THE 
 
 STEREOMETRICON. 
 
 Originator: C. BAILLAEGE, M.S.,, \^. 
 . . ' ''■( '•''■''' 
 
 Member ok the Sooisty for the Genbralizatioh ob- Edpoation in France, and 
 
 OF 8BVEBAL LBARNHD ANU SCIENTIFIC SOCIETIES ; CUBVAI.IEK OF THE 
 
 Urdkr qf St. Sauvkur ok Montk-Kkalk, Itaxv ; etc., etc. 
 
 MKASUREMENT OP ALL SOLIDS BY ONE AND THE SAME RULE. 
 V. . , UNIVERSAL APPLICATION OF THE PUISMOIDAL FORMULA. ,,; „ 
 
 Thibteen MtDALa and sierKNTEEN Diplomas and Letters awarded the Author 
 
 ,«S";'i i) '■ JfRot Russia, France, Italy, Beloium, Japan, Krc, ■•..^,-5 ,'*j,-V >;. 
 
 
 Promoter: THOMAS WIIITTY, '''^^'■^S^^^t' 
 
 u ]^ ' ■' - - ^i^'ij 
 ,^■ ' /'V^iOFESSOR AT ST. DENIS. ACADEJIV,. .MONTREAL- ,i!t'5-vf:J^;" 
 
 Comprises 200 /solids representative of all conceivable elementary forms, as of 
 .' , ' ' . ."■, , ' '. the Component parts yf Compound bodies. I' , ,, ^„,' . 
 
 Name and Je«c;(iption of each solid. What it is representative or suggestive of, or 
 'l'^'^^'^'fif^:^t'^''!^: ' V that of which it forms a Component part. v'v*- ,;-''^ 
 
 Nature and name of opposite bases and of middle section as of lateralfaces and 
 
 remaiqder of bounding Area, including every species of Plane, 
 '. -' ^1^.-: • . J-'^'S^ ' /^Spherical, Spheroidal", and Conoidal figures. ' A-.r^si-;, ''. ^'-f/ / 
 
 Division t, classes T toX : plane faced Solids and SoliJs of single curvature. 
 Division II, classes XI to XX : Soli.ls of double curvature. 
 
 PRINTED m- JOHN L(:)VELL ■&' SO>T. ' 
 
 1880. 
 
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 THE 
 
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 STEREOMETRICON 
 
 Oeiginator: C. BAILLARGE, M.S. '^ ' 
 
 Member of the Sooiwty for the Generalization op Education in France, and 
 
 of several learned and scientipic societies ; chevalier of the 
 
 Order op St. Sauveur de Monte-Realb, Italy ; etc., kto. 
 
 MEASUREMENT OF ALL SOLIDS BY ONE AND THE SAME RULE. 
 UNIVERSAL APPLICATION OF THE PRISMOIDAL FORMULA. 
 
 Thirteen Medals and seventeen Diplomas and Letters awarded the Author 
 PROM Russia, France, Italy, Belgium, Japan, etc. 
 
 Promoter: THOMAS WHITTY, 
 
 PROFESSOR AT ST. DENIS ACADEMY, MONTREAL. 
 
 Comprises 200 Solids representative of all conce-vable elementary forms, as of 
 the Component parts of Compound bodies. 
 
 Name and description of each solid. What it is representative or suggestive of, or 
 that of which it forms a Component part. 
 
 Nature and name of oppqsi*e .bajes and of middle ^ section as of lateral faces and 
 
 remainder of .bonnfting'Area,' including every jpecife.'i of Plane, 
 
 SpheVical,' SphttoidaiJ and Corfcudal Sig^ircs^ 
 
 Division I, classes I to X : plane faced Solids and Solids of single curvature. 
 Division II, classes XI to XX : Solids of double curvature. 
 
 Ptftttwat : 
 PRINTED BY JOHN LOVELL & SON. 
 
 1881. 
 

 » ■ «' »•• •• 
 
 * . ■ * I 
 
 
 t V • • 
 
 
THE STEREOMETRICON. 
 
 OniGiNATOB: C. BAILLARGE, M.S. 
 
 Member of the Society for the Generalization of Education in Prance and of seve- 
 ral learned and scientific Societies ; Chevalier of the Order of St. Sauveur de 
 Mont-Rual, Italy; etc., etc., etc. 
 
 Measurement of all solids by one and the same rule. 
 Universal application of the prismoidal formula. 
 Thirteen Medals and seventeen Diplomas and letters awarded the author, from 
 France, Russia, Italy, Belgium, Japan, etc. 
 
 Promoter : THOMAS WHITTY, professor at St. Denis Academy, Montreal, etc. 
 
 RULE : To the sitm of the opposite and parallel end areas add four times the 
 area oj a section midway between and pat-allel to the opposite buses ; multipli/ the 
 whole by \ part of the lenylh or heiyht or diameter of the solid, perpendicular to the 
 bases ; the result will be the solidity or volume, the capacity or contents of the body, 
 figure or vessel binder consideration. 
 
 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 any other unit of measure, tables of segments and zones of a circle, etc., 
 etc., at end of " Key." 
 
 The tableau comprises 200 models, disposed in 10 horizontal and 20 vertical 
 rows, series, families or classes. The solids may be indififerently placed, and 
 numbered from the right or left and from below upwards. 
 
 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. 
 
 Name of solid, objects of which it is Nature and name of opposite bases and 
 representative or suggestive, or of which middle section, of lateral faces and re- 
 it forms a component part. mainder of bounding surface. 
 
 Reference to " Key to Stereometri- Reference to page or paragraph of 
 
 con," for computation of contents and " Key " for calculation of areas and of 
 
 of factors necessary thereto. factors necessary thereto. 
 
 Note. — The author uses the term " trapezium " and not " trapezoid," as the termina- 
 tion "Old " conveys the idea ot'a solid aa paraboloid, hyperboloid, conoid, prismoid, etc. 
 For the same reason be uses the French " trapeziform " instead of trapezoidal. 
 
 51016 
 
1. — The cube or hexaedron— 
 one of the five platonio bodies. 
 
 Representative of a building or block 
 of buildings or of one of the component 
 parts thereof; a brick or cut stone, a 
 pedestal, a die or dado ; a pier or quay ; 
 a box, chest, package of merchandise or 
 parcel ; a cistern, bin, vat or other ves- 
 sel of capacity ; a pile of bricks, stones, 
 lumber, books, etc., etc., etc. 
 
 " Key to Ster.," page 61, par. (7S). 
 
 2. — A right isosceles triangular 
 prism. 
 
 On end, a triangular block or build- 
 ing ; on its base, a ridge roof; on one of 
 its sides, the roof of a pent house or lean- 
 to. " Key to Ster.," page 61. 
 
 3.— A right regular pentagonal 
 prism. 
 
 On end, the base or component part of 
 the shaft of an octagonal pier or column ; 
 on one of its sides, a baker's, butcher's 
 or other van ; an ambulance, etc. 
 " Key to Ster.," page 61. 
 
 4.— A right regular octagonal 
 prism. 
 
 Base or shaft of a column, a pier or 
 post, a bead, baluster, hand-rail, etc. 
 " Key to Ster.," page 61. 
 
 5. — An oblique hexagonal prism. 
 
 An inclined post or strut or the sec- 
 tion of a stair-rail, a baluster on a rake, 
 etc. Mitred section of a rail or bead. 
 See " Key to Ster.," page 64. 
 
 6.— Oblique rectangular prism. 
 
 On end, an inclined strut or post, etc.; 
 on its parallelogram base, the pier of a 
 skew bridge, portion of a mitred fillet, 
 etc. See " Key to Ster.," page 64. 
 
 7. — Oblique prism or parallelepi- 
 ped. 
 
 Section of mitred fillet on an inclined 
 or oblique surface, etc. 
 
 8.— A right rectangular trapezi- 
 form prism, or a prism of which 
 the base or section is a rectan- 
 gular trapezium. 
 
 On end, a pier or block of that shape ; 
 on its larger parallel face or base, the 
 
 Each of its three pairs of opposite and 
 parallel faces or of its six faces or bases 
 and middle sections, perfect and equal 
 squares. For developed surface, see 
 "Key to Ster.," page 131. 
 
 Representative of the floor, ceiling, 
 walls or partitions of a rectangular room 
 or apartment, or of the bases and sides 
 of the various objects mentioned under 
 the name of the solid. See " Key to 
 Ster.," page 60. . 
 
 Its opposite and parallel bases and 
 middle section, eoual right-angled 
 isosceles triangles. Its sides or la- 
 teral faces, rectangles- For areas, see 
 " Key to Ster.," pages 19, 22 and 60. 
 Sides suggestive of those of objects al- 
 luded to. 
 
 Its opposite and parallel bases and 
 middle section, regular and equal 
 pentagons ; sides or lateral faces, rec- 
 tangles. 
 
 Areas suggestive of those of objects 
 mentioned in adjoining column. " Key," 
 pages 35 and 19. 
 
 Its parallel and opposite bases and 
 section, regular and equal octagons 
 its sides or lateral faces, rectangles. 
 " Key," pages 35, 19. 
 
 Its parallel bases and section, sym- 
 nretrical and equal hexagons ; its 
 sides, parallelograms. " Key to Ster.," 
 pages 26, 19 and 63. Compute half of 
 sym. hex. as a trapezium. 
 
 Two of its three pairs of opposite and 
 parallel faces or bases and sections, 
 equal rectangles ; the other bases and 
 section, equal parallelograms." Key 
 to Ster.," page 63. 
 
 Each of its three pairs of parallel faces 
 or bases and sections, equal paral- 
 lelograms. 
 
 Its opposite and parallel bases and 
 section : on end, equal rectangular 
 trapeziums ; its lateral faces, rectan- 
 gles I on either of its parallel sides or 
 faces: its bases, rectangles ; its later- 
 al faces, rectangles and trapeziums 
 
partially flat roof of 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. 
 
 9.— A right trapezifoim prism. 
 
 On end, the splayed opening of a door 
 or window or loop-hole in a wall ; with 
 broader base, a partially flat roof; on 
 its lesser j)arallel base, a bin or troujrh 
 or other vessel or vehicle of cujjacily, 
 section of a ditch excavation or of a 
 railroad embankment on level ground, 
 a scow or pontoon. 
 
 10.— A right or oblique polygonal 
 compound prism, decompos- 
 able into right or oblique trian- 
 gular prisms or frusta of prisms. 
 An excavation or filling, etc. 
 A spoil bank or a borrowing pit. 
 
 See " Key to Ster.," pages 60 and 29. 
 May be treated indifferently as a prism 
 or prismoid. 
 
 On end, its bases and section, tiape- 
 ziums, and sides, rectangles ; on 
 
 eiilier of its parallel faces, its bases 
 and section, rectangles ; its sides, 
 rectangles and trapeziums, 
 
 N.B. — Its solid contents, like those of 
 Nos. 2 and 8, may be computed either as 
 prisms or prismoids. 
 
 Rule for solid content: multiply one- 
 third the sum of the three vertical edges 
 or depths of each of the component tri- 
 angular prisms, or frusta of triangular 
 prisms by the area of a section perpen- 
 dicular to sides or horizontal, and add 
 the results. 
 
 Page 67, rule II., " Key to Ster." 
 
 CLASS II. 
 
 Prisms, Frusta and Ungulae of Prisms. 
 
 11.— A right regular triangular 
 prism. 
 
 On end, a triangular building, pier or 
 lock; on one of its sides, the gable of 
 a wall, the roof of a gabled house, etc. 
 "Key to Stereometricon," page 61. 
 
 12.— Lateral wedge or ungula of 
 a right hexagonal prism, by a 
 plane through edge of base. 
 
 Portion of a mitred bead or hand-rail, 
 end of stair baluster under band-rail, 
 ridge roof of an octagonal tower against 
 a wall ; base of a chimney stack on a 
 sloped roof or gable. 
 
 13.— Lateral ungula of a right 
 hexagonal priem, by a plane 
 through opposite angles of the 
 ■olid. 
 
 Base of a chimney stack, vase or or- 
 nament on a fcloped roof or gable, etc. 
 
 N.B. — This solid and the last, are not 
 prismoids according to the definition 
 thereof, page 103, par. (206), " Key to 
 Ster. ; " but the upper half, folded over 
 
 Its parallel bases and section, equal 
 equilateral triangles ; its faces, rec- 
 tangles. Compute as prismoid with 
 rectangular bases, the upper base then 
 being au arris or line. 
 
 One of its jiarallel bases a regular 
 hexagon ; its middle base a half hex- 
 agon or trapezium ; itd upper base a 
 line ; its lateral faces a line, a rect- 
 angle, triangles and trapeziums ; 
 its sloped face a symmetrical hexa- 
 gon or 2 trapeziums, base to base 
 
 One of its opposite and parallel bases, 
 a regular hexagon ; the other, a 
 point; its middle section a half hex- 
 agon or two rectangular trape- 
 ziums base to base ; its lateral faces, 
 trapeziums and triangles ; its plane 
 of section, a symmetrical hexagon, 
 which, for area, regard as two equal tra- 
 peziums liase to base, compute and add. 
 See " Key to Ster.," page 29. 
 
6 
 
 and applied to the louver half, evidently 
 completes the prism, and hence tiie soli- 
 dity is exactly obtained by the prismoi- 
 Ual formula, as it is of a like frustutu of 
 a cylinder or of an iingula thereof by a 
 plane through edge of base. 
 
 Or the symmetrical hexagon may be 
 decomposed into a rectangle and two 
 equal triangles, for computation of area 
 
 14. — Central w^edge or ungula of One of its parallel bases, a hexa- 
 
 a right hexatjonal prism; a 
 prismoid. 
 
 A wedge, the ridge roof of a tower, 
 the base of a chimney slacic, vase or 
 ornament between two gables. 
 
 15. — An oblique trapeziform 
 prism. 
 
 The partially flat roof to a dormer 
 window, the roof of a building abutting 
 against anotlier roof, the splayed oi)en- 
 JDg of a basement window, a mitred 
 portion of a batten or moulding, section 
 of a ditch excavation, or of an embank- 
 ment on a slope. 
 
 gon ; the other, a line ; its middle sec- 
 tion, a symmetrical hexagon or two 
 trapeziums, base to bise ; its later- 
 al faces, triangles and trapeziums. 
 
 See " Key to yter./' page '29. 
 
 Treated as a prismoid: its opposite 
 and parallel bases, unequal rectan- 
 gles; its lateral faces, trapeziums. 
 
 The factors of its middle section arith- 
 metical means between tiiose of its 
 opposite and parallel bases. 
 
 16. — An oblique triangular prism. Treated as a prismoid : one of its op- 
 
 The roof of a dormer window or of a posite and parallel bases, a rectangle ; 
 
 wing to a house with a sloped roof, a the other, a line ; its lateral faces, 
 
 mitred moulding or hllet, etc. equal triangles and parallelograms. 
 
 17. — Frustum of a right triangu- As a prismoid : one of its parallel 
 
 lar prism. bases, a rectangle; its opposite base, a 
 
 Ridge roof of a building against a line ; its middle section, a rectangle. 
 
 wall, a mitred moulding, etc. 
 
 18.— Irregular frustum of an ob- Considered as a prismoid : one base, a 
 lique triangular prism. trapezium, the other, a line ; its mid- 
 Ridge roof of a building of irregular die section, a trapezium; its ends, 
 
 plan abutting on the unequally sloped non-parallel triangles ; its sides, tra- 
 
 roof of another building, etc. peziums. 
 
 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 circle or 
 ellipse, see " Key to Ster.," jmges 38, 44, 
 51, 53, 57, tables 11., III., IV., Vlil. 
 
 Parallel bases and section mixti- 
 linear figures, decomposable into a 
 rectangle and the segment or half of a 
 circle or ellispis ; the lateral face, a 
 continuous rectangle. 
 
 Note. — The segment of a circle or 
 ellipse may be equal to, less or greater 
 than a semi-circle. 
 
 20.— Regular frustum of an ob- 
 lique triangular prism. 
 A ridge roof, mitred fillet, etc. 
 
 As a prismoid : one base, a rectan- 
 gle ; the other, a line ; the middle sec- 
 tion, a rectangle. 
 
CLASS Til. 
 
 Frusta of Prisms, Prismoids, Wedges. 
 
 21.— T h e dodecahedron, o r 
 twelve-sided solid, one of the 
 five Platonic bodies. 
 
 Assemblage of twelve equal pyramids 
 with pentagonal bases, their apices or 
 summits meeting in the centre of the 
 Bolid or of the circumscribed sphere. 
 
 The capital or intermediate section of 
 a pentagonal shaft or column, a finial or 
 other ornament. 
 
 22.— A rectangular wedge, the 
 bead or heel broader than the 
 blade or edge. 
 
 The frustum of a triangular prism, 
 or may be treated as a prismoid, using 
 either of its three jinirs of parallel bases. 
 
 An inclined plane, a low pent roof, an 
 ordinary wedge, etc. 
 
 The six pairs of parallel bases or 
 twelve component faces of the solid, 
 equal and regular pentagons ; the 
 middle section a regular decagon, the 
 side of which is equal to half the diago- 
 nal of the pentagon, for area of which 
 see " Key to Ster.," page 36, rule II ; or 
 compute one of the component pyramids 
 and multii)ly by twelve. For developed 
 surface, see " Key to Ster," page 132. 
 
 On end : its opposite and parallel 
 bases, a rectangle and a line ; its mid- 
 dle base or section, a rectangle. On 
 one of eillier of its other two pairs of pa- 
 rallel bases : one base, a trapezium, the 
 other, a line ; the middle section a 
 trapezium ; side faces, a rectangle 
 and trian ;les. 
 
 23. — A rectangular wedge or in- 
 clined plane, the head or heel 
 of equal breadth with the 
 edge or blade. 
 
 A, right triangular prism. Body of a 
 dormer window or base of a chimney 
 stack on a low or steep roof, etc. 
 
 Each of its throe pairs of parallel 
 bases, a rectangle and a line ; its 
 middle sections, rectangles, respective- 
 ly equal to half the corresponding base. 
 May also be treated as a triangular 
 prism, with bases and section equal 
 triangles. 
 
 24.— An isosceles wedge, the edge As a prismoid : one of its pairs of pa- 
 
 or blade broader than the heel. 
 
 May also be considered, the frustum of 
 a triangular prism or a prismoid with 
 three pairs of parallel bases. 
 
 rallel bases, a rectangle and a line; 
 
 middle section, a rectangle ; each other 
 I)air of parallel bases, a trapezium 
 and a line ; middle section, a tra 
 
 pezium. 
 
 25. — Frustum of a right rectangu- 
 lar trapeziform prism, or a 
 prismoid. 
 
 A roof, partially flat, abutting against 
 vertical wall at one end and in rear, 
 against a sloped roof at the other, etc. 
 
 As a prismoid : its opposite and paral- 
 lel bases, rectangles ; the longer side of 
 the one corresponding to the shorter side 
 of the other; its middle section, a rec- 
 tangle ; all 
 ziums. 
 
 its lateral faces, trape- 
 
 26. — Irregular frustum of an ob- As a prismoid : its opposite and paral- 
 
 lique trapeziform prism. lei bases and middle section, trape- 
 
 A roof between two others not paral- ziums ; its lateral faces, trapeziums. 
 
 lei, irregular section of a ditch or em- Factors of middle section arithmetic 
 
 bankment. means between those of the bases. 
 
8 
 
 27.— Frustum of a right isosceles 
 trapeziform priam, a prismoid. 
 
 On its larger base, a roof, section of an 
 embankment, etc. ; on its leaser base, a 
 bin or vessel of capacity ; the capital of 
 a pilaster, a corbel ; on end, a splayed 
 opening in a wall. 
 
 28.— Frustum of an isosceles tri- 
 angular prism, a prismoid. 
 
 'Ilidge roof with ends unequally sloped, 
 mitred moulding, etc. 
 
 29.— Frustum cf a trapeziform 
 prism, a prismoid. 
 
 A flat roof, etc. ; on its lesser parallel 
 base, a bin or reservoir, a vehicle of ca- 
 pacity, a scow, a pontoon ; on end or its 
 parallel faces vertical, the splayed open- 
 ing of a window. 
 
 30. — A prismoid on a mixtilinear 
 base. 
 
 The roof of a building, circular at one 
 end or coved ceiling of a room ; on its 
 lesser base, a bathing tub, etc. ; vertical- 
 ly, the spl' ed opening of a circular 
 headed wir «v in a wall. 
 
 As a prismoid : its opposite and pa- 
 rallel bases and middle section, rect- 
 angles ; lateral faces, trapeziums. 
 
 In all such solids, the halfway factors 
 need never be measured, as they are al- 
 ways means between the parallel bases 
 of the trapezium faces. 
 
 As a prismoid : one of its opposite and 
 jiarallel bases, a rectangle ; the other, 
 a Una ; its middle section, a rectan- 
 gle. " Key to Ster.," page 19. 
 
 As a prismoid : its opposite paral- 
 lel bases and middle section, rectan- 
 gles ; its lateral faces, trapeziums. 
 Factors of intermediate section or mid- 
 dle base, arithmetic means between 
 those of the end bases. 
 " Key to Ster.," page 29. 
 
 Its opposite and parallel bases and 
 middle section, mixtilinear figures ; 
 the one a rectangle and a semi-cir- 
 cle ; the other two, rectangles and 
 semi-ellipses; its arched end deve- 
 loped, a sort of trapezium with curved 
 bases ; its area equal to half sum of 
 bases by mean breadth or heigh' 
 
 CLASS IV. 
 
 Prismolds, etc. 
 
 31— The icosahedron, or twenty- 
 sided solid ; one of the five pla- 
 tonio bodies. 
 
 An assemblage of twenty equal pyra- 
 mids on triangular bases, their apices 
 or summits meeting in a common point, 
 the centre of the solid or of the circum- 
 scribed or inscribed sphere. 
 
 A finial or other ornament, etc. More 
 expeditious to treat it for solidity by 
 computing one of its component pyra- 
 mids, and multiplying the result by 
 twenty. 
 
 32.— A prismoid, both its bases, 
 lines. Irregular triangular pyra- 
 mid. 
 
 Dormer or gablct abutting on a slojjcd 
 roof. C/omponent section of No. 79. Sec 
 " Key to St^r.," page 165, par. (212). 
 
 The ten pairs of parallel bases or 
 twenty component faces of the solid are 
 equal equilateral triangles. Its 
 middle section, a regular dodecagon. 
 Its middle section i)arallel to two oppo- 
 site apices or to the bases of any two 
 opposite pentagonal pyramids of the 
 solid, a regular decagon, whose side 
 is equal to half that of one of the edges 
 of the solid. For developed surface, see 
 " Key to Ster.," page 133. 
 
 Its opposite base?- considering the 
 solid as a prismoid resting on one of its 
 parallel edges— lines ; its middle section 
 a rectangle. See " Key to Ster.," page 
 lUI, par, (208). 
 
33. — A priimold on a trapeziform 
 base. 
 
 A cutting or embankment, etc. 
 
 One of its parallel bases, a trape- 
 zium ; the other, a line ; its middlfr 
 section, a trapezium. 
 
 34. — A railroad prlamold on a side 
 ■lope. 
 
 Section of a railroad cutting or em- 
 bankment on ground, sloping laterally 
 or in one direction only. 
 
 3S.— A railroad priamoid on a 
 grade and side slope, or on 
 ground sloping both laterally 
 and longitudinally. 
 Its narrow base upwards, an embank- 
 ment ; the same downwards, a cutting 
 Or excavation. 
 
 Its end sections or bases and middle 
 parallel section equ<t 1 quadrilaterals 
 For area of which see " Key to Ster.," 
 page 30. 
 
 This prismoid is a prism on an irregu- 
 lar base, and may be so treated. 
 
 Its opposite and parallel end bases and 
 middle section, quadrilaterals, the fac- 
 tors of the middle section being all 
 arithmetic means between those of the 
 corresponding end areas. 
 
 36. — A square or rectangular 
 prismoldal stick of timber. 
 A squared log, a tapering post, the 
 shaft of a chimney or high tower, a re- 
 ducer between rectangular 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. 
 
 Its end bases and middle section, 
 squares or rectangles. 
 
 Timber is usually measured by multi- 
 plying 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 prismoldal stick of waney 
 timber. 
 A log of waney timber ; on end, the 
 shaft of a chimney, a higli tower, a ta- 
 pering post. 
 
 Its opposite bases and middle section, 
 symmetrical octagons, for area of 
 which see " Key to Ster.," page 176, 
 par. (272), or squares or rectangles with 
 chamfered corners or angles. 
 
 38.— A concavo-convex prismoid Its opposite bases, a rectangle and 
 or curved wedge. a line ; its middle section, a rectan- 
 
 A corbel, spandrel, finial, etc. ; a brake, gle ; its developed faces, trapeaioms 
 
 a cam, etc. 
 
 " Key to Ster.," par. (141). sides, miztiiinear triangles. 
 
 39.— A reoto-oonoave prismoid,or 
 frustum of a curved wedge. 
 A corbel, spandrel, buttress, etc. May 
 
 be decomposed, as also No. 38, i^^to two 
 
 sections for more exact computation of 
 
 solid contents. 
 
 40.— Frustum of a rectangular tra- 
 peziform prism, a prismoid. 
 A fiat roof in a rectangular corner ; on 
 
 its lesser base, an angular corbel, a sink, 
 
 oiitem, bio, etc. 
 
 Its opposite and parallel bases and 
 middle section, rectangles ; its deve- 
 loped faces, trapeziums; its lateral 
 faces, mixtilinear trapeziums. 
 For areas see " Key to Ster.," page 57. 
 
 As a prismoid, its opposite and paral- 
 lel bases and middle section, rectan- 
 gles ; its lateral faces, trapeziums. 
 
 « Key to Ster.," page 104, par. (141). 
 
.f r\ 
 
 
 10 
 
 CLASS V. 
 
 »f>>;Vi«.: 
 
 Prismoids, etc. 
 
 41 —The octahedron or eight- 
 sided figure ; one of the five 
 Platonic bodies. 
 
 Assemblage of eight equal pyramids 
 on triangular bases, their apices meet- 
 ing in a common point, the centre of the 
 Solid ; or two quadrangular pyramids, 
 base to base. 
 
 Its four pairs of parallel bases or eight 
 component faces, equilateial tri- 
 angles ; its middle section, a regular 
 hexagon ; its middle section through 
 opposite apices and perpendicular to in- 
 tervening arris or edge, a lozenge ; 
 through four apices, 
 developed surface see 
 page 132. 
 
 a square. For 
 " Key to Ster.,'> 
 
 42. — A prismoid, one of its bases Its opposite and parallel bases, a 
 
 a square, the other an octagon. 
 
 Base or capital of a column, roo: of a 
 square tower, a tower, pier, vessel of ca- 
 pacity, component section of a steeple, 
 etc. 
 
 43— A prismoid, its opposite 
 bases, a square and a circle. 
 
 Base or capital of a column, roof of a 
 square tower, a tower, pier, vessel of ca- 
 pacity, a lighthouse, a section of a 
 steeple or belfry, a reducer between a 
 square and circular conduit. 
 
 square and an octagon ; the middle 
 section, a symmetrical octagon ; its 
 lateral faces, triangles and trape- 
 ziums. For area of symmetrical oct- 
 agon, see " Key to Ster.," par. (272). 
 
 One of its opposite and parallel basses 
 a square ; the other, a circle ; the 
 middle section, a mixtilinear figure 
 or a square w^ith rounded corners 
 
 Its lateral surface capable of devel- 
 opment into a plane trapexiform fi 
 gure, one base circular, the other poly 
 gonal. 
 
 44. — A prismoid, its bases unequal Its opposite bases unequal squares 
 
 squares set diagonally. 
 
 Representative of the same objects as 
 solids, Nos. 42 and 43. 
 
 45. — A prismoid, its bases a hex- 
 agon and a rectangle. 
 
 Representative of nearly the same ob- 
 jects as the three last solids. 
 
 set diagonally to each other ; the mid- 
 dle section, a symmetrical octagon ; 
 its lateral faces, triangles. 
 
 One of its bases, a hexagon ; the 
 other a rectangle ; its middle section 
 a symmetrical octagon ; its lateral 
 faces, rectangles and triangles. 
 
 46.— The lateral frustum of a rect- 
 angular prolate spindle. 
 Roof of a square tower, component 
 
 part of a steeple, etc. 
 
 47. — A prismoid, its bases, an el- 
 lipsis and a square. 
 
 A reducer between an elliptic and 
 square conduit, a roof, etc. 
 
 Its parallel bases and section, squares ; 
 its lateral surface, mixtilinear fig- 
 ures capable of development into plane 
 surfaces. For area of these see " Key 
 to Ster.," page 57. 
 
 Its middle section, a mixtilinear 
 figure or approximate oval. Its 
 lateral surface developed, a curved 
 trapezium, one base curved, the other 
 polygonal. See " Key to Ster.," p. 166. 
 
11 
 
 48.— A prismoid, ita bases a sym- 
 metrical hexagon and a line. 
 
 Kidge roof, copiag or finial to a post, 
 danel ornament, etc. 
 
 Its middle base, a symmetrical oo- 
 tagon ; its lateral surface, triangles. 
 For symmetrical hexagon, area equal to 
 double that of half the figure, which is a 
 trapezium. 
 
 49.— A prismoid, its bases, a sym- Its middle section or base, a symme- 
 metrioal hexagon and a lozenge, metrical decagon ; its lateral faces, 
 Flat roof, ornament, etc. ; on its less- triangles. Area of hexagon, double 
 er base, a fancy basket, a disk, etc. that of component trapezium. 
 
 50.— A groined ceiling or the half Its base and middle section, squares ; 
 
 of a rectangular oblate spindle. 
 
 A roof, panel ornament, etc. For 
 more exact computation of contents, de- 
 compose into two parts. 
 
 its opposite base, a point ; its lateral 
 faces, mlztilinear figures. 
 
 For areas of mixtilinear figures s«« 
 " Key to Ster.," page 57. 
 
 CLASS VI. 
 
 Pyramids and Frusta of Pyramids. 
 
 51.— The tetrahedron, or four- 
 sided figure ; one of the five pla- 
 tonic bodies. A regular trian- 
 gular pyramid. 
 
 Apex roof of a triangular building, fi- 
 nial or other ornament, the component 
 element of the icosahedron and octahe- 
 dron. 
 
 Its base and middle section, equila- 
 teral triangles, the lesser equal in area 
 to one-quarter the greater, its upper or 
 opposite base, a point ; its faces, tri- 
 angles. For development of surface 
 see " Key to Ster.," page 131. For area 
 of bases and faces, see page 36, rule It. 
 
 52.— A regular square or rectan- One of its parallel bases, a square ; 
 
 gular pyramid. the other, a point ; its middle section, 
 
 The spire of a steeple, a pinnacle, roof a square, of which the area is one quar- 
 
 of square tower, a bin, a vessel of capa- ter that of the base. Lateral faces, 
 
 city, a finial or other ornament, etc. isosceles triangles. 
 
 S3.— A pyramid, two of its faces 
 perpendicular to bas6. The un- 
 gula 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, trian- 
 gles ; apex, a point. Factors of middle 
 section half those of the base. 
 Affords a demonstration of the theorem 
 that in right-angled spherical triangles 
 the sines of the sides are as the sines of 
 the angles. 
 
 54.— Frustum of a right triangular 
 pyramid. 
 Roof, base or capital of a post or co- 
 lumn, 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, tra- 
 pexiums. Factors of section arithmetic 
 means between those of bases. 
 
 . , , .. -.-„ -A-- ■ 
 
 fp " ■ 
 
12 
 
 55.— Fnutnm of an obliqne trian- 
 gular pyramid. 
 
 That roof of triangular building abut- 
 ting against a sloped or battered wall ; 
 portion of a ditch excavation, compo- 
 nent portion of other solids. 
 
 56.— FrnBtam of a right rectangu- 
 lar pjrramld. 
 
 Flat roof to tow«r ; reducer between 
 conduits of varied size, component por- 
 tion <^ an obelisk, oapitid or base of a 
 |)Mt or column, a bin, vat or other vessel 
 of capacity, the body of a lantern, etc., 
 «te. 
 
 Ita bases and middle paralM aeetioa, 
 ■imilar triangles ; lateral faces, tra- 
 peziums ; factors of section, arithme- 
 tic means between those of the baseii 
 For areas see " Key to Ster.," pages 19, 
 22 and 29. 
 
 Its opposite bases and middle section, 
 squares or rectangles whose factors^ 
 or sides ar^ each equal to half the sum 
 of the corresponding sides of the bases, 
 or arithmetic means between them. For 
 areas se* " Key to Ster.," pages 19 and 
 29. 
 
 57. — A regular octangular or octa- 
 gonal pyramid. 
 Roof of a tower, spire of a steeple, ihil- 
 
 al or other ornament, a funnel, strainer 
 
 or filler, etc. 
 
 Its base and middle section, similar 
 octagons ; lesser area one-quarter of 
 the greater ; its upper base or opposite 
 one, an apex or a point ; lateral faces, 
 isosceles triangles. 
 
 5d.*^The frttsttttu of a regular oc- 
 tagonal pyramid. 
 
 On its broader base, a roof, tower, pier, 
 quay, component part of a steeple, etc. ; 
 base of & column, lamp or vase, etc. ; on 
 ts lesser base, a vat, bin, vase, or other 
 tessel of capacity ; the body of a lan- 
 tern, etc., etc. 
 
 59.— Irregular and oblique pyra- 
 mid on a quadrilatetal base. 
 Apex roof of an irregularly shaped 
 
 building a gainst a battered wall or roof, 
 
 a roof saddle, etc. 
 
 60.— Frustum of a pyramid with 
 non parallel bases. 
 
 Decomposable into the frnstum of a 
 pyramid with parallel bases, and an 
 irregular pyramid, by a plane paral- 
 lel to the base and passing through the 
 nearest corner or point of the upper, or 
 non parallel base. 
 
 Its opposite and parallel bases and 
 middle section, regular octagons ; 
 factors of section means to those of the 
 bases ; its lateral faces, trapesiums. 
 For expeditious mode of arriving at area 
 of octagon, see " Key to Ster.," page 176 
 or page 26, rule II. Developed surface a 
 regular polygonal sector or trapezium. 
 
 Its base, a quadrilateral or irre- 
 gular trapezium ; its summit or 
 apex, a point. Middle section similar 
 to base and equal in area to one-quarter 
 that of base. 
 
 When decomposed for computation of 
 solid contents : bases and section of 
 frustum, similar triangles ; bases 
 and section of component pyrainid or up- 
 per portion, similar quadrilaterals. 
 This pyramid has its base in one of the 
 lateral faces of the proposed solid. 
 
 ' ' CLASS VII. i > 
 
 > ; ;. . Cylinder, Frusta and Ungulae. 
 
 61. — A right cylinder or infinitary Its pariallel bases and middle sectioo, 
 prism. equal circles ; its lateral surface de- 
 
 A tower or circular apartment ; a bin, veloped in a plane, a rectangle ; its 
 
13 
 
 ▼tt, tub, bucket, pail, vase, drinking 
 Vessel, cauldron or other vessel of capa- 
 city ; a road or other roller ; the cylin- 
 der of a steam or other engine ; a gaso- 
 meter, the barrel of a pump, etc., etc., 
 «tc. 
 
 height, that of the cylinder ; its length , 
 the circumference of the solid. 
 
 For areas of circles calculated to 
 eighths, tenths and twelfths of unity, see 
 tables II., in., IV. at end of " Key to 
 Ster." 
 
 €2.— Frustum of lateral ungula or 
 wedge of a right cylinder. 
 May represent a cylindrical window 
 or opening in a sloped roof abutting to 
 A vertical wall or surface, the liquid in a 
 closed cylindrical vessel held obliquely, 
 base to chimney or vase partly on a 
 horizontal, partly on a gabled wall. 
 
 €3.— A rectangular circular ring ; 
 
 The difference between two concen- 
 tric cylinders, or a solid annulus. 
 
 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 base, a circle ; its opposite base, a 
 semi- circle or other segment ; its 
 middle section, a segment greater 
 than a semi-circle ; its plane of. 
 section the segment of an ellipsis ; its 
 cylindrical surtiace decomposable by lines 
 parallel to bases into trapeziums. 
 For areas of segments, see t-tble VIII., 
 " Key to Ster.," pages 63, 38, 44. 
 
 Its bases and parallel section, con- 
 centric annul! ; its interior and exte- 
 rior surfaces, continuous rectangles- 
 The area of annulus equal to the dif- 
 ference of the inner and outer circles, of 
 to the breadth of annulus into half the 
 sum of its circumferences. See " Key to 
 Ster.," page 39. 
 
 €4.— Central ungula or 'wedge of Its base, a circle ; its opposite base. 
 
 a right cylinder. 
 
 Ridge roof of a tower, a wedge, loop 
 hole in a wall, component portion of 
 .compound solid, a finial or other orna- 
 ment, a strainer, etc. 
 
 €5.— Frustum of central dredge or 
 ungula of cylinder No. 64. 
 
 Flat roof of tower or other building, 
 base or capital of rectangular pillar, 
 vessel of capacity, component portion 
 of compound solid, base of chimney, 
 stack or vase between two gables. 
 
 66— Lateral ungula of right cy- 
 linder or recto-cylindrical 
 wedge. 
 
 Lunette or arched headway of a 
 door or window, etc., in a sloped roof, 
 component of a compound solid, the 
 liquid in an inclined cylindrical vessel, 
 base of a salient chimney shaft over a 
 roof, etc., etc. 
 
 a line ; its middle section, the zone or 
 a circle ; its sloped faces, each a st-mi- 
 ellipsls. Its cylindrical surface decom- 
 posable into trapeziums by arcs par- 
 rallel to base. See tables II., III., IV. 
 
 IX., of " Key to Ster.," also pages 38, 
 46, 53. 
 
 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 seg- 
 ment of equal ellipses. Its cylin- 
 drical surface decomposable into trape>- 
 ziums parallel to bases. See " Key to 
 Ster.," page 51. 
 
 Its base, a semi-circle ; its inter 
 mediate base or middle section paral- 
 lel to base, also a segment ; its op- 
 posite base, a point ; its plane of 
 section or sloped face, a semi-ellipsis. 
 Its curved surface developed an ap- 
 proximate parabola, trapeziums, 
 etc. See " Key to Ster.," pages 38, 44» 
 61, toblesll.. III., IV., VIU. 
 
H 
 
 67.— Fnutum of lateral wedge 
 or nngala of a r'ght cylinder. 
 
 Lunette to arched opening in a sloped 
 roof or ceiling, abutting on a Tertical 
 wall or surface ; liquid in an inclined 
 closed cjlindrical vessel ; base of en- 
 gaged column against a battered wall, 
 etc. 
 
 68. — Irregular ungula or wedge of 
 right cylinder. 
 
 Lunette to a jartially circular open- 
 ing in an inclined ceiling, etc. Compo- 
 nent portion of a compound solid. For 
 areas, see " Ki'y to Ster.," pages 44, 46, 
 53, articles (61) and (62), tables VIII. 
 and IX. 
 
 69.— Concavo-convex prismoid 
 or cylindro-cylindrical solid or 
 concave frustum of a wedge or 
 ungula of right cylinder. 
 
 Deposit of sediment in a cylindrical 
 sewer, section of additional excavation 
 or filling, or difference between two lu- 
 nettes. 
 
 70.— Frustum of an oblique cylin- 
 der. 
 
 May be decomposed into an oblique 
 cylinder and the ungula of one by 
 
 a ])lane parallel to base, and passing 
 through nearest point of other base. 
 
 Its parallel bases and middle section, 
 segments of a circle, less than 
 more than, and equal to half ; 
 sloped face, the ezcentrio soneof an 
 ellipsis ; cylindrical surface, trape- 
 zium parallel to base. For areas ot 
 segment, see "Key to Ster.," page 44, 
 
 rule 1., rule II., ta>>le VIII. ; for zone of 
 ellipsis, see page 53, art. (62;. 
 
 Its base, the segment of a circle 
 greater than half; its opposite base, 
 a line ; its middle section, an eccen- 
 tric zone of a circle ; on'j of its side 
 faces, the segmsnt of an ellipsis ; the 
 other plane face, an eccentric zone 
 of an ellipsis. 
 
 One of its bases, the lune of a circle 
 greater than a semi-circle ; the other 
 tlie lune of a circle less than a 
 demi-circle ; the middle section, a 
 lune equal or thereabouts to a semi- 
 circle. Its side surfaces, convex and 
 concave approximate trapeziums. 
 For areas of luuos, see " Key to Ster.," 
 page 47. 
 
 When decomposed, its bases and sec- 
 tion ellipses; the base of ungula, an 
 ellipsis equal to each of tliose of the 
 inclined cylinder ; its middle section 
 half an ellipsis. For ungulie', see 
 Nos. 72, 73, 75. 
 
 CLASS VIII. 
 
 Oblique Cylinder, Frusta, UngulaeCylindroids, etc. 
 
 71.— Oblique cylinder or infini- 
 tary prism. 
 
 Mitred section of conduit, hand rail, 
 moulding ; inclined column, post, 
 strut or brace, etc. ; inclined cylindri- 
 cal opening in a wall, etc. 
 
 7 2.— Obtuse frustum or ungula of 
 oblique cylinder. 
 
 Obliqtie lunette inclined u])wards or 
 arched headway to a circular or elliptical 
 opening in a sloped roof or ceiling. 
 Component mitred portion of hand-rail, 
 bead molding, etc. 
 
 Its parallel bases and section, equal 
 ellipses ; its lateral surface capable of 
 development into a plane mixtilineal 
 figure. See " Key to Ster.," fig. n 
 [age 57. For area of ellipsis, see page 
 51 of same. 
 
 One of its opposite bases, an ellipsis 
 of slight eccentricity ; its apposite 
 base, a point; its middle section, a 
 semi-ellipsls equal to half of base ; ita 
 plane of section or lateral face, an el- 
 lip3is of greater eccentricity ; it^ 
 lateral cylindrical face developed, a fig- 
 ure like m page 57 of " Key." 
 
15 
 
 73.— Acute frastum or angola of 8&me as No. 72. For dereloped cy- 
 obliqae cylinder. lindrical surface, see fig. h, page 57 ){ 
 
 RepresentatiTe of same as No. 72, but " Key to Stereometriuon." 
 
 inclined downwards. For area of ellipsis, "Key to SUr," 
 
 pages 51 and 53. 
 
 74.— Concave ungula or frustum 
 of oblique cylinder. 
 
 Representative of same as No. 73, but 
 in arched roof or ceiling instead of 
 sloped roof. 
 
 75.— Frustum, ungula or wedge of 
 right cylinder. 
 
 Base of chimney shaft on sloped roof, 
 or same as No. 72 not inclined. 
 
 Same as No. 73, with curved instead 
 of plane section. Its cylindrical surface 
 developed similar to fig. h, page 57 of 
 " Key ; " its curved or concave section 
 developed an oval or fig. like a, page 
 57, " Key to Ster." 
 
 Slime as No. 72. For developed cy- 
 lindrical surface, see tig. g ; for ellipsis, 
 fig. b , page 57, " Key to Ster." 
 
 76.— A cylindroid; its bases, a c'r- 
 cle and an ellipsis ; infinitary 
 prismoid. 
 
 Base or capital of elliptic column, re- 
 ducer or connecting link between a cir- 
 cular and an elliptic conduit ; a tub, vat 
 or other vessel of capacity ; a hat with 
 elliptic or oval head and a circular 
 crown, etc. 
 
 77.— Cylindroid or infinitary 
 prismoid ; its bases, an ellipsis 
 and a circle. 
 
 Same as No. 76, or frustum of a conic 
 metallic vessel, which has become flat- 
 tened or battered at one end. 
 
 Its middle section, an ellipsis of 
 which the conjugate or lesser diameter 
 or axis is an arithmetic mean between 
 thoje of the opposite bases. For area 
 of circle, see tables II., Ill, IV., and of 
 ellipses, page 51, "Key toSter." Lateral 
 surface developed, a plane trapezi- 
 form fig ; its greater base, convex ; less- 
 er, concave ; its area, equal periphery of 
 middle section into mean height. 
 
 Its lateral surface developes into a 
 plane trapeziform figure, with 
 greater periphery convex ; and lesser 
 concave. Area equal to periphery of 
 middle section into mean height. 
 
 78.— Cylindroid ; its bases ellip- 
 ses at right anglesto each other. 
 
 Capital or base of elliptic column, 
 connecting link between conduits ; me- 
 tallic envelope or tube flattened at ends 
 in opposite directions. 
 
 79.— Cylindroid or prismoid ; its 
 bases an ellipsis and a line. 
 
 Ridge roof to elliptical building or 
 tower; a hut, camping tent, a strainer 
 or filter ; a finial or other ornament. 
 
 Factors of middle section, arithmetic 
 means between those of the bases. Lat- 
 eral surfice developed, a plane tra- 
 peziform figure of area equal to peri- 
 phery of middle section into mean 
 height, page 51 of " Key." 
 
 Middle section, a mixtilineal figure 
 with factors, arithmetic means between 
 those of bases. For area of middle sec- 
 lion, page 57 of " Key." Lateral surface 
 developed, a plane trapeziform fig. ; 
 its base, convex ; its opposite base, 
 angular. Area equal circumference of 
 middle section into mean height. 
 
16 
 
 80.— A compound Eolid; a cylin- For cylinder, see N'o. 61, class VII. 
 
 der and a cone. 
 
 A tower or other building, a hut, tent, 
 or camp with couical roof; a buy rick, 
 canister, fiuial ; reversed : a cauldron, 
 cistern, tub, filter, etc., etc. 
 
 for cone, see No. 81, class IX. The de- 
 veloped surface of a right cone is the 
 sector of a circle. For area, see 
 " Key to Ster." page 42. 
 
 CLASS IX. 
 
 Right and inclined Cone, Frusta, Anguiae, etc. 
 
 81.— A right cone or infinitary 
 pyramid. 
 
 Roof of tower, spire, finial or other 
 ornameut, pile of shot or shells, cornet, 
 filter or strainer, funnel, etc. 
 
 82.— Frustum of a right cone, 
 considered as a prismoid. 
 
 A tower, quay, pier, base or capital of 
 a column. Hat roof to tower, component 
 portion of a spire, a salting tub, etc. ; 
 reveroed : a butter firkin, a tub or vat 
 in a brewery or distillery, etc , a drink- 
 ing goblet, bucket, pail, dish, basket, 
 lamp shade ; a vessel of capacity, the 
 plug of a stop cock, etc., etc. 
 
 83. — Inclined or oblique cone. 
 
 Loop hole in a wall, the liquid or 
 fluid substance in a conical vessel in- 
 clined to the horizon ; a finial or orna- 
 ment adapted to a raking cornice or pe- 
 diment, etc. 
 
 84.— Frustum of inclined cone. 
 
 Unequally f played circular opening in 
 a wall ; a coal scuttle ; reducer or con- 
 necting link between two conduits of 
 different diameters laid eccentrically, 
 etc. 
 
 Its base, a circle ; its opposite base, 
 a point ; its middle section, a circle 
 equal in area to one quarter that of base. 
 Its lateral surface developed, the sector 
 of a circle. For area of circle, see tables 
 II., III., IV., " Key to Ster." 
 
 Its opposite and parallel bases and 
 middle section, circles ; its lateral sur- 
 face developed, the sector of a circu- 
 lar ring, or a curved trapezium. 
 
 The diameter of middle section an arith- 
 metic mean between those of the oppo- 
 site bases. For area of bases and sec- 
 tion see " Key to Ster.," page 38 ; for 
 lateral surface, page 43. Tables of 
 areas of circles to eighths, tenths and 
 twelfths, II., III., IV. 
 
 Its base and middle section, similar 
 ellipses — the latter equal in area to 
 one quarter the former ; the upper base, 
 an apex or point ; lateral surface deve- 
 loped an irregular sector, which, for 
 computation of area, divide into trian- 
 gles. 
 
 Its opposite and parallel bases and 
 middle section, similar ellipses ; its 
 lateral surface developed portion of an 
 eccentric annulus, art. 39, page 43 
 of " Key to Ster." Diameters of middle 
 section, arithmetic means between those 
 of bases. 
 
 85. — Flat or lo^<r cone. Its base, a circle ; opposite base or 
 
 Roof to tower or circular construction i apex, a point; its middle section, a 
 cover of box, basket, cauldron, etc. ; finial circle equal in area to one quarter that 
 or other ornament ; a Chinese hat, a pile of base; its lateral face developed in a 
 of shot or sheels, a sun shade ; reversed : plane, the sector of a circle, 
 s spinning top, bottom of a cauldron or For area of circle, see tables II., III., 
 reservoir, a funnel, strainer or filter, etc. IV., of " Key to Ster. ; " for sector, see 
 
 page 42 of same. 
 
IT 
 
 86 — FiuBtum of a low or surbased 
 cone. 
 
 Flat roof to a pavillion, tower, etc. ; R 
 hat, the cover of a vessel of capacity ; an 
 unfinished or truncated pile of shot or 
 shells ; a lamp shade ; a finial or other 
 ornameDt ; the bottom, base, top or other 
 component section of a compound solid, 
 as of No. 100 ; reversed : a dish, pan, 
 saucer, cauldron, cistern, etc. 
 
 87.— Parabolic conic ungula by a 
 plane parallel to side of cone- 
 Lunette to a circular headed opening 
 in a wall and sloped ceiling ; liquid in a 
 closed conic vessel inclined to the hor- 
 izon. 
 
 N.B.— For ratio of cord of middle sec- 
 tion, or segment to that of base, see" Key 
 to Sler.," page 143, where it is shown 
 that the squares of the cords are propor- 
 tional to the abscissae. 
 
 Its opposite bases and paralled middle 
 section or intermediate base, circles ; 
 diameter of middle section, an arithme- 
 tic mean between those of the opposite 
 bases; the lateral area developed in a 
 plane, the sector of a circular an- 
 nulus. 
 
 For areas of circles, see tables If., III., 
 IV". of" Key to Ster.," sector, page 43 of 
 same. 
 
 The base, the segment of a circle ; 
 the opposite base, a point; the middle 
 section, the segment o( a circle ; the 
 plane of section a parabola. For areas 
 of segment, see "Key to Ster.," page 44 
 and table VIII. ; for area of parabola, 
 page 54 of same. The lateral surface 
 developed an approximate sector of 
 a circle. The height or versed side of 
 middle section segment is half that of 
 
 88.— Frustum of parabolic conic 
 u gula by a plane parallel to 
 base of cone. 
 
 Splayed opening or embrasure to a 
 segment-shaped window or loop hole in 
 a wall ; lunette to opening in sloped 
 ceiling terminating in a vertical sur- 
 face ; liquid in a closed vessel in the 
 shape of the frustum of a cone, No. 82, 
 when inclined from the vertical. 
 
 For cord of middle segment, measure 
 solid or compute by page 143 of " Ster." 
 
 89.— Frustum of a right elongated 
 cone. 
 
 Shaft of Grecian column,tapered post. 
 
 The parallel bases and middle 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, art (66) ; the developed conic- 
 al surface, an approximate sector of 
 a circular annulusor, more correctly, 
 a trapezium with curved concen- 
 tric or parallel bases, for area of 
 which see note page 29, " Key to Ster.," 
 For area of segment, table VIII., and 
 page 44 of same. 
 
 Like No. 82, its opposite and paral'el 
 bases and middle section circles; dia- 
 meter of middle section equal to the half 
 
 high tower or chimney shaft, funnel, sum of these of the bases ; the developed 
 
 pipe reducer, speaking trumpet or horn, 
 pltig of a stopcock or tap, deep drinking 
 goblet, or other vessel of c>ipacity large 
 or small, shaft of a gun, component por- 
 tion of many compound solids, etc. 
 
 90. — A compound solid, composed 
 of or decomposable into the frus- 
 tum of a right cone and the 
 segment or half of a sphere or 
 spheroid. 
 
 lateral surface, the sector of a con- 
 centric annulus. 
 
 For areas of circles to eighths, tenths 
 and twelfths, see tables II., III., IV., ot 
 " Key to Ster. ;" for that of sector, page 
 43 of same. 
 
 For nature and areas of bases and mid- 
 dle section of the component frustum or 
 a cone and of its lateral surface, see Nos 
 82 and 89. 
 For arf as of bases and middle section 
 
May represent a piece of ordnance, a 
 deep conical vessel with hemi-spherical 
 hemi-spheroidal or segmental bottom or 
 top to it. 
 
 For hemi-sphere, hemi-spheroid, or sea:- 
 ments thereof, greater or less than half, 
 see classes 18, 19, '20. 
 
 For diameter of middle section in seg- 
 ment of spheroid, see " Key to Ster.," 
 pages 139 and 140, where AB : CD : : 
 
 4AoToB: o jtf and CD : AB : : 
 "4 Co . oD : o}f , or, the rectangle under 
 the required radius and 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. 
 
 18 
 
 of hemisphere or hemispheroid or of the 
 segment of either, greater or less than a 
 hemisphere, see tables II., III., IV. io 
 " Key to Ster." 
 
 For diameter of middle section in hemi- 
 sphere or in segment thereof, see "Bail- 
 large Geometry," par. 539 or Key to 
 Ster.," par. l.'J4, where oa == Vc'o . ol)-, 
 and oD ^ dinm. 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 measuring the solid. 
 
 CLASS X. 
 
 Conic Frusta and Ungulae, etc. 
 
 91.— Conic wedge or central un- 
 gula of a coue by planes drawn 
 from opposite edges of the 
 base to meet in the axis of the 
 cone. 
 
 Ridge roof to a tower, splayed open- 
 ing or embrasure to a long narrow ver- 
 tical loop hole in a wall ; component sec- 
 tion of compound solid as of a cone and 
 cylinder or of conses having their bases 
 or apices in opposite directions. 
 
 92.— Frustum of a conio wedge r 
 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 trun- 
 cated on opposite sides. 
 
 Arched and splayed embrasure in a 
 wall, component portion of a compound 
 solid. 
 
 The base, a circle ; the parallel up- 
 per 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 co- 
 nical faces developed, equal curvil- 
 inear triangle. 
 
 For anas, see pages 38, 46, 53 and 57, 
 and tables II., III., IV., of " Ster." For 
 area of zone, see table IX. of same. 
 
 The base, a circle ; the opposite and 
 parallel base, a zone of a circle ; the 
 middle section, a zone ; the lateral 
 plane faces, equal segments of 
 equal ellipses; the developed conical 
 surfaces resolvable into trapeziform 
 figures. 
 
 For area of trapezium, page 29, " Key 
 to Ster." 
 
 93.— Lateral elliptic ungula of a Its base, a circle ; its upper or oppo- 
 
 cone, by a plane 
 through edge of base. 
 
 Splayed embrasure to elliptic open- 
 ing in wall and through sloped roof or 
 ceiling, etc. 
 
 passing site base, a point ; its middle section 
 parallel to base, the segment of a cir- 
 cle ; its plane face an ellipsis ; its co- 
 nical surface developed a concavo- 
 f onvex figure like h, page 97 of " Key 
 to Ster." 
 
19 
 
 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 hy[)erboiR, 
 page 5.5, or figure e, pago 57 ; for 
 ellipsis, page 51 and 53. 
 
 The base, a segment of a circle ; 
 
 the upper base, a point ; the middle 
 section, a segment of a circle ; the 
 piano lateral face, the segment o^ 
 an ellipsis ; the developed conical 
 surface as in No. 87 or 91. If tlie cut- 
 ting plane be parallel to side of cone 
 the face will be a parabola ; if at an 
 angle greater than side of cone to base, 
 hyperbola; if less, an ellipsis. 
 
 95.— Central ungula of cone or 
 conic w^edge, by plcines 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 
 ellipses if cutting planes more inclined 
 to base than side of cone ; if less, hyper- 
 bolas ; if equally, parabolas. 
 
 96.— Frustum of conic wedge, 
 No. 85, by a plane parallel to the 
 base. 
 
 An embrasure; a reducer or connect- 
 ing link between a rectangular and 
 circular conduit, etc. 
 
 97. — Concave ungula of a cone 
 or a conical recto-concave 
 wedge. 
 
 Lunette of circular headed opening 
 in wall, reaching through vaulted, 
 groined or arched ceiling ; cone scnbed 
 to cylindrical surface, or toa shaft of 
 elliptical section. 
 
 98.— Portion of frustum of right 
 cone, by a plane through both 
 bases- 
 Splayed segment headed opening in 
 wall, liquid in closed tub lying on its 
 side ; base or capital of half column 
 against sloped wall ; component section 
 of base or capital of clustered, Gothic or 
 other column. 
 
 Bases and sections same as No. 91 > 
 developed conical surface, a concavo- 
 convex triangle compatible as per 
 page 57 of" Key to Ster." 
 
 The lateral plane faces, equal seg- 
 ments of equal ellipses, equal 
 parabolas or equal hyperbolas, as 
 case may be. — See No. 94. 
 
 Its base, a circle; other base and 
 middle section, zones of circles, for 
 
 areas of which see " Key to Stereome- 
 tricon, table l.X. 
 
 The base, the segment of a circle ; 
 the other base, a point or curved arris ; 
 its intermediate base or section, or its 
 bases or sections if divided for compu- 
 tation of cubical contents, segments 
 of circles. Its sides like No. 94. 
 
 Its parallel end bases and middle sec- 
 tion, 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 in- 
 clination of cutting plane. 
 
 99.— Lateral conic ungula or 
 wedge, by a plane through edge 
 of lesser base of frustum. 
 
 Its base, a circle ; opposite base, a 
 point; intermediate section a seg- 
 ment of a circle ; its plane face an 
 
20 
 
 Embrasure. liquid in inclined conical 
 vessel, section o>' conical elbow or mitre, 
 base of chimney stack to sloped roof. 
 May be treated also as lyingon its lutcr- 
 r1 plane face. 
 
 100.— A oompound solid com- 
 posed of decomposable or resolvable 
 into two conic frusta and a low 
 or flat cone. 
 
 May represent a covered dish, a bas- 
 ket or bumper, a vase, a tiuial or other 
 ornament, an urn, a cauldron on a 
 stand, etc , etc. 
 
 ellipsis, its conical surface developed a 
 concavo-convex figure like gor h, page 
 97 of Hter. but with concave base. 
 Treat on circular banu as easier of com- 
 putation. 
 
 All its areas to be used in computa- 
 tion of solid contents or capacity are 
 circles and can be measured to eighths, 
 tenths or twelfths of an inch or other 
 unity, and the areas found by more in- 
 spection in tables l[,, IH. and IV. at end 
 ot'Baillargc's ''Key to ater." 
 
 DIVISION 2. 
 
 Solids of double cuivature, or of which the surfaces are not capable of devel- 
 opment in a plane. 
 
 CLASS XI. 
 
 Concave Cones, Frusta and Ungulae. 
 
 101.— Right concave cone or 
 spindle. 
 
 Camping tent; roof of tower, pavil- 
 lion, hut, etc. ; spire, funnel, strainer, 
 trumpet ; finial or other ornament. 
 
 May be decomposed into two or more 
 frusta by planes parallel to base, to ad- 
 mit of more accurate determination of 
 solid contents. 
 
 102.— PruBtum of a right con- 
 cave cone between parallel 
 planes. 
 
 Illustrative of most of the objects 
 mentioned In So. 82, which see. 
 
 For more accurate computation of con- 
 tents, divide into two sections or more, 
 according to greater or lesser curvature 
 of the solid, and treat each section as a 
 separate pritmoid and add the results. 
 
 103.— Inclined concave cone. 
 
 Finial, or ornament on a raking cor- 
 nice; liquid in an inclined vessel, etc, 
 as for No. 101, maybe decomposed by 
 imaginary p'anes parallel to base into 
 two or more sections or slices, so that 
 
 Its base and parallel sections, circles ; 
 its upper or opposite base, an apex or 
 point. Its lateral surface not capable of 
 development in a plane or into a sector 
 of a circle as is the case with a regular 
 right coue, but may be readily and very 
 approximately computed by division in- 
 to continuous trapeziums by linos 
 parallel to circumference of base. See 
 " Key to Ster.," page 96. 
 
 Its bases and parallel section?, c'r- 
 cles. Intermediate diameters not, as in 
 No. 82, arithmetical means between those 
 of the opposite or end bases, but must 
 be measured or computed. Lateral area 
 may be conceived as made up of a series 
 of super or juxta-posed continuous 
 trapeziums. 
 
 Its base and section, approximate 
 ellipses of slight ezcentrioity or 
 ovoid figures ; its other base, a point. 
 
 In developing the lateral surface into 
 a series of continuous trapeziums, 
 the lines are not as in the right cone 
 
n 
 
 •lant Bide of each may bo Bonsibly a 
 straight line. Soo page 103, par. 139, 
 •• Key to Ster." 
 
 parallel to base or to circumferenoaB of 
 parallel aoctions, but are drawn equidis- 
 tant from the apex, thus leaving at the 
 baae a figure like h, page 67 of " Key to 
 Ster." 
 
 104.— Frustum of oblique oon- 
 oave cone between parallel 
 planes. 
 
 Representative of same as No. 84. 
 
 105.— Flat or low concave cone. 
 
 Representative of many of the objects 
 mentioned in No. 85. 
 
 Its bases and sections parallel thereto, 
 approximate ellipses or ovoid 
 flgures. See remarks to No. 102. 
 
 Its bases, a circle and a point; sec- 
 tion, a circle ; lateral area reducible to 
 continuous trapeziums, par. 126, 
 " Key to Ster." 
 
 106.— Frustum 
 cone. 
 
 Representative of objects under head 
 of No. 86. 
 
 of flat or low Its bases and section, circles, for areas 
 of which see tables II., III. and IV. of 
 " Key to Ster.," to eighths, tenths and 
 twelfths of inch or other unity. 
 
 107/— Ungula of concave cone 
 by a plane through outer edge of 
 base. 
 
 See No. 92, as to what it represents, 
 etc. 
 
 See No. 92. Lateral surface reducible 
 to trapeziums and triangles. 
 
 Base and sections, ovoidal figures; 
 areas, page 57 of Key. 
 
 108 -TTngula of concave oone 
 by a plane cutting the base. 
 
 See No. 93 as to what it represents, 
 etc. 
 
 Bases and section, segments of cir- 
 cles ; upper base, a point. Lateral sur- 
 face as No. 107. 
 
 109.— TTngula of hollow 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 segment 
 of a circle ; lateral area, trapeziums 
 and triangles. 
 
 110.— Frustum of (No. 109) un- Its base, a circle ; other base, a seg- 
 gula by a plane parallel to base. ment of a oirole ; its middle section 
 See Nos. 98, 116, 126. parallel to bases, also a segment. For 
 
 Base or capital of a column, or base areas of segments of circles, see " Key to 
 of chimney shaft, etc., on or outside of Ster.," table VIII., or rules, page 44 of 
 sloped roof or gable. same. 
 
22 
 
 CLASS XII. 
 
 Paraboloid or Parabolic Conoid, Frusta and 
 Ungulae^ etc. 
 
 111.— Right paraboloid or para- 
 bolic conoid. 
 
 Dome, but, hive, roof, linial or other 
 ornament, shade, globe, cover, hood, 
 cowl, etc ; reversed : a iilter, cauldron, 
 or other vessel of capacity, the bowl of a 
 cup or drinking goblet, etc., etc. 
 
 I ts base and middle ^tion, circles ; 
 its opposite base or apex, a point; its 
 lateral surface resolvable into a small 
 circle at apex, and continuous tra- 
 peziums. The squares of its interme- 
 diate diameters, proportional to abscis- 
 sae. See " Key to Ster.," page 96, 
 
 112.— Frustum of right parabo- 
 loid, between parallel planes. 
 
 Represents mostly the same objects as 
 the frustum of a cone. No. 82. 
 See page 142 " Key to St^r." 
 
 End and middle bases, circles ; 
 squares of diameters proportional to 
 abscissae. For areas of circles, see " Key 
 to Ster.," tables II., III. and IV. 
 
 113.— Oblique paraboloid. 
 
 " Key to Ster.," page 142. 
 
 Liquid in a parabolic vessel inclined 
 to the horizon, metal in an inclined 
 crucible, (iuial or ornament on an inclined 
 or raking molding or pediment, etc. 
 
 Its base and middle section, similar 
 ellipses ; its opposite base or other end, 
 an apex or point. For areas of ellijiees 
 see " Key to Ster., ' page 51 ; for lateral 
 area see No. 103. 
 
 114.— Frustum of oblique para- 
 boloid between parallel planes. 
 
 Represents same as frustum of in- 
 clined cone No. 84, "Key to Ster.," 
 pq,ge 142. 
 
 Its bases and middle section, similar 
 ellipses ; fot areas of which see " Key to 
 Bter., page 51. For lateral area, see No. 
 103 or reduce to trapeziums by lines 
 from base to base. - „ 
 
 lis.— Parabolic wedge or cen- 
 tral ungula of paraboloid. 
 
 See No. 91. 
 
 Lateral or paraboloidal surface cap- 
 able of approximate development. See 
 No. 91. 
 
 116.— Portion of a paraboloidal 
 frustum, by a plane through its 
 greater base and edge of other or 
 opposite base 
 
 See No. 98 as to what it represents. 
 Also, base of chimney stack, partly on a 
 horizontal and partly on an inclined 
 base, or sloped roof, etc. 
 
 Its lesser base, a circle ; opposite 
 base, the segment of a circle ; middle 
 section, also a segment, Its lateral 
 plane face, the segment of an ellip- 
 sis. This face would be a parabola if 
 angle of face equalled that of side ; if 
 greater, a hyperbola. 
 
 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 segment 
 of a circle. Its plane face an ellip- 
 sis. 
 
23 
 
 118.— Lateral angnla of parabo- 
 loid ; elliptic, parabolic or liyper- 
 bolic, according as plane of sec- 
 tion catB the base at an angle 
 less than, equal to, or greater than 
 that of the side and base. 
 
 lis base, the segment of a circle ; 
 ita middle section, a segment ; its up- 
 j)er or opposite base a point ; its plane 
 face, thic segmeatof an ellipsis, pa- 
 rabola or hyperbola, according to 
 angle of plane of section. 
 
 119.— Obtuse eliptic ungula of 
 a paraboloid, by a plane through 
 edge of lesser base of frustum. 
 
 Base of chimney stack, etc., to sloped 
 roof; base of a vase, statue, etc., on a 
 pediment ; a lunette, scoop, etc. 
 
 Its base, a circle; middle section, a 
 segment; other ba.se, a point; its 
 plane face, an ellipsis. For areas of 
 segments of circles, table Vlll. of" Key 
 to titer." Fjr area of ellipsis, pagie 51 of 
 same. 
 
 120-— Frustum of a paraboloid 
 between non-parallel bases. 
 
 " Key to Ster.," page 145. 
 
 Lunette through a vertical wall and 
 inclined ceiling, et<5. For computation 
 of solid contents decompose into a frus- 
 tu<u trith parallel bases and an ungula 
 by a plase parallel to base, through 
 nearest point of upper base. 
 
 Its factor areas, circles and a seg- 
 ment ; its plane face, an ellipsis. Per 
 areas of segmeiUs of circles, table VIIF. 
 of " Key to St«r," Area of circle, tables 
 II., III. aud IV. of same; ellipsis, page 
 61 of ^ame ; lateral area, page 95 > 
 solidity, page 145 of same. 
 
 CLASS Xlll. 
 
 Hyperbolold or Hyperbolic Conoid, Frusta and 
 Ongulaoy etc. 
 
 121.— Right hyperboloid or hy- 
 perbolic conoid. 
 
 Page 146, " Key to Ster." Represen- 
 tative 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. 
 
 122.— Frustum of right hyper- Except for diameter of middle section, 
 
 boloid. same '^s No. 112, or the diameter may 
 
 Representative of same, nearly as No. be measured directly. 
 112 and 82. 
 
 123.— Oblique hyperboloid. Same as No. 113, except for diameter 
 
 See " Key to Ster.," page 146. Repre- of middle section for which see " Key to 
 
 sentative of same, as No. 113. Ster.," page 147, line 3, or the diametei* 
 
 may be measured. 
 
 124.— Frustum of oblique by- Same aa No, 114, except for diameter 
 
 perboloid. of middle section for which see " Key 
 
 Representative of same, nearly as Nob. to Ster.," page 147, line 3, or may be had 
 
 84 and 114. by measurement. 
 
24 
 
 125.— Hyperbolold wedge or 
 central ungula. 
 
 Similar solid to No. 95 of a cone and 
 representative of same objects. 
 
 Except for diameter of middle section, 
 same as No. 91 or 95. For area of zone, 
 see " Key to Ster.," page 46 or table IX- 
 ofsame. 
 
 126.— Ungula of hyperbolold by Its base, a circle ; middle section, the 
 a plane through edge of base. segment of a circle; other base, a 
 
 For solid content, treat as prismoid or point. Plane lateral face, an ellipsia, 
 by par. 185 of " Key to Ster." its lateral surface of double curvature, 
 
 Solid similar to No. 93 of cone, or to as all such figures are not capable of 
 No. 117 of paraboloid. development, but reducible as required. 
 
 127.— FruBtum of hyperbolold 
 wedge. 
 
 Similar to No. 116 of paraboloid. Base 
 of chimney stack, etc., resting partly on 
 a sloped roof. 
 
 Bases same as in No. 116. Lateral 
 area developes into trapezium! by 
 lines parallel to bases. For areas of cir- 
 cles, segments, zones, see tables of " Key 
 to Ster." 
 
 128. — Ungula of hyperbolold by 
 a plane through base. 
 
 Similar to No. 118 of paraboloid. 
 
 Bases and section same as No. 118 of 
 paraboloid. See table VIII. of " Key to 
 Ster.," for areas of segments. 
 
 129.— Frustum of hyperbolold 
 w^edge, or of central ungula of 
 hyperbolold. 
 
 Similar to No. 92 of cone. 
 
 130.— A compound solid. Two 
 equal frusta of cone or conoid, 
 base to base. 
 
 Illustrative of a keg or cask, barrel, 
 hogshead, etc., of any size or shape. 
 
 Treat one-half of solid ns Nos. 82, 112, 
 122, and double the result. 
 
 Same as No. 92. For areas of circles to 
 eighths, tenths, and twelfths, see tables 
 II., III. and IV. of « Key to Ster." For 
 area of zone, see table IX. of same. La- 
 teral surface decomposable into trape- 
 ziums. 
 
 See " Key to Ster.," fig. on page 155, 
 for mode of measuring half-way diame- 
 ter, when the half solid is not the frus- 
 tum 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 diametera. 
 
 CLASS XIV. 
 
 Sundry Solids. 
 
 131.— Three azed spheroid. 
 
 See " Key to Ster.," page xzziz. 
 May for measurement be supposed to lie 
 or stand on either of its sides or apices. 
 
 Representative of a pebble, a beam, 
 spindle, torpedoe, a shell fish, a flattened 
 ellipsoid, etc., etc. 
 
 All its sections, ellipses ; all its paral- 
 lel sections, similar ellipses. For areas 
 of ellipses, "Ster.," page 51. Lateral 
 area, see general formula, page 95, " Key 
 to Ster." Or, as with the spheroid, sup- 
 pose the surface divided as a melon is 
 or orange into ungulae, terminating in 
 apices or poles of the fig. 
 
25 
 
 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 con- 
 oid. 
 
 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 Ster. For lateral 
 area, see page 96 of same. 
 
 133.— Circular disc with round- 
 ed edge. 
 
 Treat as a compound solid, to wit : 
 a flat or low cylinder, and a ring 
 semi-circular or segmental in sec- 
 tion. Add the results. 
 
 134.— Twisted prism. 
 
 Portion of a circular stairs rail, a 
 twisted pillar or column, spiral orna- 
 ment, etc. 
 
 135.— A compound solid. 
 Two frusta of cones, their less- 
 er bases joined. 
 
 A windlass, spool, handle, shaft, axle- 
 tree, etc. 
 
 136.— A compound solid. 
 Two frusta of hollow cones 
 joined by their lesser bases. 
 
 A windlass, spool, handle, shaft, axle- 
 tree, etc. 
 
 137.— Compound solid. 
 Two frusta of concave cones 
 joined by their greater bases. 
 A windlass, shaft, axle-tree, etc. 
 
 138.— Compound solid. 
 
 The segment or half of an elon- 
 gated or prolate spindle, No. 151, 
 and the segment or half of an ob- 
 late spindle, No. 141, or the seg- 
 ment of a sphere or spheroid, 
 classes XVII. and XIX., a buoy, etc. 
 
 For cylinder, see No. (31. For ring 
 compute area of section thereof as semi- 
 circle or segment, and multiply into 
 circumference. For area, mean circum- 
 ference of ring into circumference of sec- 
 tion. 
 
 Its bases and sections similar and 
 equal figures. The lateral surface of 
 each face can be developed in a plane, a 
 trapezium or rectangle. 
 
 Treat half the solid as the frustum of 
 a cone, and double the result, either for 
 solid content or area of .figure. 
 
 Treat one half the solid as frustum of 
 cone No. 102, and double the result. 
 
 Lateral area resolvable into continu- 
 ous trapeziums. 
 
 Treat half the solid, and double the 
 result. For areas of circles, see tables 
 II., III. and IV. of Ster. 
 
 Sections perpendicular to axis, circles ; 
 Area resolvable into continuous tra- 
 peziums, a circle and the sector of 
 a circle. The circle at apex of seg- 
 ment 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 instead of 
 spindle. 
 
 A finial or other ornament, a cul-de- 
 lampe or pendant. 
 
 Sedions perpendicular to axis, circles. 
 Lateral surface, continuous trape- 
 ziums, a circle, and the sector of a 
 circle at apex of cone. 
 
26 
 
 140.— Compound solid: the fros- 
 tnm of a ephere or spheroid and a 
 hollow cone. 
 
 A Moorish dom«, a minaret, cbimiiey 
 of a coal oil lamp, a decanter, a rase, a 
 pitcher. 
 
 Baeea and sections, circles. Lateral 
 surface resolvable into continuous 
 trapesinms. See general formula, page 
 
 95 of " Key to Ster." 
 
 » 
 
 CLASS XV. 
 
 Oblate or Flattened Spindle, Frusta, Segments, 
 
 Sundry. 
 
 141.— Oblate spindle, as two Treat one half as segment of spher 
 
 equal segments of sphere or sphe- or spheroid, and double the result. See 
 
 roid base to base. classes 17 and 19. 
 
 A quoit, etc. , 
 
 142.— Semi-oblate spindle by a Treat its two halves together as one 
 plane parallel to fixed axis. segment of sphere or spheroid. See 
 
 floating caisson to entrance of dock,etc. classes 17 and 19. 
 
 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 el- 
 lipsis, or two segments thereof, 
 base to base. Table VIIL, " Key to 
 Ster." 
 
 144.— Lateral frustum of oblate 
 spindle, between planes parallel 
 to fixed axis. 
 
 A flat-bottomed boat or other sailing 
 vessel or a caisson, etc. 
 
 The bases and section half-way be- 
 tween them, double segments of 
 circles or ellipses, for areas of which 
 see table VIIL, "Key to Ster.," and page 
 53 of same. 
 
 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.— Lateral frustum of oblate 
 spindle truncated at both ends. 
 
 A flat-bottomed boat or pontoon, a 
 scow, lighter, etc. 
 
 Bases, double segments of circles 
 or ellipses truncated at both ends. 
 
 Divide into trapeziums and compute 
 areas by page 57 " 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 apartment. On its 
 
 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 
 
27 
 
 lesser base, the bead of a shallow niche base, it being easier to compute circles 
 
 in a wall, etc. than ellipses. 
 
 148. — A compound body, a cone, Treat separately as cone No. 81, and 
 
 and the legment of a sphere or as segment of sphere, No. 173, or of sphe- 
 
 ■pheroid. roid No. 182. 
 
 A buoy, covered filter, etc. 
 
 149.— Elliptic ricg, or may be 
 called an eccentric ring. 
 
 Treat as circular or cylindrical ring, 
 taking for bases its least, its greater, and 
 its mean sections ; and for length the 
 mean of the inner and outer circumfer- 
 ences. 
 
 Compute half of solid as the lateral 
 frustum of a half-prolate spindle 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. 
 
 ISO,— Compound solid : a cy- 
 linder and the segment of a sphere 
 or spheroid. 
 
 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 circles tables 11., 
 III. and IV. of same. Half-way diame- 
 ter in segment of circle or sphere a mean 
 proportional between abscissae of dia- 
 meter. 
 
 CLASS XVI. 
 
 Prolate or Elongated Spindle, Frusta, Segments, 
 
 etc. 
 
 151.— Prolate spindle. 
 
 A shuttle, a torpedoe, a 
 sheath, case, etc, 
 
 Its sections perpendicular to axis, cir- 
 cigar, a cles. Decompose its lateral area into 
 continuous trapeziums and a sec- 
 tor. 
 
 152.— Semi-prolate spindle by a 
 plane through its greater or fixed 
 axis. 
 
 A boat or sailing vessel, a canoe, etc. 
 
 For solidity, compute planes perpen- 
 dicular to fixed axis, as segments of 
 circles, while the sections parallel 
 thereto are not so readily computed. 
 
 153. — Semi-prolate spindle by a For greater accuracy, divide into a 
 
 plane perpendicular to fixed axis, frustum and segment, compute and add 
 
 A hut, roof, filter or vessel of capacity cubical contents. Areas of bases, tables 
 
 a minaret or finial. II., m. and IV, of " Key to Ster." 
 
 154.— Middle frustum of pro- 
 late spindle between planes per- 
 pendicular to fixed axis. 
 
 A cask or keg, puncheon, hogshead, 
 etc. ; see page 155 " Key to Ster.'' 
 
 See page 149 of " Key to Ster.," and 
 for lateral surface, page 95 of same. See 
 page 155 of same. Bases and sections, 
 circles, tables II., III. and IV. of " Key 
 
 to Ster." 
 
28 
 
 155-— Bemi-mlddle 
 prolate spindls. 
 
 The liquid ia a cask lying on its side, 
 a boat with truncated ends. Compute 
 as No. 154 and take half. 
 
 fruttum of Bases and middle section, semi-cir- 
 Clea, see page 160 of " Key to Ster. ' 
 Lateral surface decomposable into tra- 
 peziums. '^ ' 
 
 156.— Lateral frustum of pro- 
 late spindle by planes parallel to 
 fixed or longer axis. 
 
 A flat-bottomed boat or other sailing 
 vessel. 
 
 Treat as prismoid, the greater base, a 
 double segment of a circle. 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 perpen- 
 dicular to fixed or larger axis of 
 solid. 
 
 The shaft of a Roman column. Com- 
 pute each frustum from centre and add 
 the results. 
 
 158.— Middle frustum of elon- 
 gated spindle by planes perpen- 
 dicular 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 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 cir- 
 cumference. 
 
 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 circumferences. 
 
 159.- A curved half-spindle or Base and sections, circles or ellipses 
 
 cone. of slight eccentricity. Lateral area 
 
 A horn, powder flask, tusk or tooth of decomposable into continuous trape- 
 
 an elephant, etc., a supporting bracket ziums and sector at apex, 
 from face of wall. 
 
 160.— Frustum of a prolate spin- Base and sections parallel thereto, cir- 
 dle between non parallel bases. cles ; base of ungula a circle ; middle 
 
 Decompose into a frustum with base of ungula, a semi-circle ; apex of 
 parallel bases and an ungula by a ungula or opposite base, a point ; la- 
 plane through nearest point of one of teral surface, continuous trapezi- 
 the bases. ums, and a tig. like h, page 57 " Key to 
 
 Ster." 
 
 CLASS xvn. 
 Sphere, Segments, Frusta and Ungulae, etc. 
 
 161.— The sphere. 
 
 A billiard or other playing ball, the 
 ball of a vane or steeple, spherical shot 
 and shell, school spheres, lamp globe or 
 well, component part of compound solid. 
 
 The opposite bases, points ; the mid- 
 dle section, a circle. The area of sur- 
 face admits of approximate development 
 into a series of equal figures in the 8hap« 
 of the longitudinal section of a prolate 
 
29 
 
 etc. Solid eontent may be had by com- 
 puting one of the component ungulae 
 and multiplying into the number there- 
 of. 
 
 162.— A hemisphere. 
 
 A dome, arched ceiling, globe, shade, 
 eorer, but, hire, etc. ; reversed : a bowl, 
 cauldron, copper, vaae, etc. 
 
 (Joatenta more easily computable as 
 half of those of a whole sphere, where 
 there is no intermediate diameter to cal- 
 culate or measure. . , , ■, it:-,.' 
 
 163.— Segment of a sphere lees 
 than a hemisphere. 
 
 Representative of same objects as No. 
 162, cover or bottom of a boiler. Solid 
 contents also equal to one of the com- 
 ponent ungulae into the number thereof. 
 
 spindle, or of doable segments of a 
 circle, base to base. 
 
 Surface equal to four great circles or 
 to four times that of a great circle. 
 
 Its base, a circle ; opposite base, a 
 point ; its middle section, a circle, the 
 half diameter of which equals the square 
 root of the rectangle 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 tl|e sphere. 
 
 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 diame- 
 ter of the sphere of which the segment 
 forms part. For lateral area see " Key to 
 Ster.," page 110, or General Formula, 
 page 95. 
 
 164.— Segment of sphere, great- Its base and section circles ; other 
 er than a hemisphere. base a point; radius of middle section 
 
 Representative of same as No. 102, and the root of rectangle of parts into which 
 of a Moorish or Turkish or horse-shoe it divides diameter of sphere. Lateral 
 dome. . area, see *' Key to Ster.," pages 1 17 and 
 
 123. 
 
 165.— Middle frustum of a 
 sphere. 
 
 Base, capital or middle section of a 
 column or post, a puncheon, hogshead, 
 crusher, roller, lamp shade, etc., etc. 
 
 166. — Lateral frustum of sphere- 
 
 Base or capital of column, coved ceil- 
 ing, cauldron, dish, soup plate, saucer, 
 etc. Radii of bases and sections propor- 
 tional to square roots of rectangles of 
 portions into which such radii or ordin- 
 ates divide the diameter of the sphere of 
 which the solid forms a part. 
 
 167.— Spherical wedge or cen- 
 tral ungula of a sphere by planes 
 from opposite edges of base of 
 hemisphere to meet in apex. 
 
 Component portion of a compound 
 solid. 
 
 Bases, equal circles ; middle sec- 
 tions, a circle ; see tables of areas of 
 circles to eighths, tenths and twelfths of 
 an inch or other unity of measure, II., 
 III. and IV. of "Key to Ster." 
 
 Bases and section, circles; lateral 
 area resolvable into continuous tra- 
 peziums ; or lateral area may be had 
 very nearly at one operation, if the frus- 
 tum be low or flat and that its lateral 
 curvature be not considerable. 
 
 Its base, a circle ; opposite base, a 
 ridge, or axis, or line ; middle section, 
 the Bone of a circle ; its plane faces, 
 circles; and lateral area resolvabl* 
 into trapesinma and triangles. 
 
30 
 
 l66.— Frtfatnm of a Apherioal 
 Wedge or central ungula between 
 parallel planes. 
 
 Component portion of compound 
 aolid. 
 
 Base, 8 circle ; other base snd middle 
 section, zonee of circles. For arvag 
 of zones, see table IX., " Kej to Ster." 
 
 169— Spherical pyramid, ob- 
 tuse-angled and triangular. 
 
 IHiisfrbtire of the tri-obtuse-adgnlar 
 Spherical triangle, and of the fact that 
 the sum of the angles of a spherical tri- 
 angle, may reach to six right angles, 
 when each of the component angles in- 
 creases to 180°. 
 
 170.~FruBtnm of sphere be- 
 tween non-parallel bases. 
 
 RIboW or connecting link between 
 two portions of a rail or bead ; base of a 
 Tase or other ornament on a raking 
 cornice. 
 
 Base, a spherical triangle having 
 three obtuse angles ; apex or oppo- 
 site base, a point; middle section, a 
 similar tri-obtuse angular spheri- 
 cal triangle, and whose area ia equal 
 to one-qnarter that of base, its factors 
 being halves of those of base, and J z | 
 
 = i- 
 
 Decompose into frastum and ungula 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, compute whole sphere, and de- 
 duct segment. 
 
 1.' I'll', 
 ■)■■.<■'.• 
 
 CLASS XVIII. 
 
 
 Spherical Ungulae, Sectors, Pyramids and Frusta. 
 
 171.— Quarter-sphere or rectan- 
 gular ungula of a sphere. 
 
 Domed roof to a semi-circular plan, 
 Tault of the apsis of a church, head of a 
 niche, "Key to Ster.," page 117. 
 
 Compute as a whole sphere, and divide 
 by 4, or treat as an ungula. 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., elbow of two 
 semi-cylindrical mouldings, etc., at an 
 obtuse angle. 
 
 .173 — Obtuse-angled ungula of 
 a sphere. 
 
 Bead of niche reaching into a sloped 
 ceiling ; elbow of two half-beads at an 
 acute angle, etc. 
 
 On its base : one base, a semi-cir- 
 cle ; opposite base, a point ; middle 
 section, the segment of a circle. On 
 end : each of its opposite bases^ points ; 
 its middle section, the sector of a cir- 
 cle. Only one area to compute, and 
 easier and quicker than a segment. 
 
 Its opposite bases, points ; its middle 
 section, the sector of a circle ; the 
 
 spherical surface, the component of a 
 hollow metallic (>r other sphere or sphe- 
 rical vessel, or of the covering for a 
 racket or other playing ball, etc. 
 
 For spherical area see "Key to Ster.," 
 page 117. 
 
 Opposite bases, points ; middle sec- 
 tions, the sector of a circle ; its plane 
 faces, semi-circles. Spherical area, 
 page 117 "Key to Ster." 
 
81 
 
 174.— Spherical sector or cone, 
 
 or, to avoid computing spherical areas, 
 may be treated as a compouad body, 
 a cone and the segment of a 
 sphere. 
 
 A buoy) a tinial or ornament, a top, 
 etc, a coirered filter. For areas of cir- 
 cles see tables 11., III. and IV. of" Key 
 tester." 
 
 175.— Frustum of a spherical 
 sector between parallel spheri- 
 cal bases. 
 
 Portion of a shell or bomb or hollow 
 sphere. To avoid computing spherical 
 areas, treat as frustum of cone, adding 
 greater and deducting lesser segment. 
 
 176.— Hexagonal spherical py- 
 ramid. 
 
 Its base illustrative of a spherical po- 
 lygon, page 127 of "Key to Ster." 
 
 Gomponent portion of a solid sphere 
 or ball ; keystone of a vault, finial or 
 other ornament ; decomposable for com- 
 putation into six equal triangular sphe- 
 rical pyramids, "Key to Ster.," page 
 129. See rule for spherical ai-eas at end 
 of this pamphlet. 
 
 177.— IC^rustum of hestagonal 
 spherical pyramid between pa- 
 rallel bases. 
 
 keystone of vault. Component por- 
 tion of hollow sphere. Surfaces illus- 
 trative of similar spherical polygons. 
 Height of solid equal slant height 
 of side. 
 
 178 — Half-quarter or one-eighth 
 of sphere or tri-reotangular sphe- 
 rical pyramid. 
 
 Termination or stop to chamfer on 
 angle of wall or pillar. 
 
 Compute whole sphere and divide by 
 eight. 
 
 .'..[■> 3l> ■jA-r'si ■ 
 
 : a^ni 
 
 Its base, a spherical segment ; the 
 
 other base, a point ; middle section, a 
 spherical segment concentric to the 
 base and equal in area one quarter of 
 base ; its height equal to radius of 
 sphere, its lateral face developed, the 
 sector of a circle. See " Key to Ster." 
 page 110. 
 
 179.— Acute equilateral trian- 
 gular spherical pyramid. 
 
 Its base illustrdtive of the equilater&l 
 ipherical triangle 
 
 Its bases and middle section parallel 
 thereto, concentric and similar 
 segments of spheres of corres- 
 pending radii. Its height, the length 
 of slant side. Solidity also equal to 
 difference between whole and partial 
 spherical sectors. 
 
 Its base, A regular six-sided 
 spherical polygon ; its middle sec- 
 tion a figure similar to the last, and 
 equal in area to one-quarter thereof; its 
 opposite base, a point, the centre of 
 the sphere of which it forms part. For 
 area of base, see " Key to Ster.," page 
 127. For area of component spherical 
 triangle of base, see page 123 of same. 
 Its plane faces equal sectors of a 
 c'role. 
 
 Its bases and middle section, similar 
 spherical polygons ; factor of mid- 
 dle section, as in tone, an arithmetic 
 mean between those of the bases. Its 
 lateral faces, equal frusta of equal 
 sectors of a circle, or ooncavo* 
 convex trapeziums. See rule at end 
 of this work. 
 
 Its base illustrative of the tri-reot- 
 angular 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. . , ^^ ^, .„^ . 
 
 Base and middle section "similar 
 equilateral spherical triangles, for 
 areas of whioh, see " Key to Ster.," page 
 123, and rule at end of this work. 
 
''- ' . . . • .". '. ■ 
 
 180.— FraBtum of triangalar Bases and middle section, similar 
 
 ■pberloal pyramid spherical triangles wtiose areas are 
 
 IllustratiTa in its bases of similar as tlie squares of the corresponding ra> 
 
 splierical triangles. Keystone of a vault dii ; or factors of middle section, arith- 
 
 to a triangular plan. metic means between those of the oppo- 
 
 " •'' ' "" ' site bases. 
 
 .-iivi'i*''; 
 
 ■.»S"j-i.'. ■ 
 
 .-I'lliS" 
 
 GLASS XlX. 
 
 
 Oblate Spheroid, Frusta and Segments. 
 
 181.— Oblate spheroid. 
 
 Representative, in a less exaggerated 
 ratio of its diameters or axes, of the 
 earth and planets which are flattened at 
 the poles or extremities of fixed axis and 
 protuberant at the equator. Au jr. ,;ige, 
 lamp-shade, or globe, or bowl. 
 
 182.-~Seml-oblate spheroid by 
 a plane perpendicular to Its Qiced 
 or lesser axis. 
 
 Elliptical ceiling, dome, cauldron, ba- 
 sin, dish, Tase, shade, globe, etc. 
 
 183.— Seml-oblate spheroid by 
 a plane parallel to its fixed or 
 lesser axis. 
 
 Dome or ceiling to an elliptic plan ; 
 glass globe or shade, dish cover, b-it, a 
 trough, cauldron, etc. 
 
 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 copper, etc. 
 
 •?*{ bf- <li»'4,«^t) mU.''.'. '»i'> . 
 
 i'.fi.'^ 
 
 Treated perpendicularly to its fixed 
 axis, its opposite bases are considered 
 points, as in the sphere, a plane touch* 
 ing the solid only in a point ; its middle 
 section, a circle. If considered parallel 
 to its fixed axis, its middle section, an 
 ellipsis. For spheroidal surface or 
 area, see No. 161. 
 
 Base, a circle ; opposite base, a 
 point ; middle section, a circle ; for 
 diameter of which, if not from direct 
 measurement, see " Key to Ster.," page 
 139, line 10 and page 140, line 20. 
 
 Equal in area and solid contents to 
 No. 182 and of easier and quicker com- 
 putation, if considered such, the factors 
 being circles instead of ellipses. As 
 it stands, its base and middle section, 
 similar ellipses. 
 
 Its base and middle section, circles ; 
 opposite base, a point. Spheroidal sur- 
 face 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 frnstnm or solid Opposite bases and middle section, 
 
 Bone of an oblate spheroid be- 
 tween planes perpendicular to 
 fixed or shorter axis. 
 Elepresentative of same as No. 165. 
 
 circles ; for areas of circles to eighths, 
 tenths and twelfths of an inch or other 
 unity, see tables II., III. and IV. of" Key 
 to Ster." Spheroidal area, see page 95 
 of same. „i., .„>♦ : 
 
ttt' 
 
 186— Middle frUBtnm or solid 
 aone nf oblate spheroid by planes 
 parallel to fixed or lesser axis of 
 •olid. ^.,, 
 
 187 —Segment of oblate sphe- 
 roid less than half, by a plane pa- 
 rallel to its fixad or lesser axis. 
 
 Representative of same us No. 183. 
 
 188 —Lateral frustum of oblate 
 ipheroid by jilanes parallel to fix 
 sd or shorter axis. 
 
 Coved ceiling of elliptic plan ; re- 
 rersed : a boat, a scow, a vessel of ca- 
 )ncity, etc. 
 
 189.— Half or segment of oblate 
 ipheroid by a plane inclined to 
 ixis of solid. 
 
 Liquid or fluid in a semi-spheroidal 
 essel inclined from the vertical. Finial 
 n a pediment or sloped surface. 
 
 190.— Frustum of oblate sphe- 
 
 lid betvtreen non-parallel bases. 
 
 Decompose into n frustum with 
 
 irallel bases, and an ungula by a 
 
 lane parallel to one base and drawn 
 
 trough nearest point of other base, or 
 
 )mpute whole spheroid and deduct seg- 
 
 ents. 
 
 .Oh 
 
 Its bagcs and middle section Bimilar 
 ellipses, for areas of which see page 
 51 of "Key to Ster." Spheroidal area, 
 page 95 of same. 
 
 ...* WO-vwrv-.l" 
 
 Its base, an ellipsis -,oppo3ite base, a 
 point ; middle s'.ction, an ellipsis si- 
 milar to base. For factors of middle 
 aection, see No. 182. 
 
 ■S 1 <■.. ■( 
 
 Its opposite parallel bases and middle 
 section, ellipses, for areas of which see 
 " Key to Ster.,'' page 51. 
 
 Its spheroidal surface decomposable 
 into continuous trapeziums of vari- 
 able heiglit. 
 
 Its base and middle section, similar 
 ellipses ; its opposite base, a point ; 
 its spheroidal surface, trapeaiuma, 
 with ellipsis at apex and a cuvvili- 
 uear triangle at base of sha[)e similar 
 to fig. h, page 57 of " Key to Ster.," or 
 lateral area may be divided and com- 
 puted as triangles. 
 
 Bases and middle section of compo- 
 nent frustum with parallel bases, ellip- 
 ses ; base of unprula, an ellipsis ; mid- 
 dle section of ungula the segment of 
 an ellipsis ; its other base, a point. 
 
 For factors of middle sections, see 
 " Key to Ster.," page 139, line 10, and 
 page 140, lino 20, where AB : CD : : 
 -^A^^JB : oM and CD : AB : : 'JCoJP 
 •.oM. 
 
 CLASS XX. 
 
 •4 fv ,. 
 
 ■V: } 
 
 Prolate Spheroid, Frusta and Segments. 
 
 1.91. — Prolate spheroid. 
 jlepresentative of a lemon, melon, cu- 
 aber, etc. ; a case, sheath, etc. 
 phe work of computation expedited 
 treating circles instead of ellipses, 
 is, areas perpendicular instead of 
 iUel to fixed axis. ^' 
 
 Its middle section perpendicular to 
 fixed or longer axis, a circle ; its oppo- 
 site end bases, points. Spheroidal sur- 
 face, continuous trapezoids, or a 
 series of double segments base to 
 base as the component ribs of a melon. 
 May treat as plane segment with length 
 of cord equal to semi-elliptical section. 
 
 'ii_ 
 *> 
 
 vj 
 
 n 
 
u 
 
 1^.— Semi-proTBte spheroid by 
 a plane parallel to dxei^laxift. 
 
 VftuKeii oelVing to elliptic \t)mo ; »e- 
 vereed : a boat or otlior sailing TesBel, a 
 eauldroB op wsflel of capacity, «t«., •t«. 
 
 193.— Semi-prolate upherold by 
 a plane perpeudicular to fixed 
 axis. 
 
 A hive, hut, roof of dome to citeHkiB 
 tower or apa-rtmcnt ; reversed : a coiiper 
 or boiler. 
 
 Fur.floMd eontentB and spheroidal' ■uv^' 
 flace tr«at pcrpendiciiar lio flacd nslay 
 where fbetors are «irclea or aeml- 
 oirobei iBBtead of ellipses. Vor areas 
 of circles, see tables IL, lEl. and IV. of 
 "KeyioSiT." ,, .^,„ ,r, 
 
 Bft-se, ft circle ; other base, a point ; 
 middle section, » circle. For nidlus of 
 middle section, see formula gWen in Noi 
 190, or at pikge 139; line 10, page 140^. 
 line 20 of " Key to Sier." Stiherolda* 
 area, see No. 191. 
 
 194._Ssgpneut of prolate sphe- B»«e mA middle section, circles ; it» 
 
 roid groa.ter tlian half, by a plane »ther base, tiu apex or point. Mssphe- 
 
 perpendicular to fixed axis. roidai »ur!t*ce resolvable into-continu- 
 
 A hut, hi.vcr dome, n caulUro» op eop- oua trapeaiuma and a circle at apex, 
 per, etc. 
 
 195.— Middte frnstmn or solid ■ End bases, equal circles ; middle 
 
 xoneofproiatQsplxeroidbyparal- section, a circle. Unlike the middle 
 
 lei planes perpendicular to fixed frustum, of » spindle, the solid extents 
 
 ■g3j£,_ of this solid are obtained exactly by 
 
 A cask, keg, barrel, puacheon, hogs- tsealmg the whole figure at oace. 
 head, etc., " Key to Ster.," page 13a 
 
 196.— Middle frustum or solid 
 zone of prolate spheroid by pa- 
 rallel plane» oblique to axis. 
 
 197.— Lateral frustum or solid 
 zone of prolate spheroid by 
 planes perpendicular to fixed 
 axis. 
 
 Coved ceiling, base of column, etc. ; 
 reversed : capital of column, dish, basin, 
 bowl, tub, hamper or basket, stew pan, 
 cauldron or other vessel of capacity,. 
 etc., etc. 
 
 198.— Ijateral frustum or solid 
 zone of prolate spheroid by 
 planes parallel to each other, and 
 to longer or fixed axis. 
 
 Coved ceiling of elliptical plan, etc. ; 
 reversed : a flat-bottomed boat, a scow, 
 a dish, basket, etc., etc. 
 
 Opposite bases and roidd>e section^ 
 similar ellipses. Spheroidal surface, 
 trapeziums of wbicli i*ke meatt 
 beight. ,..,'. 
 
 Bases and section, circles, fi)r areas 
 of which see taWes II., HI. and IV. 
 "Key to Ster." For diameter of middle 
 section, measure solid or.compute by for- 
 mula of i)age 13(), line 10 ; page 140, 
 line 20, where it is shown that tlie rect- 
 angle under the required radius, and 
 either axis of the spheroid is equal to 
 that under the square root of the rect-. 
 angle or product of the absciss* of the 
 first axis and the other axis. l^lO'i 
 
 Its parallel bases and* middle section, 
 similar ellipses ; for areas of which 
 see " Key to Ster.," page 61. Its lateral 
 area resolvable into continuous tra- 
 peziums of varying height if parallel 
 to bases, but of uniform height, if lines 
 be drawn from extremities of fixed axis. 
 
m 
 
 199.— Segment of prolate sphe- I u base and middle section, elmllar 
 rold by a plane Inclined to axla. ellipaea ; Us other base, a point ; iU 
 
 Liquid in sphemidiil Tosael inclined Hiiheroidal siiHace resolvable by circles 
 from the yerlical, a scoop, scuttle, etc. drawn from extremity of fixed axis into 
 
 a oliole, trapeziums and triangle. 
 
 200.— Frustum of prolate epbe- 
 roid between non-parailel pianos. 
 
 The one, iierpendiciilur to lixi'd axis, 
 tiie other oblique or inclined thereto. 
 
 Dpcompose Into frustum with parallel 
 bases, and an ungula. Compute separ- 
 iitcly, and add ; or compute whole eeg' 
 ment duo to frustum and deduct lessor 
 segment. 
 
 V 
 
 'he areas of Spherical Triangles and Polygons to 
 any radius or diameter. 
 
 The following will be found a new, easy and concise rule 
 
 By Mr. C. BAILLARGB, 
 
 |or finding the area of any sphericnl triangle, or of a triangle described on the Bur* 
 ice of a sphere of any given diameier. 
 
 The area of a sphere to diameter I. being = 3.141,502,653,589,793+ . 
 
 )ividing by 2, we get that of the hemisphere = 1.570,796,326,794,896,6 
 
 phis divided by 4 = area of tri-rectgl'r sph. triangle = 0.392,699,081,698,724,1 
 90 = area of 1° orof bi-rect. sph. tri. with8p.ei.= 1° = 0.004,363,323,129,985,8 
 
 " r = 0.000,072,722,052,166,43 
 
 " r ' = 0.000,001,212,034,202, 77 
 " 0.r> =0.000,000,121,203,420,277 
 « 0.01" =0.000,000,012,120,342,027,7 
 " O.OOl" =0.000,000,001,212,034,202,77 
 
 I Find the spherical excess, that is, the excess of the sum of the three spherical 
 igles over two right angles, or from the sum of the three spherical angles deduct 
 10°. Multiply the remainder, that is, the spherical excess, by the tabular number 
 pein above given : the degrees by the number set opposite to 1°, the minutes by 
 It corresponding to 1' and so on of the seconds and fractions of a second ; add 
 ese areas and multiply their sum by the square of the diameter of the sphere of 
 surface of which the given triangle forma part, the result is the area required. 
 
 'r'^. EXAMPLE. , 
 
 Jet the spherical excess of a triangle describtid on the surface of a sphere of 
 iich the diameter is an inch, a foot, or a mile, etc., be 3' — 4' — 2.235", What 
 i the area 7 
 
 60 = " of 1- or of " 
 
 « 
 
 II 
 
 60 = " of V or of " 
 
 i: 
 
 11 
 
 10= " ofO.r'orof " 
 
 K 
 
 II 
 
 10 = " of 0.01" orof " 
 
 II 
 
 II 
 
 10= " of0.0oi"orof " 
 
 II 
 
 II 
 
 i 
 
Areaoflo =0.004,303,^23,129,985,8 x 3o =0.013,089,969,389,955 
 
 " r =0000,072,722,052,166,43 X 4' =0,000,290,888,208,664 
 
 « 1" :r- 0.000,001,212,034,202, X 2'" =0.000,002,424,068,404 
 
 " O.r' =0.000,000,121,203,420, X 0.2» =0.000,000,242,406,840 
 
 « 0.01" = 0.000,000,012,120,342, X 0.03" = 0.000,000,036,361,026 
 
 "0.001" = 0.000,000,001,212,034, X 0.005" =0.000,000,006,060,170 
 
 •rff ■ Jo; ; 
 
 Area required 0.013,383,566,495,059 
 
 The answer is of course in square units or fractious of a square unit of the same 
 name with the diameter. That is, if the diameter is an inch, the area is the frac- 
 tion of a square inch ; if a mile, the fraction 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 thom as may be omitted. 
 
 If the seconds are omitted, as would be the case i'l dealing with any other trian- 
 gle 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 com- 
 ponent section of a sphere of the size of a dome, vaulted ceiling, gasometer, or 
 large copper or boiler, etc. ; and in dealing with such spheres as a billiard or other 
 playing ball, a cannon ball or shell, the ball of a vane or steeple, or any boiler, 
 copper, etc., of ordinary size, it will generally suffice to compute for degrees only. 
 Whence the following. 
 
 RoLH 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 case — 0.004,363,323. 
 
 Rule to degrees and minutes. 
 
 Proceed as by last rule for degrees. Multiply the spherical excess in minutes by 
 0.000,073, or for greater accuracy by 0.000,072,722. Add the results, and multiply 
 their sum by the square of the diameter for the required area. 
 
 "-^'"^ EXAMPLE I. 
 
 Sum of angles 140° + 92*^ + 68° = 300 ; 300 — 180 = 120= spherical excess, dia- 
 meter"= 30, page 124 of " Key to Ster." Answer area of 1" 0.004,363 
 
 Mult^ly by spherical excess 120° 
 
 We get 0.523,560 
 
 This multiplied by square of diameter 30 = 900 
 
 Required area = 471.194,000 
 
 A result correct to units and agreeing therefar with the answer given in the 
 " Key," which 13 471.24. If now greater accuracy be required, it is to be obtain- 
 ed by taking ia more decimals, tiius, say area 1° = 0.004,363,323 
 
 120 
 
 l^r*,^^ ^>- ^.i"::. S •"• V 0.523,598,760 
 
 '"'" : ." " . 1^1.238,884,000 
 
' .■'''■■■'•• . ''•'■*' 
 
 EXAMPLE H. ' ' ' ' ' ■ 
 
 " Key to Ster.," page 124. The three angles each 120°, their sum 360o, from 
 which deducting 180° we get spherical excess = 180°. Diameter 20, of which the 
 
 square = 400. ^g/r ,y.,, ; o'. : • '- -■<' 
 
 Answer.' ; Area to 1° = 0.004,363,323 
 
 180 
 
 \'. ■',...■ . . . 0.785,398,140 •*-<'' 
 
 400 
 
 -.1 .. '■ IV ' ■ ,- , I- J •■ • J 
 
 , ij Answer io " Key," 314.16 314.159,256,000 , 
 
 ;,-fe. ^..•.;<i!..v, ... . EXAMPLE m. - •'■ - 
 
 See example 4, page 125, " Key to Ster." The sum of the three angles of a tri- 
 angle 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 diameter = 7,912 English miles, or, which is the same thing, that 
 the diameter 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 
 
 The answer in the " Key," is 75.87321 square miles, which is only correct to the 
 second place of decimals from having assumed tt = 3.1416, instead of, as it should 
 be for greater accuracy, 3.141,592,653,589,793, etc. 
 
 Remark. — This unit 75.87 etc., as applied to the Terrestrial sphere, as explained 
 page 125, par. 106 ot " Key to Ster.," becomes a tabular number, which may be 
 used for computing the area of any triangle on the earth's surface, as it evidently 
 suffices to multiply the area 75.87 etc., corresponding to orie second (1") by the 
 number of seconds in the spherical excess to arrive at the result ; and as also stated 
 in " Key," the result may be had true to the tenth, thousandth, or millionth of a 
 second, or of any other fraction tliereof, by successively adding the same figures 
 7578 etc., witii the decimal point shifted to the left one place for every place of 
 decimals in the given fraction of such second : the tenth of a second giving 7.578 
 etc., square miles, the 0.01" = .7578 of a square mile, the 0.001 >" = .07578 etc., of a 
 square mile, and so on ; while, by shifting the decimal point to the right, we get suc- 
 cessively 10" = 757.8 square miles, 100" = 7578 etc., square miles, or V — 75.78 
 X 60", 1° = 75.78, etc., X 60" X 60\ 
 
 Rdlb. 
 
 To compute the area of apy spherical polygon. 
 
 Divide the polygon into triangles, compute each triangle separately by the fore- 
 going 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 there are sides less two. This will give the spherical excess. This 
 into the tabular area for degrees, miuutes, 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 
 
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 sphere on which the polygon is tracad will give the correct area of the proposed 
 figure. 
 
 For example see page 128 of " Key to Star." 
 
 It may be remarked here tfaart the area of a spherical lane or the convex snrface 
 of a spherical ungula is equal to the tabular number into twice the spherical excess 
 since ii is evident that every such lune is equivalent to two bi-rectangular spheri- 
 cal triangles of which the angle at the apex, that is the inclination of the planes 
 
 forming the ungula, is the spherical excess. 
 
 Rbhark. — The area found for any given spherical excess, on a sphere of given 
 diameter, may be reduced to that, for the same spherical 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 surface, 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 osculatory circle is of aiffereut radius ; 
 these areas being as the squares of the respective radii or diameters. 
 
 N.B. The " Key to Ster." also sets forth how the converse operation is per- 
 formed of obtaining the spherical excess from the area. 
 
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