ba^ill^irg£: TT»»»» t <»TT»T>» .THE STEREOMETRICON. XKW srSTKM OF MEASURING ALL liODILS BY ONE AND THE SAME RULE. Gi:m:ral application ofthi' prismoidal formula. NOMEXCi.A rriiK .\x:)(;i;xEiL\r, feaii-ues of each of jiie 200 MODELS OX THP: BOARD. THF, ai;e\s of sPin-:ri[t:.vL xitrvNiiLEs and i'olygons to any uadius : OR WAMKTEU, T A B L I<: S of the Areas of Circles^ Segments^ Zoii's — sec index ^ table of S'^ccific gravities. • QUEBEC PRINTED BY G. DAUVEAU 188i B2iXluXj^L I B O" Neyjv system of determining the solid contents of a I {Extract from the " Quebec Daily Mi Mr. Baillairg6's lecture on Wednesday evening last before the Literary and Historical Society of Quebec, pro- ved once more how very interesting, even in a popular sense, an otherurise dry and alistruse subject, may be- come, wlien ably handled. The lecturer showed the relationship of geometry to all the industries of life. He traced its origin from remote antiquity, its gradual developemeut uu to the preseut time. He showed how it is the basis of all our public works, and how we are indebted to it for all the construc- tive arts ; its relationship to mechanics, hydraulics, jntics, and all the physical sciences. The fairer portion of man- kind, said Mr. B., have the keenest, most appreciative perception of its advantages and beauties, as evidenced in the ever- varying combinations so cunningly devised iu their designs for needle tracery, laces and embroidery. He showed its relationship to chemistry iu crystallization and polarization ; to botany and zoology iu the laws of morphology ; to theology, and so on. Iu treating of the circle and other conic sections, he drew quite a poetical comparisou between the engineer who traces out his curves among the woods an«l waters of the earth, and the astro- nomer who sweeps out his mi^rhty circuits amidst the starry forests of the heavens. The parabola was fully illustrated in its applit-ation to the throwing of projectiles of war, also as evidenced in jets of water, the speaking trumpet, the mirror and the reflector, which , in light- houses, gathers the rays of light, as it were, into a bundle, and sends them forth together on their errand of humanity. In treating <.f the ellipse, this alm«»st magic curve which is traced out in the heavens by every planet that revolves about the sun, by every satellite about its primary, he alluded to that most beautiful of all ovals — the face of lo«'ely woman. He showed how the re-appearance of a comet may now be predicted even to the very day it heaves in sights, and though it has been absent for a ceu- tury, and how in former ages, when these phenomeua were unpredicted, they burst upon the world in unex- pected moments, 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. Baillairg^ we^t over the whole field of geometry and mensuration, both plane and spherical ; a difficult feat within the limits of a single lecture ; and kept the audience, so to say, en- tranced with interest for two whole hours, which the pre- sident, Dr. Anderson, remarked : were to him as but one ; and no doubt it must have been so to others, since Mr. Wilkie, in seconding the vote of thanks proposed by Capt. Ashe, alluded to the pleasure with which he had listened to the lecture as if, he said, it were like poetry to him, instead of the unpromising matter foreshadowed in the title. Mr. Baillairg^ next explained in detail his stereo- metrical tableau, which we hope to see soon introduced into all the schools of this Dominion. He showed how con- ducive it will be in abortening the time heretofore devoted to the study of solids and even to superficies, sphericul trigouometrj perspective, drawing the developei and iihadovvs, and the like. Mr. tunity had been afforded him of (i corroborated Mr. B.'s statement ii savine: in time, where many abstr ncrally require*! hours or days to s« be, as Mr. Baillairge asserts, so g( as has been certified by so many over their «)wn signatures,) with t mula and tableau, be performed ii say nothing of the use the model glance a knowledsre of their nome an acquaintaueeship with their va He showed how, tu the architect ji and mechanic, the uiodels are sug relative proportions of buililinga, quays, cisterns and reservoirs, eai and other vessels of capacity, cm comprising railroad and other cuti the shaft of the Greek and Ron winey timber, savv-lo 's, the ciin splayed opening of a door or wind a wail, the vault or arched ceiling billiard or the cannon ball, or. on earth, sun and planets. Mr. Bail received an «>rder for a tableau frc cation of New-Brimswick, with t iiw.o all the solioitls of that Froviu writing tO' Mr. Baillairge, from January last, to advise him of thi patent for that coimtry, says that \ the President ls of that Province ; .md Mr. V.-muier, in ^aillairge, from France, on the 10th of advise him of the granting of his leiters luntry, says that Messrs. Humbert & Noe, i secretary of the society for the eenera- ,ion in France, have intimated their in- lexteeneral meeting of having some mark ferred on him for the benefit which his covery are likely to confer on education, ting to Mr. Bnillairge, on the part of the 9au, Minister of Public Instruction, says: ivoir d^eu recommander I'adopticm dans ons d'education et dans toutes les ecules.'' iry and Laval University. M-. Maingui [1 ^tndie, plus on approftuidit cette formule corps, plus on est enchante (the more e sa simplicity, de sa clarte et surtout de i^ralite." Rev. Mr. MuQuarries, B. A. ted to see the old and tedious processes a formula so simple and so exact." New- ege. United States : " considers the ta- seful arrangement for showing the va- t oftbe applications of the formula." The •tion " will adopt Mr. Ba'llairge's sys- their coarse of instruction." Mr. Wilkie B author that '* the rule is precise and ** simple, and will greatly shorten the processes of calcu- lation. " Tlie tableau,'* siys thi« comnetent judare, " com- ** prising as it does a great variety of elementary models, " will serve admirably to educate the eye, and must great- " ly facilitate the study of solid mensuration." " Again," Pays Mr. Wilkie. " the Governujent would confer a boon " on schools of the middle and higher class by affording " access to so snggesfive a collection." There are others who, irrespective of considerations as to the comparative accuracy of the formula, or of its advantages, as applied to mere mensuration, are awake to the fact that the models are so much more suggestive to the pupil and the teacher than their me e ret>resentatioH on a blackboard or on paper, and who, in their written opini(m, have alluded especially to this feature of the prof)osed system. M. .Toly President oftbe Quebec Branch of the Montreal School of Arts and Design, in a letter on the subjects to Mr. Wearer, the Pre- sident of the Board, ami after having himself witnessed its advantages on more than one occasion, says, in his expressive style, " the difference is enormous." Professor Tousaint, of the Normal School, Dufresne, of the Mont- nuigny Academy, Boivin, of St. Hyacinthe, and many others, are of ihe same opinion ; among them MM. R. S. M. Bouchette, O'Farrell, Fletcher. St. Aubin, Ste<'-kel, Juneau, Venner. Gallagher, Lafrance. and the late Brother Anthony, &c., &c. Neither will it be forgotten that the professors oftbe Laval University, after reading the enun- ciation of Mr. B.'a formula, as given in his treatise of 1 866, •'Xpresseil Themselves thus : " Un douteinvolontaire s'em- p-ire d'abord de I'esprit, lorsqu'on lit le No. l.'>2l ; " mais " un examen attentif dcs paragraphes suivants, dissipe *' bientot ce doute et I'ou reste etonne ^ la vue d'une for- " mule, si claire, siaisee a retenir et dontl'applicatis Langevin and Labrecque, to report to the Council at its n"xt general meeting in June, and who, it may be taken for granted, after the many flattering testimonials in re- lation to the ntir "^ and many advantages of the stereo- metrical tableau .jr purposes of education, cannot but recommend and direct its adoption iu all the schools of the Dominiuu. BAILLAIRGE'S ST Honorary Member of the Society for the Genef New system of measuring all bodies, segments, frustums i Thirtee7i Medals of honor and Seventeen Diplomas from France, Italy, United- Slates of Ameri This is a Case 5 feet long, 3 feet wide and 5 inches deep, with a h exhibiting- and affording free access to soni- 200 well-finished Hardwo form, each of which being merely attached to the board, by means of v Student or Professor. Th" nse of tlio TivWoan an«l Mccoinpiiiiyiiiij Tre.itise, reduces the wliole scioiicis suid art of Meiisinatiou from tlie stiiily of a year to tluvt of a day or two, and so sim- plifies the study and teacliinuj ofSidid GeOTiietry, the Xo- iiieiichitnre of Geometrical and other forms, the (.evelope- meht of surfaces, iriometri- c;il projection and perspect- ive, plane and «'.nrved areas. Spherical Geometry and Trigonojnetry, and the men- suration of surfaces and solids, that tlie several branches hereinbefore men- tioned may now be taught even in the most, elementary schools, and iu convents, where such study could not even have been dreamed of heretofore. Each tableau is accompa- nied by a Treatise explana- tory of the mode of meas- urement by the " Prismoidal Eormula, and an exi)lana- tion of the S(did, its nature, shape, opposite bases, and middle section, its lateral surface developed, etc. Agents wanted for the sale of the Stereometricon in Ca- nada, the United States, <&^^ Pour tvouverle volume il' iiauce, ii. !s. Jw.O.,eic. Pali^les, »^onter quatrc iois la smtace an f "t> ' sixieiue vaitie »ie la li'i*!- remVlouLnienrau.o.ule. For the use of Architects, Engineers, Surveyors, Students and App Mathematics, Universities, Colleges, Seminaries, Convents and other Ec Measurers, Gaugers, Ship-builders, Contractors, Artisans and others in < STEREOMETRiCON. )RTnii: Generalization of Education in France, etc., etc. ts, frustums and ungulas of these bodies, by one and the same rule. ^rance, Italy, Belgium, Russia, Canada, Japan, etc. {Patented in Canada, in the ites of America, and in Eiuope.) leep, with a hinged Glass Cover, under Lockand Key, so as to exclude dust while hed Hardwood Models of every conceivable Elementary, Geometrical or other y means of wire, peg or nail, can be removed and replaced at pleasure, by the STATES and \\\ i^u l^ui rj. ATS-UNIS ^t6 pour la tiou eu France, i? ■ ^4- ^- ^^ To Hiul tho 8"lul content of any body. ■RULE : To tho sum of tha paialel and areas, add four tunes tho middle area au.l multiply tho whole by one siKth part ot the heiffht or length ot the body. 4 ^ Approved by tho Council o^ Public Instruction of the Province of Quebec, and already adopted and ordered by many Establish- ments in Canada and elsewhere. For information and testimonials apply (free of charge) to C. CAILLAIRGfi, QIJKBEC, CANADA. Honorary ^femher of the Society for the Generalization of Education in France, etc., etc. SUBSCRIBERS. The Archbishop of Quebec, tho Bishop of Kimouski, the Bishnp of Kingston, the Bishop of St. Hya- cintbc, the Dominion Board of Works, the Schools of Art and De- sign, the r.aval University, the Seminary. Q., the Colleges at; Otta- wa, Nicolet. Kimouski, Montma- gny. St. Michel, etc.. i'Kcolo 'Sot- male Laval. IfS Ecoles des Fr^res, the Commercial Academy, the Board of Land Surveyors, the De- partemeiit of Education. New Brunswick, the corporation of Quebec, It. Hamilton, Ksq., F. N. Martin, and (;. Koy, Civil Engi- neers, etc. La Soci6t6 pour la vul- garisation de I'Enseigiiement da Peuple, France, F. Peachy, J. Le- page, etc., Architects, 2f. Piton, T. Maguire. J. Marcotte. builders, the Council of Public Instruction, Q.. the Jacques Carrier Normal School. M. Piton Manitoba, the ('ollegps of Aylmer, L' Assomption, Ste Anne de fa Pocaliere, St. Hya- cinthe, the High School, Q.. the Morin College. Q., the Lafrance Academy, Q., Government Boards of "Works. Q , the Ursulines Con- . vent, the Convent of the Good Shepherd. Grev Nuns, Soeurs de J6sus-Marie, Q.. and M. S. W. Townsend, Hamilton, Stc, &c., &c. Etc., Etc., Etc. mts and Apprentices, Customs and Excise Officers, Professors of Geometry and and other Educational Establishments, Schools of Art and Design, Mechanics, id others in Canada and elsewhere. i BAILLAIRG .■ Honorary Member OF TH New system of measuring all be Thirteen Medals of honor and Seventeen Diplfi schools, and iu convents, wherfi such study conM not 'I'abloaii and accompanyins; Treatise, reduces tlu; 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 teachinuf ofS(did Geometry, the No- menclature of Geometrical and .)ther forms, the 1b Nor- Fr^res, ^^tfy. the -*^theDe- \ New itiov. of _ .. F. Tf. lent da ', J. Le- Piton, KJuildeis, "'iruction, "Normal jba, the ption, t. Hya- ]Q.. the lafrance ^Pl Boards %|es ('on- le Good lenrs de ^^^ Ai S. W. =1- For the use of Architects, Engineers, Survj and Mathematics, Universities, Colleges, Seminarieinics, Measurers, Gaugers, Ship- builders. Contractors THE -y STEREOMETRICON. Originator : C. BAILLAIRGfi, M. S. Hkmbkk of thk Sociktt fok thk GKXKiiAr.iZATiOM OK Educatio!! im Fraxcr, ahd OF eKVKKAI. LkaUNKI) AND SCIKSIIFIC SoCIKirKH; CHKVAI.IKR OF THB OuuEB OF St. Sauvkur i)k. Moxtk-Rkai.k, Italy ; ktc, kto. FkLI,0W OF THK RoYAL SOCIKTY, CANADA. I MEASUREMENT OF ALL SOLIDS BY ONE AND THK 8AMK BULK. UNIVERSAL APPLICATION OF THE PBISMOIDAL FORMULA. THIKTKRN MeDAI.8 and BKVKKTKEN DlPr.OMAS AND LKrrKHB AWARDKD THK AUTHOU FUou RtsaiA, Fhanck, Italy, Bklgiuh, Japan, ktc. PROMOTER : THOMAS WHITTY, PROFESSOR AT ST. DENIS ACADEMY, MONTREAL. Comprises 200 Solids rppresentative of all conceivaMe elementary funiis, as of the Cumpoueut parts uf Compound bodies. Name aud descriptioti of each solid. What it is represautative or »aggegti7e of, ur that of vvliich it forms a compoueut part. Nature and name of opposite bases and of middle section, as of lateral faces aud remainder of bounding Area, including every species of Piaae, Spherical, Spheroidal, and Conoidal figures. Division I, classes Ito X : pl-tvie faceti So'.ids and Solids of single curvature. Division II, classes XI to XX ; Sidids of oouble curvature. QUEBEC PRINTED BY C. DARVEAU ' r .1 ' / * { ., ^. J -y- I ".a \ ' IISTDEX Tlie SforooTTiptrlcon : noTnpnclatnro and gnneral feature of each of the 200 solids on the board ; see tho diiigr.ira at the beginning of thU pnniphli't , ' 5 Tlio AvoMs of Spherical Tri:nigles & Poly^joiis to .my radius or dia- lucttT : a iiaper read before the lloyal Socit'ty of Canada in 1833. 55 On the general applieatlDn of tlie pii.^m )i'.lal formula : a paper read be- fore the Royal Society of Canada in lQi2 61 TABLES I. Siinarcs and Square Roots of numbers from 1 to IGOO 4 II. Circumferences and areas of circles of diameter ^'j to 150, advan- ciujr by ^ 11 III. Circumferences and areas of circles of diameter ^'^^ to lOJ, advan- cing ^'y iff ' -- 19 IV. Circumferences and areas of circles of diameter 1 to 50 feet, ad- vancing by 1 inch or ^^ 25 V. Sides of Squares equal in area to a circle of diameter 1 to 100 ad- vancing by i 29 VI. Lengths of circular arcs to diameter 1 diviled into 1000 equal l)arts .- 31 VII Lengths of semi-elliptic arcs to transverse diameter 1 divided into lOOO equal parts 33 VIII. Areas of tho segments of a circle to diameter I divided into 1000 equal parts 37 IX. Areas of the zanes of a circle to diameter 1 divided into 1000 equal parts - - 3D X. Specific gravities or weights of bodies of all kinds : solid, fluid, liquid and gazeous 22 .54 THE STEREOMETRICON Orioinatob : C. BAILLAIEGfi, M .S. .\ \i^'. Member of the Society for the Greaeralisation of Education in France and of several learned and scientific Societies : Chevalier af the Orderof St. Sauveur de Monte. Realo, Italy ; Fellow of the Royal Society of Canada, etc., etc.. etc. Measurement of all solids by one and the same rule. Universal application of the prismoidal formula. TLirteen Medals and seventeen Diplomas and letters awarded the author, from Franc:;, Russia, Italy, Belgium, Japan, etc. Pbomotbr: THOMAS WHITTY, professor at St. Denis Academy, Montreal, etc. RULE : To the sum of the opposite and parallel end areas, add four times the area of a section midway between and parallel to the opposite bases ; multiply the whole by ^ part of the length or height or diamett, of the solid, perpendicular to the bases ; the result will be tfie solidity or volume, the capacity or contents of the body, figure or vessel under consideration. ^S . ; ^ For application of the rule and examples of all kinds fully worked out, see " Key to Stereometricon." For areas of all kinds, plane, and of single and double curvature, see also "Key to Stereometricon," with tables of areas of circles to eighths, tenths and twelfths of an inch, or of any other unit of measure, tables of segments and zones of a circle, etc., etc., at end of " Key." — 4 — The tabfean comprises 200 models, disposed in 10 horizontal and 20 vertical rows, series, families or classes. The solids may be indif- ferently placed, and numbered from the right or left and from below upwards or the contrary. The solids are representative of all conceivable elementary forms and figures, as of the component parts of all compound bodies. DIVISION I. Plane faced solids and solids of single curvature, or of which the surfaces are capable of being developed in a plane. CLASS I. Prisms. NoTK.— The author uses the term "trapeziu n" and not "trapezoid," as the termination "oid" conveys the idea of a solid as paraboloid, hyperbol)id, conoid, prismoid, etc. For the same reason he uses the French " trapeziform " instead of trapezoidal. Name of solid, object of which it Nature and name of opposite bases is representative or suggestive, or and middle section, lateral faces of which it forms a component part, and remainder of bounding surface. Reference to " Key to Stereome- Reference to page or paragraph of tricon," for computation of contents " Key " for calculation of areas and and of factors necessary thereto. of factors necessary thereto. ^- ? §: X — The cube or hexaedron — Each of its three pairs of opposite one of the five platonic bo- and parallel faces or of its six faces dies or bases and middle sections, per- Representative of any other rec- feet and equal squares. For de- tangular prism, of a building or veloped surface. See "Key to Ster.,' block of buildings or of one of the page 131. component parts thereof ; a brick or Representative of the floor, ceiling^ cut stone, a pt^Jestal, a die or dado ; wall oi partitions of a rectancrnlar a pier or r\ury '^ox, chest, pack- room or apartment, or of the bases age of m "cha^idise or parcel ; a and sides of the various objects cistern, bin, at or other vessel of mentioned under the name of the capacity; a pile of bricks, stones, solid. lumber, books, etc., etc., etc. See " Key to Ster.," page 60. " Key to Ster.," p. 61, par. (78). fi— Aright isosceles triangular Its opposite and parallel bases prism and middle section, equal right- On end, a triangular block or angled isosceles triangles. Its building; on its base, a ridge roof; sides or lateral faces rectangles. on one of its sides, the roof of a pent- For areas, see " Key to Ster.," pages house or lean-to. "Key to Ster. p. 6 1 . 19, 22 and 60. Sides suggestive of those of objects alluded to. 8 —A right regular pentagonal Its opposite and parallel bases and prism. middle section, regular and equal On end, the base or component pentagons ; sides or lateral faces, part of the shaft of a pentagonal pier rectangles. or column; on one of its sides, a Areas suggestive of those of ob- baker's, butcher's or other van ; an jects mentioned in adjoining co- ambulance, etc. "Key," page 61. lumu. "Key," pages 35 and 19. 4— A right regular octagonal Its parallel and opposite bases and prism. section, regular and equal oota- Base or shaft of a column, a pier gons ; its sides or lateral faces, reo- or post, a bead, baluster, hand-rail, tangles. "Key," pages 36, 19. etc. "Key to Ster., " page 61. 5 — Oblique hexagonal prism Its parallel bases and section, An inclined post or strut or the symmetrical ai.d equal hexa- section of a stair-rail, a baluster on gons ; its sides, parallelograms, a rake, etc. Mitred section of a rail " Key," pp. 26j 19 and 63. compute or bead. "Key to Ster,," page 64. half of sym. hex. as a trapezium. — 6 — 6— Oblique rectangular prism. Two of its three pairs of opposite On end, an inclined strut or post, and paralled faces or bases and etc ; on its parallelogram base, the sections, equal rectangles ; the pier of a skew bridge, portion of a other bases and section, equal pa- . mitred fillet, etc. rallelograms. " Key," page 63, See "Key to Ster.," page 64. 7— Oblique prism or parallelo- pipedon. Section of mitred fillet on an in- clined or oblique surface, etc. Each of its three pairs of parallel faces or bases and sections, equal parallelograms. 8— A righ rectangular trapezi- form prism, or a prism of -which the base or section is a rectangular trapezium, On end, a pier or block of that shape ; on its larger parallel face or base, the partially flat roof ot" a pent-house or lean-to ; the base of a rectangular stack of chimneys on a sloped roof or gable, a corbel, etc. See "Key to Ster.," page 61. Its opposite and parallel bases and section ; on end, equal rec- tangular trap'-ziums; its lateral fcices, rectangles ; on either of its parallel sides or faces: its bases, rectangles ; its lateral fiices, rec- tangles and trapeziunas See "Key to Ster.," j figes 60 and 29. May be treated indifferently as a prism or prismoid. Q — A right trapeziform prism. On end, the splayed opening of a door or window or loop-hole in a wall; on broader base, a partially flat roof; on its lesser parallel base, a bin or through or other vessel of capacity, section of a ditch excava- tion or of a railroad embaukmeut on level ground, a scow or pontoon. On end, its bases and section, trapeziums, and sides, rectan- gles ; on either of its parallel faces, its bases and section, rectangles ; its sides, rectangles and trape- ziums. N. B. Its solid contents, like those of Nos. 2 and 8, may be com- puted either as prisms or prismoids. 10 A right or oblique polygo- Eule for solid contents : multiply nal ooxnpound prism, deoom- one-third the sum of the three vert- - 7 — • . posable into right or oblique ical edges or depths of each of the triangular prisms or frusta of component triangular prisms, or prisms frusta of triangular prisms by the An excavation or filling, etc. area of a section perpendicular to A spoil bank or a borrowing pit. sides or horizontal, and add the Each frustum or component part results. Page 67, rule II, "Key." may be treated as a prismoid, one of its sides being the base. CLASS ir. Prisms, Prasta and Ungulae of Prisms. ^ •. 11 — A right regular trian3;ular Its parallel bases and section, prism. equal equilateral triangles ; its On end, a triangular building, faces, rectangles. Compute as pier or block ; on one of its sides, the prismoid with rectangular bases, gable of a wall, the roof of a gabled the upper base then being an arris house, etc. or line. -., "Key to Ster.," page 61. t; -^ ^ 12 — Lateral "wedge or ung^la One of its parallel bases a regu- of a right hexagonal prism, lar hexagon ; its middle base a by a plane through edge of half hexagon or trapezium ; its base, upper base a line; its lateral faces Portion of a mitred bead or hand- a line, a rectangle, triangles and rail, end of stair baluster under trapeziums ; its sloped face a hand-rail, ridge roof of an octagonal symmetrical hexagon or 2 tower against a wall; base of a trapeziums, base to base. chimney stack on a sloped roof or gable. — 8 — 13 — Lateral ungula of a right One of its opposite and parallel hexagonal prism, by a plane bases, a regular hexagon ; the ■ through opposite angles of other, a point ; its middle section the solid. a half hexagon or t-wo rectan- Bas'e of a chimney stack, vase or gular trapeziums base to base ; ornament on a sloped roof or gable, its lateral faces, trapeziums and etc. triangles ; its plane of section, a N. B. — This solid and the last, symmetrical hexagon, which, are not prismoids according to the for area, regard as two equal tra- definition thereof, page 163, par. peziums base to base, compute and (206), " Key to Ster. ; " but the up- add. per half, folded over and applied to See " Key to Ster.," page 29. the lower half, evidently completes Or the symmetrical hexagon may the prism, and hence the solidity is be decomposed into a rectangle and exactly obtained by the prismoidal two equal triangles, for computa- formula, as it is of a like frustum of tion of area. a cylinder or of an uugula thereof by a plane through edge of base. 14~Gentral -wedge or ungula One of its parallel bases, a hexa- of a right hexagonal prism j gon ; the other, a line ; its middle a prismoid. section, a symmetrical hexagon A wedge, the ridge roof of a or t"Wo trapeziums, base to tower, the base of a chimney stack, base ; its lateral faces, triangles vase or ornament between two and trapeziums. gables. See " Key to Ster.," page 29. 15 — An oblique trapeziform Treated as a prismoid : its oppo- prismL. site and parallel bases, unequal The partially flat roof to a dormer rectangles ; its lateral faces, tra- window, the roof of a building abut- peziums. ting against another roof, the splay- The factors of its middle section ed opening of a basement window, arithmetical means between those mitred portion of a batten or moul- of its opposite and parallel bases, ding, section of a ditch excavation, or of an embankment on a slope. — 9 — 16 — An oblique triangular Treated as a prisraoid : one of prism. its opposite and parallel bases, a. The roof of a dor ler window or rectangle ; the other, a line ; its of a wing to a house with a sloped lateml fuces, equal triangles and loof, a mitred moulding or fillet, etc. parallelograms. 17 — Frustum of a right trian- gular prism. Eidge roof of a building against a wall, a mitred moulding, etc. As a prismoid : one of its parallel bases, a rectanglr^ ; its opposite base, a line ; its middle section, a rectangle. 18 — Irregular frustum of an oblique triangular prism. Eidge roof of a building of irre- gular plan abutting on the unequal- ly sloped roof of another building, etc. Considered as a prismoid : one base, a trapezium, the other, a line ; its middle section, a trape- zium , its ends, non - parallel triangl3S ; its sides, trapeziums. 19 — A right prism on a mixti- linear base. On end, the unsplayed opening of a door or window in a wall, etc. Note, for area of segment of cir- cle or ellipse, " Key," pages 33, 44, 61, 53, 57, tables II, III, IV, VIIL <:\' Parallel bases and section mix- tilinear figures, decomposable into a rectangle and the segment or half of a circle or ellispis ; the lateral face, a continuous rect- angle. Note. — The segment of a circle or ellipse may be equal to, less or greater than a semi-circle. 20— Regular firustrum of an As a prismoid : one mse, a oblique triangular prism. rectangle ; the other, a line ; A ridge roo^ mitred fillet, etc the middle section, a rectangle. — 10 — . r CLASS III. ' ■ Pmsta of Prisms, Prismoids, Wedges. 21 — T h e dodecahedron, or The six pairs of parallel bases or t"welve-sided solid, one of the twelve component faces of the solid, five platonic bodies. equal and regular pentagons ; Assemblage of twelve equal py- the middle section a regular ramids with pentagonal bases, their decagon, the side of which is apices or summits meeting in the equal to half the diagonal of the centre of the solid or of the cir- pentagon, tor area of which see cumscribed sphere. " Key to Ster.," page 36, rule II ; The capital or intermediate sec- or compute one of the component tion of a pentagonal shaft or column, pyramids and multiply by twelve. a finial or other ornament. For developed surface, see " Key to Ster," page 132. 522 —A rectangular -wedge, the On end : its opposite and parallel head or heel broader than bases, a rectangle and a line ; its the blade or edge. middle base or section, a rectangle. The frustum of a triangular prism, On one of either of its other two or may be treated as a prismoid, pairs of parallel bases ; one base, a using either of its three pairs of trapezium, the other, a line ; the parallel bases. middle section a trapezium ; side An inclined plane, a low pent faces, a rectangle and triangles. roof, an ordinary wedge, etc. 23 -A rectangular "wedge or Each of its three pairs of parallel inclined plane the head or bases, a rectangle and a line ; its heel of equal breadth with middle sections, rectangles, res- the edge or blade. pectively equal to half the corres- A right triangular prism, body ponding bases. May also be treated of a dormer window or base of a as a triangular prism, with bases chimney stack on a low or steep and section equal triangles. roof, etc. — 11 — 24 —An isosceles "wedge, the edge or l»Ude liroader than the heel. ' - May also be considered, the frug- tum of a triangular prism or pris- inoid with three pairs of parallel bases. As a prismoid : one of its pairs of parallel bases, a rectangle and a line ; middle section, a rectangle ; each other pair of parallel bases, a trapezium and a line ; middle sec- tion, a trapezium. 25— Frustum of a right rec- tangular trapeziform prism, or a prismoid. A roof, partially flat, abutting against a vertical wall at one end and in rear, against a sloped roof at the other, etc. As a prismoid : its opposite and parallel bases, rectangles ; the longer side of the one corresponding to the shorter side of the other ; its middle section, a rectangle ; all its lateral faces, trapeziums. 26— Irregular firustum of an ob- lique trapeziform prism. A roof between two others not parallel, irregular section of a ditch or embankment As a prismoid : its opposite and parallel bases and middle section, trapeziums ; its lateral faces, tra- peziums. Factors of middle section arith- metic means between those of the bases. 27 — Frustum of a right isos- celes trapeziform prism, a prismoid. On its larger base, a roof, section of an embankment, etc.; on its lesser base, a bin or vessel of ca- pacity ; the capital of a pilaster, a corbel ; on end, a splayed opening in a wall. As a prismoid : its opposite and parallel bases and middle section, rectangles ; lateral faces, trape- ziums. In all such solids, the half way fectors need never be measured, as they are always means between the parallel bases of the trapezium faces. 28 — Frustum of an isosceles As a prismoid: one of its opposite triangular prism, a prismoid. and parallel bases, a rectangle ; the Ridge roof with ends unequally other, a line ; its middle section, a sloped, mitred moulding, etc. rectangle. " Key," page 19. — 12 — 28 — Frustum of a trapezlforzn As a prismoid : its opposite paral- prism, a prismoid. lei bases and middle section reo- A flat roof, etc. ; on its lesser tangles ; its lateral faces, trape- parallel base, a bin or reservoir, a ziums. Factors of intermediate sec- vehicle of capacitj, a scow, a pon- tion or middle base, arithmetic toon ; on end or its parallel faces means between those of the end vertical, the splayed opening of a bases, window. "Key to Ster.," page 29. 80 — A prismoid on a mixtili- Its opposite and parallel bases r.sar base. and middle section, mixtilinear The roof of a building, circular figures ; the one a rectangle and at one end or coved celling of a a semi-oirole ; the other two, reo- room ; on its lesser base, a bathing tangles and semi-ellipses ; its tub, etc. ; vertically, the splayed arched end developed, a sort of tra- opening of a circular headed window pezium with curved bases ; its area in a wall. equal to half sum of bases by mean breadth or height. CLASS IV. Prismoids, etc. 31 — The ioosahedron, or twen- The ten pairs of parallel bases or ty-sided solid; one of the twenty component faces of the solid five platonio bodies. are equal equilateral triangles. An assemblage of twenty equal Its middle section, a regular do- pyramids on triangular bases, their decagon. Its middle section pa- apices or summits meeting in a rallel to two opposite apices or to common point, the centre of the the bases of any two opposite pen- solid or of the circumscribed or tagonal pyraTnids of the solid, a inscribed sphere. regular decagon, whose side is -13 — A finial or other ornament, etc. equal to half that of one of the More expeditious to treat it for edges of the solid. For developed solidity by computing one of its surface, see " Key to Ster," p. 133. component pyramids, and multiply- ing the result by twenty. 32 — A prismoid, both its bases, lines. Irregular triangular pyramid. Dormer or gablet abutting on a sloped roof. Component section of Its opposite bases — considering the solid as a prismoid resting on one of its parallel edges — lines ; its middle section a rectangle. See "Key to Ster.," page 164, No. 79. " Key " p. 165, par. (212). par, (208). 33— A prismoid on a trapezi- One ofitsparalledbaseSjatrape- form base. zium ; the other, a line ; its middle A cutting or embankment, etc. section, a trapezium. 34 — A railroad prismoid on a Its end sections or bases and middle side slope. parallel section equal quadrila- Section of a railroad cutting or em- terals, for area of which see " Key bankment on ground, sloping late- to Ster.," page 30. rally or in one direction only. This prismoid is a prism on an irregular base, and may be so 4.,;. : , treated. 35— A railroad prismoid on a grade and side slope, or on ground sloping both lateral- ly and longitudinally. Its narrow base upwards, an em- bankment ; the same downwards, a cutting or excavation. Its opposite and parallel end bases and middle section, quadrilaterals, the factors of the middle section being all arithmetic means between those of the corresponding end areas. ■'*■.':■•■■---•;-:'■ '^^ 38— A square or rectangular Its end bases and middle section prismoidal stick of timber, squares or rectangles. A squared log, a tapering post. Timber is usually measured by — 14 — the shaft of a chimney or high tower, a reducer between rectangu- lar conduits of unequal size, etc. Note. — 25 per cent, of the whole or true content is 33J per cent., or one-third of the erroneous result. multiplying its middle section into its length. This gives an erroneous result ; the more tapering the timber is, the more so. If it tapered to a point the error would be 25 per cent., or one-quarter of the whole in defect. 37— A prismoidal stick of -wa- ney timber A log of waney timber ; on end, the shaft of a chimney, a high tower, a tapering post Its opposite bases and middle section, symmetrical octagons, for area of which see ' Key," p. 176, par. (272), or squares or rectangles with chamfered corners or angles. 38— A concavo-convex pris- moid or curved viredge. A corbel, spandrel, finial, etc. ; a brake, a cam, etc. " Key to Ster.," par. (141). Its opposite bases, a rectangle and a line ; its middle section, a rec- tangle ; its developed faces, trape- ziums ; sides, mixtilinear tri- angle. 39— A recto-concave prismoid, or frustun of a curved wedge. A corbel, spandrel, buttress, etc. May be decomposed, as also No. 38, into two sections for more exact computation of solid contents. Its opposite and parallel bases and middle section, rectangles ; its de- veloped faces trapeziums ; its late- ral faces mixtilinear trapeziums For areas see "Key," page 57. 40— Frustum of a rectangular trapeziform prism, a prismoid A flat roof in a rectangular cor- ner ; on its lesser base, an angular corbel, a sink, cistern, bin, etc. ^.,^ As a prismoid, its opposite and parallel bases and middle section, rectangles ; its lateral faces, tra- peziums. ; " Key," page 104, par. (141), ■4*^«-;^ ; «. :.,! -\. . - f j "y} ??'-»-j^^- — 15 — CLASS V. , '' . Frismoids, etc. 41 — The ootahedron or eight- Its four pairs of parallel bases or sided figure ; one of the five eight component faces, equilateral platonie bodies. triangles ; its middle section, a Assemblage of eight equal p3n:a- regular hexagon ; its middle sec. mids on triangular bases, their apices tion through opposite apices and meeting in a common point, the perpendicular to intervening arris centre of the solid ; or two quadran- or edge a lozenge; through four gular pyramids, base to base. apices, a square. For developed surface see " Key to Ster," page 132. 42 — A prisinoid,one of its bases Its opposite and parallel bases, a a square,the other an octagon square and an ootagon ; the mid. Base or capital of a column, roof die section, a symmetrical oota- of a square tower, a tower, pier, gon ; its lateral faces, triangles and vessel of capacity, component sec- trape2uums. For area of symme- tion of a steeple, etc. trical octagon, see "Key," par. (272) 43 — A prismoid, its opposite One of its opposite and parallel bases, a square and a eircle. bases a square ; the other, a oir- Base or capital of a column, roof of ole ; the middle section, a mixti. a square tower, a tower, pier, vessel linear figure or a square "with of capacity, a lighthouse, a section roimded corners, of steeple or belfry, a reducer be- Its lateral surface capable of de- tween a square and circular conduit, velopment into a plane trapezi- form figure, one bfise circular, the other polygonal. 44 — A prismoid, its bases one- Its opposite bases unequal qual squares set diagonally, squares set diargonally to each Kepresentative of the same ob- other; the middle section, a sym- jects as solids, Nos. 42 and 43. metrical oOtMgon ; its lateral faces> triangles. -■■;., :,-:,■■■-"' ■ .-■ -16- 45 — A prisxnoid its bases a hex- One of its bases, a hexagon ; agon and a rectangle. ' other a rectangle ; its middle sec- Eepresentative of nearly the tion a symmetrical octagon ; its same objects as the three last solids, lateral faces, rectangles and tri- angles. 46 — The lateral frustum of a Its parallel bases and section, rectangular prolate spindle, squares ; its lateral surface, mix- Roof of a square tower, compo- tilinear fig-ures capable of de' 1- nent part of a steeple, etc. opment into plane surfaces, i'or area of these see " Key," page 57. 47 — A prismoid, its bases, an Its middle section, a mixtili- ellipsis and a square. near figure or approximate oval. A reducer between an elliptic Its lateral surface developed, a and square conduit, a roof, etc. curved trapezium, one base curved, the other polygonal. See "Key to Ster., " page 166. 48 — A prismoid, its bases a Its middle base, a symmetrical symmetrical hexagon and a octagon ; its bteral surface, trian- line. gles. For symmetrical hexagon, Eidge roof, coping or finial to a area equal to double that of half post, panel ornament, etc. the figure, which is a trapezium. 49— A prismoid, its bases, a Its middle section or base, a symmetrical hexagon and a symmetrical decagon ; its lateral lozenge faces, triangles. Area of hexagon, Flat roof, ornament, etc. ; on its double that of component trapezium. lesser base, a fancy basket, a disk,etc. 50— A groined ceiling or the Its base and middle section, half of a rectangular oblate squares ; its opposite base, a point ; spindle. its lateral flEtces, mixtilinear fi- A roof, panel ornament, etc. For gures. more exact computation of contents, For areas of mixtilinear figures decompose into two parts. Bee " Key to Ster.," page 57. — 17 — CLASS VI. *iif P3n:aimds and Frusta of Pyramids. 61— The tetrahedoD, or four- sided figure ; one of the five platonic hodies. A regular triangular pyramid. Apex roof of a triangular building, finial or other ornament, the com- ponent element of the icosahedron and octahedron. Its base and middle section, equilateral triangles, the lesser equal in area to one-quarter the greater, its upper or opposite base, a point ; its faces, triangles. For development of surface see " Key to Ster.," page 131. For area of bases and faces, see page 36, rule II. 52— A regular square or rec- tangular pyramid. "-•' The spire of a steeple, a pinnacle, roof of square tower, a bin, a vessel of capacity, a finial or other orna- ment, etc. One of its parallel bases, a square ; the other, a point ; its middle section, a square, of which the area is one quarter that of the base. Lateral faces, isosceles tri- angles. 63 —A pyramid, t"WO of its faces perpendicular to base. The ungula of a rectangular ^ prism on either of its bases. An apex roof, section of cutting or embankment, component portion of other solids, a roof saddle. Its base and middle section, tri- angles ; apex, a point. Factors of middle section half those of the base* Affords a demonstration of the theorem that in right-angled spheri- cal triangles the sines of the sides are as the sines of the angles. 64 — Frustum of a right trian- gular pyramid. Eoof, base or capital of a post or column, base of a table-lamp or vase, a vessel of capacity, component section of other solids. Its parallel bases and middle section similar triangles ; lateral faces, trapeziums. Factors of section arithmetic means between those of bases. — 18 — 65 — Frustum of an oblique Its bases and middle parallel triangular pyramid. section, similar triangles ; lateral Flat roof of triangular building faces, trapeziums; factors of abutting against a sloped or battered section, arithmetic means between wall ; portion of a ditch excavation, those of the bases. For areas see component portion of other solids. " Key to Ster.," pages 19, 22 and 29. £6 — Frustum of a right rectan- Its opposite bases and middle gular pyramid. section, squares or rectangles Flat roof to tower ; reducer bet- whose factors or sides are each ween conduits of varied size, com- equal to half the sum of the corres- ponent portion of an obelisk, capital ponding sides of the bases, or or base of a post or column, a bin, arithmetic means between them, vat or other vessel of capacity, the For areas see " Key to Ster.," pages body of a lantern, etc., etc. 19 and 29. 57 — A regular octangular or Its base and middle section, octagonal p3n:amid. similar octagons ; lesser area Koof of a tower, spire of a steeple, one-quarter of the greater ; its finial or other ornament, a funnel, upper base or opposite one, an apex strainer or filter, etc. or a point ; lateral faces, isosceles triangles. ^ ■_^.-; -*Vl ',t ^-, . , "■■-■'''3,' 58 — The firustuxn of a regular Its opposite and parallel bases octagonal pyramid. and middle section, regular octa- On its broader base, a roof, tower, gons ; factora of section means to pier, quay, component part of a those of the bases ; its lateral faces, steeple, etc.; base of a column, lamp trapeziums. For expeditious or vase, etc.; on its lesser base, a mode of arriving at area of octagon, vat, bin, vase, or other vessel of see "Key to Ster.," page 176 or capacity ; the body of a lantern, page 26, rule II. Developed surface etc, etc. ^^ •' a regular polygonal sector or tra- ^ A; k rc a-i2- pezium. ■:-:;■._ _19 — 59 — Irregular and oblique Its base, a quadrilateral or pyramid on a quadrilateral irregular trapezium ; its sum- base, mit or apex, a point. Middle sec- Apex roof of an irregularly tion similar to base and equal in shaped building against a battered area to one-quarter that of base. wall or roof, a roof saddle, etc. 60 — Frustum of a pyramid When decomposed for computa. ■with non parallel bases. tion of solid contents : bases and Decomposable into the frustum section of frustum, similar trian- of a pyramid with parallel bases, gles ; bases and section of compo- and an irregular pyramid, by a nent pyramid or upper portion^ plane parallel to the ba:>e and similar quadrilaterals. This passing through the nearest corner pyramid has its base in one of the or point of the upper, or non lateral faces of the solid, parallel base. CLASS VII. Cylinder, Frusta and Ungulae. /^^ 61 — A right cylinder or infini- Its parallel bases and middle tary prism. section, equal circles ; its lateral A tower or circular apartment ; a surface developed in a plane, a bin, vat, tub, bucket, pail, vase, rectangle ; its height, that of the drinking vessel, cauldron or other cylinder ; its length, the circum- vessel of capacity ; a road or other ference of the solid, roller : the cylinder of a steam or Foi areas of circles calculated to other engine ; a gasometer, the barrel eighths, tenths and twelfths of unity, of a pump, etc., etc., etc. see tables II., III., IV. at end of : > . "KeytoSter. " 62 — Frustum of lateral ungnla Its base, a circle ; its opposite or "wedge of a right cylinder, base, a semi -circle or other seg- May represent a cylindrical win- ment ; its middle section, a seg- — 20 dow or opening in a sloped roof ment greater than a semi-cir- abutting to a vertical wall or sur- cle ; its plane of section the seg- face, the liquid in a closed cylindri- ment of an ellipsis ; its cylindrical cal vessel held obliquely, base to surface decomposable by lines pa- chimney or vase partly on a hori- rallel to bases into trapeziums- zontal, partly on a gabled wall. For areas of segments, see table VIII., " Key," pages 53, 38, 44. 63 — A rectangular circular ring ; The difference between two con- centric cylinders, or a solid aunulus. Horizontal section of a tower wall, cross section of a brick, iron or other conduit, section of a boiler, vat, tub, or other vessel of capacity, etc., etc. Its bases and parallel section, concentric annuli : its interior and exterior surfaces continuous rectangles. The area of annulus equal to the difference of the inner and outer circles, or to the breadth of annulus into half the sum of its circumferences. See " Key," p. 39. 64 — Central ungula or -wedge of a right cylinder. Kidge roof oi a tower, a wedge, loop hole in a wali component portion of compound solid, a finial or other ornament, a strainer, etc. Its base, a circle ; its opposite base, a line ; its middle section, the zone or a circle ; its sloped faces, each a semi-ellipsis. Its cylindri- cal surface decomposable into tra- peziums by arcs parallel to base. See tables II., III. IV., IX., of " Key to Ster.,* also pages 38, 46, 53. 65 — Frustum of central wedge or ungula of oylinder No 64 Flat roof of tower or other buil- ding, base or capiial of rectangular pillar, vessel of capacity, component portion of compound polid, base of chimney stack or vase between two gables. Its greater base a circle ; its lesser base, the central zone of a circle ; its intermediate base, the zone'of a circle : its lateral faces equal segment of equal ellipses. Its cylindrical surface decomposable into trapeziums parallel to bases. See "Key to Ster.," page 51. — 21 — 66 — Lateral ungula of rij jht Its base, a semi-circle ; its inter- cylinder or recto-cylindrical mediate base ur middle section pa- ■wedge. rallel to base also a segment > its Lunette or arched headway of a opposite base, a point ; its plane of door or window, etc., in a sloped section or sloped face, a semi-ellip- roof, component of a compound sis. Its curved surface developed solid, the liquid in an inclined cy- an approximate parabola, tra- lindrical vessel, base of a salient peziums. etc. See " Key," pages chimney shaft over a roof, etc., etc. 38, 44, 51, tables II., III., IV., VIII. 67 — Frustum of lateral wedge Its parallel bases and middle or ungula of a right cylinder, section, segments of a circle, less Lunette to arched opening in a than, more than, and equal to sloped roof or ceiling abutting on a half ; sloped face, the excentrio vertical wall or surface ; liquid in zone of an ellipsis ; cylindrical an inclined closed cylindrical ves- surface, trapezium parallel to base, sel ; base of engaged column against For areas of segment, see " Key,' a battered wall, etc. page 44, rule I., rule II., table VIII.; for zone of ellipsis,see p. 53, art. (62). 68 — Irregular ungula o*" wedge 1st base, the segment of a cir- of right cylinder cle greater than half; its op- Lunette to a partially circular posite base, a line ; its middle sec- opening in an inclined ceiling, etc. tion, an eccentric zone of a cir- Compoueut portion of a compound cle ; one of its side faces, the seg- solid. For areas, see " Key to Ster.," ment of an ellipsis ; the other pages 44, 46, 53, articles (61) and plane face, an eccentric zone of (62), tables VIII. and IX. an ellipsis. 69 — Concavo-convex prismoid One of its bases, the lune of a or cylindro-cylindrical solid circle greater than a senai-cir- or concave frustuni of a cle ; the other the lune of a cir- wedge or ungula of right cle less than a semi circle ; the cylinder . middle section, a lune equal or Deposit of sediment in a cylin- thereabouts to a semi-circle. Its drical sewer, section of additional side surfaces, convex and concave — 22 — excavation or filling, or difference approximate trapezsiums. Tot between two lunettes. areas of lunes, see " Key," page 47. 70— •Frustum of an oblique When decomposed, its bases and cylinder. section ellipses ; the base of ungu- May be decomposed into an la, an ellipsis equal to each of those oblique cylinder and the an^ula of the inclined cylinder ; its middle of one by a plane parallel to base, section half an ellipsis. For uii- and passing through nearest point gulae, see Nos. 72, 73, 75. of other base. , CLASS VIII. ObHiue Cylinder, Frusta, Ungulae, Cjlindroids, etc. 71 — Oblique cylinder or infini- Its parallel bases and section tary prism eqnal ellipses ; its lateral surface Mitred section of conduit, hand capable ofdevelopment into a plane rail, moulding ; inclined column, niixtilineal figure. See " Key to post, strut or brace, etc. ; inclined Ster.," fig. n. page 57. For area of cylindrical opening in a wall, etc. ellipsis, see page 51 of same. 72 — Obtuse frustirai or ungula One of its opposite bases, an of oblique cylinder. ellipsis of sligh eccentricity j Oblique lunette inclined upwards its opposite base, a point ; its mid- or arched headway to a circular or die section, a semi-ellipsis equal elliptical opening in a sloped roof to half of base ; its plane of section or ceiling. Component mitred por- or lateral face, an ellipsis of tion of hand-rail, bead molding, etc. greater eccentricity ; its lateral cylindrical face developed, a figure } * ^:^^; -'i like m page 57 of " Key. " 73 — Acute frustum or ungula Same as No. 72. For developed of oblique cylinder. cylindrical surface, see fig. h. page Kepresentative of same as No. 67 of " Key to Stereometricon." 72, 1) t inclined downwards. — 23 — For area ot ellii^^iis, " K^y to Ser." pages 51 and 53. 74 Concave un^u^a or fVus- t ! m of oblique cylinder. Representative of same as No. 73, but in arch roof or ceiling in- stead of sloped roof. Same as No. 73, with curved instead cf plane section. Its cylin- drical surface developed similar to fig. h, page 57 of " Key ; " its cur- ved or concave section developed an oval or fig. like a, p. 57, " Key." 75_Frustum, ungula or wedge Same as No. 72. For developed of right cylinder. cylindrical surface, see fig. g; for Base of chimney shaft on sloped ellipsis, fig. b. p. 57, "Key." roof, or same as No. 72 not inclined. 76 — A cylindroid ; its bases, a circle and an elipsis ; infini- tary prismoid, Base or capital of elliiitic column, reducer or connecting l^nk between a circular and an ul-U-'tic conduit; a tub, vat or other v.?ss-^l of capa- city ; a hat with elliplic or oval head and a circular crown, etc. Its middle section, an ellipsis of which the conjugate or lesser dia- meter or axis is an arithmetic mean between those of the opposite bases. For area of circle, see table II, III, IV, and of ellipses, p. 51, " Key." Lateral surface develofied, a plane trapeziform fig ; its greater base, convex; lesser, concave; its area» equal to periphery of middle section into mean height. 77— Cylindroid or infinitary prismoid ; its bases, an elip- sis and a circle. Same as No. 76, or frustum of a conic metallic vessel, which has become fiattened or battered at one end. Its lateral surface developes into a plane trapeziform figure, with greater periphery convex ; and les- ser concave. Area equal to peri- phery of middle section into mean height. — 24 — 7?^— Cyllndrold ; its bases ellip- Factors of middle section, arith- ses at right angles to each metic means between those of the other. bases. Lateral surface developed, a Capital or base of elliptic column, plane trapeziform figure of connecting link btitween conduits ; area equal to periphery of middle metallic envelope or tube flatten- section into mean height, page 51 ed at ends in opposite directions. of " Key." 79_Cylindroid or prismoid ; Middle section, a mixtilineal its bases an ellipsis and a figure with factors, arithmetic line. means between those of bases. For Kidgeroofto elliptical building or area of middle section, page 57 of tower ; a hut, camping tent, a strai- " Key." Lateral surface developed, ner of filter ; a finial or other orna- a plane trapeziform figure ; its ment. base, convex ; its opposite base, an- ,, gular. Area equal circumference of middle section mean height. 80— A compound solid; a cy- For cylinder, see No. 61, class Under and a cone. VII; for cone, see No. 81, class IX. A tower or other building, a hut, The developed surface of a right tent, or camp with conical roof ; a cone is the sector of a cercle. hay rick, canister, finial ; reversed : For area, see " Key," page 42. a cauldron, cistern, tub, filter, etc. . i-- v, :;';•-■"■; -"'"'■ CLASS IX. /. : . ^''- ■,,■'■; '■"-'-■■'' Right and inclined Cone, Frusta, Ungnlae, etc. 81— A right cone or infinitary Its base, a circle ; its opposite pyramid. ;,;,,. base, a point ; its middle section Eoof of tower, spire, finial or a circle equal in area to one other ornament, pile of shot or shells, quarter that of the base. Its lateral cornet, filter or strainer, funnel, etc. surface developed, the sector of a circle. For area of circle, see tables II, III, IV, « Key to Ster." — 25 — 82— Fru-^trm of a right cone, Its opposite and parallel bases considertd as a prismoid and middle section, circles; its la- A tower, quay, })ier, base or ca- teral surface developed, the sector pital of a column, tiat roof of tower, of a circular ring, or a curved comf)oiient portion of a spire, a trapezium. The diameter of mid- salting tub, etc. , reversed : a butter die section an arithmetic mean be- iirkin, a tub or vat in a brewery or tween those of the opposite bases, distillery, etc., a drinking goblet, For area of bases and section see bucket, pail, dish, basket, lamp " Key to Ster.," page 38, for lateral shade ; a vessel of capacity, the plug surface, page 43. Tables of areas of of a stop cock, etc., etc. circles to eighths, tentiis & twelfths, 11,111, IV. 83 — Inclined or oblique cone. Its base and middle section, sim- Loop hole in a wall, the liquid ilar ellipses — the latter equal in or fluid substaiiice in a conical ves- area to one quarter the former ; the sel inclined to the horizon ; a finial upper base, an apex or point ; la- or ornament adapted to a raking teral surface developed an irregu- cornice or pediment, etc. lar sector, which, for computation of area, divide into triangles. 84 — Frustum of inclined cone. Its opposite and parallel bases Unequally splayed cireular open- and middle section, similar ellip- ing in a wall ; a coal scuttle : re- ses ; its lateral surface developed ducer or connecting link between portion of an eccentric annulus, two conduits of different diameters art. 39, page 33, of "Key to Ster.," laid eccentrically etc. Diameters of middle section, arith- metic me£His between those of bases* 85 —Flat or Iotw cone. Its base, a circle ; opposite base Roof to tower or circular con- or apex, a point ; its middle sec- struction ; cover of a box, basket, tion, a circle equal in area to one cauldron, etc, ; finial or other orna- quarter that of base ; its lateral face ment ; a Chinese hat, a pile of shot developed in a plane, the sector or shells, a sun shade ; reversed : a of a circle. — 2G — spinninj? top, bottom of cauldron or For area of circle, see tables II, reservoir, a funnel, stainer or filter, ill, IV, of "Key to Ster. ;" for etc. sector, see page 42 of same. 86 - Frustum of a lovr or ^ sui "based cone. Flat roof to a pavillion, tower, etc. ; a hat, the cover of a vessel of capacity ; an uu finished or trun- cated i)ile of shot or shlay(^d oi)ening or embrasure to a si'ij;iui' lit- shaped window or loop The parallel bases and miudle section, segments of a circle ; the lateral plane face or figure, the zone of a parabola, for area of which see " Key to Ster,," page 55,. hole in a wall ; lunette to opening art, (66) ; the developed conical — 27 — in sloped ceiling terminating in a surface, an approximate sector vertical surface ; liquid in a closed of a circular annulus or, more vessel in the shape of the frustum correctly, a trapezium "with of a cone, No. 82, when inclined curved concentric or parallel from the vertical. bases, for area of which see note For chord of middle segment, page 29, " Key to Ster.," For area measure solid or compute by page of segment, table VIII, and page 143 of " Ster." 44 of same. 89— Frustum of a right elon- Like No. 82, its opposite and gated cone. parallel bases and middle section Shaft of Grecian column, tapered circles ; diameter of middle section post, high tower or chimney shaft, equal to the half sum of those of funnel, pipe reducer, speaking the bases ; the developed lateral trumpet or horn, plug of a stopcock surface, the sector of a concen- or tap, deep drinking goblet, or trie annulus. other vessel of capacity large or For areas of circles to eighths, small, shaft of a gun, component tenths and twelfths, see tables II., portion of many compound solids, III., IV., of " Key to Ster. ; " for etc. that of sector, page 43 of same. 80 — A compound solid, com- For nature and areas of bases posed of or decomposable into and middle section of ; he component the frustum of a right cone frustum or a cone and of its lateral and the segment or half of a surface, see Nos. 82 and 89. sphere or spheroid. For areas of bases and middle May represent a piece of ord- section of hemisphere or hemisphe- nance, a deep conical vessel with roid or of the segment of either, hemi-spherical, hemi-spheroidal or greater or less than a hemisphere, segmental bottom or top to it. see tables II., III., IV. in " Key to For hemi-sphere, hemi-spheroid, Ster." or segments thereof, greater or less For diameter of middle section than half, see classes 18, 19, 20. in hemisphere or in segment For diameter of middle section thereof, see " Baillairge Geometry," in segment of spheroid, see " Key par. 539 or "Key to Ster.," par. 154, to Ster.," pages 139 and 140, where where oa = y Co . oD., and oD^= — 28 — AB : CD : : \/Ao . oB : o M and CD : AB : : y VoTojT: o M., or, the rectangle under the required radius nnd either axis of the spheroid is equal to that under the square root of the rectangle or product of the abscissas of the first axis and the other axis. diam. AB minus versed sine oC; or, the square of the half cord equals the rectangle under the versed sine and remainder of the diameter ; or, may be obtained directly by mea- suring the solid. r CLASS X. 4>: Conic Frusta and Ungulae, etc, 91 -Conic wel|;e or central ungula of a cone by planes dravrn from opposite edges of the base to meet in the axis of the cone. Ptidge roof to a tower, splayed opening or embrasure to a long naiTow vertical loop hole in a \tall ; component section of com])Ound solid of a cone and cylinder or of cones having their bases or apices in opposite directions. The base, a circle ; the parallel upper base, an arris or line ; the middle section parallel to bases the zone of a circle ; the lateral plane faces equal segments of equal ellipses, each greater than half; the curved or conical tai-es developed, equal curvilinear triangles For areas, see pages 38, 46, 53 and 57, and tables II , 111, IV., of " Ster." For area of zone, see table IX, of same. 92 Frustum of a conic -wedge or of the central ungula of a cone by a plane parallel to base ; or, may be considered the frustum of a right cone, laterally and equally truncated on op- posite sides. Arched and splayed embrasure in a wall, component portion of a compound solid. The base, a circle ; the opposite and parallel base, a zone of a circle ; the middle section, a zone ; the lateral plane faces, equal seg- ments of equal ellipses the developped conical surfaces resol- vable into trapeziform figures. For area of tiapezium, page 29, " Key to Ster." — 29 — 93— Lateral elliptic nngula of a cone, by a plane passing through edge ot base. Splayed embrasure to elliptic opening in wall and ihrongh sloped roof or Ceiling; etc. Its base, a circle ; its upper or opposite base, a point ; its middle section parallel to base, the seg- ment of a circle ; its plane face an ellipsis ; its conical surface developed a concavo - convex figure like h, page 97 of " Key to Ster." 94 — Lateral elliptic conic un- gula, by a plane passing "Within the base. The liquid in an inclined conical vessel, lunette head of opening in sloped roof or ceiling ; base of struc- ture rising from an inclined surface, roof, pediment, etc. For area of parabola see key to Ster., page 54 ; for area of hyper- bola, page 55, or iigure e, page 57 ; for ellipsis, page 51 and 53. The base, a segment ot a cir- cle ; the upper base, a point ; the middle section, a segment of a circle ; the plane lateral face, the segment of an ellipsis ; the de- veloped conical surface as in No. 87 or 94. If the cutting plane be pa- rallel to side of cone the face will be a parabola ; if at an angle greater than side of cone to base, a hyperbola ; if less, an ellipsis. 95— Central ungula of cone or conic "Wedge, by planes through opposite edges of upper or lesser base and meeting in the axis of the cone. An embrasure, etc., etc. The plane lateral faces, segments of elHpses if cutting planes more inclined to base than side of cone ; if less, hyperbolas ; if equally, pa- rabolas. Bases and sections same as "No. 91 ; developed conical surface, a concavo-convex triangle cora- putible as per page 57 of "Key." The lateral plane faces, equal segments of equal ellipses* equal parabolas or equal hy- perbolas, as case may be. — See No. 94. — 30 — 96 Pru'^tUTi of conic "wedge, Its base, a circle ; other base and No 85, by a plane par llel to middle section, zones of circles, the base. "■ for areas of which see " Key to An embrasure; a reducer or con- Stereometricon, table IX, necting link between a rectangular :f-i and circular conduit, etc. 97 — Concave ungnla of a cone or a conical recto-concave 'wedge. Lunette of circular headed open- ing in wall, reaching through vault- ed, groined or arched ceiling ; cone scribed to cylindrical surface, or to a shaft of elliptical section. The base, the segment of a cir- cle ; the other base, a point or curved arris ; its intermediate base or section, or its bases or sections if divided for computation of cubi- cal contents, segments of circles. Its sides like No. 94. 98— Portion of firustum of right cone, ty a plane through both bases. Splayed segment headed opening in wall, liquid in closed tub lying on its side ; base or capita] of half column against sloped wall ; com- ponent section of base or capital of clustered, gothic or other column. Its parallel end bases and mid- dle section, segments of circles ; its conical surface developed a figure of trapezium form, having parallel or concentric arcs of circles for its bases ; its plane face, the zone of an ellipsis or of a para- bola or hyperbola according to inclination of cutting plane. 99. — Lateral conic ungula or wedge, by a plane through edge of lesser base of frustum Embrasure, liquid in inclined co- nical vessel, section of conical elbow or mitre, base of chimney stack to sloped roof. May be treated also as lying on its lateral plane face. Its base, a circle ; opposite base, a point ; intermediate section a segment of a circle ; its plane face an ellipsis, its conical surface developed a concavo-convex figure like g or h, page 97 of Ster. but with concave base. Treat on circular base as easier of computation. — 31 — 100 — A compound solid com- All its areas to be nsed in com- posed of, decomposable or resolv- putation of solid contents or capa- able into two conic frusta and city are circles, and can be mea- ; a lo"w or flat cone. * y v^ sured to eighths, tenths or tweltths May represent a covered dish, a of an inch or other unity, and the basket or hamper, a vase, a finial or areas found by mere inspection in other ornament, an urn, a cauldron tables II., III. and IV. at end of on a stand, etc., etc. Baillarg^'s " Key to Ster. " DIVISION 2. Solids of double curvature, or of which the surfaces are not capable of development in a plane. '. CLASS XI. ■ '; - ■ -. J/f-s':, Concave Cones, Frusta and Ungulae. « 101 — Right concave cone or Its base and parallel sections, spindle. circles ; its upper or opposite base. Camping tent ; roof of tower, pa- an apex or point. Its lateral surface villon, hut, etc. ; spire, funnel, not capable of development in a ' strainer, trumpet ; finial or other plane or into a sector of a circle as ornament. is the case with a regular right cone. May be decomposed into two or but may be readily and very ap- more frusta by planes parallel to proximately computed by division base, to admit of more accurate de- into continuous trapeziums by termination of sohd contents. lines parallel to circumference of base. See " Key to Ster.," page 96. 102 — Frustum of a right con- Its bases and parallel sections, cave cone hetTxreen parallel circles. Intermediate diameters planes. not, as in No. 82, arithmetical means — 32— . lllnstrativeof most of the objects between Lhose of the opposite or mentioned in No. 82, which see. end bases, but must be measured or For more accurate computation computed. Lateral area may be of contents, divide into two sections conceived as made up of a series of or more, according to greater or super or juxta-posed continuous lesser curvature of the solid, and trapeziums. treat each section as a separate - cv^^r prismoid and add the results. ; .■ - - :: ^ 103 — Inclined concave cone. Its base and section, approxi- rinial, or ornament on a raking mate ellipses of slight excen- cornice ; liquid in an inclined ves- tricity or ovoid figures ; its other sel, etc., as for No. 101, may be base, a point. decomposed by imaginary planes In developing the lateral surface parallel to base into two or more into a series of continuous trape- sections or slices, so that slant side ziums, the lines are not as in the of each may be sensibly a straight right cone parallel to base or to line. See p. 103, par. 139 " Key. " circumferences of parallel sections but are drawn equidistant from the ,, • apex, thus leaving at the base a ' ^^. figure like h, page. 57 of "Key." 104 — Frustum of oblique con- Its bases and sections parallel cave cone between parallel thereto, approximate ellipses or planes. ^ ovoid figures. See remarks to Kepresentative of same as No. 84. No. 102 .. * • ,*; 105 Flat or lo-w concave cone. Its bases, a circle and a point ; Kepresentative of many of the section, a circle ; lateral area objects mentioned in No. 85. ^ reducible to continuous trape- ziums, par. 126, " Key to Ster." 106 — Frustum of flat or low Its bases and section, circles, cone. for arejs of which see tables II., Eepresentative of objects under III. and IV. of ** Key to Ster.," to nead of No. 86. eighths, tenths and twelfths of inch or other unity. — 33 — 107 — Ungula of concave cone by a plane through outer edge of base. See No. 92, as to what it repre- sents, etc. See No. 92. Lateral surface reducible to trapeziums and triangles. Base and sections, ovoid figures ; areas, page 57 of Key. 108 — Ungula of concave cone Base" and section, segments of by a plane cutting the base, circles ; upper base, a point. See No. 93 as to what it repre- Lateral surface as No. 107- sents, etc. ? * 109— Ungula of hoUovr cone by a plane through edge of lesser base of frustum. See No. 99, base of chimney stack to a sloped roof. » : < . '^ Base, a circle ; opposite base, a point ; middle section, the seg- ment of a circle ; lateral area, trapeziums and triangles. 110— Frustum of (No. 109) un- gula by a plane parallel to base. ■••/■>-■■■"; See Nos. 98, 116, 126. ^V- Base or capital of a column, or base of chimney shaft, etc., on or outside of sloped roof or gable. Its base, a circle ; other base, a segment of a circle ; its middle section parallel to bases, also a segment. For areas of segments of circles, see " Key to Ster.," table VIII., or rules, page 44 of same. • ■ ^-^'•- ■ :: CLASS XIL (■■ C : :,:■;;/, ■ ParaTDoloid or Paracolic Conoid, Frusta and ^' *^ . ^ Ungnlae, etc. Ill — Right paraboloid or para- Its base and middle section, bolic conoid. circles ; its opposite base or apex, Dome, hut, hive, roof, finial or a point ; its lateral surface resol- other ornament, shade, globe, cover, vablu into a small circle at apex, hood, cowl, etc. ; reversed : a filter, and continuous trapeziums. The — 34 — cauldron, or other vessel of capacity, squares of its intermediate diame- thc bowl of a cup or drinking ters, proportional to abscissae. See goblet, etc., etc. . . " Key to Ster.," page 96. 112 Frustum of right parabo- End and middle bases, circles ; 1 jid bet"ween parallel planes, squares of diameters proportional liepreseuts mostly the same to abscissae. For areas of circles, objects as the frustum of a cone, see " Key to Ster.," tables II., III., No. 82. and IV. ; - > VT ■ See page 142 " Key to Ster." 113 oblque paraboloid. Its base and middle section, " Key to Ster.," page 142. similar ellipses ; its opposite base Liquid in a parabolic vessel or other end, an apex or point. For inclined to the horizon, metal in an areas of ellipses see " Key to Ster.," inclined crucible, finial or ornament page 51 ; for lateral area see No. on an inclined or raking molding 103. ■'.';! or pediment, etc. .^^^ : ; 114 Frustum of oblique para- Its bases and middle section, boloid bet"ween parallel similar ellipses; for areas of planes. which see " Key to Ster., page 51. Represents same as frustum of For lateral area, see No. 103 or inclined cone No. 84, "Key to reduce to trapeziums by lines Ster.," page 142. from base to base. 115— Parabolic wedge or cen- tral ungula of paraboloid. See No. 91. Lateral or paraboloidal surface capable of approximate develop- ment. See No. 91. 116 Portion of a paraboloidal Its lesser base, a circle ; opposite f^ustiun, by a plane through base, the segment of a circle ; its greater base and edge of middle section, also a segment, other or opposite base. Its lateral plane face, the se£fment — 35 — See No. 98 as to what it repre- of an ellipsis. This face would be sents. Also, base of chimney stack, a parabola if angle of face equalled partly on a horizontal and partly that of side ; if greater, a hyperbola, on an inclined base, or sloped roof, etc. • 117 — Lateral ungula of parabo- loid Very similar to No. 92, as to what it represents. Its base, a circle ; opposite base, a point ; middle section, the seg- ment of a circle. Its plane face an ellipsis. 118— Lateral ungula of parabo- loid; elliptic, parabolic or hyperbolic, according as pla- ne of section cuts the base at an angle less than, equal to, or greater than that of the side and base. Its base, the segment of circle ; its middle section, a segment ; its upper or opposite base, a point ; its plane face, the segment of an el- lipsis, parabola or hjrperholai according to angle of plane of sec- tion. 119— Obtuse eliptic ungula of a paraboloidjby a plane through edge of lesser base of frus- tum. Base of chimney stack, etc., to eloped roof ; base of vase, statue, etc., on a pediment; a lunette, scoop, etc. , , Its base, a circle ; middle section, a segment ; other base, a point ; its plane face, an ellipsis. For a- reas of segments of circles, table VIII of " Key to Ster." For area of ellipsis, page 51 of same. 120— Frustum af a paraboloid between non-parallele bases. " Key to Ster.," page 145. Lunette through a vertical wall and inclined ceiling, etc. For com- putation of solid contents decom- Its factor areas, circles and a segment ; its plane face, an ellip- sis. For areas of segments of circles, table VIII of "Key." Area of circle, tables II, III and IV, of same; ellipsis, page 51 of same; — 36 — pose into a frustum with parallel lateral area, page 95 ; solidity, page bases, and an ungula by a plane 145 of same. parallel to base, through nearest point of upper base. ; . . ' ^^>v'J . ^ CLASS XIII. Hperboloid or Hypertolic Conoid, Frusta and Ungnlae, eto. 121— Right hyperboloid or hy- perbolic conoid. Page 146, "KeytoSter." Repre- sentative of same as No. 111. For intermediate diameter or that of middle section, see "Key to Ster.," page 147, 3rd line, or by direct measurement. ■* 19.9. — Frustum of right hyper- Except for diameter of middle boloid. 5^ i section, same as No. 112, or the Representative of same, nearly diameter may be measured directly. as Nos. 112 and 82. ^--^ . 123 — Oblique hyperboloid. See " Key to Ster.," p. 146. Re- presentative of same, as No. 113. Same as No. 113, except for dia- meter of middle section for which see "Key to Ster.," page 147, line 3, or the diameter may be mea- sured. :a - - • 124— Frustum of oblique hy- Same as No. 114, except for dia- perboloid. meter of middle section for which Representative of same, nearly see " Key to Ster.," page 147, line as Nos. 84 and 114. 3, or may be had by measurement. 125 — Hjrperboloid wedge or Except for diameter of middle central ungula. section, same as No. 91 or 95. For Similar solid to No. 95 of a cone area of zone, see " Key to Ster.," and representative of same objects, page 46 or table IX of same. 37 — 1526— TTngulaof h3rperboloidby Its base, a circle ; middle sect- a plane through edge of base ion, the segment of a circle ; For solid content, treat as pris- other base, a point. Plane lateral moid or by par. 185 of "Key." face, an ellipsis, its lateral surface Solid similar to No. 93 of cone, of double curvature, as all such or to No. 117 of paraboloid. figures are, not capable of [develop- ment, but reducible as required. 127 — Frustum of hyperboloid Bases same as in No. 116. La- ■wedge. teral area developes into trapezi- Similar to No. 116 of paraboloid, urns by lines parallel to bases. For Base of chimney stack, etc., resting areas of circles, segments, zones, partly on a sloped roof. see tables of " Key to Ster." 128 — Ungula of hyperboloid by Bases and section same as No. a plane through base. 118 of paraboloid. See table VIII, Similar to No. 118 of paraboloid, of " Key to Ster.," for areas of seg- ments. 129--Prustum of hyperboloid Same as No. 92. For area of -wedge, or of central ungula circles to eighths, tenths & twelfths of hyperboloid. see tables II, III, and IV of " Key Similar to No. 92 of cone. to Ster." For area of zone, see table IX, of same. Lateral surface de- composable into trapeziums. 130 — A compound solid : two equal firasta of cone or co- noid, base to base. Illustrative of a keg or cask, barrel, hogshead, etc., of any size or shape. Treat one-half of solid as Nos. 92, 112, 122, and double the result. See "Key to Ster.," fig. on page 155, for mode of measuring half- way diameter, when the half solid is not the frustum of a cone, but that of a conoid or of an ellipsoid or spheroid. When of a cone middle diameter equal to arithmetic mean of end diameters. — 38 — CLASS XIV. Sundry Solids. 131 — Three axed spheroid. See " Key to Ster.," page xxxix. May for measurement be supposed to lie or stand on either of its sides or apices. Kepresentative of a pebble, a bean, spindle, torpedoe, a shell fish, a flattened ellipsoid, etc., etc. AU its sections, ellipses ; all its parallel sections, similar ellipses. For areas of ellipses, " Ster.," page 51. Lateral area, see general for- mula, page 95, "Key to Ster." Or, as with the spheroid, suppose the surface divided as a melon is or orange into ungulae, terminating in apices or poles of the fig. - 132— An ovoid or solid of the shape of an egg. Divide into two or three sections and treat separately as conoid, segment of sphere or spheroid, and frustum of conoid. All parallel areas perpendicular to longer or fixed axis, circles, which find ready calculated for all sized diameters to eighths, tenths and twelfths of an inch, or other unity of measure, tables II., III., and IV., of Key to Ster. For late- ral area, see page 96 of same. 133^ Circular disc -with round- ed edge. Treat as a compound fiolid, to wit : a flat or low cylinder, and a ring semi-circular or seg- mental in section. Add the results. For cylinder, see No. 61. For ring compute area of section thereof as semi-circle or seg- ment, and multiply into circum- ference. For area, mean circum- ference of ring into circumference of section. 134— T-wisted prism. Portion of a circular stair rail, a twisted pillar or column, spiral oruameAt, etc. Its bases and sections similar and equal figures. The lateral surface of each face can be deve- loped in a plane, a trapezium o\^ rectangle. — 39 — 135— A compound solid. Two frusta of cones, their lesser basses joined. A windlass, spool, handle, shaft, axle-tree, etc. Treat half the solid as the frus- tum of a coue, and double the result, either for solid content or area of figure. 136— A compound solid. Two frusta of hollow cones Joined by their lesser bases. A windlass, spool, handle, shaft, axle-tree, etc. Treat one half the solid as frus- tum of cone No. 102, and double the result. Lateral area resolvable into con- tinuous trapeziums. 131— Compound solid. Two frusta of concave cones joined by their greater bases A windlass, shaft, axle-tree, etc. Treat half the solid, and double the result. For areas of circles, see tables II., III. and IV. of Ster. 138 - Compound solid. The segment or half of an elongated or prolate spindle, No. 151, and the segment or half of an oblate spindle. No. 141, or the segment of a sphere or spheroid, classes XVII, and XIX., a buoy, etc. Sections perpendicular to axis, circles ; Area resolvable into con- tinuous trapeziums, a circle and the sector of a circle. The circle at apex of segment of sphere or spheroid ; the sector at apex of spindle. See page 55 of " Key to Ster." 139— Compound solid like the • last with hollow cone in- stead of spindle. A finial or other ornament, a ciQ-de-lampe or pendant. Sections perpendicular to axis, circles. Lateral surface, conti- nuous trapeziums, a circle, and the sector of a circle at apex of cone. 140 — Compound solid : the Bases and sections, circles. - frustum of a sphere or sphe- Lateral surface resolvable into — 40 — roid and a hollow cone. continuous trapeziums. See A Moorish dome, a minaret, general formula, page 95 of " Key chimney of a coal oil lamp, a to Ster." . decanter, a vase, a pitcher. . , CLASS XV. Otlate or flattened Spindle, Frusta, Segments, Sundry. ' 141 — Oblate spindle, as t"wo Treat one half as segment of equal segments of sphere or sphere or spheroid, and double the spheroid base to base. result. See classes 17 and 19. A quoit, etc. ^ 142 — Semi-oblate spindle by a Treat its two halves together as plane parallel to fixed axis, one segment of sphere or spheroid. Floating caisson to entrance of See classes 17 and 19. dock, etc. 143 Middle frustum of oblate spindle. Fixed caisson or coffer-dam. Treat as prismoid. ' : ? .. The bases and middle section each a double segment of a circle or ellipsis, or tvro seg- ments thereof, base to base. Table VIIL, "Key to Ster." 144__Lateral frustum of oblate The bases and section half-way spindle, bet-ween planes pa- between them, double segments rallel to fixed axis. of circles or ellipses, for areas A flau-bottomed boat or other of which see table VIII., " Key to sailing vetjsei or a caisson, etc. - Ster.," and page 53 of same. — 41 — 145 — Lateral frustum of oblate spindle truncated at one end. A flat-bottomed boat or other sailing vessel. Bases and middle section, double segments, base to base, of circles or ellipses truncated at one end. For areas, see page 57 ^ " Key to Ster." 146-'Lateralfirustuin of oblate Bases, double segments of spindle truncated at both circles or ellipses truncated at ends. both ends. Divide into trapeziums ' A flat-bottomed boat or pontoon, and compute areas by page 57 a scow, lighter, etc. " Key to Ster." 147 — Quarter of an oblate sphe- roid, No. 181. The arched ceiling, roof or vault of the apsis of a church or half- groined ceiling of a circular apart- ment. On its lesser base, the head of a shallow niche in a wall, etc. Its base and middle section, semi-circles, if treated on its broader base ; if on its lesser face, its base and middle section, semi- ellipses. On whatever base it stands, treat as if on broader base, it being easier to compute circles than ellipses. 148 — A compound body, a cone, Treat separately as cone No. 81, and the segment of a sphere and as segment of sphere. No. 173, or spheroid. or of spheroid No. 182. A buoy, covered filter, etc 149— Elliptic ring, or may be called an eccentric ring. Treat as circular or cylindrical ring, taking for bases, its least, its greater, and its mean sections ; a: id for length the mean of the inner and outer circumferences. Compute half of solid as the la- teral frustum of a half-prolate spin- dle or the frustum of an elongated cone-. The solid may be conceived to be formed of the middle frustum of an elongated spindle bent till its ends meet. — 42 — 150 Compound solid : a cylin- der and the segment of a spare or speroid. A mortar, a tower with domed roof, a hall or room with groined ceiling, a hut, hive, hood. For area of sphere or spheroid, see page 95 " Key to Ster.,"or page 105, 110, 124, Ex. 3. Areas of cir. cles tables II., III. and IV. of same. Half-way diameter in segment of circle or sphere a mean proportio- nal between abscissae of diameter. CLASS XVI. Prolate or Elongated Spindle, Frusta, Segments, etc. 151 — Prolate spindle. Its sections perpendicular to axis, A shuttle, a torpedoe, a cigar, a circles. Decompose its lateral area sheath, case, etc. into continuous trapeziums and ' ... ' »■ " a sector. 152 — Semi-prolate spindle by a For solidity, compute planes per- plane through its greater or pendicular to fixed axis, as seg- fixed axis. ments of circles, semi-circles, A boat or saiHng vessel, a canoe, while the sections parallel thereto etc. are not so readily computed. 153 — Semi-prolate spindle by a For greater accuracy, divide into plane perpendicular to fixed a frustum and segment, compute axis. and add cubical contents. Areas of A hut, roof, filter or vessel of ca- bases, tables II., III. and IV, of pacity, a minaret or finial. " Key to Ster. " « 154 — Middle frustum of pro- See page 149 of " Key to Ster.," late spindle bet'ween planes and for lateral surface, page 95 of perpendicular to fixed axis. same. See page 155 of same. Bases A cask or keg, puncheon, hogs- and sections, circles, tables II., III. head, etc. ; see page 155 " Key." and IV. of Key to Ster." — 43 — 155 — Semi-middle frustum of Bases and middle section, semi- prolate spindle. circles, see page 160 of " Key to The liquid in a cask lying on its Ster. " Lateral surface decomposa- side, a boat with truncated ends. . ble into trapeziums. Compute as No. 154 and take half. 156 — Lateral frustum of pro- late spindle by planes paral- lel to fixed or longer axis A flat-bottomed boat or other sailing vessel. Treat as prismoid, the greater base, a double segment of a cir- cle. The other base and section, oval figures for areas of which see page 57 of " Key to Ster." 157 — Eccentric frustum of a prolate spindle by planes perpendicular to fixed or larger axis of solid. The shaft of a Eoman column. Compute each frustum from centre and add the results. Its bases and • sections, circles, for areas of which to eighths, tenths and twelfths of inch or other unit of measure, see tables II., III. and IV., " Key to Ster. " Its lateral surface decomposable into continuous trapeziums, or nearly equal to length of side into mean circumference. 158— Middle frustum of elon- gated spindle by planes per- pendicular to fixed or longer axis. - :---'^''- The shaft of a windlass, a drum or pulley, a cigar, torpedoe, etc. Its bases and sections, circles, for areas of which see " Key to Ster.," page 38, or tables II., III. and IV. of same. Lateral area equal nearly length of curved side into mean of circum- ferences. 159 — A curved halfspindle or Base and sections circles or cone. ellipses of slight eccentricity. A horn, powder flask, tusk or Lateral area decomposable into con- tooth of an elephant, etc., a sup- tinuous trapeziums and sector porting bracket from face of wall, at apex. — 44— - :■ •:-■:-:- 180 — Frustum of a prolate spin- Base and sections parallel thereto, die bet'ween non parallel circles, base of ungula a circle ; . bases. middle base of ungula, a semi-cir- Decompose into a frustum cle ; apex of ungula or opposite ■with parallel bases and an un- base, a point ; lateral surface, con- gula by a plane through nearest tinuous trapeziums, and a fig. point of one of the bases. like h, page 57 " Key to Ster." CLASS XVII. Sphere, Segments, Frusta and Ungulae, etc. 161 — The sphere. A billiard or other playing ball, the ball of a vane or steeple, sphe- rical shot and shell, school spheres, lamp globe or well, component part of compound solid, etc. Solid con- tent mav be had hy 'computing one of the component ungulae and mul- tiplying into number thereof. The opposite bases, points ; the middle section, a circle. The area of surface admits of approximate development into a series of equal figures in the shape of the longitu- dinal section of a prolate spindle, or of double segments of a cir- cle, base to base. Surface equal to four great cir- cles or to four times that of a great circle. 162. — A hemisphere. A dome, arched celling, globe, shade, cover, hut.hive, etc. ; revers- ed : a bowl, cauldron, copper, vase, etc. r . Contents more easily computable as half of those of a whole sphere, where there is no intermediate dia- meter to calculate or measure. Its base, a circle ; opposite base, a point ; its middle section, a cir- cle, the half diameter of which equals the square root of the rec- tangle under the versed and su- versed sines or portions of the dia- meter of the sphere. The lateral area equal to two great circles of the sphere. — 45 — 163. — Segment of a sphere less than a hemisphere. Eepresentative of same objects as No. 162, cover or bottom of a boiler. Solid contents also equal to one of the component ungulae into the number thereof. Base and section, circles ; other base, a point ; radius of middle section for area thereof, equal to root of rectangle of parts into which it divides the diameter of the sphere of which the segment forms part. For later\l area see " KeytoSter.," page 110, or General Formula, page 95. j- •;> 164. — Segment of sphere, great- Its base and section circles ; er than a hemisphere. other base a point ; radius of middle Eepresentative of same as No. section the root of rectangle of 162, and of a Moorish or Turkish parts into which it divides diameter or horse-shoe dome. of sphere. Lateral area, see " Key- to Ster.," pagee 117 and 123. 165. — Middle frustum of a Bases, f'qaal circles ; middle sphere. sections, a circle ; see tables of Base, capital or middle section of areas of circles to eighths, tenths, a colunm or post, a puncheon, hogs- and twelfths of an inch or other head, -i usher, roller, lamp shade, unity of measure, II., III., and IV. etc., etc. of " Key to Ster. " 166. — Lateral frustum of sphere. Base or capital of column, coved ceihng, cauldron, dish, soup plate, saucer, etc. Eadii of bases and sec- tions proportional to square roots of rectangles of portions into which such radii or ordinates divide the diameter of which the solid forms a part. Bases and section, circles; lateral area resolvable into continuous trapeziums ; or lateral area may be had very nearly at one operation, if the frustum be low or Hat and that its lateral curvature be not considerable.- — 46 — 167. — Sherical "wedge or oen- Its base, a circle ; opposite base, tral ungula of a sphere by a ridge, or axis, or line ; middle planes from opposite edges section, the zone of a circle; its of base of hemisphere to plane faces, circles; and lateral meet in apex. area resolvable into trapeziums Component portion of a com- and triangles pound solid. 168. — Frustum of a spherical Base, a circle ; other base and wedge or central ungula middle section, zones of circles. bet"ween parallel planes. For areas of zones, see table IX.^ Component portion of compound " Key to Ster. " solid. 169 — Spherical pyramid, ob- tuse-angled and triangular. Illustrative of the tri-obtuse- angular spherical triangle, and of the fact that the sum of the angles of a spherical triangle, may reach to six right angles, when each of the component angles increases to 180°. Base, a spherical triangle having three obtuse angles \ apex or opposite base, a point ; middle section, a similar tri- obtuse angular spherical trian- gle, and whose area is equal to one-quarter that of base, its factors being halves of those of base, and i X i = i 170.— Frustum of sphere be- tween non-parallel bases. Elbow or connecting link between two portions of a rail or bead ; base of a vase or other ornament on a raking cornice. Decompose into frustum and un- gula of a sphere by a plane parallel to one of the bases and passing through nearest point of other base or more readily and exactly, com- pute whole sphere, and deduct seg- ment. — 47 — CLASS XVIII. Spherical Ungulae, Sectors, Pyramids and Frusta. 171 — Qnarter-sphere or rectan- gular ungula of a sphere. Domed roof to a semi-circular plan, vault of the apsis of a church, head of a niche, " Key to Ster., " page 117. On its base : one base, a semi- circle ; opposite base, a point ; middle section, the segment of a circle. On end : each of its oppo- site bases, points ; its middle sec- tion, the sector of a circle. Only Compute as a whole sphere, and one area to compute, and easier and divide by 4, or treat as an ungula. quickei than a segment. See opposite par. 172. — Acute>angled spherical ungula. Component portion of the ball of a vane or steeple ; natural section of an orange, or of a ribbed melon, section of a buoy, cauldron, etc., etc., elbow of two semi-cylindrical mouldings, etc., at an obtuse angle. Its opposite bases, points ; its middle section, the sector of a circle ; the spherical surface, the component of a hollow metallic or other sphere or spherical vessel, or of the covering for a racket or other playing ball, etc. For spherical area see " Key to Ster.," page 117. .. 173. — Obtuse- angled ungula of Opposite bases points ; middle a sphere. sections, the sector of a circle > Head of niche reaching into a its plane faces, semi-circles. Sphe- sloped ceiling; elbow of two half- rical area, page 117 "Key to Ster." beads at an acute angle, etc. — 48 — 174 -Spherical sector or cone, Its base, a spherical segment ; or, to avoid computing spherical the other base, a point ; middle - areas, may be treated as a com- section, a spherical segment con- pound body, a cone and the centric to the base and equal in area segment of a sphere.. one quarter of base ; its height equal A buoy, a finial or ornament, a to radius of sphere, its lateral face top, etc., a covered filter. For areas developed, the sector of a circle. of circles see tables II, III and IV, See " Key to Ster.," page 110. of « Key to Ster." 175 — Frustum of a spherical Its bases and middle section pa- sector bet"ween parallel rallel thereto, concentric and si- spherical bases. milar segments of spheres of Portion of a shell or bomb or corresponding radii. Its height hollow sphere. To avoid comput- the length of slant side. Solidity ing spherical areas, treat as frus- also equal to difference between turn of cone, adding greater and whole and partial spherical sectors, deducting lesser segment. . ,^ ,^,. 176 — Hexagonal spherical py- Its base, a regular six-sided ramid. spherical polygon; its middle Its base illustrative of a spherical section a figure similar to the last, polygon, page 127 of *' Key." and equal in area to one-quarter Component portion of a solid thereof; its opposite base, a pointj sphere or ball j keystone of a vault, the centre of the sphere of which it finial or other ornament ; decompo- forms part. For area of base, see sable for computation into six equal " Key to Ster.," page 127. For area triangular spherical pyramids, "Key of component spherical triangle of to Ster.," page 129. See rule for base, see page 123 of same. Its spherical areas at end of this pam- plane faces equal sectors cf a phlet. ^.-:^:;,:J;: ^..•>r-:^?^^---r..>':- circle. 177 prustiun of hexagonal Its bases and middle section, si- spherical pyramid between milar spherical polygons; factor parallel bases. of middle section, as in cone, an Keystone of vault. Component arithmetic mean between those of — 49 — portion of hollow sphere. Surfaces the bases. Its lateral faces, equal illustrative of similar spherical po- frusta of equal sectors '^f a cir- lygons. Height of solid equal slant cle, or ccncavo - convex trape- height of side. ziums. See rule at end of this work. 178 — Half- quarter or one- eighth of sphere or tri-rec- tangular spherical pyramid Termination or stop to chamfer on angle of wall or pillar. Compute whole sphere and di- vide by eight. Its base illustrative of the tri- rectangular spherical triangle, page 123 of "Key." May compute for solid contents as the half of an ungula where only one area is required, that of a sector of a circle. See rule at end of this work. 179 — Acute equilateral trian- gular spherical pyramid. Its base illustrative of the equi- lateral spherical triangle. Base and middle section similar equilateral spherical triangles, for areas of which, see " Key to Ster.," page 123, and rule at end of this work. 180 — Frustum of triangular Bases and middle section, simi- spherical pyramid. lar spherical triangles whose Illustrative in its bases of simi- areas are as the squares of the cor- lar spherical triangles. Keystone of responding radii ; or factors of a vault to a triangular plan. middle section, arithmetic means ■ between those of the opposite bases. '■■:':■--'/ - CLASS XIX. Otlate Spheroid, Frusta and Segments. 181 — Oblate spheroid. Treated perpendicularly to its Eepresentative, in a less exag- fixed axis, its opposite bases are gerated ratio of its diameters or axes, considered points, as in the sphere, of the Earth and planets which are a plane touching the sohd only in — 50 — jflattened at the poles or extremities a point ; its middle section, a of fixed axis and protuberant at the circle. If considered parallel to its eciuatoi'. An orange, lamp-shade, or fixed axis, its middle section, an ellipsis. For spheroidal surface or area, see N. 161. globe, or bowl. 182 — Semi-oblate spheroid by Base, a circle ; opposite base, a a plane perpendicular to its point ; middle section, a circle ; fixed or lesser axis. for diameter of which, if not from Elliptical celling, dome, cauldron, direct measurement, see " Key to basin, dish, vase, shade, globe, etc. Ster.," page 139, line 10 and page 140, line 20. 183 — Semi-oblate spheroid by Equal in area and solid contents a plane parallel to its fixed to No. 182 and of easier and quic- or lesser axis ker computation, if considered Dome or ceiling to an elliptic such, the factors being circles plan; glass globe or shade, dish instead of ellipses. As it stands, cover, hut, a trough, cauldron, etc. its base and middle section, simi- lar ellipses. ^ 184 — Segment of oblate sphe- roid, greater than half by a plane perpendicular to fixed axis, Turkish, Moorish or horse-shoe, dome or ceiling ; a cauldron or cop- per, etc. Its base and middle section, circles ; opposite base, point. Spheroidal surface continuous trapeziums and a circle at apex. For areas of circles, see tables II., III. and IV. of " Key to Ster." For factors of middle section, see No. 182. 185— Middle frustum or solid Opposite bases and middle sec- zone of an oblate spheroid be- tion, circles ; for areas of circles tween planes perpendicular to to eighths, tenths and twelfths of fixed or shorter axis. an inch or other unity, see tables Kepresentative of same as No. II., III. and IV. of " Key to Ster." 165. Spheroidal area, see page 95 of same. — 51 — 186 -Middle frustum or solid zone of oblate spheroid by planes parallel to fixed or lesser axis of solid. Its bases and middle section si- milar ellipses, for areas of which see page 51 of "Key to Ster/* Spheroidal area, page 95 of same. 187— Segment of oblate sphe- Its base, an ellipsis ; opposite roid less than half, by a base, a point ; middle section, an plane parallel to its fixed or ellipsis similar to base. For fac- lesser axis. tors of middle section, see No. 182. Kepresentative of same as as No. 183. ' '-.:,■-:.. 188— Lateral frustum of oblate Its opposite parallel bases and spheroid by planes parallel middle section, ellipses, for areas to fixed or shorter axis. of which see "Key to Ster." p. 61. Coved ceiling of elliptic plan ; re- Its spheroidal surface decompos- versed : a boat, a scow, a vessel of able into continuous trapeziums capacity, etc. of variable height. 189 — Halt or segment of oblate Its base and middle section, si- ST)her"id by a plane irclined milar ellipses ; its opposite base, o axis of solid a point ; its spheroidal surface tra- Liquid or fluid in a serai-sphe- peziums, with ellipsis at apex roidal vessel inclined from the ver- and a curvilinear triangle at tical. Finial on a pediment or base of shape similar to fig. h. page sloped surface. 57 of " Key to Ster.," or lateral ' area may be divided and computed ^ * as triangles. 190 — Frustum of oblate sphe- roid betxyeen non-parallel bases. Decompose into a frustum 'with peu-allel bases, and an ungula by a plane parallel to one base and drawn through nearest point of Bases and middle section of com- ponent frustum with parallel bases, ellipses ; base of ungula, an ellip- sis ; middle section of ungula the segment of an ellipsis ; its other base, a point. For factors of middle sections, — 62 ^ other base, or compute whole sphe- see " Key to Ster.," page 139, line 10 roid and deduct stgmeuts. and j^age 140, line 20, where AB : y, , CD:.\/Ao^B'.oMa7idGD: AB:: <>.-.-■. .■ r: . ■. '•■ VCo.oB: oM. .-,,:-■ iV/-'-^-- . CLASS XX. Prolate Spheroid, Frusta and Segments. 191 — Prolate spheroid Its middle section perpendicular Representative of a lemon, melon, to tixed or longer axis, a circle; cucumber, etc. ; a case, sheath, etc. its opposite end bases, points. The work of computation expe- Spheroidal surface, continuous dited by treating circles instead of trapezoids, or a series of double elhpses ; that is, areas perpendicular segments base to base as the instead of parallel to fixed axis. component ribs a of melon. May treat as plane segment with length of cord equal to semi-eUiptical sec- tion. 192 — Semi-prolate spheroid by For solid contents and spheroidal a plane parallel to fixed axis, surface, treat perpendicular to fixed Vaulted ceiling to elliptic plan ; axis, where factors are circles or reversed : a boat or other sailing semi-circles instead of ellipses. vessel, a cauldron or vessel of ca- For areas of circles, see tables II., pacity, etc., etc. III. and IV. of " Key to Ster." 193 —Semi-prolate spheroid by Base, a circle ; other base, a a plane perpendicular to point ; middle section, a circle. fixed axis. For radius of middle section, see A hive, hut, roof or dome to cir- formula given in No. 190, or at cular tower or apartment ; reversed : page 139, line 10, page 140, line a copper or boiler, 20 of " Key to Ster." Spheroidal area, see Xo. 191. — 53 — 194 — Segment of prolate sphe- Base and middle section, oiroles ; roid greater than half, by a its other base, an apex or point, plane perpendicular to fixed Its spheroidal surface resolvable axis. into continuous trapeziums and A hut, hive, dome, a cauldron or a circle at apex, copper, etc. __ — — .■ -^-" '■ • 195— Middle frustum or solid End bases, equal circles ; mid- zone of prolate spheroid by die section, a circle. Unlike the parallel planes perpendicu- middle frustum of a spindle, the lar to fixed axis. solid contents of this solid are ob- A cask, keg, barrel, puncheon, tained exactly by treating the whole hogshead, etc., "Key." page 138. figure at once. 196 — Middle frustum or solid Opposite bases and middle sec- zone of prolate spheroid by tiou, similar ellipses. Spheroidal parallel planes oblique to surface, trapeziums of which take axis. ^^ mean height. A boss on raking strut, etc. i 197 — Lateral frustum or solid Bases and section, circles, for zone of prolate spheroid by areas of which see tables II., III. planes perpendicular to fixed and IV. " Key to Ster." For dia- axis. meter of middle section, measure Coved ceiling, base of column, solid or compute by formula of etc. ; reversed: capital of column, page 139, line 10; page 140. line dish, basin, bowl, tub, hamper or 20, where it is shown that the rect- basket,, stew pan, cauldron or other angle under the required radius, vess"..;! of capacity, etc., etc. and either axis of the spheroid, is f equal to that under the square root of the rectangle or product of the abscissae of the first axis and the ".'"p. ^ other axis. — 54 — 188— Lateral firustuxn or solid Its parallel bases and middle zone of prolate spheroid by section, similar ellipses ; for areas planes parallel to eaoh other, of which see " Key to Ster." page and to longer or fixed axis. 51. Its lateral area resolvable into Coved ceiling of elliptical plan, continuous trapeziums of vary- etc. ; reversed : a flat-bottomed boat, ing height if parallel to bases, but a sc^w, a dish, basket, etc., etc. of uniform height, if lines be drawn from extremities of fixed axis. 199— Segment of prolate sphe- Its base and middle section, sl- roid by a plane inclined to milar ellipses ; its other base, a axis. point ; its spheroidal surface re- Liquid in spheroidal vessel in- solvable by circles drawn from ex- clined from the vertical, a scoop, tremity of fixed axis, into a circle, scuttle, etc. trapeziums and a triangle. 200 — Frustum of prolate sphe- Decompose into frustum with roid bet"ween non-parallel paraUel bases, and an ungula. Com- planes. pute separately, and add ; or com- The one, perpendicular to fixed pute whole segment due to frustuia axis, the other oblique or inclined and deduct lesser segment. thereto. SPHERICAL TRIANGLES & POLYGONS TO ANY EADIUS OE DIAMETEK. Head before the mathematical, physical and chemical section of the Royal Society of Canada, May 22nd 1883. Last year I laid before this section of the Eoyal 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 while he who had gone through a course of mathe- matics would, in three months thereafter or out of college, have complete- ly forgotten or have inextricably mixed up in his mind the numeroua 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. — 56 — What I then did for the measurement 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- tom or roof of a gasometer, boiler, or of one of the constituent sections thereof, descending even to the surface of the ball of a spire, a shell, a cannon or a billard ball. ■ - ■ TO THIS END : -■ [ - The area of a sphere to diameter I. being =3.141,592,653,580,793+ Dividing by,', wc get that of the hemisphere =l,570,71)(),3-2(5,794,896,5 This divided by 4=area of tri-rectgrr sph. triangle =0,3'J:>,699,0dl,698,724,l ' • -;-90=area of L" or of bi-rect. sph. tri. with sp. ex=lo =0,004,36:5,323,129,985,8 H-60= " of r or of « " '• " 1' =0,000,072,722,052,166,43 r-60= " of 1" or of " " « « 1" =0,000,001,212,034,202,77 -elO= " of0.1"orof " « « « 0.1" =0,0it0,000,121,203,420,277 -1-10=" of 0.01" or of" « « « 0.01" =0,000,000,012,120,342,027,7 -i-10= " of 0.001" or of " " " « O.OOr =0,000,000,001,212,034,202,77 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° = 0.004,363,323,129,985,8 X 3 = 0.013,089,969,389,955 «« 1' = 0.000,072,722,052,166,43 X 4 = 0.000,-290,888,208,664 « 1" = 0.000,001,212,034,202 X 2 =0.000,002,424,068,404 « 0.1" = 0. 000,000, I2l,20,i,420 X 2 = 0.000,000,242,406,840 « 0.01" = 0.000,000,0 12,120,342 X 3 =0.000,000,036,361,026 « 0.001" = 0.000,000,001,212,034 X 5 =0.000,000,006,060,170 Area required 0.013,383,566,495,059 — 57- The answer is of course in square units or fractions of a sqnnre unit of the same name with the diameter. That is, if the dianu'ter is an inch, the area is the fraction of a square inch ; if a mile, the franction of a square mile, and so on. Remark. — 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 otlier 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 sufdce to compute for degrees only. Whence the following RULE TO DEGREES ONLY. 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. KULE 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 Eequired area = 471.194,000 . -.- V. ,\y- - _58— ''■::<_. -'r -'--.■ A result correct to units. If now greater accuracy be required, it is be obtained by taking in more decimals ; thus,say area 1°= 0.004,363,323 120 • 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 0.785,398,140 400 314.159,256,000 EXAMPLE III. 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 given 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 Remark. — 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., coiTesponding 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 — 59 — 75.87 etc., with the decimal point shifted to ihe left, mi^ olace for every place of decimals in the given fraction of such second : the tenth of a second gi\dng 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" = 7o8.7 square miles, 100" = 7587. etc., square miles, or 1 =^ 75.87 X bU (num- ber of seconds in a minute), 1°= 75.87 x 60 x 60 (number of seconds in a degree). RULE. To compute the area of any sph'^rioal "olygon. Divide the polygon into triauglis, cmupuic . a.h triangle separaliely by the foregoing rules for triangles and add the results. OR, From the sum of all the interior angles of the polygon subtract as many times two right angles as then- are siiles 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 uugula 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 the OBculatory circle is of different radius ; these areas being as the squares of the respective radii or diameters. y fe. •»' ('i^i-:-^,'f- ■■ ON THE APPLICATION OF THE PRISMOIDAL FORMULA TO THE MEASUEEMENT OF ALL SOLIDS By CHS. BAILLAIRG^, M. A., Member of the Society for the Generalization of Education in France, and of several learned a.d sc.entiflc Societies, Cheval.er of the Order of St. Sauveur de Monte-Keaie Italv • &c Kecepient of 13 medala of honor and IT diplomas and letters from Uuasia France Italy, Belgium. Japan, &c. Member of the Koyal Society of Canada. Kead before the mathematical section of the Society on Saturday the 28th of May. ' « Cette formule V=^(B + B' + 4M) (Says « the late Eevd, N « Maingui of the Laval University) que Mr. BaiUarg^ travaille k « vulgariser, a rimmense avantage de pouvoir remplacer toutes les " autres formules de stereom^trie," The prismoidal formula reads thus: « To the sum of the opposite and parallel end areas of a prismoid, add four times tJie middle area and multiply the whole into one sixth the length, or height of the solid." •■ •' Tlie following letter from the Minister of Education, Eussia, may be considered interesting in its bearings on the subject matter of this communication MINISTEKE DE L'INSTRUCTION PUBLIQUE. Saint-Petersburg, le U f^vrier 1877. No. 1823. A M. BAILLAIEGfe, Architecte h Quebec, Monsieur, Le comit^ scientifique du minist^re de I'lnstruction Putlique, (de Eussie,) reconnaissant I'incontestable utilite de votre " Tableau Ster^o- m^trique " pour I'enseignement de la g^om^trie en general de meme que pour son application pratique k d'autres sciences, eprouve un plaisir tout particulier k joindre aux suffrages des savants de I'Europe et de I'Am^- rique sa complete approbation, en vous informant que le susdit tableau, avec toutes ses applications, sera recommande aux dcoles primaires et moyennes, pour en completer les cabinets et les collections mathema- tiques, et inscrit dans les catalogues des ouvrages approuv^s par le minist^re de I'lnstruction Publique. .. i^ .' "^ Agr^ez, monsieur, I'assurance de ma haute consideration. .. . v., -- Le chef du d^partement au miniature de I'lnstruction Publique, E. DE Bradkek. — 63 — 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, Eussia, 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 Russian Empire." Should the Royal 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 soUd having parallel end areas bounded by parallel sides. This parallelism of the sides or edges of the opposite bases or end areas does not imply, not does it exclude any proportionality between such sides or edges. . ■ . ■ : r ^ _ ; , _ ^;;/5 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. 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. ;, ..^ , , , ,..., ; >„ ^v It need hardly be said that the prism and cylinder are prismoids, whose opposite edges are equal as well as parallel in the same way as for the frusta of the pyramid and cone the opposite edges are propor- tional while parallel. — 64 — Kow, nine tentlis 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, in the brewers vat, the salting tub, the butter firkin, the commom 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. 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 leam, 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. r 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 ; a long and tedious operation, one known only to the few, most difficult to retain, forgotten as soon as learnt and therefore useless. , ,. - , , ., u With the formula proposed ou 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 hues 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. Now, of almost all the solide herein above alluded to, the opposite — 65 — bases and middle section are circles and the operation can be further expedited by taking the areas ready made, to inches and even hnes or less, from tables prepared for the purpose. The labour then reduces to the mere arithmetic of adding the areas 80 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. 5. •'-.;• v.? Vj I would have but one formula applicable 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 this 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 to the result. ^ • ^ . : -/,: vi-^ 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 you have had time to do so, the reasoning process is going on within your 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 ;,-■■;-■; , , —66— :.':-■,' 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 case, 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 the same thing as six time 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 sphere, 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 diameter 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 planes passing in any direction through the centre, to any frustum 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 inclined, as well to any parallel frustum of eituer. -67 — This 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. Maingui, professor of Mathematics at the Laval University, aS well by the Revd. M. Billion, of the Seminary of St. Sulpice— Montreal j by His Grace, bishop Langevin of Rimouski, and by many other ma- thematicians fully adequate to the task. M. Maingui says (page IX of his pamphlet and as already quoted from the french version) : " This formula "" -r- 'Ms that " which Mr. Baillairge is endeavouring to introduce ; it has the im- " mense advantage of replacing all other stereometrical formulae." This is the only formula which will 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 wherewith to charge his memory, remember it all his life and be ever ready to apply it? ' - • ■ In the case of spindles and the masurement of their middle frusta — the representatives of casks of all varieties and sizes, — the prismoidal foimula 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 lengtly, too abstruse, and the wine merchant will teU 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 having symmetrical and equal halves — the half way diameter between — 68 — the head and hung, the very element by which the cask varies ita capa- city, enters as a factor into the occupation, while the guaging rod can take no note of it. .1*. It remains but to say that in the case of hoofi j,nd 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 lules applicable to these ungulae can never be remembered, nor are or will they ever be applied in practice. Hather 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. „ .^- >5- ? Compound bodies must of course be treated separately or in parts. Thus, a gun oi mortar, as made up of a cylinder or the frustum of a cone and the segment or half of a sphere or spheroid ; a morish or tur- kish dome, as the frustum of a spheroid surmounted by a hollow cone ; a roofed tower, as a cone and cyhnder, a cone and frustum of a cone or two conic frusta as the case may be and 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 angula, subject to the percentage of error already noticed in the volume of the angula ; while, by cubing the whole conoid on 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. \u--: ^:<'6 "'^ m-v- ^/':-:K^'j inVi,- -^Ji re nu ^.-:^h _ There are a class of solid forms where it would appear at first sight that a departure from the prismoidal formula becomes necessary • 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 intrades 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, spheroidal or — 69 — cylindrical surfaces of the opposite bases, and the equally curved surface of the middle section ; while, when the faces are not parrallel or the thickness of varying dimensions, as well when the faces are everywhere aquidistant, 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 sj)herical 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; when any such area can in a few minutes be made up the mere multiplication and adilition of the elemental quantities given in the text, and any portion of the earths snrfiue 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 and 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 obsorbent or of sand or sawdust etc., when the contiary. Again may the specific gravities of bodies be applied, or their weights to making out their, volumes by sim[)le rule of (hree, or the reverse process of weighing them by ratio when their volumes are ascertained. Finally the quantities anil respective weights of the separate subs- tances which enter into amalgams or alloys are obtainable as taught by a comparison of their wei«;hts in air and water, th.it is of the amalgam itself and of its unalloyed constituents. The whole field of solid meusuration is thus gone over in these few pages, instead of the volume required to contain the niiny separate and varied formulae which the old process of com[)Utation gives rise to and renders indispensable. The whole I say is gone over in as many minutes as the oil process requires hours or even days. 'S-' I TABLES OF I. Squares and Square Eoots of numbers from 1 to 1600. II. Circumferences and areas of circles of diameter bV to 150 advancing by ^. III. Circumferences and areas of circles of diameter iV to 100 ' advancing by tV. • IV. Circumferences and areas of circles of diameter 1 to 50 feet, advancing by 1 inch. v. Sides of Squares equal in area to a circle of a diameter 1 to 100 advancing by a ^. Vv ,; . VI. Lengths of circular arcs, to diameter 1 diviled into 1000 equal parts. VII. Lengths of semi-elliptic arcs to transverse diameter 1 divided into 1000 equal parts. VIII. Areas of the segments of a circle to diameter 1 divided into 1000 equal parts. IX. Areas of the zones of a circle to a diameter 1 divided into 1000 equal parts. X. Specific gravities or weights of bodies of all kinds solid, fluid, liquid and gazeous. TABl,E OF SQUARES, SQUARE ROOTS No. Square. Sqre. root. No. Square. Sqre. root. No. Square. Sqre. root. 1 1 1.0000000 61 3721 7.8102497 121 14641 11.0000000 2 4 1.4142136 62 3844 7.8740079 122 14834 11.0453610 :i 9 1.7320508 ()3 3969 7.9372539 123 15129 11.0905365 4 16 2.0000000 64 4096 8.0000000 12^. 15376 11.1355287 5 25 2.2360680 65 4225 8.0622577 125 15625 11.1803399 6 36 2.4494897 66 4356 8.1240384 126 15876 11.2249722 7 49 2.6457513 67 4488 8.1853528 127 16129 11.2694277 8 61 2 w.'-'»->71 68 4624 82162113 128 16384 11.3137085 ■J f^l ;;.uuO(.ouo 69 4761 8.3066239 129 16641 11.3578167 10 100 3.16-J27:7 70 4900 8.3666003 130 KiOOO 11.4017543 11 121 3.3166248 71 5041 8.4261498 131 17161 11.4455231 12 1-14 3. 4641016 72 5184 8.4852814 132 17424 11.4891253 i:{ 169 3.60.">5513 73 5329 8. .5440037 133 17689 11.5325626 4 196 3.7 416574 74 5476 8.6023253 134 17956 11.5758369 15 225 3.8-.'29'-'33 75 5625 8.6602540 135 18225 11.6189500 16 256 4 OUOOOOO 76 .5776 8.7177979 136 18496 11.6619038 17 2H9 4.1-J3105() 77 5929 8.7749644 137 18769 11.7046999 18 ■ 324 4.2426407 78 6084 8.83i7609 138 19044 11.7473401 19 361 4.35.-'.')989 79 6241 8.8881944 139 19321 11.7898261 20 400 4.47-.'1360 80 6400 8.9442719 140 19600 11.8321596 21 441 4..".>25757 81 6561 9.0000000 141 19881 11.8743421 22 484 4.69041.')8 82 6724 9.0553851 142 20164 11.9163753 23 529 4.795>315 83 6889 9.1104336 143 20349 11.9582607 24 576 4>it8979.') 84 70.56 9.1651514 144 20736 12.0000000 25 625 5.0(100000 85 7225 9.2195445 145 21025 12.0415946 26 676 5.0990195 86 73yt) 9.2736185 146 21316 12.0830460 27 729 5.1961524 87 7569 .3273791 147 21(;09 12.1243557 28 784 5.-i9l5026 fiS 7744 .) 3.-'08315 148 21904 12.1655251 29 841 5.3r'51618 89 7921 9.4339811 149 22201 12.2065556 30 900 5.47722.->6 90 8100 9.4868330 150 22500 12.2474487 31 961 5.5677(544 31 8281 9.5393920 151 22801 12.2882057 32 1024 5.6.".68542 92 8464 9.5916634 152 23104 12.3288280 33 1089 5.744r.626 93 8649 9.6436.508 153 23409 12.3693169 34 11 ->6 5.S3()9519 94 8836 9.6953597 154 23716 12.4096736 :i5 1225 5.9160798 95 9025 9.7467943 155 24025 12.4498996 36 129«; 6.0()0()()00 96 92 It) 9.7979.590 156 24336 12.4899960 37 1369 6 0827 6-^5 97 9409 9.8488.578 157 24649 12.5299i;;i 38 1444 6.1644140 98 9604 9.8994949 158 24964 12.5698051 39 1521 6.2449980 99 9H01 9.949x744 159 2.5281 12.6095202 40 1600 6 3-J45553 100 10000 10.0000000 160 2.5()00 12.6491106 41 1681 6.4031242 101 10201 10.0498756 161 2.5921 12.688.5775 42 1764 6.4807407 102 10404 10.0995049 162 26244 12.727i»221 43 1849 6.5.')74385 103 10609 10.14-i8916 163 26.569 12.7671453 44 1936 6.6332496 104 10816 10 1980390 164 26896 12.8062485 45 2025 6.7082039 105 11025 10.2469508 165 27225 12.84.52326 46 2116 5.78-i3300 106 11236 10.2956301 166 27556 12.8840987 47 2209 6.8556546 107 11449 10.3440804 167 27889 12.9228480 48 2304 6.9282032 108 11664 10.3923048 168 28224 12.9614814 49 2401 7.0O()(MJ00 109 11881 10.4403065 169 28561 13.0000000 50 2500 7.07l0ti7S 110 12100 10.4880885 170 28900 13.0384048 51 2601 7.1414284 111 1232 1 10.53.56.538 171 29241 13.0766968 52 2704 7.2illO-J6 112 12544 10,5.«'300.52 172 29.5S4 13.1118770 53 2809 7.280 1U99 113 12769 10.()301458 173 29929 13.1529464 54 2916 7.34Mti9-.' 114 12996 10.6770783 174 :{()-.»76 13.1909000 55 3025 7.416U»8.-> 115 13225 10.7238053 175 30625 13.2287566 5(i 3136 7.4-'3:;i4-^ 116 13156 10.7703->96 176 30976 13.2664992 57 3249 7.549H344 117 136-9 l(t.Hl66538 177 31329 13 30 41 3^ < 58 3364 7.6157731 118 13924 10.8627805 178 31684 13.3116641 59 3481 7.681i457 119 14161 10.9087121 179 3J011 13.37'.'0>^-^2 60 3600 y7. 7459667 120 14400 10.9544512 180 32400 13,4164079 1 OF NUMBERS FROM 1 TO 1600. 1 No. Square. Sqre. root. No. Square. Sqre. root. No. Square. Sqre. root. 1 isi 327(;i 13.45:56240 -241 58081 15.5-241747 301 90001 17.349.3516 i 1^2 33124 13.4907370 -242 .58504 15.556:5492 ■M)2 91204 17.3781472 1^:^ 331r'9 13.5277493 '243 5<)049 15.58-^4573 ;io:5 91809 17.40689.52 lrt4 33^56 13. .".64 6600 244 595:56 15.0-204994 304 92410 17.4:5.55958 1*') 34225 1:5.6014705 245 60025 15.(5524758 305 93025 17.4642492 isi; 345.-0 13.6:581817 246 6051(5 15.(584:5*71 :506 9:56:50 17.49285.57 lb7 34!)()9 13.6747943 247 OI1109 15.710-2:5:56 307 94249 17..5214155 LS8 :{5344 1:3.711:5092 248 61504 15.7480157 308 94,-(54 17.5499^288 lyy 35721 l:i.7477271 '249 62,)01 15.7797:338 309 9.5481 17.5783958 IIH) 3(;i00 13.7840488 250 02500 15.811:58*3 310 90100 17.60(581(59 191 364H1 1:5.8-202750 •251 63001 15 8429795 311 90721 17.6:551921 11(2 3t;-fj4 13.8504065 2.52 6:5504 15.8745079 312 97344 17.66:55217 li);} :!7216 13.^9-244(10 •25:5 04009 15.90597:57 313 9~969 17.(5918060 li)4 376: 56 13.9-28:58*3 •254 (54510 15.9:57:5775 314 9859(5 17.7200451 IDa :!S025 13.961-2400 255 05025 15.9(587194 315 99225 17.748-2:593 11)6 :'>84I6 14.0000000 •25(> 055:50 16.0000000 310 99850 17.7763888 I'JT 36.-S()9 14.0:556088 257 6«i(J49 16.031-J195 317 100489 17.80449:38 hi6 :59-J04 14.071-2473 258 6(5564 16.002:5784 318 1011-24 17.8:5-25545 i;)L> :596Ul 14.1067:560 259 07081 16.0934709 319 101701 17.8605711 'iOO 40000 14.1421:350 200 67(500 10.1-245155 320 10-2400 17.88854:58 '^(ti 40401 14.1774409 •201 08121 16.1554944 321 lo:S041 17.91047-29 •2in 40H{i4 14.2126704 202 68044 16.1,-64141 :522 10:5084 17.944:5.584 w.i 41209 14.-2478068 203 69169 16.217-2747 323 104:5-29 17.97-22008 •J04 41616 14.'282s569 264 69696 10.2480708 324 104970 18.0000000 ViU.') 42025 14.:!178211 205 70225 16.27H.-200 ■25 105(;-25 18.0277504 aot; 424:56 14.3527001 200 70756 ie.:5095004 .520 106270 18.05.54701 •J07 42849 14.:5874946 267 7P289 10.3401:546 327 1069-29 18.0831413 •iUH 43264 14.4-2-2-2051 2m 718-24 10.:5707055 328 107584 18.1107703 i>09 4:5681 l4.4.-.08:3-23 209 72:561 16.4012195 329 108241 18.138:5571 210 44100 14.491:5767 270 7-2900 16.431(57(57 :330 108900 18.1059021 L>11 44521 14.525H390 '271 73441 10.462(t770 331 1095(51 18.1934054 •Jl-i 44944 14..5602198 '272 73984 10.4924-2-25 3:52 110-224 18.-2-208672 •jia 45:!69 14.5945195 27:5 745-J9 16.5-J-27116 333 110*89 18.'248'2876 •iI4 45796 14.0287:588 274 •75076 l6.55-2iM54 334 111.5.56 18.-2750069 ■>ir) 46225 14.0(528783 275 75(i-25 16.5831-240 335 12-225 18.30:500.52 . 2i6 46650 14.69(59385 276 76176 10.61:52477 :S30 112896 18.3:503028 ■J 17 47089 14.7:50i»199 277 76729 10.0433170 3:57 li:55f>9 18.3.575598 t>ls 47524 14.7(548-2:51 •278 77284 lti.(5783:5-20 3:58 114-244 18:5847763 •21!) 47961 14.79804.-*0 279 77841 16.7U:!'293l 3:59 114921 18.41195'26 2-JO 484(»0 14.8:523970 280 78400 16.73:;2005 340 11.5(500 18.4:5908*9 •221 4.^841 14..-()()0(587 '281 78961 10.7(5:50.540 341 116281 18.46618.53 222 492-4 14. -'99(5(544 282 79524 10.792S."..5(i :542 1 16964 18.49:52420 22:{ 49729 14.9:531845 28:5 80089 16.*-2-ji;038 313 117(i49 18 .5-202592 224 50176 14.9(560295 284 80t ;.-.(> 16.8.V2-2995 344 118:5:50 18.5472:570 225 50625 15.0000000 285 Hl-2-25 1(5.881 94 :;o 345 1190-25 18.5741756 •22(5 51076 15.03:5-2904 286 81796 10.9115:545. 346 119716 18.6010752 •>27 5l5-i'.» 15.0(«)5192 287 8'2:561> 16 9410743 347 1-20409 18.6279:560 tss 519-4 15.0990(589 288 8-2044 10.97056-27 :548 121104 18.6547.581 22!» 5-2441 15.1:5-27400 289 8:5521 17.0000000 :549 121801 18.681,5417 2:w 52900 15.16.57509 290 84100 17.0-29:;-64 350 122500 18.708-2869 •->:5l 5:5361 15. 19^(1842 291 84(5^^ I 17.05*7-2--'l 351 12:3201 18.7:549940 1 2:!2 5:5824 1 5.-23 154()2 292 85-2()4 17.0-80075 352 12:5i»04 I8.761(i630 i '2:53 54289 15.264:5375 •29:5 85819 17.117'24-28 :553 124009 18.7882912 2:u 54750 15.-ii>70.'>H5 294 861:50 17.1464282 .354 1-2: .3 16 18.8148877 ! 2:55 552-25 15.:5-'97097 295 h7025 17.175.-m;40 3.-.5 126()-25 18.83144:57 2m 55(596 15. :5. ■.•2-29 15 'HiM 87610 17.J01().505 ;556 1267:56 18."^t;79(;-23 237 56169 ]5.:594H043 "297 88209 17.2:5:;()879 :557 127449 18.89444:56 2:58 5<)().l4 15.427-2486 '298 88804 17.'2<52i;765 :558 1'28H54 1 8.9*208879 239 57121 15.4596248 '21.»9 89401 17.'>916165 :559 1288«1 18.947-29.53 240 57000 15.4919:534 300 90000 17.:5-205081 :500 1-29000 18.97:50660 TABLE OF SQUAllES, SQUARE ROOTS No. Square. Sqre. root. No. Square. Sqre. root. No. Square. Sqre. root. 361 l:;03v!l 19.0000000 4-21 177-241 20.51 --2-45 481 231:^.61 21.9318122 j :',u'2 131lt44 19.0-26-2976 4-2-2 17.--0-4 20.54-J(5:5-6 4.82 2:5-2:1-24 21.9544984 j 363 131769 19.05-255.-9 42:5 1 7.-^11-2.-' 20.5()()96:5S 48:5 2:5:;2-9 21.977-2610 364 13-2496 19.0787.- !0 4-24 179776 20.. 591-2(503 4-4 2342.->6 22.0000000 1 3()5 13:!-2Jr) 19.10497;'.-2 4-25 l-i)i;-25 •>o.(;i.">5-28l 1 485 2:5..-j25 •22. 0^227 155 ! 36() 133966 19.1311-265 4-26 ].-!1476 20.6:597674 4S6 2:!()llt6 22 0454077 1 :5t)7 134689 19.157-2441 4-27 182:5-29 20.6(5:597H3 4.«^7 2:57 1(59 -2'2. 06-0765 368 13;") 1'24 19.1833-261 4-28 183184 -20.6.8.-: 1609 ! 488 23;- 144 •22.0907-220 369 13t)I6l 19.-20937-27 4-29 1.-4011 •20.71-2:5152 1 4S9 2:39121 •22.113:3444 i 370 13()900 19.-235:1^4 1 4:50 18l'.t(iO '20.7:564414 i 490 240100 22.13594:56 1 371 1376U 19.-26i:!60:{ 431 185761 20.7605:595 i 491 2410S1 •22. 158'! 98 i 37-2 138384 19.-287:50l5 432 l,866-24 20.7.-46097 492 24-20(54 •22.18107:30 37:5 1391-29 19.3i:!-2079 433 1-71-9 20>i"^0.-6."-20 493 24:5049 •22.20:360:53 374 139.-'76 19. 3:5907 9() 424 18h:;.")() -20..^:!-26(ii)7 494 2440:56 •22. •2*26 1108 37r> 14 (I*;-.*;") I9.:i649167 435 1-9-2-25 20.8.')6(>5:!6 495 2450-25 •22.248.')955 3ro 1 11376 19.:}907191 4:^6 190( )'.)() •20..-84.M:5(t 496 246016 22.-2710575 377 14-21-9 19.41(51.878 4:57 19(1969 •jO.904.^450 497 247009 •J2. -29:54968 37 -i 14-2-81 19.14-2-2-2-21 43S 191 >M4 20.9-281195 498 -24.-'004 22.31.591:36 379 143()41 19.46792-2;', 4:;9 19-2721 20.9523-268 499 249001 22.:538:5079 380 144100 li>.4.-':!."..-'-7 440 19:5()(M) •20.9765770 500 250000 22.3606798 3«l 145161 19.. M 9-2-21:! 441 1941.-1 21.0000000 501 251001 22.:!830'293 38i 14.59J4 19.54 48-J()3 44-2 195:!61 2 1.02: '.79(50 502 252004 22. 405:5;" 65 383 14r,()89 19.570:!8.").-< 44:5 196249 21.0475().V2 503 •25:5009 •22.4276615 384 1474..6 19.59.59179 i 444 197i:;(5 21.071:5075 504 254016 •22.4499443 :;8.") M^-2-25 19. 6-2 14 169 445 1980-25 21.09.' 0-231 505 252025 22.4722051 386 14-^9'.M) 19.646-.8-27 446 l9.-^91t; 21.11S7I21 506 256036 22.49414:58 3-^7 I49:()9 l9.67-23l:i6 447 199809 21.14-2:5745 r,07 257049 •22.5166605 388 l.-)0514 I9.f)97''!56 448 20(1704 21.1(5(50 105 508 25^064 •22.5:38.8553 389 1.") 13-21 l9.7-2:50>-29 449 •201601 2l.H9(5-201 509 259041 •22..5t;iO'2,c!:5 390 1 5-2 100 19.74-1177 45(t •2(i-2.")00 21.21:5-20:54 510 2()0100 •22.;-).S:51796 i 391 1.V2881 19.77:57199 451 20:5401 2t.'2:5(57()06 511 2(51^21 '22.6053091 :'.9J ir)3()(i 1 19.79-9-99 452 204:504 21.-2(50-2916 512 2(52144 22.6-274170 393 151149 19.8-24-2-276 453 •20.V209 21. •28:579(57 51:5 2631(59 •22.649." 0:53 394 155-236 19.-194:::!2 454 •206116 2l.:50727.-)- 514 2(54196 •22.671.->6-!l 39."> l.")6.-, 19.87461 »(i9 4.55 2(»70-25 21.: 5:507-290 515 2(55225 •22.69:36114 396 15C)8I6 19.H997487 45(5 2079: !(5 2l.:5.">015()5 516 26(5256 •22.71563:14 i 397 157609 l<>.9-24-*588 457 20.-8 19 2l.:577.v>-:; 517 2(57289 •22.7376340 1 39-^ 15S4IU 19.9499:573 45 16483() '20.1494 117 4(56 2l71.Vi 2l.(!.-^70:53l 526 27(567(5 •22. 9:54 6'^; »9 ' 407 165649 -20. 174-24 lit 467 •jl.'^nS,* 2l.6l01.-i2'' 527 2777-29 •22.9.564.-106 ' 408 16tU(>4 •20.1990099 4(58 219021 2l.6:5:!:!07r 528 •27.-<7.-<4 22.97.^2506 409 1(;7-2H1 •20. •2^2374 ■'4 469 219961 2l.6.".t54(t78 529 '279841 23 0000000 410 168 KfO •20.-2181567 470 220901 1 21.6791^:54 .-):50 2c<09(!0 2:5.0217-2-^9 411 16-9-21 •2(l.-273i:;49 471 221.^11 21.70-2.">:514 531 2809(51 •23.0434:572 41i 1697 14 -20,'29778:51 47-2 •2227,-<4 21.7-25.')(510 5:52 •28:5021 •23.06.il-252 413 170.'.69 -20.:5-2-24O14 47:5 2-2:5721 » 2l.74.<)6:!2 .5:5:! 2*^ 10-19 23.'^-i6792.-< 414 17139(5 •20.3169899 474 22 1(576 21.7715111 n:;4 2^5156 •23.1084400 415 17-2-2-25 •20.371548-^ 475 225(525 21.7944947 5:}5 2.-'()22:) 2:5. 1:500(570 . 416 173056 '20.:596O781 47(5 2-.'6."»7(5 21.8174-242 :.'M •287296 •23.15167:58 417 173'J89 •2(1. 4^205779 477 227.-)29 2l..-'40:5'297 ; 5:57 2S.^;!6.> •23.17:5-2.;05 418 1747yt •20.44504.-'3 478 2-284"^ 1 21.86:52111 53r< 2-'9444 2:5.1948270 419 175561 •20.4694895 479 2-294 11 21.^^8()0().S) 1 5:59 •29(t52l •2:5.2i(;;57:;5 4-20 176400 •20.49:59015 480 '230400 2l.90890r23 j , 1 ; 540 j •291(500 •23.*2:57900l : I OF NUMBERS FUOM 1 TO 1600. 5 No. Square. Sqre. root. No. Square. Sqre. root. No. Square. Sqre. root. 541 292081 23.2.594007 001 361201 24.51.5:5013 601 4:;«5921 25.7099203 ol-i 29:57 ()4 23.280.>^935 002 :>02404 24.5356883 002 43-244 25.729:1607 543 294849 23.:502:!0U4 00:5 :5o:)009 24..55605.83 003 4:19.509 25.74.>'7864 544 2959:5(5 23.323.-07t) 604 :J0481() 24.,5704115 064 440.-i90 25.7681975 545 297025 23.:i452:551 605 36(5025 24. .5907478 605 442225 25.78759:19 546 298116 23.3000429 600 3072:50 24. '5170(573 00(5 443555 25.80097.58 547 299209 23.3880311 007 308449 24.0:57:5700 607 4448:59 25.82(53431 548 :500:504 23.409:1998 608 :509()04 24.(5.57(5560 (508 44(5224 25.84.50900 549 301401 23.4307490 009 370^81 24.0779254 (509 447.501 25.. -650:543 550 ."^02500 23.4520788 610 372100 24.0981781 670 44>900 25.884:1.582 551 3o;iooi 23.473:5892 611 37:5:121 24.7184142 671 450241 25.903(5(577 552 :!()4704 23.4940802 612 :!74544 24.7:580:W8 072 451.584 25.9229028 55:i :!05SU9 23.51 59.')20 613 :57.5709 24.7588:)68 67:5 452929 25.94224:15 554 :;oo9io 23.,5:S72040 014 37(5990 24.77902:54 (574 454-J70 25.9015100 555 :508O25 23.5584:180 015 378225 24.79919:55 675 455025 25.9>07<52l 550 :509i:50 23., 5790.522 610 37945(5 24.819:5473 070 4.50970 20.0000000 557 :510249 2:i.600>474 017 :5f-O089 24.8:594847 077 458:129 20.01922:57 558 311:504 2:5.62202:5(5 618 381924 24.8590058 078 459084 26.01584:531 559 312481 23.6431. -^08 019 383161 24.8797106 079 401041 20.0.57(52.-4 5(iU :5l:5(ioo 2:5.604:1191 020 :5844()0 24.8997992 680 402400 2(5.090^^096 501 314721 2:5.6854:580 021 385(541 24.919.-'710 6(^1 40:1701 26.07.597(57 50-J 315844 23.7005:592 622 386.-^84 24.9:599278 082 405124 2(5.1151297 5(>:i 310909 23.7270210 62:5 388129 24.9099679 08:5 4004.89 20.i:5426s7 .5(i4 313U9t) 23.7480-42 024 389370 24.97:;9920 084 4078.")6 2().15:5;5.>^:57 5()5 319225 23.7097280 625 390025 25.0000000 685 469225 26.17-.'.5047 5i;t) 320:5r)t) 23.7807545 ti2() '581870 25.0199920 (58(5 470.590 26.191001T 507 • :V21489 23.8117018 027 ;i93l29 25.0399681 087 471969 20.210(5848 508 :«2024 23.8:527500 628 :59i:!84 25.(1599282 088 47:!:54i 20.2-.:9754» 509 32:5704 23.8537209 029 395(541 25.0798724 689 474721 2(5.24.^8095 570 32 t9(H> 23.8740728 o:;o 390900 25.0998008 0'.)0 470100 •.().26T8511 571 :52(504 1 23.895000:5 ' 031 398101 25.11971:54 691 477 481 20.28687^9 57:> 327184 23.910.5215 6:!2 399424 25.1:;90102 692 47.^''t)l 20.305-929 57;i 328:529 23.9:574184 0:13 40(I(M» 25.1591913 693 480249 20. 152489:12 574 :529470 23.9.582971 0:54 4019.50 25.179:1.500 094 481():!6 20.34:1^797 575 3:i0025 23.9791576 635 40:i225 25.19920(53 095 48:5025 20.:502852» 570 :531770 24.0(100000 0:50 404490 25.2190404 096 484410 20.;5818119 577 3:52929 24.020.^243 6:57 405709 25.2:5-8589 097 485809 20.4(.f07.570 578 :534084 24.041(i:500 6:58 407044 25.25-0019 098 487204 20.419()890 ry79 3:5-.241 24.0024188 639 408:521 25,278449:5 099 4.8.>^(501 20.43-0081 580 3:50400 24.0831891 040 409(i00 25.29.-i2213 7(10 490000 20.4.575131 5H1 3:575()1 24.10:59410 641 410881 25.3179778 701 491401 20.4704046 58v; 338724 24.1240702 642 4121(54 25.:5:577189 702 4t>2r04 2(5.4952^20 58:5 3:59889 24.14.5:5929 643 4i;il49 25.:5574447 703 494209 20.5141472 584 :54i(;56 24.1000919 644 414;:5() 25.:577I551 704 49.iOtO 20..-.:i2998:5 585 342225 24.18077:12 64;) 410025 25.:!9()-5{t2 705 497(t25 20.5518:161 59(i 34:5390 24.2074:5(i9 040 417310 25.4i(;.5:50l 706 4984:50 20. .570(5005 5?7 344509 24.2280829 647 418009 25.4:101947 707 499819 20.5894716 588 1545744 24.2487113 048 419904 25.4.55-441 708 501264 20.(5082094 589 34t)921 24.269:5222 649 421201 25:4754784 709 .502081 20.02705:59 590 318100 24.2899150 650 422500 25.49.'>:!281 59;{ :i51049 24:5515913 <;53 420409 25.5.5:58047 713 5081509 26.7020598 .594 352s:;!j 24:5721152 654 427710 25.. 57:542:57 714 5097iM5 2().72077.-<4 595 354025 24.:5926218 655 429025 25.5929078 715 51 1225 •' ..7:-5i?4H:l9 596 35.5210 24.4131112 050 4:5o:!:50 25.0.524909 710 5120.5- Jl. 758 1 7(53 597 350409 24.43:5.58:14 057 4:51049 25.0320112 717 51 4089 26.77(58.557 598 357004 24.4.5403.><5 058 432901 25.(5515107 718 515.5-J4 26.7955220 599 :558801 24.47447(55 059 434281 25.0709953 719 51(5961 20.8141754 600 360000 24.4948974 600 435600 25.6904652 720 518400 26.8:528159' ■ TABLE OF SQUARES, SQUARE ROOTS No. 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 7()2 763 761 765 7()() 767 768 769 770 771 772 773 //4 775 77(5 777 778 779 780 Square. 519841 521284 522729 524176 525625 527076 528529 529984 531441 532900 534361 535824 537-289 538756 54! .J5 541696 543169 544644 546121 547600 549081 550564 552049 553536 555025 566516 558U09 55<)504 561001 562500 564001 565504 567009 568516 570025 571536 573049 574564 576081 577600 579121 580644 582169 5>3(596 585225 585756 588-,'89 589824 591:561 58290(» 594441 595984 597529 591H)76 600625 (■)02176 603729 605284 606841 608400 Sqre. root. 26.8514442 •J6.8700.577 26.8886593 26.9072481 26.9258240 26.9443872 26.9629375 26.9814751 27.0000000 27.0185122 27.0370117 27.05549.-^5 27.07;iSt727 27.0924344 27.1108834 27.1293199 27.1477439 27.1661554 27.1845544 27.2029410 27.2213152 27.2396769 27.2580263 27.27(53634 27.2946^81 27.3130006 27.3313007 27.3495887 27.3678644 27.3861279 27.4043792 27.422<)184 27.4408455 27.4590604 27.4772633 27.49.")4542 27.5136330 27.5317998 27.5499546 27.5680975 27. 68622^4 27.6043475 27.6224546 27.6405499 27.6.58()334 27.6767050 27.6947()48 27.7128129 27.7308492 27.748f-(739 •*7.766S,<68 27.7849880 27.8020775 27.820^555 27.8388218 27.f<.")677()6 27.8747197 27.892()514 27.9105715 27.9284801 No. 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 82(i 827 828 829 830 831 832 833 8:!4 835 836 837 838 839 840 Square. 609961 611524 613089 61465C) 616225 617796 »il9369 620944 622521 624100 625681 627624 628849 630436 632025 633616 635209 636804 638401 640000 641601 643204 644809 64641() 648025 649635 651249 652864 654481 656100 657721 659344 660969 662596 664225 665856 667489 669124 670761 672400 674041 675()84 677329 678976 680625 682276 683929 685584 '687241 688900 69(»561 69-J224 693889 6955.56 697225 698,<96 700569 702244 703921 705600 Sqre. root. 27.9463772 27.9642629 27.9821372 28.0000000 2,-!. 0178515 28.0356915 28.0535203 28.0713377 28.0- 438 28. 10(, 1)386 28. 1247222 28.1424946 28.1602.557 28.17K»0.56 28.19.57444 28.2134720 28.2311884 28.2488938 28.2661881 28.2842712 28.30194:54 28.319<)045 28.:5:^72546 28.35489:« 28.:J7252:" 28.3901:391 28.4077454 28.425:'>408 28.4429253 28.4604989 28.4780617 28.49561:^7 28.5131549 28.5:^06852 28.5482048 28.5657137 28.58:52119 28.6006993 28.61817(i0 28.6:^56421 28.6530976 28.6705424 2H.6879766 28.7054002 28.7228130 28.7402157 28.7507677 28.7749.^91 28.792:5601 28.b097206 28.8270706 28.844410-J 2K.H617:594 28.8790.5S2 28.896:5666 28.9i:!6()46 28.9309523 28.9482297 28.9654967 28.98275:« No. 841 H42 843 844 845 846 847 848 849 850 851 8.52 853 854 855 856 857 858 859 860 861 862 86:; 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 864 8.S5 88(5 887 888 889 890 891 892 89:? 894 895 896 897 898 899 900 Square. Sqre. root. 707281 708964 710(549 712:536 714025 71.5716 717409 719104 720801 722500 724201 25' 904 /-,-/609 729316 731025 7:W7:56 731449 736164 7:57H81 739600 74l:i2l 74:1044 744769 74(5496 748225 749956 751(589 753424 755161 756900 758641 760:584 7(52129 763876 765625 7(57376 769129 770884 772641 774400 77(5161 777924 779(589 78145(5 78.3225 784996 786769 788544 790:^21 792100 79:5881 795664 797449 7992:56 801025 80281(5 804609 80(5404 808201 810000 29.0000000 29.0172:563 29.0344(523 29.0516781 29.0688837 29.0860791 29.1032644 29.1204:396 29.1376046 29.1547595 29.1719043 29.1890390 29.20(516:37 29.22:32784 29.240:3830 29.257 till 29.2745(523 29.291(5370 29.3087018 29.:5257566 29.3428015 29.3598365 29.:3764616 29.:?938769 29.4108823 29.4278779 29.4*448637 29.4618:597 29.47880.59 29.4957624 29.5127091 29.5296461 29.54(5.5734 29.56:34910 29.580.3989 29.5972972 29.61418.58 29.6310648 29.6479342 29.(5647939 29.6816442 29.(5984848 29.7153159 29.7:521:375 29.7488496 29.7657521 29.7825452 29.799:3289 29.81610:30 29.8328678 29.8496231 29.8663690 •>9.8S31056 29.8998:328 29.91(5.5506 29.93:32591 29.9499583 29.9666481 29.98:33687 30.0000000 ■'.- OF NUMBERS FROM 1 TO 1600. 1 No. Square. Sqre. root. No. Square. Sqre. root. No. Square. Sqre. root. 901 811801 30.0106621 961 92:5521 31.0000000 i ! 1021 1042441 31.9530906 yo-i 81:5004 30.0:5:5:5148 962 925444 31.0161248 1 1022 1044484 31.9t5f57:;47 U(j:i 815409 3(t.0499584 9ti3 927369 31.0:522413 10j:5 104()529 31 984:5712 904 817210 :50.0005928 964 929-96 31.04.S5494 1024 1048576 32.00000(»(» 905 b 19025 30.0-32179 965 931225 :!1. 0644491 1025 1050625 32.0156212 1 90(i 8208:56 30.099p:5::9 966 9:5:5156 31.0.-^05405 1026 1052676 32.0312:548 1 907 822049 :50. 11 (54407 967 9:55089 31.09662:56 1027 1054729 :52. 0408407 1 90d 824401 :50.i:5:5o:!83 968 9:57024 31.1l269-!4 1028 1050784 :52. 062 4:591 n 909 820-.'81 30.1490269 969 93-961 31.1287648 1029 I0.5f^841 :32. 07^029- 1) 910 828100 ;50.10ti2O63 970 940900 31.14482:50 lo:!0 10(;0i)00 ::2.09:i0l31 » 911 829921 :50. 1827705 971 942841 31.1608729 1031 1069! 01 :52. 1091877 ' ::\'^ 8:51744 :50. 199:5:577 972 944784 31.1709145 10:52 1065024 32.1247568 9l:J 8:5:5509 30.2158899 973 940729 :il. 1929479 103:5 1007089 :52. 1403 173 911 8::5:597 ;50.2:524:329 974 948ti76 31.20.-'9731 10:14 1069156 32.155^704 915 8:57225 :50.24«9609 975 950625 31.2249900 10:55 1071225 32.1714159 91(j 1^:59050 :50.2051919 976 952576 31.2409987 lo:56 107:5296 ::2. 18095:59 917 840889 30.2-20079 977 954529 31.250999'J 1037 1075:5(59 32.2024844 918 842724 :30.2985148 978 956484 31.2729915 10: !8 1077444 32.2180074 1 919 844501 :i0.3150!28 979 958441 31.2^89757 10:59 1079521 32.2:5:55229 9::^0 840400 30.:5:!15018 980 900400 31.:50I9517 1040 1081000 :52.2490310 9-21 848241 :50 3479818 981 9()2:561 31.: 1209 195 1041 108:5081 32.264531(5 9'2-J 850084 :50.:!044529 982 964:524 31.:5:;08792 1042 1085764 :52.2K)0248 9-S.i 851929 30.:!809151 98:5 966289' :51.3.- -:508 1043 1087849 32.2955105 9-21 85:5770 30 3973683 984 968250 31.:50-7743 1044 10898:56 :52.:: 109888 9-25 855025 ;50.4K58127 985 970225 31.;5847097 1045 1092025 32.:5264598 9-.'(5 857470 :50.43024.-'l 986 972196 31.4000:569 1046 1094116 :52.:54 192:53 9-J7 859:529 30.4406747 987 974169 31.4165561 1047 109(i209 :52.:557:5794 928 801184 30.40:50924 988 976144 31.4:524673 1048 1098:504 32.:57 28281 9-29 8():;041 :50.4795013 989 978121 31.44K!704 1049 1100401 :52.:5-rt2695 9:50 ri()4900 30.4959014 990 980100 31.461J6.'.4 1050 1102500 :52.4037O;55 9:51 800701 :50.5122926 991 982081 31 4901525 1051 1104001 :52.4 191:501 9:3-j 80>024 30.52S075't 992 9H4064 3l.49(io:!15 : 1052 1 106704 :52.4:545495 9:{:j 8704-9 :50.54:.0487 99:5 986049 31.5119025 • 1053 1108899 :52.4499615 9:51 872:5.56 30.56141:56 994 9880:56 31.5277*i55 i 1054 1110916 32.4(55:5(502 9:i5 874225 30.5777097 995 990025 31.543620*) 1055 111:5025 :52.4S07635 9:!C) 87009*; 30.5941171 996 992016 31.5.594577 10.')6 11151:56 32.4901536 9:57 ?^77'.)09 30. () 104557 997 994009 31.575:500-^ 10.-7 1117249 :!2. 51 15:564 9:58 879.->44 :5062O7Hr.6 998 990004 31.5911:580 10.58 1119:564 :52.52(;9119 9:59 881721 :50.64:51009 999 1998001 31.60.')90l:! 10.59 11214S1 32.5422802 940 88:5000 30.6594194 1000 1000000 31.6227700 lOiiO 1 12:5000 :52. 55764 12 941 885181 :50. 675723:5 1001 1000201 31.0:58.5840 1001 1 125721 :52. 5729949 942 887:504 :50.0920185 1002 1004004 31.()5l:!-:!0 100 J 1127H14 :52.588:3415 94:5 889249 30.708:5051 1003 100()009 31.1)701 752 106:; 1129969 32.60:56807 944 8911:56 :50.7245.-:50 1004 100-016 31.o-.".m;m) 10<)4 1 1:52090 :52.(;i90129 945 89:5025 :50. 7408523 1005 1010025 31.7017;!I9 1065 11:54225 :52.634:5:577 940 894910 :50 7571 1:50 1000 10100:56 :!i.7i:5o:!o 1060 11:50:556 :52. 6496554 917 896^08 ;50.773:5051 1007 1014049 31 7:5:520:5:5 1067 11: '.8489 32.6049659 94rt 898704 30.7896080 1008 101(;064 31.7490157 1008 1140624 32.(;802693 949 900001 30.80581:56 1009 10180S1 31 7617003 10(59 1142761 :52. 0955654 950 902500 30.8220700 1010 1020IO(» 31.7-04972 1070 1144900 32.7108544 951 904401 :50. 8:582879 1011 10 JO 121 :!1.7962.'»i2 1071 1147041 :52. 7261:563 952 906:5(»4 :50.H544972 1012 1024144 :i!.811<.»4r4 1072 1149184 :52.74]4111 95:J 908209 :50.H706981 1013 1026169 31.>'27tS(;0'.t 1073 1151:J29 32.7566787 954 910116 :50.8HO-'904 1014 102HI96 3I.84:5:50(;6 1074 115:5476 :52. 7719:192 955 912025 :50.90:5074:5 1015 10:HI225 :5i.><.'i90iiif; 1075 1155625 :52.7871926 950 9l:!9:50 :50.9192477 1010 10:52256 :5 1.-7475 49 1076 1157776 32..«024:598 957 915819 :50. 9:554 166 1017 1031289 :;i.H«t()4:574 1077 1159929 32.8172782 958 917761 :50.95 15751 1018 10:56:524 31.9061123 1078 1162084 :32.8:529io:5 959 919081 :;0.9677251 1019 1o:sh:!61 :!1. 9217794 ! 1079 1164241 :52. 848 1354 900 921600 30.9838668 1020 1040400 31.9374:588 1080 1166400 32.863:5535 8 TABLE OF SQUARES, SQUARE ROOTS Nc. Square. Sqre. root. No. Square. Sqre. root. No. Square. Sqre. root. 1081 116^s561 32.8785644 1141 1:501881 :53.77H6915 1201 1442401 :54. 6554469 1082 1170724 :52.r^9:57684 1142 1:504614 :53. 7934905 1202 1444804 34.6(59871(5 10r*;{ 1172889 :52.90«965:J 1143 1:506449 33.8082830 1203 1447209 .34.6842904 1034 1175056 32.924155:5 1144 i:5087:;6 :5:5.82:«i69i 1204 1449616 :54. 6987031 1085 1177225 : 12.9:59:5:5^2 1145 ; 31 1025 ;53.n:57rt4H6 1205 1452025 34.7131099 1086 in9:59«i :52. 9545141 1146 1313316 :53.8526218 1206 1454436 34.7275107 1087 1181569 :52.969(W:50 1147 1315609 :5:!.867:5884 1207 1456849 ;54.74 19055 loss 118:5744 :52.9M8450 1148 1317904 :53. 882 1487 1208 1459264 34.7562944 108'J 1185921 3:!.00OU0()O 1149 1:520201 3:5.8969025 1209 14(51681 34.770(5773 lOUO llbr-lOO :5:5.O151480 1150 K5225()0 33.911(5499 1210 14(54100 34.7850543 1091 1190281 33.0:;02891 1151 i:524>^01 :53.92(5:!909 1211 1466521 34.7994253 1092 1192464 :53. 04 542:53 1152 1:527104 :53.94 11255 1212 1468944 ;54. 81:57904 109:{ 1191649 3:;.-< 1605505 1153 1329409 3:5.95585:57 1213 1471:569 :54.8281495 1094 ]196s:56 3:5.0756708 1154 1:53 17 16 :53. 9705755 1214 1473796 34.8425028 1095 119:t(i25 :5:5.o;»(i7Hi2 1155 1:5:54 (»25 :53.9-5291(l 1215 1476225 34,8568.501 I09t) 1201216 :!3 105^907 1156 13:5(5:5:56 :i4. 0000000 121(5 1478656 :54,8711915 1097 120:;409 :53. 1209903 11.57 l:53-'t)49 :54.0 147027 1217 1481089 34.8855271 109S 1205601 :53.i:5()08:;o 1158 1340964 :5 4.029:5990 1218 148:5524 34.?^998,5(57 1099 1207-01 ;53. 151 1689 1 159 134:'.281 :54.O44Of^90 1219 1485961 34.9141805 1100 1210000 :i3.1(;()2479 11(50 1345t)0() 34.0587727 1220 14884(»0 34.9281984 1101 1212201 :{3.181:52(»0 1161 1:547921 :54. 0734501 1221 1490841 34.9428984 1102 1214404 :5:r. 196:5-53 11 ()2 1:550244 :54. 0881211 1222 1493284 :54.9428104 1103 l21()t;09 :5:!.2 11 44:58 1163 13525(59 34.10278.-8 122:5 1495729 :54.95711(56 1104 l21.S-'i() :;:5.2266955 1164 1:554896 34.1174442 1224 1498176 34.9714169 1105 1221025 :5:!.241540:! 1 165 1357225 :!4. 1:5209(5:5 1225 1500(525 .34.9857114 1106 I22:5.::ui :5:5. 25657-3 1 1(56 1:55955(5 31.14(57422 1226 1503076 35.0000000 1107 1225449 :>:;.2716095 j 1167 1:561889 ;54. 161:5817 1227 1505529 :55. 0142828 1108 1-J276t;4 3:;.2ri()():5:!9 | 11(58 1364224 :54. 17(50 l.-.O .1 »28 1507984 35.0285598 1109 1229--81 :!3.:!(»l(i616 j 1169 1:566561 :54. 1906420 1229 1510441 35.0428:509 1110 12:52100 :53,31()(i(rj5 1170 1:5(58900 34.2052(527 12:50 1512900 :55. 0570963 1111 12:54:!21 :!:5.:{:5i()6ii() 1171 l::712il :54.2--'98773 1231 1515:561 :55.07i:55.58 1112 12:5(5541 :53.:;(()()6io 1172 137:5584 :54. 2:544855 1232 1517824 :55.0>S5609() Hi:} 12:58769 :5:5.;;6l()5i6 1173 l:;75929 31.2490875 12:53 1520289 :55 0998575 1114 124099() :5:i.:57(5()3^5 1174 1:578276 34 26:5(;m:54 12:54 1522756 35.1140997 1115 12^:5225 :53.:59161.-.7 1175 13-^0(525 34.27827:50 12:55 152.5225 :35. 128:5:561 lilt; I24545ii 3;5.40t;5,-'7113 13-23 1759:5-29 :56.:57:?o(>70 l:5.S5 191-2(5.-9 37.1887079 l'i64 i.')y7»i9t) 35. .55-27777 13-24 17.5-2976 :5(5. 3-^6- 108 1:;h4 19154.56 37.2021505 1 •,'»;.") leooM-jf) 35.5f)6,^3-5 13-25 1755625 :56.4005494 1:5.-5 191.-225 37.2155881 1 •,'()() lti(l-J75ii 35..5>(»-'.t:'>7 13-26 175?S276 :;(5. 4 14-28-29 i:586 192099(5 37.2^290209 lv"i7 KKt.VJS') :;5.. 59 19434 1327 17609-29 36.4-280112 i:!-7 192:!769 37.2^224489 1 vK).-^ lt)07tt-.'4 35(i(Kt.-76 1328 176:55-4 :;6.441 7:543 i:58,i 19-26544 37.-2558720 l"2«)i» lt)l():!()l 35.(;-2:!(l-J6-2 1329 17(5624 1 ;5(5. 4.5545-23 1:5.-9 19-29:521 37.269-2903 l-JTO IHl-i'.XtO 35.6370.593 V.VM\ 176-900 :5( 5. 4 69 1(550 1:590 19321 (JO 37.28^270:57 1-J71 i6ir)in :;5.651(»-()9 13;! I 1771..61 :;6 4828727 1:591 19:541-81 37.29611-24 l-JTi. ii;i7'.t-4 35.t)6510'.IO 13:!2 1774-224 :56. 49(55752 1:592 19376(54 37.3095162 1-J7:{ 1 i)-20:,2[) :;5.(;79 1-255 1333 177(5.S<9 :!(5.5 10-27-25 1:593 1940419 37.:«-29152 1-J74 1G-J:'.07() ;;5.69:!1366 1334 17795.56 :{6.52:59(547 1:594 1943-2:5(5 37.336:5094 i5>7r) Hj-j.")*;".*.') 35.70714-21 1335 178-2-2-25 3(5.5:57(5548 1 :595 194(5025 37.:M9(i988 I'iHj I(j-i8l7«) 35.7-2114-22 1336 1784>9ti ;J6.55i:5:5s-! 1:596 1948.-16 37.3630834 l-i7l it;307-.'i» 35.7:>5i:;67 1337 17?7569 ;5(5.5(5:.0106 1:597 1951(509 37.3764632 \-27f* lH33->4 35.749 i -258 133S 1790244 :56.5786823 1:598 1954404 37.3898382 l--'7<) 1»)3.".8U 35.71131095 1339 179-2921 :5(5.592:!4'«9 i:!99 19.57-201 37.4032084 1-M) 1(;384(K» 35.7770876 1340 179. ,(500 :5( 5. 60(50 104 1400 19(50000 37.4165738 1-,'81 i()4(Hitil 35.7910(i03 l::41 17982-^1 :;6.(511t(56(58 1401 1962801 37.4299345 U&2 KM 35-24 35.80.50-276 1342 180(J964 ;56.6:5:53I81 1402 19(55604 37.443-2904 r.v:5 I(i4<5(l8i» 35.8189^94 1343 180:5649 :{(S.(51(;9(544 140:5 19(58409 37.4566416 l-,'>4 l(M8()5t) 35..-^3-29157 1344 1806:;: 56 :5(5.6(50605(5 1404 1971216 37.4699880 Itid") U)r>iv!->.") 35.846S'.)66 1345 1.-090-25 :5(5.ti74-2416 1405 19740-25 37.48:5:5296 l<)r>371>6 :'..5.r6081-21 1346 1811716 :i6.(;.«'787-J6 140(5 19768:J6 37.4966665 l-,'d7 I6r)63t;9 35.87478-2-2 i:i47 l'^144(»9 :?6.7014986 1407 1979649 37.5099987 l::iAS 1658944 35 8887169 1348 1817104 :56.7151I95 1408 1982464 37.523:5261 l-J-^9 I6(>ir)-ji 35.90J6461 1349 1-19-01 ;;6. 7287:5.53 1409 1985-281 37.53(5(5487 l-2'.tO l()t)4lU0 35.!>!(m699 1350 18-2-2500 :56. 7 4-255461 1410 19.-8100 :}7. 5499(567 l-ilH ItihtiC.al 35.9301--8^ 1351 1825-201 :{6.75.59519 1411 1990921 37.5(5:52799 1-J1I2 ii;()9-it;4 35.iM44015 1352 18-27904 :5(5. 76955-26 1412 199:!744 37.5765855 12'J-S ltiVlH49 35.958309-2 1353 ]8:}(i609 :56.78314.'<3 1413 1996569 37.58989-22 \-2[)i lti744::ii 35.97-2-2115 13.54 ]8:;:5316 3(5.79!57:51;0 1414 1999:596 37.(5031913 l-2[)b 16770-25 35.9861084 13." 5 18:<60-25 :56.f« 103-246 1415 200-2225 37.6164857 1-Ji»6 Iti79tu0 36.0000000 1356 18:587:56 36.'e2:;905:5 1416 200.5056 "7.6297754 lv!97 16^-i-.'09 36.01 388t;-2 1 1357 1841449 36.8374.-^09 1417 20078.89 37.6430604 l-^!)8 lt;8».-(t4 36.0-277671 1358 1844164 :56.8510515 1418 2010724 37.6563407 Iv'iCj 1687401 36.01161-26 1359 1846881 :;6.8646172 1419 20135(51 37.6696164 l:itiO biyoodo 36.05551-28 13(;0 1849600 36.8781778 1420 2016400 37.6828874 1301 169-J601 36.069:!776 1361 I852:wi 36.8917:535 1421 2019-241 37.6961536 i:;o-> l*)95-.'04 36.0-:!-2:!7l 1362 18.5.5044 :?3.9052842 14-22 2022084 37.7094153 1303 1694809 3i;.09709l3 1363 1857769 36.9188-299 1423 20249-29 37,72-267-22 i:?04 1700416 ;!6. 110910-2 1364 18(50496 :56. 9:5-2:57 06 14-24 20-27776 37.7:^59245 1 ;!().') 17030-25 36.1-247837 1365 18():i22.5 :56.9459064 1425 20:50625 37.7491722 i:5Ut) 1705636 36.1386-2-20 13()6 18H5956 3(5.9594:572 1426 20:53476 37.76-24152 1:W7 1708->49 36.1524.550 13()7 1866689 36.97-296:51 14-27 2036:5-29 37.7756535 i:kw 1710864 36.166-2-!-26 1368 18714-24 :56.9864840 14-28 20:59184 37.7888873 i:!oy 1713481 36.18010.50 1369 1874161 :i7.000(»000 ;4'29 •2042041 37.8021163 1310 1716100 3(5.19392-21 1370 187(5900 37.01:55110 14:50 2044900 37.8153408 KUl 17187-21 36.-2077340 i 1371 1879641 37,0270172 1431 -2047761 37.8285606 1312 17-,M344 36.-2-2 15406 i 1372 188-2:5.-^4 37.0405184 1432 2050624 37.8417759 1313 17-23969 36. -23534 19 | 137:! 1885129 37.0540146 1133 2053489 37.8549864 1314 17-J«)596 36.'249i379 i:?74 18-7M7(5 37.0(575060 14:54 2056356 37.86819-24 131;") 17-29-2-!5 36.-2626-2rf7 1:575 1890(525 :}7. 0899924 1435 2059225 37.881:^9:58 1316 1731'^56 36.-2767 U3 | 1:576 1893.576 37.0944740 1436 2062096 37.8945906 1317. 17344.89 36.-2904946 ; 1:577 1K>6129 :i7. 1079506 1437 •2064959 37.90778-28 131H 1 17371-24 36.304-2697 : 1:578 l.-'98"'.-^4 :57. 1214-22! 14:58 2067844 37.9209704 1319 1739761 36.3l8o;;96 : 1:579 1901(541 :'>7.i:U8,89:! 14:59 2()/'j721 37.9341535 13'i0 174-2400 36.33l!:'U42 ]:i8U 1904100 :57.14f:5.M2 1 1440 •207:5600 37.9473319 10 TABLE OP SQUARES, SQUARE ROOTS I-: — — No. Square, Sqre. root. No. Square. Sqre. root. No. Square. Sqre. root. 1441 207(>481 37.9605058 1495 22:55025 38 6652299 1548 2396304 39.3446311 1442 2079364 37.97:567.M 1496 22:58016 38.6781.59:5 1549 2:599401 :59.:557:5:573 1443 2082249 37.98689:58 1497 2-.M10U9 :58. 6910843 1550 2402500 :i9.:5700394 1444 2085136 38.()00()0(I0 149>< 22110(14 :;,-'. 70 nto50 1551 •2405601 :i9.: 18-27373 1445 2088025 38.01315.")«) 1499 2247001 :5f-. 7169214 1552 2408704 39.39.54312 1446 2090915 :58.o.6:!()()7 15(10 225(1(100 :5-^.7298:!:!5 15.)3 -2411^09 159.408 1210 1447 209:1809 :5rt.03;t4..:;2 l.idl 2-J5;5O01 :58.7 4274i2 1554 -241491() :59.4208067 1448 2096704 :58.(t.".25>t52 15()-J •J.;. 6004 :58.7. 56447 1 555 •241c!0-25 39.43:54883 1449 2099f.01 :58. 0657:526 1.-.03 •rJ.)9(iO.! 3.-<.76-.i4:i;» 15.56 24211:56 39,4461658 1450 2102500 38.07 rtH6.M5 1 50 4 2v:()-^ol(i ::8.7.-l 1:5.-^9 1.5.57 2424-249 :59.458-:593 1451 2105401 38.09199:5".! 1. ")().') 22r..-)0-J5 :5.-'. 794:129 4 1558 24-27:5(;4 :59. 4715087 145-2 2108:i04 ;58.10.'>117.- l.'.OO •J2ti.H>3i :'.«.H).!2 !.'>.- i 559 24:!(t4>l :59.4841740 1453 2111209 :58. 11 8:5:171 1.07 22710 49 3-i.82(J0978 1560 24;5:i6UO :59. 4968:153 1454 2114116 :58.1313.')19 l.-)ttS •.•,'71064 :58.8:;297.57 1561 •24:51.721 :{9...Uit49-25 1455 2117025 3,-!. 144462-.: i.-.on -277081 :}8.-'4:.H4i>l i:.62 •24:59844 39.5-221 4..7 1456 21199:i6 ;58.1."):568l 15 1 •J-^-Ol(»0 ;5>'.8.5>7184 1 .5(5:5 2442969 :59.5347948 1457 2122849 38.1 7066. t3 151 1 •j2f^;;l.:l :>8.871.'"'^::4 15()4 244609(i :59..5474:599 1458 2125764 38. 1^:57 ii62 1..12 2286144 :i-.88 4 44 4-' 1565 2449^2-25 39.5600809 1459 2128f.S4 38.1968.'.8.". 1 1513 2-.-'91()9 :5H ,-i. 92:50009 15()8 245H(;-24 :{9.5979797 146-i 21:57444 :{H.2:561085 1..16 •J-.;'.t-25() 3s.i):{.-,8l47 1569 •2461761 :59 610(j046 1463 214U:569 38.249 18J9 j 1517 2:5ol-i>9 :iS.r,i.>i94 1571 •24(3H041 ;5i).6.1584-24 1465 2146225 38.-J7531S-1 1519 2:5073til 3,S.'.t7 4:5.-05 1572 •2471184 ;59.64,-^45..2 1466 2149156 :58.2'-8:5794 1.520 2310400 .'•KA)!*. 1774 1573 247 4:519 3:>.66l0640 1467 2152089 :58.:!0 14:560 1521 231:5441 :>9.ooooooo I..74 2 477475 3i».67.16{i."'8 1468 2] 55024 :58.:n44H-^i 15J2 2316184 :!9.ol2-^l.-i4 1.57.. 24^06-25 : 19. 6862- .96 1469 2157961 :58.:5275358 1..23 2319.529 :5:( (i-J. 6:526 1576 2 4~!;;776 :59. 1.98- 665 \ 1470 2160900 :i8.:54(»5T90 1.524 2:522576 :5it.O.;,-il426 1577 24-(>929 :59.71145i»3 \ 1471 216:5841 38.35:56178 1525 2:525()'J5 :5i».(i.',i24-:! 1..78 24.H)08J :J9.7j4(4.^1 1472 2166784 38.3661)522 1526 2:;2>(;76 :!9.064ol99 1579 •249:5241 :59.7:!6,)3j9 ' 1473 21()9729 :58.:5796H21 1527 2:5:! 1 7 -^9 ::ii.()7';.- 1.5H2 2502724 :59.774:56:!6 1476 2178576 :i8.41874.-)4 15:50 2340900 ;'.9. 11.52144 1.5.S5 •25051-9 :59. 7669: 1-25 1477 2181529 :58.4317577 1531 2:54:5961 5tt. 12791 (51 15.-^4 250i)056 .•59.799497() . 1478 2184484 38.4447656 i.5:« 2:54702! 19. 1 1 ^716 I5S5 251-2-2^25 3:).>'l-2o5S5 1479 2187441 :58.4577691 1533 2:5500-^. ' .9. i.5;;54:59 1 ..S(, ■251.5:596 ;59.'^i-.'4 11.55 1480 219(>400 :58. 4707681 l.-.:54 •j:5.-.:5l.".(i !9 ((;6:!120 1.5-^7 •251-5(i9 :i.>..-'376646 ' 1481 2193:361 38.48:57627 1.-):^) 2:55t;-.>J5 :'>9 179(»7()0 15M.S •2521744 :59.84'.)7177 1482 2196:J24 38.4967530 i.-.:i6 2:!59-J96 9 191-^:5-">9 1 . .S9 •2521921 :}9.:5()^22()28 1483 2199289 38.5097:590 15:57 2:!C,2:5(i;» ;;'.».-.'04V.>15 15;m) •25J-^I00 ;59. -74-040 1484 2202256 38.5227206 15:58 2:5(^5444 :5'.). •217:54:51 1.591 •2.5:! 1^281 :i9>-'7:54l3 1485 2205225 38.5:556977 15:59 2:568; .21 :t9.2:5oo',»()5 1.5'.I2 •25:'.-i4(S4 :59.H9.»8747 [ 1486 2208196 38.5486705 1540 2:571600 :59.^24-j.'<:5:J7 1.513 25:!7()49 :19.91-'4041 ir 1487 2211169 :58.56163S'J 1541 2:37 4 •■."'I :59.-j.".5:)7-2-' 159 4 •254(K5t; :59.9-24'.t295 1488 2214144 :58.57460:50 1542 2:57 7764 :5'.> •i6H:5(t7H 15;»5 25440-25 :i9.;»:i745l 1 1489 2217121 38.5875t;27 154:5 2:5-0-^19 :59.2-'io:'>87 1 .">9t) •2.547-216 :i9.749.-'(>-<7 p 39.9624-^24 1490 2220100 :58.600r>l81 1544 2:n:59;!6 :!9.-i9376.">4 1.597 •2.550109 1491 2223081 38.61:^4691 1.-.45 2:5^'7()25 :;9.3()61SH) 1.598 -2.55:5ti04 :59.9749922 1492 2226004 :38.6264l.->8 1546 2:590116 :59.:5 19-2065 1599 -2.5.56-^41 :59.9-^749-'0 1493 2229049 38.6393582 1547 2:59:5209 :59 :53l9-.!08 ItioO -2560000 40.0000000 1494 2232036 38.65->2962 •-i tabtje: II. a. AREAS OF CIRCLES, FROM ^ TO 150, [Advancing by an Eighth.'] Diam. Area. Diam. Area. Diam. Area. Diam. Area. Diam. Area. bV .00019 4. 12.5664 10. 78.54 16. 201.062 22. 380.134 ,u .00077 ■K 13.364 ■H 80.5157 ■A 204.21(5 ■A 384.465 Uf •*8 14.1862 ■A 82.5161 •M 207.394 ft 388.822 tV .00307 15.0331 ■A 84.5409 ■A 210.597 393.203 i .01-2t>7 ■I 2 15.9043 16.8001 :M 86.59 88.6(543 'A ■A 2l3,r25 217.073 % 397.608 402.038 tV .027H1 ■¥ 17.7205 ■% 90.7628 ■A 220.:;.')3 -A 406.493 i .04909 yA 18.6655 ■'A 92.8858 ■A 2J3.654 o,-« 440.972 5. 19.()35 11. 95.03:i4 17. 226.ii8l 23. 41(5.477 1^ .0767 ■H 20.629 ■A 97.2055 ■A 230.33 •A 420.004 f .11045 •a 21 6475 ■A 99.4022 •A 233.705 '\^ 4-.'4.557 •% 22.6907 ■A I0l.(5-J34 ■A 237.104 ■A 429.135 A .15033 ■h 23.7.->83 •A 103.8091 ■A 240.528 A 433.731 i . 19635 •fa 24.8505 ■A 106. 1394 5 120.276 ■A 2ti.->.182 •A 46t5.638 ■H 33.1831 •A 122.718 ■A 26^. 603 ■A 471.436 ! i*^ .5184H ■Vb 34.4717 ■A 125.184 •S' 272.447 ■A 476.-J59 i 7 .60132 ■¥ 3;). 7-^47 •/4, 127. ()76 ■i •-'76. 1 i 7 ■% 481.106 1 ? fA 37.1-J21 7^ 130. 192 ■A 279.811 ■A 4.-'5.978 ; u .690-J9 1 . 38.4-^46 13. 132 733 19. •J83.529 25. 490.875 ! 1. .7^.")4 % 39.8713 ■A l:!5.-4»7 ■A 287.272 ■A 495.796 \ i H .99402 ■'A 41.2825 137. --86 A 291.039 v'8 500.741 i 1.2271 ■% 42.7184 140.5 3/ •73 •J94.P31 505.711 1.4S48 ■% 44.17ti7 M 143.139 ■h 298.(54.-' ■A 510.706 .1.; 1.7671 % 45.6636 ■^^ 145.802 ■A 3(fJ.4^9 •A 515.725 •S 2.0739 ■% 47.173 ■i 14«'.489 •S 306.355 ■A 520.769 •li 2.4052 ■A 48.707 ■A 151.201 ■A 310.245 ■A 525.837 •% 2.7611 8. 50.2656 14. 153.938 20. 314.16 26. 530.93 2. 3.1416 •A 51.8486 ■A 156.699 ■A 318.099 ■A 53(5.047 •^^ 3.5465 •k 53.45(52 ■A 159.485 ■A 322.0(53 ■A 541.189 •¥ 3.976 ■H 55.0885 •A 162.295 ■4 326.051 4 546.356 551.547 •% 4.4302 ■'i 56.7451 A 165.13 ■A 330 (1(54 ■A •M 4.9087 'A 58.4264 •A 1(57.989 ■A :',34.101 'A 5.56.762 •¥. 5.4159 '% 60.1321 ■A 170.873 •i 338. 163 4 562.002 a 5.9395 •A 61.8(525 ■A 173.782 ■Ji 342.25 , •'A 567.267 '% 6.4918 9. 63.6174 15. 176.715 21. 346.361 27. 572.557 3. 7.0686 ■A 65.3968 ■Vb 179.672 •A 3.50.497 ■A 577.87 •<^ 7.6699 ■Vi 67.-J007 •A 182.(554 ■A 354.(;.")7 ■A 583.208 8.2957 ■K 69.0293 •A 185.661 ■}t 3.58. SI 1 A 588.571 8.9462 •>l 70.8823 -A 188.692 ■A 3(53.051 4 593,95'! -^2/ 9.6211 ■'A 72.7599 •% 191.748 ■A 367.2'i4 4 599.376 •1^ 10.. 3206 ■H 74.6(52 •A 194.828 ■A 371.543 4 604.807 •1^ •>8 11.0446 11.7932 ■j| 76.5887 •A 1 197.933 ■A 375.8J6 A 610.268 12 AREAS OF CIRCLES. TABLE.— (Continued.) Diam. Area. Diam. Area. 1 Diam. Area. 1 IDiiun. Area. Diam. Area. 28. 61. "..7. ".4 X). 9ti-J.ll5 \vi. I3,-'.-..4l 49. I -^8.-., 74 56. '2463.01 ■H ()VJI.-Jii3 ■% 9«)>.lt'.t9 ■H 1393.7 •is 189.-). 37 ■A •2474.02 «)-.'ti.7lH •>4 97;).9()8 ■U 14(C).98 ■'/i 190.-.. 03 ■A 2485.05 ():{-2.;;;>7 '% lH-^.,-'lv> ■H i4n».-jy % 1914.7 ■A 2496. 1 1 • K «.!7.'.M1 . >., '.H'.t."^ .,'■. 14l8.ii3 •>a 19-,*l,4-i ■A -2.507.19 1 (i4:i..')i;i v^i 99().7-3 ■h 14-.'()98 ■% 1934.15 ■A -24lS.:i ()4'.t l-'t* ■}4 loo:!. 7 9 -H 1 I3.-..31; ■fi 1943.91 ■¥ 2r-29.43 <).">4.H:!!> ■Js lOlo.r-j-J •Js 1413.77 v8 19.')3.69 ■A 2540.54 •21>. «iti(»..VJ| ■Ml 1017.878 43. 14.VJ.-il 50. 19»)3.5 57. 2.-)5l.76 •>8 Hrttl.J-.'? ■H IOi4.9.VJ •/8 I4()(».t;.-, ■H 1973.33 ■A 2562.97 <)T1. !».')'' ■% i;)3-i.oi)r) .•4 n»)9.i3 ■fi 1983. 18 ■A 2574.2 ()77.714 A 10.;9.19.") •% 1477. (;3 ■% 1993 05 ■A 2585.45 .1.. ().-i;{.4'.»4 ■}^ I(t4ti.34;» .'.. I4'^t;.i7 .'.i •>00->.97 ■A •2596.73 v^^B tl-i'.t.-i'.H •^8 lof);;..-)-.'-!' ■)i 1 l'.»4.7"J •'A 201-2.89 ■A •2608.83 •4 6'.>r).i-j>! A i(»i;ii.73-j ■h l.-)(»3.3 ■U •JO-22.85 ■¥ 2619 36 ■fs 70U.'.t81 ■fi 101)7.9(3 •>8 1511.9 ■}i 203-2. 8-2 o-^ 2630.71 30. 71 »).«(> 37. l(i7.-).-,>13 44. l;V.'0.r,3 r.i. 204-2.8-2 58. 2(542.09 •^8 7iv;.7()-i v^8 10-:-j.49 ■}i 1.VJ9.18 ■Vs •20.-.-2.85 •A 2653.49 •il 7i8.i;;» :;i l(i'!'.t.79-.' ■''A l."):!7.8(> 20b-2.9 ■A 2664.91 •II 7-.' I. till 10;t7.il8 •?« 1. ')!(). ;">.") 207-2.98 ■H 2676.36 .l" 730. CIS .1.. i lOl.UUt .'." l.").".r).t>,-j .% 2083.08 ■A 2687. . ■% 73ti.t;ii> •^8 1U1.>'44 ■fi l.')(;4.03 ■'A -2093.2 •1^ 2699.33 ■S 74J(H1 •4 ni9.-i44 ■■'i i:)7-i>l % 2103.35 ■u •2710.86 •Jt 74S.(3iM ■% 11 -,'(>.()(;-< Js ir.si.(ii ■% 21 13.. 52 ■A 27^22.4 31. 7:>l.7>)'.» 38. 1131. IH 4.-,. i.V.)fl.43 5-2. 21-23.72 59. 2733.98 ■H 7tJ0.8().S •M llU..-)9i ■}i l.->l>9.-i8 ■% 2133 94 ■A 2745.57 M '♦)t).l»l»2 .l.< I14i>.0-S» ■A lti(»s.ir> ■H 2144.19 ■A 2757.2 •:4 773.14 4 iir>i;.(;ii> 3* l()17.i>4 21. -.4. 46 ■A 2766.84 .i" 779313 . k< iit;4.i.-)9 ■ H 1()J.').97 .!•.< 2164.76 ■A 2780.51 •:^ 7-?.')..') I •?? U7I.731 ■% l():;4.9-2 ■fs -2175.08 -% 2792.21 •A 7iH.73-> •¥ 117 9. 327 •% lt)J3.si» % 2185.42 ■¥ 2803.93 4 7:>7.97,S 1 •% 1 H0.;u8 i % l(i.VJ.s8 ■A 219.-).79 n^ 2815.67 3>. b01..'>4.") 39. 1 L94..V.I3 40. Kit)!. 91 53. 2-2t)6.19 60. 2827.44 •3-8 8io.r,4.-) ■)8 120J.-J<)3 % lt)7().'.».-) ■A •2-2 16.61 ■'4 2839.23 SlG.'^H.') •1-1 l-.'tt9.'.l.')-' i Ki^O.IM ■H 22-27.05 ■A 285] .05 8'i3.-20U , .% I-217.(J77| llir^H.l •2-237.. 52 ■A 2862.89 ■H 8-Jl>..->78 .K I-II'kVI \ .k. i()9-.-»3 .'.< 2-248.01 ■A 2874.76 ■^4 .<3.n.<)7-2 I ■% li33.1-J-( i ■% 17(17.37 ■''i •2258.53 ■y^ 2886.65 ■A .^i'i.3:t(» ■% l-'IO.'.H) ■% 17 It).. -.4 ■% 2-269. (t7 ■% •289s. 57 4 H48.833 I ■% 1 -.MS. 7 98 ■% 17-J.->.73 % •2-279.64 y^ •2910.51 33. H.V).3(»l 1 40. l'i.')().t)J 47. 173 4.9.-) r.4. 2-290.23 61. •29-22.47 ■H 8.:;9 4 17.-.:!. 1.-) •2311.48 •A •294t).48 874. <) l-i^O.:!l 17ti-j.73 •23-2-2.14 "4 2958.52 .u K81.41.-) ! .u l-i8H.-J.'> ■ 1 .,' 177-J.i).-> .!.< •2332. "!3 ■\i 2970.58 '% S'18.00.") ■% l-.'9<).->l >4 17>1.39 A •2343.55 •'A 2982.67 ■i 8i»4.<)-i ■% 1304. -J ■r: l7.Mt.7ti ■% •2354.-28 ■4 •2994.78 •fi itOl.'J.VJ ■% 131-2.-J1 7> •/8 ISUll. 1 1 .y^ 2365.05 a -^ 3006.92 34. <»07.i)-2-.' 41. 13-iO.-j(; 48. lS(l9..-)f> 55. 2375.83 62. 3019.08 •% 914. Gl ■}i 13-i8.3-> •H lSls.99 ■A 23S6.65 Y^ 3o:u.-26 1^ 9-il.3-^3 ■H I33(i,4 ■H 18.'S 4f) A 2397.48 ■A ■A 3043.47 9-i8.0t) ■% 134 4.r,l ■H i S57.93 ■fi •i4,()5 ■ H 1847.4:) k' 2419 22 ■A 3067.97 4 94l.()09 •% 13H0.81 ■4 l>^.-> ■fx- •2441. '07 ■4- 3092.56 ■fi 9r)'>.:.'r)5 ■% 1377.-21 ■fi 187(;. 13 ■fi 2452.03 ■A 3104.89 AREAS OF CIRCLES. T A B I .E . — ( Cow Hnued. ) 13 Dlam. Area. Diiim. Area. Diam. Areji. Dia m. Area. Dii m. Area. (j:5. _1Z ■?! 17 -'5 70 3'-^-<.46 77. 4tM6.64 84 5541.78 91 6503.9 til 1 t . ••» ■'A •78 .1., : 186-'. -23 il 4671.77 ^ 55.58.29 •^ 6521.78 .8 :{i \-2J)i 3876. 4«)86.9'2 H 5574.82 ^ 6,539.68 :{).'.4.47 3.-89.8 :g 4702.1 H 5591.37 f^ 6557.61 .9:i :i903.63 4717.31 H 5607.95 ^ 6575.56 :{179.4l 1 •/8 71. :',917.49 •fi 4732.54 ^l .5624.56 4. 6.593.54 ri :5 lit 1.91 393i.:;7 •% 4747.79 K 5641.18 ¥ 66]]. .55 64 78 :5-i()4.44 :{'2l7. 394.'>.-J7 3'.'5i>.-2 •^ 78. 47«)3.07 4778.37 85 % 5657.84 5674.51 92 % 6629.57 6647.63 Vq :Vir>[>.i>yi .1., 3973.15 ■>i 4793.7 H 5691. -22 H 6665.7 1^ :w,')i 81 3i)--7.13 4001.13 1 - 4809.05 4824.43 2 5707.94 5724.69 i 6683.8 6701. y3 /a 1 , :!'J(i7.4t) 4015.16 .1., 4S39.83 ^2 5741.47 % 6720.08 5i 3? :5'j~o. 1- ;-2. J(i-2'.».-Jl ■% 48.55.26 fs 575H.-i7 1^ 6738.25 :{j'.i-j.>4 4n4;;.-2l> ■% 4870.71 Va. 5775.1 |4 6756.45 :;;;( 5..'<) 40..7.39 •fa 4886. 18 % 5791.94 % 6774.68 65 ■/c :{.;i'.>..il 4071... I 79. 4901.68 86 5808.82 93 ()792.92 1-^ 3:{:;i <»;» 4(K').(iti •M 4917.21 % 5825.72 <^ 6811.2 i| 3.ii:'>.8 4l>'.t9.S{ 4932.75 -H 5842.64 ^ 6829.49 ■s't :;3.)(!.7i 41 14.01 4 ■}■- 4918.33 % 5^59.59 % 6847..-2 'm. :5:{()9.5i'' 4l-J8.-jr, 41(63.92 •)l 5rt76.5r) K 6r66 16 '^A :!:{8-'.4:{ ■ ■§ '0 414-2.51 ■fs l;t79.55 % 5893.55 t^ 6884..53 ■/a .3^ :;:i'.i.').;;:j 41. 56. 78 •/.I 4l»95. 19 % 5910.58 4> 6902.93 •/4 r.Kt-'.-it) 4171.08 7* •T'a 50 10.^7 ■% 5927.62 % 6921.35 6f> :{i->l.-,* 4185.4 80. 502i'.56 87 5944. 6i» 94 ■ , ti939.79 H :5i;u 17 •74 4199.74 ■H 5012,-28 ■H 5961.. ; 4 6958.26 V 8 :U47.17 :54t;u.i.) 4'J14.1I 42-28.5; ^1 50.58.02 5073.79 5978.9 59ii6.05 6976.76 (i995. 28 ■/8 1-, •.U7A.->i 4-242. '.•;', >1 5089.5!) •ie 6013.22 h 7013.82 78 :'.48ti.;J •% 74. 4-57.37 •^ 5105.41 'A (i030.41 1^ 7032.39 :}199.4 4-27 l.M ■% 51 21. -25 U 6047 63 ii 70.50.98 67 :;r.i-j..v.i :5.VJ...6") 4 jx ;.;!:! 4:!00.-5 •Ja 81. 5137.12 5153.01 88 % 6064.87 60s2. 14 95 % 70ti9.59 7088.24 \Ji !<; :i.'):;S..-^:{ •1^ •% 4315.3.1 •>- 516-.9:! H 6099.43 H 710().9 •0 :\:>:,'2An :{:)65.ji :{57-'.4H 4:!-2'.».'.t6 4311.55 43.51t.l7 51.-4.87 5-24 'A 72.57.11 :56.'.8.44 •4 4447.37 ■H 5313 28 'A ()2.56. 1 5 ¥. 7275.1)9 P„ :?67l.85 44(i-2.1() 4 53-29.44 % 627;'.. 'k'>'.) •>8 4476.98 ■H 5345. (i3 '., 6.;9l.-25 % 7313.84 3.)98.76 44;»l.8l ■H 5361.81 ^ 6308.84 4 7332.8 3: 1-2. -24 •1^ 4506.i;7 ■i 5378.08 H 63-26.44 <^ 73. > 1.79 3'i-25.75 4.5-21.56 7/ •78 5394.34 • fi .•,314.08 y^ 7370.79 69 373;t.-29 76. 4.".3t;,47 83. 54 10.. -,2 90 63t.l.74 97. 7:i8.».,83 i 37.v2.x5 •>8 4551.4 •M 542i;.'J3 H 6379.42 H 7408.89 3766.43 •H 45t.6.36 •>l 5443.-2.i H 6.J97.I3 ¥ 71-27.97 3780.04 •% 4.581.35 3.2 •78 5459.62 % 6411.86 h 7417.08 ^ o 'A 3793.68 •>" 45'.t»i.36 •3:> 547«i.Ol % 613262 H 71 61!. 21 3807.34 '% 4611.39 •% 5492.41 % 6450 4 li 74-5.:56 ;}8-*1.02 ■% 46-2i;.45 •4 550-^.84 % 6168.21 ¥' 7.')04.r):) -k 3834.73 ■X 4641.53 .% 5525.3 % 6486 04 h 7523.75 14 AREAS OF CIRCLES. T\JiLE~{Continued).— {Advancing by a Quarter nnd a half] Diam. Area. 1 Diiim. Area. Diam. Areii. 10207. Oii Diam. Area. 11S82.32 1 Difim. ! Area. ii'^. 7.')1-J.9r( 105 h(>.'i0.03 Ii4. 123. 139. 15171. 71 ■H 75i^.v4 ■U 8;0(t.3-J •% 1025 1.>- ■Va 11030.67 ■}2 15284.0>! 7r).-^i.:)i ■h 8741.7 'h 10296.79 ■h 1 1979.2 140. 153.t3.«4 7HU0.H- -A 87f-3.18 ■% 10341.8 ■h 1 21127.66 v'o 1550.3.0^ ■^> 7620.15 106. 8>24.75 115. li)3;^ti.91 124. 12076.31 141. 1561 4. .',3 •^ imd.b ■% 8866.43 ■M l:^36.8 767f<.2;^ •% r!"50.07 ■% 10522. r;4 ■h I222V.S4 ■'A 1594-:.52 76i>7.71 107. 8902.04 1116. 10..68.34 125. 12271.87 143. 16060.54 ■% 7717.16 •Va 9U34.11 -H 10613.04 .1., I237tt.25 ■^^ 16173.15 7736.6:{ ■h 9{t76.28 ■% 106.-)0.64 12(). ~ 12469.01 144. l62(-6.05 77r)'o.i:{ ■h 91;K53 1 Si 10705.44 ■H 12568.17 •>2 16399..34 ■H 777.').66 108. OltiO.iU 1117. 10751.34 127. 12667.72 14.5. 16.M3.03 •% 7795.2 ■H 9-J03.37 ' -^ '0707.34 ^2 12767.66 ■}2 16627.il "4 1(»0. 7f^l4.7ri h 9245.ii2 : y^ )()r-4;i.43 128. 12H()7.99 146. H;741.59 7f«:?4.;{rt '% 9-J8.i.58 i ..^4 10-r0.ti2 •>2 129()8.71 •3-2 16856.44 7ri.")4. 100. 9331.34 118. 1 0935.0 129. 13069.84 147. 16971.71 •K 7i»9:{.:i2 ■H 9374.10 ■Va 109.S2.3 ■}2 13171.35 .i.f 17087.36 .>o 79:i2.74 ■h 9U7.14 ■% 11028.78 130. 13273.26 148. 17203.4 -fi 7972.21 3/ •74 9460.19 ■% 11075.37 ■H 13371.55 K 17319.83 101. 8011.87 110. 9.>03.34 119. 11122.06 131. 13178.25 149. ": 17436.67 •^ 8or.i.r)8 ■H 9546.69 ■Va 11168.83 •>^ l3.-)t^l.33 •H 175.53.89 •3 80;)1 39 A 9o89.'.'3 •>o 11215.71 132. 136«4.81 150. 17(571.5 •^ 8131.3 ■% 9633.37 ■% 11262.69 v^ i.378.-.t)7 ■'A 177c9.51 102. 8171.3 HI. 9676.91 120. 11309.7 6 133 13902.94 •K 8211.41 .1 9720.73 •% 1 135ti.03 •>2 13997.54 •S 825 1. (il 1 •/Si 9764.29 ■h 11404.2 134. 14102.64 ■^ 8291.91 3> •74 980.>.12 ■% 11 451. .57 M 1420.-.0: 103. 8332.31 112. 98.52.(16 121. 11499.04 135. " 14313.01 •>^ 6372.8 i ■H 9896.{)0 ■Va 11546.61 ■'A 1442(1.14 ■K 8413.4 ■h 9940.22 '% 11594.27 136. 14526.76 ■^ 8464.09 •^ 99>'4.45 Si 11612.09 ■H 14633.76 lOJ. 8494.89 113. 10028.77 122. 116^0.89 13V. 14741.17 •^ 8.")35.7a •H 100:3.2 H 11737.85 ■H 14H4.'^.96 •>•> 8576.77 .h 10117.72 .^■i' 117H5.9:. 138. 149.57.16 •^ 8617.^5 % 10162.34 .% Ile34.06 1 M 15065.73 1 1 To Compute the Area of a Diameter greaterthan any in the preceding Table. Rule — Divide thp dimension by two, three, four, etc., if practicable to do so. until it is reduced to a diameter to He found in ihe table. Take the tabular aiea for the diameter, multiply it by the square of the diviser, and the product will give the area required. Example — What is the area for a diameter of 1050 ? 1050-4-7=150 ; tab. area, 150=17071 5, whichx 7 ^=865903.5. area re jMirei. ;- To Compute the Area of an Integer and a Fraction not given in the Table. ^iTLS — Double, treble, or quadruple the dimension given, until the fraction is increased to a whole number, or to one of thuse in the table, as i. i, etc., provided it is practicable tc do so. T ike the area lor this niameter; and if it i.s double ot that for which the an'ais required, take one fourth of It ; if treble, take one 9th. of it and if quadruple, takj one sixteenth of it, etc., etc. Example — Required the area for a circle of 2. ^''^ inches. 3 Iff X2=45, area for which = 15.0331, wliich -5-4= 3,708 ins. TABLM II. l>. ClfiCUMPEBENCES OP CIECLES, PHOM ^ TO 150. [Advancing by an Eighth.} Diaii t u i I. ■38 ■M ■% •>_ 3/ .1^ ■$ ■'A Ciicuin. .04iK)y .O'JSIT ,19»i3r. .*.J9-.'7 .589 .7854 .9«i7r) 1.1781 1 :!744:> l.i")708 1.7(5715 l.%:;5 •^.1;VJ85 ■l.l\>i'd •2.94.'..'.-) :i. I 4 1(5 :<.:".:n;{ :{.y.'7 4.:{197 4.7 iv: f). IO..I 5.4978 5.-905 6.-i8;;2 (5.(5759 7.(KJH(5 7.4(5i:{ 7..-54 H.2t(i7 8.6:!94 9.(j:{-.'I 9.4-248 9.8175 i(».-l()2 l(t.()0«9 l(>.99:)6 I1.:5ws:{ 11.781 I i. 17.17 Diam. % H ¥ I 1.^ 3/ 7'' >8 Ciicum. Diam. 12.5(564 10. 1-2.959J ■H 1:<.:?518 •¥ l:V7445 ■% H.i:{7-i ■% 14.5-i99 ■'4 14.9-2-26 ■% 15.31.5:! ■% 15.708 11. 1(5 1007 ■ M l(i.4;»:!4 M l(5..-;ft6l ■4 17. -2788 .y> 17.(5715 % l8.0(il-2 % If-f. 4.5(59 ■% 18.849G \-z. 19.-24-J:} ■% I9.«:r> % •20.0^77 % •20.4J04 •% :2(l.8l:?l •% •21 -2058 ■% 21.59."'5 •% ■21 99 1-2 13. •22.:{8:{9 % •22.77<5t5 >4 •23.1(593 % •23 5(5^2 .% •23.9.547 ■% 24.3174 ■% 21.7401 .% •25.13-28 14. •25.5255 % •25.sn8^2 H 26.3109 •% •2(5.7036 % •27.0963 ■% •27.489 3i •27.8817 ■% •28.-2; 41 15. •28.6()71 % 29.0598 ■H •29.4.5-25 .% •29..-^4.52 ■% 30-2379 ■t^ 30.6306 ■\ 31.6233 •% Circum. ' Diam. 31.416 31.8087 3-2. -20 14 32 59U 32.9f56.-5 33.3795 :{3.772i 34.1649 31.5.57(5 34.9503 35.343 36.1-284 36 5211 3(5.9138 37.3065 37 6992 38 0919 38.484(5 38.8773 :j9.-27 :!9.6(5-2'/ 40.0554 40.4481 40.810- 41.-2335 41.(5262 42. (t 189 42.4 1 16 42.-043 43.197 43. .-.897 43.9824 44.3751 44.7678 45.1605 45.5532 45.9459 4(5.33i-«6 46.7313 47.124 47.51(57 47.9094 4-'.302l 4?^. (5948 49.0875 49 4H02 49.8729 16 17 18 19 -20 21 % % % % % !^ H >4 K h n J? Ys H H H % % Circmn. 50.265(5 50.6.583 51.051 51.4437 51.8364 52.-2-291 5-2.6;; 18 53.0145 53 4072 53.7999 54.19-26 54.5853 54.978 55.3707 55.7634 56.1561 56.54-8 56.9415 57.3342 57.7-269 58.1196 .58.51-23 58.905 59.2977 59.(5901 60.08:51 (50.4758 60.86-5 (51.-2612 61.(5.5:J9 62.0466 (52.4:593 62.8:?2 6:i.-2247 (53.6174 64.0101 64.4028 64.79.55 65. iH-^a 65.. 5809 65.97:56 66.:{(563 66.759 67.1517 (57 r '44 67.9:i;i 68. :;-298 68.72-25 Diam. 22 23 •24 -25 •26 K Yt % % 'J ¥■ % K P I H y^ Circum. 69.1152 69.5079 69.9066 70.-2933 70.686 71.0787 71.4714 71.8641 72.2568 .72.6495 73,04-22 73.4319 73.8-276 74.-2^203 74.613 75.0057 75.:,984 75.7911 76.1838 76.5765 76.9692 77-3619 77.7546 78.1473 78.9327 79.:V2o4 79.7181 80.1102 80.5035 80.8962 81. -2889 81.6816 8-2 0743 82.467 82.8597 83.^2524 83.6451 84.0:578 84.4305 84.8232 85.2159 85.6086 86.0013 86.394 86.7867 87.1794 87.5721 16 CniCUMFERENCES OF CIRCLES. TABT.E.— (Conifinucrf.) Diani. Ciiciim. Diam. Ciicum. Dmm. Ciicum. Diam. 1 Ciicum. Dia im. Ciicum. •2'i. ciT.JMUs 3.') lt)'.».9.'it) : 42 l:! 1.947 49. ir.{.938 56 175.93 •<« «5;.:i;.7.) ^8 1 Id .WW \ % 132 :4 K 154.3:;! '% 176.:{22 ■k n-^.VM-i ^4 lh).7ll •>4 13j.7:i.5 ■H 154.724 ■Ya 176.715 Sy. U:>9 •% lli.l:u •% 13.5. 125 ■% 155.117 H 177.108 '% 89..-):}5tJ •% lll.i.27 ■% 1:;3.518 •K 1.55.509 'A 177.5 ■% 8S>.9-283 •% 111.919 •% 133.911 ■^B 155.902 H 177.893 •1^ 90.3-il •% 112.312 ■il 134.303 ■% 156. -295 K 178.286 '% 90.7137 ■% 112.705 ■% 134.696 ■% 156.687 % 178.679 •29. 91.10t)4 36. 113.01W 43. 1; 15. 089 5(t. I..7 08 57 179.071 '% 91.4991 ■% 1)3.49 ■% 135.4^1 ■% 1.57.473 H 179.464 -¥ 91.H91rt ■K 113.'-K{ H l:!5.874 ■H 1.57.865 H 179.8.57 % 9'-i.-JH4."> % m.276 , % i.3i).j67 ■H 1.58.258 H 1«0.249 ■h 9-.'.h:7-> % ll4.()ti< 1 M 136.66 ■Yi l.5-<.651 Ys 180.642 ■H 9;i.0«9'.t 'A ll.).(t.il 1 H 137.0.'.2 ■% 159(144 % 181.035 i^ 9:{.4r)-2H X 11...454 % 137.415 ■% 159.4:{6 yA 181.427 ■% 9:i.8.)r)3 % n...84t) % 137.K}.-! Vb 159.82:? Yb 181.82 :iO. 94.248 37. 116 239 44- 13«.23 51. 16(t.222 58 1V2.213 •H 94.(1407 H IK) 632 ■% 138.623 •M 160.614 H 182.606 •H 95.o:{:{4 K 117.025 % 139 016 ■Va 161.007 Ya 182.998 ■% 95.4-J<)l % 117.417 i % 139.408 .% 161.4 H I83.:i9l 'Vi 9:).8I88 ■Vi 117.81 •K I39.rt0l ■H 161.792 % 183.784 •1^ 9t).-J115 % 1 1-.2U3 % 140.194 ■% l()2.1rt5 % 184.176 ■4 9«.604-> h nf<..-.95 % 141 •..■>H7 Va 162.578 Ya 184.569 o/^ 9B.St9t)9 ■h Jl,-.986 ■% 140.979 % 1(52.971 % 184.962 31. 97.:{896 ;;8 119 301 ' 45. 111.372 52. lH:!.:i6:; 59 lr^5.:{51 •% 97.7823 h 119.77 1 % 141.765 ■% 16:5.7.56 H ln5.747 ■H 98.17;> ■fi 120. I6t) i 142.1.57 164.149 H 1-^6.14 H 98.5H77 % !-i0...59 % 142.55 164. .54 1 % l.--6.53:$ •K 98.9604 ^ 12().9.'.2 M 1 12.943 •% 1()4.931 >2^ 186.925 H 99.3.-):{l 'A 121.314 % 14:{.33t; ■% 165.327 % 1.-7.318 ■X 99.74r>H % 1: 1.7:57 ■^ 143.728 ■h U)5.719 Ya 1H7.711 % 100.1:^85 ■'A 12.; 13 ■'A 144.121 % 16().112 Yb 188.1(»3 3-2. 100..5312 39 12... -.22 46. 141511 .53 1(i6.505 60 188.496 •% 100.9239 ■Vb l22.1)l.-> ■% 141.906 ■% 166.89!? H 188.rt.-«9 :¥, 101.31t5(> ■Va I2:!..i08 ■H 145.299 ■Va 167.29 H 189.2^1 10i.70.»3 ■% 123.701 ■% 145 692 % 167.6-'3 H 189.674 ■% 102.102 ■% 124.093 y^ 14().084 -y^ 168. 07() H 190.007 ■H 102.4947 ■% 121.4S6 ■% 14.1.477 ■% 16H.468 % 190.46 •^ 102.8S74 ■% 124. r:9 ■% lit; -7 ■% l(i-r6I Ya l9(t.8.52 ■ys 103.2801 ■'A 12.-..271 ■% 147.263 ■% 169.254 % 191.245 •.V3. 103.073 40. 125.664 47. 147.655 54. 169.(546 61 191.53a '% 104. 00« H 126057 M 14.-.(tl8 ■H •70.039 H 192.03 % 104.4,58 ■H 126.44;» ■% 148.441 ■ Ya 170.4:12 Ya li>2.423 104.8r)l ■% 126.842 ■% 1 4.-'. 8:53 •% 170.-i25 % 192.816 -% 10.5.244 A 127.235 ■% 149.226 ■y^ 171.217 % 193.208 •% 105.03H 4 127.627 % 149.619 ■% 171.61 % 193.601 •¥ 100.0'.'9 ■% 12>.02 ■% l.O.oll % 172.003 Ya 19:5.994 o ^ 10H.422 % !28.4r! ■% 150404 _.% 172.:i96 % 194.:{87 34. 106.814 41. 128.806 18. 150.797 55. 172.788 . 62. 194.779 ■% 107.207 H 1-9.198 ■M 151.19 ■% 173.1^1 H 195.172 t 107.6 H 129.591 M 151.5H2 •H 173.573 H 195.565 107.5n)3 4 I29.W4 ■K 151.975 4 173.966 H. 195.9.57 '% 108. 3H5 ■H 130.376 -H i:.2.3()8 •K 174.3.59 Y2 196. :15 '% 108.778 ■^ 130 769 ■% 152.76 •^ 174.7.52 Vb 196.743 '¥ 109.171 4 131.1(2 ■fi I. -.3. 1.53 ■H 17.5.144 Ya 197.1:55 -% 109.563 ■% 131 5.-.4 ■% 153 516 •% 175.537 Yb 197.528 CntCUMFERENCES OF GUCLES. TABLE.—iContinued.) 17 Diaiii. Circum. 63. 64. ■% '■H ■h ■% •74 65. ()6. •7a ■% '•% •>4 ■% ■h CT. 6'i. 69. •M .% ■% ■H ■% I :| ■>^ •K '% •% 197.L»-J1 liK:il4 lys.Tue i'.ty.Ui>9 iyy.-49-j 199.^rt4 2UU.-.>77 •-iOO.C)? i>41.06-i 201.4;-);-. •^Ul.C!4ri 202.241 2U2.6:i:{ 20:5, 026 20:}.419 203..- U 204.204 204.597 2(M.9.''9 20.S.3O2 2IM.775 20t). Itid •-:06.;-)6 •^06.y53 207.346 207.73f< 20ti.l31 20ei.r)24 20.-i.916 20i».309 209.; 02 210.095 2i0.4s7 2i0.p>,-i 211.273 21l.oti5 212.0.">8 212.451 212.843 213.-.>36 213.629 214.022 214.414 214.807 215.2 215.;')92 215.985 216 378 216.77 217.163 217.;->;')6 217.948 218.341 218.734 219.127 219.519 Dium. 71 73 /o )8 % 'A 4 H h H % % % ¥ f A % I Circum. 219.912 220.305 2-0.697 221.09 •^21.483 221. ",76 222.2t;8 222.r.t)l 223.051 2v:3.146 223.M39 224.232 224.624 32.) 017 225.41 225.803 226.195 226.5f<8 226.981 227.373 227.766 228.159 22,-^.:.51 228.944 229.337 229.73 230. 122 230.51;-> 230.908 231.3 231.t;'.'3 232.0^6 232.478 23-.'.87 I 233.2()4 233.657 234.049 234. 443 234.«:!5 235.227 235.62 236.013 236.405 236.79s 237.191 23;.;-)84 237.. <76 238.369 23-<.762 239.154 239.547 239.94 240.332 240.725 241.118 241511 Diam. 80 81 82 83 Va Va X I Circum. 2)1 903 2 4-:. 296 242 6."'9 213.(181 243.474 243. .-67 244 259 •-'44.6.2 245.045 24;). 438 245.Ki 246.223 24ti.616 247.008 247.401 247.794 24r*.l.-6 24^^.579 24-.972 249.365 249.7;-)7 2;-.().15 2:)0.543 2;50.935 251.:!28 251.72! 2.52.113 2:)2.506 2.52. >99 2. 52. 2. '2 253. (i84 254.077 254.17 254.862 25.->.255 255 ti48 256.04 2.".t>.433 2.56.-26 257.219 2;-)7.61l 2;58.004 258.:!97 2i-)8.7f*9 2.59.1-2 259.575 2.-.9.967 260.36 260.753 261.146 261.53rt 261.931 262.3J4 262.716 2(i3.109 263.502 Diam. 84. ■K •If e6. •H Va % >8 89. H ■A ■¥ ■H •H ■% Circum. 90. 1 -.■■X 263.894 264. 2n7 264. 6n 26;) 073 2f.5.465 265.. -5.'' 2(16. -.'51 2()6.(i43 ■.;67.h:;6 267.429 267.821 26f«.2ll 2()8.6U7 26-.9it9 26!».:!92 269.785 27().I7H 270.:.7 2T(t.'.)63 2:1.3:6 271.7).-' •J72.ni 272.-534 272.926 273.319 27:: 7 12 274 1(15 374.4'.i7 ■.;74.89 275.2.-^3 2:."..()75 27t).()i;"' •J76. Kil 27().-..3 277.24(1 277.629 27 .-'.032 278.424 •j7«..-^!7 279 21 279.602 279.995 280.388 280.781 281.173 281.5()6 281 .9:59 2e2.35l 282.744 2.-'3.137 2-^3. .529 283.922 284.315 284.708 285. 1 285.493 Diam. 91. 5Z 92. 93. 94 •% 78 ■% 95. ■Va 90. 97. ^8 •% ■Va ■Vs M I Circum. H I 285.886 286.278 286.671 287.064 287.456 287.849 288.242 288.634 289.027 289,42 289.813 290.205 290.. 598 290.991 281.383 281.776 292.169 292.;)62 292.954 293.347 293 74 294.132 294.;525 294.918 291.31 295.703 29().096 296.489 296.881 297.274 297.667 299.059 299.452 298.845 299.237 299.63 300.023 300.416 300.808 301.201 301.;-)94 301.986 302 379 302.772 3(13.164 303.5:57 303.95 304.343 304.735 305.128 305.:-)21 305.913 306.306 306.699 307.091 307.484 18 emCUMFERENCES OF CIRCLES. TABLE.— {Continued.) Diam. l»H. 99. loo. 101. •I loy. •1^ 103. 104. •4 Circum. 307.877 30H.'i7 308.662 309.0:m 3U9.44H 30i),«4 310.233 310.626 3U.0lci 311.411 311.H04 312.196 3l2.r).'S9 312.9.-^2 313.375 313.767 314.16 314.945 315.731 316.516 317.302 31d.M87 318.«72 319.65H 320.443 321.229 322.014 322.799 323.5,-^5 324.37 32.=.. 156 325.941 326.726 327 .5 1 i 328.297 329.083 Diam. 105. 106. Circum. 107 •4 108. •Va 109. 110. •k 111. ■Va 112. •>4 113. .% 1 / 321). ^68 330.(>.)3 331.439 332.224 333.01 33;;. 7 95 334. .58 3;i5.366 3:Ui. l.M 336.937 3:^7.722 33.-'.507 339.293 340 078 340.8ti4 341.649 342.434 343.22 314.005 344.791 345.576 34(i.3til 347.147 347.i(32 34rt.718 349.503 350-^88 .350.074 351.859 352.645 3,>3.43 Diam. Ii4. 354. :r)5. 355. 356. 357. 215 001 786 572 357 115. •k 116. 117. 118. 119. 120. 121. •>4 '-Va ■4 Va '■Ya Va ■Va Va ■Va % ■Va 122. Circum. Diam. 35.-'.142 123. 35«.9:8 ■Va 35y.713 • k 3(;o.499 Va 361.284 124. 362.069 ■Va 3ti2.ft55 .\^ 36:<.64 ■Va 364.426 125. 365.211 ■'4 365.996 126. 36t>.782 ■^ 367.56/ 127. 368.353 y? 369.138 128. 369.923 ■% 370.70<> 129. 371.494 ■y^ 372.28 130. 373.065 •>2 373.N5 131. 374.i;:'.6 ■% 3:5.421 132. .57t)..07 ■% 376 992 133 377.777 ■y9 37.-^.563 134. 379.348 •H 380.134 135. 380.919 ■Yz 381.704 136. 3,^2.49 ■H 3rt3.-<;75 137. 384.061 M 384.846 138 385.631 ■y^ Circum. 3(56.417 387. -.^02 387.9r'» 39i'.773 389.558 3110.344 391.129 391.915 39-J.7 394.271 395.842 397.412 398.983 400 ..54 402.125 403.696 405.-i(i6 406.. -'37 4(if<.40.'S 409.979 411.55 413.12 414 691 416.262 417.833 419401 4-J0.974 42J.545 424.116 425 687 427.2.58 428.H->8 4:!0.399 431.'.»7 433.541 435.112 Diiim. ■'A 139. 14o! 1 I4l/ 1 143!" .1 144.' 145'. 146. 147! 118! 149. 15o!' •>2 M Vr, Circum. % 43ti.6p2 4 38. -.'53 439.824 441.395 4 12 966 444.536 446 107 447.67rt 449.249 4.'.0 r^'l 452 39 453.961 4.')5.532 457 103 4.5H.(i74 460.244 4i;l.-15 463.3nt) 464 9.57 4H6.5-ja 46.r U9J-' 4ti9.669 471 24 472.811 To Compute ihe rirc nm of r Diam: tpr gpfatcp th;in any in the prpp, il n? Table. RuLK — Divide the dimention by two, three, four, etc., if practicable to do ao. until it is reduced to a diiimeter to be found in the t ible Take the tabular circumference for this dmontion. mult ply by 2, 3, 4, 5, etc, according a» it W:is divided, and tlie product will give the c rcurnference rei| ired Example — What is he > ircumfereiice fir a diameter ot 1050 ? 1050-^7=150; tab. circum. 150 =47!, 239, wh.ch X 7=3299.073, circum. required. Toi'ompiite the lippumfer ncf lop an Inli'Sf ami Fra lion not give '■ in Ih** Table. RuLB. — Double, treple, or qiiadr pie ih" dim ntion given, until the fraction is increase I to a' whole nurcber or to one f those in the able, as i, {, etc , provi led it is nractical to do so. ' Take the circiirafer -nces for i is di;im ter ; ami if it i< double of th u for which the circiimference is requir d. tak one hi If of it ; if t cble. ta'e one third fit ; and if quadruple, o .e fourth of 11 ExA.MPLK. — llequired the circ mfcr tice of i il875 inches 2.21875 X 2 =4.4375 =4 ' , which < 2 =8. | ; tab c rcum =27 8817, which-4-4=6 9704 ins. To('ompiile Ihe Circnm of a Ilium Irr in Pec an832 .2 21-J372 I6.:;3.i3 •J 81.713 :5j.oii;; M .UTdi).-^*) .94248 .3 2-J.Oiil- lii.i.504 Si ^:\.:>-S.i 3:.:!.58 .4 A'2Mti l.-J56t) .4 22.9022 1() '.'616 . i h4.:m^8 :!2.(i726 .5 .VM:\'i 1 ..■)70rt .5 •i3.7:8:» 17.-J78.i ..5 f^<) 5903 :5-.'.;».-()- .6 .28-'74 1.8.^5 .6 •-'4 (i3iil l7..".9-.",» .6 HH..'t75 :5:5 :!oo',i .7 .384^5 2.1991 .7 25.5 1:6 i7.'.»o;i . # 89.9 J04 :5:5.615l .8 .r)0-.'()i> v'..".l.;:! .8 •J(i42(H 18.-'21-i 8 9l.(;o9 33.;»-j<»-i 1, .9 .6;!«i: 2.8274 .9 •-'7 :i:;'.i7 ls..,:{.,i .9 93 :!l;!3 :54 24:54 ;! 1. .7.<.-.4 3.1416 6. •-'8.2: 14 IS.S496 11. 95.0334 31.5576;, .1 .9.".0:! 3.45.)7 .1 •J9.2247 19.U»:;7 .1 96 7r.91 :i4..-7I7 .2 1,1309 3 7(i.)9 2 3(t. 1907 19.4779 2 98..5-JII5 35.18.59 .3 i.:;-.'73 4.0-4 .3 :;i i;25 19.792 .3 100. .,-77 :55.501 .4 l.r):!93 4,3982 .4 3'.'.169;» ■.•0.J06J ■ j 1 02.07 05 :i5.8!42 .5 i.7i;7i 4.7124 .5 3:.!s:a V0.1-J04 ..5 lo:?..-(;91 :56.1284 .6 2.01 06 5.(»265 .6 31.21-.' •J0.7::45 .6 1 05.68:: 1 :t6.44-j5 .7 2 2t;i)8 5.3407 .7 35.".'5i;6 21.0487 . / I07.5i:u :i6.7567 .6 2.5446 5.6518 ,H 36.3168 •ji.3t;.8 .8 !0'.>.:r>9 37.(»708 ,9 2.rt:{.')J 5.9<)9 .9 37.3'.fJ8 •.'I 677 .9 1 1 1 .2204 :57.:{84 2. 3.1416 6.V-32 7. 3« 4846 •Jl.9;il2 12. li:!.(»i»7ti 37.6992 .1 3.475 .3 4.1.-)47 7.2256 .3 41.8539 22.9336 .*> 11H.823I .38.61:6 .4 4.r>-i:{;) 7..-.::.98 .4 43.0OS5 23.2478 .4 I24.76::l 38.955- .f> 4.9087 7.854 .5 44.1787 23.. 562 .5 122 7187 :59.27 .(i r>.30i»3 8.1681 .6 45.3647 23 c-<76l .6 124.6901 :5: 1.5841 .7 5.72.');') 8.4823 .7 46..'>()«:5 24.1903 .7 126.6771 39.8983 .8 6. 1 r>75 8.7964 .8 47.7837 24.5044 .8 12"'6799 40.2124 .9 6.60.')2 9.1105 .9 49.0168 24.8186 .9 1:10.6984 40.526') 3. 7.0686 9.4248 8. 50.J6.-)6 25.132-i 13. 13-.'.7326 40.W40:< .1 7.r)476 9.7389 .1 51.53 25.4469 .1 1:14.7S.'4 41.1.549 1 .2 8.0424 10.0531 .2 b'i.'*W2 25.7611 .2 1:56.84- 41.4691 ; .3 8.r>r)3 10.3672 .3 54 1662 26.07.52 .3 1.38.9294 41.7H3i ! .4 9.0792 10.6814 .4 55.4178 2- -..3891 .4 111.0264 41.0974 ; .5 9.6211 10.9956 .5 56.7451 26.7036 .5 14:5.1:591 42.4116 .6 10.1787 1 1 3097 .6 58.0881 27 0177 .6 145.26)75 42.7257 .7 lOwiVil. 1 1 6-.'39 59.4469 27 3319 .7 147.4117 43.03,t9 .8 11.3411 11.93S .8 60.8J13 27.616 .8 149..57J5 43:554 .9 11.94..9 12.2.->22 .9 62.2115 27.9602 .9 151.7171 4:5.6'.8.' 4. 12.r,664 12.5»i64 9. 63.6174 28.2714 14 1.5:!.9:5<4 43.9824 .1 13.20»r, 12. '•805 •I \ 65.0::89 28.5H,S5 .1 l,')ii.l 1.53 44.2965 ; .2 13.8r>44 13.1917 .2 66.4762 28.90-27 158.:!fi8 44.6107 .3 14.522 1 3 5088 .3 67.9.^92 29 2168 .3 160 6061 44 9248 .4 15.2().->3 I3.s-i3 .4 69.:{;»79 29.5:51 4 162.. -'605 45.2:19 .5 ir).9043 14.1372 .5 70.88:^3 29 >^4.".2 .5 165 1303 45.5.5:12 .6 16.619 11.45; 3 .6 72.3824 30.1.593 .6 167.11.58 45.867:5 ^ .# 17.3494 i4.765.T .7 73 39"'2 :J0.4735 .7 169.717 46.1815 .8 18.01(56 '.5.0796 .8 75.4298 30.7876 .8 172.034 46 4956 .\f 18.8574 15.3938 .9 76.977 31 10!8 .9 174.3()66 46.8098 20 AREAS AND CJIRCUMFERENCES OF CIRCLES. TAB LE.— ( Continued. ) Diam. Area. Ciicum. Diiim. Area. Ciicum. Diam Aifti. (ileum. 1.-). I7t>.71.') 47.124 .6 3:!: •.:9J3 64.7161 ^•> 53'.t. 12,19 82.3(199 .1 i:y U71> 47.43-1 330.536 ()5.0311 3 .54.!.-.5:;3 82.(i24 *■ .2 181.1i>rt 47.7523 .H 33;t.7954 65.34.52 .4 547.3.I-J3 82. :{8J .3 lH3.K-,42 4rt.Oh64 .9 343.070.'. 65 6:.94 5 541.5471 KV2.,24 .4 IrH. •.'()..■! 4i-.3Hl6 21. :;46.3t;i4 65.;t7:;6 .6 555.7176 83. 5. i( 1.5 .5 l-'^.();cj3 IH (iiMn .1 ;ii;i.6ii7i> ()6.287 7 .7 5..9.'.'o;i.'' (-3.f-07 .« r.tl.lol'.t 49.00.-59 2 352.1';«01 66.1 '1019 8 5()I.10. 6 -4.1948 » ./ I'Jii.iV.KW 49 3231 .3 35(i.:;-.'H| 66.916 .9 56-. 3232 ^4...09 .H iyt).(i()7j 4y.tp372 .4 359 ().~ 1 7 67.2;;02 27. .572 5. .66 84 ;<-.;32 .9 l9H.;-,.-)r>y 4:».1'.'I4 .5 3(.3.(I5I1 67..-.444 .1 57().-ii5C. H5.1373 IG. 201 (l(i-Jl .50 26.. 6 .6 3(i6. 4362 ()7 rt5-5 .2 5-1.0703 85.4515 .1 •^03..-)H;!ri 50.5711? .7 361t K!7 (;f*.1727 3 585.3. 03 85. 7 ().'.*> .2 2()ti.l2(i:5 .•.o.rticiy .« 373.2.5:i4 6-.48.;8 .4 5 :'.6)69 86 0798 .3 2(irt (;:-,".> 51.20(< .9 376.6-56 (.8.^01 .5 593.95."<7 h;.394 .4 211. --Mil 51 5224 22. 3-o.i3::«. (i9.1152 .6 .59f*.28C.3 *6.7lt8l .5 213sv;5l 5I.KM4 "a 38;!..5'.t72 69.4293 .7 602.6j'.'5 87.0223 .6 21ti.4->lrt 52.1505 .2 3-7 {»765 (i'J.7 435 .8 6ii6.98.-<5 87.3361 .7 •-•1U.(M()2 .")2.4647 .3 39(1.5751 7 0.0;.7ii .9 6I1.;;()32 87 .6506 .« 2-il (1712 52.7788 .4 394.0823 70.3718 •JH. 6 15. 7. .36 .87.9618 .9 2-.M 3H 53.(t9.{ .5 3l)7.6(W7 70. (H() .1 620. 1596 88.2789 17. 2v .9 411.-716 71.9426 .5 637.9411 89.5:!56 .4 2:!7.7.-'77 54.603ci 23. 415 47(56 7J.25()8 .6 642.42.'.: 89.8497 .5 2J 56.2346 .5 431!. 7371 73.8276 .1 9«>'..0845 91.42(»5 18. •jr)4.4():M> 56.54.-^8 .6 437.4363 74.1417 .2 669.6(.34 91.7347 .1 257:5018 56.8629 .7 441.1511 74. 4. -.59 .3 «)74 •j.".8 92.04-8 M 2t;o. ir.ort 57.1,71 .8 444.rtdl9 74.768 .4 678 8t;K{ 92.:i()3 .3 263.0226 57 4912 .9 448.62."^3 75.0d82 .5 &-3.4943 92 3772 .4 26.').y(»5 57.H)54 24. 4:.2.3904 75.3i 93.(iii»6 274.()46r) 58.7479 .3 463.7708 76.3408 .9 702 1.554 93.93:i8 .8 277.r>917 59.062 .4 467.5957 76.6523 30. 706. '-6 94.248 .9 280.r):)27 5i).37t52 .5 4;1.43()3 76.96t>2 .1 711.5.-02 94. .562 1 19. 283.5294 59.6904 .6 475.2926 77.2e33 .2 716.3162 94.8763 .1 286..V2I7 60.0(i45 .7 479. 1646 77.5i>75 .3 721.0678 95.1904 .2 2'^9.529-i 60 3187 .8 4^3.0524 77.9116 .4 725.8352 95.5046 .3 2-.12.5.536 60.6:'.28 .9 4rf6.".'558 78.2258 .5 730.6183 95.8 ia8 .4 295.5931 60.947 25. 490.875 78.54 .6 735.4171 96.1329 .5 21W.64.S{ 61.2612 .1 494.rt098 78.8541 .7 740.2316 96.1471 .6 301.7192 61. .5753 .2 49-S.7604 78.1(i93 .8 715.0(il8 96.7612 .7 301 '-06 61.8895 .3 502.7266 79.4H24 .9 749.9077 97.0754 .8 3l)7.<)0-(2 •i2.2()36 .4 50t).7O86 79.796<) 31. 754.7694 97.3896 .9 311 0252 62.517H .5 510.7063 ^0.1108 .1 759.6467 97.7U37 20. 3N.)6 62.832 .6 514.7196 8*i.424S •> • • 764.5397 98.0179 .1 317.3094 •63.1461 • < 518.74H,8 80.T:{;U .3 769.4485 98.332 .2 320.4746 63.41)03 .« 522.7936 8l.0;'.32 .4 774.;:729 9-'.6452 .3 323 1)554 637744 .9 526.854 1 Kl.3()74 .5 779.3131 98.9(i04 .4 326.H52 64.0H86 26. 530.9304 f 1.6^ 16 .6 781.2689 99.2745 .5 330.0643 t)4.4028 .1 :)55.022:'. 81.9976 .7 789.2406 99.5887 AREAS AND ClRCTTM?RRE5Cra OP ClftdtlSS, TABLV:.— (Continued.) 21 Diam. Are; I. Circum. Diam. Area. Circum. Diam. Area. Circum. .8 79A.'2'Z7rt 99.9028 .4 1098..5H0-,' 117 4958 43. 14.5-2.-2016 l:55 0-^MH .9 7y'.».-j:<0rt 100.217 .5 i 104.40-7 117.nl .1 I4.'.8.;t608 1:55.40-29 32. 804.2-4'.>tJ 100.5312 .6 1110.:J07l 1 18.) -241 .2 1405.7148 1:55.7171 .1 80.».2rt4 1oo.h4.'>:j • • nio.-j.^ii 118 4:583 .3 1472.-5:585 1:50.0:5:52 .2 Hl4.:W4l 10 1.1.595 .8 1122.2109 1 18.7.-)-24 ,4 1479.:548 i:50.:5».54 .3 81'.'.39y9 101.4730 .9 ll2rt. 1.504 119.0000 .5 14-0.1731 1:5(5.0.596 .4 «-J4.481.T 101.7478 :{8. 11:54 1170 119. tHO^ .6 14.»3.0l:59 1:50.9737 .5 H2y..")7'^7 102.102 .1 1140 > (40 1 19.0949 ,7 I4.n'.."'705 l:!7.-2879 .6 834.H9I7 102.4101 2 1140.0"<7 l-20.Mii;ii ,8 !.50t5.742: 1:57.602 .7 S3y.'i-i03 102.7:U)3 .3 1 1.52.0954 1-20 :5-2:52 .9 151:5.0-2.-7 13;.91(J2 .« 84 -4.9(147 J 03.0444 .4 115H.li;»4 l-20.(;:574 44 1.5-20.53 n l3-'.*2:!04 .9 850. 124^ lo:{.3.')eO ..5 1104.1. ".9 1 1-20.9510 .1 I5J7.4537 l:»-'.5415 33. 855.3UU«) 103.0728 .6 117 0.2 145 121.-20.57 .2 1.5;;-! 3.8M< l:5H.-.5H7 .1 8G0.492 103.9869 .7 1170.2-.57 |-.'l..57;'9 3 1.5:!1.:5:5<»0 1:59.17-28 .2 rtti5.69y2 104.30! 1 .a llrt2.:5725 121..S94 .4 l54f<.;!(K)I 1:59. l-'7 .3 870.9-J22 104.0151 .9 11;^H4051 l-.'2.-J0-'2 .5 i.'..)5.-2''8:5 l;;9.-oi-2 .4 87b.h)UH 104.9294 3l\ 1194.54:54 1-22.. 5-224 .0 1.502.28(52 140 11.5:! .5 881.4151 105.2430 .1 12(K».72T3 l-22.-i;;o5 ,7 I..09.2998 1 40. 1-295 .6 8>-^.6H5l 105.5577 .2 1200.877 1-2:!. 1.507 .8 1570 :5292 140 7i:;o .7 891.9709 105.rt7l9 .3 121:5.0421 1-234. .4-1 .9 1.5.-';i.:!742 141. ('578 .8 897.-.>7->3 100. 1-0 .4 1219.224:! 12:5.779 4.5. l.V.K).i:!5 1 1 ! :572 .9 9U-i.5'^95 100.5002 .5 1225.4211:5 l-24.09:!2 .1 1.)97..".I14 1 1 1 t)-01 34. 907 ifi->4 I00.dl44 .0 1231.0:528 124 40:3 •) lOOl.OiKiO 14 2. 00(13 .1 9i:).-J709 107. 1285 .7 I237.>i)l 1-24.7215 Si 101 1.7! N 14 2.3144 .2 9l8.ti:{.Vi 107.4-J72 .8 1244.121 125.0:5.50 .4 101:5.8:55 14.'.(i280 .3 9J4.0il5 107.7506 .9 125(1. :5u40 l-2.).:549"^ .5 1025.9713 142.94-28 ] .4 929.4109 lO.i.07 1 40. 1250.64 1-25.004 .0 l(.:53. 1-293 1 4:!.-2.5(>9 .5 9a4.rt2v;3 h>fi.\ir*:>2 .1 I2r.2 931 1-2...97-1 .7 IO40.:!02 li:5..5711 ^'{ 940.-J494 10.-S.0993 •_> 1209.2:5.<.-S l-2(). -292:5 .8 1047.4.-40 143.8-52 .7 945 (iiCi-i 109.0352 .3 127.'..5t)0J l-20.OO()4 .9 l0.54.J)8-5 144.1994 .8 951.1508 109.3070 .4 12«1.^9H4 l-2t'). 9-2(16 4(). 1001 '.fOfU 144.51:50 .9 956 &>b 109.0418 .5 12-iH.2523 1-27.-23 18 ,1 l(i09. 1:599 144.8-277 35. 902.115 lUt>.C5.)0 .6 i -294.02 1;» lJ7.r)48;> /2 1670.:5-i»l 14.5.141.) .1 907.0200 110.2701 .7 i:5o:.oii7l 127.80:51 !3 li 183 0511 145.1.50 .2 973,142 110..-.-43 .8 I:5o7.40n2 1-28.1:72 .4 1090.9:! 17 145.7702 .3 978.079 110.8984 .9 l313..-^-J49 12-.4914 .5 1098.-2:!! 1 1 ir..O-44 .4 984.2318 111.2120 41. i:!->o.-i571 1-2.-!. 80.50 .6 170.'.. -4:52 llO.:59-5 i .5 9'"i9.800.5 HI. >20S .1 i:;j().7().55 129.1197 .7 1712.87! 14ti.71-27 .6 995.:iS45 111.8109 .2 i:{:5:!.ltu»:5 I29.4:;2:5 .8 17-20.2141 l47.0-2u8 .7 l000.9«43 112 1551 .3 1:::51>.01H'.) 1-29.74'* ,9 17-27 57:10 147.311 .8 looo.o 112.4092 .4 1310.14U l:5(».(Mi-22 47. 17:54.91-^0 147.(15.52 .9 1012.2313 112.7fS34 .5 1:5.52.0.551 l:5(i.:5704 .1 1712.:5:!92 147.9093 30. 10l7.87.-^4 113.0970 .6 i;j.59.1-18 l:50.0',t05 /> 1749 7455 11 -'.-28:55 .1 10v!3 54n 113 4117 l:!«)5.7242 13l.(t047 !3 I757.1(i75 i48..5i»76 .2 10-'9.2195 113.7-259 .8 1:572.28-22 1:51.:5I88 .4 1704.0045 !4M.i)ll8 .3 1 034.9 i:U 114.04 .9 l;57f'..-<.50 131 032 ..5 1772.()5H7 1I9.-J26 .4 1040.02:!5 1!4.:{542 42. r.5H5.1450 l:;l 9472 .0 1779.5-279 14.»..5:5,!l .5 1040.:U91 114. 00-^4 .1 i:592.05(W 132.21113 .7 17-'7.0l-27 149.8.543 .6 1052.0904 114 9825 .2 l:59-'.07!7 1:52.. -.7.55 .8 1794.51:53 1.50.10.84 .7 1057.8474 1 15.2907 !3 ll0.5,:{('-'3 l:52.88,tO 9 l-'02.0-29() 150.4^-2.5 .8 1003.02 115.0UH .4 1411.9007 1:53. -20:59 4-^. l-iO'.»..-.r,16 150.7968 .9 1009.4081 11.5.;«25 .5 1418.()-i.-^7 l:53..M8 1 1817.1092 I51.U09 37. 1075. -J 126 110.2392 .0 14-25.:;r2.-. 1:53. -;52l •1 18-21.07-20 151.4-251 .1 1 OS 1.0324 1 10.55:'.:! • / 14:5-J.(U19 1:54.1103 .3 18:52.-2518 151.7:592 2 10-0.8079 110.^)75 .S 14:58,7271 l:{4.4004 .4 1 -i:59.8400 152.05:54 .3 1092.7191 117.1816 .9 U 45. 4.58 134.7746 .5 1847.4.570 152.3676 22 ABEAS AND CIRCUMFERENCES OF CIRCLES. TABLE.— (Continued.) Diiitii. A re; I. Circum. Diam. Area, Circum. Diam. Area. Circum. .»'. K..-..o-:w hVJd^l? .2 •.':{(l7.-2'224 170 '2747 .8 2f^0''.6218 187.H576 .7 1 >').'. 7-.'..;; i.V.'.. '.tiV.t .:{ .:li...7t4 170.588:5 .9 -Jr'l-.023 1-.-!. 1^*18 .« H7(t :i-j;' l.v.{.:{i .4 •.•;{j».-2-i:j lio.'.io:5 (10. •2^27 44 H-.496 .!» l>;rt.u;)ti;! !.'):< »ijj-.' ..5 •2:!:V2 >:u:< 171.2172 .1 •2'-:56 h;-2() l:rr.HlOl •I'J 1 'r^;> 7 1.") I l..:{.'.':{-» .() 2:ui.4o:{ 171.5:54:5 .2 •2-4(5.321 I89.r24:i .1 l>ii;;.-iani I5J.-2.VJ.". • t •.':n9.;(rt74 17l.84."i5 .3 28.-|5.7K) 189.4:584 .'2 ii»t»i 1:01; I.'i4.."irt'>7 .r( 2:r>8.5rt76 172.1596 .4 :28<55.-26l8 lK.t.75^.'6 :.\ l'.)(H.'.l(»(i-^ l.".1.8>t)8 .9 •2:{(i7.-2o;!4 17-2.47:5^ 5 •2ri74.7«5(»:5 190.0(56^ A i.ti().ti").-i; 15... hi.-) 55. •2:J75.HX. 17-2. 7 8ri .6 •2''8 4.-J(51.- l;i0.:5809 .;"> I'.iji <'ji;:< 15.'... 092 .1 •j:in4.4-^2^2 17;« 1021 .7 289:5.7984 190.69.M .-.':U l..«.4..1f. .4 J 110 51H2 I7i ol I'i (11 29^2-2.4 731 191 6:576 •U 1 ;».').').()").!« |. lb. 7 5.58 .5 2I19.-J^2-':'. i:4.:!"'-^8 .1 293-.'.( 16:51 191.9517 r)0. I'M ;;{..'. 157.08 .6 •2427. ..5 11 1 7 4. 1 17^29 •2941.(5(5-5 192. -.659 .1 l;»7l.:{riM l5:.:",tMl . 1 •.'i:{''.(59.5"' 1 7 1 977 1 !:1 .'95l.-2-'97 192.58 o VJlUS.i '4 l.'.7.7ii8;! .r •2445.-i:.2H 175.:!092 .4 •2; '(50 9-J65 i93.''942 !:1 19-^7. l:!-Jti L.>.0J-24 .9 •245 J. •2-2.57 ir.-,.6l.54 .5 •29a)..".791 19:!. ■.'0-4 .« l'.t9:) ojio l.')H.:};U)() 5(5. •24(.;:.oi4i 175.9-29.5 .6 •2:'f'0.2474 l9:i.5-2--'5 .;") •.'90J. '.•'><);; i:.8.6.">()- .1 •247l.^l«7 17(5. •24:57 .7 •29ff9.93l4 I9:5.K5t57 .tt •JO 11 (.90(17 15- ui;i<) •2 24-(I.(.:J-7 176..5.,79 .8 •2999.(5:5 1.14.1508 .7 •J(tli-I.^li-J-' 1.. 9. -2 791 !:5 •24-9 47 15 176.-72 .9 :5(t(t9.:5l64 194.465 ,h •j()-i;.-:ui; 15.).5.':?2 .4 •24.ir<.:i2.5'.l 177.1^62 62. 3019.077(5 194 ; 792 .!» •2o:{ 1.^77 1..9.9074 .5 -2.-.()7.1'.f:{l 177.5001 .1 :5()-28.82l4 195.09:53 f)!. •J04-J.."j-.':>4 l.i02J6 .(5 -251-,).(I7() 177..- 145 .2 :50:5n.5869 1;»5.4()75 1 •i(l^^().^4 4:! UiO ..:5.)7 ,7 -2.52 1.'. '7. ;6 178.1-2-7 ..{ :504-.:i65i 195.7216 .'.i •^0.').S..-^>4 1(50.8499 .fi "25 ;:!.>H.>^,-i 17r'.4 428 .4 :}(i.--'.l.v9i 196.0:5.58 a •.>o(ii.'.»'j:>;{ IGl ICl .!» •25r2.8l>8 178. 7. .7 .5 :5(K57.9»i.-7 196.:<5 .1 •j()7i.;>9."):'. }(;i.47rt-2 57. •2.'.5 1.7(5 1(5 17;).o;i2 .6 :5077 7.M1 19(5 61.41 .5 •j-)h;;.();71 l(i 1,7924 .1 •2.-|60. 7-2*5 179.:5S5:5 .7 :50-'7 6311 li>6.97.-<3 . ... •2.-)7"< (>9.V.> 18ir.or,;() .9 31o7.:5641 197.60(5(5 .rt •ii(i7.4it;ti 1()2.7;U8 .4 •2587.7045 l8o.:5-2;8 (5:5. 31I7.^2.V26 19J. 9-208 .9 •jii.'> T)*)!.: 1():5 049 .5 •2.5'.»6.7^287 l.ri'.642 .1 31-27. '..■.64 19-s. -2:549 r>2. •>i-.'.{.:-jiii l(i:;.:5():{-> .6 2605. 7 6S7 180.. '.■.61 .2 3i:i7.07.58 198..5491 .1 •.'|:!1 >97i"> )('•:{. ():7:{ . 1 •2614.824:5 IH|.'280:5 .3 3147.0114 198.^(5:52 .> •214(1.089:? 1():V99:?.'> .8 •25 5(;2:{ J', 26()9.4.-l^<2 18:5.1:..52 .9 :5^20(5.9.->31 •200.7482 .ti •il!<9..'.ti9r) l(!5.,'»7til .4 2('i78.6.5:?f 18:5.4(^94 (54. 32lt599-i4 •201.0(5-24 .9 •>l:i7..-71J lt;6.iy(«) .5 •2«5K:.8.;5i 18:5.78:16 .1 3227.059:5 •201.:i765 5S. •-i-.'(K).l-r:() 165.. 5(148 .6 •2697.o:i2l 184.0977 .•2 :5-23:.i:!(5 •201.6907 .1 •2-2l4..'>-Jl(i I6ti8189 .7 •27ti6.2449 = 84.4119 .3 3247.2284 •202 ( (048 .2 •^•i-.'-'.-7U4 lH7.i:{..l .8 •27l5 47:;:i I.H4 7'2() .4 3257.3:5(55 202.319 ,:{ •2^:51. •.>:?.') lt;7.447-2 .9 •27 ^24. 7175 18...0402 ..5 :5'267.460:5 202.(.:i:52 .4 •2-j.:9.<)ir.-i 1()7.7614 59. 27:i:{.9774 185.:5514 .6 :5277.51)98 •202.947:5 .r> •^•J4H.(tlll l(i8.075(> 1 274:5."25^29 1,-5.6(585 .7 3287.755 203.^2615 .6 •i-i.".t). i'2-J7 ?G-.;i897 /2 27..-2.5142 I85.98-J7 .8 :5-297.92i5 203.57.56 .7 •.'•J()4.8.01 1 6-5.7049 .:i •2761.8512 186.-2t59(5 .9 :i: 108. 11-26 203.r'.-^98 .8 •i-i7:i.-,>9:{l !n'.'.018 4 •2771. 17:59 l-(5.iill 65. :i;il>'.3i5 •204.204 .9 :iJKI.7r)19 I69.:'.:i^22 .5 27^0.51 •2:5 lr6. 9-25-2 I :5:52-.5:54 •204.5! 81 54. 2-i.'0.'iStU 169.(5464 .6 2789.8(564 187.-2:{9;J •> 3:i:59.7668 •204.8:523 .1 2-298.7165 169.9605 .7 2799.2^62 187.5535 .3 3349.0162 205.1464 AREAS AND CIRCUMFERENCES OF CIRCLES. Z5 TAB\E.~ (Continued.) Diam. .4 Area. Ciicum. Diiim. Area. Ciicum. Dium. i Area. Circum. 3:{r>;>.2.«'l4 m').46or. 71. 39.59.2014 2-i3.053(i .6 4C.<)8.:5816 240.64(ih .5 3:Mi'.».:)»)2:; 2(».').774H .1 3;»7ti.;i619 22:! :{(;77 .7 4620.4218 240.9607 .6 :{:{7i».-.>y 20<;.tt~'^9 .2 39rti.,-.:{rtl 22:{.6n:9 .8 46. .2.4776 24l.-.'74H .7 :j;W(».17I2 2(16.403 i .3 3'.t'.)2.73()l 2-j:{.i»'.'t) 9 4644 .5492 241.6987 .8 :{4()(i 4:n>2 206.7172 .4 4tM)39373 2J4.310-J 77. 46.56.6: :66 241.90:52 .9 :{41(>.f*42U •J07.03;4 .5 4015. hill 2-M.<.244 .1 4(;ii8.7:59)i 242.2173 m. :M2120J4 2(ir.:u.')6 .6 40'.'6.4002 224 93-^5 .2 46.-0. rt5-^3 242.5:515 .1 :{j:5i.r)77r. 2{)7.6:.97 .7 4037. ♦i.55 225.2527 .3 4692.9927 242.-4.56 .2 :{-i4i.w.{;» 207.1>::il> .8 4018.9-'.54 2-.'r..:.«>68 .4 47(15.1429 24:5.1.598 .3 :i4.")2.:;749 208.288 .9 41160.21 16 2-.'5.o-l .5 4717.;5087 243 474 .4 :}4t)-i.7y7l 20.-*. 6022 72. 4(171.5136 220.1 952 .6 4729.4903 243.75V-1 .;') :547:5.j:{5l 2U8.ltJti4 .1 408.'.8332 2-.'6..".093 . 4 4741. (1875 244.1023 .6 ;)4.->3.»W'!8 201) r>;{0.'. .2 1094.1645 226.82:J5 .8 475:5.9605 244.4 i64 .7 :i41»4.l64 20y.544«i .3 4!(»5.5)-^ 2J7 13:6 .9 476»i.l292 244.7:506 .8 :«t)4 .6-1:52 209.-5-8 .4 4116.0793 227.4518 78. 477rt.:i7:i6 245.0 148 .9 :ir>ir>.i4:{ 210.173 .5 4lJ8.2i587 2-J7.7tifi .1 4790.6:5:56 245.:5589 67. 3.')2.'>.6«i06 •J 10.4^7 2 .6 4l31).(i5J4 'Zr>!i 0801 .2 4p02.9()1)4 245.67:51 .1 :{r):«).l9-J(? 210.rt(»13 .7 4l5l.(J6ti7 22^ :{94:{ .3 4815.201 245.9872 .2 :{o4ti.7404 •.ill. NT).-) .8 4lti2 4943 2-Jrt 7081 .4 48J7.5082 246.:5014 .3 ■.\:>:^7 :m:i 211.4-29(; .9 4l7:<.i);!76 2-il). 022(5 .5 48:59.8311 246.615H .4 Iir)H7.MS.>7 211.7438 73. 4lrt5.39b6 2J9.:5:i68 .6 485:. 1697 246.9297 .5 3r)78.4787 212.0.58 .1 4196.8712 229.«)5(»9 .7 4864.-524 1 247.2439 .6 3r,8y.o.ii)r> 212 3721 .2 4208.::614 2-.'9.965l .8 4876.8973 247.548 .7 :{^.yi^.7l;V.> 212.tW63 .3 4219.0678 2:W2792 .9 48h;».2799 247.8722 .8 :it)io.3."..-Ji 2 13. 00' 4 .4 4231 3896 2:{0..59:{4 79. 4901 6814 248.1864 .9 :w-ji.ui6 2l3.:;i4ti .5 4-42.9271 2:!0.9()76 .1 4914.01)85 248.5005 68. :{():{ i.e-'ittt 2!:{.6-J8.-i .6 42.. 4. 4803 2:51 2217 .2 4926.5.514 248.8147 .i :5642.:{788 2t3.;)429 .7 4.:(i6.t)49:> 23l.5:!.5'.i .3 49:589-^2 249.1288 .2 3H.):{.(to38 214.2.571 .8 42;7.6;!39 2:!l.-5 .4 4951.4443 249.443 .3 3tk»:{.'S((4 214 ..712 .9 428, •.234.5 2:!2 1642 .5 496:5.9243 249.7.572 .4 3H74.r)4l 211.8454 74. 4300.8.>()4 2:52.4784 .6 4976.484 2.50.0713 .5 3Hs5.-il»;U 215 1996 .1 4;il-.'.-|.-'-,'l 2.;2.7925 .7 4988.9314 250.3855 .6 3691 i. 01)6 215.5137 .2 43-24. 1 •j;)6 •J3:!.l((67 .8 5001.4.586 250.6996 .7 370ii.844.') 2:5.8si79 .3 4335.792-' 2:5.;.42()- 1 .9 5014.0014 251.01:18 .0 3717.6437 216.142 .4 4347.4717 2:54.7:55 60. 5026.56 251.:{280 .9 37J8.4r)87 216.4562 .5 43.59.16ti3 .1:54.0492 .1 50:i9. 1:542 251.6421 69. 373y.-^894 216.7704 .6 4370.8766 2:54.:5633 .2 5051.7242 251.9.563 .1 37r)0. 1357 217.0^45 .7 43rt2.602b 2:{4.6775 .3 5064.3258 252.2704 o 3760.9;t78 217.3987 .8 4394.3448 234 9916 .4 5076.9552 252.5846 is ;]77 1.6756 217.7128 .9 4406.1018 2;{5.3058 .5 5089.5883 2.52.8988 .4 .3782.7H91 218.027 75. 4417.875 235.62 .6 5102.2411 1 53.2129 .5 3793.6783 218 3412 .1 4429. 66:« 2:J5.9341 .7 5114.909^ ' 253.5271 .6 3804.6032 218.6.553 .2 4441.4684 2:56.2483 .8 5127.59:{» 253.8412 •^ .« 38)5.54;{& 218.9695 .3 4453.2886 23ti.5624 .9 51^0.2937 254 1554 .8 :iH26r)002 219.2836 .4 4465.1246 236.8766 81. 5153.0094 254.4696 .if 384:.4722 219.5978 .5 4476.9763 237.190O .1 5165.7407 254.7837 70. 3848.46 219.912 .6 4488.8437 237..5049 .2 5178.4877 2;.5.0979 .1 :«.-»y.4s>.-,2 220.2261 .7 4500.7268 2:57.8191 .3 5191.2505 255.412 .2 :W70.4d26 220.5403 .8 4512.6256 2:58.i:5:J2 .4 5204.0285 255.7262 .3 3881.5174 220.8544 .9 4524.5401 2:58.4474 .5 5216.8231 256.0404 .4 3892.56S 221.1686 76 4536 4704 238.7616 .6 5229. 6:« 256.:i.545 .5 39' (3 6343 2214828 .1 4.548.4163 2.39.0757 .7 5242.4586 256.6687 .6 3!)14.7I63 221.7969 .2 4560.37-7 2:J9.38i»9 .8 52.55.2998 256.9828 .7 :i9-.i5.Hl4 222.1111 .3 4572.3553 2:59.704 1 .9 5268.1568 257.297 .8 3936.9274 222.42.52 .4 4584.:i583 240.0182 1 82. 5281.0286 257.6112 .9 3948.0565 222.7394 .5 4596.3571 240.3324 .1 5293.918 257.9253 24 ABEAS AND CIRCUMFEREIfCES OF CIRCLES, TABLE— (Continued.) Diam. .2 Area. Circum. Diam. Area. Circum. Diam. Area. Circum. r):{0«>-^2l 2r>8.2395 .8 (5054.5149 275.8324 .4 6851.4840 29:5.42.54 .3 r):U9.74:{U 25H..V)3H .9 6068.3224 275 14(56 .5 (W06.1631 29:5.7:596 .4 r>:!:w.()7:.'. 2:.8.-i;40 88. 6o82.l:«70 27(..4(508 .6 68.S0.rt579 •294.<»5:n' .5 r);M:..«-,'rt7 25;». 182 .1 601»5.y(584 270.77 19 .7 6''95.5(585 '294:5679 .6 r>:{:.8.r)iir)7 •2.')9.J90l .2 0109.^15 2r7.0«9l .8 (5908.-2947 •21M.(5H2 .7 r):}7i.r)98:{ •J.59.810:', .3 61-23.6774 277.40:i2 .9 69-25. 0;i(>7 294.9962 .8 5:w4.r)7«y 200.1-244 .4 6137.5554 277.7174 94. (59:59.7944 •295.3104 .9 r):5y7..')9(»M 2()(t.4380 .5 6151.4491 278.0:516 .1 6954.r.077 295.6245 83. 5410.6'.'()(; 20»».7.V.'H .6 01(55.3585 278.:{457 .2 69(59. :55<5'-' 2t(5.93-(7 .1 54-.':{.tiG») 201.0009 .7 bl79.2e37 278.(5.-,99 .3 6it84. 1(514 •2'.»6.24:{0 .2 i')4:{»).7-.'7"-i •itii :mi .8 0193.2245 -27r<.975 .4 099S.9821 •2".»6 507 .3 5441>.MH-.' 201.0'.».2 .9 0207.1811 279.2H82 .5 701:5.8183 •29«i.8Hl2 .4 (.4l)2.8'«W -2(;2.00'.»1 89. 6-221.1534 279 0024 .6 70-28.6702 297.1953 .r> 5470.00.-) I 20-2.3-230 .1 (5-235.1413 279.9105 7013.50-i5 -297.5095 .6 54-^9. 1-4M 2(i2.<):{70 2 6-249. 145 280.2:507 .8 7058.418 297.^2:56 .7 5r>o2 'i«j8;» -202.95 lit M (5203.1(544 2-'0.5448 .9 7073.:J-202 '298.1:578 .8 5:)l5.4-.>4;5 2t;3.-20» .4 (5-277. 199.-, 280.8.')9 95. 7088.'2:55 '298.452 .9 55'^8.5958 203.5802 .5 (5-2i' 1.203.-) 2Hi.;7:v2 .1 710;J.1(554 '29H.7661 84. 5.')4i.7rt-^'4 203.8i)44 .0 O30.-).3l0ci 2^1.4H73 .2 7118.1110 299.0723 .1 5r.54.9847 204. 20<) .7 6319.399 28l.882.-> .3 71:53.07:54 •299.:5944 .y i>bt}6.W.\2 2(»4.5-2J7 .8 (53:53.497 2H2.1ir)6 .4 7148.051 •299.7080 .3 55«1.4:57-J 204 83on .9 ():U7.6813 2rt2.4-298 .5 7163 0443 :500. 0-228 .4 5594.(i8til> 205 151 90. 6:5(51.74 282.744 .6 7178.05:53 :5()o,:5:569 .5 r>tio7.9:.2:} 2()5.4052 .1 6375.885 •28:5.0581 .7 7193.078 :50O65ll .6 ;)0-jJ.'2:{:{4 265.779:! .2 0:!90 04.")8 2rt3.3723 .8 7'208.1184 :500.9()52 .7 5G:{4.56-'ii 2(i6.0lt35 M (5404.-2-2-22 2«3.68(54 .9 7-22:5. 1745 :50l.-2794 .8 5(547.84-^8 200.407ti .4 (5418.4144 2M ()0t;(5 96. 72:58.2464 :50i.r)9:50 .9 5»il) 1.171 200 7218 .5 64:{2.6-22;{ 284.3148 .1 7-25:5.3:5:59 :50l.9O77 85. 5«)74.5I5 207.03(5 .6 6440. H474 •2H4. (5-289 .2 7-2(58.4371 :!02-22l9 .1 56-*7.Hr4G 2t 57. 3501 .7 (54(51. 08.-)2 284.9431 .3 7-2S3.5561 :502.5:5(5 .2 5701.25 2(57 .(504 ;5 .8 6475.3402 2rt.-).-2572 .4 7298.6907 30'2.8.")02 .3 5714.t)41 2(57.9784 .9 64rt9.01(J9 2-5.5714 .5 73 13.8411 -.503.1044 .4 57r.'8.0478 208.29-20 91. 6.-)(»:{.«^(574 285.rtH.-)6 .6 7:529.0072 :50:5.4785 .5 5741.4703 2()8.(5(»0.x .1 (5518.1995 280.1997 .7 7:544.189 :503. 79-27 .6 5754.9085 268.i>-2(i9 ') 05:V2.5173 286 5139 .8 73.->9.:5804 :504. 10(58 .7 57(58. 36-.'4 ■269.-2351 .:{ (554(5.^^9' »9 2-6.829 .9 7:574.5996 :504.4-.'! .8 578l.8:!2 •269..')492 .4 «V>01.-2081 287.14-22 97. 7:5<9. 8-286 :50» ::i.52 .9 5795.:517:{ 2(59.8034 .5 6575.5(5.-) 1 2-7.4.-.64 .1 7405.07:52 :505.0493 86. 5f 08.8184 270.1770 .6 65h<).9458 •2f?7.7705 .2 74-20. :5:5:55 :i05.:5(5:]5 .1 58-2-i.3:551 270.4917 .7 66(»4.:52-22 288.0847 .3 74:55 6095 :5O5.(5770 ^2 5835.8G75 270.8059 .8 6(5 IH 7542 •288.:59r>8 .4 7450.9013 :505.9918 !3 5&49.4157 271.12 .9 6(5:5:5.182 2-8.713 .5 74(56.2087 :50i;.:50(5 .4 586-i.97i»5 271.4342 ' 92. (5647.0:550 289.0272 .6 7481.5319 :5O6 6-i0l .5 5876.5591 271.74M4 .1 (56(52.0848 2H9.3H3 .7 7496.8707 :}06.93(53 .6 5890.1541 272.0()»i5 .2 6(570.5597 •289.(5:)55 .8 7512.2-253 :507.2484 .4 5903.7(i54 272.37(57 .3 0(591.0101 289.9096 .9 75-27.. 5950 :507. 5(5-26 .8 5917.392 27-2.6908 .4 6705.5507 290.2838 98. 7.542.9816 :507.^708 .9 5931.0344 273.005 .5 67-20.0787 290.598 .1 7558.:58:;2 :508.1',t()9 87. 5944.69:i(i 273.3192 .6 67:!4.6l(55 -290.8.3(J44 273.(5333 .7 6749.1699 291.-2-2(53 .3 7589.--'3:58 :;0S.8l<»2 , .2 597-2.0559 273.987.-, .8 0763.7:591 •291.5404 .4 7(504.682(5 30i).r.534 1 .3 5985.7091 274. -261 (5 .9 6778.:524 291.8546 .5 7(5-20.1471 :509 4470 ; .4 5999.4821 274..5758 93. 6792.9-246 •292.1(588 .6 76:55.627:5 :509.7017 ! .5 6013.2187 274.89 .1 6807.5408 •2i>2. 48-29 .7 765l.l9t53 310.0709 .6 6020.9711 275.-2041 .2 68-22.173 •292.7971 .8 7(566.9349 310.:5H5 .7 6040.7391 275.5183 .3 6836.8296 293.1112 .9 1 7682.1623 310.7072 AREAS AND CIRCUMFERENCES OF CIRCLES. S5 TXBl.E.— (Continued.) Diatn. Area. Circum. 1 Diara. Area. Circum. Dium. Area. Circum. W. 761)7. :o:)4 311 0184 .4 7760.0347 312.275 .8 7H22.61.-4 3i:5.51l6 .1 77l:!.-i»i4l 311.:{:!25 .5 7T7.').6.'>«i:'. 3IJ..>'92 .9 7H:{r!.299H 313.H458 .2 7728.r:5:5t) 311.6467 .6 779l.2i>:j.; 3I2.90:13 100. 7.-54. 314.16 .3 7744.4^S8 31l.%0i 7 T«(»6.91 ut the art*a orcircumfierence as for a Whole Number by removing the decimal point, it' for the area, iwo plaws to the right ; and if for the circum- {| ferenc< ;, one place III U8TRATI0N.- -The area of 96.7 is 7344.18!) ; hence lor 9t>7 it is 734418.9 ; and the circum-II feireaw iof96.7ia3( )3.7927, and for 967 it 18 3037.9^7 TJ^BI^V. IIIX. ABEAS AMD CIECUMPERENCBS OP CIBCLES PROM 1 TO 50 FEET. (Advancing by an Inch.) - Diam. Area. Circum. Diiim. Area. Circum. Dium. Area Circum. Feet. Feet. Ins. Feet. Feet. Ins. Feet. Feet. Ins. 1ft. .7854 3 m 'Aft. 7.0686 . y ='' 1 oA 19.635 15 8}i I .9217 3 4% 1 7.4666 9 81^ 1 20.2947 15 11^ 2 1.069 3 8 7.8;5: 9 11>^ 2 20.96:>6 16 2g 3 1.2271 3 11 3 '<,295r 10 Jla :< 21.6475 16 5^ 4 1.39fi2 4 -^H 4 rt.72t)5 10 5% 4 22.34 16 9 5 1.5761 4 5% 5 9.16rt3 10 S^ 5 23.0437 17 H 6 1.7671 4 6^ 6 9.6211 10 11% 6 23.7583 17 32 17 6^ 7 1.9689 4 11^ 7 10.0846 11 3 7 24. 48:^ 8 2.1816 5 2% 8 10..v>;»l 11 6)^ 8 25.2199 17 9^ 9 2.4052 5 5% 9 11.04 46 U 9% 9 25.9672 18 S 10 2.6:598 5 9 10 11.5-IO;t 12 }4 i 10 26.7251 18 :{Jg 11 2.8852 6 2>i 11 12.0481 12 :{%! 11 27.4943 18 7>| 2/'. 3.1416 6 3% 4A 12.5664 12 63^4 6ft 28.2744 18 10>| 1 3.4087 6 6)4 1 13. 09.. 2 1-^ 9J8: 1 29.0649 19 \H 2 3.6869 6 9% 2 13.6:5:.3 13 1 2 29.ri668 19 4% 3 3.976 7 H 3 14.1^62 13 i}i ;5 ;50.6:96 19 7i| 4 4.276 7 3J^ 4 14.7479 13 7ii\ 4 31.5029 19 10^ 5 4.5869 7 7 5 15.3-^W) 1:5 lOig I 5 32.:«76 20 ig 20 4% 20 8)1 20 llH 21 2% 21 8^ 6 4.9087 7 lOH 6 15.'.)()43 14 l^ 6 33.1831 7 5.2413 8 1% 7 16.4986 14 45^ 7 34.0:591 8 5.585 8 4^ 8 17.1041 l-l '^^ 8 34.9065 9 5.9:595 8 7% 9 17.720.. 14 11 9 :i5.7H47 10 6.:!049 8 10^ 10 18.:i476 15 2^ 10 :56.6T35 11 6.6813 y IVb 11 i.'.a.-58 15 51^ 11 37.5736 se AREAS AND dKCUMFERENCES OF CIRCLES. TABLE.— (Conftnoerf. ) Diam. 7/t. 1 3 4 5 6 7 8 9 10 11 8/<. 1 y 3 4 5 6 7 8 9 10 11 9/<. 1 2 3 4 5 6 < 8 9 10 11 10/t. I 2 3 4 5 6 7 9 10 11 ll/K. 1 2 3 4 5 6 Areiu Feet. 3."j.4846 : 40.3388 41.2rt25 42.2367 43.2J2J 44.1787 4o.l6r)6 46.1H38 47.173 48.19-J6 49.2236 r.1,3178 f.2.3816 5:5.4r)62 54.r)412 r)5.^377 56.74r)I 57.^628 58.992 60.1321 61.2826 62.4445 63ol74 64.W006 65.99r)l 67.2007 68.4 lt)6 69.644 70.8823 72.1309 73.391 74.662 75.9433 77.2362 78.i>4 79.854 81.1795 82.516 83.8627 85.2211 86.5903 e7.9«>97 89.360.i 90.7«;27 92,1749 93.5986 95.0334 96.4783 97.9347 99.4021 100.8797 102.3689 103.8691 Circum. Feet. Ins., H 11^ 21 22 22 22 23 23 23 23 24 24 24 24 25 25 25 25 26 26 26 26 27 27 27 28 28 28 26 29 29 29 29 30 30 30 30 3». 31 31 31 32 :52 32 :?2 33 33 33 34 34 34 34 35 35 35 35 36 ll^« i 23^" 9^ H «•% y>2 'A 3^ 5 2% 5>o •-^ 11% H ^% % '% lit' Di'im. 7 d 9 10 n 12A 1 'I 3 4 5 6 7 8 9 10 II 13/i!. 1 2 3 4 5 6 7 8 9 in 11 Wft. 1 2 3 4 5 6 7 8 9 10 11 l.-./i. 1 2 3 4 5 6 7 8 ;> 10 II 16/)!. 1 Aieii. Feet. I05.::794 106.;'013 io.-'.4;;42 lUi».9772 111.5319 113.0976 114.6732 IJ6.2»)07 117.rt.59 119.4674 121.0876 122.71^7 I24.359rt 126.0127 127.6765 129.3.")(»4 131.036 132.732«) 134.4391 136.1574 137.8S67 139.626 141.3771 143.1391 144.9211 146.6949 148.4rt96 150.2943 152.1109 153.93-4 155,7758 157.625 159.4852 161.35.-.3 163,2373 165.1*13 167.0331 168.9479 170.8735 172.8091 174.7565 176.715 17-^.6-32 l-0.6()3l 182.6..45 lfr4. 65.5.5 lJ^ti.6684 lS?<.6'.t23 4,0.72() 192.77 1() 194. -282 i9(i,-'.»4t; 19?^. 97 3 201.(1624 203.1615 Circum. Feet. Ins. :«) ^% 36 '% 36 lO'^B 37 '^/^ 37 .^K 37 ^% 37 n\^ :{8 '^% 38 '-% \\6 ^% 39 :i9 ■^H 39 «% 39 91^, 40 'A 40 :i% 40 ^% 40 10 41 1^ 41 4% 41 :i-o 41 i^»% 42 1^8 42 •»% 42 8 42 llfft' 43 ^M 43 -h 43 «^ 43 11% 44 ->8 44 6 44 y^ 44 H 45 SH 45 45 4t> 46 46 46 47 47 47 47 48 4^ 48 48 49 19 49 50 51 1 50 y% 4 IlK IK *\ '1% 2% 5^- ^^ Di.'im. 2 3 4 5 6 7 8 9 10 11 17/i; 1 *.» .4 3 4 5 6 7 8 9 10 11 18/it. 1 2 3 4 5 6 7 8 9 10 11 v^ft. 1 2 3 4 5 6 7 8 9 10 11 20/K. 1 2 3 4 5 6 7 8 Area. Circum. Feet. 205.2726 2(»7.;{'.>46 209.5264 211.6703 21:>..-!251 215.9896 21-!. 1662 220.3537 222.551 224.7603 226.9«06 2-i9.2l05 231.4525 2;};;.7055 2:5.5.96-2 23y.243 240.5287 242.8241 24.5. l:U6 247.45 249.7781 252.11-4 254.461)6 256.rt303 259.20:i3 261.. 5872 263.9807 2t)<).3S64 26-!.b031 271.2293 273.6678 276.1171 278.5761 281.0472 2S3 5294 286.021 288.5249 291.0397 293.5641 296 1107 298.6483 301.20.54 303.7747 .306.355 308.9448 311.5469 314.16 316.7824 319.4173 322.063 324.7182 327.3&58 330,0643 332.7522 335.4.525 Feet. Ins. 50 9>^ 51 \^ 51 'i% 51 6>^ 51 10 52 1^ 52 41^ 52 7% 52 I 01^ 5 4% 53 8 .53 11^ 04 'Z% o\ 5% r>4 8K 54 11^ 05 2% ;)5 6 55 9>^ 56 y^ 56 :\% nb 6>^ 56 9% 57 4 57 7^ n7 log 58 1% 08 4^-^ 58 7% 58 10% 58 2 59 5>^ 59 Hl^ 5y iiK 6it ^ly 60 5% 60 8% 60 U% 60 3)^ 61 6W «>1 >2^ 61 3% 62 6% 62 9% 62 1>^ 63 4^ 63 73^ 63 11>^ «3 \% 64 4% 64 7% 64 \\% AREAS AND cmCTTMFERENCES OF CIRCLIS. 2T TABLE.— {Continued.) Diam 1) 10 H -n/t. 1 •J 3 4 5 6 7 8 9 10 II 2-2/t. 1 2 3 4 5 6 4 8 9 10 11 23/lt. 1 2 3 4 5 6 7 8 9 10 11 24/!!. 1 2 3 4 5 6 7 8 9 10 11 2r,yi. 1 •> 3 Area Feet. :MO.f'rt44 3J3.t.i;4 :i4ti.3(il4 :i4'j.iiJ7 :{.') 1.^-04 :ir)7.44:w ;?t30.-J417 :5(W.05ii 3H8.T011 :571.r>4:}'2 374.;iy47 3T7.-.'587 3.-0.i:53t) :ia3.oir7 385.9144 :i88.ri-2-^ 3^1.7389 394.t)(J83 397.6087 400.5583 403.W04 406.4935 409.4759 412.4707 415.4766 4184915 421.5192 424.5577 427.6055 430.H658 433.7371 436.8175 439.9106 443.0146 446.1->78 449.2536 452.3904 455.5362 458.6948 461.8642 465.0428 468.2341 471.4363 474.6476 477.8716 481.1065 484.3.")06 487.6073 490.875 494. 1516 497.4411 500.7415 Ci.cuin Feet. Ins. t-5 53 R 6;, .->4 65 115 m 57,„ 66 9 66 i,i 67 3% 67 ti>^ 67 9^^ 68 3/ 68 m 6s 7 68 H)i^ m 1% 69 4}4 69 7% 69 10% 70 1% 70 5 70 8>^ 70 11)^ 71 2^ 71 5% 71 8% 71 11^^ 72 3^" 7'i 6U 72 90 73 y, 73 3% 73 6g 73 9>^ 74 1 74 4^ •74 7>^ 74 10^ 7o \% 75 4% 7;> il 76 51^ 76 8>^ 76 11.^ 77 2^ 77 9 ' 7H K 78 31^ 78 6K 78 9^,' 79 % 79 3J^ Di itii. ;> t> 7 '.) 10 h 26 y<. 1 2 3 4 5 6 7 8 9 10 II 27/i!. 1 2 3 4 5 6 7 8 9 10 11 28/c. 1 2 3 4 5 6 7 8 5» 10 11 29/(!. 1 2 3 4 5 6 7 d 9 10 A.ea. Feet. 5(»l.0..1 .M', :.\',:\2 .,!'). ;<)():! :)14.(l4.-^4 517.4034 .V.'0.769-> 5-.'4.l441 5v:.53H 53(1.9304 534.3:579 53/ .7583 541.l-i)() 54 4.1 29.t 548. ('i'3 551.5471 555.0-JOl 5.")8.5)»59 56J.0027 56.").5084 569.027 572.5.)66 576.0949 579.6463 5H3.'J085 586.779ti 590.3()37 593.95-57 597.5625 601.1793 604. R07 60'i.4436 612.931 615.7536 6J9.42-J8 623.10;. 6-^6.7982 H30.r.002 634.v;l5'2 637.9411 641.6758 645.4-J35 649. 1-^21 652.9495 6.56.73 660.5214 664.3J14 668. 1346 r>7l.95^7 675.7915 679.6375 6-3.4943 687.:;598 691 --'385 3/ % 11.^/8 3 ^% % 3>^ 9% 4M ^h 11% \% */^ 11 •-^ "^ «% 11>2 9 ¥^ 3W «% 9>^ 1^ 4% 10% Di iin. Area II 'Mft. 1 2 3 4 5 6 7 8 9 10 11 31/i! 1 2 3 4 5 6 7 8 9 10 n 32/«. 1 2 3 4 5 6 7 8 9 10 11 33//. 1 2 3 4 5 6 7 8 9 10 11 Uft. 1 2 3 4 Feet. 70j.!t:i7: 71 ().-(; 7ii).:;:i9 714.7:>5 7 1 -.69 7-j2 t)537 7j6.r,:;it5 730.6183 734.6147 73"'.t;242 74-J.6I47 74H 67:'.i-i 750.7 161 754 7(>9l 7.",f<.831 1 762.906-.' 7(it).992l 771.08()6 775.1914 779.3131 783.4403 787..580.-i 791.732*i 795.8922 800.0()54 804.2496 808.4422 812.6481 816.865 821.0904 825.32t)l 829.5787 833.8:}68 838.1082 842.3095 846.6813 850.9855 855.3006 ^^59.624 863.9608 868 3087 872.6649 877.0346 881.4151 885.804 890.2061 894.6196 899 0113 903.4763 907.9224 912.3767 916.(^415 921.3232 Jt2.5.-103 930.:51(t8 Oin uni Feet Ins. !'3 117^ 94 '8 94 6 91 «tj^ : 95 % 95 3U 95 6^1 95 9^ 96 ■'A 96 4 9) 714 96 1034 97 IJ-o 97 4% 97 "!% 97 K'% 98 2 98 0^ 98 ^% 98 111^ 99 2% 99 5% 99 «% loo 100 ^% loo ^ 101 % 101 m 101 6% 101 10 102 ^H 102 4% 102 ^% 102 10% 103 i^ 103 4% 103 8 103 11^ 104 2J4 104 fi'4 104 H% 104 11% 105 2% 105 6 105 9J^ 106 .^ 106 106 6% KKi 9% 107 0% 107 4 107 71^ 107 \(\% 108 1% t8 AKEAS AND CIRCUMFERENCES OP CIRCLES. TABLE.— (Continued.) Diam. 6 7 8 9 10 11 35/i!. 1 y 3 4 5 6 7 S 9 10 11 36/it. 1 2 3 4 5 8 9 10 11 37/it. .t. 2 3 4 5 6 7 H 9 10 II 1 2 3 4 5 6 7 « 9 10 n Area. Feet. 934.8223 93y.34v!l 943. (-753 9484195 9r.2.972 957.538 9t)2.115 9()«.770l 971.2989 975.9085 980.5264 985.1579 989.8U03 994.4509 999.1151 1003.7902 1008 473(5 1013.1705 1017.B784 1022.5944 1027.324 1032.0646 1030.8134 1041.5758 1046.3491 l051.i:{U6 10.'....9J57 1060.7317 101)5.5459 1070.37:{8 1075.2126 lU~0.0594 lU84.9vi01 10,-97915 1094 6711 l()99.5ii44 1104.4687 1109.3'- [ 1114.. 071 1119.241 1121.1H91 Il2;i.l478 il::4.ll76 i!39.(i95;{ ll44.ii>tW Il49.(t892 11. "4.09.17 ll.".9.lj:i9 1164 1591 Ilti9.-J02:! 1174.2592 1179 3271 1181.103 1189 4927 il94.;)9.{4 Circum. Feet. Ins. 108 108 108 109 109 109 109 110 110 110 HI HI ill 111 112 112 112 112 113 113 113 113 114 114 114 J 14 115 115 115 115 116 116 116 117 117 117 117 IH 118 118 118 119 119 119 119 1-^0 120 120 120 121 121 121 in 122 122 4^ 2 ^% 11% -'% y% 4 iS^ '\ 11>8 •;% il>8 '^% h '^% 4^ l^il % -% 11% 61^ Diam. 1 2 3 4 5 e 7 8 9 10 11 4oy<. 1 3 4 ' 5 6 7 8 9 10 II 41/i!. 1 3 4 5 6 7 8 9 l"t 11 42/;;. 1 2 3 4 5 6 7 8 9 10 11 43/if. 1 2 3 4 5 6 7 Area. Feet. 1199.7195 1204.8244 1209.9577 1215.099 1220.2..42 1225.4203 1 -.'30. 5943 223.").7^^22 1240.981 1246.1878 1251.4084 1256.()4 l261.^7:t4 1267.1327 1272.397 1277.6092 12^2.9553 12'^8.«2;V2:i 1293.5;.72 I298.87ti 1304.2057 1305.5433 1314.8949 13v:0.v;574 I325.(i-J76 1331.0119 1336.4071 1341. .-101 1:547.2271 l:{.")2.6551 1:558.0908 1:563.5406 1:569.0012 1:574.4697 l:579.95Jl 1:585.4456 1:590.2467 1:596.4619 1401.9.-8 1107.5219 141:5.6098 14)8.6287 1424.1952 1429.7759 14:55.:5675 1440.9668 1446.5802 1452.2046 1457.8:565 146:5.4827 1469.1397 1474.8044 14^0.48:53 14-6.1731 1491.8705 Circum. Feet. Ins. 122 12:5 123 123 123 124 124 124 124 125 125 125 125 126 126 126 126 127 127 127 128 128 128 128 129 129 129 129 130 i:50 1:50 l;50 131 131 131 131 132 i:52 132 132 1:53 133 i:j:5 1:54 1:54 134 134 135 135 1:J5 1.35 1:56 1.36 136 136 9>^ •IK '% 10>.4 f •/8 11 2W -% 8>2 11^8 i*>8 9 !>>8 \^ '■% I'Vs 1% % 1>8 '^^ %\ ^K ^%\ 1.1% fi%: %| :^%! 9% 1 10% 1% 4^ Diam. 8 9 10 11 1 2 3 4 5 6 8 9 10 11 45/if. 1 2 :5 4 5 6 7 8 9 10 11 46A 1 2 3 4 5 6 7. 8 9 10 11 47/;. 1 2 3 4 5 6 7 8 9 10 11 48/t. 1 2 Area. Circum. Ffiet. 1497.5821 1503.:5046 1509.0:548 1514.7791 1520..-):544 1526.2971 15:52.0742 15:57.8622 154;5.6.")78 1549.4776 15.55.-.!883 1561.1165 I5t;6.9591 1572.8125 1578.6735 1584.5488 1.590.4:^5 1596.:52.-^6 1602.2:566 1608.1555 1614.0819 1620.0226 1625.9743 16:)1.9:!:54 16:57.9068 1643..8912 1649.. -831 1655.8.<92 16.11.9064 1667.9:508 1673.9698 16800196 1686.0769 1692.14-5 1698.2311 1704.:521 1710.4254 1716..->407 1722.66:54 1728.9005 147 Feet. Ins. 137 137 137 137 138 138 138 139 1:59 139 1:59 140 140 140 141 141 141 141 141 142 142 142 142 143 143 14:5 143 144 144 144 145 145 145 145 146 146 146 146 147 2% 8% 11% ■-^^ '»% 9 % '% fi% 9% 3% 7% 10% 4% ^^ 1% 5 «% IIM •^% h% ^% 11% 3 6% y% 17:54.9486 1741.10:59 1747.2738 1753.4545 1759.6426 1765.84.52 1772.0587 1778.2795 1784.5148 1790.761 1797.0145 1803.2826 1809.5616 1815.8477 1822.1485 147 147 148 148 148 148 149 149 149 150 150 150 150 151 151 3% 6% 9% 1% 4% '0% IJ-a 4% li^ 2% H% UK 2% 5% 8% 3W 6% 3% SIDES OF EQUAL SQUARES. TABLE.— (ConHnued.) 29 Diam. Area. Circum. Diam. Area. Circum. Diam. Area. Circum. Feet. Feet. Ins. Feet. Feet Ins. Feet. Feet Ins. 3 18-iH.4«0-2 151 6% 11 1X79.3355 153 81^ 7 1930.9188 155 9W 156 ^ 4 !«:{4.77»1 151 1U>^ 49/^. 1885.74.54 153 1V4 8 1937.3159 5 !841.17-J7 152 ii^ 1 1892.1724 154 2% 9 1943.914 156 3>^ 6 1847.4571 I'o-i 4% '2 1898.5041 154 5>^ 10 1950.4:»2 156 65-^ 7 18:.:{.80rt7 152 7^4 3 1905.0367 154 8^ 11 1956.9691 156 9% 8 I. ->(>(). 175 15'i 105^ 4 I911.4lt(i5 154 11% 5oy?. 1963.5 157 % 9 iHtlti.fwVJl 15:; 1^ 5 J 9 17.9609 155 2% 10 l»7-i.y3t)5 153 3>g 6 1924.4263 155 6 TABLE OP THE SIDES OP SQUARES-EQUAL IM ABEATO A CIRCLE OP ANY DIAICETEB. FROM 1 TO 100. Diam Side of Sq 1. ■'A 5:^ 6. .88()2 1.1078 l.:;293 1.55U9 1.7724 1 9m 2.2ir,6 2.43ri 2.65h: 2.f*f<02 3.1018 3.:'.233 3..0449 3 7665 3.988 4.2096 4.43il 4.6527 4.8742 5.0958 5.3174 5.5389 5.7605 5.9-^2 6.2036 6.4251 6.6467 6.8683 Diam. Side of Sq. Diam. 8. 9. ■H 10. 11. •y^ 13. 14. •Va 'Va 7.0898 7.3114 7.5329 7.7545 7.976 8.1976 8.4192 8.6407 8.8623 9.08:{8 9.3054 9.5269 9.7485 9.97 10.1916 10.4132 i 0.6347 10.>'563 11.0778 11.2994 I ! .5209 11.7425 11.9641 l2.18.-)6 12.4072 12.6287 12.8503 13.0718 15. 16. 18. ■Va ■% ■Va ■Va ■Va 19. 20. ■Va I Side of Sq. Diam. Side of Sq 21. •Va 13.2934 13.515 13.7365 13.9581 14.1795 14.4012 14.6227 14.8443 15.06.59 15.2874 15 509 15.7;:05 15.9521 16.1736 16.3952 16.6168 16.8383 17.0.-)99 17.2814 17.503 17.7245 17.9461 18.1677 18.3892 18.6i09 18.8323 in o:.39 19.2754 22. ■Va 23. 24. '•Va 25. ■Va Ya 26. •Va 27. ■Va 28. 19.497 19.7185 19.9401 20.1617 2U.:W32 20.6048 •J0.8263 21.0479 21.2694 21.491 21.7126 21.9341 .'2.1557 22.3772 22.5988 22.8203 23.04 19 23.2634 23.485 23.7066 23 9281 24.1497 24 3712 24.5928 24.8144 25.03.-)9 25.2575 25.459 Diam. 29. "6 ■Va 30. 31. 32. ■Va ■Va ■Va % I Va Side of Sq. 33. :J4. ■Va ■Va 35. •Va 2 .7006 25.9221 26.1437 26.3653 26.5868 26.8084 27.0299 27.2515- 27.473 27.6947 27.9161 28.1377 28.3593 28.5808 28.8024 29.0239 29.24.55 29.467 29.()886 29 9102 30.1317 30.:J533 30.5748 30.7964 31.0179 31.2395 31.4611 31.6826 30 LENGTHS OF CIRCULAR ARCS. TAB LE.~(Conttnued. ) Diam. Side of Sq. Diam. Side of Sq. Diam. Side of Sq J Diam. Side of Sq. Diam. Side of Sq. 36. 31.9042 49. 43.4251 62. 54.9461 75. 66.467 88. 77.9)58 H 32. l-^.")? •Va 43.6467 -Va 55,1676 ■h 66.6886 y 78.2095 A 32.3473 '% 43.8682 ■h 55.3892 y t)6.9 1 04 y 78.4316 % 32.r)t)ti8 rA 44.0898 ■H 55.6107 ■U 67.(317 y 78.6526 37. 3VI.7904 50. 44.3113 63. 55.8323 76. 6:.:5532 89. 78.8742 M 33.0112 ■Va 44.5329 ■H 56.0538 ■H 67.. 57 48 y 79.0; '57 % 33.23:J5 ■% 44. 7.545 4 56.2754 y 67.7964 y 79.3173 Va, 33.45.-)l ■% 44.976 ■% 56.497 _u 6''.0179 y 79.5389 ■38 33.<>7H6 51. 45.1976 64. 56.7185 7<. 68.2395 90 79.7604 % ;;3.dy82 •Va 45. ' 9! 'Va 56.9401 y 68.461 y 79 982 % 34.1197 -'A 45.6407 •y^ 57.1616 y 68.6820 y 80.2035 \ 34.3413 ■Va 45.r622 -Va 57.3832 ■h 68.904 1 y 80.4251 39 34.r)628 52. 46.0838 65. 57.6047 78 69.1257 91. 80.6467 Va 3 i. 7884 ■Va 46.3054 •^ 57.8263 y 69.3473 y •'0.8682 'A 35.006 ■% 46.5J69 ■A 58 0179 K 69..5t) -8 y 81.0898 % 35 2275 •Va 46.7485 'Va 58.2691 ■^l 69.7904 y -1.3113 40 35.4491 53. 46.97 m. 58.491 7 '. 70.(»iI9 92. 81.5329 H 35.6706 ■^A 47.1916 yA :)8.7125 ■'4 70 2335 y 81.7544 A 25.892i ■% 47 4131 ■H 58.9341 .1., 70.455 ■A -1.976 % 36.1137 '% 47.6347 ■Va 59.15.56 ?4 70.6766 y 82.1975 41 3«).3:i53 54. 47.K562 67. 59.3772 r-u. 7o.8;i81 93. 8i.4191 Va. 36.5569 ■Va 48. ,-778 ■% 59.. 5988 ■H 71.1197 y 82.6407 ■ A 36.7784 ■% 48.2994 •A 59 8203 1/ 71.3413 ■A 82 8622 % 37. ■Va 48.5209 ■Va 60.0419 •?4 7 1.. -628 y 83.0-38 42 37.2215 55. 48.7425 68. 60.2()34 1 7!.7i<44 94. 83 3053 Va 37.4431 •Va 48.964 •K 60.485 .I4 72.0059 y rt3.5269 % 37.6649 •H 49 1856 •K 6i'.70ii.") >^ ; ; 2.227.-. y n.1.7484 A 37.8862 •% 49.4071 •Va 6(/.928l ■H 72 4191 ■A 83.970 43 38.1078 56. 49.6287 69. 61.1497 &i. 72.671/6 95. e4.l9i6 Va 38.3293 ■ Va 49.- 503 Va 61.3712 y 72.8.'21 y 84.4131 % 3rt.5;.09 A 5(1.0718 •y^ 61...92-< y 7.;. 1137 y 84.6317 ^ 38.7724 J/a 5(».2934 •Va til. ."^143 y 73.:{3.>3 ■A -4.8.5«!2 44 38.994 ot . 50. ."1149 70. 62.03.59 8.!. 73.55(>8 m. 85.0778 Va 39 2155 M 50.7365 •K 62.2574 y 73 7784 y 85.2993 % 39.4371 ■¥ 5U.958 •>2 62.179 y 73...9it9 y 8.>.5209 Va 3J.65-7 ■Va 51.179t) ■% t)2.700i» y 71.221.. y 85.7425 45 3i».8802 58. 51 4(»12 71. 62.9221 rt4. 74.4431 97. 85 9616 :» 40.1018 ■Va 5L.6227 ■yA (i3. 1 ):;7 y 74.6647 y 86.185 40.3233 ■H 51.8443 ■% 63.36.52 y 74.'^8.)2 y 86.4071 % 40 5449 •Va 52.06.58 ■Va !>;. 5-6^ % 75. 1077 y 86.6289 46 40.7664 59. 52.2874 72. ..3..-'o.-!;; 85. 75 3293 98. 86.8502 •M 40.9.-^8 ■Va 52.50-9 •M 64 029.* .14 75.5.508 y 87.0718 y^ 41.2096 ■'4 52.7305 •>2 64.25 1 y 75.7724 y 87.2933 ■Va 41.4311 n^ 52.9521 Va 64.47.;0 .'4 75.9934 y 87.5449 47 41.9527 60. 53.1736 73. 64. 91 76.21.55 99. 87.7364 ■Va 41.^742 ■Va 53.3952 •>4 64.9 itii H 76.4371 y 87.958 •>2 42.0958 % 53.6167 ■% ()5.1377 A 76.65-6 y 88.1796 .% 42.3173 .•% 53.8383 ■Va 65.3.592 y 76.8rt02 y 68.4011 48 42..->839 61. 54.0598 74. 65..580H t 77.1017 100. 88.6227 'Va 42 7604 1 54.2814 yA 65.802:1 ■H 77.3233 y 88.8442 14 42.982 54.503 •K 66 0239 ■A 77.5449 y 89.0658 'Va 43.2036 ■yA 54.7245 •Va 66.2455 i y 77.7664 y 89.28?4 TABLK VI. TABLE OP THE LENGTHS OP CIRCLAB ARCS. The Diameter of a Circle assumed to be Unity, and divided into 1000 equal Parts. Hght. Length. : r-1 H'ght. Length. H'ght. Length. H'ght. Length. H'ght. Length. .1 1.02645 .148 1.0^743 .196 1.09949 .244 1.15186 .292 1.21381 .101 1. 02098 .149 1.0.5819 .197 1.10U48 .245 1.15:508 .293 1.2152 .10-2 l.<>27.")2 .15 1.05896 .198 L. 10147 .246 1.15429 .294 1.216,58 .103 1.0'>806 .J5l 1.0.5973 .199 1.10247 .247 1.1.-.549 .295 1.21794 .104 1.0286 .152 1.06051 .2 1.10348 .248 1.1,567 .296 1.21926 .105 1.0-2914 .153 1.0613 .201 1.10447 .249 1.1.5791 .297 1.22061 .lOt) 1.0297 .154 1.06209 .202 1.10548 .25 1.15912 .298 1.22203 .107 1.03026 .l.n5 1.06288 .203 1.1026 .303 1.22918 .112 1.03312 .16 1.0.i693 .208 1.11165 .•-'56 1.16649 .:504 1.23061 .113 1.03371 .161 1.06775 .209 1.1126S' .257 1.16774 .305 1.23205 .114 1.0:i43 .162 1.06'-.58 .21 1.11:574 .2.")ft l.)()8'.t9 .306 1.23349 .115 1.0349 .163 1.06941 .211 1.114/9 .2.59 1.17024 .307 1.2:5494 .116 1.03551 .164 1.07025 .212 1.115H4 .•J6 1.1715 .308 1.23636 .117 1.03611 .165 1.0710'.» .213 1.11692 .261 1.17275 .309 1.2:578 .118 l.03()72 .166 1.07194 .214 1.11796 .■1&2 1.17401 .31 1.2:5921 .J 19 1.03734 .167 1.07279 .215 1.1J9(I4 .2t):5 1.17,527 .311 1.2407 .12 1.03797 .168 1.07365 .216 1.12011 .264 1.17655 .312 1.24216 .121 l.0:i86 .169 1.07451 .217 1.12)18 .265 1.17784 .313 1.2436 .122 1.0:5923 .17 1.075:^7 .218 1.12225 .266 1.17912 .314 1.24506 .123 1.03987 .17] 1.07624 .219 1.12334 , .267 l.ln04 .315 1.246,54 .124 1.04051 .172 1.07711 .22 1.12445 .268 1.18162 .316 1.24801 .125 1.04116 .173 1.07799 .221 1.125.56 .269 1.18294 .317 1.24946 .126 1.04181 .174 1.07888 .222 1.12663 .27 1.18428 .318 1.25095 .127 1.04247 .175 1.07977 .223 1.12774 .271 1.18,557 .319 1.25243 .128 1.01313 .176 1.08066 .224 1.12«85 .272 1.18688 .:52 1.25391 .129 1.0438 .177 1.08156 .225 1.12997 .273 1.18819 .:521 1.25539 .13 1.04447 .178 1.08246 .226 1.13108 .274 1.18969 .322 1.25686 .131 1.04515 .179 1.08337 .227 1.13219 .275 1. 19082 .323 1.25836 .132 1.04584 .18 1.08428 .228 l.i:«3i .276 1.19214 .324 1.25987 .133 1.04652 .181 1.08519 .229 1.13444 .277 1.19345 .325 1.26137 .134 1.04722 .182 1.08611 .23 1.13.557 .278 l.t9477 .326 1.26286 .135 1.04792 .183 1.08704 .231 1.1367] .279 1.1961 .327 1.26437 .136 1.04862 .184 1.08797 .2:^2 1.13786 .28 1.19743 .328 1.26588 .137 1.04932 .IfeS 1.0889 .2:53 1.1:590:5 .2ol 1.19887 ..329 1.2674 .138 1.05003 .186 1.08984 .234 1.1402 .282 1.20011 .33 1.26892 .139 1.05075 .187 1.09079 .235 1.141:56 .283 1.20146 .331 1.27044 .14 1.05147 .188 1.09174 ,2;]6 1.14247 .284 1.20282 .332 1.27196 .141 1.0.522 .189 1.09269 .237 1.14:563 .255 1.20419 .333 1.27349 .142 1.05293 ,19 1.09:i65 .238 1.1448 .286 l.20.->.58 .334 1.27502 .143 1.05367 .191 1.09461 .239 1.1459T .287 1.20696 .3:55 1.27656 .144 1.05441 .192 1.09557 .24 1.14714 .28^ l,2(i8-i.>< .336 1.2781 .145 1.05516 .193 1.09654 .241 1.14H3i .2-9 ■..2091 '.7 .337 1.27964 .146 1.05591 .194 1.097,52 .242 1.14949 .29 1.21202 .:538 1.28118 .147 1.05667 .195 1.0985 .243 1.15067 .21)1 1.212:39 .339 1.28273 32 LENGTHS OF CIRCULAR ARCS. T ABLE— ( Continued. ) H'ght. Length. H'ght. Length. H'ght. Length. H'ght. Length. H'ght. Length. .34 1.284-28 .373 1.3373 .406 1.39372 .439 . 1.5327 .472 1.51571 .341 1.-28583 .374 1.33896 .407 1.39548 .44 1.4.5512 .473 1.51764 .342 1. '28739 .375 1.34063 .408 1.39724 .441 1.4.5697 .474 1.51958 .343 1.'28895 .376 1.342-i9 .409 1.399 .442 1.45883 .475 1.52152 .344 1.29052 .377 1.34396 .41 1.40077 .443 1.46069 .476 1.52346 .345 1.29209 .378 1.34.5b3 .411 1.40254 .414 1.46255 .477 1.. 52541 .346 1.29366 .379 1.34731 .412 1.40432 .445 1.46441 .478 1.52736 .347 1.29.523 .38 1.34899 .413 1.406 .446 1.46628 .479 1.52931 .348 1.29681 .381 1.35068 .414 1.40788 .447 1.46H15 .48 1.53126 .349 1.29839 .382 1.35237 .415 i.4ii966 .448 1.47002 .481 1.53322 ,35 1.29997 .383 1.3.5406 .416 1.41145 .449 1.47189 .482 1.. 53518 .351 l.oOJ.')6 .384 1.35575 .417 1.41324 .45 1.47377 .483 1.53714 .35-2 1.30315 .385 1.35714 .418 1.41503 .451 1.47.565 .484 1.5391 .353 1.30474 .386 l.;i5914 .419 1.41682 .452 1.47753 .485 1..54106 .354 1.30()34 .387 1.36084 .42 1.41861 .453 1.47942 .486 1.54302 .355 1.3(»794 .388 1.36254 .421 1.42041 .454 1.48131 .487 1.54499 .3.56 1.30954 .389 1.3642.-> .422 1 .42222 .455 1.4832 .488 1.54696 .357 1.31115 .39 1 .36596 .423 1.42402 .456 1.48509 .489 1.54893 ..358 1.31276 .391 1.36767 .424 1.42583 .4.57 1.48699 .49 1.5509 .359 1.31347 .392 1.36939 .425 1.42764 .458 1.488.'.9 .491 1.55288 .36 1.31599 .393 1.37111 .426 1.42942 .459 1.49079 .492 1.55486 .361 1.31761 .394 1.37283 .427 1.43127 .46 1.49268 .493 1.55685 Mm 1.31923 .395 1.374r)5 .428 1.43309 .461 1.4946 .494 1.55854 .363 1.32086 .396 1.37628 .429 1.43491 .462 1 .49651 .495 1.56083 .364 1.32249 .397 1.37801 .43 1.43673 .463 1.49842 .496 1.56282 .365 1.32413 .398 1.37974 .431 1.43856 .464 1.50033 .497 1.56481 .366 1.32577 .399 1.38148 .432 1.44039 .465 1.50224 .498 1.5668 .367 1.32741 .4 1.38322 .433 1.44222 .466 1.. 504 16 .499 1.56879 .368 1.32905 .401 1.38496 .434 1.44405 .467 1.50608 .5 1.57079 .369 1.33069 .402 1.38671 .435 1.44589 .468 1.508 .37 1.33234 .403 1.38846 .436 1.44773 .469 1.50992 .371 1.33399 .404 1.39021 .437 1.44957 .47 1.51185 ■-" . .37-2 1.33564 .405 1.39196 .438 1.45142 .471 1.51378 To Ascertain the Length of an Arc of a Circle by the preceding Table. Rule. — Divide the height by the base, find the quotient in the column of heights, and take the length of that height from the next righthand column Multijly the length thus obtained by the base of the arc, and the product will give the lenth of the arc. Example. — What is the length of an arc of a circle, the base or span of it being 100 feet, and the height 25 feet ? 25_t_i00=.25; and .25 per table, =1.1 5912, 1.04362 .1.^. 1.0933 .198 1 14762 .246 1.20632 .294 1.26^67 .103 1.04462 .151 1.09148 .199 1.14888 .247 1.20758 .295 1.27 .104 1.04r.62 .152 I 09558 2 1.15014 .248 1.20884 .296 1.27133 .105 1.04662 .153 1.09669 .201 1.15131 .249 1.2101 .297 1.27267 .10(> 1.04762 154 1.0978 .202 1.1.5248 .25 1.21136 .298 1.27401 .107 1.04862 155 1.09891 .203 1.15366 .251 1.21263 .299 1.27535 .108 1.04962 .156 l.lOOO-i .204 l.l54c«4 .252 1.21.39 .3 1.27669 .10 J 1.05063 .157 l.lOliS .205 1.15602 .253 1.21517 .301 1.27803 .11 1.05164 .158 1.10224 .206 1.1572 .254 1.21644 .302 1.27937 .111 1.05265 .159 1.10335 .207 1.15838 .2r,5 1.21772 .303 1.28071 .llii 1.05366 .16 1.10^47 .208 1.15957 .256 1.219 .304 1.28205 .113 1.05467 .161 1.1056 .209 1.16076 .2.57 1.22028 .305 1.28339 .114 1.05568 .162 1.10672 .21 1.16196 .258 1.221.56 .306 1.28474 .115 1.0.5669 .163 1.10784 .211 1.16316 .259 1.22284 .307 1.28609 .116 1.0577 .164 1.10896 .212 1.164:]6 .26 1.22412 .308 1.28744 .117 1.05872 .165 1.11008 .213 1.165.57 .261 1.22541 .309 1.28879 .118 1.05974 .166 1.111:<; .214 1.16678 .262 1.2267 .31 1.29014 .119 1.06076 .167 M1232 .215 1.16799 .263 1.22799 ..311 1.29149 .12 1.06178 .168 1.11344 .216 1.1692 .264 1.22928 .312 1.29285 .121 1.0628 .169 1.11456 .217 1 17041 .265 1.230,57 .313 1.29421 .122 1.06382 .17 1.11569 .218 1.17163 .266 1.23186 .314 1 .29557 .123 1.06484 .171 1.11682 .219 1.17285 .267 1.23315 .315 1.29603 .124 1.065ri6 .172 1.11795 22 1.17407 .268 1.23445 .316 1.29829 .125 1.06689 .173 1.11908 .221 1.17529 .269 1.23575 .317 1 .29965 .126 1.06792 .174 1.12021 .222 1.17651 .27 1.23705 .318 1.30102 .127 1.06895 .175 1.12134 .223 1.17774 .271 1.23835 .319 1.30239 .128 1.06998 .176 1.12247 .224 1.17897 .272 1.23966 .32 1.30376 .129 1.07001 .177 1.1236 .225 1.1802 .273 1 .24097 .321 1.30513 .13 1.07204 .178 1.12473 .226 . 1.18143 .274 124228 .322 1.3065 .131 1.07308 .179 1.12.586 .227 1.18266 .275 1.24359 .323 1.30787 .132 1.07412 .18 1. 12699 .228 1.1839 .276 1.2448 .324 1.30924 .133 1.07516 .181 1. 12813 .229 1.18514 .277 1.24612 .325 1.31061 .134 1.07221 .182 1.12927 .23 1.18638 .278 1.24744 .326 1.31198 .135 1.07726 .183 1.13041 .231 1.18762 .279 1.24876 .327 1.31335 .136 1.07831 .184 1.13155 .232 1.18886 .28 1.2501 .328 1.31472 .137 1.07937 .185 1.13269 .233 1.1901 .281 1.25142 329 1.3161 .138 1.08043 .186 M3383 .234 1.19134 .282 1.25274 .33 1.31748 .139 1.08149 .187 1.13497 .2:J5 1.19258 .283 1.25406 .331 1.31886 .14 1.08255 .188 1.13611 •236 1.19382 .284 1.255;i8 .332 1.32024 .141 1.08362 .189 1.13726 .237 1.19506 .285 1.2567 .333 1.32162 .142 1.08469 .19 1.13841 .238 1.1963 .286 1 .25803 .334^ 1.323 .143 1.08576 .191 1.139.56 .239 1.19755 .287 1.25936 .335 1.32438 .144 1.08684 .192 1.14071 .24 1.1988 .288 1.26069 .336 1.32576 .145 1.08792 .193 1. 14186 .241 1.20005 .289 1.26202 .337 1.32715 .146 1.08901 .194 1.14301 .242 1.2013 .29 1.26335 .338 1.32854 .147 1.0901 .195 1.14416 .243 1.20255 .291 1.26468 .339 1.32993 34 LENGTHS OF SEMI-ELLIPTIC ARCS. ^ TABLE.— (Con/inued.) ffght. Length. ' H'pht. Length, 'h ! ght. Length. H'ght. Length. H'ght. Length. :,u 1.33132 .39t; 1.4l2il 452 1.496i'i .508 1.5KU9 .,5154 1.(570157 .341 1. 3:5272 .397 l.4l:(57 453 1.49771 .509 158474 .565 1.67245 .\U'2 1.33412 .398 1.41504 4.54 1.4" •92 4 .51 1 5>-6-*9 .566 1.67403 .343 1.3;'.552 .399 1. 41(151 4.-.5 1.50077 .51 1 l.,58784 .567 1.67.561 .344 1.33692 .4 1.41T98 4.56 I 5023 512 1.5894 .568 167719 .34.5 1.33833 401 1.41114.'. 457 1.. 50383 .513 1.59096 .5(59 1.(57877 .346 1.33974 .402 1.4209J 458 1.. 50536 .51 1 1 .592rvj .,57 1.6803(5 .347 1.34115 .403 1.42239 459 1.50689 .515 l.,59408 .571 1.68195 .348 1.34256 .104 1.42:M) 46 1.50842 516 1.. 59564 .572 1.68:^54 .349 1.34397 .405 1.42. .33 461 1.50'.'96 .5 1 7 1.597-J .573 1.68513 .3 1.34539 ,40(> 1. 42681 Wl 1.5115 .518 1.. 59876 ..574 1.68672 .351 1.34681 .407 1.4v!"<29 4<)3 1.51304 .519 1.60032 ..575 1.(58831 .352 1.34823 .408 1.4-".tT7 161 |.5i45.-s 5-.' 1 (■.0188 ..576 1.6899 .353 1.34965 .40.' 1.43I-.';. 465 1. 5161 -J .521 1.60344 .577 1.69149 .354 1.35108 .41 I.13J7:! 166 l..'.176ii 522 1.605 .,578 1.(59308 .355 1.35251 .4!l 1.424-.'] 467 1.5192 ..523 1.60(ii-6 ..579 1.69467 .356 1.35394 .412 1.425C>9 468 1. 5207 J ..524 i 60812 .,58 1.69626 .357 1.35.-i37 .413 1.43718 169 1.5-J-J29 .5-.'5 1.60968 .,581 1.69785 .358 1.3568 .414 1.43SC.7 47 1..5-.>:',->4 -..•J6 It; 1124 .582 1 :59945 .359 1.35823 .415 1.4401ti 471 1.5 .'5; .9 -5-J7 1 6128 .583 1.70105 .36 1.35967 .416 1.44 Mm 472 1.526;a 5-i8 :.6U:;6 .584 1.702(54 .361 1.36111 .417 1.44314 473 ] .52.-'49 .5-.>9 I 61592 .585 1.70424 .362 1.362,55 .418 1.44163 474 I 53004 -53 1.61748 ..586 1.70.584 .363 1.3«)399 .419 1.4-1013 475 1.53 159 .531 1 61904 .587 1-70745 .364 1.36543 .42 1.44763 476 1. 533 14 5 12 1.6206 ..588 1.70905 .365 1.36688 .421 1.41913 4T7 1..534t)9 -533 1.622)6 .589 1.71065 .366 1.36833 .422 1.4.->064 47> 1.53625 .5:!4 1.62372 .59 1.71225 .367 1.36978 .423 1.4.-)214 479 1 53781 .535 1.62.'.2H .591 1.71286 .368 1.37123 .424 1 45364 48 1.53.137 -536 1 .6-J6-4 .592 1-71.546 .369 1.37268 .425 1.45515 4'^1 1 54093 -537 1.6284 .593 1-71707 .37 1.37414 426 1.45665 4^2 1 5424.t -538 1.6-996 ..594 1 71868 .371 1.37662 .427 1.45815 4^3 1 54405 .539 1.631.52 .,595 1-72029 .372 1.3770f< .42^ I.4.V.t66 484 I 54561 -54 1,63309 .,596 1.7219 .373 1.37854 .429 1.46167 4«5 1 54718 541 1 .6:;-l65 .597 1-7235 .374 1.38 .43 1.4626H 486 1.54875 .542 1 636J3 .598 1-72511 .375 1.38146 .431 1. 461 19 48T 1.55032 -543 1.6378 .599 1-72672 .376 1.38292 .432 1.4657 488 1.55189 544 l.63i)37 .6 1-72833 .377 1.38439 .433 146721 489 1. 5.5346 -.545 1.64094 .601 1.72994 .378 1.38585 .434 1.46ri72 4.» 1 .55503 -546 1.(54251 .602 1-73155 .379 1.38732 .435 1.4/-023 491 1-5566 .547 1 .(54408 .603 1-73316 .38 1.38879 .436 1.47174 492 1.55817 ■548 1.64.565 604 1.73477 .381 1.39024 437 1.47326 493 1.55974 -549 1.(54722 .605 1-73638 .382 1.39169 .438 1.47478 494 1..56131 •5 1 61879 .606 1-7:i799 .383 1.39314 .439 1.4763 495 1 56289 -551 1 .6.503(5 .607 1-7396 .384 1.39459 44 1.47782 496 1..5t)447 -552 1.65193 .608 1-74121 .385 1.39605 .441 1.479:!4 497 1 5660ri -553 1.C.535 .609 1-74283 .386 1.29751 .442 1.48086 498 1. 56763 -554 l.6.";.507 6 1.74444 .387 1.39897 443 1.48238 499 1-56921 -.55,5 1. 6.5665 .611 174(505 .388 1.40043 .444 1.48391 5 1-57089 -556 1 .658-.i3 .612 174767 .389 1.40189 .445 1.48.544 501 1-57234 ,5.57 1.6,5981 .613 1.74929 .39 1.40335 .446 1. 48697 502 1-57389 .,558 1 66139 .614 1-75091 .391 1.40481 .447 1.4885 503 1-57544 -559 1.66297 .615 1-7.52,52 .392 1.40627 .448 1.49003 504 1-57699 -56 1.66455 .616. 1-75414 .393 1.40773 .449 1.49154 505 1578,54 -561 1.66613 .617 1-75,576 .394 1.40919 .45 1.49311 506 1-58009 •562 1.66771 .618 1.7,5738 .395 1.41065 .451 ~ — — 1.49465 507 1-58164 .563 1.66929 .619 1-759 LENGTHS OF SEMI-ELLHTIC ARCS. TABLE.— (Continued.) 35 H'Kht. Length. H'ght. Length. H'ght. . ength. H'ght Length. H'ght Length. .6-2 1.76062 .676 1.H5215 .732 1.94.5.52 .788 2.04117 .844 2.1:59,-6 .6-21 1.76224 .677 1.8.5379 .733 1.94T21 .789 2.04-29 .845 2.14155 .t^':-2 1.76386 .678 1.85544 .734 1.9489 .79 2.04162 .f46 2.14:534 .6-2:^ 1.76.548 .679 1.8.5709 .735 1.950.59 .791 2.046:15 .847 2.14513 .H24 1.7671 .68 1.8.5e74 .736 1 9.5228 792 2.04809 .848 2.14692 .6«»r> 1.76H72 .681 1.8(i(l31? ^ .737 1.9.5397 .793 2.049f^3 .849 2.14871 .6->6 1.77031 .682 1.86205 .738 1.9.5.5«i6 .794 2.051.57 .85 2.ir>05 .6-27 1.77197 .6r*3 1.8637 .739 1.95735 .795 2.0.5:{3l .851 2.15229 .6t»8 1.773.59 .684 1.8t».535 .74 1.9.5994 .796 2.0. -.505 .8:)2 2.1.5409 .6:29 1.77521 .685 1.867 .741 1.964 2.1577 .6:u 1.77847 .687 1.87031 .743 1.96414 .799 2.06(1-^7 .8.55 2.1595 .6:w 1.7H009 .»)88 1.87196 .744 1.9t)5K3 .8 2.06202 .85<) 2.1613 .>i:i:i 1.78172 .689 1.87362 .745 1.96753 .801 2.0():i77 .857 2.16:509 .634 1.78335 .6.> 1.87527 .746 l.t»6923 .802 2.06.552 .8.58 2. 16489 .63.5 1.78498 .691 1.87693 .747 1.970:<3 .803 2.06727 .859 2.16668 .636 1.7866 .692 1.87859 .748 1.97J62 .81(4 2.0ti90l .86 2. h)«48 .637 1.78823 .693 1.88024 .749 1.97432 .S05 2.07076 .H61 2.17028 .638 1.78986 .694 1.8819 .75 1.97602 .806 2.07251 .862 2.17209 .63S» 1.79149 .695 i.88:?;.6 .751 1.97772 .807 2.07427 .863 2.17389 .64 1.79312 .696 1.88522 .752 1.97943 .808 2.07tiU2 ."'(M 2.17.57 .641 1.79475 .697 1.88688 .753 1.98113 .809 2.07777 .WW) 2.17751 .642 1.79638 .698 1.88854 .754 1.98283 .81 2.079.53 .866 2.179:52 .643 1.79801 .699 1.8902 .755 1. 9*^453 .811 2.08128 .867 2.18113 .644 i.79i)64 .7 1.89186 .756 1.98623 .M2 2.0s:W4 .868 2.18294 .64.5 1.80127 .701 1.893.52 .757 1.98794 .813 2.0848 .869 2.18475 .646 1 8029 .702 1.89519 .758 1.98964 .811 2.086.56 .87 2.18656 .647 1.804.54 .703 1.89685 .759 1 99134 .K15 2.088:52 .871 2.188:57 .648 1.8061T .704 1.89^51 .76 1.99305 .816 2 09(108 .872 2.19018 .649 1.8078 .705 1.90017 .761 1.99476 .817 2.09198 .873 2.192 .65 1.80943 .706 1.90184 .762 1.99647 .818 2.09:36 .874 2 19;<82 .(•)5l 1.81107 .707 1.9035 .763 1.998i8 .819 2 095:56 .875 2.19564 .652 1.81271 .708 1 .90517 .764 199989 .82 2.09712 .876 2.19716 .6.")3 1.81435 .70it 1.90684 .765 2 0016 .821 2.09888 .877 2.19928 .654 1.81599 .71 1.90852 .766 2.00331 .822 2.10065 .878 2.20U .655 1.81763 .711 1.91019 .767 2.00502 .823 2 10242 .879 2.20292 .656 1.81928 .712 1.91189 .768 2.00673 .824 2 10419 .88 2.20474 .6.57 1.82091 .713 1.91355 .769 2.00844 .825 2. 10.596 .881 2.20656 .658 1.82255 .714 1.91523 .77 2.01016 .826 2.10773 .882 2.208:59 .6,59 1.82419 .715 1.91691 .771 2.01187 .827 2.1095 .883 2.21022 .66 1.82583 .716 1.91859 .772 2-01359 .828 2.11127 .884 2.21205 .661 1.82747 717 1.92027 .773 2.01531 .829 2.11304 .885 2.21388 .662 1.82911 .718 1.92195 .774 2.01702 .83 2.11481 .886 2.21571 .663 1.83075 .719 1.92363 .775 2 01874 .831 2.11659 .887 2.21754 .664 1.8324 .72 1.92531 .776 2.02045 .832 2.11H37 .888 2.2*937 .665 1.8:^404 .721 1.927 .777 202217 .8:» 2.12015 .889 •2.2212 .6t56 l.8;",£G8 .722 1.92868 .778 2.02389 .834 2.12193 .89 2.22303 .667 1.83733 .723 1.93036 .779 2.02561 .8:35 2.12371 .891 2.22486 .668 l.K^97 .724 1.93204 .78 2.02733 .836 2.12549 .892 2.2267 .6^9 1.84061 .725 1.93373 .781 2.02907 .837 2.12727 .893 2.22854 .6/ 1.84226 .726 1.93.541 .782 2.0308 .838 2.12905 .894 2.23038 .671 1.84391 .727 1.9371 .783 2.03252 .839 2.13083 .895 2.23222 .672 1.84556 .728 1.93678 .784 2.03125 .84 2.1:5261 .896 2.23406 .673 1.8472 .729 1.94046 .785 2.03598 .841 2.134:}9 .897 2.2359 .674 1.84885 .73 1.94215 .786 2.03771 .842 2.13618 .898 2.23774 .675 1.8505 .731 1.94383 .787 2.03944 .843 2.13797 .899 2.23958 LEN0HT8 OF SRMI-ELLIPTIO ARCS. TABLE.— (Conrmued.) H'ght Length. H'ght Length. H'gh;. Length. H'ght. Length. H'ght ■ Length. .9 2.24142 .921 2.27987 .942 2.31fc.)2 .963 2.3.581 .984 2.39823 .901 2.2433r> .922 2.2817 .y43 2.32038 .964 2.36 .985 2.40016 .902 2.24508 .923 2.28354 .!)44 2.32224 .96^5 2.36191 .9,-6 2.40-.'08 .903 2.24H91 .924 2.28537 .945 2.32411 .966 2.363HI .987 2.404 .904 2.24874 .925 2.2872 .946 2.32.'>y8 .967 2.36.571 .9H8 2.40592 .905 2.25057 .926 2.2."'903 .947 2.327W5 .9»i8 2.36762 .989 2.40784 .906 2.2524 .927 2.290«6 .948 2.32972 .969 2.36952 .119 2.401)76 .907 2.25423 .928 2.2927 .941) 2..3316 .97 2.37143 .991 2.41169 .968 2.25606 .929 2.294.53 .95 2.33348 .971 2.37334 .992 2.41362 .909 2.25789 .93 2.29636 .951 2.33537 .972 2.37525 -.993 2.41556 .91 2.25972 .931 2.29H2 .9.52 2.33726 .973 2.37716 .994 2.41749 .911 2.26l.-)5 .932 2.30004 .953 2 33915 .974 2.371)08 .995 2.4 11(43 .912 2.26338 .933 2.30188 .1)54 2.34104 .1)75 2 3-1 .996 2.4Ji:i6 .913 2.26521 .934 2.30373 .955 2.34293 .9Tt) 2.38291 .997 2.42329 .914 2.26704 .935 2.30557 .956 2.344^3 .9t7 2.3M482 .998 2.42522 .915 2.26rt88 .936 2.30741 .957 2.34673 .l»7fS 2.3H673 .999 2.42715 .916 2.27071 .937 2.30926 .958 2.34862 .979 2.38864 1. 2.42908 .917 2.27254 .938 2.31111 .9.')9 2.35051 .98 2.31(055 .918 2.27437 .939 2.3121)5 .96 2.3.5241 .;t8l 2.31»-.'47 .919 2.2762 .94 2.31471) .961 2.3.5431 .982 2.39439 .92 2.27803 .941 2.31666 .96-J 2.35621 .983 2.31)631 To AseertaiD the Length of n Semi Kiliptic Are (right Semi-Eilipse) by the preceding Table. Rule. — Divide the height by the base, find the quotient in tlie column of heights, and take the length of that height from the next iighthand column. Multiply the length thus obtained by the base of the arc, and the product will be the length of the arc. E.XAMPLK. — What is the leagth of the arc of a semi-ellipse, the base being 70 feet, and the height 30.10 feet 30.10-^70=.43 ; and .43 per table, =1.46268. Then 1.46268 X 70=102.3876 feet. When the Curve is not that of a Right Semi-Ellipse, the Height being half of the Tranverse Diameter. ' - RuiiB. — Divide half the base by .twice the height, then proceed as in the preceding example ; multiply the tabular length by twice the height, and the product will be the length required ExAMPLB. — What IS the length of the arc of a semi-ellipse, the height being 35 feet, and the base 60 feet ? 60-4-2=30, and_30-H35X2=. 428 the tabular length of which IS 1.45966. Then 1.45966X35 X 2=1 02. 1762/ee^ Note. — If in the division of a height by the base there is a remainder, proceed in the manner given for the Lengths of Circular Arcs, page 32. ■f^^^y- TABL13 Vin. TABLE OF THE AREAS OF THE SEGMENTS OF A CIRCLE. The Diameter of a Circle aaaumed to be Unity, and divided into 1000 equal Part$. Vereed Bine. Seg. Area. VerHcd Siue. 1 Seg, Area ' Versedl Sine. Seg. Area. Versed Sine. Seg. Area, Versed Sine. Seg. Area. .001 .00(104 .048 .01.3-2 .095 .0379 .142 .0()M22 .189 .10312 .(tO-J .00012 .049 .01425 .096 .03849 .143 .06892 .!.» .1039 .003 .00022 .05 .0146^ .097 .03908 .144 .069()2 .191 .10468 .004 .000:14 .051 .01512 .098 .03968 .145 .07033 . 192 .10.547 .005 .00047 .052 .0i:.56 .099 .04027 .146 .07103 .193 .10626 .006 .00062 .053 .01601 .1 .04087 .147 .07174 .194 .10705 .007 .00078 .054 .01616 .101 .04148 .148 .07245 .195 .10784 Mti .00095 .055 .01691 .102 .04208 .149 .07316 .196 .10864 .009 .00113 .056 .0173; .103 .04269 .15 .073^7 .197 .10943 .01 .00133 .057 .01783 .104 .0431 .151 .07459 .198 .11023 .011 .001.-)3 .058 .0183 .105 .04::91 .152 .07531 .199 .11102 .012 .00175 .0.59 .01W77 .106 .044:.2 .153 .07603 2 .11182 .013 .00197 .06 .0l;»24 .107 .04514 154 .07675 .201 .11262 .014 .0022 .061 .01972 .108 .04575 155 .07747 .202 .11343 .015 .00214 .062 .0202 .10;! .04638 .156 .0782 .203 .11423 .016 .00268 .063 .02068 .11 .047 .157 .07892 .204 .11503 .017 .00294 .064 .02117 .111 .04763 .1.58 .07965 .205 .11584 .018 .0032 .065 .021»S .112 .04826 .159 .08038 .2'J6 .11665 .019 .00347 .066 .02215 .113 .04889 .16 .08111 .207 .11746 1 .02 .00375 .067 .02265 .114 .04953 .161 .08185 .208 .11827 .021 .00403 .068 .02315 .115 .05016 .162 .08258 .209 .11908 .022 .00432 .069 .02336 .116 .0508 .163 .08332 .21 .1199 .023 .00462 .07 .02417 .117 .05145 .164 .08406 .211 .12071 .024 .00492 .071 .024(58 .118 .05209 .165 .0848 .212 .12153 .025 .00523 .072 .02519 .119 .05274 .166 .08.554 .213 .12235 .026 .O0."k)5 .073 .02571 .12 .05338 .167 .08629 .214 .12317 .027 .00.587 .074 .02624 .121 .05404 .168 .08704 .215 .12399 .028 .00619 .075 .02676 .122 .05469 .169 .03779 .216 .12481 .029 .001)53 .076 .02729 .123 .05534 .17 .08853 .217 .12563 .03 .00686 .077 .02782 .124 .056 .171 .08929 .218 . 12646 .031 .00721 .078 .02S35 .125 .05666 .172 .09004 .219 .12728 .032 .00756 .079 .028(?9 .126 .05733 .173 .0908 .22 .12811 .033 .00791 .08 .02943 .127 .05799 .174 .09155 .221 .12894 .034 .00827 .081 .02997 .128 .05866 .175 .09231 .222 .12977 .035 .00864 .082 .03052 .129 .05933 .176 .09307 .223 .1306 .036 .00901 .083 .03107 .13 .06 .177 .09384 .224 .13144 .037 .00938 .084 .03162 .131 .06067 .178 .0946 .225 .13227 .038 .00976 .085 .03218 .132 .06l:i5 .179 .09537 .226 .13311 .039 .01015 .086 .03274 .133 .06203 .18 .09613 .227 .13394 .04 .01054 .087 .0333 .134 .06271 .181 .0969 .228 .13478 .041 .01093 .088 .03:^87 .135 .06339 .182 .09767 .229 .13.562 .042 .01133 .089 .03444 .136 .06407 .183 .09845 .23 .13646 .043 .01173 .09 .03501 .137 .06476 .184 .09922 .231 .13731 .044 .01214 .091 .03558 .138 .06545 .185 .1 .232 .13815 .045 .01255 .092 03616 .139 .06614 .186 .10077 .233 .139 .046 .01297 .093 .03674 .14 .06()83 .187 .10155 .234 .13984 .047 .01.339 .094 .03732 .141 .06753 .188 .10233 .235 .14069 AREAS OF TIIE SEGMENTS OF A CIRCLE. T A BLE.— (Continued. ) 1 Vewd hiuu. 1' Hog Arun. .tj.lt) .I41.')4 .'2:n .142.J9 .UM .14:«4 .•j:w .I44(il» .'i4 .14494 .•24 1 .1458 .21'2 .!ii ;().'. .'U:i .14751 .'M .UK{7 .24.') . 1 4923 .246 .15(109 .247 A 'Mb .24H .15182 .249 .l.'>2()8 .2.) . 15U55 .251 .l.'>44i .2r>2 .I.-.528 .2.'>:i .1.-.615 .2.S4 .157(12 .2;').. . 15789 .2r)« .ir>e7(i .257 .l.")96J .258 .l()05l .259 .161:59 .26 .161*26 .261 .KKJU .■:6> .16402 .2(i:J . 1649 ,264 .ltw7-i .265 .1(J666 .266 . 1675.) .267 .16'-44 .268 .16iKU .269 .1702 .27 .17109 .271 .17197 .272 .17287 .27:< ,17376 .274 .17465 .275 .175.54 .276 . ! 7643 .277 .17733 .278 . 17822 .279 .17912 .28 .18002 .281 . 18092 .282 .18182 .2o3 .18272 .284 .18361 .285 .1^4.52 .286 . 18542 .287 .18633 .288 .18723 '5 ^*^'seg Ami.' ^'/^ Hcg. Area. iinc. 2H9 .1^814 2<> .l-'.Mi.. 2lt 1 .18i>95 2'.>*.> .l'.HJ86 2i>;{ .19177 291 .19268 295 . 193(; 21M; .I9J51 297 .19.-.42 298 .ll'63» 299 .19725 3 .19817 301 .19908 302 .2 303 .200;»2 304 .20184 :{U5 .20276 306 .2o;'.68 307 .2016 30-( ,20.")53 309 .20645 31 .20738 3tl .208 5 312 .20923 313 .21015 3i4 .21108 315 .21201 316 .21294 317 .21387 318 .21.i8 319 .21573 32 .21667 321 .2176 3->2 .21853 323 .21947 324 .2204 325 .22134 32(; 22228 327 .22:521 32"- .22415 329 .22.'.09 3;; .22603 331 .22697 332 .22791 333 .22c86 3;i4 .2298 335 .23074 336 .231(i9 337 .23263 338 .23359 339 .23453 34 .23.547 341 .23642 .342 ,343 .344 .315 .346 .347 .348 .319 ,:5.-) ,351 .3.52 .:i5;i .3,->4 .3.55 .356 .3.57 .X>H .3.59 .m .361 .362 .363 .364 .365 .366 .367 .3(58 .369 .37 .371 ,372 ,373 .374 .375 .37(5 ,377 .378 .379 .38 .381 .382 .38;{ ,384 .:i85 .38(5 .387 .388 .3''9 .39 .3i»l .392 .393 ,394 .23737 .23h:'.2 .23927 .24022 .24117 .24-.' 12 .24307 .21103 .•:441W .21..93 .246-9 .24784 .248rt .24;(7() .2,5071 ,25167 .25263 .25:;59 .2545.) .25551 .2.5647 .2.5743 .2.">839 .25936 .2(^032 .2C.128 .2622r. .2(;;52l .2(5-1 18 .26514 .26(511 .2670> .2f5^04 .2(5:'01 .26998 .27095 .27192 .2T2rt9 .27386 ,27483 .27580 .27677 .27775 .27872 .27969 .28067 .28164 .28262 .283.59 .284.57 .2f^.554 .2-552 .2878 8iDc. ^- ^"^ .395 .:v.m .397 .39"^ .399 .4 .401 4(r2 .103 .404 ,4 05 .406 .407 .4yf5 .409 .41 .411 .412 .413 .411 .415 .416 .417 .418 .419 .42 .421 .422 .423 .424 .425 .42(5 .427 .428 .42;> .43 .131 .432 .133 .434 .435 .4:56 .437 .439 .44 .441 .442 .443 .444 .445 446 .447 .2rt« 18 .2^915 .29043 .29141 .292::9 .29337 .2: '4 35 .29533 .2:M531 .29729 .29-27 .21K>25 ,30024 .30122 .3022 .30319 .30417 ,30515 .3(»il4 .30:12 .30811 .309119 .31008 .31107 .31205 .313iM .31403 .31.502 .31(5 .31(599 .31798 .31897 .319'.«i .32095 .32194 .322; (3 .32391 .3249 .3259 .3268i( .32788 .32887 .329H7 .33086 .331S5 .33284 .33384 .33483 .33582 .336-2 .33781 .33'-8 .3398 Sine. ^ ^"^ .448 .449 .45 .451 .4.52 .4.53 .4.54 .45.5 .4.56 .4.57 .4.58 .4.'>9 .46 .461 .462 .4(53 .464 .4(55 .466 .467 .4(58 .469 .47 .471 .472 .473 .474 .475 .476 .477 .478 .479 .48 .481 .482 .483 .484 .485 .486 .487 ,488 ,489 .49 .491 .492 .493 .494 .495 .496 .497 .498 .499 .5 .34079 .34179 .:»4278 .34378 .:M47r .:u.>57 .34676 .34776 .34H75 .3497JS .3,"i07i. .;t5!?4 .:{.5274 .3.5374 .3,-)474 .3.'k573 .3,^.«573 .35773 .35872 .35972 .3(5072 .3<)172 .3»)2:2 .36371 .36471 .3(5.571 .3(5671 .36771 .3(5871 .36971 ,37071 .3717 ,3727 ,3737 ,3747 ,3757 .3767 .3777 •3787 .3797 .3807 .3817 .3827 .3837 .3847 .:i857 .3867 .3877 .3887 .3897 .3907 .3917 .3927 AREAS OF THi: ZONES OF A CIRCLES. 39 To Aserrl'iin thn Area uf a Segment of a Circle by tlie preeediug Table. Rui I — DiTifle the hei^rht or vpned gino by the diamoter of the circl«' ; find the quotient in the coluiiiii ol vfised «ine>). Take the area noted lu the next column, niulii|ily it hy the ^•l|uare of the dianieier, and it will kivc \\u- area bx AMFLK. — Kequiitd ihe urea ot a dcgoieut, iu height being 10, and the diumeUsr of the circle fill feel. I(i_}..V)=-.2, and .2. [ler table,=-.lll82; then .ni»lX^0^='2l9.:>r,Jeel. N'lTic. — If in the division of a height i)y tiie base, the quotient has remainder after the third liict- ot de< jiiaiii, and great iicciiracy id required. Tnkf the ai«a f >r the fir.-t tliree fi.'un's. subtraet it from the next ftdlowing ar a. multiply the letiiiiiiidcr by the said fiaciioii. and add the product to the firat area ; tlie itum will be the area for Uii- who e (|iiotient. '/ W hat li the area of a sigmcnt of a circle, the diameter of which is 10 feet, and the height of it 1.675 fe.t 1.67:)-*-l<» .1575; the tabular area for .157=. 07892, and for .168=07906, the difference between which la 0()07.S. Then .5 X. 00073=0003(35. ^ Hence " .157 =.07892 .0006=. 000365 .079275, the sum by which the square of the dia- meter of the circle is to be multiplied ; and .079285 X 10 -=7.928G/«'e<. r»-r^-rvj- w-vj -i-r^-0-rn-i-L ro-0.r i^^ ■ ■■■•1' TABLE IX. i. ^ TABLE OP THE AREAS OF THE ZONES OP A CIRCLE. The Diameter of a Circle assumed to be Unity, and divided into 1000 equal Parts. H'ght. Area. H'ght. Ares. H'ght. Area. H'ght. Area. H'ght Area. .00 1 .001 .029 .02898 .057 .05688 .085 .084.-)9 .113 .11203 .00-2 .002 .03 .02998 .058 .05787 .086 .08557 .114 .113 .00:{ .003 .031 .03093 .059 .0.'>886 .087 .0H656 .115 .11398 .001 .004 .03i .03198 .06 .0.'>986 .08H .08754 .116 .11495 .oor> .00.') .033 .03298 .061 .0t=085 .0^9 .08853 .117 .11592 .OOrt .006 .034 .03397 .0-J46 .121 .11981 .01 .01 .038 .03796 .066 .0658 .094 ,09344 .122 .12078 .Oil .011 .039 .03896 .067 .0668 .095 .09443 .123 .12175 .012 .012 .04 .03996 ' .06.8 .0678 .096 .0954 .124 .12272 .013 .01:5 .041 .04095 .069 .0<)878 .097 .09639 .125 .12:^69 .914 .014 .042 .04195 .07 .06977 .098 .09737 .126 .12469 .oir, .015 .043 .042.t5 .071 .07076 .099 .09835 .127 .125<)2 .016 .016 .044 .64394 .072 .07175 .1 .09933 .128 .126.^)9 .017 .017 .045 .04494 .073 .07274 .101 .10031 .1-J9 .127.55 .OIH .Old .046 .04593 .074 .07373 .102 .10129 .13 .128.52 .019 .019 .047 .04693 .075 .07472 .103 . 1 0227 .131 .12949 .0-J .02 .048 .04793 .076 .07.').') .104 .1032.') .132 A.mh .0-21 .021 .049 .04892 .077 .07()69 .l.t-> .104'>2 .IXi .13141 .022 .022 .05 .04992 .078 .07768 .106 .1052 .134 .132:W .023 .023 .051 .0:.091 .079 .07867 .107 .10618 .1:^5 .133:i4 .024 .024 .0.V2 .0519 .08 .0796*) .108 .10715 .136 .1343 .025 .025 .053 m->d .081 .08064 .109 .10813 .137 .i:i^.27 .026 .02599 .054 .05:J89 .082 .08163 .11 .10911 .138 .1362:^ .027 .C2o9b .055 .0.^489 .083 .08262 .111 .11008 .139 .13719 .02« .02799 .056 .05588 .084 .0836 .112 .11106 .14 .13815 40 ABEA.S OF THE ZONES OF A CIRCLES. TABLE.— (Continued.) -«v. ffght. Area. H'ght. Area. H'ght. Area. H'ght. Area. H'ght Area. .141 .i.-jgu .W7 .19178 .2.53 .24175 .309 .28801 .;565 .32931 .142 .14007 .198 .1927 .254 .24261 .31 .2«88 .366 .32999 .143 .14103 .199 .19:!61 .255 .24:547 .311 .28958 .;<67 .3:3067 .144 .14198 .2 .194.53 .2.56 .24433 .312 .29036 .:}68 .331:35 .145 .14294 .201 .l'.^545 .257 .24519 .313 .29115 .369 .3:5-203 .146 .1439 .202 .196:56 .258 .24604 .314 .29192 .37 .3:527 .147 .14485 .203 .19728 .2.59 .2469 .315 .2927 .:571 .33337 .148 .14581 ,204 .19819 .26 .24775 .316 .28:348 .•572 .:33404 .149 .14677 .205 .IWl .261 .24861 .317 .29425 .:373 .:3:3471 .15 .14772 .206 .20001 .262 .24946 .3 la .29.502 .374 .3:3537 .151 .14867 .20? .20092 .263 .25021 .319 .295H .375 .3:3604 .1.52 .14962 .208 .201r'3 .^64 .265 .25116 .32 .29656 .:376 .3367 .1.53 .150.58 .209 .20274 .21201 .:321 .29733 .:377 .3:3735 .154 .151.53 .21 .20365 ,266 .'252.85 .322 .2981 .378 .:3:3801 .1.55 .15248 .211 .20156 .267 .2537 .323 .29886 .379 .3:3866 .156 .15343 .212 .20546 .268 .25455 .:{24 .29962 .38 .33931 .157 .15438 .213 .206:!7 .269 .2.5.5:^9 .:525 ..50039 .:38l .33996 .1.58 .155;?3 .214 .20727 .27 .25623 .:}26 .:50114 .3»2 .34061 .1.59 .15628 .215 .20818 .271 .25707 .:i27 .3019 .383 .341-25 .16 .15723 .216 .2(>908 .272 .2.5791 .:i28 .30266 .:384 .3419 .161 .15H17 .217 .20998 .273 .25875 .329 .30:341 .:385 .34253 .162 .15912 .218 .21088 .274 .259.59 .33 .:30416 .386 .34317 .163 .16006 .219 .21178 .275 .2<.U43 .:53l .:50491 .387 .3438 .164 .16101 .22 .21268 .276 .26126 .3:52 .30.566 .388 .34444 .165 . 16195 .221 .2i;j5» ■ .277 .26209 .:i33 .30«)41 .:389 .34507 .166 .1629 .222 .21447 .278 .2629:5 .:«4 .30715 .39 .34569 .167 .16384 .223 .21537 .279 .26376 .3.55 .3079 .;391 .34632 .168 .16478 .224 .21626 .28 .264.59 .336 .30864 .392 .34694 .169 .16.572 .225 .21716 .281 .26541 .337 .30938 .393 .34756 .17 .16667 .226 .21805 .282 .26624 .338 .31012 .394 .34818 .171 .16761 .227 .21894 .283 .26706 .:i39 .31085 .395 .34879 .172 .168,-.5 '.228 .21983 .284 .26789 .:54 .31159 .:396 .3494 .173 .16948 .229 .22072 .285 .26871 .341 .31232 .397 .3,5001 .174 .17042 .23 .22161 .286 .269.53 .:542 .31305 .398 .35062 .175 .17136 .231 .2225 .287 .270a5 .343 .31:378 .399 .35122 .176 .1723 .232 .22335 .'2f^6 .27117 .344 .:3145 .4 .:55182 .1/7 .17323 .233 .22427 .289 .27199 .345 31,523 .40 J ,3,5242 .178 .17417 .234 .22515 .29 .272'^ .346 .31595 .402 .35302 .179 .1751 .•^35 .22604 .291 .27362 .:547 .31667 .403 .3,5361 .18 .17603 .236 .22692 .292 .27443 .348 .3U)39 .404 .3542 .181 .17697 .237 .227S .293 .27524 .349 .31811 .405 .3.5479 .182 .1779 .23rt .228«)8 .294 .27605 .35 .31882 .406 .:3.5538 .183 .17883 .239 .22956 .295 .27686 .351 .319,54 .407 .35596 .184 .17976 .24 ,23044 .296 .27766 .3,52 .32025 .408 .3.5854 .185 .180<)9 .241 .2:a3l .297 .27847 .:55:5 .32096 .409 .35711 .186 .18162 .242 .23219 .298 .27927 .:i54 .:32167 .41 .3,5769 .187 .1b254 .243 .23306 .299 .28007 .:555 .32-237 .411 .358-26 .188 .18:U7 .244 .23394 .3 .2rt088 .356 .3-2:507 .412 .3,5H83 .Irt) .1844 .245 .234^1 .301 .28167 .3,57 .3-^:577 .413 .3.59:39 .19 .18532 .246 .2:5568 .302 .28247 .:i58 .:52147 .414 .35995 .191 .18625 .247 .23655 .303 .25:327 .3r)9 .:3-2517 .415 .:36051 .192 .lf^717 .248 .23742 .304 .28406 .36 .:{-25«7 .416' .36107 .193 .18809 .249 .23829 .305 .28486 .361 .3-2656 .417 .36162 .194 .18902 .25 .23915 .306 .2856.5 .:562 .327-25 .418 .:56217 .195 .18994 .251 .24002 .307 .2^-644 .3()3 .32794 .419 .36-272 .196 .19086 .252 .24089 .308 .2^723 .364 32862 .42 .36326 ABEAS OF THE ZONES OF A CIRCLE. TABLE.— (Continued.) 41 ffght. Area. H'ght Area. H'ght Area. H'ght. Area. H'ght. Area. .421 .3638 .4:w .37202 .453 .37931 .469 .38.'>49 .485 .39026 .422 .36434 .438 .3725 .454 .37973 .47 .3H5-'3 .486 .3905 .423 .:i6488 .439 .37293 .455 .38014 .471 .38617 .487 .39073 .424 .36541 .44 .37346 .456 .38056 .472 .3865 .488 .39095 .425 .36594 .441 .37:J93 .457 .38096 .473 .38683 .489 .39117 .426 .36646 .442 .W44 .458 .38137 .474 .38715 .49 .39137 .427 .36698 .443 .37487 .459 .38177 .475 .38747 .491 .391.56 .428 .3675 .444 .37533 .46 .38216 .476 .38778 .492 .39175 .429 .36802 .445 .37579 .461 .38255 .477 .38808 .493 .:^9192 .43 .36853 .446 .37624 .462 .38294 .478 .3n838 .494 .39208 .431 .36904 .447 .37669 .463 .383}2 .479 .38667 .495 .39223 .432 .36954 .448 .37714 .464 .38369 .48 .3889.% .496 .39236 .433 .37005 .449 .37758 .465 .38406 .481 .38923 .497 .39248 .434 .370.^4 .45 .37802 .466 .38443 .482 .3895 .498 .39258 .435 .37104 .451 .37845 .467 .38479 .483 .38976 .499 .39266 .436 .37153 .4.o2 .37888 .468 .38514 .484 .39001 .5 .3927 J%w Table is computed ordy^or Zones, the longest chord ^ which is diameter^ . To Aseertain the Area of a Zone by the preeeding Table. Rule 1. — When the Zone is Lexs than a Semicircle, Divide the height by the diameter, and find the quotient in the column of height. Takeout thearea opposite to it in tlie next column on the right hand and multiply it by the square of the longest chord ; the product will be the area of the zone^ ExAKPLB. — Required the area of a zone the diameter of which is 50, and its height 15. 15-^^0=. 3 ; and .3; as per table,==. 28088. Hence .28088X502=702.2 area. RuLB 2. — When the Zone is Greater than a Semicircle : Take the height on each s'de of the dia- meter of the circle, and ascertain, by Rule I, their respective areas ; add the areas of these two portions together, and the sum will be the area of the zone. Example. — Required the area of a zone, the diameter of the circle being 50, and the heights of the zone on each side of the diameter of the circle 20 and 15 respectively. 20-1-50=. 4; .4, as pertable,=. 35182 ; and .35182Xo0-'=879.55. 15-s-50=.3; .3, as per table, =.28088; and .28088X502=702.2. '^' Hence 879.654-702.2=1581.75 area. Rule 3. — When the longest chord of the zone is les^ thin diameter, Take the height or distance from the diam. to each of the chords respectively ; find the area corresponding to each height and deduct the lesser from the greater area ; the result will be the area required. Note. — When, in the division of a height by the chord, the quotient has a remainder after the third place of decimals, and great accuracy is required. Take the area for the first three figures, subtract it from the next following area, multiply the remainder by the said fraction, and add the product to the first area ; the sum will be the area for the whole quotient. Example. — What is the areacrf'a zone of a circle, the greater chord being 100 feet, and the breadth of it 14 feet 3 inches ? 14feet 3incl:es=14.25 and 14.25-t-100=.1425; the tabular area for . 142= 14007, and for 14S=a .14103, the difference between which is .00096. r Then .5X.00096=i.00048. , rr;^ y ItT:??^- Hence .142 =.1400Y .0005 =.00048 .14055, the Bom bj which the square of the greater chord.L3 to be multiplied ; .14055 X lOOa =1405.5/e«<. and VA.Bt^E SPECIFIC GRAVITIES. "" The Specific Gravity of a body is the proportion it bears to the weight of another body of known density. If a body float on a fluid, the part immersed is to the whole body as the specific gravity of the body is to the Hpecific gravity of the fluid. ' '^ When a body is immersed in a fluid, it loses such a portion of its own weight as is equal to th U of the flui ' . Example. — A compound of gold (spec. ffrav.=lB.888) and silver (spec, grao.^^ 10.535) has a specific gravity of 14 ; what is the proportion of each metaL '' 18.888—14=4.888X10.535=51.495 . .„ . ' '~ .^f :^^'' C"^^^, '♦•""" 14—10.535=3.465X18.888=65.447 '*^'''0*« ^-a v'^^'^^.' ; -H"'" !:^ '^ • 65.447+51.495 : 65.447 :: 14 : 7.835 ^ro^c^. "''^^^^/ vCii^v, •*--» ... 65.447+51.495 : 51.495 :: 14: 6.165 ai'iuer. """'*. ,, v To compute the Weights of the Ingredients, that of the M«' . is^fi*: •;..,„. compound being given. - ■••* t.^i-=.y "^ " RuT-K.— As the specific gravity of the compound is to the weight of the compunnd, so are eacli of the proportions to the weiglit of its material. > "^ ;. f ,^ Example. — The weight, as above, being 28 lbs., wbatare the weights of the 'ingredients? _^ ■ --14^28 •• I 7-835: 15.67 ^0^ ""* ...J'^j^i^ Proof of Spirituous Iiiquors. '^^'^ "^ A cubic inch of proof spirits weighs 2.S4 grains; than, u an immersed cubic inch of any lieavy body weighs 234 grains less in spirits than air, it shows that the spirit in which it was weighed is proof. If it lose less of its weight, the spirit is above proof; and if it lose more, it is below proof. Illustration. — A cubic inch of glass weighing 700 grains weighs 500 grains when weighed m a certain spirit ; what is the proof of it ? 700 — 500^200=grains=weiffht lost in the spirit. Then 200 : 234 : : 1. : 1.17= ratio of proof qf spirits compared to proof tphittf or 1.=».17 cAove proof. Solids. Rule. — Divide the specific gravity of the snbstance by 16, and tbe quotient will give the weight of a cubic foot of it in pounds. 44 SPECmO GRAVITIES. i^^' OP DIFFERENT BODIES AND SUBSTANCES. METALS. Alluminu..! Antimony Arsenic Barium BiRmuth Brass, copper 84 " tin 16 " copper 67 \ zinc 33 J plate wire Bronze, gun metal Boron ■ Bromine < Cadmium... Calcium Chromium Cinnabar Cobalt ~ Columbium Gold, pure, cast " hammered - ? " 22 carats fine « 20 carats fine Copper, cast " plates ..•«• " wire Iridium " hammered Iron, cast " gun metal... hot blast cold " wrought bars.... " wire..,. rolled plate Lead) cast > « rolled Liithium. ..<)...... ......a Manganese >.. l£agDe8ium.....ww..«.*>< Mercury— 40° «< + 32« « 60° «« 2I2<» Molybdenum.. Nickel. *' cast Osmium 4( U U U t€ Speci- fic fp'a- vity. » •*.. « <.. 2560 6712 6763 470 9823 8832 7820 8380 8214 8700 2000 3000 8660 1680 5900 8098 8600 6000 19258 19361 17486 16709 8788 8698 8880 18680 2.3000 7207 7308 7065 7218 7788 7774 7704 11362 11388 690 8000 1750 16632 13698 13580 1.3370 8600 8800 8279 10000 Weight of a cu- bic inch .0926 .2428 . 20H4 .017 .355o .3194 2828 .3031 .2972 .3147 .0723 METALS. Palladium Platinum, hammered. *• native *> rolled Potassium, 59<^ Red-lead Rhodium Ruthenium elenium Silicium Silver, pure, cast •• '• hammered. Sodium 1085 Steel, plates. .3129 .067 .2134 .2929 .3111 .217 .6965 .7003 .63 .5682 .3179 soil " tempered hardened. and IT ire.... .... ......... Strontium ..*.....< Tin, Cornish, ham nierd " pure Tellurium Thalium Titanium. Speci flc gra- vity Wolfram .... Zinc, cast .. rolled. (( WOODS (DRY.) Alder., Apple Ash... Tungsten. 3146 Uranium .3212 .6756 .8319 .2607 ,264 ,2655 ,2611 .2817 .2811 .2787 .4106 .4119 0213 .2894 .0633 .5661 .4918 .4912 .4836 .311 .3183 .2994 .3613 Bamboo Bay Beech........... ••«•••••••• 11360 20337 16000 22069 865 8940 10650 8600 4600 10474 10511 970 7806 7833 7818 7847 2540 7390 7291 6110 11850 6300 17000 10150 7119 6861 7191 Birch Box, Brazilian " Dutch M..** " French Bullet-wood Butternut » Campeachy Cedar ** Indian 800 793 845 600 400 822 852 690 567 1031 912 1328 928 376 913 661 1316 Weight of a cu- bic inch. .4106 .7.356 .6787 .7982 .0313 .3241 .3862 .3111 .1627 ..3788 ..3902 .0.361 .2823 .2833 .2828 .2838 .0918 .2673 .26.37 .221 .4286 .1917 .6149 ..3671 .2575 .2482 .26 Cubic foot. 50 49.562 62.812 43.125 26. 51.375 63.25 43.125 36.437 64.437 57. 83. 58. 23.6 57.062 35.062 82.157 SPECIFIC GRAVmBS. 45 WOODS, (Dry.) t^ {Continued.) vity. (( (< Charcoal, pine.... , " freHh burnen van ••••••••••! soft wood ... triturated ... Cherry ,, Chesnut, sweet Citron Cocoa Cork Cypreas, Spanish Dog-wood Ebony, American- " Indian Elder Weight I WOODS, (Dry.) Filbert Fir (Norway Space).... Gum, blue " water. Hackmatack Hazel Hawthorn u,.... Hemlock Hickory, pig-nut , " shell-bark.. Holly... Jasmine , Juniper Lance-wood » Larch } AjcmoD ...•.•••••••. ........ Lignum-ritas Lime Linden Locust Logwood Mahogany... .M. ...... < ** Honduras... ** Spanish Maple *' bird's eye ......... Mastic Mulberry \ Oak, African ... " Canadian. 441 380 1573 2Hn 1380 715 610 726 1040 240 644 756 1331 1 209 6;»5 570 671 600 512 843 lOOO 592 860 910 368 792 690 760 770 566 720 544 560 703 13.33 804 604 728 913 720 1063 560 852 750 676 849 561 897 823 872 ot a cu- bic I'ool. 27 5621 23.75 98 312 5 26 687 125 375 17 86 44 38 45 H5 15 40 47 83 75 43 35 41 37. 32 52 62 37. •^3.75 56.875 23. 49.5 43 125 47 5 48 125 35 375 45. 34. 31. 43 83 50 {Continued ) Speci tlognk- vity ••••»••«•«•■ . 25 26 187 562 437 625 937 5 687 5 Oak, Dantzic ' En!. ,. Plunjbago , Porphyry, red. ... Porcelain, China Pumice stone Quartz Rotten-stone Red lead liesin , Speci vity Rock, crystal Ruby Salt, commun Saltpetre Sand, coarse • common damp and loose.. dried and loose. dry mortar, Ft. Rich. " Brooklyn sillicious ,.. Sapphire Shale « it II II it Slate Slate, purple Smalt Stone, Bath Engl.. «« Blue Bill " Bluestone (basalt) «« Breakneck.. N.Y. " Bristol Engl. " Caen, Normandy Common tt 177. 165. lOJ. 175 «6. 109. 155. IG^O'lOl. 2838 •J 050 1 7.j(i 2800 1750 2184 Weight oi a cu- bic lUUt. li)00 2114 2092 2410 2650 600 1H29 177(1 1176 2l'i0 2765 2;'.ou 915 2660 1981 8940 1089 27H5 4283 2130 2090 1800 1670 1392 1560 1420 1659 1716 1701 3994 2600 2900 2672 2784 2440 1961 2640 2625 2704 2510 2076 2520 118. 130. 151. 37 83. 110 73 131. 172 143 57. 166, 123. 558. 68. 170. 133. 130. 112 IU4. 87. 97. 88. 103. 107. IU6. 162. I8i 167. 174. 152. 122. 165. 164. 169. 156 129. 157. 375 57 37o 5 37o 25 875 75 75 5 062 62 5 25 812 75 187 25 812 75 062 93 125 625 5 375 5 75 66 25 33 5 25 662 062 875 75 5 Stones,Earths,&c'Sa viTy. (( « (( <( Stone, Crai;;leth..Engl. Kentish rag '' Kip'HBay...N Y. Norfolk (Parlia- ment House) . Portland. ..Etigl Sandstone, mean " Sydney Staten Wd. N.Y Sullivan Co. " Schorl Si>ar, calcareous '" Feld, blue <« >< green.,..., " " Fluor Stalactite Sulphur, native Talc, mean Ta'c, black. ... Tile Topaz, Oriental Trap , Turquoise , ••••••• Miscellaneous. A.sphaltum Atmospheric Air. Beeswax Butter Camphor Caoutchouc s Egg Fat of Beef. " Hogs " Mutton Gamboge (jum Arabic Gunpowder, loose '* shaken. ti Gutta-percha. Horn. Ice, at 32" ... Indigo Isingiads [vory Lard < solid.. , 2316 2651 2759 2304 23(18 2200 223 2976 268 3170 2735 2693 2704 340(> 2415 2033 250(1 2900 1815 4U11 2720 2750 905 16.30 « 965 942 988 903 1090 923 936 923 1222 1452 90;i 1000 1550 1800 980 1689 920 1009 1111 1825 947 ■Weight uf a cu- bic foot. 144.75 165.687 172. 744, 148, 137, 139 186, 168, 198 170, 168, 169 215, 150 127, 156 181, 113, o 812 125 937 312 5 937 062 25 25 437 170. 56.562 103.125 .07525 60.312 58.875 61.75 56.437 57.688 58.5 57.687 90 . 75 56.25 62 5 96.875 112.5 61.25 105.662 57.5 63.062 69.437 114.062 59.187 (•) .001905. 4S SPECIFIO GRAVITIES. Miscellaneous. Mastic M^'rrh Opium Soap, Castile. Spermaceti.... Starch Sugar .66 t( Tallow. Wax.... Liquids. Acid, Acetic ^' Benzoic " Citric " Concentrated " Fluoric " Muriatic " Nitric. ** Phosphoric " " eol'd.. " Sulphuric Alcohol, pure, 60° 95 per cent 80 It (< <( (( (( 50 40 25 10 5 " proof spirit, '50 per cent 60''. " proof spirit. 50 . percent 80". J Ammonia, 27.9 per ct. Spec! flc ItTA vity 1074 1.360 1.336 1071 943 950 1606 1326 972 941 964 970 1062 667 10.34 1521 1500 1200 1217 1558 2800 1849 794 816 863 934 951 970 986 992 934 875 Welfsht of a cu- bic loot. 67, 85. 83, 56. 58, 59. 100. 82. 60. 58. 60. 60. 125 5 937 937 375 375 875 25 812 25 625 Liquids. Aquafortis, double, " single.. Beer 66 41 64 95 93 75 76 97 175 115 49 51 53 58 59 60 61 62 .375 687 .625 062 ,75 062 375 562 622 937 37.0 437 625 625 .58.375 54 8911 55 .687 687 Bitumen, liquid Blood (human) Brandy, f or 5 of spirit Cider Ether, acetic " muriatic ** sulphuric Honey • ••••a ... Milk Oil, Anise-seed. " Codfish " Cotton-seed. " Lipseed " Naphta " Olive " Palm " Petroleum . " Rape " Sunflower... " Turpentine. «« Whale Spirit, rectified Tar Vinegir Water, Dead Sea . ,- 60"... 2120... distilled, 39ot ... Mediterranean ... rain sea Wine, Burgundy " Champagne " Madeira " Port u It it <( (( (( Sppci- fic gra vity. 1300 1200 10.} i 848 1054 924 1018 866 84 715 1450 1032 986 923 940 848 915 969 878 914 926 870 923 824 1015 1080 1240 999 957 998 1029 1009 1026 992 997 10.38 997 Weight of a ca- ble foot. 81.25 75. 64.625 53. 65 875 57.75 63.625 .54.125 52.812 44 687 90 625 64.5 61.625 57 681 58 75 • 53. 57 187 60.562 54.875 57.125 57.875 54.375 57.687 51 5 63.437 67.5 77.5 62.449 59.812 62 379 64.312 62.6 64.125 62. 64.. 375 62.312 62.312 Compression of the following fluids under a pressure of 16 lbs. per square inch : . Alcohol 0000216 Ether 0000158 Mercury „ 00000265 Water 00004663 * Specific gravity of proof spirit according to Ure'a Table for Sykes'a Hydrometer, 920. tl cubic inch = .252.69 Troy grains. WEIGHTS AND VOLUMES OF VAKIOUS SUBSTANCES. 49 Elastic Fluids. It Cubic Foot of Atirtosfjheric Its asBumed Uravity of 1 t« Atmospheric air, 34°. Ammonia.. • Azote , Carbonic acid •* oxyd Carbureted hydrogen. Chlorine Chloro-curbunic Cyanogen Gas, coaI....M Hydrogen , Hydrochloric acid , Hydrocyanic " ■ Muriatic acid Nitro^^en Nitric oxyd Nitrous acid , Nitrous oxyd. Oxygen I. .589 .976 1.52 .972 .559 2.47 3.389 1.815 .4 .752 .07 1.278 .942 Air weighs 527.04 Troy Grain*. tht Unit for Elastic Fluids. Phosphuretcd hydrogen 1 Sulphureted " 1 Sulphurous acid 2. Steam, * 212"... 247 972 094 638 527 102 Smoke, of bituminous coal. coke. wood Vapor of alcohol bisulphuret of carbon Vapor of bromine chloric ether ether hydrochloric ether.... iodine nitric acid spirits of turpentine.. sulphuric acid " ether sulphur. water u (( (( t( t( l( (( (( <( 77 17 21 48S3 102 105 09 613 64 1 44 586 255 675 75 763 7 586 214 623 "M^eiu^lits ancl'Voliiincts ofvarious Substanoes in Ordinary Use. <- ...'v Substances. ^.« ..... Metals. Brass^^PP""! ( zinc 33 ) gun metal... i-heetfi wire.... Copper, cast '• plate Iron, cast •* gun metal ** heavy forging '* plates *' wrougiitbar.«. Lead, cast " rolled, Mercury, fiO", Steel, plates *' soft Cubic Cubic Foot. lucli. Lbs Lbs. 488.75 2829 543.75 .3147 513.6 .297 524.16 . 3033 547.25 .3179 543 625 3167 450.437 .2607 466.5 .27 479.5 .2775 i8l.5 .2787 486.75 2816 709.5 .4106 711.75 .4119 S48.7487 .491174 487,75 . 2823 489 . 562 . 2833 Substances. Metals. Tin Zinc, cast.... " rolled. Woods. ••••••••• Ash Bay Cork Cedar Chestnut.. Hickory, pig nut. " shell-bark. Lignum-vitse Logwood Mahogany, bon- duraa Cubic Foot. Lbs. 455.687 428.812 449 437 52.812 51.375 15. 35 062 38.125 49.5 43.125 83.312 57.062 35. 66.437 Cabio luch. Lbs. .2637 .2482 .2601 Cub.Fpet in a Ton. 42.414 43.601 149.333 63.886 58 754 45.252 51.942 66.886 39.255 64. 33.714 t Equal to .0752914.3 lbs avoiidupois. * Weight of a cubic foot, 257,333 Troy grains. 50 WEIGHTS AND VOLUMES OF VARIOUS SUBSTANCES. Sul>r,Lances. Oak, ranailinn.... '* Kii;;li-li ♦* live.seuxoiied " white, (iry... '• '• upluiicl Pine, pitch r"ii» • ••••••••• " white " well Keas-oiieil " yeilow Spruce... Walnut, black, drv Willow .'. *' dry.. Miscellaneous Air Basalt, lueun Brick, tire " mean Coal, anthriicitc I " bituiiiin.,iiieari " Canriel. " Cuinberlaud... Cubic I'OOt. Cub Fpet iu a Tuu 54.5 58.25 ()G 75 5S.75 42.9:57 41.25 :{d.875 ;U.e raliseU beiuip fg-lveu. By inversion of the preceding rule, • ^ g/W X 700^»-«' \ — i52HtJ ^d, 8 and «' feprenntittg fAe weight nfnir and gag in grains per cubic fool, and d the diameter of the balloon in feet. Example. — Given the elements in the preceeding case {/597 46-f 100X7000-5-527.04— 31.62 Then :523"6 =v/l8821.09=26.6/«e<. ' ' *•" To Connputo tlie TTeitrht of Cast 3£otaI l>v tlie WtiiKht ol tUe JPatteria. ^ ' "^ • ' When the Pattern ia of \fhite Pine. Rule.— Multiply the weight of the pattern in pounds by the following multiplier, and the product will gve the weight of the ca-»titig: Iron, 14; Brass, 15; Lead, 22; Tin, 14; Zinc, 13.5. When there are Circular Cores or Prints.— Multiply the square of the diameter ol the core or print t.y its length in inches, the product by ,0175, and the result is tlie weiglit of the pattern of the core or print to be deducted from the weight of the pattern. It IK customary, in ihe making of patterns for castings, to allow for shrnkage per lineal foot of uHilern. Iron and Lead ^th of an inch, Brass and Zinc -^jth?, and Tin ^jth. 52 KEY TO THE TABLEAU _- n ;:• y '.-.fit '•■.■..-..• ir, oVf ')■ i PROBLEM. .■'■■• "■ . To determine the accurate solidity of any irregular bady of small dimensions or of a body composed of several elementary parts -with different dimensions ,, ,• ■'-"■ and forms. *- ,' (1) RVUE. If it is the capacity of any vase or vessel which we want to measure, the idea generally suggest itself of an-ivin-f at the result by detenniniiifj the number of times which such a vessel ca i f/ive place to or contain the contenis of any other vessel of an eleinentiiry form of which toe know the capacity. («) But if it is the solidity of the substance itself of ths vessel, &c., which we desire to measure, the manaer of operating does not ituaiediutaly preseut itself to the miud of any one wishing to obtain tlie result. (3) RULE. If the solidity to be measured is that of a non absorbent substance, ^i^^ immerse it in a vessel full of water or any other liquid of which we will, measure the displacement by meanf of another vend of known capacity ; or if the first vessel is large enough and it form rectangular or cylindrical and of easy gauging, we will first put in it enough liquid to cover the object to be measured ; having afterwards observed the height of the level of the water in the vessel, we will immerse in it the object in question and observe again tJie level of the liquid ; if now we suppose that each fraction of a metre, inch line or any other unit of the height of the containing vessel corresponds to a cubic metre, foot, inch, or line, &c., we will have but to count the number of such units in the height of the displaced level of the water to obtain immediately the solidity of the proposed object. (41) If the body is absorbent, we may for instance use sand or any other fluid substance, of the kind, that toe can level the surface of by means of a rod with a rectilineal edge. In this manner we woald arrive at the solidity of the most deversified bodies of the animal, vegetable or mineral kingdom and of the thousand aud cue raw or mauafactared objects wMch we have constantly under o MENSURATION OF SOLIDS 08 and of which it wonUI often bo im possible to luenBiire the ■olidities by the ordinary rules of geometry. It is well to remind also that we may arrive by a simple proportion til the solidity of a body by coinpiiring its weight with that ot another body of the same substance and of determined solidity, that is by th^ system of upe- cifie gravities which nhowsiit the sun o time how to obtain the solidity of a body from its weight : which will form the subjects of the next (iroldem. Ex. I. The weight of an irregnliir block of stone i^ 13 pounds? ounces : required to determine with the help of the given piece the weight nearly of u cubic foot of such stone. " Anf. First cube the block of stone; to that eflfi-ct get a rectangular vessel, say 10 inches square or I0«) inches in hiuizontal area, and the height of which is divided into inches and hundreths of an inch ; having poun^l into the vessel water enough to cover the stone to be cubed, I note the height of , the water which I find 8. 53 inches, I then immerse the stone in the vessel and * I note a;;ain the height of the water which is now 9. Hi) inches ; ihe diff<;rence of these heights is I.36inche8. Since the vessel is 10 x 10 inches, it is pl.iia that every inch of its height corresponds to 100 cubic inches and conse- quently, each hundredth of an inch of such a height to one cubic inch ; there- fore the observed height 1 .3G, of the displaced level of the water correspouda ' to I3«) cubic inches ; therefore the solidity of tiie stone is 13), and we will now obtain the weight of the cubic foot by making I;}6:2I5 ounces (weight of the stone) : : 1728 cubic inches (that is a cubic foot) : 2732 ouuces, or, di- '' Tiding by 16,17()i pounds, the required weight. < ' 2> In a cylendrical vessel such that each inch of its height corresponds - to 1 cubic inch of space or solidity, we have immersed a piece of silver which ' Las displaced by 73 hundreths of an inch the level of the liquid in the vase • required the solidity of the iugot of silver ? Ans. 73 of a cubic inch. 3. Having filled with water any vessel, we have immersed in it au object the solidity of which we want to know ; we have gathered in another vessel, the water overflown, the quantity of which is 3 gal. 2 quaita and i pint ; what is the solidity of the proposed object, the galloa made use of being 231 cubic inches ? Ans. I gallon + 2 quarts + | pint =231 + 115i + 14/y = |f cubic inches. 4. Required the solidity of an absorbent subs.tance placed in a vessel one foot square filled with sand; after having removed the object to be measured, we find that the uniform height of the sand in the vessel, first levelled to that effect, is .3 of a foot, the height of the vessel being 1.5 feet ? Ans. 1 .5<--3=1.2 fe6t= height of the displaced level of the sand, and as 64 KEY TO THE TABLEA.XJ the vessel is I sqnare foot in horizontal section, it follows that the solidity of the object is 1.2 cubic feet. ft. In a vessel having the form of Ihe frnstuin of a cone is a quantity of liquid of which the diameter at the surface i.s 10 iurhes : we immerse ia itaa object which increases by 9 inches tlie heiglit or depth of the liquid in the ve.seel and which gives to its displaced surface a diameter of 14 inches ; re- quired the solidity of the propo.sed bo;ly ? Aia§. The volume of water displaced which is at the same time that of the object, is that of the frustum of a cone of which the parallel bases mea- sure respectively 10 and 14 inches and of which tlio height is 9 inches 5 this 22 2 8ol.= (iri. T.) (10 +14-1- 4 times 12) x 7354x94-6 =872 x .7854 x 1.5 684.8688 X ] .5= 1027.3032 cubic inches. ., , THEOREM. '' "*o- To determine the solicity or "weight of any body or substance, . by comparing the volume or •wei^jht of such body "with that of a body or substance of the same nature of which . "we know beforehand the weight and volume. (•>) REm. The weight of a ciibie foot of water at the temperature of 40° Faliieiiluir (at wliich water nearly reaches its greatest density) is 1000 ounces avoir du ^oids nearly, or 62i pounds (engli>h w i^ht) and we denomi- nate weight or specitie gravity of any body or substance, the weight of a volume otsuch body or substance equal to iliat of the water tiken f.>r com- parison ; whence it results that if in advance we know the weight of a cubic foot, for inst mce, <»f each of the ditferent substances that we may bo called on to measure or value, us stated in table X, we wi I at once determine by a simple proportion the volume of any other weight or quantity of the same substance or the weight of any other vo.ume of such substance, by the fol- lowing rules. i -,:■' •'. (6) RULE. To determine the solidity of a body from its weight • wnfee the proportion : the specific weight of the proposed body is to ( : ) its weiyht in ounces or pounds, c&c , as ( : : ) 1 cubic foot or 17i8 cubic inches, is to (:) the solidify cfthe body infect or inches, as the case may be. Ex. 1. The weight of a shell or cast iron ball or of any fragment of such a solid is 4.^ pounds : required the solidity of the proposed body? Ads. It is seen by table X of specific gravities that the weight of cast iron is 450 pounds nearly, per cubic foot j we will then obtain the required solidity by making 450 pounds : 1728 cubic inches : : 45 pounds : 172.8 cubic inches. MENSURATION OF SOLIDS 66 2. Required the volume of a rrnrble statue the weight of which is 1000 pounds, thn spccidc gravity of ih(i mill ble from wliich the statue is drawn being 170 pounds nearly to the cubic foot ? Ans. 170 pounds : 1 cubic foot : : 1000 pounds : 5 9 cubic feet nearly. 3. A quantity of sand weighs 13 pounds : wh;it is its solidity ? AnN. Fiom table X, the specific gravity of sand is 1.520, that is, 1.520 times the weight of an equal volume of water or lo.'O ounces to the cubic foot (since the weight of a cubic foot of w.iter is 100.) ounceK) ; we will therefore make 1520 ounces : 1728 cubic inches :: (13 x 16 = ) 20S ounces : x := 1 728 X 208 =23Gi cubic inches. r526 4. The weight of a tu.^k or tooth of an elephant is 23 pounds ; what is its solidity? Ans. Ivory is 1825 ounces to (he cubic foot; we will th('ref)re obtain the solidity of the tooth by making 1825 : 1 : : (25 pounds or) 400 ounces : .22 nearly of a cubic foot, or 1825 ounces : 1728 cubic inches :: 400 ounces : 378.74 cubic inches. », .-. ,:., ^^^w*. . ? ,, . -f ' ^irt-'i^i-^^t 5. It is required to determine in advance the probable weight of a cast iron grating which must be cast according to a carved model of piue wood the weight of which is 7 pounds I Ans. We will first obtain the solidity of the pine model by making, as per rule (the |»ine being considered in this case as of 25 pounds to the cubic foot) 25 ponnds : I cubic foot : : 7 pounds; .28 of a cubic foot. Now, as the solidity of the cast iron is 450 pounds per cubic foot, wo will obtain the weight of the proposed gratiiig = 450 x .28 = 126 pounds. (7) RULE. To determine the "weight of a bot^y from its volume; make the proportion : as one cubic foot is to ( : ) the volume of the proposed body, so is ( : : ) its specific giavity to ( : ) its weight. Ex. 1. The volume of a heap of sniug it to founder or sink beyond its deck or superior surface f AnSi. Since water weighs 62.5 pounds to tbe cubic foot and the total volume of the pontoon is 20,000 cubic feet, the total weight o'' the water which the pontoon must displace before sinking to the lever of the water is 20,000 X 62.5 = 1,250,000 pounds; now the weight of the boat is but 230,000 pounds ; whence it follows that we might still without causing the bateau to founder load it with a weight equal or nearly equal to the difference between 1250,000 pounds and 230,000 that is 1020,000 pounds. (9) RULE II- If the body to be computed is heavier than "Water • fi>'>^t weigh the body in air, then in water, by means of a hydraulic balance ; the difference between the results will be the weight lost in water, or the weight of a quantity of water equal in volume to that of the body. Make now the proportion : as the weight lost in water {: ) is to the weight of the body in air (::) sois the specific gravity of water (:) to the specific gravity of the body. ]':"" \ .. ■• . .; vj; "^;^» Ex 1. A piece of tin weighs 183 pounds, its weight in water is but 158 pounds : what is the specific gravity of tin ? Alls. 183—158=25 : 183:: 1000 : 7320=required specific gravity. 2. A block of granite weighs 21 ounces iu air and only 13 ounces in water : what is the specific gravity of the granite ? I "' " " Ans. 2625 (lO) RULE HI. If the body to be computed is lighter than water ; tie to the proposed body by a thread the weight of which is relatively null, another body heavier than water, so that both of them taken together may penetrate or sink in the water ; having first weighed each body in air, and the heavier in water, weigh then in water the compound body, and from fhe weight lost by the compound body, substract the weight lost by the heavier body as weighed alone ; the remainder is the weight lost by the light body. Then : as the weight lost by the light body in water. {:) is to the weight of that body in air, {::) sois the specific gravity of water ( : ) to the specific gravity of the body. Ex. 1. To a piece of elm which in air weighs 15 grains, we have tied a piece of copper the weight of which is 18 grains in air and 16 grains in water and the compound in water w<;ighs but 6 grains : what is the specific gravity of the elm ? 58 KEY TO THE TABLEAU Ann. 18 — 16 = 2=tlie number of grains lost by the copper t»j (he water. 18+ 15 — 6=27=the number of grains lost by the compound trt the water. 27—2 =25=tlie number of grains lost by the elm in the water. ^5: 15 :: 1000 : 600 = tlie specific gravity of the elm. 3. A piece of copper, weijijliing in air 27 ounces ami in water 24 ounces, is tied to a piece of coik weighing in air (J ounces, and the coiupound weighs in water but 5 ounces : what is the specific gravity of coik? . . .^^ .•v Ans. 0.240. •. \ u-^<\X- I ■. ^ . PROBLEM. '■ -.- i-v> ^ ' ■ -V. ,i-:-^n,;?v';.-.. To determine the quantity of each ingredient or element in a compound of two substances or elements. ^ (11) RULE. Find fir>t the specific weif/ht of the compound, mixture or alloy, and of each of the component elements and multiply the difference of every two of these three specific weights by the third. Make then : the (jreatest product, (: ) is to each of the other product, (;:)«» the weight of the atloy, ( : ) is to the weight of each ingredient. -.^ , ^.-^ ,- •,., .,. ; .^^ > . _ .. ^ .- ■... . ■■ ,_^^ , Ex. 1. A mass of gold and silver weighs G'2 ounces, and its specific gravity is 16J26 j what is the quantity ()f each ingredient, the specilic gravity of gold being- 1LI640, and that of silver 11 091 ? Alls. (19640— 11091) X 1612fi=137,86l, 174. Alloy. t,,^? - (I9()40 — 16126} X 11091 = 38,973,774. Silver, i .| , (16126— 11091) X I964() = 9d,88/,4U:). Gold. ,; ... 137,861,174 : 98,838,400 :: 63 : 45 ounces, 3 penny weight,s, 19 grains of gold. 137,861,174 : 38,973,774 .. i3 : IT ounces, 16 penny wi-iglits, 5 grains of silver. ft. A mass of copper and gold weiyhs 48 ou-:ces, and its specific gravity is 17150, the specific gravity of gold is I9u40 and that of copper 9J00 : what is the quantity of each element of the mixtiiie ? An§- Gol(l=42 ounces 2 pennyweights 2 1°5^§ grains, copper =5 ounces, 17 penny weights 21 f^eia g'^'i'is. 3. .^n alloy of silver and copper weighs 60 ounces, its specific gravity being I0.)35 : required the weight of each ingredient, their respective specific gravities being 11091 and 9000 ? Ans. 46 ounces 7 penny- weights 9 1 Jftsy^ grains silver, 13 ounces 12 penny-weights 14 rWFsVff "^ copper. 4. An alloy of copper and tin weighs 1 12 poands and its specific gravity is 8784, what is the quantity of each of the ingredients of the mixture, their respective specific gravities being 9000 and 7320 ? An*. 100 pounds copper, 12 pounds tin. MENSURATION OF SOLIDS 59 5. Hf'qninMl thp •weifjlit. of poM, in n compound of quartz ami golil the specific gravit}' of which is 3300, that uf gold being 1964IJ and that of quarts 3'JOO ! Ans. 19640- 3000= 16640 x35;i0 -58,240,000= Factor for the coiiiponnd body. V'- • 19640--3500 = 16I40, 10140 x3f)00 = 48,42n,000 = r ; : Factor for the qnartz. 3500 -3003 -- 500, 500 x 19340 = 9,823,030 = Factor for the gold. 58240000 : 9820000 :: 100 : 10.86.8612638 -ounces of gold ; if this result .be correct, the weight of the qnartz must be equal to the diflfereuce between the weight of the gold and till t of the alloy, and in fact 58210000: 48420000 :: 100 : 83.13i73li2 ) ounces of quaitz; the sum of these numbirs= 100 j there fore, &.C. -■-»■ --w;J:-- ki:^j ■:..t-<.-^r !.;■,'/- -^t'^-,-'' PROBLEM. ' ^-iU-A'rui To determine the solidity of the largest piece of squared timber that may be got out of a round log", or ,y, . out of felled or standing tree. (1*2) RUEiE. MnltipJij the diameier of the tree or log ty the half- diameter , and this proiiict by the Un'jh : the result wdl he the required solidity. In fact, it is plain that the diara. AB multip'ied by the lialf-diameter OC (oi- ^ AB) sjiven for piodnct the. area of the inscribed sipiare ABCi), tint is, the area of a section, of the timber to be computed, by a plane perpirmlicnl ir to it-* kU lenirth, and thit area multiplied by the length of thsj log gives (78 T.) th*. reipiired solidity. RI^M. This rule supposes tint the diara. of thu tree is every win r.; the same or that we make use of a mean diameter, as taken at middle of the length, aiid this gt-neially done when there is not too much diflFcreiice lulween the diameters of the oppo>^ite ends; but to be precise (148, T.) Wf must as already stated (91, T.) add to the sura of the areas of the ends of the log or tree to be measured f mk- times the area of a section taken at the centre ami multiply the whole by ths sixth ^»art of the length, or which. is the same thing, multiply the sum of the areas by the whole length and take the sixili part of the result. 1.x. I. The circumference of a log, the length of which is 12 feet, is 6.23 feet, dedui-ti(»n being ma le of the bark if nece-*sary : h iw many cubic feet of wood will there be in the stiik of siiuared timber to be got out of the log ? Ana. The circ. 6.23 correspouds to a diam 2, the section of the timber ^ KEY TO THE TABLEAU Tvill th^reforp be 2 X 1=2 square feet in area, anil as the length is 12, the solidity will be 21 cubic feet. jrar.' ,^h.^. ^•■ I« fact, it is plain tliat the surface wanting at each of the four angles, corners of edges of the tim- ber, to complete tlie square A B, is the triangle abo, or a triangle equal to abo, when as it is supposed, ef z=gh=kl=ab ; now the square on a6 is woith 4 abo ; therefore, &o. REM. 1- If the sides a&, e/, &c. are not equal to each o.her, we may take one fourth of the sum of these four sides for a mean diameter a&, or for greater accuracy, we will make separately the squares of ab, ef, Sec, and the fourth «f the sum of those squares will be, or the sum of the fourths of those squares will be the quantity, nearly, to be subtracted from the square AB to obtaiu the net area of the section of the timber. KIS9I. II. Let us observe as in the last problem that if the timber is uot throughout its entire length of equal size, its ."iection must be taken at about the middle of its length, and this is generally what is done (148 T.) or, we will determine several sections of the timber and then take their mean, or MENSUKATION OF SOLIDS 61 finally we will mnke the sain of the areas of the oppositfl ends plus four times that of the inteniu'diatc^ section and afterwards multiply the v?hole by the length nud take the sixth part of the result. RI291. III. We must also observe that we may arrive at the area of any regular or symmetrical octagon or of the kind liere illu.strated by subtracting from the square of the perpendicular distance AB which separates any two of its pai allel sides, the square of one ab of the sides aiijucent to the first. Ex. 1. An eight sides piLir is .'ifeet wide or thick AB, the side ab <>t the chamfer aob is (i inches : what is the solidity of the pillar, its length or height being 10 feet ? Ans. (3 + 3— (.5 X .5) = 8.75 superficial feet, and 8.7 i x 10—87.5 cubic feet=required solidity. a. A log of timber the edges of which are waney, measures 30 inches square and .'JO feet long, the average «»f the sides ab, ef, &c., of the wane is 9 inches ; what is the solitiity of the timber / Ans. (30 X 30) minus (9 X 9) =919 square inches -area of the section of the timber = 6.382 feet very nearly, and 6.;«2x30= I7j.4f) cubic feet. 3. We h:ive reduced to 30 inches square at the large end a tree the diam, of which w;is at that point 36 inches ; at the small end the diain. 30 inches has been reduced to 25 inches ; the w.ine, sapwood or defect from a true 8(|uarea6is from 7 to 6 inches respectively at tlie two ends, such as obtained by a direct measurement of the piece of wood to be cubs'd, or by means of a sketch made from a scale of equal parts : what, is the solidity of the timber, its length being 60 feet ? Alls. Area at the large end =(30 x 30) - (7 x 7)=851 square inches, area at small end = (25x25) — (6x6) = 589 sq. f., the intermediate area /30 - 25 30 ' 25\ /7 + 6 7 + 6\ 22 (^— g- X— 2-J~(~2~^~irj =(27ix27i)-(6Jx6i) =27.5 -6.5 = 756.25-42.25=714 ; 851 +859 • 4 times 714=4296 square inches, dividing by 144 we obtain 29.83^^ square feet, multiplying by ^ of the length or by 10 we obtain 29S.33 cubic feet. Am8. Area section at the centre = 714 square iitches, 714-4-144=4933 square feet, 4. 9583 x ()0=2y7.498 ciildc feel, th;it is, equal to the ai^curate soli- dity by Jess than one foot neatly, or by ie.s.s than one 30i)th neaily, or by less than one thiid nearly of 1 per cent, suflaeieut accuracy (148. T.) iu practice. REM- IV. A conipaiison of the two answers of the last problem indi- cates suflBciently thit the ordinary practice of cullers, who take the dimen- sions of a log at the middle of its length, and afterwaids multiply the area of th.! section at th t place by the length «»f the timier, to obtain thus its soli- dity, is, cuuaideriug all things, (148 T-) suucliuued by oiiuuiuatuuces. .■ i » ; ^ INDEX ■:-,. , ..,--..•■,'■.-:/ '-.T -*r ■: 7:^'-!.-: ::( iu >..,','-':^ :;.'.v The Stcrpotncti icon : ?iompnclntnro mid jt,.iipim1 fcahire of ^adi of thf^ 200 Kolidg on the board ; see the diiigr.ira at the beginning of lliis pamphlet 5 The Areas of Spherical Triangles & Polygons to any radius or dia- meter : a paper read before the Iloyal Society of Canada iii 18i3. 55 On the general application of tlie ptisin »i.lal formula : a paper read be- fore the Koyal Society of Canada in I8:i2.... 61 TABLES I. Squares atid Square Roots of numbers from 1 t<> IGOO .■... 4 II. Circumferences and areas of circles of diameter ^'j to 150, advaii- tiiig by ^ 11 III. Circumferences and areas of circles of dimneter ^'^ to 100, advan- «""ghy I'a 19 IV. Ciicumferences and areas of circles of diameter I to 50 feet, ad- vancing by 1 inch or xV 25 V. Sides of Squares equal in area to a circle of diameter 1 to 100 ad- vaucing by \ 29 VI. Lengths of circular arcs to diameter 1 divided into 1000 equal parts - 31 VII Lengths of serai-elliptic arcs to transverse diameter 1 divided into lUOO eipial parts -. 33 VIII. Areas of the segments of a circle to diameter I divided iuto lOOOequal parts 37 IX. Areas of the zones of a circle to diameter 1 divided iuto 1000 equal parts 33 X. Si>ecific gravities or weights of bodies of all kiuds : solid, fluid, liquid and gazeous >.. 22