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 1 
 
 2 
 
 3 
 
 1 
 
 2 
 
 3 
 
 4 
 
 5 
 
 6 
 
m. 
 
 
 

 ^'fc^f 
 
 ^V^E ClRci^ 
 
 AND 
 
 STRAIGHT LINE, 
 
 l(i^^- 
 
 PART THIRD. 
 
 = ^ 
 
 slip 
 
 iii,.. 
 
THE CIRCLE 
 
 AN'D 
 
 STEAIG-HT LIIsTE. 
 
 PART THIRD. 
 
 BT 
 
 JOHN HARRIS. 
 
 MONTREAL: 
 JOHN LOVELL, ST. NICHOLAS STRKET. 
 
 MARCH, H74. 
 
 :«|* 
 
Knteml nccording to Act of Parliament in the year one thousand f ipht hundred 
 and eevcnty-iour, by John Hakius, in the otTice of the Mini.-tir of Agriculture 
 and Statistics at Ottawa. 
 
 ^^^ £i I'm 
 
 V 
 
 'b 
 
 MoNTUKAL— John Lovbll, PniNXEn. 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 l*r()\ e <i!/ tilings ; hold fast that which ia gooi,' 
 
 St. P'nil. 
 
 PART THIRD. 
 
 THEORV OF UURVATl-RE. 
 
 Tht UUimatc-Siu(\ und the arc of incrcas'mg cnrvafin-r. — 
 If ii iiiiniln'f of intiTuiodiaie nres be described with a 
 proportionally increased radius, between 31. an<l )i., 
 between h. and o., between o. and ^)., &c., &c., respec- 
 tively, it is evident that a continuous curved line drawn 
 through the terminal points oi' all those arcs, will form 
 a compounded arc of a peculiar character and which 
 fioin its forming a connection as it were between the 
 arc-length of the half-cpiadrant and the ultimate sine — i.e., 
 the sine of the ultimate fraction of the half-quadrant — 
 possesses much interest. We will here briefly indicate 
 one mode in which the change in longitudinal magnitude 
 in conse(|uence of the elimination of curvature, as the 
 fractional arc is diminished by successive bisections, may 
 be investigated. 
 
 Taking the radius eqmd to 10. The length of the half- 
 quadrant equals the sine-length increased by addition of 
 ' the sine length divided by the units of radius less one.' 
 
 S 
 
 ^: = .s 
 
 4- 
 
 K-i 
 
 We thus obtain the magnitudinal length of the arc 
 in units of radius. Now if this arc (half-quadrant) be 
 bisected and the lonijitudinal magnitude of the remain- 
 ingarc be duplicated, the half-quadrant is not reproduced 
 
 .*v 
 
6 
 
 TIIK CIRCM-: AND STKAIOIIT LINE. 
 
 but nil arc is obtaiiuHl into whitli one half the curvature of 
 tlie half-(|uadraiit does not enter and in which the other 
 half of the uirvature beloni,'inf» to the half-(|Uadraiit is 
 divided over twice the quantity of linear extension.* The 
 second arc therefore, although it is a inagnitudinal dupli- 
 cation of the one-half of the first arc, is a lesser longitu- 
 (hnal magnitude than the lirst arc, because the niagnltu- 
 (Hiial (hiplication which has restored the direct linear 
 extension has not replaced the curvature. 
 
 The process may be thus conducted: — 
 ^^ — The arc-length equals the sine-length increased by 
 S addition of the sine divided by the units in the 
 radius less one. 
 
 C . — The arc-length equals the sine-length increased by 
 J(» addition of the sine divided by the units in the 
 (duplicated) radius less one, multiplied by two. 
 
 C. — The arc-length equals the sine -length increased by 
 '32 addition of the sine divided by the units in the 
 (duplicated) radius less one, multiplied by two, multi- 
 plied by four. 
 
 (j, — The arc length equals the sine-length increased bj' 
 64 addition of the sine divided by ihe units in tho 
 (duplicated) radius let^s one, multiplied by two, multiplied 
 by four, multiplied by four. And so on, proceeding in like 
 manner. 
 
 The successive 
 divisors will 
 be therefore. 
 
 f 1st arc 
 
 ( ^-1) 
 
 2 do 
 
 (2/?-l) X 2 
 
 3 do 
 
 (2/«?-l) X 2 X 4 
 
 4 do 
 
 (2 7e-l) X 2 X 4 X 4 
 
 5 do 
 
 (2 /?-l) X 2 X 4 X 4 X 4 
 
 , 6 do 
 
 (2 7?-l)x2x4x4x4X 
 
 *If the liaU'-(ina(Jrant, instead olbii^ected, were to be partially straiglit- 
 oiied into the same form as tlie lialt'-arc, it would exceed in lengtli 
 the diipulicated half-arc by the difference of the curvature, (i.e.. 
 the difference between the curvature in the half-quadrant and in the 
 arc of 22 J degrees.) 
 
THE CIKCLE -wsl) STHAKSHT LINK. 
 
 Taking the Bucocssive einc len^'ths from tlio talilo given at 
 pa^'e 54, Part yocoml, we obtain : — 
 
 E 
 
 B 
 
 D 
 
 J) 
 
 707107 
 
 f 
 
 •785074 
 
 ;{H 
 
 705306 
 
 + 
 
 •201413 
 
 152 
 
 7S03G1 
 
 + 
 
 -051339 
 
 008 
 
 7-841.37 
 
 h 
 
 •012897 
 
 2432 
 
 7-85082 
 
 h 
 
 •0032281 
 
 1)728 
 
 7-85319 
 
 + 
 
 •0008073 
 
 38'.) 12 
 
 7-85379 
 
 ■f- 
 
 •0002018 
 
 155648 
 
 785393 
 
 -t- 
 
 •0000504 
 
 022592 
 
 7-85390 
 
 -f 
 
 •0000120 
 
 1 R- 
 
 1 
 
 l R- 
 
 •J 
 
 10 R- 
 
 8 
 
 04 R - 
 
 32 
 
 250 ye - 
 
 128 
 
 1024 R~ 
 
 512 
 
 4090 R - 
 
 2048 
 
 10384 R- 
 
 8192 
 
 05530/?- 
 
 32768 
 
 = 7850744 
 = 7855080 
 = 7 •854951 
 
 = 7-8545!)7 
 
 - 7-854055 
 
 - 7-8539!i7 
 = 7853991 
 = 7-153984 
 = 7-853978 
 
 III tins tablo, tlie divisors in the first column E, result 
 li-()ui the last column D. Under A. are tiie sine lengths. 
 Under IS. are the <|Uotient8 of the division of the sine 
 lengths by the numbers under E, which (inotients adth-d 
 to the sine lengths yive the arc-lenirths in column C. 
 
 It becomes evident that, by continuing the table, the 
 (|uantities in colunui B., which represent the curvature, 
 would beconie continually less, and the quantities under 
 A. and C. would approximate more closely. In reference 
 to column B. it may be observed that the third quotient 
 is rather greater than the fourth part of the :2nd, that 
 the 4th is a litt.e more than the fourth part of the ;ird, 
 the 5tli a very little more than the fourth part of the 
 4th and that, of the remaining quotients, each is the fourth 
 part of that preceding it, so far as the figures are carried ; 
 a greater number of decimals would show each of the suc- 
 cessive lesser quantities to slightly exceed the one-fourth 
 of the quantity preceding it, but the approximation to 
 the one-fourth becoming continually closer as the process 
 is carried further. To appreciate the meaning of this it 
 is necessary to remember that the chord of each arc, 
 which chord equals the sine of the next succeeding arc, 
 is the square root of the square of the versed sine added 
 t© the 8(iuare of the sine ; and that the versed sine be- 
 
8 
 
 THE CmrLE AND STRAIGHT LINE. 
 
 loiigititf to ejicli arc is always greattM' than the ono-lialf of 
 that bolonijiiiu; to tlie arc precediiii? it ; but m the ter- 
 minal ratlins approaches the perpendicnlar (i.e., as the 
 cnrvature becomes more nearly eliminated), the versed 
 sine of each arc approximates more and more closely t« 
 the one-half ot that belonging to the arc preceding it. 
 
 The process thus snggestetl is put forward as worthy of 
 attentive consideration, and for the purpose of illustrating 
 the subject; but we wish it to be particularly observed that 
 (for the present) it stands by itself, nothing else is based 
 upon it ; it is not put forward as a demonstration, nor can 
 we (at present) vouch for the correctness of the process 
 or for the accuracy of the figures. 
 
 Thk akc of INCREASING CURVATL'RE — This arc is dis- 
 tinguished from the arc of the arc-length (that is from 
 an arc described with the length of the arc as a radius), 
 and illustrated in Figs. 21 and 2'2. In Fig. iil, XX 
 represents the arc described with the arc-length for a 
 radius, and S. n. o. p. the arc of increasing curvature. 
 In Fig. 22, the relation of the same two arcs to the 
 construction of the circle is illustrated on a larger scale. 
 
 {Note. — With these Figures compare Figs. 20 and 25) 
 
 The TANGENTIAL LINE B. T). 
 
 (Fig. 23 is a development of Fig. 12.) 
 Fig. 23. Bisect the radius A.B. in «., and with centre 
 a., and radius a.B. describe the quadrant B.K. ; 
 bisect the quadrant in m. and bisect the half-quadrant 
 B.m. in '«. ; through "«. draw a.n. intercepting the line 
 B.D. at n. Bisect also CD. in b., and w-ith radius b.D. 
 describe the quadrant D.K.; bisect D.K. in 31. and bisect 
 the half quadrant D.M. in 'iS^.; through 'N. draw h.N. 
 intercepting the line B.D. at N. And bisect the arc B.S. 
 in jj., and through j). draw A. p. With centre C. and 
 radius CD. describe D.A. bisected by the line R.T. in Z., 
 bisect D.Z. in 'P. and through 'P. draw CP. intercepting 
 IID. at P. 
 
 *.. 
 
TIIK CIRCLE AND STRAIGHT LINK. 
 
 9 
 
 W*. now have a division of the (lefinitc line B.f). into 
 rortaiii known (|uantitivo niagnitnih-s. Ik'caiisc JJ.u. is 
 tilt* tangent of the are /?.'«., an<l JJ.p. is the tanj;«'nt of the 
 similar gn-ater arc //.';>. wliich is \\\ ce the niagnituch' 
 of li.ii., J!. p. is twiee tlie lenirtli of 7/. /<., and n.p. is 
 equal to Ji.n. At the o[)posite extremity tlie similiir 
 points D.X. and D.P. are similarly related, and X.J*. is 
 (Mpial to I).X., which iserpial to Ji.». 
 Also because ij.7 is re(|uivalent to) the ^ ne of ]>.)»., 
 B.fj. is the half of Il.W. which is (e(|Mivalent to) the 
 sin«» of the similar greater arc U.S. 
 Taking the radius A.Ii.- 10, the numerical values will 
 
 he 
 
 li.u. =-J'07H)7 
 n.p. -i'14:>l4 
 B.q. -7-07107^:.' 
 
 iAX-10-:2-0710()S 
 
 IKK, 
 D.X. ^-- 7i.p. 
 
 D.r. 
 
 •so7SG4:-i>.A'. 
 
 Having in mind the relative magnitudes of the primary 
 lines determined by trigonometry, these values present 
 themselves at once as evident; we may then obtain, from 
 and by means of these, the rpiantitive values of other 
 lines, and may also, thus (pumtitively, detenuine the 
 geometrical (magnitudinal) relation of other lines; that is 
 to sav — the inductive reasonini;, by means of which 
 further knowledge of the relation of the parts of the 
 circle to each other is to be obtained, may be based upon 
 the facts belonging to the science of 'Nundjer and Quan- 
 tity' in place of those belonging to ' Fokia and ^lagintude. 
 For example : — 
 
 Because B.n = 2-07i07, B.K=^o^ and B.W^l-OllOl 
 Therefore K.n.= W.D. = S.W. = S.T.^K.X. =2-92SiK5 
 And, 2'9289:J x 7-07107 = 20-7107.... 
 Therefore, S.W. x B.W. - B.n. x B.D. That is— 
 
 the versed sine x the sine of the hulf-quadrant = the tangent 
 to the arc of 22^" x liie tangent to the half-quadrant. 
 
 ^^ 
 
r 
 
 f 
 ¥ 
 
 mi ! 
 
 10 
 
 niK CIRCLE AXD STKAIGHT LINE. 
 
 Rerause K.p. = B.K. - B.p. = (o - 4- 14-? 146) ^ '857804 
 therefore K.p^ 10- {OK.p - (A'.//)* = '^•02893*= 8"57S04 
 
 And P.iK = -2 K.p = 1 -7 lo7if. 
 
 Hut the sqvuiie of'4-14-2136= 17-lo72 
 
 Therefore P.p. x \0^-{S.DY=iB.py^{D.P)% &c., &c. 
 
 B.P== B.K + K.P= o-8o 7804 = 2 WD^ 2Kn. 
 
 Now the tangent to the half of the bisected half- 
 
 (piadrant duplicated, = 8-28428= 2 B.p = 2 B.P. And 
 
 (2 B.p-B.W) = {K.n-2K.p.) = (8-28428- 7-07107) = 
 
 (2-92893-- 1-71572)= 1-21321, Ac, &c. 
 
 A fact of a primary cluiracter, which has not, we 
 believe, been as yet applied in the art of computation, or 
 otherwise utilized, is exhibited in the following: — 
 
 TiiKOKKM. — If the quadrant of a circle be bisected and 
 the secant to the half-quadrant be drawn through the 
 point of bisection; and if the radius be produced through 
 the centre of the circle and the produced radius be made 
 twice the length of the origintd radius — that is, equal to 
 the diameter of the circle — and a line be drawn joining 
 the extremity of the produced radius and the extremity 
 of the tangent ; then shall the sine of the arc cut off from 
 the quadrant by the line so drawn have to the tangent of 
 tiie half-quadrant the ratio of four to five. 
 
 Fig. 24. With centre A. and radius A.B. describe the 
 (|uadrant B.C., and bisect B.C. in the point S. Draw 
 the tangent B.B. and secant A.D. Join D.C. and A.C. 
 Produce the radius A.B. through A. and make E.B. the 
 production oi'A.B. double the length of ^l.D. Join U.D. 
 intersecting the quadrant at K. and bisecting the line 
 A.C. at g. Bisect the line CD. in the point d. and 
 Join B.il. intersecting the quadrant at the same point K. 
 From /iT. draw if. ^i^. perpendicular to Z?.Z). and intercept- 
 ing B.D. at Q. From Q. draw Q.II. parallel to A.D. and 
 intercepting A.B. at the })oint //. From H. draw H.P. 
 parallel to A.C. and equal in length to B.Q. Join P.K, 
 
FIG. 2J^. 
 
 / 
 
^>'< «rRAiOiri i,iM". 
 
 — ^H 
 
 csiftr^ -ifv? 'eue;th r,i' B.D., mux I-t.J>. 
 ■m>u}^ B.d.D. ih simihir to 
 . v'iWiirU.'s B Jut 8i;ri k'J:.D 
 rli^M'ofore xhe l>n»» Ji.ti. is 
 
 ,; ' > I • < V r- ■ ! I (. < ■ ."- 
 
 / 
 
 !roJ: 
 
 .<. -."U 
 
 /-'. at rv*. 
 
 /(./^ 
 
 iV '. 
 
 ^m 
 
 f^'i 
 
. k1**WI i' tU»1J«^ 
 
 
 \. * 
 
 "^ .,1 
 
 -.ii* X 
 
 ; f t. 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 11 
 
 Because E.B. is twice the length of B.D.y aiul B.D. 
 twice the length of d.I). the triangle B.d.D. is similar to 
 the triangle -fc'.D.^. But theang.es B.D.d. and E. B.D. 
 are both of them right angles, therefore the line B.d is 
 at right angles to the line E.D. Now since fl.Q. is 
 parallel toA.D. and H.P. is parallel to A.C. the triangle 
 H.P.Q., is similar to the triangle A. CD. And since the 
 line B.d. bisects CD., the same line B.d. which inter- 
 sects P.Q. at A', also bisects the line P.Q. in the point 
 K. [And the line B.K. (part of B.d.') has been shown to 
 be at right angles to the line K.D. (part of E.D.) tiiere- 
 fore the triangle K.D.Q. is similar to the triangle B.K.Q. 
 and K.Q. is the half of P.Q.I And P.Q. is (by the con- 
 struction,) equal to B.Q., therefore ^.^.is equal to twice 
 K.Q., and K.Q. is equal to twice D.Q.; therefore the line 
 B.Q. has the same ratio to the whole line B.D. as 4 : 5. 
 Draw K.e. the sine of the arc B.K. K.e. is manifestly- 
 equal to B.Q. 
 
 Problem. — Upon a given straight line it is required to 
 (Ascribe an arc containing 45 degrees, such that if the 
 tangent of the arc be divided into five equal parts, four 
 of those parts shall be together equal to the given straight 
 line. 
 
 Fig. 24. — Let B.Q. be the given straight line, it is 
 reciuired to describe the arc upon B.Q. From the 
 point Q., perpendicular to B.Q., draw Q.P., equal 
 in length to B.Q. ; and from B., perpendicular to B.Q., 
 draw ^.77., also equal to B.Q. Join 77.^. Bisect P.Q., 
 in the point K. and join B.K. Produce B.Q. through Q. 
 and produce B.H. t' -ough 77. (Through K., at right 
 angles to B.K, draw K.D. intercepting the production 
 of B.Q., at D. From 7). parallel to Q.H.* draw D.A., 
 intercepting the production of B.H. at A. With centre ^., 
 
 • Or, join B.P., and froni D. at right angles to B.P. draw JJ.A., 
 intercepting the production oi' B.H. at A. 
 
 
12 
 
 THE CIRCLE AND STliAlOHT LINE. 
 
 and radius A.B., describe the quadrant B.S.C., inter- 
 sectingD.^l. in the point S. JB.S. shall be the required arc. 
 The demonstration to this solution is substantially 
 contained in the demonstration to the theorem.) 
 
 Bisect K.Q. and, through/, the point of bisection, at right 
 angles to B.P., draw D.A. intercepting the production 
 of B.Q. at D,, and intercepting the production of B.II. 
 at A. With centre A, and radius A.B. describe the 
 quadrant B.S.C bisected by the line D.A. in the point 
 S. B.S. shall be the required arc. 
 
 Because B.Q. f. is a right angle and /. bisects ^•^. 
 QB. is equal to the one half of K.Q. But K. bisects 
 P.Q.. and P. Q. equals B.Q., therefore Q.B. is equal to 
 the one fourth part of BQ. Wherefore if B.B., the 
 tangent of the arc B.S., be divided into five equal parts, 
 four of those equal parts, shall together contain the given 
 straight line. Q.E.D. 
 
 We have now to make known a discovery which will 
 fjicilitate the investigation of the structural characteristics 
 of the circle and which may become hereafter of much 
 utility and value in simplifying trigonometrical (cyclome- 
 trical) processes. We propose for the present to treat 
 it as a fundamental and, in a measure, independent fact, 
 belonging to the plan of the circle. The discovery is 
 stated in the following: — 
 
 Theorem. — That if the radius of the half-quadrant be 
 produced through the centre of the circle until it become 
 equal to the versed sine (of the half-((uadrant) multiplied 
 by ten, and with this radius an arc be described terminated 
 by a line drawn from the centre of the greater circle — 
 i.e., from the point at the (central) extremity of the 
 radius — through the point at the extremity of the half- 
 quadrant, the tangent to the arc last described (of greater 
 magnitude), cut off by the line so drawn from the tangent 
 of the half-quadrant, will be equal to the arc-length of 
 the half-quadrant. 
 
FIG. 25. 
 
 » 
 

 ■iwLiiiVt:;. !"i 
 
 r I 
 
 I ■ > 
 
 >.X. 
 
 mi 4. 
 
 ■idil 
 
 - *f ■ u* It ■> 
 
 -V- i'*.-- T: f , --jiv ^v^ « 
 
 •^i^Jffc** 
 
 :f 
 
 :r 
 
 ^if*^ 
 
 . S*.*! 
 
'' ''f^. 
 
 mt-*v 
 
THE CIKCLE AND STHAIGHT LINE. 
 
 13 
 
 I' "'•»»■. 
 
 Fig. 2.5. — Let B.S. be the lmlf-(|iia(lrn!it of which 
 A.B. is the radius ; produce A.B. through A. and make 
 H.B. ecjual to ten times S.W. Froiu JL through S. 
 (haw li.X. intersecting B.D. in the point X. B.X. 
 shall be equal to the arc B.S.* 
 
 Bemonstmtion. (Quantitive). Th«' length of the sine, »S'.iSr. 
 the arc -length of B S., the length of the radius 11. B. 
 and of the versed sine S.W, are known, 
 we have to obtain tlie length of B.X. 
 The radius, A.B.= \0. 
 '* V. sine, S.W. =-2-9-2^}>:3. 
 *• radius, /?. R - 10 S. II'- iO-OS!):}. 
 7?..Y. = B.B - S. W = (20-2S9:]-i>-9-?S9:3) - 20'3G0:3. 
 
 Now the lesser triangle S. W.X. is similar to the greater 
 triangle 7?..Y.*S'. 
 
 Therefore W.X: X.S.: : S.W.: B.N. 
 
 W.X. : 7-07107 : : Q-'JQS'JS : 20'3(iO:3. 
 IF.X- •78507 B\MN.S. + W.X. --= B.X. 
 Therefore, 7-07107 + •7S567 = 7-So074. 
 
 And B.S. the arc - length also ec^uals 7-S5G74. 
 Wherefore the tangent B.X., of the arc of greater mag- 
 nitude is equal to the arc length B.S., of the half 
 quadrant. (Q.E.D.) 
 
 Corollary. Hence manifestly TF.jir= the difference of 
 the sine and arc-length of the half-quadrant. 
 
 The Ultimate tangent — We have objected to Legendre's 
 circumscribed polygon — which eventually almost coincides 
 with the sine — that it cannot be a simple continuous figm-e 
 but must be fragmentary and compound ; and we based 
 this conclusion primarily and directly on the manifest 
 necessity that a circumscribed polygon if continuous must 
 be greater than the circle which it surrounds. We propose 
 now to show that Legendre's circumscribed polygon is 
 in fact compounded of the minute ultimate tangents 
 
 ■K Or :— From X, through S, draw X- Y.li., intercepting the 
 production of B.A., at B. B.B. shall be ten timea the length 
 o(S.W. 
 
 'l^'Sf-*- 
 
14 
 
 TIIK ( IIMI.K AND STUAIfSHT I.IXE. 
 
 s»M>iUMt«' from oiU'li otiior hut arranjjpd together in the 
 form of a polvsroii. To show this we will consider the 
 question whether the tangent to an arc, if thaturc be an 
 indefinitely small fraction of the circle, is necessarily 
 greater than the arc itself. It is at present held that the 
 tangent is always greater than the arc by the evidence 
 of trigojiometry, thus : — Let A.B. be the arc, and A.d. 
 
 Fig. 2G. 
 
 the tangent. The arc is bisected ; and from the point ?, 
 e.B. is drawn perpendicular to U.B., intercepting R.B. 
 at B. Now the triangle d.e.B. is similar to the triangle 
 d.R.A., and the side d.c. is greater than the side B.c. of 
 the lesser triangle, in the same ratio that the side d.R. is 
 greater than the side A.R. of the greater triangle. Since 
 evidently so long as it is possible to assign any quantity 
 of magnitude to the remaining arc, it is possible to 
 imagine the same process to be performed, it is supposed 
 to follow that, however small a fraction of the circle an 
 arc may be, so long as there be an arc, the same reason- 
 ino: demonstrates the least tangent^belongins: thereto to 
 be greater than the arc, because it is thereby shown that 
 the outer tangent (as A.d.) is greater than the two 
 
THK ClUCLK AND SrUAKlIIT I.IXi:, 
 
 16 
 
 intorior tnngeiits (as A.r. iiiul r.H.) taken together; 
 nnd it is nssuniod as niaiiifest that the two interior tangents 
 taken together are always greater tiian the are. 
 
 We submit that such conclusion is sliovvn to be erro- 
 neous by the following : Let A.B. be a given straight 
 line. At each extremity of the line draw a perpendicu- 
 lar A.B. and S-f. {t. being the same point as B.) Let 
 
 Fig. 27. 
 
 the same line A.B. be supposed to be curved* into tiie 
 form of an arc containing a very small fraction of the 
 circle. 
 
 * If the cxpre.-^siuii * licmiiiig' or ' curvinji the line ' he ohjected to, 
 then, let a fractional arc equal in length to the given .straight line, 
 and bclongingtoa circle of any very great niagnituile, be applied upon 
 the straight line A.B. in suth \vi.se that the point at one extremity of 
 the straight Vu-.'! f^hall coincide with the ]ioint at one extremity ot the 
 arc, dec, &c. 
 
H\ 
 
 THE CIRCLE AND STRAIGHT LINE. 
 
 It is manifest that the line being curved will be less 
 in direct length, measuring horizontally between the 
 perpendiculars, than when it was straight, and, although 
 the same line and of the same actual length as when 
 straight, it will not now extend quite so far from A. 
 as to reach the line S.t. Let any point h., less distant 
 than S.t. from A., by a very small quantity of space, be 
 the terminal extremity of the arc. It is evident that, 
 however small the distance of the point b. from the 
 line S.t., if a line be dravv-n from the point f. through h. 
 and the line so drawn be produced until it meets the 
 production ofA.Ii. througli B. then, if the line so drawn 
 may be correctly considered the secant of the arc, the 
 tangent is A.t. and is not greater than the arc. The 
 question, therefore, assumes the fonii : — whether the line 
 T.t. drawn through the point b. at the extremity of the 
 arc, is at right angles to the extremity of the arc. If it 
 be not at right angles, the tangent may be greater than 
 the arc ; because the point /.. at the extremity of the secant 
 might then fall beyond tiie distance B. from A. On this 
 question, however, we have the evidence of trigonome- 
 try as given in the polygon computation of Legendre 
 (already quoted) and others of the like kind, and which 
 evidence is decisive because it is shown thereby that 
 ultimately the tangent-length almost coincides with the 
 sine-length, and, since the sine-length is manifestly less 
 than the arc-length, it is evident that in fact the 
 extremity of the secant to the ultimate arc does not fall 
 beyond the arc-length on the tangential line, and, since 
 the ultimate tangent-length cannot be less than the arc- 
 length, it appears safe to conclude that the tangent-length 
 actually coincides with the arc-length of the ultimate arc. 
 
 Let D.E.A.F.B. (Fig. «) be four equal divisional parts 
 of the half-quadrant D.B. Fig. b. represents one of 
 those equal divisional parts on a larger scale, the straight 
 line being equal to A.e. Now if we apply four straight 
 lines each equal to A.e. as tangents to the divisions of 
 
 
 ^:^.i^~ 
 
THE TANGENT. 
 
 Fig. a. 
 
 Fig. b. 
 
 Fig. c. 
 
 U'C. 
 
 \rts 
 
 of 
 
 Ight 
 
 Is oi 
 
 jii'i 
 
 iiiii^ 
 
 ,l,i;,:!!Tli;ii 
 
 lllllllllllllllill 
 
 Fig. d. 
 
 MIR 
 
 B 
 
 ^:'^ 
 
i^i-^ 
 
 P 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 17 
 
 the luilf-((uaclrant, as shown in Fig. a. it is evident that 
 they cannot meet each other at the extremities which are 
 not in contact with the circle. If they had sufficient 
 length to meet each other, angles would be formed at the 
 points E. and F., but, the length being insufficient, spaces 
 are left at those points. Supposing the four lines to be 
 again divided by bisection, and the eight half-lines to be 
 arranged as tangents to the circle, there will be four 
 spaces instead of two where the extremities of the 
 lines will not quite meet each other, but tliey will be 
 much more nearly in contact than before. If the lines 
 be in like manner repeatedly divided, there will be 
 eventually obtained a polygon, formed by the ultimate 
 tangents which will then be in contact, the extremities 
 of each with the extremities of those next thereto. "VVliat 
 are the conditions of that contact ? If we accepted the 
 dogma of Euclid, we should have to consider the contact 
 as complete and the polygon as a simple figure, and to 
 accept the sum of the ultimate tangent-lengths as the 
 measurement of the circle ; but we reject the dogma and 
 insist on a real circle containing area and, consequently, 
 containing breadth. 
 
 We have, a< 'ordingly, to enquire how the lines are 
 divided. Is the section oblique or perpendicular to the 
 length of the line f Now this is readily answered — because 
 the ultimate tangent-lines, derived from continued 
 bisection, are divisions of a straight line evenly divided 
 at right angles to its length. Therefore, by placing tlie 
 divisional lines of Fig. a. in the positions, relative to each 
 other, which they would occupy as tangents to the circle, 
 and considering the rectangular extremities of the one 
 with reference to the rectangular extremities of the next 
 the actual conditions of the contact between them will 
 become manifest. Fig. c. 
 
 The fragmentary tangent-lines compounding the 
 circumscribed) polygon are, tlierefore, in contact with 
 respect to the inner surface only, but, if a straight line be 
 
 ■^■ 
 
 '■r. 
 
18 
 
 THE CIRCLE AND STRAIGHT LINE. 
 
 compounded of the fragmentary lines, the contact will 
 be complete — that is, perfect with respect to both sur- 
 faces — and the original straight line will be restored. 
 
 This case may be taken by itself. The fact of the 
 ultimate arc equal in length to its tangent, and of the 
 half-quadrant greater than the sum of all its ultimate 
 tangents, presents itself for acceptance. That it is indis- 
 putable and must be accepted seems to us evident. But, 
 what then : if there be some persons to whom this fact 
 appears irreconcilable with the facts (as now taught) of 
 trigonometry 1 
 
 In concluding this treatise it may be not out of place 
 to consider a little the meaning of the familiar statement 
 (phrase) that one fact cannot contradict another. Proba- 
 bly no one at the present time with any pretension to 
 scientitic knowledge or education will directly deny or 
 expi'css any doubt as to the truth and certainty of such 
 statement in its simple form ; but the question is as to 
 absolute certainty in the strictest and most comprehen- 
 sive sense ; and in the writings of men of scientific 
 reputation, at the present time, opinions are expressed, 
 theories propounded, and statements made, which, if 
 referred inferentially to their basis, make apparent that, 
 in the minds of the writers, no clear and reasonable 
 conviction has been arrived at that, as a scientific cer- 
 tainty, one fact may not or cannot contradict another ; 
 i.e., be otherwise than in harmony with other facts. 
 We call attention here to this circumstance because we 
 believe that, in many cases, if the writer distinctly 
 understood the nature of the conclusions involved or of 
 the reasonable inferences belonging as corollaries to those 
 theoretical opinions and statements, they would ht 
 reconsidered and, being found to be um'easonable, wouM 
 not be put forth. This circumstance, which must needs 
 be regarded as of the gravest importance, may in 
 innt ;it least, be a consequence of tlie increasing neglect 
 
 I :i 
 
 ^"•■?h 
 
 ^ 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 10 
 
 of mjithematical reasoning — by which we mean, strictly 
 lawful and (therefore) correct reasoning — as a necessary 
 element of educational training. A consequence of the 
 want of such training is a non-recognition by the mind 
 of the necessity of a primary basis, either securely 
 established or undoubtinTly believed in, upon which 
 the theory or statement must be actually or piesiimably 
 based. Practically the effect displays its'?irin the opinion 
 now more or less openly expressed by scientific teachers 
 that Theology is not an absolutely necessary and funda- 
 mental part of Science, but that, on the contrary. Science 
 and Theology, if not actually antagonistic, should be kept 
 quite distinct and separate. 
 
 This separation of Science from Theology appears to 
 be concurred in, and even supported by very many of 
 those who teach that the fundamental basis of Science 
 must be found in Theology. They, nevertheless, accept 
 the title of theologians, as distinguished from, and as 
 in some degree antagonistic to that of, men of Science. 
 Moreover the dislike of the man of Science to Theology, 
 is perhaps, more than reciprocated by the theologian 
 whose belief in the primary truth of Theology is often- 
 times a belief fearful and apprehensive lest the progress 
 of Science should discover and make known some adverse 
 truth irreconcikble with the truth of Theology. 
 
 That this view of the case is adopted by all or by 
 nearly all scientific men or theologians we are not sup- 
 posing. The exceptions on both sides are probably in 
 the aggregate very many, but the question is, taken 
 all together, what is the concludion or state of feeling 
 now entertained and held by the cducateil public gener- 
 ally ; and in what direction are the doctrines now taught 
 guiding those who are to succeed us ? 
 
 Assumingthe existing state of things to be sul)stantially 
 as we have suggested, the apprehensive belief of the 
 theologian which fears the facts and discoveries of 
 science is not a belief in a scientific sense ; such a fear 
 
 <r - 
 
20 
 
 THE CIRCLE AND STRAIGHT LINE. 
 
 i 
 
 evidently includes a doubt as to whether one fact may 
 not contradict and be irreconcilable with another or with 
 some other facts. Nor can it help matters to say that 
 the truth of theology is different from the truth of science, 
 because, if it be meant thereby that the truth of the one 
 can be otherwise than in perfect harmony with the truth 
 of the other, the entertaining or admiting in any degree 
 such a supposition is the making over Theology to Meta- 
 physics which is expressly the antagonist of sound Theolo- 
 gy as being the essential basis of all sound Science. How 
 stands the case on the other side ! If the (piestion was 
 put : what is the especial principle contended for and 
 upheld by modern science 1 the answer most generally 
 concurred in would be — 'intellectual freedom,' the right 
 of each one to exercise his intellect in his own way, 
 without let or hindrance. What does that mean ? It 
 means inteUectual laicJessncss ; intellectual anarchy ; the 
 ria;ht of each one in matters which concern others as 
 well as himself iu the most important and momentous 
 sense, to do as seemeth good in his own eyes ; the demand 
 of men, who recognise the imperious necessity of laws 
 and submission to those laws in matters of comparatively 
 little importance, to be free from the restraint of all law 
 in matters of the greatest importance. It means the claim 
 of some men to be allowed to impede sound education, 
 to pollute things sacred, to contaminate all true know- 
 ledge, to teach the most pernicious doctrines, and to 
 deceive and confound those who look to them for 
 instruction. 
 
 Against this claim of so-called intellectual freedom, 
 whether it be iiesitatingly expressed, merely as a dislike 
 to mixing up theology with science, or boldly and defi- 
 antly asserted as a determination to be intellectually 
 unrestrained by any laws or rules, we distinctly protest 
 and earnestly caution those whom we may in any degree 
 influence. It may be said — let public discussion and 
 progressive education put right what is wrong, we 
 
 . f- 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 21 
 
 cannot expect to hi /e science quite perfect at any time ; 
 that which is true is stronger than that which is false — 
 increased knowledge and discussion may be relied upon 
 to eliminate error. No : such reasoning is not itself based 
 on fact ; and such a course is not to be relied on to elimi- 
 nate error ; on the contrary if allowed to proceed, its effect 
 would be to vitiate education and to deteriorate knowledge 
 n:Dre and more, until, at length, the education would be- 
 come an altogether evil education, and the knowledge an 
 essentially unsound and destructive knowledge. Truth is 
 J^tronger, much stronger, than falsehood, but a mixture of 
 that which is true and that which is false is not truth. 
 Neither is it fact tliat error can be eliminated and got rid of 
 by discussion in which sound and unsound knowledge are 
 mingled together, in which the reasoning is not strictly 
 lawful and not properly based on that which is true and 
 certain. Such discussion is the weapon with which Un- 
 truth, wearing the mask of Science, can best succeed in 
 destroying that which is true, and confounding those who 
 love the truth. A li nnan science wliich does not distinctly 
 recognize the prinuuy truths of Theology as its ultimate 
 basis, is not based on reality ; it has not and cannot have 
 any actual and secure foundation. If the Science of 
 England is not so based, no matter what seeming progress 
 may for a time be made, whenever the trial comes it will 
 be as the house built ou the shifting sand ; and, if not 
 destroyed by sudden catastrophe, will eventually become 
 a ruin, together with the civilization wliich rests upon it. 
 
 
 f;r*-: 
 
:r ^;^ 
 
 H 
 
APPENDIX. 
 
 Figs. 27, 28, 29, 30, are additional illustrations of the 
 complementary arc of absolute curvature, of the arc of 
 increasing curvature, and of the increasing sine-length. 
 
 It may be observed that in some of these figures the 
 same letter of reference, 3£., indicates the arc length and 
 the ultimate sine-length. On the scale of these figures 
 the two points fall so closely together as to appear almost 
 identical, the difierence being only about three parts in 
 eight thousand, and an attempt to show them separately 
 in the same figure would confuse the lines and letters. 
 
 It is supposed that, with the explanations which have 
 preceded them, the general puipose of these last figiu'es 
 will be sufficiently apparent without further remark. 
 
 In reference, however, to Figs. 21 and 22, where the 
 arc of increasing curvature appears as described with the 
 radius a.S. it is desirable to point out more particularly 
 the circumstance (already stated) that the curve is not 
 an arc described with a uniform radius and a fixed centre. 
 The curve may be considered as compounded of fractional 
 arcs of which the centres are not co-incident, or as be- 
 longing to an ellipse. The same remark would apply to the 
 arc of curvature. Fig. 25, if described from the same 
 centre «. with the increased radius a.X. For example (Fig. 
 22) — If the distance of a. from B. in Fig. 22, be made 
 two tenths of B.D., and B.X-Ba (-7-S.54- 2) = .'rS.54 
 be taken as the radius, and the arc be described from X. 
 as the original point, the arc will then fall inside X,S.^ 
 because the distance of S. from a. is o'SoG, consequently 
 
 ;..%• \: 
 
 ■^ 
 
 JE' 
 
 m 
 
 5^- 
 
24 
 
 Al'PKXDlX. 
 
 the radius will fall short by the dillereiice of about 2 parts 
 in six tiiousaud. Similarly, in Fig. •J-'), (taking the 
 point a. at the same distance, B.u. as before,) U.X. - 
 J5.«. = 7-8-5G74-2 = 5-So074 ; tlie arc now falls beyond 
 X.S., because the distance a.S. is as before 5"S-5GO, 
 and the radius is now 5*S.5G74. 
 
 But, if we bisect the line S.X., (theextrennty of the line 
 B.Y.X., Fig. 2-'5,) and from the point of bisection a' right 
 angles to S.X. we draw a line intercepting the line B.D., 
 the point of interception will be b. at a distance froP' B. = 
 2-0U4o2. 
 
 Therefore B.X. - B.h. = 7-8.5()74 - 2004-32 = ryS52'2-i. . 
 which is also the exact distance of the point S. at the ex- 
 tremity of the half quadrant B.S., from the point h. on the 
 line B.D. Wherefore, in tliis case, we lind the arc X.S. 
 is the true arc of a circle described from the centre h. 
 with the radius Z;.X. = o-8-522:J. . . 
 
 We have brougiit under consideration four ;ii"cs, 
 namely : — 
 
 (1) The arc of the arc-length X. X. 
 
 (2) The CO' plementaryarc(or, arcof curvature) /S.X 
 (S) The arc of the idtimate sine-length X. X. 
 
 (4) The arc of increasing curvature *S^. X. 
 
 Tlie first and third, ir.dicated in tiie Figmvs by the 
 same letter X. X., because too close to each other to be 
 clearly distinguishable, are both of tiiem arcs of a circle 
 described from the same centre B. Tlie radius of the 
 one (1) equalling 7^-3G74...&c. ; and the radius of the 
 other (3) equalling 7-8-5:39S...&c. 
 
 The second and fourth arcs we wish now again to com- 
 pare and contrast, in order to distinguish clearly between 
 them. The fourth is not the arc of a circle, but a curve 
 of a compound character; as already mentioned, it may 
 be considered as compounded of (ultimate) fractional 
 curves differing (very slightly) from each other, or (pre- 
 fei'ably) as a compound arc described from a centre, the 
 relative position of .vhich is constantly varying, the posi- 
 
 
 ■^ 
 
APPENDIX. 25 
 
 tion and variation thereof being definite relatively to each 
 uUimate fraction of the arc described. 
 
 The following table exhibits the comparison through- 
 out the greater part of the curve : — 
 
 A. are the sine-lengths of the successive arcs — obtained 
 by the duplication of the successive fractions of the 
 repeatedly bisected arc — by the terminal extremities ef 
 which * the compound arc of increasing curvature ' i& 
 formed* 
 
 J?, are the sine-lengths of the equivalent succesaive 
 
 «rca (corresnonding to t>i"i» of coknnny),) — resulting 
 
 from the continual ' unbending ' or ' straightening ' of 
 
 the primaiy arc — by the terminal extremities of which 
 
 * the complementary arc of absolute curvature ' is 
 
 ibrmed. 
 
 A. B. 
 
 / " — — > /■ * — \ 
 
 ?_ 7071068 7-071068 
 
 8 
 
 ^ 7-653668 7-65517 
 
 16 
 
 £ 7-803613 7-80606 
 
 32 
 
 -^ 7-841371 7-84405 
 
 64 
 
 -^ 7-850828 7-85356 
 
 128 
 
 ~ 7-853193 • 7-85539 
 
 256 
 
 ~ 7-853785 785655 
 
 ~ 7-853937 7-85670 
 
 — 7-853969 7-85673 
 
 2048 
 
 Finally 7-85398.. .&c. 7-85674.. .&c. 
 
 * These sine lengths agree with those of the tabl« at page 
 55 of Appendix, Part Second. 
 
 y^> 
 
 W 
 
QC, 
 
 APrFNOIX. 
 
 TTereiii we obsorvo (l)}i quite iiulepeiideiit verification 
 of tiie fact, already a)ni>ly demonstrated, that 7'85G74 is 
 the actual arc-length of the half-quadrant U.S.; and 
 moreover we observe (-J) a very interesting definite rela- 
 tionship established between tlie arc X.S. (Fig. 2ij) and 
 the half quadrant B.S. / namely the magnitudinal ratio of 
 the circles to which they reBp'"'<''''^'y belong is . . as 
 X.S. : B.S. : : radius X.h, 5-8')2:>:3 : radius A.B., 10-0000, 
 (i.e., a half quachant described with the lesser radius 
 would have that ratio to the arc B.S.,) 
 
 Note. — Referring to Fig. 25, the following will indi- 
 cate the elements of the computation : — 
 
 Bisecting the line S.X. and drawing the line d.b. in- 
 tercepting B.D. at h. we obtain X.d. as the base of a 
 triangle similar to the triangle S.X. W. ; therefore 
 
 Now since S. W. = 2-92S9;j S.X. = 3-03247 
 And X.(l = S.X.-^2^ I -51023 5, therefore 
 -7S5G74 : 3-03247 : : 1-51G235 : 5-85223 ., 
 which last number is the length of the radius h.X. 
 7S-r/u74 5352i2 ^ 200452 - 13. 0, 
 But 7-07107- 2-00452 = 500655 
 
 And V5-0G655' + 2 •92893' = 5-85223.., 
 that is, the square of ' the sine of the half-quadrant 
 less B.h.,^ added to the square of the verscU sine S.W. 
 gives the square of the radius h.X. 
 
 In like manner the sine-lengths of the successive equi- 
 valent arcs of decreasing curvature (column B.) may be 
 verified. 
 
 ) r 
 
 
 M 
 
 ^ 
 
 
i'?^. 
 
 ■;-.f 
 
 %^ 
 
 ■MM 
 
\ 
 
I 
 
 '^%Jf 
 
 ti^. 
 
 A'. 
 
 m 
 
 M| 1,^1, ,„, 
 
 v.- 
 
 l^A- 
 
 NMHta 
 
,y' 
 
 •# 
 
 ^ 
 
 
 
 # 
 
 * '*. 
 
^^ 
 
 'm 
 
 #^ 
 
 .•^. 
 
 « 
 
 m 
 
 ':*#■ 
 
 fin *-^'IIL;lll 
 
 'w^ 
 
W'^ 
 
 r*^i 
 
 # 
 
 ,tH*' 
 
 # 
 
 M 
 
 :•!!#• 
 
/ 
 
 V * 
 
ILLUSTRATION TO 
 Fig. 20 {R.) 
 
 t^fi 
 
 -M*-"---" 
 
 ffc 
 
Supplementary Illustrations. 
 
 Analytifal Jvxaniiiuitioii of tlio <niaili':int and ot'tlic arc 
 of inci'casiiiir (.•urvaturc. 
 
 1— Fi.i;-. 20 (R.) The raiUui A.B. as erjualli)i'j 10. 
 
 The radius A. "a. A.N. :. A.M. ; A.B. Conf^equcntly 
 A. 'a . A.. VI :: A.X, : A.B. And the are--, bines, tangenti^, 
 and their equal divisions, and oilier related lines, belonging 
 to each of these radii, are in the same ratios respectively to 
 the .similar lines belonging to each of the other radii ; that 
 is,... the arc 'a.l>. ; N.II. :: M.e. ; B.S., the sine 'c.b. ; 
 d.H. :: a e. : >s..S., and so on. (Observe that the radius of 
 the arc X.JI. - the cosine of B.S.) 
 
 A.B. --= 1(»00 
 A.d. -= o'OU 
 
 A.N. - 7-OT107. 
 A. a. — o-5o5535. 
 
 As ('.'a. : 'a.d. :: d.N. : N.B., and therefore, inversely, 
 as2-9289;5 : 2-0710T :: l-4644()5 ; 1 035535. 
 
 And as b.e. : e.ll. :; U.S. : 8.D. i.i' , (inversely,) 
 as4-l-4214 ; 2-!:t2803 :: 2-07107 : 1-4(U4G5. 
 
 Conse(,uenl!y 4-14214 : 2'!t2sn3 :: 2-02893 : 2-07107; 
 &c., &c. 
 
 2— Fig. 20 {11.) Thr radius A.B. - 10. 
 
 A.e. - e.g. = B.V. K.B. = K.S. = S.D. = 4-14214. 
 A T. - B.Q. - C.Jl. = D.Z. = 2 B.V. = 8-28428. 
 
 N.K.= K.ll.= X.B. (thover.sd.sinoofB,.S.)= 2-92893. 
 Z.Y. = Px.V. = &c., &(;. . = 414214. 
 
 B.R. = A.Z. = D.T. = C.Q. = B.e. = 1000 - 414214. 
 = 5-8578(3 (= Cosine of half-fiuadrantdescrilied with A. g.) 
 
 Because the triangle A.X. K., is a part of the similar 
 triangle A. B.V., N.Jv. : B.V. :: A.X. : A.B. Therefore.— 
 
 2-92893 : 4142U :: 7-07107 : 10-000. 
 2 92893 : 7-07107 :: 4-14:' .4 : 10000. 
 2-92893 X 10000 ^ 4-14.14 x 707107. 
 
 ... V 
 
 ^- 
 
 # 
 
 M^- 
 
 m 
 
 .^. 
 
The quadrant e.y. ( = g.Z.) : B.C. :: Ac, : A.B. 
 
 The arc c.I. ( = g.K.) : B.S. :: A.o. : A.B. :: 1-U214 : 
 1000. 
 
 But tlic proportion c.y. (or g.z.) : B.S. :: 8'28428 : 
 10-900, is not strictly correct, because e.y., and g.Z., arc 
 quadrants, and contain twice the proportionate amount of 
 curvatui'c contained in the half-rpiadrant B.S. (See appendix 
 to Part Third.) 
 
 Draw g.W. the tangent of the half-quadrant g.K. g.W. = 
 B.V. the tangent of the arc B.U. of 22i doij-rees. 
 
 Scholium. — Observe the relationship... e.g. : A.B. :: 
 4'14214: : 10. And the tangent to the half-quadrant, of which 
 e.g. is the radius, equals the tangent to the arc of 22^ degrees 
 of which A.B. is the radius; but the sine (X.K.) of g.K. : 
 the sine of B.U. :: A.K. : A.U. 
 
 Some of Vi.e moi-e immediate relations of these (quantitivo 
 magnitudes) numbers may be noted, — as for instance : — 
 
 U-U2U-10 - 414214. 
 
 70710V X 2 - 1414214 
 7-07 107'' = 50-0000 
 
 4-14214 X 2 = 8-28428. 
 
 C. 
 
 7-07107 - 5000 = 2-07107 (= half the tangent of-j^) 
 
 14-14214- = 200-00 1-414214^ = 2-000. 
 8-28428 X 2-07107 = 17-15732. 
 
 4-14214* = 17.15732. 
 8-28428 X 7-07107 ^ 58-5786. 
 2-92893 X 2 X 10 = 58-5786. 
 
 2-92893' = 8-5786. 
 7-653667* - 58-5786. 
 
 And so on. (The last number represents the duplicated 
 sine of-j^) 
 
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