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 I. General applicaiion oftlie Prismoidal Formula. — 11. Hints to 
 ■^ Geometers for a new edition of Euclid. — III. Simplified Solu-,; 
 I^f- tions of problems in hydrography and parting off land. — . 
 IV. The areas of spherical triangles and polygons to . 
 
 : . Y: ,N, - . >/ any radius or diameter. ,.^ ^ . ^ 
 
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 ' BIOGllAI'HICAL SKETCH OF THE AUTHOR.- 
 
 
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 ::J::';:;. quebec 
 
 C. DAUVEAU, PUINTEU 
 1884 
 
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V 
 
B^ILI^^IROl!: 
 
 PAPERS 
 
 READ BEFORE THE 
 
 ROYAL SOCIETY OF CANADA 
 
 - ^ 1882 & 1883. 
 
 I. GeDeral application of the Prismoidal Formula. — II. Hints to 
 Geometers for a new edition of Euclid. — III. Simplified Solu- 
 tions of problems in hydrography and parting off land. — 
 IV. The areas of spherical triangles and polygons to 
 
 any radius or diameter. 
 
 BIOGllAPHICAL SKETCH OF TUE AUTHOR. 
 
 ■^rv -.. '' 
 
 
 QUEBEC 
 C. DARYEAU, PRINTER 
 
 1884 
 
^H y6 5 
 
. ,n'»>J^-_^ # - 
 
 :-■ .,'-1 .'' -■; -t: :-.■_: i- , • -v -- 
 
 PREFACE 
 
 "Yk 
 
 iROYAL SOCIETY OF CANADA 
 
 Honble P. J; 0. Chattveat^, President. 
 
 • 
 
 Mr. President, ■'"'- - .: .. 
 
 The Society has just issued (1884) a quarto volume, well printed on 
 good paper — paged in sections instead of consecutively — some 700 pages 
 of french and english literature with much abstruse matter, which not more 
 than one in a thousand can ever read, also some 46 pages of comedies 
 which, however interesting and instructive the perusal thereof may prove, 
 occupy somewhat of space in a work of such a nature ; while there ap- 
 pear but two articles, one of professor Johnston, of 1882 on the "Sym- 
 inetricallnvestigationofthe Curvatures of Surfaces," the other by M. 
 BeviUe, "La mesure des distances terrestres par des observations 
 astronomiques " : barely 18 pages out of 700, on subjects pertaining to 
 inathematics. 
 
 The object of the Society, I presume, as of any society of the 
 kind is that, among others, of reporting progress and being foremost in 
 its diffusion throughout all parts of the civilized world. 
 
 I am a member of the Society and congratulate myself on being 
 one. If the choice of the Noble Lord, its founder, first patron and fir^t 
 
— 4.— 
 
 honorary president fell upon myself as one of the original members of 
 Section III, "Mathematics, Physics and Chemistry," it must have been 
 with the full knowledge of what He was doing. The Marquis of Lome 
 knew of my labors and publications. He knew them to have been 
 crowned in France, Belgium, Italy, Russia, and introduced into Japan 
 and elswhere ; but on account of their utilitarian nature, as of other 
 meritorious works. He desired for them a vaster field of action in the 
 education of the world; The British Empire, the United States of 
 America, &c., &c. The mode of arriving at this was precisely among 
 others, the creation. of a Society, which under the prestige of its Royal 
 Birth, an'.l with the help of an annual grant from the gov;;rnment of the 
 Dominion, would be enabled to pubhsh and make knowm to the rest of 
 the world, the mineral and other treasures of our vast country, as well our 
 progress in the Arts, Sciences and Literature, both French and English. 
 
 The admirable inaugural addresses of the Noble Lord himself, of Dr. 
 Dawson, the then president of the Society and of the Honble.M. Chau- 
 veau, its vice-president, contain many happy and pertinent allusions to 
 this mode of taking by the hand and making known to other countries 
 such of us as might have done something useful in the past, or of those 
 who under the potent influence of the Royal Society might do so in the 
 future, and whose labors, without which, might have remained unknown 
 and the world have been deprived of certain knowledge which it might 
 have rendered profitable for the advantage of mankind, v.^- • ■ >^-i:^ > ^*^- 
 
 The importance, the world over, has not been seen, of saving to 
 thousands an hour each day, or be it even half an hour or a quarter, in 
 the computation for instance of the contents of a tub or vat, to which, 
 there continues to be applied in practice and taught in our schools, Le- 
 gendre's method with an area mean proportional between those of the 
 opposite bases, whilst that obtained by a multiplication of factors which 
 are arithmetic means between those of the end bases simplifies the oper- 
 ation in a way to save much valuable time, render the operation easier 
 of apprehension and far less subject to error. 
 
 Believe me. M. President, this one thing alone made known through 
 the medium of the Royal Society of Canada, would have sufficed to 
 give It at once an impetus, a great prestige of public utility, with the 
 
- % 
 
 honor of being the first in the field ; since the tub or vat exists under 
 its usual shape of the erect or reversed frustum of a right cone and 
 in all possible proportions in the arts, trades and industries of all kinds, 
 in soap and potash factories, breweries and distilleries, &c., in every 
 known country of the world. 
 
 So far, of one of my papers: that read before the Society in 1882, 
 and of one single item thereof, and I might enlarge on many others in the 
 same paper, und'^.r the heading : " Application of the Prismoidal Formula 
 to the measurement of solid forms, " t.. v 
 
 J"" ■■'■■'.'.-:■--'■'' * ■ ' ■ / .- .- - 1 .i ■ 
 
 In 1883, I contributed three several papers, see page LXVII ot the 
 ** Transactions," one of which ( II ) entitled : " Simplified solutions of two 
 " of the more difficult cases in the parting off and dividing-up of land, 
 " also a case in hydrographical surveying. " \ 
 
 Nor was anything useful seen in this, and yet they are operations 
 which the world over, surveyors and hydrographers have every day to 
 perform and repeat, and all such, I am positive would have been thank- 
 ful to the Society for making known to them simplified solutions of 
 what they consider to be difficult cases. 
 
 ' Again (III), "The areas of spherical triangles and polygons to 
 ** any radius or diameter." 
 
 Will it be said that there is nothing special enough in this, nothing 
 sufficiently scientific and utilitarian to figure in the publications of the 
 Society. -♦.:,>-..:>!'-• ; .' ^v,- .v,,.--,r 
 
 On the contrary, what labor is there not to all those having to do 
 with such calculations when the old or ordinary rules are made use of to 
 arrive at the doubly curved surfaces of a portion for instance of the 
 terrestrial spheroid, as of the spherical forms pertaining to the arts and 
 trades, such as a boiler or copper, a gazometer, a dome, the ball of a 
 steeple, a shell, a cannon or a billard ball. 
 
 Then in article (I), was there still nothing pertinent, nothing sug- 
 gestive in, my : " Hints to young geometers for a new edition of Eu- 
 clid. " Could it not be seen that in the thousands of schools of the old 
 and new worlds, where this author of 2000 years ago still holds his own, 
 
these " hints " or " suggestions " might enable professors to spare their 
 pupils much valuable time ; certainly not less than some three months 
 in their study of the elements of geometry, while at the same time scru- 
 pulously conserving all the deductions and conclusions of the greek 
 author and sacrificing nothing of their logical concatenation. 
 
 No, Sir, nothing of all this has been seen. We have been content 
 to fill a 700 pages volume with articles of merit, no doubt, and which 
 have or will have their great utility, but in a narrower field of enquiry : 
 mineralogy, botany, chemistry, physics, astronomy, &c, and nothing to 
 the point has been found in subjects which address themselves to the 
 whole world and form, so to say, the basis of all education preparatory to 
 the study of other sciences. 
 
 Let us hope, M. President, that, henceforth the publication and 
 printing of our yearly record may be tiubmitted to more mature consid- 
 eration so that if hospitality be given to he who has treated a subject 
 interesting to only a portion of the community, the door may not be shut 
 against him who on the contrary has produced something interesting to 
 humanity, the world over. v.^r & v;r- ^ i^ ^i^ e^; 
 
 ■=■.*_■.-._"■-■".. ', '----" 
 
 , I am not insensible though to the truth of the proverb which 
 has it that every thing happens for the better. ;^ 
 
 [■Cf. 
 
 Providence, may be, has been willing to side with me. Had my 
 feeble efforts in the direction of education and instruction, my writing* 
 in this field been mixed up with others in a vast quarto volume ; may be 
 they might never have been noticed, the fate of most such books, and 
 on the other hand, the circulation of the " Transactions " being necessa- 
 rily limited on account of their great cost, it was perhaps better, con- 
 sidering all things, that what I complain of should have happened as it 
 has done, so as to afibrd me the opportunity, though at m v own expense 
 of an increased circulation. 
 
 The omission may be the cause of my efforts in the direc- 
 tion of progress and improvement, being better known, more widely 
 circulated and appreciated, better even than they would or could have 
 bei^^ W^v the. shield, the prestige, the ppteut influQiiQe of a Spcietj ^ 
 
— 7 — 
 
 which the transactions less numerously reproduced, would have afforded 
 me less chance of success with the numerous class of teachers and 
 pupils, aU over the world. 
 
 This more facile and succinct teaching of the geometry of lines, and of 
 the computation of areas and solid contents, has been so to say, the object 
 of my aspirations for many years past ; and now in a higher sphere, M. 
 President, let me introduce to you as worthy of filling the first vacancy 
 which may occur in section III « Mathemat -s. Physics and Chemistry" 
 M. E. Steckel, asst. engineer of the Dept of P. M. of the Dominion 
 one of our cleverest Canadian mathematicians and who has already 
 made long and serious studies in acoustics, hydraulics and other sciences, 
 and should this happen before the next meeting of the Society, I doubt 
 not but what M. Steckel would then be ready to enrich its future annals 
 with articles which would have their echo abroad. 
 
 With my sincerest wishes for the weU being of the Societv, of 
 which to make room for Mr. Steckel, I would be pleased to remain an 
 honorary member, 
 
 -■. ^■■ 
 
 I have the honor of subscribing myself, 
 M._ President. 
 ; 'Your obedient servant, 
 
 r^ CHS. BAILLAIKGfi, 
 
 \-<'-:'n\-::f^^^^^^^^^^ A. M., 
 
 S ; Fellow of the Royal Society of Canada and of several other 
 learned societies, Member of the society for the gene- 
 ralisation of Education in France. Chevalier of the 
 Order of St. Sauveur, Italy, Architect, Engineer, 
 Surveyor, &c , &c., &c. 
 13 medals of honor and 17 diplomaa from France, Belgium, 
 Italy, Ku88ia and Japan, etc. 
 
,<■ .>-ni.-->-' ' 
 
 • .: r ON THE APPLICATION ■ ■"^•-M^i:^'m--^>: 
 
 PRiSMOIDAL FORMULA 
 
 TO THE MEASUREMENT OF ALL SOLIDS 
 
 By CHS. BAILLAIRG^, M. A., ■ 
 
 i-.: j: 
 
 Member of the Society i - the Generalization of Education in France, and of several learned 
 and scientific Socieiies. Chevalier of the Order of St. Sauveur de Al onte-Reale, Italy, 
 &c. Recepient of 13 medals of honor and 17 diplomas and letters from Russia, France, 
 Italy Belgi m, Japan, &c. Member of the Ro^al Society of Canada. 
 
 Read before the mathematical section of the Society on Saturday the 28th of 
 May, 1882. 
 
 ■^ -: 
 
 « Cette formule V = 5-(B + B' + 4 M): (Says « the late Revd, IT. 
 
 " Maingui of the Laval University) que Mr. Baillairg^ travaille k 
 " vulgariser, a rimmense avantage de pouvoir remplacer toutes les 
 " autres formules de st^r^om^trie." ' 
 
 The prismoidal formula reads thus : " To the sum of the opposite 
 and parallel end areas of a prismoid, add four times the middle 
 area and multiply the whole into one sixth the length or height of the 
 solid" 
 
 • See this formula at article " St;er6ometrie " of "Le grand dictionnaire universel du 
 XlX^me 8i6cle par P. Larousse." 
 
— 9 — 
 
 The following letter from the Minister of Education, Eussia, may 
 be considered interesting in its bearings on the subject matter of this 
 communication. 
 
 MINISTEKE DE LINSTEUCTION PUBLIQUE. 
 
 hp: 
 
 • Saint-Petersburg, le ih f^vrier 1877. 
 Ko. 1823.r ;■' ':: • '^^^ ;■•':-■:- -.^.^::pv- 
 
 A. M. BaillairgI:, "'[jg') ^ .; 
 
 ''''^'" "-^ '•■-■" ■ Architecte k Quebec, ■--'■" - --^-^t^ ^-/.^^ ^W,c":*i>ail ■ ..,. 
 
 Monsieur,""" . : ■■,%.• ■.... .r..- -^r^'U-^H^'' :--aj^iM' 
 
 Le comity scientifique du ministke de I'lnstruction Publique, (de 
 Eussie,) reconnaissant I'incontestable utility de votre "Tableau St^r^o- 
 metrique " pour Tenseignement de la g^om^trie en g^ndral, de m^me 
 que pour son application pratique k d'autres sciences, ^prouve un plaisir 
 tout particulier k joindre aux suffrages des savants de TEurope et de 
 TAm^rique sa complete approbation, en vous informant que le susdit 
 tableau, avec toutes ses applications, sera recommand^ aux dcoles pri- 
 maires et moyennes, pour en completer les cabinets et les collections 
 math^matiques, et inscrit dans les catalogues des ouvrages approuves 
 par le minist^re de Tlnstruction Publique. -^ 
 
 , Agr^ez, monsieur, I'assurance de ma haute consideration. 
 
 Le chef du departement au minist^re de I'lnstruction Publique, 
 
 E. DE Bradker. 
 
ib — 
 
 The following extract from the Quebec Mercury , July 10, 1878 
 
 further corroborates its importance, 
 
 " It will be remembered that in February, 1877, Mr. Baillairg^ re- 
 ceived an official letter from the Minister of Public Instruction, of St. 
 Petersburg, Kussia, informing him that his new system of mensuration 
 had been adopted in all the primary and medium schools of that vast 
 empire. After a lapse of eighteen months, the system having been found 
 to work well, Mr. Baillairg^ has received an additional testimonial from 
 the same source informing him that the system is to be applied in all 
 the polytechnic shools of the Eussian Empire. " 
 
 Should the Eoyal Society of Canada prove instrumental in the 
 introduction of the new system throughout the remainder of the 
 civilized world, It will have shown that its creation by the Marquis of 
 Lome, the Govr. Gen. of Canada, has been in no way premature. -^ * ' 
 
 The definition of a prismoid as generally given is understood to 
 apply to a solid having parallel end areas bounded by parallel sides. 
 
 >ri This parallelism of the sides or edges of the opposite bases or end 
 areas does not imply, nor does it exclude any proportionality between 
 such sides or edges. ' -''- - ' ;. . '• - i ^ '^ 
 
 Therefore is the frustum of a pyramid a prismoid, as also that of 
 a cone which is nothing but an infinitary pyramid, or one having for its 
 base a polygon of an infinite number of sides. . ,. w 
 
 • Now let two of the parallel edges of either base of the frustum 
 approach each other until they meet or merge in a single line or arris, 
 when we have the wedge which is therefore to all intents and purposes 
 a prismoid. 
 
 Further let this edge or arris become shorter and shorter until it 
 reduces to a point and then have we the pyramid which is again a pris- 
 moid, as is the cone. 
 
 It need hardly be said that the prism a^id cylinder are prismoids, 
 whose opposite edges are equal as well as oarallel in the same way as 
 
^■■■-_ -■: — 11 — - : ^ r ' 
 
 for the frusta of the pyramid and cone the opposite edges are propor- 
 tional while parallel. 
 
 Now, nine tenths or more of all the vessels of capacity, the world 
 over, and either on a large or reduced scale, have the shape of the frustum 
 of a cone or pyramid ; the latter as evidenced in bins, troughs and 
 cisterns of all sizes, in vehicles of capacity ; the former, ia the brewers' 
 vat, the salthig tul), the butter firkin, the common wooden pail, the 
 drinking goblet, the pan or pie dish, the wash tub — of whatever shape 
 its base — the milk pan and what not else ; again the lamp shade, the 
 shaft of a gun or mortar, the buoy, quai, pier, reservoir, tower, hay-rick, - 
 hamper, basket and the like. , i 
 
 These are forms which in every-day life the otherwise untutored 
 hand and eye are called upon to estimate. Why then not teach a mode of 
 doing it which every one can learn, and not only learn but what is of 
 greater import, retain in mind or memory when mastered. . ; : * 
 
 Why continue the old routine when, as here evidenced, it is so 
 much more simple and concise, so much quicker to apply the prismoidal 
 formula to all these forms, than resort to one more difficult of apprehen- 
 sion and which to carry or work out requires tenfold the time the other 
 does. 
 
 Legendre's formula requires a geometric mean between the areas of 
 the opposite bases of the solid under consideration. This mean is far 
 less easily conceivable than the arithmetic one ; and to arrive at it the 
 end areas are to be multiplied into each other, and the square root ex- 
 tracted of their product; along and tedious operation, one known only 
 to the few, most difficult to retain, forgotten as soon as learnt and 
 therefore useless. - j; v ^^ 
 
 Wiuh the formula proposed on the contrary, the operation is one 
 which the merest child can master, the mere mechanic or the artisan 
 remember all his life and readily apply ; for he has been taught at school 
 to compute areas, that of the circle as well as others, a figure which he 
 readily sees is resolvable into triangles by lines drawn from the centre 
 to equidistant points, or not, in the circumference, and the area thence 
 equal to the circumference — sum of the bases of the component triangles 
 
— into half the radius, or height of the successive sectors which make 
 up the figure, r ^ . ^^ 
 
 Now, of almost all the solids herein above alluded to, the opposite 
 bases and middle section are circles and the operation can be further 
 expedited by taking the areas ready made, to inches and even lines or 
 less, from tables prepared for the purpose. _ - , 
 
 The labour then reduces to the mere arithmetic of adding the areas 
 so found, that is the end areas and four times the middle area, and of 
 multiplying the sum thereof into one sixth the altitude, or depth ; that 
 is, to the simplest form of arithmetic taught in the most elementary 
 schools, to wit : addition and multiplication, with division added when 
 the cubical contents in feet, inches or other unit of capacity, are to be 
 reduced, as of inches into gallons and the like. 
 
 I would have but one formula appli^ble to all bodies, and it will 
 of course be asked : why, for instance in the case of the cylinder, the 
 whole cone or pyramid, substitute the more complex for the simpler form 
 of computation. My reason for doing so has its untold importance to 
 thousands of the human race. Memory is not a gift to every one. I 
 have none of it myself or hardly any, and its absence only entails a 
 little reasoning as I am now to show. 
 
 I have seen students, only three months out of college doubtful as 
 to which of the ordinary formulae to apply, to the pyramid or cone, the 
 conoid, the spheroid. In one — the first — the volume is due to the base 
 and one third the height ; in the second, the base and one half the height ; 
 in the other, the base and two thirds the height. Any mistake is fatal 
 tothe result. - ^ • ?. - 
 
 But with the one and only one, the unique and universal formula 
 which I propose to substitute for every other, no error can obtain. Take 
 hold of the pyramid or cone : set down its upper or one end area or that 
 of its apex, equal nought (0) or zero, its other end area, whatever that 
 may be. Its middle area, you see at once is one quarter that of its base ; 
 for the middle or half way diameter is half that of the base, and the 
 areas of similar figures as the squares of their homologous or like di- 
 mensions. Now, ere you have put this down on paper; ere yoai h£^v§ 
 
.<--/:-;,: ;.- — 13— "•-: 
 
 had time to do sc, the reasoning process is going on within yonr mind 
 and in far less time than it takes me to relate it — that four times the 
 middle area plus the area of the base is equal to twice the base, and 
 that twice the base into one sixth the altitude is precisely the same thing 
 , as once the base, that is, the base into one third the altitude, and so come 
 you back to the old or ordinary rule, the simpler of the two in this c ise, 
 and without the necessity of having this formula stored in your mind as 
 a separate process. 
 
 And so with the cylinder where you see at once that the area of 
 each base and of the middle section being all equal quantities, the sum 
 of these bases and of four times the middle section is tlie same thing as 
 six times the base, and again that six times the base into one sixth the 
 altitude is the old rule of the base into the altitude, without the ne- 
 cessity of remembering it as a separate and additional formula. 
 
 But the great advantage of this one universal rule, its beauty so to 
 say is further evidenced and more strikingly in the computation of the 
 more difficult solids, that is of those which are more difficult under the 
 old or ordinary rules. ' 
 
 In the spnere, spheroid and conoids, the one area, that at the apex 
 or crown is always nought or nothing, as a plane there touches them 
 in one and only one point. The formula applied to the sphere and 
 spheroid therefore reduces to four times the middle area into one sixth 
 the altitude or diamet3r or axis perpendicular to the plane of section. 
 
 Now, let it be required to measure the liquid in a conoidal or 
 spheroidal vessel inclined to the horison or out of the vertical. This by 
 ordinary rules, becomes an operation of much time, trouble and anxiety, 
 as the size of the whole body or solid of which the portion or figure 
 under consideration forms a part, has to be made known, its factors en- 
 tering into the formula for the content required ; whereas by the pris- 
 moidal formula, no concern need be had as to the dimensions of the 
 entire body of which the figure submitted to computation is a segment 
 
 That the rule applies to all such cases, is and has been abundantly 
 proven by myself (see my treatise of 1866) as applied to any segment 
 of a sphere or spheroid, to any ungula of such solids contained between 
 
— 14— ... 
 
 • 
 
 planes passing in any direction through the centre, to any frnstiim of 
 these bodies, — lateral or central — contained between parallel planes 
 inclined in any way to the axes ; to any parabolic or hyperbolic conoid, 
 right or inchned, as well to any parallel frustum of either. > ■ 
 
 TIlis proof has been substantiated by MM. Steckel of the Dept. of 
 Dominion Public Works, Deville a member of this society, and the late 
 Eevd. M. Maiugui, professor of Mathematics at the I>aval University, as 
 well by the |ievd. M. Billion, of the Seminary of St. Sulpice — Montreal ; 
 by His Grace, bishop Langeviu of Rimouski, and by many other ma- 
 thematicians fully adequate to the task. 
 
 M. Mangui says (page IX of his pamphlet and as already quoted 
 
 from the french version) : " This formula "" tp- ^ is that 
 
 " which Mr. Baillairge is endeavouring to introduce ; it has the irr^ 
 *' Tnense advantage of replacing all other stereometrical formulae." 
 
 This is the only formula which ^vill allow of teaching stereometry 
 in all schools however elementary, and as has just been shown, the appli- 
 cation of it is the more simple, so to say, the more complex the body is, 
 since in the conoid and segment of spheroid, one of the factors at least is 
 zero, while two of them are zeros in the sphere and spheroid as in their 
 ungulae. - 
 
 Thus while the student at college or from a University after having 
 devoted much time to the acquisition of a hundred rules for the cubing 
 of as many solids, has hopelessly forgotten them in after life, the com- 
 paratively illiterate artisan, tradesman, merchant, &c. who has never fre- 
 quented ought but a village school, will, having but one rule where- 
 with to charge his memory, remember it all his life and be ever ready 
 to apply it ? : 
 
 In the case of spindles and the measurement of their middle frusta 
 '-the representatives of casks of all varieties and sizes, — the prismoidal 
 formula does not bring out the true content to within the tenth or 
 twentieth and up to the half or thereabout of one per cent ; notwith- 
 standing which, it is the only practical formula which can bring out 
 anything like a reliable result. The true formulae for casks never can, 
 nor will they ever be applied ; they are too lengLly, too abstruse, and the 
 
— 15— . 
 
 wine merchant will tell you that the nearest the guage rod can come to 
 within the truth, the guage rod founded on these formulae, is to within 
 from one to three and even four per cent. This stands to reason, as 
 when operating on the half cask — which is always done with all figures 
 ha""iiig symmetrical and equal halves— the half way dia» eter between 
 the head and bung, the very element by which the cask varies its capa- 
 city, enteri as a factor into the computation, while the guaging rod can 
 take no note of it. ' . , = 
 
 It remains but to say that in the case of hoofs and ungulae of cones 
 and cylinders, of conoids and of spheroids, when the bounding planes do 
 not pass through the centre, the prismoidal formula is still the best to 
 be employed in practice, and again brings out the volume to within one 
 half or so of one per cent. The true rules applicable to these ungulae can 
 never be remembered, nor are or will they ever be applied in practice. 
 Eather than that, the fudging or so called rule of thumb system, some 
 averaging of the dimensions is sure to be resorted to and a result arrived 
 at, where two or three to five per cent of error is considered near enough, 
 while the proposed application of the prismoidal formula would reduce 
 the error to almost nothing. ^ 
 
 Compound bodies mUst of course be treated separately or in parts. 
 Thus, a gun or mortar, as made up of a cylinder or the frustum of a 
 cone and the segment or half of a sphere or spheroid ; a moorish or tur- 
 kish dome, as the frustum of a spheroid surmounted by a hollow cone ; 
 a roofed tower, as a cone and cylinder, a cone and frustum of a cone or 
 two conic frusta as the case may be aud so of other compound forms. 
 
 Again when frusta between non parallel bases are to be treated, the 
 solid is to be divided by a plane parallel to one of its bases and passing 
 through the nearest edge or point of its opposite base, into a frustum 
 proper and an ungula, subject to the percentage of error already noticed 
 in the volume of the ungula ; while, by cubing the whole conoid or 
 segment of a spheroid of which the frustum forms a part, and then the 
 segment which is wanting to make up the whole, the true content can 
 be arrived it. 
 
 There are a class of solid forms where it would appear at first 
 sight that a departure from the prismoidal formula becomes necessary ; 
 
— 16 — 
 
 not so however as will presently be seen. I allude to the cubing of the 
 fragment of a shell for instance, or of the material forming the vaulting 
 of a dome as contained between its intrados and extrados. This is simply 
 arrived at, when the inner and outer faces are parallel or when the dome 
 or arch is of uniform thickness by applying the spherical, sphero- ^"»1 or 
 cylindrical surfaces of the opposite bases, and the equally curved surlace 
 o." the middle section ; while, when the faces are not parallel or the 
 thickness of varying dimensions, as well when the faces are everywhere 
 equidistant, the volume may be had by cubing the outer and inner com- 
 ponent pyramids and taking the difference between them. 
 
 And in the making out of such spherical areas as may enter as 
 factors into any computation, a most concise and easy rule will be found 
 at page 35 of my '* stereometricon " published in 1880 ; where any such 
 area can in a few minutes be made up by the mere multiplication and 
 addition of the elemental quantities given in the text, and any portion 
 of the earths surface thus arrived at when the radius of the osculatory 
 circle for the given latitude is known. 
 
 With irregular forms, the figure can be sliced up and treated by the 
 formula, and those forms when small and still more complex, such as 
 carving, statuary, bronzes ami the like, can be measured with minute 
 accuracy by the indirect process of the quantity of fluid of any kind dis- 
 placed, as of water when non absorbent or of sand or sawdust etc., when 
 the contrary. 
 
 Again may the specific gravities of bodies be applied, or their weights 
 to making out their volumes by simple rule of three, or the reverse 
 process of weighing them by ratio when their volumes are ascertained. 
 
 ; . :- Finally the quantities and respective weights of the separate subs- 
 tances which enter into amalgams or alloys are obtainable as taught by 
 a comparison of their weights in air and water, that is of the amalgam 
 itself and of its unalloyed constituents. 
 
 The whole field of solid mensuration is thus gone over in these few 
 pages, instead of the volume required to contain the many separate and 
 varied formulae which the old process of computation gives rise to and 
 renders indispensable. The whole I say is gone over in as many minutes 
 as the old process requires hours or even days. 
 

 
 »■:,...?■■ 
 
 HINTS TO gf-ometi:rs 
 
 FOB A 
 
 NEW EDITION OF EUCLID. 
 
 Read before the mathematical, physical and chemical section of the 
 Ro^al Society of Canada, May 22, 1883. 
 
 :'r-':^i:%f. 
 
 Euclid is, no doubt, an admirable treatise, a purely logical series of 
 propositions, a beautifully and wonderfully Mnked concatination of 
 theorems; but I fail to see how for 2000 years it can have been 
 written and rewritten without its striking one to what considerable ex- 
 tent its several propositions are reducible in number by making mere 
 axioms of some of them, corollaries of others. > aI- 
 
 1^ It is most singular how this ancient geometry holds its own 
 against the hand of time, when other sciences have been so to say, 
 ground down and reduced, generalized and simplified. - - 
 
 r^ Our veneration for the greek author must not degenerate into 
 ignorance and ungodliness ; life is too short and there are too many other 
 sciences to learn now a-days to devote a year or more to a study of tne 
 olden master. ^--^^^^^-^^^^^ . ^^^ .^ 
 
 The two hundred and odd propositions of the first six books of 
 Euclid as edited by Dr. John Play fair in 1856 may likely be reduced 
 to less than half the number, while sacrificing none of the conclusions 
 plj of which may be retained as corollaries, postulates, axioms. 
 
— 18 — ' 
 
 The fifth book, to begin with, may be altogether eliminated by a 
 different treatment of the subject which can hardly be considered as 
 stricktly geometrical, since by the substitution for instance of the term 
 ** quantity " for " magnitude, " general expressions may be arrived at 
 applicable both to geometry and arithmetic and signally simplifying and 
 facilitating the solution of a great variety of problems, as what is un- 
 derstood to be applicable to numbers is just as easily apprehended when 
 applied to the units which go to make up all geometrical magnitudes, 
 and which magnitudes can hardly be otherwise conceived than as made 
 up of such units, whether, linear, superficial, cubic or angular. 
 
 And if the book be not eliminated in toto, many of its separate 
 theorems may be so treated and their number thus reduced. 
 
 All axioms are not such self evident propositions as to require no 
 process of reasoning to render them acceptable as such. The mental 
 process may be of extremely short duration, but it exists, and it suffices 
 to extend the process within very narrow limits to embrace numerous 
 other propositions and reduce them also to axioms or to corollaries 
 thereof ; for if things which are doubles or halves of the same thing are 
 equal tc one another, it is not more difficult to conceive, that things 
 which are quadruples or quarters of the same thing are also equal to 
 one another, and thence arrive at the more general axiom that quantities 
 which q,Te equal multiples or submultiples of the same quantity are 
 equal to each other. 
 
 ' , Now, ratios between geometrical magnitudes or quantities of all 
 kinds can not be otherwise conceived than as numerical, for if their ra- 
 tios be expressed in lines or otherwise, these last present themselves to 
 the mind as made up of equal units and again convey the idea of 
 number. ^ '- ^,:- - ; ,■ .- ^-r. , v. ::^;-; v, ,.'.—■": -ly'v''--' 
 
 Equal ratios are therefore to all intents and purposes equal num- 
 bers, and what is true of the one, must be so of the other ; hence I fail 
 to see the necessity, as a separate proposition of such theorems as the 
 11th of the 5th book of Euclid, that : 
 
 " Katios which are equal to the same ratio are equal to one an- 
 other, " since as just stated this may be made a mere corollary to that 
 
— 19 — 
 
 axiom which declares that things which are equal to the same thing are 
 equal to one another. . 
 
 In the same way may propositions, 1, 7, 9, 1 5 and F. and in fact 
 most of the others of this book be considered self evident, or rendered so 
 by a much easier process of reasoning and demonstration, by consider- 
 ing all the magnitudes in their simple and uncombined state A, B, C, D, as 
 in their compound state of m A or m times A, n li or n times B, as sim- 
 ple quantities made up of so may units of geometrical magnitude, or in 
 other words as numerical quantities or numbers expressive of the con- 
 tents thereof. V ■ 
 
 Eeturning now to a consideration of the first book of Euclid ; why, 
 may I ask, were not propositions 2 and 3 made mere postulates. Again, 
 should not proposition 22 of this book be made the very first of the 
 series, and proposition 1 a mere corollary to it. To be sure, Euclid re- 
 serves it till after the 20th where he shows that any two sides of a 
 triangle must be together greater than the third side and this renders its 
 position more strictly logical, if proposition 20 be essential ; but 20 can 
 not be so considered when the very definition of a straight line, as the 
 shortest distance between two points makes it evident that the passing 
 over any two sides of a triangle to arrive at the third is a longer mode 
 of transit than going straight to the point aimed at. . ^. . , 
 
 Theorems 13, 14, 15, 20, 27, 28 of this book relating to perpendi- 
 culars and parallels may be easily deduced as corollaries from the defi- 
 nitions, '■''' ■" -r"^-. .:;.-: •'-'?; ■-■>^<,' 
 
 Proposition 30 that straight lines w^hich are parallel to the same 
 straight line are parallel to one another need be nothing more than au 
 axiom or a corollary thereto, for this very parallelism is defined to con- 
 sist of equality of distance throughout, between the lines so styled, and 
 as equals added to or taken from equals, the sums or remainders are 
 equal, the lines being equidistant must be parallel. 
 
 Proposition 34 may be deduced from 33, and so of 36 from 35, for 
 Euclid who in his 4th and 8th of this book, applies the one triangle or 
 figure to the other might equally as w^ell have done so in the case of 36 
 
— 20 — 
 
 and thus reduced the case of parallelograms on equal bases, to the pre- 
 ceding case of parallelograms on the same base. v , 
 
 The two next propositions 37 and 38 have really, no solid founda- 
 tion for being treated separately ; for, not only can the one be merged into 
 the other as of 36 and 35, and for a like reason, but both of them should 
 be made mere corollaries to 35, dependant on the axiom or self evident pro- 
 lX)sition that what is true of the wholes is equally true of the halves, 
 every triangle being the ex.>ct half in shape and dimensions of its corres- 
 ponding parallelogram, or, conversely, every parallelogram ihe double or 
 duplicate both in size and shape of its coiTesponding or component 
 
 triangle, which is amply set forth in Euclid's S-iih of this book. 
 
 • 
 
 Of the 2nd book of Euclid most of the propositions, as of those of 
 the 5th book are susceptible of numeric or algebraic demonstration and 
 may be thus greatly simplified and rendered more easy of apprehension. 
 
 Proposition 5 of this book is a fruitful one in the tiolution of many 
 problems, as where the area and periphery of a figure are given to find 
 the sides ; but, to this effect it must be shewn, which is not done, that 
 what is termed ihe line between the points of section, is in other words 
 half the difference between the lines, and by thus connecting the opera- 
 tion with the rule for finding any two quantities of which the sum and 
 difference are given, the proposition becomes suggestive which it is not 
 in its present form. 
 
 Of the 3rd book it may be said, with Clairaut : it must be because 
 Euclid had to deal with a set of obstinate sophists who were bent on 
 refusing assent to the most self-evident propositions, that He found it 
 necessary to prove as he has done of proposion 2 of this book, that the 
 chord of a circle lies wholly within the circle, as if the very definition of 
 such a line were not sufficient to locate it. : -* 
 
 Neither can it be argued that there was any necessity for theorems 
 5 and 6 of this book which are self-evident propositions. With regard to 
 23, 24, 26, 27, 28 and 29 they may be all reduced to one general propo- 
 sition with the others brought in a*^ corollaries. 
 
 There is in reality no essential difference between problems 1 and 
 
-21- 
 
 25, as the process for finding the centre from which an arc is described, 
 applies equally whatever the extent of the arc be, and even up to when 
 it becomes an entire circumference. 
 
 ; A different and more easy solution of 33 reduces its several oases 
 to one and so of 35 and 36 where by similar triangles or in other ways 
 may the several caseri be reduced to one for each of the two propositions. 
 
 What necessity there is or vras of Playfair's, additional proposition A 
 after those of Euclid, I fail to see, as from the very definition of a circle 
 and of its diameter passing through the centre, the proposition is self 
 evident ? Playfair's prop. B of this book is nothing but a repetion of 
 Euclid's 5th of book 4, for what else are the angular points of any 
 triangle, but the same as any other three points not in one and the same 
 straight line, and of his propositions C and D the same remark may be 
 made as of No. 22 and others, that they are all deducible directly as 
 corollaries from one general proposition. 
 
 Of book 4, in prop. I, the restriction that the given straight line 
 to be placed or inscribed in the circle must not be greater than the 
 diameter of the circle can hardly be warranted, as it is evident the 
 diameter or double radius, is from the nature of the cir<ile itself the 
 greatest line capable of being drawn in it. 
 
 Propositions 6 to 9 of this book may be easily reduced to less than 
 the dimensions of any one of the four. Props. 1 1 to 14 may also be 
 made one case of, with short corollaries for the others, or the four of them 
 corollaries to prop. 10 as the circle being thus divided into ten, the joining 
 of the extremities of every second radius will afford the inscribed poly- 
 gon, and the tangent at the ends of these radii, the circumscribed figure, 
 while, from the nature of the regular polygon it will also be evident 
 that any two perpendiculars to the middle of the sides, or any two 
 bisecting lines of the angles, will where they meet give the centre of the 
 inscrih 'd and circumscribed circles. 
 
 Ha\ Lig ?,1 ready remarked on book 5, it will suffice to say in relation 
 to book 6, that with regard to theorems 14 & 15 as was done with pro- 
 positions 35 to 38 of the 3rd book, they may both be made simple co- 
 rollaries founded again on the axiom that what is true of the wholes is 
 
— 22— . ■ 
 
 true of the halves ; and as to prop. 21, it is analogous to the 30th of the 
 3rd book, for as the similarity of the triangles is due to the equ .lily of 
 their corresponding angles, it is therefore plain that this proposition is 
 true as a mere axiom or corollary to the axiom that things equal to the 
 same thing are equal to each other Nos. 16 and 17 are deducible 
 directly from the 35th of the 3rd, and 31 should evidently be a corol- 
 lary to the 47th of the 1st, in corroboration again of the axiom that what 
 is true of the whole, must be true of like parts thereof, and the same 
 would also be true of circles or semi-circles described on the sides and 
 hypothenuse of any right angled triangle, as also of the similar segments 
 or other figures so described. 
 
 The investigation need be carried no further, enough having been 
 said as suggestive of the process of retrenchment ; while retaining all the 
 conclusions of the greek geometer, and the same pruning or paring may 
 be applied to the books of planes and solids and a new treatise written 
 as suggested in the heading to this paper. 
 
* 
 
 SIMPLIFIED SOLUTIONS 
 
 Of two of the more difficult cases in parting off or dividing-up 
 land, and of a case in Hydrographic Surveying. 
 
 Read before the mathematical section o' the Royal ^Society of Canada, 
 
 May 22nd, 1882. 
 
 The ordinary geometrical, as well as the algebraic solutions for 
 dividing a figure or parting off from it a portion of given area by a 
 straight line running through a given point within the figure, are both 
 of them lengthy and laborious operations ; the algebraic formula in Gil- 
 lespie's, land surveying, covering nearly three successive hnes of type. 
 
 The consequence is that when this process has to be gone through, 
 the operation is merely fudged out and the result only approximately 
 arrived at. 
 
 In the course of my simplification and reduction of the numerous 
 propositions in EucHd to a less number of separate theorems and pro- 
 blems than contained in the work of the great geometer, making many 
 of them mere corollaries, postulates and axioms, (see my treatise of 1866 
 written in the trench language) the substitution of one for and embracincr 
 Euclid's, three cases of that proposition of the 3rd book which teaches 
 that if two chords cut one another in a circle, the parts of the one are 
 the extremes of a proportion of which the parts of the other are the 
 means, led M. Bene Steckel (a former pupil of mine, now in the employ 
 of the Dominion Board of Public works, and whom 1 hope soon to see in 
 the milks of this section of the Koyal Society, he being one of the cleverest 
 
24 — 
 
 D 
 
 mathematicians in Canada), to demonstrate the lemma or preparatory 
 proposition that the complements of parallelograms about the diameter 
 of a parallelogram are mean proportionals between these parallelograms. 
 
 Such being the case, T conceived the idea of applying this auxiliary 
 
 theorem to the solution of the above mec^ioned case. 
 
 Let F the point, AFC the line 
 of partition and ABC the area to 
 be cut off. Complete the paral- 
 lelograms shown in the figure, 
 then is complement B F '. mean 
 proportional between E G and H K 
 or, which is the same thing, the 
 triangle E F H is a mean between 
 triangles A E F, F H C ; wherefore 
 the rectangle of the unknown 
 parts AEF, F H C is known, being 
 equal to the square of E F H, that 
 
 is, to the square of the number of units of area in E F H. Also the sum 
 
 of the same unknown parts is known as equal to the given area, less that 
 
 of the parallelogram B F. 
 
 Therefore can the problem be solved by the 8th of the second book 
 of Euclid, as the half difference between AEF and F H C is obtained by 
 squaring the half sum of the unknown parts and from this deducting the 
 rectargle of those parts, which leaves the square of half the difference. 
 The half sum and half difference added give area AEF which divided 
 by half the perpendicular F M gives the base A E , whence A B becomes 
 known. 
 
 Or area F H C becomes known as equal to the half sum of the 
 unknown parts less the half difference, and area F H C divided by half the 
 perpendicular let fall from F on the side B C, gives the base H C, whence 
 B C is known. 
 
 It need hardly be remarked that if the figure or area to be cut off 
 is not a triangle, the solution can always be reduced to the case of a 
 triangle by prolonging the sides between which the line is to run, 
 thereby reducing the solution to that herein above given. 
 
25 — 
 
 Another, a more usual and more important case in its commercial 
 features, is when as occurs more or less in all cities, a tract of land, a 
 block between non paralled streets, and especially where these are com- 
 mercial thoroughfares^ is to be divided into lots of equal, proportional or 
 given area by lines drawn in such a way as to cut the non parallel sides 
 proportionally to the entire length thereof.. 
 
 The case evidently reduces, for each pircel to be portioned of! to 
 that of finding the sides AD, B C in the quadrilateral A C, wherein are 
 given side A B, the angles at A and B and the ratio of A D to B C. 
 
 if 
 
 In hydrographic problems the given angles are brought into con- 
 tact with the given sides by taking advantage of that peculiarity of the 
 circle of containing equal angles in equal or the same segment. 
 
 In the quadrilateral A C the sum of the unknown angles at D and C 
 is known equal to 360° — A-fB. In trying to find a new and simplified 
 solution for this pertinent case of apportionment of usually valuable 
 land or likely to become so, I appealed to M. Steckel for an idea, when 
 he immediately conceived that of bringing the unknown angles I) and 
 C together by bisecting the figure iu E and iT and causing one of itj* 
 
— 26 — 
 
 halves to turn upon the point F through an angle of 180", thus bringing 
 angle D around to join C. The solution then stood out prominently 
 before me as will now be undestood by completing the figure. 
 
 It is plain that A' F (that is A E) having moved through 180° is 
 parallel to to B E and equal to it, wherefore B A' is parallel to G H or 
 to E F E' produced. In the polygon E B L A' E' we have the angles 
 at B and A' to find those of the two auxiliary triangles G B E, H A' E* 
 at the same points. We have the angle at L =* C -|- D and the ratio of 
 the sides B L, A* L or B C, A D in triangle B L A' to find the angles 
 A* and B equal respectively to G and H. In the triangles G B E', H A' E* 
 we now have the angles and side B E = A' E' == J A B to find areas 
 G B E, H A' E' which added to the known area of the polygon E A' 
 or quadrilateral A C gives the area of the triangle G L H. Finally in 
 G L H, we have angles and area to find G L and H L, from which 
 deducting B G and A' H we get the required sides B L = B C and 
 A' L = A D. cj. E. D. 
 
 The case in hydrogra- 
 phy I am now about to 
 present, will be found a 
 very elegant solution due 
 entirely to M. Steckel, of 
 the interpolation of a base 
 h c and the fixing of the 
 point whence the three 
 angles are taken to the 
 four points a, 6, c and d. 
 
 On ah and c c? having described circles respectively capable of the 
 angles a oh, doc, produce ao, do to h and g, join ag, d h, complete the 
 figure. 
 
 Angle aogi^ supplementary to a o c^ as is \o doJi, wherefore the 
 angle at o of the 5 sided agure adhog ia known ; angles gah^hdc are 
 
, , —27— :;-: 
 
 known as supplementary to 6 o ^ and coh, and the sum of angles^ 
 and h is therefore known in the 5 sided figure as three times 180° less 
 the sum of the other angles. 
 
 Now, angles at g and h, their sum is the half sum of those at the 
 centre afo, deo on the same base ao, do, wherefore, the triangles afo, 
 deo being isosceles, the sum 2 m plus 2 ?i of the angles at their respect- 
 ive bases becomes known and hence their half sum m plus n which 
 together with the sum aod of the three observed angles makes up the 
 angle /oe, whereby in the triangle /oe where the radii fo and eo are 
 known, the distance /e can be found and the remainder of the solution is 
 evident , 
 
 '!»■, !: ' 
 
 .!**&■ 
 
 '''■f'-A._: 
 
 '■-■■■•'.' "*- -'■ ■''*.'' -- ' 
 
 ■ >-■-;. 
 
-•f. 
 
 M:' :fJ 
 
 --^ 
 
 : » i 
 
 THE ^REJ^S OE 
 
 SPHERICAL TRIANGLES & POLYGONS 
 
 TO ANY KADIUS OR DIAMETER. 
 
 Read before the mathematical, physical and chemical section of the 
 Royal yociety of Canada, May 22nd 1883. 
 
 Last year I laid before this section of the Royal Society my pro- 
 posal to substitute in schools the prismoidal formula for all other known 
 formulae pertaining to the cubing of solid forms. 
 
 I then showed that on this sole condition, the computation of soli- 
 dities, even the most difficult by ordinary rules, as of the segments, 
 frusta and ungulae of Conoids and Spheroids was susceptible of gene- 
 ralisation and of being taught in the most elementary institutions. 
 
 I then submitted that the advantage of the proposed system con- 
 sisted in this, that >^hile he who had gone through a course of mathe- 
 matics would, in three n^onths thereafter or out of college, have complete- 
 ly forgotten or have inextricably mixed up in his mind the numerous 
 and ever varying formulae for arriving at the contents of solids ; the 
 simple artisan, on the contrary, who at an elementary school would have 
 been taught the universal formula, and who from the fact of having to 
 learn but one, could not forget it nor mix it up in his mind with any 
 others, could apply it always and everywhere during a life time without 
 the aid even of any book excepting may be, to save time, a table of the 
 areas of circles or of other figures lengthy of computation. 
 
*•':■■ — 29 — 
 
 ' What I then did for the measuremerit of solid forms I now propose 
 to do for the mensuration of areas of spherical triangles and polygons 
 on a sphere of any radius ; I mean a simple and expeditious mode of 
 getting at the doubly curved area of any portion of the terrestrial 
 spheroid as of every sphere great or small : interior or exterior surface of 
 a dome for example or of one of its component parts, as well of the bot- 
 ton or roof of a gasometer, boiler or of one of the constituents sections 
 thereof, descending even to the surface of the ball of a spire, a shell, a 
 cannon or a billard ball. -ff 
 
 , . . TO THIS END : ^ - v^ 
 
 . The area of a sphere to diameter I. being . =:?.l41,592,6o3,n81),793-l- •' ' sM 
 
 Dividing by •>, we get that of the hemisphere =l,570,7')<i,:J:i«),7y4,^!i)(),5 ,^.' ,, - . » 
 
 This divided by 4=area of tri-rectgl'r sph. triangle =(),:i'J-i,(iinM>^l,«i>8,7-'4,l ' 
 
 -H90=area of I" or of bi-rect. sph. tri. with sp. ex=lo =0,004vit>:),:?-i:?,l'29,985,S- ^ 
 
 -HbO= " ofl' or of " " <• « 1' =0,000,07 •^>,7-,%0;VJ,lH(5,4:5 • ' 
 
 ^60= " of Tor of " " « « 1" =0,000,00 1, -i I •->,O34,-i0-J,77 ; . •• 
 
 -hlO= " of 0.1" or of " « " " 0.1" =(l,O!l0,O0O,l21,-.*0:?,4-iO,'i77 
 
 -5-10= « of 0.0 1" or of" « « « 0.01" =0,000,000,0 i -J, l-^0,:54-.>,0->:,7 
 
 -i-10= « of 0.00 1" or of « '' « « 0.001" =0,000,000,00 l,-il-.>,0:J4,-iOJ,77 
 
 Find the spherical excess, that is, the excess of the sum of the 
 three spherical angles over two right angles, or from the sum of the three 
 spherical angles deduct 180°. Multiply the remainder, that is, the 
 spherical excess, by the tabular number herein above given : the degrees 
 by the number set opposite to 1°, the minutes by that corresponding to 
 1' and so on of the seconds and fractions of a second ; add these areas 
 and multiply their sum by the square of the diameter of the sphere of 
 the surface of which the given triangle forms part ; the result is the area 
 required. 
 
 EXAMPLE. " ' " 
 
 Let the spherical excess of a triangle described on the surface of a 
 sphere of which the diameter is an inch, a foot, or a mile, etc., be 3° — 
 4' — 2.235". What is the area ? 
 
 Area of 1° = 0.004,363,:V2:},l-i9,985,8 X 3 = 0.013,089,969,389,955 
 
 « r =0.000,07-2,72i,05-.>, 160,43 X 4 = 0.000,-J90,888,->08,t;64 
 
 « l" = 0.000,O01,-Jl-i,O34,-20> X 2 = 0.000,00-J,4-,>4,( H)8, 104 
 
 « 0.1" = 0.000,000,l-Jl,->03,4JO X 2 = 0.000,000,->4-.',406,8J0 
 
 ^ COL" = 0.000,000,01->,120,342 X 3 = 0.000,000,036,3ol,0i6 
 
 « 0.001" = 0.000,000,001,-i 1-^,031 X 5 =0.000,000,006,060,170 
 
 Area required 0.013,383,.'i6(i,495,059 
 
— 30 — 
 
 The answer is of course in square units or fractions of a square unit 
 of the same name with the diameter. That is, if the diameter is an inch, 
 the area is the fraction of a square inch; if a mile, the franction of a 
 square mile, and so on. , ., , ,.: -...Mt 
 
 Eemark. — If the decimals of seconds are neglected, then of course 
 the operation is simplified by the omission of the three last lines for 
 tenths, hundredths and thousandths of a second or of so many of them 
 as may be omitted. 
 
 If the seconds are omitted, as would be the case in dealing with 
 any other triangle but one on the earth's surface, on account of its size ; 
 there will in such case remain only the two upper lines for degrees and 
 minutes, which will prove of ample accuracy when dealing with any 
 triangular space, compartment, or component section of a sphere of 
 the size of a dome, vaulted ceiling, gasometer, or large copper or boiler, 
 etc ; and in dealing with such spheres as a billiard or other playing 
 ball, a cannon ball or shell, the ball of a vane or steeple, or any boiler, 
 copper, etc., of ordinary size, it will generally suf&ce to compute for 
 degrees only. Whence the following 
 
 RULE TO DEGREES ONLY. '■ 1 
 
 Multiply the spherical excess in degrees by 0.004,363 and the 
 result by the square of the diameter for the required area. For greater 
 accuracy use— 0.004,363,323. , .. .. 
 
 RULE TO DEGREES AND MINUTES. ' ' 
 
 ■?■' - 
 
 Proceed as by last rule for degrees. Multiply the spherical excess 
 in minutes by 0.000,073, or for greater accuracy by 0.000,072,722. Add 
 the results, and multiply their sum by the square of the diameter for the 
 required area. 
 
 .;; ^'^:^^^ ■"-■^-- EXAMPLE I. -■■'" -^'■"- ^■-";-^'^ ■ ■:;;■--■ 
 
 Sum of angles 140" + 92° +68° = 300 ; 300 — 180 = 120° sphe- 
 rical excess. Diameter =30. Answer area of 1° 0.004,363 
 Multiply by spherical excess 120* 
 
 We get 0.523,560 
 
 This multiplied by square of diameter 30= 900 
 
 Required area =^ 471,194,000 
 
A result correct to units. If nov greater accuracy be required, it is be 
 obtained by taking in more decimals ; thus,say area 1°= 0.004 363,323 
 
 0.523,598,760 
 900 
 
 .471.238,884,000 
 EXAMPLE II. 
 
 The three angles each 120°, their sum 360°, from which deducting 
 180° we get spherical excess = 180°. Diameter 20, of which the square 
 = 400. 
 
 Answer Area to 1°= 0.004,363.323 
 
 . 180 
 
 0.785,398,140 
 400 
 
 EXAMPLE IIL 
 
 314.159,256,000 
 
 The sum of the three angles of a triangle traced on the surface of 
 the Terrestrial sphere exceeds by (1") one second, 180°; what is the area 
 of the triangle, supposing the earth to be a perfect sphere with a diame- 
 ter = 7,912 English miles, or, which is the same thing, that the diame- 
 ter of the Terrestrial spheroid or of its osculatory circle at the aiven 
 point on its surface be 7,912 miles. .^ 
 
 . Answer. Area of 1" to diameter 1. = 0.000,001,212,034,202 
 
 Square of diameter 62,598,744 
 /_ _. ■ ■ _ 
 
 (75.871,818,730,242,288 
 Eemark.— This unit 75.87 etc., as applied to the Terrestrial sphere, 
 becomes a tabular number, which may be used for computing the area 
 of any triangle on the earth's surface, as it evidently suffices to multiply 
 the area 75.87 etc., corresponding to one second (1") by the number of 
 seconds in the spherical excess, to arrive at the result ; and the result 
 may be had true to the tenth, thousandth, or millionth of a second, or of 
 any other fraction thereof by successively adding the same figures 
 
— 32 — 
 
 75.87 etc., with the decimal point shifted to the left, one place for every 
 place of decimals in the given faction of such second : the tenth of a 
 second giving 7.587 etc., square miles, the 0.01"=: .7587 of a square 
 mile, the 0.001"= .07587 etc., of a square mile, and so on ; while, by 
 shifting the decimal point to the right, we get successively 10" = 758.7 
 square miles, 100" = 7587. etc., square miles, or 1' = 75.87 X 60 (num- 
 ber of seconds in a minute), 1°= 75.87 X 60 X 60 (number of seconds 
 in a degree). 
 
 RULE. 
 
 ■ s' 
 
 To compute the area of any spherical polygon. 
 Divide the polygon into triangles, compute each triangle separately 
 by the foregoing rules for triangles and add the results. 
 
 I OR, 
 
 From the sum of all the interior angles of the polygon subtract as 
 many times two right angles as there are sides less two. This will give 
 the spherical excess. This into the tabular area for degrees, minutes, 
 seconds and fractions of a second, as the case may be, and the sum of 
 such areas into the square of the diameter of the sphere on which the 
 polygon is traced, will give the correct area of the proposed figure. 
 
 It may be remarked here that the area of a spherical lune or the 
 convex surface of a spherical ungula is equal to the tabular number into 
 twice the spherical excess, since it is evident that every such lune is 
 equivalent to two bi-rectangular spherical triangles of which the angle 
 at the apex, that is the inclination of the planes forming the ungula, is 
 the spherical excess. 
 
 Kemark. — The area found for any given spherical excess, on a 
 sphere of given diameter, may be reduced to that, for the same spheri- 
 cal excess, on a sphere of any other diameter ; these areas being as the 
 squares of the respective diameters. 
 
 The area found for any given spherical excess on the earth's sur- 
 face, where the diameter of the osculatory circle is supposed to be 7912 
 miles, may be reduced to that for the same spherical excess where tho 
 osculatory circle is of different radius ; these areas being as the squares 
 of the respective radii or diameters. 
 
■'V* •-•» 
 
 
 
 
 BIOGRAPHICAL SKETCH 
 
 OF THE AUTHOR* . •. 
 
 Chevalier Chas. P. F. Baillairge, M.S., 
 
 ', QUEBEC. T'. '/:'■.•;■' ^".' 
 
 
 The subject of this sketch, who is a Chevalier of the Order of St. 
 Sauveur de Monte Eeale, Italy, was born in September, 1827, and for 
 the past thirty-three years has been practising his profession as an en- 
 gineer, architect and surveyor, in the city of Quebec. Since 1856 he has 
 been a member of the Board of Examiners of Land Surveyors for the 
 province, and since 1875 its chairman ; he is an honorary member of the 
 Society for the Generalization of Education in France ; and has been the 
 recipient of thirteen medals of honor and of seventeen diplomas, &c. 
 from learned societies and public bodies in France, Belgium, Italy, 
 Russia, Japan, &c. Mr. Baillairge's father who died in 1865, at the acre 
 of 68, was born in Quebec, and for over thirty years was road surveyor 
 of that city. His mother, Charlotte Janverin Horsley, who is still livincr 
 Was born in the Isle of Wight, England, and was a daughter of Lieute- 
 nant Horsley, R.N. His grandfather on the paternal side, P. Florent 
 
 •ton— HMHMIB 
 
 * From the Canadian Biographical Dictionary and Portrait Gallery of Eminent and Self Made- 
 men. American Biographical Publishing Company, Chicago, New York and Toronto, 1881. 
 
; .-34- : 
 
 Baillairg^, is of French descent, and was connected, now nearly a cen- 
 tury ago, with the restoration of the Basilica, Quebec. The wife of tho 
 latter was Mile Cureux de St. Germain, also of French descent. 
 
 Our subject married, in 184.5, Euphdmie, daughter of Mr. Jean 
 Duval, and step-daughter of the Hon. John Duval, for many years 
 Chief Justice of Lower Canada, by whom he had eleven children, four 
 of whom only survive. His wife dying in February, 1878, he, in April 
 of the following year, married Anne, eldest daughter of Captain Benja- 
 min Wilson, of the English navy, by whom he has two children. 
 
 Mr. Baillairgd was educated at the Seminary of Quebjc, but, finding 
 the curriculum of studies too lengthy, he left that institution some time 
 before the termination of the full course of ten years, and entered into a 
 joint apprenticeship as architect, engineer and surveyor. During this 
 apprenticeship he devoted himself to mathematical and natural science 
 studies, and received diplomas for his proficiency in 1848, at the age of 
 21. At that period he entered upon his profession, and for the last seven- 
 teen years has filled the post of city engineer of Quebec, is manager of 
 its water works, and since 1875 has been engineer, on the part of the 
 city, in and over the North Shore, Piles and Lake St. John Eailways. 
 
 Mr. Baillairge has held successive commissions in the militia, as 
 ensign, lieutenant, and captain ; and in 1860, and for several years there- 
 after, was hydrographic surveyor to the Quebec Board of Harbour Com- 
 missioners. In 1861, he was elected vice-president of the Association of 
 Architects and Civil Engineers of Canada. In 1858, he was elected, and 
 again in 1861 unanimously re-elected, to represent the St. Louis ward 
 in the City Council, Quebec. In 1863, he was called for two years to 
 Ottawa, to act as joint architect of the Parliament and Departmental 
 buildings, then in course of erection. Interests of considerable magnitude 
 were then at stake between the Government and the contractors, claims 
 amounting to nearly half a million of money having to be adjusted. In 
 connection with his employment by the Government, Mr. Baillairg^ 
 found, that to continue his services he must be a party to some sacrifice 
 of principle, which, rather than consent to, he was indiscreet enough to 
 tell the authorities of the time. This excess of virtue was too moral for 
 the appointing power and more than it was disposed to brook in an em- 
 
35 
 
 ployee of the Government. The difficulty wns, therefore, got over by 
 giving Mr. Baillairg^ his feuille de route^ a com}»litn(3nt to his integrity 
 of which he has ever since been justly proud. He shortly afcerwurds 
 returned to Quebec. • - ' 
 
 During his professional career, Mr. BailLiirge designed and erected 
 numerous private residences in and around Quebec, as well as many 
 public buildings, including the. Asylum and the Church of the Sisters 
 of Charity, the Laval University building, the new gaol, the music hall 
 several churches, both in the city and in the adj(jining parishes, that of 
 Ste Marie, Beauce being much admired on account of the beauty and 
 regularity of its interior. The " Monument des Braves de 17G0'*wa3 
 erected in 1860, on the Ste Foy road, after a design by him and under 
 his superintendence. The Government, the clergy and others have often 
 availed themselves of his services in arbitrations on knotty questions of 
 technology, disputed boundaries, builders claims, surveys and reports on 
 various subjects. 
 
 In 1872, Mr. Baillarge suggested, and in 1878 designed and carried 
 out what is now known as the Dufferin Terrace, Quebec, a structure 
 some 1,500 feet in length, overlooking the St. Lawrence from a height 
 of 182 feet, and built along the face of the cliff under the Citadel. This 
 terrace was inaugurated in 1878 by their Excellencies the Marquis of 
 Lome and H.K.H. the Princess Louise, who pronounced it a splendid 
 achievement. 
 
 5 In 1873, Mr. Baillairg(^ designed and built the aqueduct bridge 
 over the St. Charles, the peculiarity about which is that the structure 
 forms an arch as does the aqueduct pipe it encloses, whereby, in case of 
 the destruction of the surrounding wood- work by fire, the pipe being 
 self-supporting, the city may not be deprived of water while re-con- 
 structing the frost-protecting tunnel enclosure. 
 
 At the age of seventeen, the subject of our sketch built a double 
 cylindered steam carriage for traffic on ordinary roads. 
 
 From 1848 to 1865 he delivered a series of lectures, in the old 
 Parliament buildings and elsewhere, on astronomy, light, steam and the 
 steam engine, pneumaticS; acoustics, geometry, the a1;mosphere, and other 
 kindred subjects, under the patronage of the Canadian and other 
 
Institutes ; audi in 1872, in th^ rooms of the Literary and Histor.!cal 
 Society, Quebec, under the auspices of that Institution, he delivered ar* 
 exhaustive lecture on geometry, mensuration, and the stereornetricon 
 (a mode of cubing all solids by one and the same rule, thus reducing 
 the study and labour of a year to that of a day or an hour), which he 
 , had then but recently invented, and for which he wa» made honorary 
 member of several learrsd societies, and received the numerous medals 
 and diplomas already alluded to. 
 
 The following letter from the Ministry of Public Instruction, Russia, 
 is worthy of insertion as explanatory of the advantages of the 
 stereornetricon : 
 
 MiNISTRE DE lInSTHUCTION PuBLIQUE, 
 
 Saint- Petersbourg, le it fevrier 1877* 
 
 Ko. 1823, 
 
 A. M. BaillairgI^, 
 
 Architede d QuSec, 
 
 MoNSiKUR.— 'Le comit^ scientifique du ministete de rinstruction 
 Publique, (de Russie), reconnaissant Tincontestable utility de votre 
 " Tableau St^reom^trique " pour I'enseignement de la geometric en 
 g(^neral, de meme que pour son application pratique k d'autres sciences, 
 ^prouve un plaisir tout particulier k joindre aux suffrages des savants de 
 I'Europe et de I'Amerique sa complete approbation, en vous informant 
 ;, que le snsdit tableau, avec toutes ses applications, sera recommand^ aux 
 ^coles primaires et moyennes, pour en completer les cabinets et les 
 collections mathematiques, et inscrit dans les catalogues des ouvrages 
 approuves par le ministere de Tlnstruction Publique. 
 
 On fera, en outre, des dispositions pour faire venir de TAmt^rique k 
 Saint-Peterbbourg quelques exemplaires de vos editions, et vous ete& 
 
— 37— • 
 
 prie instamment, monsieur, d'avoir la boute d'infornier le eomit^ s*il 
 ii'exib:3 pas quelque part en Europe, un d^pot de vos ouvrages mathd- 
 inatiques. = x -. ; r i* 
 
 Agr^ez, monsieur, I'assurance de ma haute consideration. 
 
 Le chef du departe^nent au minist^re de I'lnstruction Publique. 
 
 .( E. DE Bkadkek. 
 
 And che Quebec Mercury^ of the 10th July, 1878, has the following 
 in relation to a second letter from the same source : 
 
 It will be remembered that in February, 1877, Mr. Baillairg^ 
 received an official letter from the Minister of Public Instruction, of St. 
 Petersburg, Rrssia, informing him that his new system of mensuration 
 had been adopted in all the primary and medium schools of th&t vast 
 empire. After a lapse of eighteen months, the system having been 
 found to work well, Mr. Baillairg^ has received an additionnal 
 testimonial from the same source, informing him that the system is to 
 be applied in all the polytechnic schools of the Kussian empire." ^; 
 
 Mr. Baillairge has since that time, given occasional lectures in both 
 languages on industrial art and design, and on other interesting and 
 instructive topics and is now engaged on a dictionary or dictionaries of 
 the consonances of both the French and English languages. 
 
 In 1866, he wrote his treatise on geometry and trigonometry, plane 
 and pherical, with mathematical tables — a volume of some 900 pages 
 octavo, and has since edited several works and pamphlets on like subjects. 
 
 In his work on geometry, which, by the way, is written in tlie 
 French language, Mr. Baillairg^ has, by a process explained in the 
 preface, reduced to fully half their number the two hundred and odd 
 propositions of the first six books of Euclid, while deducing and 
 retaining all the results arrived at by the great geometer. 
 
 Mr. Baillairge, moreover, shows the practical use and adaptation of 
 problems and theorems, which might otherwise appear to be of doubtful 
 utility, as of the ratio between the tangent, whole secant and part of the 
 secant without the circle, in the laying out of railroad and other curves 
 
'^----: -:-:'"--- ^ ' - ^-- - - - -" . — 38 — : ^ _ ■^ iJ 
 
 running through given points, and numerous other examples. His 
 treatment of spherics and of the affections of the sides and angles is, in 
 many respects, novel, and more easy of apprehension by the general 
 student. , . ,- ^ - i 
 
 In a note at foot of page 330, Mr. Baillairge shows the fallacy of 
 Thorpe's pretended solution of tlie trisection ~»f an angle, at which the 
 poor man had laboured for thirty-four years, and takes the then 
 Government to task for granting Mr. Thorpe a patent for the discovery. 
 
 In February 1874, he visited Europe, and it was on the loth of 
 March of that year that he received his first laurels at the " Grand 
 Conservatoire National des Arts et Metiers," Paris. 
 
 Mr. Baillairgd lately issued a report on the defects in the mode of 
 building in this province, and recommended the establishment of a Poly- 
 technic JSchool for the Province of Quebec, which is now shortly about 
 to be opened in the provincial capital under Government patronage. It 
 is due to the praiseworthy efforts of the Kev. Brother Aphraates, sup- 
 erior of the order of Christian Brothers, and of which institution Mr. 
 Baillairge, it is understood, is to be professor of technology and en- 
 gineering. 
 
 Some of Mr. Baillairg^'s annual reports on civic affairs are very 
 interesting and instructive; that of 1878, on "the municipal situation," 
 is particularly worthy of perusal. His report of 1872 was more espe- 
 cially sought after by almost every city engineer in the Canadas and 
 United States, on account of the varied information it conveyed. It may 
 also be remembered, as illustrative of the versatility of his talent and of 
 his humoristic turn of mind, that a comedy, " Le Diable Devenu Cui- 
 sinier," written by him in the French language, was, in 1873, played in 
 the '' Music Hall," and again in the " Salle Jacques Cartier," Quebec, 
 by the Maugard Company, then in the ci^^% to the great merriment of 
 all present. . ' .^, . '- ;•- . < 
 
 Nor will the members of "Le Club des 21," composed as it is of 
 the literati^ scientists and artists of Quebec, under the presidency of 
 the Count of Premio Keal, Consul-General of Spain for Canada, soon 
 forget how, in March, 1879, Mr. Baillairgt?, in a paper read at one of tjie 
 
— 39 — .: 
 
 sittings of the Club, around a well-spread board, successively portrayed 
 and hit off the peculiarities of each and every member of the club, and of 
 the count himself, while at the same time doing full justice to the 
 abilities of all. -..<•. . ^ ,^ v 
 
 Mr. Baillairg^ is a close and industrious worker, devoting fourteen 
 hours out of the twenty-four to his professional callings, and again rob- 
 bing the night for the time to pursue his literary and sientific pursuits. 
 
 In politics, if he may be said to have any, he his inclined to Liber- 
 alism, but he is of too independent a character to be tied to a party, 
 preferring to treat each question on its merits, irrespective of its pro- 
 moters, f'^' 
 
 The subject of this sketch is brother to G. F. Baillairg^, Deputy 
 Minister of Public Works of the Dominion, and grand nephew to Frs. 
 Baillairg^, an eminent painter and sculptor " de I'Academie Koyale de 
 Peinture et Scupture, France '' '^ho carved some of the statues in the 
 Basilica, and whose studio in St. ^jouis Street (the quaint old one story 
 building, now Driscoll's livery stable) was at that time so often visited 
 by Prince Ewdard, Duke of Kent, father of Queen Victoria, during his 
 sojourn in Quebec. u. ^ * 
 
 A portrait of Mr. Baillairge, but which, however, does not do him 
 justice, accompanied by a brief biographical notice, appeared in L*Opin- 
 ion Puhlique, of the 25th April 1878. The Rivista Universale, of 
 Italy, also published his portrait and a biographical sketch of Mr. Bail- 
 lairg^'s career, in February of 1878. ' ' ' ' "■" 
 
_._,: _ —40 — 
 
 Since the above was edited, in 1879, Mr. Baillairg^ has been the 
 recipient of the following additional testimonials : 
 
 ^7^ Royal Canadian Academy of Arts, s: 
 
 - ?.; - ^ . •: ;^- ; Grenville St., Toronto, Jan. 7th 1880. 
 
 Dear Sir, "" ''''''^-I'-T^- . /:V' "".'":''''' [^' .'./''-[J. 
 
 I am commanded by His Excellency the Governor General (Mar- 
 quis of Lome) to inform you that he has been pleased to nominate you 
 as an associate of the New Canadian Academy. . v^» . ^^ ^^ s*** 
 
 " ' ''^ ^ ' ^ (Signed), ' L. N. O'BKIEN", ^^^;t 
 
 President, 
 
 ' , Royal Society of Canada. , ^ . 
 
 ' Montreal, March 7th 1882. ' 
 
 I have the honor to intimate to you by request of the Governor 
 General (Marquis of Lome,) that His Excellency hopes you will allow 
 yourself to be named by him as one of the twenty original members of 
 The Mathematical, Physical and Chemical Section of the New Literary 
 and Scientific Society of Canada, the first meeting of which will be held 
 at Ottawa on the 25th of May. Should you accept be good enough to 
 state what work you wish associated with your name. 
 
 I have the honor to be 
 
 Sir , ' 
 
 r Your most obedient, 
 
 ^""\./::!:-'^-; -;:^^ . L STEKRY HUNT, 
 
 President of the Mathematical, Physical & Chemical Section. 
 C. Baillairg^, Esq. 
 
« ■ -Ho^e? cZu Oouvernementf 
 
 A Monsieur le Chevalier Baillairg^, Quebec, 
 
 MoN Cher Monsieur, .. ' ^ , w\- :> . 
 
 ■' Je vous prie d'accepter mes sinc^res remerciments pour Tenvol que 
 vous m'avez fait d*uEe serie complete de vos ceuvres scientifiques, ainsi 
 que du volume de la '* Galerie " oh se trouve votre biographie et votre 
 portrait. J'ai M tr^s sensible k cett« attention de votre part; vos 
 travaux et votre reputation qui s'est fait jour mtoe en Europe font 
 honneur, permettez*moi de vous le dire, k notre patrie et k la nationalite 
 franco- canadienne. Notre jeune pays compte encore peu d'illustrations 
 dans le monde de la science, et il doit ^tre d'autant plus fier de ceux de 
 ses enfants qui attirent aur etix I'attention des hommes dont Topinion 
 fait autoritd* 
 
 Veuillez accepter ma photographic et agr^er, Monsieur le Chevalier, 
 rhommage de la parfaite consideration avec laquelle j'ai I'honneur d'etre, 
 
 Votre ob^issant serviteur, 
 
 THfiODOKE ROBITAILLK 
 Lieutenant-Gouverneur de la Province de Quebec. 
 
 In July 1882 Mr, Baillairg^ T^as unanimously elected president of 
 the newly incorporated body of Land Surveyors and Engineers of the 
 Province of Quebec* 
 
 Hotel du Gouvernement 
 
 Quebec 18 juin 1877. 
 
 Monsieur, ; "; ' ''-'■ "' ""''^ 
 
 Permettez-moi de vous offrir mes remerciments pour I'envoi que 
 vous m'avez fait de votre ouvrage " Traits de Geometric et de Trigono- 
 metric qui vous fait tant d'honneur ainsi qu*^ notre pays. 
 
 Comme president de la Commission Canadienne a Philadelphie j'ai 
 eu occasion de faire examiner votre tableau stereometrique par les 
 
; —42 — 
 
 repr^sentants de la Grande-Bretagne, de la France, de TAlleraagne, de 
 la Russie, de TEspagne, du Portugal, de I'ltalie etj\ une seule exception 
 il ^tait connu et hautement appr^ci^ par eux tous. 
 
 Monsieur Lavoine, Ing^nieur des Fonts et Chaussees, que je counus 
 k Philadelphie, ou il avait la direction de I'exposition des modeles des 
 Travaux Publics de France, m'en park alors, de meme que durant une 
 visite qu'il me fit a Ottawa, I'automne dernier, de la mani^re la plus 
 flatteuse pour vous et pour les Canadiens. - 
 
 ' Je suis heureux, Monsieur, de ces tdmoignages qui vous honorent 
 et de savoir que vos travaux, tant de fois couronnes dans notre pays et 
 a r^tranger, viennent de I'etre encore k rExpositionUniverselle de 1876 
 k Philadelphie. 
 
 Je demeure, 
 
 Monsieur, 
 
 Votre obeissant serviteur, 
 
 L. LETELLIER. 
 
 Lieut. -Gouverneur de la Province de Quebec. 
 
 . ■ , ' • 
 
 Monsieur C. Baillairg^, 
 
 Ing^nieur Civil, Quebec. 
 
 Hotel du Gouvernement 
 
 Quebec, 18 juin 1877. 
 
 MoN CHER Monsieur, 
 
 S'il vous ^tait possible de passer k mon bureau, j'aurais le plaisirde 
 savoir que vous consentez, k entrer dans le cercle des Auteurs Cana- 
 diens, dont je desire m'entourer intimement, de temps a autres a Spencer 
 Wood. 
 
 L. LETELLIER, 
 M. C. Baillairg^, Quebec. 
 
I :n^ D E X. 
 
 Preface — Papers read before the Ptoyal Society of Canada in 
 
 1882 and 1883 3 
 
 On the application of the Prismoidal Formula to the measurement 
 
 of all solids ;. 8 
 
 Hints to Geometers for a new edition of Euclid 17 * 
 
 Simplified Solutions of two of the more difficult cases in parting off 
 or dividing-up land, and of a case in Hydrographic Sur- 
 veying ^^^^^ 23 
 
 The areas of Spherical Triangles and Polygons to any radius or 
 
 diameter 28 
 
 Biographical sketch of the author 33