---------------------------------------- ---------------------------- ,transcrib]'s ,note ,9 ! orig9al book ! au?or us$ l[]case italic lett]s & l[]case small capitals to label ! geometric diagrams4 ,^! h be5 "h transcrib$ z l[]case & upp]case italic lett]s respectively 7a,a74 ,?is file %\ld be read us+ a font t supports ! foll[+ ,unicode *>act]s3 angle1 ratio1 proportion1 !re=e1 prime1 & d\ble prime4 ---------------------------------------- ---------------------------- ---------------------------------- ,libr>y ,edition ---------------------------------- ,,! ,,complete ,,works ,,( ,,john ,,rusk9 ,,elem5ts ,,( ,,draw+ ,,& ,,p]spective ,,! ,,two ,,pa?s ,,unto ,,?is ,,la/ ,,mun]a ,,pulv]is ,,sesame ,,& ,,lilies ,,e?ics ,,( ,,! ,,du/ ---------------------------------- ,,national ,,libr>y ,,association ,,new ,,york ,,*icago ---------------------------------- ,,! ,,elem5ts ,,( ,,p]spective ,,arrang$ ,,= ,,! ,,use ,,( ,,s*ools ,,& ,,int5d$ ,,to ,,be ,,read ,,9 ,,connection ,,) ,,! ,,fir/ ,,?ree ,,books ,,( ,,euclid4 ,,cont5ts4 ,,page ,preface ix ,introduction #a ,,problem ,i4 ,to fix ! ,position ( a giv5 ,po9t #aj ,,problem ,,ii4 ,to draw a ,"r ,l9e betwe5 two giv5 ,po9ts #ac ,,problem ,,iii4 ,to f9d ! ,vani%+-,po9t ( a giv5 ,horizontal ,l9e #ag ,,problem ,,iv4 ,to f9d ! ,divid+-,po9ts ( a giv5 ,horizontal ,l9e #bc ,,problem ,v4 ,to draw a ,horizontal ,l9e1 giv5 9 ,position & ,magnitude1 by means ( its ,sight-,magnitude & ,divid+-,po9ts #bd ,,problem ,,vi4 ,to draw any ,triangle1 giv5 9 ,position & ,magnitude1 9 a ,horizontal ,plane #bg ,,problem ,,vii4 ,to draw any ,rectil9e> ,quadrilat]al ,figure1 giv5 9 ,position & ,magnitude1 9 a ,horizontal ,plane #bi ,,problem ,,viii4 ,to draw a ,squ>e1 giv5 9 ,position & ,magnitude1 9 a ,horizontal ,plane #ca ,,problem ,,ix4 ,to draw a ,squ>e ,pill>1 giv5 9 ,position & ,magnitude1 its ,base & ,top be+ 9 ,horizontal ,planes #cd ,,problem ,x4 ,to draw a ,pyramid1 giv5 9 ,position & ,magnitude1 on a ,squ>e ,base 9 a ,horizontal ,plane #cf ,,problem ,,xi4 ,to draw any ,curve 9 a ,horizontal or ,v]tical ,plane #ch ,,problem ,,xii4 ,to divide a ,circle drawn 9 ,p]spective into any giv5 ,numb] ( ,equal ,p>ts #db ,,problem ,,xiii4 ,to draw a ,squ>e1 giv5 9 ,magnitude1 )9 a l>g] ,squ>e giv5 9 ,position & ,magnitude2 ! ,sides ( ! two ,squ>es be+ ,p>allel #de ,,problem ,,xiv4 ,to draw a ,truncat$ ,circul> ,cone1 giv5 9 ,position & ,magnitude1 ! ,truncations be+ 9 ,horizontal ,planes1 & ! ,axis ( ! ,cone v]tical #dg ,,problem ,,xv4 ,to draw an ,incl9$ ,l9e1 giv5 9 ,position & ,magnitude #ej ,,problem ,,xvi4 ,to f9d ! ,vani%+-,po9t ( a giv5 ,incl9$ ,l9e #ec ,,problem ,,xvii4 ,to f9d ! ,divid+-,po9ts ( a giv5 ,incl9$ ,l9e #ee ,,problem ,,xviii4 ,to f9d ! ,sight-,l9e ( an ,incl9$ ,plane 9 :i* ,two ,l9es are giv5 9 ,position #eg ,,problem ,,xix4 ,to f9d ! ,vani%+-,po9t ( ,/eepe/ ,l9es 9 an ,incl9$ ,plane ^: ,sight-,l9e is giv5 #ei ,,problem ,,xx4 ,to f9d ! ,vani%+-,po9t ( ,l9es p]p5dicul> to ! ,surface ( a giv5 ,incl9$ ,plane #fa ,,app5dix4 ,i4 ,practice & ,obs]vations on ! prec$+ ,problems #fi ,,ii4 ,demon/rations :i* c\ld n conv5i5tly be includ$ 9 ! ,text #ii ,,preface4 ,= "s "t back ,i h felt ! want1 among ,/ud5ts ( ,draw+1 ( a writt5 code ( accurate ,p]spective ,law2 ! modes ( con/ruction 9 common use be+ v>i\s1 &1 = "s problems1 insuffici5t4 ,x w\ld h be5 desirable to draw up su* a code 9 popul> language1 s z to d away ) ! mo/ repulsive difficulties ( ! subject2 b f9d+ ?is popul>ization w\ld be impossible1 )\t elaborate figures & long explanations1 su* z ,i _h no leisure to prep>e1 ,i h arrang$ ! necess>y rules 9 a %ort ma!matical =m1 :i* any s*oolboy may read ?r\gh 9 a few days1 aft] he has ma/]$ ! fir/ ?ree & ! six? books ( ,euclid4 ,"s awkw>d compromises h be5 admitt$ betwe5 ! fir/-attempt$ popul> explanation1 & ! sev]] arrangem5t1 involv+ irregul> lett]+ & r$undant phraseology2 b ,i _c = ! pres5t d m1 & leave ! book !re=e to its trial1 hop+ t1 if x be f\nd by ma/]s ( s*ools to answ] its purpose1 ,i may h]eaft] br+ x into bett] =m4#a ,an acc\nt ( practical me?ods1 suffici5t = g5]al purposes ( sket*+1 might inde$ h be5 set d[n 9 mu* less space3 b if ! /ud5t reads ! foll[+ pages c>efully1 he w n only f9d himself able1 on occasion1 to solve p]spective problems ( a complexity great] ?an ! ord9>y rules w rea*1 b obta9 a clue to _m important laws ( pictorial effect1 no less ?an ( outl9e4 ,! subject ?us exam9$ becomes1 at lea/ to my m9d1 v curi\s & int]e/+2 b1 = /ud5ts :o are unable or unwill+ to take x up 9 ?is ab/ract =m1 ,i believe good help w be soon furni%$1 9 a s]ies ( illu/rations ( practical p]spective n[ 9 prep>ation by ,mr4 ,le ,v5geur4 ,i h n se5 ?is essay 9 an advanc$ /ate1 b ! illu/rations %[n to me 7 v cle> & good2 &1 z ! au?or has devot$ mu* ?\ght to _! arrangem5t1 ,i hope t 8 "w w be precisely :at is want$ by ! g5]al le>n]4 ,/ud5ts wi%+ to pursue ! subject into its m ext5d$ bran*es w f9d1 ,i believe1 ,cloquet's treatise ! be/ hi!rto publi%$4#b #a ,"s irregul>ities ( arrangem5t h be5 admitt$ m]ely = ! sake ( conv5i5t ref]5ce2 ! eigh? problem1 = in/ance1 ought to h be5 giv5 z a case ( ! sev5?1 b is sep>ately enunciat$ on acc\nt ( its importance4 ,sev]al con/ructions1 :i* ought to h be5 giv5 z problems1 are on ! contr>y giv5 z coroll>ies1 9 ord] to keep ! m directly connect$ problems 9 clos] sequ5ce2 ?us ! con/ruction ( rectangles & polygons 9 v]tical planes w\ld appe> by ! ,table ( ,cont5ts to h be5 omitt$1 be+ giv5 9 ! coroll>y to ,problem ,,ix4 #b ,n\veau ,trait lm5taire de ,p]spective4 ,ba*eli]1 #ahbc4 ,,! ,,elem5ts ,,( ,,p]spective4 ,,introduction4 ,:5 y beg9 to read ?is book1 sit d[n v ne> ! w9d[1 & %ut ! w9d[4 ,i hope ! view out ( x is pretty2 b1 :atev] ! view may be1 we %all f9d en\gh 9 x = an illu/ration ( ! fir/ pr9ciples ( p]spective 7or1 lit]ally1 ( 8look+ ?r\gh074 ,e pane ( y\r w9d[ may be consid]$1 if y *oose1 z a glass picture2 & :at y see ?r\gh x1 z pa9t$ on its surface4 ,& if1 hold+ y\r head /ill1 y ext5d y\r h& to ! glass1 y may1 ) a bru% full ( any ?ick color1 trace1 r\ghly1 ! l9es ( ! l&scape on ! glass4 ,b1 to d ?is1 y mu/ hold y\r head v /ill4 ,n only y mu/ n move x sideways1 nor up & d[n1 b x mu/ n ev5 move backw>ds or =w>ds2 =1 if y move y\r head =w>ds1 y w see m ( ! l&scape ?r\gh ! pane2 &1 if y move x backw>ds1 y w see less3 or consid]+ ! pane ( glass z a picture1 :5 y hold y\r head ne> x1 ! objects are pa9t$ small1 & a great _m ( !m g into a little space2 b1 :5 y hold y\r head "s di/ance back1 ! objects are pa9t$ l>g] ^u ! pane1 & few] ( !m g into ! field ( x4 ,b1 besides hold+ y\r head /ill1 y mu/1 :5 y try to trace ! picture on ! glass1 %ut "o ( y\r eyes4 ,if y d n1 ! po9t ( ! bru% appe>s d\ble2 &1 on f>!r exp]im5t1 y w obs]ve t ea* ( y\r eyes sees ! object 9 a diff]5t place on ! glass1 s t ! trac+ :i* is true to ! sight ( ! "r eye is a c\ple ( in*es 7or m1 accord+ to y\r di/ance f ! pane17 to ! left ( t :i* is true to ! sight ( ! left4 ,?us1 x is only possible to draw :at y see ?r\gh ! w9d[ rightly on ! surface ( ! glass1 by fix+ "o eye at a giv5 po9t1 & nei!r mov+ x to ! "r nor left1 nor up nor d[n1 nor backw>ds nor =w>ds4 ,e picture drawn 9 true p]spective may be consid]$ z an upright piece ( glass1#c on :i* ! objects se5 ?r\gh x h be5 ?us drawn4 ,p]spective c1 !re=e1 only be q "r1 by be+ calculat$ = "o fix$ position ( ! eye ( ! obs]v]2 nor w x ev] appe> deceptively "r unless se5 precisely f ! po9t x is calculat$ =4 ,cu/om1 h[ev]1 enables u to feel ! rightness ( ! "w on us+ bo? our eyes1 & to be satisfi$ ) x1 ev5 :5 we /& at "s di/ance f ! po9t x is design$ =4 ,suppos+ t1 in/ead ( a w9d[1 an unbrok5 plate ( cry/al ext5d$ itself to ! "r & left ( y1 & high 9 front1 & t y _h a bru% z long z y want$ 7a mile long1 suppose71 & c\ld pa9t ) su* a bru%1 !n ! cl\ds high up1 ne>ly ov] y\r head1 & ! l&scape f> away to ! "r & left1 might be trac$1 & pa9t$1 on ?is enorm\s cry/al field4#d ,b if ! field 7 s va/ 7suppose a mile high & a mile wide71 c]ta9ly1 aft] ! picture 0 done1 y w\ld n /& z ne> to x1 to see x1 z y are n[ sitt+ ne> to y\r w9d[4 ,9 ord] to trace ! upp] cl\ds ?r\gh y\r great glass1 y w\ld h _h to /ret* y\r neck q back1 & nobody likes to b5d _! neck back to see ! top ( a picture4 ,s y w\ld walk a long way back to see ! great picture--a qu>t] ( a mile1 p]haps1--& !n all ! p]spective w\ld be wrong1 & w\ld look q di/ort$1 & y w\ld discov] t y ought to h pa9t$ x f ! great] di/ance1 if y meant to look at x f t di/ance4 ,?us1 ! di/ance at :i* y int5d ! obs]v] to /& f a picture1 & = :i* y calculate ! p]spective1 ought to regulate to a c]ta9 degree ! size ( ! picture4 ,if y place ! po9t ( obs]vation ne> ! canvas1 y %\ld n make ! picture v l>ge3 vice v]s1 if y place ! po9t ( obs]vation f> f ! canvas1 y %\ld n make x v small2 ! fix+1 !re=e1 ( ?is po9t ( obs]vation det]m9es1 z a matt] ( conv5i5ce1 )9 c]ta9 limits1 ! size ( y\r picture4 ,b x does n det]m9e ?is size by any p]spective law2 & x is a mi/ake made by _m writ]s on p]spective1 to connect "s ( _! rules def9itely ) ! size ( ! picture4 ,=1 suppose t y _h :at y n[ see ?r\gh y\r w9d[ pa9t$ actually ^u its surface1 x w\ld be q optional to cut out any piece y *ose1 ) ! piece ( ! l&scape t 0 pa9t$ on x4 ,y might h only half a pane1 ) a s+le tree2 or a :ole pane1 ) two trees & a cottage2 or two panes1 ) ! :ole f>my>d & pond2 or f\r panes1 ) f>my>d1 pond1 & =egr\nd4 ,& any ( ^! pieces1 if ! l&scape ^u !m 71 z a sc5e1 pleasantly compos$1 w\ld be agreeable pictures1 ?\gh ( q diff]5t sizes2 & yet !y w\ld be all calculat$ = ! same di/ance ( obs]vation4 ,9 ! foll[+ treatise1 !re=e1 ,i keep ! size ( ! picture entirely undet]m9$4 ,i consid] ! field ( canvas z :olly unlimit$1 & on t condition det]m9e ! p]spective laws4 ,aft] we "k h[ to apply ^? laws )\t limitation1 we %all see :at limitations ( ! size ( ! picture _! results may r5d] advisable4 ,b al?\gh ! size ( ! picture is ?us indep5d5t ( ! obs]v]'s di/ance1 ! size ( ! object repres5t$ 9 ! picture is n4 ,on ! contr>y1 t size is fix$ by absolute ma!matical law2 t is to say1 suppos+ y h to draw a t[] a hundr$ feet high1 & a qu>t] ( a mile di/ant f y1 ! height :i* y ought to give t t[] on y\r pap] dep5ds1 ) ma!matical precision1 on ! di/ance at :i* y int5d y\r pap] to be plac$4 ,s1 also1 d all ! rules = draw+ ! =m ( ! t[]1 :atev] x may be4 ,h5ce1 ! fir/ ?+ to be done 9 beg9n+ a draw+ is to fix1 at y\r *oice1 ?is di/ance ( obs]vation1 or ! di/ance at :i* y mean to /& f y\r pap]4 ,aft] t is det]m9$1 all is det]m9$1 except only ! ultimate size ( y\r picture1 :i* y may make great]1 or less1 n by alt]+ ! size ( ! ?+s repres5t$1 b by tak+ 9 m1 or few] ( !m4 ,s1 !n1 be=e proce$+ to apply any practical p]spective rule1 we mu/ always h our di/ance ( obs]vation m>k$1 & ! mo/ conv5i5t way ( m>k+ x is ! foll[+3 ,illu/ration3 ,fig4 #a4 ,,plac+ ,,( ,,! ,,sight-,,po9t1 ,,sight-,,l9e1 ,,/ation-,,po9t1 ,,& ,,/ation-,,l9e4 ,i4 ,,! ,,sight-,,po9t4--,let ,,abcd1 ,fig4 #a41 be y\r %eet ( pap]1 ! l>g] ! bett]1 ?\gh p]haps we may cut out ( x at la/ only a small piece = our picture1 su* z ! dott$ circle ,,nopq4 ,?is circle is n int5d$ to limit ei!r ! size or %ape ( our picture3 y may ultimately h x r\nd or oval1 horizontal or upright1 small or l>ge1 z y *oose4 ,i only dot ! l9e to give y an idea ( :]eab\ts y w probably l to h x2 &1 z ! op]ations ( p]spective are m conv5i5tly p]=m$ ^u pap] und]nea? ! picture ?an above x1 ,i put ?is conjectural circle at ! top ( ! pap]1 ab\t ! middle ( x1 leav+ pl5ty ( pap] on bo? sides & at ! bottom4 ,n[1 z an obs]v] g5]ally /&s ne> ! middle ( a picture to look at x1 we _h bett] at fir/1 & = simplicity's sake1 fix ! po9t ( obs]vation opposite ! middle ( our conjectural picture4 ,s take ! po9t ,s1 ! c5t] ( ! circle ,,nopq2--or1 :i* w be simpl] = y 9 y\r own "w1 take ! po9t ,s at r&om ne> ! top ( y\r pap]1 & /rike ! circle ,,nopq r\nd x1 any size y l4 ,!n ! po9t ,s is to repres5t ! po9t opposite :i* y wi% ! obs]v] ( y\r picture to place 8 eye1 9 look+ at x4 ,call ?is po9t ! 8,sight-,po9t40 ,,ii4 ,,! ,,sight-,,l9e4--,?r\gh ! ,sight-po9t1 ,s1 draw a horizontal l9e1 ,,gh1 "r across y\r pap] f side to side1 & call ?is l9e ! 8,sight-,l9e40 ,?is l9e is ( great practical use1 repres5t+ ! level ( ! eye ( ! obs]v] all ?r\gh ! picture4 ,y w f9d h]eaft] t if !re is a horizon to be repres5t$ 9 y\r picture1 z ( di/ant sea or pla91 ?is l9e def9es x4 ,,iii4 ,,! ,,/ation-,,l9e4--,f ,s let fall a p]p5dicul> l9e1 ,,sr1 to ! bottom ( ! pap]1 & call ?is l9e ! 8,/ation-,l9e40 ,?is repres5ts ! l9e on :i* ! obs]v] /&s1 at a great] or less di/ance f ! picture2 & x ought to be imag9$ z drawn "r out f ! pap] at ! po9t s4 ,hold y\r pap] upright 9 front ( y1 & hold y\r p5cil horizontally1 ) its po9t aga9/ ! po9t ,s1 z if y want$ to run x ?r\gh ! pap] !re1 & ! p5cil w repres5t ! direction 9 :i* ! l9e ,,sr ought to be drawn4 ,b z all ! measurem5ts :i* we h to set ^u ?is l9e1 & op]ations :i* we h to p]=m ) x1 are j ! same :5 x is drawn on ! pap] itself1 bel[ ,s1 z !y w\ld be if x 7 repres5t$ by a wire 9 ! position ( ! level$ p5cil1 & z !y are mu* m easily p]=m$ :5 x is drawn on ! pap]1 x is always 9 practice1 s drawn4 ,,iv4 ,,! ,,/ation-,,po9t4--,on ?is l9e1 m>k ! di/ance ,,/ at y\r pleasure1 = ! di/ance at :i* y wi% y\r picture to be se51 & call ! po9t ,t ! 8,/ation-,po9t40 ,illu/ration3 ,fig4 #b4 ,9 practice1 x is g5]ally advisable to make ! di/ance ,,/ ab\t z great z ! diamet] ( y\r int5d$ picture2 & x %\ld1 = ! mo/ "p1 be m r ?an less2 b1 z ,i h j /at$1 ?is is q arbitr>y4 ,h[ev]1 9 ?is figure1 z an approximation to a g5]ally advisable di/ance1 ,i make ! di/ance ,,/ equal to ! diamet] ( ! circle ,,nopq4 ,n[1 hav+ fix$ ?is di/ance1 ,,/1 all ! dim5sions ( ! objects 9 our picture are fix$ likewise1 & = ?is reason3-- ,let ! upright l9e ,,ab1 ,fig4 #b41 repres5t a pane ( glass plac$ :]e our picture is to be plac$2 b se5 at ! side ( x1 edgeways2 let ,s be ! ,sight-po9t2 ,,/ ! ,/ation-l9e1 :i*1 9 ?is figure1 obs]ve1 is 9 its true position1 drawn out f ! pap]1 n d[n ^u x2 & ,t ! ,/ation-po9t4 ,suppose ! ,/ation-l9e ,,/ to be cont9u$1 or 9 ma!matical language 8produc$10 ?r\gh ,s1 f> beyond ! pane ( glass1 & let ,,pq be a t[] or o!r upright object situat$ on or above ?is l9e4 ,n[ ! app>5t height ( ! t[] ,,pq is measur$ by ! angle ,,qtp1 betwe5 ! rays ( light :i* come f ! top & bottom ( x to ! eye ( ! obs]v]4 ,b ! actual height ( ! image ( ! t[] on ! pane ( glass ,,ab1 betwe5 u & x1 is ! di/ance ,p,q betwe5 ! po9ts :]e ! rays trav]se ! glass4 ,evid5tly1 ! f>!r f ! po9t ,t we place ! glass1 mak+ ,,/ long]1 ! l>g] w be ! image2 & ! ne>] we place x to ,t1 ! small] ! image1 & t 9 a fix$ ratio4 ,let ! di/ance ,,dt be ! direct di/ance f ! ,/ation-po9t to ! foot ( ! object4 ,!n1 if we place ! glass ,,ab at "o-?ird ( t :ole di/ance1 ,p,q w be "o-?ird ( ! real height ( ! object2 if we place ! glass at two-?irds ( ! di/ance1 z at ,,ef1 ,p,q 7! height ( ! image at t po9t7 w be two-?irds ! height#e ( ! object1 & s on4 ,!re=e ! ma!matical law is t ,p,q w be to ,,pq z ,,/ to ,,dt4 ,i put ?is ratio cle>ly by itself t y may rememb] x3 ,p,q ,,pq ,,/ ,,dt or 9 words3 ,p da% ,q da% is to ,,pq z ,,/ to ,,dt ,9 :i* =mula1 recollect t ,p,q is ! height ( ! appe>ance ( ! object on ! picture2 ,,pq ! height ( ! object itself2 ,s ! ,sight-po9t2 ,t ! ,/ation-po9t2 ,d a po9t at ! direct di/ance ( ! object2 ?\gh ! object is seldom plac$ actually on ! l9e ,,ts produc$1 & may be f> to ! "r or left ( x1 ! =mula is /ill ! same4 ,= let ,s1 ,fig4 #c41 be ! ,sight-po9t1 & ,,ab ! glass--"h se5 look+ d[n on its upp] edge1 n sideways2--!n if ! t[] 7repres5t$ n[1 z on a map1 by ! d>k squ>e71 in/ead ( be+ at ,d on ! l9e ,,/ produc$1 be at ,e1 to ! "r 7or left7 ( ! spectator1 /ill ! app>5t height ( ! t[] on ,,ab w be z ,s,t to ,,et1 :i* is ! same ratio z t ( ,,/ to ,,dt4 ,illu/ration3 ,fig4 #c4 ,n[ 9 _m p]spective problems1 ! position ( an object is m conv5i5tly express$ by ! two measurem5ts ,,dt & ,,de1 ?an by ! s+le oblique measurem5t ,,et4 ,i %all call ,,dt ! 8direct di/ance0 ( ! object at ,e1 & ,,de its 8lat]al di/ance40 ,x is r a lic5se to call ,,dt its 8direct0 di/ance1 = ,,et is ! m direct ( ! two2 b !re is no o!r t]m :i* w\ld n cause confusion4 ,la/ly1 9 ord] to complete our k ( ! position ( an object1 ! v]tical height ( "s po9t 9 x1 above or bel[ ! eye1 mu/ be giv52 t is to say1 ei!r ,,dp or ,,dq 9 ,fig4 #b4f3 ?is ,i %all call ! 8v]tical di/ance0 ( ! po9t giv54 ,9 all p]spective problems ^! ?ree di/ances1 & ! dim5sions ( ! object1 mu/ be /at$1 o!rwise ! problem is imp]fectly giv54 ,x ought n to be requir$ ( u m]ely to draw a room or a *ur* 9 p]spective2 b to draw ?is room f ?is corn]1 & t *ur* on t spot1 9 p]spective4 ,= want ( kn[+ h[ to base _! draw+s on ! measurem5t & place ( ! object1 ,i h kn[n practic$ /ud5ts repres5t a p>i% *ur*1 c]ta9ly 9 true p]spective1 b ) a nave ab\t two miles & a half long4 ,x is true t 9 draw+ l&scapes f nature ! sizes & di/ances ( ! objects _c be accurately kn[n4 ,:51 h[ev]1 we "k h[ to draw !m rightly1 if _! size 7 giv51 we h only to assume a rational approximation to _! size1 & ! result+ draw+ w be true en\gh = all int5ts & purposes4 ,x does n 9 ! lea/ matt] t we repres5t a di/ant cottage z eighte5 feet long1 :5 x is 9 reality only sev5te52 b x matt]s mu* t we d n repres5t x z eighty feet long1 z we easily might if we _h n be5 accu/om$ to draw f measurem5t4 ,!re=e1 9 all ! foll[+ problems ! measurem5t ( ! object is giv54 ,! /ud5t mu/ obs]ve1 h[ev]1 t 9 ord] to br+ ! diagrams into conv5i5t compass1 ! measurem5ts assum$ are g5]ally v diff]5t f any likely to occur 9 practice4 ,?us1 9 ,fig4 #c41 ! di/ance ,,ds w\ld be probably 9 practice half a mile or a mile1 & ! di/ance ,,ts1 f ! eye ( ! obs]v] to ! pap]1 only two or ?ree feet4 ,! ma!matical law is h[ev] precisely ! same1 :atev] ! proportions2 & ,i use su* proportions z are be/ calculat$ to make ! diagram cle>4 ,n[1 !re=e1 ! conditions ( a p]spective problem are ! foll[+3 ,! ,sight-l9e ,,gh giv51 ,fig4 #a42 ,! ,sight-po9t ,s giv52 ,! ,/ation-po9t ,t giv52 & ,! ?ree di/ances ( ! object1#g direct1 lat]al1 & v]tical1 ) its dim5sions1 giv54 ,! size ( ! picture1 conjecturally limit$ by ! dott$ circle1 is to be det]m9$ aft]w>ds at our pleasure4 ,on ^! conditions ,i proce$ at once to con/ruction4 #c ,if ! glass 7 n upright1 b slop+1 ! objects might /ill be drawn ?r\gh x1 b _! p]spective w\ld !n be diff]5t4 ,p]spective1 z commonly taught1 is always calculat$ = a v]tical plane ( picture4 #d ,suppos+ x to h no ?ickness2 o!rwise ! images w\ld be di/ort$ by refraction4 #e ,i say 8height0 in/ead ( 8magnitude10 = a reason /at$ 9 ,app5dix ,i41 to :i* y w soon be ref]r$4 ,read on "h at pres5t4 #f ,p & ,q be+ po9ts indicative ( ! place ( ! t[]'s base & top4 ,9 ?is figure bo? are above ! sight-l9e2 if ! t[] 7 bel[ ! spectator bo? w\ld be bel[ x1 & !re=e measur$ bel[ ,d4 #g ,m accurately1 8! ?ree di/ances ( any po9t1 ei!r 9 ! object itself1 or indicative ( its di/ance40 ,,problem ,i4 ,,to ,,fix ,,! ,,position ,,( ,a ,,giv5 ,,po9t4#h ,let ,p1 ,fig4 #d41 be ! giv5 po9t4 ,illu/ration3 ,fig4 #d4 ,let its direct di/ance be ,,dt2 its lat]al di/ance to ! left1 ,,dc2 & v]tical di/ance b5ea? ! eye ( ! obs]v]1 ,,cp4 ,let ,,gh be ! ,sight-l9e1 ,s ! ,sight-po9t1 & ,t ! ,/ation-po9t4#i ,x is requir$ to fix on ! plane ( ! picture ! position ( ! po9t ,p4 ,arrange ! ?ree di/ances ( ! object on y\r pap]1 z 9 ,fig4 #d4aj ,jo9 ,,ct1 cutt+ ,,gh 9 ,q4 ,f ,q let fall ! v]tical l9e ,,qp4 ,jo9 ,,pt1 cutt+ ,,qp 9 ,p4 ,p is ! po9t requir$4 ,if ! po9t ,p is above ! eye ( ! obs]v] in/ead ( bel[ x1 ,,cp is to be measur$ upw>ds f ,c1 & ,,qp drawn upw>ds f ,q4 ,! con/ruction w be z 9 ,fig4 #e4 ,illu/ration3 ,fig4 #e4 ,& if ! po9t ,p is to ! "r in/ead ( ! left ( ! obs]v]1 ,,dc is to be measur$ to ! "r in/ead ( ! left4 ,! figures #d4 & #e41 look$ at 9 a mirror1 w %[ ! con/ruction ( ea*1 on t supposition4 ,n[ read v c>efully ! examples & notes to ?is problem 9 ,app5dix ,i4 7page #fi74 ,i h put !m 9 ! ,app5dix 9 ord] to keep ! sequ5ce ( foll[+ problems m cle>ly traceable "h 9 ! text2 b y mu/ read ! fir/ ,app5dix be=e go+ on4 #h ,m accurately1 8,to fix on ! plane ( ! picture ! app>5t position ( a po9t giv5 9 actual position40 ,9 ! head+s ( all ! foll[+ problems ! words 8on ! plane ( ! picture0 are to be und]/ood aft] ! words 8to draw40 ,! plane ( ! picture means a surface ext5d$ indef9itely 9 ! direction ( ! picture4 #i ,! s5t5ce )9 brackets w n be repeat$ 9 succe$+ /atem5ts ( problems4 ,x is always to be und]/ood4 #aj ,9 ord] to be able to d ?is1 y mu/ assume ! di/ances to be small2 z 9 ! case ( "s object on ! table3 h[ l>ge di/ances are to be treat$ y w see pres5tly2 ! ma!matical pr9ciple1 be+ ! same = all1 is be/ illu/rat$ fir/ on a small scale4 ,suppose1 = in/ance1 ,p to be ! corn] ( a book on ! table1 sev5 in*es bel[ ! eye1 five in*es to ! left ( x1 & a foot & a half 9 advance ( x1 & t y mean to hold y\r f9i%$ draw+ at six in*es f ! eye2 !n ,,ts w be six in*es1 ,,td a foot & a half1 ,,dc five in*es1 & ,,cp sev54 ,,problem ,,ii4 ,,to ,,draw ,a ,,"r ,,l9e ,,betwe5 ,,two ,,giv5 ,,po9ts4 ,illu/ration3 ,fig4 #f4 ,let ,,ab1 ,fig4 #f41 be ! giv5 "r l9e1 jo9+ ! giv5 po9ts ,a & ,b4 ,let ! direct1 lat]al1 & v]tical di/ances ( ! po9t ,a be ,,td1 ,,dc1 & ,,ca4 ,let ! direct1 lat]al1 & v]tical di/ances ( ! po9t ,b be ,,td1 ,,dc1 & ,c,b4 ,!n1 by ,problem ,i41 ! position ( ! po9t ,a on ! plane ( ! picture is a4 ,& simil>ly1 ! position ( ! po9t ,b on ! plane ( ! picture is b4 ,jo9 ab4 ,!n ab is ! l9e requir$4 ,,coroll>y ,i4 ,if ! l9e ,,ab is 9 a plane p>allel to t ( ! picture1 "o end ( ! l9e ,,ab mu/ be at ! same direct di/ance f ! eye ( ! obs]v] z ! o!r4 ,!re=e1 9 t case1 ,,dt is equal to ,d,t4 ,!n ! con/ruction w be z 9 ,fig4 #g42 & ! /ud5t w f9d exp]im5tally t ab is n[ p>allel to ,,ab4#aa ,illu/ration3 ,fig4 #g4 ,& t ab is to ,,ab z ,,ts is to ,,td4 ,!re=e1 to draw any l9e 9 a plane p>allel to t ( ! picture1 we h only to fix ! position ( "o ( its extremities1 a or b1 & !n to draw f a or b a l9e p>allel to ! giv5 l9e1 be>+ ! proportion to x t ,,ts be>s to ,,td4 ,,coroll>y ,,ii4 ,if ! l9e ,,ab is 9 a horizontal plane1 ! v]tical di/ance ( "o ( its extremities mu/ be ! same z t ( ! o!r4 ,!re=e1 9 t case1 ,,ac equals ,,bc 7,fig4 #f474 ,& ! con/ruction is z 9 ,fig4 #h4 ,illu/ration3 ,fig4 #h4 ,9 ,fig4 #h4 produce ab to ! sight-l9e1 cutt+ ! sight-l9e 9 ,v2 ! po9t ,v1 ?us det]m9$1 is call$ ! ,,vani%+-,,po9t ( ! l9e ,,ab4 ,jo9 ,,tv4 ,!n ! /ud5t w f9d exp]im5tally t ,,tv is p>allel to ,,ab4#ab ,,coroll>y ,,iii4 ,if ! l9e ,,ab produc$ w\ld pass ?r\gh "s po9t b5ea? or above ! /ation-po9t1 ,,cd is to ,,dt z ,c,d is to ,d,t2 9 :i* case ! po9t c co9cides ) ! po9t c1 & ! l9e ab is v]tical4 ,!re=e e v]tical l9e 9 a picture is1 or may be1 ! p]spective repres5tation ( a horizontal "o :i*1 produc$1 w\ld pass b5ea? ! feet or above ! head ( ! spectator4#ac #aa ,= by ! con/ruction ,,at a,t ,,bt b,t2 & !re=e ! two triangles ,,abt1 ab,t1 7hav+ a common angle ,,atb17 are simil>4 #ab ,! demon/ration is 9 ,app5dix ,,ii4 ,article ,i4 #ac ,! reflection 9 wat] ( any lum9\s po9t or isolat$ object 7su* z ! sun or moon7 is !re=e1 9 p]spective1 a v]tical l9e2 s9ce su* reflection1 if produc$1 w\ld pass "u ! feet ( ! spectator4 ,_m arti/s 7,claude among ! re/7 kn[+ some?+ ( optics1 b no?+ ( p]spective1 h be5 l$ occasionally to draw su* reflections t[>ds a po9t at ! c5t] ( ! base ( ! picture4 ,,problem ,,iii4 ,,to ,,f9d ,,! ,,vani%+-,,po9t ,,( ,a ,,giv5 ,,horizontal ,,l9e4 ,illu/ration3 ,fig4 #i4 ,let ,,ab1 ,fig4 #i41 be ! giv5 l9e4 ,f ,t1 ! /ation-po9t1 draw ,,tv p>allel to ,,ab1 cutt+ ! sight-l9e 9 ,v4 ,v is ! ,vani%+-po9t requir$4#ad ,,coroll>y ,i4 ,z1 if ! po9t b is fir/ f\nd1 ,v may be det]m9$ by x1 s1 if ! po9t ,v is fir/ f\nd1 b may be det]m9$ by x4 ,= let ,,ab1 ,fig4 #aj41 be ! giv5 l9e1 con/ruct$ ^u ! pap] z 9 ,fig4 #h42 & let x be requir$ to draw ! l9e ab )\t us+ ! po9t ,c4 ,illu/ration3 ,fig4 #aj4 ,f9d ! position ( ! po9t ,a 9 a4 7,problem ,i47 ,f9d ! vani%+-po9t ( ,,ab 9 ,v4 7,problem ,,iii47 ,jo9 a,v4 ,jo9 ,,bt1 cutt+ a,v 9 b4 ,!n ab is ! l9e requir$4#ae ,,coroll>y ,,ii4 ,we h hi!rto proce$$ on ! supposition t ! giv5 l9e 0 small en\gh1 & ne> en\gh1 to be actually drawn on our pap] ( its real size2 z 9 ! example giv5 9 ,app5dix ,i4 ,we may1 h[ev]1 n[ d$uce a con/ruction available "u all circum/ances1 :atev] may be ! di/ance & l5g? ( ! l9e giv54 ,illu/ration3 ,fig4 #aa4 ,f ,fig4 #h4 remove1 = ! sake ( cle>ness1 ! l9es ,c,d1 b,v1 & ,,tv2 &1 tak+ ! figure z "h 9 ,fig4 #aa41 draw f a1 ! l9e a,r p>allel to ,,ab1 cutt+ ,,bt 9 ,r4 ,!n a,r is to ,,ab z a,t is to ,,at4 ---- ---- z c,t is to ,,ct4 ---- ---- z ,,ts is to ,,td4 ,t is to say1 a,r is ! sight-magnitude ( ,,ab4#af ,illu/ration3 ,fig4 #ab4 ,!re=e1 :5 ! position ( ! po9t ,a is fix$ 9 a1 z 9 ,fig4 #ab41 & a,v is drawn to ! vani%+-po9t2 if we draw a l9e a,r f a1 p>allel to ,,ab1 & make a,r equal to ! sight-magnitude ( ,,ab1 & !n jo9 ,,rt1 ! l9e ,,rt w cut a,v 9 b4 ,s t1 9 ord] to det]m9e ! l5g? ( ab1 we ne$ n draw ! long & di/ant l9e ,,ab1 b only a,r p>allel to x1 & ( its sight-magnitude2 :i* is a great ga91 = ! l9e ,,ab may be two miles long1 & ! l9e a,r p]haps only two in*es4 ,,coroll>y ,,iii4 ,9 ,fig4 #ab41 alt]+ its proportions a little = ! sake ( cle>ness1 & putt+ x z "h 9 ,fig4 #ac41 draw a horizontal l9e a,r & make a,r equal to a,r4 ,?r\gh ! po9ts ,r & b draw ,r,m1 cutt+ ! sight-l9e 9 ,m4 ,jo9 ,,tv4 ,n[ ! read] w f9d exp]im5tally t ,,vm is equal to ,,vt4#ag ,illu/ration3 ,fig4 #ac4 ,h5ce x foll[s t1 if f ! vani%+-po9t ,v we lay (f on ! sight-l9e a di/ance1 ,,vm1 equal to ,,vt2 !n draw ?r\gh a a horizontal l9e a,r1 make a,r equal to ! sight-magnitude ( ,,ab1 & jo9 ,r,m2 ! l9e ,r,m w cut a,v 9 b4 ,& ?is is 9 practice g5]ally ! mo/ conv5i5t way ( obta9+ ! l5g? ( ab4 ,,coroll>y ,,iv4 ,remov+ f ! prec$+ figure ! unnecess>y l9es1 & reta9+ only ,r,m & a,v1 z 9 ,fig4 #ad41 produce ! l9e a,r to ! o!r side ( a1 & make a,x equal to a,r4 ,jo9 ,xb1 & produce ,xb to cut ! l9e ( sight 9 ,n4 ,illu/ration3 ,fig4 #ad4 ,!n z ,,xr is p>allel to ,,mn1 & a,r is equal to a,x1 ,,vn mu/1 by simil> triangles1 be equal to ,,vm 7equal to ,,vt 9 ,fig4 #ac474 ,!re=e1 on :i*ev] side ( ,v we measure ! di/ance ,,vt1 s z to obta9 ei!r ! po9t ,m1 or ! po9t ,n1 if we measure ! sight-magnitude a,r or a,x on ! opposite side ( ! l9e a,v1 ! l9e jo9+ ,r,m or ,,xn w equally cut a,v 9 b4 ,! po9ts ,m & ,n are call$ ! 8,,divid+-,,po9ts0 ( ! orig9al l9e ,,ab 7,fig4 #ab471 & we resume ! results ( ^! coroll>ies 9 ! foll[+ ?ree problems4 #ad ,! /ud5t w obs]ve1 9 practice1 t1 8 pap] ly+ flat on ! table1 he has only to draw ! l9e ,,tv on its horizontal surface1 p>allel to ! giv5 horizontal l9e ,,ab4 ,9 !ory1 ! pap] %\ld be v]tical1 b ! /ation-l9e ,,/ horizontal 7see its def9ition above1 page #e72 9 :i* case ,,tv1 be+ drawn p>allel to ,,ab1 w be horizontal also1 & /ill cut ! sight-l9e 9 ,v4 ,! con/ruction w be se5 to be f\nd$ on ! second ,coroll>y ( ! prec$+ problem4 ,x is evid5t t if any o!r l9e1 z ,,mn 9 ,fig4 #i41 p>allel to ,,ab1 occurs 9 ! picture1 ! l9e ,,tv1 drawn f ,t1 p>allel to ,,mn1 to f9d ! vani%+-po9t ( ,,mn1 w co9cide ) ! l9e drawn f ,t1 p>allel to ,,ab1 to f9d ! vani%+-po9t ( ,,ab4 ,!re=e ,,ab & ,,mn w h ! same vani%+-po9t4 ,!re=e all p>allel horizontal l9es h ! same vani%+-po9t4 ,x w be %[n h]eaft] t all p>allel incl9$ l9es also h ! same vani%+-po9t2 ! /ud5t may "h accept ! g5]al conclusion--8,all p>allel l9es h ! same vani%+-po9t40 ,x is also evid5t t if ,,ab is p>allel to ! plane ( ! picture1 ,,tv mu/ be drawn p>allel to ,,gh1 & w !re=e nev] cut ,,gh4 ,! l9e ,,ab has 9 t case no vani%+-po9t3 x is to be drawn by ! con/ruction giv5 9 ,fig4 #g4 ,x is also evid5t t if ,,ab is at "r angles ) ! plane ( ! picture1 ,,tv w co9cide ) ,,ts1 & ! vani%+-po9t ( ,,ab w be ! sight-po9t4 #ae ,i sp>e ! /ud5t ! =mality ( ! r$uctio ad absurdum1 :i* w\ld be necess>y to prove ?is4 #af ,= def9ition ( ,sight-,magnitude1 see ,app5dix ,i4 ,x ought to h be5 read be=e ! /ud5t comes to ?is problem2 b ,i ref] to x 9 case x has n4 #ag ,! demon/ration is 9 ,app5dix ,,ii4 ,article ,,ii4 p4 #aja4 ,,problem ,,iv4 ,,to ,,f9d ,,! ,,divid+-,,po9ts ,,( ,a ,,giv5 ,,horizontal ,,l9e4 ,illu/ration3 ,fig4 #ae4 ,let ! horizontal l9e ,,ab 7,fig4 #ae47 be giv5 9 position & magnitude4 ,x is requir$ to f9d its divid+-po9ts4 ,f9d ! vani%+-po9t ,v ( ! l9e ,,ab4 ,) c5t] ,v & di/ance ,,vt1 describe circle cutt+ ! sight-l9e 9 ,m & ,n4 ,!n ,m & ,n are ! divid+-po9ts requir$4 ,9 g5]al1 only "o divid+-po9t is ne$$ = use ) any vani%+-po9t1 namely1 ! "o ne>e/ ,s 79 ?is case ! po9t ,m74 ,b its opposite ,n1 or bo?1 may be ne$$ "u c]ta9 circum/ances4 ,,problem ,v4 ,,to ,,draw ,a ,,horizontal ,,l9e1 ,,giv5 ,,9 ,,position ,,& ,,magnitude1 ,,by ,,means ,,( ,,its ,,sight-,,magnitude ,,& ,,divid+-,,po9ts4 ,illu/ration3 ,fig4 #af4 ,let ,,ab 7,fig4 #af47 be ! giv5 l9e4 ,f9d ! position ( ! po9t ,a 9 a4 ,f9d ! vani%+-po9t ,v1 & mo/ conv5i5t divid+-po9t ,m1 ( ! l9e ,,ab4 ,jo9 a,v4 ,?r\gh a draw a horizontal l9e ab & make ab equal to ! sight-magnitude ( ,,ab4 ,jo9 b,m1 cutt+ a,v 9 b4 ,!n ab is ! l9e requir$4 ,,coroll>y ,i4 ,illu/ration3 ,fig4 #ag4 ,suppos+ x 7 n[ requir$ to draw a l9e ,,ac 7,fig4 #ag47 twice z long z ,,ab1 x is evid5t t ! sight-magnitude ac mu/ be twice z long z ! sight-magnitude ab2 we h1 !re=e1 m]ely to cont9ue ! horizontal l9e ab1 make bc equal to ab1 jo9 c,m1 cutt+ a,v 9 c1 & ac w be ! l9e requir$4 ,simil>ly1 if we h to draw a l9e ,,ad1 ?ree times ! l5g? ( ,,ab1 ad mu/ be ?ree times ! l5g? ( ab1 &1 jo9+ d,m1 ad w be ! l9e requir$4 ,! /ud5t w obs]ve t ! ne>] ! portions cut (f1 bc1 cd1 etc41 approa* ! po9t ,v1 ! small] !y become2 &1 :atev] l5g?s may be add$ to ! l9e ,,ad1 & successively cut (f f a,v1 ! l9e a,v w nev] be cut (f entirely1 b ! portions cut (f w become inf9itely small1 & app>5tly 8vani%0 z !y approa* ! po9t ,v2 h5ce ?is po9t is call$ ! 8vani%+0 po9t4 ,,coroll>y ,,ii4 ,x is evid5t t if ! l9e ,,ad _h be5 giv5 orig9ally1 & we _h be5 requir$ to draw x1 & divide x into ?ree equal p>ts1 we %\ld h _h only to divide its sight-magnitude1 ad1 into ! ?ree equal p>ts1 ab1 bc1 & cd1 & !n1 draw+ to ,m f b & c1 ! l9e ad w\ld h be5 divid$ z requir$ 9 b & c4 ,& suppos+ ! orig9al l9e ,,ad be divid$ irregul>ly into any numb] ( p>ts1 if ! l9e ad be divid$ into a simil> numb] 9 ! same proportions 7by ! con/ruction giv5 9 ,app5dix ,i471 &1 f ^! po9ts ( division1 l9es are drawn to ,m1 !y w divide ! l9e ad 9 true p]spective into a simil> numb] ( proportionate p>ts4 ,! horizontal l9e drawn ?r\gh a1 on :i* ! sight-magnitudes are measur$1 is call$ ! 8,,measur+-,,l9e40 ,& ! l9e ad1 :5 prop]ly divid$ 9 b & c1 or any o!r requir$ po9ts1 is said to be divid$ 8,,9 ,,p]spective ,,ratio0 to ! divisions ( ! orig9al l9e ,,ad4 ,if ! l9e a,v is above ! sight-l9e in/ead ( b5ea? x1 ! measur+-l9e is to be drawn above also3 & ! l9es b,m1 c,m1 etc41 drawn d[n to ! divid+-po9t4 ,turn ,fig4 #ag4 upside d[n1 & x w %[ ! con/ruction4 ,,problem ,,vi4 ,,to ,,draw ,,any ,,triangle1 ,,giv5 ,,9 ,,position ,,& ,,magnitude1 ,,9 ,a ,,horizontal ,,plane4 ,illu/ration3 ,fig4 #ah4 ,let ,,abc 7,fig4 #ah47 be ! triangle4 ,z x is giv5 9 position & magnitude1 "o ( its sides1 at lea/1 mu/ be giv5 9 position & magnitude1 & ! directions ( ! two o!r sides4 ,let ,,ab be ! side giv5 9 position & magnitude4 ,!n ,,ab is a horizontal l9e1 9 a giv5 position1 & ( a giv5 l5g?4 ,draw ! l9e ,,ab4 7,problem ,v47 ,let ab be ! l9e s drawn4 ,f9d ,v & ,v1 ! vani%+-po9ts respectively ( ! l9es ,,ac & ,,bc4 7,problem ,,iii47 ,f a draw a,v1 & f b1 draw b,v1 cutt+ ea* o!r 9 c4 ,!n abc is ! triangle requir$4 ,if ,,ac is ! l9e orig9ally giv51 ac is ! l9e :i* mu/ be fir/ drawn1 & ! l9e ,vb mu/ be drawn f ,v to c & produc$ to cut ab 9 b4 ,simil>ly1 if ,,bc is giv51 ,vc mu/ be drawn to c & produc$1 & ab f its vani%+-po9t to b1 & produc$ to cut ac 9 a4 ,,problem ,,vii4 ,,to ,,draw ,,any ,,rectil9e> ,,quadrilat]al ,,figure1 ,,giv5 ,,9 ,,position ,,& ,,magnitude1 ,,9 ,a ,,horizontal ,,plane4 ,illu/ration3 ,fig4 #ai4 ,let ,,abcd 7,fig4 #ai47 be ! giv5 figure4 ,jo9 any two ( its opposite angles by ! l9e ,,bc4 ,draw fir/ ! triangle ,,abc4 7,problem ,,vi47 ,& !n1 f ! base ,,bc1 ! two l9es ,,bd1 ,,cd1 to _! vani%+-po9ts1 :i* w complete ! figure4 ,x is unnecess>y to give a diagram ( ! con/ruction1 :i* is m]ely t ( ,fig4 #ah4 duplicat$2 ano!r triangle be+ drawn on ! l9e ,,ac or ,,bc4 ,,coroll>y4 ,x is evid5t t by ?is application ( ,problem ,,vi4 any giv5 rectil9e> figure :atev] 9 a horizontal plane may be drawn1 s9ce any su* figure may be divid$ into a numb] ( triangles1 & ! triangles !n drawn 9 succession4 ,m conv5i5t me?ods may1 h[ev]1 be g5]ally f\nd1 accord+ to ! =m ( ! figure requir$1 by ! use ( succe$+ problems2 & = ! quadrilat]al figure :i* occurs mo/ frequ5tly 9 practice1 namely1 ! squ>e1 ! foll[+ con/ruction is m conv5i5t ?an t us$ 9 ! pres5t problem4 ,,problem ,,viii4 ,,to ,,draw ,a ,,squ>e1 ,,giv5 ,,9 ,,position ,,& ,,magnitude1 ,,9 ,a ,,horizontal ,,plane4 ,illu/ration3 ,fig4 #bj4 ,let ,,abcd1 ,fig4 #bj41 be ! squ>e4 ,z x is giv5 9 position & magnitude1 ! position & magnitude ( all its sides are giv54 ,fix ! position ( ! po9t ,a 9 a4 ,f9d ,v1 ! vani%+-po9t ( ,,ab2 & ,m1 ! divid+-po9t ( ,,ab1 ne>e/ ,s4 ,f9d ,v1 ! vani%+-po9t ( ,,ac2 & ,n1 ! divid+-po9t ( ,,ac1 ne>e/ ,s4 ,draw ! measur+-l9e ?r\gh a1 & make ab1 ac1 ea* equal to ! sight-magnitude ( ,,ab4 7,= s9ce ,,abcd is a squ>e1 ,,ac is equal to ,,ab47 ,draw a,v & c,n1 cutt+ ea* o!r 9 c4 ,draw a,v1 & b,m1 cutt+ ea* o!r 9 b4 ,!n ac1 ab1 are ! two ne>e/ sides ( ! squ>e4 ,n[1 cle>+ ! figure ( sup]flu\s l9es1 we h ab1 ac1 drawn 9 position1 z 9 ,fig4 #ba4 ,illu/ration3 ,fig4 #ba4 ,& because ,,abcd is a squ>e1 ,,cd 7,fig4 #bj47 is p>allel to ,,ab4 ,& all p>allel l9es h ! same vani%+-po9t4 7,note to ,problem ,,iii47 ,!re=e1 ,v is ! vani%+-po9t ( ,,cd4 ,simil>ly1 ,v is ! vani%+-po9t ( ,,bd4 ,!re=e1 f b & c 7,fig4 #bb47 draw b,v1 c,v1 cutt+ ea* o!r 9 d4 ,!n abcd is ! squ>e requir$4 ,,coroll>y ,i4 ,x is obvi\s t any rectangle 9 a horizontal plane may be drawn by ?is problem1 m]ely mak+ ab1 on ! measur+-l9e1 ,fig4 #bj41 equal to ! sight-magnitude ( "o ( its sides1 & ac ! sight-magnitude ( ! o!r4 ,,coroll>y ,,ii4 ,let abcd1 ,fig4 #bb41 be any squ>e drawn 9 p]spective4 ,draw ! diagonals ad & bc1 cutt+ ea* o!r 9 ,c4 ,!n ,c is ! c5t] ( ! squ>e4 ,?r\gh ,c1 draw ef to ! vani%+-po9t ( ab1 & gh to ! vani%+-po9t ( ac1 & ^! l9es w bisect ! sides ( ! squ>e1 s t ag is ! p]spective repres5tation ( half ! side ab2 ae is half ac2 * is half cd2 & bf is half bd4 ,illu/ration3 ,fig4 #bb4 ,,coroll>y ,,iii4 ,s9ce ,,abcd1 ,fig4 #bj41 is a squ>e1 ,,bac is a "r angle2 & z ,,tv is p>allel to ,,ab1 & ,,tv to ,,ac1 ,v,,tv mu/ be a "r angle also4 ,z ! gr\nd plan ( mo/ build+s is rectangul>1 x con/antly happ5s 9 practice t _! angles 7z ! corn]s ( ord9>y h\ses7 ?r[ ! l9es to ! vani%+-po9ts ?us at "r angles2 & s t ?is law is obs]v$1 & ,,vtv is kept a "r angle1 x does n matt] 9 g5]al practice :e!r ! vani%+-po9ts are ?r[n a little m or a little less to ! "r or left ( ,s3 b x matt]s mu* t ! relation ( ! vani%+-po9ts %\ld be accurate4 ,_! position ) respect to ,s m]ely causes ! spectator to see a little m or less on "o side or o!r ( ! h\se1 :i* may be a matt] ( *ance or *oice2 b _! rectangul> relation det]m9es ! rectangul> %ape ( ! build+1 :i* is an ess5tial po9t4 ,,problem ,,ix4 ,,to ,,draw ,a ,,squ>e ,,pill>1 ,,giv5 ,,9 ,,position ,,& ,,magnitude1 ,,its ,,base ,,& ,,top ,,be+ ,,9 ,,horizontal ,,planes4 ,let ,,ah1 ,fig4 #bc41 be ! squ>e pill>4 ,!n1 z x is giv5 9 position & magnitude1 ! position & magnitude ( ! squ>e x /&s ^u mu/ be giv5 7t is1 ! l9e ,,ab or ,,ac 9 position71 & ! height ( its side ,,ae4 ,illu/ration3 ,fig4 #bc4 ,illu/ration3 ,fig4 #bd4 ,f9d ! sight-magnitudes ( ,,ab & ,,ae4 ,draw ! two sides ab1 ac1 ( ! squ>e ( ! base1 by ,problem ,,viii41 z 9 ,fig4 #bd4 ,f ! po9ts a1 b1 & c1 raise v]tical l9es ae1 cf1 bg4 ,make ae equal to ! sight-magnitude ( ,,ae4 ,n[ because ! top & base ( ! pill> are 9 horizontal planes1 ! squ>e ( its top1 ,,fg1 is p>allel to ! squ>e ( its base1 ,,bc4 ,!re=e ! l9e ,,ef is p>allel to ,,ac1 & ,,eg to ,,ab4 ,!re=e ,,ef has ! same vani%+-po9t z ,,ac1 & ,,eg ! same vani%+-po9t z ,,ab4 ,f e draw ef to ! vani%+-po9t ( ac1 cutt+ cf 9 f4 ,simil>ly draw eg to ! vani%+-po9t ( ab1 cutt+ bg 9 g4 ,complete ! squ>e gf 9 h1 by draw+ gh to ! vani%+-po9t ( ef1 & fh to ! vani%+-po9t ( eg1 cutt+ ea* o!r 9 h4 ,!n aghf is ! squ>e pill> requir$4 ,,coroll>y4 ,x is obvi\s t if ,,ae is equal to ,,ac1 ! :ole figure w be a cube1 & ea* side1 aefc & aegb1 w be a squ>e 9 a giv5 v]tical plane4 ,& by mak+ ,,ab or ,,ac long] or %ort] 9 any giv5 proportion1 any =m ( rectangle may be giv5 to ei!r ( ! sides ( ! pill>4 ,no o!r rule is !re=e ne$$ = draw+ squ>es or rectangles 9 v]tical planes4 ,also any triangle may be ?us drawn 9 a v]tical plane1 by inclos+ x 9 a rectangle & det]m9+1 9 p]spective ratio1 on ! sides ( ! rectangle1 ! po9ts ( _! contact ) ! angles ( ! triangle4 ,& if any triangle1 !n any polygon4 ,a less complicat$ con/ruction w1 h[ev]1 be giv5 h]eaft]4#ah #ah ,see page #if 7note71 aft] y h read ,problem ,,xvi4 ,,problem ,x4 ,,to ,,draw ,a ,,pyramid1 ,,giv5 ,,9 ,,position ,,& ,,magnitude1 ,,on ,a ,,squ>e ,,base ,,9 ,a ,,horizontal ,,plane4 ,illu/ration3 ,fig4 #be4 ,let ,,ab1 ,fig4 #be41 be ! f\r-sid$ pyramid4 ,z x is giv5 9 position & magnitude1 ! squ>e base on :i* x /&s mu/ be giv5 9 position & magnitude1 & its v]tical height1 ,,cd4#ai ,illu/ration3 ,fig4 #bf4 ,draw a squ>e pill>1 ,,abge1 ,fig4 #bf41 on ! squ>e base ( ! pyramid1 & make ! height ( ! pill> ,,af equal to ! v]tical height ( ! pyramid ,,cd 7,problem ,,ix474 ,draw ! diagonals ,,gf1 ,,hi1 on ! top ( ! squ>e pill>1 cutt+ ea* o!r 9 ,c4 ,!re=e ,c is ! c5t] ( ! squ>e ,,fghi4 7,prob4 ,,viii4 ,cor4 ,,ii47 ,illu/ration3 ,fig4 #bg4 ,jo9 ,,ce1 ,,ca1 ,,cb4 ,!n ,,abce is ! pyramid requir$4 ,if ! base ( ! pyramid is above ! eye1 z :5 a squ>e spire is se5 on ! top ( a *ur*-t[]1 ! con/ruction w be z 9 ,fig4 #bg4 #ai ,if1 in/ead ( ! v]tical height1 ! l5g? ( ,,ad is giv51 ! v]tical mu/ be d$uc$ f x4 ,see ! ,ex]cises on ?is ,problem 9 ! ,app5dix1 p4 #gi4 ,,problem ,,xi4 ,,to ,,draw ,,any ,,curve ,,9 ,a ,,horizontal ,,or ,,v]tical ,,plane4 ,illu/ration3 ,fig4 #bh4 ,let ,,ab1 ,fig4 #bh41 be ! curve4 ,inclose x 9 a rectangle1 ,,cdef4 ,fix ! position ( ! po9t ,c or ,d1 & draw ! rectangle4 7,problem ,,viii4 ,coroll4 ,i47#bj ,let ,,cdef1 ,fig4 #bi41 be ! rectangle s drawn4 ,illu/ration3 ,fig4 #bi4 ,if an extremity ( ! curve1 z ,a1 is 9 a side ( ! rectangle1 divide ! side ,,ce1 ,fig4 #bi41 s t ,,ac %all be 79 p]spective ratio7 to ,,ae z ,,ac is to ,,ae 9 ,fig4 #bh4 7,prob4 ,v4 ,cor4 ,,ii47 ,simil>ly det]m9e ! po9ts ( contact ( ! curve & rectangle e1 f1 g4 ,if an extremity ( ! curve1 z ,b1 is n 9 a side ( ! rectangle1 let fall ! p]p5dicul>s ,ba1 ,bb on ! rectangle sides4 ,det]m9e ! correspond5t po9ts a & b 9 ,fig4 #bi41 z y h already det]m9$ ,a1 ,b1 e1 & f4 ,f b1 ,fig4 #bi41 draw b,b p>allel to ,,cd1#ba & f a draw a,b to ! vani%+-po9t ( ,,df1 cutt+ ea* o!r 9 ,b4 ,!n ,b is ! extremity ( ! curve4 ,det]m9e any o!r important po9t 9 ! curve1 z ,p1 9 ! same way1 by lett+ fall ,pq & ,pr on ! rectangle's sides4 ,any numb] ( po9ts 9 ! curve may be ?us det]m9$1 & ! curve drawn ?r\gh ! s]ies2 9 mo/ cases1 ?ree or f\r w be en\gh4 ,practically1 complicat$ curves may be bett] drawn 9 p]spective by an exp]i5c$ eye ?an by rule1 z ! fix+ ( ! v>i\s po9ts 9 ha/e involves too _m *ances ( error2 b x is well to draw a good _m by rule fir/1 9 ord] to give ! eye its exp]i5ce4#bb ,,coroll>y4 ,if ! curve requir$ be a circle1 ,fig4 #cj41 ! rectangle :i* incloses x w become a squ>e1 & ! curve w h f\r po9ts ( contact1 ,,abcd1 9 ! middle ( ! sides ( ! squ>e4 ,illu/ration3 ,fig4 #cj4 ,draw ! squ>e1 & z a squ>e may be drawn ab\t a circle 9 any position1 draw x ) its ne>e/ side1 ,,eg1 p>allel to ! sight-l9e4 ,let ,,ef1 ,fig4 #ca41 be ! squ>e s drawn4 ,draw its diagonals ,,ef1 ,,gh2 & ?r\gh ! c5t] ( ! squ>e 7det]m9$ by _! int]section7 draw ,,ab to ! vani%+-po9t ( ,,gf1 & ,,cd p>allel to ,,eg4 ,!n ! po9ts ,,abcd are ! f\r po9ts ( ! circle's contact4 ,illu/ration3 ,fig4 #ca4 ,on ,,eg describe a half squ>e1 ,,el2 draw ! semicircle ,,kal2 & f its c5t]1 ,r1 ! diagonals ,,re1 ,,rg1 cutt+ ! circle 9 x1 y4 ,f ! po9ts x y1 :]e ! circle cuts ! diagonals1 raise p]p5dicul>s1 ,px1 ,qy1 to ,,eg4 ,f ,p & ,q draw ,,pp1 ,,qq1 to ! vani%+-po9t ( ,,gf1 cutt+ ! diagonals 9 m1 n1 & o1 p4 ,!n m1 n1 o1 p are f\r o!r po9ts 9 ! circle4 ,?r\gh ^! eight po9ts ! circle may be drawn by ! h& accurately en\gh = g5]al purposes2 b any numb] ( po9ts requir$ may1 ( c\rse1 be det]m9$1 z 9 ,problem ,,xi4 ,! di/ance ,,ep is approximately "o-sev5? ( ,,eg1 & may be assum$ to be s 9 quick practice1 z ! error involv$ is n great] ?an w\ld be incurr$ 9 ! ha/y op]ation ( draw+ ! circle & diagonals4 ,x may frequ5tly happ5 t1 9 consequ5ce ( associat$ con/ructions1 x may be inconv5i5t to draw ,,eg p>allel to ! sight-l9e1 ! squ>e be+ p]haps fir/ con/ruct$ 9 "s oblique direction4 ,9 su* cases1 ,,qg & ,,ep mu/ be det]m9$ 9 p]spective ratio by ! divid+-po9t1 ! l9e ,,eg be+ us$ z a measur+-l9e4 ,obs4 ,9 draw+ ,fig4 #ca4 ! /ation-po9t has be5 tak5 mu* ne>] ! pap] ?an is usually advisable1 9 ord] to %[ ! "* ( ! curve 9 a v di/9ct =m4 ,if ! /ud5t turns ! book s t ,,eg may be v]tical1 ,fig4 #ca4 w repres5t ! con/ruction = draw+ a circle 9 a v]tical plane1 ! sight-l9e be+ !n ( c\rse p>allel to ,,gl2 & ! semicircles ,,adb1 ,,acb1 on ea* side ( ! diamet] ,,ab1 w repres5t ord9>y semicircul> ar*es se5 9 p]spective4 ,9 t case1 if ! book be held s t ! l9e ,,eh is ! top ( ! squ>e1 ! upp] semicircle w repres5t a semicircul> ar*1 above ! eye1 drawn 9 p]spective4 ,b if ! book be held s t ! l9e ,,gf is ! top ( ! squ>e1 ! upp] semicircle w repres5t a semicircul> ar*1 bel[ ! eye1 drawn 9 p]spective4 ,if ! book be turn$ upside d[n1 ! figure w repres5t a circle drawn on ! ceil+1 or any o!r horizontal plane above ! eye2 & ! con/ruction is1 ( c\rse1 accurate 9 e case4 #bj ,or if ! curve is 9 a v]tical plane1 ,coroll4 to ,problem ,,ix4 ,z a rectangle may be drawn 9 any position r\nd any giv5 curve1 its position ) respect to ! curve w 9 ei!r case be regulat$ by conv5i5ce4 ,see ! ,ex]cises on ?is ,problem1 9 ! ,app5dix1 p4 #he4 #ba ,or to its vani%+-po9t1 if ,,cd has "o4 #bb ,( c\rse1 by divid+ ! orig9al rectangle into any numb] ( equal rectangles1 & divid+ ! p]spective rectangle simil>ly1 ! curve may be approximately drawn )\t any tr\ble2 b1 :5 accuracy is requir$1 ! po9ts %\ld be fix$1 z 9 ! problem4 ,,problem ,,xii4 ,,to ,,divide ,a ,,circle ,,drawn ,,9 ,,p]spective ,,into ,,any ,,giv5 ,,numb] ,,( ,,equal ,,p>ts4 ,let ,,ab1 ,fig4 #cb41 be ! circle drawn 9 p]spective4 ,x is requir$ to divide x into a giv5 numb] ( equal p>ts2 9 ?is case1 #bj4 ,let ,,kal be ! semicircle us$ 9 ! con/ruction4 ,divide ! semicircle ,,kal into half ! numb] ( p>ts requir$2 9 ?is case1 #aj4 ,produce ! l9e ,,eg lat]ally1 z f> z may be necess>y4 ,f ,o1 ! c5t] ( ! semicircle ,,kal1 draw radii ?r\gh ! po9ts ( division ( ! semicircle1 p1 q1 r1 etc41 & produce !m to cut ! l9e ,,eg 9 ,p1 ,q1 ,r1 etc4 ,f ! po9ts ,,pqr draw ! l9es ,,pp1 ,,qq1 ,,rr1 etc41 ?r\gh ! c5t] ( ! circle ,,ab1 ea* cutt+ ! circle 9 two po9ts ( its circumf]5ce4 ,!n ^! po9ts divide ! p]spective circle z requir$4 ,if f ea* ( ! po9ts p1 q1 r1 a v]tical 7 rais$ to ! l9e ,,eg1 z 9 ,fig4 #ca41 & f ! po9t :]e x cut ,,eg a l9e 7 drawn to ! vani%+-po9t1 z ,,qq 9 ,fig4 #ca41 ?is l9e w\ld also det]m9e two ( ! po9ts ( division4 ,illu/ration3 ,fig4 #cb4 ,if x is requir$ to divide a circle into any numb] ( giv5 unequal p>ts 7z 9 ! po9ts ,a1 ,b1 & ,c1 ,fig4 #cc471 ! %orte/ way is ?us to raise v]tical l9es f ,a & ,b to ! side ( ! p]spective squ>e ,,xy1 & !n draw to ! vani%+-po9t1 cutt+ ! p]spective circle 9 a & b1 ! po9ts requir$4 ,only notice t if any po9t1 z ,a1 is on ! ne>] side ( ! circle ,,abc1 its repres5tative po9t1 a1 mu/ be on ! ne>] side ( ! circle abc2 & if ! po9t ,b is on ! f>!r side ( ! circle ,,abc1 b mu/ be on ! f>!r side ( abc4 ,if any po9t1 z ,c1 is s mu* 9 ! lat]al arc ( ! circle z n to be easily det]m9able by ! v]tical l9e1 draw ! horizontal ,,cp1 f9d ! correspond5t p 9 ! side ( ! p]spective squ>e1 & draw pc p>allel to ,,xy1 cutt+ ! p]spective circle 9 c4 ,illu/ration3 ,fig4 #cc4 ,,coroll>y4 ,x is obvi\s t if ! po9ts ,p1 ,q1 ,r1 etc41 by :i* ! circle is divid$ 9 ,fig4 #cb41 be jo9$ by "r l9es1 ! result+ figure w be a regul> equilat]al figure ( tw5ty sides inscrib$ 9 ! circle4 ,& if ! circle be divid$ into giv5 unequal p>ts1 & ! po9ts ( division jo9$ by "r l9es1 ! result+ figure w be an irregul> polygon inscrib$ 9 ! circle ) sides ( giv5 l5g?4 ,?us any polygon1 regul> or irregul>1 inscrib$ 9 a circle1 may be inscrib$ 9 position 9 a p]spective circle4 ,,problem ,,xiii4 ,,to ,,draw ,a ,,squ>e1 ,,giv5 ,,9 ,,magnitude1 ,,)9 ,a ,,l>g] ,,squ>e ,,giv5 ,,9 ,,position ,,& ,,magnitude2 ,,! ,,sides ,,( ,,! ,,two ,,squ>es ,,be+ ,,p>allel4 ,illu/ration3 ,fig4 #cd4 ,let ,,ab1 ,fig4 #cd41 be ! sight-magnitude ( ! side ( ! small] squ>e1 & ,,ac t ( ! side ( ! l>g] squ>e4 ,draw ! l>g] squ>e4 ,let ,,defg be ! squ>e s drawn4 ,jo9 ,,eg & ,,df4 ,on ei!r ,,de or ,,dg set (f1 9 p]spective ratio1 ,,dh equal to "o half ( ,,bc4 ,?r\gh ,h draw ,,hk to ! vani%+-po9t ( ,,de1 cutt+ ,,df 9 ,i & ,,eg 9 ,k4 ,?r\gh ,i & ,k draw ,,im1 ,,kl1 to vani%+-po9t ( ,,dg1 cutt+ ,,df 9 ,l & ,,eg 9 ,m4 ,jo9 ,,lm4 ,!n ,,iklm is ! small] squ>e1 inscrib$ z requir$4#bc ,,coroll>y4 ,illu/ration3 ,fig4 #cf4 ,if1 in/ead ( "o squ>e )9 ano!r1 x be requir$ to draw "o circle )9 ano!r1 ! dim5sions ( bo? be+ giv51 inclose ea* circle 9 a squ>e4 ,draw ! squ>es fir/1 & !n ! circles )91 z 9 ,fig4 #cf4 #bc ,illu/ration3 ,fig4 #ce4 ,if ei!r ( ! sides ( ! great] squ>e is p>allel to ! plane ( ! picture1 z ,,dg 9 ,fig4 #ce41 ,,dg ( c\rse mu/ be equal to ,,ac1 & ,,dh equal to ,,bc_/#b1 & ! con/ruction is z 9 ,fig4 #ce4 ,,problem ,,xiv4 ,,to ,,draw ,a ,,truncat$ ,,circul> ,,cone1 ,,giv5 ,,9 ,,position ,,& ,,magnitude1 ,,! ,,truncations ,,be+ ,,9 ,,horizontal ,,planes1 ,,& ,,! ,,axis ,,( ,,! ,,cone ,,v]tical4 ,let ,,abcd1 ,fig4 #cg41 be ! portion ( ! cone requir$4 ,illu/ration3 ,fig4 #cg4 ,z x is giv5 9 magnitude1 its diamet]s mu/ be giv5 at ! base & summit1 ,,ab & ,,cd2 & its v]tical height1 ,,ce4#bd ,& z x is giv5 9 position1 ! c5t] ( its base mu/ be giv54 ,illu/ration3 ,fig4 #ch4 ,draw 9 position1 ab\t ?is c5t]1#be ! squ>e pill> afd1 ,fig4 #ch41 mak+ its height1 bg1 equal to ,,ce2 & its side1 ab1 equal to ,,ab4 ,9 ! squ>e ( its base1 abcd1 inscribe a circle1 :i* !re=e is ( ! diamet] ( ! base ( ! cone1 ,,ab4 ,9 ! squ>e ( its top1 efgh1 inscribe conc5trically a circle ^: diamet] %all equal ,,cd4 7,coroll4 ,prob4 ,,xiii47 ,jo9 ! extremities ( ! circles by ! "r l9es kl1 nm4 ,!n klnm is ! portion ( cone requir$4 ,,coroll>y ,i4 ,if simil> polygons be inscrib$ 9 simil> positions 9 ! circles kn & lm 7,coroll4 ,prob4 ,,xii471 & ! correspond+ angles ( ! polygons jo9$ by "r l9es1 ! result+ figure w be a portion ( a polygonal pyramid4 7,! dott$ l9es 9 ,fig4 #ch41 connect+ ! extremities ( two diamet]s & "o diagonal 9 ! respective circles1 occupy ! position ( ! ?ree ne>e/ angles ( a regul> octagonal pyramid1 hav+ its angles set on ! diagonals & diamet]s ( ! squ>e ad1 inclos+ its base47 ,if ! cone or polygonal pyramid is n truncat$1 its apex w be ! c5t] ( ! upp] squ>e1 z 9 ,fig4 #bf4 ,,coroll>y ,,ii4 ,if equal circles1 or equal & simil> polygons1 be inscrib$ 9 ! upp] & l[] squ>es 9 ,fig4 #ch41 ! result+ figure w be a v]tical cyl9d]1 or a v]tical polygonal pill>1 ( giv5 height & diamet]1 drawn 9 position4 ,,coroll>y ,,iii4 ,if ! circles 9 ,fig4 #ch41 in/ead ( be+ inscrib$ 9 ! squ>es bc & fg1 be inscrib$ 9 ! sides ( ! solid figure be & df1 ^? sides be+ made squ>e1 & ! l9e bd ( any giv5 l5g?1 ! result+ figure w be1 accord+ to ! con/ructions employ$1 a cone1 polygonal pyramid1 cyl9d]1 or polygonal pill>1 drawn 9 position ab\t a horizontal axis p>allel to bd4 ,simil>ly1 if ! circles are drawn 9 ! sides gd & ec1 ! result+ figures w be describ$ ab\t a horizontal axis p>allel to ab4 #bd ,or if ! l5g? ( its side1 ,,ac1 is giv5 in/ead1 take ae1 ,fig4 #cg41 equal to half ! excess ( ,,ab ov] ,,cd2 f ! po9t e raise ! p]p5dicul> ce4 ,) c5t] a1 & di/ance ,,ac1 describe a circle cutt+ ce 9 c4 ,!n ce is ! v]tical height ( ! portion ( cone requir$1 or ,,ce4 #be ,! direction ( ! side ( ! squ>e w ( c\rse be regulat$ by conv5i5ce4 ,,problem ,,xv4 ,,to ,,draw ,,an ,,incl9$ ,,l9e1 ,,giv5 ,,9 ,,position ,,& ,,magnitude4 ,we h hi!rto be5 exam9+ ! conditions ( horizontal & v]tical l9es only1 or ( curves inclos$ 9 rectangles4 ,illu/ration3 ,fig4 #ci4 ,illu/ration3 ,fig4 #dj4 ,we mu/1 9 conclusion1 inve/igate ! p]spective ( incl9$ l9es1 beg9n+ ) a s+le "o giv5 9 position4 ,= ! sake ( complet5ess ( sy/em1 ,i give 9 ,app5dix ,,ii4 ,article ,,iii4 ! developm5t ( ?is problem f ! second4 ,b1 9 practice1 ! position ( an incl9$ l9e may be mo/ conv5i5tly def9$ by consid]+ x z ! diagonal ( a rectangle1 z ,,ab 9 ,fig4 #ci41 & ,i %all !re=e1 ?\gh at "s sacrifice ( sy/em1 exam9e x "h "u t condition4 ,if ! sides ( ! rectangle ,,ac & ,,ad are giv51 ! slope ( ! l9e ,,ab is det]m9$2 & !n its position w dep5d on t ( ! rectangle4 ,if1 z 9 ,fig4 #ci41 ! rectangle is p>allel to ! picture plane1 ! l9e ,,ab mu/ be s also4 ,if1 z 9 ,fig4 #dj41 ! rectangle is incl9$ to ! picture plane1 ! l9e ,,ab w be s also4 ,s t1 to fix ! position ( ,,ab1 ! l9e ,,ac mu/ be giv5 9 position & magnitude1 & ! height ,,ad4 ,illu/ration3 ,fig4 #da4 ,if ^! are giv51 & x is only requir$ to draw ! s+le l9e ,,ab 9 p]spective1 ! con/ruction is entirely simple2 ?us3-- ,draw ! l9e ,,ac by ,problem ,i4 ,let ,,ac1 ,fig4 #da41 be ! l9e s drawn4 ,f a & c raise ! v]tical l9es ad1 cb4 ,make ad equal to ! sight-magnitude ( ,,ad4 ,f d draw db to ! vani%+-po9t ( ac1 cutt+ bc 9 b4 ,jo9 ab4 ,!n ab is ! incl9$ l9e requir$4 ,illu/ration3 ,fig4 #db4 ,if ! l9e is incl9$ 9 ! opposite direction1 z ,,dc 9 ,fig4 #db41 we h only to jo9 dc in/ead ( ab 9 ,fig4 #da41 & dc w be ! l9e requir$4 ,i %all h]eaft] call ! l9e ,,ac1 :5 us$ to def9e ! position ( an incl9$ l9e ,,ab 7,fig4 #dj471 ! 8relative horizontal0 ( ! l9e ,,ab4 ,,obs]vation4 ,illu/ration3 ,fig4 #dc4 ,9 g5]al1 incl9$ l9es are mo/ ne$$ = gable ro(s1 9 :i*1 :5 ! conditions are prop]ly /at$1 ! v]tical height ( ! gable1 ,,xy1 ,fig4 #dc41 is giv51 & ! base l9e1 ,,ac1 9 position4 ,:5 ^! are giv51 draw ,,ac2 raise v]tical ,,ad2 make ,,ad equal to sight-magnitude ( ,,xy2 complete ! p]spective-rectangle ,,adbc2 jo9 ,,ab & ,,dc 7z by dott$ l9es 9 figure72 & ?r\gh ! int]section ( ! dott$ l9es draw v]tical ,,xy1 cutt+ ,,db 9 ,y4 ,jo9 ,,ay1 ,,cy2 & ^! l9es are ! sides ( ! gable4 ,if ! l5g? ( ! ro( ,,aa is also giv51 draw 9 p]spective ! complete p>allelopip$ ,a,d,,bc1 & f ,y draw ,,yy to ! vani%+-po9t ( ,,aa1 cutt+ ,d,b 9 ,y4 ,jo9 ,a,y1 & y h ! slope ( ! f>!r side ( ! ro(4 ,illu/ration3 ,fig4 #dd4 ,! con/ruction above ! eye is z 9 ,fig4 #dd42 ! ro( is rev]s$ 9 direction m]ely to famili>ize ! /ud5t ) ! diff]5t aspects ( its l9es4 ,,problem ,,xvi4 ,,to ,,f9d ,,! ,,vani%+-,,po9t ,,( ,a ,,giv5 ,,incl9$ ,,l9e4 ,if1 9 ,fig4 #dc4 or ,fig4 #dd41 ! l9es ,,ay & ,a,y be produc$1 ! /ud5t w f9d t !y meet4 ,let ,p1 ,fig4 #de41 be ! po9t at :i* !y meet4 ,f ,p let fall ! v]tical ,,pv on ! sight-l9e1 cutt+ ! sight-l9e 9 ,v4 ,!n ! /ud5t w f9d exp]im5tally t ,v is ! vani%+-po9t ( ! l9e ,,ac4#bf ,complete ! rectangle ( ! base ,,ac1 by draw+ ,a,c to ,v1 & ,,cc to ! vani%+-po9t ( ,,aa4 ,jo9 ,y,c4 ,n[ if ,,yc & ,y,c be produc$ d[nw>ds1 ! /ud5t w f9d t !y meet4 ,let !m be produc$1 & meet 9 ,p4 ,produce ,,pv1 & x w be f\nd to pass ?r\gh ! po9t ,p4 ,!re=e if ,,ay 7or ,,cy71 ,fig4 #de41 be any incl9$ l9e drawn 9 p]spective by ,problem ,,xv41 & ,,ac ! relative horizontal 7,,ac 9 ,figs4 #ci1 #dj471 also drawn 9 p]spective4 ,?r\gh ,v1 ! vani%+-po9t ( ,,av1 draw ! v]tical ,,pp upw>ds & d[nw>ds4 ,produce ,,ay 7or ,,cy71 cutt+ ,,pp 9 ,p 7or ,p74 ,!n ,p is ! vani%+-po9t ( ,,ay 7or ,p ( ,,cy74 ,illu/ration3 ,fig4 #de4 ,! /ud5t w obs]ve t1 9 ord] to f9d ! po9t ,p by ?is me?od1 x is necess>y fir/ to draw a portion ( ! giv5 incl9$ l9e by ,problem ,,xv4 ,practically1 x is always necess>y to d s1 &1 !re=e1 ,i give ! problem 9 ?is =m4 ,!oretically1 z w be %[n 9 ! analysis ( ! problem1 ! po9t ,p %\ld be f\nd by draw+ a l9e f ! /ation-po9t p>allel to ! giv5 incl9$ l9e3 b !re is no practical means ( draw+ su* a l9e2 s t 9 :atev] t]ms ! problem may be giv51 a portion ( ! incl9$ l9e 7,,ay or ,,cy7 mu/ always be drawn 9 p]spective be=e ,p c be f\nd4 #bf ,! demon/ration is 9 ,app5dix ,,ii4 ,article ,,iii4 ,,problem ,,xvii4 ,,to ,,f9d ,,! ,,divid+-,,po9ts ,,( ,a ,,giv5 ,,incl9$ ,,l9e4 ,illu/ration3 ,fig4 #df4 ,let ,p1 ,fig4 #df41 be ! vani%+-po9t ( ! incl9$ l9e1 & ,v ! vani%+-po9t ( ! relative horizontal4 ,f9d ! divid+-po9ts ( ! relative horizontal1 ,d & ,d4 ,?r\gh ,p draw ! horizontal l9e ,,xy4 ,) c5t] ,p & di/ance ,,dp describe ! two arcs ,,dx & ,d,y1 cutt+ ! l9e ,,xy 9 ,x & ,y4 ,!n ,x & ,y are ! divid+-po9ts ( ! incl9$ l9e4#bg ,obs4 ,! divid+-po9ts f\nd by ! above rule1 us$ ) ! ord9>y measur+-l9e1 w lay (f di/ances on ! retir+ incl9$ l9e1 z ! ord9>y divid+-po9ts lay !m (f on ! retir+ horizontal l9e4 ,ano!r divid+-po9t1 peculi> 9 its application1 is sometimes useful1 & is to be f\nd z foll[s3-- ,illu/ration3 ,fig4 #dg4 ,let ,,ab1 ,fig4 #dg41 be ! giv5 incl9$ l9e drawn 9 p]spective1 & ,ac ! relative horizontal4 ,f9d ! vani%+-po9ts1 ,v & ,e1 ( ,ac & ,,ab2 ,d1 ! divid+-po9t ( ,ac2 & ! sight-magnitude ( ,ac on ! measur+-l9e1 or ,,ac4 ,f ,d erect ! p]p5dicul> ,,df4 ,jo9 ,,cb1 & produce x to cut ,,de 9 ,f4 ,jo9 ,,ef4 ,!n1 by simil> triangles1 ,,df is equal to ,,ev1 & ,,ef is p>allel to ,,dv4 ,h5ce x foll[s t if f ,d1 ! divid+-po9t ( ,ac1 we raise a p]p5dicul> & make ,,df equal to ,,ev1 a l9e ,,cf1 drawn f any po9t ,c on ! measur+-l9e to ,f1 w m>k ! di/ance ,,ab on ! incl9$ l9e1 ,,ab be+ ! portion ( ! giv5 incl9$ l9e :i* =ms ! diagonal ( ! v]tical rectangle ( :i* ,,ac is ! base4 #bg ,! demon/ration is 9 ,app5dix ,,ii41 p4 #ajd4 ,,problem ,,xviii4 ,,to ,,f9d ,,! ,,sight-,,l9e ,,( ,,an ,,incl9$ ,,plane ,,9 ,,:i* ,,two ,,l9es ,,are ,,giv5 ,,9 ,,position4#bh ,z 9 ord] to fix ! position ( a l9e two po9ts 9 x mu/ be giv51 s 9 ord] to fix ! position ( a plane1 two l9es 9 x mu/ be giv54 ,illu/ration3 ,fig4 #dh ,let ! two l9es be ,,ab & ,,cd1 ,fig4 #dh4 ,z !y are giv5 9 position1 ! relative horizontals ,,ae & ,,cf mu/ be giv54 ,!n by ,problem ,,xvi4 ! vani%+-po9t ( ,,ab is ,v1 & ( ,,cd1 ,v4 ,jo9 ,,vv & produce x to cut ! sight-l9e 9 ,x4 ,!n ,,vx is ! sight-l9e ( ! incl9$ plane4 ,l ! horizontal sight-l9e1 x is ( indef9ite l5g?2 & may be produc$ 9 ei!r direction z occasion requires1 cross+ ! horizontal l9e ( sight1 if ! plane cont9ues d[nw>d 9 t direction4 ,x is ! vani%+-po9t ( all horizontal l9es 9 ! incl9$ plane4 #bh ,read ! ,article on ?is problem 9 ! ,app5dix1 p4 #ig1 be=e inve/igat+ ! problem itself4 ,,problem ,,xix4 ,,to ,,f9d ,,! ,,vani%+-,,po9t ,,( ,,/eepe/ ,,l9es ,,9 ,,an ,,incl9$ ,,plane ,,^: ,,sight-,,l9e ,,is ,,giv54 ,illu/ration3 ,fig4 #di4 ,let ,,vx1 ,fig4 #di41 be ! giv5 sight-l9e4 ,produce x to cut ! horizontal sight-l9e 9 ,x4 ,!re=e ,x is ! vani%+-po9t ( horizontal l9es 9 ! giv5 incl9$ plane4 7,problem ,,xviii47 ,jo9 ,,tx1 & draw ,,ty at "r angles to ,,tx4 ,!re=e ,y is ! rectangul> vani%+-po9t correspond+ to ,x4#bi ,f ,y erect ! v]tical ,,yp1 cutt+ ! sight-l9e ( ! incl9$ plane 9 ,p4 ,!n ,p is ! vani%+-po9t ( /eepe/ l9es 9 ! plane4 ,all l9es drawn to x1 z ,,qp1 ,,rp1 ,,np1 etc41 are ! /eepe/ possible 9 ! plane2 & all l9es drawn to ,x1 z ,,qx1 ,,ox1 etc41 are horizontal1 & at "r angles to ! l9es ,,pq1 ,,pr1 etc4 #bi ,t is to say1 ! vani%+-po9t ( horizontal l9es drawn at "r angles to ! l9es ^: vani%+-po9t is ,x4 ,,problem ,,xx4 ,,to ,,f9d ,,! ,,vani%+-,,po9t ,,( ,,l9es ,,p]p5dicul> ,,to ,,! ,,surface ,,( ,a ,,giv5 ,,incl9$ ,,plane4 ,illu/ration3 ,fig4 #ej4 ,z ! incl9$ plane is giv51 "o ( its /eepe/ l9es mu/ be giv51 or may be asc]ta9$4 ,let ,,ab1 ,fig4 #ej41 be a portion ( a /eepe/ l9e 9 ! giv5 plane1 & ,v ! vani%+-po9t ( its relative horizontal4 ,?r\gh ,v draw ! v]tical ,,gf upw>ds & d[nw>ds4 ,f ,a set (f any portion ( ! relative horizontal ,,ac1 & on ,,ac describe a semicircle 9 a v]tical plane1 ,,adc1 cutt+ ,,ab 9 ,e4 ,jo9 ,,ec1 & produce x to cut ,,gf 9 ,f4 ,!n ,f is ! vani%+-po9t requir$4 ,=1 because ,,aec is an angle 9 a semicircle1 x is a "r angle2 & !re=e ! l9e ,,ef is at "r angles to ! l9e ,,ab2 & simil>ly all l9es drawn to ,f1 & !re=e p>allel to ,,ef1 are at "r angles ) any l9e :i* cuts !m1 drawn to ! vani%+-po9t ( ,,ab4 ,& because ! semicircle ,,adc is 9 a v]tical plane1 & its diamet] ,,ac is at "r angles to ! horizontal l9es trav]s+ ! surface ( ! incl9$ plane1 ! l9e ,,ec1 be+ 9 ?is semicircle1 is also at "r angles to su* trav]s+ l9es4 ,& !re=e ! l9e ,,ec1 be+ at "r angles to ! /eepe/ l9es 9 ! plane1 & to ! horizontal l9es 9 x1 is p]p5dicul> to its surface4 99 99 99 99 99 ,! prec$+ s]ies ( con/ructions1 ) ! examples 9 ! fir/ ,article ( ! ,app5dix1 put x 9 ! p[] ( ! /ud5t to draw any =m1 h[ev] complicat$1#cj :i* does n involve int]section ( curv$ surfaces4 ,i %all n proce$ to ! analysis ( any ( ^! m complex problems1 z !y are entirely useless 9 ! ord9>y practice ( arti/s4 ,= a few words only ,i mu/ ask ! read]'s fur!r pati5ce1 respect+ ! g5]al plac+ & scale ( ! picture4 ,z ! horizontal sight-l9e is drawn ?r\gh ! sight-po9t1 & ! sight-po9t is opposite ! eye1 ! sight-l9e is always on a level ) ! eye4 ,above & bel[ ! sight-l9e1 ! eye compreh5ds1 z x is rais$ or depress$ :ile ! head is held upright1 ab\t an equal space2 &1 on ea* side ( ! sight-po9t1 ab\t ! same space is easily se5 )\t turn+ ! head2 s t if a picture repres5t$ ! true field ( easy vision1 x ought to be circul>1 & h ! sight-po9t 9 its c5t]4 ,b because "s p>ts ( any giv5 view are usually m int]e/+ ?an o!rs1 ei!r ! un9t]e/+ p>ts are left out1 or some:at m ?an w\ld g5]ally be se5 ( ! int]e/+ p>ts is includ$1 by mov+ ! field ( ! picture a little upw>ds or d[nw>ds1 s z to ?r[ ! sight-po9t l[ or high4 ,! op]ation w be und]/ood 9 a mom5t by cutt+ an ap]ture 9 a piece ( pa/ebo>d1 & mov+ x up & d[n 9 front ( ! eye1 )\t mov+ ! eye4 ,x w be se5 to embrace sometimes ! l[1 sometimes ! high objects1 )\t alt]+ _! p]spective1 only ! eye w be opposite ! l[] "p ( ! ap]ture :5 x sees ! high] objects1 & vice v]s4 ,!re is no reason1 9 ! laws ( p]spective1 :y ! picture %\ld n be mov$ to ! "r or left ( ! sight-po9t1 z well z up or d[n4 ,b !re is ?is practical reason4 ,! mom5t ! spectator sees ! horizon 9 a picture high1 he tries to hold 8 head high1 t is1 9 its "r place4 ,:5 he sees ! horizon 9 a picture l[1 he simil>ly tries to put 8 head l[4 ,b1 if ! sight-po9t is ?r[n to ! left h& or "r h&1 he does n und]/& t he is to /ep a little to ! "r or left2 & if he places himself1 z usual1 9 ! middle1 all ! p]spective is di/ort$4 ,h5ce x is g5]ally unadvisable to remove ! sight-po9t lat]ally1 f ! c5t] ( ! picture4 ,! ,dut* pa9t]s1 h[ev]1 fe>lessly take ! lic5se ( plac+ x to ! "r or left2 & (t5 ) good effect4 ,! rectil9e> limitation ( ! sides1 top1 & base ( ! picture is ( c\rse q arbitr>y1 z ! space ( a l&scape w\ld be :i* 0 se5 ?r\gh a w9d[2 less or m be+ se5 at ! spectator's pleasure1 z he retires or advances4 ,! di/ance ( ! /ation-po9t is n s arbitr>y4 ,9 ord9>y cases x %\ld n be less ?an ! int5d$ greate/ dim5sion 7height or bread?7 ( ! picture4 ,9 mo/ works by ! great ma/]s x is m2 !y n only calculate on _! pictures be+ se5 at consid]able di/ances1 b !y l bread? ( mass 9 build+s1 & dislike ! %>p angles :i* always result f /ation-po9ts at %ort di/ances4#ca ,:5ev] p]spective1 done by true rule1 looks wrong1 x is always because ! /ation-po9t is too ne>4 ,det]m9e1 9 ! outset1 at :at di/ance ! spectator is likely to exam9e ! "w1 & nev] use a /ation-po9t )9 a less di/ance4 ,!re is yet ano!r & a v important reason1 n only = c>e 9 plac+ ! /ation-po9t1 b = t accurate calculation ( di/ance & obs]vance ( measurem5t :i* h be5 insi/$ on ?r\gh\t ?is "w4 ,all draw+s ( objects on a r$uc$ scale are1 if rightly execut$1 draw+s ( ! appe>ance ( ! object at ! di/ance :i* 9 true p]spective r$uces x to t scale4 ,!y are n small draw+s ( ! object se5 ne>1 b draw+s ! real size ( ! object se5 f> (f4 ,?us if y draw a m\nta9 9 a l&scape1 ?ree in*es high1 y d n r$uce all ! features ( ! ne> m\nta9 s z to come into ?ree in*es ( pap]4 ,y c\ld n d t4 ,all t y c d is to give ! appe>ance ( ! m\nta91 :5 x is s f> (f t ?ree in*es ( pap] w\ld really hide x f y4 ,x is precisely ! same 9 draw+ any o!r object4 ,a face c no m be r$uc$ 9 scale ?an a m\nta9 c4 ,x is inf9itely delicate already2 x c only be q rightly r5d]$ on its own scale1 or at lea/ on ! slightly dim9i%$ scale :i* w\ld be fix$ by plac+ ! plate ( glass1 suppos$ to repres5t ! field ( ! picture1 close to ! figures4 ,correggio & ,raphael 7 bo? fond ( ?is slightly subdu$ magnitude ( figure4 ,colossal pa9t+1 9 :i* ,correggio excell$ all o!rs1 is usually ! enl>gem5t ( a small picture 7z a colossal sculpture is ( a small /atue71 9 ord] to p]mit ! subject ( x to be disc]n$ at a di/ance4 ,! treatm5t ( colossal 7z di/+ui%$ f ord9>y7 pa9t+s w dep5d !re=e1 9 g5]al1 on ! pr9ciples ( optics m ?an on ^? ( p]spective1 ?\gh1 occasionally1 portions may be repres5t$ z if !y 7 ! projection ( ne> objects on a plane beh9d !m4 ,9 all po9ts ! subject is "o ( great difficulty & subtlety2 & its exam9ation does n fall )9 ! compass ( ?is essay4 ,la/ly1 x w foll[ f ^! consid]ations1 & ! conclusion is "o ( great practical importance1 t1 ?\gh pictures may be enl>g$1 !y _c be r$uc$1 9 copy+ !m4 ,all attempts to engrave pictures completely on a r$uc$ scale are1 = ?is reason1 nugatory4 ,! be/ t c be done is to give ! aspect ( ! picture at ! di/ance :i* r$uces x 9 p]spective to ! size requir$2 or1 9 o!r words1 to make a draw+ ( ! di/ant effect ( ! picture4 ,good pa9t+1 l nature's own "w1 is inf9ite1 & unr$uceable4 ,i wi% ?is book _h less t5d5cy t[>ds ! inf9ite & unr$uceable4 ,x has s f> exce$$ ! limits ,i hop$ to give x1 t ,i d\bt n ! read] w p>don an abruptness ( conclusion1 & be ?ankful1 z ,i am myself1 to get to an end on any t]ms4 #cj ,z 9 algebraic sci5ce1 mu* dep5ds1 9 complicat$ p]spective1 on ! /ud5t's ready inv5tion ( exp$i5ts1 & on 8 quick sight ( ! %orte/ way 9 :i* ! solution may be accompli%$1 :5 !re are sev]al ways4 #ca ,! greate/ ma/]s are also fond ( p>allel p]spective1 t is to say1 ( hav+ "o side ( _! build+s front+ !m full1 & !re=e p>allel to ! picture plane1 :ile ! o!r side vani%es to ! sight-po9t4 ,?is is almo/ always done 9 figure backgr\nds1 secur+ simple & balanc$ l9es4 ,,app5dix4 ,i4 ,,practice ,,& ,,obs]vations4 ,,ii4 ,,demon/rations4 ,i4 ,,practice ,,& ,,obs]vations ,,on ,,! ,,prec$+ ,,problems4 ,,problem ,i4 ,an example w be necess>y to make ?is problem cle> to ! g5]al /ud5t4 ,! ne>e/ corn] ( a piece ( patt]n on ! c>pet is #d feet b5ea? ! eye1 #b feet to our "r & #c feet 9 direct di/ance f u4 ,we int5d to make a draw+ ( ! patt]n :i* %all be se5 prop]ly :5 held #a foot f ! eye4 ,x is requir$ to fix ! position ( ! corn] ( ! piece ( patt]n4 ,illu/ration3 ,fig4 #ea4 ,let ,,ab1 ,fig4 #ea41 be our %eet ( pap]1 "s #c feet wide4 ,make ,,/ equal to #a foot4 ,draw ! l9e ( sight ?r\gh ,s4 ,produce ,,ts1 & make ,,ds equal to #b feet1 !re=e ,,td equal to #c feet4 ,draw ,,dc1 equal to #b feet2 ,,cp1 equal to #d feet4 ,jo9 ,,tc 7cutt+ ! sight-l9e 9 ,q7 & ,,tp4 ,let fall ! v]tical ,,qp1 !n ,p is ! po9t requir$4 ,if ! l9es1 z 9 ! figure1 fall outside ( y\r %eet ( pap]1 9 ord] to draw !m1 x is necess>y to atta* o!r %eets ( pap] to its edges4 ,?is is inconv5i5t1 b mu/ be done at fir/ t y may see y\r way cle>ly2 & sometimes aft]w>ds1 ?\gh !re are exp$i5ts = do+ )\t su* ext5sion 9 fa/ sket*+4 ,x is evid5t1 h[ev]1 t no ext5sion ( surface c\ld be ( any use to u1 if ! di/ance ,,td1 in/ead ( be+ #c feet1 7 #ajj feet1 or a mile1 z x might easily be 9 a l&scape4 ,x is necess>y1 !re=e1 to obta9 "s o!r means ( con/ruction2 to d :i* we mu/ exam9e ! pr9ciple ( ! problem4 ,9 ! analysis ( ,fig4 #b41 9 ! introductory rem>ks1 ,i us$ ! ^w 8height0 only ( ! t[]1 ,,qp1 because x 0 only to its v]tical height t ! law d$uc$ f ! figure c\ld be appli$4 ,= suppose x _h be5 a pyramid1 z ,,oqp1 ,fig4 #eb41 !n ! image ( its side1 ,,qp1 be+1 l e o!r magnitude1 limit$ on ! glass ,,ab by ! l9es com+ f its extremities1 w\ld appe> only ( ! l5g? ,q,s2 & x is n true t ,q,s is to ,,qp z ,,ts is to ,,tp4 ,b if we let fall a v]tical ,,qd f ,q1 s z to get ! v]tical height ( ! pyramid1 !n x is true t ,q,s is to ,,qd z ,,ts is to ,,td4 ,illu/ration3 ,fig4 #eb4 ,suppos+ ?is figure repres5t$1 n a pyramid1 b a triangle on ! gr\nd1 & t ,,qd & ,,qp are horizontal l9es1 express+ lat]al di/ance f ! l9e ,,td1 /ill ! rule w\ld be false = ,,qp & true = ,,qd4 ,&1 simil>ly1 x is true = all l9es :i* are p>allel1 l ,,qd1 to ! plane ( ! picture ,,ab1 & false = all l9es :i* are incl9$ to x at an angle4 ,h5ce g5]ally4 ,let ,,pq 7,fig4 #b4 9 ,introduction1 p4 #f7 be any magnitude p>allel to ! plane ( ! picture2 & ,p,q its image on ! picture4 ,!n always ! =mula is true :i* y le>n$ 9 ! ,introduction3 ,p,q is to ,,pq z ,,/ is to ,,dt4 ,n[ ! magnitude ,p da% ,q da% 9 ?is =mula ,i call ! 8,,sight-,,magnitude0 ( ! l9e ,,pq4 ,! /ud5t mu/ fix ?is t]m1 & ! mean+ ( x1 well 9 8 m9d4 ,! 8sight-magnitude0 ( a l9e is ! magnitude :i* be>s to ! real l9e ! same proportion t ! di/ance ( ! picture be>s to ! di/ance ( ! object4 ,?us1 if a t[] be a hundr$ feet high1 & a hundr$ y>ds (f2 & ! picture1 or piece ( glass1 is "o y>d f ! spectator1 betwe5 him & ! t[]2 ! di/ance ( picture be+ !n to di/ance ( t[] z #a to #ajj1 ! sight-magnitude ( ! t[]'s height w be z #a to #ajj2 t is to say1 "o foot4 ,if ! t[] is two hundr$ y>ds di/ant1 ! sight-magnitude ( its height w be half a foot1 & s on4 ,b f>!r4 ,x is con/antly necess>y1 9 p]spective op]ations1 to measure ! o!r dim5sions ( objects by ! sight-magnitude ( _! v]tical l9es4 ,?us1 if ! t[]1 :i* is a hundr$ feet high1 is squ>e1 & tw5ty-five feet broad on ea* side2 if ! sight-magnitude ( ! height is "o foot1 ! measurem5t ( ! side1 r$uc$ to ! same scale1 w be ! hundr$? "p ( tw5ty-five feet1 or ?ree in*es3 &1 accord+ly1 ,i use 9 ?is treatise ! t]m 8sight-magnitude0 indiscrim9ately = all l9es r$uc$ 9 ! same proportion z ! v]tical l9es ( ! object4 ,if ,i tell y to f9d ! 8sight-magnitude0 ( any l9e1 ,i mean1 always1 f9d ! magnitude :i* be>s to t l9e ! proportion ( ,,/ to ,,dt2 or1 9 simpl] t]ms1 r$uce ! l9e to ! scale :i* y h fix$ by ! fir/ det]m9ation ( ! l5g? ,,/4 ,!re=e1 y mu/ le>n to draw quickly to scale be=e y d any?+ else2 = all ! measurem5ts ( y\r object mu/ be r$uc$ to ! scale fix$ by ,,/ be=e y c use !m 9 y\r diagram4 ,if ! object is fifty feet f y1 & y\r pap] "o foot1 all ! l9es ( ! object mu/ be r$uc$ to a scale ( "o fiftie? be=e y c use !m2 if ! object is two ?\s& feet f y1 & y\r pap] "o foot1 all y\r l9es mu/ be r$uc$ to ! scale ( "o two-?\s&? be=e y c use !m1 & s on4 ,only 9 ultimate practice1 ! r$uction nev] ne$ be tiresome1 =1 9 ! case ( l>ge di/ances1 accuracy is nev] requir$4 ,if a build+ is ?ree or f\r miles di/ant1 a hairbread? ( accid5tal v>iation 9 a t\* makes a diff]5ce ( t5 or tw5ty feet 9 height or bread?1 if e/imat$ by accurate p]spective law4 ,h5ce x is nev] attempt$ to apply measurem5ts ) precision at su* di/ances4 ,measurem5ts are only requir$ )9 di/ances (1 at ! mo/1 two or ?ree hundr$ feet4 ,?us x may be necess>y to repres5t a ca!dral nave precisely z se5 f a spot sev5ty feet 9 front ( a giv5 pill>2 b we %all h>dly be requir$ to draw a ca!dral ?ree miles di/ant precisely z se5 f sev5ty feet 9 advance ( a giv5 mile/one4 ,( c\rse1 if su* a ?+ be requir$1 x c be done2 only ! r$uctions are some:at long & complicat$3 9 ord9>y cases x is easy to assume ! di/ance ,,/ s z to get at ! r$uc$ dim5sions 9 a mom5t4 ,?us1 let ! pill> ( ! nave1 9 ! case suppos$1 be #db feet high1 & we are requir$ to /& #gj feet f x3 assume ,,/ to be equal to #e feet4 ,!n1 z #e is to #gj s w ! sight-magnitude requir$ be to #db2 t is to say1 ! sight-magnitude ( ! pill>'s height w be #c feet4 ,if we make ,,/ equal to #b feet1 ! pill>'s height w be #a foot1 & s on4 ,& = f9e divisions into irregul> p>ts :i* _c be measur$1 ! n9? & t5? problems ( ! six? book ( ,euclid w s]ve y3 ! foll[+ con/ruction is1 h[ev]1 ,i ?9k1 m practically conv5i5t3-- ,! l9e ,,ab 7,fig4 #ec47 is divid$ by giv5 po9ts1 a1 b1 c1 into a giv5 numb] ( irregul>ly unequal p>ts2 x is requir$ to divide any o!r l9e1 ,,cd1 into an equal numb] ( p>ts1 be>+ to ea* o!r ! same proportions z ! p>ts ( ,,ab1 & arrang$ 9 ! same ord]4 ,draw ! two l9es p>allel to ea* o!r1 z 9 ! figure4 ,jo9 ,,ac & ,,bd1 & produce ! l9es ,,ac1 ,,bd1 till !y meet 9 ,p4 ,jo9 a,p1 b,p1 c,p1 cutt+ c,d 9 f1 g1 h4 ,!n ! l9e ,,cd is divid$ z requir$1 9 f1 g1 h4 ,9 ! figure ! l9es ,,ab & ,,cd are accid5tally p]p5dicul> to ,,ap4 ,!re is no ne$ = _! be+ s4 ,illu/ration3 ,fig4 #ec4 ,n[1 to return to our fir/ problem4 ,! con/ruction giv5 9 ! figure is only ! quicke/ ma!matical way ( obta9+1 on ! picture1 ! sight-magnitudes ( ,,dc & ,,pc1 :i* are bo? magnitudes p>allel ) ! picture plane4 ,b if ^! magnitudes are too great to be ?us put on ! pap]1 y h only to obta9 ! r$uction by scale4 ,?us1 if ,,ts be "o foot1 ,,td eighty feet1 ,,dc =ty feet1 & ,,cp n9ety feet1 ! di/ance ,,qs mu/ be made equal to "o eightie? ( ,,dc1 or half a foot2 & ! di/ance ,,qp1 "o eightie? ( ,,cp1 or "o eightie? ( n9ety feet2 t is to say1 n9e eigh?s ( a foot1 or ?irte5 & a half in*es4 ,! l9es ,,ct & ,,pt are ?us practically useless1 x be+ only necess>y to measure ,,qs & ,,qp1 on y\r pap]1 ( ! due sight-magnitudes4 ,b ! ma!matical con/ruction1 giv5 9 ,problem ,i41 is ! basis ( all succe$+ problems1 &1 if x is once ?or\ghly und]/ood & practic$ 7x c only be ?or\ghly und]/ood by practice71 all ! o!r problems w foll[ easily4 ,la/ly4 ,obs]ve t any p]spective op]ation :atev] may be p]=m$ ) r$uc$ dim5sions ( e l9e employ$1 s z to br+ x conv5i5tly )9 ! limits ( y\r pap]4 ,:5 ! requir$ figure is ?us con/ruct$ on a small scale1 y h only to enl>ge x accurately 9 ! same proportion 9 :i* y r$uc$ ! l9es ( con/ruction1 & y w h ! figure con/ruct$ 9 p]spective on ! scale requir$ = use4 ,,problem ,,ix4 ,! draw+ ( mo/ build+s occurr+ 9 ord9>y practice w resolve itself into applications ( ?is problem4 ,9 g5]al1 any h\se1 or block ( h\ses1 pres5ts itself "u ! ma9 conditions assum$ "h 9 ,fig4 #ed4 ,!re w be an angle or corn] some:]e ne> ! spectator1 z ,,ab2 & ! level ( ! eye w usually be above ! base ( ! build+1 ( :i*1 !re=e1 ! horizontal upp] l9es w slope d[n to ! vani%+-po9ts1 & ! base l9es rise to !m4 ,! foll[+ practical directions w1 h[ev]1 meet ne>ly all cases3-- ,illu/ration3 ,fig4 #ed4 ,let ,,ab1 ,fig4 #ed41 be any important v]tical l9e 9 ! block ( build+s2 if x is ! side ( a /reet1 y may fix ^u su* a l9e at ! division betwe5 two h\ses4 ,if its real height1 di/ance1 etc41 are giv51 y w proce$ ) ! accurate con/ruction ( ! problem2 b usually y w nei!r "k1 nor c>e1 exactly h[ high ! build+ is1 or h[ f> (f4 ,9 su* case draw ! l9e ,,ab1 z ne>ly z y c guess1 ab\t ! "p ( ! picture x ought to occupy1 & on su* a scale z y *oose4 ,divide x into any conv5i5t numb] ( equal p>ts1 accord+ to ! height y presume x to be4 ,if y suppose x to be tw5ty feet high1 y may divide x into tw5ty p>ts1 & let ea* "p /& = a foot2 if ?irty feet high1 y may divide x into t5 p>ts1 & let ea* "p /& = ?ree feet2 if sev5ty feet high1 into f\rte5 p>ts1 & let ea* "p /& = five feet2 & s on1 avoid+ ?us v m9ute divisions till y come to details4 ,!n obs]ve h[ high y\r eye rea*es ^u ?is v]tical l9e2 suppose1 = in/ance1 t x is ?irty feet high & divid$ into t5 p>ts1 & y are /&+ s z to raise y\r head to ab\t six feet above its base1 !n ! sight-l9e may be drawn1 z 9 ! figure1 ?r\gh ! second division f ! gr\nd4 ,if y are /&+ above ! h\se1 draw ! sight-l9e above ,b2 if bel[ ! h\se1 bel[ ,a2 at su* height or dep? z y suppose may be accurate 7a y>d or two m or less matt]s little at ord9>y di/ances1 :ile at great di/ances p]spective rules become ne>ly useless1 ! eye s]v+ y bett] ?an ! necess>ily imp]fect calculation74 ,!n fix y\r sight-po9t & /ation-po9t1 ! latt] ) prop] ref]5ce to ! scale ( ! l9e ,,ab4 ,z y _c1 9 all probability1 asc]ta9 ! exact direction ( ! l9e ,,av or ,,bv1 draw ! slope ,,bv z x appe>s to y1 cutt+ ! sight-l9e 9 ,v4 ,?us hav+ fix$ "o vani%+-po9t1 ! o!r1 & ! divid+-po9ts1 mu/ be accurately f\nd by rule2 =1 z be=e /at$1 :e!r y\r entire gr\p ( po9ts 7vani%+ & divid+7 falls a little m or less to ! "r or left ( ,s does n signify1 b ! relation ( ! po9ts to ea* o!r does signify4 ,!n draw ! measur+-l9e ,,bg1 ei!r ?r\gh ,a or ,b1 *oos+ always ! /eep] slope ( ! two2 divide ! measur+-l9e into p>ts ( ! same l5g? z ^? us$ on ,,ab1 & let !m /& = ! same magnitudes4 ,?us1 suppose !re are two r[s ( w9d[s 9 ! h\se front1 ea* w9d[ six feet high by ?ree wide1 & sep>at$ by int]vals ( ?ree feet1 bo? betwe5 w9d[ & w9d[ & betwe5 ti] & ti]2 ea* ( ! divisions "h /&+ = ?ree feet1 ! l9es drawn f ,,bg to ! divid+-po9t ,d fix ! lat]al dim5sions1 & ! divisions on ,,ab ! v]tical ones4 ,= o!r magnitudes x w\ld be necess>y to subdivide ! p>ts on ! measur+-l9e1 or on ,,ab1 z requir$4 ,! l9es :i* regulate ! inn] sides or returns ( ! w9d[s 7a1 b1 c1 etc47 ( c\rse are drawn to ! vani%+-po9t ( ,,bf 7! o!r side ( ! h\se71 if ,,fbv repres5ts a "r angle2 if n1 _! own vani%+-po9t mu/ be f\nd sep>ately = ^! returns4 ,b see ,practice on ,problem ,,xi4 ,illu/ration3 ,fig4 #ee4 ,int]ior angles1 su* z ,,ebc1 ,fig4 #ee4 7suppose ! corn] ( a room71 are to be treat$ 9 ! same way1 ea* side ( ! room hav+ its measurem5ts sep>ately c>ri$ to x f ! measur+-l9e4 ,x may sometimes happ5 9 su* cases t we h to c>ry ! measurem5t up f ! corn] ,b1 & t ! sight-magnitudes are giv5 u f ! l5g? ( ! l9e ,,ab4 ,= in/ance1 suppose ! room is eighte5 feet high1 & !re=e ,,ab is eighte5 feet2 & we h to lay (f l5g?s ( six feet on ! top ( ! room wall1 ,,bc4 ,f9d ,d1 ! divid+-po9t ( ,,bc4 ,draw a measur+-l9e1 ,,bf1 f ,b2 & ano!r1 g,c1 any:]e above4 ,on ,,bf lay (f ,,bg equal to "o ?ird ( ,,ab1 or six feet2 & draw f ,d1 ?r\gh ,g & ,b1 ! l9es ,gg1 ,bb1 to ! upp] measur+-l9e4 ,!n gb is six feet on t measur+-l9e4 ,make bc1 *1 etc41 equal to bg2 & draw ce1 hf1 etc41 to ,d1 cutt+ ,,bc 9 e & f1 :i* m>k ! requir$ l5g?s ( six feet ea* at ! top ( ! wall4 ,,problem ,x4 ,?is is "o ( ! mo/ important f\ndational problems 9 p]spective1 & x is necess>y t ! /ud5t %\ld entirely famili>ize himself ) its conditions4 ,9 ord] to d s1 he mu/ fir/ obs]ve ^! g5]al relations ( magnitude 9 any pyramid on a squ>e base4 ,let ,,agh1 ,fig4 #ef41 be any pyramid on a squ>e base4 ,illu/ration3 ,fig4 #ef4 ,! be/ t]ms 9 :i* its magnitude c be giv51 are ! l5g? ( "o side ( its base1 ,,ah1 & its v]tical altitude 7,,cd 9 ,fig4 #be472 =1 kn[+ ^!1 we "k all ! o!r magnitudes4 ,b ^! are n ! t]ms 9 :i* its size w be usually asc]ta9able4 ,g5]ally1 we %all h giv5 u1 & be able to asc]ta9 by measurem5t1 "o side ( its base ,,ah1 & ei!r ,,ag ! l5g? ( "o ( ! l9es ( its angles1 or ,,bg 7or ,b,g7 ! l5g? ( a l9e drawn f its v]tex1 ,g1 to ! middle ( ! side ( its base4 ,9 measur+ a real pyramid1 ,,ag w usually be ! l9e mo/ easily f\nd2 b 9 _m ar*itectural problems ,,bg is giv51 or is mo/ easily asc]ta9able4 ,obs]ve !re=e ?is g5]al con/ruction4 ,illu/ration3 ,fig4 #eg4 ,let ,,abde1 ,fig4 #eg41 be ! squ>e base ( any pyramid4 ,draw its diagonals1 ,,ae1 ,,bd1 cutt+ ea* o!r 9 its c5t]1 ,c4 ,bisect any side1 ,,ab1 9 ,f4 ,f ,f erect v]tical ,,fg4 ,produce ,,fb to ,h1 & make ,,fh equal to ,,ac4 ,n[ if ! v]tical altitude ( ! pyramid 7,,cd 9 ,fig4 #be47 be giv51 make ,,fg equal to ?is v]tical altitude4 ,jo9 ,,gb & ,,gh4 ,!n ,,gb & ,,gh are ! true magnitudes ( ,,gb & ,,gh 9 ,fig4 #ef4 ,if ,,gb is giv51 & n ! v]tical altitude1 ) c5t] ,b1 & di/ance ,,gb1 describe circle cutt+ ,,fg 9 ,g1 & ,,fg is ! v]tical altitude4 ,if ,,gh is giv51 describe ! circle f ,h1 ) di/ance ,,gh1 & x w simil>ly cut ,,fg 9 ,g4 ,x is especially necess>y = ! /ud5t to exam9e ?is con/ruction ?or\ghly1 because 9 _m complicat$ =ms ( ornam5ts1 capitals ( columns1 etc41 ! l9es ,,bg & ,,gh become ! limits or bases ( curves1 :i* are elongat$ on ! long] 7or angle7 pr(ile ,,gh1 & %ort5$ on ! %ort] 7or lat]al7 pr(ile ,,bg4 ,we w take a simple in/ance1 b mu/ previ\sly note ano!r con/ruction4 ,x is (t5 necess>y1 :5 pyramids are ! roots ( "s ornam5tal =m1 to divide !m horizontally at a giv5 v]tical height4 ,! %orte/ way ( do+ s is 9 g5]al ! foll[+4 ,illu/ration3 ,fig4 #eh4 ,let ,,aec1 ,fig4 #eh41 be any pyramid on a squ>e base ,,abc1 & ,,adc ! squ>e pill> us$ 9 its con/ruction4 ,!n by con/ruction 7,problem ,x47 ,,bd & ,,af are bo? ( ! v]tical height ( ! pyramid4 ,( ! diagonals1 ,,fe1 ,,de1 *oose ! %orte/ 79 ?is case ,,de71 & produce x to cut ! sight-l9e 9 ,v4 ,!re=e ,v is ! vani%+-po9t ( ,,de4 ,divide ,,db1 z may be requir$1 into ! sight-magnitudes ( ! giv5 v]tical heights at :i* ! pyramid is to be divid$4 ,illu/ration3 ,fig4 #ei4 ,illu/ration3 ,fig4 #fj4 ,f ! po9ts ( division1 #a1 #b1 #c1 etc41 draw to ! vani%+-po9t ,v4 ,! l9es s drawn cut ! angle l9e ( ! pyramid1 ,,be1 at ! requir$ elevations4 ,?us1 9 ! figure1 x is requir$ to draw a horizontal black b& on ! pyramid at ?ree fif?s ( its height1 & 9 bread? "o tw5tie? ( its height4 ,! l9e ,,bd is divid$ into five p>ts1 ( :i* ?ree are c\nt$ f ,b upw>ds4 ,!n ! l9e drawn to ,v m>ks ! base ( ! black b&4 ,!n "o f\r? ( "o ( ! five p>ts is measur$1 :i* simil>ly gives ! bread? ( ! b&4 ,! t]m9al l9es ( ! b& are !n drawn on ! sides ( ! pyramid p>allel to ,,ab 7or to its vani%+-po9t if x has "o71 & to ! vani%+-po9t ( ,,bc4 ,if x happ5s t ! vani%+-po9ts ( ! diagonals are awkw>dly plac$ = use1 bisect ! ne>e/ base l9e ( ! pyramid 9 ,b1 z 9 ,fig4 #ei4 ,erect ! v]tical ,,db & jo9 ,,gb & ,,dg 7,g be+ ! apex ( pyramid74 ,f9d ! vani%+-po9t ( ,,dg1 & use ,,db = division1 c>ry+ ! measurem5ts to ! l9e ,,gb4 ,9 ,fig4 #ei41 if we jo9 ,,ad & ,,dc1 ,,adc is ! v]tical pr(ile ( ! :ole pyramid1 & ,,bdc ( ! half pyramid1 correspond+ to ,,fgb 9 ,fig4 #eg4 ,illu/ration3 ,fig4 #fa4 ,we may n[ proce$ to an ar*itectural example4 ,let ,,ah1 ,fig4 #fj41 be ! v]tical pr(ile ( ! capital ( a pill>1 ,,ab ! semi-diamet] ( its head or abacus1 & ,,fd ! semi-diamet] ( its %aft4 ,let ! %aft be circul>1 & ! abacus squ>e1 d[n to ! level ,e4 ,jo9 ,,bd1 ,,ef1 & produce !m to meet 9 ,g4 ,!re=e ,,ecg is ! semi-pr(ile ( a rev]s$ pyramid conta9+ ! capital4 ,con/ruct ?is pyramid1 ) ! squ>e ( ! abacus1 9 ! requir$ p]spective1 z 9 ,fig4 #fa42 mak+ ,,ae equal to ,,ae 9 ,fig4 #fj41 & ,,ak1 ! side ( ! squ>e1 equal to twice ,,ab 9 ,fig4 #fj4 ,make ,,eg equal to ,,cg1 & ,,ed equal to ,,cd4 ,draw ,,df to ! vani%+-po9t ( ! diagonal ,,dv 7! figure is too small to include ?is vani%+-po9t71 & ,f is ! level ( ! po9t ,f 9 ,fig4 #fj41 on ! side ( ! pyramid4 ,draw ,fm1 ,fn1 to ! vani%+-po9ts ( ,,ah & ,,ak4 ,!n ,fn & ,fm are horizontal l9es across ! pyramid at ! level ,f1 =m+ at t level two sides ( a squ>e4 ,illu/ration3 ,fig4 #fb4 ,complete ! squ>e1 & )9 x inscribe a circle1 z 9 ,fig4 #fb41 :i* is left unlett]$ t its con/ruction may be cle>4 ,at ! extremities ( ?is draw v]tical l9es1 :i* w be ! sides ( ! %aft 9 its "r place4 ,x w be f\nd to be some:at small] 9 diamet] ?an ! entire %aft 9 ,fig4 #fj41 because at ! c5t] ( ! squ>e x is m di/ant ?an ! ne>e/ edge ( ! squ>e abacus4 ,! curves ( ! capital may !n be drawn approximately by ! eye4 ,!y are n q accurate 9 ,fig4 #fb41 !re be+ a subtlety 9 _! junction ) ! %aft :i* c\ld n be %[n on s small a scale )\t confus+ ! /ud5t2 ! curve on ! left spr++ f a po9t a little way r\nd ! circle beh9d ! %aft1 & t on ! "r f a po9t on ?is side ( ! circle a little way )9 ! edge ( ! %aft4 ,b = _! m accurate con/ruction see ,notes on ,problem ,,xiv4 ,,problem ,,xi4 ,x is seldom t any complicat$ curve1 except occasionally a spiral1 ne$s to be drawn 9 p]spective2 b ! /ud5t w d well to practice = "s "t any fanta/ic %apes :i* he c f9d drawn on flat surfaces1 z on wall-pap]s1 c>pets1 etc41 9 ord] to accu/om himself to ! /range & great *anges :i* p]spective causes 9 !m4 ,illu/ration3 ,fig4 #fc4 ,! curves mo/ requir$ 9 ar*itectural draw+1 aft] ! circle1 are ^? ( po9t$ ar*es2 9 :i*1 h[ev]1 all t w be g5]ally ne$$ is to fix ! apex1 & two po9ts 9 ! sides4 ,?us if we h to draw a range ( po9t$ ar*es1 su* z ,,apb1 ,fig4 #fc41 draw ! measur$ ar* to its sight-magnitude fir/ neatly 9 a rectangle1 ,,abcd2 !n draw ! diagonals ,,ad & ,,bc2 :]e !y cut ! curve draw a horizontal l9e 7z at ! level ,e 9 ! figure71 & c>ry x along ! range to ! vani%+-po9t1 fix+ ! po9ts :]e ! ar*es cut _! diagonals all along4 ,if ! ar* is cusp$1 a l9e %\ld be drawn1 at ,f to m>k ! height ( ! cusps1 & v]ticals rais$ at ,g & ,h1 to det]m9e ! int]val betwe5 !m4 ,any o!r po9ts may be simil>ly det]m9$1 b ^! w usually be en\gh4 ,figure #fc4 %[s ! p]spective con/ruction ( a squ>e ni*e ( good ,v]onese ,go?ic1 ) an uncusp$ ar* ( simil> size & curve beyond4 ,illu/ration3 ,fig4 #fd4 ,9 ,fig4 #fd4 ! m di/ant ar* only is lett]$1 z ! con/ruction ( ! ne>e/ expla9s itself m cle>ly to ! eye )\t lett]s4 ,! m di/ant ar* %[s ! g5]al con/ruction = all ar*es se5 und]nea?1 z ( bridges1 ca!dral aisles1 etc4 ,! rectangle ,,abcd is fir/ drawn to conta9 ! outside ar*2 !n ! dep? ( ! ar*1 ,aa1 is det]m9$ by ! measur+-l9e1 & ! rectangle1 abcd1 drawn = ! inn] ar*4 ,aa1 ,bb1 etc41 g to "o vani%+-po9t2 ,,ab1 ab1 etc41 to ! opposite "o4 ,9 ! ne>] ar* ano!r n>r[ rectangle is drawn to det]m9e ! cusp4 ,! p>ts :i* w\ld actually come into sight are slightly %ad$4 ,,problem ,,xiv4 ,sev]al ex]cises w be requir$ on ?is important problem4 ,i4 ,x is requir$ to draw a circul> flat-bottom$ di% n>r[] at ! bottom ?an ! top2 ! v]tical dep? be+ giv51 & ! diamet] at ! top & bottom4 ,illu/ration3 ,fig4 #fe4 ,let ab1 ,fig4 #fe41 be ! diamet] ( ! bottom1 ac ! diamet] ( ! top1 & ad its v]tical dep?4 ,take ,,ad 9 position equal to ac4 ,on ,,ad draw ! squ>e ,,abcd1 & inscribe 9 x a circle4 ,!re=e1 ! circle s inscrib$ has ! diamet] ( ! top ( ! di%4 ,f ,a & ,d let fall v]ticals1 ,,ae1 ,,dh1 ea* equal to ad4 ,jo9 ,,eh1 & describe squ>e ,,efgh1 :i* accord+ly w be equal to ! squ>e ,,abcd1 & be at ! dep? ad b5ea? x4 ,)9 ! squ>e ,,efgh describe a squ>e ,,ik1 ^: diamet] %all be equal to ab4 ,describe a circle )9 ! squ>e ,,ik4 ,!re=e ! circle s inscrib$ has its diamet] equal to ab2 & x is 9 ! c5t] ( ! squ>e ,,efgh1 :i* is v]tically b5ea? ! squ>e ,,abcd4 ,!re=e ! circle 9 ! squ>e ,,ik repres5ts ! bottom ( ! di%4 ,n[ ! two circles ?us drawn w ei!r int]sect "o ano!r1 or !y w n4 ,if !y int]sect "o ano!r1 z 9 ! figure1 & !y are bel[ ! eye1 "p ( ! bottom ( ! di% is se5 )9 x4 ,illu/ration3 ,fig4 #ff4 ,to avoid confusion1 let u take !n two int]sect+ circles )\t ! inclos+ squ>es1 z 9 ,fig4 #ff4 ,draw "r l9es1 ab1 cd1 t\*+ bo? circles ext]nally4 ,!n ! p>ts ( ^! l9es :i* connect ! circles are ! sides ( ! di%4 ,!y are drawn 9 ,fig4 #fe4 )\t any prolongations1 b ! be/ way to con/ruct !m is z 9 ,fig4 #ff4 ,if ! circles d n int]sect ea* o!r1 ! small] mu/ ei!r be )9 ! l>g] or n )9 x4 ,if )9 ! l>g]1 ! :ole ( ! bottom ( ! di% is se5 f above1 ,fig4 #fg4 a4 ,illu/ration3 ,fig4 #fg4 ,if ! small] circle is n )9 ! l>g]1 none ( ! bottom is se5 inside ! di%1 b4 ,if ! circles are above in/ead ( b5ea? ! eye1 ! bottom ( ! di% is se5 b5ea? x1 c4 ,if "o circle is above & ano!r b5ea? ! eye1 nei!r ! bottom nor top ( ! di% is se51 d4 ,unless ! object be v l>ge1 ! circles 9 ?is case w h little app>5t curvature4 ,,ii4 ,! prec$+ problem is simple1 because ! l9es ( ! pr(ile ( ! object 7ab & cd1 ,fig4 #ff47 are /raight4 ,b if ^! l9es ( pr(ile are curv$1 ! problem becomes mu* m complex3 once ma/]$1 h[ev]1 x leaves no f>!r difficulty 9 p]spective4 ,let x be requir$ to draw a flatti% circul> cup or vase1 ) a giv5 curve ( pr(ile4 ,! basis ( con/ruction is giv5 9 ,fig4 #fh41 half ( x only be+ drawn1 9 ord] t ! eye may seize its l9es easily4 ,illu/ration3 ,fig4 #fh4 ,two squ>es 7( ! requir$ size7 are fir/ drawn1 "o above ! o!r1 ) a giv5 v]tical int]val1 ,,ac1 betwe5 !m1 & ea* is divid$ into eight p>ts by its diamet]s & diagonals4 ,9 ^! squ>es two circles are drawn2 :i* are1 !re=e1 ( equal size1 & "o above ! o!r4 ,two small] circles1 also ( equal size1 are drawn )9 ^! l>g] circles 9 ! con/ruction ( ! pres5t problem2 m may be necess>y 9 "s1 none at all 9 o!rs4 ,x w be se5 t ! portions ( ! diagonals & diamet]s ( squ>es :i* are cut (f betwe5 ! circles repres5t radiat+ planes1 occupy+ ! position ( ! spokes ( a :eel4 ,n[ let ! l9e ,,aeb1 ,fig4 #fi41 be ! pr(ile ( ! vase or cup to be drawn4 ,inclose x 9 ! rectangle ,,cd1 & if any portion ( x is n curv$1 z ,,ae1 cut (f ! curv$ portion by ! v]tical l9e ,,ef1 s z to include x 9 ! small] rectangle ,,fd4 ,draw ! rectangle ,,acbd 9 position1 & ^u x con/ruct two squ>es1 z !y are con/ruct$ on ! rectangle ,,acd 9 ,fig4 #fh42 & complete ! con/ruction ( ,fig4 #fh41 mak+ ! radius ( its l>ge out] circles equal to ,,ad1 & ( its small inn] circles equal to ,,ae4 ,! planes :i* occupy ! position ( ! :eel spokes w !n ea* repres5t a rectangle ( ! size ( ,,fd4 ,! con/ruction is %[n by ! dott$ l9es 9 ,fig4 #fi42 c be+ ! c5t] ( ! upp]mo/ circle4 ,illu/ration3 ,fig4 #fi4 ,)9 ea* ( ! small] rectangles betwe5 ! circles1 draw ! curve ,,eb 9 p]spective1 z 9 ,fig4 #fi4 ,draw ! curve xy1 t\*+ & inclos+ ! curves 9 ! rectangles1 & meet+ ! upp] circle at y4#cb ,!n xy is ! cont\r ( ! surface ( ! cup1 & ! upp] circle is its lip4 ,if ! l9e xy is long1 x may be necess>y to draw o!r rectangles betwe5 ! eight pr9cipal ones2 &1 if ! curve ( pr(ile ,,ab is complex or retort$1 !re may be sev]al l9es correspond+ to ,,xy1 inclos+ ! successive waves ( ! pr(ile2 & ! out] curve w !n be an undulat+ or brok5 "o4 ,illu/ration3 ,fig4 #gj4 ,,iii4 ,all bran*$ ornam5tation1 =ms ( fl[]s1 capitals ( columns1 ma*icolations ( r\nd t[]s1 & o!r su* arrangem5ts ( radiat+ curve1 are resolvable by ?is problem1 us+ m or few] int]ior circles accord+ to ! conditions ( ! curves4 ,fig4 #gj4 is an example ( ! con/ruction ( a circul> gr\p ( eight trefoils ) curv$ /ems4 ,"o out] or limit+ circle is drawn )9 ! squ>e ,,edcf1 & ! extremities ( ! trefoils t\* x at ! extremities ( its diagonals & diamet]s4 ,a small] circle is at ! v]tical di/ance ,,bc bel[ ! l>g]1 & ,a is ! angle ( ! squ>e )9 :i* ! small] circle is drawn2 b ! squ>e is n giv51 to avoid confusion4 ,! /ems ( ! trefoils =m droop+ curves1 arrang$ on ! diagonals & diamet]s ( ! small] circle1 :i* are dott$4 ,b no p]spective laws w d "w ( ?is intricate k9d s well z ! h& & eye ( a pa9t]4 ,,iv4 ,!re is "o common con/ruction1 h[ev]1 9 :i*1 s+ul>ly1 ! h& & eye ( ! pa9t] almo/ always fail1 & t is ! fillet ( any ord9>y capital or base ( a circul> pill> 7or any simil> =m74 ,x is r>ely necess>y 9 practice to draw su* m9or details 9 p]spective2 yet ! p]spective laws :i* regulate !m %\ld be und]/ood1 else ! eye does n see _! cont\rs rightly until x is v highly cultivat$4 ,illu/ration3 ,fig4 #ga4 ,fig4 #ga4 w %[ ! law ) suffici5t cle>ness2 x repres5ts ! p]spective con/ruction ( a fillet ^: pr(ile is a semicircle1 su* z ,,fh 9 ,fig4 #fj41 se5 above ! eye4 ,only half ! pill> ) half ! fillet is drawn1 to avoid confusion4 ,q is ! c5t] ( ! %aft4 ,,pq ! ?ickness ( ! fillet1 sight-magnitude at ! %aft's c5t]4 ,r\nd ,p a horizontal semicircle is drawn on ! diamet] ( ! %aft ab4 ,r\nd ,q ano!r horizontal semicircle is drawn on diamet] cd4 ,^! two semicircles are ! upp] & l[] edges ( ! fillet4 ,!n diagonals & diamet]s are drawn z 9 ,fig4 #fh41 &1 at _! extremities1 semicircles 9 p]spective1 z 9 ,fig4 #fi4 ,! lett]s ,a1 ,b1 ,c1 ,d1 & ,e1 indicate ! upp] & ext]ior angles ( ! rectangles 9 :i* ^! semicircles are to be drawn2 b ! inn] v]tical l9e is n dott$ 9 ! rectangle at ,c1 z x w\ld h confus$ itself ) o!r l9es4 ,!n ! visible cont\r ( ! fillet is ! l9e :i* incloses & t\*es#cc all ! semicircles4 ,x disappe>s beh9d ! %aft at ! po9t ,h1 b ,i h drawn x ?r\gh to ! opposite extremity ( ! diamet] at d4 ,turn$ upside d[n ! figure %[s ! con/ruction ( a basic fillet4 ,! capital ( a ,greek ,doric pill> %\ld be drawn frequ5tly = ex]cise on ?is f\rte5? problem1 ! curve ( its e*9us be+ exquisitely subtle1 :ile ! g5]al cont\r is simple4 #cb ,?is po9t co9cides 9 ! figure ) ! extremity ( ! horizontal diamet]1 b only accid5tally4 #cc ,! engrav+ is a little inaccurate2 ! inclos+ l9e %\ld t\* ! dott$ semicircles at ,a & ,b4 ,! /ud5t %\ld draw x on a l>ge scale4 ,,problem ,,xvi4 ,x is (t5 possible to %ort5 o!r p]spective op]ations consid]ably1 by f9d+ ! vani%+-po9ts ( ! incl9$ l9es ( ! object4 ,?us1 9 draw+ ! gabl$ ro( 9 ,fig4 #dc41 if ! gable ,,ayc be drawn 9 p]spective1 & ! vani%+-po9t ( ,,ay det]m9$1 x is n necess>y to draw ! two sides ( ! rectangle1 ,a,d & ,d,b1 9 ord] to det]m9e ! po9t ,y2 b m]ely to draw ,,yy to ! vani%+-po9t ( ,,aa & ,a,y to ! vani%+-po9t ( ,,ay1 meet+ 9 ,y1 ! po9t requir$4 ,aga91 if !re be a s]ies ( gables1 or o!r figures produc$ by p>allel incl9$ l9es1 & retir+ to ! po9t ,v1 z 9 ,fig4 #gb41cd x is n necess>y to draw ea* sep>ately1 b m]ely to det]m9e _! bread?s on ! l9e ,,av1 & draw ! slopes ( ea* to _! vani%+-po9ts1 z %[n 9 ,fig4 #gb4 ,or if ! gables are equal 9 height1 & a l9e be drawn f ,y to ,v1 ! con/ruction resolves itself into a zigzag drawn alt]nately to ,p & ,q1 betwe5 ! l9es ,,yv & ,,av4 ,! /ud5t mu/ be v cauti\s1 9 f9d+ ! vani%+-po9ts ( incl9$ l9es1 to notice _! relations to ! horizontals b5ea? !m1 else he may easily mi/ake ! horizontal to :i* !y belong4 ,?us1 let ,,abcd1 ,fig4 #gc41 be a rectangul> incl9$ plane1 & let x be requir$ to f9d ! vani%+-po9t ( its diagonal ,,bd4 ,f9d ,v1 ! vani%+-po9t ( ,,ad & ,,bc4 ,draw ,,ae to ! opposite vani%+-po9t1 s t ,,dae may repres5t a "r angle4 ,let fall f ,b ! v]tical ,,be1 cutt+ ,,ae 9 ,e4 ,jo9 ,,ed1 & produce x to cut ! sight-l9e 9 ,v4 ,illu/ration3 ,fig4 #gb4 ,!n1 s9ce ! po9t ,e is v]tically "u ! po9t ,b1 ! horizontal l9e ,,ed is v]tically "u ! incl9$ l9e ,,bd4 ,illu/ration3 ,fig4 #gc4 ,s t if we n[ let fall ! v]tical ,v,p f ,v1 & produce ,,bd to cut ,v,p 9 ,p1 ! po9t ,p w be ! vani%+-po9t ( ,,bd1 & ( all l9es p>allel to x4#ce #cd ,! diagram is inaccurately cut4 ,,yv %\ld be a "r l9e4 #ce ,! /ud5t may p]haps und]/& ?is con/ruction bett] by complet+ ! rectangle ,,adfe1 draw+ ,,df to ! vani%+-po9t ( ,,ae1 & ,,ef to ,v4 ,! :ole figure1 ,,bf1 may !n be conceiv$ z repres5t+ half ! gable ro( ( a h\se1 ,,af ! rectangle ( its base1 & ,,ac ! rectangle ( its slop+ side4 ,9 ne>ly all picturesque build+s1 especially on ! ,cont95t1 ! slopes ( gables are mu* v>i$ 7frequ5tly unequal on ! two sides71 & ! vani%+-po9ts ( _! incl9$ l9es become v important1 if accuracy is requir$ 9 ! int]sections ( til+1 sides ( dorm] w9d[s1 etc4 ,obvi\sly1 also1 irregul> triangles & polygons 9 v]tical planes may be m easily con/ruct$ by f9d+ ! vani%+-po9ts ( _! sides1 ?an by ! con/ruction giv5 9 ! coroll>y to ,problem ,,ix42 & if su* triangles or polygons h o!rs conc5trically inscrib$ )9 !m1 z (t5 9 ,byzant9e mosaics1 etc41 ! use ( ! vani%+-po9ts w become ess5tial4 ,,problem ,,xviii4 ,be=e exam9+ ! la/ ?ree problems x is necess>y t y %\ld und]/& accurately :at is meant by ! position ( an incl9$ plane4 ,cut a piece ( /rong :ite pa/ebo>d into any irregul> %ape1 & dip x 9 a slop$ position into wat]4 ,h[ev] y hold x1 ! edge ( ! wat]1 ( c\rse1 w always draw a horizontal l9e across its surface4 ,! direction ( ?is horizontal l9e is ! direction ( ! incl9$ plane4 7,9 b$s ( rock geologi/s call x _! 8/rike407 ,illu/ration3 ,fig4 #gd4 ,next1 draw a semicircle on ! piece ( pa/ebo>d2 draw its diamet]1 ,,ab1 ,fig4 #gd41 & a v]tical l9e f its c5t]1 ,,cd2 & draw "s o!r l9es1 ,,ce1 ,,cf1 etc41 f ! c5t] to any po9ts 9 ! circumf]5ce4 ,n[ dip ! piece ( pa/ebo>d aga9 into wat]1 &1 hold+ x at any incl9ation & 9 any direction y *oose1 br+ ! surface ( ! wat] to ! l9e ,,ab4 ,!n ! l9e ,,cd w be ! mo/ /eeply incl9$ ( all ! l9es drawn to ! circumf]5ce ( ! circle2 ,,gc & ,,hc w be less /eep2 & ,,ec & ,,fc less /eep /ill4 ,! ne>] ! l9es to ,,cd1 ! /eep] !y w be2 & ! ne>] to ,,ab1 ! m ne>ly horizontal4 ,:51 !re=e1 ! l9e ,,ab is horizontal 7or m>ks ! wat] surface71 its direction is ! direction ( ! incl9$ plane1 & ! incl9ation ( ! l9e ,,dc is ! incl9ation ( ! incl9$ plane4 ,9 b$s ( rock geologi/s call ! incl9ation ( ! l9e ,,dc _! 8dip40 ,to fix ! position ( an incl9$ plane1 !re=e1 is to det]m9e ! direction ( any two l9es 9 ! plane1 ,,ab & ,,cd1 ( :i* "o %all be horizontal & ! o!r at "r angles to x4 ,!n any l9es drawn 9 ! incl9$ plane1 p>allel to ,,ab1 w be horizontal2 & l9es drawn p>allel to ,,cd w be z /eep z ,,cd1 & are spok5 ( 9 ! text z ! 8/eepe/ l9es0 9 ! plane4 ,b f>!r1 :atev] ! direction ( a plane may be1 if x be ext5d$ indef9itely1 x w be t]m9at$1 to ! eye ( ! obs]v]1 by a b\nd>y l9e1 :i*1 9 a horizontal plane1 is horizontal 7co9cid+ ne>ly ) ! visible horizon72--9 a v]tical plane1 is v]tical2--&1 9 an incl9$ plane1 is incl9$4 ,?is l9e is prop]ly1 9 ea* case1 call$ ! 8sight-l9e0 ( su* plane2 b x is only prop]ly call$ ! 8horizon0 9 ! case ( a horizontal plane3 & ,i h pref]r$ us+ always ! t]m 8sight-l9e10 n only because m compreh5sive1 b m accurate2 = ?\gh ! curvature ( ! e>?'s surface is s slight t practically its visible limit always co9cides ) ! sight-l9e ( a horizontal plane1 x does n ma!matically co9cide ) x1 & ! two l9es ought n to be consid]$ z !oretically id5tical1 ?\gh !y are s 9 practice4 ,x is evid5t t all vani%+-po9ts ( l9es 9 any plane mu/ be f\nd on its sight-l9e1 &1 !re=e1 t ! sight-l9e ( any plane may be f\nd by jo9+ any two ( su* vani%+-po9ts4 ,h5ce ! con/ruction ( ,problem ,,xviii4 ,,ii4 ,,demon/rations ,,:i* ,,c\ld ,,n ,,conv5i5tly ,,be ,,includ$ ,,9 ,,! ,,text4 ,i4 ,,! ,,second ,,coroll>y1 ,,problem ,,ii4 ,9 ,fig4 #h4 omit ! l9es ,,cd1 ,c,d1 & ,,ds2 &1 z "h 9 ,fig4 #ge41 f a draw ad p>allel to ,,ab1 cutt+ ,,bt 9 d2 & f d draw de p>allel to ,,bc4 ,illu/ration3 ,fig4 #ge4 ,n[ z ad is p>allel to ,,ab-- ,,ac ac ,,bc de2 b ,,ac is equal to ,,bc-- ac de4 ,n[ because ! triangles ac,v1 bc,v1 are simil>-- ac bc a,v b,v2 & because ! triangles de,t1 bc,t are simil>-- de bc d,t b,t4 ,b ac is equal to de-- a,v b,v d,t b,t2 ! two triangles abd1 b,t,v1 are simil>1 & _! angles are alt]nate2 ,,tv is p>allel to ad4 ,b ad is p>allel to ,,ab-- ,,tv is p>allel to ,,ab4 ,,ii4 ,,! ,,?ird ,,coroll>y1 ,,problem ,,iii4 ,9 ,fig4 #ac41 s9ce a,r is by con/ruction p>allel to ,,ab 9 ,fig4 #ab41 & ,,tv is by con/ruction 9 ,problem ,,iii4 also p>allel to ,,ab-- a,r is p>allel to ,,tv1 ab,r & ,tb,v are alt]nate triangles1 a,r ,,tv ab b,v4 ,aga91 by ! con/ruction ( ,fig4 #ac41 a,r is p>allel to ,,mv-- ab,r & ,mb,v are alt]nate triangles1 a,r ,,mv ab b,v4 ,& x has j be5 %[n t also a,r ,,tv ab b,v-- a,r ,,mv a,r ,,tv4 ,b by con/ruction1 a,r a,r-- ,,mv ,,tv4 ,,iii4 ,,analysis ,,( ,,problem ,,xv4 ,we proce$ to take up ! g5]al condition ( ! second problem1 be=e left unexam9$1 namely1 t 9 :i* ! v]tical di/ances ,,bc & ,,ac 7,fig4 #f4 page #ac71 z well z ! direct di/ances ,,td & ,,td are unequal4 ,9 ,fig4 #f41 "h repeat$ 7,fig4 #gf471 produce ,c,b d[nw>ds1 & make ,c,e equal to ,,ca4 ,illu/ration3 ,fig4 #gf4 ,jo9 ,,ae4 ,!n1 by ! second ,coroll>y ( ,problem ,,ii41 ,,ae is a horizontal l9e4 ,draw ,,tv p>allel to ,,ae1 cutt+ ! sight-l9e 9 ,v4 ,v is ! vani%+-po9t ( ,,ae4 ,complete ! con/ructions ( ,problem ,,ii4 & its second ,coroll>y4 ,!n by ,problem ,,ii4 ab is ! l9e ,,ab drawn 9 p]spective2 & by its ,coroll>y ae is ! l9e ,,ae drawn 9 p]spective4 ,f ,v erect p]p5dicul> ,,vp1 & produce ab to cut x 9 ,p4 ,jo9 ,,tp1 & f e draw ef p>allel to ,,ae1 & cutt+ ,,at 9 f4 ,n[ 9 triangles ,,ebt & ,,aet1 z eb is p>allel to ,,eb & ef to ,,ae2--eb ef ,,eb ,,ae4 ,b ,,tv is also p>allel to ,,ae & ,,pv to eb4 ,!re=e also 9 ! triangles a,p,v & a,v,t1 eb ef ,,pv ,,vt4 ,!re=e ,,pv ,,vt ,,eb ,,ae4 ,&1 by con/ruction1 angle ,,tpv ,,aeb4 ,!re=e ! triangles ,,tvp1 ,,aeb1 are simil>2 & ,,tp is p>allel to ,,ab4 ,n[ ! con/ruction 9 ?is problem is entirely g5]al = any incl9$ l9e ,,ab1 & a horizontal l9e ,,ae 9 ! same v]tical plane ) x4 ,s t if we f9d ! vani%+-po9t ( ,,ae 9 ,v1 & f ,v erect a v]tical ,,vp1 & f ,t draw ,,tp p>allel to ,,ab1 cutt+ ,,vp 9 ,p1 ,p w be ! vani%+-po9t ( ,,ab1 & 7by ! same pro( z t giv5 at page #ag7 ( all l9es p>allel to x4 ,illu/ration3 ,fig4 #gg4 ,next1 to f9d ! divid+-po9t ( ! incl9$ l9e4 ,i remove "s unnecess>y l9es f ! la/ figure & repeat x "h1 ,fig4 #gg41 add+ ! measur+-l9e a,m1 t ! /ud5t may obs]ve its position ) respect to ! o!r l9es be=e ,i remove any m ( !m4 ,n[ if ! l9e ,,ab 9 ?is diagram repres5t$ ! l5g? ( ! l9e ,,ab 9 reality 7z ,,ab does 9 ,figs4 #aj4 & #aa471 we %\ld only h to proce$ to modify ,coroll>y ,,iii4 ( ,problem ,,ii4 to ?is new con/ruction4 ,we %all see pres5tly t ,,ab does n repres5t ! actual l5g? ( ! incl9$ l9e ,,ab 9 nature1 nev]!less we %all fir/ proce$ z if x did1 & modify our result aft]w>ds4 ,9 ,fig4 #gg4 draw ad p>allel to ,,ab1 cutt+ ,,bt 9 d4 ,!re=e ad is ! sight-magnitude ( ,,ab1 z a,r is ( ,,ab 9 ,fig4 #aa4 ,illu/ration3 ,fig4 #gh4 ,remove aga9 f ! figure all l9es except ,,pv1 ,,vt1 ,,pt1 ab1 ad1 & ! measur+-l9e4 ,set (f on ! measur+-l9e am equal to ad4 ,draw ,,pq p>allel to am1 & ?r\gh b draw m,q1 cutt+ ,,pq 9 ,q4 ,!n1 by ! pro( already giv5 9 page #bj1 ,,pq ,,pt4 ,!re=e if ,p is ! vani%+-po9t ( an incl9$ l9e ,,ab1 & ,,qp is a horizontal l9e drawn ?r\gh x1 make ,,pq equal to ,,pt1 & am on ! measur+-l9e equal to ! sight-magnitude ( ! l9e ,,ab 9 ! diagram1 & ! l9e jo9+ m,q w cut a,p 9 b4 ,we h n[1 !re=e1 to consid] :at relation ! l5g? ( ! l9e ,,ab 9 ?is diagram1 ,fig4 #gg41 has to ! l5g? ( ! l9e ,,ab 9 reality4 ,n[ ! l9e ,,ae 9 ,fig4 #gg4 repres5ts ! l5g? ( ,,ae 9 reality4 ,b ! angle ,,aeb1 ,fig4 #gg41 & ! correspond+ angle 9 all ! con/ructions ( ! e>li] problems1 is 9 reality a "r angle1 ?\gh 9 ! diagram necess>ily repres5t$ z obtuse4 ,illu/ration3 ,fig4 #gi4 ,!re=e1 if f ,e we draw ,,ec1 z 9 ,fig4 #gi41 at "r angles to ,,ae1 make ,,ec ,,eb1 & jo9 ,,ac1 ,,ac w be ! real l5g? ( ! l9e ,,ab4 ,n[1 !re=e1 if in/ead ( am 9 ,fig4 #gh41 we take ! real l5g? ( ,,ab1 t real l5g? w be to am z ,,ac to ,,ab 9 ,fig4 #gi4 ,& !n1 if ! l9e drawn to ! measur+-l9e ,,pq is /ill to cut a,p 9 b1 x is evid5t t ! l9e ,,pq mu/ be %ort5$ 9 ! same ratio t am 0 %ort5$2 & ! true divid+-po9t w be ,q 9 ,fig4 #hj41 fix$ s t ,q,p %all be to ,,qp z am is to am2 am repres5t+ ! real l5g? ( ,,ab4 ,b amis !re=e to am z ,,ac is to ,,ab 9 ,fig4 #gi4 ,!re=e ,,pq mu/ be to ,,pq z ,,ac is to ,,ab4 ,b ,,pq equals ,,pt 7,fig4 #gh472 & ,,pv is to ,,vt 79 ,fig4 #gh47 z ,,be is to ,,ae 7,fig4 #gi474 ,h5ce we h only to sub/itute ,,pv = ,,ec1 & ,,vt = ,,ae1 9 ,fig4 #gi41 & ! result+ diagonal ,,ac w be ! requir$ l5g? ( ,,pq4 ,illu/ration3 ,fig4 #hj4 ,x w be se5 t ! con/ruction giv5 9 ! text 7,fig4 #df47 is ! simple/ means ( obta9+ ?is magnitude1 = ,,vd 9 ,fig4 #df4 7or ,,vm 9 ,fig4 #ae47 ,,vt by con/ruction 9 ,problem ,,iv4 ,x %\ld1 h[ev]1 be obs]v$1 t ! di/ance ,,pq or ,,px1 9 ,fig4 #df41 may be laid on ! sight-l9e ( ! incl9$ plane itself1 if ! measur+-l9e be drawn p>allel to t sight-l9e4 ,& ?us any =m may be drawn on an incl9$ plane z conv5i5tly z on a horizontal "o1 ) ! s+le exception ( ! radiation ( ! v]ticals1 :i* h a vani%+-po9t1 z %[n 9 ,problem ,,xx4 ,,! ,,end4 ,transcrib]'s ,note ,a h&ful ( unequivocal typographical errors has be5 correct$4 ,= increas$ cl>ity1 a few diagrams h be5 %ift$ f _! orig9al position 9 ! text4