,c ,d 79 fig4 #a71 :i* fall ^u ! /rait l9e ,e ,f1 at ! po9ts ,b & ,d1 be p>allel2 & let ! l9es reflect$ f !m be ,b ,g & ,d ,h4 ,i say1 ,b ,g & ,d ,h are also p>allel4 ,= ! angles ,a ,b ,e & ,c ,d ,e are equal by reason ( ! p>allelism ( ,a ,b & ,c ,d2 & ! angles ,g ,b ,f & ,h ,d ,f are equal to !m by supposition2 = ! l9es ,b ,g & ,d ,h are reflect$ f ! l9es ,a ,b & ,c ,d4 ,:]e=e ,b ,g & ,d ,h are p>allel4 ,sid5ote3 ,if two /rait l9es drawn f "o po9t fall ^u ano!r /rait l9e1 ! l9es reflect$ f !m1 if !y be drawn out ! o!r way1 w meet 9 an angle equal to ! angle made by ! l9es ( incid5ce4 #b4 ,if two /rait l9es drawn f ! same po9t fall ^u ano!r /rait l9e1 ! l9es reflect$ f !m1 if !y be drawn out ! o!r way1 w meet 9 an angle equal to ! angle ( ! incid5t l9es4 ,f ! po9t ,a 79 fig4 #b7 let ! two /rait l9es ,a ,b & ,a ,d be drawn2 & let !m fall ^u ! /rait l9e ,e ,k at ! po9ts ,b & ,d2 & let ! l9es ,b ,i & ,d ,g be reflect$ f !m4 ,i say1 ,i ,b & ,g ,d d conv]ge1 & t if !y be produc$ on ! o!r side ( ! l9e ,e ,k1 !y %all meet1 z 9 ,f2 & t ! angle ,b ,f ,d %all be equal to ! angle ,b ,a ,d4 ,= ! angle ( reflection ,i ,b ,k is equal to ! angle ( incid5ce ,a ,b ,e2 & to ! angle ,i ,b ,k its v]tical angle ,e ,b ,f is equal2 & !re=e ! angle ,a ,b ,e is equal to ! angle ,e ,b ,f4 ,aga91 ! angle ,a ,d ,e is equal to ! angle ( reflection ,g ,d ,k1 t is1 to its v]tical angle ,e ,d ,f2 & !re=e ! two angles ,a ,b ,d & ,a ,d ,b ( ! triangle ,a ,b ,d are "o by "o equal to ! two angles ,f ,b ,d & ,f ,d ,b ( ! triangle ,f ,b ,d2 :]e=e also ! ?ird angle ,b ,a ,d is equal to ! ?ird angle ,b ,f ,d2 :i* 0 to be prov$4 ,coroll4 ,i4 ,if ! /rait l9e ,a ,f be drawn1 x w be p]p5dicul> to ! /rait l9e ,e ,k4 ,= bo? ! angles at ,e w be equal1 by reason ( ! equality ( ! two angles ,a ,b ,e & ,f ,b ,e1 & ( ! two sides ,a ,b & ,f ,b4 ,coroll4 ,,ii4 ,if ^u any po9t betwe5 ,b & ,d !re fall a /rait l9e1 z ,a ,c1 ^: reflect$ l9e is ,c ,h1 ?is also produc$ beyond ,c1 w fall ^u ,f2 :i* is evid5t by ! demon/ration above4 ,sid5ote3 ,if two /rait p>allel l9es1 drawn n oppositely1 b f ! same p>ts1 fall ^u ! circumf]5ce ( a circle1 ! l9es reflect$ f !m1 if produc$ !y meet )9 ! circle1 w make an angle d\ble to t :i* is made by two /rait l9es drawn f ! c5tre to ! po9ts ( incid5ce4 #c4 ,if f two po9ts tak5 )\t a circle1 two /rait p>allel l9es1 drawn n oppositely1 b f ! same p>ts1 fall ^u ! circumf]5ce2 ! l9es reflect$ f !m1 if produc$ !y meet )9 ! circle1 w make an angle d\ble to t :i* is made by two /rait l9es drawn f ! c5tre to ! po9ts ( incid5ce4 ,let ! two /rait p>allels ,a ,b & ,d ,c 79 fig4 #c7 fall ^u ! circumf]5ce ,b ,c at ! po9ts ,b & ,c2 & let ! c5tre ( ! circle be ,e2 & let ,a ,b reflect$ be ,b ,f1 & ,d ,c reflect$ be ,c ,g2 & let ! l9es ,f ,b & ,g ,c produc$ meet )9 ! circle 9 ,h2 & let ,e ,b & ,e ,c be connect$4 ,i say ! angle ,f ,h ,g is d\ble to ! angle ,b ,e ,c4 ,= see+ ,a ,b & ,d ,c are p>allels1 & ,e ,b cuts ,a ,b 9 ,b1 ! same ,e ,b produc$ w cut ,d ,c some:]e2 let x cut x 9 ,d2 & let ,d ,c be produc$ h[soev] to ,i1 & let ! int]section ( ,d ,c & ,b ,f be at ,k4 ,! angle !re=e ,i ,c ,h1 be+ ext]nal to ! triangle ,c ,k ,h1 w be equal to ! two opposite angles ,c ,k ,h & ,c ,h ,k4 ,aga91 ,i ,c ,e be+ ext]nal to ! triangle ,c ,d ,e1 is equal to ! two angles at ,d & ,e4 ,:]e=e ! angle ,i ,c ,h1 be+ d\ble to ! angle ,i ,c ,e1 is equal to ! angles at ,d & ,e twice tak52 & !re=e ! two angles ,c ,k ,h & ,c ,h ,k are equal to ! two angles at ,d & ,e twice tak54 ,b ! angle ,c ,k ,h is equal to ! angles ,d & ,a ,b ,d1 t is1 ,d twice tak52 = ,a ,b & ,d ,c be+ p>allels1 ! alt]n angles ,d & ,a ,b ,d are equal4 ,:]e=e ,c ,h ,k1 t is ! angle ,f ,h ,g is also equal to ! angle at ,e twice tak52 :i* 0 to be prov$4 ,coroll4 ,if f two po9ts tak5 )9 a circle two /rait p>allels fall ^u ! circumf]5ce1 ! l9es reflect$ f !m %all meet 9 an angle1 d\ble to t :i* is made by two /rait l9es drawn f ! c5tre to ! po9ts ( incid5ce4 ,= ! p>allels ,a ,b & ,i ,c fall+ ^u ! po9ts ,b & ,c1 are reflect$ 9 ! l9es ,b ,h & ,c ,h1 & make ! angle at ,h d\ble to ! angle at ,e1 z 0 b n[ demon/rat$4 ,sid5ote3 ,if two /rait l9es drawn f ! same po9t )\t a circle fall ^u ! circumf]5ce1 & ! l9es reflect$ f !m be+ produc$ meet )9 ! circle1 !y w make an angle equal to twice t angle1 :i* is made by two /rait l9es drawn f ! c5tre to ! po9ts ( incid5ce1 toge!r ) ! angle :i* ! incid5t l9es !mselves make4 #d4 ,if two /rait l9es drawn f ! same po9t )\t a circle fall ^u ! circumf]5ce1 & ! l9es reflect$ f !m be+ produc$ meet )9 ! circle1 !y w make an angle equal to twice t angle1 :i* is made by two /rait l9es drawn f ! c5tre to ! po9ts ( incid5ce1 toge!r ) ! angle :i* ! incid5t l9es !mselves make4 ,let ! two /rait l9es ,a ,b & ,a ,c 79 fig4 #d7 be drawn f ! po9t ,a to ! circumf]5ce ( ! circle1 ^: c5tre is ,d2 & let ! l9es reflect$ f !m be ,b ,e & ,c ,g1 &1 be+ produc$1 make )9 ! circle ! angle ,h2 also let ! two /rait l9es ,d ,b & ,d ,c be drawn f ! c5tre ,d to ! po9ts ( incid5ce ,b & ,c4 ,i say1 ! angle ,h is equal to twice ! angle at ,d toge!r ) ! angle at ,a4 ,= let ,a ,c be produc$ h[soev] to ,i4 ,!re=e ! angle ,i ,c ,h1 :i* is ext]nal to ! triangle ,c ,k ,h1 w be equal to ! two angles ,c ,k ,h & ,c ,h ,k4 ,aga91 ! angle ,i ,c ,d1 :i* is ext]nal to ! triangle ,c ,l ,d1 w be equal to ! two angles ,c ,l ,d & ,c ,d ,l4 ,b ! angle ,i ,c ,h is d\ble to ! angle ,i ,c ,d1 & is !re=e equal to ! angles ,c ,l ,d & ,c ,d ,l twice tak54 ,:]e=e ! angles ,c ,k ,h & ,c ,h ,k are equal to ! angles ,c ,l ,d & ,c ,d ,l twice tak54 ,b ! angle ,c ,l ,d1 be+ ext]nal to ! triangle ,a ,l ,b1 is equal to ! two angles ,l ,a ,b & ,l ,b ,a2 & consequ5tly ,c ,l ,d twice tak5 is equal to ,l ,a ,b & ,l ,b ,a twice tak54 ,:]e=e ,c ,k ,h & ,c ,h ,k are equal to ! angle ,c ,d ,l toge!r ) ,l ,a ,b & ,l ,b ,a twice tak54 ,also ! angle ,c ,k ,h is equal to ! angle ,l ,a ,b once & ,a ,b ,k1 t is1 ,l ,b ,a twice tak54 ,:]e=e ! angle ,c ,h ,k is equal to ! rema9+ angle ,c ,d ,l1 t is1 to ! angle at ,d1 twice tak51 & ! angle ,l ,a ,b1 t is1 ! angle at ,a1 once tak52 :i* 0 to be prov$4 ,coroll4 ,if two /rait conv]g+ l9es1 z ,i ,c & ,m ,b1 fall ^u ! concave circumf]5ce ( a circle1 _! reflect$ l9es1 z ,c ,h & ,b ,h1 w meet 9 ! angle ,h1 equal to twice ! angle ,d1 toge!r ) ! angle at ,a made by ! incid5t l9es produc$4 ,or1 if ! incid5t l9es be ,h ,b & ,i ,c1 ^: reflect$ l9es ,c ,h & ,b ,m meet 9 ! po9t ,n1 ! angle ,c ,n ,b w be equal to twice ! angle ,d1 toge!r ) ! angle ,c ,k ,h made by ! l9es ( incid5ce4 ,= ! angle ,c ,n ,b is equal to ! angle ,h1 t is1 to twice ! angle ,d1 toge!r ) ! two angles ,a1 & ,n ,b ,h1 t is1 ,k ,b ,a4 ,b ! angles ,k ,b ,a & ,a are equal to ! angle ,c ,k ,h4 ,:]e=e ! angle ,c ,n ,b is equal to twice ! angle ,d1 toge!r ) ! angle ,c ,k ,h made by ! l9es ( incid5ce ,i ,c & ,h ,b produc$ to ,k4 ,sid5ote3 ,if two /rait l9es drawn f "o po9t fall ^u ! concave circumf]5ce ( a circle1 & ! angle !y make be less ?an twice ! angle at ! c5tre1 ! l9es reflect$ f !m & meet+ )9 ! circle w make an angle1 :i* be+ add$ to ! angle ( ! incid5t l9es w be equal to twice ! angle at ! c5tre4 #e4 ,if two /rait l9es drawn f "o po9t fall ^u ! concave circumf]5ce ( a circle1 & ! angle !y make be less ?an twice ! angle at ! c5tre1 ! l9es reflect$ f !m & meet+ )9 ! circle w make an angle1 :i* be+ add$ to ! angle ( ! incid5t l9es1 w be equal to twice ! angle at ! c5tre4 ,let ! two l9es ,a ,b & ,a ,c 79 fig4 #e71 drawn f ! po9t ,a1 fall ^u ! concave circumf]5ce ( ! circle ^: c5tre is ,d2 & let _! reflect$ l9es ,b ,e & ,c ,e meet 9 ! po9t ,e2 also let ! angle ,a be less ?an twice ! angle ,d4 ,i say1 ! angles ,a & ,e toge!r tak5 are equal to twice ! angle ,d4 ,= let ! /rait l9es ,a ,b & ,e ,c cut ! /rait l9es ,d ,c & ,d ,b 9 ! po9ts ,g & ,h2 & ! angle ,b ,h ,c w be equal to ! two angles ,e ,b ,h & ,e2 also ! same angle ,b ,h ,c w be equal to ! two angles ,d & ,d ,c ,h2 & 9 l mann] ! angle ,b ,g ,c w be equal to ! two angles ,a ,c ,d & ,a1 & ! same angle ,b ,g ,c w be also equal to ! two angles ,d ,b ,g & ,d4 ,:]e=e ! f\r angles ,e ,b ,h1 ,e1 ,a ,c ,d & ,a1 are equal to ! f\r angles ,d1 ,d ,c ,h1 ,d ,b ,g & ,d4 ,if1 !re=e1 equals be tak5 away on bo? sides1 namely1 on "o side ,a ,c ,d & ,e ,b ,h1 & on ! o!r side ,d ,c ,h & ,d ,b ,g1 7= ! angle ,e ,b ,h is equal to ! angle ,d ,b ,g1 & ! angle ,a ,c ,d equal to ! angle ,d ,c ,h71 ! rema9d]s on bo? sides w be equal1 namely1 on "o side ! angles ,a & ,e1 & on ! o!r ! angle ,d twice tak54 ,:]e=e ! angles ,a & ,e are equal to twice ! angle ,d4 ,coroll4 ,if ! angle ,a be great] ?an twice ! angle ,d1 _! reflect$ l9es w div]ge4 ,=1 by ! coroll>y ( ! ?ird proposition1 if ! angle ,a be equal to twice ! angle ,d1 ! reflect$ l9es ,b ,e & ,c ,e w be p>allel2 & if x be less1 !y w concur1 z has n[ be5 demon/rat$4 ,& !re=e1 if x be great]1 ! reflect$ l9es ,b ,e & ,c ,e w div]ge1 & consequ5tly1 if !y be produc$ ! o!r way1 !y w concur & make an angle equal to ! excess ( ! angle ,a above twice ! angle ,d2 z is evid5t by art4 #d4 ,sid5ote3 ,if ?r\gh any "o po9t two unequal *ords be drawn cutt+ "o ano!r1 & ! c5tre ( ! circle be n plac$ betwe5 !m1 & ! l9es reflect$ f !m concur :]esoev]1 !re _c ?r\gh ! po9t1 ?r\gh :i* ! two =m] l9es 7 drawn1 be drawn any o!r /rait l9e ^: reflect$ l9e %all pass ?r\gh ! common po9t ( ! two =m] l9es reflect$4 #f4 ,if ?r\gh any "o po9t two unequal *ords be drawn cutt+ "o ano!r1 ei!r )9 ! circle1 or1 if !y be produc$1 )\t x1 & ! c5tre ( ! circle be n plac$ betwe5 !m1 & ! l9es reflect$ f !m concur :]esoev]2 !re _c1 ?r\gh ! po9t ?r\gh :i* ! =m] l9es 7 drawn1 be drawn ano!r /rait l9e1 ^: reflect$ l9e %all pass ?r\gh ! po9t :]e ! two =m] reflect$ l9es concur4 ,let any two unequal *ords1 z ,b ,k & ,c ,h 79 fig4 #f71 be drawn ?r\gh ! po9t ,a 9 ! circle ,b ,c2 & let _! reflect$ l9es ,b ,d & ,c ,e meet 9 ,f2 & let ! c5tre n be betwe5 ,a ,b & ,a ,c2 & f ! po9t ,a let any o!r /rait l9e1 z ,a ,g1 be drawn to ! circumf]5ce betwe5 ,b & ,c4 ,i say1 ,g ,n1 :i* passes ?r\gh ! po9t ,f1 :]e ! reflect$ l9es ,b ,d & ,c ,e meet1 w n be ! reflect$ l9e ( ,a ,g4 ,= let ! ar* ,b ,l be tak5 equal to ! ar* ,b ,g1 & ! /rait l9e ,b ,m equal to ! /rait l9e ,b ,a2 & ,l ,m be+ drawn1 let x be produc$ to ! circumf]5ce 9 ,o4 ,see+ !re=e ,b ,a & ,b ,m are equal1 & ! ar* ,b ,l equal to ! ar* ,b ,g1 & ! angle ,m ,b ,l equal to ! angle ,a ,b ,g1 ,a ,g & ,m ,l w also be equal1 &1 produc+ ,g ,a to ! circumf]5ce 9 ,i1 ! :ole l9es ,l ,o & ,g ,i w 9 l mann] be equal4 ,b ,l ,o is great] ?an ,g ,f ,n1 z %all pres5tly be demon/rat$2 & !re=e also ,g ,i is great] ?an ,g ,n4 ,:]e=e ! angles ,n ,g ,c & ,i ,g ,b are n equal4 ,:]e=e ! l9e ,g ,f ,n is n reflect$ f ! l9e ( incid5ce ,a ,g1 & consequ5tly no o!r /rait l9e1 besides ,a ,b & ,a ,c1 :i* is drawn ?r\gh ! po9t ,a1 & falls ^u ! circumf]5ce ,b ,c1 c be reflect$ to ! po9t ,f2 :i* 0 to be demon/rat$4 ,x rema9s t ,i prove ,l ,o to be great] ?an ,g ,n2 :i* ,i %all d 9 ?is mann]4 ,l ,o & ,g ,n cut "o ano!r 9 ,p2 & ,p ,l is great] ?an ,p ,g4 ,see+ n[ ,l ,p4 ,p ,g 33 ,p ,n4 ,p ,o are proportionals1 !re=e ! two extremes ,l ,p & ,p ,o toge!r tak51 t is ,l ,o1 are great] ?an ,p ,g & ,p ,n toge!r tak51 t is1 ,g ,n2 :i* rema9$ to be prov$4 ,sid5ote3 ,9 equal *ords ! same is n true4 #g4 ,b if two equal *ords be drawn ?r\gh "o po9t )9 a circle1 & ! l9es reflect$ f !m meet 9 ano!r po9t1 !n ano!r /rait l9e may be drawn betwe5 !m ?r\gh ! =m] po9t1 ^: reflect$ l9e %all pass ?r\gh ! latt] po9t4 ,let ! two equal *ords ,b ,c & ,e ,d 79 ! #g? figure7 cut "o ano!r 9 ! po9t ,a )9 ! circle ,b ,c ,d2 & let _! reflect$ l9es ,c ,h & ,d ,i meet 9 ! po9t ,f4 ,!n divid+ ! ar* ,c ,d equally 9 ,g1 let ! two *ords ,g ,k & ,g ,l be drawn ?r\gh ! po9ts ,a & ,f4 ,i say1 ,g ,l w be ! l9e reflect$ f ! *ord ,k ,g4 ,= ! f\r *ords ,b ,c1 ,c ,h1 ,e ,d & ,d ,i are by supposition all equal to "o ano!r2 & !re=e ! ar* ,b ,c ,h is equal to ! ar* ,e ,d ,i2 z also ! angle ,b ,c ,h to ! angle ,e ,d ,i2 & ! angle ,a ,m ,c to its v]ticle angle ,f ,m ,d2 & ! /rait l9e ,d ,m to ! /rait l9e ,g ,m2 &1 9 l mann]1 ! /rait l9e ,a ,c to ! /rait l9e ,f ,d2 & ! *ords ,c ,g & ,g ,d be+ drawn1 w also be equal2 & also ! angles ,f ,d ,g & ,a ,c ,g1 9 ! equal segm5ts ,g ,d ,i & ,g ,c ,b4 ,:]e=e ! /rait l9es ,f ,g & ,a ,g are equal2 &1 !re=e1 ! angle ,f ,g ,d is equal to ! angle ,a ,g ,c1 t is1 ! angle ( incid5ce equal to ! angle ( reflection4 ,:]e=e ! l9e ,g ,l is reflect$ f ! incid5t l9e ,c ,g2 :i* 0 to be prov$4 ,coroll4 ,by ! v sight ( ! figure x is manife/1 t if ,g be n ! middle po9t betwe5 ,c & ,d1 ! reflect$ l9e ,g ,l w n pass ?r\gh ! po9t ,f4 ,sid5ote3 ,two po9ts be+ giv5 9 ! circumf]5ce ( a circle1 to draw two /rait l9es to !m s z t _! reflect$ l9es may conta9 any angle giv54 #h4 ,two po9ts 9 ! circumf]5ce ( a circle be+ giv5 to draw two /rait l9es to !m1 s z t _! reflect$ l9es may be p>allel1 or conta9 any angle giv54 ,9 ! circumf]5ce ( ! circle1 ^: c5tre is ,a1 79 ! #h? figure7 let ! two po9ts ,b & ,c be giv52 & let x be requir$ to draw to !m f two po9ts tak5 )\t ! circle two incid5t l9es1 s t _! reflect$ l9es may1 fir/1 be p>allel4 ,let ,a ,b & ,a ,c be drawn2 z also any incid5t l9e ,d ,c1 ) its reflect$ l9e ,c ,f2 & let ! angle ,e ,c ,d be made d\ble to ! angle ,a2 & let ,h ,b be drawn p>allel to ,e ,c1 & produc$ till x meet ) ,d ,c produc$ 9 ,i4 ,la/ly1 produc+ ,a ,b indef9itely to ,k1 let ,g ,b be drawn s t ! angle ,g ,b ,k may be equal to ! angle ,h ,b ,k1 & !n ,g ,b w be ! reflect$ l9e ( ! incid5t l9e ,h ,b4 ,i say1 ,d ,c & ,h ,b are two incid5t l9es1 ^: reflect$ l9es ,c ,f & ,b ,g are p>allel4 ,= see+ ! angle ,e ,c ,d is d\ble to ! angle ,b ,a ,c1 ! angle ,h ,i ,c is also1 by reason ( ! p>allels ,e ,c & ,h ,i1 d\ble to ! same ,b ,a ,c2 !re=e also ,f ,c & ,g ,b1 namely1 ! l9es reflect$ f ! incid5t l9es ,d ,c & ,h ,b1 are p>allel4 ,:]e=e ! fir/ ?+ requir$ is done4 ,secondly1 let x be requir$ to draw to ! po9ts ,b & ,c two /rait l9es ( incid5ce1 s t ! l9es reflect$ f !m may conta9 ! giv5 angle ,z4 ,to ! angle ,e ,c ,d made at ! po9t ,c1 let !re be add$ on "o side ! angle ,d ,c ,l equal to half ,z1 & on ! o!r side ! angle ,e ,c ,m equal to ! angle ,d ,c ,l2 & let ! /rait l9e ,b ,n be drawn p>allel to ! /rait l9e ,c ,m2 & let ! angle ,k ,b ,o be made equal to ! angle ,n ,b ,k2 :i* be+ done1 ,b ,o w be ! l9e ( reflection f ! l9e ( incid5ce ,n ,b4 ,la/ly1 f ! incid5t l9e ,l ,c1 let ! reflect$ l9e ,c ,o be drawn1 cutt+ ,b ,o at ,o1 & mak+ ! angle ,c ,o ,b4 ,i say1 ! angle ,c ,o ,b is equal to ! angle ,z4 ,let ,n ,b be produc$ till x meet ) ! /rait l9e ,l ,c produc$ 9 ,p4 ,see+1 !re=e1 ! angle ,l ,c ,m is1 by con/ruction1 equal to twice ! angle ,b ,a ,c1 toge!r ) ! angle ,z2 ! angle ,n ,p ,l1 :i* is equal to ,l ,c ,m by reason ( ! p>allels ,n ,p & ,m ,c1 w also be equal to twice ! same angle ,b ,a ,c1 toge!r ) ! angle ,z4 ,& see+ ! two /rait l9es ,o ,c & ,o ,b fall f ! po9t ,o ^u ! po9ts ,c & ,b2 & _! reflect$ l9es ,l ,c & ,n ,b meet 9 ! po9t ,p2 ! angle ,n ,p ,l w be equal to twice ! angle ,b ,a ,c toge!r ) ! angle ,c ,o ,b4 ,b ,i h already prov$ ! angle ,n ,p ,l to be equal to twice ! angle ,b ,a ,c toge!r ) ! angle ,z4 ,!re=e ! angle ,c ,o ,b is equal to ! angle ,z2 :]e=e1 two po9ts 9 ! circumf]5ce ( a circle be+ giv51 ,i h drawn1 c42 :i* 0 to be done4 ,b if x be requir$ to draw ! incid5t l9es f a po9t )9 ! circle1 s t ! l9es reflect$ f !m may conta9 an angle equal to ! angle ,z1 ! same me?od is to be us$1 sav+ t 9 ?is case ! angle ,z is n to be add$ to twice ! angle ,b ,a ,c1 b to be tak5 f x4 ,sid5ote3 ,if a /rait l9e fall+ ^u ! circumf]5ce ( a circle be produc$ till x rea* ! semidiamet]1 & t "p ( x1 :i* is int]cept$ betwe5 ! circumf]5ce & ! semidiamet]1 be equal to t "p ( ! semidiamet] :i* is betwe5 ! po9t ( conc\rse & ! c5tre1 ! reflect$ l9e w be p>allel to ! semidiamet]4 #i4 ,if a /rait l9e1 fall+ ^u ! circumf]5ce ( a circle1 be produc$ till x rea* ! semidiamet]1 & t "p ( x :i* is int]cept$ betwe5 ! circumf]5ce & ! semidiamet] be equal to t "p ( ! semidiamet] :i* is betwe5 ! po9t ( conc\rse & ! c5tre1 ! reflect$ l9e w be p>allel to ! semidiamet]4 ,let any l9e ,a ,b 79 ! #i? figure7 be ! semidiamet] ( ! circle ^: c5tre is ,a2 & ^u ! circumf]5ce ,b ,d let ! /rait l9e ,c ,d fall1 & be produc$ till x cut ,a ,b 9 ,e1 s t ,e ,d & ,e ,a may be equal2 & f ! incid5t l9e ,c ,d let ! l9e ,d ,f be reflect$4 ,i say1 ,a ,b & ,d ,f w be p>allel4 ,let ,a ,g be drawn ?r\gh ! po9t ,d4 ,see+1 !re=e1 ,e ,d & ,e ,a are equal1 ! angles ,e ,d ,a & ,e ,a ,d w also be equal4 ,b ! angles ,f ,d ,g & ,e ,d ,a are equal2 = ea* ( !m is half ! angle ,e ,d ,h or ,f ,d ,c4 ,:]e=e ! angles ,f ,d ,g & ,e ,a ,d are equal2 & consequ5tly ,d ,f & ,a ,b are p>allel2 :i* 0 to be prov$4 ,coroll4 ,if ,e ,a be great] !n ,e ,d1 !n ,d ,f & ,a ,b be+ produc$ w concur2 b if ,e ,a be less ?an ,e ,d1 !n ,b ,a & ,d ,h be+ produc$ w concur4 ,sid5ote3 ,if f a po9t )9 a circle two /rait l9es be drawn to ! circumf]5ce1 & _! reflect$ l9es meet 9 ! circumf]5ce ( ! same circle1 ! angle made by ! reflect$ l9es w be a ?ird "p ( ! angle made by ! incid5t l9es4 #aj4 ,if f a po9t )9 a circle two /rait l9es be drawn to ! circumf]5ce1 & _! reflect$ l9es meet 9 ! circumf]5ce ( ! same circle1 ! angle made by ! l9es ( reflection w be a ?ird "p ( ! angle made by ! l9es ( incid5ce4 ,f ! po9t ,b 79 ! #aj? figure7 tak5 )9 ! circle ^: c5tre is ,a1 let ! two /rait l9es ,b ,c & ,b ,d be drawn to ! circumf]5ce2 & let _! reflect$ l9es ,c ,e & ,d ,e meet 9 ! circumf]5ce ( ! same circle at ! po9t ,e4 ,i say1 ! angle ,c ,e ,d w be a ?ird "p ( ! angle ,c ,b ,d4 ,let ,a ,c & ,a ,d be drawn4 ,see+1 !re=e1 ! angles ,c ,e ,d & ,c ,b ,d toge!r tak5 are equal to twice ! angle ,c ,a ,d 7z has be5 demon/rat$ 9 ! #e? article72 & ! angle ,c ,a ,d twice tak5 is quadruple to ! angle ,c ,e ,d2 ! angles ,c ,e ,d & ,c ,b ,d toge!r tak5 w also be equal to ! angle ,c ,e ,d f\r times tak52 & !re=e if ! angle ,c ,e ,d be tak5 away on bo? sides1 !re w rema9 ! angle ,c ,b ,d on "o side1 equal to ! angle ,c ,e ,d ?rice tak5 on ! o!r side2 :i* 0 to be demon/rat$4 ,coroll4 ,!re=e a po9t be+ giv5 )9 a circle1 !re may be drawn two l9es f x to ! circumf]5ce1 s z _! reflect$ l9es may meet 9 ! circumf]5ce4 ,= x is b trisect+ ! angle ,c ,b ,d1 :i* h[ x may be done %all be %[n 9 ! foll[+ *apt]4 ---------------------------------------- -------------------------------- ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xix4 ,fig4 #a-aj ---------------------------------------- -------------------------------- ------- ,,*apt] ,,xx4 ,,( ,,! ,,dim5sion ,,( ,a ,,circle1 ,,& ,,! ,,division ,,( ,,angles ,,or ,,ar*es4 #a4 ,! dim5sion ( a circle nev] det]m9$ 9 numb]s by ,ar*im$es & o!rs4--#b4 ,! fir/ attempt = ! f9d+ out ( ! dim5sion ( a circle by l9es4--#c4 ,! second attempt = ! f9d+ out ( ! dim5sion ( a circle f ! consid]ation ( ! nature ( crook$ness4--#d4 ,! ?ird attempt2 & "s ?+s prop\nd$ to be fur!r se>*$ into4--#e4 ,! equation ( ! spiral ( ,ar*im$es ) a /rait l9e4--#f4 ,( ! analysis ( geometricians by ! p[]s ( l9es4 ,sid5ote3 ,! dim5sion ( a circle nev] det]m9$ 9 numb]s by ,ar*im$es & o!rs4 #a4 ,9 ! comp>+ ( an ar* ( a circle ) a /rait l9e1 _m & great geometricians1 ev5 f ! mo/ anci5t times1 h ex]cis$ _! wits2 & m _h done ! same1 if !y _h n se5 _! pa9s1 ?\gh und]tak5 = ! common good1 if n br\ght to p]fection1 vilifi$ by ^? t envy ! praises ( o!r m54 ,among/ ^? anci5t writ]s ^: works are come to our h&s1 ,ar*im$es 0 ! fir/ t br\ght ! l5g? ( ! p]imet] ( a circle )9 ! limits ( numb]s v little diff]+ f ! tru?2 demon/rat+ ! same to be less ?an ?ree diamet]s & a sev5? "p1 b great] ?an ?ree diamet]s & t5 sev5ty-"o p>ts ( ! diamet]4 ,s t suppos+ ! radius to consi/ ( #aj1jjj1jjj equal p>ts1 ! ar* ( a quadrant w be betwe5 #ae1gad1bhe & #ae1gjd1bbe ( ! same p>ts4 ,9 our times1 ,ludovicus ,van ,cull5 & ,willebrordus ,snellius1 ) jo9t endeav\r1 h come yet ne>] to ! tru?2 & pron\nc$ f true pr9ciples1 t ! ar* ( a quadrant1 putt+1 z be=e1 #aj1jjj1jjj = radius1 diff]s n "o :ole unity f ! numb] #ae1gjg1ifc2 :i*1 if !y _h exhibit$ _! ari?metical op]ations1 & no man _h discov]$ any error 9 t long "w ( !irs1 _h be5 demon/rat$ by !m4 ,?is is ! fur!/ progress t has be5 made by ! way ( numb]s2 & !y t h proce$$ ?us f> des]ve ! praise ( indu/ry4 ,nev]!less1 if we consid] ! b5efit1 :i* is ! scope at :i* all speculation %\ld aim1 ! improvem5t !y h made has be5 little or none4 ,= any ord9>y man may mu* soon] & m accurately f9d a /rait l9e equal to ! p]imet] ( a circle1 & consequ5tly squ>e ! circle1 by w9d+ a small ?read ab\t a giv5 cyl9d]1 ?an any geometrician %all d ! same by divid+ ! radius into #aj1jjj1jjj equal p>ts4 ,b ?\gh ! l5g? ( ! circumf]5ce 7 exactly set out1 ei!r by numb]s1 or me*anically1 or only by *ance1 yet ?is w\ld contribute no help at all t[>ds ! section ( angles1 unless happily ^! two problems1 to divide a giv5 angle accord+ to any proportion assign$1 & to f9d a /rait l9e equal to ! ar* ( a circle1 7 reciprocal1 & foll[$ "o ano!r4 ,see+ !re=e ! b5efit proce$+ f ! k ( ! l5g? ( ! ar* ( a quadrant consi/s 9 ?is1 t we may !reby divide an angle accord+ to any proportion1 ei!r accurately1 or at lea/ accurately en\gh = common use2 & see+ ?is _c be done by ari?metic1 ,i ?\ght fit to attempt ! same by geometry1 & 9 ?is *apt] to make trial :e!r x might n be p]=m$ by ! draw+ ( /rait & circul> l9es4 ,sid5ote3 ,! fir/ attempt = ! f9d+ out ( ! dim5sion ( a circle by l9es4 #b4 ,let ! squ>e ,a ,b ,c ,d 79 ! fir/ figure7 be describ$2 & ) ! radii ,a ,b1 ,b ,c1 & ,d ,c1 ! ?ree ar*es ,b ,d1 ,c ,a1 & ,a ,c2 ( :i* let ! two ,b ,d & ,c ,a cut "o ano!r 9 ,e1 & ! two ,b ,d & ,a ,c 9 ,f4 ,! diagonals !re=e ,b ,d & ,a ,c be+ drawn w cut "o ano!r 9 ! c5tre ( ! squ>e ,g1 & ! two ar*es ,b ,d & ,c ,a 9 two equal p>ts 9 ,h & ,y2 & ! ar* ,b ,h ,d w be trisect$ 9 ,f & ,e4 ,?r\gh ! c5tre ,g let ! two /rait l9es ,k ,g ,l & ,m ,g ,n be drawn p>allel & equal to ! sides ( ! squ>e ,a ,b & ,a ,d1 cutt+ ! f\r sides ( ! same squ>e 9 ! po9ts ,k1 ,l1 ,m1 & ,n2 :i* be+ done1 ,k ,l w pass ?r\gh ,f1 & ,m ,n ?r\gh ,e4 ,!n let ,o ,p be drawn p>allel & equal to ! side ,b ,c1 cutt+ ! ar* ,b ,f ,d 9 ,f1 & ! sides ,a ,b & ,d ,c 9 ,o & ,p4 ,!re=e ,o ,f w be ! s9e ( ! ar* ,b ,f1 :i* is an ar* ( #cj degrees2 & ! same ,o ,f w be equal to half ! radius4 ,la/ly1 divid+ ! ar* ,b ,f 9 ! middle 9 ,q1 let ,r ,q1 ! s9e ( ! ar* ,b ,q1 be drawn & produc$ to ,s1 s t ,q ,s be equal to ,r ,q1 & consequ5tly ,r ,s be equal to ! *ord ( ! ar* ,b ,f2 & let ,f ,s be drawn & produc$ to ,t 9 ! side ,b ,c4 ,i say1 ! /rait l9e ,b ,t is equal to ! ar* ,b ,f2 & consequ5tly t ,b ,v1 ! triple ( ,b ,t1 is equal to ! ar* ( ! quadrant ,b ,f ,e ,d4 ,let ,t ,f be produc$ till x meet ! side ,b ,a produc$ 9 ,x2 & divid+ ,o ,f 9 ! middle 9 ,z1 let ,q ,z be drawn & produc$ till x meet ) ! side ,b ,a produc$4 ,see+ !re=e ! /rait l9es ,r ,s & ,o ,f are p>allel1 & divid$ 9 ! mid/ 9 ,q & ,z1 ,q ,z produc$ w fall ^u ,x1 & ,x ,z ,q produc$ to ! side ,b ,c w cut ,b ,t 9 ! mid/ 9 4 ,^u ! /rait l9e ,f ,z1 ! f\r? "p ( ! radius ,a ,b1 let ! equilat]al triangle a ,z ,f be con/itut$2 & ^u ! c5tre a1 ) ! radius a ,z1 let ! ar* ,z ,f be drawn2 :i* ar* ,z ,f w !re=e be equal to ! ar* ,q ,f1 ! half ( ! ar* ,b ,f4 ,aga91 let ! /rait l9e ,z ,o be cut 9 ! mid/ 9 b1 & ! /rait l9e b ,o 9 ! mid/ 9 c2 & let ! bisection be cont9u$ 9 ?is mann] till ! la/ "p ,o c be ! lea/ t c possibly be tak52 & ^u x1 & all ! re/ ( ! p>ts equal to x into :i* ! /rait l9e ,o ,f may be cut1 let s _m equilat]al triangles be und]/ood to be con/itut$2 ( :i* let ! la/ be d ,o c4 ,if1 !re=e1 ^u ! c5tre d1 ) ! radius d ,o1 be drawn ! ar* ,o c1 & ^u ! re/ ( ! equal p>ts ( ! /rait l9e ,o ,f be drawn 9 l mann] s _m equal ar*es1 all ^? ar*es toge!r tak5 w be equal to ! :ole ar* ,b ,f1 & ! half ( !m1 namely1 ^? t are compreh5d$ betwe5 ,o & ,z1 or betwe5 ,z & ,f1 w be equal to ! ar* ,b ,q or ,q ,f1 & 9 sum1 :at "p soev] ! /rait l9e ,o c be ( ! /rait l9e ,o ,f1 ! same "p w ! ar* ,o c be ( ! ar* ,b ,f1 ?\gh bo? ! ar* & ! *ord be inf9itely bisect$4 ,n[ see+ ! ar* ,o c is m crook$ ?an t "p ( ! ar* ,b ,f :i* is equal to x2 & see+ also t ! m ! /rait l9e ,x c is produc$1 ! m x div]ges f ! /rait l9e ,x ,o1 if ! po9ts ,o & c be und]/ood to be mov$ =w>ds ) /rait motion 9 ,x ,o & ,x c1 ! ar* ,o c w !reby be ext5d$ by little & little1 till at ! la/ x come some:]e to h ! same crook$ness ) t "p ( ! ar* ,b ,f :i* is equal to x4 ,9 l mann]1 if ! /rait l9e ,x b be drawn1 & ! po9t b be und]/ood to be mov$ =w>ds at ! same "t1 ! ar* c b w also by little & little be ext5d$1 till its crook$ness come to be equal to ! crook$ness ( t "p ( ! ar* ,b ,f :i* is equal to x4 ,& ! same w happ5 9 all ^? small equal ar*es :i* are describ$ ^u s _m equal p>ts ( ! /rait l9e ,o ,f4 ,x is also manife/1 t by /rait motion 9 ,x ,o & ,x ,z all ^? small ar*es w lie 9 ! ar* ,b ,f1 9 ! po9ts ,b1 ,q & ,f4 ,& ?\gh ! same small equal ar*es %\ld n be co9cid5t ) ! equal p>ts ( ! ar* ,b ,f 9 all ! o!r po9ts !re(1 yet c]ta9ly !y w con/itute two crook$ l9es1 n only equal to ! two ar*es ,b ,q & ,q ,f1 & equally crook$1 b also hav+ _! cavity t[>ds ! same p>ts2 :i* h[ x %\ld be1 unless all ^? small ar*es %\ld be co9cid5t ) ! ar* ,b ,f 9 all its po9ts1 is n imag9able4 ,!y are !re=e co9cid5t1 & all ! /rait l9es drawn f ,x1 & pass+ ?r\gh ! po9ts ( division ( ! /rait l9e ,o ,f1 w also divide ! ar* ,b ,f into ! same proportions into :i* ,o ,f is divid$4 ,n[ see+ ,x b cuts (f f ! po9t ,b ! f\r? "p ( ! ar* ,b ,f1 let t f\r? "p be ,b e2 & let ! s9e !re(1 f e1 be produc$ to ,f ,t 9 g1 = s f e w be ! f\r? "p ( ! /rait l9e f g1 because z ,o b is to ,o ,f1 s is f e to f g4 ,b ,b ,t is great] ?an f g2 & !re=e ! same ,b ,t is great] ?an f\r s9es ( ! f\r? "p ( ! ar* ,b ,f4 ,& 9 l mann]1 if ! ar* ,b ,f be subdivid$ into any numb] ( equal p>ts :atsoev]1 x may be prov$ t ! /rait l9e ,b ,t is great] ?an ! s9e ( "o ( ^? small ar*es1 s _m times tak5 z !re be p>ts made ( ! :ole ar* ,b ,f4 ,:]e=e ! /rait l9e ,b ,t is n less ?an ! ar* ,b ,f4 ,b nei!r c x be great]1 because if any /rait l9e :atsoev]1 less ?an ,b ,t1 be drawn bel[ ,b ,t1 p>allel to x1 & t]m9at$ 9 ! /rait l9es ,x ,b & ,x ,t1 x w\ld cut ! ar* ,b ,f2 & s ! s9e ( "s "o ( ! p>ts ( ! ar* ,b ,f1 tak5 s (t5 z t small ar* is f\nd 9 ! :ole ar* ,b ,f1 w\ld be great] ?an s _m ( ! same ar*es2 :i* is absurd4 ,:]e=e ! /rait l9e ,b ,t is equal to ! ar* ,b ,f2 & ! /rait l9e ,b ,v equal to ! ar* ( ! quadrant ,b ,f ,d2 & ,b ,v f\r times tak51 equal to ! p]imet] ( ! circle describ$ ) ! radius ,a ,b4 ,also ! ar* ,b ,f & ! /rait l9e ,b ,t are ev]y:]e divid$ into ! same proportions2 & consequ5tly any giv5 angle1 :e!r great] or less ?an ,b ,a ,f1 may be divid$ into any proportion giv54 ,b ! /rait l9e ,b ,v1 ?\gh its magnitude fall )9 ! t]ms assign$ by ,ar*im$es1 is f\nd1 if comput$ by ! canon ( signs1 to be some:at great] ?an t :i* is exhibit$ by ! ,rudolph9e numb]s4 ,nev]!less1 if 9 ! place ( ,b ,t1 ano!r /rait l9e1 ?\gh nev] s little less1 be sub/itut$1 ! division ( angles is imm$iately lo/1 z may by any man be demon/rat$ by ?is v s*eme4 ,h[soev]1 if any man ?9k ?is my /rait l9e ,b ,v to be too great1 yet1 see+ ! ar* & all ! p>allels are ev]y:]e s exactly divid$1 & ,b ,v comes s ne> to ! tru?1 ,i desire he w\ld se>* out ! reason1 :y1 grant+ ,b ,v to be precisely true1 ! ar*es cut (f %\ld n be equal4 ,b "s man may yet ask ! reason :y ! /rait l9es1 drawn f ,x ?r\gh ! equal p>ts ( ! ar* ,b ,f1 %\ld cut (f 9 ! tang5t ,b ,v s _m /rait l9es equal to !m1 see+ ! connect$ /raight l9e ,x ,v passes n ?r\gh ! po9t ,d1 b cuts ! /rait l9e ,a ,d produc$ 9 l2 & consequ5tly require "s det]m9ation ( ?is problem4 ,conc]n+ :i*1 ,i w say :at ,i ?9k to be ! reason1 namely1 t :il/ ! magnitude ( ! ar* do? n exce$ ! magnitude ( ! radius1 t is1 ! magnitude ( ! tang5t ,b ,c1 bo? ! ar* & ! tang5t are cut alike by ! /rait l9es drawn f ,x2 o!rwise n4 ,= ,a ,v be+ connect$1 cutt+ ! ar* ,b ,h ,d 9 ,i1 if ,x ,c be+ drawn %\ld cut ! same ar* 9 ! same po9t ,i1 x w\ld be z true t ! ar* ,b ,i is equal to ! radius ,b ,c1 z x is true t ! ar* ,b ,f is equal to ! /rait l9e ,b ,t2 & draw+ ,x ,k x w\ld cut ! ar* ,b ,i 9 ! mid/ 9 i2 also draw+ ,a i & produc+ x to ! tang5t ,b ,c 9 k1 ! /rait l9e ,b k w be ! tang5t ( ! ar* ,b i1 7:i* ar* is equal to half ! radius7 & ! same /rait l9e ,b k w be equal to ! /rait l9e k ,i4 ,i say all ?is is true1 if ! prec$+ demon/ration be true2 & consequ5tly ! proportional section ( ! ar* & its tang5t proce$s hi!rto4 ,b x is manife/ by ! gold5 rule1 t tak+ ,b h d\ble to ,b ,t1 ! l9e ,x h %all n cut (f ! ar* ,b ,e1 :i* is d\ble to ! ar* ,b ,f1 b a mu* great]4 ,= ! magnitude ( ! /raight l9es ,x ,m1 ,x ,b1 & ,m ,e1 be+ kn[n 79 numb]s71 ! magnitude ( ! /rait l9e cut (f 9 ! tang5t by ! /rait l9e ,x ,e produc$ to ! tang5t1 may also be kn[n2 & x w be f\nd to be less ?an ,b h2 ,:]e=e ! /rait l9e ,x h be+ drawn1 w cut (f a "p ( ! ar* ( ! quadrant great] ?an ! ar* ,b ,e4 ,b ,i %all speak m fully 9 ! next article conc]n+ ! magnitude ( ! ar* ,b ,i4 ,& let ?is be ! fir/ attempt = ! f9d+ out ( ! dim5sion ( a circle by ! section ( ! ar* ,b ,f4 ,sid5ote3 ,! second attempt = ! f9d+ out ( ! dim5sion ( a circle f ! consid]ation ( ! nature ( crook$ness4 #c4 ,i %all n[ attempt ! same by argum5ts drawn f ! nature ( ! crook$ness ( ! circle itself2 b ,i %all fir/ set d[n "s premises necess>y = ?is speculation2 & ,fir/1 if a /rait l9e be b[$ into an ar* ( a circle equal to x1 z :5 a /ret*$ ?read1 :i* t\*e? a "r cyl9d]1 is s b[$ 9 e po9t1 t x be ev]y:]e co9cid5t ) ! p]imet] ( ! base ( ! cyl9d]1 ! flexion ( t l9e w be equal1 9 all its po9ts2 & consequ5tly ! crook$ness ( ! ar* ( a circle is ev]y:]e uni=m2 :i* ne$s no o!r demon/ration ?an ?is1 t ! p]imet] ( a circle is an uni=m l9e4 ,secondly1 & consequ5tly3 if two unequal ar*es ( ! same circle be made by ! b[+ ( two /rait l9es equal to !m1 ! flexion ( ! long] l9e1 :il/ x is b[$ into ! great] ar*1 is great] ?an ! flexion ( ! %ort] l9e1 :il/ x is b[$ into ! less] ar*1 accord+ to ! proportion ( ! ar*es !mselves2 & consequ5tly1 ! crook$ness ( ! great] ar* is to ! crook$ness ( ! less] ar*1 z ! great] ar* is to ! less] ar*4 ,?irdly3 if two unequal circles & a /rait l9e t\* "o ano!r 9 ! same po9t1 ! crook$ness ( any ar* tak5 9 ! less] circle1 w be great] ?an ! crook$ness ( an ar* equal to x tak5 9 ! great] circle1 9 reciprocal proportion to t ( ! radii ) :i* ! circles are describ$2 or1 :i* is all "o1 any /rait l9e be+ drawn f ! po9t ( contact till x cut bo? ! circumf]5ces1 z ! "p ( t /rait l9e cut (f by ! circumf]5ce ( ! great] circle to t "p :i* is cut (f by ! circumf]5ce ( ! less] circle4 ,= let ,a ,b & ,a ,c 79 ! second figure7 be two circles1 t\*+ "o ano!r1 & ! /rait l9e ,a ,d 9 ! po9t ,a2 & let _! c5tres be ,e & ,f2 & let x be suppos$1 t z ,a ,e is to ,a ,f1 s is ! ar* ,a ,b to ! ar* ,a ,h4 ,i say ! crook$ness ( ! ar* ,a ,c is to ! crook$ness ( ! ar* ,a ,h1 z ,a ,e is to ,a ,f4 ,= let ! /rait l9e ,a ,d be suppos$ to be equal to ! ar* ,a ,b1 & ! /rait l9e ,a ,g to ! ar* ,a ,c2 & let ,a ,d1 = example1 be d\ble to ,a ,g4 ,!re=e1 by reason ( ! lik5ess ( ! ar*es ,a ,b & ,a ,c1 ! /rait l9e ,a ,b w be d\ble to ! /rait l9e ,a ,c1 & ! radius ,a ,e d\ble to ! radius ,a ,f1 & ! ar* ,a ,b d\ble to ! ar* ,a ,h4 ,& because ! /rait l9e ,a ,d is s b[$ to be co9cid5t ) ! ar* ,a ,b equal to x1 z ! /rait l9e ,a ,g is b[$ to be co9cid5t ) ! ar* ,a ,c equal also to x1 ! flexion ( ! /rait l9e ,a ,g into ! crook$ l9e ,a ,c w be equal to ! flexion ( ! /rait l9e ,a ,d into ! crook$ l9e ,a ,b4 ,b ! flexion ( ! /rait l9e ,a ,d into ! crook$ l9e ,a ,b is d\ble to ! flexion ( ! /rait l9e ,a ,g into ! crook$ l9e ,a ,h2 & !re=e ! flexion ( ! /rait l9e ,a ,g into ! crook$ l9e ,a ,c is d\ble to ! flexion ( ! same /rait l9e ,a ,g into ! crook$ l9e ,a ,h4 ,:]e=e1 z ! ar* ,a ,b is to ! ar* ,a ,c or ,a ,h2 or z ! radius ,a ,e is to ! radius ,a ,f2 or z ! *ord ,a ,b is to ! *ord ,a ,c2 s reciprocally is ! flexion or uni=m crook$ness ( ! ar* ,a ,c1 to ! flexion or uni=m crook$ness ( ! ar* ,a ,h1 namely1 "h d\ble4 ,& ?is may by ! same me?od be demon/rat$ 9 circles ^: p]imet]s are to "o ano!r triple1 quadruple1 or 9 :atsoev] giv5 proportion4 ,! crook$ness !re=e ( two equal ar*es tak5 9 sev]al circles are 9 proportion reciprocal to t ( _! radii1 or l ar*es1 or l *ords2 :i* 0 to be demon/rat$4 ,let ! squ>e ,a ,b ,c ,d be aga9 describ$ 79 ! ?ird figure71 & 9 x ! quadrants ,a ,b ,d1 ,b ,c ,a & ,d ,a ,c2 & divid+ ea* side ( ! squ>e ,a ,b ,c ,d 9 ! mid/ 9 ,e1 ,f1 ,g & ,h1 let ,e ,g & ,f ,h be connect$1 :i* w cut "o ano!r 9 ! c5tre ( ! squ>e at ,i1 & divide ! ar* ( ! quadrant ,a ,b ,d into ?ree equal p>ts 9 ,k & ,l4 ,also ! diagonals ,a ,c & ,b ,d be+ drawn w cut "o ano!r 9 ,i1 & divide ! ar*es ,b ,k ,d & ,c ,l ,a into two equal p>ts 9 ,m & ,n4 ,!n ) ! radius ,b ,f let ! ar* ,f ,e be drawn1 cutt+ ! diagonal ,b ,d 9 ,o2 & divid+ ! ar* ,b ,m 9 ! mid/ 9 ,p1 let ! /rait l9e ,e a equal to ! *ord ,b ,p be set (f f ! po9t ,e 9 ! ar* ,e ,f1 & let ! ar* a b be tak5 equal to ! ar* ,o a1 & let ,b a & ,b b be drawn & produc$ to ! ar* ,a ,n 9 c & d2 & la/ly1 let ! /rait l9e ,a d be drawn4 ,i say ! /rait l9e ,a d is equal to ! ar* ,a ,n or ,b ,m4 ,i h prov$ 9 ! prec$+ article1 t ! ar* ,e ,o is twice z crook$ z ! ar* ,b ,p1 t is to say1 t ! ar* ,e ,o is s mu* m crook$ ?an ! ar* ,b ,p1 z ! ar* ,b ,p is m crook$ ?an ! /rait l9e ,e a4 ,! crook$ness !re=e ( ! *ord ,e a1 ( ! ar* ,b ,p1 & ( ! ar* ,e ,o1 are z #j1 #a1 #b4 ,also ! diff]5ce betwe5 ! ar*es ,e ,o & ,e ,o1 ! diff]5ce betwe5 ! ar*es ,e ,o & ,e a1 & ! diff]5ce betwe5 ! ar*es ,e ,o & ,e b1 are z #j1 #a1 #b4 ,s also ! diff]5ce betwe5 ! ar*es ,a ,n & ,a ,n1 ! diff]5ce betwe5 ! ar*es ,a ,n & ,a c1 & ! diff]5ce betwe5 ! ar*es ,a ,n & ,a d1 are z #j1 #a1 #b2 & ! /rait l9e ,a c is d\ble to ! *ord ,b ,p or ,e a1 & ! /rait l9e ,a d d\ble to ! *ord ,e b4 ,aga91 let ! /rait l9e ,b ,f be divid$ 9 ! mid/ 9 ,q1 & ! ar* ,b ,p 9 ! mid/ 9 ,r2 & describ+ ! quadrant ,b ,q ,s 7^: ar* ,q ,s is a f\r? "p ( ! ar* ( ! quadrant ,b ,m ,d1 z ! ar* ,b ,r is a f\r? "p ( ! ar* ,b ,m1 :i* is ! ar* ( ! semiquadrant ,a ,b ,m7 let ! *ord ,s e equal to ! *ord ,b ,r be set (f f ! po9t ,s 9 ! ar* ,s ,q2 & let ,b e be drawn & produc$ to ! ar* ,a ,n 9 f2 :i* be+ done1 ! /rait l9e ,a f w be quadruple to ! *ord ,b ,r or ,s e4 ,& see+ ! crook$ness ( ! ar* ,s e1 or ( ! ar* ,a c1 is d\ble to ! crook$ness ( ! ar* ,b ,r1 ! excess ( ! crook$ness ( ! ar* ,a f above ! crook$ness ( ! ar* ,a c w be subduple to ! excess ( ! crook$ness ( ! ar* ,a c above ! crook$ness ( ! ar* ,a ,n2 & !re=e ! ar* ,n c w be d\ble to ! ar* c f4 ,:]e=e ! ar* c d is divid$ 9 ! mid/ 9 f1 & ! ar* ,n f is ( ! ar* ,n d4 ,& 9 l mann] if ! ar* ,b ,r be bisect$ 9 ,v1 & ! /rait l9e ,b ,q 9 ,x1 & ! quadrant ,b ,x ,y be describ$1 & ! /rait l9e ,y g equal to ! *ord ,b ,v be set (f f ! po9t ,y 9 ! ar* ,y ,x1 x may be demon/rat$ t ! /rait l9e ,b g be+ drawn & produc$ to ! ar* ,a ,n1 w cut ! ar* f d into two equal p>ts1 & t a /rait l9e drawn f ,a to ! po9t ( t section1 w be equal to eight *ords ( ! ar* ,b ,v1 & s on p]petually2 & consequ5tly1 t ! /rait l9e ,a d is equal to s _m equal *ords ( equal p>ts ( ! ar* ,b ,m1 z may be made by inf9ite bisections4 ,:]e=e ! /rait l9e ,a d is equal to ! ar* ,b ,m or ,a ,n1 t is1 to half ! ar* ( ! quadrant ,a ,b ,d or ,b ,c ,a4 ,coroll4 ,an ar* be+ giv5 n great] ?an ! ar* ( a quadrant 7= be+ made great]1 x comes aga9 t[>ds ! radius ,b ,a produc$1 f :i* x rec$$ be=e7 if a /rait l9e d\ble to ! *ord ( half ! giv5 ar* be adapt$ f ! beg9n+ ( ! ar*1 & by h[ mu* ! ar* t is subt5d$ by x is great] ?an ! giv5 ar*1 by s mu* a great] ar* be subt5d$ by ano!r /rait l9e1 ?is /rait l9e %all be equal to ! fir/ giv5 ar*4 ,suppos+ ! /rait l9e ,b ,v 79 fig4 #a7 be equal to ! ar* ( ! quadrant ,b ,h ,d1 & ,a ,v be connect$ cutt+ ! ar* ,b ,h ,d 9 ,i1 x may be ask$ :at proportion ! ar* ,b ,i has to ! ar* ,i ,d4 ,let !re=e ! ar* ,a ,y be divid$ 9 ! mid/ 9 o1 & 9 ! /rait l9e ,a ,d let ,a p be tak5 equal1 & ,a q d\ble to ! drawn *ord ,a o4 ,!n ^u ! c5tre ,a1 ) ! radius ,a q1 let an ar* ( a circle be drawn cutt+ ! ar* ,a ,y 9 r1 & let ! ar* ,y r be d\bl$ at t2 :i* be+ done1 ! drawn /rait l9e ,a t 7by :at has be5 la/ demon/rat$7 w be equal to ! ar* ,a ,y4 ,aga91 ^u ! c5tre ,a ) ! radius ,a t let ! ar* t u be drawn cutt+ ,a ,d 9 u2 & ! /rait l9e ,a u w be equal to ! ar* ,a ,y4 ,f ! po9t u let ! /rait l9e u s be drawn equal & p>allel to ! /rait l9e ,a ,b1 cutt+ ,m ,n 9 x1 & bisect$ by ,m ,n 9 ! same po9t x4 ,!re=e ! /rait l9e ,a x be+ drawn & produc$ till x meet ) ,b ,c produc$ 9 ,v1 x w cut (f ,b ,v d\ble to ,b s1 t is1 equal to ! ar* ,b ,h ,d4 ,n[ let ! po9t1 :]e ! /rait l9e ,a ,v cuts ! ar* ,b ,h ,d1 be ,i2 & let ! ar* ,d ,i be divid$ 9 ! mid/ 9 y2 & 9 ! /rait l9e ,d ,c1 let ,d z be tak5 equal1 & ,d d\ble to ! drawn *ord ,d y2 & ^u ! c5tre ,d ) ! radius ,d let an ar* ( a circle be drawn cutt+ ! ar* ,b ,h ,d 9 ! po9t n2 & let ! ar* n m be tak5 equal to ! ar* ,i n2 :i* be+ done1 ! /rait l9e ,d m w 7by ! la/ =ego+ coroll>y7 be equal to ! ar* ,d ,i4 ,if n[ ! /rait l9es ,d m & ,c ,v be equal1 ! ar* ,b ,i w be equal to ! radius ,a ,b or ,b ,c2 & consequ5tly ,x ,c be+ drawn1 w pass ?r\gh ! po9t ,i4 ,moreov]1 if ! semicircle ,b ,h ,d be+ complet$1 ! /rait l9es ,i & ,b ,i be drawn1 mak+ a "r angle 79 ! semicircle7 at ,i1 & ! ar* ,b ,i be divid$ 9 ! mid/ at i1 x w foll[ t ,a i be+ connect$ w be p>allel to ! /rait l9e ,i1 & be+ produc$ to ,b ,c 9 k1 w cut (f ! /rait l9e ,b k equal to ! /rait l9e k ,i1 & equal also to ! /rait l9e ,a cut (f 9 ,a ,d by ! /rait l9e ,i4 ,all :i* is manife/1 suppos+ ! ar* ,b ,i & ! radius ,b ,c to be equal4 ,b t ! ar* ,b ,i & ! radius ,b ,c are precisely equal1 _c 7h[ true soev] x be7 be demon/rat$1 unless t be fir/ prov$ :i* is conta9$ 9 art4 #a1 namely1 t ! /rait l9es drawn f ,x ?r\gh ! equal p>ts ( ,o ,f 7produc$ to a c]ta9 l5g?7 cut (f s _m p>ts also 9 ! tang5t ,b ,c sev]ally equal to ! sev]al ar*es cut (f2 :i* !y d mo/ exactly z f> z ,b ,c 9 ! tang5t1 & ,b ,i 9 ! ar* ,b ,e2 insomu* t no inequality betwe5 ! ar* ,b ,i & ! radius ,b ,c c be discov]$ ei!r by ! h& or by ratioc9ation4 ,x is !re=e to be fur!r enquir$1 :e!r ! /rait l9e ,a ,v cut ! ar* ( ! quadrant 9 ,i 9 ! same proportion z ! po9t ,c divides ! /rait l9e ,b ,v1 :i* is equal to ! ar* ( ! quadrant4 ,b h[ev] ?is be1 x has be5 demon/rat$ t ! /rait l9e ,b ,v is equal to ! ar* ,b ,h ,d4 ,sid5ote3 ,! ?ird attempt2 & "s ?+s prop\nd$ to be fur!r se>*$ into4 #d4 ,i %all n[ attempt ! same dim5sion ( a circle ano!r way1 assum+ ! two foll[+ lemmas4 ,lemma ,i4 ,if to ! ar* ( a quadrant1 & ! radius1 !re be tak5 9 cont9ual proportion a ?ird l9e ,z2 !n ! ar* ( ! semiquadrant1 half ! *ord ( ! quadrant1 & ,z1 w also be 9 cont9ual proportion4 ,= see+ ! radius is a mean proportional betwe5 ! *ord ( a quadrant & its semi*ord1 & ! same radius a mean proportional betwe5 ! ar* ( ! quadrant & ,z1 ! squ>e ( ! radius w be equal z well to ! rectangle made ( ! *ord & semi*ord ( ! quadrant1 z to ! rectangle made ( ! ar* ( ! quadrant & ,z2 & ^! two rectangles w be equal to "o ano!r4 ,:]e=e1 z ! ar* ( a quadrant is to its *ord1 s reciprocally is half ! *ord ( ! quadrant to ,z4 ,b z ! ar* ( ! quadrant is to its *ord1 s is half ! ar* ( ! quadrant to half ! *ord ( ! quadrant4 ,:]e=e1 z half ! ar* ( ! quadrant is to half ! *ord ( ! quadrant 7or to ! s9e ( #de degrees71 s is half ! *ord ( ! quadrant to ,z2 :i* 0 to be prov$4 ,lemma ,,ii4 ,! radius1 ! ar* ( ! semiquadrant1 ! s9e ( #de degrees1 & ! semiradius1 are proportional4 ,= see+ ! s9e ( #de degrees is a mean proportional betwe5 ! radius & ! semiradius2 & ! same s9e ( #de degrees is also a mean proportional 7by ! prec$5t lemma7 betwe5 ! ar* ( #de degrees & ,z2 ! squ>e ( ! s9e ( #de degrees w be equal z well to ! rectangle made ( ! radius & semiradius1 z to ! rectangle made ( ! ar* ( #de degrees & ,z4 ,:]e=e1 z ! radius is to ! ar* ( #de degrees1 s reciprocally is ,z to ! semiradius2 :i* 0 to be demon/rat$4 ,let n[ ,a ,b ,c ,d 79 fig4 #d7 be a squ>e2 & ) ! radii ,a ,b1 ,b ,c & ,d ,a1 let ! ?ree quadrants ,a ,b ,d1 ,b ,c ,a & ,d ,a ,c1 be describ$2 & let ! /rait l9es ,e ,f & ,g ,h1 drawn p>allel to ! sides ,b ,c & ,a ,b1 divide ! squ>e ,a ,b ,c ,d into f\r equal squ>es4 ,!y w !re=e cut ! ar* ( ! quadrant ,a ,b ,d into ?ree equal p>ts 9 ,i & ,k1 & ! ar* ( ! quadrant ,b ,c ,a into ?ree equal p>ts 9 ,k & ,l4 ,also let ! diagonals ,a ,c & ,b ,d be drawn1 cutt+ ! ar*es ,b ,i ,d & ,a ,l ,c 9 ,m & ,n4 ,!n ^u ! c5tre ,h ) ! radius ,h ,f equal to half ! *ord ( ! ar* ,b ,m ,d1 or to ! s9e ( #de degrees1 let ! ar* ,f ,o be drawn cutt+ ! ar* ,c ,k 9 ,o2 & let ,a ,o be drawn & produc$ till x meet ) ,b ,c produc$ 9 ,p2 also let x cut ! ar* ,b ,m ,d 9 ,q1 & ! /rait l9e ,d ,c 9 ,r4 ,if n[ ! /rait l9e ,h ,q be equal to ! /rait l9e ,d ,r1 & be+ produc$ to ,d ,c 9 ,s1 cut (f ,d ,s equal to half ! /rait l9e ,b ,p2 ,i say !n ! /rait l9e ,b ,p w be equal to ! ar* ,b ,m ,d4 ,= see+ ,p ,b ,a & ,a ,d ,r are l triangles1 x w be z ,p ,b to ! radius ,b ,a or ,a ,d1 s ,a ,d to ,d ,r2 & !re=e z well ,p ,b1 ,a ,d & ,d ,r1 z ,p ,b1 ,a ,d 7or ,a ,q7 & ,q ,h are 9 cont9ual proportion2 & produc+ ,h ,o to ,d ,c 9 ,t1 ,d ,t w be equal to ! s9e ( #de degrees1 z %all by & by be demon/rat$4 ,n[ ,d ,s1 ,d ,t & ,d ,r are 9 cont9ual proportion by ! fir/ lemma2 & by ! second lemma ,d ,c4 ,d ,s33 ,d ,r4 ,d ,f are proportionals4 ,& ?us x w be1 :e!r ,b ,p be equal or n equal to ! ar* ( ! quadrant ,b ,m ,d4 ,b if !y be equal1 x w !n be1 z t "p ( ! ar* ,b ,m ,d :i* is equal to ! radius1 is to ! rema9d] ( ! same ar* ,b ,m ,d2 s ,a ,q to ,h ,q1 or s ,b ,c to ,c ,p4 ,& !n w ,b ,p & ! ar* ,b ,m ,d be equal4 ,b x is n demon/rat$ t ! /rait l9es ,h ,q & ,d ,r are equal2 ?\gh if f ! po9t ,b !re be drawn 7by ! con/ruction ( fig4 #a7 a /rait l9e equal to ! ar* ,b ,m ,d1 !n ,d ,r to ,h ,q1 & also ! half ( ! /rait l9e ,b ,p to ,d ,s1 w always be s equal1 t no inequality c be discov]$ betwe5 !m4 ,i w !re=e leave ?is to be fur!r se>*$ into4 ,= ?\gh x be almo/ out ( d\bt1 t ! /rait l9e ,b ,p & ! ar* ,b ,m ,d are equal1 yet t may n be receiv$ )\t demon/ration2 & means ( demon/ration ! circul> l9e admitte? none t is n gr\nd$ ^u ! nature ( flexion1 or ( angles4 ,b by t way ,i h already exhibit$ a /rait l9e equal to ! ar* ( a quadrant 9 ! fir/ & second aggression4 ,x rema9s t ,i prove ,d ,t to be equal to ! s9e ( #de degrees4 ,9 ,b ,a produc$ let ,a ,v be tak5 equal to ! s9e ( #de degrees2 & draw+ & produc+ ,v ,h1 x w cut ! ar* ( ! quadrant ,c ,n ,a 9 ! mid/ 9 ,n1 & ! same ar* aga9 9 ,o1 & ! /rait l9e ,d ,c 9 ,t1 s t ,d ,t w be equal to ! s9e ( #de degrees1 or to ! /rait l9e ,a ,v2 also ! /rait l9e ,v ,h w be equal to ! /rait l9e ,h ,i1 or ! s9e ( #fj degrees4 ,= ! squ>e ( ,a ,v is equal to two squ>es ( ! semiradius2 & consequ5tly ! squ>e ( ,v ,h is equal to ?ree squ>es ( ! semiradius4 ,b ,h ,i is a mean proportional betwe5 ! semiradius & ?ree semiradii2 &1 !re=e1 ! squ>e ( ,h ,i is equal to ?ree squ>es ( ! semiradius4 ,:]e=e ,h ,i is equal to ,h ,v4 ,b because ,a ,d is cut 9 ! mid/ 9 ,h1 !re=e ,v ,h & ,h ,t are equal2 &1 !re=e1 also ,d ,t is equal to ! s9e ( #de degrees4 ,9 ! radius ,b ,a let ,b ,x be tak5 equal to ! s9e ( #de degrees2 = s ,v ,x w be equal to ! radius2 & x w be z ,v ,a to ,a ,h ! semiradius1 s ,v ,x ! radius to ,x ,n ! s9e ( #de degrees4 ,:]e=e ,v ,h produc$ passes ?r\gh ,n4 ,la/ly1 ^u ! c5tre ,v ) ! radius ,v ,a let ! ar* ( a circle be drawn cutt+ ,v ,h 9 ,y2 :i* be+ done1 ,v ,y w be equal to ,h ,o 7= ,h ,o is1 by con/ruction1 equal to ! s9e ( #de degrees7 & ,y ,h w be equal to ,o ,t2 &1 !re=e1 ,v ,t passes ?r\gh ,o4 ,all :i* 0 to be demon/rat$4 ,i w "h add c]ta9 problems1 ( :i* if any analy/ c make ! con/ruction1 he w !reby be able to judge cle>ly ( :at ,i h n[ said conc]n+ ! dim5sion ( a circle4 ,n[ ^! problems are no?+ else 7at lea/ to s5se7 b c]ta9 symptoms accompany+ ! con/ruction ( ! fir/ & ?ird figure ( ?is *apt]4 ,describ+1 !re=e1 aga91 ! squ>e ,a ,b ,c ,d 79 fig4 #e7 & ! ?ree quadrants ,a ,b ,d1 ,b ,c ,a & ,d ,a ,c1 let ! diagonals ,a ,c & ,b ,d be drawn1 cutt+ ! ar*es ,b ,h ,d & ,c ,i ,a 9 ! middle 9 ,h & ,i2 & ! /rait l9es ,e ,f & ,g ,l1 divid+ ! squ>e ,a ,b ,c ,d into f\r equal squ>es1 & trisect+ ! ar*es ,b ,h ,d & ,c ,i ,a1 namely1 ,b ,h ,d 9 ,k & ,m1 & ,c ,i ,a 9 ,m & ,o4 ,!n divid+ ! ar* ,b ,k 9 ! mid/ 9 ,p1 let ,q ,p ! s9e ( ! ar* ,b ,p1 be drawn & produc$ to ,r1 s t ,q ,r be d\ble to ,q ,p2 &1 connect+ ,k ,r1 let x be produc$ "o way to ,b ,c 9 ,s1 & ! o!r way to ,b ,a produc$ 9 ,t4 ,also let ,b ,v be made triple to ,b ,s1 & consequ5tly1 7by ! second article ( ?is *apt]7 equal to ! ar* ,b ,d4 ,?is con/ruction is ! same ) t ( ! fir/ figure1 :i* ,i ?\ght fit to r5ew dis*>g$ ( all l9es b su* z are necess>y = my pres5t purpose4 ,9 ! fir/ place1 !re=e1 if ,a ,v be drawn1 cutt+ ! ar* ,b ,h ,d 9 ,x1 & ! side ,d ,c 9 ,z1 ,i desire "s analy/ w\ld1 if he c1 give a reason :y ! /rait l9es ,t ,e & ,t ,c %\ld cut ! ar* ,b ,d1 ! "o 9 ,y1 ! o!r 9 ,x1 s z to make ! ar* ,b ,y equal to ! ar* ,y ,x2 or if !y be n equal1 t he w\ld det]m9e _! diff]5ce4 ,secondly1 if 9 ! side ,d ,a1 ! /rait l9e ,d a be tak5 equal to ,d ,z1 & ,v a be drawn2 :y ,v a & ,v ,b %\ld be equal2 or if !y be n equal1 :at is ! diff]5ce4 ,?irdly1 draw+ ,z b p>allel & equal to ! side ,c ,b1 cutt+ ! ar* ,b ,h ,d 9 c1 & draw+ ! /rait l9e ,a c1 & produc+ x to ,b ,v 9 d2 :y ,a d %\ld be equal & p>allel to ! /rait l9e a ,v1 & consequ5tly equal also to ! ar* ,b ,d4 ,f\r?ly1 draw+ e ,k ! s9e ( ! ar* ,b ,k1 & tak+ 79 e ,a produc$7 e f equal to ! diagonal ,a ,c1 & connect+ f ,c2 :y f ,c %\ld pass ?r\gh a 7:i* po9t be+ giv51 ! l5g? ( ! ar* ,b ,h ,d is also giv57 & c2 & :y f e & f c %\ld be equal2 or if n1 :y unequal4 ,fif?ly1 draw+ f ,z1 ,i desire he w\ld %[1 :y x is equal to ,b ,v1 or to ! ar* ,b ,d2 or if !y be n equal1 :at is _! diff]5ce4 ,six?ly1 grant+ f ,z to be equal to ! ar* ,b ,d1 ,i desire he w\ld det]m9e :e!r x fall all )\t ! ar* ,b ,c ,a1 or cut ! same1 or t\* x1 & 9 :at po9t4 ,sev5?ly1 ! semicircle ,b ,d g be+ complet$1 :y g ,i be+ drawn & produc$1 %\ld pass ?r\gh ,x1 by :i* po9t ,x ! l5g? ( ! ar* ,b ,d is det]m9$4 ,& ! same g ,i be+ yet fur!r produc$ to ,d ,c 9 h1 :y ,a d1 :i* is equal to ! ar* ,b ,d1 %\ld pass ?r\gh t po9t h4 ,eigh?ly1 ^u ! c5tre ( ! squ>e ,a ,b ,c ,d1 :i* let be k1 ! ar* ( ! quadrant ,e i ,l be+ drawn1 cutt+ e ,k produc$ 9 i1 :y ! drawn /rait l9e i ,x %\ld be p>allel to ! side ,c ,d4 ,n9?ly1 9 ! sides ,b ,a & ,b ,c tak+ g l & ,b m sev]ally equal to half ,b ,v1 or to ! ar* ,b ,h1 & draw+ m n p>allel & equal to ! side ,b ,a1 cutt+ ! ar* ,b ,d 9 o1 :y ! /rait l9e :i* connects ,v l %\ld pass ?r\gh ! po9t o4 ,t5?ly1 ,i w\ld "k ( him :y ! /rait l9e :i* connects a ,h %\ld be equal to ,b m2 or if n1 h[ mu* x diff]s f x4 ,! analy/ t c solve ^! problems )\t kn[+ fir/ ! l5g? ( ! ar* ,b ,d1 or us+ any o!r kn[n me?od ?an t :i* proce$s by p]petual bisection ( an angle1 or is drawn f ! consid]ation ( ! nature ( flexion1 %all d m ?an ord9>y geometry is able to p]=m4 ,b if ! dim5sion ( a circle _c be f\nd by any o!r me?od1 !n ,i h ei!r f\nd x1 or x is n at all to be f\nd4 ,f ! kn[n l5g? ( ! ar* ( a quadrant1 & f ! proportional division ( ! ar* & ( ! tang5t ,b ,c1 may be d$uc$ ! section ( an angle into any giv5 proportion2 z also ! squ>+ ( ! circle1 ! squ>+ ( a giv5 sector1 & _m ! l propositions1 :i* x is n necess>y "h to demon/rate4 ,i w1 !re=e1 only exhibit a /rait l9e equal to ! spiral ( ,ar*im$es1 & s dismiss ?is speculation4 ,sid5ote3 ,! equation ( ! spiral ( ,ar*im$es ) a /rait l9e4 #e4 ,! l5g? ( ! p]imet] ( a circle be+ f\nd1 t /rait l9e is also f\nd1 :i* t\*es a spiral at ! end ( its fir/ conv]sion4 ,= ^u ! c5tre ,a 79 fig4 #f7 let ! circle ,b ,c ,d ,e be describ$2 & 9 x let ,ar*im$es' spiral ,a ,f ,g ,h ,b be drawn1 beg9n+ at ,a & end+ at ,b4 ,?r\gh ! c5tre ,a let ! /rait l9e ,c ,e be drawn1 cutt+ ! diamet] ,b ,d at "r angles2 & let x be produc$ to ,i1 s t ,a ,i be equal to ! p]imet] ,b ,c ,d ,e ,b4 ,!re=e ,i ,b be+ drawn w t\* ! spiral ,a ,f ,g ,h ,b 9 ,b2 :i* is demon/rat$ by ,ar*im$es 9 8 book ,de ,spiralibus4 ,& = a /rait l9e equal to ! giv5 spiral ,a ,f ,g ,h ,b1 x may be f\nd ?us4 ,let ! /rait l9e ,a ,i1 :i* is equal to ! p]imet] ,b ,c ,d ,e1 be bisect$ 9 ,k2 & tak+ ,k ,l equal to ! radius ,a ,b1 let ! rectangle ,i ,l be complet$4 ,let ,m ,l be und]/ood to be ! axis1 & ,k ,l ! base ( a p>abola1 & let ,m ,k be ! crook$ l9e !re(4 ,n[ if ! po9t ,m be conceiv$ to be s mov$ by ! conc\rse ( two mov5ts1 ! "o f ,i ,m to ,k ,l ) velocity encreas+ cont9ually 9 ! same proportion ) ! times1 ! o!r f ,m ,l to ,i ,k uni=mly1 t bo? ^? motions beg9 toge!r 9 ,m & end 9 ,k2 ,galilu has demon/rat$ t by su* motion ( ! po9t ,m1 ! crook$ l9e ( a p>abola w be describ$4 ,aga91 if ! po9t ,a be conceiv$ to be mov$ uni=mly 9 ! /rait l9e ,a ,b1 & 9 ! same "t to be c>ri$ r\nd ^u ! c5tre ,a by ! circul> motion ( all ! po9ts betwe5 ,a & ,b2 ,ar*im$es has demon/rat$ t by su* motion w be describ$ a spiral l9e4 ,& see+ ! circles ( all ^! motions are conc5tric 9 ,a2 & ! int]ior circle is always less ?an ! ext]ior 9 ! proportion ( ! times 9 :i* ,a ,b is pass$ ov] ) uni=m motion2 ! velocity also ( ! circul> motion ( ! po9t ,a w cont9ually increase proportionally to ! times4 ,& ?us f> ! g5]ations ( ! p>abolical l9e ,m ,k1 & ( ! spiral l9e ,a ,f ,g ,h ,b1 are l4 ,b ! uni=m motion 9 ,a ,b concurr+ ) circul> motion 9 ! p]imet]s ( all ! conc5tric circles1 describes t circle1 ^: c5tre is ,a1 & p]imet] ,b ,c ,d ,e2 &1 !re=e1 t circle is 7by ! coroll4 ( art4 #a1 *ap1 ,,xvi7 ! aggregate ( all ! velocities toge!r tak5 ( ! po9t ,a :il/ x describes ! spiral ,a ,f ,g ,h ,b4 ,also ! rectangle ,i ,k ,l ,m is ! aggregate ( all ! velocities toge!r tak5 ( ! po9t ,m1 :il/ x describes ! crook$ l9e ,m ,k4 ,&1 !re=e ! :ole velocity by :i* ! p>abolical l9e ,m ,k is describ$1 is to ! :ole velocity ) :i* ! spiral l9e ,a ,f ,g ,h ,b is describ$ 9 ! same "t1 z ! rectangle ,i ,k ,l ,m is to ! circle ,b ,c ,d ,e1 t is to ! triangle ,a ,i ,b4 ,b because ,a ,i is bisect$ 9 ,k1 & ! /rait l9es ,i ,m & ,a ,b are equal1 !re=e ! rectangle ,i ,k ,l ,m & ! triangle ,a ,i ,b are also equal4 ,:]e=e ! spiral l9e ,a ,f ,g ,h ,b1 & ! p>abolical l9e ,m ,k1 be+ describ$ ) equal velocity & 9 equal times1 are equal to "o ano!r4 ,n[1 9 ! fir/ article ( *ap4 ,,xviii1 a /rait l9e is f\nd out equal to any p>abolical l9e4 ,:]e=e also a /rait l9e is f\nd out equal to a giv5 spiral l9e ( ! fir/ revolution describ$ by ,ar*im$es2 :i* 0 to be done4 ,sid5ote3 ,( ! analysis ( geometricians by ! p[]s ( l9es4 #f4 ,9 ! six? *apt]1 :i* is ( ,me?od1 t :i* ,i %\ld !re h spok5 ( ! analytics ( geometricians ,i ?\ght fit to def]1 because ,i c\ld n !re h be5 und]/ood1 z n hav+ !n s mu* z nam$ l9es1 sup]ficies1 solids1 equal & unequal1 c4 ,:]e=e ,i w 9 ?is place set d[n my ?\ghts conc]n+ x4 ,analysis is cont9ual reason+ f ! def9itions ( ! t]ms ( a proposition we suppose true1 & aga9 f ! def9itions ( ! t]ms ( ^? def9itions1 & s on1 till we come to "s ?+s kn[n1 ! composition :]e( is ! demon/ration ( ! tru? or falsity ( ! fir/ supposition2 & ?is composition or demon/ration is t we call ,syn!sis4 ,analytica1 !re=e1 is t art1 by :i* our reason proce$s f some?+ suppos$1 to pr9ciples1 t is1 to prime propositions1 or to su* z are kn[n by ^!1 till we h s _m kn[n propositions z are suffici5t = ! demon/ration ( ! tru? or falsity ( ! ?+ suppos$4 ,syn!tica is ! art itself ( demon/ration4 ,syn!sis1 !re=e1 & analysis1 diff] 9 no?+1 b 9 proce$+ =w>ds or backw>ds2 & ,logi/ica compreh5ds bo?4 ,s t 9 ! analysis or syn!sis ( any "q1 t is to say1 ( any problem1 ! t]ms ( all ! propositions ought to be conv]tible2 or if !y be enunciat$ hypo!tically1 ! tru? ( ! consequ5t ought n only to foll[ out ( ! tru? ( its antec$5t1 b contr>ily also ! tru? ( ! antec$5t mu/ necess>ily be inf]r$ f ! tru? ( ! consequ5t4 ,= o!rwise1 :5 by resolution we are arriv$ at pr9ciples1 we _c by composition return directly back to ! ?+ s\ght =4 ,= ^? t]ms :i* are ! fir/ 9 analysis1 w be ! la/ 9 syn!sis2 z = example1 :5 9 resolv+1 we say1 ^! two rectangles are equal1 & !re=e _! sides are reciprocally proportional1 we mu/ necess>ily 9 comp\nd+ say1 ! sides ( ^! rectangles are reciprocally proportional1 & !re=e ! rectangles !mselves are equal2 :i* we c\ld n say1 unless rectangles h _! sides reciprocally proportional1 & rectangles are equal1 7 t]ms conv]tible4 ,n[ 9 e analysis1 t :i* is s\ght is ! proportion ( two quantities2 by :i* proportion1 a figure be+ describ$1 ! quantity s\ght = may be expos$ to s5se4 ,& ?is exposition is ! end & solution ( ! "q1 or ! con/ruction ( ! problem4 ,& see+ analysis is reason+ f some?+ suppos$1 till we come to pr9ciples1 t is1 to def9itions1 or to !orems =m]ly kn[n2 & see+ ! same reason+ t5ds 9 ! la/ place to "s equation1 we c !re=e make no end ( resolv+1 till we come at la/ to ! causes !mselves ( equality & inequality1 or to !orems =m]ly demon/rat$ f ^? causes2 & s h a suffici5t numb] ( ^? !orems = ! demon/ration ( ! ?+ s\ght =4 ,& see+ also1 t ! end ( ! analytics is ei!r ! con/ruction ( su* a problem z is possible1 or ! detection ( ! impossibility !re(2 :5soev] ! problem may be solv$1 ! analy/ mu/ n /ay1 till he come to ^? ?+s :i* conta9 ! effici5t cause ( t :]e( he is to make con/ruction4 ,b he mu/ ( necessity /ay1 :5 he comes to prime propositions2 & ^! are def9itions4 ,^! def9itions !re=e mu/ conta9 ! effici5t cause ( 8 con/ruction2 ,i say ( 8 con/ruction1 n ( ! conclusion :i* he demon/rates2 = ! cause ( ! conclusion is conta9$ 9 ! premis$ propositions2 t is to say1 ! tru? ( ! proposition he proves is drawn f ! propositions :i* prove ! same4 ,b ! cause ( 8 con/ruction is 9 ! ?+s !mselves1 & consi/s 9 motion1 or 9 ! conc\rse ( motions4 ,:]e=e ^? propositions1 9 :i* analysis ends1 are def9itions1 b su* z signify 9 :at mann] ! con/ruction or g5]ation ( ! ?+ proce$s4 ,= o!rwise1 :5 he goes back by syn!sis to ! pro( ( 8 problem1 he w come to no demon/ration at all2 !re be+ no true demon/ration b su* z is sci5tifical2 & no demon/ration is sci5tifical1 b t :i* proce$s f ! k ( ! causes f :i* ! con/ruction ( ! problem is drawn4 ,to collect !re=e :at has be5 said into few words2 ,,analysis is ratioc9ation f ! suppos$ con/ruction or g5]ation ( a ?+ to ! effici5t cause or coeffici5t causes ( t :i* is con/ruct$ or g5]at$4 ,& ,,syn!sis is ratioc9ation f ! fir/ causes ( ! con/ruction1 cont9u$ ?r\gh all ! middle causes till we come to ! ?+ itself :i* is con/ruct$ or g5]at$4 ,b because !re are _m means by :i* ! same ?+ may be g5]at$1 or ! same problem be con/ruct$1 !re=e nei!r d all geometricians1 nor do? ! same geometrician always1 use "o & ! same me?od4 ,=1 if to a c]ta9 quantity giv51 x be requir$ to con/ruct ano!r quantity equal1 !re may be "s t w inquire :e!r ?is may n be done by means ( "s motion4 ,= !re are quantities1 ^: equality & inequality may be argu$ f motion & "t1 z well z f congru5ce2 & !re is motion1 by :i* two quantities1 :e!r l9es or sup]ficies1 ?\gh "o ( !m be crook$1 ! o!r /rait1 may be made congru\s or co9cid5t4 ,& ?is me?od ,ar*im$es made use ( 9 8 book ,de ,spiralibus4 ,also ! equality or inequality ( two quantities may be f\nd out & demon/rat$ f ! consid]ation ( weight1 z ! same ,ar*im$es did 9 8 quadrature ( ! p>abola4 ,besides1 equality & inequality are f\nd out (t5 by ! division ( ! two quantities into p>ts :i* are consid]$ z indivisable2 z ,cavall]ius ,bonav5tura has done 9 our "t1 & ,ar*im$es (t54 ,la/ly1 ! same is p]=m$ by ! consid]ation ( ! p[]s ( l9es1 or ! roots ( ^? p[]s1 & by ! multiplication1 division1 addition1 & subtraction1 z also by ! extraction ( ! roots ( ^? p[]s1 or by f9d+ :]e /rait l9es ( ! same proportion t]m9ate4 ,= example1 :5 any numb] ( /rait l9es1 h[ _m soev]1 are drawn f a /rait l9e & pass all ?r\gh ! same po9t1 look :at proportion !y h1 & if _! p>ts cont9u$ f ! po9t reta9 ev]y:]e ! same proportion1 !y %all all t]m9ate 9 a /rait l9e4 ,& ! same happ5s if ! po9t be tak5 betwe5 two circles4 ,s t ! places ( all _! po9ts ( t]m9ation make ei!r /rait l9es1 or circumf]5ces ( circles1 & are call$ plane places4 ,s also :5 /rait p>allel l9es are appli$ to "o /rait l9e1 if ! p>ts ( ! /rait l9e to :i* !y are appli$ be to "o ano!r 9 proportion duplicate to t ( ! contigu\s appli$ l9es1 !y w all t]m9ate 9 a conical section2 :i* section1 be+ ! place ( _! t]m9ation1 is call$ a solid place1 because x s]ves = ! f9d+ out ( ! quantity ( any equation :i* consi/s ( ?ree dim5sions4 ,!re are !re=e ?ree ways ( f9d+ out ! cause ( equality or inequality betwe5 two giv5 quantities2 namely1 fir/1 by ! computation ( motions2 = by equal motion1 & equal "t1 equal spaces are describ$2 & pond]ation is motion4 ,secondly1 by indivisibles3 because all ! p>ts toge!r tak5 are equal to ! :ole4 ,& ?irdly1 by ! p[]s3 = :5 !y are equal1 _! roots also are equal2 & contr>ily1 ! p[]s are equal1 :5 _! roots are equal4 ,b if ! "q be mu* complicat$1 !re _c by any ( ^! ways be con/itut$ a c]ta9 rule1 f ! supposition ( :i* ( ! unkn[n quantities ! analysis may be/ beg92 nor out ( ! v>iety ( equations1 t at fir/ appe>1 :i* we 7 be/ to *oose2 b ! success w dep5d ^u dext]ity1 ^u =m]ly acquir$ sci5ce1 & _m times ^u =tune4 ,= no man c ev] be a good analy/ )\t be+ fir/ a good geometrician2 nor d ! rules ( analysis make a geometrician1 z syn!sis do?2 :i* beg9s at ! v elem5ts1 & proce$s by a logical use ( ! same4 ,= ! true tea*+ ( geometry is by syn!sis1 accord+ to ,euclid's me?od2 & he t ha? ,euclid = 8 ma/]1 may be a geometrician )\t ,vieta1 ?\gh ,vieta 0 a mo/ admirable geometrician2 b he t has ,vieta = 8 ma/]1 n s1 )\t ,euclid4 ,& z = t "p ( analysis :i* works by ! p[]s1 ?\gh x be e/eem$ by "s geometricians1 n ! *iefe/1 to be ! be/ way ( solv+ all problems1 yet x is a ?+ ( no great ext5t2 x be+ all conta9$ 9 ! doctr9e ( rectangles1 & rectangl$ solids4 ,s t al?\gh !y come to an equation :i* det]m9es ! quantity s\ght1 yet !y _c sometimes by art exhibit t quantity 9 a plane1 b 9 "s conic section2 t is1 z geometricians say1 n geometrically1 b me*anically4 ,n[ su* problems z ^!1 !y call solid2 & :5 !y _c exhibit ! quantity s\ght = ) ! help ( a conic section1 !y call x a l9e>y problem4 ,& !re=e 9 ! quantities ( angles1 & ( ! ar*es ( circles1 !re is no use at all ( ! analytics :i* proce$ by ! p[]s2 s t ! anci5ts pron\nc$ x impossible to exhibit 9 a plane ! division ( angles1 except bisection1 & ! bisection ( ! bisect$ p>ts1 o!rwise ?an me*anically4 ,= ,pappus1 7be=e ! #ca/ proposition ( 8 f\r? book7 di/+ui%+ & def9+ ! sev]al k9ds ( problems1 says t "s are plane1 o!rs solid1 & o!rs l9e>y4 ,^?1 !re=e1 :i* may be solv$ by /rait l9es & ! circumf]5ces ( circles1 7t is1 :i* may be describ$ ) ! rule & compass1 )\t any o!r in/rum5t71 are fitly call$ plane2 = ! l9es1 by :i* su* problems are f\nd out1 h _! g5]ation 9 a plane4 ,b ^? :i* are solv$ by ! us+ ( "s "o or m conic sections 9 _! con/ruction1 are call$ solid1 because _! con/ruction _c be made )\t us+ ! sup]ficies ( solid figures1 namely1 ( cones4 ,!re rema9s ! ?ird k9d1 :i* is call$ l9e>y1 because o!r l9es besides ^? already m5tion$ are made use ( 9 _! con/ruction1 c4 ,& a little aft] he says1 ( ?is k9d are ! spiral l9es1 ! quadratrices1 ! con*oeides1 & ! cissoeides1 ,& geometricians ?9k x no small fault1 :5 = ! f9d+ out ( a plane problem any man makes use ( conics1 or new l9es4 ,n[ he ranks ! trisection ( an angle among solid problems1 & ! qu9quesection among l9e>y4 ,b :at6 are ! anci5t geometricians to be blam$1 :o made use ( ! quadratrix = ! f9d+ out ( a /rait l9e equal to ! ar* ( a circle8 ,& ,pappus himself1 0 he faulty1 :5 he f\nd out ! trisection ( an angle by ! help ( an hyp]bole8 ,or am ,i 9 ! wrong1 :o ?9k ,i h f\nd out ! con/ruction ( bo? ^! problems by ! rule & compass only8 ,nei!r !y1 nor ,i4 ,= ! anci5ts made use ( ?is analysis :i* proce$s by ! p[]s2 & ) !m x 0 a fault to d t by a m remote p[]1 :i* might be done by a ne>]2 z be+ an argum5t t !y did n suffici5tly und]/& ! nature ( ! ?+4 ,! virtue ( ?is k9d ( analysis consi/s 9 ! *ang+ & turn+ & toss+ ( rectangles & analogisms2 & ! skill ( analy/s is m]e logic1 by :i* !y are able me?odically to f9d out :atsoev] lies hid ei!r 9 ! subject or pr$icate ( ! conclusion s\ght =4 ,b ?is do? n prop]ly belong to algebra1 or ! analytics speci\s1 symbolical1 or cossick2 :i* are1 z ,i may say1 ! bra*ygraphy ( ! analytics1 & an art nei!r ( tea*+ nor le>n+ geometry1 b ( regi/]+ ) brevity & cel]ity ! inv5tions ( geometricians4 ,= ?\gh x be easy to disc\rse by symbols ( v remote propositions2 yet :e!r su* disc\rse des]ve to be ?\ght v pr(itable1 :5 x is made )\t any ideas ( ! ?+s !mselves1 ,i "k n4 ---------------------------------------- -------------------------------- ,illu/ration3 ,c4 ,,xx4 ( ! translation4 ,fig4 #a-b ,illu/ration3 ,c4 ,,xx4 ( ! translation4 ,fig4 #c ,illu/ration3 ,c4 ,,xx4 ( ! translation4 ,fig4 #d ,illu/ration3 ,c4 ,,xx4 ( ! translation4 ,fig4 #e-f ---------------------------------------- -------------------------------- ------- ,,*apt] ,,xxi4 ,,( ,,circul> ,,motion4 #a4 ,9 simple motion1 e /rait l9e tak5 9 ! body mov$ is s c>ri$1 t x is always p>allel to ! places 9 :i* x =m]ly 04--#b4 ,if circul> motion be made ab\t a re/+ c5tre1 & 9 t circle !re be an epicycle1 ^: revolution is made ! contr>y way1 9 su* mann] t 9 equal times x make equal angles1 e /rait l9e tak5 9 t epicycle w be s c>ri$1 t x w always be p>allel to ! places 9 :i* x =m]ly 04--#c4 ,! prop]ties ( simple motion4--#d4 ,if a fluid be mov$ ) simple circul> motion1 all ! po9ts tak5 9 x w describe _! circles 9 times proportional to ! di/ances f ! c5tre4--#e4 ,simple motion dissipates het]og5e\s & congregates homog5e\s bodies4--#f4 ,if a circle made by a mov5t mov$ ) simple motion be comm5surable to ano!r circle made by a po9t :i* is c>ri$ ab\t by ! same mov5t1 all ! po9ts ( bo? ! circles w at "s "t return to ! same situation4--#g44 ,if a sph]e h simple motion1 its motion w m dissipate het]og5e\s bodies by h[ mu* x is m remote f ! poles4--#h4 ,if ! simple circul> motion ( a fluid body be h9d]$ by a body :i* is n fluid1 ! fluid body w spread itself ^u ! sup]ficies ( t body4--#i4 ,circul> motion ab\t a fix$ c5tre ca/e? (f by ! tang5t su* ?+s z lie ^u ! circumf]5ce & /ick n to x4--#aj4 ,su* ?+s1 z are mov$ ) simple circul> motion1 beget simple circul> motion4--#aa4 ,if t :i* is s mov$ h "o side h>d & ! o!r side fluid1 its motion w n be p]fectly circul>4 ,sid5ote3 ,9 simple motion1 e /rait l9e tak5 9 ! body mov$ is s c>ri$1 t x is always p>allel to ! places 9 :i* x =m]ly 04 #a4 ,i h already def9$ simple motion to be t1 9 :i* ! sev]al po9ts tak5 9 a mov$ body d 9 sev]al equal times describe sev]al equal ar*es4 ,& !re=e 9 simple circul> motion x is necess>y t e /rait l9e tak5 9 ! mov$ body be always c>ri$ p>allel to itself2 :i* ,i ?us demon/rate4 ,fir/1 let ,a ,b 79 ! fir/ figure7 be any /rait l9e tak5 9 any solid body2 & let ,a ,d be any ar* drawn ^u any c5tre ,c & radius ,,ca4 ,let ! po9t ,b be und]/ood to describe t[>ds ! same p>ts ! ar* ,b ,e1 l & equal to ! ar* ,a ,d4 ,n[ 9 ! same "t 9 :i* ! po9t ,a transmits ! ar* ,a ,d1 ! po9t ,b1 :i* by reason ( its simple motion is suppos$ to be c>ri$ ) a velocity equal to t ( ,a1 w transmit ! ar* ,b ,e2 & at ! end ( ! same "t ! :ole ,a ,b w be 9 ,d ,e2 & !re=e ,a ,b & ,d ,e are equal4 ,& see+ ! ar*es ,a ,d & ,b ,e are l & equal1 _! subt5d+ /rait l9es ,,ad & ,,be w also be equal2 & !re=e ! f\r-sid$ figure ,a ,b ,d ,e w be a p>allelogram4 ,:]e=e ,a ,b is c>ri$ p>allel to itself4 ,& ! same may be prov$ by ! same me?od1 if any o!r /rait l9e be tak5 9 ! same mov$ body 9 :i* ! /rait l9e ,a ,b 0 tak54 ,s t all /rait l9es1 tak5 9 a body mov$ ) simple circul> motion1 w be c>ri$ p>allel to !mselves4 ,coroll4 ,i4 ,x is manife/ t ! same w also happ5 9 any body :i* ha? simple motion1 ?\gh n circul>4 ,= all ! po9ts ( any /rait l9e :atsoev] w describe l9es1 ?\gh n circul>1 yet equal2 s t ?\gh ! crook$ l9es ,a ,d & ,b ,e 7 n ar*es ( circles1 b ( p>abolas1 ellipses1 or ( any o!r figures1 yet bo? !y1 & _! subt5ses1 & ! /rait l9es :i* jo9 !m1 w\ld be equal & p>allel4 ,coroll4 ,,ii4 ,x is also manife/1 t ! radii ( ! equal circles ,a ,d & ,b ,e1 or ! axis ( a sph]e1 w be s c>ri$1 z to be always p>allel to ! places 9 :i* !y =m]ly 74 ,= ! /rait l9e ,b ,f drawn to ! c5tre ( ! ar* ,b ,e be+ equal to ! radius ,a ,c1 w also be equal to ! /rait l9e ,f ,e or ,c ,d2 & ! angle ,b ,f ,e w be equal to ! angle ,a ,c ,d4 ,n[ ! int]section ( ! /rait l9es ,c ,a & ,b ,e be+ at ,g1 ! angle ,c ,g ,e 7see+ ,b ,e & ,a ,d are p>allel7 w be equal to ! angle ,d ,a ,c4 ,b ! angle ,e ,b ,f is equal to ! same angle ,d ,a ,c2 & !re=e ! angles ,c ,g ,e & ,e ,b ,f are also equal4 ,:]e=e ,a ,c & ,b ,f are p>allel2 :i* 0 to be demon/rat$4 ,sid5ote3 ,if circul> motion be made ab\t a re/+ c5tre1 & 9 t circle !re be an epicycle ^: revolution is made ! contr>y way1 9 su* mann] t 9 equal times x make equal angles1 e /rait l9e tak5 9 t epicycle w be s c>ri$1 t x w always be p>allel to ! places 9 :i* x =m]ly 04 #b4 ,let !re be a circle giv5 79 ! second figure7 ^: c5tre is ,a1 & radius ,a ,b2 & ^u ! c5tre ,b & any radius ,b ,c let ! epicycle ,c ,d ,e be describ$4 ,let ! c5tre ,b be und]/ood to be c>ri$ ab\t ! c5tre ,a1 & ! :ole epicycle ) x till x be co9cid5t ) ! circle ,f ,g ,h1 ^: c5tre is ,i2 & let ,b ,a ,i be any angle giv54 ,b 9 ! "t t ! c5tre ,b is mov$ to ,i1 let ! epicycle ,c ,d ,e h a contr>y revolution ^u its own c5tre1 namely f ,e by ,d to ,c1 accord+ to ! same proportions2 t is1 9 su* mann]1 t 9 bo? ! circles1 equal angles be made 9 equal times4 ,i say ,e ,c1 ! axis ( ! epicycle1 w be always c>ri$ p>allel to itself4 ,let ! angle ,f ,i ,g be made equal to ! angle ,b ,a ,i2 ,i ,f & ,a ,b w !re=e be p>allel2 & h[ mu* ! axis ,a ,g has dep>t$ f its =m] place ,a ,c 7! measure ( :i* progression is ! angle ,c ,a ,g1 or ,c ,b ,d1 :i* ,i suppose equal to x7 s mu* 9 ! same "t has ! axis ,i ,g1 ! same ) ,b ,c1 dep>t$ f its own =m] situation4 ,:]e=e1 9 :at "t ,b ,c comes to ,i ,g by ! motion f ,b to ,i ^u ! c5tre ,a1 9 ! same "t ,g w come to ,f by ! contr>y motion ( ! epicycle2 t is1 x w be turn$ backw>ds to ,f1 & ,i ,g w lie 9 ,i ,f4 ,b ! angles ,f ,i ,g & ,g ,a ,c are equal2 & !re=e ,a ,c1 t is1 ,b ,c1 & ,i ,f1 7t is ! axis1 ?\gh 9 diff]5t places7 w be p>allel4 ,:]e=e1 ! axis ( ! epicycle ,e ,d ,c w be c>ri$ always p>allel to itself2 :i* 0 to be prov$4 ,coroll4 ,f h5ce x is manife/1 t ^? two annual motions :i* ,cop]nicus ascribes to ! e>?1 are r$ucible to ?is "o circul> simple motion1 by :i* all ! po9ts ( ! mov$ body are c>ri$ always ) equal velocity1 t is1 9 equal times !y make equal revolutions uni=mly4 ,?is1 z x is ! mo/ simple1 s x is ! mo/ frequ5t ( all circul> motions2 be+ ! same :i* is us$ by all m5 :5 !y turn any?+ r\nd ) _! arms1 z !y d 9 gr9d+ or sift+4 ,= all ! po9ts ( ! ?+ mov$ describe l9es :i* are l & equal to "o ano!r4 ,s t if a man _h a rul]1 9 :i* _m p5s' po9ts ( equal l5g? 7 fa/5$1 he might ) ?is "o motion write _m l9es at once4 ,sid5ote3 ,prop]ties ( simple motion4 #c4 ,hav+ %[n :at simple motion is1 ,i w "h also set d[n "s prop]ties ( ! same4 ,fir/1 :5 a body is mov$ ) simple motion 9 a fluid m$ium :i* ha? no vacuity1 x *anges ! situation ( all ! p>ts ( ! fluid ambi5t :i* resi/ its motion2 ,i say !re are no p>ts s small ( ! fluid ambi5t1 h[ f> soev] x be cont9u$1 b d *ange _! situation 9 su* mann]1 z t !y leave _! places cont9ually to o!r small p>ts t come into ! same4 ,= 79 ! same second figure7 let any body1 z ,k ,l ,m ,n1 be und]/ood to be mov$ ) simple circul> motion2 & let ! circle1 :i* e po9t !re( describes1 h any det]m9$ quantity1 suppose t ( ! same ,k ,l ,m ,n4 ,:]e=e ! c5tre ,a & e o!r po9t1 & consequ5tly ! mov$ body itself1 w be c>ri$ sometimes t[>ds ! side :]e is ,k1 & sometimes t[>ds ! o!r side :]e is ,m4 ,:5 !re=e x is c>ri$ to ,k1 ! p>ts ( ! fluid m$ium on t side w g back2 &1 suppos+ all space to be full1 o!rs on ! o!r side w succe$4 ,& s x w be :5 ! body is c>ri$ to ! side ,m1 & to ,n1 & e way4 ,n[ :5 ! ne>e/ p>ts ( ! fluid m$ium g back1 x is necess>y t ! p>ts next to ^? ne>e/ p>ts g back also2 & suppos+ /ill all space to be full1 o!r p>ts w come into _! places ) succession p]petual & inf9ite4 ,:]e=e all1 ev5 ! lea/ p>ts ( ! fluid m$ium1 *ange _! places1 c4 ,:i* 0 to be prov$4 ,x is evid5t f h5ce1 t simple motion1 :e!r circul> or n circul>1 ( bodies :i* make p]petual returns to _! =m] places1 ha? great] or less =ce to dissipate ! p>ts ( resi/+ bodies1 z x is m or less swift1 & z ! l9es describ$ h great] or less magnitude4 ,n[ ! greate/ velocity t c be1 may be und]/ood to be 9 ! lea/ circuit1 & ! lea/ 9 ! greate/2 & may be s suppos$1 :5 !re is ne$4 ,sid5ote3 ,if a fluid be mov$ ) simple circul> motion1 all ! po9ts tak5 9 x w describe _! circles 9 times proportional to ! di/ances f ! c5tre4 #d4 ,secondly1 suppos+ ! same simple motion 9 ! air1 wat]1 or o!r fluid m$ium2 ! p>ts ( ! m$ium1 :i* adh]e to ! mov$ body1 w be c>ri$ ab\t ) ! same motion & velocity1 s t 9 :at "t soev] any po9t ( ! mov5t f9i%es its circle1 9 ! same "t e "p ( ! m$ium1 :i* adh]es to ! mov5t1 %all also describe su* a "p ( its circle1 z is equal to ! :ole circle ( ! mov5t2 ,i say1 x %all describe a "p1 & n ! :ole circle1 because all its p>ts receive _! motion f an int]ior conc5tric mov5t1 & ( conc5tric circles ! ext]ior are always great] ?an ! int]ior2 nor c ! motion impr9t$ by any mov5t be ( great] velocity ?an t ( ! mov5t itself4 ,f :5ce x foll[s1 t ! m remote p>ts ( ! fluid ambi5t %all f9i% _! circles 9 times1 :i* h to "o ano!r ! same proportion ) _! di/ances f ! mov5t4 ,= e po9t ( ! fluid ambi5t1 z long z x t\*e? ! body :i* c>ries x ab\t1 is c>ri$ ab\t ) x1 & w\ld make ! same circle1 b t x is left beh9d s mu* z ! ext]ior circle exce$s ! int]ior4 ,s t if we suppose "s ?+1 :i* is n fluid1 to float 9 t "p ( ! fluid ambi5t :i* is ne>e/ to ! mov5t1 x w toge!r ) ! mov5t be c>ri$ ab\t4 ,n[ t "p ( ! fluid ambi5t1 :i* is n ! ne>e/ b almo/ ! ne>e/1 receiv+ its degree ( velocity f ! ne>e/1 :i* degree _c be great] ?an x 0 9 ! giv]1 do? !re=e 9 ! same "t make a circul> l9e1 n a :ole circle1 yet equal to ! :ole circle ( ! ne>e/4 ,!re=e 9 ! same "t t ! mov5t describes its circle1 t :i* do? n t\* x %all n describe its circle2 yet x %all describe su* a "p ( x1 z is equal to ! :ole circle ( ! mov5t4 ,& aft] ! same mann]1 ! m remote p>ts ( ! ambi5t w describe 9 ! same "t su* p>ts ( _! circles1 z %all be sev]ally equal to ! :ole circle ( ! mov5t2 &1 by consequ5t1 !y %all f9i% _! :ole circles 9 times proportional to _! di/ances f ! mov5t2 :i* 0 to be prov$4 ,sid5ote3 ,simple motion dissipates het]og5e\s & congregates homog5e\s bodies4 #e4 ,?irdly1 ! same simple motion ( a body plac$ 9 a fluid m$ium1 congregates or ga!rs into "o place su* ?+s z naturally float 9 t m$ium1 if !y be homog5e\s2 & if !y be het]og5e\s1 x sep>ates & dissipates !m4 ,b if su* ?+s z be het]og5e\s d n float1 b settle1 !n ! same motion /irs & m+les !m disord]ly toge!r4 ,= see+ bodies1 :i* are unlike to "o ano!r1 t is1 het]og5e\s bodies1 are n unlike 9 t !y are bodies2 = bodies1 z bodies1 h no diff]5ce2 b only f "s special cause1 t is1 f "s int]nal motion1 or motions ( _! smalle/ p>ts 7= ,i h %[n 9 *ap4 ,,ix1 art4 #i1 t all mutation is su* motion71 x rema9s t het]og5e\s bodies h _! unlik5ess or diff]5ce f "o ano!r f _! int]nal or specifical motions4 ,n[ bodies :i* h su* diff]5ce receive unlike & diff]5t motions f ! same ext]nal common mov5t2 & !re=e !y w n be mov$ toge!r1 t is to say1 !y w be dissipat$4 ,& be+ dissipat$ !y w necess>ily at "s "t or o!r meet ) bodies l !mselves1 & be mov$ alike & toge!r ) !m2 & aft]w>ds meet+ ) m bodies l !mselves1 !y w unite & become great] bodies4 ,:]e=e homog5e\s bodies are congregat$1 & het]og5e\s dissipat$ by simple motion 9 a m$ium :]e !y naturally float4 ,aga91 su* z be+ 9 a fluid m$ium d n float1 b s9k1 if ! motion ( ! fluid m$ium be /rong en\gh1 w be /irr$ up & c>ri$ away by t motion1 & consequ5tly !y w be h9d]$ f return+ to t place to :i* !y s9k naturally1 & 9 :i* only !y w\ld unite1 & out ( :i* !y are promiscu\sly c>ri$2 t is1 !y are disord]ly m+l$4 ,n[ ?is motion1 by :i* homog5e\s bodies are congregat$ & het]og5e\s are scatt]$1 is t :i* is commonly call$ f]m5tation1 f ! ,lat9 f]v]e2 z ! ,greeks h _! 1 :i* signifies ! same1 f f]veo4 ,= see?+ makes all ! p>ts ( ! wat] *ange _! places2 & ! p>ts ( any ?+1 t is ?r[n into x1 w g sev]al ways accord+ to _! sev]al natures4 ,& yet all f]v\r or see?+ is n caus$ by fire2 = new w9e & _m o!r ?+s h also _! f]m5tation & f]v\r1 to :i* fire contributes little1 & sometimes no?+4 ,b :5 9 f]m5tation we f9d heat1 x is made by ! f]m5tation4 ,sid5ote3 ,if a circle made by a mov5t mov$ ) simple motion1 be comm5surable to ano!r circle made by a po9t :i* is c>ri$ ab\t by ! same mov5t1 all ! po9ts ( bo? ! circles w at "s "t return to ! same situation4 #f4 ,f\r?ly1 9 :at "t soev] ! mov5t1 ^: c5tre is ,a 79 fig4 #b7 mov$ 9 ,k ,l ,n1 %all1 by any numb] ( revolutions1 t is1 :5 ! p]imet]s ,b ,i & ,k ,l ,n be comm5surable1 h describ$ a l9e equal to ! circle :i* passes ?r\gh ! po9ts ,b & ,i2 9 ! same "t all ! po9ts ( ! float+ body1 ^: c5tre is ,b1 %all return to h ! same situation 9 respect ( ! mov5t1 f :i* !y dep>t$4 ,= see+ x is z ! di/ance ,b ,a1 t is1 z ! radius ( ! circle :i* passes ?r\gh ,b ,i is to ! p]imet] itself ,b ,i1 s ! radius ( ! circle ,k ,l ,n is to ! p]imet] ,k ,l ,n2 & see+ ! velocities ( ! po9ts ,b & ,k are equal1 ! "t also ( ! revolution 9 ,i ,b to ! "t ( "o revolution 9 ,k ,l ,n1 w be z ! p]imet] ,b ,i to ! p]imet] ,k ,l ,n2 & !re=e s _m revolutions 9 ,k ,l ,n1 z toge!r tak5 are equal to ! p]imet] ,b ,i1 w be f9i%$ 9 ! same "t 9 :i* ! :ole p]imet] ,b ,i is f9i%$2 & !re=e also ! po9ts ,l1 ,n1 ,f & ,h1 or any ( ! re/1 w 9 ! same "t return to ! same situation f :i* !y dep>t$2 & ?is may be demon/rat$1 :atsoev] be ! po9ts consid]$4 ,:]e=e all ! po9ts %all 9 t "t return to ! same situation2 :i* 0 to be prov$4 ,f h5ce x foll[s1 t if ! p]imet]s ,b ,i & ,l ,k ,n be n comm5surable1 !n all ! po9ts w nev] return to h ! same situation or configuration 9 respect ( "o ano!r4 ,sid5ote3 ,if a sph]e h simple motion1 its motion w m dissipate het]og5e\s bodies by h[ mu* x is m remote f ! poles4 #g4 ,9 simple motion1 if ! body mov$ be ( a sph]ical figure1 x ha? less =ce t[>ds its poles ?an t[>ds its middle to dissipate het]og5e\s1 or to congregate homog5e\s bodies4 ,let !re be a sph]e 7z 9 ! ?ird figure7 ^: c5tre is ,a & diamet] ,b ,c2 & let x be conceiv$ to be mov$ ) simple circul> motion2 ( :i* motion let ! axis be ! /rait l9e ,d ,e1 cutt+ ! diamet] ,b ,c at "r angles 9 ,a4 ,let n[ ! circle1 :i* is describ$ by any po9t ,b ( ! sph]e1 h ,b ,f = its diamet]2 & tak+ ,f ,g equal to ,b ,c1 & divid+ x 9 ! middle 9 ,h1 ! c5tre ( ! sph]e ,a w1 :5 half a revolution is f9i%$1 lie 9 ,h4 ,& see+ ,h ,f & ,a ,b are equal1 a circle describ$ ^u ! c5tre ,h ) ! radius ,h ,f or ,h ,g1 w be equal to ! circle ^: c5tre is ,a & radius ,a ,b4 ,& if ! same motion be cont9u$1 ! po9t ,b w at ! end ( ano!r half revolution return to ! place f :5ce x began to be mov$2 & !re=e at ! end ( half a revolution1 ! po9t ,b w be c>ri$ to ,f1 & ! :ole hemisph]e ,d ,b ,e into t hemisph]e 9 :i* are ! po9ts ,l1 ,k & ,f4 ,:]e=e t "p ( ! fluid m$ium1 :i* is contigu\s to ! po9t ,f1 w 9 ! same "t g back ! l5g? ( ! /rait l9e ,b ,f2 & 9 ! return ( ! po9t ,f to ,b1 t is1 ( ,g to ,c1 ! fluid m$ium w g back z mu* 9 a /rait l9e f ! po9t ,c4 ,& ?is is ! effect ( simple motion 9 ! middle ( ! sph]e1 :]e ! di/ance f ! poles is greate/4 ,let n[ ! po9t ,i be tak5 9 ! same sph]e ne>] to ! pole ,e1 & ?r\gh x let ! /rait l9e ,i ,k be drawn p>allel to ! /rait l9e ,b ,f1 cutt+ ! ar* ,f ,l 9 ,k1 & ! axis ,h ,l 9 ,m2 !n connect+ ,h ,k1 ^u ,h ,f let ! p]p5dicul> ,k ,n be drawn4 ,9 ! same "t !re=e t ,b comes to ,f ! po9t ,i w come to ,k1 ,b ,f & ,i ,k be+ equal & describ$ ) ! same velocity4 ,n[ ! motion 9 ,i ,k to ! fluid m$ium ^u :i* x works1 namely1 to t "p ( ! m$ium :i* is contigu\s to ! po9t ,k1 is oblique1 :]eas if x proce$$ 9 ! /rait l9e ,h ,k x w\ld be p]p5dicul>2 & !re=e ! motion :i* proce$s 9 ,i ,k has less p[] ?an t :i* proce$s 9 ,h ,k ) ! same velocity4 ,b ! motions 9 ,h ,k & ,h ,f d equally ?ru/ back ! m$ium2 & !re=e ! "p ( ! sph]e at ,k moves ! m$ium less ?an ! "p at ,f1 namely1 s mu* less z ,k ,n is less ?an ,h ,f4 ,:]e=e also ! same motion ha? less p[] to disp]se het]og5e\s1 & to congregate homog5e\s bodies1 :5 x is ne>]1 ?an :5 x is m remote f ! poles2 :i* 0 to be prov$4 ,coroll4 ,x is also necess>y1 t 9 planes :i* are p]p5dicul> to ! axis1 & m remote ?an ! pole itself f ! middle ( ! sph]e1 ?is simple motion h no effect4 ,= ! axis ,d ,e ) simple motion describes ! sup]ficies ( a cyl9d]2 & t[>ds ! bases ( ! cyl9d] !re is 9 ?is motion no endeav\r at all4 ,sid5ote3 ,if a simple circul> motion ( a fluid body be h9d]$ by a body :i* is n fluid1 ! fluid body w spread itself ^u ! sup]ficies ( t body4 #h4 ,if 9 a fluid m$ium mov$ ab\t1 z ha? be5 said1 ) simple motion1 !re be conceiv$ to float "s o!r sph]ical body :i* is n fluid1 ! p>ts ( ! m$ium1 :i* are /opp$ by t body1 w endeav\r to spread !mselves e way ^u ! sup]ficies ( x4 ,& ?is is manife/ en\gh by exp]i5ce1 namely1 by ! spread+ ( wat] p\r$ out ^u a pavem5t4 ,b ! reason ( x may be ?is4 ,see+ ! sph]e ,a 79 fig4 #c7 is mov$ t[>ds ,b1 ! m$ium also 9 :i* x is mov$ w h ! same motion4 ,b because 9 ?is motion x falls ^u a body n liquid1 z ,g1 s t x _c g on2 & see+ ! small p>ts ( ! m$ium _c g =w>ds1 nor c !y g directly backw>ds aga9/ ! =ce ( ! mov5t2 x rema9s1 !re=e1 t !y diffuse !mselves ^u ! sup]ficies ( t body1 z t[>ds ,o & ,p2 :i* 0 to be prov$4 ,sid5ote3 ,circul> motion ab\t a fix$ c5tre ca/e? (f by ! tang5t su* ?+s z lie ^u ! circumf]5ce /ick n to x4 #i4 ,comp\nd$ circul> motion1 9 :i* all ! p>ts ( ! mov$ body d at once describe circumf]5ces1 "s great]1 o!rs less1 accord+ to ! proportion ( _! sev]al di/ances f ! common c5tre1 c>ries ab\t ) x su* bodies1 z be+ n fluid1 adh]e to ! body s mov$2 & su* z d n adh]e1 x ca/e? =w>ds 9 a /rait l9e :i* is a tang5t to ! po9t f :i* !y are ca/ (f4 ,= let !re be a circle ^: radius is ,a ,b 79 fig4 #d72 & let a body be plac$ 9 ! circumf]5ce 9 ,b1 :i* if x be fix$ !re1 w necess>ily be c>ri$ ab\t ) x1 z is manife/ ( itself4 ,b :il/ ! motion proce$s1 let u suppose t body to be unfix$ 9 ,b4 ,i say1 ! body w cont9ue its motion 9 ! tang5t ,b ,c4 ,= let bo? ! radius ,a ,b & ! sph]e ,b be conceiv$ to consi/ ( h>d matt]2 & let u suppose ! radius ,a ,b to be /rick5 9 ! po9t ,b by "s o!r body :i* falls ^u x 9 ! tang5t ,d ,b4 ,n[1 !re=e1 !re w be a motion made by ! conc\rse ( two ?+s1 ! "o1 endeav\r t[>ds ,c 9 ! /rait l9e ,d ,b produc$1 9 :i* ! body ,b w\ld proce$1 if x 7 n reta9$ by ! radius ,a ,b2 ! o!r1 ! ret5tion itself4 ,b ! ret5tion alone cause? no endeav\r t[>ds ! c5tre2 &1 !re=e1 ! ret5tion be+ tak5 away1 :i* is done by ! unfix+ ( ,b1 !re w rema9 b "o endeav\r 9 ,b1 namely1 t 9 ! tang5t ,b ,c4 ,:]e=e ! motion ( ! body ,b unfix$ w proce$ 9 ! tang5t ,b ,c2 :i* 0 to be prov$4 ,by ?is demon/ration x is manife/1 t circul> motion ab\t an unmov$ axis %akes (f & puts fur!r f ! c5tre ( its motion su* ?+s z t\*1 b d n /ick fa/ to its sup]ficies2 & ! m1 by h[ mu* ! di/ance is great] f ! poles ( ! circul> motion2 & s mu* ! m also1 by h[ mu* ! ?+s1 t are %ak5 (f1 are less driv5 t[>ds ! c5tre by ! fluid ambi5t1 = o!r causes4 ,sid5ote3 ,su* ?+s z are mov$ ) simple circul> motion1 beget simple circul> motion4 #aj4 ,if 9 a fluid m$ium a sph]ical body be mov$ ) simple circul> motion1 & 9 ! same m$ium !re float ano!r sph]e ^: matt] is n fluid1 ?is sph]e also %all be mov$ ) simple circul> motion4 ,let ,b ,c ,d 79 fig4 #e7 be a circle1 ^: c5tre is ,a1 & 9 ^: circumf]5ce !re is a sph]e s mov$1 t x describes ) simple motion ! p]imet] ,b ,c ,d4 ,let also ,e ,f ,g be ano!r sph]e ( consi/5t matt]1 ^: semidiamet] is ,e ,h1 & c5tre ,h2 & ) ! radius ,a ,h let ! circle ,h ,i be describ$4 ,i say1 ! sph]e ,e ,f ,g w1 by ! motion ( ! body 9 ,b ,c ,d1 be mov$ 9 ! circumf]5ce ,h ,i ) simple motion4 ,= see+ ! motion 9 ,b ,c ,d 7by art4 #d ( ?is *apt]7 makes all ! po9ts ( ! fluid m$ium describe 9 ! same "t circul> l9es equal to "o ano!r1 ! po9ts ,e1 ,h & ,g ( ! /rait l9e ,e ,h ,g w 9 ! same "t describe ) equal radii equal circles4 ,let ,e ,b be drawn equal & p>allel to ! /rait l9e ,a ,h2 & let ,a ,b be connect$1 :i* w !re=e be equal & p>allel to ,e ,h2 & !re=e also1 if ^u ! c5tre ,b & radius ,b ,e ! ar* ,e ,k be drawn equal to ! ar* ,h ,i1 & ! /rait l9es ,a ,i1 ,b ,k & ,i ,k be drawn1 ,b ,k & ,a ,i w be equal2 & !y w also be p>allel1 because ! two ar*es ,e ,k & ,h ,i1 t is1 ! two angles ,k ,b ,e & ,i ,a ,h are equal2 &1 consequ5tly1 ! /rait l9es ,a ,b & ,k ,i1 :i* connect !m1 w also be equal & p>allel4 ,:]e=e ,k ,i & ,e ,h are p>allel4 ,see+1 !re=e1 ,e & ,h are c>ri$ 9 ! same "t to ,k & ,i1 ! :ole /rait l9e ,i ,k w be p>allel to ,e ,h1 f :5ce x dep>t$4 ,&1 !re=e1 see+ ! sph]e ,e ,f ,g is suppos$ to be ( consi/5t matt]1 s z all its po9ts keep always ! same situation1 x is necess>y t e o!r /rait l9e1 tak5 9 ! same sph]e1 be c>ri$ always p>allel to ! places 9 :i* x =m]ly 04 ,:]e=e ! sph]e ,e ,f ,g is mov$ ) simple circul> motion2 :i* 0 to be demon/rat$4 ,sid5ote3 ,if t :i* is s mov$ h "o side h>d & ! o!r side fluid1 its motion w n be p]fectly circul>4 #aa4 ,if 9 a fluid m$ium1 ^: p>ts are /irr$ by a body mov$ ) simple motion1 !re float ano!r body1 :i* ha? its sup]ficies ei!r :olly h>d1 or :olly fluid1 ! p>ts ( ?is body %all approa* ! c5tre equally on all sides2 t is to say1 ! motion ( ! body %all be circul>1 & conc5tric ) ! motion ( ! mov5t4 ,b if x h "o side h>d1 & ! o!r side fluid1 !n bo? ^? motions %all n h ! same c5tre1 nor %all ! float+ body be mov$ 9 ! circumf]5ce ( a p]fect circle4 ,let a body be mov$ 9 ! circumf]5ce ( ! circle ,k ,l ,m ,n 79 fig #b47 ^: c5tre is ,a4 ,& let !re be ano!r body at ,i1 ^: sup]ficies is ei!r all h>d or all fluid4 ,also let ! m$ium1 9 :i* bo? ^! bodies are plac$1 be fluid4 ,i say1 ! body at ,i w be mov$ 9 ! circle ,i ,b ab\t ! c5tre ,a4 ,= ?is has be5 demon/rat$ 9 ! la/ article4 ,:]e=e let ! sup]ficies ( ! body at ,i be fluid on "o side1 & h>d on ! o!r4 ,& fir/1 let ! fluid side be t[>ds ! c5tre4 ,see+1 !re=e1 ! motion ( ! m$ium is su*1 z t its p>ts d cont9ually *ange _! places1 7z ha? be5 %[n 9 art #e72 if ?is *ange ( place be consid]$ 9 ^? p>ts ( ! m$ium :i* are contigu\s to ! fluid sup]ficies1 x mu/ ne$s be t ! small p>ts ( t sup]ficies ent] into ! places ( ! small p>ts ( ! m$ium :i* are contigu\s to !m2 & ! l *ange ( place w be made ) ! next contigu\s p>ts t[>ds ,a4 ,& if ! fluid p>ts ( ! body at ,i h any degree at all ( t5acity 7= !re are degrees ( t5acity1 z 9 ! air & wat]7 ! :ole fluid side w be lift$ up a little1 b s mu* ! less1 z its p>ts h less t5acity2 :]eas ! h>d "p ( ! sup]ficies1 :i* is contigu\s to ! fluid "p1 has no cause at all ( elevation1 t is to say1 no endeav\r t[>ds ,a4 ,secondly1 let ! h>d sup]ficies ( ! body at ,i be t[>ds ,a4 ,by reason1 !re=e1 ( ! said *ange ( place ( ! p>ts :i* are contigu\s to x1 ! h>d sup]ficies mu/1 ( necessity1 see+ by supposition !re is no empty space1 ei!r come ne>] to ,a1 or else its smalle/ p>ts mu/ supply ! contigu\s places ( ! m$ium1 :i* o!rwise w\ld be empty4 ,b ?is _c be1 by reason ( ! suppos$ h>dness2 &1 !re=e1 ! o!r mu/ ne$s be1 namely1 t ! body come ne>] to ,a4 ,:]e=e ! body at ,i has great] endeav\r t[>ds ! c5tre ,a1 :5 its h>d side is next x1 ?an :5 x is av]t$ f x4 ,b ! body 9 ,i1 :ile x is mov+ 9 ! circumf]5ce ( ! circle ,i ,b1 has sometimes "o side1 sometimes ano!r1 turn$ t[>ds ! c5tre2 &1 !re=e1 x is sometimes ne>]1 sometimes fur!r (f f ! c5tre ,a4 ,:]e=e ! body at ,i is n c>ri$ 9 ! circumf]5ce ( a p]fect circle2 :i* 0 to be demon/rat$4 ---------------------------------------- -------------------------------- ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xxi4 ,fig4 #a-e ---------------------------------------- -------------------------------- ------- ,,*apt] ,,xxii4 ,,( ,,o!r ,,v>iety ,,( ,,motion4 #a4 ,endeav\r & pressure h[ !y diff]4--#b4 ,two k9ds ( m$iums 9 :i* bodies are mov$4--#c4 ,propagation ( motion1 :at x is4--#d4 ,:at motion bodies h1 :5 !y press "o ano!r4--#e4 ,fluid bodies1 :5 !y are press$ toge!r1 p5etrate "o ano!r4--#f4 ,:5 "o body presse? ano!r & do? n p5etrate x1 ! action ( ! press+ body is p]p5dicul> to ! sup]ficies ( ! body press$4--#g4 ,:5 a h>d body1 press+ ano!r body1 p5etrates ! same1 x do? n p5etrate x p]p5dicul>ly1 unless x fall p]p5dicul>ly ^u x4--#h4 ,motion sometimes opposite to t ( ! mov5t4--#i4 ,9 a full m$ium1 motion is propagat$ to any di/ance4--#aj4 ,dilatation & contraction :at !y are4--#aa4 ,dilatation & contraction suppose mutation ( ! smalle/ p>ts 9 respect ( _! situation4--#ab4 ,all traction is pulsion4--#ac4 ,su* ?+s z be+ press$ or b5t re/ore !mselves1 h motion 9 _! int]nal p>ts4--#ad4 ,?\gh t :i* c>rie? ano!r be /opp$1 ! body c>ri$ w proce$4--#ae1 #af4 ,! effects ( p]cussion n to be comp>$ ) ^? ( weight4--#ag1 #ah4 ,motion _c beg9 fir/ 9 ! int]nal p>ts ( a body4--#ai4 ,action & reaction proce$ 9 ! same l9e4--#bj4 ,habit1 :at x is4 ,sid5ote3 ,endeav\r & pressure h[ !y diff]4 #a4 ,i h already 7*apt] ,,xv4 art4 #b7 def9$ endeav\r to be motion ?r\gh "s l5g?1 ?\gh n consid]$ z l5g?1 b z a po9t4 ,:e!r1 !re=e1 !re be resi/ance or no resi/ance1 ! endeav\r w /ill be ! same4 ,= simply to endeav\r is to g4 ,b :5 two bodies1 hav+ opposite endeav\rs1 press "o ano!r1 !n ! endeav\r ( ei!r ( !m is t :i* we call pressure1 & is mutual :5 _! pressures are opposite4 ,sid5ote3 ,two k9ds ( m$iums 9 :i* bodies are mov$4 #b4 ,bodies mov$1 & also ! m$iums 9 :i* !y are mov$1 are ( two k9ds4 ,= ei!r !y h _! p>ts coh]5t 9 su* mann]1 z no "p ( ! mov$ body w easily yield to ! mov5t1 except ! :ole body yield also1 & su* are ! ?+s we call h>d3 or else _! p>ts1 :ile ! :ole rema9s unmov$1 w easily yield to ! mov5t1 & ^! we call fluid or s(t bodies4 ,= ! words fluid1 s(t1 t\gh1 & h>d1 9 ! same mann] z great & little1 are us$ only comp>atively2 & are n diff]5t k9ds1 b diff]5t degrees ( quality4 ,sid5ote3 ,propagation ( motion1 :at x is4 #c4 ,to d1 & to suff]1 is to move & to be mov$2 & no?+ is mov$ b by t :i* t\*e? x & is also mov$1 z has be5 =m]ly %[n4 ,& h[ great soev] ! di/ance be1 we say ! fir/ mov5t move? ! la/ mov$ body1 b m$iately2 namely s1 z t ! fir/ move? ! second1 ! second ! ?ird1 & s on1 till ! la/ ( all be t\*$4 ,:5 !re=e "o body1 hav+ opposite endeav\r to ano!r body1 move? ! same1 & t move? a ?ird1 & s on1 ,i call t action propagation ( motion4 ,sid5ote3 ,:at motion bodies h :5 !y press "o ano!r4 #d4 ,:5 two fluid bodies1 :i* are 9 a free & op5 space1 press "o ano!r1 _! p>ts w endeav\r1 or be mov$1 t[>ds ! sides2 n only ^? p>ts :i* are !re :]e ! mutual contact is1 b all ! o!r p>ts4 ,= 9 ! fir/ contact1 ! p>ts1 :i* are press$ by bo? ! endeav\r+ bodies1 h no place ei!r =w>ds or backw>ds 9 :i* !y c be mov$2 & !re=e !y are press$ out t[>ds ! sides4 ,& ?is expressure1 :5 ! =ces are equal1 is 9 a l9e p]p5dicul> to ! bodies press+4 ,b :5soev] ! =emo/ p>ts ( bo? ! bodies are press$1 ! h9d]mo/ also mu/ be press$ at ! same "t2 = ! motion ( ! h9d]mo/ p>ts _c 9 an in/ant be /opp$ by ! resi/ance ( ! =emo/ p>ts1 b proce$s = "s "t2 & !re=e1 see+ !y mu/ h "s place 9 :i* !y may be mov$1 & t !re is no place at all = !m =w>ds1 x is necess>y t !y be mov$ into ! places :i* are t[>ds ! sides e way4 ,& ?is effect foll[s ( necessity1 n only 9 fluid1 b 9 consi/5t & h>d bodies1 ?\gh x be n always manife/ to s5se4 ,= ?\gh f ! compression ( two /ones we _c ) our eyes disc]n any swell+ outw>ds t[>ds ! sides1 z we p]ceive 9 two bodies ( wax2 yet we "k well en\gh by reason1 t "s tum\r mu/ ne$s be !re1 ?\gh x be b little4 ,sid5ote3 ,fluid bodies1 :5 !y are press$ toge!r1 p5etrate "o ano!r4 #e4 ,b :5 ! space is enclos$1 & bo? ! bodies be fluid1 !y w1 if !y be press$ toge!r1 p5etrate "o ano!r1 ?\gh diff]5tly1 accord+ to _! diff]5t endeav\rs4 ,= suppose a holl[ cyl9d] ( h>d matt]1 well /opp$ at bo? ends1 b fill$ fir/1 bel[ ) "s heavy fluid body1 z quicksilv]1 & above ) wat] or air4 ,if n[ ! bottom ( ! cyl9d] be turn$ upw>ds1 ! heavie/ fluid body1 :i* is n[ at ! top1 hav+ ! greate/ endeav\r d[nw>ds1 & be+ by ! h>d sides ( ! vessel h9d]$ f ext5d+ itself sideways1 mu/ ( necessity ei!r be receiv$ by ! light] body1 t x may s9k ?r\gh x1 or else x mu/ op5 a passage ?r\gh itself1 by :i* ! light] body may asc5d4 ,= ( ! two bodies1 t1 ^: p>ts are mo/ easily sep>at$1 w be ! fir/ divid$2 :i* be+ done1 x is n necess>y t ! p>ts ( ! o!r suff] any sep>ation at all4 ,& !re=e :5 two liquors1 :i* are enclos$ 9 ! same vessel1 *ange _! places1 !re is no ne$ t _! smalle/ p>ts %\ld be m+l$ ) "o ano!r2 = a way be+ op5$ ?r\gh "o ( !m1 ! p>ts ( ! o!r ne$ n be sep>at$4 ,n[ if a fluid body1 :i* is n enclos$1 press a h>d body1 its endeav\r w inde$ be t[>ds ! int]nal p>ts ( t h>d body2 b be+ exclud$ by ! resi/ance ( x1 ! p>ts ( ! fluid body w be mov$ e way accord+ to ! sup]ficies ( ! h>d body1 & t equally1 if ! pressure be p]p5dicul>2 = :5 all ! p>ts ( ! cause are equal1 ! effects w be equal also4 ,b if ! pressure be n p]p5dicul>1 !n ! angles ( ! incid5ce be+ unequal1 ! expansion also w be unequal1 namely1 great] on t side :]e ! angle is great]1 because t motion is mo/ direct :i* proce$s by ! directe/ l9e4 ,sid5ote3 ,:5 "o body presse? ano!r & do? n p5etrate x1 ! action ( ! press+ body is p]p5dicul> to ! sup]ficies ( ! body press$4 #f4 ,if a body1 press+ ano!r body1 d n p5etrate x1 x w nev]!less give to ! "p x presse? an endeav\r to yield1 & rec$e 9 a /rait l9e p]p5dicul> to its sup]ficies 9 t po9t 9 :i* x is press$4 ,let ,a ,b ,c ,d 79 fig4 #a7 be a h>d body1 & let ano!r body1 fall+ ^u x 9 ! /rait l9e ,e ,a1 ) any incl9ation or )\t incl9ation1 press x 9 ! po9t ,a4 ,i say ! body s press+1 & n p5etrat+ x1 w give to ! "p ,a an endeav\r to yield or rec$e 9 a /rait l9e p]p5dicul> to ! l9e ,a ,d4 ,= let ,a ,b be p]p5dicul> to ,a ,d1 & let ,b ,a be produc$ to ,f4 ,if !re=e ,a ,f be co9cid5t ) ,a ,e1 x is ( itself manife/ t ! motion 9 ,e ,a w make ,a to endeav\r 9 ! l9e ,a ,b4 ,let n[ ,e ,a be oblique to ,a ,d1 & f ! po9t ,e let ! /rait l9e ,e ,c be drawn1 cutt+ ,a ,d at "r angles 9 ,d1 & let ! rectangles ,a ,b ,c ,d & ,a ,d ,e ,f be complet$4 ,i h %[n 79 ! #h? article ( *apt] ,,xvi7 t ! body w be c>ri$ f ,e to ,a by ! conc\rse ( two uni=m motions1 ! "o 9 ,e ,f & its p>allels1 ! o!r 9 ,e ,d & its p>allels4 ,b ! motion 9 ,e ,f & its p>allels1 :]e( ,d ,a is "o1 contributes no?+ to ! body 9 ,a to make x endeav\r or press t[>ds ,b2 & !re=e ! :ole endeav\r1 :i* ! body ha? 9 ! incl9$ l9e ,e ,a to pass or press ! /rait l9e ,a ,d1 x ha? x all f ! p]p5dicul> motion or endeav\r 9 ,f ,a4 ,:]e=e ! body ,e1 aft] x is 9 ,a1 w h only t p]p5dicul> endeav\r :i* proce$s f ! motion 9 ,f ,a1 t is1 9 ,a ,b2 :i* 0 to be prov$4 ,sid5ote3 ,:5 a h>d body1 press+ ano!r body1 p5etrates ! same1 x do? n p5etrate x p]p5dicul>ly1 unless x fall p]p5dicul>ly ^u x4 #g4 ,if a h>d body fall+ ^u or press+ ano!r body p5etrate ! same1 its endeav\r aft] its fir/ p5etration w be nei!r 9 ! incl9$ l9e produc$1 nor 9 ! p]p5dicul>1 b sometimes betwixt bo?1 sometimes )\t !m4 ,let ,e ,a ,g 79 ! same fig4 #a7 be ! incl9$ l9e produc$2 & fir/1 let ! passage ?r\gh ! m$ium1 9 :i* ,e ,a is1 be easi] ?an ! passage ?r\gh ! m$ium 9 :i* ,a ,g is4 ,z soon !re=e z ! body is )9 ! m$ium 9 :i* is ,a ,g1 x w f9d great] resi/ance to its motion 9 ,d ,a & its p>allels1 ?an x did :il/ x 0 above ,a ,d2 & !re=e bel[ ,a ,d x w proce$ ) sl[] motion 9 ! p>allels ( ,d ,a1 ?an above x4 ,:]e=e ! motion :i* is comp\nd$ ( ! two motions 9 ,e ,f & ,e ,d w be sl[] bel[ ,a ,d ?an above x2 & !re=e also1 ! body w n proce$ f ,a 9 ,e ,a produc$1 b bel[ x4 ,see+1 !re=e1 ! endeav\r 9 ,a ,b is g5]at$ by ! endeav\r 9 ,f ,a2 if to ! endeav\r 9 ,f ,a !re be add$ ! endeav\r 9 ,d ,a1 :i* is n all tak5 away by ! imm]sion ( ! po9t ,a into ! l[] m$ium1 ! body w n proce$ f ,a 9 ! p]p5dicul> ,a ,b1 b beyond x2 namely1 9 "s /rait l9e betwe5 ,a ,b & ,a ,g1 z 9 ! l9e ,a ,h4 ,secondly1 let ! passage ?r\gh ! m$ium ,e ,a be less easy ?an t ?r\gh ,a ,g4 ,! motion1 !re=e1 :i* is made by ! conc\rse ( ! motions 9 ,e ,f & ,f ,b1 is sl[] above ,a ,d ?an bel[ x2 & consequ5tly1 ! endeav\r w n proce$ f ,a 9 ,e ,a produc$1 b beyond x1 z 9 ,a ,i4 ,:]e=e1 if a h>d body fall+1 c42 :i* 0 to be prov$4 ,?is div]g5cy ( ! /rait l9e ,a ,h f ! /rait l9e ,a ,g is t :i*1 ! writ]s ( optics commonly call$ refraction1 :i*1 :5 ! passage is easi] 9 ! fir/ ?an 9 ! second m$ium1 is made by div]g+ f ! l9e ( incl9ation t[>ds ! p]p5dicul>2 & contr>ily1 :5 ! passage is n s easy 9 ! fir/ m$ium1 by dep>t+ fur!r f ! p]p5dicul>4 ,sid5ote3 ,motion sometimes opposite to t ( ! mov5t4 #h4 ,by ! #f? !orem x is manife/1 t ! =ce ( ! mov5t may be s plac$1 z t ! body mov$ by x may proce$ 9 a way almo/ directly contr>y to t ( ! mov5t1 z we see 9 ! motion ( %ips4 ,= let ,a ,b 79 fig4 #b7 repres5t a %ip1 ^: l5g? f ! pr[ to ! poop is ,a ,b1 & let ! w9d lie ^u x 9 ! /rait p>allel l9es ,c ,b1 ,d ,e & ,f ,g2 & let ,d ,e & ,f ,g be cut 9 ,e & ,g by a /rait l9e drawn f ,b p]p5dicul> to ,a ,b2 also let ,b ,e & ,e ,g be equal1 & ! angle ,a ,b ,c any angle h[ small soev]4 ,!n betwe5 ,b ,c & ,b ,a let ! /rait l9e ,b ,i be drawn2 & let ! sail be conceiv$ to be spread 9 ! same l9e ,b ,i1 & ! w9d to fall ^u x 9 ! po9ts ,l1 ,m & ,b2 f :i* po9ts1 p]p5dicul> to ,b ,i1 let ,b ,k1 ,m ,q & ,l ,p be drawn4 ,la/ly1 let ,e ,n & ,g ,o be drawn p]p5dicul> to ,b ,g1 & cutt+ ,b ,k 9 ,h & ,k2 & let ,h ,n & ,k ,o be made equal to "o ano!r1 & sev]ally equal to ,b ,a4 ,i say1 ! %ip ,b ,a1 by ! w9d fall+ ^u x 9 ,c ,b1 ,d ,e1 ,f ,g1 & o!r l9es p>allel to !m1 w be c>ri$ =w>ds almo/ opposite to ! w9d1 t is to say1 9 a way almo/ contr>y to ! way ( ! mov5t4 ,= ! w9d t bl[s 9 ! l9e ,c ,b w 7z ha? be5 %[n 9 art4 #f7 give to ! po9t ,b an endeav\r to proce$ 9 a /rait l9e p]p5dicul> to ! /rait l9e ,b ,i1 t is1 9 ! /rait l9e ,b ,k2 & to ! po9ts ,m & ,l an endeav\r to proce$ 9 ! /rait l9es ,m ,q & ,l ,p1 :i* are p>allel to ,b ,k4 ,let n[ ! measure ( ! "t be ,b ,g1 :i* is divid$ 9 ! middle 9 ,e2 & let ! po9t ,b be c>ri$ to ,h 9 ! "t ,b ,e4 ,9 ! same "t1 !re=e1 by ! w9d bl[+ 9 ,d ,m & ,f ,l1 & z _m o!r l9es z may be drawn p>allel to !m1 ! :ole %ip w be appli$ to ! /rait l9e ,h ,n4 ,also at ! end ( ! second "t ,e ,g1 x w be appli$ to ! /rait l9e ,k ,o4 ,:]e=e ! %ip w always g =w>d2 & ! angle x makes ) ! w9d w be equal to ! angle ,a ,b ,c1 h[ small soev] t angle be2 & ! way x makes w 9 e "t be equal to ! /rait l9e ,e ,h4 ,i say1 ?us x w\ld be1 if ! %ip might be mov$ ) z great cel]ity sideways f ,b ,a t[>ds ,k ,o1 z x may be mov$ =w>ds 9 ! l9e ,b ,a4 ,b ?is is impossible1 by reason ( ! resi/ance made by ! great quantity ( wat] :i* presse? ! side1 mu* exce$+ ! resi/ance made by ! mu* small] quantity :i* presse? ! pr[ ( ! %ip2 s t ! way ! %ip makes sideways is sc>ce s5sible2 &1 !re=e1 ! po9t ,b w proce$ almo/ 9 ! v l9e ,b ,a1 mak+ ) ! w9d ! angle ,a ,b ,c1 h[ acute soev]2 t is to say1 x w proce$ almo/ 9 ! /rait l9e ,b ,c1 t is1 9 a way almo/ contr>y to ! way ( ! mov5t2 :i* 0 to be demon/rat$4 ,b ! sail 9 ,b ,i mu/ be s /ret*$ z t !re be left 9 x no bosom at all2 = o!rwise ! /rait l9es ,l ,p1 ,m ,q & ,b ,k w n be p]p5dicul> to ! plane ( ! sail1 b fall+ bel[ ,p1 ,q & ,k1 w drive ! %ip backw>ds4 ,b by mak+ use ( a small bo>d = a sail1 a little waggon ) :eels = ! %ip1 & ( a smoo? pavem5t = ! sea1 ,i h by exp]i5ce f\nd ?is to be s true1 t ,i c\ld sc>ce oppose ! bo>d to ! w9d 9 any obliquity1 ?\gh nev] s small1 b ! waggon 0 c>ri$ =w>ds by x4 ,by ! same #f? !orem x may be f\nd1 h[ mu* a /roke1 :i* falls obliquely1 is weak] ?an a /roke fall+ p]p5dicul>ly1 !y be+ l & equal 9 all o!r respects4 ,let a /roke fall ^u ! wall ,a ,b obliquely1 z = example1 9 ! /rait l9e ,c ,a 79 fig4 #c47 ,let ,c ,e be drawn p>allel to ,a ,b1 & ,d ,a p]p5dicul> to ! same ,a ,b & equal to ,c ,a2 & let bo? ! velocity & "t ( ! motion 9 ,c ,a be equal to ! velocity & "t ( ! motion 9 ,d ,a4 ,i say1 ! /roke 9 ,c ,a w be weak] ?an t 9 ,d ,a1 9 ! proportion ( ,e ,a to ,d ,a4 ,= produc+ ,d ,a h[soev] to ,f1 ! endeav\r ( bo? ! /rokes w 7by art4 #f7 proce$ f ,a 9 ! p]p5dicul> ,a ,f4 ,b ! /roke 9 ,c ,a is made by ! conc\rse ( two motions 9 ,c ,e & ,e ,a1 ( :i* t 9 ,c ,e contributes no?+ to ! /roke 9 ,a1 because ,c ,e & ,b ,a are p>allels2 &1 !re=e1 ! /roke 9 ,c ,a is made by ! motion :i* is 9 ,e ,a only4 ,b ! velocity or =ce ( ! p]p5dicul> /roke 9 ,e ,a1 to ! velocity or =ce ( ! /roke 9 ,d ,a1 is z ,e ,a to ,d ,a4 ,:]e=e ! oblique /roke 9 ,c ,a is weak] ?an ! p]p5dicul> /roke 9 ,d ,a1 9 ! proportion ( ,e ,a to ,d ,a or ,c ,a2 :i* 0 to be prov$4 ,sid5ote3 ,9 a full m$ium1 motion is propagat$ to any di/ance4 #i4 ,9 a full m$ium1 all endeav\r proce$s z f> z ! m$ium itself rea*e?2 t is to say1 if ! m$ium be inf9ite1 ! endeav\r w proce$ inf9itely4 ,= :atsoev] endeav\re? is mov$1 & !re=e :atsoev] /&e? 9 its way x make? x yield1 at lea/ a little1 namely1 s f> z ! mov5t itself is mov$ =w>ds4 ,b t :i* yielde? is also mov$1 & consequ5tly make? t to yield :i* is 9 its way1 & s on successively z long z ! m$ium is full2 t is to say1 inf9itely1 if ! full m$ium be inf9ite2 :i* 0 to be prov$4 ,n[ al?\gh endeav\r ?us p]petually propagat$ d n always appe> to ! s5ses z motion1 yet x appe>s z action1 or z ! effici5t cause ( "s mutation4 ,= if !re be plac$ be=e our eyes "s v little object1 z = example1 a small gra9 ( s&1 :i* at a c]ta9 di/ance is visible2 x is manife/ t x may be remov$ to su* a di/ance z n to be any long] se51 ?\gh by its action x /ill "w ^u ! organs ( sight1 z is manife/ f t :i* 0 la/ prov$1 t all endeav\r proce$s inf9itely4 ,let x be conceiv$ !re=e to be remov$ f our eyes to any di/ance h[ great soev]1 & a suffici5t numb] ( o!r gra9s ( s& ( ! same bigness add$ to x2 x is evid5t t ! aggregate ( all ^? s&s w be visible2 & ?\gh none ( !m c be se5 :5 x is s+le & sev]$ f ! re/1 yet ! :ole heap or hill :i* !y make w manife/ly appe> to ! sight2 :i* w\ld be impossible1 if "s action did n proce$ f ea* sev]al "p ( ! :ole heap4 ,sid5ote3 ,dilatation contraction :at !y are4 #aj4 ,betwe5 ! degrees ( h>d & s(t are ^? ?+s :i* we call t\gh1 t\gh be+ t :i* may be b5t )\t be+ alt]$ f :at x 02 & ! b5d+ ( a l9e is ei!r ! adduction or diduction ( ! extreme p>ts1 t is1 a motion f /raitness to crook$ness1 or contr>ily1 :il/ ! l9e rema9s /ill ! same x 02 = by draw+ out ! extreme po9ts ( a l9e to _! greate/ di/ance1 ! l9e is made /rait1 :i* o!rwise is crook$4 ,s also ! b5d+ ( a sup]ficies is ! diduction or adduction ( its extreme l9es1 t is1 _! dilatation & contraction4 ,sid5ote3 ,dilatation contraction suppose mutation ( ! smalle/ p>ts 9 respect ( _! situation4 #aa4 ,dilatation & contraction1 z also all flexion1 supposes necess>ily t ! int]nal p>ts ( ! body b[$ d ei!r come ne>] to ! ext]nal p>ts1 or g fur!r f !m4 ,= ?\gh flexion be consid]$ only 9 ! l5g? ( a body1 yet :5 t body is b[$1 ! l9e :i* is made on "o side w be convex1 & ! l9e on ! o!r side w be concave2 ( :i* ! concave1 be+ ! int]ior l9e1 w1 unless some?+ be tak5 f x & add$ to ! convex l9e1 be ! m crook$1 t is1 ! great] ( ! two4 ,b !y are equal2 &1 !re=e1 9 flexion !re is an accession made f ! int]ior to ! ext]ior p>ts2 &1 on ! contr>y1 9 t5sion1 f ! ext]ior to ! int]ior p>ts4 ,& z = ^? ?+s :i* d n easily suff] su* transposition ( _! p>ts1 !y are call$ brittle2 & ! great =ce !y require to make !m yield1 makes !m also ) sudd5 motion to leap asund]1 & break 9 pieces4 ,sid5ote3 ,all traction is pulsion4 #ab4 ,also motion is di/+ui%$ into pulsion & traction4 ,& pulsion1 z ,i h already def9$ x1 is :5 t :i* is mov$ goes be=e t :i* move? x4 ,b contr>ily1 9 traction ! mov5t goes be=e t :i* is mov$4 ,nev]!less1 consid]+ x ) great] att5tion1 x seeme? to be ! same ) pulsion4 ,= ( two p>ts ( a h>d body1 :5 t :i* is =emo/ drives be=e x ! m$ium 9 :i* ! motion is made1 at ! same "t t :i* is ?ru/ =w>ds ?ru/e? ! next1 & ?is aga9 ! next1 & s on successively4 ,9 :i* action1 if we suppose t !re is no place void1 x mu/ ne$s be1 t by cont9ual pulsion1 namely1 :5 t action has gone r\nd1 ! mov5t w be beh9d t "p1 :i* at ! fir/ seem$ n to be ?ru/ =w>ds1 b to be drawn2 s t n[ ! body1 :i* 0 drawn1 goes be=e ! body :i* gives x motion2 & its motion is no long] traction1 b pulsion4 ,sid5ote3 ,su* ?+s z be+ press$ or b5t re/ore !mselves1 h motion 9 _! int]nal p>ts4 #ac4 ,su* ?+s z are remov$ f _! places by =cible compression or ext5sion1 &1 z soon z ! =ce is tak5 away1 d pres5tly return & re/ore !mselves to _! =m] situation1 h ! beg9n+ ( _! re/itution )9 !mselves1 namely1 a c]ta9 motion 9 _! int]nal p>ts1 :i* 0 !re1 :51 be=e ! tak+ away ( ! =ce1 !y 7 compress$1 or ext5d$4 ,= t re/itution is motion1 & t :i* is at re/ _c be mov$1 b by a mov$ & a contigu\s mov5t4 ,nor do? ! cause ( _! re/itution proce$ f ! tak+ away ( ! =ce by :i* !y 7 compress$ or ext5d$2 = ! remov+ ( imp$im5ts ha? n ! efficacy ( a cause1 z has be5 %[n at ! end ( ! #crd article ( *ap4 ,,xv4 ,! cause !re=e ( _! re/itution is "s motion ei!r ( ! p>ts ( ! ambi5t1 or ( ! p>ts ( ! body compress$ or ext5d$4 ,b ! p>ts ( ! ambi5t h no endeav\r :i* contributes to _! compression or ext5sion1 nor to ! sett+ ( !m at lib]ty1 or re/itution4 ,x rema9s !re=e t f ! "t ( _! compression or ext5sion !re be left "s endeav\r or motion1 by :i*1 ! imp$im5t be+ remov$1 e "p resumes its =m] place2 t is to say1 ! :ole re/ores itself4 ,sid5ote3 ,?\gh t :i* c>rie? ano!r be /opp$1 ! body c>ri$ w proce$4 #ad4 ,9 ! c>riage ( bodies1 if t body1 :i* c>ries ano!r1 hit ^u any ob/acle1 or be by any means sudd5ly /opp$1 & t :i* is c>ri$ be n /opp$1 x w g on1 till its motion be by "s ext]nal imp$im5t tak5 away4 ,= ,i h demon/rat$ 7*ap4 ,,viii1 art4 #ai7 t motion1 unless x be h9d]$ by "s ext]nal resi/ance1 w be cont9u$ et]nally ) ! same cel]ity2 & 9 ! #g? article ( *ap4 ,,ix1 t ! action ( an ext]nal ag5t is ( no effect )\t contact4 ,:5 !re=e t1 :i* c>rie? ano!r ?+1 is /opp$1 t /op do? n pres5tly take away ! motion ( t :i* is c>ri$4 ,x w !re=e proce$1 till its motion be by little & little ext+ui%$ by "s ext]nal resi/ance3 :i* 0 to be prov$2 ?\gh exp]i5ce alone _h be5 suffici5t to prove ?is4 ,9 l mann]1 if t body :i* c>rie? ano!r be put f re/ into sudd5 motion1 t :i* is c>ri$ w n be mov$ =w>ds toge!r ) x1 b w be left beh9d4 ,= ! contigu\s "p ( ! body c>ri$ ha? almo/ ! same motion ) ! body :i* c>ries x2 & ! remote p>ts w receive diff]5t velocities accord+ to _! diff]5t di/ances f ! body t c>ries !m2 namely1 ! m remote ! p>ts are1 ! less w be _! degrees ( velocity4 ,x is necess>y1 !re=e1 t ! body1 :i* is c>ri$1 be left accord+ly m or less beh9d4 ,& ?is also is manife/ by exp]i5ce1 :5 at ! />t+ =w>d ( ! horse ! rid] falle? backw>ds4 ,sid5ote3 ,! effects ( p]cussion n to be comp>$ ) ^? ( weight4 #ae4 ,9 p]cussion1 !re=e1 :5 "o h>d body is 9 "s small "p ( x /rick5 by ano!r ) great =ce1 x is n necess>y t ! :ole body %\ld yield to ! /roke ) ! same cel]ity ) :i* ! /rick5 "p yields4 ,= ! re/ ( ! p>ts receive _! motion f ! motion ( ! "p /rick5 & yield+1 :i* motion is less propagat$ e way t[>ds ! sides1 ?an x is directly =w>ds4 ,& h5ce x is1 t sometimes v h>d bodies1 :i* be+ erect$ c h>dly be made to /&1 are m easily brok5 ?an ?r[n d[n by a viol5t /roke2 :51 nev]!less1 if all _! p>ts toge!r 7 by any weak motion ?ru/ =w>ds1 !y w\ld easily be ca/ d[n4 #af4 ,?\gh ! diff]5ce betwe5 trusion & p]cussion consi/ only 9 ?is1 t 9 trusion ! motion bo? ( ! mov5t & mov$ body beg9 bo? toge!r 9 _! v contact2 & 9 p]cussion ! /rik+ body is fir/ mov$1 & aft]w>ds ! body /rick52 yet _! effects are s diff]5t1 t x seems sc>ce possible to comp>e _! =ces ) "o ano!r4 ,i say1 any effect ( p]cussion be+ prop\nd$1 z = example1 ! /roke ( a beetle ( any weight assign$1 by :i* a pile ( any giv5 l5g? is to be driv5 into e>? ( any t5acity giv51 x seems to me v h>d1 if n impossible1 to def9e ) :at weight1 or ) :at /roke1 & 9 :at "t1 ! same pile may be driv5 to a dep? assign$ into ! same e>?4 ,! cause ( :i* difficulty is ?is1 t ! velocity ( ! p]cuti5t is to be comp>$ ) ! magnitude ( ! pond]ant4 ,n[ velocity1 see+ x is comput$ by ! l5g? ( space transmitt$1 is to be acc\nt$ b z "o dim5sion2 b weight is z a solid ?+1 be+ measur$ by ! dim5sion ( ! :ole body4 ,& !re is no comp>ison to be made ( a solid body ) a l5g?1 t is1 ) a l9e4 ,sid5ote3 ,motion _c beg9 fir/ 9 ! int]nal p>ts ( a body4 #ag4 ,if ! int]nal p>ts ( a body be at re/1 or reta9 ! same situation ) "o ano!r = any "t h[ little soev]1 !re _c 9 ^? p>ts be g5]at$ any new motion or endeav\r1 :]e( ! effici5t cause is n )\t ! body ( :i* !y are p>ts4 ,= if any small "p1 :i* is compreh5d$ )9 ! sup]ficies ( ! :ole body1 be suppos$ to be n[ at re/1 & by & by to be mov$1 t "p mu/ ( necessity receive its motion f "s mov$ & contigu\s body4 ,b by supposition1 !re is no su* mov$ & contigu\s "p )9 ! body4 ,:]e=e1 if !re be any endeav\r or motion or *ange ( situation 9 ! int]nal p>ts ( t body1 x mu/ ne$s arise f "s effici5t cause t is )\t ! body :i* conta9s !m2 :i* 0 to be prov$4 #ah4 ,9 h>d bodies1 !re=e1 :i* are compress$ or ext5d$1 if1 t :i* compresse? or ext5de? !m be+ tak5 away1 !y re/ore !mselves to _! =m] place or situation1 x mu/ ne$s be t t endeav\r or motion ( _! int]nal p>ts1 by :i* !y 7 able to recov] _! =m] places or situations1 0 n ext+ui%$ :5 ! =ce by :i* !y 7 compress$ or ext5d$ 0 tak5 away4 ,!re=e1 :5 ! la? ( a cross-b[ b5t do?1 z soon z x is at lib]ty1 re/ore itself1 ?\gh to him1 t judges by s5se1 bo? x & all its p>ts seem to be at re/2 yet he1 t judg+ by reason do? n acc\nt ! tak+ away ( imp$im5t = an effici5t cause1 nor conceives t )\t an effici5t cause any ?+ c pass f re/ to motion1 w conclude t ! p>ts 7 already 9 motion be=e !y began to re/ore !mselves4 ,sid5ote3 ,action & reaction proce$ 9 ! same l9e4 #ai4 ,action & reaction proce$ 9 ! same l9e1 b f opposite t]ms4 ,= see+ reaction is no?+ b endeav\r 9 ! pati5t to re/ore itself to t situation f :i* x 0 =c$ by ! ag5t2 ! endeav\r or motion bo? ( ! ag5t & pati5t or reag5t w be propagat$ betwe5 ! same t]ms2 yet s1 z t 9 action ! t]m1 f :i*1 is 9 reaction ! t]m to :i*4 ,& see+ all action proce$s 9 ?is mann]1 n only betwe5 ! opposite t]ms ( ! :ole l9e 9 :i* x is propagat$1 b also 9 all ! p>ts ( t l9e1 ! t]ms f :i* & to :i*1 bo? ( ! action & reaction1 w be 9 ! same l9e4 ,:]e=e action & reaction proce$ 9 ! same l9e1 c4 ,sid5ote3 ,habit1 :at x is4 #bj4 ,to :at has be5 said ( motion1 ,i w add :at ,i h to say conc]n+ habit4 ,habit1 !re=e1 is a g5]ation ( motion1 n ( motion simply1 b an easy conduct+ ( ! mov$ body 9 a c]ta9 & design$ way4 ,& see+ x is atta9$ by ! weak5+ ( su* endeav\rs z div]t its motion1 !re=e su* endeav\rs are to be weak5$ by little & little4 ,b ?is _c be done b by ! long cont9uance ( action1 or by actions (t5 repeat$2 & !re=e cu/om begets t facility1 :i* is commonly & rightly call$ habit2 & x may be def9$ ?us3 ,,habit is motion made m easy & ready by cu/om2 t is to say1 by p]petual endeav\r1 or by it]at$ endeav\rs 9 a way diff]+ f t 9 :i* ! motion proce$$ f ! beg9n+1 & oppos+ su* endeav\rs z resi/4 ,& to make ?is m p]spicu\s by example1 we may obs]ve1 t :5 "o t has no skill 9 music fir/ puts 8 h& to an in/rum5t1 he _c aft] ! fir/ /roke c>ry 8 h& to ! place :]e he w\ld make ! second /roke1 )\t tak+ x back by a new endeav\r1 &1 z x 7 beg9n+ aga91 pass f ! fir/ to ! second4 ,nor w he be able to g on to ! ?ird place )\t ano!r new endeav\r2 b he w be =c$ to draw back 8 h& aga91 & s successively1 by r5ew+ 8 endeav\r at e /roke2 till at ! la/1 by do+ ?is (t51 & by comp\nd+ _m int]rupt$ motions or endeav\rs into "o equal endeav\r1 he be able to make 8 h& g readily on f /roke to /roke 9 t ord] & way :i* 0 at ! fir/ design$4 ,nor are habits to be obs]v$ 9 liv+ creatures only1 b also 9 bodies inanimate4 ,= we f9d t :5 ! la? ( a cross-b[ is /rongly b5t1 & w\ld if ! imp$im5t 7 remov$ return aga9 ) great =ce2 if x rema9 a long "t b5t1 x w get su* a habit1 t :5 x is loos$ & left to its own fre$om1 x w n only n re/ore itself1 b w require z mu* =ce = ! br++ ( x back to its fir/ po/ure1 z x did = ! b5d+ ( x at ! fir/4 ---------------------------------------- -------------------------------- ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xxii4 ,fig4 #a-c ---------------------------------------- -------------------------------- ------- ,,*ap4 ,,xxiii4 ,,( ,,! ,,c5tre ,,( ,,equipond]ation2 ,,( ,,bodies ,,press+ ,,d[nw>ds ,,9 ,,/rait ,,p>allel ,,l9es4 #a4 ,def9itions & suppositions4--#b4 ,two planes ( equipond]ation are n p>allel4--#c4 ,! c5tre ( equipond]ation is 9 e plane ( equipond]ation4--#d4 ,! mom5ts ( equal pond]ants are to "o ano!r z _! di/ances f ! c5tre ( ! scale4--#e1 #f4 ,! mom5ts ( unequal pond]ants h _! proportion to "o ano!r comp\nd$ ( ! proportions ( _! weights & di/ances f ! c5tre ( ! scale4--#g4 ,if two pond]ants h _! weights & di/ances f ! c5tre ( ! scale 9 reciprocal proportion1 !y are equally pois$2 & contr>ily4--#h4 ,if ! p>ts ( any pond]ant press ! beams ( ! scale e :]e equally1 all ! p>ts cut (f1 reckon$ f ! c5tre ( ! scale1 w h _! mom5ts 9 ! same proportion ) t ( ! p>ts ( a triangle cut (f f ! v]tex by /rait l9es p>allel to ! base4--#i4 ,! diamet] ( equipond]ation ( figures1 :i* are defici5t accord+ to comm5surable proportions ( _! altitudes & bases1 divides ! axis1 s t ! "p tak5 next ! v]tex is to ! o!r "p ( ! complete figure to ! defici5t figure4--#aj4 ,! diamet] ( equipond]ation ( ! complem5t ( ! half ( any ( ! said defici5t figures1 divides t l9e :i* is drawn ?r\gh ! v]tex p>allel to ! base1 s t ! "p next ! v]tex is to ! o!r "p z ! complete figure to ! complem5t4--#aa4 ,! c5tre ( equipond]ation ( ! half ( any ( ! defici5t figures 9 ! fir/ r[ ( ! table ( art4 #c1 *ap4 ,,xvii1 may be f\nd out by ! numb]s ( ! second r[4--#ab4 ,! c5tre ( equipond]ation ( ! half ( any ( ! figures ( ! second r[ ( ! same table1 may be f\nd out by ! numb]s ( ! f\r? r[4--#ac4 ,! c5tre ( equipond]ation ( ! half ( any ( ! figures 9 ! same table be+ kn[n1 ! c5tre ( ! excess ( ! same figure above a triangle ( ! same altitude & base is also kn[n4--#ad4 ,! c5tre ( equipond]ation ( a solid sector is 9 ! axis s divid$1 t ! "p next ! v]tex be to ! :ole axis1 want+ half ! axis ( ! portion ( ! sph]e1 z #c to #d4 ,,def9itions4 ,sid5ote3 ,def9itions4 ,i4 ,a scale is a /rait l9e1 ^: middle po9t is immovable1 all ! re/ ( its po9ts be+ at lib]ty2 & t "p ( ! scale1 :i* rea*es f ! c5tre to ei!r ( ! weights1 is call$ ! beam4 ,,ii4 ,equipond]ation is :5 ! endeav\r ( "o body1 :i* presses "o ( ! beams1 resi/s ! endeav\r ( ano!r body press+ ! o!r beam1 s t nei!r ( !m is mov$2 & ! bodies1 :5 nei!r ( !m is mov$1 are said to be equally pois$4 ,,iii4 ,weight is ! aggregate ( all ! endeav\rs1 by :i* all ! po9ts ( t body1 :i* presses ! beam1 t5d d[nw>ds 9 l9es p>allel to "o ano!r2 & ! body :i* presses is call$ ! pond]ant4 ,,iv4 ,mom5t is ! p[] :i* ! pond]ant has to move ! beam1 by reason ( a det]m9$ situation4 ,v4 ,! plane ( equipond]ation is t by :i* ! pond]ant is s divid$1 t ! mom5ts on bo? sides rema9 equal4 ,,vi4 ,! diamet] ( equipond]ation is ! common section ( ! two planes ( equipond]ation1 & is 9 ! /rait l9e by :i* ! weight is hang$4 ,,vii4 ,! c5tre ( equipond]ation is ! common po9t ( ! two diamet]s ( equipond]ation4 ,,suppositions4 ,sid5ote3 ,suppositions4 ,i4 ,:5 two bodies are equally pois$1 if weight be add$ to "o ( !m & n to ! o!r1 _! equipond]ation ceases4 ,,ii4 ,:5 two pond]ants ( equal magnitude1 & ( ! same species or matt]1 press ! beam on bo? sides at equal di/ances f ! c5tre ( ! scale1 _! mom5ts are equal4 ,also :5 two bodies endeav\r at equal di/ances f ! c5tre ( ! scale1 if !y be ( equal magnitude & ( ! same species1 _! mom5ts are equal4 ,sid5ote3 ,two planes ( equipond]ation are n p>allel4 #b4 ,no two planes ( equipond]ation are p>allel4 ,let ,a ,b ,c ,d 79 fig4 #a7 be any pond]ant :atsoev]2 & 9 x let ,e ,f be a plane ( equipond]ation2 p>allel to :i*1 let any o!r plane be drawn1 z ,g ,h4 ,i say1 ,g ,h is n a plane ( equipond]ation4 ,= see+ ! p>ts ,a ,e ,f ,d & ,e ,b ,c ,f ( ! pond]ant ,a ,b ,c ,d are equally pois$2 & ! weight ,e ,g ,h ,f is add$ to ! "p ,a ,e ,f ,d1 & no?+ is add$ to ! "p ,e ,b ,c ,f1 b ! weight ,e ,g ,h ,f is tak5 f x2 !re=e1 by ! fir/ supposition1 ! p>ts ,a ,g ,h ,d & ,g ,b ,c ,h w n be equally pois$2 & consequ5tly ,g ,h is n a plane ( equipond]ation4 ,:]e=e1 no two planes ( equipond]ation are p>allel2 :i* 0 to be prov$4 ,sid5ote3 ,! c5tre ( equipond]ation is 9 e plane ( equipond]ation4 #c4 ,! c5tre ( equipond]ation is 9 e plane ( equipond]ation4 ,= if ano!r plane ( equipond]ation be tak51 x w n1 by ! la/ article1 be p>allel to ! =m] plane2 & !re=e bo? ^? planes w cut "o ano!r4 ,n[ t section 7by ! #f? def9ition7 is ! diamet] ( equipond]ation4 ,aga91 if ano!r diamet] ( equipond]ation be tak51 x w cut t =m] diamet]2 & 9 t section 7by ! #g? def9ition7 is ! c5tre ( equipond]ation4 ,:]e=e ! c5tre ( equipond]ation is 9 t diamet] :i* lies 9 ! said plane ( equipond]ation4 ,sid5ote3 ,! mom5ts ( equal pond]ants are to "o ano!r z _! di/ances f ! c5tre ( ! scale4 #d4 ,! mom5t ( any pond]ant appli$ to "o po9t ( ! beam1 to ! mom5t ( ! same or an equal pond]ant appli$ to any o!r po9t ( ! beam1 is z ! di/ance ( ! =m] po9t f ! c5tre ( ! scale1 to ! di/ance ( ! latt] po9t f ! same c5tre4 ,or ?us1 ^? mom5ts are to "o ano!r1 z ! ar*es ( circles :i* are made ^u ! c5tre ( ! scale ?r\gh ^? po9ts1 9 ! same "t4 ,or la/ly ?us1 !y are z ! p>allel bases ( two triangles1 :i* h a common angle at ! c5tre ( ! scale4 ,let ,a 79 fig4 #b7 be ! c5tre ( ! scale2 & let ! equal pond]ants ,d & ,e press ! beam ,a ,b 9 ! po9ts ,b & ,c2 also let ! /rait l9es ,b ,d & ,c ,e be diamet]s ( equipond]ation2 & ! po9ts ,d & ,e 9 ! pond]ants ,d & ,e be _! c5tres ( equipond]ation4 ,let ,a ,g ,f be drawn h[soev]1 cutt+ ,d ,b produc$ 9 ,f1 & ,e ,c 9 ,g2 & la/ly1 ^u ! common c5tre ,a1 let ! two ar*es ,b ,h & ,c ,i be describ$1 cutt+ ,a ,g ,f 9 ,h & ,i4 ,i say1 ! mom5t ( ! pond]ant ,d to ! mom5t ( ! pond]ant ,e is z ,a ,b to ,a ,c1 or z ,b ,h to ,c ,i1 or z ,b ,f to ,c ,g4 ,= ! effect ( ! pond]ant ,d1 9 ! po9t ,b1 is circul> motion 9 ! ar* ,b ,h2 & ! effect ( ! pond]ant ,e1 9 ! po9t ,c1 circul> motion 9 ! ar* ,c ,i2 & by reason ( ! equality ( ! pond]ants ,d & ,e1 ^! motions are to "o ano!r z ! quicknesses or velocities ) :i* ! po9ts ,b & ,c describe ! ar*es ,b ,h & ,c ,i1 t is1 z ! ar*es !mselves ,b ,h & ,c ,i1 or z ! /rait p>allels ,b ,f & ,c ,g1 or z ! p>ts ( ! beam ,a ,b & ,a ,c2 = ,a ,b4 ,a ,c33 ,b ,f4 ,c ,g33 ,b ,h4 ,c ,i4 are proportionals2 & !re=e ! effects1 t is1 by ! #d? def9ition1 ! mom5ts ( ! equal pond]ants appli$ to sev]al po9ts ( ! beam1 are to "o ano!r z ,a ,b & ,a ,c2 or z ! di/ances ( ^? po9ts f ! c5tre ( ! scale2 or z ! p>allel bases ( ! triangles :i* h a common angle at ,a2 or z ! conc5tric ar*es ,b ,h & ,c ,i2 :i* 0 to be demon/rat$4 ,sid5ote3 ,! mom5ts ( unequal pond]ants h _! proportion to "o ano!r comp\nd$ ( ! proportions ( _! weights & di/ances f ! c5tre ( ! scale4 #e4 ,unequal pond]ants1 :5 !y are appli$ to sev]al po9ts ( ! beam1 & hang at lib]ty1 t is1 s z ! l9e by :i* !y hang be ! diamet] ( equipond]ation1 :atsoev] be ! figure ( ! pond]ant1 h _! mom5ts to "o ano!r 9 proportion comp\nd$ ( ! proportions ( _! di/ances f ! c5tre ( ! scale1 & ( _! weights4 ,let ,a 79 fig4 #c7 be ! c5tre ( ! scale1 & ,a ,b ! beam2 to :i* let ! two pond]ants ,c & ,d be appli$ at ! po9ts ,b & ,e4 ,i say1 ! proportion ( ! mom5t ( ! pond]ant ,c to ! mom5t ( ! pond]ant ,d1 is comp\nd$ ( ! proportions ( ,a ,b to ,a ,e1 & ( ! weight ,c to ! weight ,d2 or1 if ,c & ,d be ( ! same species1 ( ! magnitude ,c to ! magnitude ,d4 ,let ei!r ( !m1 z ,c1 be suppos$ to be bigg] ?an ! o!r1 ,d4 ,if1 !re=e1 by ! addition ( ,f1 ,f & ,d toge!r be z "o body equal to ,c1 ! mom5t ( ,c to ! mom5t ( ,f ,d w be 7by ! la/ article7 z ,b ,g is to ,e ,h4 ,n[ z ,f ,d is to ,d1 s let ,e ,h be to ano!r ,e ,i2 & ! mom5t ( ,f ,d1 t is ( ,c1 to ! mom5t ( ,d1 w be z ,b ,g to ,e ,i4 ,b ! proportion ( ,b ,g to ,e ,i is comp\nd$ ( ! proportions ( ,b ,g to ,e ,h1 t is1 ( ,a ,b to ,a ,e1 & ( ,e ,h to ,e ,i1 t is1 ( ! weight ,c to ! weight ,d4 ,:]e=e unequal pond]ants1 :5 !y are appli$1 c4 ,:i* 0 to be prov$4 #f4 ,! same figure rema9+1 if ,i ,k be drawn p>allel to ! beam ,a ,b1 & cutt+ ,a ,g 9 ,k2 & ,k ,l be drawn p>allel to ,b ,g1 cutt+ ,a ,b 9 ,l1 ! di/ances ,a ,b & ,a ,l f ! c5tre w be proportional to ! mom5ts ( ,c & ,d4 ,= ! mom5t ( ,c is ,b ,g1 & ! mom5t ( ,d is ,e ,i1 to :i* ,k ,l is equal4 ,b z ! di/ance ,a ,b f ! c5tre is to ! di/ance ,a ,l f ! c5tre1 s is ,b ,g1 ! mom5t ( ! pond]ant ,c1 to ,l ,k1 or ,e ,i ! mom5t ( ! pond]ant ,d4 ,sid5ote3 ,if two pond]ants h _! weights & di/ances f ! c5tre ( ! scale 9 reciprocal proportion1 !y are equally pois$2 & contr>ily4 #g4 ,if two pond]ants h _! weights & di/ances f ! c5tre 9 reciprocal proportion1 & ! c5tre ( ! scale be betwe5 ! po9ts to :i* ! pond]ants are appli$1 !y w be equally pois$4 ,& contr>ily1 if !y be equally pois$1 _! weights & di/ances f ! c5tre ( ! scale w be 9 reciprocal proportion4 ,let ! c5tre ( ! scale 79 ! same ?ird figure7 be ,a1 ! beam ,a ,b2 & let any pond]ant ,c1 hav+ ,b ,g = its mom5t1 be appli$ to ! po9t ,b2 also let any o!r pond]ant ,d1 ^: mom5t is ,e ,i1 be appli$ to ! po9t ,e4 ,?r\gh ! po9t ,i let ,i ,k be drawn p>allel to ! beam ,a ,b1 cutt+ ,a ,g 9 ,k2 also let ,k ,l be drawn p>allel to ,b ,g1 ,k ,l w !n be ! mom5t ( ! pond]ant ,d2 & by ! la/ article1 x w be z ,b ,g1 ! mom5t ( ! pond]ant ,c 9 ! po9t ,b1 to ,l ,k ! mom5t ( ! pond]ant ,d 9 ! po9t ,e1 s ,a ,b to ,a ,l4 ,on ! o!r side ( ! c5tre ( ! scale1 let ,a ,n be tak5 equal to ,a ,l2 & to ! po9t ,n let !re be appli$ ! pond]ant ,o1 hav+ to ! pond]ant ,c ! proportion ( ,a ,b to ,a ,n4 ,i say1 ! pond]ants 9 ,b & ,n w be equally pois$4 ,= ! proportion ( ! mom5t ( ! pond]ant ,o1 9 ! po9t ,n1 to ! mom5t ( ! pond]ant ,c 9 ! po9t ,b1 is by ! #e? article1 comp\nd$ ( ! proportions ( ! weight ,o to ! weight ,c1 & ( ! di/ance f ! c5tre ( ! scale ,a ,n or ,a ,l to ! di/ance f ! c5tre ( ! scale ,a ,b4 ,b see+ we h suppos$1 t ! di/ance ,a ,b to ! di/ance ,a ,n is 9 reciprocal proportion ( ! weight ,o to ! weight ,c1 ! proportion ( ! mom5t ( ! pond]ant ,o1 9 ! po9t ,n1 to ! mom5t ( ! pond]ant ,c1 9 ! po9t ,b1 w be comp\nd$ ( ! proportions ( ,a ,b to ,a ,n1 & ( ,a ,n to ,a ,b4 ,:]e=e1 sett+ 9 ord] ,a ,b1 ,a ,n1 ,a ,b1 ! mom5t ( ,o to ! mom5t ( ,c w be z ! fir/ to ! la/1 t is1 z ,a ,b to ,a ,b4 ,_! mom5ts !re=e are equal2 & consequ5tly ! plane :i* passes ?r\gh ,a w 7by ! fif? def9ition7 be a plane ( equipond]ation4 ,:]e=e !y w be equally pois$2 z 0 to be prov$4 ,n[ ! conv]se ( ?is is manife/4 ,= if !re be equipond]ation & ! proportion ( ! weights & di/ances be n reciprocal1 !n bo? ! weights w always h ! same mom5ts1 al?\gh "o ( !m h m weight add$ to x or its di/ance *ang$4 ,coroll4 ,:5 pond]ants are ( ! same species1 & _! mom5ts be equal2 _! magnitudes & di/ances f ! c5tre ( ! scale w be reciprocally proportional4 ,= 9 homog5e\s bodies1 x is z weight to weight1 s magnitude to magnitude4 ,sid5ote3 ,if ! p>ts ( any pond]ant press ! beams ( ! scale e :]e equally1 all ! p>ts cut (f1 reckon$ f c5tre ( ! scale1 w h _! mom5ts 9 ! same proportion ) t ( ! p>ts ( a triangle cut (f f ! v]tex by /rait l9es p>allel to ! base4 #h4 ,if to ! :ole l5g? ( ! beam !re be appli$ a p>allelogram1 or a p>allelopip$um1 or a prisma1 or a cyl9d]1 or ! sup]ficies ( a cyl9d]1 or ( a sph]e1 or ( any portion ( a sph]e or prisma2 ! p>ts ( any ( !m cut (f ) planes p>allel to ! base w h _! mom5ts 9 ! same proportion ) ! p>ts ( a triangle1 :i* has its v]tex 9 ! c5tre ( ! scale1 & = "o ( its sides ! beam itself1 :i* p>ts are cut (f by planes p>allel to ! base4 ,fir/1 let ! rectangl$ p>allelogram ,a ,b ,c ,d 79 figure #d7 be appli$ to ! :ole l5g? ( ! beam ,a ,b2 & produc+ ,c ,b h[soev] to ,e1 let ! triangle ,a ,b ,e be describ$4 ,let n[ any "p ( ! p>allelogram1 z ,a ,f1 be cut (f by ! plane ,f ,g1 p>allel to ! base ,c ,b2 & let ,f ,g be produc$ to ,a ,e 9 ! po9t ,h4 ,i say1 ! mom5t ( ! :ole ,a ,b ,c ,d to ! mom5t ( its "p ,a ,f1 is z ! triangle ,a ,b ,e to ! triangle ,a ,g ,h1 t is1 9 proportion duplicate to t ( ! di/ances f ! c5tre ( ! scale4 ,=1 ! p>allelogram ,a ,b ,c ,d be+ divid$ into equal p>ts1 inf9ite 9 numb]1 by /rait l9es drawn p>allel to ! base2 & suppos+ ! mom5t ( ! /rait l9e ,c ,b to be ,b ,e1 ! mom5t ( ! /rait l9e ,f ,g w 7by ! #g? article7 be ,g ,h2 & ! mom5ts ( all ! /rait l9es ( t p>allelogram w be s _m /rait l9es 9 ! triangle ,a ,b ,e drawn p>allel to ! base ,b ,e2 all :i* p>allels toge!r tak5 are ! mom5t ( ! :ole p>allelogram ,a ,b ,c ,d2 & ! same p>allels d also con/itute ! sup]ficies ( ! triangle ,a ,b ,e4 ,:]e=e ! mom5t ( ! p>allelogram ,a ,b ,c ,d is ! triangle ,a ,b ,e2 & = ! same reason1 ! mom5t ( ! p>allelogram ,a ,f is ! triangle ,a ,g ,h2 & !re=e ! mom5t ( ! :ole p>allelogram to ! mom5t ( a p>allelogram :i* is "p ( ! same1 is z ! triangle ,a ,b ,e to ! triangle ,a ,g ,h1 or 9 proportion duplicate to t ( ! beams to :i* !y are appli$4 ,& :at is "h demon/rat$ 9 ! case ( a p>allelogram may be und]/ood to s]ve = t ( a cyl9d]1 & ( a prisma1 & _! sup]ficies2 z also = ! sup]ficies ( a sph]e1 ( an hemisph]e1 or any portion ( a sph]e4 ,= ! p>ts ( ! sup]ficies ( a sph]e h ! same proportion ) t ( ! p>ts ( ! axis cut (f by ! same p>allels1 by :i* ! p>ts ( ! sup]ficies are cut (f1 z ,ar*im$es has demon/rat$2 & !re=e :5 ! p>ts ( any ( ^! figures are equal & at equal di/ances f ! c5tre ( ! scale1 _! mom5ts also are equal1 9 ! same mann] z !y are 9 p>allelograms4 ,secondly1 let ! p>allelogram ,a ,k ,i ,b n be rectangl$2 ! /rait l9e ,i ,b w nev]!less press ! po9t ,b p]p5dicul>ly 9 ! /rait l9e ,b ,e2 & ! /rait l9e ,l ,g w press ! po9t ,g p]p5dicul>ly 9 ! /rait l9e ,g ,h2 & all ! re/ ( ! /rait l9es :i* are p>allel to ,i ,b w d ! l4 ,:atsoev] !re=e ! mom5t be :i* is assign$ to ! /rait l9e ,i ,b1 z "h1 = example1 x is suppos$ to be ,b ,e1 if ,a ,e be drawn1 ! mom5t ( ! :ole p>allelogram ,a ,i w be ! triangle ,a ,b ,e2 & ! mom5t ( ! "p ,a ,l w be ! triangle ,a ,g ,h4 ,:]e=e ! mom5t ( any pond]ant1 :i* has its sides equally appli$ to ! beam1 :e!r !y be appli$ p]p5dicul>ly or obliquely1 w be always to ! mom5t ( a "p ( ! same 9 su* proportion z ! :ole triangle has to a "p ( ! same cut (f by a plane :i* is p>allel to ! base4 ,sid5ote3 ,! diamet] ( equipond]ation ( figures :i* are defici5t accord+ to comm5surable proportions ( _! altitudes & bases1 divides ! axis1 s t ! "p tak5 next ! v]tex is to ! o!r "p z ! complete figure to ! defici5t figure4 #i4 ,! c5tre ( equipond]ation ( any figure1 :i* is defici5t accord+ to comm5surable proportions ( ! altitude & base dim9i%$1 & ^: complete figure is ei!r a p>allelogram or a cyl9d]1 or a p>allelopip$um1 divides ! axis1 s1 t ! "p next ! v]tex1 to ! o!r "p1 is z ! complete figure to ! defici5t figure4 ,= let ,c ,i ,a ,p ,e 79 fig4 #e7 be a defici5t figure1 ^: axis is ,a ,b1 & ^: complete figure is ,c ,d ,f ,e2 & let ! axis ,a ,b be s divid$ 9 ,z1 t ,a ,z be to ,z ,b z ,c ,d ,f ,e is to ,c ,i ,a ,p ,e4 ,i say1 ! c5tre ( equipond]ation ( ! figure ,c ,i ,a ,p ,e w be 9 ! po9t ,z4 ,fir/1 t ! c5tre ( equipond]ation ( ! figure ,c ,i ,a ,p ,e is some:]e 9 ! axis ,a ,b is manife/ ( itself2 & !re=e ,a ,b is a diamet] ( equipond]ation4 ,let ,a ,e be drawn1 & let ,b ,e be put = ! mom5t ( ! /rait l9e ,c ,e2 ! triangle ,a ,b ,e w !re=e 7by ! ?ird article7 be ! mom5t ( ! complete figure ,c ,d ,f ,e4 ,let ! axis ,a ,b be equally divid$ 9 ,l1 & let ,g ,l ,h be drawn p>allel & equal to ! /rait l9e ,c ,e1 cutt+ ! crook$ l9e ,c ,i ,a ,p ,e 9 ,i & ,p1 & ! /rait l9es ,a ,c & ,a ,e 9 ,k & ,m4 ,moreov]1 let ,z ,o be drawn p>allel to ! same ,c ,e2 & let x be1 z ,l ,g to ,l ,i1 s ,l ,m to ano!r1 ,l ,n2 & let ! same be done 9 all ! re/ ( ! /rait l9es possible1 p>allel to ! base2 & ?r\gh all ! po9ts ,n1 let ! l9e ,a ,n ,e be drawn2 ! ?ree-sid$ figure ,a ,n ,e ,b w !re=e be ! mom5t ( ! figure ,c ,i ,a ,p ,e4 ,n[ ! triangle ,a ,b ,e is 7by ! #i? article ( *apt] ,,xvii7 to ! ?ree-sid$ figure ,a ,n ,e ,b1 z ,a ,b ,c ,d ,a ,i ,c ,b is to ,a ,i ,c ,b twice tak51 t is1 z ,c ,d ,f ,e ,c ,i ,a ,p ,e is to ,c ,i ,a ,p ,e twice tak54 ,b z ,c ,i ,a ,p ,e is to ,c ,d ,f ,e1 t is1 z ! weight ( ! defici5t figure is to ! weight ( ! complete figure1 s is ,c ,i ,a ,p ,e twice tak5 to ,c ,d ,f ,e twice tak54 ,:]e=e1 sett+ 9 ord] ,c ,d ,f ,e ,c ,i ,a ,p ,e4 #b ,c ,i ,a ,p ,e4 #b ,c ,d ,f ,e2 ! proportion ( ,c ,d ,f ,e ,c ,i ,a ,p ,e to ,c ,d ,f ,e twice tak5 w be comp\nd$ ( ! proportion ( ,c ,d ,f ,e ,c ,i ,a ,p ,e to ,c ,i ,a ,p ,e twice tak51 t is1 ( ! proportion ( ! triangle ,a ,b ,e to ! ?ree-sid$ figure ,a ,n ,e ,b1 t is1 ( ! mom5t ( ! complete figure to ! mom5t ( ! defici5t figure1 & ( ! proportion ( ,c ,i ,a ,p ,e twice tak5 to ,c ,d ,f ,e twice tak51 t is1 to ! proportion reciprocally tak5 ( ! weight ( ! defici5t figure to ! weight ( ! complete figure4 ,aga91 see+ by supposition ,a ,z4 ,z ,b33 ,c ,d ,f ,e4 ,c ,i ,a ,p ,e are proportionals2 ,a ,b4 ,a ,z33 ,c ,d ,f ,e ,c ,i ,a ,p ,e4 ,c ,d ,f ,e w also1 by comp\nd+1 be proportionals4 ,& see+ ,a ,l is ! half ( ,a ,b1 ,a ,l4 ,a ,z33 ,c ,d ,f ,e ,c ,i ,a ,p ,e4 #b ,c ,d ,f ,e w also be proportionals4 ,b ! proportion ( ,c ,d ,f ,e ,c ,i ,a ,p ,e to #b ,c ,d ,f ,e is comp\nd$1 z 0 b n[ %[n1 ( ! proportions ( mom5t to mom5t1 c41 & !re=e ! proportion ( ,a ,l to ,a ,z is comp\nd$ ( ! proportion ( ! mom5t ( ! complete figure ,c ,d ,f ,e to ! mom5t ( ! defici5t figure ,c ,i ,a ,p ,e1 & ( ! proportion ( ! weight ( ! defici5t figure ,c ,i ,a ,p ,e to ! weight ( ! complete figure ,c ,d ,f ,e2 b ! proportion ( ,a ,l to ,a ,z is comp\nd$ ( ! proportions ( ,a ,l to ,b ,z & ( ,b ,z to ,a ,z4 ,n[ ! proportion ( ,b ,z to ,a ,z is ! proportion ( ! weights reciprocally tak51 t is to say1 ( ! weight ,c ,i ,a ,p ,e to ! weight ,c ,d ,f ,e4 ,!re=e ! rema9+ proportion ( ,a ,l to ,b ,z1 t is1 ( ,l ,b to ,b ,z1 is ! proportion ( ! mom5t ( ! weight ,c ,d ,f ,e to ! mom5t ( ! weight ,c ,i ,a ,p ,e4 ,b ! proportion ( ,a ,l to ,b ,z is comp\nd$ ( ! proportions ( ,a ,l to ,a ,z & ( ,a ,z to ,z ,b2 ( :i* proportions t ( ,a ,z to ,z ,b is ! proportion ( ! weight ,c ,d ,f ,e to ! weight ,c ,i ,a ,p ,e4 ,:]e=e 7by art4 #e ( ?is *apt]7 ! rema9+ proportion ( ,a ,l to ,a ,z is ! proportion ( ! di/ances ( ! po9ts ,z & ,l f ! c5tre ( ! scale1 :i* is ,a4 ,&1 !re=e1 7by art4 #f7 ! weight ,c ,i ,a ,p ,e %all hang f ,o 9 ! /rait l9e ,o ,z4 ,s t ,o ,z is "o diamet] ( equipond]ation ( ! weight ,c ,i ,a ,p ,e4 ,b ! /rait l9e ,a ,b is ! o!r diamet] ( equipond]ation ( ! same weight ,c ,i ,a ,p ,e4 ,:]e=e 7by ! #g? def9ition7 ! po9t ,z is ! c5tre ( ! same equipond]ation2 :i* po9t1 by con/ruction1 divides ! axis s1 t ! "p ,a ,z1 :i* is ! "p next ! v]tex1 is to ! o!r "p ,z ,b1 z ! complete figure ,c ,d ,f ,e is to ! defici5t figure ,c ,i ,a ,p ,e2 :i* is t :i* 0 to be demon/rat$4 ,coroll4 ,i4 ,! c5tre ( equipond]ation ( any ( ^? plane ?ree-sid$ figures1 :i* are comp>$ ) _! complete figures 9 ! table ( art4 #c1 *ap4 ,,xvii1 is to be f\nd 9 ! same table1 by tak+ ! d5om9ator ( ! fraction = ! "p ( ! axis cut (f next ! v]tex1 & ! num]ator = ! o!r "p next ! base4 ,= example1 if x be requir$ to f9d ! c5tre ( equipond]ation ( ! second ?ree-sid$ figure ( f\r means1 !re is 9 ! conc\rse ( ! second column ) ! r[ ( ?ree-sid$ figures ( f\r means ?is fraction #eg1 :i* signifies t t figure is to its p>allelogram or complete figure z #eg to unity1 t is1 z #eg to #gg1 or z #e to #g2 &1 !re=e ! c5tre ( equipond]ation ( t figure divides ! axis1 s t ! "p next ! v]tex is to ! o!r "p z #g to #e4 ,coroll4 ,,ii4 ,! c5tre ( equipond]ation ( any ( ! solids ( ^? figures1 :i* are conta9$ 9 ! table ( art4 #g ( ! same *ap4 ,,xvii1 is exhibit$ 9 ! same table4 ,= example1 if ! c5tre ( equipond]ation ( a cone be s\ght =1 ! cone w be f\nd to be #ac ( its cyl9d]2 &1 !re=e1 ! c5tre ( its equipond]ation w s divide ! axis1 t ! "p next ! v]tex to ! o!r "p w be z #c to #a4 ,also ! solid ( a ?ree-sid$ figure ( "o mean1 t is1 a p>abolical solid1 see+ x is #bd1 t is ( its cyl9d]1 w h its c5tre ( equipond]ation 9 t po9t1 :i* divides ! axis1 s t ! "p t[>ds ! v]tex be d\ble to ! "p t[>ds ! base4 ,sid5ote3 ,! diamet] ( equipond]ation ( ! complem5t ( ! half ( any ( ! said defici5t figures1 divides t l9e :i* is drawn ?r\gh ! v]tex p>allel to ! base1 s t ! "p next ! v]tex is to ! o!r "p z ! complete figure to ! complem5t4 #aj4 ,! diamet] ( equipond]ation ( ! complem5t ( ! half ( any ( ^? figures :i* are conta9$ 9 ! table ( art4 #c1 *ap4 ,,xvii1 divides t l9e :i* is drawn ?r\gh ! v]tex p>allel & equal to ! base1 s t ! "p next ! v]tex w be to ! o!r "p1 z ! complete figure to ! complem5t4 ,= let ,a ,i ,c ,b 79 ! same fig4 #e7 be ! half ( a p>abola1 or ( any o!r ( ^? ?ree-sid$ figures :i* are 9 ! table ( art4 #c1 *ap4 ,,xvii1 ^: axis is ,a ,b1 & base ,b ,c1 hav+ ,a ,d drawn f ! v]tex1 equal & p>allel to ! base ,b ,c1 & ^: complete figure is ! p>allelogram ,a ,b ,c ,d4 ,let ,i ,q be drawn at any di/ance f ! side ,c ,d1 b p>allel to x2 & let ,a ,d be ! altitude ( ! complem5t ,a ,i ,c ,d1 & ,q ,i a l9e ord9ately appli$ 9 x4 ,:]e=e ! altitude ,a ,l 9 ! defici5t figure ,a ,i ,c ,b is equal to ,q ,i ! l9e ord9ately appli$ 9 its complem5t2 & contr>ily1 ,l ,i ! l9e ord9ately appli$ 9 ! figure ,a ,i ,c ,b is equal to ! altitude ,a ,q 9 its complem5t2 & s 9 all ! re/ ( ! ord9ate l9es & altitudes ! mutation is su*1 t t l9e1 :i* is ord9ately appli$ 9 ! figure1 is ! altitude ( its complem5t4 ,&1 !re=e1 ! proportion ( ! altitudes decreas+ to t ( ! ord9ate l9es decreas+1 be+ multiplicate accord+ to any numb] 9 ! defici5t figure1 is submultiplicate accord+ to ! same numb] 9 its complem5t4 ,= example1 if ,a ,i ,c ,b be a p>abola1 see+ ! proportion ( ,a ,b to ,a ,l is duplicate to t ( ,b ,c to ,l ,i1 ! proportion ( ,,ad to ,a ,q 9 ! complem5t ,a ,i ,c ,d1 :i* is ! same ) t ( ,b ,c to ,l ,i1 w be subduplicate to t ( ,c ,d to ,q ,i1 :i* is ! same ) t ( ,a ,b to ,a ,l2 & consequ5tly1 9 a p>abola1 ! complem5t w be to ! p>allelogram z #a to #c2 9 a ?ree-sid$ figure ( two means1 z #a to #d2 9 a ?ree-sid$ figure ( ?ree means1 z #a to #e1 c4 ,b all ! ord9ate l9es toge!r 9 ,a ,i ,c ,d are its mom5t2 & all ! ord9ate l9es 9 ,a ,i ,c ,b are its mom5t4 ,:]e=e ! mom5ts ( ! complem5ts ( ! halves ( defici5t figures 9 ! table ( art4 #c ( *ap4 ,,xvii1 be+ comp>$1 are z ! defici5t figures !mselves2 &1 !re=e1 ! diamet] ( equipond]ation w divide ! /rait l9e ,a ,d 9 su* proportion1 t ! "p next ! v]tex be to ! o!r "p1 z ! complete figure ,a ,b ,c ,d is to ! complem5t ,a ,i ,c ,d4 ,coroll4 ,! diamet] ( equipond]ation ( ^! halves may be f\nd by ! table ( art4 #c ( *ap4 ,,xvii1 9 ?is mann]4 ,let !re be prop\nd$ any defici5t figure1 namely1 ! second ?ree-sid$ figure ( two means4 ,?is figure is to ! complete figure z #ce to #a1 t is #c to #e4 ,:]e=e ! complem5t to ! same complete figure is z #b to #e2 &1 !re=e1 ! diamet] ( equipond]ation ( ?is complem5t w cut ! /rait l9e drawn f ! v]tex p>allel to ! base1 s t ! "p next ! v]tex w be to ! o!r "p z #e to #b4 ,&1 9 l mann]1 any o!r ( ! said ?ree-sid$ figures be+ prop\nd$1 if ! num]ator ( its fraction f\nd out 9 ! table be tak5 f ! d5om9ator1 ! /rait l9e drawn f ! v]tex is to be divid$1 s t ! "p next ! v]tex be to ! o!r "p1 z ! d5om9ator is to ! rema9d] :i* t subtraction leaves4 ,sid5ote3 ,! c5tre ( equipond]ation ( ! half ( any ( ! defici5t figures 9 ! fir/ r[ ( ! table ( art4 #c1 *apt] xvii1 may be f\nd out by ! numb]s ( ! second r[4 #aa4 ,! c5tre ( equipond]ation ( ! half ( any ( ^? crook$-l9$ figures1 :i* are 9 ! fir/ r[ ( ! table ( art4 #c ( *ap4 ,,xvii1 is 9 t /rait l9e :i*1 be+ p>allel to ! axis1 divides ! base accord+ to ! numb]s ( ! fraction next bel[ x 9 ! second r[1 s t ! num]ator be answ]able to t "p :i* is t[>ds ! axis4 ,= example1 let ! fir/ figure ( ?ree means be tak51 ^: half is ,a ,b ,c ,d 79 fig4 #f71 & let ! rectangle ,a ,b ,e ,d be complet$4 ,! complem5t !re=e w be ,b ,c ,d ,e4 ,& see+ ,a ,b ,e ,d is to ! figure ,a ,b ,c ,d 7by ! table7 z #e to #d1 ! same ,a ,b ,e ,d w be to ! complem5t ,b ,c ,d ,e z #e to #a4 ,:]e=e1 if ,f ,g be drawn p>allel to ! base ,d ,a1 cutt+ ! axis s t ,a ,g be to ,g ,b z #d to #e1 ! c5tre ( equipond]ation ( ! figure ,a ,b ,c ,d w1 by ! prec$5t article1 be some:]e 9 ! same ,f ,g4 ,aga91 see+1 by ! same article1 ! complete figure ,a ,b ,e ,d1 is to ! complem5t ,b ,c ,d ,e z #e to #a1 !re=e if ,b ,e & ,a ,d be divid$ 9 ,i & ,h z #e to #a ! c5tre ( equipond]ation ( ! complem5t ,b ,c ,d ,e w be some:]e 9 ! /rait l9e :i* connects ,h & ,i4 ,let n[ ! /rait l9e ,l ,k be drawn ?r\gh ,m ! c5tre ( ! complete figure1 p>allel to ! base2 & ! /rait l9e ,n ,o ?r\gh ! same c5tre ,m1 p]p5dicul> to x2 & let ! /rait l9es ,l ,k & ,f ,g cut ! /rait l9e ,h ,i 9 ,p & ,q4 ,let ,p ,r be tak5 quadruple to ,p ,q2 & let ,r ,m be drawn & produc$ to ,f ,g 9 ,s4 ,r ,m !re=e w be to ,m ,s z #d to #a1 t is1 z ! figure ,a ,b ,c ,d to its complem5t ,b ,c ,d ,e4 ,:]e=e1 see+ ,m is ! c5tre ( ! complete figure ,a ,b ,e ,d1 & ! di/ances ( ,r & ,s f ! c5tre ,m be 9 proportion reciprocal to t ( ! weight ( ! complem5t ,b ,c ,d ,e to ! weight ( ! figure ,a ,b ,c ,d1 ,r & ,s w ei!r be ! c5tres ( equipond]ation ( _! own figures1 or ^? c5tres w be 9 "s o!r po9ts ( ! diamet]s ( equipond]ation ,h ,i & ,f ,g4 ,b ?is la/ is impossible4 ,= no o!r /rait l9e c be drawn ?r\gh ! po9t ,m t]m9at+ 9 ! /rait l9es ,h ,i & ,f ,g1 & reta9+ ! proportion ( ,m ,r to ,m ,s1 t is1 ( ! figure ,a ,b ,c ,d to its complem5t ,b ,c ,d ,e4 ,! c5tre1 !re=e1 ( equipond]ation ( ! figure ,a ,b ,c ,d is 9 ! po9t ,s4 ,n[1 see+ ,p ,m ha? ! same proportion to ,q ,s :i* ,r ,p ha? to ,r ,q1 ,q ,s w be #e ( ^? p>ts ( :i* ,p ,m is f\r1 t is1 ( :i* ,i ,n is f\r4 ,b ,i ,n or ,p ,m is #b ( ^? p>ts ( :i* ,e ,b or ,f ,g is #f2 &1 !re=e1 if x be z #d to #e1 s #b to a f\r?1 t f\r? w be #b4 ,:]e=e ,q ,s is #b ( ^? p>ts ( :i* ,f ,g is #f4 ,b ,f ,q is #a2 &1 !re=e1 ,f ,s is #c4 ,:]e=e ! rema9+ "p ,g ,s is #b4 ,s t ,f ,g is s divid$ 9 ,s1 t ! "p t[>ds ! axis is 9 proportion to ! o!r "p1 z #b to #c1 t is z #e to #g2 :i* answ]e? to ! fraction #eg 9 ! second r[1 next "u ! fraction #de 9 ! fir/ r[4 ,:]e=e draw+ ,s ,t p>allel to ! axis1 ! base w be divid$ 9 l mann]4 ,by ?is me?od x is manife/1 t ! base ( a semip>abola w be divid$ into #c & #e2 & ! base ( ! fir/ ?ree-sid$ figure ( two means1 into #d & #f2 & ( ! fir/ ?ree-sid$ figure ( f\r means1 into #f & #h4 ,! fractions1 !re=e1 ( ! second r[ d5ote ! proportions1 into :i* ! bases ( ! figures ( ! fir/ r[ are divid$ by ! diamet]s ( equipond]ation4 ,b ! fir/ r[ beg9s "o place high] ?an ! second r[4 ,sid5ote3 ,! c5tre ( equipond]ation ( ! half ( any ( ! figures ( ! second r[ ( ! same table may be f\nd out by ! numb]s ( ! f\r? r[4 #ab4 ,! c5tre ( equipond]ation ( ! half ( any ( ! figures 9 ! second r[ ( ! same table ( art4 #c1 *ap4 ,,xvii1 is 9 a /rait l9e p>allel to ! axis1 & divid+ ! base accord+ to ! numb]s ( ! fraction 9 ! f\r? r[1 two places l[]1 s z t ! num]ator be answ]able to t "p :i* is next ! axis4 ,let ! half ( ! second ?ree-sid$ figure ( two means be tak52 & let x be ,a ,b ,c ,d 79 fig4 #g72 ^: complem5t is ,b ,c ,d ,e1 & ! rectangle complet$ ,a ,b ,e ,d4 ,let ?is rectangle be divid$ by ! two /rait l9es ,l ,k & ,n ,o1 cutt+ "o ano!r 9 ! c5tre ,m at "r angles2 & because ,a ,b ,e ,d is to ,a ,b ,c ,d z #e to #c1 let ,a ,b be divid$ 9 ,g1 s t ,a ,g to ,b ,g be z #c to #e2 & let ,f ,g be drawn p>allel to ! base4 ,also because ,a ,b ,e ,d is 7by art4 #i7 to ,b ,c ,d ,e z #e to #b1 let ,b ,e be divid$ 9 ! po9t ,i1 s t ,b ,i be to ,i ,e z #e to #b2 & let ,i ,h be drawn p>allel to ! axis1 cutt+ ,l ,k & ,f ,g 9 ,p & ,q4 ,let n[ ,p ,r be s tak51 t x be to ,p ,q z #c to #b1 & let ,r ,m be drawn & produc$ to ,f ,g 9 ,s4 ,see+1 !re=e1 ,r ,p is to ,p ,q1 t is1 ,r ,m to ,m ,s1 z ,a ,b ,c ,d is to its complem5t ,b ,c ,d ,e1 & ! c5tres ( equipond]ation ( ,a ,b ,c ,d & ,b ,c ,d ,e are 9 ! /rait l9es ,f ,g & ,h ,i1 & ! c5tre ( equipond]ation ( !m bo? toge!r 9 ! po9t ,m2 ,r w be ! c5tre ( ! complem5t ,b ,c ,d ,e1 & ,s ! c5tre ( ! figure ,a ,b ,c ,d4 ,& see+ ,p ,m1 t is ,i ,n1 is to ,q ,s1 z ,r ,p is to ,r ,q2 & ,i ,n or ,p ,m is #c ( ^? p>ts1 ( :i* ,b ,e1 t is ,f ,g1 is #ad2 !re=e ,q ,s is #e ( ! same p>ts2 & ,e ,i1 t is ,f ,g1 #d2 & ,f ,s1 #i2 & ,g ,s1 #e4 ,:]e=e ! /rait l9e ,s ,t be+ drawn p>allel to ! axis1 w divide ! base ,a ,d into #e & #i4 ,b ! fraction #ei is f\nd 9 ! f\r? r[ ( ! table1 two places bel[ ! fraction 9 ! second r[4 ,by ! same me?od1 if 9 ! same second r[ !re be tak5 ! second ?ree-sid$ figure ( ?ree means1 ! c5tre ( equipond]ation ( ! half ( x w be f\nd to be 9 a /rait l9e p>allel to ! axis1 divid+ ! base accord+ to ! numb]s ( ! fraction #faj1 two places bel[ 9 ! f\r? r[4 ,& ! same way s]ves = all ! re/ ( ! figures 9 ! second r[4 ,9 l mann]1 ! c5tre ( equipond]ation ( ! ?ird ?ree-sid$ figure ( ?ree means w be f\nd to be 9 a /rait l9e p>allel to ! axis1 divid+ ! base1 s t ! "p next ! axis be to ! o!r "p z #g to #ac1 c4 ,coroll4 ,! c5tres ( equipond]ation ( ! halves ( ! said figures are kn[n1 see+ !y are 9 ! int]section ( ! /rait l9es ,s ,t & ,f ,g1 :i* are bo? kn[n4 ,sid5ote3 ,! c5tre ( equipond]ation ( ! half ( any ( ! figures 9 ! same table be+ kn[n1 ! c5tre ( ! excess ( ! same figure above a triangle ( ! same altitude & base is also kn[n4 #ac4 ,! c5tre ( equipond]ation ( ! half ( any ( ! figures1 :i* 79 ! table ( art4 #c1 *ap4 ,,xvii7 are comp>$ ) _! p>allelograms1 be+ kn[n2 ! c5tre ( equipond]ation ( ! excess ( ! same figure above its triangle is also kn[n4 ,= example1 let ! semip>abola ,a ,b ,c ,d 79 fig4 #h7 be tak51 ^: axis is ,a ,b2 ^: complete figure is ,a ,b ,e ,d2 & ^: excess above its triangle is ,b ,c ,d ,b4 ,its c5tre ( equipond]ation may be f\nd out 9 ?is mann]4 ,let ,f ,g be drawn p>allel to ! base1 s t ,a ,f be a ?ird "p ( ! axis2 & let ,h ,i be drawn p>allel to ! axis1 s t ,a ,h be a ?ird "p ( ! base4 ,?is be+ done1 ! c5tre ( equipond]ation ( ! triangle ,a ,b ,d w be ,i4 ,aga91 let ,k ,l be drawn p>allel to ! base1 s t ,a ,k be to ,a ,b z #b to #e2 & ,m ,n p>allel to ! axis1 s t ,a ,m be to ,a ,d z #c to #h2 & let ,m ,n t]m9ate 9 ! /rait l9e ,k ,l4 ,! c5tre1 !re=e1 ( equipond]ation ( ! p>abola ,a ,b ,c ,d is ,n2 & !re=e we h ! c5tres ( equipond]ation ( ! semip>abola ,a ,b ,c ,d1 & ( its "p ! triangle ,a ,b ,d4 ,t we may n[ f9d ! c5tre ( equipond]ation ( ! rema9+ "p ,b ,c ,d ,b1 let ,i ,n be drawn & produc$ to ,o1 s t ,n ,o be triple to ,i ,n2 & ,o w be ! c5tre s\ght =4 ,= see+ ! weight ( ,a ,b ,d to ! weight ( ,b ,c ,d ,b is 9 proportion reciprocal to t ( ! /rait l9e ,n ,o to ! /rait l9e ,i ,n2 & ,n is ! c5tre ( ! :ole1 & ,i ! c5tre ( ! triangle ,a ,b ,d2 ,o w be ! c5tre ( ! rema9+ "p1 namely1 ( ! figure ,b ,d ,c ,b2 :i* 0 to be f\nd4 ,coroll4 ,! c5tre ( equipond]ation ( ! figure ,b ,d ,c ,b is 9 ! conc\rse ( two /rait l9es1 :]e( "o is p>allel to ! base1 & divides ! axis1 s t ! "p next ! base be or #iae ( ! :ole axis2 ! o!r is p>allel to ! axis1 & s divides ! base1 t ! "p t[>ds ! axis be 1 or #abbd ( ! :ole base4 ,= draw+ ,o ,p p>allel to ! base1 x w be z ,i ,n to ,n ,o1 s ,f ,k to ,k ,p1 t is1 s #a to #c1 or #e to #ae4 ,b ,a ,f is #eae1 or ( ! :ole ,a ,b2 & ,a ,k is #fae1 or 2 & ,f ,k #aae2 & ,,kp #cae2 & !re=e ,a ,p is #iae ( ! axis ,a ,b4 ,also ,a ,h is 1 or #hbd2 & ,a ,m 1 or #ibd ( ! :ole base2 & !re=e ,o ,q be+ drawn p>allel to ! axis1 ,m ,q1 :i* is triple to ,h ,m1 w be #cbd4 ,:]e=e ,a ,q is #abbd1 or ( ! base ,a ,d4 ,! excesses ( ! re/ ( ! ?ree-sid$ figures 9 ! fir/ r[ ( ! table ( art4 #c1 *ap4 ,,xvii1 h _! c5tres ( equipond]ation 9 two /rait l9es1 :i* divide ! axis & base accord+ to ^? fractions1 :i* add #d to ! num]ators ( ! fractions ( a p>abola #iae & #abbd2 & #f to ! d5om9ators1 9 ?is mann]3-- ,9 a p>abola1 ! axis #iae1 ! base #abbd4 ,9 ! fir/ ?ree-sid$ figure1 ! axis #acba1 ! base #afcj4 ,9 ! second ?ree-sid$ figure1 ! axis #agbg1 ! base #bjcf1 c4 ,& by ! same me?od1 any man1 if x be wor? ! pa9s1 may f9d out ! c5tres ( equipond]ation ( ! excesses above _! triangles ( ! re/ ( ! figures 9 ! second & ?ird r[1 c4 ,sid5ote3 ,! c5tre ( equipond]ation ( a solid sector is 9 ! axis s divid$1 t ! "p next ! v]tex be to ! :ole axis1 want+ half ! axis ( ! portion ( ! sph]e1 z #c to #d4 #ad4 ,! c5tre ( equipond]ation ( ! sector ( a sph]e1 t is1 ( a figure comp\nd$ ( a "r cone1 ^: v]tex is ! c5tre ( ! sph]e1 & ! portion ( ! sph]e ^: base is ! same ) t ( ! cone1 divides ! /rait l9e :i* is made ( ! axis ( ! cone & half ! axis ( ! portion toge!r tak51 s t ! "p next ! v]tex be triple to ! o!r "p1 or to ! :ole /rait l9e z #c to #d4 ,= let ,a ,b ,c 79 fig4 #i7 be ! sector ( a sph]e1 ^: v]tex is ! c5tre ( ! sph]e ,a2 ^: axis is ,a ,d2 & ! circle ^u ,b ,c is ! common base ( ! portion ( ! sph]e & ( ! cone ^: v]tex is ,a2 ! axis ( :i* portion is ,e ,d1 & ! half !re( ,f ,d2 & ! axis ( ! cone1 ,a ,e4 ,la/ly1 let ,a ,g be #cd ( ! /rait l9e ,a ,f4 ,i say1 ,g is ! c5tre ( equipond]ation ( ! sector ,a ,b ,c4 ,let ! /rait l9e ,f ,h be drawn ( any l5g?1 mak+ "r angles ) ,a ,f at ,f2 & draw+ ! /rait l9e ,a ,h1 let ! triangle ,a ,f ,h be made4 ,!n ^u ! same c5tre ,a let any ar* ,i ,k be drawn1 cutt+ ,a ,d 9 ,l2 & its *ord1 cutt+ ,a ,d 9 ,m2 & divid+ ,m ,l equally 9 ,n1 let ,n ,o be drawn p>allel to ! /rait l9e ,f ,h1 & meet+ ) ! /rait l9e ,a ,h 9 ,o4 ,see+ n[ ,b ,d ,c is ! sph]ical sup]ficies ( ! portion cut (f ) a plane pass+ ?r\gh ,b ,c1 & cutt+ ! axis at "r angles2 & see+ ,f ,h divides ,e ,d1 ! axis ( ! portion1 into two equal p>ts 9 ,f2 ! c5tre ( equipond]ation ( ! sup]ficies ,b ,d ,c w be 9 ,f 7by art4 #h72 & = ! same reason ! c5tre ( equipond]ation ( ! sup]ficies ,i ,l ,k1 ,k be+ 9 ! /rait l9e ,a ,c1 w be 9 ,n4 ,& 9 l mann]1 if !re 7 drawn1 betwe5 ! c5tre ( ! sph]e ,a & ! out]mo/ sph]ical sup]ficies ( ! sector1 ar*es inf9ite 9 numb]1 ! c5tres ( equipond]ation ( ! sph]ical sup]ficies1 9 :i* ^? ar*es are1 w\ld be f\nd to be 9 t "p ( ! axis1 :i* is int]cept$ betwe5 ! sup]ficies itself & a plane pass+ along by ! *ord ( ! ar*1 & cutt+ ! axis 9 ! middle at "r angles4 ,let x n[ be suppos$ t ! mom5t ( ! out]mo/ sph]ical sup]ficies ,b ,d ,c is ,f ,h4 ,see+ !re=e ! sup]ficies ,b ,d ,c is to ! sup]ficies ,i ,l ,k 9 proportion duplicate to t ( ! ar* ,b ,d ,c to ! ar* ,i ,l ,k1 t is1 ( ,b ,e to ,i ,m1 t is1 ( ,f ,h to ,n ,o2 let x be z ,f ,h to ,n ,o1 s ,n ,o to ano!r ,n ,p2 & aga91 z ,n ,o to ,n ,p1 s ,n ,p to ano!r ,n ,q2 & let ?is be done 9 all ! /rait l9es p>allel to ! base ,f ,h t c possibly be drawn betwe5 ! base & ! v]tex ( ! triangle ,a ,f ,h4 ,if !n ?r\gh all ! po9ts ,q !re be drawn ! crook$ l9e ,a ,q ,h1 ! figure ,a ,f ,h ,q ,a w be ! complem5t ( ! fir/ ?ree-sid$ figure ( two means2 & ! same w also be ! mom5t ( all ! sph]ical sup]ficies1 ( :i* ! solid sector ,a ,b ,c ,d is comp\nd$2 & by consequ5t1 ! mom5t ( ! sector itself4 ,let n[ ,f ,h be und]/ood to be ! semidiamet] ( ! base ( a "r cone1 ^: side is ,a ,h1 & axis ,a ,f ,:]e=e1 see+ ! bases ( ! cones1 :i* pass ?r\gh ,f & ,n & ! re/ ( ! po9ts ( ! axis1 are 9 proportion duplicate to t ( ! /rait l9es ,f ,h & ,n ,o1 c41 ! mom5t ( all ! bases toge!r1 t is1 ( ! :ole cone1 w be ! figure itself ,a ,f ,h ,q ,a2 & !re=e ! c5tre ( equipond]ation ( ! cone ,a ,f ,h is ! same ) t ( ! solid sector4 ,:]e=e1 see+ ,a ,g is ( ! axis ,a ,f1 ! c5tre ( equipond]ation ( ! cone ,a ,f ,h is 9 ,g2 & !re=e ! c5tre ( ! solid sector is 9 ,g also1 & divides ! "p ,a ,f ( ! axis s t ,a ,g is triple to ,g ,f2 t is1 ,a ,g is to ,a ,f z #c to #d2 :i* 0 to be demon/rat$4 ,note1 t :5 ! sector is a hemisph]e1 ! axis ( ! cone vani%e? into t po9t :i* is ! c5tre ( ! sph]e2 & !re=e x adde? no?+ to half ! axis ( ! portion4 ,:]e=e1 if 9 ! axis ( ! hemisph]e !re be tak5 f ! c5tre ( half ! axis1 t is1 ( ! semidiamet] ( ! sph]e1 !re w be ! c5tre ( equipond]ation ( ! hemisph]e4 ---------------------------------------- -------------------------------- ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4,,xxiii4 ,fig4 #a-i ---------------------------------------- -------------------------------- ,,*apt] ,,xxiv4 ,,( ,,refraction ,,& ,,reflection4 #a4 ,def9itions4--#b4 ,9 p]p5dicul> motion !re is no refraction4--#c4 ,?+s ?r[n out ( a ?9n] into a ?ick] m$ium are s refract$ t ! angle refract$ is great] ?an ! angle ( incl9ation4--#d4 ,endeav\r1 :i* f "o po9t t5de? e way1 w be s refract$1 z t ! s9e ( ! angle refract$ w be to ! s9e ( ! angle ( incl9ation1 z ! d5sity ( ! fir/ m$ium is to ! d5sity ( ! second m$ium1 reciprocally tak54--#e4 ,! s9e ( ! refract$ angle 9 "o incl9ation is to ! s9e ( ! refract$ angle 9 ano!r incl9ation1 z ! s9e ( ! angle ( t incl9ation is to ! s9e ( ! angle ( ?is incl9ation4--#f4 ,if two l9es ( incid5ce1 hav+ equal incl9ation1 be ! "o 9 a ?9n]1 ! o!r 9 a ?ick] m$ium1 ! s9e ( ! angle ( incl9ation w be a mean proportional betwe5 ! two s9es ( ! refract$ angles4--#g4 ,if ! angle ( incl9ation be semirect1 & ! l9e ( incl9ation be 9 ! ?ick] m$ium1 & ! proportion ( _! d5sities be ! same ) t ( ! diagonal to ! side ( a squ>e1 & ! sep>at+ sup]ficies be plane1 ! refract$ l9e w be 9 ! sep>at+ sup]ficies4--#h4 ,if a body be c>ri$ 9 a /rait l9e ^u ano!r body1 & d n p5etrate ! same1 b be reflect$ f x1 ! angle ( reflection w be equal to ! angle ( incid5ce4--#i4 ,! same happ5s 9 ! g5]ation ( motion 9 ! l9e ( incid5ce4 ,,def9itions4 ,sid5ote3 ,def9itions4 ,i4 ,refraction is ! break+ ( t /rait l9e1 9 :i* a body is mov$ or its action w\ld proce$ 9 "o & ! same m$ium1 into two /rait l9es1 by reason ( ! diff]5t natures ( ! two m$iums4 ,,ii4 ,! =m] ( ^! is call$ ! l9e ( incid5ce2 ! latt] ! refract$ l9e4 ,,iii4 ,! po9t ( refraction is ! common po9t ( ! l9e ( incid5ce1 & ( ! refract$ l9e4 ,,iv4 ,! refract+ sup]ficies1 :i* also is ! sep>at+ sup]ficies ( ! two m$iums1 is t 9 :i* is ! po9t ( refraction4 ,v4 ,! angle refract$ is t1 :i* ! refract$ l9e makes 9 ! po9t ( refraction ) t l9e1 :i* f ! same po9t is drawn p]p5dicul> to ! sep>at+ sup]ficies 9 a diff]5t m$ium4 ,,vi4 ,! angle ( refraction is t :i* ! refract$ l9e makes ) ! l9e ( incid5ce produc$4 ,,vii4 ,! angle ( incl9ation is t :i* ! l9e ( incid5ce makes ) t l9e1 :i* f ! po9t ( refraction is drawn p]p5dicul> to ! sep>at+ sup]ficies4 ,,viii4 ,! angle ( incid5ce is ! complem5t to a "r angle ( ! angle ( incl9ation4 ,& s1 79 fig4 #a7 ! refraction is made 9 ,a ,b ,f4 ,! refract$ l9e is ,b ,f4 ,! l9e ( incid5ce is ,a ,b4 ,! po9t ( incid5ce & ( refraction is ,b4 ,! refract+ or sep>at+ sup]ficies is ,d ,b ,e4 ,! l9e ( incid5ce produc$ directly is ,a ,b ,c4 ,! p]p5dicul> to ! sep>at+ sup]ficies is ,b ,h4 ,! angle ( refraction is ,c ,b ,f4 ,! angle refract$ is ,h ,b ,f4 ,! angle ( incl9ation is ,a ,b ,g or ,h ,b ,c4 ,! angle ( incid5ce is ,a ,b ,d4 ,,ix4 ,moreov] ! ?9n] m$ium is und]/ood to be t 9 :i* !re is less resi/ance to motion1 or to ! g5]ation ( motion2 & ! ?ick] t :]e9 !re is great] resi/ance4 ,x4 ,& t m$ium 9 :i* !re is equal resi/ance ev]y:]e1 is a homog5e\s m$ium4 ,all o!r m$iums are het]og5e\s4 ,sid5ote3 ,9 p]p5dicul> motion !re is no refraction4 ,if a body pass1 or !re be g5]ation ( motion f "o m$ium to ano!r ( diff]5t d5sity1 9 a l9e p]p5dicul> to ! sep>at+ sup]ficies1 !re w be no refraction4 ,= see+ on e side ( ! p]p5dicul> all ?+s 9 ! m$iums are suppos$ to be l & equal1 if ! motion itself be suppos$ to be p]p5dicul>1 ! incl9ations also w be equal1 or r none at all2 & !re=e !re c be no cause f :i* refraction may be inf]r$ to be on "o side ( ! p]p5dicul>1 :i* w n conclude ! same refraction to be on ! o!r side4 ,:i* be+ s1 refraction on "o side w de/roy refraction on ! o!r side2 & consequ5tly ei!r ! refract$ l9e w be ev]y:]e1 :i* is absurd1 or !re w be no refract$ l9e at all2 :i* 0 to be demon/rat$4 ,coroll4 ,x is manife/ f h5ce1 t ! cause ( refraction consi/e? only 9 ! obliquity ( ! l9e ( incid5ce1 :e!r ! incid5t body p5etrate bo? ! m$iums1 or )\t p5etrat+1 propagate motion by pressure only4 ,sid5ote3 ,?+s ?r[n out ( a ?9n] into a ?ick] m$ium are s refract$ t ! angle refract$ is great] ?an ! angle ( incl9ation4 #c4 ,if a body1 )\t any *ange ( situation ( its int]nal p>ts1 z a /one1 be mov$ obliquely out ( ! ?9n] m$ium1 & proce$ p5etrat+ ! ?ick] m$ium1 & ! ?ick] m$ium be su*1 z t its int]nal p>ts be+ mov$ re/ore !mselves to _! =m] situation2 ! angle refract$ w be great] ?an ! angle ( incl9ation4 ,= let ,d ,b ,e 79 ! same fir/ figure7 be ! sep>at+ sup]ficies ( two m$iums2 & let a body1 z a /one ?r[n1 be und]/ood to be mov$ z is suppos$ 9 ! /rait l9e ,a ,b ,c2 & let ,a ,b be 9 ! ?9n] m$ium1 z 9 ! air2 & ,b ,c 9 ! ?ick]1 z 9 ! wat]4 ,i say ! /one1 :i* be+ ?r[n1 is mov$ 9 ! l9e ,a ,b1 w n proce$ 9 ! l9e ,b ,c1 b 9 "s o!r l9e1 namely1 t1 ) :i* ! p]p5dicul> ,b ,h makes ! refract$ angle ,h ,b ,f great] ?an ! angle ( incl9ation ,h ,b ,c4 ,= see+ ! /one com+ f ,a1 & fall+ ^u ,b1 makes t :i* is at ,b proce$ t[>ds ,h1 & t ! l is done 9 all ! /rait l9es :i* are p>allel to ,b ,h2 & see+ ! p>ts mov$ re/ore !mselves by contr>y motion 9 ! same l9e2 !re w be contr>y motion g5]at$ 9 ,h ,b1 & 9 all ! /rait l9es :i* are p>allel to x4 ,:]e=e1 ! motion ( ! /one w be made by ! conc\rse ( ! motions 9 ,a ,g1 t is1 9 ,d ,b1 & 9 ,g ,b1 t is1 9 ,b ,h1 & la/ly1 9 ,h ,b1 t is1 by ! conc\rse ( ?ree motions4 ,b by ! conc\rse ( ! motions 9 ,a ,g & ,b ,h1 ! /one w be c>ri$ to ,c2 & !re=e by add+ ! motion 9 ,h ,b1 x w be c>ri$ high] 9 "s o!r l9e1 z 9 ,b ,f1 & make ! angle ,h ,b ,f great] ?an ! angle ,h ,b ,c4 ,& f h5ce may be d]iv$ ! cause1 :y bodies :i* are ?r[n 9 a v oblique l9e1 if ei!r !y be any ?+ flat1 or be ?r[n ) great =ce1 w1 :5 !y fall ^u ! wat]1 be ca/ up aga9 f ! wat] into ! air4 ,= let ,a ,b 79 fig4 #b7 be ! sup]ficies ( ! wat]2 into :i*1 f ! po9t ,c1 let a /one be ?r[n 9 ! /rait l9e ,c ,a1 mak+ ) ! l9e ,b ,a produc$ a v little angle ,c ,a ,d2 & produc+ ,b ,a indef9itely to ,d1 let ,c ,d be drawn p]p5dicul> to x1 & ,a ,e p>allel to ,c ,d4 ,! /one !re=e w be mov$ 9 ,c ,a by ! conc\rse ( two motions 9 ,c ,d & ,d ,a1 ^: velocities are z ! l9es !mselves ,c ,d & ,d ,a4 ,& f ! motion 9 ,c ,d & all its p>allels d[nw>ds1 z soon z ! /one falls ^u ,a1 !re w be reaction upw>ds1 because ! wat] re/ores itself to its =m] situation4 ,if n[ ! /one be ?r[n ) suffici5t obliquity1 t is1 if ! /rait l9e ,c ,d be %ort en\gh1 t is1 if ! endeav\r ( ! /one d[nw>ds be less ?an ! reaction ( ! wat] upw>ds1 t is1 less ?an ! endeav\r x ha? f its own gravity 7= t may be71 ! /one w by reason ( ! excess ( ! endeav\r :i* ! wat] ha? to re/ore itself1 above t :i* ! /one ha? d[nw>ds1 be rais$ aga9 above ! sup]ficies ,a ,b1 & be c>ri$ high]1 be+ reflect$ 9 a l9e :i* goes high]1 z ! l9e ,a ,g4 ,sid5ote3 ,endeav\r1 :i* f "o po9t t5de? e way1 w be s refract$1 z t ! s9e ( ! angle refract$ w be to ! s9e ( ! angle ( incl9ation1 z ! d5sity ( ! fir/ m$ium is to ! d5sity ( ! second m$ium1 reciprocally tak54 #d4 ,if f a po9t1 :atsoev] ! m$ium be1 endeav\r be propagat$ e way into all ! p>ts ( t m$ium2 & to ! same endeav\r !re be obliquely oppos$ ano!r m$ium ( a diff]5t nature1 t is1 ei!r ?9n] or ?ick]2 t endeav\r w be s refract$1 t ! s9e ( ! angle refract$1 to ! s9e ( ! angle ( incl9ation1 w be z ! d5sity ( ! fir/ m$ium to ! d5sity ( ! second m$ium1 reciprocally tak54 ,fir/1 let a body be 9 ! ?9n] m$ium 9 ,a 7fig4 #c71 & let x be und]/ood to h endeav\r e way1 & consequ5tly1 t its endeav\r proce$ 9 ! l9es ,a ,b & ,a b2 to :i* let ,b b ! sup]ficies ( ! ?ick] m$ium be obliquely oppos$ 9 ,b & b1 s t ,a ,b & ,a b be equal2 & let ! /rait l9e ,b b be produc$ bo? ways4 ,f ! po9ts ,b & b1 let ! p]p5dicul>s ,b ,c & b c be drawn2 & ^u ! c5tres ,b & b1 & at ! equal di/ances ,b ,a & b ,a1 let ! circles ,a ,c & ,a c be describ$1 cutt+ ,b ,c & b c 9 ,c & c1 & ! same ,c ,b & c b produc$ 9 ,d & d1 z also ,a ,b & ,a b produc$ 9 ,e & e4 ,!n f ! po9t ,a to ! /rait l9es ,b ,c & b c let ! p]p5dicul>s ,a ,f & ,a f be drawn4 ,a ,f !re=e w be ! s9e ( ! angle ( incl9ation ( ! /rait l9e ,a ,b1 & ,a f ! s9e ( ! angle ( incl9ation ( ! /rait l9e ,a h1 :i* two incl9ations are by con/ruction made equal4 ,i say1 z ! d5sity ( ! m$ium 9 :i* are ,b ,c & b c is to ! d5sity ( ! m$ium 9 :i* are ,b ,d & b d1 s is ! s9e ( ! angle refract$1 to ! s9e ( ! angle ( incl9ation4 ,let ! /rait l9e ,f ,g be drawn p>allel to ! /rait l9e ,a ,b1 meet+ ) ! /rait l9e b ,b produc$ 9 ,g4 ,see+ !re=e ,a ,f & ,b ,g are also p>allels1 !y w be equal2 & consequ5tly1 ! endeav\r 9 ,a ,f is propagat$ 9 ! same "t1 9 :i* ! endeav\r 9 ,b ,g w\ld be propagat$ if ! m$ium 7 ( ! same d5sity4 ,b because ,b ,g is 9 a ?ick] m$ium1 t is1 9 a m$ium :i* resi/s ! endeav\r m ?an ! m$ium 9 :i* ,a ,f is1 ! endeav\r w be propagat$ less 9 ,b ,g ?an 9 ,a ,f1 accord+ to ! proportion :i* ! d5sity ( ! m$ium1 9 :i* ,a ,f is1 ha? to ! d5sity ( ! m$ium 9 :i* ,b ,g is4 ,let !re=e ! d5sity ( ! m$ium1 9 :i* ,b ,g is1 be to ! d5sity ( ! m$ium1 9 :i* ,a ,f is1 z ,b ,g is to ,b ,h2 & let ! measure ( ! "t be ! radius ( ! circle4 ,let ,h ,i be drawn p>allel to ,b ,d1 meet+ ) ! circumf]5ce 9 ,i2 & f ! po9t ,i let ,i ,k be drawn p]p5dicul> to ,b ,d2 :i* be+ done1 ,b ,h & ,i ,k w be equal2 & ,i ,k w be to ,a ,f1 z ! d5sity ( ! m$ium 9 :i* is ,a ,f is to ! d5sity ( ! m$ium 9 :i* is ,i ,k4 ,see+ !re=e 9 ! "t ,a ,b1 :i* is ! radius ( ! circle1 ! endeav\r is propagat$ 9 ,a ,f 9 ! ?9n] m$ium1 x w be propagat$ 9 ! same "t1 t is1 9 ! "t ,b ,i 9 ! ?ick] m$ium f ,k to ,i4 ,!re=e1 ,b ,i is ! refract$ l9e ( ! l9e ( incid5ce ,a ,b2 & ,i ,k is ! s9e ( ! angle refract$2 & ,a ,f ! s9e ( ! angle ( incl9ation4 ,:]e=e1 see+ ,i ,k is to ,a ,f1 z ! d5sity ( ! m$ium 9 :i* is ,a ,f to ! d5sity ( ! m$ium 9 :i* is ,i ,k2 x w be z ! d5sity ( ! m$ium 9 :i* is ,a ,f or ,b ,c to ! d5sity ( ! m$ium 9 :i* is ,i ,k or ,b ,d1 s ! s9e ( ! angle refract$ to ! s9e ( ! angle ( incl9ation4 ,& by ! same reason x may be %[n1 t z ! d5sity ( ! ?9n] m$ium is to ! d5sity ( ! ?ick] m$ium1 s w ,k ,i ! s9e ( ! angle refract$ be to ,a ,f ! s9e ( ! angle ( incl9ation4 ,secondly1 let ! body1 :i* endeav\re? e way1 be 9 ! ?ick] m$ium at ,i4 ,if1 !re=e1 bo? ! m$iums 7 ( ! same d5sity1 ! endeav\r ( ! body 9 ,i ,b w\ld t5d directly to ,l2 & ! s9e ( ! angle ( incl9ation ,l ,m w\ld be equal to ,i ,k or ,b ,h4 ,b because ! d5sity ( ! m$ium1 9 :i* is ,i ,k1 to ! d5sity ( ! m$ium1 9 :i* is ,l ,m1 is z ,b ,h to ,b ,g1 t is1 to ,a ,f1 ! endeav\r w be propagat$ fur!r 9 ! m$ium 9 :i* ,l ,m is1 ?an 9 ! m$ium 9 :i* ,i ,k is1 9 ! proportion ( d5sity to d5sity1 t is1 ( ,m ,l to ,a ,f4 ,:]e=e1 ,b ,a be+ drawn1 ! angle refract$ w be ,c ,b ,a1 & its s9e ,a ,f4 ,b ,l ,m is ! s9e ( ! angle ( incl9ation2 & !re=e aga91 z ! d5sity ( "o m$ium is to ! d5sity ( ! diff]5t m$ium1 s reciprocally is ! s9e ( ! angle refract$ to ! s9e ( ! angle ( incl9ation2 :i* 0 to be demon/rat$4 ,9 ?is demon/ration1 ,i h made ! sep>at+ sup]ficies ,b b plane by con/ruction4 ,b ?\gh x 7 concave or convex1 ! !orem w\ld nev]!less be true4 ,= ! refraction be+ made 9 ! po9t ,b ( ! plane sep>at+ sup]ficies1 if a crook$ l9e1 z ,p ,q1 be drawn1 t\*+ ! sep>at+ l9e 9 ! po9t ,b2 nei!r ! refract$ l9e ,b ,i1 nor ! p]p5dicul> ,b ,d1 w be alt]$2 & ! refract$ angle ,k ,b ,i1 z also its s9e ,k ,i1 w be /ill ! same !y 74 ,sid5ote3 ,! s9e ( ! refract$ angle 9 "o incl9ation is to ! s9e ( ! refract$ angle 9 ano!r incl9ation1 z ! s9e ( ! angle ( t incl9ation is to ! s9e ( ! angle ( ?is incl9ation4 #e4 ,! s9e ( ! angle refract$ 9 "o incl9ation is to ! s9e ( ! angle refract$ 9 ano!r incl9ation1 z ! s9e ( ! angle ( t incl9ation to ! s9e ( ! angle ( ?is incl9ation4 ,= see+ ! s9e ( ! refract$ angle is to ! s9e ( ! angle ( incl9ation1 :atsoev] t incl9ation be1 z ! d5sity ( "o m$ium to ! d5sity ( ! o!r m$ium2 ! proportion ( ! s9e ( ! refract$ angle1 to ! s9e ( ! angle ( incl9ation1 w be comp\nd$ ( ! proportions ( d5sity to d5sity1 & ( ! s9e ( ! angle ( "o incl9ation to ! s9e ( ! angle ( ! o!r incl9ation4 ,b ! proportions ( ! d5sities 9 ! same homog5e\s body are suppos$ to be ! same4 ,:]e=e refract$ angles 9 diff]5t incl9ations are z ! s9es ( ! angles ( ^? incl9ations2 :i* 0 to be demon/rat$4 ,sid5ote3 ,if two l9es ( incid5ce1 hav+ equal incl9ation1 be "o 9 a ?9n] ! o!r 9 a ?ick] m$ium1 ! s9e ( ! angle ( incl9ation w be a mean proportional betwe5 ! two s9es ( ! refract$ angles4 #f4 ,if two l9es ( incid5ce1 hav+ equal incl9ation1 be ! "o 9 a ?9n]1 ! o!r 9 a ?ick] m$ium1 ! s9e ( ! angle ( _! incl9ation w be a mean proportional betwe5 ! two s9es ( _! angles refract$4 ,= let ! /rait l9e ,a ,b 79 fig4 #c7 h its incl9ation 9 ! ?9n] m$ium1 & be refract$ 9 ! ?ick] m$ium 9 ,b ,i2 & let ,e ,b h z mu* incl9ation 9 ! ?ick] m$ium1 & be refract$ 9 ! ?9n] m$ium 9 ,b ,s2 & let ,r ,s1 ! s9e ( ! angle refract$1 be drawn4 ,i say1 ! /rait l9es ,r ,s1 ,a ,f1 & ,i ,k are 9 cont9ual proportion4 ,= x is1 z ! d5sity ( ! ?ick] m$ium to ! d5sity ( ! ?9n] m$ium1 s ,r ,s to ,a ,f4 ,b x is also z ! d5sity ( ! same ?ick] m$ium to t ( ! same ?9n] m$ium1 s ,a ,f to ,i ,k4 ,:]e=e ,r ,s4 ,a ,f 33 ,a ,f4 ,i ,k are proportionals2 t is1 ,r ,s1 ,a ,f1 & ,i ,k are 9 cont9ual proportion1 & ,a ,f is ! mean proportional2 :i* 0 to be prov$4 ,sid5ote3 ,if ! angle ( incl9ation be semirect1 & ! l9e ( incl9ation be 9 ! ?ick] m$ium1 & ! proportion ( _! d5sities be ! same ) t ( ! diagonal to ! side ( a squ>e1 & ! sep>at+ sup]ficies be pla91 ! refract$ l9e w be 9 ! sep>at+ sup]ficies4 #g4 ,if ! angle ( incl9ation be semirect1 & ! l9e ( incl9ation be 9 ! ?ick] m$ium1 & ! proportion ( ! d5sities be z t ( a diagonal to ! side ( its squ>e1 & ! sep>at+ sup]ficies be pla91 ! refract$ l9e w be 9 t sep>at+ sup]ficies4 ,= 9 ! circle ,a ,c 7fig4 #d7 let ! angle ( incl9ation ,a ,b ,c be an angle ( #de degrees4 ,let ,c ,b be produc$ to ! circumf]5ce 9 ,d2 & let ,c ,e1 ! s9e ( ! angle ,e ,b ,c1 be drawn1 to :i* let ,b ,f be tak5 equal 9 ! sep>at+ l9e ,b ,g4 ,b ,c ,e ,f w !re=e be a p>allelogram1 & ,f ,e & ,b ,c1 t is ,f ,e & ,b ,g equal4 ,let ,a ,g be drawn1 namely ! diagonal ( ! squ>e ^: side is ,b ,g1 & x w be1 z ,a ,g to ,e ,f s ,b ,g to ,b ,f2 & s1 by supposition1 ! d5sity ( ! m$ium1 9 :i* ,c is1 to ! d5sity ( ! m$ium 9 :i* ,d is2 & s also ! s9e ( ! angle refract$ to ! s9e ( ! angle ( incl9ation4 ,draw+ !re=e ,f ,d1 & f ,d ! l9e ,d ,h p]p5dicul> to ,a ,b produc$1 ,d ,h w be ! s9e ( ! angle ( incl9ation4 ,& see+ ! s9e ( ! angle refract$ is to ! s9e ( ! angle ( incl9ation1 z ! d5sity ( ! m$ium1 9 :i* is ,c1 is to ! d5sity ( ! m$ium 9 :i* is ,d1 t is1 by supposition1 z ,a ,g is to ,f ,e1 t is z ,b ,g is to ,d ,h2 & see+ ,d ,h is ! s9e ( ! angle ( incl9ation1 ,b ,g w !re=e be ! s9e ( ! angle refract$4 ,:]e=e ,b ,g w be ! refract$ l9e1 & lye 9 ! pla9 sep>at+ sup]ficies2 :i* 0 to be demon/rat$4 ,coroll4 ,x is !re=e manife/1 t :5 ! incl9ation is great] ?an #de degrees1 z also :5 x is less1 provid$ ! d5sity be great]1 x may happ5 t ! refraction w n ent] ! ?9n] m$ium at all4 ,sid5ote3 ,if a body be c>ri$ 9 a /rait l9e ^u ano!r body1 & d n p5etrate x1 b be reflect$ f x1 ! angle ( reflection w be equal to ! angle ( incid5ce4 #h4 ,if a body fall 9 a /rait l9e ^u ano!r body1 & d n p5etrate x1 b be reflect$ f x1 ! angle ( reflection w be equal to ! angle ( incid5ce4 ,let !re be a body at ,a 79 fig4 #e71 :i* fall+ ) /rait motion 9 ! l9e ,a ,c ^u ano!r body at ,c1 passe? no fur!r1 b is reflect$2 & let ! angle ( incid5ce be any angle1 z ,a ,c ,d4 ,let ! /rait l9e ,c ,e be drawn1 mak+ ) ,d ,c produc$ ! angle ,e ,c ,f equal to ! angle ,a ,c ,d2 & let ,a ,d be drawn p]p5dicul> to ! /rait l9e ,d ,f4 ,also 9 ! same /rait l9e ,d ,f let ,c ,g be tak5 equal to ,c ,d2 & let ! p]p5dicul> ,g ,e be rais$1 cutt+ ,c ,e 9 ,e4 ,?is be+ done1 ! triangles ,a ,c ,d & ,e ,c ,g w be equal & l4 ,let ,c ,h be drawn equal & p>allel to ! /rait l9e ,a ,d2 & let ,h ,c be produc$ indef9itely to ,i4 ,la/ly let ,e ,a be drawn1 :i* w pass ?r\gh ,h1 & be p>allel & equal to ,g ,d4 ,i say ! motion f ,a to ,c1 9 ! /rait l9e ( incid5ce ,a ,c1 w be reflect$ 9 ! /rait l9e ,c ,e4 ,= ! motion f ,a to ,c is made by two coeffici5t or concurr5t motions1 ! "o 9 ,a ,h p>allel to ,d ,g1 ! o!r 9 ,a ,d p]p5dicul> to ! same ,d ,g2 ( :i* two motions t 9 ,a ,h works no?+ ^u ! body ,a aft] x has be5 mov$ z f> z ,c1 because1 by supposition1 x do? n pass ! /rait l9e ,d ,g2 :]eas ! endeav\r 9 ,a ,d1 t is 9 ,h ,c1 worke? fur!r t[>ds ,i4 ,b see+ x do? only press & n p5etrate1 !re w be reaction 9 ,h1 :i* cause? motion f ,c t[>ds ,h2 & 9 ! meantime ! motion 9 ,h ,e rema9s ! same x 0 9 ,a ,h2 & !re=e ! body w n[ be mov$ by ! conc\rse ( two motions 9 ,c ,h & ,h ,e1 :i* are equal to ! two motions x _h =m]ly 9 ,a ,h & ,h ,c4 ,:]e=e x w be c>ri$ on 9 ,c ,e4 ,! angle !re=e ( reflection w be ,e ,c ,g1 equal1 by con/ruction1 to ! angle ,a ,c ,d2 :i* 0 to be demon/rat$4 ,n[ :5 ! body is consid]$ b z a po9t1 x is all "o :e!r ! sup]ficies or l9e 9 :i* ! reflection is made be /rait or crook$2 = ! po9t ( incid5ce & reflection ,c is z well 9 ! crook$ l9e :i* t\*e? ,d ,g 9 ,c1 z 9 ,d ,g itself4 ,sid5ote3 ,! same happ5s 9 ! g5]ation ( motion 9 ! l9e ( incid5ce4 #i4 ,b if we suppose t n a body be mov$1 b "s endeav\r only be propagat$ f ,a to ,c1 ! demon/ration w nev]!less be ! same4 ,= all endeav\r is motion2 & :5 x ha? rea*$ ! solid body 9 ,c1 x presse? x1 & endeav\re? fur!r 9 ,c ,i4 ,:]e=e ! reaction w proce$ 9 ,c ,h2 & ! endeav\r 9 ,c ,h concurr+ ) ! endeav\r 9 ,h ,e1 w g5]ate ! endeav\r 9 ,c ,e1 9 ! same mann] z 9 ! rep]cussion ( bodies mov$4 ,if !re=e endeav\r be propagat$ f any po9t to ! concave sup]ficies ( a sph]ical body1 ! reflect$ l9e ) ! circumf]5ce ( a great circle 9 ! same sph]e w make an angle equal to ! angle ( incid5ce4 ,= if endeav\r be propagat$ f ,a 79 fig4 #f7 to ! circumf]5ce 9 ,b1 & ! c5tre ( ! sph]e be ,c1 & ! l9e ,c ,b be drawn1 z also ! tang5t ,d ,b ,e2 & la/ly if ! angle ,f ,b ,d be made equal to ! angle ,a ,b ,e1 ! reflection w be made 9 ! l9e ,b ,f1 z ha? be5 newly %[n4 ,:]e=e ! angles1 :i* ! /rait l9es ,a ,b & ,f ,b make ) ! circumf]5ce1 w also be equal4 ,b x is "h to be not$1 t if ,c ,b be produc$ h[soev] to ,g1 ! endeav\r 9 ! l9e ,g ,b ,c w proce$ only f ! p]p5dicul> reaction 9 ,g ,b2 & t !re=e !re w be no o!r endeav\r 9 ! po9t ,b t[>ds ! p>ts :i* are )9 ! sph]e1 besides t :i* t5ds t[>ds ! c5tre4 ,& "h ,i put an end to ! ?ird "p ( ?is disc\rse2 9 :i* ,i h consid]$ motion & magnitude by !mselves 9 ! ab/ract4 ,! f\r? & la/ "p1 conc]n+ ! ph5om5a ( nature1 t is to say1 conc]n+ ! motions & magnitudes ( ! bodies :i* are p>ts ( ! _w1 real & exi/5t1 is t :i* foll[s4 ---------------------------------------- -------------------------------- ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xxiv4 ,fig4 #a-f ---------------------------------------- -------------------------------- ,,"p ,,iv4 ,,physics1 ,,or ,,! ,,ph5om5a ,,( ,,nature4 ------- ,,*apt] ,,xxv4 ,,( ,,s5se ,,& ,,animal ,,motion4 #a4 ,! connexion ( :at ha? be5 said ) t :i* foll[e?4--#b4 ,! inve/igation ( ! nature ( s5se1 & ! def9ition ( s5se4--#c4 ,! subject & object ( s5se4--#d4 ,! organs ( s5se4--#e4 ,all bodies are n indu$ ) s5se4--#f4 ,b "o phantasm at "o & ! same "t4--#g4 ,imag9ation ! rema9s ( pa/ s5se1 :i* also is memory4 ,( sleep4--#h4 ,h[ phantasms succe$ "o ano!r4--#i4 ,dreams1 :5ce !y proce$4--#aj4 ,( ! s5ses1 _! k9ds1 _! organs1 & phantasms prop] & common4--#aa4 ,! magnitude ( images1 h[ & by :at x is det]m9$4--#ab4 ,pleasure1 pa91 appetite & av]sion1 :at !y are4--#ac4 ,delib]ation & w1 :at4 ,sid5ote3 ,! connexion ( :at ha? be5 said ) t :i* foll[e?4 #a4 ,i h1 9 ! fir/ *apt]1 def9$ philosophy to be k ( effects acquir$ by true ratioc9ation1 f k fir/ _h ( _! causes & g5]ation2 & ( su* causes or g5]ations z may be1 f =m] k ( _! effects or appe>ances4 ,!re are1 !re=e1 two me?ods ( philosophy2 "o1 f ! g5]ation ( ?+s to _! possible effects2 & ! o!r1 f _! effects or appe>ances to "s possible g5]ation ( ! same4 ,9 ! =m] ( ^! ! tru? ( ! fir/ pr9ciples ( our ratioc9ation1 namely def9itions1 is made & con/itut$ by ourselves1 :il/ we cons5t & agree ab\t ! appellations ( ?+s4 ,& ?is "p ,i h f9i%$ 9 ! =ego+ *apt]s2 9 :i*1 if ,i am n deceiv$1 ,i h affirm$ no?+1 sav+ ! def9itions !mselves1 :i* ha? n good coh]5ce ) ! def9itions ,i h giv52 t is to say1 :i* is n suffici5tly demon/rat$ to all ^?1 t agree ) me 9 ! use ( words & appellations2 = ^: sake only ,i h writt5 ! same4 ,i n[ ent] ^u ! o!r "p2 :i* is ! f9d+ out by ! appe>ances or effects ( nature1 :i* we "k by s5se1 "s ways & means by :i* !y may be1 ,i d n say !y are1 g5]at$4 ,! pr9ciples1 !re=e1 ^u :i* ! foll[+ disc\rse dep5ds1 are n su* z we ourselves make & pron\nce 9 g5]al t]ms1 z def9itions2 b su*1 z be+ plac$ 9 ! ?+s !mselves by ! ,au?or ( ,nature1 are by u obs]v$ 9 !m2 & we make use ( !m 9 s+le & p>ticul>1 n univ]sal propositions4 ,nor d !y impose ^u u any necessity ( con/itut+ !orems2 _! use be+ only1 ?\gh n )\t su* g5]al propositions z h be5 already demon/rat$1 to %[ u ! possibility ( "s production or g5]ation4 ,see+1 !re=e1 ! sci5ce1 :i* is "h taught1 ha? its pr9ciples 9 ! appe>ances ( nature1 & ende? 9 ! atta9+ ( "s k ( natural causes1 ,i h giv5 to ?is "p ! title ( ,,physics1 or ! ,ph5om5a ( ,nature4 ,n[ su* ?+s z appe>1 or are %[n to u by nature1 we call ph5om5a or appe>ances4 ,( all ! ph5om5a or appe>ances :i* are ne> u1 ! mo/ admirable is app>ition itself1 2 namely1 t "s natural bodies h 9 !mselves ! patt]ns almo/ ( all ?+s1 & o!rs ( none at all4 ,s t if ! appe>ances be ! pr9ciples by :i* we "k all o!r ?+s1 we mu/ ne$s ackn[l$ge s5se to be ! pr9ciple by :i* we "k ^? pr9ciples1 & t all ! k we h is d]iv$ f x4 ,& z = ! causes ( s5se1 we _c beg9 our se>* ( !m f any o!r ph5om5on ?an t ( s5se itself4 ,b y w say1 by :at s5se %all we take notice ( s5se8 ,i answ]1 by s5se itself1 namely1 by ! memory :i* = "s "t rema9s 9 u ( ?+s s5sible1 ?\gh !y !mselves pass away4 ,= he t p]ceives t he ha? p]ceiv$1 rememb]s4 ,9 ! fir/ place1 !re=e1 ! causes ( our p]ception1 t is1 ! causes ( ^? ideas & phantasms :i* are p]petually g5]at$ )9 u :il/ we make use ( our s5ses1 are to be enquir$ into2 & 9 :at mann] _! g5]ation proce$s4 ,to help :i* inquisition1 we may obs]ve fir/ ( all1 t our phantasms or ideas are n always ! same2 b t new ones appe> to u1 & old ones vani%1 accord+ z we apply our organs ( s5se1 n[ to "o object1 n[ to ano!r4 ,:]e=e !y are g5]at$1 & p]i%4 ,& f h5ce x is manife/1 t !y are "s *ange or mutation 9 ! s5ti5t4 ,sid5ote3 ,! inve/igation ( ! nature ( s5se1 & ! def9ition ( s5se4 #b4 ,n[ t all mutation or alt]ation is motion or endeav\r 7& endeav\r also is motion7 9 ! int]nal p>ts ( ! ?+ t is alt]$1 ha? be5 prov$ 79 art4 #i1 *ap4 ,,viii7 f ?is1 t :il/ ev5 ! lea/ p>ts ( any body rema9 9 ! same situation 9 respect ( "o ano!r1 x _c be said t any alt]ation1 unless p]haps t ! :ole body toge!r ha? be5 mov$1 ha? happ5$ to x2 b t x bo? appe>e? & is ! same x appe>$ & 0 be=e4 ,s5se1 !re=e1 9 ! s5ti5t1 c be no?+ else b motion 9 "s ( ! int]nal p>ts ( ! s5ti5t2 & ! p>ts s mov$ are p>ts ( ! organs ( s5se4 ,= ! p>ts ( our body1 by :i* we p]ceive any ?+1 are ^? we commonly call ! organs ( s5se4 ,& s we f9d :at is ! subject ( our s5se1 namely1 t 9 :i* are ! phantasms2 & p>tly also we h discov]$ ! nature ( s5se1 namely1 t x is "s int]nal motion 9 ! s5ti5t4 ,i h %[n besides 79 *ap4 ,,ix1 art4 #g7 t no motion is g5]at$ b by a body contigu\s & mov$3 f :5ce x is manife/1 t ! imm$iate cause ( s5se or p]ception consi/s 9 ?is1 t ! fir/ organ ( s5se is t\*$ & press$4 ,= :5 ! utt]mo/ "p ( ! organ is press$1 x no soon] yields1 b ! "p next )9 x is press$ also2 &1 9 ?is mann]1 ! pressure or motion is propagat$ ?r\gh all ! p>ts ( ! organ to ! inn]mo/4 ,& ?us also ! pressure ( ! utt]mo/ "p proce$s f ! pressure ( "s m remote body1 & s cont9ually1 till we come to t f :i*1 z f its f\nta91 we d]ive ! phantasm or idea t is made 9 u by our s5se4 ,& ?is1 :atsoev] x be1 is t we commonly call ! object1 ,s5se1 !re=e1 is "s int]nal motion 9 ! s5ti5t1 g5]at$ by "s int]nal motion ( ! p>ts ( ! object1 & propagat$ ?r\gh all ! m$ia to ! inn]mo/ "p ( ! organ4 ,by :i* words ,i h almo/ def9$ :at s5se is4 ,moreov]1 ,i h %[n 7art4 #b1 *ap4 ,,xv7 t all resi/ance is endeav\r opposite to ano!r endeav\r1 t is to say1 reaction4 ,see+1 !re=e1 !re is 9 ! :ole organ1 by reason ( its own int]nal natural motion1 "s resi/ance or reaction aga9/ ! motion :i* is propagat$ f ! object to ! inn]mo/ "p ( ! organ1 !re is also 9 ! same organ an endeav\r opposite to ! endeav\r :i* proce$s f ! object2 s t :5 t endeav\r inw>ds is ! la/ action 9 ! act ( s5se1 !n f ! reaction1 h[ little soev] ! duration ( x be1 a phantasm or idea ha? its be+2 :i*1 by reason t ! endeav\r is n[ outw>ds1 do? always appe> z some?+ situate )\t ! organ4 ,s t n[ ,i %all give y ! :ole def9ition ( s5se1 z x is drawn f ! explication ( ! causes !re( & ! ord] ( its g5]ation1 ?us3 ,,s5se is a phantasm1 made by ! reaction & endeav\r outw>ds 9 ! organ ( s5se1 caus$ by an endeav\r inw>ds f ! object1 rema9+ = "s "t m or less4 ,sid5ote3 ,! subject & object ( s5se4 #c4 ,! subject ( s5se is ! s5ti5t itself1 namely1 "s liv+ creature2 & we speak m correctly1 :5 we say a liv+ creature see?1 ?an :5 we say ! eye see?4 ,! object is ! ?+ receiv$2 & x is m accurately said1 t we see ! sun1 ?an t we see ! light4 ,= light & col\r1 & heat & s\nd1 & o!r qualities :i* are commonly call$ s5sible1 are n objects1 b phantasms 9 ! s5ti5ts4 ,= a phantasm is ! act ( s5se1 & diff]s no o!rwise f s5se ?an fi]i1 t is1 be+ a do+1 diff]s f factum esse1 t is1 be+ done2 :i* diff]5ce1 9 ?+s t are done 9 an in/ant1 is none at all2 & a phantasm is made 9 an in/ant4 ,= 9 all motion :i* proce$s by p]petual propagation1 ! fir/ "p be+ mov$ moves ! second1 ! second ! ?ird1 & s on to ! la/1 & t to any di/ance1 h[ great soev]4 ,& 9 :at po9t ( "t ! fir/ or =emo/ "p proce$$ to ! place ( ! second1 :i* is ?ru/ on1 9 ! same po9t ( "t ! la/ save "o proce$$ into ! place ( ! la/ yield+ "p2 :i* by reaction1 9 ! same in/ant1 if ! reaction be /rong en\gh1 makes a phantasm2 & a phantasm be+ made1 p]ception is made toge!r ) x4 ,sid5ote3 ,! organs ( s5se4 #d4 ,! organs ( s5se1 :i* are 9 ! s5ti5t1 are su* p>ts !re(1 t if !y be hurt1 ! v g5]ation ( phantasms is !reby de/roy$1 ?\gh all ! re/ ( ! p>ts rema9 entire4 ,n[ ^! p>ts 9 ! mo/ ( liv+ creatures are f\nd to be c]ta9 spirits & membranes1 :i*1 proce$+ f ! pia mat]1 involve ! bra9 & all ! n]ves2 also ! bra9 itself1 & ! art]ies :i* are 9 ! bra92 & su* o!r p>ts1 z be+ /irr$1 ! he>t also1 :i* is ! f\nta9 ( all s5se1 is /irr$ toge!r ) !m4 ,= :5soev] ! action ( ! object rea*e? ! body ( ! s5ti5t1 t action is by "s n]ve propagat$ to ! bra92 & if ! n]ve lead+ ?i!r be s hurt or ob/ruct$1 t ! motion c be propagat$ no fur!r1 no s5se foll[s4 ,also if ! motion be int]cept$ betwe5 ! bra9 & ! he>t by ! defect ( ! organ by :i* ! action is propagat$1 !re w be no p]ception ( ! object4 ,sid5ote3 ,all bodies are n endu$ ) s5se4 #e4 ,b ?\gh all s5se1 z ,i h said1 be made by reaction1 nev]!less x is n necess>y t e ?+ t reacte? %\ld h s5se4 ,i "k !re h be5 philosoph]s1 & ^? le>n$ m51 :o h ma9ta9$ t all bodies are endu$ ) s5se4 ,nor d ,i see h[ !y c be refut$1 if ! nature ( s5se be plac$ 9 reaction only4 ,&1 ?\gh by ! reaction ( bodies inanimate a phantasm might be made1 x w\ld nev]!less cease1 z soon z ev] ! object 7 remov$4 ,= unless ^? bodies _h organs1 z liv+ creatures h1 fit = ! reta9+ ( su* motion z is made 9 !m1 _! s5se w\ld be su*1 z t !y %\ld nev] rememb] ! same4 ,& !re=e ?is ha? no?+ to d ) t s5se :i* is ! subject ( my disc\rse4 ,= by s5se1 we commonly und]/& ! judgm5t we make ( objects by _! phantasms2 namely1 by comp>+ & di/+ui%+ ^? phantasms2 :i* we c\ld nev] d1 if t motion 9 ! organ1 by :i* ! phantasm is made1 did n rema9 !re = "s "t1 & make ! same phantasm return4 ,:]e=e s5se1 z ,i "h und]/& x1 & :i* is commonly s call$1 ha? necess>ily "s memory adh]+ to x1 by :i* =m] & lat] phantasms may be comp>$ toge!r1 & di/+ui%$ f "o ano!r4 ,s5se1 !re=e1 prop]ly s call$1 mu/ necess>ily h 9 x a p]petual v>iety ( phantasms1 t !y may be disc]n$ "o f ano!r4 ,= if we %\ld suppose a man to be made ) cle> eyes1 & all ! re/ ( 8 organs ( sight well dispos$1 b endu$ ) no o!r s5se2 & t he %\ld look only ^u "o ?+1 :i* is always ( ! same col\r & figure1 )\t ! lea/ appe>ance ( v>iety1 he w\ld seem to me1 :atsoev] o!rs may say1 to see1 no m ?an ,i seem to myself to feel ! bones ( my own limbs by my organs ( feel+2 & yet ^? bones are always & on all sides t\*$ by a mo/ s5sible membrane4 ,i might p]haps say he 7 a/oni%$1 & look$ ^u x2 b ,i %\ld n say he saw x2 x be+ almo/ all "o = a man to be always s5sible ( "o & ! same ?+1 & n to be s5sible at all ( any ?+4 ,sid5ote3 ,b "o phantasm at "o & ! same "t4 #f4 ,& yet su* is ! nature ( s5se1 t x does n p]mit a man to disc]n _m ?+s at once4 ,= see+ ! nature ( s5se consi/s 9 motion2 z long z ! organs are employ$ ab\t "o object1 !y _c be s mov$ by ano!r at ! same "t1 z to make by bo? _! motions "o s9c]e phantasm ( ea* ( !m at once4 ,& !re=e two sev]al phantasms w n be made by two objects work+ toge!r1 b only "o phantasm comp\nd$ f ! action ( bo?4 ,besides1 z :5 we divide a body1 we divide its place2 & :5 we reckon _m bodies1 we mu/ necess>ily reckon z _m places2 & contr>ily1 z ,i h %[n 9 ! sev5? *apt]2 s :at numb] soev] we say !re be ( times1 we mu/ und]/& ! same numb] ( motions also2 & z (t z we c\nt _m motions1 s (t we reckon _m times4 ,= ?\gh ! object we look ^u be ( div]s col\rs1 yet ) ^? div]s col\rs x is b "o v>i$ object1 & n v>iety ( objects4 ,moreov]1 :il/ ^? organs :i* are common to all ! s5ses1 su* z are ^? p>ts ( e organ :i* proce$ 9 m5 f ! root ( ! n]ves to ! he>t1 are vehem5tly /irr$ by a /rong action f "s "o object1 !y are1 by reason ( ! contumacy :i* ! motion1 !y h already1 gives !m aga9/ ! reception ( all o!r motion1 made ! less fit to receive any o!r impression f :atsoev] o!r objects1 to :at s5se soev] ^? objects belong4 ,& h5ce x is1 t an e>ne/ /udy+ ( "o object1 takes away ! s5se ( all o!r objects = ! pres5t4 ,= /udy is no?+ else b a possession ( ! m9d1 t is to say1 a vehem5t motion made by "s "o object 9 ! organs ( s5se1 :i* are /upid to all o!r motions z long z ?is la/e?2 accord+ to :at 0 said by ,t]5ce1 8,populus /udio /upidus 9 funambulo animum occup>at40 ,= :at is /upor b t :i* ! ,greeks call 1 t is1 a cessation f ! s5se ( o!r ?+s8 ,:]e=e at "o & ! same "t1 we _c by s5se p]ceive m ?an "o s+le object2 z 9 read+1 we see ! lett]s successively "o by "o1 & n all toge!r1 ?\gh ! :ole page be pres5t$ to our eye2 & ?\gh e sev]al lett] be di/9ctly writt5 !re1 yet :5 we look ^u ! :ole page at once1 we read no?+4 ,f h5ce x is manife/1 t e endeav\r ( ! organ outw>ds1 is n to be call$ s5se1 b t only1 :i* at sev]al times is by vehem5ce made /rong] & m pr$om9ant ?an ! re/2 :i* deprives u ( ! s5se ( o!r phantasms1 no o!rwise ?an ! sun deprives ! re/ ( ! />s ( light1 n by h9d]+ _! action1 b by obscur+ & hid+ !m ) 8 excess ( brightness4 ,sid5ote3 ,imag9ation1 ! rema9s ( pa/ s5se2 :i* also is memory4 ,( sleep4 #g4 ,b ! motion ( ! organ1 by :i* a phantasm is made1 is n commonly call$ s5se1 except ! object be pres5t4 ,& ! phantasm rema9+ aft] ! object is remov$ or pa/ by1 is call$ fancy1 & 9 ,lat9 imag9atio2 :i* ^w1 because all phantasms are n images1 do? n fully answ] ! signification ( ! ^w fancy 9 its g5]al acceptation4 ,nev]!less ,i may use x safely en\gh1 by und]/&+ x = ! ,greek 4 ,,imag9ation !re=e is no?+ else b s5se decay+1 or weak5$1 by ! abs5ce ( ! object4 ,b :at may be ! cause ( ?is decay or weak5+8 ,is ! motion ! weak]1 because ! object is tak5 away8 ,if x 71 !n phantasms w\ld always & necess>ily be less cle> 9 ! imag9ation1 ?an !y are 9 s5se2 :i* is n true4 ,= 9 dreams1 :i* are ! imag9ations ( ^? t sleep1 !y are no less cle> ?an 9 s5se itself4 ,b ! reason :y 9 m5 wak+ ! phantasms ( ?+s pa/ are m obscure ?an ^? ( ?+s pres5t1 is ?is1 t _! organs be+ at ! same "t mov$ by o!r pres5t objects1 ^? phantasms are ! less pr$om9ant4 ,:]eas 9 sleep1 ! passages be+ %ut up1 ext]nal action do? n at all di/urb or h9d] int]nal motion4 ,if ?is be true1 ! next ?+ to be consid]$1 w be1 :e!r any cause may be f\nd out1 f ! supposition :]e( x w foll[1 t ! passage is %ut up f ! ext]nal objects ( s5se to ! int]nal organ4 ,i suppose1 !re=e1 t by ! cont9ual action ( objects1 to :i* a reaction ( ! organ1 & m especially ( ! spirits1 is necess>ily consequ5t1 ! organ is we>i$1 t is1 its p>ts are no long] mov$ by ! spirits )\t "s pa92 & consequ5tly ! n]ves be+ ab&on$ & gr[n slack1 !y retire to _! f\nta91 :i* is ! cavity ei!r ( ! bra9 or ( ! he>t2 by :i* means ! action :i* proce$$ by ! n]ves is necess>ily int]cept$4 ,= action ^u a pati5t1 t retires f x1 makes b little impression at ! fir/2 & at la/1 :5 ! n]ves are by little & little slack5$1 none at all4 ,& !re=e !re is no m reaction1 t is1 no m s5se1 till ! organ be+ refre%$ by re/1 & by a supply ( new spirits recov]+ /r5g? & motion1 ! s5ti5t awake?4 ,& ?us x seems to be always1 unless "s o!r pret]natural cause int]v5e2 z heat 9 ! int]nal p>ts f lassitude1 or f "s disease /irr+ ! spirits & o!r p>ts ( ! organ 9 "s extraord9>y mann]4 ,sid5ote3 ,h[ phantasms succe$ "o ano!r4 #h4 ,n[ x is n )\t cause1 nor s casual a ?+ z _m p]haps ?9k x1 t phantasms 9 ?is _! great v>iety proce$ f "o ano!r2 & t ! same phantasms sometimes br+ into ! m9d o!r phantasms l !mselves1 & at o!r times extremely unlike4 ,= 9 ! motion ( any cont9u$ body1 "o "p foll[s ano!r by cohesion2 & !re=e1 :il/ we turn our eyes & o!r organs successively to _m objects1 ! motion :i* 0 made by e "o ( !m rema9+1 ! phantasms are r5ew$ z (t5 z any "o ( ^? motions comes to be pr$om9ant above ! re/2 & !y become pr$om9ant 9 ! same ord] 9 :i* at any "t =m]ly !y 7 g5]at$ by s5se4 ,s t :5 by l5g? ( "t v _m phantasms h be5 g5]at$ )9 u by s5se1 !n almo/ any ?\ght may arise f any o!r ?\ght2 insomu* t x may seem to be a ?+ indiff]5t & casual1 :i* ?\ght %all foll[ :i*4 ,b = ! mo/ "p ?is is n s unc]ta9 a ?+ to wak+ z to sleep+ m54 ,= ! ?\ght or phantasm ( ! desir$ end br+s 9 all ! phantasms1 t are means conduc+ to t end1 & t 9 ord] backw>ds f ! la/ to ! fir/1 & aga9 =w>ds f ! beg9n+ to ! end4 ,b ?is supposes bo? appetite1 & judgm5t to disc]n :at means conduce to ! end1 :i* is gott5 by exp]i5ce2 & exp]i5ce is /ore ( phantasms1 aris+ f ! s5se ( v _m ?+s4 ,= & mem9isse1 fancy & memory1 diff] only 9 ?is1 t memory suppose? ! "t pa/1 :i* fancy do? n4 ,9 memory1 ! phantasms we consid] are z if !y 7 worn out ) "t2 b 9 our fancy we consid] !m z !y are2 :i* di/9ction is n ( ! ?+s !mselves1 b ( ! consid]ations ( ! s5ti5t4 ,= !re is 9 memory some?+ l t :i* happ5s 9 look+ ^u ?+s at a great di/ance2 9 :i* z ! small p>ts ( ! object are n disc]n$1 by reason ( _! remot5ess2 s 9 memory1 _m accid5ts & places & p>ts ( ?+s1 :i* 7 =m]ly p]ceiv$ by s5se1 are by l5g? ( "t decay$ & lo/4 ,! p]petual aris+ ( phantasms1 bo? 9 s5se & imag9ation1 is t :i* we commonly call disc\rse ( ! m9d1 & is common to m5 ) o!r liv+ creatures4 ,= he t ?9ke?1 comp>e? ! phantasms t pass1 t is1 take? notice ( _! lik5ess or unlik5ess to "o ano!r4 ,& z he t obs]ves readily ! lik5esses ( ?+s ( diff]5t natures1 or t are v remote f "o ano!r1 is said to h a good fancy2 s he is said to h a good judgm5t1 t f9ds out ! unlik5esses or diff]5ces ( ?+s t are l "o ano!r4 ,n[ ?is obs]vation ( diff]5ces is n p]ception made by a common organ ( s5se1 di/9ct f s5se or p]ception prop]ly s call$1 b is memory ( ! diff]5ces ( p>ticul> phantasms rema9+ = "s "t2 z ! di/9ction betwe5 hot & lucid1 is no?+ else b ! memory bo? ( a heat+1 & ( an enlight5+ object4 ,sid5ote3 ,dreams1 :5ce !y proce$4 #i4 ,! phantasms ( m5 t sleep1 are dreams4 ,conc]n+ :i* we are taught by exp]i5ce ^! five ?+s4 ,fir/1 t = ! mo/ "p !re is nei!r ord] nor coh]5ce 9 !m4 ,secondly1 t we dream ( no?+ b :at is comp\nd$ & made up ( ! phantasms ( s5se pa/4 ,?irdly1 t sometimes !y proce$1 z 9 ^? t are dr[sy1 f ! int]ruption ( _! phantasms by little & little1 brok5 & alt]$ ?r\gh sleep9ess2 & sometimes also !y beg9 9 ! mid/ ( sleep4 ,f\r?ly1 t !y are cle>] ?an ! imag9ations ( wak+ m51 except su* z are made by s5se itself1 to :i* !y are equal 9 cle>ness4 ,fif?ly1 t :5 we dream1 we admire nei!r ! places nor ! looks ( ! ?+s t appe> to u4 ,n[ f :at ha? be5 said1 x is n h>d to %[ :at may be ! causes ( ^! ph5om5a4 ,= z = ! fir/1 see+ all ord] & coh]5ce proce$s f frequ5t look+ back to ! end1 t is1 f consultation2 x mu/ ne$s be1 t see+ 9 sleep we lose all ?\ght ( ! end1 our phantasms succe$ "o ano!r1 n 9 t ord] :i* t5ds to any end1 b z x happ5e?1 & 9 su* mann]1 z objects pres5t !mselves to our eyes :5 we look indiff]5tly ^u all ?+s be=e u1 & see !m1 n because we w\ld see !m1 b because we d n %ut our eyes2 = !n !y appe> to u )\t any ord] at all4 ,! second proce$s f ?is1 t 9 ! sil5ce ( s5se !re is no new motion f ! objects1 & !re=e no new phantasm1 unless we call t new1 :i* is comp\nd$ ( old ones1 z a *im]a1 a gold5 m\nta91 & ! l4 ,z = ! ?ird1 :y a dream is sometimes z x 7 ! cont9uation ( s5se1 made up ( brok5 phantasms1 z 9 m5 di/emp]$ ) sickness1 ! reason is manife/ly ?is1 t 9 "s ( ! organs s5se rema9s1 & 9 o!rs x faile?4 ,b h[ "s phantasms may be reviv$1 :5 all ! ext]ior organs are b5umb$ ) sleep1 is n s easily %[n4 ,nev]!less t1 :i* ha? already be5 said1 conta9s ! reason ( ?is also4 ,= :atsoev] /rikes ! pia mat]1 revive? "s ( ^? phantasms t are /ill 9 motion 9 ! bra92 & :5 any int]nal motion ( ! he>t rea*e? t membrane1 !n ! pr$om9ant motion 9 ! bra9 makes ! phantasm4 ,n[ ! motions ( ! he>t are appetites & av]sions1 ( :i* ,i %all pres5tly speak fur!r4 ,& z appetites & av]sions are g5]at$ by phantasms1 s reciprocally phantasms are g5]at$ by appetites & av]sions4 ,= example1 heat 9 ! he>t proce$s f ang] & fight+2 & aga91 f heat 9 ! he>t1 :atsoev] be ! cause ( x1 is g5]at$ ang] & ! image ( an enemy1 9 sleep4 ,& z love & beauty /ir up heat 9 c]ta9 organs2 s heat 9 ! same organs1 f :atsoev] x proce$s1 (t5 cause? desire & ! image ( an unresi/+ beauty4 ,la/ly1 cold do? 9 ! same mann] g5]ate fe> 9 ^? t sleep1 & cause? !m to dream ( gho/s1 & to h phantasms ( horror & dang]2 z fe> also cause? cold 9 ^? t wake4 ,s reciprocal are ! motions ( ! he>t & bra94 ,! f\r?1 namely1 t ! ?+s we seem to see & feel 9 sleep1 are z cle> z 9 s5se itself1 proce$s f two causes2 "o1 t hav+ !n no s5se ( ?+s )\t u1 t int]nal motion :i* makes ! phantasm1 9 ! abs5ce ( all o!r impressions1 is pr$om9ant2 & ! o!r1 t ! p>ts ( our phantasms :i* are decay$ & worn out by "t1 are made up ) o!r fictiti\s p>ts4 ,to conclude1 :5 we dream1 we d n wond] at /range places & ! appe>ances ( ?+s unkn[n to u1 because admiration requires t ! ?+s appe>+ be new & unusual1 :i* c happ5 to none b ^? t rememb] =m] appe>ances2 :]eas 9 sleep1 all ?+s appe> z pres5t4 ,b x is "h to be obs]v$1 t c]ta9 dreams1 especially su* z "s m5 h :5 !y are betwe5 sleep+ & wak+1 & su* z happ5 to ^? t h no k ( ! nature ( dreams & are )al sup]/iti\s1 7 n h]et(ore nor are n[ acc\nt$ dreams4 ,= ! app>itions m5 ?\ght !y saw1 & ! voices !y ?\ght !y he>d 9 sleep1 7 n believ$ to be phantasms1 b ?+s subsi/+ ( !mselves1 & objects )\t ^? t dream$4 ,= to "s m51 z well sleep+ z wak+1 b especially to guilty m51 & 9 ! night1 & 9 hall[$ places1 fe> alone1 help$ a little ) ! /ories ( su* app>itions1 ha? rais$ 9 _! m9ds t]rible phantasms1 :i* h be5 & are /ill deceitfully receiv$ = ?+s really true1 "u ! names ( gho/s & incorporeal sub/ances4 ,sid5ote3 ,( ! s5ses1 _! k9ds1 _! organs & phantasms1 prop] & common4 #aj4 ,9 mo/ liv+ creatures !re are obs]v$ five k9ds ( s5ses1 :i* are di/+ui%$ by _! organs1 & by _! diff]5t k9ds ( phantasms2 namely1 sight1 he>+1 smell1 ta/e1 & t\*2 & ^! h _! organs p>tly peculi> to ea* ( !m sev]ally1 & p>tly common to !m all4 ,! organ ( sight is p>tly animate1 & p>tly inanimate4 ,! inanimate p>ts are ! ?ree hum\rs2 namely1 ! wat]y hum\r1 :i* by ! int]position ( ! membrane call$ uvea1 ! p]=ation :]e( is call$ ! apple ( ! eye1 is conta9$ on "o side by ! fir/ concave sup]ficies ( ! eye1 & on ! o!r side by ! cili>y processes1 & ! coat ( ! cry/all9e hum\r2 ! cry/all9e1 :i*1 hang+ 9 ! mid/ betwe5 ! cili>y processes1 & be+ almo/ ( sph]ical figure1 & ( a ?ick consi/5ce1 is enclos$ on all sides ) its own transp>5t coat2 & ! vitre\s or glassy hum\r1 :i* fille? all ! re/ ( ! cavity ( ! eye1 & is some:at ?ick] !n ! wat]y hum\r1 b ?9n] ?an ! cry/all9e4 ,! animate "p ( ! organ is1 fir/1 ! membrane *oroeides1 :i* is a "p ( ! pia mat]1 sav+ t x is cov]$ ) a coat d]iv$ f ! m>r[ ( ! optic n]ve1 :i* is call$ ! ret9a2 & ?is *oroeides1 see+ x is "p ( ! pia mat]1 is cont9u$ to ! beg9n+ ( ! m$ulla sp9alis )9 ! scull1 9 :i* all ! n]ves :i* are )9 ! head h _! roots4 ,:]e=e all ! animal spirits t ! n]ves receive1 ent] into !m !re2 = x is n imag9able t !y c ent] into !m any:]e else4 ,see+ !re=e s5se is no?+ else b ! action ( objects propagat$ to ! fur!/ "p ( ! organ2 & see+ also t animal spirits are no?+ b vital spirits purifi$ by ! he>t1 & c>ri$ f x by ! art]ies2 x foll[s necess>ily1 t ! action is d]iv$ f ! he>t by "s ( ! art]ies to ! roots ( ! n]ves :i* are 9 ! head1 :e!r ^? art]ies be ! plexus reti=mis1 or :e!r !y be o!r art]ies :i* are ins]t$ into ! sub/ance ( ! bra94 ,&1 !re=e1 ^? art]ies are ! complem5t or ! rema9+ "p ( ! :ole organ ( sight4 ,& ?is la/ "p is a common organ to all ! s5ses2 :]eas1 t :i* rea*e? f ! eye to ! roots ( ! n]ves is prop] only to sight4 ,! prop] organ ( he>+ is ! tympanum ( ! e> & its own n]ve2 f :i* to ! he>t ! organ is common4 ,s ! prop] organs ( smell & ta/e are n]v\s membranes1 9 ! palate & tongue = ! ta/e1 & 9 ! no/rils = ! smell2 & f ! roots ( ^? n]ves to ! he>t all is common4 ,la/ly1 ! prop] organ ( t\* are n]ves & membranes disp]s$ ?r\gh ! :ole body2 :i* membranes are d]iv$ f ! root ( ! n]ves4 ,& all ?+s else belong+ alike to all ! s5ses seem to be adm9i/]$ by ! art]ies1 & n by ! n]ves4 ,! prop] phantasm ( sight is light2 & "u ?is "n ( light1 col\r also1 :i* is no?+ b p]turb$ light1 is compreh5d$4 ,:]e=e ! phantasm ( a lucid body is light2 & ( a col\r$ body1 col\r4 ,b ! object ( sight1 prop]ly s call$1 is nei!r light nor col\r1 b ! body itself :i* is lucid1 or enlight5$1 or col\r$4 ,= light & col\r1 be+ phantasms ( ! s5ti5t1 _c be accid5ts ( ! object4 ,:i* is manife/ en\gh f ?is1 t visible ?+s appe> (t5times 9 places 9 :i* we "k assur$ly !y are n1 & t 9 diff]5t places !y are ( diff]5t col\rs1 & may at "o & ! same "t appe> 9 div]s places4 ,motion1 re/1 magnitude1 & figure1 are common bo? to ! sight & t\*2 & ! :ole appe>ance toge!r ( figure1 & light or col\r1 is by ! ,greeks commonly call$ 1 & 1 & 2 & by ! ,lat9s1 species & imago2 all :i* names signify no m b appe>ance4 ,! phantasm1 :i* is made by he>+1 is s\nd2 by smell1 od\r2 by ta/e1 sav\r2 & by t\*1 h>dness & s(tness1 heat & cold1 wetness1 oil9ess1 & _m m1 :i* are easi] to be di/+ui%$ by s5se ?an words4 ,smoo?ness1 r\ghness1 r>ity1 & d5sity1 ref] to figure1 & are !re=e common bo? to t\* & sight4 ,& z = ! objects ( he>+1 smell1 ta/e1 & t\*1 !y are n s\nd1 od\r1 sav\r1 h>dness1 c41 b ! bodies !mselves f :i* s\nd1 od\r1 sav\r1 h>dness1 c4 proce$2 ( ! causes ( :i*1 & ( ! mann] h[ !y are produc$1 ,i %all speak h]eaft]4 ,b ^! phantasms1 ?\gh !y be effects 9 ! s5ti5t1 z subject1 produc$ by objects work+ ^u ! organs2 yet !re are also o!r effects besides ^!1 produc$ by ! same objects 9 ! same organs2 namely c]ta9 motions proce$+ f s5se1 :i* are call$ animal motions4 ,= see+ 9 all s5se ( ext]nal ?+s !re is mutual action & reaction1 t is1 two endeav\rs oppos+ "o ano!r1 x is manife/ t ! motion ( bo? ( !m toge!r w be cont9u$ e way1 especially to ! conf9es ( bo? ! bodies4 ,& :5 ?is happ5s 9 ! int]nal organ1 ! endeav\r outw>ds w proce$ 9 a solid angle1 :i* w be great]1 & consequ5tly ! idea great]1 ?an x w\ld h be5 if ! impression _h be5 weak]4 ,sid5ote3 ,! magnitude ( images1 h[ & by :at x is det]m9$4 #aa4 ,f h5ce ! natural cause is manife/1 fir/1 :y ^? ?+s seem to be great]1 :i*1 ct]is p>ibus1 are se5 9 a great] angle3 secondly1 :y 9 a s]5e cold night1 :5 ! moon do? n %9e1 m ( ! fix$ />s appe> ?an at ano!r "t4 ,= _! action is less h9d]$ by ! s]5ity ( ! air1 & n obscur$ by ! great] light ( ! moon1 :i* is !n abs5t2 & ! cold1 mak+ ! air m press+1 helpe? or /r5g!ne? ! action ( ! />s ^u our eyes2 9 s mu* z />s may !n be se5 :i* are se5 at no o!r "t4 ,& ?is may suffice to be said 9 g5]al conc]n+ s5se made by ! reaction ( ! organ4 ,=1 z = ! place ( ! image1 ! deceptions ( sight1 & o!r ?+s ( :i* we h exp]i5ce 9 ourselves by s5se1 see+ !y dep5d = ! mo/ "p ^u ! fabric itself ( ! eye ( man1 ,i %all speak ( !m !n :5 ,i come to speak ( man4 ,sid5ote3 ,pleasure1 pa91 appetite1 & av]sion1 :at !y are4 #ab4 ,b !re is ano!r k9d ( s5se1 ( :i* ,i w say some?+ 9 ?is place1 namely1 ! s5se ( pleasure & pa91 proce$+ n f ! reaction ( ! he>t outw>ds1 b f cont9ual action f ! out]mo/ "p ( ! organ t[>ds ! he>t4 ,= ! orig9al ( life be+ 9 ! he>t1 t motion 9 ! s5ti5t1 :i* is propagat$ to ! he>t1 mu/ necess>ily make "s alt]ation or div]sion ( vital motion1 namely1 by quick5+ or slack5+1 help+ or h9d]+ ! same4 ,n[ :5 x helpe?1 x is pleasure2 & :5 x h9d]e?1 x is pa91 tr\ble1 grief1 c4 ,& z phantasms seem to be )\t1 by reason ( ! endeav\r outw>ds1 s pleasure & pa91 by reason ( ! endeav\r ( ! organ inw>ds1 seem to be )92 namely1 !re :]e ! fir/ cause ( ! pleasure or pa9 is2 z :5 ! pa9 proce$s f a w\nd1 we ?9k ! pa9 & ! w\nd are bo? 9 ! same place4 ,sid5ote3 ,pleasure1 pa91 appetite1 & av]sion1 :at !y are4 ,n[ vital motion is ! motion ( ! blood1 p]petually circulat+ 7z ha? be5 %[n f _m infallible signs & m>ks by ,doctor ,h>vey1 ! fir/ obs]v] ( x7 9 ! ve9s & art]ies4 ,:i* motion1 :5 x is h9d]$ by "s o!r motion made by ! action ( s5sible objects1 may be re/or$ aga9 ei!r by b5d+ or sett+ /rait ! p>ts ( ! body2 :i* is done :5 ! spirits are c>ri$ n[ into ^!1 n[ into o!r n]ves1 till ! pa91 z f> z is possible1 be q tak5 away4 ,b if vital motion be help$ by motion made by s5se1 !n ! p>ts ( ! organ w be dispos$ to guide ! spirits 9 su* mann] z conduce? mo/ to ! pres]vation & augm5tation ( t motion1 by ! help ( ! n]ves4 ,& 9 animal motion ?is is ! v fir/ endeav\r1 & f\nd ev5 9 ! embryo2 :i* :ile x is 9 ! womb1 move? its limbs ) volunt>y motion1 = ! avoid+ ( :atsoev] tr\ble? x1 or = ! pursu+ ( :at please? x4 ,& ?is fir/ endeav\r1 :5 x t5ds t[>ds su* ?+s z are kn[n by exp]i5ce to be pleasant1 is call$ appetite1 t is1 an approa*+2 & :5 x %uns :at is tr\blesome1 av]sion1 or fly+ f x4 ,& little infants1 at ! beg9n+ & z soon z !y are born1 h appetite to v few ?+s1 z also !y avoid v few1 by reason ( _! want ( exp]i5ce & memory2 & !re=e !y h n s great a v>iety ( animal motion z we see 9 ^? t are m gr[n4 ,= x is n possible1 )\t su* k z is d]iv$ f s5se1 t is1 )\t exp]i5ce & memory1 to "k :at w prove pleasant or hurtful2 only !re is "s place = conjecture f ! looks or aspects ( ?+s4 ,& h5ce x is1 t ?\gh !y d n "k :at may d !m good or h>m1 yet sometimes !y approa* & sometimes retire f ! same ?+1 z _! d\bt prompts !m4 ,b aft]w>ds1 by accu/om+ !mselves by little & little1 !y come to "k readily :at is to be pursu$ & :at to be avoid$2 & also to h a ready use ( _! n]ves & o!r organs1 9 ! pursu+ & avoid+ ( good & bad4 ,:]e=e appetite & av]sion are ! fir/ endeav\rs ( animal motion4 ,consequ5t to ?is fir/ endeav\r1 is ! impulsion into ! n]ves & retraction aga9 ( animal spirits1 ( :i* x is necess>y !re be "s receptacle or place ne> ! orig9al ( ! n]ves2 & ?is motion or endeav\r is foll[$ by a swell+ & relaxation ( ! muscles2 & la/ly1 ^! are foll[$ by contraction & ext5sion ( ! limbs1 :i* is animal motion4 ,sid5ote3 ,delib]ation & w1 :at4 #ac4 ,! consid]ations ( appetites & av]sions are div]s4 ,= see+ liv+ creatures h sometimes appetite & sometimes av]sion to ! same ?+1 z !y ?9k x w ei!r be = _! good or _! hurt2 :ile t vicissitude ( appetites & av]sions rema9s 9 !m1 !y h t s]ies ( ?\ghts :i* is call$ delib]ation2 :i* la/e? z long z !y h x 9 _! p[] to obta9 t :i* please?1 or to avoid t :i* displease? !m4 ,appetite1 !re=e1 & av]sion are simply s call$ z long z !y foll[ n delib]ation4 ,b if delib]ation h gone be=e1 !n ! la/ act ( x1 if x be appetite1 is call$ w2 if av]sion1 unwill+ness4 ,s t ! same ?+ is call$ bo? w & appetite2 b ! consid]ation ( !m1 namely1 be=e & aft] delib]ation1 is div]s4 ,nor is t :i* is done )9 a man :il/ he wille? any ?+1 diff]5t f t :i* is done 9 o!r liv+ creatures1 :il/1 delib]ation hav+ prec$$1 !y h appetite4 ,nei!r is ! fre$om ( will+ or n will+1 great] 9 man1 ?an 9 o!r liv+ creatures4 ,= :]e !re is appetite1 ! entire cause ( appetite ha? prec$$2 &1 consequ5tly1 ! act ( appetite c\ld n *oose b foll[1 t is1 ha? ( necessity foll[$ 7z is %[n 9 *apt] ,,ix1 article #e74 ,& !re=e su* a lib]ty z is free f necessity1 is n to be f\nd 9 ! w ei!r ( m5 or bea/s4 ,b if by lib]ty we und]/& ! faculty or p[]1 n ( will+1 b ( do+ :at !y w1 !n c]ta9ly t lib]ty is to be all[$ to bo?1 & bo? may equally h x1 :5soev] x is to be _h4 ,aga91 :5 appetite & av]sion d ) cel]ity succe$ "o ano!r1 ! :ole s]ies made by !m ha? its "n sometimes f "o1 sometimes f ! o!r4 ,= ! same delib]ation1 :il/ x incl9es sometimes to "o1 sometimes to ! o!r1 is f appetite call$ hope1 & f av]sion1 fe>4 ,= :]e !re is no hope1 x is n to be call$ fe>1 b hate2 & :]e no fe>1 n hope1 b desire4 ,to conclude1 all ! passions1 call$ passions ( ! m9d1 consi/ ( appetite & av]sion1 except pure pleasure & pa91 :i* are a c]ta9 fruition ( good or evil2 z ang] is av]sion f "s imm95t evil1 b su* z is jo9$ ) appetite ( avoid+ t evil by =ce4 ,b because ! passions & p]turbations ( ! m9d are innum]able1 & _m ( !m n to be disc]n$ 9 any creatures besides m52 ,i w speak ( !m m at l>ge 9 t section :i* is conc]n+ man4 ,z = ^? objects1 if !re be any su*1 :i* d n at all /ir ! m9d1 we are said to contemn !m4 ,& ?us mu* ( s5se 9 g5]al4 ,9 ! next place ,i %all speak ( s5sible objects4 ------- ,,*apt] ,,xxvi4 ,,( ,,! ,,_w ,,& ,,( ,,! ,,/>s4 #a4 ,! magnitude & duration ( ! _w1 inscrutable4--#b4 ,no place 9 ! _w empty4--#c4 ,! argum5ts ( ,lucretius = vacuum1 invalid4--#d4 ,o!r argum5ts = ! e/abli%+ ( vacuum1 invalid4--#e4 ,six suppositions = ! salv+ ( ! ph5om5a ( nature4--#f4 ,possible causes ( ! motions annual & diurnal2 & ( ! app>5t direction1 /ation1 & retrogradation ( ! planets4--#g4 ,! supposition ( simple motion1 :y likely4--#h4 ,! cause ( ! ecc5tricity ( ! annual motion ( ! e>?4--#i4 ,! cause :y ! moon ha? always "o & ! same face turn$ t[>ds ! e>?4--#aj4 ,! cause ( ! tides ( ! ocean4--#aa4 ,! cause ( ! precession ( ! equ9oxes4 ,sid5ote3 ,! magnitude & duration ( ! _w1 inscrutable4 #a4 ,consequ5t to ! contemplation ( s5se is ! contemplation ( bodies1 :i* are ! effici5t causes or objects ( s5se4 ,n[ e object is ei!r a "p ( ! :ole _w1 or an aggregate ( p>ts4 ,! greate/ ( all bodies1 or s5sible objects1 is ! _w itself2 :i* we behold :5 we look r\nd ab\t u f ?is po9t ( ! same :i* we call ! e>?4 ,conc]n+ ! _w1 z x is "o aggregate ( _m p>ts1 ! ?+s t fall "u inquiry are b few2 & ^? we c det]m9e1 none4 ,( ! :ole _w we may inquire :at is its magnitude1 :at its duration1 & h[ _m !re be1 b no?+ else4 ,= z = place & "t1 t is to say1 magnitude & duration1 !y are only our own fancy ( a body simply s call$1 t is to say1 ( a body indef9itely tak51 z ,i h %[n be=e 9 *apt] ,,vii4 ,all o!r phantasms are ( bodies or objects1 z !y are di/+ui%$ f "o ano!r2 z col\r1 ! phantasm ( col\r$ bodies2 s\nd1 ( bodies t move ! s5se ( he>+1 c4 ,! que/ions conc]n+ ! magnitude ( ! _w are :e!r x be f9ite or inf9ite1 full or n full2 conc]n+ its duration1 :e!r x _h a beg9n+1 or be et]nal2 & conc]n+ ! numb]1 :e!r !re be "o or _m2 ?\gh z conc]n+ ! numb]1 if x 7 ( inf9ite magnitude1 !re c\ld be no controv]sy at all4 ,also if x _h a beg9n+1 !n by :at cause & ( :at matt] x 0 made2 & aga91 f :5ce t cause & t matt] _h _! be+1 w be new que/ions2 till at la/ we come to "o or _m et]nal cause or causes4 ,& ! det]m9ation ( all ^! ?+s belonge? to him t pr(esse? ! univ]sal doctr9e ( philosophy1 9 case z mu* c\ld be kn[n z c be s\ght4 ,b ! k ( :at is inf9ite c nev] be atta9$ by a f9ite inquir]4 ,:atsoev] we "k t are m51 we le>n x f our phantasms2 & ( inf9ite1 :e!r magnitude or "t1 !re is no phantasm at all2 s t x is impossible ei!r = a man or any o!r creature to h any conception ( inf9ite4 ,& ?\gh a man may f "s effect proce$ to ! imm$iate cause !re(1 & f t to a m remote cause1 & s asc5d cont9ually by "r ratioc9ation f cause to cause2 yet he w n be able to proce$ et]nally1 b we>i$ w at la/ give ov]1 )\t kn[+ :e!r x 7 possible = him to proce$ to an end or n4 ,b :e!r we suppose ! _w to be f9ite or inf9ite1 no absurdity w foll[4 ,= ! same ?+s :i* n[ appe>1 might appe>1 :e!r ! ,creator _h pleas$ x %\ld be f9ite or inf9ite4 ,besides1 ?\gh f ?is1 t no?+ c move itself1 x may rightly be inf]r$ t !re 0 "s fir/ et]nal mov5t2 yet x c nev] be inf]r$1 ?\gh "s us$ to make su* inf]5ce1 t t mov5t 0 et]nally immoveable1 b r et]nally mov$4 ,= z x is true1 t no?+ is mov$ by itself2 s x is true also t no?+ is mov$ b by t :i* is already mov$4 ,! que/ions !re=e ab\t ! magnitude & beg9n+ ( ! _w1 are n to be det]m9$ by philosoph]s1 b by ^? t are lawfully au?oriz$ to ord] ! wor%ip ( ,god4 ,= z ,almighty ,god1 :5 he _h br\ght 8 p into ,juda1 all[$ ! prie/s ! fir/ fruits res]v$ to himself2 s :5 he _h deliv]$ up ! _w to ! disputations ( m51 x 0 8 pleasure t all op9ions conc]n+ ! nature ( inf9ite & et]nal1 kn[n only to himself1 %\ld1 z ! fir/ fruits ( wisdom1 be judg$ by ^? ^: m9i/ry he meant to use 9 ! ord]+ ( religion4 ,i _c !re=e comm5d ^? t boa/ !y h demon/rat$1 by reasons drawn f natural ?+s1 t ! _w _h a beg9n+4 ,!y are contemn$ by idiots1 because !y und]/& !m n2 & by ! le>n$1 because !y und]/& !m2 by bo? des]v$ly4 ,= :o c comm5d him t demon/rates ?us8 ,if ! _w be et]nal1 !n an inf9ite numb] ( days1 or o!r measures ( "t1 prec$$ ! bir? ( ,abraham4 ,b ! bir? ( ,abraham prec$$ ! bir? ( ,isaac2 & !re=e "o inf9ite is great] ?an ano!r inf9ite1 or "o et]nal ?an ano!r et]nal2 :i*1 he says1 is absurd4 ,?is demon/ration is l 81 :o f ?is1 t ! numb] ( ev5 numb]s is inf9ite1 w\ld conclude t !re are z _m ev5 numb]s z !re are numb]s simply1 t is to say1 ! ev5 numb]s are z _m z all ! ev5 & odd toge!r4 ,!y1 :i* 9 ?is mann] take away et]nity f ! _w1 d !y n by ! same means take away et]nity f ! ,creator ( ! _w8 ,f ?is absurdity !re=e !y run into ano!r1 be+ =c$ to call et]nity nunc /ans1 a /&+ /ill ( ! pres5t "t1 or an abid+ n[2 &1 :i* is mu* m absurd1 to give to ! inf9ite numb] ( numb]s ! "n ( unity4 ,b :y %\ld et]nity be call$ an abid+ n[1 r ?an an abid+ !n8 ,:]e=e !re mu/ ei!r be _m et]nities1 or n[ & !n mu/ signify ! same4 ,) su* demon/rators z ^!1 t speak 9 ano!r language1 x is impossible to ent] into disputation4 ,& ! m51 t reason ?us absurdly1 are n idiots1 b1 :i* makes ! absurdity unp>donable1 geometricians1 & su* z take ^u !m to be judges1 imp]t95t1 b sev]e judges ( o!r m5's demon/rations4 ,! reason is ?is1 t z soon z !y are entangl$ 9 ! words inf9ite & et]nal1 ( :i* we h 9 our m9d no idea1 b t ( our own insuffici5cy to compreh5d !m1 !y are =c$ ei!r to speak some?+ absurd1 or1 :i* !y love worse1 to hold _! peace4 ,= geometry ha? 9 x some:at l w9e1 :i*1 :5 new1 is w9dy2 b aft]w>ds ?\gh less pleasant1 yet m :olesome4 ,:atsoev] !re=e is true1 "y geometricians ?9k demon/rable2 b eld] n4 ,:]e=e ,i purposely pass ov] ! que/ions ( inf9ite & et]nal2 cont5t+ myself ) t doctr9e conc]n+ ! beg9n+ & magnitude ( ! _w1 :i* ,i h be5 p]suad$ to by ! holy ,scriptures & fame ( ! miracles :i* confirm !m2 & by ! cu/om ( my c\ntry1 & rev]5ce due to ! laws4 ,& s ,i pass on to su* ?+s z x is n unlawful to dispute (4 ,sid5ote3 ,no place 9 ! _w empty4 #b4 ,conc]n+ ! _w x is fur!r que/ion$1 :e!r ! p>ts !re( be contigu\s to "o ano!r1 9 su* mann] z n to admit ( ! lea/ empty space betwe52 & ! disputation bo? = & aga9/ x is c>ri$ on ) probability en\gh4 ,= ! tak+ away ( vacuum1 ,i w in/ance 9 only "o exp]im5t1 a common "o1 b ,i ?9k unansw]able4 ,let ,a ,b 79 fig4 #a7 repres5t a vessel1 su* z g>d5]s use to wat] _! g>d5s )al2 ^: bottom ,b is full ( little holes2 & ^: m\? ,a may be /opp$ ) "o's f+]1 :5 !re %all be ne$4 ,if n[ ?is vessel be fill$ ) wat]1 ! hole at ! top ,a be+ /opp$1 ! wat] w n fl[ out at any ( ! holes 9 ! bottom ,b4 ,b if ! f+] be remov$ to let 9 ! air above1 x w run out at !m all2 & z2 soon z ! f+] is appli$ to x aga91 ! wat] w sudd5ly & totally be /ay$ aga9 f runn+ out4 ,! cause :]e( seems to be no o!r b ?is1 t ! wat] _c by its natural endeav\r to desc5d drive d[n ! air bel[ x1 because !re is no place = x to g into1 unless ei!r by ?ru/+ away ! next contigu\s air1 x proce$ by cont9ual endeav\r to ! hole ,a1 :]e x may ent] & succe$ into ! place ( ! wat] t fl[e? out1 or else1 by resi/+ ! endeav\r ( ! wat] d[nw>ds1 p5etrate ! same & pass up ?r\gh x4 ,by ! fir/ ( ^! ways1 :ile ! hole at ,a rema9s /opp$1 !re is no possible passage2 nor by ! second1 unless ! holes be s great t ! wat]1 fl[+ out at !m1 c by its own weight =ce ! air at ! same "t to asc5d into ! vessel by ! same holes3 z we see x does 9 a vessel ^: m\? is wide en\gh1 :5 we turn sudd5ly ! bottom upw>ds to p\r out ! wat]2 = !n ! air be+ =c$ by ! weight ( ! wat]1 ent]s1 z is evid5t by ! sobb+ & resi/ance ( ! wat]1 at ! sides or circumf]5ce ( ! orifice4 ,& ?is ,i take = a sign t all space is full2 = )\t ?is1 ! natural motion ( ! wat]1 :i* is a heavy body1 d[nw>ds1 w\ld n be h9d]$4 ,sid5ote3 ,! argum5ts ( ,lucretius = vacuum invalid4 #c4 ,on ! contr>y1 = ! e/abli%+ ( vacuum1 _m & speci\s argum5ts & exp]im5ts h be5 br\ght4 ,nev]!less !re seems to be some?+ want+ 9 all ( !m to conclude x firmly4 ,^! argum5ts = vacuum are p>tly made by ! foll[]s ( ! doctr9e ( ,epicurus2 :o taught t ! _w consi/s ( v small spaces n fill$ by any body1 & ( v small bodies t h )9 !m no empty space1 :i* by reason ( _! h>dness he calls atoms2 & t ^! small bodies & spaces are e :]e int]m+l$4 ,_! argum5ts are ?us deliv]$ by ,lucretius4 ,& fir/ he says1 t unless x 7 s1 !re c\ld be no motion4 ,= ! (fice & prop]ty ( bodies is to )/& & h9d] motion4 ,if1 !re=e1 ! univ]se 7 fill$ ) body1 motion w\ld ev]y:]e be h9d]$1 s z to h no beg9n+ any:]e2 & consequ5tly !re w\ld be no motion at all4 ,x is true t 9 :atsoev] is full & at re/ 9 all its p>ts1 x is n possible motion %\ld h beg9n+4 ,b no?+ is drawn f h5ce = ! prov+ ( vacuum4 ,= ?\gh x %\ld be grant$ t !re is vacuum1 yet if ! bodies :i* are int]m+l$ ) x1 %\ld all at once & toge!r be at re/1 !y w\ld nev] be mov$ aga94 ,= x has be5 demon/rat$ above1 9 *ap4 ,,ix1 art4 #g1 t no?+ c be mov$ b by t :i* is contigu\s & already mov$4 ,b suppos+ t all ?+s are at re/ toge!r1 !re c be no?+ contigu\s & mov$1 & !re=e no beg9n+ ( motion4 ,n[ ! d5y+ ( ! beg9n+ ( motion1 do? n take away pres5t motion1 unless beg9n+ be tak5 away f body also4 ,= motion may be ei!r co-et]nal1 or concreat$ ) body4 ,nor do? x seem m necess>y t bodies 7 fir/ at re/1 & aft]w>ds mov$1 ?an t !y 7 fir/ mov$1 & re/$1 if ev] !y re/$ at all1 aft]w>ds4 ,nei!r do? !re appe> any cause1 :y ! matt] ( ! _w %\ld1 = ! admission ( motion1 be int]m+l$ ) empty spaces r ?an full2 ,i say full1 b )al fluid4 ,nor1 la/ly1 is !re any reason :y ^? h>d atoms may n also1 by ! motion ( int]m+l$ fluid matt]1 be congregat$ & br\ght toge!r into comp\nd$ bodies ( su* bigness z we see4 ,:]e=e no?+ c by ?is argum5t be conclud$1 b t motion 0 ei!r coet]nal1 or ( ! same duration ) t :i* is mov$2 nei!r ( :i* conclusions consi/e? ) ! doctr9e ( ,epicurus1 :o all[s nei!r to ! _w nor to motion any beg9n+ at all4 ,! necessity1 !re=e1 ( vacuum is n hi!rto demon/rat$4 ,& ! cause1 z f> z ,i und]/& f !m t h disc\rs$ ) me ( vacuum1 is ?is1 t :il/ !y contemplate ! nature ( fluid1 !y conceive x to consi/1 z x 71 ( small gra9s ( h>d matt]1 9 su* mann] z meal is fluid1 made s by gr9d+ ( ! corn2 :5 nev]!less x is possible to conceive fluid to be ( its own nature z homog5e\s z ei!r an atom1 or z vacuum itself4 ,! second ( _! argum5ts is tak5 f weight1 & is conta9$ 9 ^! v]ses ( ,lucretius3 ,corporis (ficium e/ quoniam prem]e omnia deorsum2 ,contra autem natura manet s9e pond]e inanis2 ,ergo1 quod magnum e/ que1 leviusque videtur1 ,nimirum plus esse sibi decl>at inanis4--,i4 #cfc-ff4 ,t is to say1 see+ ! (fice & prop]ty ( body is to press all ?+s d[nw>ds2 & on ! contr>y1 see+ ! nature ( vacuum is to h no weight at all2 !re=e :5 ( two bodies ( equal magnitude1 "o is light] ?an ! o!r1 x is manife/ t ! light] body ha? 9 x m vacuum ?an ! o!r4 ,to say no?+ ( ! assumption conc]n+ ! endeav\r ( bodies d[nw>ds1 :i* is n rightly assum$1 because ! _w ha? no?+ to d ) d[nw>ds1 :i* is a m]e fiction ( ours2 nor ( ?is1 t if all ?+s t5d$ to ! same l[e/ "p ( ! _w1 ei!r !re w\ld be no coalesc5ce at all ( bodies1 or !y w\ld all be ga!r$ toge!r into ! same place3 ?is only is suffici5t to take away ! =ce ( ! argum5t1 t air1 int]m+l$ ) ^? 8 atoms1 _h s]v$ z well = 8 purpose z 8 int]m+l$ vacuum4 ,! ?ird argum5t is drawn f ?is1 t lightn+1 s\nd1 heat & cold1 d p5etrate all bodies1 except atoms1 h[ solid soev] !y be4 ,b ?is reason1 except x be fir/ demon/rat$ t ! same ?+s _c happ5 )\t vacuum by p]petual g5]ation ( motion1 is altoge!r invalid4 ,b t all ! same ?+s may s happ51 %all 9 due place be demon/rat$4 ,la/ly1 ! f\r? argum5t is set d[n by ! same ,lucretius 9 ^! v]ses3 ,duo de concursu corpora lata ,si cita dissiliant1 nempe a] omne necesse e/1 ,int] corpora quod fu]at1 possidat inane4 ,is porro quamvis circum cel]antibus auris ,confluat1 haud pot]it tam5 uno tempore totum ,compl]i spatium2 nam primum quemque necesse e/ ,occupet ille locum1 de9de omnia possideantur4--,i4 #che-ia4 ,t is1 if two flat bodies be sudd5ly pull$ asund]1 ( necessity ! air mu/ come betwe5 !m to fill up ! space !y left empty4 ,b ) :at cel]ity soev] ! air fl[ 91 yet x _c 9 "o in/ant ( "t fill ! :ole space1 b fir/ "o "p ( x1 !n successively all4 ,:i* nev]!less is m repugnant to ! op9ion ( ,epicurus1 ?an ( ^? t d5y vacuum4 ,= ?\gh x be true1 t if two bodies 7 ( inf9ite h>dness1 & 7 jo9$ toge!r by _! sup]ficies :i* 7 mo/ exactly plane1 x w\ld be impossible to pull !m asund]1 9 reg>d x c\ld n be done b by motion 9 an in/ant2 yet1 if z ! greate/ ( all magnitudes _c be giv51 nor ! swifte/ ( all motions1 s nei!r ! h>de/ ( all bodies2 x might be1 t by ! application ( v great =ce1 !re might be place made = a successive fl[+ 9 ( ! air1 namely1 by sep>at+ ! p>ts ( ! jo9$ bodies by succession1 beg9n+ at ! out]mo/ & end+ at ! inn]mo/ "p4 ,he ought1 !re=e1 fir/ to h prov$1 t !re are "s bodies extremely h>d1 n relatively z comp>$ ) s(t] bodies1 b absolutely1 t is to say1 inf9itely h>d2 :i* is n true4 ,b if we suppose1 z ,epicurus do?1 t atoms are indivisible1 & yet h small sup]ficies ( _! own2 !n if two bodies %\ld be jo9$ toge!r by _m1 or b "o only small sup]ficies ( ei!r ( !m1 !n ,i say ?is argum5t ( ,lucretius w\ld be a firm demon/ration1 t no two bodies made up ( atoms1 z he supposes1 c\ld ev] possibly be pull$ asund] by any =ce :atsoev]4 ,b ?is is repugnant to daily exp]i5ce4 ,sid5ote3 ,o!r argum5ts = ! e/abli%+ ( vacuum1 invalid4 #d4 ,& ?us mu* ( ! argum5ts ( ,lucretius4 ,let u n[ consid] ! argum5ts :i* are drawn f ! exp]im5ts ( lat] writ]s4 ,i4 ,! fir/ exp]im5t is ?is3 t if a holl[ vessel be ?ru/ into wat] ) ! bottom upw>ds1 ! wat] w asc5d into x2 :i* !y say x c\ld n d1 unless ! air )9 7 ?ru/ toge!r into a n>r[] place2 & t ?is 7 also impossible1 except !re 7 little empty places 9 ! air4 ,also1 t :5 ! air is compress$ to a c]ta9 degree1 x c receive no fur!r compression1 its small p>ticles n suff]+ !mselves to be p5t into less room4 ,?is reason1 if ! air c\ld n pass ?r\gh ! wat] z x asc5ds )9 ! vessel1 might seem valid4 ,b x is suffici5tly kn[n1 t air w p5etrate wat] by ! application ( a =ce equal to ! gravity ( ! wat]4 ,if !re=e ! =ce1 by :i* ! vessel is ?ru/ d[n1 be great] or equal to ! endeav\r by :i* ! wat] naturally t5de? d[nw>ds1 ! air w g out t way :]e ! resi/ance is made1 namely1 t[>ds ! edges ( ! vessel4 ,=1 by h[ mu* ! deep] is ! wat] :i* is to be p5etrat$1 s mu* great] mu/ be ! depress+ =ce4 ,b aft] ! vessel is q "u wat]1 ! =ce by :i* x is depress$1 t is to say1 ! =ce by :i* ! wat] rise? up1 is no long] increas$4 ,!re is !re=e su* an equilibration betwe5 !m1 z t ! natural endeav\r ( ! wat] d[nw>ds is equal to ! endeav\r by :i* ! same wat] is to be p5etrat$ to ! increas$ dep?4 ,,ii4 ,! second exp]im5t is1 t if a concave cyl9d] ( suffici5t l5g?1 made ( glass1 t ! exp]im5t may be ! bett] se51 hav+ "o end op5 & ! o!r close %ut1 be fill$ ) quicksilv]1 & ! op5 end be+ /opp$ ) "o's f+]1 be toge!r ) ! f+] dipp$ into a di% or o!r vessel1 9 :i* also !re is quicksilv]1 & ! cyl9d] be set upright1 we %all1 ! f+] be+ tak5 away to make way = ! desc5t ( ! quicksilv]1 see x desc5d into ! vessel "u x1 till !re be only s mu* rema9+ )9 ! cyl9d] z may fill ab\t tw5ty-six in*es ( ! same2 & ?us x w always happ5 :atsoev] be ! cyl9d]1 provid$ t ! l5g? be n less ?an tw5ty-six in*es4 ,f :5ce !y conclude t ! cavity ( ! cyl9d] above ! quicksilv] rema9s empty ( all body4 ,b 9 ?is exp]im5t ,i f9d no necessity at all ( vacuum4 ,= :5 ! quicksilv] :i* is 9 ! cyl9d] desc5ds1 ! vessel "u x mu/ ne$s be fill$ to a great] height1 & consequ5tly s mu* ( ! contigu\s air mu/ be ?ru/ away z may make place = ! quicksilv] :i* is desc5d$4 ,n[ if x be ask$ :i!r t air goes1 :at c be answ]$ b ?is1 t x ?ru/e? away ! next air1 & t ! next1 & s successively1 till !re be a return to ! place :]e ! propulsion fir/ began4 ,& !re1 ! la/ air ?us ?ru/ on w press ! quicksilv] 9 ! vessel ) ! same =ce ) :i* ! fir/ air 0 ?ru/ away2 & if ! =ce ) :i* ! quicksilv] desc5ds be great en\gh1 :i* is great] or less z x desc5ds f a place ( great] or less height1 x w make ! air p5etrate ! quicksilv] 9 ! vessel1 & g up into ! cyl9d] to fill ! place :i* !y ?\ght 0 left empty4 ,b because ! quicksilv] ha? n 9 e degree ( height =ce en\gh to cause su* p5etration1 !re=e 9 desc5d+ x mu/ ( necessity /ay some:]e1 namely1 !re1 :]e its endeav\r d[nw>ds1 & ! resi/ance ( ! same to ! p5etration ( ! air1 come to an equilibrium4 ,& by ?is exp]im5t x is manife/1 t ?is equilibrium w be at ! height ( tw5ty-six in*es1 or !reab\ts4 ,,iii4 ,! ?ird exp]im5t is1 t :5 a vessel ha? z mu* air 9 x z x c naturally conta91 !re may nev]!less be =c$ into x z mu* wat] z w fill ?ree qu>t]s ( ! same vessel4 ,& ! exp]im5t is made 9 ?is mann]4 ,into ! glass bottle1 repres5t$ 79 figure #b7 by ! sph]e ,f ,g1 ^: c5tre is ,a1 let ! pipe ,b ,a ,c be s fitt$1 t x may precisely fill ! m\? ( ! bottle2 & let ! end ,b be s ne> ! bottom1 t !re may be only space en\gh left = ! free passage ( ! wat] :i* is ?ru/ 9 above4 ,let ! upp] end ( ?is pipe h a cov] at ,d1 ) a sp\t at ,e1 by :i* ! wat]1 :5 x asc5ds 9 ! pipe1 may run out4 ,also let ,h ,c be a cock1 = ! op5+ or %utt+ ( ! passage ( ! wat] betwe5 ,b & ,d1 z !re %all be occasion4 ,let ! cov] ,d ,e be tak5 (f1 & ! cock ,h ,c be+ op5$1 let a syr+e full ( wat] be =c$ 92 & be=e ! syr+e be tak5 away1 let ! cock be turn$ to h9d] ! go+ out ( ! air4 ,& 9 ?is mann] let ! injection ( wat] be repeat$ z (t5 z x %all be requisite1 till ! wat] rise )9 ! bottle2 = example1 to ,g ,f4 ,la/ly1 ! cov] be+ fa/5$ on aga91 & ! cock ,h ,c op5$1 ! wat] w run swiftly out at ,e1 & s9k by little & little f ,g ,f to ! bottom ( ! pipe ,b4 ,f ?is ph5om5on1 !y argue = ! necessity ( vacuum 9 ?is mann]4 ,! bottle1 f ! beg9n+1 0 full ( air2 :i* air c\ld nei!r g out by p5etrat+ s great a l5g? ( wat] z 0 inject$ by ! pipe1 nor by any o!r way4 ,( necessity1 !re=e1 all ! wat] z high z ,f ,g1 z also all ! air t 0 9 ! bottle be=e ! wat] 0 =c$ 91 mu/ n[ be 9 ! same place1 :i* at fir/ 0 fill$ by ! air alone2 :i* 7 impossible1 if all ! space )9 ! bottle 7 =m]ly fill$ ) air precisely1 t is1 )\t any vacuum4 ,besides1 ?\gh "s man p]haps may ?9k ! air1 be+ a ?9 body1 may pass ?r\gh ! body ( ! wat] conta9$ 9 ! pipe1 yet f t o!r ph5om5on1 namely1 t all ! wat] :i* is 9 ! space ,b ,f ,g is ca/ out aga9 by ! sp\t at ,e1 = :i* x seems impossible t any o!r reason c be giv5 besides ! =ce by :i* ! air frees itself f compression1 x foll[s1 t ei!r !re 0 9 ! bottle "s space empty1 or t _m bodies may be toge!r 9 ! same place4 ,b ?is la/ is absurd2 & !re=e ! =m] is true1 namely1 t !re 0 vacuum4 ,?is argum5t is infirm 9 two places4 ,= fir/1 t is assum$ :i* is n to be grant$2 & 9 ! second place1 an exp]im5t is br\ght1 :i* ,i ?9k is repugnant to vacuum4 ,t :i* is assum$ is1 t ! air c h no passage out ?r\gh ! pipe4 ,nev]!less1 we see daily t air easily asc5ds f ! bottom to ! sup]ficies ( a riv]1 z is manife/ by ! bubbles t rise2 nor do? x ne$ any o!r cause to give x ?is motion1 ?an ! natural endeav\r d[nw>ds ( ! wat]4 ,:y1 !re=e1 may n ! endeav\r upw>ds ( ! same wat]1 acquir$ by ! injection1 :i* endeav\r upw>ds is great] ?an ! natural endeav\r ( ! wat] d[nw>ds1 cause ! air 9 ! bottle to p5etrate 9 l mann] ! wat] t presse? x d[nw>ds2 especially1 see+ ! wat]1 z x rise? 9 ! bottle1 do? s press ! air t is above x1 z t x g5]ate? 9 e "p !re( an endeav\r t[>ds ! ext]nal sup]ficies ( ! pipe1 & consequ5tly make? all ! p>ts ( ! enclos$ air to t5d directly t[>ds ! passage at ,b8 ,i say1 ?is is no less manife/1 ?an t ! air :i* rise? up f ! bottom ( a riv] %\ld p5etrate ! wat]1 h[ deep soev] x be4 ,:]e=e ,i d n yet see any cause :y ! =ce1 by :i* ! wat] is inject$1 %\ld n at ! same "t eject ! air4 ,& z = _! argu+ ! necessity ( vacuum f ! rejection ( ! wat]2 9 ! fir/ place1 suppos+ !re is vacuum1 ,i dem& by :at pr9ciple ( motion t ejection is made4 ,c]ta9ly1 see+ ?is motion is f )9 outw>ds1 x mu/ ne$s be caus$ by "s ag5t )9 ! bottle2 t is to say1 by ! air itself4 ,n[ ! motion ( t air1 be+ caus$ by ! ris+ ( ! wat]1 beg9s at ! bottom1 & t5ds upw>ds2 :]eas ! motion by :i* x ejecte? ! wat] ought to beg9 above1 & t5d d[nw>ds4 ,f :5ce !re=e ha? ! enclos$ air ?is endeav\r t[>ds ! bottom8 ,to ?is "q ,i "k n :at answ] c be giv51 unless x be said1 t ! air desc5ds ( its own accord to expel ! wat]4 ,:i*1 because x is absurd1 & t ! air1 aft] ! wat] is =c$ 91 ha? z mu* room z its magnitude requires1 !re w rema9 no cause at all :y ! wat] %\ld be =c$ out4 ,:]e=e ! ass]tion ( vacuum is repugnant to ! v exp]im5t :i* is "h br\ght to e/abli% x4 ,_m o!r ph5om5a are usually br\ght = vacuum1 z ^? ( wea!r-glasses1 olipyles1 w9d-guns1 c4 :i* w\ld all be v h>d to be salv$1 unless wat] be p5etrable by air1 )\t ! int]mixture ( empty space4 ,b n[1 see+ air may ) no great endeav\r pass ?r\gh n only wat]1 b any o!r fluid body ?\gh nev] s /ubborn1 z quicksilv]1 ^! ph5om5a prove no?+4 ,nev]!less1 x might 9 reason be expect$1 t he t w\ld take away vacuum1 %\ld )\t vacuum %[ u su* causes ( ^! ph5om5a1 z %\ld be at lea/ ( equal1 if n great] probability4 ,?is !re=e %all be done 9 ! foll[+ disc\rse1 :5 ,i come to speak ( ^! ph5om5a 9 _! prop] places4 ,b fir/1 ! mo/ g5]al hypo!ses ( natural philosophy are to be premis$4 ,& see+ t suppositions are put = ! true causes ( app>5t effects1 e supposition1 except su* z be absurd1 mu/ ( necessity consi/ ( "s suppos$ possible motion2 = re/ c nev] be ! effici5t cause ( any?+2 & motion suppose? bodies moveable2 ( :i* !re are ?ree k9ds1 fluid1 consi/5t1 & mix$ ( bo?4 ,fluid are ^?1 ^: p>ts may by v weak endeav\r be sep>at$ f "o ano!r2 & consi/5t ^? = ! sep>ation ( ^: p>ts great] =ce is to be appli$4 ,!re are !re=e degrees ( consi/5cy2 :i* degrees1 by comp>ison ) m or less consi/5t1 h ! names ( h>dness or s(tness4 ,:]e=e a fluid body is always divisible into bodies equally fluid1 z quantity into quantities2 & s(t bodies1 ( :atsoev] degree ( s(tness1 into s(t bodies ( ! same degree4 ,& ?\gh _m m5 seem to conceive no o!r diff]5ce ( fluidity1 b su* z arise? f ! diff]5t magnitudes ( ! p>ts1 9 :i* s5se du/1 ?\gh ( diamonds1 may be call$ fluid2 yet ,i und]/& by fluidity1 t :i* is made su* by nature equally 9 e "p ( ! fluid body2 n z du/ is fluid1 = s a h\se :i* is fall+ 9 pieces may be call$ fluid2 b 9 su* mann] z wat] seems fluid1 & to divide itself into p>ts p]petually fluid4 ,& ?is be+ well und]/ood1 ,i come to my suppositions4 ,sid5ote3 ,six suppositions = ! salv+ ( ! ph5om5a ( nature4 #e4 ,fir/1 !re=e1 ,i suppose t ! imm5se space1 :i* we call ! _w1 is ! aggregate ( all bodies :i* are ei!r consi/5t & visible1 z ! e>? & ! />s2 or invisible1 z ! small atoms :i* are dissem9at$ ?r\gh ! :ole space betwe5 ! e>? & ! />s2 & la/ly1 t mo/ fluid e!r1 :i* s fills all ! re/ ( ! univ]se1 z t x leaves 9 x no empty place at all4 ,secondly1 ,i suppose ) ,cop]nicus1 t ! great] bodies ( ! _w1 :i* are bo? consi/5t & p]man5t1 h su* ord] among/ !mselves1 z t ! sun ha? ! fir/ place1 ,m]cury ! second1 ,v5us ! ?ird1 ! ,e>? ) ! moon go+ ab\t x ! f\r?1 ,m>s ! fif?1 ,jupit] ) 8 att5dants ! six?1 ,saturn ! sev5?2 & aft] ^!1 ! fix$ />s h _! sev]al di/ances f ! sun4 ,?irdly1 ,i suppose t 9 ! sun & ! re/ ( ! planets !re is & always has be5 a simple circul> motion4 ,f\r?ly1 ,i suppose t 9 ! body ( ! air !re are c]ta9 o!r bodies int]m+l$1 :i* are n fluid2 b )al t !y are s small1 t !y are n p]ceptible by s5se2 & t ^! also h _! prop] simple motion1 & are "s ( !m m1 "s less h>d or consi/5t4 ,fif?ly1 ,i suppose ) ,kepl] t z ! di/ance betwe5 ! sun & ! e>? is to ! di/ance betwe5 ! moon & ! e>?1 s ! di/ance betwe5 ! moon & ! e>? is to ! semidiamet] ( ! e>?4 ,z = ! magnitude ( ! circles1 & ! times 9 :i* !y are describ$ by ! bodies :i* are 9 !m1 ,i w suppose !m to be su* z %all seem mo/ agreeable to ! ph5om5a 9 "q4 ,sid5ote3 ,possible causes ( ! motions annual & diurnal2 & ( ! app>5t direction1 /ation1 & retrogradation ( ! planets4 #f4 ,! causes ( ! diff]5t seasons ( ! ye>1 & ( ! sev]al v>iations ( days & nights 9 all ! p>ts ( ! sup]ficies ( ! e>?1 h be5 demon/rat$1 fir/ by ,cop]nicus1 & s9ce by ,kepl]1 ,galileus1 & o!rs1 f ! supposition ( ! e>?'s diurnal revolution ab\t its own axis1 toge!r ) its annual motion ab\t ! sun 9 ! ecliptic accord+ to ! ord] ( ! signs2 & ?irdly1 by ! annual revolution ( ! same e>? ab\t its own c5tre1 contr>y to ! ord] ( ! signs4 ,i suppose ) ,cop]nicus1 t ! diurnal revolution is f ! motion ( ! e>?1 by :i* ! equ9octial circle is describ$ ab\t x4 ,& z = ! o!r two annual motions1 !y are ! effici5t cause ( ! e>?'s be+ c>ri$ ab\t 9 ! ecliptic 9 su* mann]1 z t its axis is always kept p>allel to itself4 ,:i* p>allelism 0 = ?is reason introduc$1 le/ by ! e>?'s annual revolution its poles %\ld seem to be necess>ily c>ri$ ab\t ! sun1 contr>y to exp]i5ce4 ,i h1 9 art4 #aj1 *ap4 ,,xxi1 demon/rat$1 f ! supposition ( simple circul> motion 9 ! sun1 t ! e>? is s c>ri$ ab\t ! sun1 z t its axis is !reby kept always p>allel to itself4 ,:]e=e1 f ^! two suppos$ motions 9 ! sun1 ! "o simple circul> motion1 ! o!r circul> motion ab\t its own c5tre1 x may be demon/rat$ t ! ye> ha? bo? ! same v>iations ( days & nights1 z h be5 demon/rat$ by ,cop]nicus4 ,= if ! circle a b c d 79 fig4 #c7 be ! ecliptic1 ^: c5tre is e1 & diamet] a e c2 & ! e>? be plac$ 9 a1 & ! sun be mov$ 9 ! little circle f g h i1 namely1 accord+ to ! ord] f1 g1 h1 & i1 x ha? be5 demon/rat$1 t a body plac$ 9 a w be mov$ 9 ! same ord] ?r\gh ! po9ts ( ! ecliptic a1 b1 c1 & d1 & w always keep its axis p>allel to itself4 ,b if1 z ,i h suppos$1 ! e>? also be mov$ ) simple circul> motion 9 a plane t passe? ?r\gh a1 cutt+ ! plane ( ! ecliptic s z t ! common section ( bo? ! planes be 9 a c1 ?us also ! axis ( ! e>? w be kept always p>allel to itself4 ,= let ! c5tre ( ! e>? be mov$ ab\t 9 ! circumf]5ce ( ! epicycle1 ^: diamet] is l a k1 :i* is a "p ( ! /rait l9e l a c2 !re=e l a k1 ! diamet] ( ! epicycle1 pass+ ?r\gh ! c5tre ( ! e>?1 w be 9 ! plane ( ! ecliptic4 ,:]e=e see+ t by reason ( ! e>?'s simple motion bo? 9 ! ecliptic & 9 its epicycle1 ! /rait l9e l a k is kept always p>allel to itself1 e o!r /rait l9e also tak5 9 ! body ( ! e>?1 & consequ5tly its axis1 w 9 l mann] be kept always p>allel to itself2 s t 9 :at "p soev] ( ! ecliptic ! c5tre ( ! epicycle be f\nd1 & 9 :at "p soev] ( ! epicycle ! c5tre ( ! e>? be f\nd at ! same "t1 ! axis ( ! e>? w be p>allel to ! place :]e ! same axis w\ld h be51 if ! c5tre ( ! e>? _h nev] gone out ( ! ecliptic4 ,n[ z ,i h demon/rat$ ! simple annual motion ( ! e>? f ! supposition ( simple motion 9 ! sun2 s f ! supposition ( simple motion 9 ! e>? may be demon/rat$ ! mon?ly simple motion ( ! moon4 ,= if ! names be b *ang$1 ! demon/ration w be ! same1 & !re=e ne$ n be repeat$4 ,sid5ote3 ,! supposition ( simple motion1 :y likely4 #g4 ,t :i* makes ?is supposition ( ! sun's simple motion 9 ! epicycle f g h i probable1 is fir/1 t ! p]iods ( all ! planets are n only describ$ ab\t ! sun1 b s describ$1 z t !y are all conta9$ )9 ! zodiac1 t is to say1 )9 ! latitude ( ab\t sixte5 degrees2 = ! cause ( ?is seems to dep5d ^u "s p[] 9 ! sun1 especially 9 t "p ( ! sun :i* respects ! zodiac4 ,secondly1 t 9 ! :ole compass ( ! heav5s !re appe>s no o!r body f :i* ! cause ( ?is ph5om5on c 9 probability be d]iv$4 ,besides1 ,i c\ld n imag9e t s _m & su* v>i\s motions ( ! planets %\ld h no dep5dance at all ^u "o ano!r4 ,b1 by suppos+ motive p[] 9 ! sun1 we suppose motion also2 = p[] to move )\t motion is no p[] at all4 ,i h !re=e suppos$ t !re is 9 ! sun = ! gov]n+ ( ! prim>y planets1 & 9 ! e>? = ! gov]n+ ( ! moon1 su* motion1 z be+ receiv$ by ! prim>y planets & by ! moon1 makes !m necess>ily appe> to u 9 su* mann] z we see !m4 ,:]eas1 t circul> motion1 :i* is commonly attribut$ to !m1 ab\t a fix$ axis1 :i* is call$ conv]sion1 be+ a motion ( _! p>ts only1 & n ( _! :ole bodies1 is insuffici5t to salve _! appe>ances4 ,= see+ :atsoev] is s mov$1 ha? no endeav\r at all t[>ds ^? p>ts :i* are )\t ! circle1 !y h no p[] to propagate any endeav\r to su* bodies z are plac$ )\t x4 ,& z = !m t suppose ?is may be done by magnetical virtue1 or by incorporeal & immat]ial species1 !y suppose no natural cause2 nay1 no cause at all4 ,= !re is no su* ?+ z an incorporeal mov5t1 & magnetical virtue is a ?+ altoge!r unkn[n2 & :5soev] x %all be kn[n1 x w be f\nd to be a motion ( body4 ,x rema9s1 !re=e1 t if ! prim>y planets be c>ri$ ab\t by ! sun1 & ! moon by ! e>?1 !y h ! simple circul> motions ( ! sun & ! e>? = ! causes ( _! circulations4 ,o!rwise1 if !y be n c>ri$ ab\t by ! sun & ! e>?1 b t e planet ha? be5 mov$1 z x is n[ mov$1 ev] s9ce x 0 made1 !re w be ( _! motions no cause natural4 ,= ei!r ^! motions 7 concreat$ ) _! bodies1 & _! cause is sup]natural2 or !y are coet]nal ) !m1 & s !y h no cause at all4 ,= :atsoev] is et]nal 0 nev] g5]at$4 ,i may add besides1 to confirm ! probability ( ?is simple motion1 t z almo/ all le>n$ m5 are n[ ( ! same op9ion ) ,cop]nicus conc]n+ ! p>allelism ( ! axis ( ! e>?1 x seem$ to me to be m agreeable to tru?1 or at lea/ m h&some1 t x %\ld be caus$ by simple circul> motion alone1 ?an by two motions1 "o 9 ! ecliptic1 & ! o!r ab\t ! e>?'s own axis ! contr>y way1 nei!r ( !m simple1 nor ei!r ( !m su* z might be produc$ by any motion ( ! sun4 ,i ?\ght be/ !re=e to reta9 ?is hypo!sis ( simple motion1 & f x to d]ive ! causes ( z _m ( ! ph5om5a z ,i c\ld1 & to let su* alone z ,i c\ld n d$uce f !nce4 ,x w p]haps be object$1 t al?\gh by ?is supposition ! reason may be giv5 ( ! p>allelism ( ! axis ( ! e>?1 & ( _m o!r appe>ances1 nev]!less1 see+ x is done by plac+ ! body ( ! sun 9 ! c5tre ( t orb :i* ! e>? describes ) its annual motion1 ! supposition itself is false2 because ?is annual orb is ecc5tric to ! sun4 ,9 ! fir/ place1 !re=e1 let u exam9e :at t ecc5tricity is1 & :5ce x proce$s4 ,sid5ote3 ,! cause ( ! ecc5tricity ( ! annual motion ( ! e>?4 #h4 ,let ! annual circle ( ! e>? a b c d 79 fig4 #c7 be divid$ into f\r equal p>ts by ! /rait l9es a c & b d1 cutt+ "o ano!r 9 ! c5tre e2 & let a be ! beg9n+ ( ,libra1 b ( ,capricorn1 c ( ,aries & d ( ,canc]2 & let ! :ole orb a b c d be und]/ood1 accord+ to ,cop]nicus1 to h e way s great di/ance f ! zodiac ( ! fix$ />s1 t x be 9 comp>ison ) x b z a po9t4 ,let ! e>? be n[ suppos$ to be 9 ! beg9n+ ( ,libra at a4 ,! sun1 !re=e1 w appe> 9 ! beg9n+ ( ,aries at c4 ,:]e=e1 if ! e>? be mov$ f a to b1 ! app>5t motion ( ! sun w be f c to ! beg9n+ ( ,canc] 9 d2 & ! e>? be+ mov$ =w>ds f b to c1 ! sun also w appe> to be mov$ =w>ds to ! beg9n+ ( ,libra 9 a2 :]e=e c d a w be ! summ] ar*1 & ! w9t] ar* w be a b c4 ,n[1 9 ! "t1 ( ! sun's app>5t motion 9 ! summ] ar*1 !re are numb]$ #ahf days2 &1 consequ5tly1 ! e>? makes 9 ! same "t ! same numb] ( diurnal conv]sions 9 ! ar* a b c2 &1 !re=e1 ! e>? 9 its motion ?r\gh ! ar* c d a w make only #agh diurnal conv]sions4 ,:]e=e ! ar* a b c ought to be great] ?an ! ar* c d a by #h days1 t is to say1 by almo/ s _m degrees4 ,let ! ar* a r1 z also c s1 be ea* ( !m an ar* ( two degrees & #aaf4 ,:]e=e ! ar* r b s w be great] ?an ! semicircle a b c by #d degrees1 & great] ?an ! ar* s d r by #h degrees4 ,! equ9oxes1 !re=e1 w be 9 ! po9ts r & s2 & !re=e also1 :5 ! e>? is 9 r1 ! sun w appe> 9 s4 ,:]e=e ! true place ( ! sun w be 9 t1 t is to say1 )\t ! c5tre ( ! e>?'s annual motion by ! quantity ( ! s9e ( ! ar* a r1 or ! s9e ( two degrees & #af m9utes4 ,n[ ?is s9e1 putt+ #ajj1jjj = ! radius1 w be ne> #cehj p>ts !re(4 ,& s mu* is ! ecc5tricity ( ! e>?'s annual motion1 provid$ t t motion be 9 a p]fect circle2 & s & r are ! equ9octial p>ts4 ,& ! /rait l9es s r & c a1 produc$ bo? ways till !y rea* ! zodiac ( ! fix$ />s1 w fall /ill ^u ! same fix$ />s2 because ! :ole orb a b c d is suppos$ to h no magnitude at all 9 respect ( ! great di/ance ( ! fix$ />s4 ,suppos+ n[ ! sun to be 9 c1 x rema9s t ,i %[ ! cause :y ! e>? is ne>] to ! sun1 :5 9 its annual motion x is f\nd to be 9 d1 ?an :5 x is 9 b4 ,& ,i take ! cause to be ?is4 ,:5 ! e>? is 9 ! beg9n+ ( ,capricorn at b1 ! sun appe>s 9 ! beg9n+ ( ,canc] at d2 & !n is ! mid/ ( summ]4 ,b 9 ! mid/ ( summ]1 ! nor!rn p>ts ( ! e>? are t[>ds ! sun1 :i* is almo/ all dry l&1 conta9+ all ,europe & mu* ! greate/ "p ( ,asia & ,am]ica4 ,b :5 ! e>? is 9 ! beg9n+ ( ,canc] at d1 x is ! mid/ ( w9t]1 & t "p ( ! e>? is t[>ds ! sun1 :i* conta9s ^? great seas call$ ! ,s\? ,sea & ! ,indian ,sea1 :i* are ( f> great] ext5t ?an all ! dry l& 9 t hemisph]e4 ,:]e=e by ! la/ article ( *apt] ,,xxi1 :5 ! e>? is 9 d1 x w come ne>] to its fir/ mov5t1 t is1 to ! sun :i* is 9 t2 t is to say1 ! e>? is ne>] to ! sun 9 ! mid/ ( w9t] :5 x is 9 d1 ?an 9 ! mid/ ( summ] :5 x is 9 b2 &1 !re=e1 dur+ ! w9t] ! sun is 9 its ,p]igum1 & 9 its ,apogum dur+ ! summ]4 ,& ?us ,i h %[n a possible cause ( ! ecc5tricity ( ! e>?2 :i* 0 to be done4 ,i am1 !re=e1 ( ,kepl]'s op9ion 9 ?is1 t he attributes ! ecc5tricity ( ! e>? to ! diff]5ce ( ! p>ts !re(1 & supposes "o "p to be affect$1 & ano!r disaffect$ to ! sun4 ,& ,i diss5t f him 9 ?is1 t he ?9ks x to be by magnetic virtue1 & t ?is magnetic virtue or attraction & ?ru/+ back ( ! e>? is wr\ght by immat]iate species2 :i* _c be1 because no?+ c give motion b a body mov$ & contigu\s4 ,= if ^? bodies be n mov$ :i* are contigu\s to a body unmov$1 h[ ?is body %\ld beg9 to be mov$ is n imag9able2 z has be5 demon/rat$ 9 art4 #g1 *ap4 ,,ix1 & (t5 inculcat$ 9 o!r places1 to ! end t philosoph]s might at la/ ab/a9 f ! use ( su* unconceivable connexions ( words4 ,i diss5t also f him 9 ?is1 t he says ! similitude ( bodies is ! cause ( _! mutual attraction4 ,= if x 7 s1 ,i see no reason :y "o egg %\ld n be attract$ by ano!r4 ,if1 !re=e1 "o "p ( ! e>? be m affect$ by ! sun ?an ano!r "p1 x proce$s f ?is1 t "o "p ha? m wat]1 ! o!r m dry l&4 ,& f h5ce x is1 z ,i %[$ above1 t ! e>? comes ne>] to ! sun :5 x %9es ^u t "p :]e !re is m wat]1 ?an :5 x %9es ^u t :]e !re is m dry l&4 ,sid5ote3 ,! cause :y ! moon ha? always "o & ! same face turn$ t[>ds ! e>?4 #i4 ,?is ecc5tricity ( ! e>? is ! cause :y ! way ( its annual motion is n a p]fect circle1 b ei!r an elliptical1 or almo/ an elliptical l9e2 z also :y ! axis ( ! e>? is n kept exactly p>allel to itself 9 all places1 b only 9 ! equ9octial po9ts4 ,n[ see+ ,i h said t ! moon is c>ri$ ab\t by ! e>?1 9 ! same mann] t ! e>? is by ! sun2 & t ! e>? goe? ab\t ! sun 9 su* mann] z t x %[s sometimes "o hemisph]e1 sometimes ! o!r to ! sun2 x rema9s to be enquir$1 :y ! moon has always "o & ! same face turn$ t[>ds ! e>?4 ,suppose1 !re=e1 ! sun to be mov$ ) simple motion 9 ! little circle f g h i1 79 fig4 #d7 ^: c5tre is t2 & let be ! annual circle ( ! e>?2 & a ! beg9n+ ( ,libra4 ,ab\t ! po9t a let ! little circle l k be describ$2 & 9 x let ! c5tre ( ! e>? be und]/ood to be mov$ ) simple motion2 & bo? ! sun & ! e>? to be mov$ accord+ to ! ord] ( ! signs4 ,^u ! c5tre a let ! way ( ! moon m n o p be describ$2 & let q r be ! diamet] ( a circle cutt+ ! globe ( ! moon into two hemisph]es1 :]e( "o is se5 by u :5 ! moon is at ! full1 & ! o!r is turn$ f u4 ,! diamet] !re=e ( ! moon q o r w be p]p5dicul> to ! /rait l9e t a4 ,:]e=e ! moon is c>ri$1 by reason ( ! motion ( ! e>?1 f o t[>ds p4 ,b by reason ( ! motion ( ! sun1 if x 7 9 p x w\ld at ! same "t be c>ri$ f p t[>ds o2 & by ^! two contr>y mov5ts ! /rait l9e q r w be turn$ ab\t2 &1 9 a quadrant ( ! circle m n o p1 x w be turn$ s mu* z makes ! f\r? "p ( its :ole conv]sion4 ,:]e=e :5 ! moon is 9 p1 q r w be p>allel to ! /rait l9e m o4 ,secondly1 :5 ! moon is 9 m1 ! /rait l9e q r w1 by reason ( ! motion ( ! e>?1 be 9 m o4 ,b by ! work+ ( ! sun's motion ^u x 9 ! quadrant p m1 ! same q r w be turn$ s mu* z makes ano!r qu>t] ( its :ole conv]sion4 ,:51 !re=e1 ! moon is 9 m1 q r w be p]p5dicul> to ! /rait l9e o m4 ,by ! same reason1 :5 ! moon is 9 n1 q r w be p>allel to ! /rait l9e m o2 &1 ! moon return+ to o1 ! same q r w return to its fir/ place2 & ! body ( ! moon w 9 "o entire p]iod make also "o entire conv]sion ^u h] own axis4 ,9 ! mak+ ( :i*1 x is manife/1 t "o & ! same face ( ! moon is always turn$ t[>ds ! e>?4 ,& if any diamet] 7 tak5 9 t little circle1 9 :i* ! moon 7 suppos$ to be c>ri$ ab\t ) simple motion1 ! same effect w\ld foll[2 = if !re 7 no action f ! sun1 e diamet] ( ! moon w\ld be c>ri$ ab\t always p>allel to itself4 ,:]e=e ,i h giv5 a possible cause :y "o & ! same face ( ! moon is always turn$ t[>ds ! e>?4 ,b x is to be not$1 t :5 ! moon is )\t ! ecliptic1 we d n always see ! same face precisely4 ,= we see only t "p :i* is illum9at$4 ,b :5 ! moon is )\t ! ecliptic1 t "p :i* is t[>ds u is n exactly ! same ) t :i* is illum9at$4 ,sid5ote3 ,! cause ( ! tides ( ! ocean4 #aj4 ,to ^! ?ree simple motions1 "o ( ! sun1 ano!r ( ! moon1 & ! ?ird ( ! e>?1 9 _! own little circles f g h i1 l k1 & q r1 toge!r ) ! diurnal conv]sion ( ! e>?1 by :i* conv]sion all ?+s t adh]e to its sup]ficies are necess>ily c>ri$ ab\t ) x1 may be ref]r$ ! ?ree ph5om5a conc]n+ ! tides ( ! ocean4 ,:]e( ! fir/ is ! alt]nate elevation & depression ( ! wat] at ! %ores1 twice 9 ! space ( tw5ty-f\r h\rs & ne> ^u fifty-two m9utes2 = s x has con/antly cont9u$ 9 all ages4 ,! second1 t at ! new & full moons1 ! elevations ( ! wat] are great] ?an at o!r times betwe54 ,& ! ?ird1 t :5 ! sun is 9 ! equ9octial1 !y are yet great] ?an at any o!r "t4 ,= ! salv+ ( :i* ph5om5a1 we h already ! f\r above-m5tion$ motions2 to :i* ,i assume also ?is1 t ! "p ( ! e>? :i* is call$ ,am]ica1 be+ high] ?an ! wat]1 & ext5d$ almo/ ! space ( a :ole semicircle f nor? to s\?1 gives a /op to ! motion ( ! wat]4 ,?is be+ grant$1 9 ! same #d? figure1 :]e l b k c is suppos$ to be 9 ! plane ( ! moon's mon?ly motion1 let ! little circle l d k e be describ$ ab\t ! same c5tre a 9 ! plane ( ! equ9octial4 ,?is circle !re=e w decl9e f ! circle l b k c 9 an angle ( almo/ #bh degrees2 = ! greate/ decl9ation ( ! ecliptic is #bc1 to :i* add+ #e = ! greate/ decl9ation ( ! moon f ! ecliptic1 ! sum w be #bh degrees4 ,see+ n[ ! wat]s1 :i* are "u ! circle ( ! moon's c\rse1 are by reason ( ! e>?'s simple motion 9 ! plane ( ! same circle mov$ toge!r ) ! e>?1 t is to say1 toge!r ) _! own bottoms1 nei!r outgo+ nor outgone2 if we add ! diurnal motion1 by :i* ! o!r wat]s :i* are "u ! equ9octial are mov$ 9 ! same ord]1 & consid] )al t ! circles ( ! moon & ( ! equ9octial int]sect "o ano!r2 x w be manife/1 t bo? ^? wat]s1 :i* are "u ! circle ( ! moon1 & "u ! equ9octial1 w run toge!r "u ! equ9octial2 & consequ5tly1 t _! motion w n only be swift] ?an ! gr\nd t c>ries !m2 b also t ! wat]s !mselves w h great] elevation :5soev] ! e>? is 9 ! equ9octial4 ,:]e=e1 :atsoev] ! cause ( ! tides may be1 ?is may be ! cause ( _! augm5tation at t "t4 ,aga91 see+ ,i h suppos$ ! moon to be c>ri$ ab\t by ! simple motion ( ! e>? 9 ! little circle l b k c2 & demon/rat$1 at ! #d? article ( *apt] ,,xxi1 t :atsoev] is mov$ by a mov5t t ha? simple motion1 w be mov$ always ) ! same velocity2 x foll[s1 t ! c5tre ( ! e>? w be c>ri$ 9 ! circumf]5ce l b k c ) ! same velocity ) :i* ! moon is c>ri$ 9 ! circumf]5ce m n o p4 ,:]e=e ! "t1 9 :i* ! moon is c>ri$ ab\t 9 m n o p1 is to ! "t1 9 :i* ! e>? is c>ri$ ab\t 9 l b k c1 z "o circumf]5ce to ! o!r1 t is1 z a o to a k4 ,b a o is obs]v$ to be to ! semidiamet] ( ! e>? z #ei to #a2 & !re=e ! e>?1 if a k be put = its semidiamet]1 w make fifty-n9e revolutions 9 l b k c 9 ! "t t ! moon makes "o mon?ly circuit 9 m n o p4 ,b ! moon makes h] mon?ly circuit 9 little m ?an tw5ty-n9e days4 ,:]e=e ! e>? %all make its circuit 9 ! circumf]5ce l b k c 9 twelve h\rs & a little m1 namely1 ab\t tw5ty-six m9utes m2 t is to say1 x %all make two circuits 9 tw5ty-f\r h\rs & almo/ fifty-two m9utes2 :i* is obs]v$ to be ! "t betwe5 ! high-wat] ( "o day & ! high-wat] ( ! day foll[+4 ,n[ ! c\rse ( ! wat]s be+ h9d]$ by ! s\!rn "p ( ,am]ica1 _! motion w be int]rupt$ !re2 & consequ5tly1 !y w be elevat$ 9 ^? places1 & s9k d[n aga9 by _! own weight1 twice 9 ! space ( tw5ty-f\r h\rs & fifty-two m9utes4 ,& ?us ,i h giv5 a possible cause ( ! diurnal reciprocation ( ! ocean4 ,n[ f ?is swell+ ( ! ocean 9 ^? p>ts ( ! e>?1 proce$ ! fl[+s & ebb+s 9 ! ,atlantic1 ,spani%1 ,briti%1 & ,g]man seas2 :i* ?\gh !y h _! set times1 yet ^u sev]al %ores !y happ5 at sev]al h\rs ( ! day4 ,& !y receive "s augm5tation f ! nor?1 by reason t ! %ores ( ,*9a & ,t>t>y1 h9d]+ ! g5]al c\rse ( ! wat]s1 make !m swell !re1 & dis*>ge !mselves 9 "p ?r\gh ! /rait ( ,anian into ! ,nor!rn ,ocean1 & s into ! ,g]man ,sea4 ,z = ! spr+ tides :i* happ5 at ! "t ( ! new & full moons1 !y are caus$ by t simple motion1 :i* at ! beg9n+ ,i suppos$ to be always 9 ! moon4 ,= z1 :5 ,i %[$ ! cause ( ! ecc5tricity ( ! e>?1 ,i d]iv$ ! elevation ( ! wat]s f ! simple motion ( ! sun2 s ! same may "h be d]iv$ f ! simple motion ( ! moon4 ,= ?\gh f ! g5]ation ( cl\ds1 !re appe> 9 ! sun a m manife/ p[] ( elevat+ ! wat]s ?an 9 ! moon2 yet ! p[] ( increas+ moi/ure 9 vegetables & liv+ creatures appe>s m manife/ly 9 ! moon ?an 9 ! sun2 :i* may p]haps proce$ f ?is1 t ! sun raise? up great]1 & ! moon less] drops ( wat]4 ,nev]!less1 x is m likely1 & m agreeable to common obs]vation1 t ra9 is rais$ n only by ! sun1 b also by ! moon2 = almo/ all m5 expect *ange ( wea!r at ! "t ( ! conjunctions ( ! sun & moon ) "o ano!r & ) ! e>?1 m ?an 9 ! "t ( _! qu>t]s4 ,9 ! la/ place1 ! cause :y ! spr+ tides are great] at ! "t ( ! equ9oxes ha? be5 already suffici5tly decl>$ 9 ?is article1 :]e ,i h demon/rat$1 t ! two motions ( ! e>?1 namely1 its simple motion 9 ! little circle l b k c1 & its diurnal motion 9 l d k e1 cause necess>ily a great] elevation ( wat]s :5 ! sun is ab\t ! equ9oxes1 ?an :5 he is 9 o!r places4 ,i h !re=e giv5 possible causes ( ! ph5om5on ( ! fl[+ & ebb+ ( ! ocean4 ,sid5ote3 ,cause ( ! precession ( ! equ9oxes4 #aa4 ,z = ! explication ( ! ye>ly precession ( ! equ9octial po9ts1 we mu/ rememb] t1 z ,i h already %[n1 ! annual motion ( ! e>? is n 9 ! circumf]5ce ( a circle1 b ( an ellipsis1 or a l9e n consid]ably diff]5t f t ( an ellipsis4 ,9 ! fir/ place1 !re=e1 ?is elliptical l9e is to be describ$4 ,let ! ecliptic 79 fig4 #e7 be divid$ into f\r equal p>ts by ! two /rait l9es a b & 1 cutt+ "o ano!r at "r angles 9 ! c5tre c4 ,& tak+ ! ar* b d ( two degrees & sixte5 m9utes1 let ! /rait l9e d e be drawn p>allel to a b1 & cutt+ 9 f2 :i* be+ done1 ! ecc5tricity ( ! e>? w be c f4 ,see+ !re=e ! annual motion ( ! e>? is 9 ! circumf]5ce ( an ellipsis1 ( :i* is ! great] axis1 a b _c be ! less] axis2 = a b & are equal4 ,:]e=e ! e>? pass+ ?r\gh a & b1 w ei!r pass above 1 z ?r\gh g1 or pass+ ?r\gh 1 w fall betwe5 c & a2 x is no matt] :i*4 ,let x pass !re=e ?r\gh g2 & let g l be tak5 equal to ! /rait l9e 2 & divid+ g l equally 9 i1 g i w be equal to f1 & i l equal to f 2 & consequ5tly ! po9t i w cut ! ecc5tricity c f into two equal p>ts2 & tak+ i h equal to i f1 h i w be ! :ole ecc5tricity4 ,if n[ a /rait l9e1 namely1 ! l9e i 1 be drawn ?r\gh i p>allel to ! /rait l9es a b & e d1 ! way ( ! sun 9 summ]1 namely1 ! ar* g 1 w be great] ?an 8 way 9 w9t]1 by #h degrees4 ,:]e=e ! true equ9oxes w be 9 ! /rait l9e i 2 & !re=e ! ellipsis ( ! e>?'s annual motion w n pass ?r\gh a1 g1 b1 & l2 b ?r\gh 1 g1 & l4 ,:]e=e ! annual motion ( ! e>? is 9 ! ellipsis g l2 & _c be1 ! ecc5tricity be+ salv$1 9 any o!r l9e4 ,& ?is p]haps is ! reason1 :y ,kepl]1 aga9/ ! op9ion ( all ! a/ronom]s ( =m] "t1 ?\ght fit to bisect ! ecc5tricity ( ! e>?1 or1 accord+ to ! anci5ts1 ( ! sun1 n by dim9i%+ ! quantity ( ! same ecc5tricity1 7because ! true measure ( t quantity is ! diff]5ce by :i* ! summ] ar* exce$s ! w9t] ar*71 b by tak+ = ! c5tre ( ! ecliptic ( ! great orb ! po9t c ne>] to f1 & s plac+ ! :ole great orb z mu* ne>] to ! ecliptic ( ! fix$ />s t[>ds 1 z is ! di/ance betwe5 c & i4 ,= see+ ! :ole great orb is b z a po9t 9 respect ( ! imm5se di/ance ( ! fix$ />s1 ! two /rait l9es & a b1 be+ produc$ bo? ways to ! beg9n+s ( ,aries & ,libra1 w fall ^u ! same po9ts ( ! sph]e ( ! fix$ />s4 ,let !re=e ! diamet] ( ! e>? m n be 9 ! plane ( ! e>?'s annual motion4 ,if n[ ! e>? be mov$ by ! sun's simple motion 9 ! circumf]5ce ( ! ecliptic ab\t ! c5tre i1 ?is diamet] w be kept always p>allel to itself & to ! /rait l9e g l4 ,b see+ ! e>? is mov$ 9 ! circumf]5ce ( an ellipsis )\t ! ecliptic1 ! po9t n1 :il/ x passe? ?r\gh 1 w g 9 a less] circumf]5ce ?an ! po9t m2 & consequ5tly1 z soon z ev] x beg9s to be mov$1 x w lose its p>allelism ) ! /rait l9e 2 s t m n produc$ w at la/ cut ! /rait l9e g l produc$4 ,& contr>ily1 z soon z m n is pa/ 1 ! e>? mak+ its way 9 ! int]nal elliptical l9e l 1 ! same m n produc$ t[>ds m1 w cut l g produc$4 ,& :5 ! e>? ha? almo/ f9i%$ its :ole circumf]5ce1 ! same m n %all aga9 make a "r angle ) a l9e drawn f ! c5tre i1 a little %ort ( ! po9t f :i* ! e>? began its motion4 ,& !re ! next ye> %all be "o ( ! equ9octial po9ts1 namely1 ne> ! end ( 2 ! o!r %all be opposite to x ne> ! end ( 4 ,& ?us ! po9ts 9 :i* ! days & nights are made equal d e ye> fall back2 b ) s sl[ a motion1 t1 9 a :ole ye>1 x makes b #ea fir/ m9utes4 ,& ?is relapse be+ contr>y to ! ord] ( ! signs1 is commonly call$ ! precession ( ! equ9oxes4 ,( :i* ,i h f my =m] suppositions d$uc$ a possible cause2 :i* 0 to be done4 ,accord+ to :at ,i h said conc]n+ ! cause ( ! ecc5tricity ( ! e>?2 & accord+ to ,kepl]1 :o = ! cause !re( suppose? "o "p ( ! e>? to be affect$ to ! sun1 ! o!r "p to be disaffect$2 ! apogum & p]igum ( ! sun %\ld be mov$ e ye> 9 ! same ord]1 & ) ! same velocity1 ) :i* ! equ9octial po9ts are mov$2 & _! di/ance f !m %\ld always be ! quadrant ( a circle2 :i* seems to be o!rwise4 ,= a/ronom]s say1 t ! equ9oxes are n[1 ! "o ab\t #bh degrees gone back f ! fir/ /> ( ,aries1 ! o!r z mu* f ! beg9n+ ( ,libra2 s t ! apogum ( ! sun or ! aphelium ( ! e>? ought to be ab\t ! #bh? degree ( ,canc]4 ,b x is reckon$ to be 9 ! #g? degree4 ,see+1 !re=e1 we h n suffici5t evid5ce ( ! 7t s x is17 x is 9 va9 to seek = ! 7:y x is s47 ,:]e=e1 z long z ! motion ( ! apogum is n obs]vable by reason ( ! sl[ness !re(1 & z long z x rema9s d\btful :e!r _! di/ance f ! equ9octial po9ts be m or less ?an a quadrant precisely2 s long x may be lawful = me to ?9k !y proce$ bo? ( !m ) equal velocity4 ,also1 ,i d n at all m$dle ) ! causes ( ! ecc5tricities ( ,saturn1 ,jupit]1 ,m>s1 & ,m]cury4 ,nev]!less1 see+ ! ecc5tricity ( ! e>? may1 z ,i h %ewn1 be caus$ by ! unlike con/itution ( ! sev]al p>ts ( ! e>? :i* are alt]nately turn$ t[>ds ! sun1 x is cr$ible also1 t l effects may be produc$ 9 ^! o!r planets f _! hav+ _! sup]ficies ( unlike p>ts4 ,& ?is is all ,i %all say conc]n+ ,sid]eal ,philosophy4 ,&1 ?\gh ! causes ,i h "h suppos$ be n ! true causes ( ^! ph5om5a1 yet ,i h demon/rat$ t !y are suffici5t to produce !m1 accord+ to :at ,i at fir/ prop\nd$4 ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xxvi4 ,fig4 #a-e ------- ,,*apt] ,,xxvii4 ,,( ,,light1 ,,heat1 ,,& ,,( ,,col\rs4 #a4 ,( ! imm5se magnitude ( "s bodies1 & ! unspeakable littl5ess ( o!rs4--#b4 ,( ! cause ( ! light ( ! sun4--#c4 ,h[ light heate?4--#d4 ,! g5]ation ( fire f ! sun4--#e4 ,! g5]ation ( fire f collision4--#f4 ,! cause ( light 9 gl[-worms1 rott5 wood1 & ! ,bolognan /one4--#g4 ,! cause ( light 9 ! concussion ( sea wat]4--#h4 ,! cause ( flame1 sp>ks1 & colliquation4--#i4 ,! cause :y wet hay sometimes burns ( its own accord2 also ! cause ( lightn+4--#aj4 ,! cause ( ! =ce ( gunp[d]2 & :at is to be ascrib$ to ! coals1 :at to ! brim/one1 & :at to ! nitre4--#aa4 ,h[ heat is caus$ by attrition4--#ab4 ,! di/9ction ( light into fir/1 second1 c4--#ac4 ,! causes ( ! col\rs we see 9 look+ ?r\gh a prisma ( glass1 namely1 ( r$1 yell[1 blue1 & violet col\r4--#ad4 ,:y ! moon & ! />s appe> r$d] 9 ! horizon ?an 9 ! mid/ ( ! heav54--#ae4 ,! cause ( :it5ess4--#af4 ,! cause ( blackness4 ,sid5ote3 ,( ! imm5se magnitude ( "s bodies1 & ! unspeakable littl5ess ( o!rs4 #a4 ,besides ! />s1 ( :i* ,i h spok5 9 ! la/ *apt]1 :atsoev] o!r bodies !re be 9 ! _w1 !y may be all compreh5d$ "u ! "n ( int]sid]eal bodies4 ,& ^! ,i h already suppos$ to be ei!r ! mo/ fluid !r1 or su* bodies ^: p>ts h "s degree ( cohesion4 ,n[1 ^! diff] f "o ano!r 9 _! sev]al consi/5cies1 magnitudes1 motions1 & figures4 ,9 consi/5cy1 ,i suppose "s bodies to be h>d]1 o!rs s(t] ?r\gh all ! sev]al degrees ( t5acity4 ,9 magnitude1 "s to be great]1 o!rs less1 & _m unspeakably little4 ,= we mu/ rememb] t1 by ! und]/&+1 quantity is divisible into divisibles p]petually4 ,&1 !re=e1 if a man c\ld d z mu* ) 8 h&s z he c ) 8 und]/&+1 he w\ld be able to take f any giv5 magnitude a "p :i* %\ld be less ?an any o!r magnitude giv54 ,b ! ,omnipot5t ,creator ( ! _w c actually f a "p ( any ?+ take ano!r "p1 z f> z we by our und]/&+ c conceive ! same to be divisible4 ,:]e=e !re is no impossible smallness ( bodies4 ,& :at h9d]s b t we may ?9k ?is likely8 ,= we "k !re are "s liv+ creatures s small t we c sc>ce see _! :ole bodies4 ,yet ev5 ^! h _! "y ones2 _! little ve9s & o!r vessels1 & _! eyes s small z t no microscope c make !m visible4 ,s t we _c suppose any magnitude s little1 b t our v supposition is actually exce$$ by nature4 ,besides1 !re are n[ su* microscopes commonly made1 t ! ?+s we see ) !m appe> a hundr$ ?\s& times bigg] ?an !y w\ld d if we look$ ^u !m ) our b>e eyes4 ,nor is !re any d\bt b t by augm5t+ ! p[] ( ^! microscopes 7= x may be augm5t$ z long z nei!r matt] nor ! h&s ( workm5 are want+7 e "o ( ^? hundr$ ?\s&? p>ts might yet appe> a hundr$ ?\s& times great] ?an !y did be=e4 ,nei!r is ! smallness ( "s bodies to be m admir$ ?an ! va/ greatness ( o!rs4 ,= x belongs to ! same ,inf9ite ,p[]1 z well to augm5t inf9itely z inf9itely to dim9i%4 ,to make ! great orb1 namely1 t ^: radius rea*e? f ! e>? to ! sun1 b z a po9t 9 respect ( ! di/ance betwe5 ! sun & ! fix$ />s2 &1 on ! contr>y1 to make a body s little1 z to be 9 ! same proportion less ?an any o!r visible body1 proce$s equally f "o & ! same ,au?or ( ,nature4 ,b ?is ( ! imm5se di/ance ( ! fix$ />s1 :i* = a long "t 0 acc\nt$ an incr$ible ?+1 is n[ believ$ by almo/ all ! le>n$4 ,:y !n %\ld n t o!r1 ( ! smallness ( "s bodies1 become cr$ible at "s "t or o!r8 ,= ! ,maje/y ( ,god appe>s no less 9 small ?+s ?an 9 great2 & z x exce$e? human s5se 9 ! imm5se greatness ( ! univ]se1 s also x do? 9 ! smallness ( ! p>ts !re(4 ,nor are ! fir/ elem5ts ( compositions1 nor ! fir/ beg9n+s ( actions1 nor ! fir/ mom5ts ( times m cr$ible1 ?an t :i* is n[ believ$ ( ! va/ di/ance ( ! fix$ />s4 ,"s ?+s are ackn[l$g$ by mortal m5 to be v great1 ?\gh f9ite1 z see+ !m to be su*4 ,!y ackn[l$ge also t "s ?+s1 :i* !y d n see1 may be ( inf9ite magnitude4 ,b !y are n pres5tly nor )\t great /udy p]suad$1 t !re is any mean betwe5 inf9ite & ! greate/ ( ^? ?+s :i* ei!r !y see or imag9e4 ,nev]!less1 :5 aft] m$itation & contemplation _m ?+s :i* we wond]$ at be=e are n[ gr[n m famili> to u1 we !n believe !m1 & transf] our admiration f ! creatures to ! ,creator4 ,b h[ little soev] "s bodies may be1 yet ,i w n suppose _! quantity to be less ?an is requisite = ! salv+ ( ! ph5om5a4 ,& 9 l mann] ,i %all suppose _! motion1 namely1 _! velocity & sl[ness1 & ! v>iety ( _! figures1 to be only su* z ! explication ( _! natural causes requires4 ,& la/ly1 ,i suppose1 t ! p>ts ( ! pure !r1 z if x 7 ! fir/ matt]1 h no motion at all b :at !y receive f bodies :i* float 9 !m1 & are n !mselves fluid4 ,sid5ote3 ,( ! cause ( ! light ( ! sun4 #b4 ,hav+ laid ^! gr\nds1 let u come to speak ( causes2 & 9 ! fir/ place let u inquire :at may be ! cause ( ! light ( ! sun4 ,see+1 !re=e1 ! body ( ! sun do? by its simple circul> motion ?ru/ away ! ambi5t e!real sub/ance sometimes "o way sometimes ano!r1 s t ^? p>ts1 :i* are next ! sun1 be+ mov$ by x1 d propagate t motion to ! next remote p>ts1 & ^! to ! next1 & s on cont9ually2 x mu/ ne$s be t1 not)/&+ any di/ance1 ! =emo/ "p ( ! eye w at la/ be press$2 & by ! pressure ( t "p1 ! motion w be propagat$ to ! inn]mo/ "p ( ! organ ( sight1 namely1 to ! he>t2 & f ! reaction ( ! he>t1 !re w proce$ an endeav\r back by ! same way1 end+ 9 ! endeav\r outw>ds ( ! coat ( ! eye1 call$ ! ret9a4 ,b ?is endeav\r outw>ds1 z has be5 def9$ 9 *apt] ,,xxv1 is ! ?+ :i* is call$ light1 or ! phantasm ( a lucid body4 ,= x is by reason ( ?is phantasm t an object is call$ lucid4 ,:]e=e we h a possible cause ( ! light ( ! sun2 :i* ,i und]took to f9d4 ,sid5ote3 ,h[ light heate?4 #c4 ,! g5]ation ( ! light ( ! sun is accompani$ ) ! g5]ation ( heat4 ,n[ e man kn[s :at heat is 9 himself1 by feel+ x :5 he gr[s hot2 b :at x is 9 o!r ?+s1 he kn[s only by ratioc9ation4 ,= x is "o ?+ to gr[ hot1 & ano!r ?+ to heat or make hot4 ,& !re=e ?\gh we p]ceive t ! fire or ! sun heate?1 yet we d n p]ceive t x is itself hot4 ,t o!r liv+ creatures1 :il/ !y make o!r ?+s hot1 are hot !mselves1 we inf] by reason+ f ! l s5se 9 ourselves4 ,b ?is is n a necess>y inf]5ce4 ,= ?\gh x may truly be said ( liv+ creatures1 t !y heat1 !re=e !y are !mselves hot2 yet x _c f h5ce be truly inf]r$ t fire heate?1 !re=e x is itself hot2 no m ?an ?is1 fire cause? pa91 !re=e x is itself 9 pa94 ,:]e=e1 t is only & prop]ly call$ hot1 :i* :5 we feel we are necess>ily hot4 ,n[ :5 we gr[ hot1 we f9d t our spirits & blood1 & :atsoev] is fluid )9 u1 is call$ out f ! int]nal to ! ext]nal p>ts ( our bodies1 m or less1 accord+ to ! degree ( ! heat2 & t our sk9 swelle?4 ,he1 !re=e1 t c give a possible cause ( ?is evocation & swell+1 & su* z agrees ) ! re/ ( ! ph5om5a ( heat1 may be ?\ght to h giv5 ! cause ( ! heat ( ! sun4 ,x ha? be5 %[n1 9 ! #e? article ( *apt] ,,xxi1 t ! fluid m$ium1 :i* we call ! air1 is s mov$ by ! simple circul> motion ( ! sun1 z t all its p>ts1 ev5 ! lea/1 d p]petually *ange places ) "o ano!r2 :i* *ange ( places is t :i* !re ,i call$ f]m5tation4 ,f ?is f]m5tation ( ! air1 ,i h1 9 ! #h? article ( ! la/ *apt]1 demon/rat$ t ! wat] may be drawn up into ! cl\ds4 ,& ,i %all n[ %[ t ! fluid p>ts may1 9 l mann]1 by ! same f]m5tation1 be drawn out f ! int]nal to ! ext]nal p>ts ( our bodies4 ,= see+ t :]esoev] ! fluid m$ium is contigu\s to ! body ( any liv+ creature1 !re ! p>ts ( t m$ium are1 by p]petual *ange ( place1 sep>at$ f "o ano!r2 ! contigu\s p>ts ( ! liv+ creature mu/1 ( necessity1 endeav\r to ent] into ! spaces ( ! sep>at$ p>ts4 ,= o!rwise ^? p>ts1 suppos+ !re is no vacuum1 w\ld h no place to g into4 ,& !re=e t1 :i* is mo/ fluid & sep>able 9 ! p>ts ( ! liv+ creature :i* are contigu\s to ! m$ium1 w g fir/ out2 & into ! place !re( w succe$ su* o!r p>ts z c mo/ easily transpire ?r\gh ! pores ( ! sk94 ,& f h5ce x is necess>y t ! re/ ( ! p>ts1 :i* are n sep>at$1 mu/ altoge!r be mov$ outw>ds1 = ! keep+ ( all places full4 ,b ?is motion outw>ds ( all p>ts toge!r mu/1 ( necessity1 press ^? p>ts ( ! ambi5t air :i* are ready to leave _! places2 & !re=e all ! p>ts ( ! body1 endeav\r+ at once t way1 make ! body swell4 ,:]e=e a possible cause is giv5 ( heat f ! sun2 :i* 0 to be done4 ,sid5ote3 ,! g5]ation ( fire f ! sun4 #d4 ,we h n[ se5 h[ light & heat are g5]at$2 heat by ! simple motion ( ! m$ium1 mak+ ! p>ts p]petually *ange places ) "o ano!r2 & light by ?is1 t by ! same simple motion action is propagat$ 9 a /rait l9e4 ,b :5 a body ha? its p>ts s mov$1 t x s5sibly bo? heats & %9es at ! same "t1 !n x is t we say fire is g5]at$4 ,n[ by fire ,i d n und]/& a body di/9ct f matt] combu/ible or gl[+1 z wood or iron1 b ! matt] itself1 n simply & always1 b !n only :5 x %9e? & heate?4 ,he1 !re=e1 t r5d]s a cause possible & agreeable to ! re/ ( ! ph5om5a1 namely1 :5ce1 & f :at action1 bo? ! %9+ & heat+ proce$1 may be ?\ght to h giv5 a possible cause ( ! g5]ation ( fire4 ,let1 !re=e1 ,a ,b ,c 79 ! fir/ figure7 be a sph]e1 or ! portion ( a sph]e1 ^: c5tre is ,d2 & let x be transp>5t & homog5e\s1 z cry/al1 glass1 or wat]1 & object$ to ! sun4 ,:]e=e1 ! =emo/ "p ,a ,b ,c w1 by ! simple motion ( ! sun1 by :i* x ?ru/s =w>ds ! m$ium1 be wr\ght ^u by ! sunbeams 9 ! /rait l9es ,e ,a1 ,f ,b1 & ,g ,c2 :i* /rait l9es may1 9 respect ( ! great di/ance ( ! sun1 be tak5 = p>allels4 ,& see+ ! m$ium )9 ! sph]e is ?ick] ?an ! m$ium )\t x1 ^? beams w be refract$ t[>ds ! p]p5dicul>s4 ,let ! /rait l9es ,e ,a & ,g ,c be produc$ till !y cut ! sph]e 9 ,h & ,i2 & draw+ ! p]p5dicul>s ,a ,d & ,c ,d1 ! refract$ beams ,e ,a & ,g ,c w ( necessity fall1 ! "o betwe5 ,a ,h & ,a ,d1 ! o!r betwe5 ,c ,i & ,c ,d4 ,let ^? refract$ beams be ,a ,k & ,c ,l4 ,& aga91 let ! l9es ,d ,k ,m & ,d ,l ,n be drawn p]p5dicul> to ! sph]e2 & let ,a ,k & ,c ,l be produc$ till !y meet ) ! /rait l9e ,b ,d produc$ 9 ,o4 ,see+1 !re=e1 ! m$ium )9 ! sph]e is ?ick] ?an t )\t x1 ! refract$ l9e ,a ,k w rec$e fur!r f ! p]p5dicul> ,k ,m ?an ,k ,o w rec$e f ! same4 ,:]e=e ,k ,o w fall betwe5 ! refract$ l9e & ! p]p5dicul>4 ,let1 !re=e1 ! refract$ l9e be ,k ,p1 cutt+ ,f ,o 9 ,p2 & = ! same reason ! /rait l9e ,l ,p w be ! refract$ l9e ( ! /rait l9e ,c ,l4 ,:]e=e1 see+ ! beams are no?+ else b ! ways 9 :i* ! motion is propagat$1 ! motion ab\t ,p w be s mu* m vehem5t ?an ! motion ab\t ,a ,b ,c1 by h[ mu* ! base ( ! portion ,a ,b ,c is great] ?an ! base ( a l portion 9 ! sph]e1 ^: c5tre is ,p1 & ^: magnitude is equal to t ( ! little circle ab\t ,p1 :i* compreh5de? all ! beams t are propagat$ f ,a ,b ,c2 & ?is sph]e be+ mu* less ?an ! sph]e ,a ,b ,c1 ! p>ts ( ! m$ium1 t is1 ( ! air ab\t ,p1 w *ange places ) "o ano!r ) mu* great] cel]ity ?an ^? ab\t ,a ,b ,c4 ,if1 !re=e1 any matt] combu/ible1 t is to say1 su* z may be easily dissipat$1 be plac$ 9 ,p1 ! p>ts ( t matt]1 if ! proportion be great en\gh betwe5 ,a ,c & a l portion ( ! little circle ab\t ,p1 w be fre$ f _! mutual cohesion1 & be+ sep>at$ w acquire simple motion4 ,b vehem5t simple motion g5]ates 9 ! behold] a phantasm ( lucid & hot1 z ,i h be=e demon/rat$ ( ! simple motion ( ! sun2 & !re=e ! combu/ible matt] :i* is plac$ 9 ,p w be made lucid & hot1 t is to say1 w be fire4 ,:]e=e ,i h r5d]$ a possible cause ( fire2 :i* 0 to be done4 ,sid5ote3 ,! g5]ation ( fire f collision4 #e4 ,f ! mann] by :i* ! sun g5]ate? fire1 x is easy to expla9 ! mann] by :i* fire may be g5]at$ by ! collision ( two fl9ts4 ,= by t collision "s ( ^? p>ticles ( :i* ! /one is compact$1 are viol5tly sep>at$ & ?r[n (f2 & be+ )al swiftly turn$ r\nd1 ! eye is mov$ by !m1 z x is 9 ! g5]ation ( light by ! sun4 ,:]e=e !y %9e2 & fall+ ^u matt] :i* is already half dissipat$1 su* z is t9d]1 !y ?or\ghly dissipate ! p>ts !re(1 & make !m turn r\nd4 ,f :5ce1 z ,i h newly %[n1 light & heat1 t is to say fire1 is g5]at$4 ,sid5ote3 ,! cause ( light 9 gl[-worms1 rott5 wood1 & ! ,bolognan /one4 #f4 ,! %9+ ( gl[-worms1 "s k9ds ( rott5 wood1 & ( a k9d ( /one made at ,bologna1 may h "o common cause1 namely1 ! expos+ ( !m to ! hot sun4 ,we f9d by exp]i5ce t ! ,bologna /one %9es n1 unless x be s expos$2 & aft] x has be5 expos$ x %9es b = a little "t1 namely1 z long z x reta9s a c]ta9 degree ( heat4 ,& ! cause may be t ! p>ts1 ( :i* x is made1 may toge!r ) heat h simple motion impr9t$ 9 !m by ! sun4 ,:i* if x be s1 x is necess>y t x %9e 9 ! d>k1 z long z !re is suffici5t heat 9 x2 b ?is ceas+1 x w %9e no long]4 ,also we f9d by exp]i5ce t 9 ! gl[-worm !re is a c]ta9 ?ick hum\r1 l ! cry/all9e hum\r ( ! eye2 :i* if x be tak5 out & held long en\gh 9 "o's f+]s1 & !n be c>ri$ into ! d>k1 x w %9e by reason ( ! w>m? x receiv$ f ! f+]s2 b z soon z x is cold x w cease %9+4 ,f :5ce1 !re=e1 c ^! creatures h _! light1 b f ly+ all day 9 ! sun%9e 9 ! hotte/ "t ( summ]8 ,9 ! same mann]1 rott5 wood1 except x gr[ rott5 9 ! sun%9e1 or be aft]w>ds long en\gh expos$ to ! sun1 w n %9e4 ,t ?is do? n happ5 9 e worm1 nor 9 all k9ds ( rott5 wood1 nor 9 all calc9$ /ones1 ! cause may be t ! p>ts1 ( :i* ! bodies are made1 are diff]5t bo? 9 motion & figure f ! p>ts ( bodies ( o!r k9ds4 ,sid5ote3 ,! cause ( light 9 ! concussion ( sea wat]4 #g4 ,also ! sea wat] %9e? :5 x is ei!r da%$ ) ! /rokes ( o>s1 or :5 a %ip 9 its c\rse breaks /rongly ?r\gh x2 b m or less1 accord+ z ! w9d bl[s f diff]5t po9ts4 ,! cause :]e( may be ?is1 t ! p>ticles ( salt1 ?\gh !y nev] %9e 9 ! salt-pits1 :]e !y are b sl[ly drawn up by ! sun1 be+ "h beat5 up into ! air 9 great] quantities & ) m =ce1 are )al made to turn r\nd1 & consequ5tly to %9e1 ?\gh weakly4 ,i h1 !re=e1 giv5 a possible cause ( ?is ph5om5on4 ,sid5ote3 ,! cause ( flame1 sp>ks1 colliquation4 #h4 ,if su* matt] z is comp\nd$ ( h>d little bodies be set on fire1 x mu/ ne$s be1 t1 z !y fly out 9 great] or less quantities1 ! flame :i* is made by !m w be great] or less4 ,& if ! e!real or fluid "p ( t matt] fly out toge!r ) !m1 _! motion w be ! swift]1 z x is 9 wood & o!r ?+s :i* flame ) a manife/ mixture ( w9d4 ,:51 !re=e1 ^! h>d p>ticles by _! fly+ out move ! eye /rongly1 !y %9e bright2 & a great quantity ( !m fly+ out toge!r1 !y make a great %9+ body4 ,= flame be+ no?+ b an aggregate ( %9+ p>ticles1 ! great] ! aggregate is1 ! great] & m manife/ w be ! flame4 ,i h1 !re=e1 %[n a possible cause ( flame4 ,& f h5ce ! cause appe>s evid5tly1 :y glass is s easily & quickly melt$ by ! small flame ( a c&le bl[n1 :i* w n be melt$ )\t bl[+ b by a v /rong fire4 ,n[1 if f ! same matt] !re be a "p brok5 (f1 namely1 su* a "p z consi/e? ( _m ( ! small p>ticles1 ( ?is is made a sp>k4 ,= f ! break+ (f x ha? a viol5t turn+ r\nd1 & f h5ce x %9es4 ,b ?\gh f ?is matt] !re fly nei!r flame nor sp>ks1 yet "s ( ! smalle/ p>ts ( x may be c>ri$ out z f> z to ! sup]ficies1 & rema9 !re z a%es2 ! p>ts :]e( are s extremely small1 t x _c any long] be d\bt$ h[ f> nature may proce$ 9 divid+4 ,la/ly1 ?\gh by ! application ( fire to ?is matt] !re fly little or no?+ f x1 yet !re w be 9 ! p>ts an endeav\r to simple motion2 by :i* ! :ole body w ei!r be melt$1 or1 :i* is a degree ( melt+1 s(t5$4 ,= all motion has "s effect ^u all matt] :atsoev]1 z has be5 %[n at art4 #c1 *ap4 ,,xv4 ,n[ if x be s(t5$ to su* a degree1 z t ! /ubbornness ( ! p>ts be exce$$ by _! gravity1 !n we say x is melt$2 o!rwise1 s(t5$ & made pliant & ductile4 ,aga91 ! matt] hav+ 9 x "s p>ticles h>d1 o!rs e!real or wat]y2 if1 by ! application ( fire1 ^! latt] be call$ out1 ! =m] w !reby come to a m full contact ) "o ano!r2 &1 consequ5tly1 w n be s easily sep>at$2 t is to say1 ! :ole body w be made h>d]4 ,& ?is may be ! cause :y ! same fire makes "s ?+s s(t1 o!rs h>d4 ,sid5ote3 ,! cause :y wet hay sometimes burns ( its own accord2 also ! cause ( lightn+4 #i4 ,x is kn[n by exp]i5ce t if hay be laid wet toge!r 9 a heap1 x w aft] a "t beg9 to smoke1 & !n burn z x 7 ( itself4 ,! cause :]e( seems to be ?is1 t 9 ! air1 :i* is enclos$ )9 ! hay1 !re are ^? little bodies1 :i*1 z ,i h suppos$1 are mov$ freely ) simple motion4 ,b ?is motion be+ by degrees h9d]$ m & m by ! desc5d+ moi/ure1 :i* at ! la/ fills & /ops all ! passages1 ! ?9n] p>ts ( ! air asc5d by p5etrat+ ! wat]2 & ^? h>d little bodies1 be+ s ?ru/ toge!r t !y t\* & press "o ano!r1 acquire /rong] motion2 till at la/ by ! increas$ /r5g? ( ?is motion ! wat]y p>ts are fir/ driv5 outw>ds1 f :5ce appe>s vap\r2 & by ! cont9u$ increase ( ?is motion1 ! smalle/ p>ticles ( ! dri$ hay are =c$ out1 & recov]+ _! natural simple motion1 !y gr[ hot & %9e1 t is to say1 !y are set on fire4 ,! same also may be ! cause ( lightn+1 :i* happ5s 9 ! hotte/ "t ( ! ye>1 :5 ! wat] is rais$ up 9 greate/ quantity & c>ri$ highe/4 ,= aft] ! fir/ cl\ds are rais$1 o!rs aft] o!rs foll[ !m2 & be+ congeal$ above1 !y happ51 :il/ "s ( !m asc5d & o!rs desc5d1 to fall "o ^u ano!r 9 su* mann]1 z t 9 "s places all _! p>ts are jo9$ toge!r1 9 o!rs !y leave holl[ spaces betwe5 !m2 & into ^! spaces1 ! e!real p>ts be+ =c$ out by ! compressure ( ! cl\ds1 _m ( ! h>d] little bodies are s p5t toge!r1 z !y h n ! lib]ty ( su* motion z is natural to ! air4 ,:]e=e _! endeav\r gr[s m vehem5t1 till at la/ !y =ce _! way ?r\gh ! cl\ds1 sometimes 9 "o place1 sometimes 9 ano!r2 &1 break+ ?r\gh ) great noise1 !y move ! air viol5tly1 & /rik+ our eyes1 g5]ate light1 t is to say1 !y %9e4 ,& ?is %9+ is t we call lightn+4 ,sid5ote3 ,! cause ( ! =ce ( gunp[d]2 & :at is to be ascrib$ to ! coals1 :at to ! brim/one1 & :at to ! nitre4 #aj4 ,! mo/ common ph5om5on proce$+ f fire1 & yet ! mo/ admirable ( all o!rs1 is ! =ce ( gunp[d] fir$2 :i* be+ comp\nd$ ( nitre1 brim/one & coals1 beat5 small1 ha? f ! coals its fir/ tak+ fire2 f ! brim/one its n\ri%m5t & flame1 t is to say1 light & motion1 & f ! nitre ! vehem5ce ( bo?4 ,n[ if a piece ( nitre1 be=e x is beat51 be laid ^u a burn+ coal1 fir/ x melts1 &1 l wat]1 qu5*e? t "p ( ! coal x t\*e?4 ,!n vap\r or air1 fly+ out :]e ! coal & nitre jo91 bl[e? ! coal ) great swiftness & vehem5ce on all sides4 ,& f h5ce x comes to pass1 t by two contr>y motions1 ! "o1 ( ! p>ticles :i* g out ( ! burn+ coal1 ! o!r1 ( ^? ( ! e!real & wat]y sub/ance ( ! nitre1 is g5]at$ t vehem5t motion & inflammation4 ,&1 la/ly1 :5 !re is no m action f ! nitre1 t is to say1 :5 ! volatile p>ts ( ! nitre are fl[n out1 !re is f\nd ab\t ! sides a c]ta9 :ite sub/ance1 :i* be+ ?r[n aga9 into ! fire1 w gr[ r$-hot aga91 b w n be dissipat$1 at lea/ unless ! fire be augm5t$4 ,if n[ a possible cause ( ?is be f\nd out1 ! same w also be a possible cause :y a gra9 ( gunp[d] set on fire do? exp& itself ) su* vehem5t motion1 & %9e4 ,& x may be caus$ 9 ?is mann]4 ,let ! p>ticles1 ( :i* nitre consi/e?1 be suppos$ to be "s ( !m h>d1 o!rs wat]y1 & ! re/ e!real4 ,also let ! h>d p>ticles be suppos$ to be sph]ically holl[1 l small bubbles1 s t _m ( !m gr[+ toge!r may con/itute a body1 ^: little cav]ns are fill$ ) a sub/ance :i* is ei!r wat]y1 or e!real1 or bo?4 ,z soon1 !re=e1 z ! h>d p>ticles are dissipat$1 ! wat]y & e!real p>ticles w necess>ily fly out2 & z !y fly1 ( necessity bl[ /rongly ! burn+ coals & brim/one :i* are m+l$ toge!r2 :]eupon !re w foll[ a great expansion ( light1 ) vehem5t flame1 & a viol5t dissipation ( ! p>ticles ( ! nitre1 ! brim/one & ! coals4 ,:]e=e ,i h giv5 a possible cause ( ! =ce ( fir$ gunp[d]4 ,x is manife/ f h5ce1 t = ! r5d]+ ( ! cause :y a bullet ( lead or iron1 %ot f a piece ( ordnance1 flies ) s great velocity1 !re is no necessity to introduce su* r>efaction1 z1 by ! common def9ition ( x1 makes ! same matt] to h sometimes m1 sometimes less quantity2 :i* is inconceivable4 ,= e ?+ is said to be great] or less1 z x ha? m or less quantity4 ,! viol5ce ) :i* a bullet is ?ru/ out ( a gun1 proce$s f ! swiftness ( ! small p>ticles ( ! fir$ p[d]2 at lea/ x may proce$ f t cause )\t ! supposition ( any empty space4 ,sid5ote3 ,h[ heat is caus$ by attrition4 #aa4 ,besides1 by ! attrition or rubb+ ( "o body aga9/ ano!r1 z ( wood aga9/ wood1 we f9d t n only a c]ta9 degree ( heat1 b fire itself is sometimes g5]at$4 ,= su* motion is ! reciprocation ( pressure1 sometimes "o way1 sometimes ! o!r2 & by ?is reciprocation :atsoev] is fluid 9 bo? ! pieces ( wood is =c$ hi!r & ?i!r2 & consequ5tly1 to an endeav\r ( gett+ out2 & at la/ by break+ out makes fire4 ,sid5ote3 ,! di/9ction ( light into fir/1 second1 c4 #ab4 ,n[ light is di/+ui%$ into1 fir/1 second1 ?ird1 & s on inf9itely4 ,& we call t fir/ light1 :i* is 9 ! fir/ lucid body2 z ! sun1 fire1 c43 second1 t :i* is 9 su* bodies1 z be+ n transp>5t are illum9at$ by ! sun2 z ! moon1 a wall1 c43 & ?ird1 t :i* is 9 bodies n transp>5t1 b illum9at$ by second light1 c4 ,sid5ote3 ,! causes ( ! col\rs we see 9 look+ ?r\gh a prisma ( glass1 namely1 ( r$1 yell[1 blue1 violet col\r4 #ac4 ,col\r is light1 b tr\bl$ light1 namely1 su* z is g5]at$ by p]turb$ motion2 z %all be made manife/ by ! r$1 yell[1 blue & purple1 :i* are g5]at$ by ! int]position ( a diaphan\s prisma1 ^: opposite bases are triangul>1 betwe5 ! light & t :i* is enlight5$4 ,sid5ote3 ,discussion ( ,figure #bg4b ,= let !re be a prisma ( glass1 or ( any o!r transp>5t matt] :i* is ( great] d5sity ?an air2 & let ! triangle ,a ,b ,c be ! base ( ?is prisma4 ,also let ! /rait l9e ,d ,e be ! diamet] ( ! sun's body1 hav+ oblique position to ! /rait l9e ,a ,b2 & let ! sunbeams pass 9 ! l9es ,d ,a & ,e ,b ,c4 ,& la/ly1 let ! /rait l9es ,d ,a & ,e ,c be produc$ indef9itely to ,f & ,g4 ,see+ !re=e ! /rait l9e ,d ,a1 by reason ( ! d5sity ( ! glass1 is refract$ t[>ds ! p]p5dicul>2 let ! l9e refract$ at ! po9t ,a be ! /rait l9e ,a ,h4 ,& aga91 see+ ! m$ium bel[ ,a ,c is ?9n] ?an t above x1 ! o!r refraction1 :i* w be made !re1 w div]ge f ! p]p5dicul>4 ,let !re=e ?is second refract$ l9e be ,a ,i4 ,also let ! same be done at ! po9t ,c1 by mak+ ! fir/ refract$ l9e to be ,c ,k1 & ! second ,c ,l4 ,see+ !re=e ! cause ( ! refraction 9 ! po9t ,a ( ! /rait l9e ( ,a ,b is ! excess ( ! resi/ance ( ! m$ium 9 ,a ,b above ! resi/ance ( ! air1 !re mu/ ( necessity be reaction f ! po9t ,a t[>ds ! po9t ,b2 & consequ5tly ! m$ium at ,a )9 ! triangle ,a ,b ,c w h its motion tr\bl$1 t is to say1 ! /rait motion 9 ,a ,f & ,a ,h w be mix$ ) ! transv]se motion betwe5 ! same ,a ,f & ,a ,h1 repres5t$ by ! %ort transv]se l9es 9 ! triangle ,a ,f ,h4 ,aga91 see+ at ! po9t ,a ( ! /rait l9e ,a ,c !re is a second refraction f ,a ,h 9 ,a ,i1 ! motion ( ! m$ium w aga9 be p]turb$ by reason ( ! transv]se reaction f ,a t[>ds ,c1 repres5t$ likewise by ! %ort transv]se l9es 9 ! triangle ,a ,h ,i4 ,& 9 ! same mann] !re is a d\ble p]turbation repres5t$ by ! transv]se l9es 9 ! triangles ,c ,g ,k & ,c ,k ,l4 ,b z = ! light betwe5 ,a ,i & ,c ,g1 x w n be p]turb$2 because1 if !re 7 9 all ! po9ts ( ! /rait l9es ,a ,b & ,a ,c ! same action :i* is 9 ! po9ts ,a & ,c1 !n ! plane ( ! triangle ,c ,g ,k w\ld be ev]y:]e co9cid5t ) ! plane ( ! triangle ,a ,f ,h2 by :i* means all w\ld appe> alike betwe5 ,a & ,c4 ,besides1 x is to be obs]v$1 t all ! reaction at ,a t5ds t[>ds ! illum9at$ p>ts :i* are betwe5 ,a & ,c1 & consequ5tly p]turbe? ! fir/ light4 ,& on ! contr>y1 t all ! reaction at ,c t5ds t[>ds ! p>ts )\t ! triangle or )\t ! prisma ,a ,b ,c1 :]e !re is none b second light2 & t ! triangle ,a ,f ,h %[s t p]turbation ( light :i* is made 9 ! glass itself2 z ! triangle ,a ,h ,i %[s t p]turbation ( light :i* is made bel[ ! glass4 ,9 l mann]1 t ,c ,g ,k %[s ! p]turbation ( light )9 ! glass2 & ,c ,k ,l t :i* is bel[ ! glass4 ,f :5ce !re are f\r div]s motions1 or f\r diff]5t illum9ations or col\rs1 ^: diff]5ces appe> mo/ manife/ly to ! s5se 9 a prisma1 ^: base is an equilat]al triangle1 :5 ! sunbeams t pass ?r\gh x are receiv$ ^u a :ite pap]4 ,= ! triangle ,a ,f ,h appe>s r$ to ! s5se2 ! triangle ,a ,h ,i yell[2 ! triangle ,c ,g ,k gre51 & approa*+ to blue2 & la/ly1 ! triangle ,c ,k ,l appe>s purple4 ,x is !re=e evid5t t :5 weak b fir/ light passe? ?r\gh a m resi/+ diaphan\s body1 z glass1 ! beams1 :i* fall ^u x transv]sely1 make r$ness2 & :5 ! same fir/ light is /rong]1 z x is 9 ! ?9n] m$ium bel[ ! /rait l9e ,a ,c1 ! transv]se beams make yell[ness4 ,also :5 second light is /rong1 z x is 9 ! triangle ,c ,g ,k1 :i* is ne>e/ to ! fir/ light1 ! transv]se beams make gre5ness2 & :5 ! same second light is weak]1 z 9 ! triangle ,c ,k ,l1 !y make a purple col\r4 ,sid5ote3 ,:y ! moon & ! />s appe> r$d] 9 ! horizon ?an 9 ! mid/ ( ! heav54 #ad4 ,f h5ce may be d$uc$ a cause1 :y ! moon & />s appe> bigg] & r$d] ne> ! horizon ?an 9 ! mid-heav54 ,= betwe5 ! eye & ! app>5t horizon !re is m impure air1 su* z is m+l$ ) wat]y & e>?y little bodies1 ?an is betwe5 ! same eye & ! m elevat$ "p ( heav54 ,b vision is made by beams :i* con/itute a cone1 ^: base1 if we look ^u ! moon1 is ! moon's face1 & ^: v]tex is 9 ! eye2 & !re=e1 _m beams f ! moon mu/ ne$s fall ^u little bodies t are )\t ! visual cone1 & be by !m reflect$ to ! eye4 ,b ^! reflect$ beams t5d all 9 l9es :i* are transv]se to ! visual cone1 & make at ! eye an angle :i* is great] ?an ! angle ( ! cone4 ,:]e=e1 ! moon appe>s great] 9 ! horizon1 ?an :5 %e is m elevat$4 ,& because ^? reflect$ beams g transv]sely1 !re w be g5]at$1 by ! la/ article1 r$ness4 ,a possible cause !re=e is %[n1 :y ! moon z also ! />s appe> great] & r$d] 9 ! horizon1 ?an 9 ! mid/ ( heav54 ,! same also may be ! cause1 :y ! sun appe>s 9 ! horizon great] & ( a col\r m deg5]at+ to yell[1 ?an :5 he is high] elevat$4 ,= ! reflection f ! little bodies betwe51 & ! transv]se motion ( ! m$ium1 are /ill ! same4 ,b ! light ( ! sun is mu* /rong] ?an t ( ! moon2 & !re=e1 by ! la/ article1 8 spl5d\r mu/ ne$s by ?is p]turbation deg5]ate into yell[ness4 ,b = ! g5]ation ( ^! f\r col\rs1 x is n necess>y t ! figure ( ! glass be a prisma2 = if x 7 sph]ical x w\ld d ! same4 ,= 9 a sph]e ! sunbeams are twice refract$ & twice reflect$4 ,& ?is be+ obs]v$ by ,des ,c>tes1 & )al t a ra9b[ nev] appe>s b :5 x ra9s2 z also1 t ! drops ( ra9 h _! figures almo/ sph]ical2 he ha? %[n f !nce ! cause ( ! col\rs 9 ! ra9b[2 :i* !re=e ne$ n be repeat$4 ,sid5ote3 ,! cause ( :it5ess4 #ae4 ,:it5ess is light1 b light p]turb$ by ! reflections ( _m beams ( light com+ to ! eye toge!r )9 a little space4 ,= if glass or any o!r diaphan\s body be r$uc$ to v small p>ts by contusion or concussion1 e "o ( ^? p>ts1 if ! beams ( a lucid body be f any "o po9t ( ! same reflect$ to ! eye1 w repres5t to ! behold] an idea or image ( ! :ole lucid body1 t is to say1 a phantasm ( :ite4 ,= ! /ronge/ light is ! mo/ :ite2 & !re=e _m su* p>ts w make _m su* images4 ,:]e=e1 if ^? p>ts lie ?ick & close toge!r1 ^? _m images w appe> confus$ly1 & w by reason ( ! confus$ light repres5t a :ite col\r4 ,s t f h5ce may be d$uc$ a possible cause1 :y glass beat51 t is1 r$uc$ to p[d]1 looks :ite4 ,also :y wat] & sn[ are :ite2 !y be+ no?+ b a heap ( v small diaphan\s bodies1 namely1 ( little bubbles1 f ^: sev]al convex sup]ficies !re are by reflection made sev]al confus$ phantasms ( ! :ole lucid body1 t is to say1 :it5ess4 ,= ! same reason1 salt & nitre are :ite1 z consi/+ ( small bubbles :i* conta9 )9 !m wat] & air2 z is manife/ 9 nitre1 f ?is1 t be+ ?r[n into ! fire x viol5tly bl[s ! same2 :i* salt also do?1 b ) less viol5ce4 ,b if a :ite body be expos$1 n to ! light ( ! day1 b to t ( ! fire or ( a c&le1 x w n at ! fir/ sight be easily judg$ :e!r x be :ite or yell[2 ! cause :]e( may be ?is1 t ! light ( ^? ?+s1 :i* burn & flame1 is almo/ ( a middle col\r betwe5 :it5ess & yell[ness4 ,sid5ote3 ,! cause ( blackness4 #af4 ,z :it5ess is light1 s blackness is ! privation ( light1 or d>kness4 ,&1 f h5ce x is1 fir/1 t all holes1 f :i* no light c be reflect$ to ! eye1 appe> black4 ,secondly1 t :5 a body ha? little em95t p>ticles erect$ /raight up f ! sup]ficies1 s t ! beams ( light :i* fall ^u !m are reflect$ n to ! eye b to ! body itself1 t sup]ficies appe>s black2 9 ! same mann] z ! sea appe>s black :5 ruffl$ by ! w9d4 ,?irdly1 t any combu/ible matt] is by ! fire made to look black be=e x %9es4 ,= ! endeav\r ( ! fire be+ to dissipate ! smalle/ p>ts ( su* bodies z are ?r[n into x1 x mu/ fir/ raise & erect ^? p>ts be=e x c "w _! dissipation4 ,if1 !re=e1 ! fire be put out be=e ! p>ts are totally dissipat$1 ! coal w appe> black2 = ! p>ts be+ only erect$1 ! beams ( light fall+ ^u !m w n be reflect$ to ! eye1 b to ! coal itself4 ,f\r?ly1 t burn+ glasses d m easily burn black ?+s ?an :ite4 ,= 9 a :ite sup]ficies ! em95t p>ts are convex1 l little bubbles2 & !re=e ! beams ( light1 :i* fall ^u !m1 are reflect$ e way f ! reflect+ body4 ,b 9 a black sup]ficies1 :]e ! em95t p>ticles are m erect$1 ! beams ( light fall+ ^u !m are all necess>ily reflect$ t[>ds ! body itself2 &1 !re=e1 bodies t are black are m easily set on fire by ! sun beams1 ?an ^? t are :ite4 ,fif?ly1 t all col\rs t are made ( ! mixture ( :ite & black proce$ f ! diff]5t position ( ! p>ticles t rise above ! sup]ficies1 & _! diff]5t =ms ( asp]ity4 ,=1 accord+ to ^! diff]5ces1 m or few] beams ( light are reflect$ f sev]al bodies to ! eye4 ,b 9 reg>d ^? diff]5ces are innum]able1 & ! bodies !mselves s small t we _c p]ceive !m2 ! explication & precise det]m9ation ( ! causes ( all col\rs is a ?+ ( s great difficulty1 t ,i d>e n und]take x4 ---------------------------------------- -------------------------------- ,illu/ration3 ,vol4 ,i4 ,lat4 ,eng4 ,c4 ,,xxvii4 ,fig4 #a-b ,fig #a ,fig #b ---------------------------------------- -------------------------------- ------- ,,*apt] ,,xxviii4 ,,( ,,cold1 ,,w9d1 ,,h>d1 ,,ice1 ,,re/itution ,,( ,,bodies ,,b5t1 ,,diaphan\s1 ,,lightn+ ,,& ,,?und]2 ,,& ,,( ,,! ,,heads ,,( ,,riv]s4 #a4 ,:y brea? f ! same m\? sometimes heats & sometimes cools4--#b4 ,w9d1 & ! incon/ancy ( w9ds1 :5ce4--#c ,:y !re is a con/ant1 ?\gh n a great w9d1 f ea/ to we/1 ne> ! equator4--#d4 ,:at is ! effect ( air p5t 9 betwe5 ! cl\ds4--#e4 ,no *ange f s(t to h>d1 b by motion4--#f4 ,:at is ! cause ( cold ne> ! poles4--#g44 ,! cause ( ice2 & :y ! cold is m remiss 9 ra9y ?an 9 cle> wea!r4 ,:y wat] do? n freeze 9 deep wells z x do? ne> ! sup]ficies ( ! e>?4 ,:y ice is n s heavy z wat]2 & :y w9e is n s easily froz5 z wat]4--#h4 ,ano!r cause ( h>dness f ! full] contact ( atoms2 also1 h[ h>d ?+s are brok54--#i4 ,a ?ird cause ( h>dness f heat4--#aj4 ,a f\r? cause ( h>dness f ! motion ( atoms enclos$ 9 a n>r[ space4--#aa4 ,h[ h>d ?+s are s(t5$4--#ab4 ,:5ce proce$ ! spontane\s re/itution ( ?+s b5t4--#ac4 ,diaphan\s & opac\s1 :at !y are1 & :5ce4--#ad4 ,! cause ( lightn+ & ?und]4--#ae4 ,:5ce x proce$s t cl\ds c fall aga9 aft] !y are once elevat$ & froz54--#af4 ,h[ x c\ld be t ! moon 0 eclips$1 :5 %e 0 n diametrically opposite to ! sun4--#ag4 ,by :at means _m suns may appe> at once4--#ah4 ,( ! heads ( riv]s4 ,sid5ote3 ,:y brea? f ! same m\? sometimes heats & sometimes cools4 #a4 ,z1 :5 ! motion ( ! ambi5t e!real sub/ance makes ! spirits & fluid p>ts ( our bodies t5d outw>ds1 we ackn[l$ge heat2 s1 by ! endeav\r inw>ds ( ! same spirits & hum\rs1 we feel cold4 ,s t to cool is to make ! ext]ior p>ts ( ! body endeav\r inw>ds1 by a motion contr>y to t ( calefaction1 by :i* ! int]nal p>ts are call$ outw>ds4 ,he1 !re=e1 t w\ld "k ! cause ( cold1 mu/ f9d by :at motion or motions ! ext]ior p>ts ( any body endeav\r to retire inw>ds4 ,to beg9 ) ^? ph5om5a :i* are ! mo/ famili>4 ,!re is almo/ no man b kn[s1 t brea? bl[n /rongly1 & :i* comes f ! m\? ) viol5ce1 t is to say1 ! passage be+ /rait1 w cool ! h&2 & t ! same brea? bl[n g5tly1 t is to say1 ?r\gh a great] ap]ture1 w w>m ! same4 ,! cause ( :i* ph5om5on may be ?is1 ! brea? go+ out ha? two motions2 ! "o1 ( ! :ole & direct1 by :i* ! =emo/ p>ts ( ! h& are driv5 inw>ds2 ! o!r1 simple motion ( ! small p>ticles ( ! same brea?1 :i*1 7z ,i h %[n 9 ! #crd article ( ! la/ *apt]1 cause? heat4 ,accord+1 !re=e1 z ei!r ( ^! motions is pr$om9ant1 s !re is ! s5se sometimes ( cold1 sometimes ( heat4 ,:]e=e1 :5 ! brea? is s(tly brea!d out at a l>ge passage1 t simple motion :i* cause? heat prevaile?1 & consequ5tly heat is felt2 & :51 by compress+ ! lips1 ! brea? is m /rongly bl[n out1 !n is ! direct motion preval5t1 :i* makes u feel cold4 ,=1 ! direct motion ( ! brea? or air is w9d2 & all w9d cools or dim9i%e? =m] heat4 ,sid5ote3 ,w9d1 & ! incon/ancy ( w9ds1 :5ce4 #b4 ,& see+ n only great w9d1 b almo/ any v5tilation & /irr+ ( ! air1 do? refrig]ate2 ! reason ( _m exp]im5ts conc]n+ cold _c well be giv5 )\t f9d+ fir/ :at are ! causes ( w9d4 ,n[1 w9d is no?+ else b ! direct motion ( ! air ?ru/ =w>ds2 :i*1 nev]!less1 :5 _m w9ds concur1 may be circul> or o!rwise indirect1 z x is 9 :irlw9ds4 ,:]e=e1 9 ! fir/ place we are to enquire into ! causes ( w9ds4 ,w9d is air mov$ 9 a consid]able quantity1 & t ei!r 9 ! mann] ( waves1 :i* is bo? =w>ds & also up & d[n1 or else =w>ds only4 ,suppos+1 !re=e1 ! air bo? cle> & calm = any "t h[ little soev]1 yet1 ! great] bodies ( ! _w be+ s dispos$ & ord]$ z has be5 said1 x w be necess>y t a w9d pres5tly arise some:]e4 ,=1 see+ t motion ( ! p>ts ( ! air1 :i* is made by ! simple motion ( ! sun 9 8 own epicycle1 cause? an exhalation ( ! p>ticles ( wat] f ! seas & all o!r moi/ bodies1 & ^? p>ticles make cl\ds2 x mu/ ne$s foll[1 t1 :il/ ! p>ticles ( wat] pass upw>ds1 ! p>ticles ( air1 = ! keep+ ( all spaces full1 be jo/l$ out on e side1 & urge ! next p>ticles1 & ^! ! next2 till hav+ made _! circuit1 !re comes cont9ually s mu* air to ! h9d] p>ts ( ! e>? z !re w5t wat] f be=e x4 ,:]e=e1 ! asc5d+ vap\rs move ! air t[>ds ! sides e way2 & all direct motion ( ! air be+ w9d1 !y make a w9d4 ,& if ?is w9d meet (t5 ) o!r vap\rs :i* arise 9 o!r places1 x is manife/ t ! =ce !re( w be augm5t$1 & ! way or c\rse ( x *ang$4 ,besides1 accord+ z ! e>?1 by its diurnal motion1 turns sometimes ! dri]1 sometimes ! moi/] "p t[>ds ! sun1 s sometimes a great]1 sometimes a less1 quantity ( vap\rs w be rais$2 t is to say1 sometimes !re w be a less1 sometimes a great] w9d4 ,:]e=e1 ,i h r5d]$ a possible cause ( su* w9ds z are g5]at$ by vap\rs2 & also ( _! incon/ancy4 ,f h5ce x foll[s t ^! w9ds _c be made 9 any place1 :i* is high] ?an t to :i* vap\rs may asc5d4 ,nor is t incr$ible :i* is report$ ( ! highe/ m\nta9s1 z ! ,peak ( ,t5]iffe & ! ,&es ( ,p]u1 namely1 t !y are n at all tr\bl$ ) ^! incon/ant w9ds4 ,& if x 7 c]ta9 t nei!r ra9 nor sn[ 7 ev] se5 9 ! highe/ tops ( ^? m\nta9s1 x c\ld n be d\bt$ b t !y are high] ?an any place to :i* vap\rs use to asc5d4 ,sid5ote3 ,:y !re is a con/ant1 ?\gh n a great w9d1 f ea/ to we/1 ne> ! equator4 #c4 ,nev]!less1 !re may be w9d !re1 ?\gh n t :i* is made by ! asc5t ( vap\rs1 yet a less & m con/ant w9d1 l ! cont9u$ bla/ ( a pair ( bell[s1 bl[+ f ! ea/4 ,& ?is may h a d\ble cause2 ! "o1 ! diurnal motion ( ! e>?2 ! o!r1 its simple motion 9 its own epicycle4 ,= ^! m\nta9s be+1 by reason ( _! height1 m em95t ?an all ! re/ ( ! p>ts ( ! e>?1 d by bo? ^! motions drive ! air f ! we/ ea/w>ds4 ,to :i*1 ?\gh ! diurnal motion contribute b little1 yet see+ ,i h suppos$ t ! simple motion ( ! e>?1 9 its own epicycle1 makes two revolutions 9 ! same "t 9 :i* ! diurnal motion makes b "o1 & t ! semidiamet] ( ! epicycle is d\ble to ! semidiamet] ( ! diurnal conv]sion1 ! motion ( e po9t ( ! e>? 9 its own epicycle w h its velocity quadruple to t ( ! diurnal motion2 s t by bo? ^! motions toge!r1 ! tops ( ^? hills w s5sibly be mov$ aga9/ ! air2 & consequ5tly a w9d w be felt4 ,= :e!r ! air /rike ! s5ti5t1 or ! s5ti5t ! air1 ! p]ception ( motion w be ! same4 ,b ?is w9d1 see+ x is n caus$ by ! asc5t ( vap\rs1 mu/ necess>ily be v con/ant4 ,sid5ote3 ,:at is ! effect ( air p5t 9 betwe5 ! cl\ds4 #d4 ,:5 "o cl\d is already asc5d$ into ! air1 if ano!r cl\d asc5d t[>ds x1 t "p ( ! air1 :i* is int]cept$ betwe5 !m bo?1 mu/ ( necessity be press$ out e way4 ,also :5 bo? ( !m1 :il/ ! "o asc5ds & ! o!r ei!r /ays or desc5ds1 come to be jo9$ 9 su* mann] z t ! e!real sub/ance be %ut )9 !m on e side1 x w by ?is compression also g out by p5etrat+ ! wat]4 ,b 9 ! meantime1 ! h>d p>ticles1 :i* are m+l$ ) ! air & are agitat$1 z ,i h suppos$1 ) simple motion1 w n pass ?r\gh ! wat] ( ! cl\ds1 b be m /raitly compress$ )9 _! cavities4 ,& ?is ,i h demon/rat$ at ! #d? & #e? articles ( *apt] ,,xxii4 ,besides1 see+ ! globe ( ! e>? floate? 9 ! air :i* is agitat$ by ! sun's motion1 ! p>ts ( ! air resi/$ by ! e>? w spread !mselves e way ^u ! e>?'s sup]ficies2 z ,i h %[n at ! #h? article ( *apt] ,,xxi4 ,sid5ote3 ,no *ange f s(t to h>d1 b by motion4 #e4 ,we p]ceive a body to be h>d1 f ?is1 t1 :5 t\*+ x1 we w\ld ?ru/ =w>ds t "p ( ! same :i* we t\*1 we _c d x o!rwise ?an by ?ru/+ =w>ds ! :ole body4 ,we may inde$ easily & s5sibly ?ru/ =w>ds any p>ticle ( ! air or wat] :i* we t\*1 :il/ yet ! re/ ( its p>ts rema9 to s5se unmov$4 ,b we _c d s to any "p ( a /one4 ,:]e=e ,i def9e a h>d body to be t :]e( no "p c be s5sibly mov$1 unless ! :ole be mov$4 ,:atsoev] !re=e is s(t or fluid1 ! same c nev] be made h>d b by su* motion z makes _m ( ! p>ts toge!r /op ! motion ( "s "o "p1 by resi/+ ! same4 ,sid5ote3 ,:at x ! cause ( cold ne> ! poles4 #f4 ,^? ?+s premis$1 ,i %all %[ a possible cause :y !re is great] cold ne> ! poles ( ! e>?1 ?an fur!r f !m4 ,! motion ( ! sun betwe5 ! tropics1 driv+ ! air t[>ds t "p ( ! e>?'s sup]ficies :i* is p]p5dicul>ly "u x1 makes x spread itself e way2 & ! velocity ( ?is expansion ( ! air gr[s great] & great]1 z ! sup]ficies ( ! e>? comes to be m & m /rait5$1 t is to say1 z ! circles :i* are p>allel to ! equator come to be less & less4 ,:]e=e ?is expansive motion ( ! air drives be=e x ! p>ts ( ! air1 :i* are 9 its way1 cont9ually t[>ds ! poles m & m /rongly1 z its =ce comes to be m & m unit$1 t is to say1 z ! circles :i* are p>allel to ! equator are less & less2 t is1 s mu* ! m1 by h[ mu* !y are ne>] to ! poles ( ! e>?4 ,9 ^? places1 !re=e1 :i* are ne>] to ! poles1 !re is great] cold ?an 9 ^? :i* are m remote f !m4 ,n[ ?is expansion ( ! air ^u ! sup]ficies ( ! e>?1 f ea/ to we/1 do?1 by reason ( ! sun's p]petual accession to ! places :i* are successively "u x1 make x cold at ! "t ( ! sun's ris+ & sett+2 b z ! sun comes to be cont9ually m & m p]p5dicul> to ^? cool$ places1 s by ! heat1 :i* is g5]at$ by ! sup]v5+ simple motion ( ! sun1 t cold is aga9 remitt$2 & c nev] be great1 because ! action by :i* x 0 g5]at$ is n p]man5t4 ,:]e=e ,i h r5d]$ a possible cause ( cold 9 ^? places t are ne> ! poles1 or :]e ! obliquity ( ! sun is great4 ,sid5ote3 ,! cause ( ice2 & :y ! cold is m remiss 9 ra9y ?an 9 cle> wea!r4 ,:y wat] do? n freeze 9 deep wells1 z x do? ne> ! sup]ficies ( ! e>?4 ,:y ice is n s heavy z wat]2 & :y w9e is n s easily froz5 z wat]4 #g4 ,h[ wat] may be congeal$ by cold1 may be expla9$ 9 ?is mann]4 ,let ,a 79 figure #a7 repres5t ! sun1 & ,b ! e>?4 ,a w !re=e be mu* great] ?an ,b4 ,let ,e ,f be 9 ! plane ( ! equ9octial2 to :i* let ,g ,h1 ,i ,k1 & ,l ,c be p>allel4 ,la/ly1 let ,c & ,d be ! poles ( ! e>?4 ,! air1 !re=e1 by its action 9 ^? p>allels1 w rake ! sup]ficies ( ! e>?2 & t ) motion s mu* ! /rong]1 by h[ mu* ! p>allel circles t[>ds ! poles gr[ less & less4 ,f :5ce mu/ arise a w9d1 :i* w =ce toge!r ! upp]mo/ p>ts ( ! wat]1 & )al raise !m a little1 weak5+ _! endeav\r t[>ds ! c5tre ( ! e>?4 ,& f _! endeav\r t[>ds ! c5tre ( ! e>?1 jo9$ ) ! endeav\r ( ! said w9d1 ! upp]mo/ p>ts ( ! wat] w be press$ toge!r & coagulat$1 t is to say1 ! top ( ! wat] w be sk9n$ ov] & h>d5$4 ,& s aga91 ! wat] next ! top w be h>d5$ 9 ! same mann]1 till at l5g? ! ice be ?ick4 ,& ?is ice1 be+ n[ compact$ ( little h>d bodies1 mu/ also conta9 _m p>ticles ( air receiv$ into x4 ,z riv]s & seas1 s also 9 ! same mann] may ! cl\ds be froz54 ,= :51 by ! asc5d+ & desc5d+ ( sev]al cl\ds at ! same "t1 ! air int]cept$ betwe5 !m is by compression =c$ out1 x rakes1 & by little & little h>d5s !m4 ,& ?\gh ^? small drops1 :i* usually make cl\ds1 be n yet unit$ into great] bodies1 yet ! same w9d w be made2 & by x1 z wat] is congeal$ into ice1 s w vap\rs 9 ! same mann] be congeal$ into sn[4 ,f ! same cause x is t ice may be made by art1 & t n f> f ! fire4 ,= x is done by ! m+l+ ( sn[ & salt toge!r1 & by bury+ 9 x a small vessel full ( wat]4 ,n[ :ile ! sn[ & salt1 :i* h 9 !m a great deal ( air1 are melt+1 ! air1 :i* is press$ out e way 9 w9d1 rakes ! sides ( ! vessel2 & z ! w9d by its motion rakes ! vessel1 s ! vessel by ! same motion & action congeals ! wat] )9 x4 ,we f9d by exp]i5ce1 t cold is always m remiss 9 places :]e x ra9s1 or :]e ! wea!r is cl\dy1 ?+s be+ alike 9 all o!r respects1 ?an :]e ! air is cle>4 ,& ?is agree? v well ) :at ,i h said be=e4 ,= 9 cle> wea!r1 ! c\rse ( ! w9d :i*1 z ,i said ev5 n[1 rakes ! sup]ficies ( ! e>?1 z x is free f all int]ruption1 s also x is v /rong4 ,b :5 small drops ( wat] are ei!r ris+ or fall+1 t w9d is repell$1 brok51 & dissipat$ by !m2 & ! less ! w9d is1 ! less is ! cold4 ,we f9d also by exp]i5ce1 t 9 deep wells ! wat] freeze? n s mu* z x do? ^u ! sup]ficies ( ! e>?4 ,= ! w9d1 by :i* ice is made1 ent]+ into ! e>? by reason ( ! laxity ( its p>ts1 m or less1 lose? "s ( its =ce1 ?\gh n mu*4 ,s t if ! well be n deep1 x w freeze2 :]eas if x be s deep1 z t ! w9d :i* cause? cold _c rea* x1 x w n freeze4 ,we f9d moreov] by exp]i5ce1 t ice is light] ?an wat]4 ,! cause :]e( is manife/ f t :i* ,i h already %[n1 namely1 t air is receiv$ 9 & m+l$ ) ! p>ticles ( ! wat] :il/ x is congeal+4 ,la/ly1 w9e is n s easily congeal$ z wat]1 because 9 w9e !re are p>ticles1 :i*1 be+ n fluid1 are mov$ v swiftly1 & by _! motion congelation is ret>d$4 ,b if ! cold prevail aga9/ ?is motion1 !n ! out]mo/ p>ts ( ! w9e w be fir/ froz51 & aft]w>ds ! inn] p>ts2 :]e( ?is is a sign1 t ! w9e :i* rema9s unfroz5 9 ! mid/ w be v /rong4 ,sid5ote3 ,ano!r cause ( h>dness f ! full] contact ( atoms4 ,also h[ h>d ?+s are brok54 #h4 ,we h se5 "o way ( mak+ ?+s h>d1 namely1 by congelation4 ,ano!r way is ?us4 ,hav+ already suppos$ t innum]able atoms1 "s h>d] ?an o!rs & t h sev]al simple motions ( _! own1 are int]m+l$ ) ! e!real sub/ance2 x foll[s necess>ily f h5ce1 t by reason ( ! f]m5tation ( ! :ole air1 ( :i* ,i h spok5 9 *apt] ,,xxi1 "s ( ^? atoms meet+ ) o!rs w cleave toge!r1 by apply+ !mselves to "o ano!r 9 su* mann] z is agreeable to _! motions & mutual contacts2 &1 see+ !re is no vacuum1 _c be pull$ asund]1 b by s mu* =ce z is suffici5t to ov]come _! h>dness4 ,n[ !re are innum]able degrees ( h>dness4 ,z = example1 !re is a degree ( x 9 wat]1 z is manife/ f ?is1 t ^u a plane x may be drawn any way at pleasure by "o's f+]4 ,!re is a great] degree ( x 9 clammy liquors1 :i*1 :5 !y are p\r$ out1 d 9 fall+ d[nw>ds dispose !mselves into "o cont9u$ ?read2 :i* ?read1 be=e x be brok51 w by little & little dim9i% its ?ickness1 till at la/ x be s small1 z t x seems to break only 9 a po9t2 & 9 _! sep>ation ! ext]nal p>ts break fir/ f "o ano!r1 & !n ! m int]nal p>ts successively "o aft] ano!r4 ,9 wax !re is yet a great] degree ( h>dness4 ,= :5 we w\ld pull "o "p ( x f ano!r1 we fir/ make ! :ole mass sl5d]]1 be=e we c pull x asund]4 ,& h[ mu* ! h>d] any?+ is :i* we w\ld break1 s mu* ! m =ce we mu/ apply to x4 ,:]e=e1 if we g on to h>d] ?+s1 z ropes1 wood1 metals1 /ones1 c41 reason prompte? u to believe t ! same1 ?\gh n always s5sibly1 w necess>ily happ52 & t ev5 ! h>de/ ?+s are brok5 asund] 9 ! same mann]1 namely1 by solution ( _! cont9uity begun 9 ! out]mo/ sup]ficies1 & proce$+ successively to ! inn]mo/ p>ts4 ,9 l mann]1 :5 ! p>ts ( bodies are to be sep>at$1 n by pull+ !m asund]1 b by break+ !m1 ! fir/ sep>ation w necess>ily be 9 ! convex sup]ficies ( ! b[$ "p ( ! body1 & aft]w>ds 9 ! concave sup]ficies4 ,= 9 all b[+ !re is 9 ! convex sup]ficies an endeav\r 9 ! p>ts to g "o f ano!r1 & 9 ! concave sup]ficies to p5etrate "o ano!r4 ,?is be+ well und]/ood1 a reason may be giv5 h[ two bodies1 :i* are contigu\s 9 "o common sup]ficies1 may by =ce be sep>at$ )\t ! introduction ( vacuum2 ?\gh ,lucretius ?\ght o!rwise1 believ+ t su* sep>ation 0 a /rong e/abli%m5t ( vacuum4 ,= a m>ble pill> be+ made to hang by "o ( its bases1 if x be long en\gh1 x w by its own weight be brok5 asund]2 & yet x w n necess>ily foll[ t !re %\ld be vacuum1 see+ ! solution ( its cont9uity may beg9 9 ! circumf]5ce1 & proce$ successively to ! mid/ !re(4 ,sid5ote3 ,a ?ird cause ( h>dness1 f heat4 #i4 ,ano!r cause ( h>dness 9 "s ?+s may be 9 ?is mann]4 ,if a s(t body consi/ ( _m h>d p>ticles1 :i* by ! int]mixture ( _m o!r fluid p>ticles coh]e b loosely toge!r1 ^? fluid p>ts1 z ha? be5 %[n 9 ! la/ article ( *apt] ,,xxi1 w be exhal$2 by :i* means ea* h>d p>ticle w apply itself to ! next to x accord+ to a great] sup]ficies1 & consequ5tly !y w coh]e m closely to "o ano!r1 t is to say1 ! :ole mass w be made h>d]4 ,sid5ote3 ,a f\r? cause ( h>dness1 f ! motion ( atoms enclos$ 9 a n>r[ space4 #aj4 ,aga91 9 "s ?+s h>dness may be made to a c]ta9 degree 9 ?is mann]4 ,:5 any fluid sub/ance ha? 9 x c]ta9 v small bodies int]m+l$1 :i*1 be+ mov$ ) simple motion ( _! own1 contribute l motion to ! p>ts ( ! fluid sub/ance1 & ?is be done 9 a small enclos$ space1 z 9 ! holl[ ( a little sph]e1 or a v sl5d] pipe1 if ! motion be vehem5t & !re be a great numb] ( ^! small enclos$ bodies1 two ?+s w happ52 ! "o1 t ! fluid sub/ance w h an endeav\r ( dilat+ itself at once e way2 ! o!r1 t if ^? small bodies c n[h]e get out1 !n f _! reflection x w foll[1 t ! motion ( ! p>ts ( ! enclos$ fluid sub/ance1 :i* 0 vehem5t be=e1 w n[ be mu* m vehem5t4 ,:]e=e1 if any "o p>ticle ( t fluid sub/ance %\ld be t\*$ & press$ by "s ext]nal mov5t1 x c\ld n yield b by ! application ( v s5sible =ce4 ,:]e=e ! fluid sub/ance1 :i* is enclos$ & s mov$1 ha? "s degree ( h>dness4 ,n[1 great] & less degree ( h>dness dep5ds ^u ! quantity & velocity ( ^? small bodies1 & ^u ! n>r[ness ( ! place bo? toge!r4 ,sid5ote3 ,h[ h>d ?+s are s(t5$4 #aa4 ,su* ?+s z are made h>d by sudd5 heat1 namely su* z are h>d5$ by fire1 are commonly r$uc$ to _! =m] s(t =m by mac]ation4 ,= fire h>d5s by evaporation1 & !re=e if ! evaporat$ moi/ure be re/or$ aga91 ! =m] nature & =m is re/or$ toge!r ) x4 ,& su* ?+s z are froz5 ) cold1 if ! w9d by :i* !y 7 froz5 *ange into ! opposite qu>t]1 !y w be unfroz5 aga91 unless !y h gott5 a habit ( new motion or endeav\r by long cont9uance 9 t h>dness4 ,nor is x en\gh to cause ?aw+1 t !re be a cessation ( ! freez+ w9d2 = ! tak+ away ( ! cause do? n de/roy a produc$ effect2 b ! ?aw+ also mu/ h its prop] cause1 namely1 a contr>y w9d1 or at lea/ a w9d opposite 9 "s degree4 ,& ?is we f9d to be true by exp]i5ce4 ,=1 if ice be laid 9 a place s well enclos$ t ! motion ( ! air _c get to x1 t ice w rema9 un*ang$1 ?\gh ! place be n s5sibly cold4 ,sid5ote3 ,:5ce proce$s ! spontane\s re/itution ( ?+s b5t4 #ab4 ,( h>d bodies1 "s may manife/ly be b[$2 o!rs n1 b are brok5 9 ! v fir/ mom5t ( _! b5d+4 ,& ( su* bodies z may manife/ly be b5d$1 "s be+ b5t1 d1 z soon z ev] !y are set at lib]ty1 re/ore !mselves to _! =m] po/ure2 o!rs rema9 /ill b5t4 ,n[ if ! cause ( ?is re/itution be ask$1 ,i say1 x may be 9 ?is mann]1 namely1 t ! p>ticles ( ! b5d$ body1 :il/ x is held b5t1 d nev]!less reta9 _! motion2 & by ?is motion !y re/ore x z soon z ! =ce is remov$ by :i* x 0 b5t4 ,= :5 any ?+ is b5t1 z a plate ( /eel1 &1 z soon z ! =ce is remov$1 re/ores itself aga91 x is evid5t t ! cause ( its re/itution _c be ref]r$ to ! ambi5t air2 nor c x be ref]r$ to ! removal ( ! =ce by :i* x 0 b5t2 = 9 ?+s t are at re/ ! tak+ away ( imp$im5ts is n a suffici5t cause ( _! future motion2 !re be+ no o!r cause ( motion1 b motion4 ,! cause !re=e ( su* re/itution is 9 ! p>ts ( ! /eel itself4 ,:]e=e1 :il/ x rema9s b5t1 !re is 9 ! p>ts1 ( :i* x consi/e?1 "s motion ?\gh invisible2 t is to say1 "s endeav\r at lea/ t way by :i* ! re/itution is to be made2 & !re=e ?is endeav\r ( all ! p>ts toge!r is ! fir/ beg9n+ ( re/itution2 s t ! imp$im5t be+ remov$1 t is to say1 ! =ce by :i* x 0 held b5t1 x w be re/or$ aga94 ,n[ ! motion ( ! p>ts1 by :i* ?is done1 is t :i* ,i call$ simple motion1 or motion return+ into itself4 ,:5 !re=e 9 ! b5d+ ( a plate ! ends are drawn toge!r1 !re is on "o side a mutual compression ( ! p>ts2 :i* compression is "o endeav\r opposite to ano!r endeav\r3 & on ! o!r side a divulsion ( ! p>ts4 ,! endeav\r !re=e ( ! p>ts on "o side t5ds to ! re/itution ( ! plate f ! middle t[>ds ! ends2 & on ! o!r side1 f ! ends t[>ds ! middle4 ,:]e=e ! imp$im5t be+ tak5 away1 ?is endeav\r1 :i* is ! beg9n+ ( re/itution1 w re/ore ! plate to its =m] po/ure4 ,& ?us ,i h giv5 a possible cause :y "s bodies1 :5 !y are b5t1 re/ore !mselves aga92 :i* 0 to be done4 ,z = /ones1 see+ !y are made by ! accretion ( _m v h>d p>ticles )9 ! e>?2 :i* p>ticles h no great coh]5ce1 t is to say1 t\* "o ano!r 9 small latitude1 & consequ5tly admit _m p>ticles ( air2 x mu/ ne$s be t1 9 b5d+ ( !m1 _! int]nal p>ts w n easily be compress$1 by reason ( _! h>dness4 ,& because _! coh]5ce is n firm1 z soon z ! ext]nal h>d p>ticles are disjo9$1 ! e!real p>ts w necess>ily break out1 & s ! body w sudd5ly be brok54 ,sid5ote3 ,diaphan\s & opac\s1 :at !y are1 & :5ce4 #ac4 ,^? bodies are call$ diaphan\s1 ^u :i*1 :il/ ! beams ( a lucid body d "w1 ! action ( e "o ( ^? beams is propagat$ 9 !m 9 su* mann]1 z t !y /ill reta9 ! same ord] among/ !mselves1 or ! inv]sion ( t ord]2 & !re=e bodies1 :i* are p]fectly diaphan\s1 are also p]fectly homog5e\s4 ,on ! contr>y1 an opac\s body is t1 :i*1 by reason ( its het]og5e\s nature1 do? by innum]able reflections & refractions 9 p>ticles ( diff]5t figures & unequal h>dness1 weak5 ! beams t fall ^u x be=e !y rea* ! eye4 ,& ( diaphan\s bodies1 "s are made su* by nature f ! beg9n+2 z ! sub/ance ( ! air1 & ( ! wat]1 & p]haps also "s p>ts ( /ones1 unless ^! also be wat] t has be5 long congeal$4 ,o!rs are made s by ! p[] ( heat1 :i* congregates homog5e\s bodies4 ,b su*1 z are made diaphan\s 9 ?is mann]1 consi/ ( p>ts :i* 7 =m]ly diaphan\s4 ,sid5ote3 ,! cause ( lightn+ & ?und]4 #ad4 ,9 :at mann] cl\ds are made by ! motion ( ! sun1 elevat+ ! p>ticles ( wat] f ! sea & o!r moi/ places1 ha? be5 expla9$ 9 *apt] ,,xxvi4 ,also h[ cl\ds come to be froz51 ha? be5 %[n above at ! #g? article4 ,n[ f ?is1 t air may be enclos$ z x 7 9 cav]ns1 & p5t toge!r m & m by ! meet+ ( asc5d+ & desc5d+ cl\ds1 may be d$uc$ a possible cause ( ?und] & lightn+4 ,= see+ ! air consi/s ( two p>ts1 ! "o e!real1 :i* has no prop] motion ( its own1 z be+ a ?+ divisible into ! lea/ p>ts2 ! o!r h>d1 namely1 consi/+ ( _m h>d atoms1 :i* h e "o ( !m a v swift simple motion ( its own3 :il/ ! cl\ds by _! meet+ d m & m /rait5 su* cavities z !y int]cept1 ! e!real p>ts w p5etrate & pass ?r\gh _! wat]y sub/ance2 b ! h>d p>ts w 9 ! meantime be ! m ?ru/ toge!r1 & press "o ano!r2 & consequ5tly1 by reason ( _! vehem5t motions1 !y w h an endeav\r to reb\nd f ea* o!r4 ,:5soev]1 !re=e1 ! compression is great en\gh1 & ! concave p>ts ( ! cl\ds are1 = ! cause ,i h already giv51 congeal$ into ice1 ! cl\d w necess>ily be brok52 & ?is break+ ( ! cl\d produce? ! fir/ clap ( ?und]4 ,aft]w>ds ! air1 :i* 0 p5t 91 hav+ n[ brok5 ?r\gh1 makes a concussion ( ! air )\t1 & f h5ce proce$s ! ro>+ & murmur :i* foll[s2 & bo? ! fir/ clap & ! murmur t foll[s x make t noise :i* is call$ ?und]4 ,also1 f ! same air break+ ?r\gh ! cl\ds & ) concussion fall+ ^u ! eye1 proce$s t action ^u our eye1 :i* cause? 9 u a p]ception ( t light1 :i* we call lightn+4 ,:]e=e ,i h giv5 a possible cause ( ?und] & lightn+4 ,sid5ote3 ,:5ce x proce$s t cl\ds c fall aga91 aft] !y are once elevat$ & froz54 #ae4 ,b if ! vap\rs1 :i* are rais$ into cl\ds1 d run toge!r aga9 into wat] or be congeal$ into ice1 f :5ce is x1 see+ bo? ice & wat] are heavy1 t !y are su/a9$ 9 ! air8 ,or r1 :at may ! cause be1 t be+ once elevat$1 !y fall d[n aga98 ,= !re is no d\bt b ! same =ce :i* c\ld c>ry up t wat]1 c\ld also su/a9 x !re4 ,:y !re=e be+ once c>ri$ up1 do? x fall aga98 ,i say x proce$s f ! same simple motion ( ! sun1 bo? t vap\rs are =c$ to asc5d1 & t wat] ga!r$ into cl\ds is =c$ to desc5d4 ,= 9 *apt] ,,xxi1 article #aa1 ,i h %[n h[ vap\rs are elevat$2 & 9 ! same *apt]1 article #e1 ,i h also %[n h[ by ! same motion homog5e\s bodies are congregat$1 & het]og5e\s dissipat$2 t is to say1 h[ su* ?+s1 z h a l nature to t ( ! e>?1 are driv5 t[>ds ! e>?2 t is to say1 :at is ! cause ( ! desc5t ( heavy bodies4 ,n[ if ! action ( ! sun be h9d]$ 9 ! rais+ ( vap\rs1 & be n at all h9d]$ 9 ! ca/+ ( !m d[n1 ! wat] w desc5d4 ,b a cl\d _c h9d] ! action ( ! sun 9 mak+ ?+s ( an e>?ly nature desc5d to ! e>?1 ?\gh x may h9d] x 9 mak+ vap\rs asc5d4 ,= ! l[] "p ( a ?ick cl\d is s cov]$ by its upp] "p1 z t x _c receive t action ( ! sun by :i* vap\rs are c>ri$ up2 because vap\rs are rais$ by ! p]petual f]m5tation ( ! air1 or by ! sep>at+ ( its smalle/ p>ts f "o ano!r1 :i* is mu* weak] :5 a ?ick cl\d is int]pos$1 ?an :5 ! sky is cle>4 ,& !re=e1 :5soev] a cl\d is made ?ick en\gh1 ! wat]1 :i* w\ld n desc5d be=e1 w !n desc5d1 unless x be kept up by ! agitation ( ! w9d4 ,:]e=e ,i h r5d]$ a possible cause1 bo? :y ! cl\ds may be su/a9$ 9 ! air1 & also :y !y may fall d[n aga9 to ! e>?2 :i* 0 prop\nd$ to be done4 ,sid5ote3 ,h[ x c\ld be t ! moon 0 eclips$1 :5 %e 0 n diametrically opposite to ! sun4 #af4 ,grant+ t ! cl\ds may be froz51 x is no wond] if ! moon h be5 se5 eclips$ at su* "t z %e ha? be5 almo/ two degrees above ! horizon1 ! sun at ! same "t appe>+ 9 ! horizon2 = su* an eclipse 0 obs]v$ by ,m/l91 at ,tub+51 9 ! ye> #aeij4 ,= x might happ5 t a froz5 cl\d 0 !n int]pos$ betwe5 ! sun & ! eye ( ! obs]v]4 ,& if x 7 s1 ! sun1 :i* 0 !n almo/ two degrees bel[ ! horizon1 might appe> to be 9 x1 by reason ( ! pass+ ( 8 beams ?r\gh ! ice4 ,& x is to be not$ t ^?1 t attribute su* refractions to ! atmosph]e1 _c attribute to x s great a refraction z ?is4 ,:]e=e n ! atmosph]e1 b ei!r wat] 9 a cont9u$ body1 or else ice1 mu/ be ! cause ( t refraction4 ,sid5ote3 ,by :at means _m suns may appe> at once4 #ag4 ,aga91 grant+ t !re may be ice 9 ! cl\ds1 x w be no long] a wond] t _m suns h sometimes appe>$ at once4 ,= look+-glasses may be s plac$1 z by reflections to %[ ! same object 9 _m places4 ,& may n s _m froz5 cl\ds s]ve = s _m look+-glasses8 ,& may !y n be fitly dispos$ = t purpose8 ,besides1 ! numb] ( appe>ances may be increas$ by refractions also2 & !re=e x w\ld be a great] wond] to me1 if su* ph5om5a z ^! %\ld nev] happ54 ,& 7 x n = t "o ph5om5on ( ! new /> :i* 0 se5 9 ,cassiopea1 ,i %\ld ?9k comets 7 made 9 ! same mann]1 namely1 by vap\rs drawn n only f ! e>? b f ! re/ ( ! planets also1 & congeal$ into "o cont9u$ body4 ,= ,i c\ld v well f h5ce give a reason bo? ( _! hair1 & ( _! motions4 ,b see+ t /> rema9$ sixte5 :ole mon?s 9 ! same place among/ ! fix$ />s1 ,i _c believe ! matt] ( x 0 ice4 ,:]e=e ,i leave to o!rs ! disquisition ( ! cause ( comets2 conc]n+ :i* no?+ t ha? hi!rto be5 publi%$1 besides ! b>e hi/ories ( !m1 is wor? consid]+4 ,sid5ote3 ,( ! heads ( riv]s4 #ah4 ,! heads ( riv]s may be d$uc$ f ra9-wat]1 or f melt$ sn[s1 v easily2 b f o!r causes1 v h>dly1 or n at all4 ,= bo? ra9-wat] & melt$ sn[s run d[n ! desc5ts ( m\nta9s2 & if !y desc5d only by ! outw>d sup]ficies1 ! %[]s or sn[s !mselves may be acc\nt$ ! spr+s or f\nta9s2 b if !y ent] ! e>? & desc5d )9 x1 !n1 :]esoev] !y break out1 !re are _! spr+s4 ,& z ^! spr+s make small /reams1 s1 _m small /reams runn+ toge!r make riv]s4 ,n[1 !re 0 nev] any spr+ f\nd1 b :]e ! wat] :i* fl[$ to x1 0 ei!r fur!r1 or at lea/ z f> f ! c5tre ( ! e>?1 z ! spr+ itself4 ,& :]eas x has be5 object$ by a great philosoph]1 t 9 ! top ( ,m\nt ,c5is1 :i* p>ts ,savoy f ,pi$mont1 !re spr+s a riv] :i* runs d[n by ,susa2 x is n true4 ,= !re are above t riv]1 = two miles l5g?1 v high hills on bo? sides1 :i* are almo/ p]petually cov]$ ) sn[2 f :i* innum]able little /reams runn+ d[n d manife/ly supply t riv] ) wat] suffici5t = its magnitude4 ------- ,,*ap4 ,,xxix4 ,,( ,,s\nd1 ,,od\r1 ,,sav\r1 ,,& ,,t\*4 #a44 ,! def9ition ( s\nd1 & ! di/9ctions ( s\nds4--#b4 ,! cause ( ! degrees ( s\nds4--#c4 ,! diff]5ce betwe5 s\nds acute & grave4--#d4 ,! diff]5ce betwe5 cle> & ho>se s\nds1 :5ce4--#e4 ,! s\nd ( ?und] & ( a gun1 :5ce x proce$s4--#f4 ,:5ce x is t pipes1 by bl[+ into !m1 h a cle> s\nd4--#g4 ,( reflect$ s\nd4--#h4 ,f :5ce x is t s\nd is uni=m & la/+4--#i4 ,h[ s\nd may be help$ & h9d]$ by ! w9d4--#aj4 ,n only air1 b o!r bodies h[ h>d soev] !y be1 convey s\nd4--#aa4 ,! causes ( grave & acute s\nds1 & ( conc5t4--#ab4 ,ph5om5a = smell+4--#ac4 ,! fir/ organ & ! g5]ation ( smell+4--#ad4 ,h[ x is help$ by heat & by w9d4--#ae4 ,:y su* bodies are lea/ smelt1 :i* h lea/ int]mixture ( air 9 !m4--#af4 ,:y odor\s ?+s become m odor\s by be+ bruis$4--#ag4 ,! fir/ organ ( ta/+2 & :y "s sav\rs cause nause\sness4--#ah4 ,! fir/ organ ( feel+2 & h[ we come to ! k ( su* objects z are common to ! t\* & o!r s5ses4 ,sid5ote3 ,! def9ition ( s\nd1 & ! di/9ction ( s\nds4 #a4 ,,s\nd is s5se g5]at$ by ! action ( ! m$ium1 :5 its motion rea*e? ! e> & ! re/ ( ! organs ( s5se4 ,n[1 ! motion ( ! m$ium is n ! s\nd itself1 b ! cause ( x4 ,= ! phantasm :i* is made 9 u1 t is to say1 ! reaction ( ! organ1 is prop]ly t :i* we call s\nd4 ,! pr9cipal di/9ctions ( s\nds are ^!2 fir/1 t "o s\nd is /rong]1 ano!r weak]4 ,secondly1 t "o is m grave1 ano!r m acute4 ,?irdly1 t "o is cle>1 ano!r ho>se4 ,f\r?ly1 t "o is prim>y1 ano!r d]ivative4 ,fif?ly1 t "o is uni=m1 ano!r n4 ,six?ly1 t "o is m durable1 ano!r less durable4 ,( all :i* di/9ctions ! memb]s may be subdi/+ui%$ into p>ts di/+ui%able almo/ inf9itely4 ,= ! v>iety ( s\nds seems to be n mu* less ?an t ( col\rs4 ,z vision1 s he>+ is g5]at$ by ! motion ( ! m$ium1 b n 9 ! same mann]4 ,= sight is f pressure1 t is1 f an endeav\r2 9 :i* !re is no p]ceptible progression ( any ( ! p>ts ( ! m$ium2 b "o "p urg+ or ?ru/+ on ano!r propagate? t action successively to any di/ance :atsoev]2 :]eas ! motion ( ! m$ium1 by :i* s\nd is made1 is a /roke4 ,= :5 we he>1 ! drum ( ! e>1 :i* is ! fir/ organ ( he>+1 is /rick52 & ! drum be+ /rick51 ! pia mat] is also %ak51 & ) x ! art]ies :i* are ins]t$ into x2 by :i* ! action be+ propagat$ to ! he>t itself1 by ! reaction ( ! he>t a phantasm is made :i* we call s\nd2 & because ! reaction t5de? outw>ds1 we ?9k x is )\t4 ,sid5ote3 ,! cause ( ! degrees ( s\nds4 #b4 ,& see+ ! effects produc$ by motion are great] or less1 n only :5 ! velocity is great] or less1 b also :5 ! body ha? great] or less magnitude ?\gh ! velocity be ! same2 a s\nd may be great] or less bo? ^! ways4 ,& because nei!r ! greate/ nor ! lea/ magnitude or velocity c be giv51 x may happ5 t ei!r ! motion may be ( s small velocity1 or ! body itself ( s small magnitude1 z to produce no s\nd at all2 or ei!r ( !m may be s great1 z to take away ! faculty ( s5se by hurt+ ! organ4 ,f h5ce may be d$uc$ possible causes ( ! /r5g? & weakness ( s\nds 9 ! foll[+ ph5om5a4 ,! fir/ :]e( is ?is1 t if a man speak ?r\gh a trunk :i* ha? "o end appli$ to ! m\? ( ! speak]1 & ! o!r to ! e> ( ! he>]1 ! s\nd w come /rong] ?an x w\ld d ?r\gh ! op5 air4 ,& ! cause1 n only ! possible1 b ! c]ta9 & manife/ cause is ?is1 t ! air :i* is mov$ by ! fir/ brea? & c>ri$ =w>ds 9 ! trunk1 is n diffus$ z x w\ld be 9 ! op5 air1 & is consequ5tly br\ght to ! e> almo/ ) ! same velocity ) :i* x 0 fir/ brea!d out4 ,:]eas1 9 ! op5 air1 ! fir/ motion diffuse? itself e way into circles1 su* z are made by ! ?r[+ ( a /one into a /&+ wat]1 :]e ! velocity gr[s less & less z ! undulation proce$s fur!r & fur!r f ! beg9n+ ( its motion4 ,! second is ?is1 t if ! trunk be %ort1 & ! end :i* is appli$ to ! m\? be wid] ?an t :i* is appli$ to ! e>1 ?us also ! s\nd w be /rong] ?an if x 7 made 9 ! op5 air4 ,& ! cause is ! same1 namely1 t by h[ mu* ! wid] end ( ! trunk is less di/ant f ! beg9n+ ( ! s\nd1 by s mu* ! less is ! diffusion4 ,! ?ird1 t x is easi] = "o1 t is )9 a *amb]1 to he> :at is spok5 )\t1 ?an = him1 t /&s )\t1 to he> :at is spok5 )94 ,= ! w9d[s & o!r inlets ( ! mov$ air are z ! wide end ( ! trunk4 ,& = ?is reason "s creatures seem to he> ! bett]1 because nature has be/[$ ^u !m wide & capaci\s e>s4 ,! f\r? is ?is1 t ?\gh he1 :i* /&e? ^u ! sea-%ore1 _c he> ! collision ( ! two ne>e/ waves1 yet nev]!less he he>s ! ro>+ ( ! :ole sea4 ,& ! cause seems to be ?is1 t ?\gh ! sev]al collisions move ! organ1 yet !y are n sev]ally great en\gh to cause s5se2 :]eas no?+ h9d]s b t all ( !m toge!r may make s\nd4 ,sid5ote3 ,! diff]5ce betwe5 s\nds acute & grave4 #c4 ,t bodies :5 !y are /rick5 d yield "s a m grave1 o!rs a m acute s\nd1 ! cause may consi/ 9 ! diff]5ce ( ! times 9 :i* ! p>ts /rick5 & =c$ out ( _! places return to ! same places aga94 ,= 9 "s bodies1 ! re/itution ( ! mov$ p>ts is quick1 9 o!rs sl[4 ,& ?is also may be ! cause1 :y ! p>ts ( ! organ1 :i* are mov$ by ! m$ium1 return to _! re/ aga91 sometimes soon]1 sometimes lat]4 ,n[1 by h[ mu* ! vibrations or ! reciprocal motions ( ! p>ts are m frequ5t1 by s mu* do? ! :ole s\nd made at ! same "t by "o /roke consi/ ( m1 & consequ5tly ( small] p>ts4 ,= :at is acute 9 s\nd1 ! same is subtle 9 matt]2 & bo? ( !m1 namely acute s\nd & subtle matt]1 consi/ ( v small p>ts1 t ( "t1 & ?is ( ! matt] itself4 ,! ?ird di/9ction ( s\nds _c be conceiv$ cle>ly en\gh by ! names ,i h us$ ( cle> & ho>se1 nor by any o!r t ,i "k2 & !re=e x is ne$ful to expla9 !m by examples4 ,:5 ,i say ho>se1 ,i und]/& :isp]+ & hiss+1 & :atsoev] is l to ^!1 by :at appellation soev] x be express$4 ,& s\nds ( ?is k9d seem to be made by ! =ce ( "s /rong w9d1 rak+ r ?an /rik+ su* h>d bodies z x falls ^u4 ,on ! contr>y1 :5 ,i use ! ^w cle>1 ,i d n und]/& su* a s\nd z may be easily & di/9ctly he>d2 = s :isp]s w\ld be cle>2 b su* z is made by some:at t is brok51 & su* z is clam\r1 t9kl+1 ! s\nd ( a trumpet1 c4 & to express x significantly 9 "o ^w1 noise4 ,& see+ no s\nd is made b by ! conc\rse ( two bodies at ! lea/1 by :i* conc\rse x is necess>y t !re be z well reaction z action1 t is to say1 "o motion opposite to ano!r2 x foll[s t accord+ z ! proportion betwe5 ^? two opposite motions is div]sifi$1 s ! s\nds :i* are made w be diff]5t f "o ano!r4 ,& :5soev] ! proportion betwe5 !m is s great1 z t ! motion ( "o ( ! bodies be ins5sible if comp>$ ) ! motion ( ! o!r1 !n ! s\nd w n be ( ! same k9d2 z :5 ! w9d falls v obliquely ^u a h>d body1 or :5 a h>d body is c>ri$ swiftly ?r\gh ! air2 = !n !re is made t s\nd :i* ,i call a ho>se s\nd1 9 ,greek 4 ,!re=e ! brea? bl[n ) viol5ce f ! m\? makes a hiss+1 because 9 go+ out x rakes ! sup]ficies ( ! lips1 ^: reaction aga9/ ! =ce ( ! brea? is n s5sible4 ,& ?is is ! cause :y ! w9ds h t ho>se s\nd4 ,also if two bodies1 h[ h>d soev]1 be rubb$ toge!r ) no great pressure1 !y make a ho>se s\nd4 ,& ?is ho>se s\nd1 :5 x is made1 z ,i h said1 by ! air rak+ ! sup]ficies ( a h>d body1 seeme? to be no?+ b ! divid+ ( ! air into innum]able & v small files4 ,= ! asp]ity ( ! sup]ficies do?1 by ! em95ces ( its innum]able p>ts1 divide or cut 9 pieces ! air t slides ^u x4 ,sid5ote3 ,! diff]5ce betwe5 cle> ho>se s\nds1 :5ce4 #d4 ,noise1 or t :i* ,i call cle> s\nd1 is made two ways2 "o1 by two ho>se s\nds made by opposite motions2 ! o!r1 by collision1 or by ! sudd5 pull+ asund] ( two bodies1 :]eby _! small p>ticles are put into commotion1 or be+ already 9 commotion sudd5ly re/ore !mselves aga92 :i* motion1 mak+ impression ^u ! m$ium1 is propagat$ to ! organ ( he>+4 ,& see+ !re is 9 ?is collision or divulsion an endeav\r 9 ! p>ticles ( "o body1 opposite to ! endeav\r ( ! p>ticles ( ! o!r body1 !re w also be made 9 ! organ ( he>+ a l opposition ( endeav\rs1 t is to say1 ( motions2 & consequ5tly ! s\nd aris+ f !nce w be made by two opposite motions1 t is to say1 by two opposite ho>se s\nds 9 "o & ! same "p ( ! organ4 ,=1 z ,i h already said1 a ho>se s\nd suppose? ! s5sible motion ( b "o ( ! bodies4 ,& ?is opposition ( motions 9 ! organ is ! cause :y two bodies make a noise1 :5 !y are ei!r sudd5ly /rick5 aga9/ "o ano!r1 or sudd5ly brok5 asund]4 ,sid5ote3 ,! s\nd ( ?und] & ( a gun1 :5ce x proce$s4 #e4 ,?is be+ grant$1 & see+ )al t ?und] is made by ! vehem5t eruption ( ! air out ( ! cavities ( congeal$ cl\ds1 ! cause ( ! great noise or clap may be ! sudd5 break+ asund] ( ! ice4 ,= 9 ?is action x is necess>y t !re be n only a great concussion ( ! small p>ticles ( ! brok5 p>ts1 b also t ?is concussion1 by be+ communicat$ to ! air1 be c>ri$ to ! organ ( he>+1 & make impression ^u x4 ,& !n1 f ! fir/ reaction ( ! organ proce$s t fir/ & greate/ s\nd1 :i* is made by ! collision ( ! p>ts :il/ !y re/ore !mselves4 ,& see+ !re is 9 all concussion a reciprocation ( motion =w>ds & backw>ds 9 ! p>ts /rick52 = opposite motions _c ext+ui% "o ano!r 9 an in/ant1 z ,i h %[n 9 ! #aa? article ( *apt] ,,viii2 x foll[s necess>ily t ! s\nd w bo? cont9ue1 & gr[ weak] & weak]1 till at la/ ! action ( ! reciprocat+ air gr[ s weak1 z to be imp]ceptible4 ,:]e=e a possible cause is giv5 bo? ( ! fir/ fi]ce noise ( ! ?und]1 & also ( ! murmur t foll[s x4 ,! cause ( ! great s\nd f a dis*>g$ piece ( ordnance is l t ( a clap ( ?und]4 ,= ! gunp[d] be+ fir$ do?1 9 its endeav\r to g out1 attempt e way ! sides ( ! metal 9 su* mann]1 z t x enl>ge? ! circumf]5ce all along1 & )al %ort5e? ! axis2 s t :il/ ! piece ( ordnance is 9 dis*>g+1 x is made bo? wid] & %ort] ?an x 0 be=e2 & !re=e also pres5tly aft] x is dis*>g$ its wid5ess w be dim9i%$1 & its l5g? increas$ aga9 by ! re/itution ( all ! p>ticles ( ! matt]1 ( :i* x consi/e?1 to _! =m] position4 ,& ?is is done ) su* motions ( ! p>ts1 z are n only v vehem5t1 b also opposite to "o ano!r2 :i* motions1 be+ communicat$ to ! air1 make impression ^u ! organ1 & by ! reaction ( ! organ create a s\nd1 :i* la/e? = "s "t2 z ,i h already %[n 9 ?is article4 ,i note by ! way1 z n belong+ to ?is place1 t ! possible cause :y a gun recoils :5 x is %ot (f1 may be ?is2 t be+ fir/ swoll5 by ! =ce ( ! fire1 & aft]w>ds re/or+ itself1 f ?is re/itution !re proce$s an endeav\r f all ! sides t[>ds ! cavity2 & consequ5tly ?is endeav\r is 9 ^? p>ts :i* are next ! bree*2 :i* be+ n holl[1 b solid1 ! effect ( ! re/itution is by x h9d]$ & div]t$ into ! l5g?2 & by ?is means bo? ! bree* & ! :ole gun is ?ru/ backw>ds2 & ! m =cibly by h[ mu* ! =ce is great]1 by :i* ! "p next ! bree* is re/or$ to its =m] po/ure1 t is to say1 by h[ mu* ! ?9n] is t "p4 ,! cause1 !re=e1 :y guns recoil1 "s m "s less1 is ! diff]5ce ( _! ?ickness t[>ds ! bree*2 & ! great] t ?ickness is1 ! less !y recoil2 & contr>ily4 ,sid5ote3 ,:5ce x is t pipes1 by bl[+ into !m1 h a cle> s\nd4 #f4 ,also ! cause :y ! s\nd ( a pipe1 :i* is made by bl[+ into x1 is nev]!less cle>1 is ! same ) t ( ! s\nd :i* is made by collision4 ,= if ! brea?1 :5 x is bl[n into a pipe1 d only rake its concave sup]ficies1 or fall ^u x ) a v %>p angle ( incid5ce1 ! s\nd w n be cle>1 b ho>se4 ,b if ! angle be great en\gh1 ! p]cussion1 :i* is made aga9/ "o ( ! holl[ sides1 w be rev]b]at$ to ! opposite side2 & s successive rep]cussions w be made f side to side1 till at la/ ! :ole concave sup]ficies ( ! pipe be put into motion2 :i* motion w be reciprocat$1 z x is 9 collision2 & ?is reciprocation be+ propagat$ to ! organ1 f ! reaction ( ! organ w arise a cle> s\nd1 su* z is made by collision1 or by break+ asund] ( h>d bodies4 ,9 ! same mann] x is ) ! s\nd ( a man's voice4 ,= :5 ! brea? passe? out )\t int]ruption1 & do? b lightly t\* ! cavities ?r\gh :i* x is s5t1 ! s\nd x make? is a ho>se s\nd4 ,b if 9 go+ out x /rike /rongly ^u ! l>ynx1 !n a cle> s\nd is made1 z 9 a pipe4 ,& ! same brea?1 z x comes 9 div]s mann]s to ! palate1 ! tongue1 ! lips1 ! tee?1 & o!r organs ( spee*1 s ! s\nds into :i* x is articulat$ become diff]5t f "o ano!r4 ,sid5ote3 ,( reflect$ s\nd4 #g4 ,i call t prim>y s\nd1 :i* is g5]at$ by motion f ! s\nd+ body to ! organ 9 a /rait l9e )\t reflection2 & ,i call t reflect$ s\nd1 :i* is g5]at$ by "o or m reflections1 be+ ! same ) t we call e*o1 & is it]at$ z (t5 z !re are reflections made f ! object to ! e>4 ,& ^! reflections are made by hills1 walls1 & o!r resi/+ bodies1 s plac$ z t !y make m or few] reflections ( ! motion1 accord+ z !y are !mselves m or few] 9 numb]2 & !y make !m m or less frequ5tly1 accord+ z !y are m or less di/ant f "o ano!r4 ,n[ ! cause ( bo? ^! ?+s is to be s\ght = 9 ! situation ( ! reflect+ bodies1 z is usually done 9 sight4 ,= ! laws ( reflection are ! same 9 bo?1 namely1 t ! angles ( incid5ce & reflection be equal to "o ano!r4 ,if1 !re=e1 9 a holl[ elliptic body1 ^: inside is well poli%$1 or 9 two "r p>abolical solids1 :i* are jo9$ toge!r by "o common base1 !re be plac$ a s\nd+ body 9 "o ( ! burn+ po9ts1 & ! e> 9 ! o!r1 !re w be he>d a s\nd by _m degrees great] ?an 9 ! op5 air2 & bo? ?is1 & ! burn+ ( su* combu/ible ?+s1 z be+ put 9 ! same places are set on fire by ! sun-beams1 are effects ( "o & ! same cause4 ,b1 z :5 ! visible object is plac$ 9 "o ( ! burn+ po9ts1 x is n di/9ctly se5 9 ! o!r1 because e "p ( ! object be+ se5 9 e l9e1 :i* is reflect$ f ! concave sup]ficies to ! eye1 makes a confusion 9 ! sight2 s nei!r is s\nd he>d articulately & di/9ctly :5 x comes to ! e> 9 all ^? reflect$ l9es4 ,& ?is may be ! reason :y 9 *ur*es :i* h ar*$ ro(s1 ?\gh !y be nei!r elliptical nor p>abolical1 yet because _! figure is n mu* diff]5t f ^!1 ! voice f ! pulpit w n be he>d s articulately z x w\ld be1 if !re 7 no vault+ at all4 ,sid5ote3 ,f :5ce x is t s\nd is uni=m & la/+4 #h4 ,conc]n+ ! uni=mity & duration ( s\nds1 bo? :i* h "o common cause1 we may obs]ve1 t su* bodies z be+ /rick5 yield an unequal or h>% s\nd1 are v het]og5e\s1 t is to say1 !y consi/ ( p>ts :i* are v unlike bo? 9 figure & h>dness1 su* z are wood1 /ones1 & o!rs n a few4 ,:5 ^! are /rick51 !re foll[s a concussion ( _! int]nal p>ticles1 & a re/itution ( !m aga94 ,b !y are nei!r mov$ alike1 nor h !y ! same action ^u "o ano!r2 "s ( !m recoil+ f ! /roke1 :il/ o!rs :i* h already f9i%$ _! recoil+s are n[ return+2 by :i* means !y h9d] & /op "o ano!r4 ,& f h5ce x is t _! motions are n only unequal & h>%1 b also t _! reciprocations come to be quickly ext+ui%$4 ,:5soev]1 !re=e1 ?is motion is propagat$ to ! e>1 ! s\nd x makes is unequal & ( small duration4 ,on ! contr>y1 if a body t is /rick5 be n only suffici5tly h>d1 b h also ! p>ticles ( :i* x consi/e? l to "o ano!r bo? 9 h>dness & figure1 su* z are ! p>ticles ( glass & metals1 :i* be+ fir/ melt$ d aft]w>ds settle & h>d52 ! s\nd x yielde? w1 because ! motions ( its p>ts & _! reciprocations are l & uni=m1 be uni=m & pleasant1 & be m or less la/+1 accord+ z ! body /rick5 ha? great] or less magnitude4 ,! possible cause1 !re=e1 ( s\nds uni=m & h>%1 & ( _! long] or %ort] duration1 may be "o & ! same lik5ess & unlik5ess ( ! int]nal p>ts ( ! s\nd+ body1 9 respect bo? ( _! figure & h>dness4 ,besides1 if two plane bodies ( ! same matt] & ( equal ?ickness1 d bo? yield an uni=m s\nd1 ! s\nd ( t body1 :i* ha? ! greate/ ext5t ( l5g?1 w be ! longe/ he>d4 ,= ! motion1 :i* 9 bo? ( !m ha? its beg9n+ f ! po9t ( p]cussion1 is to be propagat$ 9 ! great] body ?r\gh a great] space1 & consequ5tly t propagation w require m "t2 & !re=e also ! p>ts :i* are mov$1 w require m "t = _! return4 ,:]e=e all ! reciprocations _c be f9i%$ b 9 long] "t2 & be+ c>ri$ to ! e>1 w make ! s\nd la/ ! long]4 ,& f h5ce x is manife/1 t ( h>d bodies :i* yield an uni=m s\nd1 ! s\nd la/e? long] :i* comes f ^? t are r\nd & holl[1 ?an f ^? t are plane1 if !y be l 9 all o!r respects4 ,= 9 circul> l9es ! action1 :i* beg9s at any po9t1 ha? n f ! figure any end ( its propagation1 because ! l9e 9 :i* x is propagat$ returns aga9 to its beg9n+2 s t ! figure h9d]s n b t ! motion may h inf9ite progression4 ,:]eas 9 a plane1 e l9e ha? its magnitude f9ite1 beyond :i* ! action _c proce$4 ,if1 !re=e1 ! matt] be ! same1 ! motion ( ! p>ts ( t body ^: figure is r\nd & holl[1 w la/ long] ?an ( t :i* is plane4 ,also1 if a /r+ :i* is /ret*$ be fa/5$ at bo? ends to a holl[ body1 & be /rick51 ! s\nd w la/ long] ?an if x 7 n s fa/5$2 because ! trembl+ or reciprocation :i* x receives f ! /roke1 is by reason ( ! connection communicat$ to ! holl[ body2 & ?is trembl+1 if ! holl[ body be great1 w la/ ! long] by reason ( t greatness4 ,:]e=e also1 = ! reason above m5tion$1 ! s\nd w la/ ! long]4 ,sid5ote3 ,h[ s\nd may be help$ & h9d]$ by ! w9d4 #i4 ,9 he>+ x happ5s1 o!rwise ?an 9 see+1 t ! action ( ! m$ium is made /rong] by ! w9d :5 x bl[s ! same way1 & weak] :5 x bl[s ! contr>y way4 ,! cause :]e( _c proce$ f any?+ b ! diff]5t g5]ation ( s\nd & light4 ,= 9 ! g5]ation ( light1 none ( ! p>ts ( ! m$ium betwe5 ! object & ! eye are mov$ f _! own places to o!r places s5sibly di/ant2 b ! action is propagat$ 9 spaces imp]ceptible2 s t no contr>y w9d c dim9i%1 nor fav\rable w9d encrease ! light1 unless x be s /rong z to remove ! object fur!r (f or br+ x ne>] to ! eye4 ,= ! w9d1 t is to say ! air mov$1 do? n by its int]position betwe5 ! object & ! eye "w o!rwise ?an x w\ld d1 if x 7 /ill & calm4 ,=1 :]e ! pressure is p]petual1 "o "p ( ! air is no soon] c>ri$ away1 b ano!r1 by succe$+ x1 receives ! same impression1 :i* ! "p c>ri$ away _h receiv$ be=e4 ,b 9 ! g5]ation ( s\nd1 ! fir/ collision or break+ asund] beate? away & drive? out ( its place ! ne>e/ "p ( ! air1 & t to a consid]able di/ance1 & ) consid]able velocity2 & z ! circles gr[ by _! remot5ess wid] & wid]1 s ! air be+ m & m dissipat$1 ha? also its motion m & m weak5$4 ,:5soev] !re=e ! air is s /rick5 z to cause s\nd1 if ! w9d fall ^u x1 x w move x all ne>] to ! e>1 if x bl[ t way1 & fur!r f x if x bl[ ! contr>y way2 s t accord+ z x bl[s f or t[>ds ! object1 s ! s\nd :i* is he>d w seem to come f a ne>] or remot] place2 & ! reaction1 by reason ( ! unequal di/ances1 be /r5g!n$ or debilitat$4 ,f h5ce may be und]/ood ! reason :y ! voice ( su* z are said to speak 9 _! bellies1 ?\gh x be utt]$ ne> h&1 is nev]!less he>d1 by ^? t suspect no?+1 z if x 7 a great way (f4 ,= hav+ no =m] ?\ght ( any det]m9$ place f :i* ! voice %\ld proce$1 & judg+ accord+ to ! greatness1 if x be weak !y ?9k x a great way (f1 if /rong ne>4 ,^! v5triloqui1 !re=e1 by =m+ _! voice1 n z o!rs by ! emission ( _! brea?1 b by draw+ x inw>ds1 d make ! same appe> small & weak2 :i* weakness ( ! voice deceives ^?1 t nei!r suspect ! artifice nor obs]ve ! endeav\r :i* !y use 9 speak+2 & s1 in/ead ( ?9k+ x weak1 !y ?9k x f> (f4 ,sid5ote3 ,n only air1 b o!r bodies1 h[ h>d soev] !y be1 convey s\nd4 #aj4 ,z = ! m$ium1 :i* conveys s\nd1 x is n air only4 ,= wat]1 or any o!r body h[ h>d soev]1 may be t m$ium4 ,= ! motion may be propagat$ p]petually 9 any h>d cont9u\s body2 b by reason ( ! difficulty1 ) :i* ! p>ts ( h>d bodies are mov$1 ! motion 9 go+ out ( h>d matt] makes b a weak impression ^u ! air4 ,nev]!less1 if "o end ( a v long & h>d beam be /rick51 & ! e> be appli$ at ! same "t to ! o!r end1 s t1 :5 ! action goe? out ( ! beam1 ! air1 :i* x /rike?1 may imm$iately be receiv$ by ! e>1 & be c>ri$ to ! tympanum1 ! s\nd w be consid]ably /rong4 ,9 l mann]1 if 9 ! night1 :5 all o!r noise :i* may h9d] s\nd cease?1 a man lay 8 e> to ! gr\nd1 he w he> ! s\nd ( ! /eps ( pass5g]s1 ?\gh at a great di/ance2 because ! motion1 :i* by _! tread+ !y communicate to ! e>?1 is propagat$ to ! e> by ! upp]mo/ p>ts ( ! e>? :i* receive? x f _! feet4 ,sid5ote3 ,! causes ( grave & acute s\nds1 & ( conc5t4 #aa4 ,i h %[n above1 t ! diff]5ce betwe5 grave & acute s\nds consi/e? 9 ?is1 t by h[ mu* ! %ort] ! "t is1 9 :i* ! reciprocations ( ! p>ts ( a body /rick5 are made1 by s mu* ! m acute w be ! s\nd4 ,n[ by h[ mu* a body ( ! same bigness is ei!r m heavy or less /ret*$1 by s mu* ! long] w ! reciprocations la/2 & !re=e heavi] & less /ret*$ bodies1 if !y be l 9 all o!r respects1 w yield a grav] s\nd ?an su* z be light] & m /ret*$4 ,z = ! conc5t ( s\nds1 x is to be consid]$ t ! reciprocation or vibration ( ! air1 by :i* s\nd is made1 aft] x ha? rea*$ ! drum ( ! e>1 impr9te? a l vibration ^u ! air t is inclos$ )9 x2 by :i* means ! sides ( ! drum )9 are /rick5 alt]nately4 ,n[ ! conc5t ( two s\nds consi/s 9 ?is1 t ! tympanum receives its s\nd+ /roke f bo? ! s\nd+ bodies 9 equal & equally frequ5t spaces ( "t2 s t :5 two /r+s make _! vibrations 9 ! same times1 ! conc5t !y produce is ! mo/ exquisite ( all o!r4 ,= ! sides ( ! tympanum1 t is to say ( ! organ ( he>+1 w be /rick5 by bo? ^? vibrations toge!r at once1 on "o side or o!r4 ,= example1 if ! two equal /r+s ,a ,b & ,c ,d be /rick5 toge!r1 & ! latitudes ( _! vibrations ,e ,f & ,g ,h be also equal1 & ! po9ts ,e1 ,g1 ,f & ,h be 9 ! concave sup]ficies ( ! tympanum1 s t x receive /rokes f bo? ! /r+s toge!r 9 ,e & ,g1 & aga9 toge!r 9 ,f & ,h1 ! s\nd1 :i* is made by ! vibrations ( ea* /r+1 w be s l1 t x may be tak5 = ! same s\nd1 & is call$ unison2 :i* is ! greate/ concord4 ,aga91 ! /r+ ,a ,b reta9+ /ill its =m] vibration ,e ,f1 let ! /r+ ,c ,d be /ret*$ till its vibration h d\ble ! swiftness x _h be=e1 & let ,e ,f be divid$ equally 9 ,i4 ,9 :at "t !re=e ! /r+ ,c ,d makes "o "p ( its vibration f ,g to ,h1 9 ! same "t ! /r+ ,a ,b w make "o "p ( its vibration f ,e to ,i2 & 9 :at "t ! /r+ ,c ,d ha? made ! o!r "p ( its vibration back f ,h to ,g1 9 ! same "t ano!r "p ( ! vibration ( ! /r+ ,a ,b w be made f ,i to ,f4 ,b ! po9ts ,f & ,g are bo? 9 ! sides ( ! organ1 & !re=e !y w /rike ! organ bo? toge!r1 n at e /roke1 b at e o!r /roke4 ,& ?is is ! ne>e/ concord to unison1 & makes t s\nd :i* is call$ an eigh?4 ,aga91 ! vibration ( ! /r+ ,a ,b rema9+ /ill ! same x 01 let ,c ,d be /ret*$ till its vibration be swift] ?an t ( ! /r+ ,a ,b 9 ! proportion ( #c to #b1 & let ,e ,f be divid$ into ?ree equal p>ts 9 ,k & ,l4 ,9 :at "t !re=e ! /r+ ,c ,d makes "o ?ird "p ( its vibration1 :i* ?ird "p is f ,g to ,h1 ! /r+ ,a ,b w make "o ?ird "p ( its vibration1 t is to say1 two-?irds ( ,e ,f1 namely1 ,e ,l4 ,& 9 :at "t ! /r+ ,c ,d makes ano!r ?ird "p ( its vibration1 namely ,h ,g1 ! /r+ ,a ,b w make ano!r ?ird "p ( its vibration1 namely f ,l to ,f1 & back aga9 f ,f to ,l4 ,la/ly1 :il/ ! /r+ ,c ,d makes ! la/ ?ird "p ( its vibration1 t is f ,g to ,h1 ! /r+ ,a ,b w make ! la/ ?ird "p ( its vibration f ,l to ,e4 ,b ! po9ts ,e & ,h are bo? 9 ! sides ( ! organ4 ,:]e=e1 at e ?ird "t1 ! organ w be /rick5 by ! vibration ( bo? ! /r+s toge!r1 & make t concord :i* is call$ a fif?4 ,a-------,b ,c-------,d ,g ,e ,k ,i ,l ,h ,f ,sid5ote3 ,ph5om5a ( smell+4 #ab4 ,= ! f9d+ out ! cause ( smells1 ,i %all make use ( ! evid5ce ( ^! foll[+ ph5om5a4 ,fir/1 t smell+ is h9d]$ by cold1 & help$ by heat4 ,secondly1 t :5 ! w9d bl[e? f ! object1 ! smell is ! /rong]2 &1 contr>ily1 :5 x bl[e? f ! s5ti5t t[>ds ! object1 ! weak]2 bo? :i* ph5om5a are1 by exp]i5ce1 manife/ly f\nd to be true 9 dogs1 :i* foll[ ! track ( bea/s by ! sc5t4 ,?irdly1 t su* bodies1 z are less p]vi\s to ! fluid m$ium1 yield less smell ?an su* z are m p]vi\s2 z may be se5 9 /ones & metals1 :i*1 comp>$ ) plants & liv+ creatures1 & _! p>ts1 fruits & excretions1 h v little or no smell at all4 ,f\r?ly1 t su* bodies1 z are ( _! own nature odor\s1 become yet m odor\s :5 !y are bruis$4 ,fif?ly1 t :5 ! brea? is /opp$1 at lea/ 9 m51 no?+ c be smelt4 ,six?ly1 t ! s5se ( smell+ is also tak5 away by ! /opp+ ( ! no/rils1 ?\gh ! m\? be left op54 ,sid5ote3 ,! fir/ organ & ! g5]ation ( smell+4 #ac4 ,by ! fif? & six? ph5om5on x is manife/1 t ! fir/ & imm$iate organ ( smell+ is ! inn]mo/ cuticle ( ! no/rils1 & t "p ( x1 :i* is bel[ ! passage common to ! no/rils & ! palate4 ,= :5 we draw brea? by ! no/rils we draw x into ! lungs4 ,t brea?1 !re=e1 :i* conveys smells is 9 ! way :i* passe? to ! lungs1 t is to say1 9 t "p ( ! no/rils :i* is bel[ ! passage ?r\gh :i* ! brea? goe?4 ,=1 no?+ is smelt1 nei!r beyond ! passage ( ! brea? )91 nor at all )\t ! no/rils4 ,& see+ t f diff]5t smells !re mu/ necess>ily proce$ "s mutation 9 ! organ1 & all mutation is motion2 x is !re=e also necess>y t1 9 smell+1 ! p>ts ( ! organ1 t is to say ( t int]nal cuticle & ! n]ves t are ins]t$ into x1 mu/ be div]sely mov$ by diff]5t smells4 ,& see+ also1 t x ha? be5 demon/rat$1 t no?+ c be mov$ b by a body t is already mov$ & contigu\s2 & t !re is no o!r body contigu\s to ! int]nal membrane ( ! no/rils b brea?1 t is to say attract$ air1 & su* little solid invisible bodies1 if !re be any su*1 z are int]m+l$ ) ! air2 x foll[s necess>ily1 t ! cause ( smell+ is ei!r ! motion ( t pure air or e!real sub/ance1 or ! motion ( ^? small bodies4 ,b ?is motion is an effect proce$+ f ! object ( smell1 &1 !re=e1 ei!r ! :ole object itself or its sev]al p>ts mu/ necess>ily be mov$4 ,n[1 we "k t odor\s bodies make od\r1 ?\gh _! :ole bulk be n mov$4 ,:]e=e ! cause ( od\r is ! motion ( ! invisible p>ts ( ! odor\s body4 ,& ^! invisible p>ts d ei!r g out ( ! object1 or else1 reta9+ _! =m] situation ) ! re/ ( ! p>ts1 are mov$ toge!r ) !m1 t is to say1 !y h simple & invisible motion4 ,!y t say1 !re goes some?+ out ( ! odor\s body1 call x an effluvium2 :i* effluvium is ei!r ( ! e!real sub/ance1 or ( ! small bodies t are int]m+l$ ) x4 ,b1 t all v>iety ( od\rs %\ld proce$ f ! effluvia ( ^? small bodies t are int]m+l$ ) ! e!real sub/ance1 is altoge!r incr$ible1 = ^! consid]ations2 fir/1 t c]ta9 ungu5ts1 ?\gh v little 9 quantity1 d nev]!less s5d =? v /rong od\rs1 n only to a great di/ance ( place1 b also = a great cont9uance ( "t1 & are to be smelt 9 e po9t bo? ( t place & "t2 s t ! p>ts issu$ out are suffici5t to fill t5 ?\s& times m space1 ?an ! :ole odor\s body is able to fill2 :i* is impossible4 ,secondly1 t :e!r t issu+ out be ) /rait or ) crook$ motion1 if ! same quantity %\ld fl[ f any o!r odor\s body ) ! same motion1 x w\ld foll[ t all odor\s bodies w\ld yield ! same smell4 ,?irdly1 t see+ ^? effluvia h great velocity ( motion 7z is manife/ f ?is1 t noisome od\rs proce$+ f cav]ns are pres5tly smelt at a great di/ance7 x w\ld foll[1 t1 by reason !re is no?+ to h9d] ! passage ( ^? effluvia to ! organ1 su* motion alone 7 suffici5t to cause smell+2 :i* is n s2 = we _c smell at all1 unless we draw 9 our brea? ?r\gh our no/rils4 ,smell+1 !re=e1 is n caus$ by ! effluvium ( atoms2 nor1 = ! same reason1 is x caus$ by ! effluvium ( e!real sub/ance2 = s also we %\ld smell )\t ! draw+ 9 ( our brea?4 ,besides1 ! e!real sub/ance be+ ! same 9 all odor\s bodies1 !y w\ld always affect ! organ 9 ! same mann]2 &1 consequ5tly1 ! od\rs ( all ?+s w\ld be alike4 ,x rema9s1 !re=e1 t ! cause ( smell+ mu/ consi/ 9 ! simple motion ( ! p>ts ( odor\s bodies )\t any efflux or dim9ution ( _! :ole sub/ance4 ,& by ?is motion !re is propagat$ to ! organ1 by ! int]m$iate air1 ! l motion1 b n /rong en\gh to excite s5se ( itself )\t ! attraction ( air by respiration4 ,& ?is is a possible cause ( smell+4 ,sid5ote3 ,h[ smell+ is help$ by heat & by w9d4 #ad4 ,! cause :y smell+ is h9d]$ by cold & help$ by heat may be ?is2 t heat1 z ha? be5 %[n 9 *apt] ,,xxi1 g5]ate? simple motion2 & !re=e also1 :]esoev] x is already1 !re x w increase x2 & ! cause ( smell+ be+ increas$1 ! smell itself w also be increas$4 ,z = ! cause :y ! w9d bl[+ f ! object makes ! smell ! /rong]1 x is all "o ) t = :i* ! attraction ( air 9 respiration do? ! same4 ,=1 he t draws 9 ! air next to him1 draws ) x by succession t air 9 :i* is ! object4 ,n[1 ?is motion ( ! air is w9d1 &1 :5 ano!r w9d bl[e? f ! object1 w be increas$ by x4 ,sid5ote3 ,:y su* bodies are lea/ smelt1 :i* h ! lea/ int]mixture ( air 9 !m4 #ae4 ,t bodies :i* conta9 ! lea/ quantity ( air1 z /ones & metals1 yield less smell ?an plants & liv+ creatures2 ! cause may be1 t ! motion1 :i* cause? smell+1 is a motion ( ! fluid p>ts only2 :i* p>ts1 if !y h any motion f ! h>d p>ts 9 :i* !y are conta9$1 !y communicate ! same to ! op5 air1 by :i* x is propagat$ to ! organ4 ,:]e1 !re=e1 !re are no fluid p>ts z 9 metals1 or :]e ! fluid p>ts receive no motion f ! h>d p>ts1 z 9 /ones1 :i* are made h>d by accretion1 !re c be no smell4 ,& !re=e also ! wat]1 ^: p>ts h little or no motion1 yielde? no smell4 ,b1 if ! same wat]1 by se$s & ! heat ( ! sun1 be toge!r ) p>ticles ( e>? rais$ into a plant1 & be aft]w>ds press$ out aga91 x w be odor\s1 z w9e f ! v9e4 ,& z wat] pass+ ?r\gh plants is by ! motion ( ! p>ts ( ^? plants made an odor\s liquor2 s also ( air1 pass+ ?r\gh ! same plants :il/ !y are gr[+1 are made odor\s airs4 ,& ?us also x is ) ! juices & spirits1 :i* are br$ 9 liv+ creatures4 ,sid5ote3 ,:y odor\s ?+s become m odor\s1 :5 bruis$4 #af4 ,t odor\s bodies may be made m odor\s by contrition proce$s f ?is1 t be+ brok5 into _m p>ts1 :i* are all odor\s1 ! air1 :i* by respiration is drawn f ! object t[>ds ! organ1 do? 9 its passage t\* ^u all ^? p>ts1 & receive _! motion4 ,n[1 ! air t\*e? ! sup]ficies only2 & a body hav+ less sup]ficies :il/ x is :ole1 ?an all its p>ts toge!r h aft] x is r$uc$ to p[d]1 x foll[s t ! same odor\s body yielde? less smell :il/ x is :ole1 ?an x w d aft] x is brok5 into small] p>ts4 ,& ?us mu* ( smells4 ,sid5ote3 ,! fir/ organ ( ta/+2 & :y "s sav\rs cause nause\sness4 #ag4 ,! ta/e foll[s2 ^: g5]ation ha? ?is diff]5ce f t ( ! sight1 he>+1 & smell+1 t by ^! we h s5se ( remote objects2 :]eas1 we ta/e no?+ b :at is contigu\s1 & do? imm$iately t\* ei!r ! tongue or palate1 or bo?4 ,f :5ce x is evid5t1 t ! cuticles ( ! tongue & palate1 & ! n]ves ins]t$ into !m are ! fir/ organ ( ta/e2 & 7because f ! concussion ( ! p>ts ( ^!1 !re foll[e? necess>ily a concussion ( ! pia mat]7 t ! action communicat$ to ^! is propagat$ to ! bra91 & f !nce to ! f>!/ organ1 namely1 ! he>t1 9 ^: reaction consi/e? ! nature ( s5se4 ,n[1 t sav\rs1 z well z od\rs1 d n only move ! bra9 b ! /oma* also1 z is manife/ by ! loa?+s t are caus$ by !m bo?2 !y1 t consid] ! organ ( bo? ^! s5ses1 w n wond] at all2 see+ ! tongue1 ! palate & ! no/rils1 h "o & ! same cont9u$ cuticle1 d]iv$ f ! dura mat]4 ,& t effluvia h no?+ to d 9 ! s5se ( ta/+1 is manife/ f ?is1 t !re is no ta/e :]e ! organ & ! object are n contigu\s4 ,by :at v>iety ( motions ! diff]5t k9ds ( ta/es1 :i* are innum]able1 may be di/+ui%$1 ,i "k n4 ,i might ) o!rs d]ive !m f ! div]s figures ( ^? atoms1 ( :i* :atsoev] may be ta/$ consi/e?2 or f ! div]se motions :i* ,i might1 by way ( supposition1 attribute to ^? atoms2 conjectur+1 n )\t "s likelihood ( tru?1 t su* ?+s z ta/e sweet h _! p>ticles mov$ ) sl[ circul> motion1 & _! figures sph]ical2 :i* makes !m smoo? & pleas+ to ! organ2 t bitt] ?+s h circul> motion1 b vehem5t1 & _! figures full ( angles1 by :i* !y tr\ble ! organ2 & t s\r ?+s h /rait & reciprocal motion1 & _! figures long & small1 s t !y cut & w\nd ! organ4 ,& 9 l mann] ,i might assign = ! causes ( o!r ta/es su* sev]al motions & figures ( atoms1 z might 9 probability seem to be ! true causes4 ,b ?is w\ld be to revolt f philosophy to div9ation4 ,sid5ote3 ,! fir/ organ ( feel+2 & h[ we came to ! k ( su* objects z are common to ! t\* & to o!r s5ses4 #ah4 ,by ! t\*1 we feel :at bodies are cold or hot1 ?\gh !y be di/ant f u4 ,o!rs1 z h>d1 s(t1 r\gh1 & smoo?1 we _c feel unless !y be contigu\s4 ,! organ ( t\* is e "o ( ^? membranes1 :i* be+ cont9u$ f ! pia mat] are s diffus$ ?r\gh\t ! :ole body1 z t no "p ( x c be press$1 b ! pia mat] is press$ toge!r ) x4 ,:atsoev] !re=e presse? x1 is felt z h>d or s(t1 t is to say1 z m or less h>d4 ,& z = ! s5se ( r\gh1 x is no?+ else b innum]able p]ceptions ( h>d & h>d succe$+ "o ano!r by %ort int]vals bo? ( "t & place4 ,= we take notice ( r\gh & smoo?1 z also ( magnitude & figure1 n only by ! t\*1 b also by memory4 ,= ?\gh "s ?+s are t\*$ 9 "o po9t1 yet r\gh & smoo?1 l quantity & figure1 are n p]ceiv$ b by ! flux ( a po9t1 t is to say1 we h no s5se ( !m )\t "t2 & we c h no s5se ( "t )\t memory4 ------- ,,*apt] ,,xxx4 ,,( ,,gravity4 #a4 ,a ?ick body do? n conta9 m matt]1 unless also m place1 ?an a ?94--#b4 ,t ! desc5t ( heavy bodies proce$s n f _! own appetite1 b f "s p[] ( ! e>?4--#c4 ,! diff]5ce ( gravities proce$e? f ! diff]5ce ( ! impetus ) :i* ! elem5ts1 :]e( heavy bodies are made1 d fall ^u ! e>?4--#d4 ,! cause ( ! desc5t ( heavy bodies4--#e4 ,9 :at proportion ! desc5t ( heavy bodies is accel]at$4--#f4 ,:y ^? t dive d n1 :5 !y are "u wat]1 feel ! weight ( ! wat] above !m4--#g4 ,! weight ( a body t floate?1 is equal to ! weight ( s mu* wat] z w\ld fill ! space1 :i* ! imm]s$ "p ( ! body takes up )9 ! wat]4--#h4 ,if a body be light] ?an wat]1 !n h[ big soev] t body be1 x may float ^u any quantity ( wat]1 h[ little soev]4--#i4 ,h[ wat] may be lift$ up & =c$ out ( a vessel by air4--#aj4 ,:y a bladd] is heavi] :5 bl[n full ( air1 ?an :5 x is empty4--#aa4 ,! cause ( ! ejection upw>ds ( heavy bodies f a w9d-gun4--#ab4 ,! cause ( ! asc5t ( wat] 9 a wea!r-glass4--#ac4 ,! cause ( motion upw>ds 9 liv+ creatures4--#ad4 ,t !re is 9 nature a k9d ( body heavi] ?an air1 :i* nev]!less is n by s5se di/+ui%able f x4--#ae4 ,( ! cause ( magnetical virtue4 ,sid5ote3 ,a ?ick body do? n conta9 m matt]1 unless also m place1 ?an a ?94 #a4 ,9 *apt] ,,xxi ,i h def9$ ?ick & ?91 z t place requir$1 s1 z t by ?ick 0 signifi$ a m resi/+ body1 & by ?91 a body less resi/+2 foll[+ ! cu/om ( ^? t h be=e me disc\rs$ ( refraction4 ,n[ if we consid] ! true & vulg> signification ( ^? words1 we %all f9d !m to be names collective1 t is to say1 names ( multitude2 z ?ick to be t1 :i* takes up m p>ts ( a space giv51 & ?9 t1 :i* conta9s few] p>ts ( ! same magnitude 9 ! same space1 or 9 a space equal to x4 ,?ick !re=e is ! same ) frequ5t1 z a ?ick troop2 & ?9 ! same ) unfrequ5t1 z a ?9 rank1 ?9 ( h\ses2 n t !re is m matt] 9 "o place ?an 9 ano!r equal place1 b a great] quantity ( "s nam$ body4 ,= !re is n less matt] or body1 indef9itely tak51 9 a des]t1 ?an !re is 9 a city2 b few] h\ses1 or few] m54 ,nor is !re 9 a ?ick rank a great] quantity ( body1 b a great] numb] ( soldi]s1 ?an 9 a ?94 ,:]e=e ! multitude & paucity ( ! p>ts conta9$ )9 ! same space d con/itute d5sity & r>ity1 :e!r ^? p>ts be sep>at$ by vacuum or by air4 ,b ! consid]ation ( ?is is n ( any great mom5t 9 philosophy2 & !re=e ,i let x alone1 & pass on to ! se>* ( ! causes ( gravity4 ,sid5ote3 ,t ! desc5t ( heavy bodies proce$s n f _! own appetite1 b f "s p[] ( ! e>?4 #b4 ,n[ we call ^? bodies heavy1 :i*1 unless !y be h9d]$ by "s =ce1 are c>ri$ t[>ds ! c5tre ( ! e>?1 & t by _! own accord1 = aught we c by s5se p]ceive to ! contr>y4 ,"s philosoph]s !re=e h be5 ( op9ion1 t ! desc5t ( heavy bodies proce$$ f "s int]nal appetite1 by :i*1 :5 !y 7 ca/ upw>ds1 !y desc5d$ aga91 z mov$ by !mselves1 to su* place z 0 agreeable to _! nature4 ,o!rs ?\ght !y 7 attract$ by ! e>?4 ,to ! =m] ,i _c ass5t1 because ,i ?9k ,i h already cle>ly en\gh demon/rat$ t !re c be no beg9n+ ( motion1 b f an ext]nal & mov$ body2 & consequ5tly1 t :atsoev] ha? motion or endeav\r t[>ds any place1 w always move or endeav\r t[>ds t same place1 unless x be h9d]$ by ! reaction ( "s ext]nal body4 ,heavy bodies1 !re=e1 be+ once ca/ upw>ds1 _c be ca/ d[n aga9 b by ext]nal motion4 ,besides1 see+ inanimate bodies h no appetite at all1 x is ridicul\s to ?9k t by _! own innate appetite !y %\ld1 to pres]ve !mselves1 n und]/&+ :at pres]ves !m1 =sake ! place !y are 91 & transf] !mselves to ano!r place2 :]eas man1 :o ha? bo? appetite & und]/&+1 _c1 = ! pres]vation ( 8 own life1 raise himself by leap+ above ?ree or f\r feet f ! gr\nd4 ,la/ly1 to attribute to creat$ bodies ! p[] to move !mselves1 :at is x else ?an to say t !re be creatures :i* h no dep5dance ^u ! ,creator8 ,to ! latt]1 :o attribute ! desc5t ( heavy bodies to ! attraction ( ! e>?1 ,i ass5t4 ,b by :at motion ?is is done1 ha? n z yet be5 expla9$ by any man4 ,i %all !re=e 9 ?is place say some:at ( ! mann] & ( ! way by :i* ! e>? by its action attracte? heavy bodies4 ,sid5ote3 ,! diff]5ce ( gravities proce$e? f ! diff]5ce ( ! impetus ) :i* ! elem5ts1 :]e( heavy bodies are made1 d fall ^u ! e>?4 #c4 ,t by ! supposition ( simple motion 9 ! sun1 homog5e\s bodies are congregat$ & het]og5e\s dissipat$1 has already be5 demon/rat$ 9 ! #e? article ( *apt] ,,xxi4 ,i h also suppos$1 t !re are int]m+l$ ) ! pure air c]ta9 little bodies1 or1 z o!rs call !m1 atoms2 :i* by reason ( _! extreme smallness are invisible1 & diff]+ f "o ano!r 9 consi/5ce1 figure1 motion1 & magnitude2 f :5ce x comes to pass t "s ( !m are congregat$ to ! e>?1 o!rs to o!r planets1 & o!rs are c>ri$ up & d[n 9 ! spaces betwe54 ,& see+ ^?1 :i* are c>ri$ to ! e>?1 diff] f "o ano!r 9 figure1 motion1 & magnitude1 !y w fall ^u ! e>?1 "s ) great]1 o!rs ) less impetus4 ,& see+ also t we compute ! sev]al degrees ( gravity no o!rwise ?an by ?is _! fall+ ^u ! e>? ) great] or less impetus2 x foll[s1 t we conclude ^? to be ! m heavy t h ! great] impetus1 & ^? to be less heavy t h ! less impetus4 ,our inquiry !re=e mu/ be1 by :at means x may come to pass1 t ( bodies1 :i* desc5d f above to ! e>?1 "s are c>ri$ ) great]1 o!rs ) less impetus2 t is to say1 "s are m heavy ?an o!rs4 ,we mu/ also inquire1 by :at means su* bodies1 z settle ^u ! e>?1 may by ! e>? itself be =c$ to asc5d4 ,sid5ote3 ,! cause ( ! desc5t ( heavy bodies4 #d4 ,let ! circle made ^u ! c5tre ,c 79 fig4 #b7 be a great circle 9 ! sup]ficies ( ! e>?1 pass+ ?r\gh ! po9ts ,a & ,b4 ,also let any heavy body1 z ! /one ,a ,d1 be plac$ any:]e 9 ! plane ( ! equator2 & let x be conceiv$ to be ca/ up f ,a ,d p]p5dicul>ly1 or to be c>ri$ 9 any o!r l9e to ,e1 & suppos$ to re/ !re4 ,!re=e1 h[ mu* space soev] ! /one took up 9 ,a ,d1 s mu* space x takes up n[ 9 ,e4 ,& because all place is suppos$ to be full1 ! space ,a ,d w be fill$ by ! air :i* fl[s into x fir/ f ! ne>e/ places ( ! e>?1 & aft]w>ds successively f m remote places4 ,^u ! c5tre ,c let a circle be und]/ood to be drawn ?r\gh ,e2 & let ! plane space1 :i* is betwe5 ! sup]ficies ( ! e>? & t circle1 be divid$ into plane orbs equal & conc5tric2 ( :i* let t be ! fir/1 :i* is conta9$ by ! two p]imet]s t pass ?r\gh ,a & ,d4 ,:il/ !re=e ! air1 :i* is 9 ! fir/ orb1 fille? ! place ,a ,d1 ! orb itself is made s mu* less1 & consequ5tly its latitude is less ?an ! /rait l9e ,a ,d4 ,:]e=e !re w necess>ily desc5d s mu* air f ! orb next above4 ,9 l mann]1 = ! same cause1 !re w also be a desc5t ( air f ! orb next above t2 & s by succession f ! orb 9 :i* ! /one is at re/ 9 ,e4 ,ei!r !re=e ! /one itself1 or s mu* air1 w desc5d4 ,& see+ air is by ! diurnal revolution ( ! e>? m easily ?ru/ away ?an ! /one1 ! air1 :i* is 9 ! orb t conta9s ! /one1 w be =c$ fur!r upw>ds ?an ! /one4 ,b ?is1 )\t ! admission ( vacuum1 _c be1 unless s mu* air desc5d to ,e f ! place next above2 :i* be+ done1 ! /one w be ?ru/ d[nw>ds4 ,by ?is means !re=e ! /one n[ receives ! beg9n+ ( its desc5t1 t is to say1 ( its gravity4 ,fur!rmore1 :atsoev] is once mov$1 w be mov$ cont9ually 7z ha? be5 %[n 9 ! #ai? article ( *apt] ,,viii7 9 ! same way1 & ) ! same cel]ity1 except x be ret>d$ or accel]at$ by "s ext]nal mov5t4 ,n[ ! air1 :i* is ! only body t is int]pos$ betwe5 ! e>? ,a & ! /one above x ,e1 w h ! same action 9 e po9t ( ! /rait l9e ,e ,a1 :i* x ha? 9 ,e4 ,b x depress$ ! /one 9 ,e2 & !re=e also x w depress x equally 9 e po9t ( ! /rait l9e ,e ,a4 ,:]e=e ! /one w desc5d f ,e to ,a ) accel]at$ motion4 ,! possible cause !re=e ( ! desc5t ( heavy bodies "u ! equator1 is ! diurnal motion ( ! e>?4 ,& ! same demon/ration w s]ve1 if ! /one be plac$ 9 ! plane ( any o!r circle p>allel to ! equator4 ,b because ?is motion ha?1 by reason ( its great] sl[ness1 less =ce to ?ru/ (f ! air 9 ! p>allel circles ?an 9 ! equator1 & no =ce at all at ! poles1 x may well be ?\ght 7= x is a c]ta9 consequ5t7 t heavy bodies desc5d ) less & less velocity1 z !y are m & m remote f ! equator2 & t at ! poles !mselves1 !y w ei!r n desc5d at all1 or n desc5d by ! axis2 :i* :e!r x be true or false1 exp]i5ce mu/ det]m9e4 ,b x is h>d to make ! exp]im5t1 bo? because ! times ( _! desc5ts _c be easily measur$ ) suffici5t exactness1 & also because ! places ne> ! poles are inaccessible4 ,nev]!less1 ?is we "k1 t by h[ mu* ! ne>] we come to ! poles1 by s mu* ! great] are ! flakes ( ! sn[ t falls2 & by h[ mu* ! m swiftly su* bodies desc5d z are fluid & dissipable1 by s mu* ! small] are ! p>ticles into :i* !y are dissipat$4 ,sid5ote3 ,9 :at proportion ! desc5t ( heavy bodies is accel]at$4 #e4 ,suppos+1 !re=e1 ?is to be ! cause ( ! desc5t ( heavy bodies1 x w foll[ t _! motion w be accel]at$ 9 su* mann]1 z t ! spaces1 :i* are transmitt$ by !m 9 ! sev]al times1 w h to "o ano!r ! same proportion :i* ! odd numb]s h 9 succession f unity4 ,= if ! /rait l9e ,e ,a be divid$ into any numb] ( equal p>ts1 ! heavy body desc5d+ w1 by reason ( ! p]petual action ( ! diurnal motion1 receive f ! air 9 e "o ( ^? times1 9 e sev]al po9t ( ! /rait l9e ,e ,a1 a sev]al new & equal impulsion2 & !re=e also 9 e "o ( ^? times1 x w acquire a sev]al & equal degree ( cel]ity4 ,& f h5ce x foll[s1 by t :i* ,galileus ha? 9 8 ,dialogues ( ,motion demon/rat$1 t heavy bodies desc5d 9 ! sev]al times ) su* diff]5ces ( transmitt$ spaces1 z are equal to ! diff]5ces ( ! squ>e numb]s t succe$ "o ano!r f unity2 :i* squ>e numb]s be+ #a1 #d1 #i1 #af1 c4 _! diff]5ces are #c1 #e1 #g1 c42 t is to say1 ! odd numb]s :i* succe$ "o ano!r f unity4 ,aga9/ ?is cause ( gravity :i* ,i h giv51 x w p]haps be object$1 t if a heavy body be plac$ 9 ! bottom ( "s holl[ cyl9d] ( iron or adamant1 & ! bottom be turn$ upw>ds1 ! body w desc5d1 ?\gh ! air above _c depress x1 mu* less accel]ate its motion4 ,b x is to be consid]$ t !re c be no cyl9d] or cav]n1 b su* z is support$ by ! e>?1 & be+ s support$ is1 toge!r ) ! e>?1 c>ri$ ab\t by its diurnal motion4 ,= by ?is means ! bottom ( ! cyl9d] w be z ! sup]ficies ( ! e>?2 & by ?ru/+ (f ! next & l[e/ air1 w make ! upp]mo/ air depress ! heavy body1 :i* is at ! top ( ! cyl9d]1 9 su* mann] z is above explicat$4 ,sid5ote3 ,:y ^? t dive1 d n1 :5 !y are "u wat]1 feel ! weight ( ! wat] above !m4 #f4 ,! gravity ( wat] be+ s great z by exp]i5ce we f9d x is1 ! reason is dem&$ by _m1 :y ^? t dive1 h[ deep soev] !y g "u wat]1 d n at all feel ! weight ( ! wat] :i* lies ^u !m4 ,& ! cause seems to be ?is1 t all bodies by h[ mu* ! heavi] !y are1 by s mu* ! great] is ! endeav\r by :i* !y t5d d[nw>ds4 ,b ! body ( a man is heavi] ?an s mu* wat] z is equal to x 9 magnitude1 & !re=e ! endeav\r d[nw>ds ( a man's body is great] ?an t ( wat]4 ,& see+ all endeav\r is motion1 ! body also ( a man w be c>ri$ t[>ds ! bottom ) great] velocity ?an s mu* wat]4 ,:]e=e !re is great] reaction f ! bottom2 & ! endeav\r upw>ds is equal to ! endeav\r d[nw>ds1 :e!r ! wat] be press$ by wat]1 or by ano!r body :i* is heavi] ?an wat]4 ,& !re=e by ^! two opposite equal endeav\rs1 ! endeav\r bo? ways 9 ! wat] is tak5 away2 & consequ5tly1 ^? t dive are n at all press$ by x4 ,coroll4 ,f h5ce also x is manife/1 t wat] 9 wat] ha? no weight at all1 because all ! p>ts ( wat]1 bo? ! p>ts above1 & ! p>ts t are directly "u1 t5d t[>ds ! bottom ) equal endeav\r & 9 ! same /rait l9es4 ,sid5ote3 ,! weight ( a body t floate?1 is equal to ! weight ( s mu* wat] z w\ld fill ! space :i* ! imm]s$ "p ( ! body takes up )9 ! wat]4 #g4 ,if a body float ^u ! wat]1 ! weight ( t body is equal to ! weight ( s mu* wat] z w\ld fill ! place :i* ! imm]s$ "p ( ! body takes up )9 ! wat]4 ,let ,e ,f 79 fig4 #c7 be a body float+ 9 ! wat] ,a ,b ,c ,d2 & let ! "p ,e be above1 & ! o!r "p ,f "u ! wat]4 ,i say1 ! weight ( ! :ole body ,e ,f is equal to ! weight ( s mu* wat] z ! space ,f w receive4 ,= see+ ! weight ( ! body ,e ,f =ce? ! wat] out ( ! space ,f1 & place? x ^u ! sup]ficies ,a ,b1 :]e x presse? d[nw>ds2 x foll[s1 t f ! resi/ance ( ! bottom !re w also be an endeav\r upw>ds4 ,& see+ aga91 t by ?is endeav\r ( ! wat] upw>ds1 ! body ,e ,f is lift$ up1 x foll[s1 t if ! endeav\r ( ! body d[nw>ds be n equal to ! endeav\r ( ! wat] upw>ds1 ei!r ! :ole body ,e ,f w1 by reason ( t inequality ( _! endeav\rs or mom5ts1 be rais$ out ( ! wat]1 or else x w desc5d to ! bottom4 ,b x is suppos$ to /& s1 z nei!r to asc5d nor desc5d4 ,:]e=e !re is an equilibrium betwe5 ! two endeav\rs2 t is to say1 ! weight ( ! body ,e ,f is equal to ! weight ( s mu* wat] z ! space ,f w receive2 :i* 0 to be prov$4 ,sid5ote3 ,if a body be light] ?an wat]1 !n h[ big soev] t body be1 x may float ^u any quantity ( wat]1 h[ little soev]4 #h4 ,f h5ce x foll[s1 t any body1 ( h[ great magnitude soev]1 provid$ x consi/ ( matt] less heavy ?an wat]1 may nev]!less float ^u any quantity ( wat]1 h[ little soev]4 ,let ,a ,b ,c ,d 79 fig4 #d7 be a vessel2 & 9 x let ,e ,f ,g ,h be a body consi/+ ( matt] :i* is less heavy ?an wat]2 & let ! space ,a ,g ,c ,f be fill$ ) wat]4 ,i say1 ! body ,e ,f ,g ,h w n s9k to ! bottom ,d ,c4 ,= see+ ! matt] ( ! body ,e ,f ,g ,h is less heavy ?an wat]1 if ! :ole space )\t ,a ,b ,c ,d 7 fall ( wat]1 yet "s "p ( ! body ,e ,f ,g ,h1 z ,e ,f ,i ,k1 w\ld be above ! wat]2 & ! weight ( s mu* wat] z w\ld fill ! space ,i ,g ,h ,k w\ld be equal to ! weight ( ! :ole body ,e ,f ,g ,h2 & consequ5tly ,g ,h w\ld n t\* ! bottom ,d ,c4 ,z = ! sides ( ! vessel1 x is no matt] :e!r !y be h>d or fluid2 = !y s]ve only to t]m9ate ! wat]2 :i* may be done z well by wat] z by any o!r matt] h[ h>d soev]2 & ! wat] )\t ! vessel is t]m9at$ some:]e1 s z t x c spread no f>!r4 ,! "p !re=e ,e ,f ,i ,k w be extant above ! wat] ,a ,g ,c ,f :i* is conta9$ 9 ! vessel4 ,:]e=e ! body ,e ,f ,g ,h w also float ^u ! wat] ,a ,g ,c ,f1 h[ little soev] t wat] be2 :i* 0 to be demon/rat$4 ,sid5ote3 ,h[ wat] may be lift$ up & =c$ out ( a vessel by air4 #i4 ,9 ! #d? article ( *apt] ,,xxvi1 !re is br\ght = ! prov+ ( vacuum ! exp]im5t ( wat] enclos$ 9 a vessel2 :i* wat]1 ! orifice above be+ op5$1 is eject$ upw>ds by ! impulsion ( ! air4 ,x is !re=e dem&$1 see+ wat] is heavi] ?an air1 h[ t c be done4 ,let ! second figure ( ! same1 *apt] ,,xxvi be consid]$1 :]e ! wat] is ) great =ce inject$ by a syr+e into ! space ,f ,g ,b4 ,9 t injection1 ! air 7b pure air7 goe? ) ! same =ce out ( ! vessel ?r\gh ! inject$ wat]4 ,b z = ^? small bodies1 :i* =m]ly ,i suppos$ to be int]m+l$ ) air & to be mov$ ) simple motion1 !y _c1 toge!r ) ! pure air1 p5etrate ! wat]2 b rema9+ beh9d are necess>ily ?ru/ toge!r into a n>r[] place1 namely into ! space :i* is above ! wat] ,f ,g4 ,! motions !re=e ( ^? small bodies w be less & less free1 by h[ mu* ! quantity ( ! inject$ wat] is great] & great]2 s t by _! motions fall+ ^u "o ano!r1 ! same small bodies w mutually compress ea* o!r1 & h a p]petual endeav\r ( rega9+ _! lib]ty1 & ( depress+ ! wat] t h9d]s !m4 ,:]e=e1 z soon z ! orifice above is op5$1 ! wat] :i* is next x w h an endeav\r to asc5d1 & w !re=e necess>ily g out4 ,b x _c g out1 unless at ! same "t !re ent] 9 z mu* air2 & !re=e bo? ! wat] w g out1 & ! air ent] 91 till ^? small bodies :i* 7 left )9 ! vessel h recov]$ _! =m] lib]ty ( motion2 t is to say1 till ! vessel be aga9 fill$ ) air1 & no wat] be left ( suffici5t height to /op ! passage at ,b4 ,:]e=e ,i h %[n a possible cause ( ?is ph5om5on1 namely1 ! same ) t ( ?und]4 ,= z 9 ! g5]ation ( ?und]1 ! small bodies enclos$ )9 ! cl\ds1 by be+ too closely p5t toge!r1 d by _! motion break ! cl\ds1 & re/ore !mselves to _! natural lib]ty2 s "h also ! small bodies enclos$ )9 ! space :i* is above ! /rait l9e ,f ,g1 d by _! own motion expel ! wat] z soon z ! passage is op5$ above4 ,& if ! passage be kept /opp$1 & ^! small bodies be m vehem5tly compress$ by ! p]petual =c+ 9 ( m wat]1 !y w at la/ break ! vessel itself ) great noise4 ,sid5ote3 ,:y a bladd] is heavi] :5 bl[n full ( air1 ?an :5 x is empty4 #aj4 ,if air be bl[n into a holl[ cyl9d]1 or into a bladd]1 x w increase ! weight ( ei!r ( !m a little1 z _m h f\nd by exp]i5ce1 :o ) great accurat5ess h tri$ ! same4 ,& x is no wond]1 see+1 z ,i h suppos$1 !re are int]m+l$ ) ! common air a great numb] ( small h>d bodies1 :i* are heavi] ?an ! pure air4 ,=1 ! e!real sub/ance1 be+ on all sides equally agitat$ by ! motion ( ! sun1 ha? an equal endeav\r t[>ds all ! p>ts ( ! univ]se2 &1 !re=e1 x ha? no gravity at all4 ,sid5ote3 ,! cause ( ! ejection upw>ds ( heavy bodies f a w9d-gun4 #aa4 ,we f9d also by exp]i5ce1 t1 by ! =ce ( air enclos$ 9 a holl[ cannon1 a bullet ( lead may ) consid]able viol5ce be %ot out ( a gun ( late inv5tion1 call$ ! w9d-gun4 ,9 ! end ( ?is cannon !re are two holes1 ) _! valves on ! inside1 to %ut !m close2 "o ( !m s]v+ = ! admission ( air1 & ! o!r = ! lett+ ( x out4 ,also1 to t end :i* s]ves = ! receiv+ 9 ( air1 !re is jo9$ ano!r cannon ( ! same metal & bigness1 9 :i* !re is fitt$ a ramm] :i* is p]=at$1 & ha? also a valve op5+ t[>ds ! =m] cannon4 ,by ! help ( ?is valve ! ramm] is easily drawn back1 & lette? 9 air f )\t2 & be+ (t5 drawn back & return$ aga9 ) viol5t /rokes1 x =ce? "s "p ( t air into ! =m] cannon1 s long1 till at la/ ! resi/ance ( ! enclos$ air is great] ?an ! =ce ( ! /roke4 ,& by ?is means m5 ?9k !re is n[ a great] quantity ( air 9 ! cannon ?an !re 0 =m]ly1 ?\gh x 7 full be=e4 ,also1 ! air ?us =c$ 91 h[ mu* soev] x be1 is h9d]$ f gett+ out aga9 by ! a=esaid valves1 :i* ! v endeav\r ( ! air to get out do? necess>ily %ut4 ,la/ly1 t valve be+ op5$ :i* 0 made = ! lett+ out ( ! air1 x pres5tly breake? out ) viol5ce1 & drive? ! bullet be=e x ) great =ce & velocity4 ,z = ! cause ( ?is1 ,i c\ld easily attribute x1 z mo/ m5 d1 to cond5sation1 & ?9k t ! air1 :i* _h at ! fir/ b its ord9>y degree ( r>ity1 0 aft]w>ds1 by ! =c+ 9 ( m air1 cond5s$1 & la/ ( all1 r>ifi$ aga9 by be+ let out & re/or$ to its natural lib]ty4 ,b ,i _c imag9e h[ ! same place c be always full1 &1 nev]!less1 conta9 "s times a great]1 sometimes a less quantity ( matt]2 t is to say1 t x c be full] ?an full4 ,nor c ,i conceive h[ fulness c ( itself be an effici5t cause ( motion4 ,= bo? ^! are impossible4 ,:]e=e we mu/ seek out "s o!r possible cause ( ?is ph5om5on4 ,:il/1 !re=e1 ! valve :i* s]ves = ! lett+ 9 ( air1 is op5$ by ! fir/ /roke ( ! ramm]1 ! air )9 do? ) equal =ce resi/ ! ent]+ ( ! air f )\t2 s t ! endeav\rs betwe5 ! int]nal & ext]nal air are opposite1 t is1 !re are two opposite motions :il/ ! "o goe? 9 & ! o!r come? out2 b no augm5tation at all ( air )9 ! cannon4 ,= !re is driv5 out by ! /roke z mu* pure air1 :i* passe? betwe5 ! ramm] & ! sides ( ! cannon1 z !re is =c$ 9 ( air impure by ! same /roke4 ,& ?us1 by _m =cible /rokes1 ! quantity ( small h>d bodies w be increas$ )9 ! cannon1 & _! motions also w gr[ /rong] & /rong]1 z long z ! matt] ( ! cannon is able to endure _! =ce2 by :i*1 if x be n brok51 x w at lea/ be urg$ e way by _! endeav\r to free !mselves2 & z soon z ! valve1 :i* s]ves to let !m out1 is op5$1 !y w fly out ) viol5t motion1 & c>ry ) !m ! bullet :i* is 9 _! way4 ,:]e=e1 ,i h giv5 a possible cause ( ?is ph5om5on4 ,sid5ote3 ,! cause ( ! asc5t ( wat] 9 a wea!r-glass4 #ab4 ,wat]1 contr>y to ! cu/om ( heavy bodies1 asc5de? 9 ! wea!r-glass2 b x do? x :5 ! air is cold3 = :5 x is w>m x desc5de? aga94 ,& ?is organ is call$ a !rmomet] or !rmoscope1 because ! degrees ( heat & cold are measur$ & m>k$ by x4 ,x is made 9 ?is mann]4 ,let ,a ,b ,c ,d 79 fig4 #e7 be a vessel full ( wat]1 & ,e ,f ,g a holl[ cyl9d] ( glass1 clos$ at ,e & op5 at ,g4 ,let x be heat$1 & set upright )9 ! wat] to ,f2 & let ! op5 end rea* to ,g4 ,?is be+ done1 z ! air by little & little gr[s cold]1 ! wat] w asc5d sl[ly )9 ! cyl9d] f ,f t[>ds ,e2 till at la/ ! ext]nal & int]nal air com+ to be bo? ( ! same temp]1 x w nei!r asc5d high] nor desc5d l[]1 till ! temp] ( ! air be *ang$4 ,suppose x1 !re=e1 to be settl$ any:]e1 z at ,h4 ,if n[ ! heat ( ! air be augm5t$1 ! wat] w desc5d bel[ ,h2 & if ! heat be dim9i%$1 x w asc5d above x4 ,:i*1 ?\gh x be c]ta9ly kn[n to be true by exp]i5ce1 ! cause1 nev]!less1 ha? n z yet be5 discov]$4 ,9 ! six? & sev5? articles ( *apt] ,,xxviii1 :]e ,i consid] ! cause ( cold1 ,i h %[n1 t fluid bodies are made cold] by ! pressure ( ! air1 t is to say1 by a con/ant w9d t presse? !m4 ,= ! same cause x is1 t ! sup]ficies ( ! wat] is press$ at ,f2 & hav+ no place1 to :i* x may retire f ?is pressure1 besides ! cavity ( ! cyl9d] betwe5 ,h & ,e1 x is !re=e necess>ily =c$ ?i!r by ! cold1 & consequ5tly x asc5de? m or less1 accord+ z ! cold is m or less increas$4 ,& aga91 z ! heat is m int5se or ! cold m remiss1 ! same wat] w be depress$ m or less by its own gravity1 t is to say1 by ! cause ( gravity above explicat$4 ,sid5ote3 ,cause ( motion upw>ds 9 liv+ creatures #ac4 ,also liv+ creatures1 ?\gh !y be heavy1 c by leap+1 swimm+ & fly+1 raise !mselves to a c]ta9 degree ( height4 ,b !y _c d ?is except !y be support$ by "s resi/+ body1 z ! e>?1 ! wat] & ! air4 ,= ^! motions h _! beg9n+ f ! contraction1 by ! help ( ! muscles1 ( ! body animate4 ,= to ?is contraction !re succe$e? a di/5sion ( _! :ole bodies2 by :i* di/5sion1 ! e>?1 ! wat]1 or ! air1 :i* supporte? !m1 is press$2 & f h5ce1 by ! reaction ( ^? press$ bodies1 liv+ creatures acquire an endeav\r upw>ds1 b su* z by reason ( ! gravity ( _! bodies is pres5tly lo/ aga94 ,by ?is endeav\r1 !re=e1 x is1 t liv+ creatures raise !mselves up a little way by leap+1 b to no great purpose3 b by swimm+ & fly+ !y raise !mselves to a great] height2 because1 be=e ! effect ( _! endeav\r is q ext+ui%$ by ! gravity ( _! bodies1 !y c r5ew ! same endeav\r aga94 ,t by ! p[] ( ! s\l1 )\t any antec$5t contraction ( ! muscles or ! help ( some?+ to support him1 any man c be able to raise 8 body upw>ds1 is a *ildi% conceit4 ,= if x 7 true1 a man might raise himself to :at height he pleas$4 ,sid5ote3 ,t !re is 9 nature a k9d ( body heavi] ?an air1 :i* nev]!less is n by s5se di/+ui%able f x4 #ad4 ,! diaphan\s m$ium1 :i* surr\nds ! eye on all sides1 is invisible2 nor is air to be se5 9 air1 nor wat] 9 wat]1 nor any?+ b t :i* is m opac\s4 ,b 9 ! conf9es ( two diaphan\s bodies1 "o ( !m may be di/+ui%$ f ! o!r4 ,x is n !re=e a ?+ s v ridicul\s = ord9>y p to ?9k all t space empty1 9 :i* we say is air2 x be+ ! "w ( reason to make u conceive t ! air is any?+4 ,= by :i* ( our s5ses is x1 t we take notice ( ! air1 see+ we nei!r see1 nor he>1 nor ta/e1 nor smell1 nor feel x to be any?+8 ,:5 we feel heat1 we d n impute x to ! air1 b to ! fire3 nor d we say ! air is cold1 b we ourselves are cold2 & :5 we feel ! w9d1 we r ?9k some?+ is com+1 ?an t any ?+ is already come4 ,also1 we d n at all feel ! weight ( wat] 9 wat]1 mu* less ( air 9 air4 ,t we come to "k t to be a body1 :i* we call air1 x is by reason+2 b x is f "o reason only1 namely1 because x is impossible = remote bodies to "w ^u our organs ( s5se b by ! help ( bodies int]m$iate1 )\t :i* we c\ld h no s5se ( !m1 till !y come to be contigu\s4 ,:]e=e1 f ! s5ses alone1 )\t reason+ f effects1 we _c h suffici5t evid5ce ( ! nature ( bodies4 ,= !re is und]gr\nd1 9 "s m9es ( coals1 a c]ta9 matt] ( a middle nature betwe5 wat] & air1 :i* nev]!less _c by s5se be di/+ui%$ f air2 = x is z diaphan\s z ! pure/ air2 &1 z f> z s5se c judge1 equally p5etrable4 ,b if we look ^u ! effect1 x is l t ( wat]4 ,= :5 t matt] breaks out ( ! e>? into "o ( ^? pits1 x fills ! same ei!r totally or to "s degree2 & if a man or fire be !n let d[n 9 x1 x ext+ui%es !m 9 almo/ z little "t z wat] w\ld d4 ,b = ! bett] und]/&+ ( ?is ph5om5on1 ,i %all describe ! #f? figure4 ,9 :i* let ,a ,b repres5t ! pit ( ! m9e2 & let "p !re(1 namely ,c ,b1 be suppos$ to be fill$ ) t matt]4 ,if n[ a light$ c&le be let d[n into x bel[ ,c1 x w z sudd5ly be ext+ui%$ z if x 7 ?ru/ into wat]4 ,also1 if a grate fill$ ) coals ?or\ghly k9dl$ & burn+ nev] s brightly1 be let d[n1 z soon z ev] x is bel[ ,c1 ! fire w beg9 to gr[ pale1 & %ortly aft]1 los+ its light1 be ext+ui%$1 no o!rwise ?an if x 7 qu5*$ 9 wat]4 ,b if ! grate be drawn up aga9 pres5tly1 :il/ ! coals are /ill v hot1 ! fire w1 by little & little1 be k9dl$ aga91 & %9e z be=e4 ,!re is1 inde$1 betwe5 ?is matt] & wat] ?is consid]able diff]5ce1 t x nei!r wette?1 nor /icke? to su* ?+s z are put d[n into x1 z wat] do?2 :i*1 by ! moi/ure x leave?1 h9d]e? ! k9dl+ aga9 ( ! matt] once ext+ui%$4 ,9 l mann]1 if a man be let d[n bel[ ,c1 he w pres5tly fall into a great difficulty ( brea?+1 & imm$iately aft] into a swoon1 & die unless he be sudd5ly drawn up aga94 ,!y1 !re=e1 t g d[n into ^! pits1 h ?is cu/om1 t z soon z ev] !y feel !mselves sick1 !y %ake ! rope by :i* !y 7 let d[n1 to signify !y are n well1 & to ! end t !y may spe$ily be pull$ up aga94 ,= if a man be drawn out too late1 void ( s5se & motion1 !y dig up a turf1 & put 8 face & m\? into ! fre% e>?2 by :i* means1 unless he be q dead1 he comes to himself aga91 by little & little1 & recov]s life by brea?+ out1 z x 71 ( t suffocat+ matt]1 :i* he _h suck$ 9 :il/ he 0 9 ! pit2 almo/ 9 ! same mann] z !y t are dr[n$ come to !mselves aga9 by vomit+ up ! wat]4 ,b ?is do? n happ5 9 all m9es1 b 9 "s only2 & 9 ^? n always1 b (t54 ,9 su* pits z are subject to x1 !y use ?is rem$y4 ,!y dig ano!r pit1 z ,d ,e1 close by x1 ( equal dep?1 & jo9+ !m bo? toge!r ) "o common *annel1 ,e ,b1 !y make a fire 9 ! bottom ,e1 :i* c>ries out at ,d ! air conta9$ 9 ! pit ,d ,e2 & ?is draws ) x ! air conta9$ 9 ! *annel ,e ,b2 :i*1 9 l mann]1 is foll[$ by ! noxi\s matt] conta9$ 9 ,c ,b2 &1 by ?is means1 ! pit is = t "t made heal?ful4 ,out ( ?is hi/ory1 :i* ,i write only to su* z h _h exp]i5ce ( ! tru? ( x1 )\t any design to support my philosophy ) /ories ( d\btful cr$it1 may be collect$ ! foll[+ possible cause ( ?is ph5om5on2 namely1 t !re is a c]ta9 matt] fluid & mo/ transp>5t1 & n mu* light] ?an wat]1 :i*1 break+ out ( ! e>?1 fills ! pit to ,c2 & t 9 ?is matt]1 z 9 wat]1 bo? fire & liv+ creatures are ext+ui%$4 ,sid5ote3 ,( ! cause ( magnetical virtue4 #ae4 ,ab\t ! nature ( heavy bodies1 ! greate/ difficulty arise? f ! contemplation ( ^? ?+s :i* make o!r heavy bodies asc5d to !m2 su* z jet1 amb]1 & ! load/one4 ,b t :i* tr\bles m5 mo/ is ! load/one1 :i* is also call$ ,lapis ,h]culeus2 a /one1 ?\gh o!rwise despicable1 yet ( s great p[] t x take? up iron f ! e>?1 & holds x susp5d$ 9 ! air1 z ,h]cules did ,antu4 ,nev]!less1 we wond] at x some:at ! less1 because we see jet draw up /raws1 :i* are heavy bodies1 ?\gh n s heavy z iron4 ,b z = jet1 x mu/ fir/ be excit$ by rubb+1 t is to say1 by motion to & fro2 :]eas ! load/one ha? suffici5t excitation f its own nature1 t is to say1 f "s int]nal pr9ciple ( motion peculi> to itself4 ,n[1 :atsoev] is mov$1 is mov$ by "s contigu\s & mov$ body1 z ha? be5 =m]ly demon/rat$4 ,& f h5ce x foll[s evid5tly1 t ! fir/ endeav\r1 :i* iron ha? t[>ds ! load/one1 is caus$ by ! motion ( t air :i* is contigu\s to ! iron3 also1 t ?is motion is g5]at$ by ! motion ( ! next air1 & s on successively1 till by ?is succession we f9d t ! motion ( all ! int]m$iate air take? its beg9n+ f "s motion :i* is 9 ! load/one itself2 :i* motion1 because ! load/one seems to be at re/1 is invisible4 ,x is !re=e c]ta91 t ! attractive p[] ( ! load/one is no?+ else b "s motion ( ! smalle/ p>ticles !re(4 ,suppos+1 !re=e1 t ^? small bodies1 ( :i* ! load/one is 9 ! b[els ( ! e>? compos$1 h by nature su* motion or endeav\r z 0 above attribut$ to jet1 namely1 a reciprocal motion 9 a l9e too %ort to be se51 bo? ^? /ones w h "o & ! same cause ( attraction4 ,n[ 9 :at mann] & 9 :at ord] ( work+ ?is cause produce? ! effect ( attraction1 is ! ?+ to be enquir$4 ,& fir/ we "k1 t :5 ! /r+ ( a lute or viol is /rick51 ! vibration1 t is1 ! reciprocal motion ( t /r+ 9 ! same /rait l9e1 cause? l vibration 9 ano!r /r+ :i* ha? l t5sion4 ,we "k also1 t ! dregs or small s&s1 :i* s9k to ! bottom ( a vessel1 w be rais$ up f ! bottom by any /rong & reciprocal agitation ( ! wat]1 /irr$ ) ! h& or ) a /aff4 ,:y1 !re=e1 %\ld n reciprocal motion ( ! p>ts ( ! load/one contribute z mu* t[>ds ! mov+ ( iron8 ,=1 if 9 ! load/one !re be suppos$ su* reciprocal motion1 or motion ( ! p>ts =w>ds & backw>ds1 x w foll[ t ! l motion w be propagat$ by ! air to ! iron1 & consequ5tly t !re w be 9 all ! p>ts ( ! iron ! same reciprocations or motions =w>ds & backw>ds4 ,& f h5ce also x w foll[1 t ! int]m$iate air betwe5 ! /one & ! iron w1 by little & little1 be ?ru/ away2 & ! air be+ ?ru/ away1 ! bodies ( ! load/one & ! iron w necess>ily come toge!r4 ,! possible cause !re=e :y ! load/one & jet draw to !m1 ! "o iron1 ! o!r /raws1 may be ?is1 t ^? attract+ bodies h reciprocal motion ei!r 9 a /rait l9e1 or 9 an elliptical l9e1 :5 !re is no?+ 9 ! nature ( ! attract$ bodies :i* is repugnant to su* a motion4 ,b :y ! load/one1 if ) ! help ( cork x float at lib]ty ^u ! top ( ! wat]1 %\ld f any position :atsoev] s place itself 9 ! plane ( ! m]idian1 z t ! same po9ts1 :i* at "o "t ( its be+ at re/ respect ! poles ( ! e>?1 %\ld at all o!r times respect ! same poles1 ! cause may be ?is2 t ! reciprocal motion1 :i* ,i suppos$ to be 9 ! p>ts ( ! /one1 is made 9 a l9e p>allel to ! axis ( ! e>?1 & has be5 9 ^? p>ts ev] s9ce ! /one 0 g5]at$4 ,see+ !re=e1 ! /one1 :il/ x rema9s 9 ! m9e1 & is c>ri$ ab\t toge!r ) ! e>? by its diurnal motion1 do? by l5g? ( "t get a habit ( be+ mov$ 9 a l9e :i* is p]p5dicul> to ! l9e ( its reciprocal motion1 x w aft]w>ds1 ?\gh its axis be remov$ f ! p>allel situation x _h ) ! axis ( ! e>?1 reta9 its endeav\r ( return+ to t situation aga92 & all endeav\r be+ ! beg9n+ ( motion1 & no?+ int]v5+ t may h9d] ! same1 ! load/one w !re=e return to its =m] situation4 ,=1 any piece ( iron t has = a long "t re/$ 9 ! plane ( ! m]idian1 :5soev] x is =c$ f t situation & aft]w>ds left to its own lib]ty aga91 w ( itself return to lie 9 ! m]idian aga92 :i* return is caus$ by ! endeav\r x acquir$ f ! diurnal motion ( ! e>? 9 ! p>allel circles :i* are p]p5dicul> to ! m]idians4 ,if iron be rubb$ by ! load/one drawn f "o pole to ! o!r1 two ?+s w happ52 "o1 t ! iron w acquire ! same direction ) ! load/one1 t is to say1 t x w lie 9 ! m]idian1 & h its axis & poles 9 ! same position ) ^? ( ! /one2 ! o!r1 t ! l poles ( ! /one & ( ! iron w avoid "o ano!r1 & ! unlike poles approa* "o ano!r4 ,& ! cause ( ! =m] may be ?is1 t iron be+ t\*$ by motion :i* is n reciprocal1 b drawn ! same way f pole to pole1 !re w be impr9t$ 9 ! iron also an endeav\r f ! same pole to ! same pole4 ,= see+ ! load/one diff]s f iron no o!rwise ?an z ore f metal1 !re w be no repugnance at all 9 ! iron to receive ! same motion :i* is 9 ! /one4 ,f :5ce x foll[s1 t see+ !y are bo? affect$ alike by ! diurnal motion ( ! e>?1 !y w bo? equally return to _! situation 9 ! m]idian1 :5soev] !y are put f ! same4 ,also1 ( ! latt] ?is may be ! cause1 t z ! load/one 9 t\*+ ! iron do? by its action impr9t 9 ! iron an endeav\r t[>ds "o ( ! poles1 suppose t[>ds ! ,nor? ,pole2 s reciprocally1 ! iron by its action ^u ! load/one do? impr9t 9 x an endeav\r t[>ds ! o!r pole1 namely t[>ds ! ,s\? ,pole4 ,x happ5s !re=e 9 ^! reciprocations or motions =w>ds & backw>ds ( ! p>ticles ( ! /one & ( ! iron betwixt ! nor? & ! s\?1 t :il/ 9 "o ( !m ! motion is f nor? to s\?1 & ! return f s\? to nor?1 9 ! o!r ! motion w be f s\? to nor?1 & ! return f nor? to s\?2 :i* motions be+ opposite to "o ano!r1 & communicat$ to ! air1 ! nor? pole ( ! iron1 :il/ ! attraction is work+1 w be depress$ t[>ds ! s\? pole ( ! load/one2 or contr>ily1 ! nor? pole ( ! load/one w be depress$ t[>ds ! s\? pole ( ! iron2 & ! axis bo? ( ! load/one & ( ! iron w be situate 9 ! same /rait l9e4 ,! tru? :]e( is taught u by exp]i5ce4 ,z = ! propagation ( ?is magnetical virtue1 n only ?r\gh ! air1 b ?r\gh any o!r bodies h[ h>d soev]1 x is n to be wond]$ at1 see+ no motion c be s weak1 b t x may be propagat$ inf9itely ?r\gh a space fill$ ) body ( any h>dness :atsoev]4 ,= 9 a full m$ium1 !re c be no motion :i* do? n make ! next "p yield1 & t ! next1 & s successively )\t end2 s t !re is no effect :atsoev]1 b to ! production !re( some?+ is necess>ily contribut$ by ! sev]al motions ( all ! sev]al ?+s t are 9 ! _w4 ,sid5ote3 ,conclusion4 ,& ?us mu* conc]n+ ! nature ( body 9 g5]al2 ) :i* ,i conclude ?is my fir/ section ( ! ,elem5ts ( ,philosophy4 ,9 ! fir/1 second1 & ?ird p>ts1 :]e ! pr9ciples ( ratioc9ation consi/ 9 our own und]/&+1 t is to say1 9 ! legitimate use ( su* words z we ourselves con/itute1 all ! !orems1 if ,i be n deceiv$1 are rightly demon/rat$4 ,! f\r? "p dep5ds ^u hypo!ses2 :i* unless we "k !m to be true1 x is impossible = u to demon/rate t ^? causes1 :i* ,i h !re explicat$1 are ! true causes ( ! ?+s ^: productions ,i h d]iv$ f !m4 ,nev]!less1 see+ ,i h assum$ no hypo!sis1 :i* is n bo? possible & easy to be compreh5d$2 & see+ also t ,i h reason$ aright f ^? assumptions1 ,i h )al suffici5tly demon/rat$ t !y may be ! true causes2 :i* is ! end ( physical contemplation4 ,if any o!r man f o!r hypo!ses %all demon/rate ! same or great] ?+s1 !re w be great] praise & ?anks due to him ?an ,i dem& = myself1 provid$ 8 hypo!ses be su* z are conceivable4 ,= z = ^? t say any?+ may be mov$ or produc$ by itself1 by species1 by its own p[]1 by sub/antial =ms1 by incorporeal sub/ances1 by in/9ct1 by antip]i/asis1 by antipa?y1 sympa?y1 occult quality1 & o!r empty words ( s*oolm51 _! say+ s is to no purpose4 ,& n[ ,i proce$ to ! ph5om5a ( man's body2 :]e ,i %all speak ( ! optics1 & ( ! dispositions1 affections1 & mann]s ( m51 if x %all please ,god to give me life1 & %[ _! causes4 ,illu/ration3 ,vol4 #a4 ,lat4 ,eng4 ,c4 ,,xxviii ,,xxx4 ,fig4 #a-f ,,end ,,( ,,vol4 ,i4 ,,london3 ,c4 ,,ri*>ds1 ,,/4 ,,m>t9',s ,,lane1 ,,*>+ ,,cross4 ---------------------------------------- -------------------------------- ,transcrib]'s ,note ,figures #ah4b & #bg4b are n m5tion$ by "n 9 ! text4 ,figure #ad4aj is m5tion$ only 9 ,*apt] ,,xiii4 ,h[ev]1 ,figure #b ( ,*apt] ,,xviii w\ld appe> to be ! subject ( a discussion on p4 #bgj4 ,a sid5ote has be5 add$1 & includes a ref]5ce to ! figure4 ,likewise ,figure #b ( ,*apt] ,,xxvii1 depict+ a #8prisma'1 is discuss$ 9 detail on p4 #dei et seq44 ,aga91 a sid5ote has be5 int]polat$ at ! op5+ ( t description1 & includes a ref]5ce to ! figure4 ,! diacritics us$ 9 ! ,greek phrases is foll[$ z pr9t$1 ?\gh !y sometimes seem non-/&>d 7i4e41 r ?an 74 ,9 ea* *apt]'s summ>y1 ! article numb]s are prefix$ ) a da% if !y appe>$ midl9e4 ,da%es h be5 add$ to ! text 9 ! case :]e ! article numb] happ5$ to appe> 9 ! fir/ column4 ,on r>e occasions1 !re are errors 9 ! numb]+ ( ! articles 9 ! *apt] summ>y4 ,= example1 9 ,*apt] ,,xvii1 article #g 0 misnumb]$ z a second #8#f'4 ,9 article #ah ( ,*apt] ,,xiii1 a ref]5ce to article #aj ( t *apt] us$ ! ,roman #8,x' :i* is ! only in/ance ( t usage4 ,x is n[ ref]r$ to z article #aj4 ,o!r errors deem$ mo/ likely to be ! pr9t]'s h be5 correct$1 & are not$ "h4 ,! ref]5ces are to ! page & l9e 9 ! orig9al4 #eb4ag e man is a liv+ creature ,re/or$4 #gc4ad 9 ! fir/ place1 we we are to se>* ,remov$4 #bee4aa see+ ! proproportion ,remov$4 #bfg4bb ,& !re=e if x x be aga9 z ,r to ,t ,remov$4 #cjg4bd ! spiral ( ,ar*i_/emides ,replac$4 #cbi4bi ) simple motion ! ! p]imet] ,remov$4 #cci4aa be cut 9 ,e & & ,g by a /rait l9e ,remov$4 #cgb4ab to ! base ,f ,h t t c possibly be ,remov$4 drawn #dfd4bc ! sea appe>s black :5 ruffl$ ,ins]t$4 #dci4b w make fifty-n9e revolutions ,ins]t$4 ,*apt] ,,xiv ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,illu/ration3 ,fig4 #c4 ,illu/ration3 ,fig4 #d4 ,illu/ration3 ,fig4 #e4 ,illu/ration3 ,fig4 #f4 ,illu/ration3 ,fig4 #g4 ,illu/ration3 ,fig4 #h4 ,illu/ration3 ,fig4 #i4 ,illu/ration3 ,fig4 #aj4 ,,*apt] ,,xvi ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,illu/ration3 ,fig4 #c4 ,illu/ration3 ,fig4 #d4 ,illu/ration3 ,fig4 #e4 ,illu/ration3 ,fig4 #f4 ,illu/ration3 ,fig4 #g4 ,illu/ration3 ,fig4 #h4 ,illu/ration3 ,fig4 #i4 ,illu/ration3 ,fig4 #aj4 ,illu/ration3 ,fig4 #aa4 ,*apt] ,,xvii ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,illu/ration3 ,fig4 #c4 ,illu/ration3 ,fig4 #d4 ,illu/ration3 ,fig4 #e4 ,illu/ration3 ,fig4 #f4 ,illu/ration3 ,fig4 #g4 ,illu/ration3 ,fig4 #h4 ,*apt] ,,xviii ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,*apt] ,,xix ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,illu/ration3 ,fig4 #c4 ,illu/ration3 ,fig4 #d4 ,illu/ration3 ,fig4 #e4 ,illu/ration3 ,fig4 #f4 ,illu/ration3 ,fig4 #g4 ,illu/ration3 ,fig4 #h4 ,illu/ration3 ,fig4 #i4 ,illu/ration3 ,fig4 #aj4 ,*apt] ,,xx ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,illu/ration3 ,fig4 #c4 ,illu/ration3 ,fig4 #d4 ,illu/ration3 ,fig4 #e4 ,illu/ration3 ,fig4 #f4 ,*apt] ,,xxi ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,illu/ration3 ,fig4 #c4 ,illu/ration3 ,fig4 #d4 ,illu/ration3 ,fig4 #e4 ,*apt] ,,xxii ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b4 ,illu/ration3 ,fig4 #c4 ,*apt] ,,xxiii ,illu/ration3 ,fig4 #a4 ,illu/ration3 ,fig4 #b ,illu/ration3 ,fig4 #c4 ,illu/ration3 ,fig4 #d ,illu/ration3 ,fig4 #e4 ,illu/ration3 ,fig4 #f ,illu/ration3 ,fig4 #g4 ,illu/ration3 ,fig4 #h ,illu/ration3 ,fig #i4 ,,*apt] ,,xxiv4 ,illu/ration3 ,fig #a4 ,illu/ration3 ,fig #b4 ,illu/ration3 ,fig #c4 ,illu/ration3 ,fig #d4 ,illu/ration3 ,fig #e4 ,illu/ration3 ,fig #f4 ,,*apt] ,,xxvi4 ,illu/ration3 ,fig #a4 ,illu/ration3 ,fig #b4 ,illu/ration3 ,fig #c4 ,illu/ration3 ,fig #d4 ,illu/ration3 ,fig #e4 ,*apt] ,,xxvii4 ,illu/ration3 ,fig #a4 ,illu/ration3 ,fig #b4 ,*apt] ,,xxx4 ,illu/ration3 ,fig #a4 ,illu/ration3 ,fig #b4 ,illu/ration3 ,fig #c4 ,illu/ration3 ,fig #d4 ,illu/ration3 ,fig #e4 ,illu/ration3 ,fig #f4