GEOMRTRY, MENSURATION 
 
 AN1> THE 
 
 STEREOMETRICAL 
 
 TABLEAU 
 
 LECTURE READ 
 BEFORE TDE ftUEBEC LITERARY AND HISTORICAL SOCIETY 
 
 20TH MARCH 1 872. 
 
 BY CHS. BAIL.LAIRGI: ESQ. 
 
 CIVIL ENGINEER, ETC. 
 
 QUEBEC : 
 
 C. DARVEAU, PRINTER AND PUBLISHER, 
 
 N° 8, Mountain HiU. 
 
 1873 
 
LECTTJllE 
 
 {Exlract fyom 
 
 Mr. Baillargii's Iwtttre, on V.'cdriosday 'venini": j i 
 last, bi'loie the Littiaiy and IJistuiiciil iSoiiity ol' ; ^ 
 Quebec, proved once more how veiy interesting, evt n ( 
 
 iu a pojiiilav .sense, an otlifrw isc diy imd abstni.sc sub- I 
 
 jecl. may beconie, ubcn aldy baiidlid. < 
 
 The lecture showed the I'ehitionshij) of geometry to i 
 
 all the iudustiies ot'liCe. lie traced its oiigiii Troni 1 
 
 reniolt- aiitiiiuity, its gradual dcvflopiniu nl ii|> to llie | < 
 present tinje. lie .showed how it i.s the basis of all j ; 
 our public works, and jiow we are indebted to it for i 
 
 all the con.stiuctive arts; its K'latiouship to niecha- j 1 
 nics, hydraulics. oj)iics, aud all ihe physical sciemes. j 1 
 'J'he fairei' portion of mankjud, said .Mr. li., lia>«' tiio : i 
 keenest, uiost appreciative i)ercei>tioM of its advau- | 1 
 tages and beauties, as evidenced iu the ever-varyiug ; 
 oojubiiiatious so (luniingly devised in their doigns | 
 for needle traci-ry, hu-es ami embroidery. Jle showed 1 
 
 its relationship to chemistry in crystallization and ! i 
 l>olarization ; to botany ami zoology in the laws of i ( 
 morphology: totheologv. anil so on. in treat iug of i ( 
 the circle and othei' conic sections, he drew (jiiite a ( 
 
 poetical C(mii)ari.son between the engineer who traces 
 out his curves among tlie woods and waters of the 
 earth, and the astionomer who sweeps out his mighty 
 circuits anndst the starry forests of the heavens. The 
 l»arabola was fully illustrated in its application to the 
 tlirowing of projectiles of war, also as evidenced in 
 jets of water, tlie siieaking truni])et, the nniror and 
 the rellectoi', which, in ligiit-houses, gathers the ra\ ,< 
 of light, as it were, into a bundle, anil sends them oil' 
 together on their errand of humanity. In ticating of 
 the elli])se, this almost magic curve which is traced 
 out iu tlie lieavens by every planet that revolves 
 about the si;n, by every satellit(^ about its primary, 
 he alluded to that most l»eautifid of all ovals- the 
 face ol lovely wctman. lie showed how the re-ap- 
 pearance of a comet may now be i)redicted evin to the 
 very day it heave.s in .sight, and though it has becTi 
 ab.sent for a centuiy. and how in formii' ages, when 
 these ])h<"iiomemi weie unpredicled, they burst upon 
 the world iu uiu'\£>ected uuHuents, carrying terror 
 everywhere and giving rise to the utmost anxiety and 
 consternation, as if the end of all things were at hand. 
 In a word, Mr. Baillarge went over tlie whole held of 
 geouKstiy iin.d mensuration, both ])lane ami .s])erical ; 
 a ditticidt feat within the limits of a single lecture; 
 and kept tli(^ audience, so to say, eiitranced ^^ ith in- 
 terest lor two w hole hours, w Inch the ])resi(U'nt. \)x. 
 Anderson, reuuirked were to him as but one ; and no 
 doubt it uuist have been so to otheis, siiuc .Mr. \\ ilkie, 
 in .seconding the vote of thanks ]»r<»])o.s(d by (apt. 
 Ashe, alluded to the jdeasure Avith which he hiid liste- 
 ned to the lecture as if, he said, it were like poetry to 
 him, instea<l of the unpromising matter foK shadowed 
 in the title. Mr. Uaillarge next ex]>lained in detail 
 his steriometrical tableau, which we hope to see soon 
 lutrochucd into all the schools of this Donunion. He 
 Hhov.<d](»u c(»nduci\c it will be in shoiieiiiiig the 
 tini«' hcictofore devoted to the .study of .solids and 
 eveu to that of plane and couve-x .superiicie.s, Hi)herical 
 
MR 
 
 ba.illaikg^e:. 
 
 m the '' (Jucbec Daily iMcrcury'' of iGth March, 1S72. 
 
 tvi^'ouoTiicti'}, .renmf'trirnl r^iojcctinii. j'or.^pcftivc dvit- 
 
 ■^\ill,i;-, lllf tlcvrlopillfllt of MliliHTS, sllildcs ;|ll(i A\\\- 
 
 d<i\\>. iiml ilic ]jkc. Mv. Wilkic, so lin- ;i.s oj.i.orimiitv 
 liiKl Itccii iitloidcd liiiii ol" jdoviii.y til,' ciilciiljitioiis, 
 (•<)ii«)l)oi;ti((l Ml'. IJ.'s .statciiKiil iu icliitioii l<> tlir 
 iMiiiiciisc ,saviii,i;- in lime, \vlicr( inniiy iiUstnisc pio- 
 lilciiis wliicli ;;viHT;illy ]V(|iiiiv(l lioiiis dr dnvs to solve, 
 rail now (ii'tlK! rule be, iis Mr. i:!iillar,i;('« 'iisscits. so 
 ficiK'iallx apiiliraiile, and, a.-, liiis liccii ('crdiicd l>v so 
 iiiaiiv pci-.-diis ill (cstiiiioiiials over ilicii' own si"iia- 
 lurcs,) witli lln- licip oft lie new loi inula and hihlT-au, 
 1»c pel foi mcd iu as many iiiinntcs; to say iiotliinu of 
 tlic rise tlic models ave in inipa;tiii!L!; nt a ^lanee a 
 Ixiiowled.^*' ot" (iieir noineiielatnre or" names, 'liiid an 
 a( (|uaiiitanee,sliip with their varied shapes and li<iUTes. 
 lie showed how, to the arehitcri a'ld eiiuine(V, the 
 liiiihler and mechanic, the models ave sii^oj.stive of 
 tlie lorms and relative propoilions of 1)iiildini;s, roofs, 
 domes, jiiers and quays, cisterns' ami lesei vttiTs, caul- 
 drons, vals, casks, tnhs and other vessi !s of capacity, 
 eatlnvorks ofall kinds, c(»ni]irisiii^ railroad iind otiier 
 cuttini^s and eniliankments, the sjiafl ol'the (jreek ami 
 IJoimin column, s(piare and waney tiinher, s;iw-lo;'s, 
 th(> campin-;' tent, tln^ s(piai( or splayed opeiiin;;- of a 
 door or window, iiich oi- lo(.i)li<»Ie iii a wail, th(' vault 
 or arched ceiliiio' of a ciiiD'ch or hall, the billiard or 
 the cannon ball, or, on a lai;i;er scale, the moon, earth, 
 sun and ])1aiiets. Mr. liaillar.nt', we may add. has 
 received an order for a tableau Imm tlie Minister of 
 Education of New Urniiswick, with the view of iutro- 
 (liicin.ii- it into all tlie schools of tliat rrovineo ; and 
 Mr. A'annier, in wiitimj;' to Mr. IJaillair^e, from France, 
 on the 10th of .laniiary last, to advise him of the 
 <j;raiitiiii;' of his letteis-j)alent for that country, says 
 that Messrs. Humbert &; Xoe, the Presidem and secre- 
 tary of liie soei<'ty lor the v'( 'leralizatiou of education 
 in I'lance, liu\-e intimated their intention, at tiieir 
 next .i^eneral meelini;-, of having some mark of distinc- 
 tion eoiil'erred on him for the beiieiit which his imcn- 
 tion and discoveiy are likel_\ to confer on i'«lueation. 
 Mr, viiard, in writin.u' to Mr. Iiaillair;;!', (Ui the part of 
 the Hon. Mr. Chaiiveau, Minister oi Public Instruction, 
 say : " 11 se fera un devoir d'eii recommaiider I'aih)])- 
 " ti<ui dans toutes les maisons d'education et dans 
 '' tontes les ec(»les. " From tlu^ Seminary and Laval 
 Fniversity, Mr. .Maiii^ui writes: " I'lus on etudie, 
 " plus on a]»profoiidit cette forinnh* du cuba,i;<> (h's 
 " corjKS, itliis on est eiichante (the more one marvels) 
 '• de sa simpli<'ite, de sa clai te et snrtout de sa grande 
 '< oeiieralite. '' Kev. Mr. Mctiiiarries, 15. A. "shall be 
 " di'li.uhted to hoc the old and tedious ])rocesses 
 '* snperstMied by a formiihi so simple and s(» exact. "' 
 Jsewton, of \ale (,'olleye, i'uited States: "considers 
 " the tableau a must useful arra;i}>enu'ut for showing' 
 '"the variety and extent of the applications of the 
 " j'oi nulla. " The C'olleg'c rAssomption "will adopt 
 " Mr. Uaillairge's system as part of their course of 
 " iiisiructi(Ui. " Mr. Wilkic has written tst the auth.or 
 that " the rule is jwecise and .simple, and w ill greatly 
 " shorten the processes of calculation. The tableau," 
 
 says this coiii[)etent Judge, "comprising as it does a 
 " <.'i'eat varii'ix of * element, iry models, v ill serve 
 •• a<liiiirabl\ to educat*' the eye, and must greatly 
 '■ faciiitate'ihe stiidv of solid mensurati<ui." "Again, 
 says Mr. Uilkie, "the (iovermeut would coub'r a 
 "'boon on .schools (.f the middle and higher class by 
 " alfordinu access to so sngg<'stive a colleetum. 
 There areddiers who, irrespective of consuU'rations 
 as t(» the cmnparative accuracy of the formula, or ot 
 itsadvanti-es, as ai.plied io mere mensuration, are 
 awake to tlu; fact that the models are so iuuch more 
 suiigestivc to the i)iipil and the teacher than their , 
 iiinv repiiseiitation <.ii a blackluml or on paper, and 
 wlio, in tlicir written opinions, have alluded especially 
 to this feature of the proposed system. M. .lolv I'le- 
 sident of the Quebec Hrancli of the Montreal School 
 of Arts and I )esii;n, in a letter on the subject to .Mr. 
 Weaver, the l'resi«lent of the Hoard, and after having 
 himself witnessed its advantages on more than one 
 (Kcasion,savs,in hisexpressive style, "the diiference is 
 eiiormoii^."' professor Tonssiiint, of the N«n'mal School, 
 Dufresiie, of the .Montmaguy Academy, Hoiviu, ()f St. 
 llvacinfhe, and manv Others, are of the same «>1>H}V,*'!' 
 among them MM. K. S. M. IW.m-hette, OT'y.'H' '*/«*- 
 cher. St. Aiibin, Steckel, .Juneau, Venner, (.alhigher, 
 Lafraiice, and the late Hn^ther Anthony, cVc, &c. 
 Neither will it be forgotten tiiat the; professors ot the 
 Laval Lniversitv, after reading the enunciation ot Mr. 
 15.\s tbrmula, as given in his treatise of IHtiti, expressed 
 themselves thus: " Un <h»iite iuvoloutaire s'emi»ai'e 
 " d'abord de I'esprit, hnsinron lit le No. 1521 ; niam 
 •' un exameu atteutif des ]»aragra plies suivants, dissipo 
 " bieiitot ce donte et Ton reste etonue a hi viu;^ d uiie 
 " formule, si elaire, si aisee a retenir et dont rappli- 
 " ,ati(ui est si geiierale. " Mr. Fletcher, of the Crown 
 Lands Department, savs: "I have compared, in the 
 " case of several solids, the results obtained by your 
 " mode of computation with those resulting from the 
 "ordinary and more lengthy processes, ami ccmgra- 
 " tulateyoii sincerely on your enunciation of a formula 
 " so bri('f and simjile in "its character, and so pri'cise 
 " and satisfactory in its results. " Mr. Uaillairge also 
 took occasion during his lecture to allude, in other 
 relations, to his treatise on geometry and mensuration, 
 in which he showed he has introduced many important 
 inoditicatious in the usual mode of treating the subject 
 of idane and snherical u'eometry and trigonometry. 
 In conclusion, w'e must add that the Council of Public 
 Instruction, at its last meeting, api)ointed a C<mimitt<'e, 
 composed of tlu' Lord llishop of (Quebec, and of nishoi)8 
 Langovin and Laroc<iue, to report to the Couucd at 
 its next general nu'cting iu .June, and who, it may be 
 taken for granted, after the many tlatteriiig testimo- 
 uials in relation to the utility and many advantages 
 of the stereometrical tableau for purp(»ses of education, 
 cannot but re<'ommend and direct its adoption in all 
 tlie schools <»f the l>omiiiiou. 
 
 AVe learn with phasiire that Mr. 1 }a ilia rge has bccu 
 invited to repeat this lecture iu Montreal. 
 
M A 11/11 1/11 A 11® dilP-'P^l ^l^'^i'l^lPillMflP^^^lP /%\ 1 
 
 ii 
 
 Honorary Member of the Society for the Generalization of Education in France, etc, 
 (Patented in Canada, in the United States of America, and in Europe.) 
 
 This is a Case 5 feet long, 3 feet wide and 5 inches deep, with a hino^ed Glass Cover, under Lock and Key, s( 
 exhibiting, and affording free access to some 200 well-finished Hardwood Models of every conceivable Elements 
 form, each of which being neatly attached to the board, by means of a wire-peg or nail, can be removed and re 
 Student or Professor. 
 
 The use of the Tableau 
 and accompanying^ Treatise, 
 reduces the whole science 
 and art of Mensuration from 
 the study of a year to that 
 of a day or two, and so sim- 
 plifies the study and teaching 
 of Solid Geometry, the No- 
 menclature of Geometrical 
 and other forms, the develop- 
 ment of surfaces, geometri- 
 cal projection and perspect- 
 ive, plane and curved areas 
 and Spherical Geometry, 
 and Trigonometry, and men- 
 suration of surfaces and 
 solids, thai the several 
 branches hereinbefore men- 
 tioned may now be taught 
 even in the most elementary 
 schools, and in convents, 
 where such study could not 
 even have been dreamed of 
 heretofore. 
 
 Eaeh Tableau is accom- 
 panied by a Treatise explan- 
 atory of the mode of meas- 
 urement by the " Prismoidal 
 Formula, " and an explana- 
 tion of the solid, its nature, 
 shape, opposite bases, and 
 middle section. 
 
 Agents tca/nted for the sale 
 of the Tableau m Canada, 
 the United States, &c. 
 
 Pouf tronver le volume 
 d'ua oorpB^eioonque 
 
 REQLE:— A la sommo 
 des surfaces lies oxtrfe- 
 
 mit^^s «arall6le3, ajou- 
 terquatro foia la sur- 
 face nu centre, et mul- 
 tiplier ie tout par a 
 Bixi^ine partio de la 
 hauteur ou longueur 
 du solide. 
 
 
 Toflnd the solid 
 of any bod: 
 
 " on Prance. 
 
 xx*-*-^ RULE:— To the 
 
 Patented in CANADA .the^U^l | 
 
 Honorary Member of the Society for the Gon- 
 Uonoiary ^^^^j.^^^j^^ ^f Education 
 
 in France. 
 
 add lour tiiiU'S i 
 die area, and n 
 the whole by oi 
 part of the ht 
 length of the b 
 
 en France. 
 
 
 ^ f^#© 
 
 ™ 1 w 
 
 ^ ^ ^ © ® ^, 
 
 4k ^ 
 
 For the use of Architects, Engineers. Snrv(!yors, Students and Apprentices, Customs and Excise Officers, Pi 
 Mathematics, Universities, Colleges, Scminai-ies, Convents and otIuM Educational Esta])Hshments, Schools of Ai 
 Measurers, Gangers, Ship-builders, Contractori, Artizans and others in Canada, and elsewhere. ' < 
 
MAI 
 
 m^ 
 
 ANCE, ETC., ETC. 
 
 GEOMETRY, MENSURATION 
 
 ANO THE 
 
 nd Key, so as to exclude dust while 
 ; Elementary, Geometrical or other 
 ed and replaced at pleasure, by the 
 
 STEREOMETRICAL 
 
 To find the solid oomen' 
 ol any body, 
 
 ^ULE : — To the sura of 
 tho iiariilk'li'nil iiroiis, 
 add :oiir tuiius llio iiiiJ- 
 die area, and multiply 
 the whole by one !<ixtb 
 oart of tho hei^'ht or 
 length of the body. 
 
 ^^ ® 
 
 Approved by the Council of Pub- 
 lic Instruction of the Province of 
 Quebec, nnd already adopted and 
 ord'Ted by many I'Mucational and 
 other E?tabli>hcui(!nts in Canada 
 and elsewhere. For information 
 j and testimonials apply to 
 
 C. BAILLARGE, 
 
 QUE3R0. 
 
 Canada. 
 
 Bonornrij Member of the Society 
 for the Gen':raUznHon of Edu- 
 cation in France, etc., etc, 
 
 SUBSCRIBERS. 
 
 The Atchbi!-ho|) of Quotiee, the 
 Bishop of Riinouski, the Bishop of 
 Kingston, tho Bishop ofSt. Hya- 
 ointhe, the rominion Board of 
 Works, tho Schools of Arts aud De- 
 sign, tho Lavfil University, the 
 Seminary, Q., tho College at Otta- 
 wa, Nicolet, l-iimouski. Montma- 
 gny. St. Michel, etc,, I'Ecole Nor- 
 male Laval, les Ecolos des Fr5res, 
 the Commercial Academy, the 
 Board of Land Surveyors, the De- 
 partment of Education, Now Brun- 
 swick, the Corporation.of Quebec, 
 R. Uamilton, Esq., F.N. Martin, 
 and U Roy, Civil Engineers, etc., 
 Ln Sociftd pour la vulgarisation de 
 I'Enseignemont du Peuple, France 
 F. Peachy, J. Le))ago, etc.. Ar- 
 chitects, N. Piton, T. Maguire, J. 
 Marcotte, builder,*, the Council of 
 Public Instruction, Q, the Jacques 
 Carticr Normal School, M. Piton 
 Manitoba, tho Colleges of Aylmer, 
 L'Assomption . Ste. Anne do la Po. 
 oat'ore, St. Hyncinthe, the High 
 School, Q., the .Morin College, Q., 
 the Lnfranee Academy, Q., Gover- 
 nment Boards of Works, Q., the 
 Ursulines f^onvent, the Convent of 
 the Good Shepherd, Grey Nuns, 
 Soeurs do la Coner<5:iation, Soeurs 
 do ,l(;sns-Marie, Q., and >T. S. W. 
 Townsond, Hamilton, &o. &c. &o. 
 
 TABLEAU 
 
 Etc., Etc 
 
 Etc. 
 
 Officers, Professors of Geometry and 
 hools of Art and Design, Mechanics, 
 
 LECTU5E READ 
 BEFORE TDE fimUC LITERARY AND HISTORICAL SOCIETY 
 
 20TH MARCH 1872. 
 
 <« BY CHS. baiIjLairgI: esq. 
 
 CIVIL ENGINEER, ETC. 
 
 QUEBEC : 
 
 C. DARVEAU, PRINTER AND PUBLISHER, 
 
 N° 8, Mountain HiU. 
 
 1873 
 
. I.A' i 
 
I'APKU rv. 
 
 GEOMETRY, MENSURATION, 
 
 AND THE 
 
 STEREOMETRICAL TABLEAU. 
 
 By CHARLES BAILLAIHGf, Esq., Civil Engineer, &c. 
 
 (Read before the Society, March 20th, 1872. J " 
 
 No apology, I presume, iicctl be oifercd for my selection of the 
 subjci't of wliicli I am about to treat ; for, tliougli, at first sight, it 
 may api)ear to be devoid of interest and practical utility, it is, on the 
 coutraiy, fraught with paramount importance on account of its rela- 
 tioutsliip to almost all the industries of life. 
 
 Geometry is the basis of our public works ; to its precepts we are 
 indebted for all tlie constructive tuts, our public edifices, our dwelling- 
 liouses, tlie foitiflcaticms of ouv citi<;s, our ports, canals and roads, and 
 the wcmdrous archit(H'.ture of the many floating palaces and towers 
 that thread our streams and rivers and plough the mighty seas. The 
 g(^omet(n' it is who measures and designs, in iheir true proportions, 
 the diverse parts of states and teriitories, and who, by thus bringing 
 together their several points, enables the eye to appreciate the con- 
 sequences of their relative iiositions ; he it is who directs the use of 
 our engines of war, and to his calculations are made subordinate the 
 movements of armies. Geometry makes the astronomer and guides 
 the navigator, and all sciences are allied to it. It is the foundation 
 of mechanics, hydraulics, and optics j and all the physical sciences 
 .are constantly indebted to it. 
 
 In a less elevated sphere, geometry teaches us to measure and 
 design or represent our fields, our gardens, and domestic buildings ; 
 it enables us to estimate and compare their products and expendi- 
 tUT-e ; it deteniiines inaccessil)le heights and distances, g-uides the 
 draughtsman's hand, an«l presents an infinity of detached usages, of 
 ten applicable in domestic economy. 
 
4 GEOMETRY, MENSURATION AND 
 
 The fniror portion of mniikiiul liavr flic keenest, most ii[»pre('iii- 
 tive perception of its ndvantagcs and beantiew, as evidenced in tlio 
 ever-varying combinations so cunningly devieed in their dcHigns for 
 patch-work, laces, and embroidery, &c. 
 
 Geometry measures extension, com;)aring jiortions of space witli 
 each othor. Its elements are lines, surfaces, und volumes or solids. 
 A portion of space, such as might be filled by a solid body is itself 
 called in geometry a solid. The boundaries of this geometrical solid 
 are called surfaces, -which last may also be conceived as separating 
 Bpace from space as well as bounding it. They constitute a zero of 
 solidity, but have a magnitude of their own, called supei-ficiea. It is 
 sometimes snggestive and advantageous, in reasoning about solids 
 and their mode of measurement, to consider them as made up of an 
 infinite number of suri'aces or superficies overlying one another like 
 the leaves of a book. If a surface be limited in extent, the boundary 
 on any side is a line, which has neither solidity nor superficial area, 
 having magnitude in length only. Surfaces or supei-ficies may bo 
 conceived to bo made up of an infinity of lines in juxta-i)osition to 
 to each other. If the line is limited in extent, its extremities are 
 points. A point, therefore, is a zero, not only in solidity and super- 
 ficies, but in length also, having no magnitude or proportion, and 
 retaiinng only order or position as the sole element of its existence. 
 A lino may be conceived as made uj) of an infinite series of points 
 following each other in close contact or succession. In the iM)sition 
 of points, the diflPercuce in direction of a fir.^t and second point from 
 a third is called an angle. We have here all the elemental concep- 
 tions of geometry, viz., a point, a line, a surface, a solid, an angle. 
 From these definitions as data, a vast amount of geometrical science 
 may be deduced by the laws of \og\c. 
 
 The relation of geometry to other sciences is twofold, giving and 
 receiving. To mechanics it gives the only possibility of understand- 
 ing the laws of motion j and from mechanics it receives the conception 
 of moving points, lines, and surfaces, and thus gen<^rating lines, sur- 
 faces, and solids. To chemistry it gives the only means of investi- 
 gating crystallization, j)olarization, &c., and from chemistry receives 
 new ideas concerning the symmetry of jdanes. It holds a like rela- 
 tion to botany and zoology iu the laws of fonn and morphology. In 
 the study of the humaiu mind, geometry proposes the question of the 
 foundations of belief, by giving the first examjdes of demonstration ; 
 and from the inquiries thus aroused, geometry has received from age 
 to ago the new conceptions which have been the base of many new 
 methods of investigation and i)roof. To theology g<,'onietry gives 
 definite conceptions of the order and wisdom of the natural creation, 
 and from theology has been stimulat(!d to many fresh exertions in 
 the investigation of these theological (luestlons. 
 
THE STEREOMETIUOAI. TAIir,EAir. 
 
 The Iiislorv <»!' (jcomc^ti-y ii^ divided Uy CIimsIcm, in liis vidiiiihlc 
 " Ap^'r^ill liinti>i'i(jit(' den mt'iliodcs en tjcowelrie,^^ into five ])ciiotls. 'I'lio 
 first is that of tlu' GrrT'k gcoiiu'tiy, JMstinn' about 1000 years, or till 
 A.I). 'mO, Tlieii, after a pause of 100(1 yciar- theseeond period bc^iiii 
 in the revival of ancient j;e»»iii(!try about ir»r)0. A third period was 
 marked in the be;;inning of the 17th century by Descaites' co-ordi- 
 nates and the analytical iSfcometry, The fourtli ])eriod was inau,<4U- 
 rated in 1G84 by the suldime invention of tlie ditferential cal- 
 culus. Th(^ liftli era is marked iu our own century by ,Mon,i;<''s 
 *' Descriptive (Jeometry,'' by which he develo[>ed (he idea of reduci:i,n' 
 the ]>roblenis of solid ;;'eometry to ])roblems in a plane. One beauti- 
 ful (^\'am]»le of this branch of sciojice may be found in lijiear jierspec- 
 tivo, which simply projects the lutints of a solid upon a ]»Iane by 
 etraiyht lines t)f light from the eye. ; 
 
 Since Chasles' '' Apcvi'ii UhforUiue'''' was ]niblished, a sixth period 
 has been introduced by tjie •■ui)lica'' • i of" II(iiiiillnn\s OiKilmiioiis.'- 
 
 (jreek .geometry, it is said, beiiiui witli 'I'halcs jind Tytlia.iioras, 
 who obtained (jieir first i(h'as frtnn K;nyi)t and from India. Diodorus, 
 Herodotus and Stralto ait' of opinion that th(> science of mensuration 
 had its rise among the Egyj)tians, whom Ihoy represent as constrained 
 on ncof)uiit of the removal or defacing of the hnid-maiks by tlu; 
 annual inundation of (Iu- Nile, to devisee some method of ascertaining 
 tlie ancient Itondaries, after the waters had .su!»sided. indeed, tlu; 
 scieJice must liave been neaily coeval with tlie existence of man, for 
 wo are told in Holy Writ that Cain built a city ; to do a\ hicli, it is 
 evident, would require scune knowledge of a measuring unit, Avhich is 
 the tirst jtrinciple of mensuratitm. IJy the Sisme infallible testimony 
 we find tliat the arts and sciences were cnltivated to a consiih-rable 
 extent long before the tlood. Jnbal was the father of all such as 
 handled tlie harp and organ, and Tubal-CJain an instructor of every 
 artificer in brass and irmi. It is also more than probable that Noah 
 was well acqnainted with geometry as jnartised in his day, i'ov it 
 does not ajipear that he found any dilliculty in building tlu^ ark. Jie 
 this as it nnij', it is well known that Kgypfc was for many ages the 
 mother and nurse of the arts taud sciences. From this country they 
 were conveyed into Greece by Tliales about (iOO years befcn-e the 
 Christian era. Euclid, about :?00 years before Clirist, established a 
 mathematical school at Alexandria, where Archimeih's, VpoUonius, 
 Ptolemy, Tlieon, &c., received instruction from the '' jnince of geo- 
 meters. " Tt is, however, reasonable to suppose that befoie Kuclid\s 
 time there existed treatises on g(M)metry, for Produs athrms tliat 
 Euclid inqtroved .nauy things in the elements of Eudoxus and in 
 those of Theatt'tus, and established, by tlu' most funi and cov.vincing 
 denumstrations, such luopositions as were but superficially explained- 
 
6 GEOMETRY, MENSURATION, AND 
 
 Tlio I'vtlmgovo.in sohoo' (Icnioiirttiatod <!!e iiicomii)07isnrabili1y 
 of llu' (liagouiil of a Kquavo willi its suIl", and investigated tlic live 
 V('<;'nlar solids. Tli(>y had sonic knowlcd.2,<i of triangles and circles, 
 and were proltaldy accinainted with the fact that the circle and sphere 
 are the laru'est tignres of the same p(oimeter and surface. Alumtii 
 ceiitnry after Pythagoras, tlu^ great Plato and his disciples conuneneed 
 a course of rapid and astonishing discoveries. The anceiut analytic 
 mode of geometrical reasoning consisted in assuming the truth of the 
 theorem to be proved, and tlien shewing that this implied the truth 
 of those propositions only which were already known to l>e true. In 
 niodeiii days, the algel>raic metiiod, since it allows tlic introduction 
 of unknown tjuantities, has taken the name of analytic. Conic sec- 
 tions emhruco, as is well Known, the study of the cui'ves generated 
 l»y intersecting a cone by a plan<' surface ; and most marvellous, so 
 to say, are some of them. The circle, the most beautifnl of all, we 
 .see exemplified in the thousand-and-one forms of every-day life. 
 The ellipse, this almost magic curve, is traced out in the heuA'cus by 
 every planet that revovles aV>out tlie sun. by e'veiy satellite about its 
 primary. It has two centres or foci, the sun, or jtrimary in one of 
 them. It approaciies iu tliese cases nearly to a circle, or lias little 
 eccentricity, Avhile, as in the case of comets, it is lengtluMU'd out 
 almost indefinitely. It is pioduccd whenever the intersecting plane 
 cuts the cone in a direction oblicpu' to its axis, tlii' anghi tbrnuMl by 
 this section and the c<me's base being at the same time less thau that 
 included between the b;ise and side. In imitation of the Divine 
 Architect, man has made it subservient to his re<iuirements ; we .^ee 
 it iu the arclies of all sizes that s])an our rivers and «'i'owii our struc- 
 tures, great and small. The artist applies it as a fitting frame to that 
 most beautiful <tf ovals, the face of lovely woman. It has very pecu- 
 liar properties : tor instance, the sum of any two lines or radii drawn 
 from its centi'es to any point in tlie circumference oi' iterimeter of the 
 figure, is a ccmstaut (]uantity ; and a ray of light <u' sound or heat 
 thrown outer radiatcid from one of its foci towards the circumference, 
 is rellected to tiui other focus. Ycm have all heard of the whispering 
 gallery in St. Paul's Cathedral : it is merely an elliptical sjtace 
 surrounded by walls, in which, when two peisons stand in the op]>o- 
 site foci, and though c<mij)aratively far apart, ihey can talk to each 
 other in the merest whisper, and without the least danger of being 
 overluiai'd by any other person within the gallery, We have, next, 
 th(^ hyperbola, a peculiar cnrve, in which not the smu is constant, 
 but the diit'erence of any two radii or lines drawn from the foci to a 
 point in the curve. We obtain it in the cone when cut by a plane at 
 an angle greater than tliat which the side makes with the base. Tr 
 is, in a popnl.-i)' sense, a somewliaf p.uadoxical (igure, iuasnuich a> 
 
THE BTERKOMETRICAL TABLEAU. 7 
 
 tlioiigli it \yo pro(lnc(.'(l ever so far, mid in so jloiiio- it approaoli more, 
 juid iiuMv to two certain line.s called its ussymptotes, and tliougli tlie 
 line?, lie close by, yet can it never reach tbeni, even thongli it sliould 
 continue onward nutil the end of time. It hats not, in practi(;e, the 
 importance of the otiier conic sections, Imt has some uses, snc]i as that 
 of expressing or atfording an illustration of the varying pressure of 
 steam while working expansively within a cylinder or other vessel. 
 
 Come we now to the parabohi, a marvellous curve, indeed. How 
 singular tliat this ligurc, wliich every cone presents when cut by a 
 plane parallel to its side, should Ite that in whidi certain comets visit 
 our system — those tliat enter it but once, and leave it again, never 
 more to return ; for it is a curve which, like the hyperbola, but unlike 
 the circle or ellipsis, returns not back \ipon itself, but of wliich the 
 opposite sides or branches continually separate more amUnore, never 
 to meet again. This is, indeed, a most fascinating curve. How very 
 strange that it should happen to be the self-sauu' tigure whidi a stono 
 describes wlien, thrown obliquely into space, it falls again ! An ac- 
 quaintance with the theory of this nu)st useful arc guides the gunner 
 in throwing his balls and shells into the ftntress of the enemy ; for, 
 see you, such a curve as they describe wIh'U ascending into space, the 
 seLf-samo curve they uiake again while falling to their destination— 
 the whole arc or curve a true parabola, and each portion thereof tlui 
 exact half or counterpart of its fellow. Yet, have I not done telling 
 you about this most interesting section. Every jet of water or other 
 liquid issuing under lucssure from tlie sich- of a reservoir or cistein, 
 or from the pipe of a tire-engine, describes this curve ; and hence the. 
 distance of projection can be in advance calculated and ascertained. 
 But it has still other applications, to wit : in the construction of the 
 speaking-trumpet, Avhere the mouth lies in the focus of the curve (for 
 it will already have been guessed from the description that it has but 
 one), and the rays of sound strildng upon the sides ofthe tube are 
 projected forth together in a pencil or bundle, so to say, parallel to 
 one another, and direct towards the object spoken to. See, again^ 
 this curved line in the mirror which collects the i)arallel rays of the 
 sun, or other source of light or heat, and reflects them one and all 
 from the surface to the centre or focus, wherein a light or fire may be 
 thereby kindled ; and again, in the rellector which, in liglit-houses, 
 gathers the div(!rging rays proceeding from the focus, and sends them 
 off together on their errand of humanity. But I, too, am wandering, 
 I find ; and though the curve return not upon itself, I nmst not further 
 follow it in its erratic course. 
 
 Within 150 years after Plato's tinie, this study of thv conic secticms 
 had been pushed by Appolonius arid otlu^'s to a degree which has 
 scarcely been aupassed by any subsequent geometer. 
 
8 GEOMETRY, MENSURATION, AND 
 
 rjpoTiKti'ical loci aiT lines and siniaccs defined by tlie fact lliai 
 every iioiiit. in I lie line oi' sutface I'ldlils one and the sani(! eoiuliliou 
 of positioji. Thus, ilio h»cus of a point equally lenioved from any 
 two given points, is the peipcndieular drawn froju the centre of tho 
 ''rejoining tliese tAVo points ; the locus of tlu> vertices ()r all triangles 
 jiaving the same base and o(pial areas is a line i»arallel to the base ; 
 the locus of the veitices of all triangles })aving the same base, and 
 the same ratio between their sides, is the circumfen-nce of a circle 
 liaving i(s centre in the base produced, and such as to (Mit it in tho 
 required ratio. The investigation of such loci has been, from IMatos' 
 day to the i)resent, one of the most fruitful of all sources of geometri- 
 cal knowledge. Just before the time of Apollonius, P^udid introduced 
 into Geometry- a devicj^ of reasoning which was exceedingly useful in 
 cases where neither synthesis — that is, direct proof — nor tho analytic 
 mode is readily applicabh;, tlie rcdnctio ad absitrdnin ; it consists in 
 assuming the contrary of your proposition to be true, and then 
 shewing that this iiiiplies the truth of what is known to be false. 
 Contemporary with Apollonius was Archimtides, who introduced into 
 g<'ometry the fruitful idea of exhausti(m. l?y calculating inscribed and 
 circumscribed ]>olygons about a circle, and increasing the ninaber of 
 sides until the dilference between the external and internal polygons 
 became^ exceedingly small, Archimed(>s arrived at the first known 
 ratio between the diameter and circumference of a circle, which he 
 found to be as 1 t(» 3}. llipi)arclius, before Christ, and Ptolemy, after 
 Christ, a])plied nnitliematics to astronomy. Vieta, the inventor of 
 algebra, applied it to geometry. Keider introduced the idea of the 
 iidinitesimal, thus perfecting the Archimedean exhaustion, and led to 
 the solution of questions of maxinui nml minima. INIeanwliile, New- 
 tim's Fluxions and Leibnitz's Differential Calculus had come into use, 
 and many fnu^ discoveries were made in regard to curves in general j 
 and so fruitful the results, that, as is now well known by every one, the 
 time of an eclipse of the sun may be calculated and foretold for years 
 in advance, and that to the very niinute— nay, to within a second 
 almost — of its actual occurrence. Even the reappciirance of a comet 
 may be predicted to the very day it heaves in sight, and though it 
 has been absent for a century. In former ages, when these pheno- 
 mena were unprediited, they burst upon the world in unexpected 
 moments, carrying terror everywhere, and giving rise to the utmost 
 anxiety and ccmsternation, as if the end of all things were at hand. 
 
 Although the elements of Euclid are the groundwork of every 
 mathematical education, yet, many valuable rules are reduced from 
 the higher branches of amdysis, and which appear to have little or 
 no dependence upon geonu'try. The dil!erential analysis has admitt- 
 ed us to the knowledge of trutlis which would astonish mathenuiticians 
 of former ages ; and to Newton we are principally indebted for dis- 
 
THK STKRKOMETRICAL TAnf.EAU. y 
 
 •I'OYcvu'S Avliicli linvt' ificatly ndviiix'cd Mk' jut of incusuvatioii. Aiti- 
 i'M'crs (fall kinds arc indebted to ^itcoiiictry and niciisuration for tlio 
 *'staMisli('iu<'iit of their various occupations ; and the ]>crfectiou and 
 conse<iuent value of tliei'' labours depend entirely on the near 
 a]»i>roach they ujake to the standard of gectinetrical accuracy. All 
 the Ljreat anil in.i;'enious devices of mankind owe tlieir ori<i;in to this 
 snlilime science. By its means the architect draws his plan and 
 erect his editice. ^^'l!en bridi^es are to be built over' lurffo streams, 
 tlie ni(\st exact ac(ptaintance with jreonietry is re(piired. In tho 
 constiiiction of ships, of every kind, ,ii;eon)etrical knowledge is 
 icfpiisite ; and tlie sailor well knows the use of a j^'ood astrono- 
 mer on boaril his vessel to ^uide it through the trackless ocean. 
 The sublime science of astronomy is built ui)on .!»-eonA(*trical know- 
 ledge : ami the telescopic oliservcr keeps up, as it Aveve, a con- 
 versation with the heavens. In .slunt, all the ele<'ancies of life, 
 and most of its conveniences, owe their existence to tlie <feome- 
 Irical art. The veiidable world abminds with productions shaped 
 witii autre than human cunning- ; the beautiful tracery observable 
 in the petals of some tloAvers is really astonishin<>', and the* nmst 
 exact proportion of the parts is always ]ireserved. In the mi- 
 neial world, a sinnlar tiiith forces itself upon the iina^imition ; 
 and wlH'i'cver the eye of man has lieen allowed to |»enetrate, the same 
 neonu'trical havmony is found amon^' all the parts of created matter. 
 And w.iat is the f(uii)(hition, the gioundwcuk, upon Avhich this science 
 of ,';'eonu'tiy is built ? Why, so to say, on but oiu' or two elementary 
 pi'opositious, which, if the truths implied of them did not exist, the 
 Avhole scienc<' must tail. One of these fundamental theorems is, that 
 in every ]»lane triangle the sum of the an,y,les is constant and equal 
 to two rinht anf>les, or to li^O decrees, wliereby, when any two of 
 them are known, the third is fouml with the utuiost facility ; and 
 thus are we enabled to arrive at the distances which separate us from 
 inaccessible objects, or wliicli divide those objects from each other. 
 It is thus that the surveyor and engineer, by the hell) *^^ ^ base-line 
 of measured lennlh, and the angles observed at each extremity 
 thereof, can, by an easy f^eouu'l .ical construction, or l»y arithmetical 
 ealculati<»n, arrive at the exact brea<lth of a river which it is intemled 
 to l»rid;^e. Thus, also, <loes the astronomer, by adopting a broader 
 base, arrive at the diameter of the eartli we inhabit, wiihout the 
 nt'cessity of <j;oin^ rcmnd it. The earth, in its t\irn, is nnide the basis 
 for computing; the <listance of the moon, the sun, and the jdanets ; 
 and when, as in the case of the tixed stars, this basis fails,— when the 
 astronomer at ojtjiosite ends of the earth, or with a base-line of sonic 
 eight thousand miles in extent, fails to elicit any ditt'erence between 
 the sum (tf the (d)served an,u.les and two right angles, he bides his 
 time, and, taking one of them on any convenient day, he takes the 
 
10 GFO^FF-TRY, JIKNSURATION AND 
 
 KOfoiul that (liiy six luoinlis, wlini, with the ciirlh in its yciiily revo- 
 lution about the suu, he sliall have aiiivitl al the o[»j»(>site end (»f llie 
 eartlrs oihit, and have thuw wM'uied ji base-line <»(' nearly 2(ll),0<)(>,()(K> 
 of miles in extent —thou^li, escn so, he is sorely died, and has ahno.>l 
 suj)erhuniaii <lif'ti(idtieH t() encounter in solving this v,reat ]»i'ohh'n» of 
 the distance of tlie svars : for, will it he Ixdieved .' — s<» tremendous, 
 8o iiie(»nceival>le is this distance, thai the thiid ;in.u,le of tlie trian.i;le 
 — the one opposite to this innuense base of i2(M) millions of miles— 
 this third au;;le, 1 say, is lait the fraction of a second, or of th,- i',|,,tii 
 part of a decree. And yet, strange contradiction, so ^('ry small is 
 this enormous distance of the nean-st lixed star, and from Avhicii li,'j,ht, 
 thou<;h it travels with tiie inconcei\able velocity of 2()I>.(*0U ndh-s in 
 every second of t'lne, re<[uires three years to icach us,— sosnndl is it, 
 in t'OUii)aris()n with this boundless uni\ cise, that there are stars ten 
 times — aye, 10,(M)() times — more remote, ami iiirther still, beyond the 
 BXiace pe?ietratin!i poweis of the most potent telescopes, siu-h as that 
 of Lord Kosse, Avhich is not less than six feet in diameter, and Miore 
 than ()0 feet in length. 
 
 And if this property of the triangle did not exist-— if the sum of 
 the three angles were not a c<uistant quantity, — then shoidd we pro- 
 bably have remained ignorant, and torever — not, j)eiha|is, of the size 
 of the earth itself, which may be girdled and sul'inilted to diicet 
 measurement, but of the distances and sizes ol'all o!»j«!cts exterior to 
 the earth or (uit of our immediate reach, and man deprived ofoiu; 
 grand and inexhaustible source of enjoynuiit. 
 
 There is anotlier property of triangles nn which I must dwell for 
 a moment : it is this, that when similai- or e(|uiangular, their homtdo- 
 gous or corresponding sides are piojtortional ; and from this |»ropeily 
 it occurs that in every right-angled triangle the s(piare of the liyjx)- 
 thenuse is ecpuil t > the sum of the squares ujion th(^ other two sides. 
 These two i)roposi!i(nis or theorems, tog'('ther with the moie im[»or- 
 tant one already alluded to. are at the foumlation, so lo say, of all 
 geometrical science ; and all the other theorems and problems of 
 
 geometry dej)end intimately upon these for their very existence or 
 solution. 
 
 Because the sides and altittides of sinular triangles are propor- 
 tional, it follows that their areas are as the sipiares of any of their 
 corresponding dimensions. That is, if the base of a triangle be 
 doable that of another, so is its altitude also double ; and as twice 
 two are four, the aica of the sec!)nd is four-tinu's that of (he lirst ; or 
 if the base of the one be three-limes (hat of (he other, ho will its 
 altitude, and the area .'i-times .'{, or 9-times that of the first. Hence, 
 while the lim-al dimensions increase as the natural numbers 1, 2, 3, 
 &c., the superficies increase as the squares, 1, 4, 9, &c., tliereof ; and 
 
THK STKREOMRTinCAL TABLKAU. 11 
 
 this iitTmils n way of dividiiii,' iuiy trinnefiilnr fii^nivo ov fjpnoo into 
 ]»u!'tioiis wliicli shall he (-(iiiiil to ciu'Ii other, ot bciir to oacli other any 
 i('(|nir(Ml ratio. And what iis true of simihiv triangles is also tru(M)f 
 rill oilier similar lifi'iires— that is, of . such as are made up of. aveeai)al»le 
 of heiii;;' divided into, an e(|nal number of similar and similarly 
 situated t riauules. Aiijain, because «'very rectilineal fi,i;ur(> niay be 
 divided into as many tiianyles as the fi.n'uie has sides, less bytwo.aiid 
 as the sum of the ani^'h's of each of tin' constitnent tri.mji'les is eiiual 
 to two ri.^'ht angles, therefore does it follow t'lat the siuii of the 
 interior aiiji'les of any (juadrilateral or four-sided iiu;iire is 4 ri<;ht 
 an;.':les ; (»f any iieiirauon, the sum of Hm' an.u'les is e(|na] to (! ri.yht 
 aiii;k's ; in an octagon, 12 light angles ; and so on. 'I'liis inqiortant 
 |)ro]!erty enables the land-surveyor or engineer, after measuring the 
 angles of any tract of land, to test the accuracy of his triaugulatiou, 
 and detect an error, if there be one, since the sum of all th<' angles, 
 when taken togeth(>r, and whatever fractions of a degree or minute 
 they may severally i'onlaiii, must, of necessity, make up an exact 
 nnml>er, and that, an even number of right angles— !;i, 4, (). S, 10, or 
 IJO, as the case may be,— but never •'{, ."), 7, !•, or any other odd 
 number. 
 
 1 have alluded to tln^ ciicle as being the most beautiful of all 
 figures, but have as yet said nothing of some very useful properties 
 with which it is endowed. F(»i' instance, an angle at lis centre is 
 measui'ed by the arc which it subtends ; this we all know, and that it 
 slnudd be so does jiot a])pear at all strange : it s(><'ms, on the contrary, 
 that it should not be otherwise, and hence it suiRees to define the 
 thing, o)' to enunciate the proposition, to have it credited. Hut it is 
 Hingular that wlien the ap<'X. of the angle isinoi'on the circumference, 
 such angle is imt one-half— not less, not more — of the corresponding 
 angle at the v'cntre; and out of this arises the very curious and useful 
 property that all angles in the same segment are<>qual to each otliei' ; 
 also, that (>very angle in a seuii-circle is a right angle, llem'e the 
 possiI)ility of drawing a tangent to a circle from any point without it ; 
 hence, iilso, can a right angle be most easily and readily laid out by 
 the draughtsman on his boai'd or ]»ai>ei', or by the surveyor in the 
 lield. Hence, again, can the very })retty and useful ])robleui be 
 solved of liiiding a nu'an ))roporti(Mial or a geometrical mean between 
 any two given Hik^s. a giaphic nuMh' of extracting tlie stpiare ritot of 
 the ])i()duc.t of any two given numbers, (U' of linding the side of a 
 s(|uare eipial in area to that of a given rectangle. I5ut, in the same 
 way as a geonntric mean may be found between any two lines, so 
 can <'ither of these last be determiiu'd when the other is known and 
 the mean between it and its fellow ; and in this way is the (iigiueer 
 <'nabled to lind the radius of a railroad curve, for these curves are 
 usually of vast extent, and, unlike the circle on a board or sheet of 
 
12 GKO.MKTIIY. MENSURVrrON, ANt> 
 
 impor, the oontiv t';inii(»t be mmmi nor foiiiid, nor can the radius he 
 inea.sun'd ; or it'you coimc across a jxirt'on ol' a slinii|), and liavc a. 
 curiosity lo know the size of tree cut IVoni it, (haw any chord acro.sH. 
 it, bisect it, scjuare its halt, (hvide the i>ro(hict by the versed sine or 
 hei^^ht or breadth of the segment at its centre, and there will conu^ 
 the rest of the dhuueter. Tins is not all ; for on a scale 1(1.00(1 times 
 more vast, or even millions, does the astcoiiomoi' comitnle, and al- 
 most in the self-same way, fron» a knowledi^c cd' a ndntite itortion of 
 the air or orbit, those tremendous circles, eccentric thou,uh they be. 
 which satellites sweep out around their primaries planets around the 
 sun. Yes, and as the enj;ineer, without a center or a ladiiis, can 
 follow out his curved track amou^' the woods ami. waters of the 
 earth, also does the astronomer tra«'e out his mi;;hty circuits tliroui;ii 
 the starry forests of the daik blue heavens. Aye, even is the erratic 
 comet in this way foUoweil up with eye intent upon its ever-\ aryini^ 
 direction iinu)n;'' the stars, and from the minutest portion of its cir- 
 cuit eau the ellii)lic lij;iive be computed whicli nill enable tlie tinu' 
 of its ])eriodic return to Ite iMi-dictcd to a certainty ; and the sanu' 
 courses of observation will also tell if the path auu)n;i,' the ))lau»'ts be 
 not (dlii»tic, biit lather parabolic, or that of sonu' strange uietecn* 
 wliich is approachiu^j; this world's precints for tlie fust time, and wil! 
 leave it, nt^ver to return — unless, to be sure, the palii described 
 should approach very nearly to the elliptic, and, as in political as- 
 tronomy, the iutlueuce or attraction of some j^reat planet so swerv(> 
 it on its way, so alter its direction, as to brin^- abiuit the phenonu-na 
 of a trausf(U'mation. 
 
 I have just now said that the an<j;le at the circiunfereuce of a 
 circle is half the anule at the centric on the same aic. and this pro- 
 perty eau be tuiiu'd to u,reat account. In surveyin,y the coast of any 
 c«Mintrv, with a view to layiiiu,' down on charts and ]na))s its shoals 
 and breakers, th(Oiydi'OL:;rapher takes his ansiles fron» points imme- 
 diately over those of which (he ])(>sitions are to b<' asceitained to 
 three (h- more points on shore, of which the distances apart aie known. 
 Now, were the an<iies Just alluded to adjacent to any one of thi^ 
 nu'asured distances, the solution of the problem would reduce to 
 that of deteiiainin;^' the third an,i:;le and otlu'r two sides of a triangle, 
 of which two an;;les and a side were known. Hut tiie elenuMits are 
 not ill contact; they are not adjacent: how, then, can (hey be 
 bi()U<;ht together for the purposes of calculation '.' \N liy, by this very 
 beautiful property of tlu^ circle Just enunciated — that an an<;le 
 havinif its vertex in the circumference is e(pial to any other 
 allele similarly s'.tuated and bearing' upon the same arc, whereby, if 
 a circle be described around three of the points eoiioerned in the 
 problem,— and, as wh' all know, a circle can be so described, — and 
 if the two angles, takcu from thu hydi'ajrapUical poiui uudur cuii- 
 
TMK '■T!:RF,()MK'I':Ui'.\r, T.\ VJ.I' AT. llj 
 
 Bidi'iMtioii, arc nnd!" to travel round tliv circle (ill the ajx'X of 
 cacli of tliciii arrive at llic opposite exiiiiuilic-i of the liase, tlie 
 \v!io!e diiii.iilly will have heeii made to disapp:-. r, and the proldoii 
 he reduced to that of solving- a ca^e in phnie tri,:;(tiioiiietr,v. la 
 this way did a I'oiiiier pmiil of mine. Mr. li. Steck( 1. now ol" (he 
 Deuutment of I'nhljr Works at Ottawa, solve, in a most in.i;('nious 
 an I simp'.;- ni iiiacr, t lie |tioh!em of the i:it;'r|> )laiio,i of a hase-liae, 
 or of lindiuii' the nid<nown porlioa. !> ( ', of a str.ii.y,'lit line, A l> (' D, 
 to whicii tiii'<'e angles had l)een taken from a liflli point, i'. of whith 
 the position was to he ascertained. This solntion is ^iveii at i>a,^(' 
 •JVl of my treari;e on (Ji'onr.'try, etc., pnhlislied in l.-^iii!; and at 
 })i:;i' '177 of the same wiU'k is a nr>st inj,v'iuons solution of a rather 
 dillic dl pro')]' ni in th"divi-ion of lands — that of a (|nadrilateral into 
 (■([1 d Ol' proportional areas, with sid 's ah;) [)r.)p )rtional to those <»f 
 the wimle finnre. 
 
 'I'liere are yet a few other properties of the circle which \ must 
 m)lice ere I take leave of tiu- sui»ject. Thus, tw(» tan.ijents drawn to 
 a circle from any point without it. are equ.d, and ian^fui is p.'rpen- 
 dicnlar to tliit railius w iiich is drawn to its p;)int of contact ; and 
 from these eircnm-iranees, ;iiid tin' t'a"t alr<Mdv (Miiraeiated, tjiat tin' 
 sum of the tiire<' aii.i^les of any trianide is e([ual to twori.iiiit anj^les, it 
 Iti'com's possdde to calculate tiie diameter of the earth l>y m:'i<'ly ob- 
 servini;' tin- an:^le of d 'pression of tin- horizon from the top of .i nn)an- 
 taiu or other (d'vated sit a ition, t!ie ir'i.!,iit of wlii.di ahove tin- earth's 
 surface is kiiown. A eircle can always l>e described capable of con- 
 taining,' a iH'iveii annie on a ,i;iveii base; hence, for instance, if the, 
 hei'^iit of the fl iLj-stalf o!i the citadd b(> known, and the an^le it sub- 
 ten Is tVnm the o;»i)>dre siile of the rivi'i', t »;j;'th:'r with t!ie distance 
 across, the lieixht of r!i ' cit nl d itself m ly be eom[)Uted. 'I'lie yeome- 
 t lic.d soluiion or construction of tins problem, as m my others of 
 practical utility, is ii'i veil at l)a^es ^J.'iv* to UiJl of my treatise. Out of 
 tile fact tliat an an.i^le at tin- circumfi'r -lu'e is half of that at the 
 centre, there arises also the condition that to iirseribe a four-sided 
 linni-e in ;i circle, its oppusite an.yles, taken toi^ether. must be e(pial 
 to two >ii;'Iit iUi.u'les ; and because any tw<» liin-s or chonls which lait 
 one another in a circle ha\*e tlie pirts-of tiie one proportional to those 
 of the other, the diameter or radius c;ui be found of a circle of which 
 a zone, or portion included between any two parallel ehords, foruM 
 part. .\nd, a^iaia, out of tin- circuui^tance llnl if t a'o lines be drawn 
 from any paint without u circle to tlieopjtositeor concavj' side thereof 
 these lines are reciprocally i»roportional to their .segments situated 
 witiiout tlu' circle, there arises one of the modes of solvin;;' that case 
 of plaiu' trigonometry wherein the three sides of :i triangle are yivoii 
 to tiud the angles ; and a tanu'ent drawn from the same exterior point 
 to Llic cii'clu is u ^'oouiolriuul mcuu, or a uicau pruijui liouul bcLwocu 
 
14 OK.OMlTltV, MKNSniJATroV, AM) 
 
 eitluT of flio nforctiifin iitiH (I liiics iiiid its exterior pint ; w lieneo tliern 
 arises ii mode of niiuiiiiu' ji r.iilro.nl <Mirve, lor iii>t,iiiee, t!iroii'j,Ii niiy 
 two n'iven i»')iiiN i'.ud tiiii,';eiil to l!ie str,ii;^Iit or emved portion of 
 anoiliei' road. 
 
 llow iiinch more wliicli I luive not time to reliite ? -and vet liow 
 strange, (liiif ol" I iiis tlie most i('L;nl:ir of all fi'^ur<-s, and wliieli man 
 lias niide snbservieiir lo s » miny piiri)oses of praetieal utility, — 'his 
 tiy'nrc, wliieli il is so very e.isy to trace out, neither Ins the eir.'nin- 
 ference nor the area yet heen i'oinnl. Archimedes, as I have already 
 said, foiind an approximate ratio i\ty the diameter to 1 he circnnderence, 
 that of I to '.V. ; M-'tins, a ration of Ii:{ to ;]").'); und other nnthema- 
 ticiann that of 1 to :{. M! "iit'J. etc. In I ');iO. Cenlen. who livol in th(^ 
 time of Metins, extend'd the cdcal ition to '.Vi decimisls, Avhich were 
 on<;raveii on his toiub. lie arrived at this result by r ilcuhitin^ the 
 ehor<ls of sin'cessive arcs, each of which was the half of the ])vecedin,<? 
 o]ie ; tile last arc in this eas" Iteino- i lie side ot a polygon of .'it!. H',»;], 
 488,147,41!>,I():{.2:{:{ -nearly .-{r !>iHions of times 4l;> million of sides. 
 The moile of calculation was thereafter iijreatly simplifn'd by Snell, 
 who, with tin' help of a poly iron of only .i,'J!::3,HS() sides, carried the; 
 ai»pi'o\imation to 5") pla.ee; of iiii'nres. The comimtation was dnrini; 
 tile last century continued Ity other mathemilicians, who snccessively 
 carriecl the nuinlier of li,:;iiies to 7't, KM), I2S and 1 40 decimals. 
 
 Xotwithstandin;.!,' that l/inihert. in I7<)I, and Lencndre, in his 
 elements of ireonietry, have jiiovrd tliat the ratio of the diameter of 
 a circle to its circumference cannot l»i' exju'essed in numbers, the 
 desir(^ to satisfv those wlio still hoped to liiid this ratio led other ma- 
 thematicians to eontiiuu' adding- to these fi/fiires. In Hlti, ij(l(> 
 decim lis had been olttaine I, aiid'^Kt the foliowinj; year. In IH.")! the 
 number w.is extended to MITi; theti to •'}'>(». Shanks carried it to 
 527, ijnd in Irf.").'} to (507 places «>f decimals. Wiien it became evident 
 that the arirhnictical «'Xj»i'ession for this ratio was out of the (juestion, 
 many perscni continued to hope lor som<' .u,eon)ehical solution to tln^ 
 far-famed i>rol)lem : bvtt it is <;-enerally re('o,<;;nized at present that 
 this method is iminaclicalde ; and it must be admitted that there has 
 resulted but triiliiii;' advantage, if any, from th<' enormous time and 
 troubh devoted to this famous ])roposition. The Fieiich Acaih'my 
 of Sciences, in 177."), and, soon after, the lioyal Society of London, 
 with the view of dis!'ourau,inu,' siicli futile and fruitless researches, 
 r<'fused to take further notice of any comnmnieatiou relating; to tlie 
 quadrature ol' the circle, the trisection of an an.ule. the dn])licati(Mi of 
 the (aihe, oi' perpetual motion. An approximation of (iOO li^ures, or 
 even less, is e(piivalent to iierfeet and abscdute accuracy ; for, let it 
 be remarked : it sultices to take in 17 decimidsonly to avoid an eiror 
 of the thousandth i)art of an inch on the six ]iundre<l millions of 
 miles which constitute the length of the orbit of the earth arouud the 
 
j TiiK sTKfU'MMF.i ii;('.\r, '1'.\iu,F':att. • 15 
 
 sun. Ton (IccIiiinN '.vill afford llic (inunit'fn'iicc of tlic cirtli to 
 witliiii !iii iiK'li, and Vi dcciniiils, to within tiic tlionsandtli jt;iit ofaii 
 inch, or less than a liair's Ijicadth; and, at any rate, thclgiK's 
 already found nioii' tlian snlHcc for the ]>iii|>ost' of (li'tcrniiniii!;' with 
 iil)so!iifc aciMiracy not only llic dinn'nsions and distanci's of tlic jtla- 
 lu'ts, but also of the most distant starsi or nohnla- tlnit man «'i!n dis- 
 cover witli the Indp of the most powerful and space-peiietratinij; 
 telescopes, or of tlxtse which he miuiit discover wi(.Ii optical instru- 
 lueutH 10,000 times more powerful than those wliich h(i already 
 l)ossesses. 
 
 Alludint;', a.n'ain, to the celebrated problem of the tris'-ction of an 
 ani;lc, 1 must once nH)re talce occasion to protest, thus publicly, as j 
 have already done, thoiij>h witliout result, at ])i!,!;'(; 'i'-V) of my treatise 
 of IHtiiJ, not anainst th( i)retended and ridicubms solution ol' this ]U'(»- 
 blem by a certain Mi'. Tliorpe, (•!" Ottawa, after, as h<' says, — jyoor 
 man ! — dev<Uin,!4' .'U years of his life to its discovery, but against the 
 ;4()veiiiement ol' ("anada — the I'atent-ollice — for sanctioning' the [U'c- 
 tended solution by the issue ol" Fjctters-Patent cori'oborative of tlu* 
 same, and thus setting; the opinions of t!ie ^'overnmciit ollicials 
 before those numerous sdranlyi of fiuropc aiul other countries, who 
 have, and had lon;^ before that linu'. declared the .i;eom"trical solution 
 of this pr(»l)lem to be inipossibh' — lhou<;h thei'e are, of course, as with 
 the circle, uMxIes of ajjiu'oximatinu' to tlie true solution to witliin 
 limits as nai'row as any tli'it can be assi'^r.ed. 
 
 A few more reiimrks on sonu' properties of certain plaue ii,^•ures, 
 and, 1 shall have done with this portion of my subject. 
 
 In ])arallelo,nrams (I ueed not remind you wliat they are; their 
 etynndooy is su,y<;'ostive euouu'h of that), Mr. Steckel shewed that 
 each of tlu> compbunents about the diameter is a mean ]U'v)]>orti(nial 
 between the com])()m'nt ])aralleloi;rams about tlu> same, a ])roperty 
 which T fortunately conceived tlu' id<'a of applyin;^ (see ])a,n'e 1!>0 of 
 my treatise) to the solution of a ])robb'm of fre([ueut occni'rence in 
 the division of lands by a strai,:;lit line runnin,^,' throu.i>li a ^iven point. 
 This ])roblem was previously a matter of some diiliculty wlu'tluT 
 alu'cbraically or <j;eonu"trically considered (see ]ia<;'e 510 to r>'22 of 
 *' t!illes])ie's Land-surveyijig," where tlie Ibrinuhi runs over three 
 liiu's of type). 
 
 The re,i;ular hexa;j;<m or six-sided polygon is the only figure, ex- 
 clusive of the square and equilateral triani!,le whidi will fit to<j:ether 
 without leaving- a sjjace between, as d.-es tii" octagon, tW instance — 
 a fact which nuiy be seen exem]>lilied in i)ait(U'-hang'ings, in the 
 l)atterns of oil-cloths, and in marble and mosaic tilings. Now, the 
 very hov kiH)ws its geometry so well, that it builds its cells in hexa- 
 gons. The S(iuare or triangle would have fuliilled the condition of 
 
18 (5i;n'\tRTRY. MV.N'STJUATION, AND 
 
 lc;iviM ,f no iiit'Tslic!', no lo ;■* «»t' >]>)(•!• Ix'twc 'u tli ■ I'clls ; Itiit iiciilicr 
 wo'.iM li.ivi- been so well ailipti'il to tin- iiliiio^f circiil.tr sliapi' ol' the 
 insect's iiodv jis i> llic lic\.iv,oii. .Mv voiim; tViciuls will, of coiirso, 
 «u.i>',:j('st tint tlic circiiliir iiistcul ol' tlic lii'\iii;(»n;il would liiivc Ixni 
 even ii hctlcr slinpc for tlic Ium- to dwell or move in. (iijinted ; but 
 (•iirle. like oi'tMuon- and other li^nres. will not (It eaeh other wlth«)nt 
 loss ot" space ; and thei'e is another, and. no do'iht. mach more iin- 
 ])ortant consida'ation to the be<' : it is that A\ith tln' hexan'on each 
 coni])oni'nt wall or ]iartition answers for two nil.joiiiin,u' cells, whereas 
 Avith the '.i'cle or c,\ Under a whole one would have been reqnired for 
 each tiny animal, and the necessary <piantity of wax therel»y lU'arly 
 doubled. 
 
 There is, th.'re has loiii;' Iceu. a ti-nden.'v towards uTmaalizatioii 
 in the exact and ot]i"i sciences, and wilh more reason now than ever. 
 Two tlntusand years aii'o. when Muclid lived, sreim and electricity 
 were unkn(»wn ; pneumatics, optics and chemistry W( I'c not practised ; 
 photo.nrapliy was not dreamed of. In those {\,\y!<, one could afford to 
 devote years to the s(de study of m itln-matics. We cannot do so now : 
 life is too short, and tluac are too many ihinus to learn. Imbued it 
 Avas with this idea that f wrote my treatise of l.'^tid. Nobody. ajtiJa- 
 rently, had dai'eil before me to lay his sa( rile^ions hands upon the 
 venerable teachim^s of the jjcim-e and patriarch of yconu'ter ; neither 
 have r dom' so; bat what I have don<' (and that I was not far wron,i>' 
 in actini;- so. Mill, I think, be admitted) has been to lednce, by nn)re 
 than oiu'-half, the separat(> and demonstrable propositions of the 
 Greek geomet<'V, while retainin;:',' the whole of his conclusions. 
 
 The fifth book 1 have cliiuimiled altouether. 1 have r» moved it 
 from the elements, and ^iven all its teaching's i'.; my " principles, " 
 niakiuix axioms ol'sonie and corollaiies of others of Euclid's proposi- 
 tions ; for. I hold that t(» conceive and admit the truth of r.n axiom, 
 there takes place within the numl a certain jnocess of reasoidng, 
 liowever short it be. For instance, equal ratios are equal quantities, 
 and quantities which are equal to the saau' or to etiual (piantities 
 are ecpial to each other. It, therefore, follows, as a mere corollary 
 of tills axiom, that " llatios which are ecjual to the same or (<» e(pial 
 Tiitios are e(pial to one another ; " hence, I do m>t see the necessity 
 of making this a demonstiable inoy)osition. A^ain have I nnide an 
 axiom of proposition F of I'layfaii's Fuclid, and Justiliably so, I 
 take it; for quantities which are made nj) of the same or of equal 
 qnai.tities are e(pial to (me anotlier ; and since ratios are quantities 
 — numerical ones — therefiue an; '' ratios which are composed or 
 made nj) of the same or eipial ratios equal to each other." Of Euclid's 
 2nd and livd pro])ositmns, book !., I have made postulates. (Jfhis 
 22nd I have macU' my 1st, and thence deduced his 1st, as a simple 
 consequence thereof. Why, for instance, n. «,ke a theorem of the 
 
THE STEREOMKTUICAL TAIJLEAU. 17 
 
 Minnciiition tliiif t\v«>lii)0>tit;ir!illcl tn i\ .'Jrd iirc uniMllcl Iooih^ nnotlier, 
 or thill twi) tri;mi;l»'s similar lo :i tliiid ari^ similiii- to one aiiotlicr? — 
 lor, what coiistitiitcs this i»arall(lisiii of tlic lines, tills siiiiilaiity of 
 till' tiiaiiinlcs, l»ut ('(pialily u\' dislaiici' in the llrst and ('(|iiality of 
 aii;;ii1ar sp.icc in tlic second ? — In nee have I made of the foinier ji 
 foiollaiv to niv <lelinilion of paralh-l lines, and of the lattei' a corol- 
 lary to my delinilion of similar li;;ui'es. Of Knclicrs .T)th and ;{(ilh of 
 tiie 1st liook I have made l»nt one ]>rop()sition ; for Knclid himself, 
 u lio in his llli and Hth of ;he sann' hook ]>laces his li;i;nres the on(^ 
 upon the oliiei' to prove their e(pnility, mi;ilit in the Hame way havo 
 snperposcd tiie eipial bases of his paialleloL!,riinis, so as then to consi- 
 der them as one and the sann' l»ase, which would have allowed him 
 to maiie of the second i>roposition a mere corollary of the lirwt. 
 Amain, with Kuclid's two next propositi<»ns of (he same book, his .'{7th 
 and 'i-^i]\, and after his own assertion in his axioms that "thin;;s 
 which are halves or donbles of the same thin^, <>i' of e<pnd thinj^s, are 
 equal,"' why did ]n' not reduce them to nu're coiollaries of his .'}.'{rd ami 
 .'Mill .' I have in the second book added a lemma, which shews how 
 important a proposition is the liftli of that book, and how frnilful in 
 resnlts. I canm)t hnt a.^ree with Clairanlt in saying' that if Enclid 
 considered it necessary to denionstiate such a self-evident propo- 
 sition as that a line joininji,- two ])oints in the circiimferenee lies eu- 
 tiirly within the circle, it mnst be because lu', had to answer and. 
 confute the objections of oltstiuate sophists who made it a point to 
 refuse their assent to the most evident truths ; for as Avell min'ht it bo 
 attemitted to l)e proven that the diagonal of u sipum; lies within and 
 not without the ligiire, or that the centre of a circle is within it. 
 What dillerence is tlieic between linding the centre of a circle or of 
 a portion only of its circumference ? And again, what difterenco 
 between circumscril)ing a circle about a triangle and making one 
 pass through three given p«»ints ? Why, then, did Euclid or his com- 
 mentators nnike of these i)roblems as many diiferent propositions, 
 Avheii they really constitute but one and th(i same operation ? A 
 difterent solution of Euclid's ;}-"}i<l of the 'hd book allows of reducing 
 its three several cases to one ; and soof his.'Mth and.'?(ith of the same 
 book. Simili". ])roce'.ses have been followed out b^- me in reducing in 
 number the demonstrable propositions of the 4th book; and in the 
 5th, which, as already stated, I have put annmg the principles, the 
 substitulion for " magnitude " of the woid '* (luantity, " with its 
 Mgnilication delin<'d, to c(»mpriseuunn'rical as well as other quantities, 
 has aUowed of my leasoning on numbers, and giving, as I have done 
 the mode of arriving at the numerical and practical solution of the 
 many problems propounded in my work. Tn Euclid's sixth, why 
 should 14 and I.') be separate theorems, in vi<'W of axiom two, which 
 sets forth that what is true of the whole is true of the half? These 
 
18 nEOMF.TRT, MKNSTTRATION, AND 
 
 citations will Hiiilicc to give an idea of tii(> imoccsm of rctluctioii iiiiil 
 geiicializali(»ii followed out by iiic in tlic gconit-liy of lini'H and 
 wufact'H ; and in the same way huv(^ I nuKlill«;d the ordinary <lenioii- 
 strations of Molid ;>eonu'try, and of plaiu' and sphericiil tri;;(tnoni('try. 
 Nor is my treatise less strictly lo;;i(al in all its teaeliin^is than that of 
 Euelid, every proiiosition deiiendin;j; for its demonstration or si>lutii)n 
 ou thoMC that came before, and in no way on those that follow. 
 
 MENSURATION. 
 
 ,' AREAS. 
 
 Every trian<j;le, it is evident, is tlio half, th(» exaet half, of'itH 
 corresponding i)arallelogram. Now, the parallelogram is, in area, 
 equal to the rectangle of the same base and altitude; for, if the 
 oblique or triangular i)ortiou be eut from om- end and added to the 
 other, tlie figure becomes a rectangle ; and as the area of a ;e<'tangle 
 is eqiuil to the jnoduct of the number of units in its ba-^v and altitude, 
 it follows that the area of any triangh; is eciual to half the product of 
 its length and breadth. This, then, may be a(h)i)ted as an element 
 into which all plane tigures can be <li\ided, and their component 
 areas made np separattily and put together. In the case of the regu- 
 lar polygons, tlie computation of their areas becomes simplilied, as 
 they can be divided from the centre into as many eipuil triangles as 
 there are sides. Tlie area of the trapezium is half the ]»roduet of its 
 altitude into the sum of its parallel sides. 
 
 A sector of a circle is nothing but a triangle of equal altitude 
 throughout, or having a circular base, every point of which is (Mjui- 
 (listaut from the apex ; and its area is, therefore, equal to the half- 
 product of its base and altitude ; for its arched base may bo conceived 
 to be divided into a number of parts, sucli tliat each of tlu>m shall be 
 ■without sensible error, a strfiight line, and hence the rule ; for it is 
 evidently the same thing to compute separately and take the sum of 
 the component triangles of the sector, or to add together their conti- 
 guous bases and multiply, once for all, by the altitude or radius. 
 Again, the whole circle is but made up of contiguous sectors or 
 triangles, whence it follows that the area of any circle is equal to the 
 half-product of its circumference and radius. Next, wo have to con- 
 sider among plaiie figures the segment of a circle, or that which is 
 included between a chord and its corresponding arc ; and this is 
 evidently equal to the area of the sector, less the area of the triangle 
 formed by the chord and radii. Now, the lune, a figure like tlic new 
 moon, and hence its name, formed of two non-concentric arcs of the 
 
THE STRllROMETItinAIi tahleau. 19 
 
 'flnmo or (liffcrcnt nidii : fli(^ nrca '»f tliiw fifjurc is tlio diffcronoo of its 
 two OMihitoiicnl s(';;iiiciit>*, s(»(liii< ii iiH'ic repetition of tli<' jtroccKs just 
 (Icscrilx'd will inciisiire ilssiipcrlicits. TIic /t>n('. or i>orti<ni ofacirclo 
 Ix'twct'U two paralli'l clioids ciiii also In; conccivj'd as tlic dillVrenco 
 botwoi'ii two HCffincnts, or as made ui)ot'a trapcziuiu and two scj;- 
 iiifints, and its urea fomul accord in«>ly. Concentric and eccentric 
 rin<fs arc, of conrsc, eqind in area to tiie difTerence of llieir component 
 circles. Of the ellipse, the, area is e(|nal to the product of its diame- 
 ters into decimal ./riol ; tor it is found, in the manner liereinabovo 
 set forth, that of a circle whose, diameter is one, the area is .7854 : in 
 other wimls, the area of the <'ircl(i is about 78 J percent of itscircum- 
 8cribin<i: sipiare, so that this .".rea is mon' quic^kly arrived at by 
 squiirin;;- the diameter iind reduciii<; the result in the required ratio ; 
 and the ellipse bein^f analogous to the circle, its area is found in a 
 correspoudin^if manner. The area of the i)arabola, I may add, is just 
 i of its circumscribing rectangle. 
 
 Tliere is a more general mode of arriving at tlie areas of all piano 
 ligures; it is by dividing them into a number of trapeziums by a 
 series of equidistiiiit parallel lines or ordinates, and of multii)lying 
 the common breadth or distance between the lines by the sum or 
 combin«id length of all of them except the first and last, of which 
 oiie-hiilf only must be taken; and this applies to every imaginable 
 iigure, in which, the; closer the ordinates, the uiore accurate, of course, 
 the area. There are some few tigures bounded by curvilineal lines, 
 of which it is, nevertheless, easy to conii)ute the areas; for instance, 
 where a convex portion thereof is compensated by a corresponding 
 concavity, as in tlie develojyped surfaces of int(!rsccting vaults or 
 arched ceilings, or, on a smaller scale, the developed n\Tii of the 
 elbow of a pipe of cylinder of any kind. Finally, there is a mode of 
 measuring the surfaces of very irregularly-outlined figures by a sys- 
 tem of compensating lines which are drawn so as to include such 
 portions of tiie sujx'rticial sjtace outside the ligun^ as may make up 
 for, or be equivalent to, the portions left witliout the lines. 
 
 SOLIDS. 
 
 Tlie measurement of solids comprises that of their surfaces as 
 well as that of their volumes or solidities. Solids may be classitied 
 as prisms and prismoids, cylinders and cylindroids, pyramids, cones 
 and conoids, spindles, spheres and spheroids. Prisms are solids which 
 have two equal and parallel ends or bases, and of which all the other 
 faces ar(! paralleh)gTams ; so that the prism is equal in diameter or 
 breadth throughout its whole extent, and dilferent in this respect 
 from the pyraaiiid or prismoid of wliich the sides incline or taper 
 
20 OEOMETRY, MENSURATION AND 
 
 towardh one end of the iigui'c. A pvism iiiiiy liiive a tiiaiiulo tbi its. 
 base or any otlier figure, and is called, after tlie iiafiivc of siicli l)iis(', 
 trianfi'ular, (|nadiini,n'ular, pentii^^oiuil, and so fortli. 'Die cvliiidcv is 
 nothing but a pi'isni havinj;' a ciivuhir base or jioly^'oii ol" an iiirinite 
 nmnber of sides. Ani(nt,i>' prisnis, tiie i)arall(]oi>iiK'(loii is tliat of 
 which the opposite faces are parallel, as the name iiiii)lies. as in the 
 cube; in conseqnence of wliicli, any side or fiice of tliis or similav 
 solids may bo assumed as the base. We all known wliat a pariinii<l 
 is, and so of a cone, Avhich is only a i)yra]iiid Avitli a circular base. A 
 conoid is a species of cone roinidcd oft at tlie top like a pear, or lik(? 
 the ice-cone at the .Montmorency Falls, It may be conceived to be 
 generated or traced out in space by the revolution of a parabola, or 
 other like figure, about its axis ; and it goes by the niiine of the gene- 
 rating cm've, as parabolic, hypcrltolic conoid. The frustum of a 
 pyramid, c(»ne, or conoid, is that portion of the solid which remains 
 after the apex has been removed ; and it is said to l)e contained bet- 
 ween parallel bases when the cutting plane is i);irallel to that (Ui 
 Avhich the solid stands.. All these solids nniy be right or inclined ; 
 and if so, they require to be so (lesignatcil, as a right ])ent;ig()nal 
 prism, an inclined octagonal pynunid, an oblicpu' con(> or conoid. 
 The spindle, as its popular nanu' implies, is a well-known form, being 
 circular in its cross-section, and tapering from its centre towiirds the 
 ends : it may be gem-rated in space by the revolution of an arc of a 
 circle, (U'ofan ellipse, or by a p;iraI)ola or hyperbola, around a line 
 Avhich is called the axis of the si)indle, and, lilce the conoid, derives 
 its distinctive name from that of tlie generating curve, as a circular 
 spindle, an elliptic, parabolic or hyperbolic spiinUe, — not that the 
 spindle itself, as a whole, is a A'ery important solid, but that its 
 middle frustum is the gecnnetrical representation of almost every 
 form of cask, the world over. The spher(>, that most beautiful otall 
 solid forms, and which, as I lun'e already stated, contains within 
 itself more space or volmne than any other hgure of equal superficies ; 
 the spheroid or llattened sphere, the ligure of this earth of ours, and 
 of the moon, and sun, and i»lanets, winch, one iind all, are flattened 
 at the poles ami iirotuberant at the e([uator ; and, linally, the prolate, 
 or elongated spheroid, make n\) the varied classes or families of solids, 
 or space-enclosing ligures, with their frustums, segnuMits, ungulas or 
 hoofs, and otlun- sections which the linrita of a lectun; will iu>t alhuv 
 me more fully to dehne. 
 
 THE STKHEKKOJtKTIUCAI- TABLKAU. 
 
 1 come now to tluMUore immediate object of this lecture, that 
 which has been, perhaps, to some; extent, instrumental in jnocuring 
 for me the honour of an invitation by the Literaiy and Historical 
 
TIIF, STKREO:\IETRI0AL TABF.KATT. 21 
 
 Society of Qiu-ltcc to roixl ii pMi^cr '.villiin tlio cliissic i)i'('ciiicts of its 
 liistoi'ic lialls; mimI may I liopc I slmll liMVc ticiitcd tlic siiltjcct in :i 
 w'iiy to wiiir.nil tlic coiivtcsy iuid hr of sonic iiitciot to tlic \ i ry 
 ■liuinci'ous iiiKJ liii;'lily ii|>|ii'ccialivc audiciicc, tiic cHlr. llic iii istoriacy, 
 so to say, of llic cdiiciitcd or well-read ])ortioii of llie coinniiin.ily, 
 \\ liieli lias on this eveniuu lionored lue with its attendance and kind 
 and tlatteiin,!;' attention. I alluih', of eonrse, t«» the Slcrcoiiu'li-icdl 
 Tdhlvdn Avliicli you see ju've liefoi'c yon, and of which it behoves me 
 to say sonu'thin;^', even at the risk of apj)! aiin.i;- i)aitial to myself. 
 This Tableau, or board. \vlii<-h is mtule iii» of sonu' 'JOO models, each 
 of whicli can be removed and replaced at pleasnre, an<l ])ut into tln^ 
 hands of the pn]»il for examination, comprises almost all the elemen- 
 tary foiins which it is possible to conceive. .\mon<i' them, those 
 which J have jnst now enumerated, as [uisnis and prismoids, cylin- 
 ders an<l cylindroids, both ri.u'li and oblhpu', ar.d the frusta and 
 iinyidae of these bodies; pyrami<ls, cones and conoi<ls, ri;;ht and 
 oblifjue with their frusta and an^uidae or hoofs; the sphere, with its 
 subdivisions into hemisphei'c, ([uarter, lnilf-(iuarter or tii-i'ectan,uidar 
 pyramid, se<>'ments, zones, fiu>ta and un^nlae, and many other sec- 
 tions of tliis s(»]id ; the prohite and oblate spheroid, with their many 
 sections ami subdivisions; s])indles and their sections, including,' 
 nu)(U'1s of casks of all varieties; the five rej;ular polyhedrons or so- 
 called plat<niic bodies, though known before the time of Plato: there 
 are also a host of other varied forms, such as con.xcx ami concave 
 cones and other solitls, and frusta and annulae of the sanu' ; concen- 
 tric and eccentric rin^s, and a ceitaiu number of coni]tound liiiures, 
 nnnle ny of, or capabh' of bein^' subdivided into, the elenu'Utary solids 
 just enumerated. 
 
 ."\Iy object in the p!('|)aratiou of tin's Tableau has been to _<;enei'a- 
 lize and make easy and popular the study of solid Ibrms, and tho 
 mode of measuring their surfaces and their solidities or volumes. 
 The lati'ral faces or sides and the opposite or i)arallel bases ov .ends 
 of these s(»lids all'ord all the plane li^ures, from the triangle, s(piai'e, 
 and polygon, to the circle, I'llipse, ami paiabola, etc., with their com- 
 ponent sectors, segments, zones, and lunes. Of tlu'se you have 
 already learned to estimate the snperiicies. The nnxlels also alford 
 exam[des of conv<'x superficies, inclusive of the spherical triangle and 
 polygon, the spherical zone ami lune and segnu'iit, and of all the 
 compoiuMit parts thereof, 'i'he mode of aiTiving at the areas of those 
 is t(» assimilate them to (dane ligures by a division of the concave or 
 convex nurface by ecpiidistant <»rdiiiates drawn from the extremity of 
 the lixed axis of the solid, or that around which the generating curve 
 is sui»posed to have rotated in sweeping out. (he solid, 'i'he distance 
 betweeu the ordiuates, each of which, from the description just giveu, 
 
22 nr.oMF.TiiY, mensuratton and 
 
 is, of ronrsc, n ciiclc or i)oi(ioii of ;i ciiclc, i.-; iiiiido so sjiinll as to 
 iillow of llic iiitcrvcniii^- arc hciiii^- considered a stiai,)j,"Ii lino or tlierc- 
 jilioiits. 'I'lie finures fliiis traced out upon (he solid under conside- 
 ration are eitln'i' continuous /ones or jMutions thereof, and arc to be 
 consiihred as trai»('ziuins— tlie liist continuous, tlie ot iters not so, or 
 only parily so. Now, to jict the area of ,i trai»e/.iuni, which, as 
 already defined, is a plane liyiire with t\v(» paiailcl sides, — and these 
 last may, of coui'se, he circular or curved as well as strai,i;ht, — the 
 half-sum of such :(les is multiplied into the perpendicular distance 
 between them, or th(> heij;ln oi' breadth of the tiyiiic ; and the same 
 rule apjilied to the curved area to be computed is stat<'d thus: to the 
 half-sum of the lengths ol' tli(> end ordinates— that is, of the tirst and 
 last— add the sum of all the other ordinates or arcs and circles, and 
 multii»Iy the whole by the distance between the ordimitos or by tlio 
 breadth of the component zone. Tlie combiiu'd len,nth of the circles 
 or circumferences is «>asily obtainable from a multiplication of the 
 sum of their diameter; by 31, tu, mon^ correctly, 3.1 4U». This ride, 
 as ai>plie(l Ity nu' (l»a,nt' ()(»!• of my j;e(MiU'tiy) to a hemisphere of 'M'>i 
 units in diametei. and with only four ordinates or live se_i;nu'nts or 
 zones, hrin,L;s out tiie result witiiin less than one per «'ent of the 
 truth ; wiiile w il h nine ordinates. the result is erroneous to the extt'Ut 
 ofoidy the sixth part of one percent ; and with 1!> ordinates, or 20 
 sections, the |', ol one per cent, or within j,,',,,, of the true content. 
 Not tliat I insist, however, on this mo(h' of nu'asurenu'Ut for convex 
 or concave siiperticies, where there are other rules which, as in the 
 case of tlie perfect sjthere or spheroid, or sei!,nients of (hose bodies, 
 brin;; out their curved areas exact; but tlu? j^n-at and manifest 
 advanta.i;-e of this general system is, that its a<'curacy is independent 
 of the sliape of (lie body to he measured; wliiU', if the rule for a 
 sphere, lor instance, were applied to a Ixxly liot strictly spherical, 
 the result nd.inht pr<»ve erroneous to a far greater extent than if ai'ri- 
 ved at by the system of e»iuidistant ordinates, — to say nolhinu' of the 
 great advantaj>'e to the inactical measurer of havin.t'' to store h.s me- 
 juory with but one general rule api»licable to all eases, and this rule 
 the same as that for i)lane figure.s, whereby the whole range of areas 
 or superlicies, whether plane or convex, becomes submitted to one 
 and the same formula. To wit: a suhdivision by ecpiidistant |)arallel 
 lines into ligures, every one of which is a trapezium -whether conti- 
 iHums or non-continuous, it matters not. it now remains to compute 
 the volumes or cubical contents of the solids of the Tahlvau, and, as 
 already stated, they comprise all known cdenientary forms; and here 
 it is that I lay special claim to tin' introdmrtioii of a system of niensu- 
 ration which is not approximately accurate as applied to the great 
 mujurity of gcumclricul Ibriusj, but ol' wkicU tlic absolute accuracy is 
 
THE STKREOMKTRinAL TAr.LEATT, 23 
 
 juovcd mikI uii(lt>iil)t<(l. 'riic rule is siiiijilv w li;il it |>iir[i()its to Itc, 
 as ])iiiil('(l ill tlic lu'ild i>t' tlic tdlilcait, i.e. : " 'I'o the sum ol'llic ]t;iiiilicl 
 end areas add foiir-tiiucs the middle aica, and multiply tlie wlxde l»y 
 I part of the height or leii.nlli of the solid." 'I'lic word •' jxindlrr' is 
 introduced as a reiidiider that the opposite ends or hases must l»e 
 eoiitaiiied Ix'tween i»;iiallrl phiiies, or, if not so origin, illy, that they 
 must l»e made so liy snlidi\ ision or de(omi»ositioii ol'the solid into its 
 constituent elements. 'J'he whole dillieulty is, therefore, r«'dueed, bv 
 my system, to measuriu,!? the areas of the o]»posi(e hases and middle 
 seetion, the remainder of the work Iteiii^- a mere iniilti[»lieation ; so 
 that the jiroposed formula renders this l)raneh of study of sueh easy 
 and general applieatioii that the ait or scienee may now be tau;nht in 
 a few lessons wliore it formerly re([uired months, or even years. 
 Take up, for instance, the sciiinent of a conoid or spheioid cut nlf 
 by a plane inclined in any way to the axis of the solid, a (inure such 
 as would be presented by the space occupied by any liijuid or lluid 
 substance in a vessel of this shaite when inclined to the horizon ; 
 look at the preliminary labour retinired by the old rules ol' (indiui*' 
 out the axis or diameters of the entire solid of wiiich the segment 
 under considera(i<ui forms a part, ami these factois ar<' necessary as 
 elenu'iits in the computation. .My system dispenses with all this, and 
 the solid, whatever it may be, is taken hold of and submitted to 
 direct measurement, without in any \va\ imiuiriun about the sj/c or 
 form of body of which it is a section. Ibit for another reascm is this 
 study exdmled from general education, because, with onliuary rides, 
 the higher calculus is often indis[)ensal>le ; and as even when it has 
 been tauiiht and learnt, it is as soiui forgotten, therefore can these 
 ordinary rules be of little or no use to tiic juactical ineasinfr, even 
 when supplied with all tiie necessary books and data Ibr working 
 out his prol>h'ms. 
 
 It may be objected that, for the prism and cylinder, for instance, 
 tlu? ordinary ride is even more simple than the prismoidal one. Of 
 course it is ; but it flows of itself directly ami immediately from the 
 fornnda. Take up a prism : I have delined it to be of (Mpial breadth 
 throughout ; then is its nnddle section, or any other, when made 
 jiarallel to the l)ase or end. eipial in area to such base; ami the aigu- 
 ment (H't'urs that six-tinu-s this area into (UH>-sixth the altitu(h' reduces 
 to the nH)re simple enunciation of once the area into tin- whole alti- 
 tude. Again, in tin- «'ase of tli(> pyramid (u-eone, the half-way dianu'ter 
 or breadth is Just one-half of what it is at the base ; ami as the half of 
 one-Jndf, or the product of A x |, is \, therefoic is the half-way area 
 a <puirter of that at the base. The ]>upil who liasalr«'ady learnt this, 
 BOOS it at a ghuice, or recalls it to his memory, ami reasons thus : 
 four-times the mnldle nveu is ecpnd to the base ; ami twice the base 
 (for the upper area hero is zero) into ^ tlie altitude i« identical with 
 
21 aEOMETllY, MENSURATIfK'J, AND 
 
 tlic oidiunr.v rule of (Hicc I he hnsc into J tlii' alliludc ov \ tlic itiddiict 
 oftlic base jiiid altitiulc. 'IMicic is ouc iiioic cast' in wiiicli the «tl(loi- 
 ordinary rule is a|»})ait'ntly nidic sini[>li' ilian the ucncral roniiula ; 
 it is wlnM wlic have to do willi a paraboloid, ol" wliicli tiic volume is 
 just (»i!t'-Iiair of its col Tcsiiondiny cyliiidtn'. 
 
 r>ul licic \vc have done with flu* coiuparalivc advaida,u,'os oftlic 
 old rules, and in all otlier ca.ses the foiiiuda is exccediuiily inoi'c 
 simple. Take, f(»r instance, the frustum of a pyianiid ; and, liist of 
 all, how I<n(^w you that it is one, e\cepl by measiiiini;- its ujtper ami 
 undei' edges, and conipaiini;- their respective len,i;'ths to !ind out^ 
 which you must do — that due proportionality exists between them ; 
 else is tiie body net the fiu>lum of a pyramid, and, therefore, not 
 subjecl to the lule ? Uut, granted even that it is the ii.i;ure you do 
 take it for, see yoa the trouble (»f ,<;-ettini;a nuMU pr<iportional between 
 the areas of its opposite )»ascs, whicii includes a U'n/;thy multiplica- 
 lion of those areas and a laboriturs extraction of thesijuaic root of the 
 pioduct, which very lew persons know how to work out ? ilow 
 luuch more sinijile to airive at the arithmetical mean of the o]>itosito 
 diamctcis ami ilie middle area therefrom .' -and if the liinure be the 
 frustum of a cone, the three diameters are squared, the s([uare of the 
 middle owr taken four-times, and the whole niultiplie<l toj;'elher. t!iat 
 is, their sum. by decimal •7>^r)4, and the result by ,', the altitmh' of the 
 frustum ; and as this calculation has to be repeated every day, in all 
 j)artsofihe world, in computing' the contents of tubs and vats of all 
 imaginable sorts and sizes, tlie saving in time and trouble is certainly 
 most worthy of consideriition. 
 
 Ibit suppose this jirism or cylin<ler, this |>,\ranud or cone, this 
 conoid, or tiiis trust um, to Ik? not Iridy such a tigure ; let it ditfer liut 
 ever «o slightly from what it should be to enable ii to be submitted 
 to oidinary rales, — theJi, if such lules bi' made use of in computing 
 its contents, adieu to all accuracy, since the very element by which 
 the body ditfers fi'oni its geometrical prototype -that is. its interme- 
 diate diameter — is not taken the least notice- of; while the juismoidal 
 formula, on the contrary, takes in this ev<'r-varyiug element, this 
 half-wa_\ breadth between lite top and l)ot torn in a tub or vat, between 
 the bung and head, as in a i-ask, and gives a result, in IK) cases our of 
 100, mine (rue than any (»lhcr system whcit- this impcutant and indis- 
 l)ensable element of variation is not attended to. With regard to 
 the sphere or spheroid, each of its opposite bases is a zero of sui»erti- 
 cies, as a plane can touch either oi' tiiem only in one point, and the 
 sum of the areas in this ease is four-timi's the middle area. And how 
 correct this is you shall directly sec, for, by oi'dinaiy rules you are 
 taught to multi]»ly the cun\ex area ol" tlu^ si>Iiere by i of the radius; 
 but- this convex area is precisely ecxuiil to Ibiu-tiiues the middle 
 
THE STEREOMETlirCAL TABLEAU. 26 
 
 Boi'tioii, or to four great circles oftlu; splicro, and i tlic riidius is the 
 same thing as J^ tlie dianu'ter or altitude, — so that here again, you 
 see, as in the case of the piisiu or cylinder, the ])yraniid or cone, the 
 ])roof direct of the accuracy of the rule. Now, take uj> a heiuisphere : 
 the half-way area is easily shewn to be just J of that at the base ; 
 and as foar-tinies 5 iU'e three, and 3 and 1 iire four, four great circles 
 into I the idtitnde of the half- s})here gives, of eourse, halfthe solidity 
 just obtained, or that of the heniis])liere under consiih-ration. The 
 same is true of the llattened or of the elongated s)>here or s]»heroid 
 and ellijisoid, and of the half ihei'cof, and whether the cutting ]daue 
 or base be perpendicular or not to either axis of the solid ; and the 
 areas which enter as elements into the computation of the cubical 
 contents aie ahvays ellipses, and, what is more, tiiey ar<^ similar or 
 propoitional elli])ses; so that, from knowingany onettf the diameters 
 of the middle st'ction, the aiea, can be directly found by a rule-of- 
 three, since, as already shewn, the areas of similar tigures are j)ro- 
 portional to t]\i' sijuarcs of any of their corresponding dimensions. 
 The exactitnde of the Ibrniula. as applied to any other segment or 
 zone of a s])here, is fully demonstrated at paragraph 152!) of my 
 Mensuration ; its very near approach to trntli, in the case of spindles 
 or their frusta, at paragraphs lo^] and l.)74 ; and its absolute accura- 
 cy in thi! case of any segment of a spheroid, the right or inclined 
 paraboloid, or hyperboloid, at i)aragraphs 1500 to 15(J7 of nty work. 
 
 Now, there may be some curiosity to know how the idea occurred 
 to me of treating every solid as a i)rismoid by this one and undeviat- 
 ing formula ; it is this: taking up tlie ordinary prismoid, Ilind its 
 detinition to read thus : — " Any solid having for its oi)i)osite bases 
 '' parallel rectangles ; and, by extension, any solid having for its 
 '' parallel bases plane Hgures with iiarallel sides. " Now, ]deaKO 
 observe tlial the only condition ex])rcssed or imjdied in this detinition 
 of a prismoid is the ])arallelism of the sides, and nothing nu)re. .Such 
 ])arall«'lism does iH)t exclude the propcutiimality of the sides ; there- 
 fore is the frustum of a ]>yramid, to me, a prismoid; ami this is what 
 no one belong me, that I ani aware of, at least, ajipears to have con- 
 eeived ; for in no treatise iiave 1 ever se(!n the i>risnn)idal formula 
 Hpi)lied to the frustum of a pyramid or cone. Look, again, at tlie 
 rectangular ])rismoul, and as no ratio of 'lue sides is implied, let the 
 ratio be inhnite ; or. in other words, let one of the jtarallel sides 
 a]>proaeh towards the other until they meet and form a single line, 
 or edge, or arris ; and then have vre tlie wedge, wliieli is. tlierefoic, 
 another prismoid. Again, let this edge, or line, or arris, become, 
 shorter, and still shorter, until it dwin<lles to a jioint; and then have 
 we a pyramid, which is also a prismoid, to all intents and |)urposes. 
 ^Viid since a line or edge may tn-comc a mere point, so, eonverseiy. 
 
26 GEOMETRY, MENSURATION, AND 
 
 may a point bpconio a lino, whi'i-cby a prisiuoid, oii;;iiially sqiian- or 
 reetaii^mlar, may liavc; one or both of its opposite bail's niodilicd into 
 an almost iiiliiiito variety of similar <»r <lissiniilar lintiics, ns I liavo 
 shewn at i)a<jes 713 to 7IH of my treatise; and tlie moic f>eiieral 
 enunciation is thereafter arrifed at, that a prismoi*! may have tor its 
 parallel bases any two (i;;iires, whetiier eciiial or iineiiiial, similar or 
 dissimilar; any li^iire an<l a line j>arallel to the jdane thereof, as in 
 the wed^e ; any ii<iure and a point, as in the eone and jyyraniid j 
 any two lines not parallel, but situated in jtarallel planes. Other 
 definitions nuiy be <i;iven nnne «'oneise than this, moie technical and 
 8cieutitic, as tiiat " tlie prismoid is swept out in space l»y the revolu- 
 tion of a straight line around two jiarallel planes of any form what- 
 ever, and iiTes]»ective of the relative velocities of the two extremities 
 of the generating line ; " but the former gives the best i(U'a of the 
 foiTD of Holid, as it delines the figure of its j-nds or bases. 
 
 The Tableau you will find upon inspection to offer a variety of 
 forms; for instance, one base a s(juare, the other also a scjuare of 
 greater or less size, but turned diagonally as regai'ds the other, tiu' 
 middle base an octagon. Again have we |)rismoids or cylindroids of 
 whicli one bast^ is a circle, the othei- an ellipse, or two ellipses of 
 equal or of different size, the longer dianu'ter of the one corres]tonding 
 to the shorter diameter of the other, and other forms may l>e con- 
 ceived in almost endh'ss variety ; and of all, without exception, the 
 formula gives the true cubical contents, each of the models exhibiting 
 at a glance, by means of the pencil-line to be seen upon it, the 
 nature and dimensions of the middle section. A woid in relation to 
 the regular polyhedrons which are also among the m(»dels on the 
 board. Of these bodies there are but live, strange eixmgh to say ; 
 and yet, the conclusion is immediate and inevitable, for it takes at 
 least three idanes to make a solid angle; and as the sum of these 
 plane angles must be less than 3<i()* , or four right angh's, as other- 
 wise the solid angle woubl then cease to exist and become a plane 
 surface, it suHices to examine which are the regular polygons, whose 
 angles, taken in threes and fours and fives, etc., makc^ up an angle 
 less than four right angles. The (Mpiilateral triangle can be put 
 together in 3's and 4's and 5's, which affords the tetrahedron, and 
 icosahedron. It cannot be taken in sixes, as six-tinu's ()0- ^;{(J()S ■ 
 and therefore caJi we have no other regular solid with its faces 
 equilateral triangles than the three .iust enumerated. Tlie right 
 angle can be taken in 3's only: two would not enclose a s])ace, and 
 four woubl tbrm a plane ; h(;nce is the jierfect cube the only solid 
 that can be formed with s(puire faces. Lastly, the jientagon, of 
 which the angle is 108 ^ , supplies the dodecahedron, or l::i-sided 
 figure, the sum of its three plane angles being liii ^ . The regular 
 
THE STEREOMETRICAL TABLEAU. 27 
 
 hoxaf,'on (Munot Itc made to answer, as its angle is 120® , and tlireo 
 of tlu'in would make uj) four nj;ht angles, as you see tliey do wlien 
 lodking at their e.\(|uisite arrangement in the beehive, or the not less 
 Ixaiitifiil symmetry disphiyed in tlie yest of tlie eommon wasp. A 
 fortiori, then, can heptagons or any other i)olygon not be used to 
 build tliese solids witli ; and lienee, again, as just stated, can there 
 be but tiv<', and oidy five, of these jdatonic forms. As to their mode 
 of measurement, tlu- cube or lu^xahedron is a mere prism, the tetra- 
 hedron a mere pyramid, tlie octahedron two pyiamids base to base, 
 and (he two last, tin- 12 and 2()-side(l figures, mad<Mip of as many 
 l)yran\ids, having their common apex in the centre of the solid or of 
 its in\aginary circumscribing s]there, whereby will the computation 
 of ont^ of these component pyramids afford the volume of the 
 Mhole. 
 
 As to the nn'asui'enu'nt of comjmund figure like the fmstmn of 
 a cone or pyramid, or of any other body, between bases that are not 
 paralhd, the solid is resolved or decomposed by a section parallel to 
 the base and ])assing through the lowermost edge of the frustum's 
 njtper base into two jiortions, one of which is a frustum proper, the 
 other ;i hoof, or nngula, or i)yramid, or some other figure, as the case 
 may be. The connnon buoy is thus resolved into a cone and segment 
 of a si)here ; a gun or mortar, the frustum of a cone or cylinder, with 
 a hemisjdiere or segment of a spheroid ; the Turkish or Moorish 
 dome, or ])innacle, or s])ire, the middle frustum of a sphere or 
 spheroid, suiinounted by a ludlow or concave eone, and so on. If 
 a spherical cone be ]noposed, it is evident that it can be conceived 
 and treated under two asi>ects — first, as a cone proper, with the ad- 
 dition of a segment of that sjdiere of which the cone forms part; or 
 (and this a)»]>]ies to any s])herical pyramid, or frustum of such 
 pyramid or slttll, or hollow si»here, or any portion of a shell,) to the 
 sum of its end-areas, si»herical though they be, add four-times the 
 parallel and middh' area, and their sum into J^ the altitude will be 
 the true <*oiitent. Of course, I need hardly remark that in dealinjr 
 with a hollow sphere or shell, the content is imue quickly arrived at 
 by apply I v the foiniuhi direct to and taking the diiference of the 
 inner and o iter or component spheres. 
 
 I ha « said that the formula applies exactly .and demonstrably to 
 the great majority of solids. From this it is, of course, inferred that 
 there are some exceptions, as in the case of hoofs, ungulas and spin- 
 dles, but ill the same way as the cask or middle frustum of a spindle 
 is measured to within almost jierfect accuracy — say to within tin; 
 quarter or j^ or jV of one ]»er cent, or a half-pint on a hogshead (see 
 pages 707, 70S, and 70!) of my treatise), by working iijiou its half, or 
 by taking the bung diameter as that of one of its ends or bases, and 
 
28 OKOMKTRY. MENSURATION, AND 
 
 f »i' the mi(l<ll<' iirca tlial wliicli is iif its (iiiiirtor, or Iialf-way lictwccn 
 the iiead and hiiiiii, -iu IIh' same way, I say, as (his is (lone, so, iu 
 the hoof aiul uii^uhi of any solid may almt»st absolute Jiccuiiicy l>o 
 attained by u subdivision of the body into iiarallel sliees, two ov 
 three of tiieni jncnerally suflicinji,-, or four or live when tlie minutest 
 aeciiraey is insisted on, Just as we ai»i>r(»aeh nearer and more near to 
 the eircninferenee of any circli^ of wliieh the diameter is known, l»y 
 taking iu more deeimals. And in the mine manner may hoUow or 
 coucjivc cones or cylinders, ci- if they be couvex or swollen out, or 
 other bodies be deeomjjosed and measured, which are n<»t true geome- 
 trical ligr.res, and thus the one, and oidy one, most simple formuhi 
 lUaintained, made use of, and applied in every conceivable case, 
 without the necessity of learning or remembering any other. The 
 subdivisions, the decomi>osing planes of sectioji, may be made equi- 
 distant, and the gran<l result thus arrived at, that one universal rule, 
 (me formula, will nu-asurci all solids and all surfaces; for, though F 
 have not yet allu<h'(l to tlu' fact, it is demonstrable — ^moreover, it is 
 shewn — that when idane ligures, surfaces, are cut up by equidistant 
 ordinates, their areas, the urea of each of them is equal to the sum of 
 its bases and four-times the middle base or section into J the altitude 
 of tlu^ figure. Now, if any s(did be divided by a serii's of equidistant 
 planes, and any surface^ Ity a similar series «)f equidistant lines into 
 portions of equal altitude, and if intennediary or half-way sections 
 be conceived in either case, we get the uni\ ersal formula that their 
 contents, soli«l or suiu'riicial, as the case may be, are equal to the 
 sum of the extreme bases, together with twice the sum of all tlu* 
 other end-bases, and four-times the sum of all the half-way -sections 
 into ,\ the common altitude ; the bases and sections being, of course, 
 superficial or linear, according as the ligure is a space-enclosing one, 
 or a mere surface, plane or curved though it be. I have already 
 alluded to the tendency towanls geneializalion in almost everything: 
 in physics and in cliemistry are all phentmiena subntitted, so to say, 
 to some general, some universal law or rule. The mechanical powers 
 may be said to constitute but one, and are, at any rate, similar in 
 this, subjected to the law, that what they give in power they consume 
 in time or space. The one measuring unit, the metre, is fiucing its 
 way thi'onghout the world, and iu its wake must follow one unit for 
 all suifaces, one for all ca])acities. So are coin and currency beco- 
 nnng simplified. Complete the scheme, and to the sameness and 
 identity of the unit add (me general uuKle of computation ; then will 
 not only the unification of the currency be brought about, as projjosed 
 in Fiance, and the project most ably abetted in this Dominion by 
 (uir fellow-countryman, K. H. M. Jiouchette, Esq., but it shall be the 
 more nniversal scheme, the one grand unification and generalization 
 of all science, one nniversal language, wliereby nations may commune 
 
THE STEREOMETRICAL TABLEAU. 29 
 
 coiivcrw with one iinotlior, — some day, mir rcli-'ion, oik- slicidicnl, 
 iiiul one (lock, — tlio inillcnimn, indeed. IJiit do ,V(»ii sec the iiiniieiiKc 
 iidviiiita^'cs of this <;('iiendiz:itioii ? See .yon ils untold <-onse(|ii( iices ? 
 Loolv at tlie time now neeessaril.v (h-voted to tiie tik re redintioii and 
 t lanslal ion of all tlieseever-var,vin<;' rules and units; tlie.v fill a lliou- 
 Maiul volumes, while tiie time and frouble »levot«'d to their eompila- 
 tion mijuht bo so much more prolitaltly (nuployed, iiow-a-days, in the 
 study <tf new seieiH-es and niiW {iits comhuive to the <;reatei' liappi- 
 lU'HS and welfare of niaidciiid. 
 
 Now tliat w(Onive, so to say, f-lanced at all tlie lii-ures on llu' 
 '* T(il>lf((ii,^^ with their more oi' less icuularity of outline, their plane 
 !ind curved surfaces, tln^ (juestion still arises: " How are irregular 
 bodies of all kinds to be measured, such as statuary, bronzes, ♦•arvinj-', 
 and the like ?'' Ami that this lecture, so fai' as the limits of a lecture 
 will allow, may be complete in itself, and <<;{) over the whole ^rouml 
 foreshadowed in its title, it behoves me, in a few words, to supply 
 tlu! necessary intbnnation. it is very simple, and. in fact, more so, 
 to arrive at the exact cubical c<mtents of a very irre<;ular body than 
 of one of more exact form ; foi', with the latter an attempt is always 
 made at direct measurement, while with the former a mechanir;il 
 process is followed, which solves the problem in a manner most expc- 
 tlitions and most satisfactory. Take n|t a statue or otln^i- carved 
 liji;nre, or the capital of an Toidc or Corinthian column, a square (u- 
 cylindrical vessel capable of con tain! ny- it. I'oui' water into this ves- 
 sel until it reach to aliove the top of the object t(» l»e measured, and 
 mark the heij-lit at which the water stands; then remove the olijcrt, 
 and a{:tain note how hij;h the water stands, when the dillcrencc will 
 imuH'diately att'ord the vobune sought. If the substance of the bod,\ 
 or of the containiui;" vessel be of an al)sorbcnt nature, use sand oi 
 some like substance instead <»f water. There are still other modes ot 
 arriving;' at correct conclusions. T!u' s[»ecilie /gravity (»f any biuiy is 
 its weij;ht compared with that of water. Suppose, tli«ii, thai lalihs 
 liave been prepared wherein the ratio of weight of every substance is 
 j;'iveu to its (Mpiivalent of water, or its absdlute weight without ic^aid 
 to the ('(luivalent: take uj) from ott' the public hi,uiiway a shapele.,> 
 stone, and weigh it; compare its W( i,i;ht by lule-of-thice with llial 
 of a cubic foot, or inch, or yar<l of the sauu' sultstaucc, and hence do 
 you arrive directly at its cubiciil contents. Converse ly, if the ucijuht 
 be reepured of some object which cannot be sulnuitted to direct com- 
 ])Utation by i)utting in a l>alance, but if its vobune can i»c arrived at, 
 then also can its weight beascertaiiu'd by a simple lule of proportion. 
 
 Now, let it be retpnred t(t lind the component (|iiantilies of some 
 C(mi|)ouiMl body OI iunalgam. l-oi' instance, you have a mixture ot 
 copper ami zinc fused together and solidilied into one comi)act bod^ , 
 
30 GEOMETRY, MKNSURATION, AND 
 
 wiflioiit :i tnicc nfcitlii r of tli<' coiwtidiciits ; the wi'inlit of (mcIi of 
 tlif icspcclivr iiU'lals ciiii lie siilniiillcil to dirrrt tiilriiliitioli, the fiir- 
 tois or cltiiii Ills ('iitM'iii;,' into tlic roiuiicd formula Ix-iiij;' iiincly tlu' 
 N])<-«-itic \V('i;nlit MS well of llic compoiiiKl as of i \\v iii<;rcili('iits of w liidi 
 it iMiiiail('ii|). And so of )i mass itf ipiart/. ainl ,i;ol(| ; ami llioiiuli 
 little or none of the precious metal may l»e visiide to tlie eve, the 
 wei^iiit of the latter eaii he arrived at with comparative facility. We 
 are told that iliero, Kin;;' of Syracuse, /nave to some .clever artiliceru 
 <liiaiitity of <>old wherewith to falnicate a ckumi; hut snspecfiii<;, 
 when the crown was linisln'd, that the Jewellei' had purloined a poi- 
 ti(»n of the ^old and siihstitnted silver in its stead, he sni»mitted the 
 <liiestioii to Archimedes to ]uop(»mid. Specilic ji'iavities were not 
 then known; Itiit our |)liilosopher, while in his l>atli, it seems, was 
 c'o.nitatiii^ how he iiiii^lit liest solve the propositimi preptuiixled hy 
 the Kinji', when, iiotiiinj;- the <lilference in wei;;Iit of his own body 
 when immersed in water to what it was in air, he conceived the 
 hai»l>y idea of sulmiill iii.u' the crow n to a like |>rocess of computation, 
 and, aft<'r weiu'hiii.u the crown itself in water, and then pure ;;(»ld and 
 silver, found, l»y an easy calculation, that, as the Kin^ had rightly 
 jfiiessed, the crown was in reality made up of ^old and silver instead 
 of U'old alone. So yiad was (uii' i»liilosoplier of the <liscovere<l he had 
 mad<', that lie ran tliioiijih the streets of the «Mty, cryinj;-: " Eureka I 
 Kureka!'"— " I have found it ; I iiave found it." 
 
 lieferriny. a.ij;ain, t(»the" 7W/>^y(/(," the word .S7cnv)H(c/>vrf(/ wotdd 
 seem to imply that it is intended (uily or altogether for purposes of 
 mensuration. Such, however, is mtt the case, as it w ill immediately 
 be evident that is must also be of j;reat utility in acipiiring (u- im- 
 ])artin,u a knowledge of the nomenclature of solid foiiiis, an ac- 
 (piaiiilanceshi]» with their varied sliajtes and li.!;iir<'s, wliiili. without 
 such help, would re(piire a previous familiarity with the principles 
 and teachiniis ofdiawiiit; ami jierspective. 'To the architect, the 
 eu'^iiieer, and the biiihh r, the models are su^'^iestive of the forms 
 and relative [import ions of blocks of buildiii;u;s, roofs, domes, j»iers, 
 and (jiiays; cisterns, reservoirs, and cauldrons ; vats, casks, and other 
 vessels of "cai)acity ; earthworks of all kinds, comjuisin^' railroad and 
 other cuttinj;'s and embankments ; the shaft of the ( Jreek or Komaii 
 eoliimn ; square and waney timber and saw -loi^s ; the campiii<;-tent ; 
 the square or s]»layed oiienin;* of a door or window, or niche or 
 loophole in a wall; the quarter of a sphere or spheroid, the half- 
 segment, the vault or arched ceilini;- of the apsis of a church or hall ; 
 th<,' whole s]»liere or sjdii'roid, the billiard or the cannon ball ; or, on 
 a larger scale, the «'arth, moon, sun. and i)laiiets. The iimdels must 
 also pi'o\- "I yreat h"1p in t";icliiii'f |»!Ts|icrti\(' •''■■'/•' ■ ; ,iiid ;'><• 
 geumeliicui £ii«.>jocLiuu ui' tsuiidis ou u pluiic ; uLao, liicir iskudcis uu«i 
 
THE STERKOMKTUICAL TABLEAU, 81 
 
 tlic sliiiilouH Hiy prujccl. .\;;;iiii. (lie Mil of ^><iiit(tri<;il (Icvcl<t|»iii<iit 
 of Hiiitiicvs is tlicitW.v imicli ('inilitiit('(l. Tlu'ii' is ;iIm> (Ik- pdliir 
 triaii;iiic, and otlirr lines iH'cossiir.v lor tlif sludy of splicricMl tii^i^oiio- 
 iiiflr.v. 
 
 W'lit'tlicr my attciiipt to reduce l<Mtiie sitiiple and miironii system 
 tlie present niiiltit'old and complex lules lor lindin;; tlie contents ot' 
 solids, or tlie capacily of >pace-enclosin,i; areas, shall prove siiccessfid,- 
 time alone can tell; for, ihon^li it has ilie merit of Iteiii;; new, it 
 also has tJK' disadvanla;;es of n»»velty, as Mr. Scoli-Kir»ell said of 
 the iii]i;cnions screw-propeller inveiilcd sonn- years a^o l»y C^oia- 
 mandei' Aslie, e\-pve>ident of this Society. 'I'iie scieiitilic world, as 
 well as the political, has its conservatives. We have not yet well 
 learnt the ad\anta;^'es of decimal ariiliinetic; nor has the tiresonu- 
 computation of pounds, shillin;i;s and pence lie< n yet aliandoned for 
 the nn>re e\i»ediiions dollar, where the nnic shifting of a point works 
 womlers. It takes much tinu' to work out such a revolution a ge- 
 neration, so to say ; Inil that 1 shall not have to wait so Ion,!;' lor an 
 interpretei' 1 coiiljdently hope, Jud,i;in,u at least fi<un the many 
 Hatterin^' testiuntnials 1 haxc already received in relation to the 
 multiplied advanta,n('s of my invent ion or discoveiy. 
 
 'J'he Council of I'uhlic Instructiiui, at its last general nwetin^', 
 appointed a committcj', composed of tin; Loril llishop of (.Quebec 
 llisJKtp l.aii;;i'vin, of Uimouski -himself a thorough master of the 
 art, — and IWshop Laroc<iue, of St. llyacinthe, to report ii[»on the 
 subject ; ami who, 1 take it, after havin<ji suhndtted l(» them the very 
 favourahle opinions of so nniny of our best matin inaticians. and of 
 other competent Jud^i's, such as 1). Wilkie, K. S. .M. IJouchetle, the 
 l>rofessors of the Laval rniversity, Ili.u'h School, .Morrin C'oile;.;'e, and 
 other educational estahlishnn'nts of Canada elsewlieie, can hardly 
 fail to recojumemi the introduction of the '^ Tabled ii" into all the 
 schools of this Dominion. 
 
 I have hut re<eived a letter from the Minister of Kducatioii of 
 Xew Ihiinswick, askin>;' nu' to send a " 'l(thl('((ii.'" with the view, says 
 he, of introdiiciiif'' it into all the schools of that Province. Not, 
 however, that 1 in any way intend to cctnline myself to Canada. On 
 the contrary, 1 have already patented the •' Tdhltutu"' in the United 
 States, where I h()])e, of <-(UUse, t<» introdinc it ; ami Mr. \'annier, 
 in writinj;- to me from Frant-e on the Kith of .Fanuary last, to advise 
 me of the inrantinf>- of my letters-patent for tiiat <"ountry, adds that 
 MM. Humbert and Xoe. the president and secrtlary of the Society 
 for the (jieneralization of Education in France, iiave intimated rheir 
 intention, at their next general meetinij;, of having' sonu' mark of 
 distinction conferred on me for the benellts which my systi'ui is 
 likely to confer on education. 
 
?,2 OKOMETRY, MRNSURATION AND 
 
 The Iloiildc. Mr. Cliinivciui, Minister of Education, niul otlicrwiso 
 Avcll <|iiiililinl tojnd^i', will lUiikc it IiIh duty, so Hiiys liin letter ou 
 the sul)ie(-t, to reeoninieud Uh adoption in all edueationul estalilisli- 
 inentH and in every school, so confident is he of its practical utility : 
 " Si' tera un (h'voir (Vvu recoinniander Tadoption dans toutes les 
 " maisons d'cilucation et dans toutes les ecoles, certain qu'ii est dc 
 '' sou utilite pratique." 
 
 From tlu^ Seminnry, M. Mainj,nii writes: — "Plus on etudie, plus 
 ** (Ml approl'ondit cette forninle du cuha^c iles corps, ])lus on est 
 " enchante" (the more one nnirvelsat) '* de sa siniplicite, desaclarte, 
 " et surtout de sa ^rande ;j,eneralite." Biuclow, M. A., " believing it. 
 " to be of universal use, shall heartily lend himself t«» the introduction 
 'S>f my system. " Mcliuarrie, H. A., " shall be delighted to m'v tlie 
 " old tedious process superseded by a formula so sini|de aiul so 
 ''exact." L>. Wilkie says: — "The rule is piccise and simple, and, 
 " bein<x a])j>lical»le to almost any variety of solid, will greatly shorten 
 " the processes of calculation. I have," he adds, "proved its accu- 
 " racy, as applied to several bodies. The Tahhau, comi)risiny a 
 " <;i'eat variety of elementary forms, will servo adnurably to educate 
 " the eye, ami must greatly facilitate the study of uu'usuration. The 
 << oovernment would confer a boon on schocds of the nnddle and 
 " hiii'her class l)y allordinj;' access to so su<;^estivc a collection. '? 
 I'rofessor Newton, of Yale Ct)llef?e, Massachusetts, considers tlio 
 TdhUdu a very useful an'an<,'ement for shewing the variety and extent 
 of the ajtplications of the formula. The ('olle,i>e I'Assomptiou " will 
 "adopt my vstem as part of their «'ourse of instruction. " Kev. T. 
 iJoivin, of St. Ilyacinthe, says : — " Votre decouvertc^ est precieuse, et 
 jo rec(unnuuide tortement l'a(h)ptiou do votre tahlcmi. " 
 
 Tlu'ie are others who, irresi>ective of considerations as to th^ 
 comparative accuracy of the fornuda, or of its advantages as api)lied 
 to nu'r(^ mensuration, Iiav(^ seen how far the models are uww su_in-,i;-es- 
 tiv(^ to the ]tiipil and the teacher than t!ie meie representalion 
 thereof on the black-board or <m pajier, and who, in their written 
 oi»inions, have alluded esjtecially to this feature of the proposed 
 system. Mr. Joly, President of tlu^ Quebec branch of tlio Montreal 
 ydiocd of Alts, in a letter to Mr. Weaver on the subject, aiul after 
 havin.ii!," himself witnessed it on more than one occasion, says, in his 
 exiu'cssive style:—" The dilference is enormous. " The professors at 
 tlic! Normal School are of the sa:ue oi)inion ; and others there arc 
 who variously estimate the saving in tinu', by my system, at from 
 twelvi^ to eighteen months. 
 
 The ]»rismoi(ial formula is not new: it has been long known, and 
 sometinies used to compute the oidinary rectangular prisnu)id, as 
 well as the familiar forms of railroad and caual cuttings and embaulc- 
 
TIIK STKUKOMKTUICAL TAIJLKATI. 33 
 
 monts; but no oiw. wciiis to have coMccivcd (licMdcii of applying.' it; 
 «'V('ii l(» tln' rnistimi ol'a (•(tiic or pyinniid, to wliicli, 1 imisl iiccciSHii- 
 I'ily iiiiVr, it was not kii<»\vii to ii|)ply, flsc, simple as it is, and ho 
 riiiicli iiioiv sitiii»l(' ami direct than liC^iendic's tin irein, it must have 
 found ils way ere lliis into treatises on meuHuration and t)i(; like. 
 Ncitlier lias it ever been empb>y('d, tliat I know of, to eomi>utc tlio 
 se;Hineii( of a sphere or s|»lieroid, nor l(» many otiier well-known 
 forms; so that, in this respect. I nniy lay claim as if to the discovery 
 and as well for a lar^c nMml»er of other solids to which it never wan 
 attempted to apply it. And even if the idea of so doinj^ has at any 
 time HU;i;;osted itself to others, as sometimes hinted at, tbey do not 
 appear tct have )>iit it to the te«t, oi' to have arrived at any Jiseful 
 I'oiicliisiou in relation to it, any more than the tirst man who, on 
 seeing sicam issue undei' pressure from the no/zle of a tea-kettle, 
 conceived the idea that such an a;,'ent could be made to work the 
 wonders that \v(^ know of; nor was steam ever made available in 
 |)ractice till Watt invented the steam on^nne, or electriiiity till Morso 
 put up a teleiiiaph. (iranted, however, that this formula was disco- 
 vered before my time, and that T have mirely di.senga<!;ed it from the 
 dust of years, or re-discovered it, still is tlieie, perhaps, some little 
 merit to b(^ claimed. Adams and Leverrier both own to the disco- 
 very of the ]danet Neptune; ami Leibnitz was not robbed of the 
 iumor of his int((gral and ditferential calculus, though Newton had 
 by .{^'eral years preceded him in the Held by the discovery of 
 '' fluxions. " • 
 
 f'- 
 
 BAILLAIRGE'S STEIIEOMETRTCAL TABLEAU. 
 
 Our engraving is a perspective Anew of the above named educa- 
 tional device, which has b(^en patented for its inventor, Mr. C. Bail- 
 lairge, of Qiudx'c, in the Unite<l States, Canada, and Europe. It con- 
 sists of a board, about tivefec^t lonj; and three feet Avide, with some 
 two hundred wooden models, comprising, so to say, all tlu* elemen- 
 tary forms, their segments, and sections, and nuinerous other solids, 
 einvido and comi»ound. 
 
 The tableau is set in an appropriate frame, Avith glass coA^ering, 
 BO as to exhibit the models while (^vcluding the dust. The front can 
 be opened at i)leasure so as to afford access to the models, each of 
 which is merely sup])oited on tlu^ board by a round nail or wire, 
 Avhich admits of its easy removal and re]»lacement by teacher or 
 ]>ui)il. The instruction conveyed by this tableau, appealing, as it 
 does, to the nneducated eye and mind, is, the inventoi- thinks, des- 
 tined to bo of great use in developing the intelligence of the untaught 
 
S4 GEOMETRY, MENSURATION, AND 
 
 masses of mankind. Ho r-xpocts to introdiifo it into all the ('(luca- 
 tional institutions of tlio United States and cIscwIictc, as it is now 
 being disseminated in Canada j and lie has n(Mloiil»t that the taldeaii 
 will also find its place in the stndio of the engiiiee)' and arcliitect, t<» 
 whom the models -will be sn<r^festive of various forms and relative 
 proportioiiB Avlueh cannot fail to aid tliem in their i»ursuits. Tluj 
 rapid success attained by a scliool in Quebec, in ijiensiiration of all 
 kinds of surfaces and yet higher mathematics, inciuch'iig conic sec- 
 tions, was attributed to the use of tin's tabh\'iu. Every tableau is 
 inscribed with a rule for finding the solid contents of any body, calli^d 
 " tlie prismoidal formula." This formula has been shown, by Mr. 
 Baillairge in his treatise on geometry and niensnration publislied iu 
 1856, to be less restrictive than supi)osed, and he has added to the 
 known solids, measurable thereby, a long list of others discovered by 
 him, the whole of Avhich are given in the tableau. Each tabh-aii is 
 also accompanied by a printed treatise^ ex [)Iauatory of every use to 
 which the models can be put. Mr. Bailhiirge is iu posses^siou of a 
 mass of testimonials, from high officials and othi-r distinguislied men, 
 both in Canada and Europe, together with reports of various educa- 
 tional and other institutions, all highly complimentary to him and 
 his invention. 
 
 Dr. Wilkie, of Quebec, thinks " the government would confer a 
 boon on schools of the middle and higher classes by atlbrdiiig access 
 to so suggestive a collection 5" and Professor Newton, of ^'ale College, 
 considers the tableau/' of great use for showing the variety and ex* 
 tent of applications of the prismoidal formula." 
 
 ticicnfific American, June 1st. 1872. 
 
 No. 2272-71. MINISTRY OF PUBLIC INSTRUCTION. 
 
 Quebec, this 18fch September 1871. 
 
 C. Baillairge, Esq., Quebec. 
 
 SiK, — I am instructed by the Honorable I\Iiuister of Public Ins- 
 truction to acknowledge receipt of your letter of the 8th instant, trans- 
 mitting copy of the prospectus of youi- " Stereometrical Tableau." 
 
 He will make it a duty to recommeml its adoption iu all educa- 
 tional establishments and in all schools, persuaded as he is of its 
 practical utility. The tableau and accomi»anying formula reduce to 
 an operation of the most simple kind the measurement of every des- 
 cription of solid, which recpiired according to the old method a cal- 
 culation long and often very difiicult for persons especially who were 
 not in the daily practice thereof. 
 
 I have the honor to be. Sir, yonr obedient servant, 
 
 Louis Giaud, tSicrctdii/. 
 
THE STEREOMETRIC AL TABLEAU. ' 85 
 
 Sir. — That this fovnmla is uiatliematically correct, as applied to all 
 the solids enumerated hy you in your prospectus, there is of course 
 no doubt. You have fully demonstrated this in your valuable work 
 on Geometry and Mensuration published in 186G. Mr. Steckel has 
 not been slow in showing this in a most concise manner in his letter 
 to you on tlie subject, to say nothing of the letters of the liR. MM. 
 Methot and Maingui on part of tin; professors of mathematics of the 
 Quebec Seminary and Laval University, Avherc the expressions, 
 ''6tonue" and "enchante" sufficiently show the high estimation iu 
 which your discovery is held by these competent judges; but, in my 
 opinion, you do not sufficiently insist on the great value, the manifest 
 and manifold advantages of your rule im applied to spindles, the 
 middle frusta of which are met with every day and in every part of 
 the civilized world under the tlicmsand and one forms of casks of 
 every conceivable size? and variety, and the necessity of measuring 
 Avhich witii promptness, on account of their number, and with accu- 
 racy, on account of the generally valuable nature of their contents, 
 renders some simple, easy and commodious rule, like the cue now 
 proposed by you, of the lirst importance to all manlcind. 
 
 Now, Sir, that your rule embraces these valuable requisites, let 
 me compare it, in its working and in its results, with the rules laid 
 <l()wn by some of our best mathematicians and authors such as lion- 
 iiycajtle for instance, see Kev. E. C. Tyson's edition of his mensura- 
 tion, page 147. 
 
 Problem XXVII (for example). 
 
 "To tind the soh'dity of the middle frustum of an elliptic spindle ; 
 " its lengtii, its diameters at th(! middle and end being given ; also 
 " the diameter which is half way between the middle and end dia- 
 " meter being known." 
 
 Rule, "1° From the sum of three times the square of the middle 
 " diameter, and the square of the end diameter, take four times the 
 ** sciuare of the diametei- between the mid<Ile and end, and from four 
 " times the last diameter take tlie sum of the least diameter and three 
 " times that of the niiihlle, and J of the quotient arising from divid- 
 " ing the former ditieience by the latter will give the central distance. 
 
 '< 2° Find the axes of the ellipse by Problem II, and the suea of 
 " the elliptical segment, whose cord is the length of the iiustum, by 
 ^' Problem V, 
 
 " y^ Divide three times the area thus found by the length of the 
 " frustum, and from the <iuotient subtract the difference between the 
 ^' middle diauK'tei' and that of tho end, and multiply the remainder by 
 " eight times the central distance. 
 
36 QEOMETUY, HrENSTTI'ATION', AND 
 
 " 4° Then fVom (lie sum of tlit' siiiiavc of llic Iciist (liiuiictci', ami 
 " twice the sqiiai'o of that in the niichUc, take the proiliict last fouiitl, 
 " and tliis dittViciice niultiplicd by the Icii^^th and the product again 
 " by .2(J171>!), Sec, Mill ^ivc the solidity UM[uiicd." 
 
 Ilcre, the mind is absohitely bewihhM'cd at tlic, move recital of 
 tlie multifarious o])(Mations to be ])erformed (not less than ii7 in num- 
 ber) and the m( re results of each of tluse o|»erutions, irrespective of 
 the details of the multipli('ati(»iis, divisions and other eomi)ulatioiis 
 necessary to arrive .at them, lake up two whole pages of the book. 
 
 Applied, say to a cask of 28 inches in length, bung diameter 24 
 inches, head diameter 21.0 inches, ami diameter half way between 
 head and bung 2;}.4()!)()I) inches, the result, as fully woiked out at 
 page 148, 140 of said book, gives 11,854J|^ cubic iuclus, very nearl}-, 
 or 51 gallons and 5 half-pints. 
 
 Now, the same example. Sir, by your formula, brings out 11,855.2 
 cubic in., which ditlers from the last result by only .0000045 <)r less 
 than half an inch on nearly I2(t00 imhes, or the 240th part uf one per 
 cent in excess, the 14th i>art of a gill. 
 
 Not only then, is your fonnula in this case to be considered in 
 every respect as accuiate as that of Bonnycastle, but it is really moie 
 so in practice; for, even if the error in excess attained the maximum 
 of .005 or i of one per cent, where is the practical measurer or ganger 
 who, for the sake of a (|uart on a .50 gallon keg or half a gallon on a 
 hogsilu'ad, would, could devote hours of his time to calculate by tiie 
 old method what can be done with greater accuracy and in less tlian 
 2 minutes by the new ; for, every merchant will tell yon that in piac- 
 tical cask gauging then! is generally an error in excess or in defect of 
 from one to two gallons on a hogshead. 
 
 And even this e<.mparative accuracy of tin; old ruh', can only bo 
 arrived at by taking in all the decimals, which no one would be likely 
 to do, on account of tiu' immense lab(»ur «»f the computations ; wheie- 
 as, by the new formula, by reason of its great simplicity and concise- 
 ness, all the decinnils may easily be taken in and no harm can result 
 at ^onie of the last decimals being neglectc«l, since the result as shown 
 above is, and, f(»r convex forms, always is, though ever so slightly, in 
 excess of the true ctmteiit. 
 
 I am wrong howev<'r in assuming that the nniximum error in 
 cask gauging by your rule is .005 or the half of one pvv cent ; njjitlier 
 do you say so in your pros]»ectus, and on the contrary you show 
 most satisfactorily at pagtw08, 70!> of your said tic^atise, in the nu- 
 merous <'xamples given by you and fully woiked out and eomi>ared 
 in each cas(vwilh the results given by Honnycastle's rides, that the 
 juaxinnim error i» excess does, in your first ami 2iid examples, n.it 
 
THE STEREOJIETRICAL TABLEAU. 37 
 
 oxoeod \ 1)1' Olio ju'r cent or one quart on Ji hojuslii-ad ; in ex. 8 it is ^ 
 1)1" 1 jK'v cent J Ex. 10{>ives tlie maxiiiiniii error as }f of 1 per cent; Ex. 
 5 gives ^ of 1 per cent ; Ex. 4 and 12 (2), ^^^ of 1 ]ter oeut ; Ex. [), ^\ of 
 1 per cent ; Ex. 2 and 12 (I and ti), ^V <»f ' I't'i" t'*'Ut ; and ex 7, ^ij of 
 1 per cent, and tlii'se exainjilcs cover all varieties and sizes of circu- 
 lar, elliptic, and paraltolic casks, llu;» is of the three varieties gene- 
 rally met Avith in practice. 
 
 But in dwelling on ihe fiuinula, I liiid I have as yet said nothing 
 of the all important " Stereoinetrical Tableau" without* which, as you 
 pertinently remark, the rule would he almost as useless in teaching 
 mensuration in schools, if not in the ]»iactice of it, as steam without 
 the steam engine (»r «'h'ctricity without the tch-giaph. 
 
 There are many otiier advantages, apart from the mere mensu- 
 raticm of Itodies, which your tal»h\'iu possesses, as enumerated iiy you 
 in your ])rospectus and which it is useless for mc to dwell u[)on, as I 
 fully ciuicur in all that you claim for it ; though I tliink you might 
 have fill tlier insisted of the advantage t)f such a taldeau in the studio 
 of the a|)i>rentice, nay even of the professional architect, who will, 
 among the models, find that of almost every conceivable shape or 
 proportion of roof, dome, &.c,, which he nniy be called upon <o design j 
 the Civil Engineer, every descrii»tii)n of luismoid to be met with in 
 the cuttings or embankments for railroads, canals, docks, <Jcc., or in 
 the piers or abutments of luidges or other stiuctui'cs; the nu'clianical 
 eugiiieer, every variety of boiler, cop[»er or other vessel and the com- 
 ponent parts of all sorts of machinery, 
 
 Quebec, 0th Uecember 187 J. J. Gallagiieu. 
 
 1(), St. George St., Battery. Dec. 20 18*1. 
 
 Mv DKAU Sin, — It is with the utmost satisfacticm that I have re- 
 viewed th(.' Prospectus of the Stereometrical Tableau, which, in con- 
 juncticm with the Photograjih, you were so polite as to send me. 1 
 have bet-n unable from want of leisure to go into so thorough an ex- 
 amination as I should desire, and whirh your invenlion merits. IJut 
 in the instances which I have subjected to critical analysis, I have 
 found the rule to work most admirably— ctmibining comprehensive- 
 ness, utility with simplicity and great exactness. It will render ji 
 study heretofore charged with ditlicully and abstruseness at once 
 easy and acceptable— modtniizing that which was ancient and wiiich 
 from its multitudinous formuhu had become an isolated branch of 
 Mathematics. Believing it to be of universal use, 1 shall heartily lend 
 myself to the introduction of your system. 
 
 Very respectfully. Your obedient servant, 
 
 IIUKATIO a, N. BlUKJ.OW, M. A. 
 
 C. Baillaiuoik, Est,?. 
 
38 GEOMETRY, MENSURATION, AND 
 
 Ser. 2, No. 2r)5. Education Office, Province of New Brunswicli, 
 
 Frt'd(!iicton, Jamiiiry iioth 1872. 
 
 C. Baillaiuge, Esq., Qu('l)oc, Dear Sih.— T am instmotcd by tlio 
 Roavd of Education for thi.s Proviiico to apply to you for a wet of your 
 S'cn'oincfric Tithlcan and your t<'xt-l>o(»k on Practical Matlicuiatics. 
 Tlio Moard desire tlicsc articles for inspection, witii a view of prcjs- 
 cribing them for general use in all the Schools of this I'roviucte, siiould 
 tliey be deemed suitable for the i»urpose. Should there be any charge 
 for tiiese articles, the same will be mvi, I>y this Department. 
 
 Your Obdt. Scrvt., TiiEonouE 11. Kaud. 
 
 ^Ir. Bai]laiij;e'sStereometrical Tableau seems to me to be a very 
 
 useful arran,uemenl for siiowinjj: tlu' variety and <'xtent of the ai>pli- 
 
 oations of the I'l-iffnioiilid FontinUi, Wliere demonstrations are ,;;iven 
 
 in the study of Mensuration of Solids, it will aid a teacher in ilhistra- 
 
 ting th(^ rules, but it would i)robabIy be much more valuable to those 
 
 who tiy to teach that study without intioducing denionstrations of 
 
 the rules. 
 
 11. A. NicwTON, Prof, of Math, iu Y. College. 
 
 Yale College, Feb. 5th 1872. 
 
 No. 12X7. Subj. fH).'). Ilef. 20814. Dcj^tarfmcnt of r,tl>lic Worls. 
 
 Ottawa, Feby. 7th i;572. 
 
 Sir, — In reply to your letter of the 2(5 ulto., I am directiid by tho 
 Minister to request you to fnrnisii the I)e[tartnient with onct of your 
 ^'Tableau Stereometii{[Ue "' at the price of Fifty <lollars, together 
 with your account for the same. 
 
 I have the honor to be, Sir, your obt. servant, 
 Ciis. Baillairoe, Esq., F. Braix, 
 
 Architect, Sec, Quebec. ISccretari/. 
 
 New Haven, Feb. 7th 1872. 
 
 Ciis. Baillairge, Esq., Dear Sir, — I have been much interested 
 
 in looking (»vei' the papers descriptive of your nsefid, va]ual)le and 
 
 (as it vciy plainly ai»i»ears) univ«'rsal ajiplication (»f a ruU' for tho 
 
 mensuration of solids. 1 sinceiely cimgratulatc^ you on tlu^ Huccess 
 
 Aviiich your discovery has nu't with in all <piarters in which it has at 
 
 present been introduced. It must have been a great labour to work 
 
 it out to its present state of jterfection and you liave tln^ satisfaction 
 
 of knowing that you are a benefactor and staunch pilot in that sea of 
 
 dilliculty : Geometry. 
 
 Voui' very sincerely, E. B. Barber. 
 
THE STKREOMETRICAL TABLEAU. 89 
 
 Hiuh School, Quebec, 8tli Fel). 1872. 
 
 Ciis, nAn,i,.vii;(.i:, Ks(^, Mv Dkau Siit, — I 1k',l> (o iirknowlcd^c? 
 uitli iiumy fliiiiiks Hie icccipt <if ii iiuii)l)ci' of pjiixTs expliiuatoiy ot' 
 y<»iir new toniiiilii lor liiidiii^- tlic coiitciits ol" solid bodies. 
 
 'riie iiilc is precise iuid siiii|»I(', miuI beiii.n' iippliciible to abuost 
 any vaiiely ol" solid, will yieally sliorteu the processes of calculation. 
 
 I liave proved its accuracy as applied (o several bodies. 
 
 'J'iie 'J'ableau coiuprisinn' a ^reat variety of eleiueidary models 
 will seiv(! a(bnirab!y to educate the eye aud must greally I'acilitato 
 the study of solid meusmatiou. 
 
 'I'lie Govenuiu'Ut would confer iv boon ou scluxds of the uii(Ull(> 
 aud higher class by atlbr(liii;n- a( cess to so su^7;'estive a collection. 
 
 I liave the honor to be, uiy dear sir, your obedient servant, 
 
 . J). VVii.KiK, Kector. 
 
 Quebec, Hth Janiuuy 1872. 
 
 Cns. iJAiLi.AiKOi:, Esq., Dkau Siit, — I?i'in,i;- absent from homo 
 when your lavour of the 1st ult. arrivi'd and haviui; returned only a 
 few days a^i-o I have found it impossible to .i;ive to vour Stereomelri- 
 cal Tableau that attention which the subject uu'rits. 1 have liow- 
 ever in the case of a few solids compared your formula with the ordi- 
 nary methods of computation and found it eijually ccnrect. I shall 
 be deli,i;hted to see the old tedious processes HUi»erseded by a formula 
 80 simple and so exact. 
 
 1 have the honor to be, Dear Sir, y<Hir obedient servant, 
 
 A. N. McQu^uiUiE, B. A.* 
 
 Quelx'O, 27 Dec, 1871. ' 
 
 CliAULKs IJAiLLAiudK, EsQ., Qi'iciucc. Dkau Sru,— I be";' to ack- 
 nowledj;e w ith nnuiy thanks the receipt of the papers and photograms 
 explanatory of your Stereouu'trical 'J'ableau. 
 
 I h;'ve com])ared, in the case of several solids, the I'csults obtiuned 
 by your mode of « (»mi»ntation with those resulting from the ordinary 
 and more lengthy processes, and con;;ratulate y(»u sinceily on your 
 enunciation of a foiniula so brief and simple in its character and so 
 precise and satisfactory in its results. 
 
 I remain. Dear Sir, Your very odedient servant, 
 
 E. 'i'. Fij:Tcni-.i£, 
 InsiKctor of Surveys Ik^tt. of Crown Landn. 
 
 * Professor of MathematioF, etc., at tbo Morin College.— C. B. 
 
40 GEOMETRY, ,>1ENSURATI0N AND 
 
 QiK'bco, .S('i)tnnlm' 1872. 
 l\To\>;iKriJ. .T':ii Ic |)liii.sii- dc voiis iiiiKonccr <|ii(' lo Conscil do 
 riiistiiutioii I'liltliqiic viciit d'appioiivcr votic TiiUlcau ct suis lu;U- 
 reux dc voiw t'U tV'licitcr. 
 
 IJicn siiirrroiiiciit votio lout devour, 
 C. Eaillaikge, Ecr. P. J. O. Ciiauvkau. 
 
 TZ/c Janunnj Session of the Board of Ernmhirrn for Loud Sttrvcyors 
 for tin- Proriinc of Quvhec. \S7)i. 
 
 KXTIiArr FliOM TIIK MIMTKS. 
 
 Moved l»y tlic rn'sidtiit, Adoli)lK' I^aUiic, Kstj., and .seconded l>y 
 E. T. Flctolicr, V.sq., and resolved : 
 
 ^' That the Boar<l ot'Exaiuiiieis for Land Surveyors, haviiif? taken 
 into consideration the Stereonietri<iil Tahh'au ofCharh's Iiaillair,n(?, 
 Es(|.. Civil Engineer. and tlie "\ cry neat and precise fornmia conuictcd 
 therewith, desiic to record liicii' opinion ol'lhc utility and ini]>orta!iC(^ 
 of this formula, and coiiuidc wholly witli tiie opinions expressed hy 
 those to wiioin it lias hccii already submitted, and further they would 
 roconiinend tiiat flic Hoard he ])i'o\ iih'd with one of tiiese Tableaux. 
 Quebec, .January tind, \f^7)l. Ai.kxanpkk 8i:wki-1., 
 
 StTretorji of the Boord of SKrveyors, 
 
 No. 2^272-71. Miiiislcre de r Instriiefion Vuhliipie. 
 
 Qut'bcc, ce 7 .Sej>tenibie 1H72. 
 C. H.Mi.LAiKdK, EcuvKK. QiKUKO. iMoNsiKUi}, — .I'ai riioiineur de 
 vous transmcttri , sur Taiitre feuillct, co|»ie de la resolution adoptee 
 j)ar le Conscil de rinstruction Publicpn', approuvant votre *'Tal>lcau 
 Stereonictriiiuc pour toiscr tons Ics corps. scu,ineiits. troncs et onylets 
 de ces corps,"' aiiisi (pie votre '* Nouveaii traite dc j^'Mnnetrie et de tri- 
 gonometric rectilignc et spherique," suivi dii " Toise des surfaces et 
 des volumes." 
 
 J'ai I'honneui" d'eti'c, Monsieur, votre obt. serA'itenr, 
 
 Loi' IS (i i.vim:), Seeretaire-A rehirit^fc. 
 
 J^uis, le ler Aout 1H72. 
 
 A Aronsieiir naillair^e, Architcete, etc., a Quebec, (Canada). 
 
 MoNsir.rn' — J'ai 1 homicur (h- viuis donner avis ([ue le C'onseil 
 Sui»erieur vient de vous admettie k faire partic de la .Societe de Vul- 
 garisation pourrEnseignementdu peupic a titre de jMembrcTitulaire. 
 
 Nous somincs lieureux d'liue decision qui assure sV notre Q•]uvr<^ 
 votre precieux c<»nconrs et nous cspeions Ics mcillciirs cHets {\o votre 
 propagaude active en fax cur de rinstruction et de TEducation po])u- 
 
 laires. 
 
 A'eiiillcz agreer. Monsieur et ties honore Collegue, I'as- 
 burance, de iiU'S sentiments ch' liaiite consideration. 
 
 Lv .Secretaire (Jt'iH'r.il Ibndaleui. 
 
 (Siglie) At (i. Ill .MliEUT. 
 
THE STEREOMETRICAL TABLEAU 41 
 
 Quebec, i)th January 1872. 
 G. W. Weaver, Esq., 
 
 President of Board of Arts d- Manufactures. 
 
 My Pkau Sir, — Our cvoiiiu^jf class bo;j;an last week, T iiin liap])y 
 to say every tliiufj^ looks promising and we havebeen fortunate enough, 
 to securer again Mr. C. Haillairge's invaluable services, as teacher, for 
 this -winter. 
 
 We have i)rocnre(l from him, for our school, the " Stereomctrirnl 
 Tableau'''^ which is his invention, and T am so delighted with it, that 
 I send you a i)Ii»»tographic icinesentation of it, and a number of let- 
 ters and other documents printed which will s(;rvc to exjilain the ta- 
 bleau , and show at tho same time how nseful it is consitlered by the 
 most eminent authorities in this country. 
 
 Y(m ought to get tin; tablesui for your schools, at Montr(>al. You 
 ehowed me last year when I visited youi" schools several wooden mo- 
 dels of geometrical figures ; T was struck with their usefulness at the 
 time, and thought of procuring some for our schools, but there are 
 only a few of them and their price is very liigh. Mr. l^aillairge's Ta- 
 bleau costs oidy fifty dollars, and it contains two hundred geometrical 
 figures. I fancy the collection embraces every variety of iigure that 
 can ever be reipiired for practical use. 
 
 They are solid figures made of wood, each fixed on a nnil, S(» that 
 they can be taken olf by the teacher for demonstration and handed 
 to the pujiils ; who are enabled to understand and master their divers 
 shapes and forms Avith much greater ease, than if they saw th(;m 
 drawn on a blaclv board, or in a book ; the ditterencc is enormous. 
 
 In addition to the gre.at help they afford, for the study of geome- 
 try, these figures are very useful as models for earthworks, piers, 
 reservoirs, castings, roofs, domes, columns, cauldroues, &c., &c., &;c. 
 
 The tablean is most useful too for the working out of that won- 
 derfully simple rule, which has been applied by Mr. Bjullairge, for 
 the fust time, to the measurement of the solid contents of all bodies. 
 It was known previous to his discovery to apply to a certain number 
 of b(»lies, but he has found out that it applied to all without exvcption. 
 You will find that rule in the papers I send you, and in his treatise 
 on gi'ometry. I will soon let you know, what progress the school is 
 making and remain, my dear Sir, 
 
 Yours truly (Signed,) H. G. Joly.* 
 
 • President Quebec School of Arts. 
 
42 OKOMFTKY, MENSURATION AND 
 
 Ertrmt from an Addnss hif (liv i)uj)il)i lo ('lis. lUtHUtlnji', Esq., I'ln- 
 J'cssor (it tlic Svlnud i>l' AitK, \',Mli AjH'U IS7'.J. 
 
 We «U'(.'iii it not out of pliirc to rciiiMrk lliiil in oiii- o|»iiiioii tlm 
 word " sTi;i:i:()Mi:ri;i(Ai, '' Avliicli you liiiv«' prclixcd. us (juiilitnlivc of 
 llic uses Ihiit .your '' TAlii.KAi' " ciin lie iipl>Ii«<l to, is not su<i-;fcstiv<i 
 I'non.uli ol' tlir niiiiiy ndvantiiycs wliicli sncli a viiricd collection of 
 uiodrls i>i(scnts; for, n(»t only in it ot" jtiiranionnt importance and 
 utility, MS illustrative of your systt'iu of nn-nsuration, by (uie and tliti 
 same iuvariable tormula, implied in tlu; title at the head of the hoard ; 
 hut, we hesitate not to say that to the usi' of tlu! '' taui.kai" we an; 
 indeltted in an eminent decree for tlu' sinyulaily rapid pro^icss we 
 have been enabled to inake since t lie Itli of danuary last (only Ml) les- 
 sons) not only intieonu'try itrop.ei' and in Ihe Mi'nsuraticm of surfaces 
 and solids, Itoth plane and spherical ; Itut also in the study of f^eonie- 
 trical projection and perspective, shades and sha(h>ws, the (h'velop- 
 ment of surfaces and tlui lines of penetration of divers solids, &c., \:c. 
 
 Extract from the " QiwIkt (;«,-<///<;" March 2^ii<l 1872. 
 
 Mli. r.ArLLAJIlGE'S LECTURE. 
 
 'rh(> lecture on (leometry, (h^livered by (). Uai1hiir,i>'e, K,s<|.. before 
 the Jjiteiary and iiistoricai .Society, in ihe Mtnin C."olle;;c, on the eve- 
 iiin;;- of Wednesihiy hist, was a rare scientilic treat, lost to many win), 
 (htubtless, thinking- the subject adry one, did not attend. TIk' au- 
 dience, thouiih not as large as nught be ex])ected, comprised the elite 
 of the scientilic and well icad men of this community. 'I'hat tlu' sub- 
 ject, as handled l>y Mr. Baillairge, was not a «by one, may be infencd 
 from his showing, during its course, that it not only applied to that 
 most attractive .of all ovals, the female countenance, but that the keeu 
 appreciation of its charms by the fairer jtortion of nmnkind was clearly 
 evidenced in the beautiful and ever vaiying tiacery of their laces, 
 endtroidery, iV.c. Neither was it wanting in poeti*- imagination, as 
 illustratcMl by the lecturer, in comi)aring the curves traced out by the 
 engineer anndst tlie woods ami waters of (he earth, to the mighty 
 circuits of the comets anddst the stiirry forests of tiie dark blue hea- 
 vens. That part of the lecture touching upon conic sections was 
 especially interesting, owing t(» the lucid maimer in which their prin- 
 ciples weie described as applied in the throwing of [Mojecliles, jets 
 of water, mirrors, reiU'ctois, &c. The lecturer exhibited his Strrco- 
 iHctrical Talilcaii, which is now attracting so much attention in this as 
 well as other «'ountiies, and denn)nstrat<'d, to the peifect satisfactiou 
 of his hearer.-., that it wa-i'ii!!y ( iitiMcd !'» :dl tli ■ a<lvaiila';is claimed 
 for it. At the close of the n-ading, (."apt. Ashe, U, N., in proposing a 
 
THE 8TERE0METRICAL TARLEAU, 43 
 
 vofo of thanks to Uw Icrtiircr, Avliilst t'laiiuiii<; for Dr. Simpson tlie 
 (lis,'«ivery of tlio prisnioitlal formula, as known to apply to certain 
 bodies (a faet alliitled to in tlu^ lecture), nevertlieless hi/>lily compli- 
 menteil Mr. Uaillair^e, giving him dui^ meed of i)raise for his general 
 aitplicalion of the foniiula to all known solids. 1). Wilkie, Esq., Rec- 
 tor of the nii;]i School, than wliom no more comi»eteiit judge of the 
 sultject could be fouiMJ in our midst, in seconding the vote of thanks, 
 ]io)»ed that a lecture so interesting and instructive wouldbe published, 
 so as to bring it within th(> n^iich of all, and gave expression to his 
 most un(|ua1ified admiration of the high talents of Mr. Haillairge and 
 his devotion to the scicMice of tigures. He considered tiie production 
 of the Stctromvf rival Tahlcau of vast imimrtance to the cdiicational 
 system, by re<lucing the work of a year to that of a day or two — so to 
 say. He also comj>limented the lecturer on the happy way in which 
 lie treated the subject, making that which many Avere wont to consi- 
 (h'r dry even [)oetical. 'Vlw Tresident, Dr. Anderson, in putting the 
 motion before the nu'cting, said that in his recollection, he had never 
 been so pleased or gratified Avith a lecture as that AAiiich they had 
 just heard. That it Avas most flattering to Mr. Baillairge to knoAv 
 that he had kept an audience entranced for tAvo Avhole hours Avith 
 such unflagging interest, that tlie two hours had passed as though but 
 one. The Tresident concluded ) is renmrks by stating that though 
 Mr. Simpson had made the discoA'ery alluded to, it had seldom if ever 
 been practically applied, and that, therefore, ^Ir. BaiUairge should be 
 considci(Ml tlu' ical disco\er, a iact «'arried out by the lecturer himself 
 Avhen stating that *' the formula would be nothing AA-ithout the tableau, 
 any more than steam Avithout the steam engine or electricity without 
 the telegraph." 
 
 The UrsuUne Convent. 
 
 We learn w ith satisfaction and legitimate pride that this admir- 
 abl(^ instituti(»n has ordered one of Mr. BaiUairge's Stereometrical 
 Tableau, and that this gentleman, during a single sitting of a few 
 hours devotion, generously granted him by the Nuns, managed to 
 rendi-r them tlioroughly conversant Avith his system of nomenclature 
 and mensuration. It seems almost incredibh', and yet, Ave are as- 
 sured that the Reverend Superior (the talented Miss Cimon, of St. 
 I'aul's liay) accompanied by Sister St. Croix and Sister St. Raphael, 
 at once mastered and perfectly understood Mr. BaiUairge's system in 
 all its details. Paying comparatively little attention to the more or- 
 dinary forms of wliich the mode of measurement Avas to them appar- 
 ent at a glance, tlie.v sch'cted for their questions the more complex 
 tbrms, such as the sections of the sphere and spheroids and the nu 
 
44 GEOMETRY, MENSURATION AND 
 
 nicroiia and vmicd piisinoids, to be found amon^' llic '200 niodcli^ of 
 the tahlcau. The education ^dvrn l>y the, K<li<;ious Ladies (o tiu^ir 
 pupils ccMnprises tho geometry of lines and surfaeeH, tnul from this to 
 the mensuration of solids by Mr. Haillairge's system, there is but one 
 step, a simphi addition of certain surfaces and the multiplication of 
 their sum by one sixth i)ait of the height or length of the body un- 
 der consideration. The Nuns intend to commence imnudiately the 
 teaclung of this branch to their i)upils, wlu> will be cxannned on the 
 tableau at the next examination which takes place in June. The 
 m)ble example thus given by the Ursulines has been rapidly followed 
 by another important educati()nal establislnnent, the Convent of the 
 Religious La<lies of Jesus-Marie, on the Cap Kouge road, and we are, 
 moreover, informed that the Samrs de la (Congregation St. Roch, to- 
 gether with the Nuns of the Good She]dierd and the Sisters of Charity 
 intend forthwith to add to the already varied progrannne of their 
 tuition, this study of Stereotomy, wliicli, up to the present time, 
 could not be even dreanu'd of, but which Mr. Maillairge's system now 
 renders possible by reducing, as it docs, the study of a year to that of 
 a day or two, so to say. 
 
 COPY. WoucKSTEU Free Institlte. 
 
 Worcester, Mass,, July 24 1873. 
 
 This certifios that I have carefidly examined Baillairge's models 
 to illustrate the applications of the Prismoidal Formula, and c<msider 
 them eminently calculated to be useful in all schools where mensii- 
 ratiou is taught. 
 
 (Signed,) C. 0.- THOMPSON, 
 True copy. Principal Wr. Free Inst. 
 
 EXTRAIT d'uNE LKTTRE DE V. VANNIER. 
 
 Paris, 8 JniUct 1873. 
 
 ** H est bion decide que votre tableau aura la premiere recom- 
 pense de la societe libre d'instruction et d'education popuhiire; vous 
 en serez averti ofticiellement i\ la r<;ntree des v.acancesT, fln octobre, 
 et vous serez en meme temps convcxpie pour la distribution solen- 
 nelle des recompenses qui aura lieu dans le courant du mois de Mars 
 prochain. 
 
>r