,) ! respectful complim5ts ( ! au?or4 ,address1 ,mr4 ,r4 ,,ball>d1 ,malv]n1 ,,engl&4 ,illu/ration3 ,plumb ,l9e ov] ,pyramids ,,! ,,solution ,,( ,,! ,,pyramid ,,problem ,,or1 ,,pyramid ,,discov]ies4 ,,) ,a ,,new ,,!ory ,,z ,,to ,,_! ,,anci5t ,,use4 ,,by ,,rob]t ,,ball>d1 ,m4 ,,in/4 ,c4,e41 ,,engl&2 ,m4 ,,am]1 ,,soc4 ,c4,e4 ,,*ief ,,eng9e] ,,( ,,! ,,c5tral ,,& ,,nor!rn ,,railway ,,division ,,( ,,! ,,colony ,,( ,,que5sl&1 ,,au/ralia4 ,,new ,,york3 ,,john ,,wiley ,,sons4 #ahhb4 ,copyright1 #ahhb1 ,by ,,john ,,wiley ,,sons4 ,,press ,,( ,j4 ,j4 ,,little ,,co41 ,,nos4 #aj ,,to #bj ,,a/or ,,place1 ,,new ,,york4 ,,note4 ,9 prep>+ ?is "w = publication ,i h receiv$ valuable help f ! foll[+ fri5ds 9 ,que5sl&3-- ,e4 ,a4 ,deliss]1 ,l4,s4 & ,c4,e41 ,bogantungan1 :o assi/$ me 9 my calculations1 & furni%$ _m useful sugge/ions4 ,j4 ,brunton ,/eph5s1 ,brisbane1 :o p]suad$ me to publi% my !ory1 & :o also und]took ! "w ( correction = ! press4 ,j4 ,a4 ,cl>ke1 ,arti/1 ,brisbane1 :o contribut$ to ! ,illu/rations4 ,lyne ,br[n1 ,em]ald1--7photographs74 ,f4 ,ro!ry1 ,em]ald1--7models74 &--,a4 ,w4 ,voysey1 ,em]ald1--7maps & diagrams74 ,,cont5ts4 ,,page #a4 ,! ,gr\nd ,plan ( ! ,gzeh ,gr\p #i ,plan ,ratios connect$ into ,natural ,numb]s #ac #b4 ,! ,orig9al ,cubit ,measure ( ! ,gzeh ,gr\p #ae #c4 ,! ,exact ,measure ( ! ,bases ( ! ,pyramids #ai #d4 ,! ,slopes1 ,ratios & ,angles ( ! ,?ree ,pr9cipal ,pyramids ( ! ,gzeh ,gr\p #bj #e4 ,! ,exact ,dim5sions ( ! ,pyramids #bc #f4 ,geometrical ,peculi>ities ( ! ,pyramids #bh #f;,a4 ,! ,cas+ ,/ones ( ! ,pyramids #cb #g4 ,peculi>ities ( ! ,triangles #c1 #d1 #e & #bj1 #ba1 #bi #ce #h4 ,g5]al ,obs]vations #ch #i4 ,! ,pyramids ( ,egypt ! ,!odolites ( ! ,egyptian ,l& ,surveyors #da #aj4 ,h[ ! ,pyramids 7 made use ( #dd #aa4 ,description ( ! anci5t ,portable ,survey ,in/rum5t #ea ,table to expla9 ,figure #fj #fd #ab4 ,prim>y ,triangles & _! ,satellites2--or ! ,anci5t ,sy/em ( ,"r-angl$ ,trigonometry unfold$ by a ,/udy ( ! ,plan ( ! ,pyramids ( ,gzeh #fd ,table ( "s ,prim>y ,triangles & _! ,satellites #gd #ac4 ,! ,size & ,%ape ( ! ,pyramids indicat$ by ! ,plan #gg #ad4 ,a ,simple ,in/rum5t = lay+ (f ,prim>y ,triangles #gi #ad;,a4 ,g5]al ,obs]vations #hj #ae4 ,prim>y ,triangulation #hb #af4 ,! ,p5tangle or ,five-po9t$ ,/> ! ,geometric ,symbol ( ! ,great ,pyramid #ia ,table %[+ ! comp>ative ,measures ( ,l9es #if #ag4 ,! mann] 9 :i* ! ,slope ,ratios ( ! ,pyramids 7 arriv$ at #ajd ,,li/ ,,( ,,works ,,consult$4 ,p5ny ,cyclopdia4 7,knight1 ,london4 #ahcc47 ,%>pe's ,egypt4 ,our ,inh]itance 9 ! ,great ,pyramid4 ,piazzi ,smy?4 ,! ,pyramids ( ,egypt4 ,r4 ,a4 ,proctor4 7,article 9 ,g5tleman's ,magaz9e4 ,feb4 #ahhj47 ,traite de la ,gr&eur et de la ,figure de la ,t]re4 ,cass9i4 7,am/]dam4 #agbc47 ,pyramid ,facts & ,fancies4 ,j4 ,bonwick4 ,illu/ration3 ,g5]al ,plan ( ,gzeh ,gr\p ,,! ,,solution ,,( ,,! ,,pyramid ,,problem4 ,) ! firm conviction t ! ,pyramids ( ,egypt 7 built & employ$1 among o!r purposes1 = "o special1 ma91 & important purpose ( ! greate/ utility & conv5i5ce1 ,i f9d x necess>y be=e ,i c e/abli% ! !ory ,i advance1 to endeavor to det]m9e ! proportions & measures ( "o ( ! pr9cipal gr\ps4 ,i take t ( ,gzeh z be+ ! "o af=d+ mo/ data1 & z be+ probably "o ( ! mo/ important gr\ps4 ,i %all fir/ try to set =? ! results ( my inve/igations into ! peculi>ities ( con/ruction ( ! ,gzeh ,gr\p1 & aft]w>ds %[ h[ ! ,pyramids 7 appli$ to ! national "w = :i* ,i believe !y 7 design$4 #a4 ,,! ,,gr\nd ,,plan ,,( ,,! ,,gizeh ,,gr\p4 ,i f9d t ! ,pyramid ,*eops is situat$ on ! acute angle ( a "r-angl$ triangle--sometimes call$ ! ,py?agorean1 or ,egyptian triangle--( :i* base1 p]p5dicul>1 & hypot5use are to ea* o!r z #c1 #d1 & #e4 ,! ,pyramid call$ ,myc]9us1 is situate on ! great] angle ( ?is triangle1 & ! base ( ! triangle1 measur+ ?ree1 is a l9e due ea/ f ,myc]9us1 & jo9+ p]p5dicul> at a po9t due s\? ( ,*eops4 7,see ,figure #a47 ,fig4 #a4 ,*eops1 ,myc]9us ,i f9d t ! ,pyramid ,*eops is also situate at ! acute angle ( a "r-angl$ triangle m beautiful ?an ! s-call$ triangle ( ,py?agoras1 because m practically useful4 ,i h nam$ x ! #bj1 #ba1 #bi triangle4 ,base1 p]p5dicul>1 & hypot5use are to ea* o!r z tw5ty1 tw5ty-"o1 & tw5ty-n9e4 ,! ,pyramid ,cephr5 is situate on ! great] angle ( ?is triangle1 & base & p]p5dicul> are z be=e describ$ 9 ! ,py?agorean triangle ^u :i* ,myc]9us is built4 7,see ,fig4 #b47 ,fig4 #b4 ,*eops1 ,cephr5 ,fig4 #c4 ,*eops1 ,cephr51 ,myc]9us ,figure #c repres5ts ! comb9ation1--,a be+ ,*eops1 ,f ,cephr51 & ,d ,myc]9us4 ,l9es ,,dc1 ,,ca1 & ,,ad are to ea* o!r z #c1 #d1 & #e2 & l9es ,,fb1 ,,ba1 & ,,af are to ea* o!r z #bj1 #ba1 & #bi4 ,! l9e ,,cb is to ,,ba1 z #h to #g2 ! l9e ,,fh is to ,,dh1 z #if to #ee2 & ! l9e ,,fb is to ,,bc1 z #e to #f4 ,! ,ratios ( ! fir/ triangle multipli$ by =ty-five1 ( ! second multipli$ by f\r1 & ! o!r ?ree sets by twelve1 "o1 & sixte5 respectively1 produce ! foll[+ connect$ l5g?s 9 natural numb]s = all ! l9es4 ,,dc #ace ,,ca #ahj ,,ad #bbe ----------- ,,fb #hj ,,ba #hd ,,af #aaf ----------- ,,cb #if ,,ba #hd ----------- ,,fh #if ,,dh #ee ----------- ,,fb #hj ,,bc #if ,figure #d connects ano!r pyramid ( ! gr\p--x is ! "o to ! s\?w>d & ea/w>d ( ,*eops4 ,9 ?is connection1 ,a ,y ,z ,a is a #c1 #d1 #e triangle1 & ,b ,y ,z ,o ,b is a squ>e4 ,l9es ,,ya to ,,ca are z #a to #e ,,cy to ,,yz z #c to #a ,,fo to ,,zo z #h to #c & ,,da to ,,az z #ae to #d4 ,i may also po9t out on ! same plan t call+ ! l9e ,,fa radius1 & ! l9es ,,ba & ,,fb s9e & co-s9e1 !n is ,,ya equal 9 l5g? to v]s$ s9e ( angle ,,afb4 ,?is connects ! #bj1 #ba1 #bi triangle ,,fab ) ! #c1 #d1 #e triangle ,,azy4 ,i h n suffici5t data at my disposal to enable me to connect ! rema9+ elev5 small pyramids to my satisfaction1 & ,i consid] ! f\r are suffici5t = my purpose4 ,fig4 #d4 ,at level ( ,at level ( ,own base ,cephr5's base ,( ^! natural ,*eops #ef #eb numb]s ! bases ,cephr5 #eb #eb ( ! pyramids are ,myc]9us #bf #bg z foll[s4 ,i n[ e/abli% ! foll[+ li/ ( measurem5ts ( ! plan 9 connect$ natural numb]s4 7,see ,figure #d47 ,plan ,ratios connect$ into ,natural ,numb]s4 ---------------------------------------- --------- ,,by #a #dh;,,bc #f #if;,,dc #de #ace;,,fb #e #hj #dh #af #c #af ,,yz #a #dh;,,fb #e #hj;,,bc #cb #if;,,by #c #dh ---------------------------------------- --------- ,,dn #fa #ahc;,,dn #fa #ahc;,,cy #c #add;,,fh #if #if #c #c #dh #a ,,nr #fj #ahj,,nz #dh #add;,,bc #b #if;,,dh #ee #ee ---------------------------------------- --------- ,,cy #af #add,,pn #fa #adfd;,,je #c #gb,,yx #g #fc #i #bd #bd #i ,,dc #ae #ace,,pa #dh #aaeb;,,ex #b #dh;,,ay #d #cf ---------------------------------------- --------- ,,ba #ba #hd;,,ca #d #ahj;,,bc #cb #if;,,ea #g #aje #d #de #c #ae ,,fb #bj #hj;,,dc #c #ace;,,eb #ba #fc;,,az #d #fj ---------------------------------------- --------- ,,cb #h #if,,yz #d #dh;,,fo #cb #abh;,,ab #g #hd #ab #ab #d #ab ,,ba #g #hd ,,ay #c #cf,,or #ba #hd;,,bo #d #dh ---------------------------------------- --------- ,,ed #h #abj;,,ba #d #hd;,,ft #hd #hd;,,bc #h #if #ae #ba #a #ab ,,ae #g #aje;,,eb #c #fc,,/ #ee #ee;,,ac #ae #ahj ---------------------------------------- --------- ,,vw #ee #ee ,,ge #d #if,,vw #ee #ee,,nd #fa #ahc #a #bd #a #c ,,fw #dh #dh;,,dg #c #gb,,sv #cf #cf,,no #cb #if ---------------------------------------- --------- ,,sj #g #hd;,,hn #d #abh;,,bj #de #ace,,pa #dh #aaeb #ab #cb #c #bd ,,su #f #gb;,,fh #c #if;,,ab #bh #hd;,,az #be #fj ---------------------------------------- --------- ,,gx #b #add;,,gu #e #ahj;,,eo #cg #aaa,,sr #fa #ahc #gb #cf #c #c ,,dg #a #gb;,,dg #b #gb;,,ay #ab #cf,,rz #ab #cf ---------------------------------------- --------- ,,su #b #gb;,,hw #add #add;,,ht #cf #ahj;,,fh #if #if #cf #a #e #a ,,sv #a #cf;,,dh #ee #ee;,,dh #aa #ee;,,fe #ag #ag ---------------------------------------- --------- ,,tw #cf #cf;,,fo #h #abh;,,da #ae #bbe;,,ea #aje #aje #a #af #ae #a ,,tu #ag #ag,,oz #c #dh;,,az #d #fj;,,ef #ag #ag ---------------------------------------- --------- ,,sr #fa #ahc;,,jb #de #ace;,,ac #ae #ahj,,: #add #add #c #c #ab #a ,,ro #bh #hd;,,by #af #dh;,,cn #d #dh;,,hg #ag #ag ---------------------------------------- --------- ,,yw #bj #hj;,,fw #dh #dh,,yv #ae #ace,,? #ahj #ahj #d #a #i #a ,,ay #i #cf;,,fe #ag #ag;,,ay #d #cf;,,hg #ag #ag ---------------------------------------- --------- ,,my #i #ajh;,,ac #bj #ahj,,vz #fa #ahc #ab #i #c ,,zy #d #dh;,,cg #g #fc,,zo #af #dh ---------------------------------------- --------- ,,ac #i #ahj;,,ea #ce #aje;,,eu #hd #hd #bj #c #a ,,* #d #hj;,,ay #ab #cf;,,fe #ag #ag ---------------------------------------- --------- ,,nz #ab #add;,,cy #c #add;,,ca #e #ahj #ab #dh #cf ,,za #e #fj,,yz #a #dh;,,ay #a #cf ---------------------------------------- --------- ,! above connect$ natural numb]s multipli$ by eight become ,r4,b4 cubits4 ,r4,b4,c4 7,?us1 ,,by #dh #h #chd ,,gx #add #h #aaeb74 #b4 ,,! ,,orig9al ,,cubit ,,measure ,,( ,,! ,,gizeh ,,gr\p4 ,mr4 ,j4 ,j4 ,wild1 9 8 lett] to ,"l ,br\gham writt5 9 #ahej1 call$ ! base ( ,cephr5 sev5 seconds4 ,i e/imate ! base ( ,cephr5 to be j sev5 ?irtie?s ( ! l9e ,,da4 ,! l9e ,,da is !re=e ?irty seconds ( ! ,e>?'s ,pol> circumf]5ce4 ,! l9e ,,da is !re=e #cjccaahfbe ,briti% feet1 & ! base ( ,cephr5 #gjggbg ,briti% feet4 ,i appli$ a v>iety ( ,cubits b f\nd none to "w 9 )\t fractions on ! beautiful set ( natural dim5sions :i* ,i _h work$ out = my plan4 7,see table ( connect$ natural numb]s47 ,i ultimately arriv$ at a cubit z ! anci5t measure :i* ,i h call$ ! ,r4,b4 cubit1 because x closely resembles ! ,royal ,babylonian ,cubit ( #eaca metre1 or #afhccii ,briti% feet4 ,! diff]5ce is #a_/fjj ( a foot4 ,i arriv$ at ! ,r4,b4 cubit 9 ! foll[+ mann]4 ,tak+ ! pol> axis ( ! e>? at five hundr$ million geometric in*es1 ?irty seconds ( ! circumf]5ce w be #cfcfajbfjh--;geometric in*es1 or #cfcigdbce ,briti% in*es1 at n9e hundr$ & n9ety-n9e to ! ?\s&--& #cjcjjhee geometric feet1 or #cjccaahfbe ,briti% feet4 ,n[ divid+ a second into sixty p>ts1 !re are #ahjj ,r4,b4 cubits 9 ! l9e ,,da2 & ! l9e ,,da be+ ?irty seconds1 measures #cfcigdbce ,briti% in*es1 :i* divid$ by #ahjj makes "o ( my cubits #bjbbjgijh ,briti% in*es1 or #afhejff ,briti% feet4 ,simil>ly1 #cfcfajbfjh geometric in*es divid$ by #ahjj makes my cubit #bjbjjeg geometric in*es 9 l5g?4 ,i h !re=e def9$ ?is cubit z foll[s3--,"o ,r4,b4 cubit is equal to #bj4bjjf geo4 in*es1 #bjbbjh ,brit4 in*es1 & #afhe ,brit4 feet4 ,i n[ con/ruct ! foll[+ table ( measures4 ---------------------------------------- --- ,r4 ,b4 ,,ple?ra ,,or ,,cubits4 ,,seconds4 ,,/adia4 ,,m9utes4,,degrees4 ---------------------------------------- --- #fj #a #cfj #f #a #cfjj #fj #aj #a #bafjjj #cfjj #fjj #fj #a #gggfjjjj #abifjjj #bafjjj #bafjj #cfj ---------------------------------------- --- ,?us !re are sev5ty-sev5 million1 sev5 hundr$ & sixty ?\s& ,r4,b4 cubits1 or two hundr$ & sixte5 ?\s& /adia1 to ! ,pol> circumf]5ce ( ! e>?4 ,?us we h obta9$ a p]fect set ( natural & conv5i5t measures :i* fits ! plan1 & fits ! circumf]5ce ( ! e>?4 ,& ,i claim = ! ,r4,b4 cubit t x is ! mo/ p]fect anci5t measure yet discov]$1 be+ ! measure ( ! plan ( ! ,pyramids ( ,gzeh4 ,! same =gott5 wisdom :i* divid$ ! circle into ?ree hundr$ & sixty degrees1 ! degree into sixty m9utes1 & ! m9ute into sixty seconds1 subdivid$ ^? seconds1 = e>? measurem5ts1 into ! sixty p>ts repres5t$ by sixty ,r4,b4 cubits4 ,we are aw>e t ?irds & f\r?s 7 us$ 9 anci5t a/ronomical calculations4 99 99 99 99 99 ,! read] w n[ obs]ve t ! cubit measures ( ! ma9 ,py?agorean triangle ( ! plan are obta9$ by multiply+ ! orig9al #c1 #d & #e by #cfj2 & t ! entire dim5sions are obta9$ 9 ,r4,b4 cubits by multiply+ ! la/ column ( connect$ natural numb]s 9 ! table by eight1--?us-- ,r4 ,b4 ,,cubits4 ,,dc #c #cfj #ajhj ,,ca #d #cfj #addj ,,da #e #cfj #ahjj or1 ,,natural ,r4 ,b4 ,,numb]s4 ,,cubits4 ,,dc #ace #h #ajhj ,,ca #ahj #h #addj ,,da #bbe #h #ahjj c41 c4 7,see ,figure #e1 p4 #ah47 ,accord+ to ,cass9i1 a degree 0 #fjj /adia1 a m9ute #aj /adia2 & a mod]n ,italian mile1 9 ! ye> #agbc1 0 equal to "o & a qu>t] anci5t ,roman miles2 & "o & a qu>t] anci5t ,roman miles 7 equal to t5 /adia or "o m9ute4 7,cass9i1 ,traite de la gr&eur et de la ,figure de la ,t]re4 ,am/]dam1 #agbc47 ,fig4 #e ,r4,b4,cubits4 ,,fb #fdj ,,eb #ejd ,,ba #fgb ,,ba #fgb ,,af #ibh ,,ae #hdj ,,dc #ajhj ,,no #gfh ,,ca #addj ,,( #ajbd ,,ad #ahjj ,,fn #abhj ,,dg #egf ,,ay #bhh ,,ge #gfh ,,zy #chd ,,ed #ifj ,,za #dhj ,r4,b4 ,cub4 ,at level ( ,cephr5's ,*eops' ,base #dbj ,base :i* is ! plane ,cephr5's ,base #dbj or level ( ! plan4-- ,myc]9us' ,base #bah ,dufeu also made a /adium ! six hundr$? "p ( a degree4 ,he made ! degree #aajhbgfh metres1 :i* multipli$ by #cbhjhda gives #cfcfjgiif ,briti% feet2 & #cfcfjgiif divid$ by #fjj equals #fjfjaccbg feet to 8 /adium4 ,i make ! /adium #fjffbcgf ,briti% feet4 ,!re be+ #cfj cubits to a /adium1 ,dufeu's /adium divid$ by #cfj1 gives #afhcc ,briti% feet1 :i* is ! exact measure giv5 = a ,royal ,babylonian ,cubit1 if r$uc$ to metres1 viz43 #jeaca ( a metre1 & !re=e probably ! orig9 ( ! measure call$ ! ,royal ,babylonian cubit4 ,accord+ to ?is measure1 ! ,gzeh plan w\ld be ab\t #a_/ajaa small] ?an if measur$ by ,r4,b4 cubits4 #c4 ,,! ,,exact ,,measure ,,( ,,! ,,bases ,,( ,,! ,,pyramids4 ,a /adium be+ #cfj ,r4,b4 cubits1 or six seconds--& a ple?ron #fj ,r4,b4 cubits1 or "o second1 ! base ( ! ,pyramid ,cephr5 is sev5 ple?ra1 or a /adium & a ple?ron1 equal to sev5 seconds1 or f\r hundr$ & tw5ty ,r4,b4 cubits4 ,myc]9us' base is ackn[l$g$ to be half ! base ( ,cephr54 ,piazzi ,smy? makes ! base ( ! ,pyramid ,*eops #iacaje pyramid 7or geometric7 in*es1 :i* divid$ by #bjbjjf gives #debja ,r4,b4 cubits4 ,i call x #deb cubits1 & accept x z ! measure :i* exactly fits ! plan4 ,i h n suffici5t data to det]m9e ! exact base ( ! o!r & small] pyramid :i* ,i h m>k$ on my plan4 ,! bases1 !n1 ( ,myc]9us1 ,cephr51 & ,*eops1 are #baj1 #dbj & #deb cubits1 respectively4 ,b 9 plan ! bases %\ld be r$uc$ to "o level4 ,i h !re=e drawn my plan1 or horizontal section1 at ! level or plane ( ! base ( ,cephr51 at :i* level or plane ! bases or horizontal sections ( ! pyramids are--,myc]9us1 #bah cubits1 ,cephr51 #dbj cubits1 & ,*eops1 #dbj cubits4 ,i %all %[ h[ ,i arrive at ?is by-&-by1 & %all also %[ t ! horizontal section ( ,*eops1 correspond+ to ! horizontal section ( ,cephr5 at ! level ( ,cephr5's base1 occurs1 z x %\ld d1 at ! level ( "o ( ! c\rses ( masonry1 viz4--! top ( ! t5? c\rse4 #d4 ,,! ,,slopes1 ,,ratios1 ,,& ,,angles ,,( ,,! ,,?ree ,,pr9cipal ,,pyramids ,,( ,,! ,,gizeh ,,gr\p4 ,be=e ent]+ on ! description ( ! exact slopes & angles ( ! ?ree pr9cipal pyramids1 ,i mu/ premise t ,i 0 guid$ to my conclusions by mak+ full use ( ! comb9$ evolutions ( ! two wond]ful "r-angl$ triangles1 #c1 #d1 #e1 & #bj1 #ba1 #bi1 :i* seem to run ?r\gh ! :ole design z a sort ( dom9ant4 ,f ! fir/ ,i 0 firmly conv9c$ t 9 su* skilful workman%ip "s v simple & easily appli$ templates mu/ h be5 employ$1 & s x turn$ out4 ,build]s d n m>k a dim5sion on a plan :i* !y _c measure1 nor h a hidd5 measure ( any importance )\t "s cle> out] way ( e/abli%+ x4 ,?is made me g /raight = ! slant ratios4 ,:5 ! ,pyramids 7 cas$ f top to bottom ) poli%$ m>ble1 !re 7 only two feasible measures1 ! bases & ! apo!ms2#a & = t reason ,i conjectur$ t ^! w\ld be ! def9ite plan ratios4 ,footnote #a3 ,! ,apo!m is a p]p5dicul> f ! v]tex ( a pyramid on a side ( ! base4--,*amb]s' ,practical ,ma!matics1 p4 #aef4 99 99 99 99 99 ,figures #f1 #g & #h %[ ! exact slope ratios ( ,*eops1 ,cephr51 & ,myc]9us1 measur$ z %[n on ! diagrams--viz41 ,*eops1 #ba to #cd1 ,cephr51 #bj to #cc1 & ,myc]9us1 #bj to #cb--t is1 half base to apo!m4 ,fig4 #f ,*eops4 ,note4 ,! ,ratios ( ,bases to ,altitudes are v ,fig4 #g ,cephr54 ne>ly z foll[s1 viz3-- ,fig4 #h ,myc]9us4 ,*eops #cc to #ba or #ccj to #baj ,cephr5 #cb to #ba or #cbj to #baj ,myc]9us #cb to #bj or #ccf to #baj ,! ratios ( base to altitude are1 ,*eops1 #cc to #ba1 ,cephr51 #cb to #ba1 & ,myc]9us1 #cb to #bj3 n exactly1 b ne> en\gh = all practical purposes4 ,= ! sake ( comp>ison1 x w be well to call ^! ratios #ccj to #baj1 #cbj to #baj1 & #ccf to #baj1 respectively4 99 99 99 99 99 ,fig4 #i4 ,*eops4 ,fig4 #aj4 ,*eops4 ,figures #i & #aj are m]idional & diagonal sections1 %[+ ratios ( ,*eops1 viz41 half base to apo!m1 #ba to #cd exactly2 half base to altitude1 #e to #g ne>ly1 & #ahc to #bcc1 ne>] /ill 7be+ ! ratio ( ,piazzi ,smy?74 ,! ratio ( ,sir ,f4 ,james1 half diagonal #aj to altitude #i is also v ne>ly correct4 ,my altitude = ,*eops is #dhdhhg ,briti% feet1 & ! half base #chjha ,briti% feet4 ,! ratio ( #g to #e gives #dhdff1 & ! ratio ( #bcc to #ahc gives #dhdhe = ! altitude4 ,my half diagonal is #echedfe1 & ratio #aj to #i1 gives #dhdfi ,briti% feet = ! altitude4 ,i h m5tion$ ! above to %[ h[ v ne>ly ^! ratios agree ) my exact ratio ( #ba to #cd half base to apo!m4 99 99 99 99 99 ,fig4 #aa4 ,cephr54 ,fig4 #ab4 ,cephr54 ,figures #aa & #ab %[ ! ratios ( ,cephr51 viz41 half base to apo!m1 #bj to #cc exactly1 & half base1 altitude1 & apo!m respectively1 z #hj1 #aje1 & #acb1 v ne>ly4 ,also half diagonal1 altitude1 & edge1 practically z #dca1 #djj1 & #ehh4 99 99 99 99 99 ,fig4 #ac4 ,myc]9us4 ,fig4 #ad4 ,myc]9us4 ,figures #ac & #ad %[ ! ratios ( ,myc]9us1 viz41 half base to apo!m1 #bj to #cb exactly1 & half base1 altitude1 & apo!m respectively1 z #bj1 #be1 & #cb v ne>ly4 ,also full diagonal to edge z #big to #aih1 ne>ly4 ,a peculi>ity ( ?is pyramid is1 t base is to altitude z apo!m is to half base4 ,?us1 #dj 3 #be 33 #cb 3 #bj2 t is1 half base is a f\r? proportional to base1 apo!m1 & altitude4 #e4 ,,! ,,exact ,,dim5sions ,,( ,,! ,,pyramids4 ,r4,b4 ,cub4 ,brit4 ,ft4 ,fig4 #ae4 ,*eops4 #deb #gfafb #bhggfg #dhdhhg ,fig4 #af4 ,*eops4 #cfeijdg #fafedi #dcjjeh #gbdfdg #fcibbdd #ajggjic ,figures #ae to #bj inclusive1 %[ ! l9e> dim5sions ( ! ?ree pyramids1 also _! angles4 ,! base angles are1 ,*eops1 #ea #ea #bj2 ,cephr51 #eb #da #da2 & ,myc]9us1 #ea #ai #d4 ,r4,b4 ,cub4 ,brit4 ,ft4 ,fig4 #ag4 ,cephr54 #dbj #gjggj #bgefa #dfddj ,fig4 #ah4 ,cephr54 #cdfej #ehche #djeaf #fhbfi #eicig #ajjjhd ,r4,b4 ,cub4 ,brit4 ,ft4 ,fig4 #ai4 ,myc]9us4 #baj #cec4he #afh #bhc4jh ,fig4 #bj4 ,myc]9us3 #aca4ad #bbj4ig #aih4aj #cccgihe #bifihdh #ejjdb ,9 ,*eops1 my dim5sions agree ) ,piazzi ,smy?--9 ! base ( ,cephr51 ) ,vyse & ,p]r+--9 ! height ( ,cephr51 ) ,sir ,g>dn] ,wilk9son1 ne>ly--9 ! base ( ,myc]9us1 !y agree ) ! usually accept$ measures1 & 9 ! height ( ,myc]9us1 !y exce$ ,jas4 ,j4 ,wild's measure1 by n q "o ( my cubits4 ,9 my angles ,i agree v ne>ly ) ,piazzi ,smy?1 = ,*eops1 & ) ,agnew1 = ,cephr51 diff]+ ab\t half a degree f ,agnew1 = ,myc]9us1 :o took ?is pyramid to repres5t ! same relation ( ,pi t ,p4 ,smy? ascribes to ,*eops 7viz43 #ea #ea #adc71 :ile he gave ,*eops ab\t ! same angle :i* ,i ascribe to ,myc]9us4 ,i %all n[ %[ h[ ,i make ,cephr5 & ,*eops ( equal bases ( #dbj ,r4,b4 cubits at ! same level1 viz4--t ( ,cephr5's base4 ,john ,james ,wild made ! bases ( ,*eops1 ,cephr51 & ,myc]9us1 respectively1 #hj1 #ajj1 & #ajdij cubits above "s po9t t he call$ ,nile ,level4 ,8 cubit 01 ,i believe1 ! ,memphis1 or ,nilometric cubit--b at any rate1 he made ! base ( ,cephr5 #dab ( !m4 ,i !re=e divid$ ! recogniz$ base ( ,cephr5--viz41 #gjgge ,briti% feet--by #dab1 & got a result ( #agagh ,briti% feet = 8 cubit4 ,!re=e1 8 measures multipli$ by #agagh & divid$ by #afhe w turn 8 cubits into ,r4,b4 cubits4 ,i ?us make ,*eops1 ,cephr51 & ,myc]9us1 respectively1 #haef1 #ajaic1 & #ajfic ,r4,b4 cubits above ! datum t ,j4 ,j4 ,wild calls ,nile ,level4 ,accord+ to ,bonwick's ,facts & ,fancies1 p4 #ca1 high wat] ,nile w\ld be #ach ft4 bel[ base ( ,*eops 7or #hbai ,r4,b4 cubits74 ,piazzi ,smy? makes ! pavem5t ( ,*eops #ageb ,briti% in*es 7or #hffd ,r4,b4 cubits7 above av]age ,nile ,level1 b1 by scal+ 8 map1 8 high ,nile ,level appe>s to agree ne>ly ) ,wild4 ,x is ! relative levels ( ! ,pyramids1 h[ev]1 t ,i require1 no matt] h[ mu* above ,nile ,level4 ,cephr5's base ( #dbj cubits be+ #ajaic cubits1 & ,*eops' base ( #deb cubits be+ #haef cubits above ,wild's datum1 ! diff]5ce 9 level ( _! bases is1 #bjcg cubits4 ,! ratio ( base to altitude ( ,*eops be+ #ccj to #baj1 !re=e #bjcg cubits divid$ by #baj & multipli$ by #ccj equals #cb cubits2 & #deb cubits m9us #cb cubits1 equals #dbj4 ,simil>ly1 ! base ( ,myc]9us is #e cubits above ! base ( ,cephr51 & ! ratio ( base to altitude #cb to #bj2 !re=e1 #e cubits divid$ by #bj & multipli$ by #cb equals #h cubits to be add$ to ! #baj cubit base ( ,myc]9us1 mak+ x #bah cubits 9 bread? at ! level ( ,cephr5's base4 ,?us1 a horizontal section or plan at ! level ( ,cephr5's base w\ld meet ! slopes ( ! ,pyramids s t !y w\ld on plan appe> z squ>es ) sides equal to #bah1 #dbj1 & #dbj ,r4,b4 cubits1 = ,myc]9us1 ,cephr51 & ,*eops1 respectively4 ,fig4 #ba4 ,r4,b4 ,cub4 ,apex ( ,cephr5 above ,base ,*eops #bieih ,apex ( ,*eops above ,base ,*eops #bhggg ,apex ( ,myc]9us above ,base ,*eops #aefea ,base ,cephr5 above base ( ,*eops #bjcg ,base ,myc]9us above base ( ,*eops #becg ,piazzi ,smy? makes ! top ( ! t5? c\rse ( ,*eops #dad pyramid in*es above ! pavem5t2 & #dad divid$ by #bjbjjf equals #bjdi ,r4,b4 cubits4 ,b ,i h already prov$ t ,*eops' #dbj cubit base measure occurs at a level ( #bjcg cubits above pavem5t2 !re=e is ?is level ! level ( ! top ( ! t5? c\rse1 = ! diff]5ce is only #jab ,r4,b4 cubits1 or #b in*es4 99 99 99 99 99 ,i wi% "h to note z a matt] ( int]e/1 b n z affect+ my !ory1 ! foll[+ measures ( ,piazzi ,smy?1 turn$ into ,r4,b4 cubits1 viz43-- ,,pyr4 ,,in*es4 ,r4,b4 ,,cubits4 ,k+'s ,*amb] floor1 above pavem5t #agjb #hdbe ,*eops' ,base1 z be=e /at$ #iacaje #debja ,k+'s ,*amb]1 ,true ,l5g?1 #dabacb #bjdj ,true ,fir/ ,height1 #bcjchi #aadj ,true ,bread?1 #bjfjff #ajbj ,he makes ! pres5t summit plat=m ( ,*eops #edde pyramid in*es above pavem5t4 ,my calculation ( #bfihj ,r4,b4 cub4 7,see ,fig4 #ba7 is equal to #edej pyramid in*es--?is is ab\t #ah cubits bel[ ! !oretical apex4 99 99 99 99 99 ,figure #ba repres5ts ! comp>ative levels & dim5sions ( ,myc]9us1 ,cephr51 & ,*eops4 ,! foll[+ peculi>ities are noticeable3--,t ,*eops & ,cephr5 are ( equal bases at ! level ( ,cephr5's base2--t1 at ! level ( ,*eops' base1 ! latt] is only half a cubit l>g]2--t1 f ! level ( ,myc]9us' base1 ,*eops is j d\ble ! height ( ,myc]9us2--& t f ! level ( ,cephr5's base1 ,cephr5 is j d\ble ! height ( ,myc]9us2 measur+ 9 ! latt] case1 h[ev]1 only up to ! level plat=m at ! summit ( ,cephr51 :i* is said to be ab\t eight feet wide4 ,! pres5t summit ( ,cephr5 is #bcjg cubits above ! pres5t summit ( ,*eops1 & ! complet$ apex ( ,cephr5 w\ld be #hba cubits above ! complet$ apex ( ,*eops4 ,9 ! summit plat=ms ,i h be5 guid$ by ,p4 ,smy?'s e/imate ( height defici5t1 #cfc pyr4 in*es1 = ,*eops1 & ,i h tak5 #h feet base = ,cephr5's summit plat=m4 #f4 ,,geometrical ,,peculi>ities ,,( ,,! ,,pyramids4 ,9 any pyramid1 ! apo!m is to half ! base z ! area ( ! f\r sides is to ! area ( ! base4 ,?us--,ratio apo!m to half base ,myc]9us #cb to #bj ,cephr5 #cc to #bj ,*eops #cd to #ba ,,area ,,( ,,! ,,f\r ,,sides4 ,,area ,,( ,,! ,,base4 ,myc]9us #gjefj #ddajj ,cephr5 #biajfj #agfdjj ,*eops #ccjgggij #bjdcjd ,all 9 ,r4,b4 cubits4 ,!re=e--#cb 3 #bj 3 3 #gjefj 3 #ddajj #cc 3 #bj 3 3 #biajfj 3 #agfdjj #cd 3 #ba 3 3 #ccjgggij 3 #bjdcjd #b;,h]odotus /ates t ! area ( ea* ( ! f\r faces ( ,*eops 0 equal to ! area ( a squ>e ^: base 0 ! altitude ( a ,pyramid2 or1 9 o!r words1 t altitude 0 a mean proportional to apo!m & half base2 ?us--area ( "o face equals ! f\r? ( #ccjgggij or #hbfiddge ,r4,b4 cubits1 & ! squ>e root ( #hbfiddge is #bhgef4 ,b ! correct altitude is #bhggg1 s ! error is #jba1 or #d ,briti% in*es4 ,i h !re=e ! au?ority ( ,h]odotus to support ! !ory :i* ,i %all subsequ5tly set =?1 t ?is pyramid 0 ! expon5t ( l9es divid$ 9 mean & extreme ratio4 ,by tak+ ! dim5sions ( ! ,pyramid f :at ,i may call its work+ level1 t is1 ! level ( ! base ( ,cephr51 ?is peculi>ity %[s m cle>ly1 z also o!rs to :i* ,i %all ref]4 ,?us--base ( ,*eops at work+ level1 #dbj cubits1 & apo!m #cdj cubits2 base area is1 !re=e1 #agfdjj cubits1 & area ( "o face is 7#dbj cubits1 multipli$ by half apo!m1 or #agj cubits7 #gadjj cubits4 ,n[ ! squ>e root ( #gadjj w\ld give altitude1 or side ( squ>e equal to altitude1 #bfgbjgghd cubits3 b ! real altitude is 7#cdj-baj7 #gaejj #bfgcidhci4 ,s t ! error ( ,h]odotus's proposition is ! diff]5ce betwe5 #gad & #gae4 ,footnote #b3 ,proctor is responsible = ?is /atem5t1 z ,i am quot+ f an essay ( 8 9 ! ,g5tleman's ,magaz9e4 ,r4 ,b4 ,?is leads to a consid]ation ( ! prop]ties ( ! angle =m$ by ! ratio apo!m #cd to half base #ba1 peculi> to ! pyramid ,*eops4 7,see ,figure #bb47 ,fig4 #bb4 ,diagram illu/rat+ relations ( ratios ( ! pyramid ,*eops4 ,call+ apo!m #cd1 radius2 & half base #ba1 s9e--,i f9d t-- ,radius is ! squ>e root ( #aaef ,s9e #dda ,co-s9e #gae ,tang5t #gac ,secant #ahfi & ,co-v]s$-s9e #afi ,s x foll[s t ! area ( "o ( ! faces1 #gad1 is a mean betwe5 ! squ>e ( ! altitude or co-s9e1 #gae1 & ! squ>e ( ! tang5t1 #gac4 ,?us ! read] w notice t ! peculi>ities ( ! ,pyramid ,*eops lie 9 ! regul> relations ( ! squ>es ( its v>i\s l9es2 :ile ! peculi>ities ( ! o!r two pyramids lie 9 ! relations ( ! l9es !mselves4 ,myc]9us & ,cephr51 born1 z "o may say1 ( ^? two noble triangles #c1 #d1 #e1 & #bj1 #ba1 #bi1 exhibit 9 _! l9eal developm5ts ratios s ne>ly p]fect t1 = all practical purposes1 !y may be call$ correct4 ,?us--,myc]9us1 #cbj #be #ajbe1 & #cb #ajbd4 & ,cephr51 #dhj #aje #agdbe1 & #acb #agdbd4 or #edjj #dca #cdegfa1 & #ehh #cdegdd4 ,see diagrams1 ,figures #aa to #ad inclusive4 ,9 ! ,pyramid ,*eops1 altitude is v ne>ly a mean proportional betwe5 apo!m & half base4 ,apo!m be+ #cd1 & half base #ba1 !n altitude w\ld be 7#cd-ba7 #gae #bfgcidhci1 &-- #ba 3 #bfgcidhci 33 #bfgcidhci 3 #cd1 ne>ly4 ,"h1 ( c\rse1 ! same diff]5ce comes 9 z occurr$ 9 consid]+ ! assumption ( ,h]odotus1 viz41 ! diff]5ce betwe5 #gae & #gad2 because if ! altitude 7 #gad1 !n w\ld x be exactly a mean proportional betwe5 ! half base & ! apo!m2 7?us1 #ba 3 #bfgbjgg 33 #bfgbjgg 33 #cd47 ,footnote #c3 ,half base to altitude4 ,footnote #d3 ,half base to altitude4 ,footnote #e3 ,half diagonal ( base to altitude4 ,9 ,*eops1 ! ratios ( apo!m1 half base & edge are1 #cd1 #ba1 & #dj1 v ne>ly1 ?us1 #cd #ba #aeig1 & #dj #afjj4 ,! dim5sions ( ,*eops 7f ! level ( ! base ( ,cephr57 to be :at ,piazzi ,smy? calls a ,pi pyramid1 w\ld be-- ,half base #baj ,r4,b4 cubits4 ,altitude #bfgchjcjd1 c4 ,apo!m #cciihhegc1 c4 ,altitude be+ to p]imet] ( base1 z radius ( a circle to circumf]5ce4 ,my dim5sions ( ! pyramid !re=e 9 :i*-- ,half base #baj ,r4,b4 cubits4 ,altitude #bfgcidhci c4 ,apo!m #cdj come ab\t z ne> to ! ratio ( ,pi z x is possible to come1 & provide simple l9es & templates to ! workm5 9 con/ruct+ ! build+2 & ,i ent]ta9 no d\bt t on ! simple l9es & templates t my ratios provide1 7 ^! ?ree pyramids built4 #f;,a4 ,,! ,,cas+ ,,/ones ,,( ,,! ,,pyramids4 ,figures #bc1 #bd1 & #be1 repres5t ord9>y cas+ /ones ( ! ?ree pyramids1 & ,figures #bf1 #bg1 & #bh1 repres5t angle or quo9 cas+ /ones ( ! same4 ,! cas+ /one ( ,*eops1 f\nd by ,colonel ,vyse1 is repres5t$ 9 ,bonwick's ,pyramid ,facts & ,fancies1 page #af1 z measur+ f\r feet ?ree in*es at ! top1 eight feet ?ree in*es at ! bottom1 f\r feet elev5 in*es at ! back1 & six feet ?ree in*es at ! front4 ,tak+ f\r feet elev5 in*es z ,radius1 & six feet ?ree in*es z ,secant1 !n ! ,tang5t is ?ree feet t5 in*es & ?ree t5?s4 ,fig4 #bc4 ,*eops ,cas+ ,/one4 ,fig4 #bd4 ,cephr5 ,cas+ ,/one4 ,fig4 #be4 ,myc]9us ,cas+ ,/one4 ,fig4 #bf4 ,*eops ,angle or ,quo9 ,/one4 ,fig4 #bg4 ,cephr5 ,angle or ,quo9 ,/one4 ,fig4 #bh4 ,myc]9us ,angle or ,quo9 ,/one4 ,?us1 9 in*es 77#ge-ei77 #dfcj in*es2 !re=e ! incl9ation ( ! /one mu/ h be5--slant height #ge in*es to #dfcj in*es horizontal4 ,n[1 #dfcj is to #ge1 z #ba is to #cd4 ,!re=e1 ,col4 ,vyse's cas+ /one agrees exactly ) my ratio = ! ,pyramid ,*eops1 viz41 #ba to #cd4 7,see ,figure #bi47 ,fig4 #bi4 ,col4 ,vyse's ,cas+ ,/one4 #ge 3 #dfc 33 #cd 3 #ba ,?is /one mu/ h be5 out ( plumb at ! back an in* & sev5 t5?s2 p]haps to give room = gr\t+ ! back jo9t ( ! m>ble cas+ /one to ! lime/one body ( ! "w3 or1 because1 z x is n a necessity 9 good masonry t ! back ( a /one %\ld be exactly plumb1 s long z ! error is on ! "r side1 ! build]s might n h be5 p>ticul> 9 t respect4 ,fig4 #ei4 7,temple ( ,*eops1 /&+ at angle ( wall47 ,figure #ei repres5ts su* a template z ! masons w\ld h us$ 9 build+ ,*eops1 bo? = dress+ & sett+ ! /ones4 7,! c\rses are drawn out ( proportion to ! template47 ,! o!r pyramids mu/ h be5 built by ! aid ( simil> templates4 ,su* l>ge blocks ( /one z 7 us$ 9 ! cas+ ( ^! pyramids c\ld n h be5 completely dress$ be=e sett+2 ! back & ends1 & ! top & bottom b$s 7 probably dress$ (f truly1 & ! face r\ghly scabbl$ (f2 b ! true slope angle c\ld n h be5 dress$ (f until ! /one _h be5 truly set & b$d$1 o!rwise !re w\ld h be5 great dang] to ! %>p arises4 99 99 99 99 99 ,i %all n[ record ! peculi>ities ( ! #c1 #d1 #e or ,py?agorean triangle1 & ! "r-angl$ triangle #bj1 #ba1 #bi4 #g4 ,,peculi>ities ,,( ,,! ,,triangles #c1 #d1 #e1 ,,& #bj1 #ba1 #bi4 ,fig4 #cj to #ce4 ,,peculi>ities ,,( ,,! ,,triangles ,! #c1 #d1 #e triangle conta9s #cf #eb #aafe & ! complem5t or great] angle #ec #g #dhce ,radius #e #fj :ole numb]s4#f ,co-s9e #d #dh ,s9e #c #cf ,v]s$ s9e #a #ab ,co-v]s$ s9e #b #bd ,tang5t #c #de ,secant #f #ge ,co-tang5t #f #hj ,co-secant #h #ajj ,tang5t ,secant ,diamet] or #b ,radius ,co-tan ,co-sec #c ,radius ,s9e 3 ,v]s$-s9e 33 #c 3 #a ,co-s9e 3 ,co-v]s$ s9e 33 #b 3 #a ,figure #cj illu/rates ! prec$+ description4 ,figure #ca %[s ! #ca triangle1 & ! #ba triangle built up on ! s9e & co-s9e ( ! #c1 #d1 #e triangle4 ,! #ca triangle conta9s #ah #bf #ehb & ! #ba triangle #bf #cc #edai2 ! latt] has be5 frequ5tly notic$ z a pyramid angle 9 ! gall]y incl9ations4 ,figure #cb %[s ^! two triangles comb9$ ) ! #c1 #d1 #e triangle1 on ! circumf]5ce ( a circle4 ,footnote #f3 #fj #c #d #e ,! #bj1 #ba1 #bi triangle conta9s #dc #cf #ajae & ! complem5t1 #df #bc #dihe4 ,express$ 9 :ole numb]s-- ,radius #bi #abahjg ,s9e #bj #hdjj ,co-s9e #ba #hhbj ,v]s$ s9e #h #ccfj ,co-v]s$ s9e #i #cghj ,tang5t #aafjj ,co-tang5t #abghi ,secant #afhbj ,co-sec #agffa ,tang5t ,secant #b radius ,co-tan ,co-sec #b radius ,s9e 3 ,v]s$ s9e 33 #e 3 #b ,co-s9e 3 ,co-v]s$ s9e 33 #g 3 #c ,footnote #g3 #abahj #bj #ba #bi ,x is noticeable t :ile ! multipli] requir$ to br+ radius #e & ! re/ into :ole numb]s1 = ! #c1 #d1 #e triangle is twelve1 9 ! #bj1 #ba1 #bi triangle x is #dbj1 ! key measure = ! bases ( ! two ma9 pyramids 9 ,r4,b4 cubits4#h ,footnote #h3 #ab #c #d1 & #dbj #bj #ba ,i am l$ to believe f /udy ( ! plan1 & consid]ation ( ! :ole numb]s 9 ?is #bj1 #ba1 #bi triangle1 t ! ,r4,b4 cubit1 l ! ,memphis cubit1 0 divid$ into #bhj p>ts4 ,! :ole numb]s ( radius1 s9e1 & co-s9e divid$ by #bhj1 give a v pretty measure & s]ies 9 ,r4,b4 cubits1 viz41 #dc1 #cj1 & #ca1 or #hg1 #fj1 & #fc1 or #agd1 #abj & #abf2--;all exce$+ly useful 9 "r-angl$ measurem5ts4 ,notice t ! "r-angl$ triangle #agd1 #abj1 #abf1 9 ! sum ( its sides am\nts to #dbj4 ,figure #cc illu/rates ! #bj1 #ba1 #bi triangle4 ,figure #cd %[s ! #eb & #gc triangles built up on ! s9e & co-s9e ( ! #bj1 #ba1 #bi triangle4 ,! #eb triangle conta9s #ba #dh #ejh & ! #gc triangle #bc #aa #edih4 ,figure #ce %[s h[ ^! two triangles are comb9$ ) ! #bj1 #ba1 #bi triangle on ! circumf]5ce1 & ,figure #cf gives a g5]al view & id5tification ( ^! six triangles :i* occupi$ an important position 9 ! trigonometry ( a p :o did all _! "w by "r angles & proportional l9es4 ,fig4 #cf4 ,ratios ( ,lead+ ,triangles4 #h4 ,,g5]al ,,obs]vations4 ,x mu/ be admitt$ t 9 ! details ( ! build+ ( ! ,pyramids ( ,gzeh !re are traces ( o!r measures ?an ,r4 ,b4 cubits1 b t ! orig9al cubit ( ! plan 0 #afhe ,briti% feet ,i feel no d\bt4 ,x is a p]fect & beautiful measure1 fit = su* a noble design1 &1 repres5t+ z x does ! sixtie? "p ( a second ( ! ,e>?'s pol> circumf]5ce1 x is & 0 a measure = all "t4 ,x may be object$ t ^! anci5t geometricians c\ld n h be5 aw>e ( ! measure ( ! ,e>?'s circumf]5ce2 & wisely s1 7 x n = two di/9ct answ]s t arise4 ,! fir/ be+1 t s9ce ,i ?9k ,i h %[n t ,py?agoras nev] discov]$ ! ,py?agorean triangle1 b t x mu/ h be5 kn[n & practically employ$ ?\s&s ( ye>s be=e 8 era1 9 ! ,egyptian ,colleges :]e he obta9$ 8 ,m4,a4 degree1 s 9 ! same way x is probable t ,erato/h5es1 :5 he w5t to "w to prove t ! e>?'s circumf]5ce 0 fifty times ! di/ance f ,sy5e to ,alex&ria1 may h obta9$ ! idea f 8 ready access to ! ill-fat$ ,alex&rian ,libr>y1 9 :i* p]haps "s record ( ! le>n+ ( ! build]s ( ! ,pyramids 0 /or$4 ,& !re=e ,i claim t !re is no reason :y ! pyramid build]s %\ld n h kn[n z mu* ab\t ! circumf]5ce ( ! e>? z ! mod]n _w t has calmly /ood by 9 its ignorance & p]mitt$ ^? magnific5t &1 z ,i %all prove1 useful edifices to be /ripp$ ( _! beautiful g>m5ts ( poli%$ m>ble4 ,my second answ] is t ! correct cubit measure may h be5 got by its inv5tors 9 a v>iety ( o!r ways2 = in/ance1 by obs]vations ( %ad[s ( heav5ly bodies1 )\t any k ev5 t ! e>? 0 r\nd2 or x may h be5 evolv$ l ! ,briti% in*1 :i* ,sir ,john ,h]s*el tells u is )9 a ?\s&? "p ( be+ "o five hundr$ million? ( ! e>?'s pol> axis4 ,i d\bt if ! circumf]5ce ( ! e>? 0 consid]$ by ! inv5tor ( ! ,briti% in*4 ,x 0 a peculi>ity ( ! ,h9doo ma!maticians t !y tri$ to make out t all !y knew 0 v old4 ,mod]n savants appe> to take ! opposite /& = any little in=mation !y happ5 to possess4 ,! cubit :i* is call$ ! ,royal ,babylonian cubit & /at$ to measure ,o#eaca metre1 diff]s s slightly f my cubit1 only ! six-hundr$? "p ( a foot1 t x may fairly be said to be ! same cubit1 & x w be = antiqu>ies to trace ! connection1 z ?is may ?r[ "s light on ! id5tity ( ! build]s ( ! ,pyramids ( ,gzeh4 ,few good ,engli% two-foot rules agree bett] ?an ^! two cubits d4 ,:ile ,i 0 grop+ ab\t 9 ! d>k se>*+ = ?is bright ne$le1 ,i tri$ on ! plan _m likely anci5t measures4 ,= a long "t ,i work$ 9 ,memphis or ,nilometric cubits1 :i* ,i made #agabf ,briti% feet2 !y seem to v>y f #agj to #agb1 & al?\gh ,i made good use ( !m 9 id5tify+ o!r p's measures1 /ill !y 7 evid5tly n 9 accordance ) ! design2 b ! ,r4,b4 cubit ( #afhe ,briti% feet works z truly into ! plan ( ! ,pyramids )\t fractions z x does into ! circumf]5ce ( ! e>?4 ,"h ,i might1 to prev5t o!rs f fall+ into "o ( my errors1 po9t out a rock on :i* ,i 0 agr\nd = a long "t4 ,i took ! base ( ! ,pyramid ,*eops1 det]m9$ by ,piazzi ,smy?1 f ,bonwick's ,pyramid ,facts & ,fancies 7a valuable little ref]5ce book71 z #gfc4ha ,briti% feet1 & ! altitude z #dhf4befg2 & !n f ,piazzi ,smy?'s ,inh]itance1 page #bg1 ,i confirm$ ^! figures1 & s work$ on !m = a long "t1 b f\nd always a great flaw 9 my "w1 & at la/ adopt$ a fre% base = ,*eops1 feel+ sure t ,mr4 ,smy?'s base 0 wrong3 = ,i 0 absolutely gr\nd$ 9 my conviction t at a c]ta9 level1 ,*eops' & ,cephr5's measures bore c]ta9 relations to ea* o!r4 ,i subsequ5tly f\nd 9 ano!r "p ( ,mr4 ,smy?'s book1 t ! correct measures 7 #gfa4fe & #dhd4ia ,briti% feet = base & altitude1 :i* 7 exactly :at ,i want$1 & enabl$ me to be 9 accordance ) him 9 t pyramid :i* he appe>s to h made 8 p>ticul> /udy4 ,= ! in=mation ( ^? :o may wi% to comp>e my measures1 :i* are ! results ( an ev5 or regul> circumf]5ce )\t fractions1 ) ,mr4 ,smy?'s measures1 :i* are ! results ( an ev5 or regul> diamet] )\t fractions1 x may be well to /ate t !re are j ab\t #ii ,r4,b4 cubits 9 #hj ( ,piazzi ,smy?'s cubits ( #be pyramid in*es ea*4 #i4 ,,! ,,pyramids ,,( ,,egypt1 ,,! ,,!odolites ,,( ,,! ,,egyptian ,,l& ,,surveyors4 ,ab\t tw5ty-?ree ye>s ago1 on my road to ,au/ralia1 ,i 0 cross+ f ,alex&ria to ,cairo1 & saw ! pyramids ( ,gzeh4 ,i wat*$ !m c>efully z ! tra9 pass$ along1 notic$ _! cle> cut l9es aga9/ ! sky1 & _! con/antly *ang+ relative position4 ,i !n felt a /rong conviction t !y 7 built = at lea/ "o useful purpose1 & t purpose 0 ! survey ( ! c\ntry4 ,i said1 ,"h be ! ,!odolites ( ! ,egyptians4 ,built by sci5tific m51 well v]s$ 9 geometry1 b unacqua9t$ ) ! use ( glass l5ses1 ^! great /one monum5ts are s suit$ 9 %ape = ! purposes ( l& survey+1 t ! practical eng9e] or surveyor mu/1 aft] consid]ation1 admit t !y may h be5 built ma9ly = t purpose4 ,n only might ! c\ntry h be5 survey$ by ^! great in/rum5ts1 & ! l& allott$ at p]iodical times to ! p2 b !y1 rema9+ always 9 "o position1 7 !re to correct & readju/ b\nd>ies de/roy$ or confus$ by ! annual inundations ( ! ,nile4 ,! ,pyramids ( ,egypt may be consid]$ z a great sy/em ( l&m>ks = ! e/abli%m5t & easy readju/m5t at any "t ( ! b\nd>ies ( ! hold+s ( ! p4 ,! ,pyramids ( ,gzeh appe> to h be5 ma9 m>ks2 & ^? ( ,ab\sir1 ,sakkrah1 ,da%[1 ,li%t1 ,meyd\n1 c41 ) ! great pyramids 9 ,lake ,mo]is1 subord9ate m>ks1 9 ?is sy/em1 :i* 0 probably ext5d$ f ,*aldea ?r\gh ,egypt into ,e?iopia4 ,! pyramid build]s may p]haps h made ! entombm5t ( _! ,k+s "o ( _! exot]ic objects1 play+ on ! morbid vanity ( _! rul]s to induce !m to ! "w1 b 9 ! m9ds ( ! build]s be=e ev] !y built mu/ h be5 plant$ ! int5tion to make use ( ! /ructures = ! purposes ( l& survey+4 ,! l& ( ,egypt 0 valuable & ma9ta9$ a d5se population2 e ye> x 0 mo/ly subm]g$1 & ! b\nd>ies de/roy$ or confus$4 ,e soldi] _h six to twelve acres ( l&2 ! prie/s _h _! slice ( ! l& too2 aft] e w> a reallotm5t ( ! l&s mu/ h tak5 place1 p]haps e ye>4 ,:ile ! wat] 0 ly+ on ! l&1 x s s(t5$ ! gr\nd t ! /one b\nd>y m>ks mu/ h requir$ frequ5t readju/m5t1 z !y w\ld h be5 likely to fall on "o side4 ,by ! aid ( _! great /one !odolites1 ! surveyors1 :o belong$ to ! prie/ly ord]1 7 able to readju/ ! b\nd>ies ) great precision4 ,t all sci5ce 0 compris$ 9 _! secret my/]ies may be "o reason :y no hi]oglyphic record ( ! sci5tific uses ( ! pyramids rema9s4 ,x is possible t at ! "t ( ,diodorus & ,h]odotus1 7& ev5 :5 ,py?agoras visit$ ,egypt17 !ology may h s smo!r$ sci5ce1 t ! uses ( ! pyramids may h be5 =gott5 by ! v prie/s to :om 9 =m] times ! k belong$2 b a respectful retic5ce :i* has be5 notic$ 9 "s ( ^! old writ]s on pyramid & o!r prie/ly matt]s w\ld r lead u to believe t an initiation into ! my/]ies may h seal$ _! lips on subjects ab\t :i* !y might o!rwise h be5 m explicit4 ,! clos+ ( "o pyramid ov] ano!r 9 br++ any ( _! _m l9es into true ord]1 mu/ ev5 n[ be v p]fect2--b n[ we c only imag9e ! beauties ( ^! great in/rum5tal wond]s ( ! _w :5 ! cas+ /ones 7 on !m4 ,we c picture ! rosy lights ( "o1 & ! bright :ite lights ( o!rs2 _! cle> cut l9es aga9/ ! sky1 true z ! hairs ( a !odolite2 & ! sombre d>kness ( ! contra/+ %ades1 br++ out _! angles ) />tl+ di/9ctness4 ,"u ! influ5ce ( ! ,ea/]n sun1 ! faces mu/ h be5 a v blaze ( light1 & c\ld h be5 se5 at enorm\s di/ances l great mirrors4 ,i decl>e t ! pyramids ( ,gzeh 9 all _! poli%$ glory1 be=e ! de/roy] /ripp$ !m ( _! beautiful g>m5ts1 7 9 e respect adapt$ to fla% ar\nd cle>ly def9$ l9es ( sight1 ^u :i* ! l&s ( ! nation c\ld be accurately ?read$4 ,! v ?\ght ( ^! mighty !odolites ( ! old ,egyptians fills me ) wond] & rev]5ce4 ,:at p]fect & beautiful in/rum5ts !y 76 nev] out ( adju/m5t1 always correct1 always ready2 no magnetic deviation to all[ =4 ,no wond] !y took ! tr\ble !y did to build !m s correctly 9 _! s m>vell\sly suitable positions4 ,if ,a/ronom]s agree t obs]vations ( a pole /> c\ld h be5 accurately made by pe]+ up a small gall]y on ! nor? side ( "o ( ! pyramids only a few hundr$ feet 9 l5g?1 ,i feel t ,i %all h little difficulty 9 satisfy+ !m t accurate measurem5ts to po9ts only miles away c\ld h be5 made f angul> obs]vations ( ! :ole gr\p4 #aj4 ,,h[ ,,! ,,pyramids ,,7 ,,made ,,use ,,(4 ,x appe>s f :at ,i h already set =? t ! plan ( ! ,pyramids "u consid]ation is geometrically exact1 a p]fect set ( measures4 ,i %all n[ %[ h[ ^! edifices 7 appli$ to a ?or\ghly geometrical purpose 9 ! true mean+ ( ! ^w--to measure ! ,e>?4 ,i %all %[ h[ true /raight l9es c\ld be ext5d$ f ! ,pyramids 9 giv5 directions useful 9 "r-angl$ trigonometry1 by direct obs]vation ( ! build+s1 & )\t ! aid ( o!r in/rum5ts4 ,& ,i %all %[ h[ by ! aid ( a simple in/rum5t angles c\ld be exactly obs]v$ f any po9t4 ,?is ,survey !ory does n /& or fall on ! m]its ( my !ory ( ! ,gzeh plan4 ,let x be prov$ t ?is gr\p is n built on ! exact sy/em ( triangulation set =? by me1 x is /ill a fact t its plan is 9 a simil> %ape1 & any su* %ape w\ld enable a surveyor acqua9t$ ) ! plan to lay d[n accurate surveys by obs]vations ( ! gr\p ev5 %\ld x n occupy ! precise l9es assum$ by me4 ,& "h ,i mu/ /ate t al?\gh ! l9es ( ! plan z laid d[n h]e9 agree ne>ly ) ! l9es z laid d[n 9 ,piazzi ,smy?'s book1 9 ! ,p5ny ,cyclopdia1 & 9 an essay ( ,proctor's 9 ! ,g5tleman's ,magaz9e1 /ill ,i f9d t !y d n agree at all satisfactorily ) a map ( ! ,pyramids 9 ,%>p's ,egypt1 said to be copi$ f ,wilk9son's map4 ,we w1 h[ev]1 = ! "t1 & to expla9 my survey !ory1 suppose ! plan !ory to be correct1 z ,i firmly believe x is4 ,& !n1 suppos+ x may be prov$ t ! respective positions ( ! pyramids are slightly diff]5t to ^? t ,i h allott$ to !m on my plan1 x w only make a simil> slight diff]5ce to ! l9es & angles :i* ,i %all "h %[ c\ld be laid out by _! aid4 ,let u 9 ! fir/ place compreh5d cle>ly ! %ape ( ! l& ( ,egypt4 ,a sector or fan1 ) a long h&le--! fan or sector1 ! delta2 & ! h&le ( ! fan1 ! ,nile ,valley1 runn+ ne>ly due s\?4 ,! ,pyramids ( ,gzeh are situate at ! angle ( ! sector1 on a rocky em95ce :5ce !y c all be se5 = _m miles4 ,! summits ( ! two high ones c be se5 f ! delta1 & f ! ,nile ,valley to a v great di/ance2 h[ f>1 ,i am unable to say2 b ,i %\ld ?9k t :ile ! gr\p c\ld be made g5]al use ( = a radius ( fifte5 miles1 ! summits ( ,cephr5 & ,*eops c\ld be made use ( = a di/ance ( ?irty miles2 tak+ into consid]ation ! g5]al fall ( ! c\ntry4 ,x mu/ be admitt$ t if m]idian obs]vations ( ! /> ,alpha ( ! ,dragon c\ld be made ) accuracy by peep+ up a small hole 9 "o ( ! pyramids1 !n surely might ! surveyors h c>ri$ true nor? & s\? l9es up ! ,nile ,valley z f> z ! summit ( ,*eops 0 visible1 by plumb+ 9 ! /> & ! apex ( ! pyramid by ! aid ( a /r+ & a /one4 ,true ea/ & we/ l9es c\ld h be5 made to int]sect su* nor? & s\? l9es f ! v>i\s gr\ps ( pyramids along ! riv] banks1 by ^: aid also su* l9es w\ld be prolong$4 ,next1 suppos+ t _! a/ronom]s _h be5 aw>e ( ! latitude ( ,*eops1 & ! annual nor?+ & s\?+ ( ! sun1 /raight l9es c\ld h be5 laid out 9 v>i\s sectoral directions to ! nor?-ea/w>d & nor?-we/w>d ( ,*eops1 across ! delta1 z f> z ! extreme apex ( ! pyramid 0 visible1 by obs]vations ( ! sun1 ris+ or sett+ ov] 8 summit4 7,t ! ,dog-/> 0 obs]v$ 9 ?is mann] f ! nor?-we/1 ,i h little d\bt47 ,fig4 #cg4 ,sun above ,pyramid & s9k+ beh9d ,pyramid ,= ?is purpose1 surveyors w\ld be /ation$ at suitable di/ances ap>t ) _! /r+s & _! /ones1 ready to cat* ! sun simultane\sly1 & at ! v mom5t he became transfix$ ^u ! apex ( ! pyramid1 & 01 z x 71 swall[$ by x4 7,see ,figure #cg47 ,! k ( ! pyramid slope angle f diff]5t po9ts ( view w\ld enable ! surveyor to place himself 9 read9ess ne>ly on ! l9e4 ,surely su* l9es z ^! w\ld be z true & z p]fect z we c\ld lay out n[adays ) all our mod]n in/rum5tal appliances4 ,a /r+ & a /one "h1 a clean-cut po9t ( /one tw5ty miles away1 & a great ball ( fire beh9d t po9t at a di/ance ( n9ety odd million miles4 ,! error 9 su* a l9e w\ld be v trifl+4 ,su* obs]vations z la/ m5tion$ w\ld h be5 probably ext5d$ f ,cephr5 = long l9es1 z be+ ! high] pyramid above ! e>?'s surface1 & may h be5 made f ! moon or />s4 ,9 ^? days 0 ! sun ! intimate fri5d ( man4 ,! moon & />s 7 8 h&-maid5s4 ,h[ _m ( u c po9t to ! spot ( ! sun's ris+ or sett+8 ,we1 ) our clocks1 & our wat*es1 & our compasses1 r>ely obs]ve ! sun or />s4 ,b 9 a l& & an age :5 ! sun 0 ! only clock1 & ! pyramid ! only compass1 ! movem5ts & positions ( ! heav5ly bodies 7 kn[n to all4 ,^! p 7 famili> ) ! />s1 & kept a wat* ^u _! movem5ts4 ,h[ _m ( our vaunt$ educat$ population c\ld po9t out ! ,dog-/> 9 ! heav5s8--b ! :ole ,egyptian nation hail$ 8 ris+ z ! beg9n+ ( _! ye>1 & z ! h>b+] ( _! annual bless+1 ! ris+ ( ! wat]s ( ! ,nile4 ,fig4 #ch4 ,f ! ,nor? ,we/ ,be>+ #cae ,sun 9 ! ,we/4 ,fig4 #ci4 ,f ! ,s\? ,ea/ ,be>+ #ace ,sun 9 ! ,we/4 ,fig4 #dj4 ,f ! ,nor? ,ea/ ,be>+ #de ,sun 9 ! ,ea/4 ,fig4 #da4 ,f ! ,s\? ,we/ ,be>+ #bbe ,sun 9 ! ,ea/4 ,x is possible !re=e t ! l& surveyors ( ,egypt made full use ( ! heav5ly bodies 9 _! surveys ( ! l&2 & :ile we are pitifully lay+ out our new c\ntries by ! circumf]5t] & ! compass1 we presume to speak slight+ly ( ! suppos$ d>k hea!n days1 :5 ! l& ( ,egypt 0 survey$ by means ( ! sun & ! />s1 & ! !odolites 7 built ( /one1 ) v]tical limbs five hundr$ feet 9 height1 & horizontal limbs ?ree ?\s& feet 9 diamet]4 ,imag9e half a doz5 su* in/rum5ts z ?is 9 a di/ance ( ab\t sixty miles 7= ea* gr\p ( pyramids 0 effectually su* an in/rum5t71 & we c =m "s conception ( ! p]fection ( ! surveys ( an almo/ prehi/oric nation4 ,! c5tre ( ,lake ,mo]is1 9 :i* ,h]odotus tells u two pyramids /ood #cjj feet above ! level ( ! lake1 appe>s f ! maps to be ab\t ,s4 #bh ,w41 or ,s4 #bi ,w4 f ,gzeh1 di/ant ab\t #eg miles1 & ! ,meidn gr\p ( pyramids appe>s to be ab\t #cc miles due s\? ( ,gzeh4 ,figures #ch1 #ci1 #dj & #da1 %[ t nor?-we/1 s\?-ea/1 nor?-ea/1 & s\?-we/ l9es f ! pyramids c\ld be ext5d$ by simply plumb+ ! angles4 ,^! l9es w\ld be run 9 sets ( two's & ?ree's1 accord+ to ! numb] ( pyramids 9 ! gr\p2 & _! kn[n di/ances ap>t at t angle w\ld *eck ! correctness ( ! "w4 ,a spl5did l9e 0 ! l9e be>+ #dc #cf #ajae1 or #bbc #cf #ajae f ,*eops & ,cephr51 ! pyramids cov]+ ea* o!r1 ! l9e ( hypot5use ( ! great #bj1 #ba1 #bi triangle ( ! plan4 ,?is ,i call ! #bj1 #ba l9e4 7,see ,figure #db47 ,figure #dc repres5ts ! #c1 #d1 #e triangle l9e f ! summits ( ,myc]9us & ,*eops 9 true l9e be>+ #baf #eb' #aafe4 ,?is ,i call ! s\? #d1 we/ #c l9e4 ,! next l9e is :at ,i call ! #b1 #a l9e1 & is illu/rat$ by figure #dd4 ,x is "o ( ! mo/ p]fect ( ! s]ies1 & be>s ,s4 #bf #cc' #edi ,w4 f ! apex ( ,cephr54 ,?is l9e demon/rates cle>ly :y ,myc]9us 0 cas$ ) r$ granite4 ,n 9 memory ( ! beautiful & rosy-*eek$ ,nitocris1 z "s ( ! tomb !ory p say1 b = a less romantic b m useful object2 simply because1 f ?is qu>t]1 & r\nd ab\t1 ! l9es ( ! pyramids w\ld h be5 confus$ if ,myc]9us _h n be5 ( a diff]5t color4 ,! #b1 #a l9e is a l9e 9 :i* ,myc]9us w\ld h be5 absolutely lo/ 9 ! slopes ( ,cephr5 b = 8 r$ color4 ,!re is n a fact t m cle>ly e/abli%es my !ory1 & ! wisdom & =e?\ght ( ^? :o plann$ ! ,gzeh pyramids1 ?an ?is r$ pyramid ,myc]9us1 & ! #b1 #a l9e4 ,hekeyan ,bey1 speaks ( ?is pyramid z ( a ruddy complexion2 ,john ,greaves quotes f ! ,arabic book1 ,morat ,alzeman1 & ! less] :i* is col\r$2 & an ,arabic writ] :o dates ! ,pyramids ?ree hundr$ ye>s be=e ! ,flood1 & _c f9d among ! le>n$ m5 ( ,egypt any c]ta9 relation conc]n+ !m nor any memory ( !m among/ m51 also expatiates ^u ! beauties ( ! col\r$ sat9 cov]+ ( ?is "o p>ticul> pyramid4 ,fig4 #db4 ,s\? #ba4 ,we/ #bj4 ,be>+ #bbc4cf'4ajae4 ,fig4 #dc4 ,s\? #d4 ,we/ #c4 ,be>+ #baf4eb'4aafe4 ,fig4 #dd4 ,s\? #b4 ,we/ #a4 ,be>+ #bjf4cc'4edah4 ,fig4 #de4 ,s\? #if4 ,we/ #ee4 ,be>+ #bji4dh'4cbha4 ,fig4 #df4 ,s\? #c4 ,we/ #a4 ,be>+ #aih4bf'4ehb4 ,fig4 #dg4 ,s\? #e4 ,we/ #b4 ,be>+ #bja4dh'4e4 ,fig4 #dh4 ,s\? #g4 ,we/ #c4 ,be>+ #bjc4aa'4ee4 ,figure #de repres5ts ! l9e s\? #if1 we/ #ee1 f ,cephr51 be>+ #bji #dh' #cbha2 ! apex ( ,cephr5 is imm$iately above ! apex ( ,myc]9us4 ,figure #df is ! ,s4 #c ,w4 #a l9e1 be>+ #aih #bf' #e4hb2 "h ! d>k slope angle ( ! pyramids ) ! sun to ! ea/w>d occupies half ( ! app>5t half base4 ,figure #dg is ! ,s4 #e1 ,w4 #b l9e1 be>+ #bja #dh' #e2 "h ,cephr5 & ,myc]9us are 9 outside slope l9e4 ,figure #dh is ! ,s4 #g ,w4 #c l9e1 be>+ #bjc #aa' #ee2 "h ! inside slope ( ,cephr5 spr+s f ! c5tre ( ! app>5t base ( ,myc]9us4 ,i mu/ cont5t myself ) ! prec$+ examples ( a few pyramid l9es1 b mu/ h said en\gh to %[ t f e po9t ( ! compass _! appe>ance 0 di/9ctly m>k$ & def9itely to be det]m9$ by surveyors acqua9t$ ) ! plan4 #aa4 ,,description ,,( ,,! ,,anci5t ,,portable ,,survey ,,in/rum5t4 ,i mu/ n[ comm5ce ) a s+le pyramid1 %[ h[ approximate obs]vations c\ld be made f x1 & !n ext5d ! !ory to a gr\p ) ! obs]vations !reby r5d]$ m p]fect & delicate4 ,we w suppose ! surveyor to be /&+ look+ at ! pyramid ,cephr52 he kn[s t its base is #dbj cubits1 & its apo!m #cdf cubits4 ,he has provid$ himself ) a model 9 wood1 or /one1 or metal1 & "o ?\s&? ( its size--!re=e 8 model w be ,o4#db cubit base1 & ,o4#cdfe cubit apo!m--or1 9 r\nd numb]s1 eight & half in*es base1 & sev5 in*es apo!m4 ,?is model is fix$ on ! c5tre ( a c>d or disc1 graduat$ f ! c5tre to ! circumf]5ce1 l a compass c>d1 to ! v>i\s po9ts ( ! compass1 or divisions ( a circle4 ,! model pyramid is fa/5$ due nor? & s\? on ! l9es ( ?is c>d or disc1 s t :5 ! nor? po9t ( ! c>d po9ts nor?1 ! nor? face ( ! model pyramid faces to ! nor?4 ,! surveyor also has a table1 :i*1 ) a pair ( plumb l9es or mason's levels1 he c erect q level3 ?is table is also graduat$ f ! c5tre ) divisions ( a circle1 or po9ts ( ! compass1 & x is l>g] ?an ! c>d or disc atta*$ to ! model4 ,?is table is made s t x c revolve ^u its /&1 & c be clamp$4 ,we w call x ! l[] limb4 ,!re is a p9 9 ! c5tre ( ! l[] limb1 & a hole 9 ! c5tre ( ! disc be>+ ! model1 :i* c be ?us plac$ ^u ! c5tre ( ! table1 & becomes ! upp] limb4 ,! upp] limb c be clamp$ to ! l[] limb4 ,! fir/ process w be to clamp bo? upp] & l[] limbs toge!r1 ) ! nor? & s\? l9es ( bo? 9 unison1 !n revolve bo? limbs on ! /& till ! nor? & s\? l9e po9ts /raight = ! pyramid 9 ! di/ance1 :i* is done by ! aid ( sights erect$ at ! nor? & s\? po9ts ( ! p]imet] ( ! l[] limb4 ,:5 ?is is adju/$1 clamp ! l[] limb & release ! upp] limb2 n[ revolve ! upp] limb until ! model pyramid exactly cov]s ! pyramid 9 ! di/ance1 & %[s j ! same %ade on "o side & light on ! o!r1 :5 view$ f ! sights ( ! clamp$ l[] limb--& ! l9es1 angles1 & %ades ( ! model co9cide ) ! l9es1 angles1 & %ades ( ! pyramid obs]v$2--n[ clamp ! upp] limb4 ,n[ does ! model /& really due nor? & s\?1 ! same z ! pyramid 9 ! di/ance2 x ?r[s ! same %ades1 & exhibits ! same angles :5 se5 f ! same po9t ( view2 j z mu* ( x is 9 %ade & z mu* ( x is 9 light z ! pyramid "u obs]vation2 !re=e x mu/ be /&+ due nor? & s\?1 because ,cephr5 himself is /&+ due nor? & s\?1 & ! upp] limb reads (f on ! l[] limb ! angle or be>+ obs]v$4 ,s f> we possess an in/rum5t equal to ! mod]n circumf]5t]1 & yet we h only br\ght "o pyramid into "w4 ,if ,i h %[n t su* an op]ation z ! above is practically feasible1 if ,i h %[n t angles c be tak5 ) mod]ate accuracy by obs]v+ "o pyramid ( #dbj cubits base1 h[ mu* m accurate w ! obs]vation be :5 ! surveyor's plane table be>s a gr\p ( pyramids :i* occupy a repres5tative space ( ab\t #adjj cubits :5 view$ f ! s\? or nor?1 & ab\t #agfj cubits :5 view$ f ! ea/ or we/4 ,if situat$ a mile or two s\? ( ! ,gzeh gr\p our surveyor c\ld also tie 9 & p]fect 8 "w by sights to ! ,skk>ah gr\p ) ,skk>ah models2 & s on1 up ! ,nile ,valley1 he w\ld f9d e few miles gr\ps ( pyramids by aid ( :i* he w\ld be enabl$ to tie 8 "w toge!r4 ,if ! ,gzeh gr\p ( pyramids is plac$ & %ap$ 9 ! mann] ,i h describ$1 x mu/ be cle> t an exact model & plan1 say a ?\s&? ( ! size1 c\ld be v easily made--! plan be+ at ! level ( ! base ( ,cephr5 :]e ! bases ( ! two ma9 pyramids are ev52--& if !y are n exactly s plac$ & %ap$1 x may be admitt$ t _! position & dim5sions 7 kn[n to ! surveyors or prie/s1 s t su* models c\ld be con/ruct$4 ,x is probable1 !re=e1 t ! in/rum5t us$ 9 conjunction ) ^! pyramids1 0 a ma*9e con/ruct$ 9 a simil> mann] to ! simple ma*9e ,i h describ$1 only in/ead ( !re be+ b "o model pyramid on ! disc or upp] limb1 x bore ! :ole gr\p2 & ! small] pyramids 7 :at we may call v]ni] po9ts 9 ?is great circle1 enabl+ ! surveyor to m>k (f kn[n angles ) great accuracy by notic+ h[1 z he work$ r\nd ! gr\p ( pyramids1 "o or o!r ( ! small] ones 0 cov]$ by its neighb\rs4#i ,footnote #i3 ,see g5]al plan ( ,gzeh ,gr\p op4 page #a4 ,! imm5sity ( ! ma9 pyramids w\ld require ! small] ones to be us$ = surveys 9 ! imm$iate neighb\rhood1 z ! surveyor might easily be too close to get accurate obs]vations f ! ma9 pyramids4 ,! upp] limb1 !n1 0 a disc or circul> plate be>+ ! model ( ! gr\p4 ,*eops w\ld be situat$ 9 ! c5tre ( ! circle1 & obs]vations w\ld be tak5 by br++ ! :ole model gr\p into ev5 l9e & ev5 light & %ade ) ! ,gzeh gr\p4 ,i believe t ) a reasonable-siz$ model occupy+ a circle ( six or sev5 feet diamet]1 su* z a c\ple ( m5 c\ld c>ry1 v accurate be>+s c\ld h be5 tak51 & probably 7 tak54 ,! pyramid %ape is ! v %ape ( all o!rs to employ = su* purposes4 ,a cone w\ld be useless1 because ! lights & %ades w\ld be s(t5$ (f & its angles f all po9ts w\ld be ! same4 ,o!r solids ) p]p5dicul> angles w\ld be useless1 because al?\gh !y w\ld v>y 9 wid? f diff]5t po9ts ( view !y w\ld n pres5t t ev] *ang+ angle t a pyramid does :5 view$ f diff]5t directions4 ,aft] famili>ity ) ! models :i* ,i h made use ( 9 prosecut+ ^! inve/igations1 ,i f9d t ,i c judge ) great accuracy f _! appe>ance only ! be>+ ( ! gr\p f any po9t at :i* ,i /&4 ,i make bold to say t ! pocket compass ( ! ,egyptian surveyor 0 a little model ( ! gr\p ( pyramids 9 8 di/rict1 & he _h only to hold x up on 8 h& & turn x r\nd 9 ! sun till its %ades & angles correspond$ ) ! appe>ance ( ! gr\p1 to tell z well z we c\ld tell by our compasses1 p]haps bett]1 8 be>+ f ! l&m>ks t gov]n$ 8 surveys4 ,! ,great ,circle ( ,gold describ$ by ,diodorus 7,diod4 ,sic4 lib4 ,x41 "p #b1 cap4 #a7 z hav+ be5 employ$ by ! ,egyptians1 & on :i* 0 m>k$ among/ o!r ?+s1 ! position ( ! ris+ & sett+ ( ! />s1 & /at$ by him to h be5 c>ri$ (f by ,cambysses :5 ,egypt 0 conqu]$ by ! ,p]sians1 is suppos$ by ,cass9i to h be5 also employ$ = f9d+ ! m]idian by obs]vation ( ! ris+ & sett+ ( ! sun4 ,?is in/rum5t & o!rs describ$ by writ]s on ,egypt w\ld h be5 9 practice v simil> to ! in/rum5t :i* ,i h describ$ z hav+ be5 probably employ$ = t]re/rial obs]vations4 ,! table or disc compris+ ! l[] limb ( ! in/rum5t1 might h be5 support$ ^u a small /& ) a circul> hole 9 ! c5tre1 s arrang$ t ! in/rum5t c\ld be ei!r set up alone & support$ by its own tripod1 or re/$ fairly on ! top ( any ( ^? curi\s /one b\nd>y m>ks :i* 7 made use (1 n only to m>k ! corn]s ( ! diff]5t hold+s1 b to %[ ! level ( ! ,nile inundations4 7,see ,figure #di1 copi$ f ,%>pe's ,egypt1 vol4 ,i41 p4 #f47 ,! peculi> %ape ( ! top ( ^! /one l&m>ks1 or sacr$ b\nd>y /ones1 appe>s suitable = su* purposes1 & x w\ld h be5 a great conv5i5ce to ! surveyor1 & conducive to accuracy1 t x %\ld be s arrang$ t ! in/rum5t %\ld be fix$ imm$iately ov] ! m>k1 z appe>s probable f ! %ape ( ! /one4 ,fig #di4 ,sacr$ b\nd>y /one4 ,a noticeable po9t 9 ?is !ory is1 t x is n 9 ! lea/ ess5tial t ! apex ( a pyramid %\ld be complete4 ,if _! summits 7 left p]man5tly flat1 !y w\ld "w 9 = survey purposes q z well1 & ,i ?9k bett]1 ?an if c>ri$ to a po9t1 & !y w\ld be m useful ) a flat top = def9$ %ad[s :5 us$ z sun dials4 ,9 ! ,gzeh gr\p1 ! summit ( ,*eops appe>s to me to h be5 left incomplete ! bett] to get ! range ) ,cephr5 = l9es d[n ! delta4 ,9 ?is sy/em ( survey+1 !re is always a beautiful connection betwe5 ! horizontal be>+s & ! app>5t or obs]v$ angles pres5t$ by ! slopes & edges ( ! pyramid4 ,?us1 9 pyramids l ^? ( ,gzeh1 :i* /& nor? & s\?1 & ^: m]idional sections conta9 less1 & ^: diagonal sections conta9 m ?an a "r angle1 ! v]tex be+ ! po9t at :i* ! angle is measur$--?is law holds3-- ,t ! smalle/ int]ior angle at ! v]tex1 conta9$ betwe5 ! inside edge & ! outside edge1 w exhibit ! same angle z ! be>+ ( ! obs]v]'s eye f ! apex ( ! pyramid :5 ! angle at ! apex conta9$ by ! outside edges appe>s to be a "r angle4 ,figures #ej to #ee inclusive illu/rate ?is beautiful law f :i* x w be se5 t ! ,gzeh surveyors possess$1 9 ?is mann] alone1 eight di/9ctly def9$ be>+s f ea* pyramid4 ,fig4 #ej4 ,*eops f po9ts be>+ ,s #ai4ab4bb ,w ,w #ai4ab4bb ,n ,n #ai4ab4bb ,e ,e #ai4ab4bb ,s ,fig4 #ea4 ,*eops f po9ts be>+ ,s #ai4ab4bb ,e ,w #ai4ab4bb ,s ,n #ai4ab4bb ,w ,e #ai4ab4bb ,n ,fig4 #eb4 ,cephr5 f po9ts be>+ ,s #bc4g4ejbd ,w ,w #bc4g4ejbd ,n ,n #bc4g4ejbd ,e ,e #bc4g4ejbd ,n ,fig4 #ec4 ,cephr5 f po9ts be>+ ,s #bc4g4ejbd ,e ,w #bc4g4ejbd ,s ,n #bc4g4ejbd ,w ,e #bc4g4ejbd ,n ,fig4 #ed4 ,myc]9us f po9ts be>+ ,s #ag4a4djd ,w ,w #ag4a4djd ,n ,n #ag4a4djd ,e ,e #ag4a4djd ,s ,fig4 #ee4 ,myc]9us f po9ts be>+ - ,s #ag4a4djd ,e ,w #ag4a4djd ,s ,n #ag4a4djd ,w ,e #ag4a4djd ,n 99 99 99 99 99 ,fig4 #ef4 ,*eops model4 ,fig4 #eg4 ,cephr5 model ,fig4 #eh4 ,myc]9us model 99 99 99 99 99 ,i recomm5d any "o desir\s to ?or\ghly compreh5d ^! matt]s1 to make a plan f my diagram1 ,figure #e1 us+ ,r4,b4 cubits = measures1 & to a suitable scale1 on a piece ( c>d-bo>d4 ,!n to cut out ( ! c>d-bo>d ! squ>es ( ! bases ( ! pyramids at ! level ( ,cephr51 viz41 #dbj1 #dbj & #bah cubits respectively1 = ! ?ree ma9 pyramids4 ,"o hundr$ cubits to ! in* is a conv5i5t scale & )9 ! limits ( a %eet ( ,ba? bo>d4 ,by /rik+ out ! models on c>d-bo>d 9 ! mann] %[n by diagrams 7see ,figures #ef1 #eg1 & #eh7 !y c be cut out ) a p5knife--cutt+ only half ?r\gh :]e ! l9es are dott$--b5t up toge!r1 & pa/$ along ! edges ) /rips ( writ+ pap] ab\t half an in* wide4 ,^! models c be dropp$ into ! squ>es cut out ( ! c>d-bo>d plan1 ?us correct+ ! error caus$ by ! ?ickness ( ! c>d-bo>d base1 & if plac$ 9 ! sun1 or at night by ! light ( "o lamp or c&le prop]ly plac$ to repres5t ! sun 9 ! ea/w>d or we/w>d1 ! cle> cut l9es & cle> contra/+ %ades w be manife/1 & ! l9es illu/rat$ by my figures c be id5tifi$4 ,:5 inspect+ ! model1 x is well to be> 9 m9d t ! eye mu/ be kept v ne>ly level ) ! table1 or ! pyramids w appe> z if view$ f a balloon4 99 99 99 99 99 ,i believe t ! /ones 7 got up to ! build+ by way ( ! nor? side ( ea* pyramid4 ,! cas+ on ! s\?1 ea/1 & we/1 0 probably built up z ! "w proce$$1 & ! :ole ( ^! ?ree faces 7 probably ?us f9i%$ & complet$ :ile !re 0 n a s+le cas+ /one set on ! nor? side4 ,!n ! "w w\ld be clos$ up until !re rema9$ no?+ b a great gap or not*1 wide at ! bottom1 & n>r[+ to ! apex4 ,! "w on ! nor? side w\ld !n be clos$ f ! sides & top1 & ! bottom cas+ /one ab\t ! c5tre ( ! nor? side1 w\ld be ! la/ /one set on ! build+4 ,^! old build]s 7 too exp]t n to h ?us made use ( all ! %ade :i* _! own build+ w\ld ?us af=d to a majority ( ! workm54 99 99 99 99 99 ,_m ( ! obelisks 7 probably m>ks on pyramid l9es ( survey4 ,! pyramid inde$ may h be5 a developm5t ( ! obelisk = ?is purpose4 ,_! slant+ sides might correspond ) "s ( ! ne>ly upright slant angles ( ! pyramids1 9 positions opposite c]ta9 l9es4 ,ref]5ce to sev]al ( my figures w %[ h[ well ?is w\ld come 94 ,h]odotus speaks ( two obelisks at ,heliopolis1 & ,bonwick tells u t ,abd al ,latif saw two !re :i* he call$ ,ph>aoh's ,ne$les4 ,an ,arab travell]1 9 #aaij1 saw a pyramid ( copp] on ! summit ( ! "o t rema9$1 b x is n[ want+4 ,ph>aoh's ,ne$les appe> to h be5 situat$ ab\t #bj miles ,,ne4 ( ! ,gzeh gr\p1 & _! slope angles might h co9cid$ ) ! app>5t slope angles ( ,cephr5 or ,*eops on ! edge ne>e/ ! obelisk4 ,! anci5t me?od ( describ+ ! m]idian by means ( ! %ad[ ( a ball plac$ on ! summit ( an obelisk po9ts to a reasonable int]pretation = ! peculi> con/ruction ( ! two pill>s1 ,ja*9 & ,boaz1 :i* are said to h be5 situat$ 9 front ( ! ,hebrew ,temple at ,j]usalem1 & ab\t :i* s mu* my/]i\s speculation has occurr$4 ,!y 7 no d\bt us$ z sun-dials = ! morn+ & aft]noon sun by ! %ad[ ( ! balls or *apit]s ?r[n ^u ! pavem5t4 ,)\t presum+ to dispute ! objects assign$ by o!rs = ! gall]ies & passages :i* h be5 discov]$ 9 ! pyramid ,*eops1 ,i v5ture to op9e t !y 7 employ$ to c>ry wat] to ! build]s4 ,!y are connect$ ) a well1 & ! well ) ! ,nile or canal4 ,:e!r ! wat] 0 slid$ up ! smoo? gall]ies 9 boxes1 or :e!r ! co*lea1 or wat] screw1 0 work$ 9 !m1 _! angles be+ suitable1 x is impossible to conjecture2 ei!r plan w\ld h be5 conv5i5t & feasible4 ,^! s+ul> *amb]s & passages may inde$ possibly h _h to d ) "s hydraulic ma*9]y ( great p[] :i* mod]n sci5ce kn[s no?+ ab\t4 ,! section ( ! pyramid1 %[+ ^! gall]ies1 9 ! pyramid books1 has a mo/ hydraulic appe>ance4 ,! trem5d\s /r5g? & regul>ity ( ! cavities call$ ! ,k+'s & ,que5's *amb]s1 ! regul>ity & ! smallness ( mo/ ( ! passages or massive /one connect+ pipes1 favor ! idea t ! *amb]s might h be5 res]voirs1 _! curi\s ro(s1 air *amb]s1 & ! gall]ies or passages1 connect+ pipes = work+ wat] "u pressure4 ,wat] rais$ ?r\gh ! passages ( ?is "o pyramid ne>e/ to ! canal1 might h be5 c>ri$ by tr\ghs to ! o!r pyramids1 :i* 7 9 all probability 9 c\rse ( con/ruction at ! same p]iod ( "t4 ,a pr(ane fri5d ( m9e ?9ks t ! s>cophagus or sacr$ c(f] 9 ! ,k+'s *amb] may h be5 us$ by ! *ief ar*itect & lead+ m5 ( ! works z a ba?1 & t ! ,k+'s *amb] 0 no?+ m or less ?an a delightful ba? room4 99 99 99 99 99 ,! foll[+ quotation f ! writ+ ( an ,arabian au?or 7,ibn ,abd ,alkokm71 is extract$ f ,bonwick's ,pyramid ,facts & ,fancies1 page #gb3--,! ,coptites m5tion 9 _! books t ^u !m 7! ,pyramids7 is an inscription engrav52 ! exposition ( x 9 ,arabicke is ?is3--',i1 ,saurid ! ,k+ built ! ,pyramids 79 su* & su* a "t71 & f9i%$ !m 9 six ye>s2 he t comes aft] me1 & says he is equal to me1 let him de/roy !m 9 six hundr$ ye>s2 & yet x is kn[n t x is easi] to pluck d[n ?an to build2 & :5 ,i _h f9i%$ !m1 ,i cov]$ !m ) satt91 & let him cov] !m ) slats4' ,! italics are my own4 ,! build] seems to h ent]ta9$ ! idea t 8 "w w\ld be p>tially de/roy$1 & aft]w>ds tempor>ily repair$ or rebuilt4 ,! fir/ "p has un=tunately come true1 & x is possible t ! la/ "p ( ! idea ( ,k+ ,saurid may be c>ri$ out1 because x w\ld n be s v exp5sive an und]tak+ = any civiliz$ nation 9 ! int]e/ ( sci5ce to re-case ! pyramids ( ,gzeh1 s t !y might be once m appli$ to l&-survey+ purposes 9 ! anci5t mann]4 ,x w\ld n be absolutely necess>y to case ! :ole ( ! pyramid faces1 s long z suffici5t cas+ 0 put on to def9e ! angles4 ,! slats us$ might be a light wood5 framework cov]$ ) ?9 metal4 ,! metal %\ld be pa9t$ :ite1 except 9 ! case ( ,myc]9us1 :i* %\ld be ( a r$di% color4 #ab4 ,,prim>y ,,triangles ,,& ,,_! ,,satellites2--,,or ,,! ,,anci5t ,,sy/em ,,( ,,"r-,,angl$ ,,trigonometry ,,unfold$ ,,by ,a ,,/udy ,,( ,,! ,,plan ,,( ,,! ,,pyramids ,,( ,,gizeh4 ,ma9 ,triangul> ,dim5sions ( ,plan are ,repres5t$ by ! ,foll[+ ,eight ,"r-angl$ ,triangles4 ,,table ,,to ,,expla9 ,,figure #fj4 ---------------------------------------- ----------------------- ,,ab #bh #hd #fgb ,,dg #c #gb #egf ,,bj #de #c #ace #h #ajhj ,,ge #d #bd #if #h #gfh ,,ja #ec #aei #abgb ,,ed #e #abj #ifj ---------------------------------------- ----------------------- ,,dc #c #ace #ajhj ,,fw #dh #dh #chd ,,ca #d #de #ahj #h #addj ,,wv #ee #a #ee #h #ddj ,,ad #e #bbe #ahjj ,,vf #gc #gc #ehd ---------------------------------------- ----------------------- ,,eb #c #fc #ejd ,,fb #bj #hj #fdj ,,ba #d #ba #hd #h #fgb ,,ba #ba #d #hd #h #fgb ,,ae #e #aje #hdj ,,af #bi #aaf #ibh -------------------------- ,,fh #c #if #gfh ,,hn #d #cb #abh #h #ajbd ,note4--,9 ! above table ! fir/ ,,xf #e #afj #abhj column is ! ,ratio1 ! second -------------------------- ! connect$ ,natural ,numb]s1 & ,,ay #c #cf #bhh ! ?ird column repres5ts ! ,,yz #d #ab #dh #h #chd l5g? ea* l9e 9 ,r4,b4 cubits4 ,,za #e #fj #dhj ---------------------------------------- ----------------------- ,fig4 #fj4 ,ref]5ce to ,fig4 #fj & ! prec$+ table1 w %[ t ! ma9 triangul> dim5sions ( ?is plan 7imp]fect z x is f ! lack ( elev5 pyramids7 are repres5t$ by f\r ma9 triangles1 viz3-- ,ratio4 ,c ,a ,d ,c 44 44 #c1 #d1 #e ,f ,b ,a ,f 44 44 #bj1 #ba1 #bi ,a ,b ,j ,a 44 44 #bh1 #de1 #ec ,f ,w ,v ,f 44 44 #dh1 #ee1 #gc ,figures #cj to #cf illu/rate ! two =m]1 & ,figures #fa & #fb illu/rate ! two latt]4 ,i w call triangles ( ?is class prim>y triangles1 z ! mo/ suitable t]m1 al?\gh x is appli$ to ! ma9 triangles ( geodetic surveys4 ,we h only to select a numb] ( su* triangles & a sy/em ( trigonometry ensues1 9 :i* base1 p]p5dicul>1 & hypot5use ( e triangle is a :ole measure )\t fractions1 & 9 :i* ! nom5clature = e angle is cle> & simple4 ,an angle ( #dc #cf #ajae w be call$ a #bj1 #ba angle1 & an angle ( #cf #eb #aafe w be call$ a #c1 #d angle1 & s on4 ,9 ! exi/+ sy/em :ole angles1 su* z #dj1 #de1 or #ej degrees1 are surr\nd$ by l9es1 mo/ ( :i* c only be describ$ 9 numb]s by int]m9able fractions4 ,9 ! anci5t sy/em1 l9es are only dealt )1 & e angle 9 ! table is surr\nd$ by l9es measur+ :ole units1 & describ$ by ! use ( a c\ple ( simple numb]s4 ,connect+ ?is ) our pres5t sy/em ( trigonometry w\ld effect a sav+ 9 calculation1 & g5]al use ( c]ta9 peculi> angles by means ( :i* all ! simplicity & beauty ( ! "w ( ! anci5ts w\ld be comb9$ ) ! excell5ces ( our mod]n in/rum5tal appliances4 ,surveyors %\ld appreciate ! advantages to be d]iv$ f lay+ out trav]ses on ! hypot5uses ( prim>y triangles1 by ! sav+ ( calculation & facility ( plott+ to be obta9$ f ! practice4 ,! key to ^! old tables is ! fact1 t 9 prim>y triangles ! "r-angl$ triangle =m$ by ! s9e & v]s$ s9e1 also by ! co-s9e & co-v]s$-s9e1 is "o 9 :i* base & p]p5dicul> are measur$ by numb]s )\t fractions4 ,^! ,i w call satellite triangles4 ,?us1 to ! prim>y triangle #bj1 #ba1 #bi1 ! ratios ( ! co-s9al & s9al satellites are respectively #g to #c1 & #b to #e4 7,see ,figure #ce47 ,to ! #dh1 #ee1 #gc triangle ! satellites are #aa1 #e & #h1 #c 7,fig4 #fb72 to ! #c1 #d1 #e triangle !y are #b1 #a & #c1 #a 7,fig4 #cj72 & to ! #bh1 #de1 #ec triangle1 !y are #i1 #e & #g1 #b 7,fig4 #fa74 ,! prim>y triangle1 #g1 #bd1 #be1 possesses z satellites ! prim>y triangle1 #c1 #d1 #e1 & ! ord9>y triangle1 #d1 #a2 & ! prim>y triangle #da1 #hdj1 #hda1 is att5d$ by ! #bj1 #ba1 #bi triangle1 z a satellite ) ! ord9>y triangle #da1 #a1 & s on4 ,fig4 #fa4 ,! #bh-de-ec ,triangle4 ,fig4 #fb4 ,! #dh-ee-gc ,triangle4 ,s9ce any ratio1 h[ev]1 ^: t]ms1 "o or bo?1 are repres5t$ by fractions1 c be trans=m$ into :ole numb]s1 x evid5tly foll[s t e conceivable relative measure ( two l9es :i* we may decide to call co-s9e & co-v]s$-s9e1 becomes a satellite to a correspond+ prim>y triangle4 ,n[1 s9ce ! angle ( ! satellite on ! circumf]5ce mu/ be half ! angle ( ! adjac5t prim>y triangle at ! c5tre1 x foll[s t 9 con/ruct+ a li/ ( satellites & _! angles1 ! angles ( ! correspond+ prim>y triangles c be f\nd4 ,= in/ance-- ,satellite #h1 #c1 conta9s #bj #cc #bagf ,satellite #b1 #g1 conta9s #ae #ef #dcdbe ,ea* ( ^! angles d\bl$1 gives ! angle ( a prim>y triangle z foll[s1 viz43-- ,! #dh1 #ee1 #gc triangle #da #f #dceb ,! #bh1 #de1 #ec triangle #ca #ec #bfhe ,! angles ( ! satellites toge!r mu/ always be #de1 because ! angle at ! circumf]5ce ( a quadrant mu/ always be #ace4 ,f ! ,gzeh plan1 z f> z ,i h develop$ x1 ! foll[+ ord] ( satellites beg9s to appe>1 :i* may be a guide to ! complete ,gzeh plan ratio1 & to ^? prim>y triangles 9 use by ! pyramid surveyors 9 _! ord9>y "w4 ---------------------------------------- ------------ #a1 #b #b1 #c #c1 #d #d1 #e #e1 #f #f1 #g #g1 #h #h1 #i #a1 #c #b1 #e #c1 #e #d1 #g #e1 #g #g1 #i #a1 #d #b1 #g #c1 #g #d1 #i #e1 #h #a1 #e #b1 #i #c1 #h #e1 #i #g1 #a #a1 #f #e1 #aa #a1 #g #c1 #aa #e1 #ac #a1 #h #c1 #ac #a1 #i #a1 #aa #a1 #ac #a1 #ae #a1 #ag ---------------------------------------- ------------ ,prim>y triangles may be f\nd f ! angle ( ! satellite1 b x is an exce$+ly r\nd-ab\t way4 ,i w1 h[ev]1 give an example4 ,let u con/ruct a prim>y triangle f ! satellite #d1 #i4 ,rad4 #d -------- #ddddddd ,tangt4 #bc #eg #dejda #i #bc #eg #dejda #b #dg #ee #cjjhc4 !re=e ! angles ( ! prim>y are #dg #ee #cjjhc4 & #db #d #biiag4 ,! natural s9e ( #db #d #biiag #fgjajbe4 ,! natural co-s9e #db #d #biiag #gdbbfhd4 ,! greate/ common measure ( ^! numb]s is ab\t #ajbgag1 !re=e-- ,radius #ajjjjjjj #ajbgag #ig ,co-s9e #gdbbfhd #ajbgag #gb ,s9e #fgjajbe #ajbgag #fe & #fe1 #gb1 #ig is ! prim>y triangle to :i* ! satellites are #d1 #i1 & #e1 #ac4 7,see ,fig4 #fc47 ,! figures 9 ! calculation d n balance exactly1 9 consequ5ce ( ! insuffici5t delicacy ( ! tables or calculations4 ,fig4 #fc4 ,! connection betwe5 prim>ies & satellites is %[n by figure #fd4 ,fig4 #fd4 ,let ! triangle ,,adb be a satellite1 #e1 #b1 :i* we w call ,,bd #bj1 & ,,ad #h4 ,let ,c be c5tre ( semi-circle ,,abe4 ,,ad 3 ,,db 33 ,,db 3 ,,de #ej 7,euc4 ,,vi4 #h7 ,,ad ,,de ,,ae #eh diamet] ,,ae #b ,,ac ,,bc #bi radius ,,ac-,,ad ,,dc #ba co-s9e & ,,db #bj s9e ,f ! prec$+ x is manife/ t-- s9e ----- v]-s dia4 v]-s ,! =mula to f9d ! prim>y triangle to any satellite is z foll[s3-- ,let ! long ratio l9e ( ! satellite or s9e be call$ a1 & ! %ort ratio l9e or v]s$-s9e be call$ b4 ,!n-- 7#a7 a s9e4 a b 7#b7 ------- radius4 #b;b a - b 7#c7 ------- co-s9e4 #b;b ,!re=e v>i\s prim>y triangles c be con/ruct$ on a side ,,db 7,fig4 #fd7 z s9e1 by tak+ diff]5t measures = ,,ad z v]s$-s9e4 ,= example-- #e s9e #e #e #a ,f ------- radius #ac ,satellite #b #a #e1 #a4 #e - #a ------- co-s4 #ab #b #a 99 99 99 99 99 #e s9e #e #bj #e #b ,f ------- radius #g #d #bi ,satellite #b #b #e1 #b4 #e #b ------- co-s4 #e #ba #b #b 99 99 99 99 99 ,f9ally arises ! foll[+ simple rule = ! con/ruction ( prim>ies to conta9 any angle--,decide ^u a satellite :i* %all conta9 half ! angle--say1 #e1 #a4 ,call ! fir/ figure a1 ! second b1 !n-- a b hypot5use4 a - b p]p5dicul>4 a #b;b base4 ,,prim>y ,,l[e/ ,,ratio4 ,?us-- #e #a #bf #ac ,satellite #e1a #e - #a #bd #ab #e #b #a #aj #e --------------- ---------------------------- &-- #e #b #bi #bi ,satellite #e1b #e - #b #ba #ba #e #b #b #bj #bj ,hav+ f\nd ! l[e/ ratio ( ! ?ree sides ( a prim>y triangle1 ! l[e/ :ole numb]s = tang5t1 secant1 co-secant1 & co-tang5t1 if requir$1 are obta9$ 9 ! foll[+ mann]4 ,take = example ! #bj1 #ba1 #bi triangle1 n[ #bj #ba #dbj1 & #bi #dbj #abahj1 a new radius in/ead ( #bi f :i* ) ! s9e #bj1 & co-s9e #ba1 increas$ 9 ! same ratio1 ! :ole canon ( ! #bj1 #ba1 #bi triangle w come out 9 :ole numb]s4 ,simil>ly 9 ! triangle #dh1 #ee1 #gc1 radius #gc #acbjj 7! product ( #dh #ee7 makes radius 9 :ole numb]s #ifcfjj1 = an ev5 canon )\t fractions4 ,?is is because s9e & co-s9e are ! two d5om9ators 9 ! fractional p>ts ( ! o!r l9es :5 work$ out at ! l[e/ ratio ( s9e1 co-s9e1 & radius4 ,aft] ,i f\nd t ! plan ( ! ,gzeh gr\p 0 a sy/em ( prim>y triangles1 ,i _h to "w out ! rule = con/ruct+ !m1 = ,i _h nev] met ) x 9 any book1 b ,i came across x aft]w>ds 9 ! ,p5ny ,encyclop$ia1 & 9 ,rank9e's ,civil ,eng9e]+4 ,! practical utility ( ^! triangles1 h[ev]1 does n appe> to h receiv$ suffici5t consid]ation4 ,i c]ta9ly nev] met ) any except ! #c1 #d1 #e1 9 ! practice ( any surveyor ( my acqua9tance4 7,= squ>+ (f a l9e no?+ c\ld be m conv5i5t ?an ! #bj1 #ba1 #bi triangle2 = in/ance1 tak+ a base ( #dj l9ks1 !n us+ ! :ole *a9 = ! two rema9+ sides ( #db & #eh l9ks47 ,table ( ,"s ,prim>y ,triangles & _! ,satellites4 ,,angle ,,( ,,prim>y ,,prim>y ,,satellite4 ,,angle ,,( ,,satellite ,,deg4 ,,m94 ,,sec4 ,,rad4 ,,co4-,s4 ,,s9e4 ,,deg4 ,,m94 ,,sec4 #b #dg #cigj #hda #hdj #da #da #a #a #bc #dihe #f #dc #ehfb #ade #add #ag #ag #a #c #ba #eica #h #dg #ejfi #he #hd #ac #ac #a #d #bc #eecd #aj #bc #aihi #fa #fj #aa #aa #a #e #aa #ciid #ab #dj #dicg #da #dj #i #i #a #f #bj #bdfh #ad #ad #eiaj #fe #fc #af #h #a #g #g #biee #af #ae #cfgc #be #bd #g #g #a #h #g #dhcf #ah #ee #bhga #cg #ce #ab #f #a #i #bg #ddce #bb #cg #aaea #ac #ab #e #e #a #aa #ah #cege #be #c #bgbg #he #gg #cf #i #b #ab #ca #dcfc #be #ei #babb #hi #hj #ci #ac #c #ab #ei #djfa #bh #d #bjid #ag #ae #h #d #a #ad #b #ajdg #cj #cj #cfdi #fe #ef #cc #aa #c #ae #ae #ahbd #ca #ec #bfhe #ec #de #bh #g #b #ae #ef #dcdb #cf #eb #aafe #e #d #c #c #a #ah #bf #ehb #da #f #dceb #gc #ee #dh #h #c #bj #cc #bagf #db #d #cjjh #ig #gb #fe #ac #e #ba #b #aejd #dc #cf #ajae #bi #ba #bj #e #b #ba #dh #ejg #df #bc #dihe #bi #bj #ba #g #c #bc #aa #edib #dg #ee #biib #ig #fe #gb #i #d #bc #eg #ddif #dh #ec #afdh #gc #dh #ee #aa #e #bd #bf #chbd #ec #g #dhce #e #c #d #b #a #bf #cc #edag #eh #f #ccae #ec #bh #de #i #e #bi #c #afeg #ei #bi #bcea #fe #cc #ef #g #d #bi #dd #dage #fa #ee #cijf #ag #h #ae #e #c #cj #eg #diec #fd #j #chgh #hi #ci #hj #h #e #cb #j #aici #fd #ef #cbgc #he #cf #gg #aa #g #cb #bh #afcf #fg #bb #dhdi #ac #e #ab #c #b #cc #da #bdbd #ga #d #cabi #cg #ab #ce #g #e #ce #cb #aefd #gc #dd #bcbg #be #g #bd #d #c #cf #eb #aafc #ge #de #jij #fe #af #fc #i #g #cg #eb #cjde #gg #ai #ajfc #da #i #dj #e #d #ch #ci #ceca #gi #cf #djaa #fa #aa #fj #f #e #ci #dh #bjje #ha #ab #ica #he #ac #hd #g #f #dj #cf #dfe #hc #af #ach #ade #ag #add #i #h #da #ch #jfi #hg #ab #bjcj #hda #da #hdj #ba #bj #dc #cf #ajae ,ref]5ce to ! plan ratio table at ! comm5cem5t1 & to ! tables "h introduc$1 w %ew t mo/ ( ! prim>y triangles m5tion$ are indicat$ on ! plan ratio table pr9cipally by ! l9es correspond+ to ! ratios ( ! satellites4 ,?us-- ,,prim>y ,,triangle ,,indicat$ ,,by #ag1 #add1 #ade4 ,triangle ,,fp1 ,,pa1 ,,af on plan4 #ac1 #hd1 #he4 ,plan ratio ( ,,sj to ,,su1 #g to #f4 #aa1 #fj1 #fa4 ,plan ratio ,,bc to ,,fb1 #f to #e1 & ,,dn to ,,nr1 #fa to #fj4 #ab1 #ce1 #cg4 ,plan ratio ,,eo to ,,ay1 #cg to #ab1 & ,,ea to ,,ay1 #ce to #ab4 #e1 #ab1 #ac4 ,plan ratio ,,cy to ,,bc1 #c to #b2 ,,je to ,,ex1 #c to #b2 ,,ca to ,,ya1 #e to #a2 & ,,nz to ,,za1 #ab to #e4 #h1 #ae1 #ag4 ,plan ratio ,,fb to ,,by1 #e to #c1 & ,,ac to ,,bc1 #ae to #h4 #cc1 #ef1 #ee4 ,plan ratio ,,yx to ,,ay1 #g to #d2 ,,ab to ,,bo1 #g to #d2 & ,,ea to ,,az1 #g to #d4 #bh1 #de1 #ec4 ,exi/s on plan1 ,,ab1 ,,bj1 ,,ja4 #c1 #d1 #e4 ,p]vades ! plan1 & is also indicat$ by plan ratio ,,gx to ,,dg1 #b to #a2 ,,su to ,,sv1 #b to #a2 & ,,cy to ,,yz1 #c to #a4 #dh1 #ee1 #gc4 ,exi/s on plan1 ,,fw1 ,,wv1 ,,vf--& is also indicat$ by plan ratio ,,fo to ,,oz1 #h to #c4 #fe1 #gb1 #ig4 ,plan ratio ,,ac to ,,*1 #i to #d2 ,,my to ,,yz1 #i to #d4 #bj1 #ba1 #bi4 ,exi/s on plan ,,fb1 ,,ba1 ,,af2 & plan ratio1 ,,gu to ,,dg1 #e to #b4 ,x seems probable t c\ld ,i add to my pyramid plan ! l9es & triangles t ! miss+ elev5 pyramids w\ld supply1 x w\ld comprise a complete table on :i* w\ld appe> indications ( all ! ratios & triangles made use ( 9 "r-angl$ trigonometry1 a ratiomet] 9 fact4 ,i firmly believe t s f> z ,i h gone x is correct--& x is possible1 !re=e1 ) ! />t t ,i h made1 = o!rs to cont9ue ! "w1 & add ! elev5 pyramids to ! plan 9 _! correct geometrical position4 ,by cont9u+ ! sy/em ( evolution by :i* ,i def9$ ! position ( ,cephr51 & ! little pyramid to ! s\?-ea/ ( ,*eops1 aft] ,i _h obta9$ ,*eops & ,myc]9us1 may be rebuilt1 at "o & ! same "t1 a skeleton ( ! trigonometrical tables ( a =gott5 civilization1 & ! plan ( ^? pyramids :i* are its only l9k ) ! pres5t age4 #ac4 ,,! ,,size ,,& ,,%ape ,,( ,,! ,,pyramids ,,indicat$ ,,by ,,! ,,plan4 ,i pursu$ my inve/igations into ! slopes & altitudes ( ! pyramids )\t ref]5ce to ! plan1 aft] once decid+ _! exact bases4 ,n[ x w be int]e/+ to note "s ( ! ways 9 :i* ! plan h9ts at ! %ape & size ( ^! pyramids1 & corroborates my "w4 ,! dim5sions ( ,*eops are indicat$ on ! plan by ! l9es ,,ea to ,,ya1 measur+ #hdj & #bhh ,r4,b4 cubits respectively1 be+ ! half p]iph]y ( its horizontal section at ! level ( ,cephr5's base1 & its own altitude f its own base4 7,see ,fig4 #e47 ,! l9e ,,ea1 9 fact1 repres5ts 9 ,r4,b4 cubits ! half p]iph]y ( ! bases ( ei!r ,*eops or ,cephr5 measur$ at ! level :i* ,i h set =? z ! plan level1 viz41 base ( ,cephr54 ,! ratio ( ,cephr5's base to ,cephr5's altitude is indicat$ on ! plan by ! ratios ( ! l9es ,,bc to ,,eb1 or ,,fo to ,,or1 viz41 #cb to #ba4 7,see ,fig4 #d47 ,! altitude ( ,myc]9us above ,cephr5's base appe>s on plan 9 ! l9e ,,ef1 measur+ #acf ,r4,b4 cubits4 ,! l9e ,,eo on plan measures #hhh cubits1 :i* w\ld be ! l5g? ( a l9e /ret*$ f ! apex ( ,*eops to ! po9t ,e1 at ! level ( ,*eops' base4 ,?is m]its consid]ation3--! l9es ,,ea & ,,ay are connect$ on plan at ! c5tre ( ,*eops1 & ! l9es ,,eo & ,,ea are connect$ on plan at ! po9t ,e4 ,n[ ! l9es ,,eo1 ,,ea & ,,ay are sides ( a prim>y triangle1 ^: ratio is #cg1 #ce1 #ab1 & ^: measure 9 cubits is #hhh1 #hdj1 & #bhh2 & if we suppose ! l9e ,,ea to be /ret*$ horizontally b5ea? ! pyramids at ! level ( ! base ( ,*eops f ,e to ,a on plan1 & ! l9e ,,ay to be a plumb l9e hang+ f ! apex ( ,*eops to ! level ( 8 base1 !n w ! l9e ,,eo j /ret* f ! po9t ,e to ! apex ( ,*eops1 & ! ?ree l9es w connect ! two ma9 pyramids by a v]tical triangle ( :i* ,,ea1 ,,ay & ,,eo =m ! base1 p]p5dicul>1 & hypot5use4 ,or1 to expla9 x 9 ano!r mann]3 let ! l9e ,,ea be a cord /ret*+ horizontally f ,a at ! c5tre ( ! base ( ,*eops to ! po9t ,e1 bo? ends be+ at ! same level2 let ! l9e ,,ay be a rod1 lift x on ! end ,a till x /&s erect1 !n is ! end ,y ! apex ( ,*eops4 ,n[1 ! l9e ,,eo w\ld j /ret* f ! top ( ! rod ,,ay to ! po9t ,e fir/ describ$4 ,x is a s+ul> co9cid5ce1 & "o t may be int]e/+ to /ud5ts ( ! int]ior ( ! ,pyramids1 t ! side ,,ep1 ( ! small #c1 #d1 #e triangle1 ,,ep1 ,,pf1 ,,fe1 9 ! c5tre ( ! plan1 measures #hafj ,r4,b4 cubits1 :i* is v ne>ly eight times ! true bread? ( ! ,k+'s *amb] 9 ,*eops1 accord+ to ,piazzi ,smy?2 = #hafj_/h #ajbj ,r4,b4 cubits1 or #bjfjdf pyramid in*es 7"o ,r4,b4 cubit be+ #bjbjjf pyramid in*es74 ,! sides ( ?is little triangle measure #hafj1 #ajhhj1 & #acf1 ,r4,b4 cubits respectively1 z c be easily prov$ f ! plan ratio table4 #ad4 ,a ,,simple ,,in/rum5t ,,= ,,lay+ ,,(f ,,prim>y ,,triangles4 ,a simple in/rum5t = lay+ (f prim>y triangles ^u ! gr\nd1 might h be5 made ) ?ree rods divid$ into a numb] ( small equal divisions1 ) holes ?r\gh ea* division1 :i* rods c\ld be p9n$ toge!r triangul>ly1 ! rods work+ z arms on a flat table1 & ! p9s act+ z po9t]s or sights4 ,"o ( ! p9s w\ld be p]man5tly fix$ 9 ! table ?r\gh ! fir/ hole ( two ( ! rods or arms1 & ! two o!r p9s w\ld be movable s z to fix ! arms into ! %ape ( ! v>i\s prim>y triangles4 ,?us ) ! two ma9 arms p9n$ to ! cross arm 9 ! #ba/ & #bi? hole f ! p]man5tly p9n$ end1 ) ! cross arm /ret*$ to tw5ty divisions1 a #bj1 #ba1 #bi triangle w\ld be ! result1 & s on4 #ad;a4 ,,g5]al ,,obs]vations4 ,i mu/ be excus$ by geometricians = go+ s mu* 9 detail into ! simple tru?s connect$ ) "r-angl$ trigonometry4 ,my object has be5 to make x v cle> to t portion ( ! public n v]s$ 9 geometry1 t ! ,pyramids ( ,egypt mu/ h be5 us$ = l& survey+ by "r-angl$ triangles ) sides hav+ :ole numb]s4 ,a re-exam9ation ( ^! pyramids on ! gr\nd ) ! ideas sugge/$ by ! prec$+ pages 9 view1 may lead to int]e/+ discov]ies4 ,= in/ance1 x is j possible t ! v accurately & beautifully work$ /ones 9 ! walls ( ! ,k+'s *amb] ( ,*eops1 may be f\nd to indicate ! ratios ( ! rectangles =m$ by ! bases & p]p5dicul>s ( ! triangulations us$ by ! old surveyors--t on ^! walls may be f\nd1 9 fact1 corroboration ( ! !ory t ,i h set =?4 ,i am l$ to believe also f ! fact t ,gzeh 0 a c5tral & comm&+ locality1 & t x 0 ! cu/om ( ^? :o prec$$ ^? ,egyptians t hi/ory tells (1 to excavate mighty cav]ns 9 ! e>?--t1 !re=e1 9 ! lime/one ^u :i* ! pyramids are built1 & und]nea? ! pyramids1 may be f\nd va/ excavations1 *amb]s & gall]ies1 t _h entrance on ! face ( ! ridge at ! level ( ,high ,nile4 ,f ?is subt]rane\s city1 occupi$ by ! prie/s & ! surveyors ( ,memphis1 access may be f\nd to e pyramid2 & :ile to ! outside _w ! pyramids might h appe>$ seal$ up z mausoleums to ! ,k+s t x may h se5 publicly int]r$ !re91 ?is v seal+ & clos+ ( ! out] gall]ies may h only r5d]$ _! my/]i\s recesses m private to ! prie/s :o ent]$ f bel[1 & :o 71 p]haps1 enabl$ to asc5d by private passages to _! v summits4 ,! rec5t discov]y ( a numb] ( regal mummies /[$ away 9 an out ( ! way cave on ! banks ( ! ,nile1 po9ts to ! unc]emoni\s mann] 9 :i* ! real rul]s ( ,k+s & p may h dealt ) _! sov]eigns1 ! pomp & circum/ance ( a public burial once ov]4 ,x is j possible t ! *amb]s 9 ! pyramids may h be5 us$ 9 connection ) _! my/]ies3 & ! small passages call$ by "s v5tilators or air passages1 seal$ z !y 7 f ! *amb] by a ?9 /one 7& !re=e no v5tilators7 may h be5 auditory passages along :i* s\nd might h be5 project$ f o!r *amb]s n yet op5$ by ! mod]ns2 s\nds :i* 7 p]haps a "p ( ! hanky panky ( ! anci5t c]emonial connect$ ) ! my/]ies or ! religion ( t p]iod4 ,d[n t well :i* exi/s 9 ! int]ior ( ,*eops1 & 9 ! lime/one f\ndations ( ! pyramid1 %\ld ,i be dispos$ to look = op5+s into ! va/ subt]rane\s *amb]s :i* ,i am conv9c$ d exi/ bel[ ! ,pyramids ( ,gzeh4 ,! prie/s ( ! ,pyramids ( ,lake ,mo]is _h _! va/ subt]ranean resid5ces4 ,x appe>s to me m ?an probable t ^? ( ,gzeh 7 simil>ly provid$4 ,& ,i g fur!r3--,out ( ^! v cav]ns may h be5 excavat$ ! lime/one ( :i* ! pyramids 7 built1 ?us kill+ two birds ) "o /one--build+ ! in/rum5ts & f9d+ cool qu>t]s bel[ = ^? :o 7 to make use ( !m4 ,9 ! b[els ( t lime/one ridge on :i* ! pyramids are built w yet be f\nd1 ,i feel conv9c$1 ample in=mation z to _! uses4 ,a good diamond drill ) two or ?ree hundr$ feet ( rods is :at is :at is want$ to te/ ?is1 & ! solidity ( ! pyramids at ! same "t4 #ae4 ,,prim>y ,,triangulation4 ,prim>y triangulation w\ld be useful to m5 ( almo/ e trade & pr(ession 9 :i* tools or in/rum5ts are us$4 ,any "o might 9 a %ort "t con/ruct a table = himself answ]+ to e degree or s 9 ! circumf]5ce ( a circle = :i* only =ty or fifty triangles are requir$4 ,x w\ld be wor? :ile = "s "o to pr9t & publi% a correct set ( ^! tables embrac+ a close division ( ! circle1 9 :i* set !re %\ld be a column %[+ ! angle 9 degrees1 m9utes1 seconds & decimals1 & also a column = ! satellite1 ?us-- ,,satellite4 ,,prim>y4 ,,angle4 #e #b #bj #ba #bi #dc #cf #ajae #g #c #ba #bj #bi #df #bc #dihe & s on4 ,su* a set ( tables w\ld be a boon to sailors1 ar*itects1 surveyors1 eng9e]s1 & all h&i-craftsm53 & ,i make bold to say1 w\ld assi/ 9 ! intricate inve/igations ( ! a/ronom]3--& ! rule = build+ ! tables is s simple1 t !y c\ld easily be a*iev$4 ,! ar*itect f ^! tables might arrange ! %ape ( 8 *amb]s1 passages or gall]ies1 s t all measures1 n only at "r angles on ! walls1 b f any corn] ( floor to ceil+ %\ld be ev5 feet4 ,! pit* ( 8 ro(s might be m v>i$1 & ! monotony ( ! build+s reliev$1 ) raft]s & tie-beams always 9 ev5 measures4 ,! "o solit>y #c1 #d1 #e ( ,vitruvius w\ld cease to be 8 /&>d = a /aircase2 & ev5 9 doors & sa%es1 & panels ( glass1 w\ld he be alive to ! p]fection ( rectitude ga9$ by ev5ly-measur$ diagonals4 ,by a slight modification ( ! compass c>d1 ! navigator ( blue wat] might /e] 8 c\rses on ! hypot5uses ( great prim>y triangles--su* tables w\ld be useful to all sailors & surveyors :o h to deal ) latitude & dep>ture4 ,= in/ance1 famili>ity ) su* tables w\ld make ev] pres5t 9 ! m9d ( ! surveyor or sailor 8 proportionate nor?+ & ea/+1 no matt] :at c\rse he 0 /e]+ betwe5 nor? & ea/1 ! prim>y embraces ! ?ree ideas 9 "o view4 ,9 design+ truss$ ro(s or bridges1 ! prim>ies w\ld be invaluable to ! eng9e]1 /ra9-calculations on diagonal & upright memb]s w\ld be simplifi$1 & ! build] w\ld f9d ! b5efit ( a measure 9 ev5 feet or in*es f c5tre ( "o p9 or connection to ano!r4 ,= e>?work slopes #c1 #d1 #e2 #bj1 #ba1 #bi2 #ba1 #bj1 #bi2 & #d1 #c1 #e w\ld be f\nd m conv5i5t ratios ?an #a to #a1 & #a to #a1 etc4 ,templates & batt]+ rules w\ld be m p]fect & correct1 & ! eng9e] c\ld prove 8 slopes & measure 8 "w at "o & ! same "t )\t ! aid ( a /aff or level2 ! slope measures w\ld reveal ! dep?1 & ! slope measures & bottom wid? w\ld be all ! measures requir$1 :ile ! top wid? w\ld prove ! correctness ( ! slopes & ! measurem5ts4 ,to ! l& surveyor1 h[ev]1 ! prim>y triangle w\ld be ! mo/ useful1 & m especially to ^? lay+ out new hold+s1 :e!r small or l>ge1 9 new c\ntries4 ,:e!r x be = a squatt]'s run1 or = a t[n allotm5t1 ! advantages ( a diagonal measure to e p>allelogram 9 ev5 miles1 *a9s1 or feet1 %\ld be ke5ly felt & appreciat$4 ,?is 01 ,i believe1 "o ( ! secrets ( ! spe$y & correct replacem5t ( b\nd>y m>ks by ! ,egyptian l& surveyors4 ,i h he>d ( a review 9 ! ,contempor>y1 ,septemb]1 #ahha1 ref]r+ to ! translation ( a papyrus 9 ! ,briti% ,museum1 by ,dr4 ,eis5lohr--,a h&book ( practical ari?metic & geometry1 etc41 su* z we might suppose w\ld be us$ by a scribe act+ z cl]k ( ! works1 or by an ar*itect to %ew ! work+ out ( ! problems he _h to solve 9 8 op]ations4 ,i %\ld l to see a translation ( ! book1 f :i* x appe>s t ! clums9ess ( ! ,egyptian me?od is v rem>kable4 ,p]haps ?is ,egyptian ,h&book may yet %ew t _! op]ations 7 n s clumsy1 z !y appe> at fir/ sight to ^? accu/om$ to ! practice ( mod]n trigonometry4 ,i may n h got ! exact hang ( ! ,egyptian me?od ( l& survey+--= ,i d n suppose t ev5 _! clumsy me?od is to be got at intuitively2 b ,i claim t ,i h %ewn h[ ! ,pyramids c\ld be us$ = t purpose1 & t ! subsidi>y in/rum5t describ$ by me 0 practicable4 ,i claim1 !re=e1 t ! !ory ,i h set up1 t ! pyramids 7 ! !odolites ( ! ,egyptians1 is s\nd4 ,t ! gr\nd plan ( ^! pyramids discloses a beautiful sy/em ( prim>y triangles & satellites ,i ?9k ,i h %[n beyond ! %ad[ ( a d\bt2 & t ?is sy/em ( geometric triangulation or "r-angl$ trigonometry 0 ! me?od practis$1 seems 9 ! prec$+ pages to be fairly e/abli%$4 ,i claim1 !re=e1 t ,i h discov]$ & describ$ ! ma9 secret ( ! pyramids1 t ,i h f\nd = !m at la/ a practical use1 & t x is no long] a m>vel h[ aft] ! annual inundation1 ea* prop]ty c\ld h be5 accurately describ$ by ! aid ( geometry4 ,i h advanc$ no?+ 9 ! %ape ( a !ory t w n /& a practical te/2 b to d x1 ! pyramids %\ld be re-cas$4 ,iron %eet+1 on iron or wood5 framework1 w\ld answ]4 ,i may be wrong 9 "s ( my conclusions1 b 9 ! ma9 ,i am satisfi$ t ,i am "r4 ,x mu/ be admitt$ t ,i h work$ "u difficulties2 a glimpse at ! pyramids ?ree & tw5ty ye>s ago1 & ! meagre libr>y ( a nomad 9 ! ,au/ralian wild]ness hav+ be5 all my advantages1 & "t at my disposal only t snat*$ f ! r>e int]vals ( leisure af=d$ by a ardu\s pr(essional life4 ,aft] fruitless wait+ = a *ance ( visit+ ,egypt & ,europe1 to sift ! matt] to ! bottom1 ,i h at la/ resolv$ to give my ideas to ! _w z !y /&2 crude necess>ily1 s ,i mu/ be excus$ if 9 "s details ,i may be f\nd errone\s2 !re is tru? ,i "k 9 ! g5]al conclusions4 ,i am presumptu\s en\gh to believe t ! ,r4,b4 cubit ( #afhe ,briti% feet 0 ! measure ( ! pyramids ( ,gzeh1 al?\gh !re may h be5 an a/ronomical #be in* cubit also4 ,x appe>s to me t no cubit measure to be dep5d$ on is ei!r to be got f a /ray measur+ /ick f\nd 9 ! jo9ts ( a ru9$ build+1 or f any l9e or dim5sions ( "o ( ! pyramids4 ,i submit t a mo/ reasonable way to get a cubit measure out ( ! ,pyramids ( ,gzeh1 0 to d z ,i did3--take !m z a :ole1 compreh5d & e/abli% ! g5]al gr\nd plan1 f9d x geometric & h>monic1 obta9 ! ratios ( all ! l9es1 e/abli% a complete set ( natural & ev5 numb]s to repres5t ! measures ( ! l9es1 & f9ally br+ ^! numb]s to cubits by a common multipli] 7:i* 9 ?is case 0 ! numb] eight74 ,aft] ! :ole proportions _h be5 ?us express$ 9 a cubit evolv$ f ! :ole proportions1 ,i e/abli%$ its l5g? 9 ,briti% feet by divid+ ! base ( ,cephr51 z kn[n1 by ! numb] ( my cubits repres5t+ its base4 ,x is pretty s\nd evid5ce ( ! !ory be+ correct t ?is te/1 ) #dbj cubits neat = ,cephr51 gave me also a neat measure = ,*eops1 f ,piazzi ,smy?'s base1 ( #deb cubits1 & t at ! same level1 ^! two pyramids become equal bas$4 ,i h paid little att5tion to ! inside measurem5ts4 ,i take x we %\ld fir/ obta9 our exot]ic k be=e v5tur+ on esotoric rese>*4 ,?us ! intricate int]nal measurem5ts ( ,*eops1 made by v>i\s enquir]s h be5 little s]vice to me1 :ile ! accurate measures ( ! base ( ,*eops by ,piazzi ,smy?1 & ,john ,james ,wild's lett] to ,"l ,br\gham1 help$ me amaz+ly1 z f ! two ,i e/abli%$ ! plan level & ev5 bases ( ,*eops & ,cephr5 at plan level--z ,i h %[n 9 ! prec$+ pages4 ,my !ory dem&$ t bo? = ! build+ ( ! pyramids & = ! con/ruction ( ! models or subsidi>y in/rum5ts ( ! surveyors1 simple slope ratios %\ld gov]n ea* build+2 be=e ,i conclude1 ,i %all %[ h[ ,i got at my slope ratios1 by evolv+ !m f ! g5]al gr\nd plan4 ,i am firmly conv9c$ t a c>eful inve/igation into ! gr\nd plans ( ! v>i\s o!r gr\ps ( pyramids w amply confirm my survey !ory--! relative positions ( ! gr\ps %\ld also be e/abli%$--mu* additional light w be !n ?r[n on ! subject4 ,let me conjure ! inve/igator to view ^! piles f a di/ance ) 8 m9d's eye1 z ! old surveyors view$ !m ) _! bodily eye4 ,approa* !m too ne>ly1 &1 l ,h5ry ,k+lake1 y w be lo/ 9 ! "o idea ( solid imm5sity4 ,common s5se tells u !y 7 built to be view$ f a di/ance4 ,mod]n surveyors /& ne> _! in/rum5ts1 & s5d _! flagm5 to a di/ance2 ! ,egyptian surveyor 0 "o ( 8 own flagm51 & 8 in/rum5ts 7 t[]+ to ! skies on ! di/ant horizon4 ,^! mighty tools w la/ out _m a g5]ation ( surveyors4 ,! mod]n a/ronom] f ! top ( an obs]vatory po9ts 8 in/rum5ts direct at ! />s2 ! ,egyptian a/ronom] f ! summit ( 8 p>ticul> pyramid direct$ 8 obs]vations to ! ris+ & sett+ ( ! />s1 or ! positions ( ! heav5ly bodies 9 respect to ! f> away gr\ps ( pyramids scatt]$ ar\nd him 9 ! di/ance2 & by comp>+ notes1 & ) ! k ( ! relative position ( ! gr\ps1 did ^! obs]v]s map out ! sky4 ,sol> & lun> %ad[s ( _! own pyramids on ! flat tr5*es prep>$ = ! purpose1 enabl$ ! a/ronom] at ea* obs]vatory to record ! annual & mon?ly flight ( "t1 :ile its h\rs 7 m>k$ by ! %ad[s ( _! obelisks1 capp$ by copp] pyramids or balls1 on ! m delicate pavem5ts ( ! c\rt-y>ds ( _! public build+s4 ,we mu/ grasp t _! cele/ial & t]re/rial surveys 7 almo/ a rev]se process to our own1 be=e we c v5ture to enquire into its details4 ,x !n becomes a mu* easi] tangle to unravel4 ,t a p>ticul> pyramid among s _m1 %\ld h be5 *os5 z a fav\r$ int]pret] ( ,div9e tru?s1 seems an unfair conclusion to ! o!r pyramids2--t ! o!r pyramids 7 r\gh & imp]fect imitations1 appe>s to my poor capacity a base & impot5t conclusion2--7z f> z ,i c le>n1 ,myc]9us1 9 its p]fection1 0 a m>vel ( ! mason's art27 b t "o p>ticul> pyramid %\ld h any?+ to d ) ! pa/ or ! future ( ! lo/ t5 tribes ( ,israel 7:oev] t fraction ( our pres5t e>?ly community may be71 seems to me ! wilde/ conclusion ( all1 except p]haps ! !ory t ?is "o pyramid po9ts to ! future ( ! ,briti% race4 ,yet 9 "o way d ,i admit t ! pyramids po9t to our future4 ,?irty-six c5turies ago1 !y1 already v5]able ) antiquity1 look$ pr\dly d[n on liv+ lab\r+ ,israel1 9 helpless slav]y1 9 ! mid/ ( an advanc$ civilization1 ( :i* ! hi/ory1 language1 & religion are n[ =gott51 or only at be/1 slightly und]/ood4 ,?irty-six c5turies h5ce1 !y may look d[n on a civilization equally /range1 9 :i* our hi/ory1 language1 & religion1 ,hebrew race1 & ,briti% race1 may h no place1 no "p4 ,if ! ?\ghts ( noble poets live1 z !y seem to d1 old ,*eops1 t m\nta9 ( massive masonry1 may 7l ! brook ( our ,laureate71 9 t dim future1 /ill be s++1 z he seems to s+ n[1 ?is idea1 ?\gh n p]haps ^! words3 ,= m5 may come1 & m5 may g1 ,b ,i g on = ev]4 ,ars longa1 vita brevis4 ,man's "w rema9s1 :5 ! workman is =gott52 fair "w & squ>e1 c nev] p]i% entirely f m5's m9ds1 s long z ! _w /&s4 ,^! pyramids 7 gr& & noble works1 & !y w n p]i% till _! reputation has be5 re-e/abli%$ 9 ! _w1 :5 !y w live 9 m5's memories to all g5]ations z symbols ( ! mighty pa/4 ,to ! m9ds ( _m n[1 z to ,josephus 9 8 day1 !y are va/ & va9 monum5ts1 records ( folly4 ,to me !y are z monum5ts ( peace1 civilization & ord]--relics ( a p liv+ "u wise & b5efic5t rul]s--evid5ces ( cultivation1 sci5ce1 & art4 #af4 ,,! ,,p5tangle ,,or ,,five ,,po9t$ ,,/> ,,! ,,geometric ,,symbol ,,( ,,! ,,great ,,pyramid4 ,f "t immemorial ?is symbol has be5 a blaz+ po9t] to gr& & noble tru?s1 & a solemn emblem ( important duties4 ,its geometric significance1 h[ev]1 has long be5 lo/ sight (4 ,x is said to h con/itut$ ! seal or signet ( ,k+ ,solomon 7#ajjj ,b4,c471 & 9 e>ly times x 0 9 use among ! ,jews1 z a symbol ( safety4 ,x 0 ! ,p5talpha ( ,py?agoras1 & ! ,py?agorean emblem ( heal? 7#ecj ,b4,c474 ,x 0 c>ri$ z ! bann] ( ,antio*us1 ,k+ ( ,syria 7surnam$ ,sot]1 or ! ,pres]v]71 9 8 w>s aga9/ ! ,gauls 7#bfj ,b4,c474 ,among ! ,cabali/s1 ! /> ) ! sacr$ "n writt5 on ea* ( its po9ts1 & 9 ! c5tre1 0 consid]$ talismanic2 & 9 anci5t times x 0 employ$ all ov] ,asia z a *>m aga9/ wit*craft4 ,ev5 n[1 ,european troops at w> ) ,arab tribes1 sometimes f9d1 "u ! clo?+1 on ! brea/s ( _! sla9 enemies1 ?is anci5t emblem1 9 ! =m ( a metal talisman1 or *>m4 ,! ,european ,ge! puts ^! words into ! m\? ( ,mephi/opheles3 ,i am h9d]$ egress by a qua9t device ^u ! ?re%old1--t five-to$ damn$ spell4 ,i %all set =? ! geometric significance ( ?is />1 z f> z my g5]al subject w>rants me1 & %[ t x is ! geometric emblem ( extreme & mean ratio1 & ! symbol ( ! ,egyptian ,pyramid ,*eops4 ,a plane geometric />1 or a solid geometric pyramid1 may be lik5$ to ! corolla ( a fl[]1 ea* sep>ate side repres5t+ a petal4 ,) its petals op5$ & expos$ to view1 ! fl[] appe>s 9 all its glori\s beauty2 b :5 clos$1 _m ( its beauties are hidd54 ,! botani/ seeks to view x flat or op5 9 its geometric symmetry1 & also clos$1 z a bud1 or 9 repose3--yet judges & appreciates ! "o /ate f ! o!r4 ,9 ! same mann] mu/ we deal ) ! five po9t$ />1 & also ) ! ,pyramid ,*eops4 ,9 deal+ ) s qua9t a subject1 ,i may be excus$1 9 pass+1 = ! qua9t conceit ( lik5+ ! int]ior gall]ies & *amb]s ( ?is pyramid to ! int]ior :orl ( a fl[]1 /am5s & pi/il1 my/]i\s & incompreh5sible4 ,figure #fg 7page #aja71 is ! five po9t$ />1 =m$ by ! unlapp+ ( ! five slant sides ( a pyramid ) a p5tagonal base4 ,figure #gj 7page #ajf71 is a /> =m$ by ! unlapp+ ( ! f\r slant sides ( ! pyramid ,*eops4 ,! p5tagon ,,gfrhq1 7,fig4 #fg7 is ! base ( ! pyramid ,p5talpha & ! triangles ,,egf1 ,,bfr1 ,,roh1 ,,hnq & ,,qag1 repres5t ! five sides1 s t suppos+ ! l9es ,,gf1 ,,fr1 ,,rh1 ,,hq & ,,qg1 to be h+es connect+ ^! sides ) ! base1 !n by lift+ ! sides1 & clos+ !m 91 ! po9ts ,a1 ,e1 ,b1 ,o1 & ,n1 w\ld meet ov] ! c5tre ,c4 ,?us d we close ! geometric fl[] ,p5talpha1 & conv]t x into a pyramid4 ,9 ! same mann] mu/ we lift ! f\r slant sides ( ! pyramid ,*eops f its /> developm5t1 7,fig4 #gj7 & close !m 91 ! f\r po9ts meet+ ov] ! c5tre ( ! base1 =m+ ! solid pyramid4 ,su* transitions po9t to ! indissoluble connection betwe5 plane & solid geometry4 ,z ! geometric emblem ( extreme & mean ratio1 ! p5tangle appe>s z an assemblage ( l9es divid$ ! "o by ! o!rs 9 extreme & mean ratio4 ,to expla9 to read]s n v]s$ 9 geometry1 :at extreme & mean ratio signifies1 ,i ref] to ,figure #fe3-- ,fig4 #fe4 ,let ,,ab be ! giv5 l9e to be divid$ 9 extreme & mean ratio1 i4e41 s t ! :ole l9e may be to ! great] "p1 z ! great] is to ! less "p4 ,draw ,,bc p]p5dicul> to ,,ab1 & equal to half ,,ab4 ,jo9 ,,ac2 & ) ,,bc z a radius f ,c z a c5tre1 describe ! arc ,,db2 !n ) c5tre ,a1 & radius ,,ad1 describe ! arc ,,de2 s %all ,,ab be divid$ 9 ,e1 9 extreme & mean ratio1 or s t ,,ab3 ,,ae33 ,,ae3 ,,eb4 7,note t ,,ae is equal to ! side ( a decagon inscrib$ 9 a circle ) radius ,,ab47 ,let x be not$ t s9ce ! division ( a l9e 9 mean & extreme ratio is effect$ by means ( ! #b1 #a triangle1 ,,abc1 !re=e1 z ! expon5t ( ?is ratio1 ano!r reason pres5ts itself :y x %\ld be s important a feature 9 ! ,gzeh pyramids 9 addition to its connection ) ! prim>y triangle #c1 #d1 #e4 ,fig4 #ff4 ,to complete ! explanation (f]$ ) figure #fe1 ,i mu/ ref] to ,fig4 #ff1 :]e 9 con/ruct+ a p5tagon1 ! #b1 #a triangle ,,abc1 is aga9 made use (4 ,! l9e ,,ab is a side ( ! p5tagon4 ,! l9e ,,bc is a p]p5dicul> to x1 & half its l5g?4 ,! l9e ,,ac is produc$ to ,f1 ,,cf be+ made equal to ,,cb2 !n ) ,b z a c5tre1 & radius ,,bf1 ! arc at ,e is describ$2 & ) ,a z a c5tre1 & ! same radius1 ! arc at ,e is int]sect$1 _! int]section be+ ! c5tre ( ! circle circumscrib+ ! p5tagon1 & ^u :i* ! rema9+ sides are laid (f4 ,we w n[ ref] to figure #fg1 9 :i* ! p5tangle appe>s z ! symbolic expon5t ( ! division ( l9es 9 extreme & mean ratio4 ,?us3 ,,mc 3 ,,mh 33 ,,mh 3 ,,hc ,,af 3 ,,ag 33 ,,ag 3 ,,gf ,,ab 3 ,,af 33 ,,af 3 ,,fb :ile ,,mn1 ,,mh or ,,xc3 ,,cd33 #b3 #a--;be+ ! geometric template ( ! "w4 ,?us e l9e 9 ?is beautiful symbol by its int]sections ) ! o!r l9es1 manife/s ! problem4 ,note also t ,,gh ,,ga ,,ae ,,af ,,dh ,,de ,i app5d a table %[+ ! comp>ative measures ( ! l9es 9 ,fig4 #fg1 tak+ radius ( ! circle z a million units4 ,fig #fg4 ,table ,%[+ ! ,comp>ative ,measures ( ,l9es4 7,fig4 #fg47 ,,me #bjjjjjj diamet]4 ,,ab #aijbaac ,,ad ,,db ,,mb #afahjcd ,,mc ,,mh ,,mp ,,pb ,,z #aechhdae ,,ep #adecjhf ,,ag ,,fb ,,af #aageegj ,,ae ,,gb ,,mc #ajjjjjj radius ,,cd ,,dx ,,* ,,cx ,,ad #ieajefe ,,db ,,ds ,,pb #hedajb ,,qs #habbihe ,,mp #gfcicb ,,* #b base ( ,*eops4 ,,ag #gbfedc ,,gh ,,xh ,,hn ,,pf ,,fb ,slant edge ( ,*eops4 slant edge ( ,p5t4 ,pyr4 ,,de #fijihc ,,dh ,,xd apo!m ( ,p5tagonal ,pyramid4 apo!m ( ,*eops4 ,,mh #fahjcd ,,mn ,,xc altitude ( ,p5tagonal ,pyramid4 side ( decagon inscr'd 9 circle4 ,,ms #ejjjjj mean proportional betwe5 ,,mh & ,,hc #dhehfh altitude ( ,*eops4 ,,op #ddijbg ,,gf ,,gd ,,df ,,hc #chaiff half base ( ,*eops4 ,,s #cfcbgae ,,hs ,,cd #cjijag half ,,mh ,,pr #bggeaf ,,gd #bbdeace ,,sp #bfcicb ,! triangle ,,dxh repres5ts a v]tical section ( ! p5tagonal pyramid2 ! edge ,,hx is equal to ,,hn1 & ! apo!m ,,dx is equal to ,,de4 ,let ,,dh be a h+e atta*+ ! plane ,,dxh to ! base1 n[ lift ! plane ,,dxh until ! po9t ,x is v]tical above ! c5tre ,c4 ,!n ! po9ts ,a1 ,e1 ,b1 ,o1 ,n ( ! five slant slides1 :5 clos$ up1 w all meet at ! po9t ,x ov] ! c5tre ,c4 ,we h n[ built a pyramid out ( ! p5tangle1 ^: slope is #b to #a1 altitude ,,cx be+ to ,,cd z #b to #a4 ,apo!m ,,dx ,,de ,altitude ,,cx ,,hm or ,,mn ,altitude ,,cx ,,* ,,cm radius4 ,apo!m ,,dx ,,cd ,,cm radius4 ,edge ,,hx ,,hn or ,,pf ,note also t ,,mp -- ,,* #b ,,op ,,hr ,let u n[ consid] ! ,p5tangle z ! symbol ( ! ,great ,pyramid ,*eops4 ,! l9e ,,mp ! base ( ,*eops4 ,! l9e ,,* half base ( ,*eops4 ,! l9e ,,hm apo!m ( ,*eops4 ,! l9e ,,hn slant edge ( ,*eops4 ,?us3 ,apo!m ( ,*eops side ( decagon4 ,apo!m ( ,*eops altitude ( p5tagonal pyramid4 ,slant edge ( ,*eops slant edge ( p5tagonal pyramid4 ,n[ s9ce apo!m ( ,*eops ,,mh & half base ( ,*eops ,,hc !n d apo!m & half base repres5t1 :5 tak5 toge!r1 extreme & mean ratio1 & altitude is a mean proportional betwe5 !m3 x hav+ already be5 /at$1 :i* also is prov$ by ! figures 9 ! table1 t ,,mc 3 ,,mh 33 ,,mh 3 ,,hc & apo?3 alt 33 alt 3 half base4 ,?us is ! f\r po9t$ /> ,*eops evolv$ f ! five po9t$ /> ,p5talpha4 ,?is is %[n cle>ly by ,fig4 #fh1 ?us3-- ,fig4 #fh4 ,)9 a circle describe a p5tangle1 ar\nd ! int]ior p5tagon ( ! /> describe a circle1 ar\nd ! circle describe a squ>e2 !n w ! squ>e repres5t ! base ( ,*eops4 ,draw two diamet]s ( ! out] circle pass+ ?r\gh ! c5tre squ>e at "r angles to ea* o!r1 & ea* diamet] p>allel to sides ( ! squ>e2 !n w ! p>ts ( ^! diamet]s betwe5 ! squ>e & ! out] circle repres5t ! f\r apo!ms ( ! f\r slant sides ( ! pyramid4 ,connect ! angles ( ! squ>e ) ! circumf]5ce ( ! out] circle by l9es at ! f\r po9ts indicat$ by ! diamet]s1 & ! /> ( ! pyramid is =m$1 :i*1 :5 clos$ z a solid1 w be a correct model ( ,*eops4 ,call+ apo!m ( ,*eops1 ,,mh #cd & half base1 ,,hc #ba z p] ,figure #f4 ,!n-- ,,mh ,,mc #ee & #ee 3 #cd 33 #cd 3 #bajah1 be+ only 9 error a few in*es 9 ! pyramid itself1 if c>ri$ into actual measures4 ,! ratio1 !re=e1 ( apo!m to half-base1 #cd to #ba1 :i* ,i ascribe to ,*eops1 is z ne> z /one & mort> c be got to illu/rate ! above proportions4 ,correctly /at$ ari?metically let ,,mh #b4 ,!n ,,hc #e - #a ,,mc #e #a & altitude ( ,*eops 7,,mh ,,mc7 ,let u n[ comp>e ! con/ruction ( ! two />s3-- ,fig4 #fi4 ,,to ,,con/ruct ,,! ,,/> ,,p5talpha ,,fig4 #fi4 ,describe a circle4 ,draw diamet] ,,mce4 ,divide ,,mc 9 mean & extreme ratio at ,h4 ,lay (f half ,,mh f ,c1 to ,d4 ,draw *ord ,,adb1 at "r angles to diamet] ,,ecm4 ,draw *ord ,,bhn1 ?r\gh ,h4 ,draw *ord ,,aho1 ?r\gh ,h4 ,connect ,,ne4 ,connect ,,eo4 ,fig4 #gj4 ,,to ,,con/ruct ,,! ,,/> ,,*eops1 ,,fig4 #gj4 ,describe a circle4 ,draw diamet] ,,mce4 ,divide ,,mc 9 mean & extreme ratio1 at ,h4 ,describe an inn] circle ) radius ,,*1 & ar\nd x describe ! squ>e a1 b1 c1 d4 ,draw diamet] ,,acb1 at "r angles to diamet] ,,ecm4 ,draw ,aa1 a,e1 ,eb1 b,b1 ,bd1 d,m1 ,mc1 & c,a4 ,! "q n[ arises1 does ?is pyramid ,*eops set =? by ! relations ( its altitude to p]imet] ( base ! ratio ( diamet] to circumf]5ce2 or1 does x set =? mean proportional1 & extreme & mean ratio1 by ! proportions ( its apo!m1 altitude1 & half-base8 ,! answ] is--f ! practical impossibility ( su* extreme accuracy 9 su* a mass ( masonry1 t x po9ts alike to all1 & may z fairly be consid]$ ! expon5t ( ! "o z ( ! o!rs4 ,piazzi ,smy? makes ,*eops #gfafe feet base1 & #dhdia feet altitude1 :i* is v ne>ly :at he calls a ,pi pyramid1 = :i* ,i reckon ! altitude w\ld be ab\t #dhdhg feet ) ! same base3 & = a pyramid ( extreme & mean ratio ! altitude w\ld be #dhdcd feet4 ,! :ole diff]5ce1 !re=e1 is only ab\t six in*es 9 a height ( ne>ly five hundr$ feet4 ,?is diff]5ce1 evid5tly beyond ! p[] ( man to discov]1 n[ t ! pyramid is a ru91 w\ld ev5 9 its p]fect /ate h be5 inappreciable4 ,x appe>s mo/ probable t ! /> ,p5talpha l$ to ! /> ,*eops1 & t ! /> ,*eops 7,fig4 #gj7 0 ! plan us$ by ! anci5t ar*itect1 & ! ratio ( #cd to #ba1 hypot5use to base1 ! template us$ by ! anci5t build]s4 ,suppose "s k+ said to 8 ar*itect1 ,make me a plan ( a pyramid1 ( :i* ! base %all be #dbj cubits squ>e1 & altitude %all be to ! p]imet] ( ! base z ! radius ( a circle to ! circumf]5ce4--,!n might ! ar*itect prep>e an elaborate plan 9 :i* ! relative dim5sions w\ld be ab\t-- ,r4 ,b4 ,,cubits ,base #dbj ,base angle #ea #ea #adc ,altitude #bfgchjcjd c4 ,apo!m #cciihhegc c4 ,! k+ !n ord]s ano!r pyramid1 ( ! same base1 ( :i* altitude is to be a mean proportional betwe5 apo!m & half-base--& apo!m & half-base tak5 z "o l9e are to be 9 mean & extreme ratio4 ,! ar*itect's plan ( ?is pyramid w be ! simple figure illu/rat$ by me 7,fig4 #gj71 & ! dim5sions ab\t-- ,r4 ,b4 ,,cubits4 ,base #dbj ,base angle #ea #di #cg-db_/dga ,altitude #bfgabcihdi c4 ,apo!m #ccighgeaec c4 ,b ! build] practically c>ries out bo? plans :5 he builds to my templates ( #cd to #ba )-- ,r4 ,b4 ,,cubits4 ,base #dbj ,base angle #ea #ea #bj ,altitude #bfgcidhci c4 ,apo!m #cdj & nei!r k+ nor ar*itect c\ld detect error 9 ! "w4 ,! read] w rememb] t ,i h previ\sly advanc$ t ! level ( ,cephr5's base 0 ! plan level ( ! ,gzeh pyramids1 & t at ?is level ! base ( ,*eops measures #dbj ,r4,b4 cubits--same z ! base ( ,cephr54 ,?is hypo!sis is support$ by ! revelations ( ! p5tangle1 9 :i* ! ratio ( #cd to #ba apo!m #cdj to half-base #baj ,r4,b4 cubits1 is s ne>ly approa*$4 ,%[+ h[ proportional l9es 7 ! ord] ( ! pyramids ( ,gzeh1 we w summ>ise ! proportions ( ! ?ree ma9 pyramids z %ewn by my dim5sions & ratios1 v ne>ly1 viz43-- ,myc]9us4 ,base 3 ,apo!m 33 ,altitude 3 ,half-,base4 z %[n by ! ratios1 7,fig4 #ac71 #dj 3 #cb 33 #be 3 #bj4 ,cephr54 ,diagonal ( ,base 3 ,edge 33 ,edge 3 ,altitude4 z %[n by ratios1 7,fig4 #ab71 #hfb 3 #ehh 33 #ehh 3 #djj4 ,*eops4 7,apo!m ,half ,base73 ,apo?4 33 ,apo?4 3 ,half ,base4 z %[n by ! ratios1 7,fig4 #i71 #ee 3 #cd 33 #cd 3 #ba4 &--,apo!m 3 ,altitude 33 ,altitude 3 ,half-,b4 ,simil> close relations to o!r />s may be f\nd 9 o!r pyramids4 ,?us3--,suppose ,,nho ( figure #fi to be ! ,,nho ( a heptangle in/ead ( a p5tangle1 !n does ,,nh repres5t apo!m1 & ,,no repres5t base ( ! pyramid ,myc]9us1 :ile ! co-s9e ( ! angle ,,nhm 7be+ ,,mh m9us v]s$ s9e7 w be equal to ! altitude ( ! pyramid4 ,! angle ,,nhm 9 ! heptangle is1 #ch #cd #agadb1 & accord+ to my plan ( ! pyramid ,myc]9us1 ! correspond+ angle is #ch #dj #ef4 7,see ,fig4 #ai47 ,?is angul> diff]5ce ( #j #f #ci w\ld only make a diff]5ce 9 ! apo!m ( ! pyramid ( eight in*es1 & ( t5 in*es 9 its altitude 7apo!m be+ #bhc ft4 #a in*1 & altitude #bba ft474 #ag4 ,,! ,,mann] ,,9 ,,:i* ,,! ,,slope ,,ratios ,,( ,,! ,,pyramids ,,7 ,,arriv$ ,,at4 ,! mann] 9 :i* ,i arriv$ at ! ,slope ,ratios ( ! ,pyramids1 viz41 #cb to #bj1 #cc to #bj1 & #cd to #ba1 = ,myc]9us1 ,cephr51 & ,*eops1 respectively 7see ,figures #h1 #g & #f71 0 z foll[s3-- ,fir/1 believ+ 9 ! connection betwe5 ! relative positions ( ! ,pyramids on plan 7see ,fig4 #c1 #d or #e71 & _! slopes1 ,i view$ _! positions ?us3-- ,myc]9us1 situate at ! angle ( ! #c1 #d1 #e triangle ,,adc1 is likely to be connect$ ) t prim>y 9 8 slopes4 ,cephr51 situate at ! angle ( ! #bj1 #ba1 #bi triangle ,,fab1 & /rung1 z x 71 on ! hypot5use ( ! #c1 #d1 #e triangle ,,dac1 is likely to be connect$ ) bo? prim>ies 9 8 slopes4 ,*eops1 situate at ! po9t ,a1 common to bo? ma9 triangles1 gov]n+ ! position ( ! o!r pyramids1 is likely to be a sort ( mean betwe5 ^! two pyramids 9 8 slope ratios4 ,reason+ ?us1 ) ! addition ( ! k ,i possess$ ( ! angul> e/imates ( ^! slopes made by ^? :o _h visit$ ! gr\nd1 & a useful />t = my ratios ga9$ by ! r$uction ( base measures already kn[n into ,r4,b4 cubits1 giv+ #dbj z a g5]al base = ,*eops & ,cephr5 at "o level1 & tak+ #baj cubits z ! base ( ,myc]9us 7half ! base ( ,cephr51 z g5]ally admitt$71 ,i _h some?+ solid & sub/antial to g ^u4 ,i comm5c$ ) ,myc]9us4 7,see ,fig4 #ga47 ,fig4 #ga4 7,myc]9us7 ,,lhnm repres5ts ! base ( ! pyramid4 ,on ! half-base ,,ac ,i describ$ a #c1 #d1 #e triangle ,,abc4 ,i !n project$ ! l9e ,,cf ,,bc to be ! altitude ( ! pyramid4 ,?us ,i erect$ ! triangle ,,bfc1 ratio ( ,,bc to ,,cf be+ #a to #a4 ,f ?is datum ,i arriv$ at ! triangles ,,bea1 ,,adc1 & ,,gkh4 ,,gk1 ,,ea1 & ,,ad1 ea* repres5t apo!m ( pyramid2 ,,cf1 & ,,cd1 altitude2 & ,,hk1 edge4 ,! l5g? ( ! l9e ,,ad be+ 7,,ac ,,cd71 ! l5g? ( ! l9e ,,hk be+ 7,,hg ,,gk71 & l9e ,,* 7half diagonal ( base7 be+ 7,,cg ,,gh74 ,^! measures r$uc$ to ,r4,b4 cubits1 call+ ! l9e ,,ac ratio #d #aje cubits1 half-base ( pyramid1 give ! foll[+ results3-- ,r4 ,b4 ,,briti% ,,cubits4 ,,feet4 ,half-base ,,la #ajejjj #agfibe ,apo!m ,,ea #afhjhb #bhcbah ,edge ,,hk #aihahc #cccicg ,altitude ,,cd #acabej #bbaaef ,half diag4 ( base ,,* #adhdibd #bejbji & ?us ,i acquir$ ! ratios3-- ,half-base 3 ,altitude 33 ,apo!m 3 ,base4 #bj 3 #be 33 #cb 3 #dj ne>ly4 ,to place ! l9es ( ! diagram 9 _! actual solid position--,let ,,ab1 ,,bc1 ,,ca & ,,hg be h+es atta*+ ! planes ,,aeb1 ,,bfc1 ,,cda & ,,hkg to ! base ,,lhnm4 ,lift ! plane ,,bcf on its h+e till ! po9t ,f is v]tical ov] ! c5tre ,c4 ,lift plane ,,cda on its h+e1 till po9t ,d is v]tical ov] ! c5tre ,c2 !n w l9e ,,cd t\* ,,cf1 & become "o l9e4 ,n[ lift ! plane ,,aeb on its h+e1 until po9t ,e is v]tical ov] ! c5tre ,c1 & plane ,,hkg on its h+e till po9t ,k is v]tical ov] ! c5tre ,c2 !n w po9ts ,e1 ,f1 ,d & ,k1 all meet at "o po9t above ! c5tre ,c1 & all ! l9es w be 9 _! prop] places4 ,! angle at ! base ( ,myc]9us1 if built to a ratio ( #d to #e 7half-base to altitude71 & n to ! m practical b ne>ly p]fect ratio ( #cb to #bj 7apo!m to half-base7 w\ld be ! complem5t ( angle ,,adc1 ?us-- #d #afe --- #h ,tan4 ,,adc #ch #ci #ce--- #e #dgg #cab ,,dac #ea #bj #bd--- #dgg b z x is probable t ! pyramid 0 built to ! ratio ( #cb to #bj1 ,i h %[n its base angle 9 ,figure #ai1 z #ea #ai #d4 ,figure #gb %[s h[ ! slopes ( ,cephr5 7 arriv$ at4 ,fig4 #gb4 7,cephr57 ,,lhnm repres5ts ! base ( ! pyramid4 ,on ! half-base ,,ac1 ,i describ$ a #c1 #d1 #e triangle ,,abc4 ,i !n project$ ! l9e ,,cf 7ratio #ba to ,,bc #bj71 ?us erect+ ! #bj1 #ba1 #bi triangle ,,bcf4 ,f ?is datum1 ,i arriv$ at ! triangles ,,bea1 ,,adc1 & ,,gkh2 ,,gk1 ,,ea & ,,ad ea* repres5t+ apo!m2 ,,cf & ,,cd1 altitude2 & ,,hk1 edge4 ,! l5g?s ( ! l9es ,,ad1 ,,hk & ,,* be+ got at z 9 ! pyramid ,myc]9us4 ,^! measures r$uc$ to cubits1 call+ ,,ac ratio #af #baj cubits 7half-base ( pyramid7 give ! foll[+ result4 ,r4 ,b4 ,,briti% ,,cubits4 ,,feet4 ,half-base #bajjj #ceche ,,la ,apo!m #cdfej #ehche ,,ea ,edge #djeaf #fhbfi ,,hk ,altitude #bgefbe #dfddc ,,cd ,half-diag4 ( base #bifihe #ejjdb ,,* ?us ,i get ! ratios (--,apo!m 3 ,half-,base 33 #cc 3 #bj1 c4 ,! planes 9 ! diagram are plac$ 9 _! correct positions1 z direct$ = ,figure #ga4 ,! angle at ! base ( ,cephr51 if built to ! ratio ( #af to #ba 7half-base to altitude71 & n to ! practical ratio ( #cc to #bj 7apo!m to half-base71 w\ld be ! complem5t ( ,,adc1 ?us-- #af #af -- #gfaijd ,tan4 ,,adc #cg #ah #ad-- #ba #df #cj ,,dac #eb #da #de-- #df b z x is probable t ! pyramid 0 built to ! ratio ( #cc to #bj1 ,i h m>k$ ! base angle 9 ,fig4 #ag1 z #eb #da #da4 ,i took ,*eops out1 fir/ z a ,pi pyramid1 & made 8 l9es to a base ( #dbj cubits1 z foll[s-- ,half-base #baj ,altitude #bfgchjcjd ,apo!m #cciihhegc 7,see ,fig4 #gc47 ,fig4 #gc4 7,*eops7 ,b to produce ! build+ ratio ( #cd to #ba1 z p] diagram ,figure #f or #i1 ,i _h to alt] x to-- ,half-base #baj ,altitude #bfgcidhci ,apo!m #cdj ,?us ! !oretical angle ( ,*eops is #ea #ea #adc1 & ! probable angle at :i* x 0 built1 is #ea #ea #bj1 z p] figure #ae4 ,*eops is !re=e ! mean or c5tre ( a sy/em--! slopes ( ,myc]9us be+ a little flatt]1 & ^? ( ,cephr5 a little /eep]1 ,*eops com+ fairly betwe5 ! two1 )9 ab\t #aj m9utes2 & ?us ! connection betwe5 ! gr\nd plan ( ! gr\p & ! slopes ( ! ?ree pyramids is exactly z "o might expect aft] exam9ation ( ,figure #c1 #d or #e4