y / I P^v0 /viA V <#£ f iJJt*' r A N APPENDIX TOT HE ELEMENTS of EUCLID, IN SEVEN^ BOOKS. CONTAINING Forty-two moveable Schemes for forming the various Kinds of Solids, and their Sections, by which the Doctrine of Solids in the Eleventh, Twelfth, and Fifteenth Books of Euclid is illuftrated, and rendered more eafy to Learners than heretofore. BOOK I. Contains the Five regular Solids. II. Shews the Infcription and Circum- fcription thereof, as fet forth in the Fifteenth Book of the Elements. III. Exhibits a great Variety of irre- gular Solids* IV. Contains fundry Sorts of Prifms." V. Various Kinds of Pyramids, and Fruftrums thereof. VI. Some difficult Proportions in the Eleventh and Twelfth Books. VII. The Cone and its feveral Sections,; ■ i 1 1 in '•r ■ I SECOND EDITION. By JOHN LODGE COWLEY, F. R, S, Profeflbr of Mathematicks in the Royal Academy at Woolwich. LONDON: Sold by T. CADELL, Bookseller in the Strand, ■ I P R E F A C E. ih THE approbation fhewn to the firft edition of this work, and the many applications made for it while out of print, are motives that have encouraged me not only to iflue this fecond edition, but alfo to make fundry additions and im- provements to it, in which the Solids contained in the firft and third books hereof, with the addition of their fe&ion-planes, are fhewn by a new and more expreflive method of exhibiting them ; the number of irregular folids is alfo considerably in- creafed by the addition of others, whofe formations, far as I know, are wholly new; and the fixth book is made more extenfively ufeful by a farther consideration of the dodrine of Planes defcribed in the definitions and propofitions of the eleventh and twelfth books of Euclid, the fame being more particularly illuftrated and explained by moveable diagrams having the fame lines and letters thereon as are in the plano- fchemes belonging to thofe books, which being duly elevated, the feveral planes and lines heretofore defcribed in piano ap- pear in the very pofitions they are to be confidered, or con- ceived in the mind of the learner, principally adapted to the fchemes given of thofe books in the edition by p; ofeffor Simfon A of PREFACE. of Glafgow, together with others fuited to thofe in the feventh and eighth books of the elements of Geometry by my prede- ceflbr; but as thefe things would augment the prefent work too much, and that thofe who have the firft Edition hereof may have an opportunity of obtaining thefe additions without detriment to their former purchafe, they are referred for a fe- cond or fupplemental volume to what is herein contained. J. L# t 0» APPENDIX to the Elements of EUCLID, BOOK L Of the Five regular Solids. * I ^O form the Solids defcribed in this book. Raife up the moveable parts of the fcheme, and fold them around the part that is fhaded. L Of the Tetraedron, Plate i. The Tetraedron is a folid bounded by four equal equilateral triangles, II. Of the Hexaedron, or Cube, Plate 2. The Hexaedron, or Cube, is bounded by fix equal fquares. III. Of the OSioedron, Plate 3. The O&oedron is bounded by eight equal equilateral triangles, IV. Of the Dodecaedron, Plate 4. The Dodecaedron is bounded by twelve equal regular pentagons. V. Of the Icofaedron, Plate 5. The Icofaedron is bounded by twenty equal equilateral triangles. Th ere 2 Of the Five regular Solids. There are particular rules for finding either the fuperficial or folid con- tent of each of thofe bodies ; but the fame may be more readily obtained by the following table. A T A B L E, mewing the fuperficial and folid content of each of the five regular folids, each fide * whereof is unity. Names. Superficies. Solidity. Tetraedron. 1,732051 0,1178511 Hexaedron. 6,ooopoo 1,0000000 Octoedron. 3,464102 0,4714045 Dodecaedron. 20,645729 7,663119 Icofaedron. 8,660254 2,181695 Ufe of the above table. To find the fuperficial content of either of the above-named fbllds, one fide thereof being given. Multiply the tabular number that flands under Superficies in a line with the respective folid, by the fquare of the given iide thereof; and For the folid content, Multiply the tabular number that flands accordingly under Solidity by the cube of the given fide, the former of thofe produces is the fuperficial^ and the latter is the folid content required. * By the word fJe is here to be underftood, the line in which any two fides of the bound- ing planes meet each other. EXAMPLE Of the Five regular Solids. 3 E X A M P L E I. What is the content fuperflcial and folid of a Tetraedron, each fide being 9 inches ? Superficies. Solidity. 1,732051 81 1732051 13856408 Area 140,296131 Sup. 0,1178511 729 10606599 2357022 * 2 49577 Area 85,9134519 Solid. EXAMPLE II. Suppofe each fide of the Hexaedron, or Cube, 9 inches ? Superficies. \ Solidity. 81 6, 486, 729 1 729 EXAMPLE III. Suppofe each fide of the Ocloedron 9 inches ? Superficies. Solidity. 3,464102 81 3464102 27712816 280,592262 >47 x 4 729 42426 9428 32998 343^5° 6 B EXAMPLE Of the Five regular Solids. EXAMPLE IV. Suppofe each fide. of the Dodecaedron 9 inches? Superficies. 20,645729 81 20645729 165165832 1672,304049 Solidity. 7,663119 729 689 68071 15326238 53641 S 33 55^^37 5 l EXAMPLE V. Suppofe each fide of the Icofaedron 9 inches ? Superficies. 8^660254 81 8660254 69282032 701,480574 2,181695 729 J 9 6 3$255 43 6 '329° 15271865 *S9° ASSESS BOOK yBMJVfcM%; ■§■ c 5 J BO O K II. Containing an illufiration of Euclid Y Fifteenth Book. Directions for folding the fchemes together. P L A T E VI. THIRST form the Tetraedron, by folding together its federal triangles ■*• around that particular one which is Shaded : Then move its vertex till the other angular point, which is moveable, is fo much elevated above the plane, that the fide which connects* thofe two points of the Tetraedron becomes parallel to the plane from which the whole was railed. The Tetraedron being in that pofition, bring the feveral parts which constitute the Hexaedron around it; thus will the four angular points of the Tetraedron. fall exa&iy in the four angles of the Cube. PLATE VII. The Odoedron being firnV formed, fold the Tetraedron over it, PLATE VIII. Firft form the Oftoedron, and raife it perpendicularly upon one of its angular points, then fold the Cube around it, and the angular points of the Octoedron will then touch the center of each fide of the Cube. PLATE IX. Form the Cube, and incline it a little forward till its" fides bifecT: thofe of the O&oedron, when folded around it. PLATE X. Form the Dodecaedron, and incline it fo that its fides may bifect thofe of the Icofaedron, when folded around it. BOOK C 6 ] BOOK III. Containing feveral irregular Solids^ PROBLEM. '"pO meafure an irregular folid of any form whatfoever. X General Rule. Procure a fmtable veffel, as a tub, ciftern, or any fuch as can be moft eafily rneafured; then put the body whofe content is required into it, and pour into the veffel fo much water as will juft cover the folid : mark the fide of the veflel where it is even with the furface of the water, then take out the folid, and obferve well how far the water defcends ; for that part of the veffel contained between the higheft afcent and loweft defcent of the water being meafured, gives the folidity of the body required. Example. Suppofe the folidity of a very irregular piece of ftone was required, and that having put it into a fquare cittern, whofe dimenfions are 50 inches, after having juft covered it with water, on taking it out' I find the water to defcend £ inches, from thence to find its content. 50 inches 50 Superficial area 2500 of the bafe of the ciftern, 8 its depth, or defcent of the water, Solidity 20000 of that part of the ciftern porTefTed by the body, and therefore the content of the body required in inches, which, divided by 1728, gives 11,516 feet, &c. the content in folid feet. N. B. To put the water into the veffel firft, and then immerfe the folid therein, may, in fome cafes, be a more preferable way of proceeding. To fold any of the figures contained in this book. Bring the contiguous parts of each figure around that which is fhaded, the reft will then join together, and form the folid required. 2 BOOK [ 7 ] BOOK IV. Containing various forts of Prifms. DEFINITION. A Prism is a folid figure comprehended by planes, among which two oppofite ones are parallel, equal, and fimilar. REMARKS. I. The fides of all prifms are parallelograms; but the ends or bafes are of various forms ; hence it is that prifms receive different names, and are accordingly called triangular, quadrangular, pentangular, hexan- gular, &c. II. When a prifm is terminated at its ends by parallelograms, it is then moll: generally termed a parallelopipedon. ' III. The method of folding together the fchemes contained in this book is fo obvious, that nothing more need here be obferved concerning it, than only to bring the contiguous parts of each figure around that particular one which is fhadcd. PROBLEM. To meafure any fort of prifm. General Rule. Meafure from the center of its bafe or end, to the middle of any one of its fides, multiply that length by half the fum of all the fides by which the prifm is bounded, and that producl is the area of the bafe ; which being multiplied by the whole length of the prifm, gives the folid con- tent. Or thus: Square the fide of the bafe, and multiply it by the affixed number in the following table belonging to the figure which "die bafe of the prifm is of, and that produa gives the area of the bafe ; which being multiplied by the length of the prifm gives the content as before. C A GENERAL I* 8 Of Prifms. A GENERAL TABLE For finding the area of regular polygons. Sides. Names. Multipliers. 3 Triangle. '433° l 3 4 Square. 1,000000 5 Pentagon. 1,720477 6 Hexagon* 2,598076 7 Heptagon. 3> 6 33959 8 Octagon. 4,828427 A 7 . 5. The parallelopipedon is bell: meafured by multiplying its length, breadth, and depth into each other. I forbear troubling the reader here with any particular examples, there being fufficient in fuch authors as have wrote on the fubjecl of menfura- tion ; for to exhibit the bodies themfelves is the only defign in which I am now engaged. BOOK [ 9 1 BO OK V. Containing various forts of Pyramids, and fruflrums thereof; •\ 1 DEFINITIONS. I. A Pyramid is a.folid having a polygon for its bafe, and compre- f ; hendedunder triangles, all meeting together in one point; which point is called its vertex. II. The fruftrum of a pyramid is that part which remains, when the top part is cut away by a plane or fedion paffing through all its fides pa- rallel to its bafe, & " To fold any of the figures contained in this book. The parts which are fhaded are fuch evident indications, as render fur- ther directions luperfluous. PROBLEM I. To meafure a pyramid. General Rule. Multiply the area of its bafe by a third part of its perpendicular alti- tude, that product is the folid content. N. B. The perpendicular altitude is meafured by a line drawn from the vertex to the center of its bafe. . PROBLEM IL To meafure the fruftrum of a pyramid. . Rule.. Multiply the areas of the bottom and top bafes together, extract the fquare root of that produft, and add thereto" the fum of both «£f that total mult.phed by one third of the fruiWs altitude, gives the folidlty 7 B O OK I [ 10 ] BOOK VI. Containing an tllujiration of fome Theorems in Euclid'.? Eleventh and Twelfth Books. Eucl. XL Prop, xxviii. THEOREM. A Plane paffing through the diameters of opposite planes of a paralle- lopipedon divides it into two equal prifms. See this illuftrated in Plates XXXIV. and XXXV. Directions for folding the fch ernes contrived for illuft rating the above theorem. PLATE XXXIV. I. Raife up one half of the figure, and fold it fo that the point E may fall upon A, and the point F upon B, and raife up the parallelogram at the end ; thus have you one of the prifms. II. Then bring over the other part of the fcheme, fo that the corner C may likewife fall upon A, and the corner D upon B; thus will there be formed the whole parallelopipedon and its fection made by the plane paffing through its diagonal A B. PLATE XXXV. I. Lift up the figure, and bring the point A to C, and the point B toD. II. Fold back the reft of the figure upon the line A B, which is pro- perly cut for that purpofe ; fo that the two parallelograms which are fe- parated by A B may lie clofe together. III. Then bring over the- remaining part of the fcheme, fo that the points E and F may coincide with C and D ; then folding together the four triangles, the whole parallelopipedon will be formed, and alfo its feclion made according to a plane paffing through the diagonal of its bafe or end. Eucl. [ *3 } BOOK VII. Containing the Conic Se&ipns. DEFINITIONS. I. A Cone is a round folid, having a circle for its bafe, and maybe conceived to be generated by the revolution of a right-angled tri- angle turning round on one of the fides, which includes the right-angle . IL The axis of the cone is that fixed right line, around which? the- triangle is luppoied to revolve. III. The extreme end or point, by which the cone is terminated is called its vertex. ' IV. The fruftrum of a cone is the part remaining, when its top is cut off by a plane or fedion palling through it parallel to its bafe. REMARKS. I. The cone and its fruftrum are meafured by the fame rules as obtain for pyramids and their fruftrums. II. As a pyramid is the third part of a prifm, having the fame bafe and altitude, fo in like manner a cone is the third part of a cylinder, having the fame bafe and altitude; but the roundnefs of thofe figures does not admit of an explanation of this truth by this mechanical method of reprefenting them. Directions for folding together the fchemes contained in this booh. PLATE XXXVIIL I. Bend the arc A B evenly round the arc E F, thereby caufW A to come to E, and B to F. & •d i L ^ Um , ab °. U l th f trian S le B C D u P° n the fide B C > & as to make BD coincide with the diameter of the bafe E F, and AC to coincide with DC ; thus will one half of the cone be formed.. III. In like manner form the other half, by bringing G to E, and H : to F, the line G K even upon E F, and the fides H I and K I even to each* other; fo will the whole cone and the fedion, through its axis, both ap- pear. PLATE ■ *2 Of Geometrical Theorems* Eucl. XII. Prop. vit. T HE OR EM. Tj^VERY pyramid is the third part of a prifm, having, the fame bafe «"' and height. See this illuftaated in a triangular prifm by PLATE XXXVII. * 'To f aid the figure contained in this plate* I. Firfl form the pyramid ABC, by bringing the point C to A, fa that the line B C may fall upon A B* II. Then form the pyramid BAD, by bringing D to B, making A D coincide with A B, and raife up the triangle at B. III. Then turn back the reft of the figure upon the line A D, fo that the triangle marked 7 may clofely adhere to that marked 6. IV. Laftly,, The vacuity that is now between the two pyramids fo* formed will be filled exactly by folding together the pyramid contained under the triangles marked 8, 9, 10, in fuch fort that E be connected with A, and F with B; thus is the whole prifm completed, and the fec- tions above defcribed clearly feen, N. B. When the lafiVmentioned pyramid is introduced into the Ipace remaining between the other two, by preffing a Httle upon that part which is the upper edge or fide of the prifm, the formation of the prifm will be rendered more perfect BOOK 14 Of the Conic SeEiions. PLATE XXXIX. I. Fold the arc A D round the bafe, fo that A may come to B, and the arc D C in like manner, making C to unite with A at the point B. II. Make the lines AF and C G unite together, which will form the fruftrum of a cone cut parallel to its bafe, and fhew the fedion to be a circle. P L A T E XL. Fold the arcs A B and B C round the arcs B D E, B F E, and make the fides A G and C H coincide every where together upwards from the point E, which will reprefent a cone cut oblique to its bafe, and fhew the fec- tioii to be an elliplis. PLATE XLL I. Form the whole part of the cone A C B, by bringing the points A atid B together, and bending it fo as to become round as near as may be. II. Bend regularly the other part of the figure, fo that the arcs D E, FE, may furround the arcs FG, FH. III. Raife up the parabola G I H, making the point I meet the coinci- dent points A and B ; thus may you fee the fedion made by cutting a cone parallel to one of its fides. PLATE XLII. The laft directions obtain here ; for by bringing E and F together, and rounding the part E D F, as was there defcribed, and the points A and B being brought to G and H, if the hyperbola G I H be then raifed up, as before directed, the fedion of a cone, cut parallel to its axis, will be thereby fubjugated to view. FINIS. n r.^ a r^ m mj mm r »n Lately Publijhed, by the fame Author, The THEORY of PERSPECTIVE, Demonftrated and illuftrated by moveable Schemes, Which ihew the feveral Planes, Lines, and Points ufed in that Art in the true Pofitions in which they are to be confidered. u f fl The Hexaedrou or Cube, / /■ w/.'.y. ■f 111 ■ I I ■ ■ I ■ ■ Of Geometrical Theorems. xx Eucl. XII. Prop. iii. THEOREM. A Triangular pyramid may be divided into two equal triangular -* * pyramids, having each the fame bafe and altitude, and into two equal triangular prifms, which two prifms are together greater than the half of the whole pyramid. Directions for folding together the fcheme which explains this propofztion. PLATE XXXVI. 1, Firft form the triangular pyramid AC B, by making the lineBC coincide with AC. II Y F °! d . back u P on the line BC the triangle numbered 4, fo that it may lie clofe to that which is marked 3. III. Then fold over the fides 5, 6, 7, bringing the line D E fo that D may fall upon A, and E upon C, which, with the quadrilateral bafe that is lhaded, forms one of the prifms. IV Then fold back the other part of the figure, fo that the fide 8 may be dole to that marked 7 ; thus will there remain a triangular vacuity at the bafe of the pyramid ACB, which muft be filled by folding together the fides marked 9 and 10, making the point F fall upon A, and the point Cj upon C ; thus will the other prifm be formed. .Y*, T*"* done > tnere remains only to form the other pyramid G I H, which is effected by folding its parts back upon the line GK, and folding round the triangles marked 12, 13, 14; the partition lines whereof are all cut on the front fide of the figure, which, being folded accordingly, comp etes the whole pyramid, and exhibits a view of the feveral fedions deicnbed in the above theorem. ;M, ■ D Eucl. I V / • : ■"II 1 i ■ ■ '! I ■ < - • ■ Cctwd/ron I Fla re. vm tl H /3 es/ . % 2 doe dron //hw/y/tr/ /■// flu He&aedrtni. r CqucCcC J.') Prop. 3. 7 ■ ' N ,' \ r ' \ \ \ TT PJ a (cm 1 h e OcfoedroB /^vvf i * < fe&dfi. Q. 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