,with the respectful compliments of the author4 ,address1 ,mr4 ,r4 ,,ballard1 ,malvern1 ,,england4 ,illustration3 ,plumb ,line over ,pyramids ,,the ,,solution ,,of ,,the ,,pyramid ,,problem ,,or1 ,,pyramid ,,discoveries4 ,,with ,a ,,new ,,theory ,,as ,,to ,,their ,,ancient ,,use4 ,,by ,,robert ,,ballard1 ,m4 ,,inst4 ,c4,e41 ,,england2 ,m4 ,,amer1 ,,soc4 ,c4,e4 ,,chief ,,engineer ,,of ,,the ,,central ,,and ,,northern ,,railway ,,division ,,of ,,the ,,colony ,,of ,,queensland1 ,,australia4 ,,new ,,york3 ,,john ,,wiley ,,sons4 #ahhb4 ,copyright1 #ahhb1 ,by ,,john ,,wiley ,,sons4 ,,press ,,of ,j4 ,j4 ,,little ,,co41 ,,nos4 #aj ,,to #bj ,,astor ,,place1 ,,new ,,york4 ,,note4 ,in preparing this work for publication ,i have received valuable help from the following friends in ,queensland3-- ,e4 ,a4 ,delisser1 ,l4,s4 and ,c4,e41 ,bogantungan1 who assisted me in my calculations1 and furnished many useful suggestions4 ,j4 ,brunton ,stephens1 ,brisbane1 who persuaded me to publish my theory1 and who also undertook the work of correction for the press4 ,j4 ,a4 ,clarke1 ,artist1 ,brisbane1 who contributed to the ,illustrations4 ,lyne ,brown1 ,emerald1--7photographs74 ,f4 ,rothery1 ,emerald1--7models74 and--,a4 ,w4 ,voysey1 ,emerald1--7maps and diagrams74 ,,contents4 ,,page #a4 ,the ,ground ,plan of the ,gzeh ,group #i ,plan ,ratios connected into ,natural ,numbers #ac #b4 ,the ,original ,cubit ,measure of the ,gzeh ,group #ae #c4 ,the ,exact ,measure of the ,bases of the ,pyramids #ai #d4 ,the ,slopes1 ,ratios and ,angles of the ,three ,principal ,pyramids of the ,gzeh ,group #bj #e4 ,the ,exact ,dimensions of the ,pyramids #bc #f4 ,geometrical ,peculiarities of the ,pyramids #bh #f;,a4 ,the ,casing ,stones of the ,pyramids #cb #g4 ,peculiarities of the ,triangles #c1 #d1 #e and #bj1 #ba1 #bi #ce #h4 ,general ,observations #ch #i4 ,the ,pyramids of ,egypt the ,theodolites of the ,egyptian ,land ,surveyors #da #aj4 ,how the ,pyramids were made use of #dd #aa4 ,description of the ancient ,portable ,survey ,instrument #ea ,table to explain ,figure #fj #fd #ab4 ,primary ,triangles and their ,satellites2--or the ,ancient ,system of ,right-angled ,trigonometry unfolded by a ,study of the ,plan of the ,pyramids of ,gzeh #fd ,table of some ,primary ,triangles and their ,satellites #gd #ac4 ,the ,size and ,shape of the ,pyramids indicated by the ,plan #gg #ad4 ,a ,simple ,instrument for laying off ,primary ,triangles #gi #ad;,a4 ,general ,observations #hj #ae4 ,primary ,triangulation #hb #af4 ,the ,pentangle or ,five-pointed ,star the ,geometric ,symbol of the ,great ,pyramid #ia ,table showing the comparative ,measures of ,lines #if #ag4 ,the manner in which the ,slope ,ratios of the ,pyramids were arrived at #ajd ,,list ,,of ,,works ,,consulted4 ,penny ,cyclopdia4 7,knight1 ,london4 #ahcc47 ,sharpe's ,egypt4 ,our ,inheritance in the ,great ,pyramid4 ,piazzi ,smyth4 ,the ,pyramids of ,egypt4 ,r4 ,a4 ,proctor4 7,article in ,gentleman's ,magazine4 ,feb4 #ahhj47 ,traite de la ,grandeur et de la ,figure de la ,terre4 ,cassini4 7,amsterdam4 #agbc47 ,pyramid ,facts and ,fancies4 ,j4 ,bonwick4 ,illustration3 ,general ,plan of ,gzeh ,group ,,the ,,solution ,,of ,,the ,,pyramid ,,problem4 ,with the firm conviction that the ,pyramids of ,egypt were built and employed1 among other purposes1 for one special1 main1 and important purpose of the greatest utility and convenience1 ,i find it necessary before ,i can establish the theory ,i advance1 to endeavor to determine the proportions and measures of one of the principal groups4 ,i take that of ,gzeh as being the one affording most data1 and as being probably one of the most important groups4 ,i shall first try to set forth the results of my investigations into the peculiarities of construction of the ,gzeh ,group1 and afterwards show how the ,pyramids were applied to the national work for which ,i believe they were designed4 #a4 ,,the ,,ground ,,plan ,,of ,,the ,,gizeh ,,group4 ,i find that the ,pyramid ,cheops is situated on the acute angle of a right-angled triangle--sometimes called the ,pythagorean1 or ,egyptian triangle--of which base1 perpendicular1 and hypotenuse are to each other as #c1 #d1 and #e4 ,the ,pyramid called ,mycerinus1 is situate on the greater angle of this triangle1 and the base of the triangle1 measuring three1 is a line due east from ,mycerinus1 and joining perpendicular at a point due south of ,cheops4 7,see ,figure #a47 ,fig4 #a4 ,cheops1 ,mycerinus ,i find that the ,pyramid ,cheops is also situate at the acute angle of a right-angled triangle more beautiful than the so-called triangle of ,pythagoras1 because more practically useful4 ,i have named it the #bj1 #ba1 #bi triangle4 ,base1 perpendicular1 and hypotenuse are to each other as twenty1 twenty-one1 and twenty-nine4 ,the ,pyramid ,cephren is situate on the greater angle of this triangle1 and base and perpendicular are as before described in the ,pythagorean triangle upon which ,mycerinus is built4 7,see ,fig4 #b47 ,fig4 #b4 ,cheops1 ,cephren ,fig4 #c4 ,cheops1 ,cephren1 ,mycerinus ,figure #c represents the combination1--,a being ,cheops1 ,f ,cephren1 and ,d ,mycerinus4 ,lines ,,dc1 ,,ca1 and ,,ad are to each other as #c1 #d1 and #e2 and lines ,,fb1 ,,ba1 and ,,af are to each other as #bj1 #ba1 and #bi4 ,the line ,,cb is to ,,ba1 as #h to #g2 the line ,,fh is to ,,dh1 as #if to #ee2 and the line ,,fb is to ,,bc1 as #e to #f4 ,the ,ratios of the first triangle multiplied by forty-five1 of the second multiplied by four1 and the other three sets by twelve1 one1 and sixteen respectively1 produce the following connected lengths in natural numbers for all the lines4 ,,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 another pyramid of the group--it is the one to the southward and eastward of ,cheops4 ,in this connection1 ,a ,y ,z ,a is a #c1 #d1 #e triangle1 and ,b ,y ,z ,o ,b is a square4 ,lines ,,ya to ,,ca are as #a to #e ,,cy to ,,yz as #c to #a ,,fo to ,,zo as #h to #c and ,,da to ,,az as #ae to #d4 ,i may also point out on the same plan that calling the line ,,fa radius1 and the lines ,,ba and ,,fb sine and co-sine1 then is ,,ya equal in length to versed sine of angle ,,afb4 ,this connects the #bj1 #ba1 #bi triangle ,,fab with the #c1 #d1 #e triangle ,,azy4 ,i have not sufficient data at my disposal to enable me to connect the remaining eleven small pyramids to my satisfaction1 and ,i consider the four are sufficient for my purpose4 ,fig4 #d4 ,at level of ,at level of ,own base ,cephren's base ,of these natural ,cheops #ef #eb numbers the bases ,cephren #eb #eb of the pyramids are ,mycerinus #bf #bg as follows4 ,i now establish the following list of measurements of the plan in connected natural numbers4 7,see ,figure #d47 ,plan ,ratios connected into ,natural ,numbers4 ---------------------------------------- --------- ,,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,,st #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,,wh #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,,th #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 ,,ch #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 ---------------------------------------- --------- ,the above connected natural numbers multiplied by eight become ,r4,b4 cubits4 ,r4,b4,c4 7,thus1 ,,by #dh #h #chd ,,gx #add #h #aaeb74 #b4 ,,the ,,original ,,cubit ,,measure ,,of ,,the ,,gizeh ,,group4 ,mr4 ,j4 ,j4 ,wild1 in his letter to ,lord ,brougham written in #ahej1 called the base of ,cephren seven seconds4 ,i estimate the base of ,cephren to be just seven thirtieths of the line ,,da4 ,the line ,,da is therefore thirty seconds of the ,earth's ,polar circumference4 ,the line ,,da is therefore #cjccaahfbe ,british feet1 and the base of ,cephren #gjggbg ,british feet4 ,i applied a variety of ,cubits but found none to work in without fractions on the beautiful set of natural dimensions which ,i had worked out for my plan4 7,see table of connected natural numbers47 ,i ultimately arrived at a cubit as the ancient measure which ,i have called the ,r4,b4 cubit1 because it closely resembles the ,royal ,babylonian ,cubit of #eaca metre1 or #afhccii ,british feet4 ,the difference is #a_/fjj of a foot4 ,i arrived at the ,r4,b4 cubit in the following manner4 ,taking the polar axis of the earth at five hundred million geometric inches1 thirty seconds of the circumference will be #cfcfajbfjh--;geometric inches1 or #cfcigdbce ,british inches1 at nine hundred and ninety-nine to the thousand--and #cjcjjhee geometric feet1 or #cjccaahfbe ,british feet4 ,now dividing a second into sixty parts1 there are #ahjj ,r4,b4 cubits in the line ,,da2 and the line ,,da being thirty seconds1 measures #cfcigdbce ,british inches1 which divided by #ahjj makes one of my cubits #bjbbjgijh ,british inches1 or #afhejff ,british feet4 ,similarly1 #cfcfajbfjh geometric inches divided by #ahjj makes my cubit #bjbjjeg geometric inches in length4 ,i have therefore defined this cubit as follows3--,one ,r4,b4 cubit is equal to #bj4bjjf geo4 inches1 #bjbbjh ,brit4 inches1 and #afhe ,brit4 feet4 ,i now construct the following table of measures4 ---------------------------------------- --- ,r4 ,b4 ,,plethra ,,or ,,cubits4 ,,seconds4 ,,stadia4 ,,minutes4,,degrees4 ---------------------------------------- --- #fj #a #cfj #f #a #cfjj #fj #aj #a #bafjjj #cfjj #fjj #fj #a #gggfjjjj #abifjjj #bafjjj #bafjj #cfj ---------------------------------------- --- ,thus there are seventy-seven million1 seven hundred and sixty thousand ,r4,b4 cubits1 or two hundred and sixteen thousand stadia1 to the ,polar circumference of the earth4 ,thus we have obtained a perfect set of natural and convenient measures which fits the plan1 and fits the circumference of the earth4 ,and ,i claim for the ,r4,b4 cubit that it is the most perfect ancient measure yet discovered1 being the measure of the plan of the ,pyramids of ,gzeh4 ,the same forgotten wisdom which divided the circle into three hundred and sixty degrees1 the degree into sixty minutes1 and the minute into sixty seconds1 subdivided those seconds1 for earth measurements1 into the sixty parts represented by sixty ,r4,b4 cubits4 ,we are aware that thirds and fourths were used in ancient astronomical calculations4 99 99 99 99 99 ,the reader will now observe that the cubit measures of the main ,pythagorean triangle of the plan are obtained by multiplying the original #c1 #d and #e by #cfj2 and that the entire dimensions are obtained in ,r4,b4 cubits by multiplying the last column of connected natural numbers in the table by eight1--thus-- ,r4 ,b4 ,,cubits4 ,,dc #c #cfj #ajhj ,,ca #d #cfj #addj ,,da #e #cfj #ahjj or1 ,,natural ,r4 ,b4 ,,numbers4 ,,cubits4 ,,dc #ace #h #ajhj ,,ca #ahj #h #addj ,,da #bbe #h #ahjj c41 c4 7,see ,figure #e1 p4 #ah47 ,according to ,cassini1 a degree was #fjj stadia1 a minute #aj stadia2 and a modern ,italian mile1 in the year #agbc1 was equal to one and a quarter ancient ,roman miles2 and one and a quarter ancient ,roman miles were equal to ten stadia or one minute4 7,cassini1 ,traite de la grandeur et de la ,figure de la ,terre4 ,amsterdam1 #agbc47 ,fig4 #e ,r4,b4,cubits4 ,,fb #fdj ,,eb #ejd ,,ba #fgb ,,ba #fgb ,,af #ibh ,,ae #hdj ,,dc #ajhj ,,no #gfh ,,ca #addj ,,of #ajbd ,,ad #ahjj ,,fn #abhj ,,dg #egf ,,ay #bhh ,,ge #gfh ,,zy #chd ,,ed #ifj ,,za #dhj ,r4,b4 ,cub4 ,at level of ,cephren's ,cheops' ,base #dbj ,base which is the plane ,cephren's ,base #dbj or level of the plan4-- ,mycerinus' ,base #bah ,dufeu also made a stadium the six hundredth part of a degree4 ,he made the degree #aajhbgfh metres1 which multiplied by #cbhjhda gives #cfcfjgiif ,british feet2 and #cfcfjgiif divided by #fjj equals #fjfjaccbg feet to his stadium4 ,i make the stadium #fjffbcgf ,british feet4 ,there being #cfj cubits to a stadium1 ,dufeu's stadium divided by #cfj1 gives #afhcc ,british feet1 which is the exact measure given for a ,royal ,babylonian ,cubit1 if reduced to metres1 viz43 #jeaca of a metre1 and therefore probably the origin of the measure called the ,royal ,babylonian cubit4 ,according to this measure1 the ,gzeh plan would be about #a_/ajaa smaller than if measured by ,r4,b4 cubits4 #c4 ,,the ,,exact ,,measure ,,of ,,the ,,bases ,,of ,,the ,,pyramids4 ,a stadium being #cfj ,r4,b4 cubits1 or six seconds--and a plethron #fj ,r4,b4 cubits1 or one second1 the base of the ,pyramid ,cephren is seven plethra1 or a stadium and a plethron1 equal to seven seconds1 or four hundred and twenty ,r4,b4 cubits4 ,mycerinus' base is acknowledged to be half the base of ,cephren4 ,piazzi ,smyth makes the base of the ,pyramid ,cheops #iacaje pyramid 7or geometric7 inches1 which divided by #bjbjjf gives #debja ,r4,b4 cubits4 ,i call it #deb cubits1 and accept it as the measure which exactly fits the plan4 ,i have not sufficient data to determine the exact base of the other and smaller pyramid which ,i have marked on my plan4 ,the bases1 then1 of ,mycerinus1 ,cephren1 and ,cheops1 are #baj1 #dbj and #deb cubits1 respectively4 ,but in plan the bases should be reduced to one level4 ,i have therefore drawn my plan1 or horizontal section1 at the level or plane of the base of ,cephren1 at which level or plane the bases or horizontal sections of the pyramids are--,mycerinus1 #bah cubits1 ,cephren1 #dbj cubits1 and ,cheops1 #dbj cubits4 ,i shall show how ,i arrive at this by-and-by1 and shall also show that the horizontal section of ,cheops1 corresponding to the horizontal section of ,cephren at the level of ,cephren's base1 occurs1 as it should do1 at the level of one of the courses of masonry1 viz4--the top of the tenth course4 #d4 ,,the ,,slopes1 ,,ratios1 ,,and ,,angles ,,of ,,the ,,three ,,principal ,,pyramids ,,of ,,the ,,gizeh ,,group4 ,before entering on the description of the exact slopes and angles of the three principal pyramids1 ,i must premise that ,i was guided to my conclusions by making full use of the combined evolutions of the two wonderful right-angled triangles1 #c1 #d1 #e1 and #bj1 #ba1 #bi1 which seem to run through the whole design as a sort of dominant4 ,from the first ,i was firmly convinced that in such skilful workmanship some very simple and easily applied templates must have been employed1 and so it turned out4 ,builders do not mark a dimension on a plan which they cannot measure1 nor have a hidden measure of any importance without some clear outer way of establishing it4 ,this made me go straight for the slant ratios4 ,when the ,pyramids were cased from top to bottom with polished marble1 there were only two feasible measures1 the bases and the apothems2#a and for that reason ,i conjectured that these would be the definite plan ratios4 ,footnote #a3 ,the ,apothem is a perpendicular from the vertex of a pyramid on a side of the base4--,chambers' ,practical ,mathematics1 p4 #aef4 99 99 99 99 99 ,figures #f1 #g and #h show the exact slope ratios of ,cheops1 ,cephren1 and ,mycerinus1 measured as shown on the diagrams--viz41 ,cheops1 #ba to #cd1 ,cephren1 #bj to #cc1 and ,mycerinus1 #bj to #cb--that is1 half base to apothem4 ,fig4 #f ,cheops4 ,note4 ,the ,ratios of ,bases to ,altitudes are very ,fig4 #g ,cephren4 nearly as follows1 viz3-- ,fig4 #h ,mycerinus4 ,cheops #cc to #ba or #ccj to #baj ,cephren #cb to #ba or #cbj to #baj ,mycerinus #cb to #bj or #ccf to #baj ,the ratios of base to altitude are1 ,cheops1 #cc to #ba1 ,cephren1 #cb to #ba1 and ,mycerinus1 #cb to #bj3 not exactly1 but near enough for all practical purposes4 ,for the sake of comparison1 it will be well to call these ratios #ccj to #baj1 #cbj to #baj1 and #ccf to #baj1 respectively4 99 99 99 99 99 ,fig4 #i4 ,cheops4 ,fig4 #aj4 ,cheops4 ,figures #i and #aj are meridional and diagonal sections1 showing ratios of ,cheops1 viz41 half base to apothem1 #ba to #cd exactly2 half base to altitude1 #e to #g nearly1 and #ahc to #bcc1 nearer still 7being the ratio of ,piazzi ,smyth74 ,the ratio of ,sir ,f4 ,james1 half diagonal #aj to altitude #i is also very nearly correct4 ,my altitude for ,cheops is #dhdhhg ,british feet1 and the half base #chjha ,british feet4 ,the ratio of #g to #e gives #dhdff1 and the ratio of #bcc to #ahc gives #dhdhe for the altitude4 ,my half diagonal is #echedfe1 and ratio #aj to #i1 gives #dhdfi ,british feet for the altitude4 ,i have mentioned the above to show how very nearly these ratios agree with my exact ratio of #ba to #cd half base to apothem4 99 99 99 99 99 ,fig4 #aa4 ,cephren4 ,fig4 #ab4 ,cephren4 ,figures #aa and #ab show the ratios of ,cephren1 viz41 half base to apothem1 #bj to #cc exactly1 and half base1 altitude1 and apothem respectively1 as #hj1 #aje1 and #acb1 very nearly4 ,also half diagonal1 altitude1 and edge1 practically as #dca1 #djj1 and #ehh4 99 99 99 99 99 ,fig4 #ac4 ,mycerinus4 ,fig4 #ad4 ,mycerinus4 ,figures #ac and #ad show the ratios of ,mycerinus1 viz41 half base to apothem1 #bj to #cb exactly1 and half base1 altitude1 and apothem respectively1 as #bj1 #be1 and #cb very nearly4 ,also full diagonal to edge as #big to #aih1 nearly4 ,a peculiarity of this pyramid is1 that base is to altitude as apothem is to half base4 ,thus1 #dj 3 #be 33 #cb 3 #bj2 that is1 half base is a fourth proportional to base1 apothem1 and altitude4 #e4 ,,the ,,exact ,,dimensions ,,of ,,the ,,pyramids4 ,r4,b4 ,cub4 ,brit4 ,ft4 ,fig4 #ae4 ,cheops4 #deb #gfafb #bhggfg #dhdhhg ,fig4 #af4 ,cheops4 #cfeijdg #fafedi #dcjjeh #gbdfdg #fcibbdd #ajggjic ,figures #ae to #bj inclusive1 show the linear dimensions of the three pyramids1 also their angles4 ,the base angles are1 ,cheops1 #ea #ea #bj2 ,cephren1 #eb #da #da2 and ,mycerinus1 #ea #ai #d4 ,r4,b4 ,cub4 ,brit4 ,ft4 ,fig4 #ag4 ,cephren4 #dbj #gjggj #bgefa #dfddj ,fig4 #ah4 ,cephren4 #cdfej #ehche #djeaf #fhbfi #eicig #ajjjhd ,r4,b4 ,cub4 ,brit4 ,ft4 ,fig4 #ai4 ,mycerinus4 #baj #cec4he #afh #bhc4jh ,fig4 #bj4 ,mycerinus3 #aca4ad #bbj4ig #aih4aj #cccgihe #bifihdh #ejjdb ,in ,cheops1 my dimensions agree with ,piazzi ,smyth--in the base of ,cephren1 with ,vyse and ,perring--in the height of ,cephren1 with ,sir ,gardner ,wilkinson1 nearly--in the base of ,mycerinus1 they agree with the usually accepted measures1 and in the height of ,mycerinus1 they exceed ,jas4 ,j4 ,wild's measure1 by not quite one of my cubits4 ,in my angles ,i agree very nearly with ,piazzi ,smyth1 for ,cheops1 and with ,agnew1 for ,cephren1 differing about half a degree from ,agnew1 for ,mycerinus1 who took this pyramid to represent the same relation of ,pi that ,p4 ,smyth ascribes to ,cheops 7viz43 #ea #ea #adc71 while he gave ,cheops about the same angle which ,i ascribe to ,mycerinus4 ,i shall now show how ,i make ,cephren and ,cheops of equal bases of #dbj ,r4,b4 cubits at the same level1 viz4--that of ,cephren's base4 ,john ,james ,wild made the bases of ,cheops1 ,cephren1 and ,mycerinus1 respectively1 #hj1 #ajj1 and #ajdij cubits above some point that he called ,nile ,level4 ,his cubit was1 ,i believe1 the ,memphis1 or ,nilometric cubit--but at any rate1 he made the base of ,cephren #dab of them4 ,i therefore divided the recognized base of ,cephren--viz41 #gjgge ,british feet--by #dab1 and got a result of #agagh ,british feet for his cubit4 ,therefore1 his measures multiplied by #agagh and divided by #afhe will turn his cubits into ,r4,b4 cubits4 ,i thus make ,cheops1 ,cephren1 and ,mycerinus1 respectively1 #haef1 #ajaic1 and #ajfic ,r4,b4 cubits above the datum that ,j4 ,j4 ,wild calls ,nile ,level4 ,according to ,bonwick's ,facts and ,fancies1 p4 #ca1 high water ,nile would be #ach ft4 below base of ,cheops 7or #hbai ,r4,b4 cubits74 ,piazzi ,smyth makes the pavement of ,cheops #ageb ,british inches 7or #hffd ,r4,b4 cubits7 above average ,nile ,level1 but1 by scaling his map1 his high ,nile ,level appears to agree nearly with ,wild4 ,it is the relative levels of the ,pyramids1 however1 that ,i require1 no matter how much above ,nile ,level4 ,cephren's base of #dbj cubits being #ajaic cubits1 and ,cheops' base of #deb cubits being #haef cubits above ,wild's datum1 the difference in level of their bases is1 #bjcg cubits4 ,the ratio of base to altitude of ,cheops being #ccj to #baj1 therefore #bjcg cubits divided by #baj and multiplied by #ccj equals #cb cubits2 and #deb cubits minus #cb cubits1 equals #dbj4 ,similarly1 the base of ,mycerinus is #e cubits above the base of ,cephren1 and the ratio of base to altitude #cb to #bj2 therefore1 #e cubits divided by #bj and multiplied by #cb equals #h cubits to be added to the #baj cubit base of ,mycerinus1 making it #bah cubits in breadth at the level of ,cephren's base4 ,thus1 a horizontal section or plan at the level of ,cephren's base would meet the slopes of the ,pyramids so that they would on plan appear as squares with sides equal to #bah1 #dbj1 and #dbj ,r4,b4 cubits1 for ,mycerinus1 ,cephren1 and ,cheops1 respectively4 ,fig4 #ba4 ,r4,b4 ,cub4 ,apex of ,cephren above ,base ,cheops #bieih ,apex of ,cheops above ,base ,cheops #bhggg ,apex of ,mycerinus above ,base ,cheops #aefea ,base ,cephren above base of ,cheops #bjcg ,base ,mycerinus above base of ,cheops #becg ,piazzi ,smyth makes the top of the tenth course of ,cheops #dad pyramid inches above the pavement2 and #dad divided by #bjbjjf equals #bjdi ,r4,b4 cubits4 ,but ,i have already proved that ,cheops' #dbj cubit base measure occurs at a level of #bjcg cubits above pavement2 therefore is this level the level of the top of the tenth course1 for the difference is only #jab ,r4,b4 cubits1 or #b inches4 99 99 99 99 99 ,i wish here to note as a matter of interest1 but not as affecting my theory1 the following measures of ,piazzi ,smyth1 turned into ,r4,b4 cubits1 viz43-- ,,pyr4 ,,inches4 ,r4,b4 ,,cubits4 ,king's ,chamber floor1 above pavement #agjb #hdbe ,cheops' ,base1 as before stated #iacaje #debja ,king's ,chamber1 ,true ,length1 #dabacb #bjdj ,true ,first ,height1 #bcjchi #aadj ,true ,breadth1 #bjfjff #ajbj ,he makes the present summit platform of ,cheops #edde pyramid inches above pavement4 ,my calculation of #bfihj ,r4,b4 cub4 7,see ,fig4 #ba7 is equal to #edej pyramid inches--this is about #ah cubits below the theoretical apex4 99 99 99 99 99 ,figure #ba represents the comparative levels and dimensions of ,mycerinus1 ,cephren1 and ,cheops4 ,the following peculiarities are noticeable3--,that ,cheops and ,cephren are of equal bases at the level of ,cephren's base2--that1 at the level of ,cheops' base1 the latter is only half a cubit larger2--that1 from the level of ,mycerinus' base1 ,cheops is just double the height of ,mycerinus2--and that from the level of ,cephren's base1 ,cephren is just double the height of ,mycerinus2 measuring in the latter case1 however1 only up to the level platform at the summit of ,cephren1 which is said to be about eight feet wide4 ,the present summit of ,cephren is #bcjg cubits above the present summit of ,cheops1 and the completed apex of ,cephren would be #hba cubits above the completed apex of ,cheops4 ,in the summit platforms ,i have been guided by ,p4 ,smyth's estimate of height deficient1 #cfc pyr4 inches1 for ,cheops1 and ,i have taken #h feet base for ,cephren's summit platform4 #f4 ,,geometrical ,,peculiarities ,,of ,,the ,,pyramids4 ,in any pyramid1 the apothem is to half the base as the area of the four sides is to the area of the base4 ,thus--,ratio apothem to half base ,mycerinus #cb to #bj ,cephren #cc to #bj ,cheops #cd to #ba ,,area ,,of ,,the ,,four ,,sides4 ,,area ,,of ,,the ,,base4 ,mycerinus #gjefj #ddajj ,cephren #biajfj #agfdjj ,cheops #ccjgggij #bjdcjd ,all in ,r4,b4 cubits4 ,therefore--#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;,herodotus states that the area of each of the four faces of ,cheops was equal to the area of a square whose base was the altitude of a ,pyramid2 or1 in other words1 that altitude was a mean proportional to apothem and half base2 thus--area of one face equals the fourth of #ccjgggij or #hbfiddge ,r4,b4 cubits1 and the square root of #hbfiddge is #bhgef4 ,but the correct altitude is #bhggg1 so the error is #jba1 or #d ,british inches4 ,i have therefore the authority of ,herodotus to support the theory which ,i shall subsequently set forth1 that this pyramid was the exponent of lines divided in mean and extreme ratio4 ,by taking the dimensions of the ,pyramid from what ,i may call its working level1 that is1 the level of the base of ,cephren1 this peculiarity shows more clearly1 as also others to which ,i shall refer4 ,thus--base of ,cheops at working level1 #dbj cubits1 and apothem #cdj cubits2 base area is1 therefore1 #agfdjj cubits1 and area of one face is 7#dbj cubits1 multiplied by half apothem1 or #agj cubits7 #gadjj cubits4 ,now the square root of #gadjj would give altitude1 or side of square equal to altitude1 #bfgbjgghd cubits3 but the real altitude is 7#cdj-baj7 #gaejj #bfgcidhci4 ,so that the error of ,herodotus's proposition is the difference between #gad and #gae4 ,footnote #b3 ,proctor is responsible for this statement1 as ,i am quoting from an essay of his in the ,gentleman's ,magazine4 ,r4 ,b4 ,this leads to a consideration of the properties of the angle formed by the ratio apothem #cd to half base #ba1 peculiar to the pyramid ,cheops4 7,see ,figure #bb47 ,fig4 #bb4 ,diagram illustrating relations of ratios of the pyramid ,cheops4 ,calling apothem #cd1 radius2 and half base #ba1 sine--,i find that-- ,radius is the square root of #aaef ,sine #dda ,co-sine #gae ,tangent #gac ,secant #ahfi and ,co-versed-sine #afi ,so it follows that the area of one of the faces1 #gad1 is a mean between the square of the altitude or co-sine1 #gae1 and the square of the tangent1 #gac4 ,thus the reader will notice that the peculiarities of the ,pyramid ,cheops lie in the regular relations of the squares of its various lines2 while the peculiarities of the other two pyramids lie in the relations of the lines themselves4 ,mycerinus and ,cephren1 born1 as one may say1 of those two noble triangles #c1 #d1 #e1 and #bj1 #ba1 #bi1 exhibit in their lineal developments ratios so nearly perfect that1 for all practical purposes1 they may be called correct4 ,thus--,mycerinus1 #cbj #be #ajbe1 and #cb #ajbd4 and ,cephren1 #dhj #aje #agdbe1 and #acb #agdbd4 or #edjj #dca #cdegfa1 and #ehh #cdegdd4 ,see diagrams1 ,figures #aa to #ad inclusive4 ,in the ,pyramid ,cheops1 altitude is very nearly a mean proportional between apothem and half base4 ,apothem being #cd1 and half base #ba1 then altitude would be 7#cd-ba7 #gae #bfgcidhci1 and-- #ba 3 #bfgcidhci 33 #bfgcidhci 3 #cd1 nearly4 ,here1 of course1 the same difference comes in as occurred in considering the assumption of ,herodotus1 viz41 the difference between #gae and #gad2 because if the altitude were #gad1 then would it be exactly a mean proportional between the half base and the apothem2 7thus1 #ba 3 #bfgbjgg 33 #bfgbjgg 33 #cd47 ,footnote #c3 ,half base to altitude4 ,footnote #d3 ,half base to altitude4 ,footnote #e3 ,half diagonal of base to altitude4 ,in ,cheops1 the ratios of apothem1 half base and edge are1 #cd1 #ba1 and #dj1 very nearly1 thus1 #cd #ba #aeig1 and #dj #afjj4 ,the dimensions of ,cheops 7from the level of the base of ,cephren7 to be what ,piazzi ,smyth calls a ,pi pyramid1 would be-- ,half base #baj ,r4,b4 cubits4 ,altitude #bfgchjcjd1 c4 ,apothem #cciihhegc1 c4 ,altitude being to perimeter of base1 as radius of a circle to circumference4 ,my dimensions of the pyramid therefore in which-- ,half base #baj ,r4,b4 cubits4 ,altitude #bfgcidhci c4 ,apothem #cdj come about as near to the ratio of ,pi as it is possible to come1 and provide simple lines and templates to the workmen in constructing the building2 and ,i entertain no doubt that on the simple lines and templates that my ratios provide1 were these three pyramids built4 #f;,a4 ,,the ,,casing ,,stones ,,of ,,the ,,pyramids4 ,figures #bc1 #bd1 and #be1 represent ordinary casing stones of the three pyramids1 and ,figures #bf1 #bg1 and #bh1 represent angle or quoin casing stones of the same4 ,the casing stone of ,cheops1 found by ,colonel ,vyse1 is represented in ,bonwick's ,pyramid ,facts and ,fancies1 page #af1 as measuring four feet three inches at the top1 eight feet three inches at the bottom1 four feet eleven inches at the back1 and six feet three inches at the front4 ,taking four feet eleven inches as ,radius1 and six feet three inches as ,secant1 then the ,tangent is three feet ten inches and three tenths4 ,fig4 #bc4 ,cheops ,casing ,stone4 ,fig4 #bd4 ,cephren ,casing ,stone4 ,fig4 #be4 ,mycerinus ,casing ,stone4 ,fig4 #bf4 ,cheops ,angle or ,quoin ,stone4 ,fig4 #bg4 ,cephren ,angle or ,quoin ,stone4 ,fig4 #bh4 ,mycerinus ,angle or ,quoin ,stone4 ,thus1 in inches 77#ge-ei77 #dfcj inches2 therefore the inclination of the stone must have been--slant height #ge inches to #dfcj inches horizontal4 ,now1 #dfcj is to #ge1 as #ba is to #cd4 ,therefore1 ,col4 ,vyse's casing stone agrees exactly with my ratio for the ,pyramid ,cheops1 viz41 #ba to #cd4 7,see ,figure #bi47 ,fig4 #bi4 ,col4 ,vyse's ,casing ,stone4 #ge 3 #dfc 33 #cd 3 #ba ,this stone must have been out of plumb at the back an inch and seven tenths2 perhaps to give room for grouting the back joint of the marble casing stone to the limestone body of the work3 or1 because1 as it is not a necessity in good masonry that the back of a stone should be exactly plumb1 so long as the error is on the right side1 the builders might not have been particular in that respect4 ,fig4 #ei4 7,temple of ,cheops1 standing at angle of wall47 ,figure #ei represents such a template as the masons would have used in building ,cheops1 both for dressing and setting the stones4 7,the courses are drawn out of proportion to the template47 ,the other pyramids must have been built by the aid of similar templates4 ,such large blocks of stone as were used in the casing of these pyramids could not have been completely dressed before setting2 the back and ends1 and the top and bottom beds were probably dressed off truly1 and the face roughly scabbled off2 but the true slope angle could not have been dressed off until the stone had been truly set and bedded1 otherwise there would have been great danger to the sharp arises4 99 99 99 99 99 ,i shall now record the peculiarities of the #c1 #d1 #e or ,pythagorean triangle1 and the right-angled triangle #bj1 #ba1 #bi4 #g4 ,,peculiarities ,,of ,,the ,,triangles #c1 #d1 #e1 ,,and #bj1 #ba1 #bi4 ,fig4 #cj to #ce4 ,,peculiarities ,,of ,,the ,,triangles ,the #c1 #d1 #e triangle contains #cf #eb #aafe and the complement or greater angle #ec #g #dhce ,radius #e #fj whole numbers4#f ,co-sine #d #dh ,sine #c #cf ,versed sine #a #ab ,co-versed sine #b #bd ,tangent #c #de ,secant #f #ge ,co-tangent #f #hj ,co-secant #h #ajj ,tangent ,secant ,diameter or #b ,radius ,co-tan ,co-sec #c ,radius ,sine 3 ,versed-sine 33 #c 3 #a ,co-sine 3 ,co-versed sine 33 #b 3 #a ,figure #cj illustrates the preceding description4 ,figure #ca shows the #ca triangle1 and the #ba triangle built up on the sine and co-sine of the #c1 #d1 #e triangle4 ,the #ca triangle contains #ah #bf #ehb and the #ba triangle #bf #cc #edai2 the latter has been frequently noticed as a pyramid angle in the gallery inclinations4 ,figure #cb shows these two triangles combined with the #c1 #d1 #e triangle1 on the circumference of a circle4 ,footnote #f3 #fj #c #d #e ,the #bj1 #ba1 #bi triangle contains #dc #cf #ajae and the complement1 #df #bc #dihe4 ,expressed in whole numbers-- ,radius #bi #abahjg ,sine #bj #hdjj ,co-sine #ba #hhbj ,versed sine #h #ccfj ,co-versed sine #i #cghj ,tangent #aafjj ,co-tangent #abghi ,secant #afhbj ,co-sec #agffa ,tangent ,secant #b radius ,co-tan ,co-sec #b radius ,sine 3 ,versed sine 33 #e 3 #b ,co-sine 3 ,co-versed sine 33 #g 3 #c ,footnote #g3 #abahj #bj #ba #bi ,it is noticeable that while the multiplier required to bring radius #e and the rest into whole numbers1 for the #c1 #d1 #e triangle is twelve1 in the #bj1 #ba1 #bi triangle it is #dbj1 the key measure for the bases of the two main pyramids in ,r4,b4 cubits4#h ,footnote #h3 #ab #c #d1 and #dbj #bj #ba ,i am led to believe from study of the plan1 and consideration of the whole numbers in this #bj1 #ba1 #bi triangle1 that the ,r4,b4 cubit1 like the ,memphis cubit1 was divided into #bhj parts4 ,the whole numbers of radius1 sine1 and co-sine divided by #bhj1 give a very pretty measure and series in ,r4,b4 cubits1 viz41 #dc1 #cj1 and #ca1 or #hg1 #fj1 and #fc1 or #agd1 #abj and #abf2--;all exceedingly useful in right-angled measurements4 ,notice that the right-angled triangle #agd1 #abj1 #abf1 in the sum of its sides amounts to #dbj4 ,figure #cc illustrates the #bj1 #ba1 #bi triangle4 ,figure #cd shows the #eb and #gc triangles built up on the sine and co-sine of the #bj1 #ba1 #bi triangle4 ,the #eb triangle contains #ba #dh #ejh and the #gc triangle #bc #aa #edih4 ,figure #ce shows how these two triangles are combined with the #bj1 #ba1 #bi triangle on the circumference1 and ,figure #cf gives a general view and identification of these six triangles which occupied an important position in the trigonometry of a people who did all their work by right angles and proportional lines4 ,fig4 #cf4 ,ratios of ,leading ,triangles4 #h4 ,,general ,,observations4 ,it must be admitted that in the details of the building of the ,pyramids of ,gzeh there are traces of other measures than ,r4 ,b4 cubits1 but that the original cubit of the plan was #afhe ,british feet ,i feel no doubt4 ,it is a perfect and beautiful measure1 fit for such a noble design1 and1 representing as it does the sixtieth part of a second of the ,earth's polar circumference1 it is and was a measure for all time4 ,it may be objected that these ancient geometricians could not have been aware of the measure of the ,earth's circumference2 and wisely so1 were it not for two distinct answers that arise4 ,the first being1 that since ,i think ,i have shown that ,pythagoras never discovered the ,pythagorean triangle1 but that it must have been known and practically employed thousands of years before his era1 in the ,egyptian ,colleges where he obtained his ,m4,a4 degree1 so in the same way it is probable that ,eratosthenes1 when he went to work to prove that the earth's circumference was fifty times the distance from ,syene to ,alexandria1 may have obtained the idea from his ready access to the ill-fated ,alexandrian ,library1 in which perhaps some record of the learning of the builders of the ,pyramids was stored4 ,and therefore ,i claim that there is no reason why the pyramid builders should not have known as much about the circumference of the earth as the modern world that has calmly stood by in its ignorance and permitted those magnificent and1 as ,i shall prove1 useful edifices to be stripped of their beautiful garments of polished marble4 ,my second answer is that the correct cubit measure may have been got by its inventors in a variety of other ways2 for instance1 by observations of shadows of heavenly bodies1 without any knowledge even that the earth was round2 or it may have been evolved like the ,british inch1 which ,sir ,john ,herschel tells us is within a thousandth part of being one five hundred millionth of the earth's polar axis4 ,i doubt if the circumference of the earth was considered by the inventor of the ,british inch4 ,it was a peculiarity of the ,hindoo mathematicians that they tried to make out that all they knew was very old4 ,modern savants appear to take the opposite stand for any little information they happen to possess4 ,the cubit which is called the ,royal ,babylonian cubit and stated to measure ,o#eaca metre1 differs so slightly from my cubit1 only the six-hundredth part of a foot1 that it may fairly be said to be the same cubit1 and it will be for antiquaries to trace the connection1 as this may throw some light on the identity of the builders of the ,pyramids of ,gzeh4 ,few good ,english two-foot rules agree better than these two cubits do4 ,while ,i was groping about in the dark searching for this bright needle1 ,i tried on the plan many likely ancient measures4 ,for a long time ,i worked in ,memphis or ,nilometric cubits1 which ,i made #agabf ,british feet2 they seem to vary from #agj to #agb1 and although ,i made good use of them in identifying other people's measures1 still they were evidently not in accordance with the design2 but the ,r4,b4 cubit of #afhe ,british feet works as truly into the plan of the ,pyramids without fractions as it does into the circumference of the earth4 ,here ,i might1 to prevent others from falling into one of my errors1 point out a rock on which ,i was aground for a long time4 ,i took the base of the ,pyramid ,cheops1 determined by ,piazzi ,smyth1 from ,bonwick's ,pyramid ,facts and ,fancies 7a valuable little reference book71 as #gfc4ha ,british feet1 and the altitude as #dhf4befg2 and then from ,piazzi ,smyth's ,inheritance1 page #bg1 ,i confirmed these figures1 and so worked on them for a long time1 but found always a great flaw in my work1 and at last adopted a fresh base for ,cheops1 feeling sure that ,mr4 ,smyth's base was wrong3 for ,i was absolutely grounded in my conviction that at a certain level1 ,cheops' and ,cephren's measures bore certain relations to each other4 ,i subsequently found in another part of ,mr4 ,smyth's book1 that the correct measures were #gfa4fe and #dhd4ia ,british feet for base and altitude1 which were exactly what ,i wanted1 and enabled me to be in accordance with him in that pyramid which he appears to have made his particular study4 ,for the information of those who may wish to compare my measures1 which are the results of an even or regular circumference without fractions1 with ,mr4 ,smyth's measures1 which are the results of an even or regular diameter without fractions1 it may be well to state that there are just about #ii ,r4,b4 cubits in #hj of ,piazzi ,smyth's cubits of #be pyramid inches each4 #i4 ,,the ,,pyramids ,,of ,,egypt1 ,,the ,,theodolites ,,of ,,the ,,egyptian ,,land ,,surveyors4 ,about twenty-three years ago1 on my road to ,australia1 ,i was crossing from ,alexandria to ,cairo1 and saw the pyramids of ,gzeh4 ,i watched them carefully as the train passed along1 noticed their clear cut lines against the sky1 and their constantly changing relative position4 ,i then felt a strong conviction that they were built for at least one useful purpose1 and that purpose was the survey of the country4 ,i said1 ,here be the ,theodolites of the ,egyptians4 ,built by scientific men1 well versed in geometry1 but unacquainted with the use of glass lenses1 these great stone monuments are so suited in shape for the purposes of land surveying1 that the practical engineer or surveyor must1 after consideration1 admit that they may have been built mainly for that purpose4 ,not only might the country have been surveyed by these great instruments1 and the land allotted at periodical times to the people2 but they1 remaining always in one position1 were there to correct and readjust boundaries destroyed or confused by the annual inundations of the ,nile4 ,the ,pyramids of ,egypt may be considered as a great system of landmarks for the establishment and easy readjustment at any time of the boundaries of the holdings of the people4 ,the ,pyramids of ,gzeh appear to have been main marks2 and those of ,abousir1 ,sakkrah1 ,dashow1 ,lisht1 ,meydoun1 c41 with the great pyramids in ,lake ,moeris1 subordinate marks1 in this system1 which was probably extended from ,chaldea through ,egypt into ,ethiopia4 ,the pyramid builders may perhaps have made the entombment of their ,kings one of their exoteric objects1 playing on the morbid vanity of their rulers to induce them to the work1 but in the minds of the builders before ever they built must have been planted the intention to make use of the structures for the purposes of land surveying4 ,the land of ,egypt was valuable and maintained a dense population2 every year it was mostly submerged1 and the boundaries destroyed or confused4 ,every soldier had six to twelve acres of land2 the priests had their slice of the land too2 after every war a reallotment of the lands must have taken place1 perhaps every year4 ,while the water was lying on the land1 it so softened the ground that the stone boundary marks must have required frequent readjustment1 as they would have been likely to fall on one side4 ,by the aid of their great stone theodolites1 the surveyors1 who belonged to the priestly order1 were able to readjust the boundaries with great precision4 ,that all science was comprised in their secret mysteries may be one reason why no hieroglyphic record of the scientific uses of the pyramids remains4 ,it is possible that at the time of ,diodorus and ,herodotus1 7and even when ,pythagoras visited ,egypt17 theology may have so smothered science1 that the uses of the pyramids may have been forgotten by the very priests to whom in former times the knowledge belonged2 but a respectful reticence which has been noticed in some of these old writers on pyramid and other priestly matters would rather lead us to believe that an initiation into the mysteries may have sealed their lips on subjects about which they might otherwise have been more explicit4 ,the closing of one pyramid over another in bringing any of their many lines into true order1 must even now be very perfect2--but now we can only imagine the beauties of these great instrumental wonders of the world when the casing stones were on them4 ,we can picture the rosy lights of one1 and the bright white lights of others2 their clear cut lines against the sky1 true as the hairs of a theodolite2 and the sombre darkness of the contrasting shades1 bringing out their angles with startling distinctness4 ,under the influence of the ,eastern sun1 the faces must have been a very blaze of light1 and could have been seen at enormous distances like great mirrors4 ,i declare that the pyramids of ,gzeh in all their polished glory1 before the destroyer stripped them of their beautiful garments1 were in every respect adapted to flash around clearly defined lines of sight1 upon which the lands of the nation could be accurately threaded4 ,the very thought of these mighty theodolites of the old ,egyptians fills me with wonder and reverence4 ,what perfect and beautiful instruments they were6 never out of adjustment1 always correct1 always ready2 no magnetic deviation to allow for4 ,no wonder they took the trouble they did to build them so correctly in their so marvellously suitable positions4 ,if ,astronomers agree that observations of a pole star could have been accurately made by peering up a small gallery on the north side of one of the pyramids only a few hundred feet in length1 ,i feel that ,i shall have little difficulty in satisfying them that accurate measurements to points only miles away could have been made from angular observations of the whole group4 #aj4 ,,how ,,the ,,pyramids ,,were ,,made ,,use ,,of4 ,it appears from what ,i have already set forth that the plan of the ,pyramids under consideration is geometrically exact1 a perfect set of measures4 ,i shall now show how these edifices were applied to a thoroughly geometrical purpose in the true meaning of the word--to measure the ,earth4 ,i shall show how true straight lines could be extended from the ,pyramids in given directions useful in right-angled trigonometry1 by direct observation of the buildings1 and without the aid of other instruments4 ,and ,i shall show how by the aid of a simple instrument angles could be exactly observed from any point4 ,this ,survey theory does not stand or fall on the merits of my theory of the ,gzeh plan4 ,let it be proved that this group is not built on the exact system of triangulation set forth by me1 it is still a fact that its plan is in a similar shape1 and any such shape would enable a surveyor acquainted with the plan to lay down accurate surveys by observations of the group even should it not occupy the precise lines assumed by me4 ,and here ,i must state that although the lines of the plan as laid down herein agree nearly with the lines as laid down in ,piazzi ,smyth's book1 in the ,penny ,cyclopdia1 and in an essay of ,proctor's in the ,gentleman's ,magazine1 still ,i find that they do not agree at all satisfactorily with a map of the ,pyramids in ,sharp's ,egypt1 said to be copied from ,wilkinson's map4 ,we will1 however1 for the time1 and to explain my survey theory1 suppose the plan theory to be correct1 as ,i firmly believe it is4 ,and then1 supposing it may be proved that the respective positions of the pyramids are slightly different to those that ,i have allotted to them on my plan1 it will only make a similar slight difference to the lines and angles which ,i shall here show could be laid out by their aid4 ,let us in the first place comprehend clearly the shape of the land of ,egypt4 ,a sector or fan1 with a long handle--the fan or sector1 the delta2 and the handle of the fan1 the ,nile ,valley1 running nearly due south4 ,the ,pyramids of ,gzeh are situate at the angle of the sector1 on a rocky eminence whence they can all be seen for many miles4 ,the summits of the two high ones can be seen from the delta1 and from the ,nile ,valley to a very great distance2 how far1 ,i am unable to say2 but ,i should think that while the group could be made general use of for a radius of fifteen miles1 the summits of ,cephren and ,cheops could be made use of for a distance of thirty miles2 taking into consideration the general fall of the country4 ,it must be admitted that if meridian observations of the star ,alpha of the ,dragon could be made with accuracy by peeping up a small hole in one of the pyramids1 then surely might the surveyors have carried true north and south lines up the ,nile ,valley as far as the summit of ,cheops was visible1 by plumbing in the star and the apex of the pyramid by the aid of a string and a stone4 ,true east and west lines could have been made to intersect such north and south lines from the various groups of pyramids along the river banks1 by whose aid also such lines would be prolonged4 ,next1 supposing that their astronomers had been aware of the latitude of ,cheops1 and the annual northing and southing of the sun1 straight lines could have been laid out in various sectoral directions to the north-eastward and north-westward of ,cheops1 across the delta1 as far as the extreme apex of the pyramid was visible1 by observations of the sun1 rising or setting over his summit4 7,that the ,dog-star was observed in this manner from the north-west1 ,i have little doubt47 ,fig4 #cg4 ,sun above ,pyramid and sinking behind ,pyramid ,for this purpose1 surveyors would be stationed at suitable distances apart with their strings and their stones1 ready to catch the sun simultaneously1 and at the very moment he became transfixed upon the apex of the pyramid1 and was1 as it were1 swallowed by it4 7,see ,figure #cg47 ,the knowledge of the pyramid slope angle from different points of view would enable the surveyor to place himself in readiness nearly on the line4 ,surely such lines as these would be as true and as perfect as we could lay out nowadays with all our modern instrumental appliances4 ,a string and a stone here1 a clean-cut point of stone twenty miles away1 and a great ball of fire behind that point at a distance of ninety odd million miles4 ,the error in such a line would be very trifling4 ,such observations as last mentioned would have been probably extended from ,cephren for long lines1 as being the higher pyramid above the earth's surface1 and may have been made from the moon or stars4 ,in those days was the sun the intimate friend of man4 ,the moon and stars were his hand-maidens4 ,how many of us can point to the spot of the sun's rising or setting8 ,we1 with our clocks1 and our watches1 and our compasses1 rarely observe the sun or stars4 ,but in a land and an age when the sun was the only clock1 and the pyramid the only compass1 the movements and positions of the heavenly bodies were known to all4 ,these people were familiar with the stars1 and kept a watch upon their movements4 ,how many of our vaunted educated population could point out the ,dog-star in the heavens8--but the whole ,egyptian nation hailed his rising as the beginning of their year1 and as the harbinger of their annual blessing1 the rising of the waters of the ,nile4 ,fig4 #ch4 ,from the ,north ,west ,bearing #cae ,sun in the ,west4 ,fig4 #ci4 ,from the ,south ,east ,bearing #ace ,sun in the ,west4 ,fig4 #dj4 ,from the ,north ,east ,bearing #de ,sun in the ,east4 ,fig4 #da4 ,from the ,south ,west ,bearing #bbe ,sun in the ,east4 ,it is possible therefore that the land surveyors of ,egypt made full use of the heavenly bodies in their surveys of the land2 and while we are pitifully laying out our new countries by the circumferenter and the compass1 we presume to speak slightingly of the supposed dark heathen days1 when the land of ,egypt was surveyed by means of the sun and the stars1 and the theodolites were built of stone1 with vertical limbs five hundred feet in height1 and horizontal limbs three thousand feet in diameter4 ,imagine half a dozen such instruments as this in a distance of about sixty miles 7for each group of pyramids was effectually such an instrument71 and we can form some conception of the perfection of the surveys of an almost prehistoric nation4 ,the centre of ,lake ,moeris1 in which ,herodotus tells us two pyramids stood #cjj feet above the level of the lake1 appears from the maps to be about ,s4 #bh ,w41 or ,s4 #bi ,w4 from ,gzeh1 distant about #eg miles1 and the ,meidn group of pyramids appears to be about #cc miles due south of ,gzeh4 ,figures #ch1 #ci1 #dj and #da1 show that north-west1 south-east1 north-east1 and south-west lines from the pyramids could be extended by simply plumbing the angles4 ,these lines would be run in sets of two's and three's1 according to the number of pyramids in the group2 and their known distances apart at that angle would check the correctness of the work4 ,a splendid line was the line bearing #dc #cf #ajae1 or #bbc #cf #ajae from ,cheops and ,cephren1 the pyramids covering each other1 the line of hypotenuse of the great #bj1 #ba1 #bi triangle of the plan4 ,this ,i call the #bj1 #ba line4 7,see ,figure #db47 ,figure #dc represents the #c1 #d1 #e triangle line from the summits of ,mycerinus and ,cheops in true line bearing #baf #eb' #aafe4 ,this ,i call the south #d1 west #c line4 ,the next line is what ,i call the #b1 #a line1 and is illustrated by figure #dd4 ,it is one of the most perfect of the series1 and bears ,s4 #bf #cc' #edi ,w4 from the apex of ,cephren4 ,this line demonstrates clearly why ,mycerinus was cased with red granite4 ,not in memory of the beautiful and rosy-cheeked ,nitocris1 as some of the tomb theory people say1 but for a less romantic but more useful object2 simply because1 from this quarter1 and round about1 the lines of the pyramids would have been confused if ,mycerinus had not been of a different color4 ,the #b1 #a line is a line in which ,mycerinus would have been absolutely lost in the slopes of ,cephren but for his red color4 ,there is not a fact that more clearly establishes my theory1 and the wisdom and forethought of those who planned the ,gzeh pyramids1 than this red pyramid ,mycerinus1 and the #b1 #a line4 ,hekeyan ,bey1 speaks of this pyramid as of a ruddy complexion2 ,john ,greaves quotes from the ,arabic book1 ,morat ,alzeman1 and the lesser which is coloured2 and an ,arabic writer who dates the ,pyramids three hundred years before the ,flood1 and cannot find among the learned men of ,egypt any certain relation concerning them nor any memory of them amongst men1 also expatiates upon the beauties of the coloured satin covering of this one particular pyramid4 ,fig4 #db4 ,south #ba4 ,west #bj4 ,bearing #bbc4cf'4ajae4 ,fig4 #dc4 ,south #d4 ,west #c4 ,bearing #baf4eb'4aafe4 ,fig4 #dd4 ,south #b4 ,west #a4 ,bearing #bjf4cc'4edah4 ,fig4 #de4 ,south #if4 ,west #ee4 ,bearing #bji4dh'4cbha4 ,fig4 #df4 ,south #c4 ,west #a4 ,bearing #aih4bf'4ehb4 ,fig4 #dg4 ,south #e4 ,west #b4 ,bearing #bja4dh'4e4 ,fig4 #dh4 ,south #g4 ,west #c4 ,bearing #bjc4aa'4ee4 ,figure #de represents the line south #if1 west #ee1 from ,cephren1 bearing #bji #dh' #cbha2 the apex of ,cephren is immediately above the apex of ,mycerinus4 ,figure #df is the ,s4 #c ,w4 #a line1 bearing #aih #bf' #e4hb2 here the dark slope angle of the pyramids with the sun to the eastward occupies half of the apparent half base4 ,figure #dg is the ,s4 #e1 ,w4 #b line1 bearing #bja #dh' #e2 here ,cephren and ,mycerinus are in outside slope line4 ,figure #dh is the ,s4 #g ,w4 #c line1 bearing #bjc #aa' #ee2 here the inside slope of ,cephren springs from the centre of the apparent base of ,mycerinus4 ,i must content myself with the preceding examples of a few pyramid lines1 but must have said enough to show that from every point of the compass their appearance was distinctly marked and definitely to be determined by surveyors acquainted with the plan4 #aa4 ,,description ,,of ,,the ,,ancient ,,portable ,,survey ,,instrument4 ,i must now commence with a single pyramid1 show how approximate observations could be made from it1 and then extend the theory to a group with the observations thereby rendered more perfect and delicate4 ,we will suppose the surveyor to be standing looking at the pyramid ,cephren2 he knows that its base is #dbj cubits1 and its apothem #cdf cubits4 ,he has provided himself with a model in wood1 or stone1 or metal1 and one thousandth of its size--therefore his model will be ,o4#db cubit base1 and ,o4#cdfe cubit apothem--or1 in round numbers1 eight and half inches base1 and seven inches apothem4 ,this model is fixed on the centre of a card or disc1 graduated from the centre to the circumference1 like a compass card1 to the various points of the compass1 or divisions of a circle4 ,the model pyramid is fastened due north and south on the lines of this card or disc1 so that when the north point of the card points north1 the north face of the model pyramid faces to the north4 ,the surveyor also has a table1 which1 with a pair of plumb lines or mason's levels1 he can erect quite level3 this table is also graduated from the centre with divisions of a circle1 or points of the compass1 and it is larger than the card or disc attached to the model4 ,this table is made so that it can revolve upon its stand1 and can be clamped4 ,we will call it the lower limb4 ,there is a pin in the centre of the lower limb1 and a hole in the centre of the disc bearing the model1 which can be thus placed upon the centre of the table1 and becomes the upper limb4 ,the upper limb can be clamped to the lower limb4 ,the first process will be to clamp both upper and lower limbs together1 with the north and south lines of both in unison1 then revolve both limbs on the stand till the north and south line points straight for the pyramid in the distance1 which is done by the aid of sights erected at the north and south points of the perimeter of the lower limb4 ,when this is adjusted1 clamp the lower limb and release the upper limb2 now revolve the upper limb until the model pyramid exactly covers the pyramid in the distance1 and shows just the same shade on one side and light on the other1 when viewed from the sights of the clamped lower limb--and the lines1 angles1 and shades of the model coincide with the lines1 angles1 and shades of the pyramid observed2--now clamp the upper limb4 ,now does the model stand really due north and south1 the same as the pyramid in the distance2 it throws the same shades1 and exhibits the same angles when seen from the same point of view2 just as much of it is in shade and as much of it is in light as the pyramid under observation2 therefore it must be standing due north and south1 because ,cephren himself is standing due north and south1 and the upper limb reads off on the lower limb the angle or bearing observed4 ,so far we possess an instrument equal to the modern circumferenter1 and yet we have only brought one pyramid into work4 ,if ,i have shown that such an operation as the above is practically feasible1 if ,i have shown that angles can be taken with moderate accuracy by observing one pyramid of #dbj cubits base1 how much more accurate will the observation be when the surveyor's plane table bears a group of pyramids which occupy a representative space of about #adjj cubits when viewed from the south or north1 and about #agfj cubits when viewed from the east or west4 ,if situated a mile or two south of the ,gzeh group our surveyor could also tie in and perfect his work by sights to the ,skkarah group with ,skkarah models2 and so on1 up the ,nile ,valley1 he would find every few miles groups of pyramids by aid of which he would be enabled to tie his work together4 ,if the ,gzeh group of pyramids is placed and shaped in the manner ,i have described1 it must be clear that an exact model and plan1 say a thousandth of the size1 could be very easily made--the plan being at the level of the base of ,cephren where the bases of the two main pyramids are even2--and if they are not exactly so placed and shaped1 it may be admitted that their position and dimensions were known to the surveyors or priests1 so that such models could be constructed4 ,it is probable1 therefore1 that the instrument used in conjunction with these pyramids1 was a machine constructed in a similar manner to the simple machine ,i have described1 only instead of there being but one model pyramid on the disc or upper limb1 it bore the whole group2 and the smaller pyramids were what we may call vernier points in this great circle1 enabling the surveyor to mark off known angles with great accuracy by noticing how1 as he worked round the group of pyramids1 one or other of the smaller ones was covered by its neighbours4#i ,footnote #i3 ,see general plan of ,gzeh ,group op4 page #a4 ,the immensity of the main pyramids would require the smaller ones to be used for surveys in the immediate neighbourhood1 as the surveyor might easily be too close to get accurate observations from the main pyramids4 ,the upper limb1 then1 was a disc or circular plate bearing the model of the group4 ,cheops would be situated in the centre of the circle1 and observations would be taken by bringing the whole model group into even line and even light and shade with the ,gzeh group4 ,i believe that with a reasonable-sized model occupying a circle of six or seven feet diameter1 such as a couple of men could carry1 very accurate bearings could have been taken1 and probably were taken4 ,the pyramid shape is the very shape of all others to employ for such purposes4 ,a cone would be useless1 because the lights and shades would be softened off and its angles from all points would be the same4 ,other solids with perpendicular angles would be useless1 because although they would vary in width from different points of view they would not present that ever changing angle that a pyramid does when viewed from different directions4 ,after familiarity with the models which ,i have made use of in prosecuting these investigations1 ,i find that ,i can judge with great accuracy from their appearance only the bearing of the group from any point at which ,i stand4 ,i make bold to say that the pocket compass of the ,egyptian surveyor was a little model of the group of pyramids in his district1 and he had only to hold it up on his hand and turn it round in the sun till its shades and angles corresponded with the appearance of the group1 to tell as well as we could tell by our compasses1 perhaps better1 his bearing from the landmarks that governed his surveys4 ,the ,great ,circle of ,gold described by ,diodorus 7,diod4 ,sic4 lib4 ,x41 part #b1 cap4 #a7 as having been employed by the ,egyptians1 and on which was marked amongst other things1 the position of the rising and setting of the stars1 and stated by him to have been carried off by ,cambysses when ,egypt was conquered by the ,persians1 is supposed by ,cassini to have been also employed for finding the meridian by observation of the rising and setting of the sun4 ,this instrument and others described by writers on ,egypt would have been in practice very similar to the instrument which ,i have described as having been probably employed for terrestrial observations4 ,the table or disc comprising the lower limb of the instrument1 might have been supported upon a small stand with a circular hole in the centre1 so arranged that the instrument could be either set up alone and supported by its own tripod1 or rested fairly on the top of any of those curious stone boundary marks which were made use of1 not only to mark the corners of the different holdings1 but to show the level of the ,nile inundations4 7,see ,figure #di1 copied from ,sharpe's ,egypt1 vol4 ,i41 p4 #f47 ,the peculiar shape of the top of these stone landmarks1 or sacred boundary stones1 appears suitable for such purposes1 and it would have been a great convenience to the surveyor1 and conducive to accuracy1 that it should be so arranged that the instrument should be fixed immediately over the mark1 as appears probable from the shape of the stone4 ,fig #di4 ,sacred boundary stone4 ,a noticeable point in this theory is1 that it is not in the least essential that the apex of a pyramid should be complete4 ,if their summits were left permanently flat1 they would work in for survey purposes quite as well1 and ,i think better1 than if carried to a point1 and they would be more useful with a flat top for defined shadows when used as sun dials4 ,in the ,gzeh group1 the summit of ,cheops appears to me to have been left incomplete the better to get the range with ,cephren for lines down the delta4 ,in this system of surveying1 there is always a beautiful connection between the horizontal bearings and the apparent or observed angles presented by the slopes and edges of the pyramid4 ,thus1 in pyramids like those of ,gzeh1 which stand north and south1 and whose meridional sections contain less1 and whose diagonal sections contain more than a right angle1 the vertex being the point at which the angle is measured--this law holds3-- ,that the smallest interior angle at the vertex1 contained between the inside edge and the outside edge1 will exhibit the same angle as the bearing of the observer's eye from the apex of the pyramid when the angle at the apex contained by the outside edges appears to be a right angle4 ,figures #ej to #ee inclusive illustrate this beautiful law from which it will be seen that the ,gzeh surveyors possessed1 in this manner alone1 eight distinctly defined bearings from each pyramid4 ,fig4 #ej4 ,cheops from points bearing ,s #ai4ab4bb ,w ,w #ai4ab4bb ,n ,n #ai4ab4bb ,e ,e #ai4ab4bb ,s ,fig4 #ea4 ,cheops from points bearing ,s #ai4ab4bb ,e ,w #ai4ab4bb ,s ,n #ai4ab4bb ,w ,e #ai4ab4bb ,n ,fig4 #eb4 ,cephren from points bearing ,s #bc4g4ejbd ,w ,w #bc4g4ejbd ,n ,n #bc4g4ejbd ,e ,e #bc4g4ejbd ,n ,fig4 #ec4 ,cephren from points bearing ,s #bc4g4ejbd ,e ,w #bc4g4ejbd ,s ,n #bc4g4ejbd ,w ,e #bc4g4ejbd ,n ,fig4 #ed4 ,mycerinus from points bearing ,s #ag4a4djd ,w ,w #ag4a4djd ,n ,n #ag4a4djd ,e ,e #ag4a4djd ,s ,fig4 #ee4 ,mycerinus from points bearing - ,s #ag4a4djd ,e ,w #ag4a4djd ,s ,n #ag4a4djd ,w ,e #ag4a4djd ,n 99 99 99 99 99 ,fig4 #ef4 ,cheops model4 ,fig4 #eg4 ,cephren model ,fig4 #eh4 ,mycerinus model 99 99 99 99 99 ,i recommend any one desirous to thoroughly comprehend these matters1 to make a plan from my diagram1 ,figure #e1 using ,r4,b4 cubits for measures1 and to a suitable scale1 on a piece of card-board4 ,then to cut out of the card-board the squares of the bases of the pyramids at the level of ,cephren1 viz41 #dbj1 #dbj and #bah cubits respectively1 for the three main pyramids4 ,one hundred cubits to the inch is a convenient scale and within the limits of a sheet of ,bath board4 ,by striking out the models on card-board in the manner shown by diagrams 7see ,figures #ef1 #eg1 and #eh7 they can be cut out with a penknife--cutting only half through where the lines are dotted--bent up together1 and pasted along the edges with strips of writing paper about half an inch wide4 ,these models can be dropped into the squares cut out of the card-board plan1 thus correcting the error caused by the thickness of the card-board base1 and if placed in the sun1 or at night by the light of one lamp or candle properly placed to represent the sun in the eastward or westward1 the clear cut lines and clear contrasting shades will be manifest1 and the lines illustrated by my figures can be identified4 ,when inspecting the model1 it is well to bear in mind that the eye must be kept very nearly level with the table1 or the pyramids will appear as if viewed from a balloon4 99 99 99 99 99 ,i believe that the stones were got up to the building by way of the north side of each pyramid4 ,the casing on the south1 east1 and west1 was probably built up as the work proceeded1 and the whole of these three faces were probably thus finished and completed while there was not a single casing stone set on the north side4 ,then the work would be closed up until there remained nothing but a great gap or notch1 wide at the bottom1 and narrowing to the apex4 ,the work on the north side would then be closed from the sides and top1 and the bottom casing stone about the centre of the north side1 would be the last stone set on the building4 ,these old builders were too expert not to have thus made use of all the shade which their own building would thus afford to a majority of the workmen4 99 99 99 99 99 ,many of the obelisks were probably marks on pyramid lines of survey4 ,the pyramid indeed may have been a development of the obelisk for this purpose4 ,their slanting sides might correspond with some of the nearly upright slant angles of the pyramids1 in positions opposite certain lines4 ,reference to several of my figures will show how well this would come in4 ,herodotus speaks of two obelisks at ,heliopolis1 and ,bonwick tells us that ,abd al ,latif saw two there which he called ,pharaoh's ,needles4 ,an ,arab traveller1 in #aaij1 saw a pyramid of copper on the summit of the one that remained1 but it is now wanting4 ,pharaoh's ,needles appear to have been situated about #bj miles ,,ne4 of the ,gzeh group1 and their slope angles might have coincided with the apparent slope angles of ,cephren or ,cheops on the edge nearest the obelisk4 ,the ancient method of describing the meridian by means of the shadow of a ball placed on the summit of an obelisk points to a reasonable interpretation for the peculiar construction of the two pillars1 ,jachin and ,boaz1 which are said to have been situated in front of the ,hebrew ,temple at ,jerusalem1 and about which so much mysterious speculation has occurred4 ,they were no doubt used as sun-dials for the morning and afternoon sun by the shadow of the balls or chapiters thrown upon the pavement4 ,without presuming to dispute the objects assigned by others for the galleries and passages which have been discovered in the pyramid ,cheops1 ,i venture to opine that they were employed to carry water to the builders4 ,they are connected with a well1 and the well with the ,nile or canal4 ,whether the water was slided up the smooth galleries in boxes1 or whether the cochlea1 or water screw1 was worked in them1 their angles being suitable1 it is impossible to conjecture2 either plan would have been convenient and feasible4 ,these singular chambers and passages may indeed possibly have had to do with some hydraulic machinery of great power which modern science knows nothing about4 ,the section of the pyramid1 showing these galleries1 in the pyramid books1 has a most hydraulic appearance4 ,the tremendous strength and regularity of the cavities called the ,king's and ,queen's chambers1 the regularity and the smallness of most of the passages or massive stone connecting pipes1 favor the idea that the chambers might have been reservoirs1 their curious roofs1 air chambers1 and the galleries or passages1 connecting pipes for working water under pressure4 ,water raised through the passages of this one pyramid nearest to the canal1 might have been carried by troughs to the other pyramids1 which were in all probability in course of construction at the same period of time4 ,a profane friend of mine thinks that the sarcophagus or sacred coffer in the ,king's chamber may have been used by the chief architect and leading men of the works as a bath1 and that the ,king's chamber was nothing more or less than a delightful bath room4 99 99 99 99 99 ,the following quotation from the writing of an ,arabian author 7,ibn ,abd ,alkokm71 is extracted from ,bonwick's ,pyramid ,facts and ,fancies1 page #gb3--,the ,coptites mention in their books that upon them 7the ,pyramids7 is an inscription engraven2 the exposition of it in ,arabicke is this3--',i1 ,saurid the ,king built the ,pyramids 7in such and such a time71 and finished them in six years2 he that comes after me1 and says he is equal to me1 let him destroy them in six hundred years2 and yet it is known that it is easier to pluck down than to build2 and when ,i had finished them1 ,i covered them with sattin1 and let him cover them with slats4' ,the italics are my own4 ,the builder seems to have entertained the idea that his work would be partially destroyed1 and afterwards temporarily repaired or rebuilt4 ,the first part has unfortunately come true1 and it is possible that the last part of the idea of ,king ,saurid may be carried out1 because it would not be so very expensive an undertaking for any civilized nation in the interest of science to re-case the pyramids of ,gzeh1 so that they might be once more applied to land-surveying purposes in the ancient manner4 ,it would not be absolutely necessary to case the whole of the pyramid faces1 so long as sufficient casing was put on to define the angles4 ,the slats used might be a light wooden framework covered with thin metal4 ,the metal should be painted white1 except in the case of ,mycerinus1 which should be of a reddish color4 #ab4 ,,primary ,,triangles ,,and ,,their ,,satellites2--,,or ,,the ,,ancient ,,system ,,of ,,right-,,angled ,,trigonometry ,,unfolded ,,by ,a ,,study ,,of ,,the ,,plan ,,of ,,the ,,pyramids ,,of ,,gizeh4 ,main ,triangular ,dimensions of ,plan are ,represented by the ,following ,eight ,right-angled ,triangles4 ,,table ,,to ,,explain ,,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--,in the above table the first ,,xf #e #afj #abhj column is the ,ratio1 the second -------------------------- the connected ,natural ,numbers1 and ,,ay #c #cf #bhh the third column represents the ,,yz #d #ab #dh #h #chd length each line in ,r4,b4 cubits4 ,,za #e #fj #dhj ---------------------------------------- ----------------------- ,fig4 #fj4 ,reference to ,fig4 #fj and the preceding table1 will show that the main triangular dimensions of this plan 7imperfect as it is from the lack of eleven pyramids7 are represented by four main 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 illustrate the two former1 and ,figures #fa and #fb illustrate the two latter4 ,i will call triangles of this class primary triangles1 as the most suitable term1 although it is applied to the main triangles of geodetic surveys4 ,we have only to select a number of such triangles and a system of trigonometry ensues1 in which base1 perpendicular1 and hypotenuse of every triangle is a whole measure without fractions1 and in which the nomenclature for every angle is clear and simple4 ,an angle of #dc #cf #ajae will be called a #bj1 #ba angle1 and an angle of #cf #eb #aafe will be called a #c1 #d angle1 and so on4 ,in the existing system whole angles1 such as #dj1 #de1 or #ej degrees1 are surrounded by lines1 most of which can only be described in numbers by interminable fractions4 ,in the ancient system1 lines are only dealt with1 and every angle in the table is surrounded by lines measuring whole units1 and described by the use of a couple of simple numbers4 ,connecting this with our present system of trigonometry would effect a saving in calculation1 and general use of certain peculiar angles by means of which all the simplicity and beauty of the work of the ancients would be combined with the excellences of our modern instrumental appliances4 ,surveyors should appreciate the advantages to be derived from laying out traverses on the hypotenuses of primary triangles1 by the saving of calculation and facility of plotting to be obtained from the practice4 ,the key to these old tables is the fact1 that in primary triangles the right-angled triangle formed by the sine and versed sine1 also by the co-sine and co-versed-sine1 is one in which base and perpendicular are measured by numbers without fractions4 ,these ,i will call satellite triangles4 ,thus1 to the primary triangle #bj1 #ba1 #bi1 the ratios of the co-sinal and sinal satellites are respectively #g to #c1 and #b to #e4 7,see ,figure #ce47 ,to the #dh1 #ee1 #gc triangle the satellites are #aa1 #e and #h1 #c 7,fig4 #fb72 to the #c1 #d1 #e triangle they are #b1 #a and #c1 #a 7,fig4 #cj72 and to the #bh1 #de1 #ec triangle1 they are #i1 #e and #g1 #b 7,fig4 #fa74 ,the primary triangle1 #g1 #bd1 #be1 possesses as satellites the primary triangle1 #c1 #d1 #e1 and the ordinary triangle1 #d1 #a2 and the primary triangle #da1 #hdj1 #hda1 is attended by the #bj1 #ba1 #bi triangle1 as a satellite with the ordinary triangle #da1 #a1 and so on4 ,fig4 #fa4 ,the #bh-de-ec ,triangle4 ,fig4 #fb4 ,the #dh-ee-gc ,triangle4 ,since any ratio1 however1 whose terms1 one or both1 are represented by fractions1 can be transformed into whole numbers1 it evidently follows that every conceivable relative measure of two lines which we may decide to call co-sine and co-versed-sine1 becomes a satellite to a corresponding primary triangle4 ,now1 since the angle of the satellite on the circumference must be half the angle of the adjacent primary triangle at the centre1 it follows that in constructing a list of satellites and their angles1 the angles of the corresponding primary triangles can be found4 ,for instance-- ,satellite #h1 #c1 contains #bj #cc #bagf ,satellite #b1 #g1 contains #ae #ef #dcdbe ,each of these angles doubled1 gives the angle of a primary triangle as follows1 viz43-- ,the #dh1 #ee1 #gc triangle #da #f #dceb ,the #bh1 #de1 #ec triangle #ca #ec #bfhe ,the angles of the satellites together must always be #de1 because the angle at the circumference of a quadrant must always be #ace4 ,from the ,gzeh plan1 as far as ,i have developed it1 the following order of satellites begins to appear1 which may be a guide to the complete ,gzeh plan ratio1 and to those primary triangles in use by the pyramid surveyors in their ordinary work4 ---------------------------------------- ------------ #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 ---------------------------------------- ------------ ,primary triangles may be found from the angle of the satellite1 but it is an exceedingly round-about way4 ,i will1 however1 give an example4 ,let us construct a primary triangle from the satellite #d1 #i4 ,rad4 #d -------- #ddddddd ,tangt4 #bc #eg #dejda #i #bc #eg #dejda #b #dg #ee #cjjhc4 therefore the angles of the primary are #dg #ee #cjjhc4 and #db #d #biiag4 ,the natural sine of #db #d #biiag #fgjajbe4 ,the natural co-sine #db #d #biiag #gdbbfhd4 ,the greatest common measure of these numbers is about #ajbgag1 therefore-- ,radius #ajjjjjjj #ajbgag #ig ,co-sine #gdbbfhd #ajbgag #gb ,sine #fgjajbe #ajbgag #fe and #fe1 #gb1 #ig is the primary triangle to which the satellites are #d1 #i1 and #e1 #ac4 7,see ,fig4 #fc47 ,the figures in the calculation do not balance exactly1 in consequence of the insufficient delicacy of the tables or calculations4 ,fig4 #fc4 ,the connection between primaries and satellites is shown by figure #fd4 ,fig4 #fd4 ,let the triangle ,,adb be a satellite1 #e1 #b1 which we will call ,,bd #bj1 and ,,ad #h4 ,let ,c be centre of semi-circle ,,abe4 ,,ad 3 ,,db 33 ,,db 3 ,,de #ej 7,euc4 ,,vi4 #h7 ,,ad ,,de ,,ae #eh diameter ,,ae #b ,,ac ,,bc #bi radius ,,ac-,,ad ,,dc #ba co-sine and ,,db #bj sine ,from the preceding it is manifest that-- sine ----- ver-s dia4 ver-s ,the formula to find the primary triangle to any satellite is as follows3-- ,let the long ratio line of the satellite or sine be called a1 and the short ratio line or versed-sine be called b4 ,then-- 7#a7 a sine4 a b 7#b7 ------- radius4 #b;b a - b 7#c7 ------- co-sine4 #b;b ,therefore various primary triangles can be constructed on a side ,,db 7,fig4 #fd7 as sine1 by taking different measures for ,,ad as versed-sine4 ,for example-- #e sine #e #e #a ,from ------- radius #ac ,satellite #b #a #e1 #a4 #e - #a ------- co-s4 #ab #b #a 99 99 99 99 99 #e sine #e #bj #e #b ,from ------- radius #g #d #bi ,satellite #b #b #e1 #b4 #e #b ------- co-s4 #e #ba #b #b 99 99 99 99 99 ,finally arises the following simple rule for the construction of primaries to contain any angle--,decide upon a satellite which shall contain half the angle--say1 #e1 #a4 ,call the first figure a1 the second b1 then-- a b hypotenuse4 a - b perpendicular4 a #b;b base4 ,,primary ,,lowest ,,ratio4 ,thus-- #e #a #bf #ac ,satellite #e1a #e - #a #bd #ab #e #b #a #aj #e --------------- ---------------------------- and-- #e #b #bi #bi ,satellite #e1b #e - #b #ba #ba #e #b #b #bj #bj ,having found the lowest ratio of the three sides of a primary triangle1 the lowest whole numbers for tangent1 secant1 co-secant1 and co-tangent1 if required1 are obtained in the following manner4 ,take for example the #bj1 #ba1 #bi triangle1 now #bj #ba #dbj1 and #bi #dbj #abahj1 a new radius instead of #bi from which with the sine #bj1 and co-sine #ba1 increased in the same ratio1 the whole canon of the #bj1 #ba1 #bi triangle will come out in whole numbers4 ,similarly in the triangle #dh1 #ee1 #gc1 radius #gc #acbjj 7the product of #dh #ee7 makes radius in whole numbers #ifcfjj1 for an even canon without fractions4 ,this is because sine and co-sine are the two denominators in the fractional parts of the other lines when worked out at the lowest ratio of sine1 co-sine1 and radius4 ,after ,i found that the plan of the ,gzeh group was a system of primary triangles1 ,i had to work out the rule for constructing them1 for ,i had never met with it in any book1 but ,i came across it afterwards in the ,penny ,encyclopedia1 and in ,rankine's ,civil ,engineering4 ,the practical utility of these triangles1 however1 does not appear to have received sufficient consideration4 ,i certainly never met with any except the #c1 #d1 #e1 in the practice of any surveyor of my acquaintance4 7,for squaring off a line nothing could be more convenient than the #bj1 #ba1 #bi triangle2 for instance1 taking a base of #dj links1 then using the whole chain for the two remaining sides of #db and #eh links47 ,table of ,some ,primary ,triangles and their ,satellites4 ,,angle ,,of ,,primary ,,primary ,,satellite4 ,,angle ,,of ,,satellite ,,deg4 ,,min4 ,,sec4 ,,rad4 ,,co4-,s4 ,,sine4 ,,deg4 ,,min4 ,,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 ,reference to the plan ratio table at the commencement1 and to the tables here introduced1 will shew that most of the primary triangles mentioned are indicated on the plan ratio table principally by the lines corresponding to the ratios of the satellites4 ,thus-- ,,primary ,,triangle ,,indicated ,,by #ag1 #add1 #ade4 ,triangle ,,fp1 ,,pa1 ,,af on plan4 #ac1 #hd1 #he4 ,plan ratio of ,,sj to ,,su1 #g to #f4 #aa1 #fj1 #fa4 ,plan ratio ,,bc to ,,fb1 #f to #e1 and ,,dn to ,,nr1 #fa to #fj4 #ab1 #ce1 #cg4 ,plan ratio ,,eo to ,,ay1 #cg to #ab1 and ,,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 and ,,nz to ,,za1 #ab to #e4 #h1 #ae1 #ag4 ,plan ratio ,,fb to ,,by1 #e to #c1 and ,,ac to ,,bc1 #ae to #h4 #cc1 #ef1 #ee4 ,plan ratio ,,yx to ,,ay1 #g to #d2 ,,ab to ,,bo1 #g to #d2 and ,,ea to ,,az1 #g to #d4 #bh1 #de1 #ec4 ,exists on plan1 ,,ab1 ,,bj1 ,,ja4 #c1 #d1 #e4 ,pervades the plan1 and is also indicated by plan ratio ,,gx to ,,dg1 #b to #a2 ,,su to ,,sv1 #b to #a2 and ,,cy to ,,yz1 #c to #a4 #dh1 #ee1 #gc4 ,exists on plan1 ,,fw1 ,,wv1 ,,vf--and is also indicated by plan ratio ,,fo to ,,oz1 #h to #c4 #fe1 #gb1 #ig4 ,plan ratio ,,ac to ,,ch1 #i to #d2 ,,my to ,,yz1 #i to #d4 #bj1 #ba1 #bi4 ,exists on plan ,,fb1 ,,ba1 ,,af2 and plan ratio1 ,,gu to ,,dg1 #e to #b4 ,it seems probable that could ,i add to my pyramid plan the lines and triangles that the missing eleven pyramids would supply1 it would comprise a complete table on which would appear indications of all the ratios and triangles made use of in right-angled trigonometry1 a ratiometer in fact4 ,i firmly believe that so far as ,i have gone it is correct--and it is possible1 therefore1 with the start that ,i have made1 for others to continue the work1 and add the eleven pyramids to the plan in their correct geometrical position4 ,by continuing the system of evolution by which ,i defined the position of ,cephren1 and the little pyramid to the south-east of ,cheops1 after ,i had obtained ,cheops and ,mycerinus1 may be rebuilt1 at one and the same time1 a skeleton of the trigonometrical tables of a forgotten civilization1 and the plan of those pyramids which are its only link with the present age4 #ac4 ,,the ,,size ,,and ,,shape ,,of ,,the ,,pyramids ,,indicated ,,by ,,the ,,plan4 ,i pursued my investigations into the slopes and altitudes of the pyramids without reference to the plan1 after once deciding their exact bases4 ,now it will be interesting to note some of the ways in which the plan hints at the shape and size of these pyramids1 and corroborates my work4 ,the dimensions of ,cheops are indicated on the plan by the lines ,,ea to ,,ya1 measuring #hdj and #bhh ,r4,b4 cubits respectively1 being the half periphery of its horizontal section at the level of ,cephren's base1 and its own altitude from its own base4 7,see ,fig4 #e47 ,the line ,,ea1 in fact1 represents in ,r4,b4 cubits the half periphery of the bases of either ,cheops or ,cephren measured at the level which ,i have set forth as the plan level1 viz41 base of ,cephren4 ,the ratio of ,cephren's base to ,cephren's altitude is indicated on the plan by the ratios of the lines ,,bc to ,,eb1 or ,,fo to ,,or1 viz41 #cb to #ba4 7,see ,fig4 #d47 ,the altitude of ,mycerinus above ,cephren's base appears on plan in the line ,,ef1 measuring #acf ,r4,b4 cubits4 ,the line ,,eo on plan measures #hhh cubits1 which would be the length of a line stretched from the apex of ,cheops to the point ,e1 at the level of ,cheops' base4 ,this merits consideration3--the lines ,,ea and ,,ay are connected on plan at the centre of ,cheops1 and the lines ,,eo and ,,ea are connected on plan at the point ,e4 ,now the lines ,,eo1 ,,ea and ,,ay are sides of a primary triangle1 whose ratio is #cg1 #ce1 #ab1 and whose measure in cubits is #hhh1 #hdj1 and #bhh2 and if we suppose the line ,,ea to be stretched horizontally beneath the pyramids at the level of the base of ,cheops from ,e to ,a on plan1 and the line ,,ay to be a plumb line hanging from the apex of ,cheops to the level of his base1 then will the line ,,eo just stretch from the point ,e to the apex of ,cheops1 and the three lines will connect the two main pyramids by a vertical triangle of which ,,ea1 ,,ay and ,,eo form the base1 perpendicular1 and hypotenuse4 ,or1 to explain it in another manner3 let the line ,,ea be a cord stretching horizontally from ,a at the centre of the base of ,cheops to the point ,e1 both ends being at the same level2 let the line ,,ay be a rod1 lift it on the end ,a till it stands erect1 then is the end ,y the apex of ,cheops4 ,now1 the line ,,eo would just stretch from the top of the rod ,,ay to the point ,e first described4 ,it is a singular coincidence1 and one that may be interesting to students of the interior of the ,pyramids1 that the side ,,ep1 of the small #c1 #d1 #e triangle1 ,,ep1 ,,pf1 ,,fe1 in the centre of the plan1 measures #hafj ,r4,b4 cubits1 which is very nearly eight times the true breadth of the ,king's chamber in ,cheops1 according to ,piazzi ,smyth2 for #hafj_/h #ajbj ,r4,b4 cubits1 or #bjfjdf pyramid inches 7one ,r4,b4 cubit being #bjbjjf pyramid inches74 ,the sides of this little triangle measure #hafj1 #ajhhj1 and #acf1 ,r4,b4 cubits respectively1 as can be easily proved from the plan ratio table4 #ad4 ,a ,,simple ,,instrument ,,for ,,laying ,,off ,,primary ,,triangles4 ,a simple instrument for laying off primary triangles upon the ground1 might have been made with three rods divided into a number of small equal divisions1 with holes through each division1 which rods could be pinned together triangularly1 the rods working as arms on a flat table1 and the pins acting as pointers or sights4 ,one of the pins would be permanently fixed in the table through the first hole of two of the rods or arms1 and the two other pins would be movable so as to fix the arms into the shape of the various primary triangles4 ,thus with the two main arms pinned to the cross arm in the #bast and #bith hole from the permanently pinned end1 with the cross arm stretched to twenty divisions1 a #bj1 #ba1 #bi triangle would be the result1 and so on4 #ad;a4 ,,general ,,observations4 ,i must be excused by geometricians for going so much in detail into the simple truths connected with right-angled trigonometry4 ,my object has been to make it very clear to that portion of the public not versed in geometry1 that the ,pyramids of ,egypt must have been used for land surveying by right-angled triangles with sides having whole numbers4 ,a re-examination of these pyramids on the ground with the ideas suggested by the preceding pages in view1 may lead to interesting discoveries4 ,for instance1 it is just possible that the very accurately and beautifully worked stones in the walls of the ,king's chamber of ,cheops1 may be found to indicate the ratios of the rectangles formed by the bases and perpendiculars of the triangulations used by the old surveyors--that on these walls may be found1 in fact1 corroboration of the theory that ,i have set forth4 ,i am led to believe also from the fact that ,gzeh was a central and commanding locality1 and that it was the custom of those who preceded those ,egyptians that history tells of1 to excavate mighty caverns in the earth--that1 therefore1 in the limestone upon which the pyramids are built1 and underneath the pyramids1 may be found vast excavations1 chambers and galleries1 that had entrance on the face of the ridge at the level of ,high ,nile4 ,from this subterraneous city1 occupied by the priests and the surveyors of ,memphis1 access may be found to every pyramid2 and while to the outside world the pyramids might have appeared sealed up as mausoleums to the ,kings that it may have seen publicly interred therein1 this very sealing and closing of the outer galleries may have only rendered their mysterious recesses more private to the priests who entered from below1 and who were1 perhaps1 enabled to ascend by private passages to their very summits4 ,the recent discovery of a number of regal mummies stowed away in an out of the way cave on the banks of the ,nile1 points to the unceremonious manner in which the real rulers of ,kings and people may have dealt with their sovereigns1 the pomp and circumstance of a public burial once over4 ,it is just possible that the chambers in the pyramids may have been used in connection with their mysteries3 and the small passages called by some ventilators or air passages1 sealed as they were from the chamber by a thin stone 7and therefore no ventilators7 may have been auditory passages along which sound might have been projected from other chambers not yet opened by the moderns2 sounds which were perhaps a part of the hanky panky of the ancient ceremonial connected with the mysteries or the religion of that period4 ,down that well which exists in the interior of ,cheops1 and in the limestone foundations of the pyramid1 should ,i be disposed to look for openings into the vast subterraneous chambers which ,i am convinced do exist below the ,pyramids of ,gzeh4 ,the priests of the ,pyramids of ,lake ,moeris had their vast subterranean residences4 ,it appears to me more than probable that those of ,gzeh were similarly provided4 ,and ,i go further3--,out of these very caverns may have been excavated the limestone of which the pyramids were built1 thus killing two birds with one stone--building the instruments and finding cool quarters below for those who were to make use of them4 ,in the bowels of that limestone ridge on which the pyramids are built will yet be found1 ,i feel convinced1 ample information as to their uses4 ,a good diamond drill with two or three hundred feet of rods is what is what is wanted to test this1 and the solidity of the pyramids at the same time4 #ae4 ,,primary ,,triangulation4 ,primary triangulation would be useful to men of almost every trade and profession in which tools or instruments are used4 ,any one might in a short time construct a table for himself answering to every degree or so in the circumference of a circle for which only forty or fifty triangles are required4 ,it would be worth while for some one to print and publish a correct set of these tables embracing a close division of the circle1 in which set there should be a column showing the angle in degrees1 minutes1 seconds and decimals1 and also a column for the satellite1 thus-- ,,satellite4 ,,primary4 ,,angle4 #e #b #bj #ba #bi #dc #cf #ajae #g #c #ba #bj #bi #df #bc #dihe and so on4 ,such a set of tables would be a boon to sailors1 architects1 surveyors1 engineers1 and all handi-craftsmen3 and ,i make bold to say1 would assist in the intricate investigations of the astronomer3--and the rule for building the tables is so simple1 that they could easily be achieved4 ,the architect from these tables might arrange the shape of his chambers1 passages or galleries1 so that all measures1 not only at right angles on the walls1 but from any corner of floor to ceiling should be even feet4 ,the pitch of his roofs might be more varied1 and the monotony of the buildings relieved1 with rafters and tie-beams always in even measures4 ,the one solitary #c1 #d1 #e of ,vitruvius would cease to be his standard for a staircase2 and even in doors and sashes1 and panels of glass1 would he be alive to the perfection of rectitude gained by evenly-measured diagonals4 ,by a slight modification of the compass card1 the navigator of blue water might steer his courses on the hypotenuses of great primary triangles--such tables would be useful to all sailors and surveyors who have to deal with latitude and departure4 ,for instance1 familiarity with such tables would make ever present in the mind of the surveyor or sailor his proportionate northing and easting1 no matter what course he was steering between north and east1 the primary embraces the three ideas in one view4 ,in designing trussed roofs or bridges1 the primaries would be invaluable to the engineer1 strain-calculations on diagonal and upright members would be simplified1 and the builder would find the benefit of a measure in even feet or inches from centre of one pin or connection to another4 ,for earthwork slopes #c1 #d1 #e2 #bj1 #ba1 #bi2 #ba1 #bj1 #bi2 and #d1 #c1 #e would be found more convenient ratios than #a to #a1 and #a to #a1 etc4 ,templates and battering rules would be more perfect and correct1 and the engineer could prove his slopes and measure his work at one and the same time without the aid of a staff or level2 the slope measures would reveal the depth1 and the slope measures and bottom width would be all the measures required1 while the top width would prove the correctness of the slopes and the measurements4 ,to the land surveyor1 however1 the primary triangle would be the most useful1 and more especially to those laying out new holdings1 whether small or large1 in new countries4 ,whether it be for a squatter's run1 or for a town allotment1 the advantages of a diagonal measure to every parallelogram in even miles1 chains1 or feet1 should be keenly felt and appreciated4 ,this was1 ,i believe1 one of the secrets of the speedy and correct replacement of boundary marks by the ,egyptian land surveyors4 ,i have heard of a review in the ,contemporary1 ,september1 #ahha1 referring to the translation of a papyrus in the ,british ,museum1 by ,dr4 ,eisenlohr--,a handbook of practical arithmetic and geometry1 etc41 such as we might suppose would be used by a scribe acting as clerk of the works1 or by an architect to shew the working out of the problems he had to solve in his operations4 ,i should like to see a translation of the book1 from which it appears that the clumsiness of the ,egyptian method is very remarkable4 ,perhaps this ,egyptian ,handbook may yet shew that their operations were not so clumsy1 as they appear at first sight to those accustomed to the practice of modern trigonometry4 ,i may not have got the exact hang of the ,egyptian method of land surveying--for ,i do not suppose that even their clumsy method is to be got at intuitively2 but ,i claim that ,i have shewn how the ,pyramids could be used for that purpose1 and that the subsidiary instrument described by me was practicable4 ,i claim1 therefore1 that the theory ,i have set up1 that the pyramids were the theodolites of the ,egyptians1 is sound4 ,that the ground plan of these pyramids discloses a beautiful system of primary triangles and satellites ,i think ,i have shown beyond the shadow of a doubt2 and that this system of geometric triangulation or right-angled trigonometry was the method practised1 seems in the preceding pages to be fairly established4 ,i claim1 therefore1 that ,i have discovered and described the main secret of the pyramids1 that ,i have found for them at last a practical use1 and that it is no longer a marvel how after the annual inundation1 each property could have been accurately described by the aid of geometry4 ,i have advanced nothing in the shape of a theory that will not stand a practical test2 but to do it1 the pyramids should be re-cased4 ,iron sheeting1 on iron or wooden framework1 would answer4 ,i may be wrong in some of my conclusions1 but in the main ,i am satisfied that ,i am right4 ,it must be admitted that ,i have worked under difficulties2 a glimpse at the pyramids three and twenty years ago1 and the meagre library of a nomad in the ,australian wilderness having been all my advantages1 and time at my disposal only that snatched from the rare intervals of leisure afforded by a arduous professional life4 ,after fruitless waiting for a chance of visiting ,egypt and ,europe1 to sift the matter to the bottom1 ,i have at last resolved to give my ideas to the world as they stand2 crude necessarily1 so ,i must be excused if in some details ,i may be found erroneous2 there is truth ,i know in the general conclusions4 ,i am presumptuous enough to believe that the ,r4,b4 cubit of #afhe ,british feet was the measure of the pyramids of ,gzeh1 although there may have been an astronomical #be inch cubit also4 ,it appears to me that no cubit measure to be depended on is either to be got from a stray measuring stick found in the joints of a ruined building1 or from any line or dimensions of one of the pyramids4 ,i submit that a most reasonable way to get a cubit measure out of the ,pyramids of ,gzeh1 was to do as ,i did3--take them as a whole1 comprehend and establish the general ground plan1 find it geometric and harmonic1 obtain the ratios of all the lines1 establish a complete set of natural and even numbers to represent the measures of the lines1 and finally bring these numbers to cubits by a common multiplier 7which in this case was the number eight74 ,after the whole proportions had been thus expressed in a cubit evolved from the whole proportions1 ,i established its length in ,british feet by dividing the base of ,cephren1 as known1 by the number of my cubits representing its base4 ,it is pretty sound evidence of the theory being correct that this test1 with #dbj cubits neat for ,cephren1 gave me also a neat measure for ,cheops1 from ,piazzi ,smyth's base1 of #deb cubits1 and that at the same level1 these two pyramids become equal based4 ,i have paid little attention to the inside measurements4 ,i take it we should first obtain our exoteric knowledge before venturing on esotoric research4 ,thus the intricate internal measurements of ,cheops1 made by various enquirers have been little service to me1 while the accurate measures of the base of ,cheops by ,piazzi ,smyth1 and ,john ,james ,wild's letter to ,lord ,brougham1 helped me amazingly1 as from the two ,i established the plan level and even bases of ,cheops and ,cephren at plan level--as ,i have shown in the preceding pages4 ,my theory demanded that both for the building of the pyramids and for the construction of the models or subsidiary instruments of the surveyors1 simple slope ratios should govern each building2 before ,i conclude1 ,i shall show how ,i got at my slope ratios1 by evolving them from the general ground plan4 ,i am firmly convinced that a careful investigation into the ground plans of the various other groups of pyramids will amply confirm my survey theory--the relative positions of the groups should also be established--much additional light will be then thrown on the subject4 ,let me conjure the investigator to view these piles from a distance with his mind's eye1 as the old surveyors viewed them with their bodily eye4 ,approach them too nearly1 and1 like ,henry ,kinglake1 you will be lost in the one idea of solid immensity4 ,common sense tells us they were built to be viewed from a distance4 ,modern surveyors stand near their instruments1 and send their flagmen to a distance2 the ,egyptian surveyor was one of his own flagmen1 and his instruments were towering to the skies on the distant horizon4 ,these mighty tools will last out many a generation of surveyors4 ,the modern astronomer from the top of an observatory points his instruments direct at the stars2 the ,egyptian astronomer from the summit of his particular pyramid directed his observations to the rising and setting of the stars1 or the positions of the heavenly bodies in respect to the far away groups of pyramids scattered around him in the distance2 and by comparing notes1 and with the knowledge of the relative position of the groups1 did these observers map out the sky4 ,solar and lunar shadows of their own pyramids on the flat trenches prepared for the purpose1 enabled the astronomer at each observatory to record the annual and monthly flight of time1 while its hours were marked by the shadows of their obelisks1 capped by copper pyramids or balls1 on the more delicate pavements of the court-yards of their public buildings4 ,we must grasp that their celestial and terrestrial surveys were almost a reverse process to our own1 before we can venture to enquire into its details4 ,it then becomes a much easier tangle to unravel4 ,that a particular pyramid among so many1 should have been chosen as a favoured interpreter of ,divine truths1 seems an unfair conclusion to the other pyramids2--that the other pyramids were rough and imperfect imitations1 appears to my poor capacity a base and impotent conclusion2--7as far as ,i can learn1 ,mycerinus1 in its perfection1 was a marvel of the mason's art27 but that one particular pyramid should have anything to do with the past or the future of the lost ten tribes of ,israel 7whoever that fraction of our present earthly community may be71 seems to me the wildest conclusion of all1 except perhaps the theory that this one pyramid points to the future of the ,british race4 ,yet in one way do ,i admit that the pyramids point to our future4 ,thirty-six centuries ago1 they1 already venerable with antiquity1 looked proudly down on living labouring ,israel1 in helpless slavery1 in the midst of an advanced civilization1 of which the history1 language1 and religion are now forgotten1 or only at best1 slightly understood4 ,thirty-six centuries hence1 they may look down on a civilization equally strange1 in which our history1 language1 and religion1 ,hebrew race1 and ,british race1 may have no place1 no part4 ,if the thoughts of noble poets live1 as they seem to do1 old ,cheops1 that mountain of massive masonry1 may 7like the brook of our ,laureate71 in that dim future1 still be singing1 as he seems to sing now1 this idea1 though not perhaps these words3 ,for men may come1 and men may go1 ,but ,i go on for ever4 ,ars longa1 vita brevis4 ,man's work remains1 when the workman is forgotten2 fair work and square1 can never perish entirely from men's minds1 so long as the world stands4 ,these pyramids were grand and noble works1 and they will not perish till their reputation has been re-established in the world1 when they will live in men's memories to all generations as symbols of the mighty past4 ,to the minds of many now1 as to ,josephus in his day1 they are vast and vain monuments1 records of folly4 ,to me they are as monuments of peace1 civilization and order--relics of a people living under wise and beneficent rulers--evidences of cultivation1 science1 and art4 #af4 ,,the ,,pentangle ,,or ,,five ,,pointed ,,star ,,the ,,geometric ,,symbol ,,of ,,the ,,great ,,pyramid4 ,from time immemorial this symbol has been a blazing pointer to grand and noble truths1 and a solemn emblem of important duties4 ,its geometric significance1 however1 has long been lost sight of4 ,it is said to have constituted the seal or signet of ,king ,solomon 7#ajjj ,b4,c471 and in early times it was in use among the ,jews1 as a symbol of safety4 ,it was the ,pentalpha of ,pythagoras1 and the ,pythagorean emblem of health 7#ecj ,b4,c474 ,it was carried as the banner of ,antiochus1 ,king of ,syria 7surnamed ,soter1 or the ,preserver71 in his wars against the ,gauls 7#bfj ,b4,c474 ,among the ,cabalists1 the star with the sacred name written on each of its points1 and in the centre1 was considered talismanic2 and in ancient times it was employed all over ,asia as a charm against witchcraft4 ,even now1 ,european troops at war with ,arab tribes1 sometimes find1 under the clothing1 on the breasts of their slain enemies1 this ancient emblem1 in the form of a metal talisman1 or charm4 ,the ,european ,gethe puts these words into the mouth of ,mephistopheles3 ,i am hindered egress by a quaint device upon the threshold1--that five-toed damned spell4 ,i shall set forth the geometric significance of this star1 as far as my general subject warrants me1 and show that it is the geometric emblem of extreme and mean ratio1 and the symbol of the ,egyptian ,pyramid ,cheops4 ,a plane geometric star1 or a solid geometric pyramid1 may be likened to the corolla of a flower1 each separate side representing a petal4 ,with its petals opened and exposed to view1 the flower appears in all its glorious beauty2 but when closed1 many of its beauties are hidden4 ,the botanist seeks to view it flat or open in its geometric symmetry1 and also closed1 as a bud1 or in repose3--yet judges and appreciates the one state from the other4 ,in the same manner must we deal with the five pointed star1 and also with the ,pyramid ,cheops4 ,in dealing with so quaint a subject1 ,i may be excused1 in passing1 for the quaint conceit of likening the interior galleries and chambers of this pyramid to the interior whorl of a flower1 stamens and pistil1 mysterious and incomprehensible4 ,figure #fg 7page #aja71 is the five pointed star1 formed by the unlapping of the five slant sides of a pyramid with a pentagonal base4 ,figure #gj 7page #ajf71 is a star formed by the unlapping of the four slant sides of the pyramid ,cheops4 ,the pentagon ,,gfrhq1 7,fig4 #fg7 is the base of the pyramid ,pentalpha and the triangles ,,egf1 ,,bfr1 ,,roh1 ,,hnq and ,,qag1 represent the five sides1 so that supposing the lines ,,gf1 ,,fr1 ,,rh1 ,,hq and ,,qg1 to be hinges connecting these sides with the base1 then by lifting the sides1 and closing them in1 the points ,a1 ,e1 ,b1 ,o1 and ,n1 would meet over the centre ,c4 ,thus do we close the geometric flower ,pentalpha1 and convert it into a pyramid4 ,in the same manner must we lift the four slant sides of the pyramid ,cheops from its star development1 7,fig4 #gj7 and close them in1 the four points meeting over the centre of the base1 forming the solid pyramid4 ,such transitions point to the indissoluble connection between plane and solid geometry4 ,as the geometric emblem of extreme and mean ratio1 the pentangle appears as an assemblage of lines divided the one by the others in extreme and mean ratio4 ,to explain to readers not versed in geometry1 what extreme and mean ratio signifies1 ,i refer to ,figure #fe3-- ,fig4 #fe4 ,let ,,ab be the given line to be divided in extreme and mean ratio1 i4e41 so that the whole line may be to the greater part1 as the greater is to the less part4 ,draw ,,bc perpendicular to ,,ab1 and equal to half ,,ab4 ,join ,,ac2 and with ,,bc as a radius from ,c as a centre1 describe the arc ,,db2 then with centre ,a1 and radius ,,ad1 describe the arc ,,de2 so shall ,,ab be divided in ,e1 in extreme and mean ratio1 or so that ,,ab3 ,,ae33 ,,ae3 ,,eb4 7,note that ,,ae is equal to the side of a decagon inscribed in a circle with radius ,,ab47 ,let it be noted that since the division of a line in mean and extreme ratio is effected by means of the #b1 #a triangle1 ,,abc1 therefore1 as the exponent of this ratio1 another reason presents itself why it should be so important a feature in the ,gzeh pyramids in addition to its connection with the primary triangle #c1 #d1 #e4 ,fig4 #ff4 ,to complete the explanation offered with figure #fe1 ,i must refer to ,fig4 #ff1 where in constructing a pentagon1 the #b1 #a triangle ,,abc1 is again made use of4 ,the line ,,ab is a side of the pentagon4 ,the line ,,bc is a perpendicular to it1 and half its length4 ,the line ,,ac is produced to ,f1 ,,cf being made equal to ,,cb2 then with ,b as a centre1 and radius ,,bf1 the arc at ,e is described2 and with ,a as a centre1 and the same radius1 the arc at ,e is intersected1 their intersection being the centre of the circle circumscribing the pentagon1 and upon which the remaining sides are laid off4 ,we will now refer to figure #fg1 in which the pentangle appears as the symbolic exponent of the division of lines in extreme and mean ratio4 ,thus3 ,,mc 3 ,,mh 33 ,,mh 3 ,,hc ,,af 3 ,,ag 33 ,,ag 3 ,,gf ,,ab 3 ,,af 33 ,,af 3 ,,fb while ,,mn1 ,,mh or ,,xc3 ,,cd33 #b3 #a--;being the geometric template of the work4 ,thus every line in this beautiful symbol by its intersections with the other lines1 manifests the problem4 ,note also that ,,gh ,,ga ,,ae ,,af ,,dh ,,de ,i append a table showing the comparative measures of the lines in ,fig4 #fg1 taking radius of the circle as a million units4 ,fig #fg4 ,table ,showing the ,comparative ,measures of ,lines4 7,fig4 #fg47 ,,me #bjjjjjj diameter4 ,,ab #aijbaac ,,ad ,,db ,,mb #afahjcd ,,mc ,,mh ,,mp ,,pb ,,as #aechhdae ,,ep #adecjhf ,,ag ,,fb ,,af #aageegj ,,ae ,,gb ,,mc #ajjjjjj radius ,,cd ,,dx ,,ch ,,cx ,,ad #ieajefe ,,db ,,ds ,,pb #hedajb ,,qs #habbihe ,,mp #gfcicb ,,ch #b base of ,cheops4 ,,ag #gbfedc ,,gh ,,xh ,,hn ,,pf ,,fb ,slant edge of ,cheops4 slant edge of ,pent4 ,pyr4 ,,de #fijihc ,,dh ,,xd apothem of ,pentagonal ,pyramid4 apothem of ,cheops4 ,,mh #fahjcd ,,mn ,,xc altitude of ,pentagonal ,pyramid4 side of decagon inscr'd in circle4 ,,ms #ejjjjj mean proportional between ,,mh and ,,hc #dhehfh altitude of ,cheops4 ,,op #ddijbg ,,gf ,,gd ,,df ,,hc #chaiff half base of ,cheops4 ,,so #cfcbgae ,,hs ,,cd #cjijag half ,,mh ,,pr #bggeaf ,,gd #bbdeace ,,sp #bfcicb ,the triangle ,,dxh represents a vertical section of the pentagonal pyramid2 the edge ,,hx is equal to ,,hn1 and the apothem ,,dx is equal to ,,de4 ,let ,,dh be a hinge attaching the plane ,,dxh to the base1 now lift the plane ,,dxh until the point ,x is vertical above the centre ,c4 ,then the points ,a1 ,e1 ,b1 ,o1 ,n of the five slant slides1 when closed up1 will all meet at the point ,x over the centre ,c4 ,we have now built a pyramid out of the pentangle1 whose slope is #b to #a1 altitude ,,cx being to ,,cd as #b to #a4 ,apothem ,,dx ,,de ,altitude ,,cx ,,hm or ,,mn ,altitude ,,cx ,,ch ,,cm radius4 ,apothem ,,dx ,,cd ,,cm radius4 ,edge ,,hx ,,hn or ,,pf ,note also that ,,mp -- ,,ch #b ,,op ,,hr ,let us now consider the ,pentangle as the symbol of the ,great ,pyramid ,cheops4 ,the line ,,mp the base of ,cheops4 ,the line ,,ch half base of ,cheops4 ,the line ,,hm apothem of ,cheops4 ,the line ,,hn slant edge of ,cheops4 ,thus3 ,apothem of ,cheops side of decagon4 ,apothem of ,cheops altitude of pentagonal pyramid4 ,slant edge of ,cheops slant edge of pentagonal pyramid4 ,now since apothem of ,cheops ,,mh and half base of ,cheops ,,hc then do apothem and half base represent1 when taken together1 extreme and mean ratio1 and altitude is a mean proportional between them3 it having already been stated1 which also is proved by the figures in the table1 that ,,mc 3 ,,mh 33 ,,mh 3 ,,hc and apoth3 alt 33 alt 3 half base4 ,thus is the four pointed star ,cheops evolved from the five pointed star ,pentalpha4 ,this is shown clearly by ,fig4 #fh1 thus3-- ,fig4 #fh4 ,within a circle describe a pentangle1 around the interior pentagon of the star describe a circle1 around the circle describe a square2 then will the square represent the base of ,cheops4 ,draw two diameters of the outer circle passing through the centre square at right angles to each other1 and each diameter parallel to sides of the square2 then will the parts of these diameters between the square and the outer circle represent the four apothems of the four slant sides of the pyramid4 ,connect the angles of the square with the circumference of the outer circle by lines at the four points indicated by the diameters1 and the star of the pyramid is formed1 which1 when closed as a solid1 will be a correct model of ,cheops4 ,calling apothem of ,cheops1 ,,mh #cd and half base1 ,,hc #ba as per ,figure #f4 ,then-- ,,mh ,,mc #ee and #ee 3 #cd 33 #cd 3 #bajah1 being only in error a few inches in the pyramid itself1 if carried into actual measures4 ,the ratio1 therefore1 of apothem to half-base1 #cd to #ba1 which ,i ascribe to ,cheops1 is as near as stone and mortar can be got to illustrate the above proportions4 ,correctly stated arithmetically let ,,mh #b4 ,then ,,hc #e - #a ,,mc #e #a and altitude of ,cheops 7,,mh ,,mc7 ,let us now compare the construction of the two stars3-- ,fig4 #fi4 ,,to ,,construct ,,the ,,star ,,pentalpha ,,fig4 #fi4 ,describe a circle4 ,draw diameter ,,mce4 ,divide ,,mc in mean and extreme ratio at ,h4 ,lay off half ,,mh from ,c1 to ,d4 ,draw chord ,,adb1 at right angles to diameter ,,ecm4 ,draw chord ,,bhn1 through ,h4 ,draw chord ,,aho1 through ,h4 ,connect ,,ne4 ,connect ,,eo4 ,fig4 #gj4 ,,to ,,construct ,,the ,,star ,,cheops1 ,,fig4 #gj4 ,describe a circle4 ,draw diameter ,,mce4 ,divide ,,mc in mean and extreme ratio1 at ,h4 ,describe an inner circle with radius ,,ch1 and around it describe the square a1 b1 c1 d4 ,draw diameter ,,acb1 at right angles to diameter ,,ecm4 ,draw ,aa1 a,e1 ,eb1 b,b1 ,bd1 d,m1 ,mc1 and c,a4 ,the question now arises1 does this pyramid ,cheops set forth by the relations of its altitude to perimeter of base the ratio of diameter to circumference2 or1 does it set forth mean proportional1 and extreme and mean ratio1 by the proportions of its apothem1 altitude1 and half-base8 ,the answer is--from the practical impossibility of such extreme accuracy in such a mass of masonry1 that it points alike to all1 and may as fairly be considered the exponent of the one as of the others4 ,piazzi ,smyth makes ,cheops #gfafe feet base1 and #dhdia feet altitude1 which is very nearly what he calls a ,pi pyramid1 for which ,i reckon the altitude would be about #dhdhg feet with the same base3 and for a pyramid of extreme and mean ratio the altitude would be #dhdcd feet4 ,the whole difference1 therefore1 is only about six inches in a height of nearly five hundred feet4 ,this difference1 evidently beyond the power of man to discover1 now that the pyramid is a ruin1 would even in its perfect state have been inappreciable4 ,it appears most probable that the star ,pentalpha led to the star ,cheops1 and that the star ,cheops 7,fig4 #gj7 was the plan used by the ancient architect1 and the ratio of #cd to #ba1 hypotenuse to base1 the template used by the ancient builders4 ,suppose some king said to his architect1 ,make me a plan of a pyramid1 of which the base shall be #dbj cubits square1 and altitude shall be to the perimeter of the base as the radius of a circle to the circumference4--,then might the architect prepare an elaborate plan in which the relative dimensions would be about-- ,r4 ,b4 ,,cubits ,base #dbj ,base angle #ea #ea #adc ,altitude #bfgchjcjd c4 ,apothem #cciihhegc c4 ,the king then orders another pyramid1 of the same base1 of which altitude is to be a mean proportional between apothem and half-base--and apothem and half-base taken as one line are to be in mean and extreme ratio4 ,the architect's plan of this pyramid will be the simple figure illustrated by me 7,fig4 #gj71 and the dimensions about-- ,r4 ,b4 ,,cubits4 ,base #dbj ,base angle #ea #di #cg-db_/dga ,altitude #bfgabcihdi c4 ,apothem #ccighgeaec c4 ,but the builder practically carries out both plans when he builds to my templates of #cd to #ba with-- ,r4 ,b4 ,,cubits4 ,base #dbj ,base angle #ea #ea #bj ,altitude #bfgcidhci c4 ,apothem #cdj and neither king nor architect could detect error in the work4 ,the reader will remember that ,i have previously advanced that the level of ,cephren's base was the plan level of the ,gzeh pyramids1 and that at this level the base of ,cheops measures #dbj ,r4,b4 cubits--same as the base of ,cephren4 ,this hypothesis is supported by the revelations of the pentangle1 in which the ratio of #cd to #ba apothem #cdj to half-base #baj ,r4,b4 cubits1 is so nearly approached4 ,showing how proportional lines were the order of the pyramids of ,gzeh1 we will summarise the proportions of the three main pyramids as shewn by my dimensions and ratios1 very nearly1 viz43-- ,mycerinus4 ,base 3 ,apothem 33 ,altitude 3 ,half-,base4 as shown by the ratios1 7,fig4 #ac71 #dj 3 #cb 33 #be 3 #bj4 ,cephren4 ,diagonal of ,base 3 ,edge 33 ,edge 3 ,altitude4 as shown by ratios1 7,fig4 #ab71 #hfb 3 #ehh 33 #ehh 3 #djj4 ,cheops4 7,apothem ,half ,base73 ,apoth4 33 ,apoth4 3 ,half ,base4 as shown by the ratios1 7,fig4 #i71 #ee 3 #cd 33 #cd 3 #ba4 and--,apothem 3 ,altitude 33 ,altitude 3 ,half-,b4 ,similar close relations to other stars may be found in other pyramids4 ,thus3--,suppose ,,nho of figure #fi to be the ,,nho of a heptangle instead of a pentangle1 then does ,,nh represent apothem1 and ,,no represent base of the pyramid ,mycerinus1 while the co-sine of the angle ,,nhm 7being ,,mh minus versed sine7 will be equal to the altitude of the pyramid4 ,the angle ,,nhm in the heptangle is1 #ch #cd #agadb1 and according to my plan of the pyramid ,mycerinus1 the corresponding angle is #ch #dj #ef4 7,see ,fig4 #ai47 ,this angular difference of #j #f #ci would only make a difference in the apothem of the pyramid of eight inches1 and of ten inches in its altitude 7apothem being #bhc ft4 #a inch1 and altitude #bba ft474 #ag4 ,,the ,,manner ,,in ,,which ,,the ,,slope ,,ratios ,,of ,,the ,,pyramids ,,were ,,arrived ,,at4 ,the manner in which ,i arrived at the ,slope ,ratios of the ,pyramids1 viz41 #cb to #bj1 #cc to #bj1 and #cd to #ba1 for ,mycerinus1 ,cephren1 and ,cheops1 respectively 7see ,figures #h1 #g and #f71 was as follows3-- ,first1 believing in the connection between the relative positions of the ,pyramids on plan 7see ,fig4 #c1 #d or #e71 and their slopes1 ,i viewed their positions thus3-- ,mycerinus1 situate at the angle of the #c1 #d1 #e triangle ,,adc1 is likely to be connected with that primary in his slopes4 ,cephren1 situate at the angle of the #bj1 #ba1 #bi triangle ,,fab1 and strung1 as it were1 on the hypotenuse of the #c1 #d1 #e triangle ,,dac1 is likely to be connected with both primaries in his slopes4 ,cheops1 situate at the point ,a1 common to both main triangles1 governing the position of the other pyramids1 is likely to be a sort of mean between these two pyramids in his slope ratios4 ,reasoning thus1 with the addition of the knowledge ,i possessed of the angular estimates of these slopes made by those who had visited the ground1 and a useful start for my ratios gained by the reduction of base measures already known into ,r4,b4 cubits1 giving #dbj as a general base for ,cheops and ,cephren at one level1 and taking #baj cubits as the base of ,mycerinus 7half the base of ,cephren1 as generally admitted71 ,i had something solid and substantial to go upon4 ,i commenced with ,mycerinus4 7,see ,fig4 #ga47 ,fig4 #ga4 7,mycerinus7 ,,lhnm represents the base of the pyramid4 ,on the half-base ,,ac ,i described a #c1 #d1 #e triangle ,,abc4 ,i then projected the line ,,cf ,,bc to be the altitude of the pyramid4 ,thus ,i erected the triangle ,,bfc1 ratio of ,,bc to ,,cf being #a to #a4 ,from this datum ,i arrived at the triangles ,,bea1 ,,adc1 and ,,gkh4 ,,gk1 ,,ea1 and ,,ad1 each represent apothem of pyramid2 ,,cf1 and ,,cd1 altitude2 and ,,hk1 edge4 ,the length of the line ,,ad being 7,,ac ,,cd71 the length of the line ,,hk being 7,,hg ,,gk71 and line ,,ch 7half diagonal of base7 being 7,,cg ,,gh74 ,these measures reduced to ,r4,b4 cubits1 calling the line ,,ac ratio #d #aje cubits1 half-base of pyramid1 give the following results3-- ,r4 ,b4 ,,british ,,cubits4 ,,feet4 ,half-base ,,la #ajejjj #agfibe ,apothem ,,ea #afhjhb #bhcbah ,edge ,,hk #aihahc #cccicg ,altitude ,,cd #acabej #bbaaef ,half diag4 of base ,,ch #adhdibd #bejbji and thus ,i acquired the ratios3-- ,half-base 3 ,altitude 33 ,apothem 3 ,base4 #bj 3 #be 33 #cb 3 #dj nearly4 ,to place the lines of the diagram in their actual solid position--,let ,,ab1 ,,bc1 ,,ca and ,,hg be hinges attaching the planes ,,aeb1 ,,bfc1 ,,cda and ,,hkg to the base ,,lhnm4 ,lift the plane ,,bcf on its hinge till the point ,f is vertical over the centre ,c4 ,lift plane ,,cda on its hinge1 till point ,d is vertical over the centre ,c2 then will line ,,cd touch ,,cf1 and become one line4 ,now lift the plane ,,aeb on its hinge1 until point ,e is vertical over the centre ,c1 and plane ,,hkg on its hinge till point ,k is vertical over the centre ,c2 then will points ,e1 ,f1 ,d and ,k1 all meet at one point above the centre ,c1 and all the lines will be in their proper places4 ,the angle at the base of ,mycerinus1 if built to a ratio of #d to #e 7half-base to altitude71 and not to the more practical but nearly perfect ratio of #cb to #bj 7apothem to half-base7 would be the complement of angle ,,adc1 thus-- #d #afe --- #h ,tan4 ,,adc #ch #ci #ce--- #e #dgg #cab ,,dac #ea #bj #bd--- #dgg but as it is probable that the pyramid was built to the ratio of #cb to #bj1 ,i have shown its base angle in ,figure #ai1 as #ea #ai #d4 ,figure #gb shows how the slopes of ,cephren were arrived at4 ,fig4 #gb4 7,cephren7 ,,lhnm represents the base of the pyramid4 ,on the half-base ,,ac1 ,i described a #c1 #d1 #e triangle ,,abc4 ,i then projected the line ,,cf 7ratio #ba to ,,bc #bj71 thus erecting the #bj1 #ba1 #bi triangle ,,bcf4 ,from this datum1 ,i arrived at the triangles ,,bea1 ,,adc1 and ,,gkh2 ,,gk1 ,,ea and ,,ad each representing apothem2 ,,cf and ,,cd1 altitude2 and ,,hk1 edge4 ,the lengths of the lines ,,ad1 ,,hk and ,,ch being got at as in the pyramid ,mycerinus4 ,these measures reduced to cubits1 calling ,,ac ratio #af #baj cubits 7half-base of pyramid7 give the following result4 ,r4 ,b4 ,,british ,,cubits4 ,,feet4 ,half-base #bajjj #ceche ,,la ,apothem #cdfej #ehche ,,ea ,edge #djeaf #fhbfi ,,hk ,altitude #bgefbe #dfddc ,,cd ,half-diag4 of base #bifihe #ejjdb ,,ch thus ,i get the ratios of--,apothem 3 ,half-,base 33 #cc 3 #bj1 c4 ,the planes in the diagram are placed in their correct positions1 as directed for ,figure #ga4 ,the angle at the base of ,cephren1 if built to the ratio of #af to #ba 7half-base to altitude71 and not to the practical ratio of #cc to #bj 7apothem to half-base71 would be the complement of ,,adc1 thus-- #af #af -- #gfaijd ,tan4 ,,adc #cg #ah #ad-- #ba #df #cj ,,dac #eb #da #de-- #df but as it is probable that the pyramid was built to the ratio of #cc to #bj1 ,i have marked the base angle in ,fig4 #ag1 as #eb #da #da4 ,i took ,cheops out1 first as a ,pi pyramid1 and made his lines to a base of #dbj cubits1 as follows-- ,half-base #baj ,altitude #bfgchjcjd ,apothem #cciihhegc 7,see ,fig4 #gc47 ,fig4 #gc4 7,cheops7 ,but to produce the building ratio of #cd to #ba1 as per diagram ,figure #f or #i1 ,i had to alter it to-- ,half-base #baj ,altitude #bfgcidhci ,apothem #cdj ,thus the theoretical angle of ,cheops is #ea #ea #adc1 and the probable angle at which it was built1 is #ea #ea #bj1 as per figure #ae4 ,cheops is therefore the mean or centre of a system--the slopes of ,mycerinus being a little flatter1 and those of ,cephren a little steeper1 ,cheops coming fairly between the two1 within about #aj minutes2 and thus the connection between the ground plan of the group and the slopes of the three pyramids is exactly as one might expect after examination of ,figure #c1 #d or #e4