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Those too Isrge to be entirety included in one sxposurs srs filmed beginning in the upper left hand comer, left to right and top to bottom, aa many framee as required. The following diagrama iiluatrate the method: Las cartaa. planchaa, tableaux, etc., pauvent Atra nim^m i dea taux de rMuction diff Arents. l^rsque le document est trop grand pour itre retiroduit en un seul cliche, ii eat fiimi A partir de Tangle supArieur gauche, de gauche A droite. et de haut an baa. an prenant le nombre d'Imagea n^cassaire. i-aa diagrammea suivants illustrent la m^thode. 1 2 3 1 2 3 4 5 6 d 5 13 2 1 OF THE OIADRATIRE OF THE CIRCLE. BT PETER FLEMING, CIVIL ENGINEER, AUTHOR OF A SYSTEM OF LAND S UnVF-TlNC; AND A MF.THOD OK MEASURING A BASE LINE if ANGULAR OBSERVATION. MONTREAL : IMIINTED FOR THE AUTHOR. 1860. I A' PREFACE. The liile to the following pages, must indicate that the mind of the writer is actuated eitlier by the most vain presumption or by the strongest conviction,— for that which is here now assumed to be discovered, remained through all preceding ages undiscovered. The Problem of the Quadrature of the Circle passed under the view of New- ton, Playfair and Leslie of Great Britain, and of Euler, Lagrange, Legendre and Laplace of the European Continent, whose discoveries and applications of Mathematical Science, have opened up, even to familiarity, the most profound laws of nature.— Never- theless, they, and all others before them, have left without any finite solution, the ancient Problems— namely, the Quadrature of the Circle, the Trisection of the Angle, and the Duplication of the Cube. — Thcrefoi-e, it must appear to be a great boldness in him who would even think of, and still more hope, to accomplish I hat, which those celebrated men, may bo considered to have laid aside, or passed over as a hopeless undertaking.— Yet it may be supposed that the attempts if any made by them, to solve these ancient Problems wen; very limited, for the solutions of them if attained, would have led to no great or useful result ; while time for this, that might have been uselessly spent, was occupied in the advancement of Ma- thematical Science, and in researches directed towards the more splendid discoveries regarding the Phenomena of the Physical world, and the laws of the universe.— Such have been those discoveries that the Laws of the stability and movements of the heavenly bodies are demonstrated to be o'lly one, and that one universal, which alone regulates their motions and retains llieiii in their orbits. Also, the mutual distances, masses and densities of them have been measured, their periods exactly determined, their anomalies aeeonnted lor, imd all linally and rigorously demonstrated. From the above view, it may appear that (he modern advancements made in the Mathematical Sciences, may have rather retarded than advanced farther attempts to solve those aneieiil Problems, and whieli may be said to have bei-n laid aside by those most prodcient in tiiese sciences; for the past liiul shown only ever failing attem|)ts ; while at the same time another and inexhaustible field of discovery and usefulness lay open before them,— henee it is not to he expected that the time, which could be applied to the latter, would be expended in seeliing lli.al, witieli the greatest of improbability made doubtful to be possible. Legendre, in the fourth (Vole lo his Geonielry, has demonstrated that the ratio of the eireumferenee and diameter of a Circle are irrational numbers; and all at- tempts heretofore made by immense labour of ealeulatioii to lind that relation in entire iir rational numbers, avail nothing, more than )btaining a useful approximation by a deeiinal of six or sevn figures. Hut diat wliieh may lie inemnmeDHurable by mun- bers, it is well known may l)e commensurable in (teometry. Therefore the ancient PMblems ari' .still open to iuvesfigniion by geoiuetry ; for wfim the lenffth of the cir- vuinfrirna' of a nivvtt Cirrlv nin tw foiinil or rvsoliu-d into a stfititfhl linr, the (|iiadiiilnre of the Circle is ai'ioniplislied. Ii i-i now the solution of ihjv I'rohlem wIlM h llle w rilei piesnmes to liij before llie pnlilie 9^'\0•^^ Whether this attempt succeeds or fails, according as it may be dotermmed by the many eminent Geometricians of the present time-the author i^. treatmg it by a number of Lemmas and Proposition, has afforded the opportunity lor detectmg any defi- eiency that may exist in the demonstrations. The figures, except the first, second and third, have all been drawn to the same scale-and instead of the Lines o the given Circle being taken from the construction of the fourth figure, as referred to in fhe text, they are taken for construction, from calculation of the tabular natural Sines and tangents of the angles, for each polygonal perimeter,-and for the convenience of tho.e who may think of constructing the figures after the third, so as to avoid inaccu- racy by Geometrical drawing, the numbers used for those figures, are here given : The numerical lengths of perimeters of Polygons used in the construction of all the figures after the third. inc'd. polygon, no. of sides. Radius of given Circle Fig. 4 1,000 3 - the half equal to Radius AC of all the figures after the third . . 5,657 8,000 cibc'd. polygon. 5,196 4 8 16 32 6,133 6,627 6,243 6,365 6,273 ^'303 PETER FLEMING. Montreal, June, 1850. SUBSCRIBERS' NAMES. Anderson, T. B., Esq., Andrew, Wm., Esq., Mathematical Professor, M'Gill College,- - - Appleton, T., Esq., Badgley, VVm., Hon., Q.C., M.P.P., Barret, Jos., Esq., Benny, Walter, Esq., Biggar, Thomas, " Brodie. Hugh, " Barron, F. H., Esq., Principal U. C. College, Cartier, Geo. E., Esq., M.P '., - Crofton, VV. C, " Toronto, - - Coffin, William P., Esq., - - - Crawford, James, M.D., - - - - Cross, A., Esq., Cassels, James, Esq., Campbell, John, " Drummond, Lewis T., Esq., M.P. P., Day, John J., Esq., Day, Honorable Mr. Justice, - - Dunkin, Christopher, Esq., - - - Dumas, A., Esq., Drolet, Cha*., " David, E. D., " Dow, Willia.,1, " Dunbar, James, " Frothinghniii, J., Esq., - - - - Ferres, J. M., " Grecnshields, Samuel, Esq., - - Greenshiekh, 'as. B., " - - Griffin, Frederick, " - - Gah, A. T., " - - Guy, His Iliinor Judge, - . . - Harrington, T. D., Esq., - - - Hcugh, Tlios., » - . - - Hart, Tlicodore, " ... Honey, T,, " - - - - Hoinii's, Benjamin, Esq., M.P. P., Kinncar, David, Esq., « « - - Kerr, Wm., " - - - - Keefer, T. C, COFIEa. 1 Monk, S. W., " Montizambert, E. L., Esq., - - - M'Culloeh, M., M.D., M'Cord, S. C, His Honor Judge, . M'Tavish, Wm., Esq., .... Monk, Jolin, " . - . . Moffat, Honorable George, - - ■ M'Gill, Honorable Peter, - - - M'Lcan, John, Esq., M'Ginn, Thos., " M'Kenzie, J. G., « M'Donald, Joscpli, Esq., - - - Murray, William, " .... Morris, Honorable William, - - OstcU, John, Esq., Pyke, Geo., Esq., Proctor, Thomas, Esq , - - . . Palsgrave, C. T., " .... Ross, James, " . . . . lloss, Josepli, " .... Rutherford, Peter, " . . . . Roljertson, Geo. R., Esq., - - . Rose, John,, " . - . . Ry;ui, 'I'lionuia, " . . - Ramsay, Hew, " . . . . Savage, Alfred, " ... Sutherland, ,lohn, " . . . . Spiers, Wm., Esq. (For the Mecha^ nies' Institute), Slayner, T. A., E.sq., . - . . . Simpson, A., " Smith, Honorable Mr. Justice, - Sluiter, Joseph, Esq., . . . - Taylor, H., " ... - Tylce, RoluTl S., " .... Tiiom.son, Johnston, " ... Watson, William, " ... Wrigiit, Tiiomas, " ... Zowski, C. S. C, " . - - I GEOMETRICAL SOLUTIONS OF THE QUADRATURE OF THE CIRCLE. LEMMA 1. With iiny riulins CA, describe the s-emiciicle ABDE, and rniike the distiinccs FIG. L AH unci HI) fiicli equal to the radium AC, also describe from A tlirough C the arc BC, ami Ironi B the are ACl), and biseet tlu; are CD in tlie point D', and join A and D. Tiieii li-oui any nunibei' of points a, 1), c, d, e, f, g, h, of the ar- BD malve on the are li(;^_tl„; are Ba' e(iiial to I lie aic Ba,— the arc HW equal to Bb, Be' oqual to Be, Scc- and from the point A as a (enter with the radius Aa, describe the arc am, andf rom C witii the distance Ca' deserilx- tiie arc a'm, intersecting the arc am in the point m. In the same manner from the centers A and C, throni^ii tlie points b and b', — c and e', &L-.— describe the intersections n, o, p, (|, r, s, t, &c.— and throui,'h the points B, m, n, o, &c., ilraw the line Hnniopipst, Sic— wiiich let be sjranted is drawn through infinity of points of intersection, and cutting the arc C D— It shall be a curved line passing through the point D'. For make liie arc BI equal to the arc BB' ; hut the are CD' is by construction equal to the arc CB', and BB' is e(|ual to CD'— hence tiie point D', must be on the intersec- tion of the arcs B'D' and ID'- conseciuently the i.pint D' must be on the line Bmno- pcn-st, &.C., hut the radii of each inleisection an^ unequal, and, each intersection is of (liH'erent radii— therefore the line Bnmop(|rst, Stc, must be a curved line, cutting CD in D'. LEMMA 2. From the point A a;.: a < enter through B, describe the arc BCA" and from D' as a FIG. 2. center (h'scribe through B, the arc IBA'A" meeting the arc BCA" in the point A"— iUKJ from A ' as a center describe the arc AD', cutting tli<" arc BA' in tin- point C, and the an BC in the point B', and Join Al, jjiissing through the intersections B and H by con>truclion. Next make tlie arcs Ba and Ba' .'([ual to each other, and Bb' equal to Bb,— Be' e(|nal to Be,— and Bd e.iiiai to Bd, icc. Also make D'a' eiiual to Da,— D'b' c(iual to D'b,— I) e' eciual to D'c', and D'd' e(|ual to D'd, &e., and through the points a and a',— b and b',- c and c', and d and d', ice, describe from C and \ as centers, the intersections ni, n, o, p, ^c In the same manner from C and A" as centers describe the intersections m'n'o'p'— it is evident, that a line drawn liirongh the points D', rn', n', ,)', p', &c., and B, m, n, o, p, Jce., nmst meet in a connnon point G, on the line A B' H— for the arcsB H and B B' are synuiielriial with the arcs D' H and D' B' to the straight line A B' FI. The points B, m,"n, o, p, ... G, and the points 1)', m', n', o', p', ... G shall be on (lie arc of a circle. For let (i be the point of meeiing of the <'.ur-e lines Bmiiop and D'm'n'o'p', &c., on the straight line A'B'III and take any point K* on the lino (iH, at less disumce .piQ. 3, from C than the half of Gil, or near to G— and through the points B, K, and D', dc- scrilx- the are BKI)',— also make the distance Gl, equal to GK, and through the points B, L, and I)', describe the arc BLD',— Next from the center A, with the dis- tance AG, describe ill.' arc Gr, meeting the are Bll in the point r and cutting the arc BK in th<' point g',— also from C, describe the are Gu, and from A through any point of the are Br, describe the are pn, cutting the arc BK in the point n'— Again make iim cijual to Bp, mhI Iimiii C as a center, describe the arc nm, cutting the arc ML in the point n", and intersecting the arc pn in the point n~also from C as a center through the |)oiiils g' and n', describe the arcs g'x and n's, and from A as a center throuifh the points g and n", describe the arcs gq and n"o. Now lei Kt; and \.V-, each lie l)iseeted, and through the points of each bisection and tlie pom ts B 1), describe circular arcs — the one arc, must be between the arc B(M) BKD' ami I he other arc between the arcs BGD' and BLD', hence it is evident tliut tlie liist arc luiisl iiiiei>eot the arc r(j between the points f,'' anil C, and tlie ollici- are must intersect the arc ii(i, between tlie points si;and G. In the same manner describe other ares in injinittim, through each point of biseetioii of tlie distance of tlie last bisection from G on lAi and IvG — and the points H and D' — it is evident liiat the nhiinale arc must pass through tiie |)oint (J, and be tiie arc HGU' — also tiiat the arcs g(J and g(T, must continually be diminislK'd and e(|nidly vanish or become nothing — but the arcs n'n and n"ii also must i'(|nally vanish, when the arcs g'G and gG vanishes — for when the arc \g' coincides with the arc iiG, and (|g, coincides with rC;, the arc sn' must coincide with the re mn, and the arc on" must coincide with the arc |)n — and the point n must be on the nllimali- arc lUilJ' — l)ut by constnietion Bp is cciualto Um — conset be on an arc of the same circle. Or let the point (i move aloni; the circular arc HG, then the points u and r would equally meet in the point H, for by constnietion Hu and lir are e(|ual to one another, and the arcs r(i and ii(;, liv and 15ii woidd e(|ually vanish — coiisC(piently llieir nllimati' ration are ((inal* — ihcrei'orc Mie aic ISni must be ccpial to the arc I5p— and llie inierseciion n nin>t be on lli< ijrcidar arc !5(ll) . FIG. 5. l.KMMA ;J. I,et AC be eipial !o one half of the perimeter of ilie eircnmscribing lrianti;le alid of the uiven circle AIM), Fig. I, and from the point (' as a center, describe tlir sinii- eircle AHDK and make the chords AIJ and HI), each ei|ual to the radius AC — next make the chord or di>tan<-e A I, ccpial to the perimeler ol the inscribed sipiare eiijh, — llie distance \i, ecjual to the perimeter ol the inscribed polvgon of eiyiit ^ides — the distance A;), equal to llii' perimeler of the pnlyi(on of sixteen sides. See., of the cir- cle .\IK', Fiu- I, — alMi make M', c(|ual to ihe perimeter of the {•ircninseribing >(|nare e'l'g'h', and \1 e([nal to liie priinielcr of the polygon of eight side: A.'J' c(jMal to the perimeler of llie polyjfon of lliirly-two sides, itc, hence the points I, 2, J, are the tir>t three of the inlinite series of the inscribeihpolytronal perimeters A I, A2, A3, 6i<'., and the puinls I', i\ .'5', I', are the first four of the inlinite -eric- dj the eirenm- scriiiing polygonal perimeters of the yixcii ciri'le AUl) Fig. I, — and let it be granted the distimce An, isi'cpial to the perimeter of the jiolygon of an infinite number of sides, or is eipial to the eircumfcrence ol the yi\en circle AlU), Fig. I. Again from the point H as a ccnicr, w ilh the distances HI', H J , H;i', &(•., and from till' point I), a- a eenier, with the di^taiice 1)1, 1)2. I);J, iitc., descrilie the intersections 1, a, b, c, &c., and through tlie points I, a, b, e, &e., draw the line labc .. n.. - by con- slnietion the points I, a, b, c, .. n, ai-e on a ( iirved line concave to the arc ill), and meetinif the arc HI) in the point ii. .M-o from the |»oint H as a ( futer, u iili the disinnees H-»', H.)', HI', &c.— and from I) as center with the ilis.imees 1)1, l)J, 1)3, kc., des- cribe the iliter>-ecrioiis ll', I)', (•',&(•., Illld draw llironuh the points a', b', e', .... n the line a'bc' .... M -by con-tniclion ;he points a', b', d' n are uii a curved line, coiicavi' to the arc iili and shall meet the arc HI) in the point n. FiM' as the laiices of the poii.ts I , ,», ,1, &(•., and J', 3', ■•', &''.i fr'^iii A, ofeijeh HPfies approacli nearer to the distance .An, the nearer must the intersections iipproiich on the curve line ab'c', ke., to the point n hence the ultimate interseetioii iiiiisl liccome inlinilcly iieiir to, or coincide with the point n — eon-^cipiciiily the line ihrniiL,'li the int'rseelions a', li', e', Sic., iiinst meet the arc HI) ill tin' point n. LF.MMA I. |.'I(J ,5 t,et it he irriiMled thai the curve liiicM iilu! ... ti iiiiil ii'h'e' ... II, MRi tlftUVIl tliroilirli Mil iiiruiilc nimibcr ol interseeiion", and snppoxe ih' chords im and ii'n draw n iiiul that the points I and -', are vnriiilile ; — and that the point I iiiomh ur approachc IowukIh i Ncwlnii'« MmhfmMlciil PrliiiM|il> « of Nniiiril l'lul(nii|ih]t, Hndk l,SiTt|(m I. the point 2, aiid llic jjoiiil 2 a|i|)niiiclirs towards tlit; jjuiiit li, in tiic ratio of the ares 12 and 23, or that the point I woidd arrive at the point 2, ecpially tliat the point 2 would arrive at the point ;3, and by eonslrnetion the varyin<>; intersection a', woidd arrive at tlie ))oint a, at the same time that the varyini,' intersection I)' would arrive at the point b. There shall he a point of variation hetween the |)oints 1 and 2, on the arc 12, and a point of variation between tl.3 points 2 and 3, through which by intersections de- scribed from B and D, the curve line a'b'c' ... n shall be varied and shall coincide with its chord line a'n. For let the point I arrive at or coincide with the point 2, and the point 2 arrive at or coincide with the point 3 — it is evident that the intersections a', b', e,' ... n, must also coincide with the intersections a, b, ... n, for the intersection a' must have moved along the arc a'a2', to the intersection a, and in the same time the intersection b' must have moved along the arc l)'b;3' to the point b — hence the curve a'b' ... n and its chord a'n, must coincirrti(in a must mo\e along the arc id^^, and the intersection b nnisl move along the arc be .1, and when the curve line ab ... n coin- cidi's with b'c' ... n, the curve line ab .. n must have clumped its concavity to the opposite side of it- clior IT, 2S, and 3H, meeiifiij the are CI) in the points T, S and II. W-o lhrou«li liie points 2,3 and I, describe the arcs J K, 3(), and It^, mi'ctini; the arc (1) ill the points Iv, t) and l^. \c\t on thi' arc IIC make the arc HI", eipial to ilie are Bl,-- 112'' eipial to lb'- aic H2, — H3 e(|nal lo tin- arc H.l. ;\lso inalii' the arc H2 iMpnil I,) H2'— 113' cipial to H3 and 111* cipial lo II I — then (rom C as a center ihidiigh the points 1", 2 iiiul 3 ' dcMribe the ars T 'I", 2 S and 3 U', ineeiini,' the rrc CI) in the poinlsT',S and H . .\lso describe the arcs 2 K , .3() , and K^ ihroiiLjh the points .>', 3 and r.— Again w ilh tlii' radius AC, thronuh ihe points I) and K, and li and K (hscribe the arcs UK and UK'-- the aii' IlK inlersecting the series of arcs IK in the poiiiis d, s, r, ... (|, o, and K, and the iic HK , miersei'ting the series of arcs I K', in the points I, II, v ... (|', o' and K -then Iroin Ihe point H, as a center, with the distances HK, Ho, ami H(|, and from the point K as a center with the dislances Kd, Ks and Kr, describe the inlerseeliouH V, p and i| also from H, wilh the distniices K'l, K'li, and K'v, de- scribe the Interseclions \ , p and (|,' and draw through Vp and ([ and ihrough V "p and (|' — Ihe curved lines \ |)(| and S pi| -wiiicli by eonslrnetion are coiiciive lo their eliiirds Vn' and Vii' , on ihi- opposite side lo lliiil of llie curved line a'b'c' ... ii. There- fore there kImiII be ail are of ihc ratlins AC, to be descrilied ihroiiuh H, thai upon inler- cepled pari of w liicli aic in the Name manner, Ihe interseclions V, p, '|, &c,, may be MKaiii (l«>»cril>ed from H and die point ol tln' aic mectini/ ihe arc ('!> so ihiit lliose inlrrwc- FlCi. I lions will l.e upon a straight line with the ultimate intersection .1' ; which arc shall meet the arc CD, tetween the points K and D.-Also there shall be another arc oi th.. radius \ tlnou«h B, upon which th,' intersections in the same manner descnbed, shall he in a straight line with tlic uhimatc intersection n"-which arc must meet the arc CU, between the points K' and C. , • 1 For the points 1, 2, 3 ... 4'3'2, have varied by construction to be the points d, s, r, ... n o K on the arc B K, and tlu' intersection, a'h'c' ... n to be tiie int.-rsections V, p, q, „' but in this variation the .-urve a'b'e ... n, has pass.'d over to the opposite side of 'i'ts chord and become tlu^ curve Vp.i ... n', and therfore (Lem. 4.) there must be an arc ofih.. radius \C through B, b.-twe.-n the arcs BK and Bl), upon wliich the chord and curve must be on a straitjht line-also by the same, there nnist be another arc between the 'ire BK' and BC, on which the curve and its chord innsi be on one straight line. jSoTK —The -^ide of the chord on whi<'li may i>e llie convexity or concavity, of a curve line drawn through inters.'ctions as above described, must evidently d.'peiid upon ,he liw of tlie .lilleren.es or distances of ihv points, from the point of ultimate intersec- „ ,,n,l ,.\m, the distan.M. <,f the point n from the centers B an.l D, and as these may hi. v.rici the chord line mav be on the <.ne side or ih.- otiier ol its curv.>, or the con- ,..,vity of' the curve .han-cl to the opposite sides. For the points ol variation upon which the y construction. I.EMMA (!. Draw through th.- points of ini.'rs.'cli..ns B, .1,.', f ... h, i, k, D', th.' curv.'d line Bd.f '■ hikD' an.l from iIm' i...int B as a .'.'iit.'r with lli.' .lislaiu'.'s Bk, Bi and Bh, &c.,— ;,','„1 f,..,m th.' p..inl I) as a ..'ut.'r, with th.' .li.taiie.'s D .1, D'.', and I) f, .l.'s.rib.' th.' i„l..rs..,'li..n-. a, b, and c &c., and through a, b, .',&.'., -Iraw ih.' line abe ... n, and l.'l it b.' grant.'d that the cnrv.' line ub.' ... n is through an Inlinii.' Miu.ib.'r ..f points, mctuig th.' curv.- lin.' Bd.f ... hikD' in the point 11, and by construction is .-..luav.' to tin' ar.' iiB, an.l conH.'.iu.nllv th.' .'hord an inu>t b.' on th.' ..pposite sid.' ..f Hi.' .'inv.' lin.- ab.' ... n, to what that..! ilir.lu.r.lsVir and Vn"(Fig.5,) is to th.'ir enrv.' lin.'s \ p.| ... n an.l V p.| „" ■ Tli.'iv >liall be an ar.' or .'iirv.- lin.' siiiiih.r an.l -'.iiial t.. iIm' . miv,' lin.' B.l.'t '"' hikD' ihroniihB, betw.'.'i. th.' enrv.' B.l.'f ... hikD' an.l Hi.' aiv ..r.urM' lin.' H.l'sr ... ,|oK (Fig. 5,) s.. that on its int.r.'.pl.d ar.' b.'tw.'.'ii th.' ares 2'K an.l IT, lii.' .urv lin.' V,»l n' an.l its .h.ml Vn\ will b ih.' sain.' straight lin.',-als.. th.T.' shall b.' an .,r,. or .'i.rv.' lin.' similar an.l .'.inal to th.' .'urv lin.' B.h'f ... hikl)', tlirough B, b.- vv.'.n th.' ar.' ..r .urv.' liii.' Biuv ... .p/K', (Fig. 5,) on its iiil.'r.'-'pt.'.l arc, b.'tw.'.'ii hcar.'s2K' an.l IT', so that th.' .'iirv.' lin.' V'p'.|' ... n," '"hI its.li.)rd Vn will c - iw t oiue to Ih' '111 th.- sami' straiglit liii.'. F.M'it i-.'vi.l.'ii!, bv Ih.' .■on-liii.tinii iliat the curve line B.l.'f ... hikl) is th.- only .•omm.Mi ar.' whi.h .'i'ln b.' .I.'s.rib.'.l int.rs.'.'tiiig tlu' s.'ri.'s of ares IK an.l l"K', tliere- f,„',' ilHT.- .'aiinol b.' anv .'omiimn inl.r.'.pl.'.l ar.' but .Ik-bu1 llu'r.' mm\ be two inler- ,...,,ledarcs(L.'m. 5.) on whi.h th.' .nrv.'s Vp.i ... 11' an.l V"p'.|' ... n " w..nhl .'omcide with th.'ir clionls— lh.'r.'f..re th.' .mi.' niiist !..■ 1 are thnrngh B, between B.l.'i ... hikl)' an.l B.l'sr ... , to that ol th.' lin.' abe ... n I., its .'ti..r.l an.-Ther.' shall be points on • Til. i..l.r..rll.„, r' ,.>.,iml li. Iiitr ..I.M'mI hi, .1,. Hii.r. , »lll».Ht d. .rrltiln. It n.i n ..lurt, Inrr-f «..lr. the arc de, and c/", through which, if arcs be dascribed in the same manner, from the points B and U' as centers, and intersecting the arcs ka, and ib, that the intersec- tions a, b, and r, or the curve lino abc ... n, shall be on the chord an. For let llic point d move on the arc de towards the point e, and the point c move towards/, in like manrsr the point of intersection a will move towards a', and the intersection b will move towards b' — and let the point d arrive at the point e, in the same time the point e arrives at the point /—the point a, must equally arrive at the point a' and the arcs da and eb, coincide with the arcs ea, and Jh' ; but the points b and c have changed sides to the chord an, for the cur\'e line abc ... n, will now coincide with the curve line a'b'c' ... u — consequently there must be points of variation passed on the arcs de, and p/, through which if intersections be described from the centers D' and B, that the curve line abc ... n will become a straight line, or that the intersections a, b, c, will be on a straight line with the point n, and coincide with the chord an. LEMMA 8. From the center C of the semicircle ABDK, describe through the point of intersec- piQ, 7, tion s, the are ss', meeting the arc IT, in thi; point «', and from the same center de- scribe through the intersection r, the are rr', meeting the arc 2S, in the point r' — and from K as a center with the distances Ks' and Kr, describe the arcs s's" and r'r". — Tlu' point s", shall be the grt^atest variation of the point t' towards the point s, on the intercepted arc t"K of the arc UK, — and the point r", sliiill he the greatest variation of the point s towards the |)oint r, and the arcs ts" and si'", shall be proportional to the arcs lij, and 2A.* For by construction, the arc 1 2 3 ... 'l'3'2' on the arc BU, is the least intercepted arc by the series of arcs IK, and the arc t'K is the greatest of all the intercepted arcs of the radius AC, which can be descriliein\< n and V — the arcs u'u", and v'v". — The arcs tu'' and uv ", must he the grealesi variations of the point t towards u, and of u to- wards v, because llie greatest intercepted arc is tK' Now let the iuv BK, move on the point B as a center, until it coincides with the are Bl), it is evident that the variable eurviiini'ar triangles ts's, and srT, must vanish or at til'.- same linn'or heroine nolliiiig ; because the urcs s's, and i-'r, must always remain concentric to the are Bl),— and therefore liie \iiriationsi's" and sr", equally vanish, when tlie arcs S'S and r'r coinciile with the ar<'s \i and 23, on wiiieh arcs, the variations t'ome to be nothini;. In tlie same manner wlien the are BK' comes to coin- eide with lln' are BC, the viirialions tu" and uv" vanish; lor the ares til' and nv, are eai'h (oneenlrie to the are BC ; eonse(|uenlly the ultimate ratios of thr variiilions on ilie ares BK and UK' are ecpial ; and the variation ts" is to the viiriaiion -r", a-^ the \ a rial inn lu", is loilie viirialion 11 v"; also tin' nil iiiiate ratios, of ts and sr, are equal, and iherefure t's is to sr as the lu'c li is to the arc L'3 ; but the ultimate ratios of Is' and sr'', and the ultimate ratios of t's and sr an- equal ; for when (he ares ts and sr cdineide with the ares M and 23, llii' variations t's" and sr ' vanish at the same time ; iherelore tin' varialion t's" is to the variation sr", as ihe are Vi is to the arc 23. ^ l.F.MMA !» Bisect the are CD, in lln' polnl D', and ihrou;,.i the points of intersection (Li'in. 1,) y^^, ,j_ B, d, e, f, ... li, i, l<. !>'» •'fi'W the curve line Bdef ... hikl)', and lei the lines Btiiv .. q'o'K', and Bl'sr ... q<>K, be each di'scribed similar and equal to tlie line Bdef „. hikD'— and bisect the di.stance lU)', by the straight line All, inlcrsecting tho curve line in the point n,— Next from the point D', with the distance D'c' and l)/',e the ares re" and f f" J The point r" shall be the variation of ihe point d towards llie iMiint e, imd the lM»iiit /', shall be Ihe \ariation of the point c towards tlie point /, on (he iiitch'epteii nrc. dk, which is conmmii to the serii s of arcs IK mid T'K— I's" shall be to sr", ns tu" Is to uv", mid M lie'' ia to el". I • Titli U, iu|)|ioilnR Uiil the iirrt B», HK', IIK mid BC, iir« ilmntr ind rqiiil to «uh nlhw, nr of ll» nm* riilliii At. 6 For let the curve lines BK' and BK be supposed each to move on the point B as a t'enter equally towards each other, they must meet on, iuid coincide with the curve line BGD' — because the arcs KD' and K'D', are equal by construction, and the curvilinear triangles ts's and sr'r, and the curvilineai- triangles lu'u and uv'v, will coincide with the curvilinear triangles de'e and f//— also the i>rcs of variation ts" and tu", will eac'i coincide with the arc dc" — and the arcs sr'and uv'' each coincide with the arc e/''— hence their ultimate ratios are ecjual (Lem. 8) ; consequently the variation t's'' is to the variation sr",— as the variation tu" is to the variation uv'',--as the varia- tion de" is to the variation ef; therefore de" and ef" must be the arcs of variation, on the common curve line, or arc dk, of the point d towards e, and of the point e towards/. FIG. 7. LEMMA 10. From the point B as a center, with the distances Bk' and Bt", describe the arcs k'k'' and it\ — the arc kk", shall be the variation of k towards j, and the arc n" shall be the variation of i towards h. For from the point A, with the distances Ao' and Aq', describe the arcs ox' and qq' ; and from thi; point B as a center, witli the distances B(i' and Bx', describe the arcs (jci" and x!x' ; Also from the point C as a center, with the distances CK and Co, describe the arcs K/ and op,' and from the jioint B, with the distances Bp' and By', describe the arcs p'p " and y'y". Now let the curve line BK and BK',eacli move upon the point B as a center, to- wards the curve line BD',lhey nuist eacli coincide witli liD', (Lem. 9,) — hence the trian- gles q(i'o' and o'.r'K' and tlie triangles qp O, and oy K, will coincide, and be similar and equal to the triangles lit'i, and tk'k ; consequently their ultimate ratios are equal, and tiie variation Ky", is to the variation op", as the variation K'x", is to the variation o'q', asthe variation kk", is to the variation n", (Lrni. i),)--tiieref()re kk" must he the variation of the point k towards t, and «i", the variation of the point i towards h. FIG. 7. THE SOLUTIONS. Cask First. — When Ihc nrc Rdef ... hikl)\ i.s a ciirre linr, through intenections described through points of equal distances on the arcs BI) and BC, [Lem. 1, Fig. 1.) SOLUTION FIKST. PRoi'osrrioN i. TIIKOUKM. From the point B with the dislanci's Bk and Bi, and from the point !)', with the distances De" and 1)'/') descrii)e thi^ intersections a and b, and thningli the points a and b, draw the straight line aim meeting the curvi; line IKiU' in the poim n. — 'i'||,, slraigiit line al)n, shidl be that upon which, by variation of the jioints d nnd <>, to the points e'' and ./"" the curve line abc ... n (fig. (J), shall coincide willi its chord an. For as tlu- curves \"p'q ... n", and Vp(| ... n', are eaili upon the opposite side of iheirchords, to the curve a'b'u' ... n on the arc BI) (tig. 5) — it is evident that intersections described through the points of variation n" and v", and s" and r", (fig. 7,) nnisl give the curve lines on tlie same side of tiieir chord as Vpq ... n' and V'pc] ... n" — and there- fore bo aWo on the opposite side to the curve line a'b'c' ... n (fig. 5) — because tlie varia- tions lu" and uv" are proportional to tu and uv — and is' and sr" an- |)ropi>rtional to Is and , and iln'relbrc as the ctirvo line described l)y interseciions through tile points u' and v" sliidl vary, the arc HK' will move towards BC on the center B, and that ilescril)ed ihroiigh the |)oints s" and r", the arc BK, nuist move towanis BI) on the renter B, (Lem. l())~so tliat the 'urves of intersections descrihi'd through the points of variations u" and v", nnd h" imd r", will come to be, each In one straight line with tiieir chords; Now it is evident from their ultimate ratios being e(|ual, that the distances K'u" will becomn eiiual to Ks", and Kv" equal to Kr'', ami BK' ti\\m[ Ui BK, iiml Bfy rqunl to Bo'; but this ran only be possible nn the curve line BI)', on which is the interi'epied are dk common to both of the si'ries of nres IK and l"K', and upon ilk are the connnon variations dr" and *;/'", (Lem. !(,); consequently it is through the points r" and J" nnd h and i, in this rnor, that ih* intPrseetioMs deserilied from the points B and D', forming the curve line iih ,,, n will only coincide with the chord nn. PROPOSITION 2. THEOREM. Join the points A and n ; the distance An shall be equal to the perimeter of the FIG. 7. polygon of an infinite number of sides, or equal to the cireumference of the given circle "^^^Fo^whh Uie distance An, from A as a center describe through the point n, the arc nnN, meeting the arc BD in the point n'nN', and CD i.i the point N. Also from he poin; C, as a center, with ,he distance Cn, describe through n, the arc n nN meeting the arc BC in the point n', and CD in the point N' ; because the carve line Bdef .. hdcD is through intersections ofequal arcs on the arcs BD and BC (Lorn. 1,) the arc Bn , must be equat to the arc Bn ; but the point n is the point of ultimate intersection by construc- tion on tin- curve line Bdef ... hikD' which is through intersections ol tlie infinite series of arcs through the points l,2,3,...n, and through the Mifinite series o. points 2' 3' 4', .. n ; hence the point n, must be on the arc nN (fig. 5.) Also the pomt n 'is 'the point of uhimate intersection of the infinite series of points 1,2,3 ... n and of the infinite series of points 2\ 3\ 4> ... n', each nterseeting Hie curve line Bdef . hikl)' ; h..nec the point n must be on the arc n'N', (fig. 5), and conscquemly the distance An, must be equal to An (fig. 5), and equal to the circumference of the circle ABD, (fig. 4) ; but the part BG of the curve li.ie Bdef ... hikD is the arc of a circle (Lem. 2, fig. 3,) having its center on the right line ABG, in the point L, hence the point n is on the arc of a circle ; and tlie distance An is the determinate length of the circumference of the circle ABD (fig. 4). SOLUTION SECOND. PROPOSITION 3. THEOREM. From the point D' wltii the distances D'e and D'f, and from B, with the distances FIG. 7. Bk" and Bi", describe the intersections a and b', and through the points a' and b', draw the straight line a'b'n, meeting the circular arc lidef ... liiG, (Lem. 2), in the point n. Tiie point n siiall be on the intersections of the arcs nN, n'N, BG and the right line abn,-and tlie distance An, shall be equal to the length of the circumference of the given circle ABD, (tig. 4.) . . For the point k", is tlie variation of k towards i, and the point i" the variation of I towards li, upon the common intercepted arc dk, (Lem. 10); and in the same manner as demonstrated, (prop. 1), that tiie curve of intersections, described tlirouglr.he points y'' and p" and r" and q" iniiy l)e in one straight line with liielr eiiords, the arcBK must move towards HI), and BK' must move towards BC, (Lem. 10) ; llien It is evident thatthedlstaneeBo;" must coineto be ecpial to By", and Bif must he equal to Bp--and K'u must be equal to K's, and K'v must be eciual to Kr, for tiieir uhimate ratios are equal ; but this can only tie possible on the common arc dk--nnd therefore, (prop. 1 ), it is only through the points oC variation k'' and i", and llie points e and /, that the curve lln<' throut,'h the intersections a'b' ... n, in tliis case, will come to be on the same straight line wltii its chord a'n ; l)nt the point n is by construction the ultimate intersection of the line a'b' ... ti on the i'lirvu line Hdr/',.. hikD' of the Intersections described from H and D', llirougii llie infinite scries of points (',/... n, and of k", t"... » on the common arc ihrongli tlie intersections, of the series of arcs IK, and 1"K ; licncc the point :i uni>l be on the arc nN, and also on the arc n'N', and consc(|neiilly on the Intersection of nN and n'N' ; but the part Bdc/"... hiG, of the ( urv<- line Bilr/'... hikD' Is a cir- cular arc, (Lcin-2.) Therefore llie dlstanci' An, niUj*t be the determinate length of the circumference of the circle ABD, (Fig. 4.) SOLUTION THIRD. PROPOSITION 4. THEOREM. Let the straight line abii, be driiwn through the points of intersection n and b, FIG. 7. and u'b'n' drawn through tiio points a' niul b', (prop. 3), intersecting eacli other in 11. The dlfiiancc An, shall bo the dctorniinnte length of the cirrumferencn tlie ]) of till le ABD, (Fig. 4.) 8 For the point n is the ultimate intersection of the intersections of the lines ab ... n and a'b' ... n meeting upon the curve line Bdef ... hikl)', (prop. 2 and 3.) Now from the points A, describe through n the arc nn, meclingthe arc BD in the point M— it is evident tiiat the distance An is equal to ilic distance An, (Fig. 5.) There- fore the distance An will be equal to the deternunatc length of the circumference of the circle ABD, (Fig. 4.) The intersection of the straight lines ab and a'b' in the point D, is independent of the nature of the curve line or arc BD'— for let BU', be any curve Itne described between the circular arc BD', and tiie cur\-c line Bdef ... hikD, and describe the arcs BK' and BK, similar and equal to such curve line, it is evident tliat when the curve lines BK' and BK, come each to coincide with the similar and equal curve line BD'— the variations de" and ef'—\ik" and «", must be the variations of the intercepted arc dk of the curve line BD'— and as the intersection n, of ab and ab', must be the ultimate intersection on dfc, the point n consequently must be upon the intersection of the arc nN, (Fig. 5), and the curve line BD' ; therefore this solution must be inde- pendent of the nature of the curve line BD'. FIG. 8. Case Second.— H'Afn the ore Bdef... hikD', is the arc of a circle of the radius AC, described, through the poims B and D'—itUersecting either of the serie. of arcs \Kor 1"A only, {Lem. 3, Fig. 5.) SOLUTION FOURTH. PROPOSITION 5. THEOREM. Bisect the arc CD in tlie point D', and with tbr; radius AC, describe through the points B and D' the arc BD', intersecting the arcs IT, 2S, 311, 4'Q, 3'0 and 2'K, in the points de/"... ink. Then from the points C, with the distance Ce and C/' describe the arc ce' meeting the arc IT, in the point e' and the arc 2S, in the poiin /'. Then from C as a center, with tlie distances Ck, Ci, Cli, &c., describe the arcs 2'K', 3'0', and 4'Q', &.c.,— and from the same cenl'T with the distances Cd, Ce, and Cf, Sic, describe the arcs 1"T', 2"S', and 3'Il', &c.— Then through the points B and K', and B and K, describe the arcs BK' and BK, meeting tiic arc CD in tin- points K' and K ; also from the point D' as a center with tlic distaiucs D'c', and I)/', descrilic the arcs e'e" and //; "the point e" shall be the variation of d towards e", and/' the variation of c towards f. For in the same manner we have liy (Fig. 7,)— The arc tu" is the variation of t towards u, and nv" is tiic variation of u towards v on the arc BK'— and t's" is the variation of t' towards s, and sr" is the variation of s towards r, on tlie are BK. Now let the arc BK', move on tlic center B towards the arc BC, till it coincides with the arc BD— the variation tn" must coincide with tiic arc of variation , draw th(^ straight line abn, meeting the circular arc BD' in the point n, nnd join .\ and n ; tiic and the arc uv" is the variation of u towards V on the are IK' : also the arc t's" is tiic variation of t' towards s, and the arc sr" \h the variation of s towards r, on tl>e arc tK— and from D' as a center with the dis- tnnces D'p" and DT' describe the arcs e'e" and ff'—ihe arc dc" is the variation of d towards r, and ef" is the variation of e towards / on the arc dk, (Lem. 9.) The demonstration of this is the same as for PropoBiiion 5, FIG 9. 10 PROPOSITION 10. THEOREM. From the point D', with tlie distances D'e" and D/", and from the point B, will* the distances Bk and Bt, describe the intersections a and b, and through the points a and b, draw the straight line abn, meding the circular arc BO' in the point n ; then from C as a center with the distance Cn., describe through n the arc n'nN', and on the arc BD make the arc Bn equal to the arc Bn— and join A and n.— The distance An is equal to the circumference of the circle VBD, (Fig. 4.) For it is dcmonstra.ed (prop. I.) that the point n, is the point of ultimate inter- section of the line ab &c., or of all ihe intersections described through the series of points of variation e" and /", &c., and through the points k and i &c— through which the curve line ab ... n shall coincide with its chord an (Lcm. 3.) ; therefore the point n must br n die arc n'N' (Fig. 5.) ; but Bn is equal to Bn (Fig. 5.) — hence An (Fig. 9.), is etiual to the circumference of the circle ABD (Fig. 4.) FIG. 9. SOLUTION EIGHTH. PROPOSITION 11. THEOREM. From the point A as a center with the distances Aq and Ao' describe the arcs qq' and o"x, and from B as a center with the distances Bq' and B.f' describe llie arcs q'q'' and x'x" — also from C as a center with the distances CK ami Co, describe the arcs Ky and op' — and from B as a center with the distances B,i/' and B|»', dL'scribe the arcs w'u" and p'p' — and with th(,' distances Bk' and Bi, describe the arcs k'k' and ii" then from B as a center witii the distances Bk" and Bi", and from 1)' as a center with the distances Dc and 1)/, descril)c the inteisections a' and b', and through the points a! and b' draw the straight line a'b'n meeting the circular arc BD' in the point n ; then from C as a center through the point n, describe die arc- niiN , and on the are BI), make the ar. Bn ecpial to the arc B«', and join A and n. — The distance An is equal to the circumference of the circle ABU (Fig. 4.) The demonstration of this evideinly must be the same as prop. 10. SOLUTION NINTH. PROPOSITION U. THEOREM. Let Ihe straight line abn bo drawn through the points of intersections a and b, FIG. 9. and Ihe straight line a'b'n drawn through the points a' and b, intersecting each other in the point n — also from the point C as a center, through n describe the arc n'niV' meeting the arc BC in tli(! point n, and liic arc CD in the point A'^, — aiiii on the arc BD make the arc Bn ecjual to the arc Bn , and join .1 and n. — The dislance .In shall be equal to the determinate length of the circumference of the Circle ABD, Fig. 4. For the point n is the ultimate intersection ol tiie intersections ol eacli ol the lines ab ... n and a'b' ... n, upon the circular arc Bdef ... litkD' (prop. 2 and 3), and the points (],(;/... h,i,k, arc the intersections of the infinite series of arcs I'T', 2"S', and 3'R', &e., and of the inlinile si.'rics of ares 2"K', 3'0' and 4"(i', &e. — hence the point n must be upon the arc n'nJV', Fig. 5, but Bn' is by construction equal to Bn, Fig. 5 — hence Bn, Fig. 9, must be ecpial to Bn Fig. 5, and An Fig. 9 is c'ip* 4 JE Fio--^. Je 7k - fh-r' (,htr/f/"'(f/f>i r,r^'/'>iA.'^/o •/'/■'I'ffmif?. /y.i>. I /;,r'0/Mr/'''^/W/r fn 'Vh/Z'^'y/'AV'-^/inir/ M(\ Fig 5 7V.i>. 'T PI o. (Irr 'Qiimr:,ftk rir''/)ih''fn /'//aw///// y/,nr/,M'.'ir. Pip^.d PI. 5. (iff 'f/n,i''fn /'/■/fminf/. .\fffrr/,M.'if\ V- i \ /V.5«. (>C( .(>>"> 'f rum ( ' March mo. Vi'd ' *7 /V.i« ('( >v ''>'«"-'^"'^'^"'^^r;:i/*;.. f^lr/ >-VN \ r Mfttr/, /-^'P y'Kj.8. /v. 5. (/'n'{>ihi(/'Wf/i' (' i r.' / 'n />'' />\ /.'/Vrmi/f// Mftrr/, /^^P «jr- "^^•^■;^$::^ P/.6. A/,nrAM7/> r\