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 ^V^E CIRc^ 
 
 AND 
 
 STRAIGHT LINE . 
 
 PART SECOND. 
 
 >S! 
 
THE CIRCLE 
 
 AND 
 
 STRAIGHT LINE. 
 
 PART SECOND. 
 
 8T 
 
 JOHN HARRIS. 
 
 MONTREAL: 
 JOHN LOVELL, ST. NICHOLAS STREET. 
 
 FEBRUARY, 1874. 
 
if 
 
 Entered according to Act of FarlianieDt in the year one thousand eight hundred 
 and eevcnty-four, by John Hauhis, iu the office of the Minister of Agricultur* 
 >nd Statistics at Ottawa. 
 
 HAH^iLTC" 
 
 f • vm^ f*- 
 
 A^Y 
 
 
 Montreal:— John Lovbll, PniuTER, 
 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 ' Prove all things ; bold fast that which is good.' 
 
 St. Paul. 
 
 PART SECOND. 
 
 The Construction of the Cirdc. 
 
 Having in the preceding exposition of the subject 
 demonstrated * tlie true relation of the circle to the straight 
 line ' in accordance with Euclid's methodical application 
 of the inductive system of reasoning, we will here give an 
 illustration of the general relationship thereby established, 
 in order — to make, in some measure, apparent the struc- 
 tural completeness and perfection of tlie circle as a reality, 
 — to show the circumstantial necessity for that particular 
 inter-relationship of tlie parts whicli lias been now demon- 
 strated to be actually existent, and thereby — to make 
 manifest the great importance of the circle as one of the 
 fundamental facts belonmnff to the Plan of Creation. 
 
 Let us take, in the first place, the construction of Fig. '2, 
 as it was left at page ^ 9, and develop that construction 
 as follows : — 
 
 Figs. 10 & 11. Construction. (Fig. 10 is a repetition 
 of Fig. 2 on a smaller scale, similar letters denoting similar 
 parts.) Produce the radius -4. jB. through A. and through 
 B. and make A.B., (Fig. 11,) the production oi A.B., ten 
 times the length of A.B. AVith centre A. and radius 
 A.B. describe the quadrant B.F. ; bisect the quadrant at M. 
 and divide the arcM.F. into ten equal parts at the points 
 of equal division 'a. ^h. 'r. ^d. 'c. /. ^g. 'h. ^i. Complete 
 *he greater Bgiire similarly to the lesser figure. 
 
 /I 
 
6 
 
 THE CIRCLE AXU STRAKHIT LINE. 
 
 Illustration of the fact tluU tlir differmci' beturen the 
 chord and arc-length «f the qnadrant is an aliquot part of 
 the chord, and of the arc-length. * 
 
 Because R.]\[. in the greater figure equals ^.7?. and 
 li.M. of the suiaUer figure taken togetlier (to wit, S.M.) 
 it follows that, however small the magnitude of a circle 
 may be, the arc-length and the chord of the (juadrant ol 
 that circle are each divisible into a certain number ol 
 equal parts, each of which i)arts is necessarily equal to 
 the diderence of the chord and arc, and eacli of wh' h 
 parts is an ali(inot part of the chord and of the arc- 
 length of the ((uadrant of any (^ther circle of which 
 the magnitudt! is an ecpiinudtiple of the magnitude 
 of the first circle, Jiowever great that other circle 
 may be. Now it is to be particularly observed that, il 
 K.M. of tiie greater figure were not equal to S.lt. and 
 R.M. of the lesser figure taken together, this would l)e 
 no longer true. In that case either S.lt. or ll.^l. of the 
 lesser figure might be evenly divided, but the equal division- 
 al parts would not be ali([uot parts of the compound parts 
 belonging to the circle of greater magnituile. The rela- 
 tive magnitude of tiie greater circle might be such that 
 R.^I. of the greater would contain aii equimultiple of 
 H.M. of the lesser ; but, then, S.^[. of the same greatei' 
 circle could not also contain an equimultiple of li.M. of 
 the lesser, nor of S.lt. of the lesser, neither could S.R. of 
 the greater contain an equinudtiple of 7]?.il/. of tiie lesser, 
 nor, \{ S.M. of the lesser an<l R.il. of the greater figure, 
 were unequal, could R. M. of the greater, however great 
 the magnitude of the greater circle might be, contain an 
 equimultiple of S.R- of the lesser. 
 
 The general theorem illustrated by this construction 
 may be thus stated. The diflerence between the chord 
 
 % 
 
 * Evidently, in this connection, the rclaiion of the hall- 
 chord to the half-quadrant is equivalent to the relation ol" 
 the chord to the quadrant. 
 
THE CIKCJ.E AND STRAIGHT LINE. 7 
 
 ami arc-length of the quadrant of a circle is an aliquot 
 part of the circle ; and if a second circle be so related to 
 the first that the magnitude of the first is an aliquot part 
 of the magnitude of the second, then is the difierence of 
 the chord and arc-length of the quadrant of the first also 
 an aliquot part of the second circle. 
 
 F((j. \2. (a) Construction. — With centre vl. and radius 
 A.B describe the quadrant jB. 6'. C ; drawjB.D. and CD. 
 and join A.D. bisecting the ((uadrant at S. Through 
 S. and at right angles to A.B. draw E.S.2\, intercept- 
 ing A.B. at B. and CD. at T. Divide R. S. into nine 
 equal parts at the points of equal division ; from S.T., 
 at the distance S.X. equal to one of the ecjual divisions of 
 B.S., draw X. X. perpendicular to B.D., intercepting 
 B.D. at X. Divide the radius A.B. ii.^o ten equal parts 
 at the points of equal division. With the point 6'. as a 
 centre and with the radius S.h. equal to one of the 
 equal divisional parts of the greater radius .1.^. describe 
 the small quadrant b.s.t., bisected by the line A.D. 
 at the point s. Draw the tangent b.d. and the sine S.r, 
 of the small arc b.s., and join t.d. 
 
 Scholium. Since the radius of the small arc is one- 
 tenth the radius of the greater arc ; the arc, the sine, 
 and the tangent, of the small arc are respectively one 
 tenth of the arc, of the sine, and of the tangent, of the 
 greater arc ; and therefore, also, tlie difference of the 
 sine and arc on the small scale is one-tenth of the difference 
 of the sine and arc on the larger scale (^i.e. — x.s. is the 
 one-tenth of X.S.) But s.t. the difference of the sine and 
 tangent on the small scale is manifestl)'^ one-tenth of S.T. 
 on the large scale, and x.t. must be also necessarily 
 one-tenth of X.T. It will follow, if the small arc &• s. 
 is straightened upon or rolled upon the line h.d., that the 
 point of contact must ftill upon the line X.X. This will 
 become more apparent by developing the coustructioif. 
 
8 THE CIKCLE AND STKAIGIIT LINE. 
 
 Prop, (in quantitive geometry.) 
 
 Theorem. — The quantity of length hy nhich the length 
 of the half-qmuhant of a circle exceeds the length of the 
 sine of the half-queulmnt is equal to the one-tenth part of 
 the length contained in the half-eptadrnnt. 
 
 Construction. — Fig. 12. (a) has been already described. 
 
 Specification. — Let^.«S^. be the half-quadrant of which 
 S.li. is the sine, and let S.X. indicate the diff. in length 
 between B.S. and S.R,] — S.X. shall equal one" of the 
 divisional parts of the arc U.S., if B.S. be divided into 
 ten equal parts. 
 
 Demonstration. — Because the radius S.b. is the one- 
 tenth of the radius A.B., the sine S.s. of the lesser 
 arc b.s. is one-tenth of B.S. the sine of the greater arc. 
 But S.X. the diff. of the arc-length and sine of the greater 
 arc is ten times greater than the similar diff. s.x. of the 
 arc-length and sine of the lesser arc; therefore s.x. + S.s.^ 
 that is S.X., is the one-tenth of B.X. Now B.X. is the 
 arc-length of B.S. ; and S.X. is the diff. between that 
 arc-length and the sine of the same arc. "Wherefore the 
 tiieorem has been demonstrated. 
 
 Scholium. — Tlie foregoing prop, may be numerically 
 exhibited in units of radius : for example : 
 The radius A.B. = 10.— The sine i?.>S'. = VoO-- 7-07 1008 
 
 * Therefore R. X = 7-850742 
 " '^ *S'.^/.= 1.— The sine ^.s. = VJ. = 0-707107 
 
 6' X. = 0-7850742 
 .s\a:-. = 0-0785074 
 Development of the Construction. Fig. 12. (b.) From the 
 point X. on the line B.D., at the distance X.V. equal to 
 X.B.J take the point V. From V. at right angles to 
 S.D. draw V.F., intersecting S.B. on the line X.X., and 
 intersecting the line b.d. at v. 
 
 Illustration. — Now, if a second arc similar and equal to 
 the greater arc B.S.C. be described in such wise that 
 i^e point F. become the extremity of the secant, and the 
 
 * Because R.X- ^ RS. divided by 9, x 10. 
 
V THE CIKCLK AND STKAIOHT LINK. 9 
 
 production of V.F. be tlic sccoiit, and the pro<1uction of 
 V.D. be the tangent, of the second greater arc, it is 
 manifest tiiat the second arc, so described, must, neces- 
 sarily, intersect the first greater arc on a prochiction of the 
 line X.X. Now S.d., the secant of the small arc, is includ- 
 ed in the secant-^. 1). of the first greater arc, and likewise, 
 if a second small arc be described similarly related to the 
 second greater arc, then will the secant of the second 
 lesser arc be included in the (production of the) line V.F., 
 the secant of the second greater arc ; therefore since 
 x.(L is one tenth oi' X.I)., x.v. must necessarily be the 
 one tenth of XT'. It becomes evident accordinclv 
 that the ])oint .r. of the lesser arc necessarily falls upon 
 the line X.X. 
 
 Fig. 12. (h.J may be further develoitcd by rom))letiug 
 the double figure. I'roduce X.T. through T. and on that 
 jiroduction make (7'.)X ecpial to ll.X. Produce A.O. 
 through C. and produce B.D. througii D. and make C. 
 (A.) and D.(B.) each equal to T. (Ji.) on the line II. (E). 
 With centre (A) and radius (A.)(B.') descrilje the arc 
 (B.){C.) intersecting the arc B. C. in the line X.X. 
 With the centre F. (on the line E.(I{.) and radius F.s. 
 equal to S.b. (equal to A.B. divided by 10), describe 
 the arc ^.?/. Join ^■.d.b. Draw n.m. the sine of the 
 arc. Join i/.r. and produce F.;;. througii s. intercept- 
 ing the line B.(B.) at Q. 
 
 The analytical value of the construction thus developed 
 may be understood by considering that the line J?.(7?.) 
 contains the sines It.S. and (R.)F. of both the arcs 
 together with the space S. F. And F (R.) also contains 
 the tangents B.T. and {T.)(M.) of both the arcs, less 
 T.(T.) Again, because the radius F.s. is equal to the 
 radius s.h., and because F.s. has the same relation to the 
 arc {B.){C.) that S.h. has to B.C. therefore y.x. is 
 equal to t.x.y etc., &c. 
 
10 
 
 THE CIRCLE AND STRAIGHT LINE. 
 
 I'a 
 
 
 Example. The radi us ^. ^. ^ 1 0. 
 R (R.) ^2 R.T.-2 X.T. equals 20 
 
 -4-2S652- 15-7134 
 7?. (7?.) = 2 R.S. + 2S.1\ equals . . ]4'14214 
 
 + 1-57134=10 -7 1.34 
 II. (i?.) = 2 B.T. + 2 aS'.X - 2 S.T. equals 
 
 20 + 1-07134 - 585776 - 15-7134 
 Jl. (B.) = R.X. + M.:i\ Which equals 
 
 R.S. R.S. R.>S. 
 
 + iS:c. = 7-S5G74/ 
 
 15-7134 
 
 h\ S. + 
 
 10 "*" 100 "*" 1000 
 
 ^,^ (R.F. (R.F. (R.F. 
 
 Ilhtstraiion hi/ the construction. Because the line x.x. 
 belonging to the small arc is included in the line X.X. 
 of the greater, the line S.X. is equal to S.s. added to 
 s.x. (i.e. the diff. of the sine and arc on the large scale 
 is equal to the sine of tlie arc added to the diff. of the 
 sine find arc on the small scale. 
 
 Quantitivc and Numerical Illustrations, to Fig. 12. 
 
 K.T. = A.D. = 
 R.S. = V50 = 
 
 10-000000 S.D, = A.D. — A.S,= 4'ir213.J 
 70710li8 S.T. = R.T. — R.S, = 2''.)2S'j32 
 
 R.S. 
 
 A.D. = A.S. + S.D. = ll'112I3a R.S. = R.S.4- -— - = 7778174 
 
 R.S, 
 R.x. =-y-X 10 = 7'85G742 
 
 Because S.s. = 
 And S.s -f s.t. = 
 
 h'.S. 
 
 And S.t. = 
 
 .T. 
 10 
 
 And R.T. = R.S. + S.T. 
 
 10 
 R.T. 
 
 To" 
 
 Therefore (1) R.s. -i-s.t. =R.t. 
 
 (2) R.T. -9 S.t. i-R.S. = R-t. 
 
 (3) R.S. + S.t. =R.t. 
 
 And 
 
 (1) 7-778174S-i--292S93 = S-0710G7S 
 
 (2) 10-9 + 7-0710078 = 8-0710078 
 
 (3) 7-0710078 + 1-0000 = 8-0710078 
 
 II 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 11 
 
 Because S. s. = 
 
 R.o. 
 
 Therefore R.s. = R.S. 
 
 R.S. 
 
 and consequently if the scale be again reduced to one 
 tenth of the lesser figure, and then again reduced to one 
 tenth of the last, and so an ' ad infinitum" R.X. must 
 evidently include the (sura of the one-tenths) one-tenth 
 of each and every of all the sines; (i. e., the one-tenth 
 of each of the sines of all the figures from the grestest 
 (R.S.) to the least imaginable). Therefore R.X. equals 
 
 RC1 R.fe. rv.o. R.iS. R.o. 1 • i.' 
 
 .S. 4- + + + ,^„ +&C., admt. 
 
 10 100 ^1000 10000 ' 
 
 7-0710678 
 
 70710678 
 
 70710678 
 
 70710678 
 
 70710678 
 
 70710678 
 
 &c., &c., «S:c., &c., ad infinitum. 
 
 7-856742 &c., &c. • 
 
 Arithmetical Illustration of the fact that x. falls on the 
 line X.X. 
 
 Because S.s. - — r-r- Therefore S.s.=- 
 
 10 
 
 11 
 
 •707 1 nr.7S , 7-0710678 _ 7-0710678 ^ -70710678 
 10 11 11 
 
 And S.s : S.x. : : S.R : R.X. Therefore. 
 
 i>,x, — S.s. -f- 
 
 S.s. 
 
 -7856742 = -70710678 + -07856742 
 
 R.X. = S.X + R.S. 7-866742 = -7856742 + 7-0710678 
 
 7-77817458 
 
 S 
 
 11 
 
 =-70710678 
 
12 
 
 S. 
 
 THE CIRCLE AND STKAIGHT LINE. 
 
 But S.s. : S.^. : : S.R. : R.X. 
 S.R. 
 
 6'. 
 
 10 
 
 R. S.= S.R. 4- S.5. = 7-77S174 
 
 S. X. - s.s. + 
 
 S.s. 
 
 •7856742 = 
 
 R.X 
 
 9 10 
 
 S. X X 10 = 7-S.5G74 = R.X = S.X. + S'?. 
 S. .r. 
 
 s. a;'. 
 
 10 
 
 And R.^. -- Rs. + s.x. 
 
 That is, 7-77S174 + -078067 = 7'856742 
 And again X.T- - R.T. - KX. = 2'143258 
 
 X.t = S.t. - S.X = 2143208 (i.e. Xt. =±i^) 
 
 10 
 
 Because the lesser (right angled) triangle S.W.D 
 
 is similar to the greater triangle A.R.S., therefore 
 
 SW. ,, _ R.8. , , 
 S.D. : : -^ : A.S. 
 
 9 
 
 rru . ■ 2-92S9:32 
 Ihat IS, ; X 
 
 9 ■ • 9 
 
 , , S.W. A.S 
 And — - — X 
 
 S.D. 
 
 10 
 
 S.R. 
 
 9 
 3-25437 
 
 4-14213 4-14213 
 
 •7856742 
 
 Quantitivc and Numerical lUustration (demonstration) 
 
 of the fact that the quantity obtained by analytical 
 
 methods, which is now sujiposed to represent the ratio 
 
 of the circumf. of a circle to the diameter is erroneous. 
 
 Scholium. TJie quantity under examination, stated as 
 
 the ratio of the circle to a unit of diameter, equals 
 
 3'141o9 ; therefore, taking the radius = 10. the arc of 45 
 
 degrees = 7-85397. 
 
 By Fig. 12. Taking the radius A-B. = 10. 
 
 And assuming the arc-length to be as stated, then 
 
 R.X = 785397 RT. = A.B. = 10 
 
 RS. = v50 = 7-071067 And therefore :— 
 
 S.X = R.X - R.S. = -78291 
 
 A • o K.S. 
 
 Agani, S. s. - -— - = 70710678 
 
 And. s. X = -Vtt = -078291 
 10. 
 
TUE CIRCLE AND STRAIGHT LINE. 
 
 13 
 
 Therefore S.X = S.s. + a.X. = -78539. 
 
 But it has been shown that S.X. = -78291, and there- 
 fore the same line has ('appears to havej two difterent 
 lengths which is impossible. 
 
 And again, further : — 
 
 R.s. = R. S. + J^^ S.s. X 11 = 7 77S1745, conse- 
 quently (R.X-R.s.) X 10 = sX X 10 = SX - -75795. 
 
 But it has been already shown that S.jtf. equals '78539, 
 and also equals '78291 — and therefore the same line S.X 
 is (apparently) demonstrated to have three different 
 lengths, which is absurd. 
 
 Again by the assumption ; R.X = 7 '85397 
 
 But S.X = 
 
 10 
 
 •785397 
 
 R.X. = S.X. + S.R. = -785397 +7 0710078 7-85040. 
 
 Therefore R..^. has (appears to have) two different 
 lengths which is impossible. Wherefore the assumption 
 is erroneous. 
 
 Fig 13. 
 
 Construction. — Describe the quadrant B.C. bisected by 
 the line A.D. in the point S- Draw the sine JR-S. and 
 the tangent B.D. of the arc BS. Join A.C- and I)C. 
 and produce MS. through S., intercepting DC. at T. 
 Divide the radius A.B. into ten equal parts at the points 
 of equal division. With centre 'S^. and radius S.h., de- 
 scribe the quadrant />.^. bisected by S.B. in the point s. ; 
 draw the sine sr. and the tangent h.d. of the arc /^..9. 
 
 Produce the line B.B. through a> , and make X.F. 
 equal toX.B.; produce ^.C. through <^., and f'lom F. 
 draw F.E. perpendicular to B.F., and intercepting the 
 production of A.C in the point E. From the point w., 
 where A.D. intersects X-X"-, join n.B., and from the 
 same point join uF- intersecting CD. at IL, and from 
 
14 
 
 THE CIRCLE AND STRAIGHT LINE. 
 
 H 
 
 the same point draw also n.I.E' at right angles to A.D. 
 intersecting CD. at 7., and intercepting the point E. at 
 the vertex of the angle A.E.F. With centre 7. and 
 radius 7.77. describe the quadrant H.K. bisected by I.n 
 in tlie point c. Draw the sine G.c. of the arc H.e., 
 and from d. through c. draw d.t. perpendicular to B.T., 
 and intercepting R.T. at t. From K. at the extremity 
 of the quadrant H.K. draw K.m. perpendicular to B.T. 
 and intercepting R.T. at m. 
 
 Scholium. — We have now three similar triangles, 
 namely : A.n.B., S.n.h-f I.n.II. Because n. is 
 a point in the line X.X., and T. is a point in the 
 radius 7.77., fnd 7?. a point in the radius A.B.; the 
 ratio of the radius 7.77. to the radius A.B. is the same as 
 the ratio of the line X.T. to the line X.R., and also 
 the ratio of the radius S.h. to the radius 7.77, the same 
 as the ratio of S.X. to T.X. 
 
 Illustration (a) of the fact that the point x. falls on the 
 line X.X. 
 
 Because the radius S.h. is the one-tenth of the radius 
 A.B.^ therefore s.x. is the one-tenth of S.X. and x.d. 
 the one-tenth of X.T. But S.t. (^i.e., S.x. + x.d.) is 
 the one-tenth of 7?.7., and the ratio of S.x. to x.t. is the 
 same as the ratio of B.X. to X.T. (because the rp,tios 
 of similar arcs each to each, and of the tangents, and also 
 of the sines of similar arcs each to each, are directly as the 
 ratios of the radii each to each, to which they respectively 
 belong.) And the ratio of B.S. to T.t. is also the same as 
 the ratio oi R.X. to X.T., therefore a;. T. is equal* to X.T. 
 and x.R. equal to X.R. Wherefore it becomes evident 
 that the point x. is included in the line X.X. 
 
 By the same reasoning applied to the secant , 
 
 Because S.h. is the one-tenth of A.B., therefore S.d. 
 is the one-tenth of -4.7)., and S.s. the one-tenth of R.S., 
 and s.d. the one-tenth of S.D. But s.n. is, therefore, 
 the one-tenth of S.n., and n.d. the one-tenth of m.7). 
 Consequently the point n., which is a point in the line 
 
 * Itecause X. r. = x.t. x 10, and sr.ff, = x.r. x 10. 
 
THE CIRCLE AND STItAIGHT LINE. 
 
 lo 
 
 X.X., is also included in tlie line X.X. Wherefore it 
 clearly appears that the line x.x. is the same line as XX 
 
 l^emonstration («) by the co .vcruction Fi.r 33 (to 
 prop. B.) That the dif. of the sine and arc-length of the 
 half-quadrant is one tenth the length of the half-quadrant 
 
 Because 8.t.= 
 
 10 
 
 s.X. =~~ 
 10 
 
 R.T. - S.t. = i-^"^- = R.S. + T.t. 
 10 
 
 and R.T. = R.S. + T.t. + S.t. - (R.S. + S.a.) + (T.t. + tx.) 
 
 and also R. T. = R.S. + T.t. + ^- + II^ 
 
 9 9 
 
 Therefore S.X equals -j-f And t.X. equals ' 
 
 10 
 
 10 
 
 For let it be supposed possible that S.X may be a mag- 
 nitude less than_?j±: then must t.X. be a mao-nitude 
 
 greater thanZ:^- (and s.X. less than *l^-\ 
 10 ^ 10 ^ 
 
 But R.T. is wholly compounded of R.X. and T.X. 
 together. (Demonst. b), and R.T. contains R.S. and T.t. 
 
 R.T. 
 
 RS. T.t. 
 
 -') and 
 
 together with - * - (i.e. R.S. + T.t.+ 
 
 R T R S T t. 
 
 -^" - -^* = -^ consequently if S.X. be any magnitude 
 
 10 
 less than 
 
 9 
 R.S. 
 
 then must the remaining magnitude, of 
 
 vvhicli, together with -^, the magnitude ?ll: is com- 
 
 y 10 
 
 T t 
 pounded, be greater than — I_L 
 
 Now T.t. is the sine of an arc similar to the arc of 
 which R. S. is the sine, and therefore it is impossible 
 that the ratio of t.X, the diff. of the sine and arc lengtii 
 
IG 
 
 THE C[RCLE AND STRAIGHT LINE. 
 
 of H.e.,- to T.t., the sine of the arc II. e., can be greater 
 than the ratio of S.X, the diff. of the sine and arc-length 
 of B.S., to R.S. the sine of the arc B.8. Wherefore it is 
 demonstrated that S.X., the diff. of the sine and arc-length 
 of the half quadrant B.S., is the one-tentli part of the arc- 
 length contained in the half quadrant B.S. Q.E.D. 
 
 Denionst. (b). That the line R.T. is wholly compounded 
 of the arc-length of B.S. together ivith the arc-length of 
 H.e. J and that the same point X' is the extremity of the arc- 
 length of each of the tivo arcs — to tvit, of the arc B.S. and 
 the arc H.e. 
 
 (If the arc-length of B.S. contained between R. and a 
 point indicated by X. in the line R.T. be taken from R.T. 
 it is manifest that a magnitude will remain which must 
 be equal to the arc-length of some arc contained between 
 T. and the point indicated by X.) The triangle I.H.n. 
 is similar to the triangle A.B.n. and the bases of the two 
 triangles together — to wit H.n. and B.n. together — 
 include all of the magnitude contained in the line R.T., 
 and, since X, is the divisional point which divides the 
 part of the line R.T. belonging to R, from the part thereof 
 belonging to T, the ratio of B..X. to T.X. must be the 
 same as the ratio of B.n. to H.n. and the same as the ratio 
 of the radius A.B. to the radius I.H. Therefore the 
 point X has a similar relation to each of the arcs, and to 
 the similar lines belonging to each of the arcs respectively 
 — to wit, to the arc H.e. and the arc B.S. . . to the sine 
 G.e. and the sine R.S. ... to the ' diff. tx, of the sine and 
 arc-length' of the arc, II. e. and the ^ dift'. S..r. of the 
 sine and arc-length ' of the arc B.S. . . to the ' dill', of 
 the arc-length and tangent' x.m. and the ' diff. of the 
 arc-length and tangent' x.T- Now, if the difTerence 
 between the sine R. S. and the arc length of the arc 
 B.S. were either less or greater than H..%'. then would 
 x.m. be not in the same ratio to x.T. as the ratio 
 of .^'.T. to X.ll. (for if it be supposed possible tliat 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 17 
 
 the tangent of the arc li.e. may be greater than T.m. 
 then will t.m. no longer have the same ratio to T.t. 
 which S. T. haL io R. S. ; nor will the ratio of t.m. 
 to S.T. be the same as the ratio of the radius I.H. to 
 the radius A. B.). Therefore, since the sine R.S. and 
 the diff. S.X., of the sine and the arc, together with the difF. 
 XT. of the arc and tlie tangent belonging to B.S. 
 wholly compound the line R.T., and since T.t. is to 
 t.^. and to x.m., respectively, in the same ratio as R.X. 
 is to S.X., and x.T. respectively, the point x. at the 
 extremity of the arc-length of each arc cannot be other 
 than the same point X. which divides the line R.T. in 
 such wise that the part Jl.X. thereof has the same ratio 
 to the part T.X. which the radius A.B. has to the radius 
 I.H. Wherefore it is demonstrated, &c., Q.E.D. 
 
 Quantitive and numerical illustrations to the construc- 
 tion. Fig. 13. 
 
 By the Construction : — 
 
 R.T. = A.B. = 10-00000 
 R.S. = V50 = 7-07107 
 
 R.T. -R.S. -S.T. = 2-9289:3 
 
 S.t. = S.b.=^^- =1-00000 
 10 
 
 1-92893 =T.t. 
 
 By Demonstration : — 
 cj V R.S. 7-07107 
 ^••*- ^~Y^ — 9 -"" 0-785674 
 
 R.X = RS. + S.X = 7-07107 + 0-785674 = 7-S5G74 
 
 Now tx. : T.t. : : S.X : R.S. 
 
 rp, e 1-92893 X -785674 „i.oir + 
 Therefore — = -214325 = t. a::. 
 
 7-07107 
 T.X. = T.t + t.x. = 1.92S93 + -214325 = 2-14325 
 And T.X. = S.T. - S..^. = 5-92893 - -7856742-1 4325 
 
 Test : 
 
18 
 
 THE CIRCLE AND STRAIGHT LINE. 
 
 Let it be possible for S.X to be less than 5:^= -jonoi 
 ^ 9 9 
 
 and let S.X. - 0-782S3 Then 
 
 R.X-R.S. + SX = 7-07107 + 0-78283 = 7-8539 
 Now tx. ■ T.t. :: S.X. : R.S. 
 
 1-92893 X -78283 
 
 Therefore 
 
 •07107 
 
 •213549 
 
 and T.X. = T.t. + t.a;= 1-92S93 + -213549 = 2-14248 
 
 But T.X. = S.T. - S.X = 2-92893- -78539 = 2-14354 
 Therefore the same hne T.X. has two different quan- 
 tities of magnitude which is impossible. In the same 
 manner it may be shown that S.X cannot be greater 
 
 than 
 
 R.S. 
 9 
 
 Wherefore S.X = 
 
 R.S . 
 9 
 
 Illustration of the relationship of the arc-length to the 
 sine of the half-quadrant. By the construction, Fig. 13. 
 
 It is evident that, since s. x. : S.x ::s.b. : A. B., if another 
 arc be described with the point 5. as a centre, and a 
 radius s.c. less than s.h. in the same ratio that 5.6. is 
 less than A.B., then s.x. must contain the sine and arc- 
 length of the arc so described with the radius s. c. ; and, 
 again, the difi'.f.x, between the sine and arc-length of 
 this small arc must contain the sine and arc-length of a 
 smaller arc, described with the point at the extremity 
 of the siue of the last small arc as a centre, and with a 
 radius less than s.c. in the same proportion that s.c. is 
 less than s.h. ; and, in the like manner, arcs may be con- 
 tinually described, each arc being less than the arc 
 preceding it in the same proportion, so long as there be 
 any assignable quantity of distance remaining between 
 the extremity of the sine of the arc last described mid 
 the point x. 
 
 Note. — The reader may, if he please, describe Fig. 1:J, 
 on a scale ten times larger ; the small arc and radius, 
 b.s. and S.b. will be then enlarged to the size of tlie 
 
THE CIRCLE AND 8TKAIGHT LINE. 
 
 19 
 
 greater arc and radius— -B.^S. and A.B., and the second 
 smaller arc, described with radius s.c, will be then, if 
 described, the same size and occupy the same relative 
 position in the enlarged figure, which the arc b.s., 
 described with the radius S.b.j occupies in our Figure 13. 
 The figure may be then again enlarged in the same pro- 
 portion as before and a tliird smaller arc be described ; 
 and so on 'ad infinitum.' Instead, however, of actually 
 describing enlarged figures, an alteration in the letters 
 denoting the parts will serve to illustrate the case, by sup- 
 posing the lesser arc to have been enlarged into the 
 greater arc, as often as may be desired. — 
 
 Quantitive and Numerical Illustration, 
 
 R.jf. + T.^T. = R.T. = 10.0000. 
 R.S. RS^ R.S. 
 10 "^ 
 
 10 
 
 Since R.JT. equals R.S. + 
 And T.X equals T.t. + 
 
 ■^100 "^1000 
 
 T.t. T.t. 
 ]00"''lOOO 
 
 + ad inf. 
 
 + ad inf. 
 
 Therefore -! 
 
 I 
 
 r 707107 
 707107 
 707107 
 707107 
 707107 
 707107 
 &c.,&c., 
 'ad infinitum' ■ 
 
 ^ + -! 
 
 J 
 
 192893 
 192893 
 192893 
 192893 
 192893 
 
 192893 
 &c., &c., 
 'ad infinitum.' 
 
 = 10.0000 
 
 Figure 11. 
 
 Quantitive and Numerical Illustration fi to the construction. 
 
 (By the construction : — A.B. the radius of the greater 
 figure (Fig. 11), equals ^.J5. the radius of the lesser figure 
 (Fig. 10) multiplied by ten.) 
 
 By the established trigonometrical relationship of the 
 parts : 
 
r 
 
 20 
 
 THE CIRCL?: AND STRAIOHl 
 
 ' LINE 
 
 
 
 Ii 
 
 1 the greater 
 Fig. 11. 
 
 
 n the lesser 
 Fig. 10. 
 
 The 
 
 radius A.B. 
 
 = 10-000000 
 
 A.B. 
 
 = 1-0000000 
 
 The 
 
 sine V 50 S.R. 
 
 = 7-07 lOOS 
 
 S.R. 
 
 = 0-7071068 
 
 The ditf. of ^ S.R. 
 the arc and ^ ,j - 
 
 = M.R.= 0-78.5G74 
 
 UM. 
 
 = 0-07S5G74 
 
 sine ) 
 
 The arc leiigtli M.F. or S.M. = 7-S.5G74 
 
 The arc length M.F. or S.M. = 0-7S-5074 
 
 Therefore : 
 
 S.M. 
 
 R.M. 
 
 10 
 
 -0-7S5G74 
 
 /S'.Jf. = 0-78.5674 
 
 Demonstration that U.S., the sine of the arc O.S. 
 {Fig. 2.) contains a certain number of equal (Vijisional parts^ 
 each of them equal to R.M. the diff. of the sine and the 
 arc-length. 
 
 Divide the radius A.M., of the arc B.M., at /. the 
 point of bisection. Witli centre/, and radius I.M., de- 
 scribe the quadrant //.ill. (r. half the length (magnitude) 
 of the quadrant B.M.F. (because the radius I.M. is 
 half o( A.M.) The point Al. which bisects the greater, 
 bisects also the lesser quadrant, and the arc 31. G., the 
 half of the lesser, is similar to 31. F., the half of the greater 
 (ju ad rant. 
 
 Now, let the arc II.3I.G. be rolled upon the straight 
 line U.W. until the point 31. becomes in contact on the 
 line . . . Since the arc is similar to the greater arc and 
 the motion is similar, and the lesser contains one half the 
 length of the greater, it is manifest that the point G. at the 
 extremity of the arc will intersect the line J/.*S'. at the 
 point of bisection of that line . . to wit, at the point (?., half 
 tiie distance of S. from 31. on the line 3I.S. But since the 
 two arcs are similar and the magnitude of the lesser is 
 one-half of the greater, the sine of the lesser is also one- 
 half the sine of the greater. Consequently the reinain- 
 i'lff half '?.>S'. of the line 31.S. must contain one-half of 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 21 
 
 the sine-length and one-half of Tt.M. the difl'eience be- 
 tween the sine and arc-length of tlic greater arc. If, 
 therefore, the sine S.Il. of the greater arc be evenly 
 divided, e.S. contains one-half of those even divisions, 
 together with one-half of R.M. But p.^S*. also contains 
 one-half of the even divisions contained in the line 3I.S. 
 (arc-length.) Therefore, each of the even divisions of 
 the sine tS.B. is an aliquot part of the line M.S., and 
 consequently Ii.3I. is an aliquot part of li.S. 
 
 But the line S.3I. is divided into ten equal parts ; 
 and Ii.3I. is known to contain either the whole or very 
 nearly the whole of one of the divisional parts. If R.3I. 
 were equal to the half of one of the ten divisional parts, 
 then would *S'.7i. be equal to nine and a half of those 
 
 S Tt 
 parts ; that is, — ^— would then equal ll.M. But since 
 
 M.M. is known to equal considerably more than the half 
 of one of the parts and is shown to be an aliquot part 
 of S.Ii.y it must equal the wliole of one of those parts. 
 Therefore, since li.M. is an aliquot part of S.R., R.3I. 
 must be one of the ten equal divisional parts of S.3I. ; 
 imdiS.ll, (equal to S.M. — il/.T?.) must evidently contain 
 nine of those equal divisions. AVherefore, the sine U.S. 
 is shown to contain nine equal divisional parts each of 
 them equal to Fi.M. the ditf. of the sine and arc -length. 
 
 (Fig. 14.) — Construction. 
 
 With centre A. and radius A.B. describe the quad- 
 rant B.M.F. Bisect the quadrant in the point M., and 
 through M. draw A.K., the secant. Draw, also, M.N', 
 the sine of the arc B.31., intercepting A.B. at iV. 
 Divide A.N. into ten equal parts, and divide also N.B. 
 into ten equal parts, at the points of equal division, 1.2.-3. 
 4.5.6.7.8.9., respectively: — 
 With radius 9.9. describe a quadrant of one-tenth less 
 
 magnitude than B.M.F. 
 
22 
 
 THE CIRCLE AND STKAIGIIT LINE. 
 
 With nulius S.8. describe a qiiaJnint of two-tenths less 
 
 niagnituile tlian B.M.F. 
 With riidiiis 7.7. describe a quadrant of tiiree-tenths less 
 
 magnitude than B.M.F. 
 
 And so on. Finally : — 
 
 With radius 1.1. describe a quadrant of nine-tenths less 
 
 magnitude than B.M.F. 
 
 To each successive arc draw the tangent an<l cotangent 
 of the arc, and also from each of the points in the line 
 M.N., where the line is intersected by the successive 
 arcs, draw a perpendicular through the line B.K. 
 
 We now obtain three distinct divisions of B.K. ; 
 
 (1) the successive tangent-lengths dividing B.K. into 
 ten equal parts ; 
 
 (•3) the successive arc-lengths dividing B.O. into ten 
 ecjual parts ; 
 
 (3) the successive sine-lengths dividing B.D. into 
 ten equal parts. 
 
 Taking the radius .-l.i. = 10. The quantitive relations 
 of the tiiree lines are B.K. = lO'OOOOOO 
 
 BO. = 7-8oG74:3 B.D. = 7-0710()S 
 And these relative proportions are precisely the same in 
 the least as in the greatest of the arcs, and however far 
 the reduction in size may be carried, so long as it be pos- 
 sible to describe a similar are, or to imagine a similar arc to 
 be described, tiiese lines pertaining to the arc, and wdiich 
 even in imagination are inseparable therefrom, must neces- 
 sarily have the same relative proportions each to each and 
 each to the arc. Referring to Fig. 11, we observe that so 
 soon as the quadrant B.M.F. commences to roll upon the 
 line B.E., the (upper half quadrant) arc M.F. commences 
 at M. to i>ass through the line M.S. ; when the rolling 
 process is completed, and the point 31. becomesin contact 
 on B.F., the entire arc has passed through the line M.S., 
 and every component part of the arc has passed through the 
 line at the same angle ; therefore if we suppose a number 
 of perpendiculars drawn through the line M.S. at any very 
 
THE CIRCLE AND STIUIGIIT LINE. 
 
 33 
 
 minute distances from each other, each jiroportionally 
 minute part of the arc in passing through tlie line M.S. 
 would form vvitii that line and tiie perpendicular (which 
 cuts the line where the minute portion of the arc com- 
 mences to pass through,) the figure H.O.S. on a scale pro- 
 portionate to the minute magnitude of the lines 
 compounding it. Now when the point M- of the 
 arc has arrived at 0. the straight * longitudinal ex- 
 tension of the arc is contained in the length of the 
 line — to wit, in (O.T.) li.S. Every portion of the 
 arc, therefore, relatively to the straight longitudinal 
 measurement thereof on the line O.E. suffers a dimi- 
 nution in length in the ratio of U.S. to 3I.S. Con- 
 sequently a manifest relationship becomes apparent be- 
 tween the horizontal and perpendicular lines 3I.S. and 
 ii.O., which must be so proportioned to each other as 
 to result in the arc length O.S. containing as its sine 
 the horizontal length It.S. Tlie quantitive relationship 
 expressed in figures makes this relationship readily ap- 
 parent, 
 
 for let A.B. the radius, - 10, then : — 
 3I.S. the arc-length, = 7-S5G74 
 U.S., the sine, = 7-07107 
 
 11. 0. the versed sine = 2-92S93 
 Now R.S. + E. 0. = A. B. 
 thus, -7071068 x 10 = 7-071068. 
 •7856742 
 
 •7856742— 
 
 10 
 
 X 10 = 7^071068 
 
 And 7-071068 + 2-92893 = 10-00000 
 
 The quantitive relationship of the other principal lines 
 which occur in these figures (Figs. 11 and 14) may be 
 briefly noticed. The secant A.K. is equal to the chord 
 
 • "Wo mean by this expression — the longitudinal space 
 between two perpendiculars, the one drawn through the 
 point at one extremity of the arc, and the other perpendi- 
 cular drawn through the opposite extremity of the arc. 
 
24 
 
 THE CIKCLE AND STRAIGHT LINE. 
 
 of the quadrant which, relatively to the radius * as 10, 
 contains V200 = 14.14214 (equal to twice the sine of the 
 arc B.M.) and, therefore, if divided into nine equal parts, 
 the quadrant contains ten equal parts, each equal to each 
 of those nine parts (Coroll. prop. D.) Again, the differ- 
 ence of the radius A.M. and secant A.K., namely M.K., 
 is in the same proportion to M.D. as A.M.: A.N., io 
 wit, as 10: 707107. Therefore, il/./iT. = 4 14214. (?". e. 
 V200-10). if.i). = 2. 92S93. . . The numerical values 
 of these quantities of magnitude in units of radius exhibit 
 their relationship, thus : U.S. + li.Cl = A.B. (or B.K.) 
 7-07107 + 2-92S93 = 10 
 
 31. K. X 31. N. 
 
 B. K 
 
 = 31. D. 
 
 7-07107 
 4-14214 X — — — - 2-92S93. 
 
 M 
 
 MATIIEJIATIUS Ax\D THE ART OF COMPUTATION. 
 
 The fundamental character of the relation of the circle 
 to the science of number and quantity is established by 
 demonstration that tlie difference of the quadrant and the 
 chord of the quadrant, (of the arc and sine of the arc) is 
 an aliquot part of the quadrant and of the chord, and 
 that tlje number of those equal parts contained in the 
 chord being nine — the quadrant contains ten ; because 
 herein we find conclusive evidence that the (so-called) 
 Arabic system of notation is not an artificial human 
 contrivance, but a great natural fact of a primary 
 character, a fundamental part of the Science of 
 
 * It may bo observed that taking the radius = 1. Tho 
 chord of tho quadrant becomes V2 and the sine of the half 
 quadrant becomes Vi- 
 
 ! '1- 
 
THE CIKCLE AND STKAIGIIT LINE. 
 
 25 
 
 Creation. "When this is well understood it will be 
 only necessary, in order to appreciate in some mea- 
 sure the immense number of facts of a secondary 
 character, belonging to the science of Kiimber and 
 Quantity, furnished by the correlation of the lines 
 compounding (or belonging to) the circle, to consider 
 the relationship of the few primary lines — exhibited in 
 their numerical values furnished above — in connection 
 witli the method shown in Fig. 14, of reducing the arc 
 B.M. from a radius equalling 10. to the one-tenth thereof 
 by a gradation of similar arcs, through the nine successive 
 intermediate magnitudes. It is to be noted tliat each of 
 these lesser arcs has its own dependent lines with the same 
 respective proportions each to each as the lines belong- 
 ing to the greater arc, and that since each lesser arc has 
 a known deiinite ratio to tlie greater arc, each* and every 
 line belonging to each lesser arc has a known definite 
 (|uantitive (numerical) ratio to each and every line 
 belonging to the greater arc. Again, this primary divi- 
 sion of the arc of given magnitude (tluit of the arc with 
 radius = 10 into units,) may be subjected to subdivi- 
 sion, and then, to further subdivision ; each division 
 turnishing an additional series of quantitive representa- 
 tives or members, each member of the series having its 
 own system of compounding lines, with their definite 
 ratios each to each, and each to the radius of that 
 member, and each member filso having a definite known 
 ratio to each of the other members of that series, and, 
 through the primary member of that series, having a defi- 
 nite known ratio to the general primary, — that is, to the 
 primary arc or circle of given magnitude ; and, througli 
 the general primary, having also, a known definite ratio 
 to each of tlie members, and to each and all the definite 
 <li visions and subdivisions of the primary circle. 
 
 Or which may be readily kaowu. 
 
2(; 
 
 TIIF, CIRCLE AND STRAIGHT LINE. 
 
 Now tlu'so n'liitioiis of (lio compounding linos belong- 
 ing to the circlo ;iro naturnl ftn'ts. Tliey niiiy Ix; justly 
 considered the materinl out of wliicli and with which the 
 instruments of ((uantitive analysis are to bo constructed, 
 to be improved, to be simplified, and to be, as far as 
 possible, perfe<'t(!d. 
 
 Tliat these facts are existent, and that they are not by 
 any means deei)ly ))uried, is w<dl known to many of those 
 whose occu[)ation it is to till the (/roinid, or whose espe- 
 cial duty it is so to do ; but many of those ]iersons prefer, 
 with a strange, and, as it would seem, with an increasing 
 perversity, to cultivate the thorns and thistles, leaving 
 the good seed as not wortli utilizing. 
 
 Amongst the things denominated 'thorns and thistles' 
 we are not to be understood as intending to include 
 the modern methods of matiiematical analysis. It is 
 true that in th(^ com[)Ound trigonometrical process 
 (under which general expression it is intended to include 
 collectively all of the several forms of ajiplied quan- 
 titive mathematics) we have nn instrument of great 
 vahie, a mechanism which is correctly adapted to per- 
 form its work with certainty and accuracy, and which is, 
 when properly used, perfectly reliable and trustworthy. 
 It is also true that this mechanism has performed a part in 
 the work of civilization, the value and importance of which, 
 in a human sense, it might be difficult to overstate or to 
 estimate too highly ; and it is again also trtie that the 
 lives of very many able men belonging to successive 
 generations have been employed in constructing, elabor- 
 ating and perfecting this mechanism, and that it has 
 been now brought, comparatively speaking, to a, state of 
 completeness and, almost, of perfection. ]iut all this 
 being admitted and fully appreciated, — it is no less true 
 that this mechanism is oidy a human contrivance, — it is 
 one form of appli(!ation only out of a number in which the 
 material may be applied ; it is the utilization only of a very 
 few facts where the ninnber, from which selection may ba 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 27 
 
 made, is almost unlimited. That the meclianism is of a 
 complicated and elaborate character requiring in its 
 higher forms a special training, a long course of study, 
 and much practice to apply it efKciently and witli snf(?ty 
 to the purpose for wiiich it is adapted, is undenijiblc. Is 
 this a recommendation or the reverse f No doubt its 
 jipplication and use had much better be left to those who 
 have had the necessary training and are therefore al)le 
 to apply it in the proper manner : but is it to the advan- 
 tage of the community that such potentiality should be 
 confined to a very few individuals ? Is there any obvious 
 natural necessity tliat it should be so ? Wlio sliall say 
 that the few facts now utilized are t!ie best adapted for 
 the purpose of such an instrument, or that tlie form of 
 mechanism now in use is the most simple and objective 
 that can be devised. The evil of separating one division 
 of knowledge from the rest as an exclusive department 
 is manifold and unquestionable, and that it is so would 
 probably be admitted in theory by some, perhaps be 
 readily admitted by niany, who are, nevertheless, quite 
 prepared to sny to those of their fellow labourers in the 
 field of sciei ce who approach too nearly their depart- 
 ment. — ' This is holy ground — go back : thou art unfit.' Is 
 it, or is it not, true that the language of mathematics is fast 
 becoming an unknown tongue to ordinarily educated men, 
 and that tliose to whom it is known can scarcely hold 
 converse with their fellows (on any scientific sub- 
 ject) in ordinary language without a feeling of condes- 
 cension and scarcely without a feeling of impropriety! 
 No nuui's knowledge is perfect, or nearly perfect,in respect 
 even to one kind or variety of knowledge, and much of 
 the value of any one kind unquestionably consists in its 
 belonging to general science, as the part belongs to 
 the whole. Is it true that the mathematician does now, 
 in some ih'ijreOj regard his fellow-worker who is unprac- 
 ticed in the calculus and non-conversant with differential- 
 methods as but little better than a publican and heathen ? 
 
28 
 
 THE ClIJCLE AND STRAIGHT LINE. 
 
 ;i 
 
 If it be true that such a result does manifest itself in 
 any considerable degree, it may be pronounced decidedly 
 unwholesome and bad — bad for science and bad for civi- 
 lization — because mathematical knowledge is a necessity 
 to science and a necessity to civilization. 
 
 AVe have now put before the public demonstration of 
 the problem which determines the quantitive ratio of 
 the perimeter of the circle to the diameter. We know 
 that examination will show the demonstration to be 
 matliematically incontestable, and we know, moreover, 
 that tlie requisite examination cannot be now much 
 longer delayed. We are mindful, liowever, that there 
 has been for a long time past a trigonometrical (supposed) 
 solution of the problem, which is looked upon by many 
 persons, educated in this department of science, as a 
 mathematical determination of the question ; so that 
 when those persons are asked to examine and consider 
 a geometrical solution of the problem, they reply by 
 requiring to have that which they consider to be adverse 
 demonstration disproved before they are willing to pay 
 any attention at all to tlie alleged geometrical demon- 
 stration. In the circumstances of this particular case 
 there is, at least to some extent, a justification of such a 
 requisition as reasonable, because mathematicians have 
 been exposed to much and continual annoyance on this 
 subject from persons who, having neglected the legitimate 
 means of qualifying themselves to investigate such a 
 subject, have not hesitated to give trouble to and waste 
 the time of other people without first taking the trouble 
 themselves to consider whether their qualifications were 
 such as to entitle them to occupy the attention of others, 
 or to justify them in endeavouring to do so. We will 
 now, therefore, attend to this requisition. 
 
 There are several methods in which the trigonome- 
 trical process is applied, from each and all of which the 
 same result (supposed solution) is obtained. But these 
 several methods belong essentially to one and the same 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 29 
 
 process, of which they are merely variations. That 
 process consists in dividing and subdividing the arc of 
 a circle, and in measuring by trigonometry the extremely 
 small portion of the arc, which would result from a long 
 continued process of red iction by definite division. The 
 application of the process being fundamentally based on 
 an assumption or assumed axiom — that, if the arc of a 
 circle be continually reduced by continued bi-section, 
 the last remaining fraction of the arc is equivalent 
 to the fraction of a straight line. The truth of this fun- 
 damental assumption we deny. In regard to the trigono- 
 metrical result, our objection is to the application of 
 that result and to the inference that a correct solution 
 of the question is thereby arrived at. Tlie most satis- 
 factory way to specify our objection and to point out 
 the precise locality of the mistake, will be to take 
 Legendre's method of applying the process by inscrib- 
 ing and circumscribing polygons in and about the circle ; 
 because Legendre's formal exhibition of the process so 
 applied is generally considered by mathematicians as 
 particularly conclusive and reliable. 
 
30 THE CIRCLE AND STIUIGIIT LINE. 
 
 Legendre's Geometry, (translated by D. Brewster). 
 Book v.— Prop. XIII. Problem. 
 
 " The surface of a regular inscribed polygon and that of 
 a similar polygon circMmscrihed, being given; to find the 
 surface of the regular inscribed and circumscribed 2>olygons 
 having double the member of sides. 
 
 Let A.B. be a side of the given inscribed polygon ; 
 E.F., parallel to A.B., a side of the circumscribed poly- 
 gon ; C, the centre of the circle. If the chord A.M., 
 and the tangents A.P., B.Q., be drawn, A.M. will be a 
 side of the inscribed polygon, having twice the number 
 of sides; and A.P. + P.M. = 2 P.M. or P.Q. will 
 be a side of the similar circumscribed polygon (Prop. VI. 
 Cor. 3.) Now, as the same construction will take place 
 4it each of the angles equal to A. O.M., it will be sufficient 
 to consider A. CM. by itself, the triangles connected 
 with it being evidently to each other as the whole 
 polygons of which they form part. Let A., then, be 
 the surface of the inscribed polygon whose side is A.B.j 
 B. that of the similar circumscribed polygon ; A', the 
 surface of the polygon whose side is A.M.^ B' . that of 
 the similar circumscribed polygon : A. and B. are given, 
 we have to find A', and B'. 
 
 First: The triangles A. CD., A. CM., having the 
 common vertex A., are to each other as their ba^es 
 CD., CM.; they are likewise to each other as the 
 polygons A. and A', of which they form part : hence 
 A: A'-.: CD. : CM. Again, the triangles CA.M., 
 CM.E., having the common vertex M., are to each 
 other as their bases CA. CE. ; they are likewise 
 to each other as the polygons A. and A', of which they 
 form part ; hence A. : A.' : : CD : CM. Again, the 
 triangles CA.M., CM.E., having the common vertex i>/., 
 are to each other as their bases CA., CE. ; they are 
 likewise to each other as the polygons A', and B. of 
 which they form part; hence A.': B. : : C.A.: CE. 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 31 
 
 But since A.I), and ME. are parallel, we have CD. : 
 CM. : : CA. : C.E,— hence A : A.' : : A.' : ^.— hence the 
 polygon A'., one of those required, is a mean proportional 
 between the two given polygons A. and 2?., and conse- 
 quently A'. = ^A. X B. 
 
 Secondly.— "The altitud*; CM. being common, the tri- 
 angle O.F.M. is to the triangle CF.E. as P.M. is to P.E.; 
 but since CP. bisects the angle M.CE. we have P.M. : 
 P.E. : : CM. : CE. (Book IV., Prop. XVII.) : : CD. : 
 CA. ::A.: A'.— hence CP.M. : CP.E. : : A. : A'.— 
 and consequently CP.31. : CP.M+ CP.E. or CM.E. : : 
 A. : A. + A'. But G.M.P.A., or 2 CM.P. and CM.E. are 
 to each other as the polygons^', and B. of which they form 
 part 5 hence B' : B. : . 2 A.: A. + A'. Now A', has been 
 already determined ; this new proportion will serve for 
 
 2 A B 
 
 determining B'. and give us B'' = ^ ^^, = and tiius by 
 
',i-2 
 
 THE CIKCLE AND STRAKJIIT LINE. 
 
 means of the polygons A. and B. it is easy to find the 
 polygons A', and B'. which shall have double the number 
 of sides. 
 
 Prop. XIV. Problem. 
 
 To find the approximate ratio of the circumference to the 
 
 diameter. 
 
 Let the radius of tiie circle be 1 ; the side of the in- 
 scribed square will be V2 (Prop. III. Sch.,) that of the 
 circumscribed square wiU be equal to ttt diameter 2 ; 
 hence the surface of the inscribed square is 2, and that 
 of the circumscribed square is 4. Let us therefore put 
 A. = 2 and 5. = 4 ; by the last proposition we shall find 
 the inscribed octagon ^' = V8^ 2-8284271, and the cir- 
 
 cumscribed octagon -g.'^ 2 + V8 ^ 3-3137085. The m- 
 
 scribed and the circumscribed octagons being thus deter- 
 mined, we shall easily, by means of them, determine the 
 polygons having twice the number of sides. We have 
 only in this case to put A = 2-8284271. 5. = 3-3137085; 
 
 2 A B 
 
 we shall find yl' = V-'l.^- = 3-0014074, and B'.= ^ ^^, ' 
 
 - 3-1825979. These polygons of 10 sides will in their 
 turn enable us to find the polygons of 32 — and the pro- 
 cess may be continued till there remains no longer any 
 difference between the inscribed and the circumscribed 
 polygon, at least so far as that place of decimals where 
 the computation stops, and so far as the seventh place, in 
 tliis example. Being arrived at this point, we shall infer 
 that the last result expresses the area of the circle, which, 
 since it must always lie between the hiscribed and cir- 
 cumscribed polygon, and since those polygons agree as 
 far as a certain place of decimals, must also agree witli 
 both as far as the same place. 
 
 We have subjoined the computation of those polygons, 
 carried on till they agree as fiir as the seventh place of 
 decimals : 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 33 
 
 Xumbe'' of 
 sides. 
 
 Inscribed 
 poll/yon. 
 
 (Jirciimicvibcd 
 yohjgon. 
 
 4 2-0000000 4-0000000 
 
 S -2-8284271 3-313708-5 
 
 ir, ..3-0(il4()74 3-182597!) 
 
 32 3-12J44ol 3-1517240 
 
 04 3- 1305485 3-1441184 
 
 128 3-1403311 3-1422230 
 
 256 3-1412772 3-1417504 
 
 512 3-1415138 3-141G321 
 
 10-24 3-1415729 3-1410025 
 
 2048 3-1415877 3-1415951 
 
 4096 3-1415914 3'1415933 
 
 8192 3-1415923 3-1415928 
 
 16384 3-1415925. 3-1415927 
 
 32768 3-1415926 3*1415926 
 
 The area of the circle, we infer, therefore, is equal to 
 3-1415926. Some doubt may exist perhaps about the 
 last decimal figure, owing to errors proceeding from the 
 parts omitted ; but the calculation has been carried on 
 with an additional figure, that the final result here given 
 might be absolutely correct even to the last decimal place. 
 
 Since the area of the circle is equal to half the circum- 
 fierence multiplied by the radius, the half circumference 
 must be 3*1415926, when the radius is 1; or the whole 
 circumference must be 3-1415926, when the diameter is 
 1 ; hence the ratio of the circumference to the diameter, 
 formally expressed by '^, is equal to 3-1415925. The 
 number 3-1416 is the one generally used." 
 
 Of these two propositions the last is a computation * 
 
 '■"^ All computations really such (i. e. which are not merely 
 explauatory or merely the numerical equivalcntH of state- 
 monti?), may be considered the solutions of propositions in 
 the science of quantity and number. If an algebraical com- 
 putation, the proposition is cjiiantitive ; if belonging to arith- 
 metic, the proposition is numerical. 
 
 C 
 
34 
 
 THE CIRCLE AND STKAIGHT LINE. 
 
 based upon the first, or it may be considered as tlie num- 
 erical equivalent and illustration of the first, exhibiting in 
 numerical units of the radius those relations of the parts 
 which the first proposition apparently establishes. The 
 first of these two propositions (prop, xiii.) is, however, 
 quantitive,and may be also correctly considered as belong- 
 ing to the science of Quantity and Number * ratlier than 
 to that of ' Magnitude and Form.' Our objection to these 
 propositions is two-fold, by which we mean that we 
 object to two assumptions involved in these propo:;itions, 
 and which two assumptions, although nearly related, may 
 be considered distinct. 
 
 Almost at the commencement of proposition xiii., we 
 find the first assumption, to which we object, distinctly 
 stated in these words : '' If the chord A. M. and the tan- 
 gents ^1. P., B. Q. be drawn, A. M. will be a side of the 
 inscribed polygon, having twice the number of sides ; 
 and A. 1'. + P. 31. =-- 2 P. M. or P. Q. will be a :ide of 
 the similar circumscribed polygon." Here then we have 
 the statement tliat A. P. + P. M. = P. Q. To justify 
 the acceptance of tliis statement it should either be sup- 
 ported by do lonstration or be in itself manifestly true. 
 (!) Is it supported by demonstration? We refer as 
 directed to Cor. 3 of Prop, vi., which reads tluis: "It 
 is plain that N. If. + 11. T. = H. T. + T. G. = H. G. 
 one of the equal sides of the polygon," «tc. Now these 
 lines are in the same case as A. P. + P. M. = P. Q. ; 
 so that we find therein not demonstration, but an asser- 
 tion that the thing stated is a manifest fact. (2) Is it 
 manifestly true? The case in the Corollary, just quoted 
 
 * It is not meant tliat these propositions of Lcgendro nvv, 
 for this reason, less reliable or of loss value, but the dis- 
 tinction is noted as a protest against calling- things which 
 belong to different divisions of science, and which are not 
 the same, by the same name. According to the title of the 
 book, they purport to be propositions in Geometi-y. 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 35 
 
 from, is that N. II. + II. T., which is an angle and 
 indirectly defined, by the preceding and following pro- 
 positions, to be a complete angnlar figure* is equivalent 
 to H. G. which is a straight line. Similarly in proposi- 
 tion XIII., the angle A. I\ + P. M. is assumed to equal 
 the straight line P. Q., beciiuse either A. P. or P. M. 
 measured separately as a straight line is apparently equal 
 to one-half tiu; straight line P. Q. It is, therefore, 
 assumed as manifest that the vertex of tiie angle has no 
 quantitive value in itself independently of its sides con- 
 sidered as straight lines. 
 
 Fig. 1. 
 
 Yw. 
 
 Let us suppose Fig. I to be a ligure compounded 
 of straight lines placed close together, and Fig. 2 to be 
 a figure likewise compounded of angle-lines placed close 
 together. In Fig. 1, if two of the lines be considered 
 relatively to each other, they appear to be similar and 
 equal magnitudes, and on consideration of the figure, — 
 that is, of all the lines compounding the figure relatively 
 to each other, — it becomes manifest that the lines are ac- 
 tually similar and equal longitudinal magnitudes, because 
 both the sides of the figure compounded by the lines 
 are perpendicular to the longitudinal extension of those 
 straight lines, and the compounded figure is rectangular : 
 therefore, if two or more of the lines be similarly divided, 
 the divisional part or parts of the one will be equal to the 
 
 * That is — not a compound fragmentary figure formed by 
 ■'jwo lines merely placed together. 
 
36 
 
 TFIK CIKCLE AND STRAIGHT LINE. 
 
 similar divisioiuil part or jiartsof the others. In Fig. -J, 
 if the two top or bottom an^le lines be considered rela- 
 tively to each other they appear to be also similar and 
 equal magnitudes, but on consideration of the hgure, 
 that is, of all the lines compounding the ligure relatively 
 to each other, it becomes manifest that the lines, althougli 
 similar, are not e(|ual, for the lower lines of the hgure 
 are evidently lesser longitudinal magnitudes than the 
 upper lines ; nevertheless, if two of the angles compared 
 together be considered as compounded of four straight 
 lines, and the four straight lines be compared with each 
 other as longitudinal magnitudes, the difference may be 
 less than any assignable quantity ; moreover, if the equal 
 lengths of the lines be equally increased, the extremely 
 minute ditt'erence uuist be proportionately diminished, andy 
 if theequal longitudinal magnitudes be in such wise inde- 
 iinitely increased, tiie ditference willbe indefinitely dimin- 
 ished ; nevertheless, the diff. is actual, and if one of the 
 angle lines compounding the figure, however extremely 
 minute may be the breadth which that line is ii.iagined 
 to possess, be (supposed) separated into two lines each 
 having half that breadth, then must neccssarih/ the outer 
 of the two halves be of greater length than the inner ; 
 and, moreover, the under surface of the outer half must 
 necessarily be in longitudinal magnitude gre; *:er than 
 the upper surface of the inner half. Before coasiUering 
 the (1) objection further, we will go on to define the 
 character of the (H) objection which, as we have stated, 
 is in some measure distinct, and we will then consider 
 the two-fold objection as one. In Fig, 3, we have 
 the arc A.B., the sine C.B. and the tangent A.d. of the 
 are. Now, there is an assumption, which we are about 
 to explain and to which we are about to object, in 
 rexpect 1o the diminution of this figure by definite divi- 
 sion of the arc, which is at the present time generally 
 adopted by mathematicians ; and which, although not 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 37 
 
 distinctly put forward as an axiom or theorem in 
 either of the two propositions, is adopted also by 
 Legendre, and forms an essential part of the foun- 
 dation* upon whicii his apparent demonstration rests. 
 Let us suppose the arc A. B. to be bisected, and 
 the sine of tlie remaining half-arc to be drawn, and the 
 tangent A. d, to be also bisected; we shall then find 
 that both the sine and tangent in the lesser resulting 
 figure are much nearer to the arc throughout tiieir length 
 
 than in Fig, 3. If the arc and the tangent of tiie lesser 
 figure be also bisected, a still nearer approach of the sine, 
 the arc, and the tangent to each other will result, and it 
 is evident that if this process of bisection were to be con- 
 
 * The coincidence of the linos'appcars to be included in, 
 and to be partly the subject of his demonstration ; but careful 
 consideration will show that the]i5roces8 itself, having regard 
 to its intended application, is i^rimarily based upon a fore- 
 gone conclusion as to such coincidence. 
 
38 
 
 THE CIKCLE AND STflAIGIIT LINE. 
 
 tinned in like manner, the deviation of either of the 
 remaining parts of the three lines from a single straight 
 line would be but very small ; for, at the 8th bisection, 
 only the 2f56th part of the tangent A. d. would remain, 
 and this would evidently very nearly coincide vvitli the 
 remaining 2oGth part of the arc. Now, the assumption 
 whir!, we wish to specify, and to which we object, is 
 that — if the process of bisection be so continued, a small 
 part of each of the three lines will eventually remain, 
 which will absolutely coincide and become essentially 
 one line. Observation of Fig. 3 shows that the two 
 extremities of the figure A. C. B. d. are not similar ; the 
 relations of the tliree lines — the sine, the arc, and the 
 tangent — are such that if the line C. B. be moved down 
 and applied upon the line A. d., a small part of the tm-ee 
 lines at the extremity^, will appear to coincide ; but at the 
 opposite extremity B.d., although ihetwo lines C.B. and 
 A.d. w ould in that case coincide, the arc A.B. would deviate 
 by the amount of its curvature from the straight line. 
 Now if the usual method of consider'ng the result of the 
 process is reversed, and, instead of considering the 
 eventual relationship of the lines (when the process has 
 been carried very nearly to the vanishing point) with 
 reference to the extremity A. and the perpendicular 
 B, A., that eventual relationship is considered with 
 reference to the opposite extremity of the remaining 
 arc and to the radius which intercepts that opposite 
 extremity — the impossibility of an absolute coincidenct? 
 of the three lines becomes at once apparent, because — 
 however minute the fraction of the arc remainina; — so 
 long as there be a)nj arc, the radius which intercepts the 
 extremity of the arc opposite from A. can never become 
 quite perpendicular, consecp'ently the sine must ne- 
 cessarily remain, until the very last, inside the arc, and 
 the tangent must remain outside. The obvious impossi- 
 
THE CIRCLE AND STKAIGIIT LINE. 
 
 39 
 
 bility of an actual coincidence of the three lines would 
 seem to have forced itself for the moment (to be again 
 immediately lost sight of) on the attention of Legendre, 
 and it is admitted in Prop. xiv. by the words "We shall 
 infer that tlie last result expresses the area of the circle, 
 which, since it must always lie between the inscribed and 
 circumscribed polygon J * and since these polygons agree as 
 far as a certain place of decimals, must also agree with 
 both as far as the same place." Now the cumulative 
 character of any disagreement has here been overlooked ; 
 a very close approximation to equality in the three lines, 
 left remaining as the result of continued bisection carried 
 to the extreme, is quite intelligible and indisputable, but a 
 very close approximation between the inscribed polygon, 
 the circumscribed polygon, and the ai'c between them, 
 if that arc contains the eighth part of a circle, becomes, 
 when the construction of the circle is correctly under- 
 stood, a manifest impossibility. If the increase of the 
 small remaining part of the arc was merely an extension 
 thereof into a greater magnitude of similar form, such as 
 would result from increasing tlie length of the radius 
 namely, the production of a larger arc similar in form to 
 the small arc, there would then be a possibility of an 
 almost absolute agreement, such as alleged ; but a circle 
 cannot be produced in such a way; the circle may be con- 
 sidered as formed by the longitudinal union of extremely 
 small arc magnitudes similar in form to each other ; con- 
 sequently, as the compounded magnitude increases, so does 
 the deviation from the straight line, and so, also, does the 
 
 * We have put these words in italics to specify the admis- 
 sion. It is to be noted that this admission is quite irre- 
 concilable with the alleged agreement or equality of the 
 lines, because such agreement would be equivalent to abso 
 kite coincidence of the lines. 
 
 J - 
 
40 
 
 THE CIRCLE AND STKAIGIIT LINE. 
 
 amount of difference between the arc and straight line, 
 continually accumulate. Tiie difference which expresses 
 the disagreement in actual length of the small lines, 
 is a part of the relation inform of those lines, which in- 
 creases proportionally to the increasing development and 
 magnitude of the figure. If the small part remaining as 
 the result of bisection be the 100th part of the half- 
 quadrant * then the difference between the sum of the 
 one hundred small sines, the sum of the one hundred 
 small tangents, and the half-quadrant itself will be one 
 hundred times greater than the corresponding difference 
 existent in the small arc between the similar lines ; 
 or, if the small arc be the one-thousandth or the one- 
 millionth part of the half-quadrant, then must the 
 difference between tlie sum of the sines and of the tan- 
 gents of all the combined small arcs, and the half-quadrant 
 itself be one thousand times, and one million times res- 
 pectively, greater than the difference between the corres- 
 ponding lines, to wit — between the sine, the tangent, and 
 the arc-length, belonging to the small arc. 
 
 * Tlie circle naturally divides itself into eight parts com- 
 pared to an inscribed or circumscribed square, because the 
 four quadrants are each divisible into two half-quadrants, 
 each of which has the same relative relation to the side of 
 the square as the relation of the other half ; so that the eight 
 parts of the circle relatively to tlie sides of the square are 
 similar and equal each to each, but if further division be 
 made, then such equality and similarity in form relatively to 
 the side of the square will be no longer obtained, because the 
 curvature of the diminished arc will be less than that of the 
 part divided off from it. The non-appreciation of this fact 
 has much to do with the erroneous conclusion supposed to 
 have bt ^n demonstrated ; it is, indeed, in the distinct and 
 thorough api^rehension of this relationshii) that an explana- 
 tion of the pre' 'je character of the fallacy is to be found. 
 
THE CIRCLE AND STKAIGIIT LINE. 
 
 41 
 
 8ome of those persons who are reasonable enough to 
 feel that to nominally define a thing by calling it an abs- 
 traction does not define it at all, prefer to adopt the expres- 
 sion 'surface' whereby to denote a line, * and define a 
 geometrical line as 'the surface of a geometrical figure.' 
 Now the surface of a thing must either belong to and be a 
 part of the thing or must be outside of it . . . in regard to 
 the circle, for instance, the surface must either belong 
 to and be a part of the area of the circle or it must be 
 outside the circle. If supposed to be within the circle 
 then must the circumscribed polygon, compounded by 
 the union of the small tangents, (if the circumscribed 
 
 * Strictly speaking, according to the presently accepted doctrine, a 
 line is the extremity oi' a f-uperliciep. It' a superficies were to be defined 
 as a real surface, compounded of real lines, the expression would not, 
 we opine, be subject to objection ; a line night be then considered as 
 one of the extremities of the surface, or as a section of, or as one of 
 the elementary constituents of the surface. As the (socalled) defini- 
 tion now stands, if we attempt to directly cognize it as an intelligible 
 idea, we find ourselves almost immediately enveloped in a network of 
 contradictions. (1) The surface is either compounded of lines, or it 
 is not compounded of them. — If it is, how can that wliich liath breadth 
 be compounded of tliat wliich is without breadth? If it is not ; what 
 then is that surface of wliich the line is the extremity, and of which, 
 nevertheless, tlie line is not a part? of what then is that surfacv com- 
 pounded, and is the surface itself a part of, or does it belong to any- 
 thing? But the difficulty (dilemma) is of a still more refined charac- 
 ter.— " The extremities of a line are points," which are negatively 
 defined to be nothing. Now, tlie relation of tlie line to the surface ie 
 defined to be similar to the relation of the point to the line : it there- 
 fore seems to follow that since tlie line lias nothing for its extremities, 
 th« surface likewise has notliing for its extremities. How are we to 
 cognize the idea of a surface without any extremities ; or, of a surface 
 which hath length and breadth together with extremities wliich, 
 according to the definition, certainly have no breadtli and probably 
 liave no length, for the supposition of a line without extremities po8- 
 seesing length is, if it have any meaning, the negative suppo' tion of 
 a non-existent line which Jiiight be possessed of length if it were 
 existent. 
 
 (See, in the Appendix, Dr. Simeon's explanatior of a superficies, 
 and illuBtration of Euclid's dogma.) 
 
41.' THE CIRCLK AND STKAIGHT LINE. 
 
 polygon be a continuous and not a fractional figure) 
 be necessarily greater than the surface. And if the sur- 
 face be outside the circle, then must the surface be greater 
 than the sine, which is always within the circle and 
 therefore less than the circumscribing polygon by which 
 the circle is surrounded. 
 
 The objections to Legendre's propositions, as a sup- 
 posed demonstration of the ratio of the circle to the dia- 
 meter, are' — 
 
 First. — Evidently, he assumes the circumscribed poly- 
 gon to bo a simple continuous figure ; because the appli- 
 cation of his propositi* > is based upon a comparison of 
 the polygon with the of he circle which is a simple 
 continuous figure, and f the polygon be not simple and 
 continuous his demonstration fails, since, in that cnse, the 
 reasonableness of the application is not shown. It is ob- 
 jected that the case is not in fact as he assumes it to be ; 
 the circumscribed polygon is not simple and continuous, 
 but compound and fragmentary . . for, if a straight line 
 be bisected and the two halves thereof be placed together, 
 so that their adjoining extremities form the vertex of an 
 angle, it is manifest that the quantity of length contained 
 in the exterior surface of that angle must be, measured as 
 a continuous surface, greater than the sum of the two 
 lines, measured separately and taken together ; because 
 the interior surface (within the angle) must be equal 
 to the lengths of the two lines taken together, and the 
 exterior surface of an angle, considered as a complete 
 figure, is evidently greater than the interior surface. 
 
 Second. — The inscribed polygon is assumed, if the 
 process of bisection be continued, to become eventually 
 coincident with the circle. It 13 objected that this is 
 impossible, for so long as any arc remains, however 
 minute that part may be, the sine of the arc must be 
 within the arc ; and, if this be admitted, (which it must 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 43 
 
 necessarily be) it then follows that, in comparing the halt 
 quadrant with the radius, the difference, because cumu- 
 lative, must be increased proportionally to the magnitude 
 of the half-quadrant compared with that of the minute 
 arc. 
 
 Some mathematicians are disposed to require in this 
 case, in addition to adverse demonstration and to reason- 
 able objection, the explanation of an apparent difficulty 
 of a particular kind. It is said : — there can be no doubt 
 that this numerical quantity found by Legendre and 
 others as the ratio of the circle to the diameter, does 
 represent the ratio of a quantity which has some actual 
 and significant relationship to the circle ; that it must 
 be so, is established by indirect evidence, for the very 
 same quantity appears as the notable result of quite a 
 number of different computations connected with the 
 circle. Explain, therefore, what this quantity really is. 
 What is the relationship of this quantity if it be not in 
 f^-ct that assigned to it by Legendre and others ? 
 
 The answer to this requisition is : — The quantity in 
 question, namely — '78539 .... is the sum of the sine- 
 lengths of all the elementary arcs contained in the half- 
 quadrant, and into which the half-quadrant may be 
 divided by the continued process of bisection. In other 
 words, 78-539 .... represents the length of the sine 
 belonging to the extremely minute (ultimate) arc, which 
 results from the continued process of bisection, multi- 
 plied by the number of those minute arcs contained in 
 the half-quadrant, when the radius of the circle is valued 
 as a unit.* 
 
 The origin of the error in Legendre's method, as well 
 
 * This fact, with its precise significance, may become more 
 distinctly apparent by consideration of the process of the continued 
 duplication of the continually bisected arc — an explanation of 
 which will be found in the Appendix. 
 
44 
 
 THE CIRCLE AND STiOAIGHT LINE. 
 
 as in the application of the process known as tluit of 
 continued bisection, is in tlie omission to observe that 
 comparison has to be made between a continuous curved 
 hne (tlie circle) and a continuous straight line (the 
 diameter.) The circle may be divided into four equal 
 parts, and the inscribed (or circumscribed) square is then 
 equally and similarly related to each of those juirts ; and, 
 further, each of the quadrants and each side of the 
 square may be bisected, and still the same equal 
 and similar relationship will exist between each of the 
 eight parts of the curvilinear figure and each half-side of 
 the S((ii;ire; but, if division be carried further, the rela- 
 tionship will no longer continue similar and equal. If 
 the half side of the scpiare were to be broken up into its 
 (ultimate) component parts, and these arranged as an 
 inscribed (or circumscribed) polygon, each minute part 
 would be related to its arc — similarly to each of the 
 other minute parts ; but the polygon, being fragmentary 
 and non-continuous, would not be in the same case as, 
 and would not admit of indiscriminate comparision 
 with, a continuous straight line. 
 
 Fur the purpose of defining the characteristic distinc- 
 tion between a straight line and an arc of a circle, and, 
 at the same time, indicating the boundary betweeil the 
 conceivable and the inconceivable — between the reality 
 of Science and unreality of Metaphysics, — we will notice 
 here the word ' infinite-' In the first place it may be of 
 service to call attention to the foolish and mischievous 
 manner in which this word is becoming more and more 
 frequently used. The word 'infinite' is properly and 
 correctly used in Mathematics and in other divisions of 
 science as the opposite to 'finite.' Outside or independ- 
 ently of its use in such sense the word * infinite' has a 
 naturdly sacred character as applying to the attributes 
 of the Creator, and it is very desirable that the use of 
 the word should be restricted to its proper and correct 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 45 
 
 application. It is now used in all sorts of literary- 
 composition, — in newspaper articles, lectures, sermons, 
 &C. — in a manner chat may be termed tmsense, and which 
 would oftentimes be absurd if it we-e not calculated to 
 have a seriously mischievous effect. Perhaps, the most 
 common mistake is its use as equivalent to the expres- 
 sions — 'indefinite,' which metaisthat which is finite, but 
 of which the limit is not, or cannot be, defined — and 
 ' immeasurable,' which means that which cannot be 
 measured in consequence of its exceeding greatness. 
 
 ' Infinite ' means boundless, unlimited, endless, con- 
 tinual extension, &c., &c. The word does not correctly 
 compound with words (adjectives) which express com- 
 parative extent, or in which measurement is implied. Such 
 a compound expression as ' infinitely great ' is, therefore, 
 (with exception of the theological sense) barbarous; ' infi- 
 nitely greater ' or ' wiser' is about equally so ; * iiifinitely 
 small' and ' infinitely less,' or ^infinitely more foolish' 
 are perhaps even worse. We make these remarks in this 
 place, primarily, for the purpose of obtaining the requisite 
 attention to the distinction betv^-een a circle described 
 (I) with a definite radius or (2) with a radius of inde- 
 finitely great length, and (3) a circle described with 
 (using the expression for illustration only) an infinite 
 radius. In the first there is included, because necessarily 
 resulting therefrom, tlie idea of a definite circle, of which 
 the magnitude is determined and defined by the definite 
 magnitude of the radius. Belonging to the second is a 
 circle the nuignitude of which is limited ; its magnitude 
 may be innnensely, perhaps immeasurably, great, and of 
 unknown greatness, but it is limited by the limited 
 length of the radius, whatever that length may be . . 
 for example, the radius, or radial distance, may be the 
 distance between the earth and the most distant star, 
 and, as the distance of the star is evidently finite, so 
 must the magnitude of the circle described with that 
 
46 
 
 THE CIRCLE AND STRAKillT LINE. 
 
 radial distance be limited accordingly. It may be here 
 noted that we can conceive or adopt as a conception and 
 distinctly cognize as an idea, anything of which we have 
 (obtained) certain knowledge as a fact . . . taking the 
 example just stated, or the similar case — of the earth 
 moving in its orbit of revolution around the sun, if 
 we consider the motion of tlie earth through a quantity 
 (an extension) of space equivalent to a few yards or feet 
 of terrestrial measurement, the motion would appear to 
 be in a straight hue. Compared to any merely terrestrial 
 motion, it would be almost absolutely in a straight line ; 
 and, considered merely in reference to any such motions, 
 its deviation from a straight line would be inconceivably 
 small, and would scarcely admit even of intelligible 
 expression as a comparison with any merely terrestrial 
 qnantity ; nevertheless, by a knowledge of the fact, and 
 of certain other facts with which to make comparison, 
 we are enabled not only to obtain the conception of, but 
 also to distinctly appr-'ciate the deviation and even to 
 determine and measure its amount witli almost perfect 
 accuracy ; we aj'e enabled to be perfectly sure that the 
 patli or hue of the eartli's motion, throughout tliat quan- 
 tity of space, is not in a straight line but in the arc of a 
 circle, having the sine of the arc within and the tangent 
 of the arc without, the boundary (perimeter) of that 
 circle. 
 
 Thirdly, (3) we have the idea of a straight line, or, so 
 to speak, we have the compound idea* of the arc of a 
 circle which has become a straight line. If it were con- 
 ceivable that a radius-vector might have infinite length, 
 it wovild then follow that the extremity of such a radius 
 would describe a straight line, and the straight line 
 
 * We have already stated that these expressions are used, 
 in this connection, only in ilhistration of the par cular ease 
 under consideration. 
 
THE CIRCLE AND STRAIGHT LINE. 
 
 47 
 
 might then be considered as tlie arc of a circle of infinite 
 mao'nitude. It may be that herein we have the nearest 
 approach that can be made by the human mind to a con- 
 ception of that which, if it can exist, does not belong 
 to the woild of humanity ; to a conception of the incon- 
 ceivable ; for, in what may be termed a compound nega- 
 tive sense, the mind does seem to be very nearly able to 
 distinctly cognize tlie supposition as a positive idea. 
 Havin^ certain knowlrdge of tlie straight line as a fact, 
 and apprehending the necessity that, if the supposition of 
 the infinite radius were admissible, the straight line must 
 have that relation to it wliich the arc of the circle has to 
 a finite radius — the shadow seems to assume substance, 
 and it is almost as though the mind were able to gi'asp 
 the unreal idea as a reality. T.-.t, in fact, there would 
 be no circle and the supposition as a positive hypothesis 
 does not belong to (human) science. To entertain and 
 to wih'iilly play with such an Uureal idea as a positive or 
 concrete conception is forbidden ; to do so would be to 
 leave the realm of science, and, contemning the guidance 
 and authority of reason, to enter the dark domain of me- 
 taphysics. We believe, however, it is admissible to enter- 
 tain the idea, thus far, negatively, for the express purpose 
 of defining the boundary and of clearly realizing the 
 essential distinction between tlie straight line and the 
 circle. 
 
 A very serious obstruction in the way of intellectual 
 progress has been now pointed out and clearly indicated. 
 The false statement at the commencement of Euclid's 
 great work is now unmasked and its actual character 
 made manifest as a monstrous deception of Untruth in 
 the place of that which it purports to be — a true defini- 
 tion of reality. Those who for the future are deceived by 
 it must be so by wilfully subjecting themselves to such 
 deception, So soon, however, as its false and deceptive 
 
48 
 
 THE CIRCLE AND STKAIUHT LINE. 
 
 character has become distinctly understood, those who 
 more particidarly represent Education atid Science, 
 should nuu-k its baneful influence in the vantage ground 
 it has so long occupied, and see to its speedy condemna- 
 tion and removal. Let 'the stumbling-block be taken 
 up out of the way.' 
 
 T' ' .-.- 
 
APPENDIX. 
 
 An ingenious ondoavour has been made by Dr. Simson, 
 (and others,) to give an intelligible explanation of the com- 
 pound dogma prefixed to Euclid's worl , by moans of the 
 figure of a solid cube or parallelopiped. The solid is sup- 
 posed ^o be evenly divided by a section at right angles- to 
 its sidv ,^. It is then explained (argUed) that the section is 
 the (jrii)erticles, and that the superficies cannot belong to 
 eitliei' \iiivt of the solid, because if oitnor one of the parts be 
 rem.fvci the surface still remains wit the other part ; and 
 hence it is inferred that the surface ha: no breadth ; &c., &c., 
 us follows : 
 
 Dr. Simsori's Explanation. — " It is necessary to consider a 
 solid, that Is, a magnitude which has length, breadth and 
 thickness, in order to understand aright the definitions of a 
 point, line, s-^d superficies ; for these all arise from a solid, 
 and exist in it. The boundary, or boundaries, which con- 
 tain a solid, are called superficies, or the boundary which is 
 common to two solids which are contiguous, or which divides 
 one solid into two contiguous parts, is called a superficies : 
 thus, if BCGF. be one of the boundaries which contain the 
 solid ABCDEFGH., or which is the common boundary of 
 this solid and the solid BKLCFNMG., and is therefore in the 
 one as well as the other solid, it is called a superficies, and 
 has no thickness ; for if it have any, this thickness must 
 either be a part of the thickness of the solid AG., or the solid 
 BM., or a part of the thickness of each of them. It cannot 
 be a part of the thickness of the solid BM. ; because, if this 
 solid be removed from the solid AG., the superficies 
 BCGF., the boimdaryofthe solid AG., remains still the same 
 as it was. iNor can it be a part of the thickness of the solid 
 AG., because if this be removed from the solid BM., the 
 
50 
 
 APPENDIX. 
 
 
 H 
 
 G 
 
 •^ 
 
 E 
 
 Vy 
 
 
 F 
 
 
 ^ 
 
 
 
 / 
 
 C 
 
 / 
 
 A 
 
 / 
 
 / 
 
 
 / 
 
 superficies BCGL. the houmlftry of the solid BM., does never- 
 theless remain ; therefore the superficies BCGF. has no thick- 
 ness, but only length and breadth. 
 
 " The V)oundary of a superficies is called a line ; or a line 
 "s the common boundary of two superficies that are contigu- 
 ous, or it is that which divides one superficies into two con- 
 tiguous parts : thuK, ifBC. bo 
 one of the boundaries which 
 contain the superficies AB 
 CD., or which is the common 
 boundary of this superficies 
 and of the superficies KBCL. 
 
 which is contiguous to it. this "D. 1 L 1 .!« 
 
 boundary BC. is called a line, 
 and has no breadth ; for, if it 
 have any, this must be part B 
 
 either of the breadth of the superficies ABCD. or of the 
 superficies KBCL., or part of each of them. It is not part 
 of the breadth of the superficies KBCL. ; for, if this super- 
 ficies be removed from the superficies ABCD., the line BC, 
 which is the boundary of the superficies ABCD., remains 
 the same as it was. Nor can the breadth that BC. is sup- 
 j)0sed to iiave be a part of the breadth of the superficies 
 ABCD.; because, if this be removed from the superficies 
 KBCL., the line BC., which is the boundary of the superficies 
 KBCL., does nevertheless remain: Therefore the line BC. 
 has no bruadth ; and, because the line BC. is a superficies, 
 and that a superficies has no thickness, as Avas shewn ; there- 
 fore a line has neither breadth nor thickness, but only 
 length. 
 
 " The boundary of a line is called a point, or a point is a 
 common boundary or extremity of two lines that are con- 
 tiguous : Thus, if B. be the extremity of the line AB. or the 
 common extremity of the two lines AB., KB., this extremity 
 is called a point, and has no length ; for if it have any, thiq 
 length must either be part of the length of the line AB. 
 or of the line KB. It is not part of the length of KB. ; for, 
 if the line KB. be removed from AB. the point B. which 
 is the extremity of the line AB., remains the same as 
 it was ; nor is it part of the length of the line AB. ; for 
 
APPENDIX. 
 
 51 
 
 if AB. be removed from the Hue KB., the point B., which is 
 the extremity of the lino KB., does nevertheless remain : 
 Therefore the point B. has no length ; and, because a point 
 is in a line and a line has neither breadth, nor thickness, 
 therefore a point has no length, breadth nor thickness. And 
 in this manner the definition uf a point, line, and superficies 
 are to bo understood-" 
 
 We object to this explanation ; that . .the boundary defined 
 is a space outside the 8oli<l : for (1) the boundary which sur- 
 rounds and contains a solid must be greater than and must 
 be outside the solid which it contains, and (2) the boundary 
 which separates and is common to the contiguous sides of 
 two solids is a space — which space must contain more or 
 loss breadth according to the greater or less distance between 
 the two solids (or two parts of the solid.) Let the distance 
 
 between them beany definite small quantity of sjiace the 
 
 two contiguous solids (or parts of the solid), may be then 
 removed to twice that distance, and the boundary which is 
 
 common to both will have twice the amount of breadth 
 
 or, the two solids may bo bronL,ut nearer together and the 
 distance between them lessened to th > one-half, the boun- 
 dary common to both will then have one-half the breadth it 
 previously had, or the distance may be lessened to the one 
 millionth part and then will the breadth of the boundary be 
 diminished to the one millionth part of the breadth contained 
 in the previous boundary, and so on. But now, if the two 
 solids (or the two parts of a solid) are brought into absolute 
 proximity and united into one solid (or into a whole undi- 
 vided solid,) the boundaiy common to the two solids and 
 which is outside each of the solids (and is a part of the boun- 
 dary containing each of the solids respectively) has disap- 
 peared, it is no longer between them for they are united ; 
 evidently the boundary of the solid (or part) ABCDEFGH,, 
 if it be outside that solid must be now a part of the other solid 
 (or part) BKLCFNMG. , and inversely the boundary of the 
 last, if it be outside the last solid itself, must be a part of the 
 
 fil'St. 
 
 The reasoning applies equally to the similarly related case 
 of the line in which the line BC. must be a breadth-contain- 
 ing space between the two contiguous superficies and be 
 common to both of them ; for if they be united, a boundary 
 
52 
 
 APBSNDDC. 
 
 in the aame sense is no longer exiatent ;, neither iff it con- 
 ceivable—for, to conceive such a boundary is to include the 
 idea of separation which necessitates the cognition of a space 
 measured by the distance (amount) of that separation. 
 And it is evident that the, space is the lino so conceived or 
 cognized. 
 
 Similarly in regard to the point B. If the point be the 
 extremity of the line AB., and be in the line ; then is it a 
 part of the line AB., and if the line AB, be removed, then is 
 the point B. removed with it ; for, to assert the contrary is 
 to assert that the same one point B. can at the same time be 
 in two or more different places^ which is absui'd. And again, 
 if it be neither in the line AB., nor in the line KB., then it 
 cannot be the extremity of either line, in the sense of 
 belonging to and ending the line, therefore it must be the 
 space, between the extremities of the line* and must contain 
 magnitude (breadth) limited and measured by the distance 
 which separates thcca extromities, (or a minute and definite 
 divisional part of that space may be taken to represent the 
 line; i.e., the line maybe considered as constituted by the 
 space, or as contained in the space and constituted! by a divi- 
 sional part of the space.) 
 
 THE ULTIMATE SINE. 
 
 The repeating j^rocess of duplicating the bisected arc. — The 
 result of this process is to obtain the sine length belonging 
 to the minute (ultimate) part of the arc which wouk' rcninin 
 after the process of bisection had been repeated a great num- 
 ber of times — that U, repeated until the vanishing-point had 
 been almost arrived at. It is, therefore, strictly speaking, a 
 inethod belonging to the same general process of which sev- 
 eral methods are silroady known and practised, amongst them 
 being that of Legeiidro which we have quoted. 
 
 The method which we are now about to explain has the 
 advantage, we think, of exhibiting tue facts from, which the 
 elements of the computation are derived, in a more simple, 
 direct, and readily intelligible form ; and perhaps, also, in a 
 more generally instructive and useful form. 
 
5011!- 
 
 th© 
 >ace 
 ion. 
 d or 
 
 I the 
 it a 
 en is 
 ryis 
 le be 
 »ain, 
 lenit 
 je of 
 e the 
 ntain 
 tance 
 finite 
 it the 
 ly the 
 idivi- 
 
 -The 
 
 ,nging 
 emain 
 t num- 
 )it had 
 dng, a 
 
 ih. HOV- 
 
 !t them 
 
 has the 
 ich the 
 simple, 
 so, in a 
 
FIO, 18, 
 
^•^:''r-*jm'>-Ht\ *^>->yy h&. t^taw tsf-^i^sr-d ; — li.e iiaif. quadrant 
 
 )^.» 
 
 ■;ff ris .( 
 
 ■.•tff. 
 
 ivx 
 
^t^t.^<M9^' 
 
 ■m. 
 
APPENDIX. 
 
 53 
 
 
 The method may be thus explained : — The half quadrant 
 (or any other definite fraction of a circle) being described 
 with a given radius of definite length — (1) The chord of ' t 
 arc is drawn ; through the terminal point of the arc, and at 
 right angles to the chord, a line is drawn (of indefinite length,) 
 which line is a part of the secant belonging to half of the 
 arc duplicated in magnitude. (2) The arc is bisected and 
 through the point of bisection, from the extremity of the 
 radius, a line is drawn which intercepts the secant of the 
 duplicated half-arc ; the line last drawn is therefore th e 
 chord of the half-arc duplicated, and the point at which that 
 line intercepts the secant is the terminal point of the half- 
 arc of duplicated magnitude : that is, if the radius of the 
 primary arc were to be duplicated * and a half-quadrant 
 described therewith, the point thus found would necessarily 
 bisect that half-quadrant. (See Fig. 18.) In like manner, a 
 line is drawn through the point thus found at right angles to 
 its chord, which line is a part of the secant belonging to the 
 second half-arc (thehalf of the duplicated half-arc)duplicated 
 in magnitude. Now in order to draw the chord of the third 
 arc, without actually describing the second arc, the chord, 
 or any part of the chord, of the second arc, may bo taken as 
 a radius and an arc described therewith intercepting the tan- 
 gent of the primary arc ; the arc so described being bisected, 
 the point of bisection would be necessarily a point in the line 
 drawn from the extremity of the radiis through the point 
 of bisection of the first duplicated half-arc (if the arc be des- 
 cribed and the line be drawn.) And the line being drawn 
 from the extremity of the radius through the point thus 
 found and produced until it intercepts the secant last draAvn, 
 the point of interception will be the terminal point of the 
 second half-arc quadrupled in magnitude, and the line join- 
 ing that point with the extremity of the radius is the chord 
 belonging to the second half-arc quadrupled. In like man- 
 ner the process may be carried on so long as any space 
 remains between the tangent of the primary arc and the 
 terminal extremity of the arc last drawn. 
 
 * Meaiiiug the radius duplicated by doubling the distance of the 
 centre, with which the arc is described, from the point at the original 
 extremity of the arc. 
 
64 
 
 APPENDIX. 
 
 The foregoing is a brief explanation of the general method 
 which, however, admits of several variations by means of 
 which the construction may be made to more completely 
 exhibit and illustrate the application of the method to the 
 purpose of measuring and investigating the lines pertaining 
 to the successive {rrcs. Fig. 20, for example, in addition to 
 the preceding explanation, which is illustrated therein, shows 
 a convenient method of obtaining the terminal parts of 
 the radii belonging to the successive arcs respectively. 
 
 In order to make the reliability and computative value of 
 the method distinctly apparent, we will now, taking the 
 radius equal to 1 0, and commencing with the half quadrant, 
 carry the computation through eight successive bisections, 
 and then compare the results with those obtained by Legen- 
 drefrom the process of inscribing and circumscribing poly- 
 gons. 
 
 (Note. The square root of ' the square of the radius less 
 the square of the sine,' deducted from the radius, gives the 
 versed sine.) 
 
 The square of the sine together with the square of the 
 versed sine give the square of the chord. 
 
 C C 
 
 The chord of (the half quadrant) -equals the sine of -j^ 
 
 duplicated; and so on. 
 
 8 
 
 16 
 
 Eadius =10. 
 
 The half. ^ 
 quadrant 
 
 R = 10 
 
 8 
 
 Sine 
 
 V.S. 
 
 T- 071068* = 50-000000 
 2-92893» = 8-578631 
 
 Chord 7 •653667* = 58-578631 
 
 Ist Bisectn. 
 R = 20 
 
 C 
 
 16 
 
 Sine 7-653667* 
 V.S. 1-5224P 
 
 58-578631 
 2-317732 
 
 Chord 7-803612* = 60-896363 
 
 2nd Bisectn 
 
 E = 40 J 
 
 3rd Bisectn. 
 E = 80 
 
 itn 1 £ 
 
 32 
 
 _C 
 64 
 
 ^ Sine 
 
 V.S. 
 
 7-803612* 
 •76859* 
 
 60-896363 
 •590745 
 
 Chord 7-84137' =- 61-487108 
 
 Sine 7-84137* 
 V. S. -38522* 
 
 61 •487108 
 •148394 
 
 Chord 7 '850827* = 61 •635502 
 
APPENDIX. 
 
 55 
 
 4th Bisectn. 
 
 5th Biscctii. 
 
 6th Bieectn. 
 
 B, = 160 128 
 
 . rSine 7 •850827' = 
 
 c |v. s. -mtr = 
 
 [chord 7-85319* - 
 
 ^Sine 7-85319' = 
 
 C 
 
 R = 320 256 
 
 V. S. 
 
 c 
 
 E-640 512 
 
 ^ 
 
 0964' 
 
 Chord 7-85379' 
 
 rSine 7-85379' 
 V. S -0474' 
 
 61-635502 
 •037249 
 
 61-672751 
 
 61-672751 
 -009293 
 
 61-682044 
 
 61-682044 
 •0022457 
 
 Chord . 7-.J553.934' = 61-6842897 
 
 The following is in correction of an error in the last part 
 of the table on j.age 55. Appendix, to Part Second. 
 
 ( Bine 
 6th Bisectn. 1 C V. S 
 
 E = 64o| 
 
 7-85379' 
 -0482' 
 
 512 
 
 7th Bisectn. ^ C 
 E= 1280 J 1024' 
 
 8th Bisectn. \ C 
 R = 2560 J 2048 
 
 rSine 
 
 V. s. 
 
 7-85393' 
 •024^ 
 
 ^ Chord 7-85396' 
 
 rSine 7-85396' 
 V. S 
 
 ^ Chord. 
 
 61-682044 
 •0023232 
 
 Chord 7-85393' = 61-6843672 
 
 61-6843672 
 -0005808 
 
 61-6849480 
 61-6849480 
 
 IZO 
 
 = i-esouozo 
 
 ■i - anubzi 
 
 l^«th 
 
 256 
 
 = 7-853193 
 
 7-85319 
 
 256th 
 
 512 
 
 = 7-853785 
 
 7-85379 
 
 512th 
 
 1024 
 
 = 7-853932 
 
 7-85393 
 
 1024th 
 
 2048 
 
 = 7-853969 
 
 7-85396 
 
 2048th 
 
 It is sufficiently evident that the two methods of computa- 
 tion are precisely equivalent in their results. To correctly 
 appreciate the true relationship and significance of that re- 
 
 * The tabulated figures of Legendre's computation will be found at page 33. 
 
54 
 
 APPENDIX, 
 
 The foregoing is a brief explanation of the general method 
 which, however, admits of several variations by means of 
 which the construction may be made to more completely 
 exhibit and illustrate the application of the method to the 
 purpose of measuring and investigating the lines pertaining 
 to the successive tn*cs. Fig. 20, for example, in addition to 
 the preceding explanation, which is illustrated therein, shows 
 a convenient method of obtaining the terminal parts of 
 the radii belonging to the successive arcs respectively. 
 
 In order to make the reliability and computative value of 
 
 ,1-v 
 
 *U- 
 
 R = 20 J 
 
 lO 
 
 3rd Bisectn, 
 E = 80 
 
 I. Chord 7-803612» = 60-896363 
 
 2nd Bisectn 1 [ 
 E = 40 J " 
 
 C 
 
 32 
 
 _C 
 
 64 
 
 Sine 7 •80361^' 
 V. S. -76859" 
 
 60-896363 
 •590745 
 
 Chord 7-84137'' = 61-487108 
 
 r Sine 7-84137* 
 V. S. -38522* 
 
 61-487108 
 •148394 
 
 Chord 7-850827* = 61 -63^502 
 
APPENDIX. 
 
 55 
 
 4th Bisectn. 
 R = 160 
 
 5th Bisectn, 
 E = 320 
 
 6th Bisectn. 
 E = 640 
 
 Chord 
 
 7 -850827 
 •1927P 
 
 7-85319' 
 
 7- 85319' 
 •0964* 
 
 » _ 
 
 Chord 7' 
 
 85379=' 
 
 85379* 
 •0474» 
 
 61-635502 
 •037249 
 
 61-672751 
 
 61-672751 
 -009293 
 
 61-682044 
 
 61-682044 
 •0022457 
 
 Chord 
 
 fSino 
 
 V. S. 
 
 853934* 
 
 ■ 853934* = 
 - 0224* = 
 
 61-6842897 
 
 7-853966* 
 7-853966* 
 
 61-6842897 
 -0005017 
 
 61-6847914 
 61-6847914 
 
 7th Bisectn. 1 C 
 
 E= 1280 I 1024 1, , 
 J t Chord 
 
 }f Sine 
 C J V. s 
 t Chord 
 
 T ^.ompare Legendre's computation * according to the 
 method of inscribing and circumscribing polygons : Since 
 he takes the whole circle and the diameter as unity, his 
 figures must be multiplied by 10 and divided by 4. 
 
 The inscribed polygon of Le- The bisected are duplicated. The 
 
 fendrp's method ; multiplying the sine lengths belong to the succes- 
 gures by 10, and dividing by 4, sive arcs of bisection when in- 
 to compare with the half quad- creased in magnitude to equal 
 rant and with R=10. the half quadrant. 
 
 The number 
 of sides. 
 
 
 The lengths. 
 
 The lengths. 
 
 The fractional parts 
 of the circle. 
 
 8 
 
 = 
 
 7-071068 
 
 7-071067 
 
 The 8th part. 
 
 16 
 
 = 
 
 7-653668 
 
 7-653667 
 
 16th " 
 
 32 
 
 = 
 
 7-803613 
 
 7-803612 
 
 32nd " 
 
 64 
 
 = 
 
 7-841371 
 
 7-841371 
 
 64th " 
 
 128 
 
 = 
 
 7-850828 
 
 7-850827 
 
 128th " 
 
 256 
 
 = 
 
 7-853193 
 
 7-85319 
 
 256th " 
 
 512 
 
 r= 
 
 7-853785 
 
 7-85379 
 
 512th " 
 
 1024 
 
 r= 
 
 7-853932 
 
 7-85393 
 
 1024th " 
 
 2048 
 
 z= 
 
 7-853969 
 
 7-85396 
 
 2048th " 
 
 It is sufficiently evident that the two methods of computa- 
 tion are precisely equivalent in their results. To correctly 
 appreciate the true relationship and significance of that re- 
 
 * The tabulated figures of Legendre's computation will be found at page 33. 
 
5« 
 
 CONCLUSION OF APPEXDIX. 
 
 Bvtlt it is necossary to |>articularly note that tho radius of each 
 successive arc is of increased magnitude; in such wise that 
 the seventh arc, resulting from the process, is oomparallo 
 with its radius which equuls 1280 ; whereas the primary arc 
 is comparable with its radius which equals 10. Therefore, 
 although the sscventh arc is absolutely ihe eame length as the 
 primary arc and may bo, in that sense, conoctly considered 
 as the same arc fi om which the greater part of its curvature 
 has been elimin Ued, yet relatively to the radius and thero- 
 foiorelati ,'ol\ tc the complete circle (or to the half quadrant) 
 the seventh arc 's the 128th part ol the primary arc divided 
 off therefrom by repeated bisection and, relatively to the tan- 
 gential straight line, dissimilar therefrom in form. It now 
 clearly appears thLt, although the length of the sine belong- 
 ing to the seventh arc approximates to the length of that 
 arc, since 1:^8 of these fractional arcs must be combined in 
 order to reproduce the ixalf-quadrant, the difference between 
 the sine and arc length of the seventh arc whatever that 
 difference may be, is sub'oct to multiplicaf" .m by 128, in 
 order to obtain the difference represented by ^ he ratio of that 
 sine-length when increased b^ 128 mognit'ides, to the arc- 
 length of the half-quiidrant. 
 
 An advantage of this method for theptirpose of quantitive 
 investigation, is that the one continued comjmtation deter- 
 mines the sine lengths belonging to each of the successive 
 arcs. The arcs are al] equal in length each to each, and as 
 the curvature is eliminated from the arc by the successive 
 bisections and duplications of magnitude, the ri*tio of the 
 oine length to the arc-length of the arc continually approaches 
 equality, (i.e. until the ttltimate limit at the vanishing 
 )ioint of the circle is reached.) Much facility in carrying 
 on the computation arises from the fact, which an examina- 
 tion of the figure will make apparent, that the chord of the 
 one arc is the sine of the arc next succeeding ; for example, 
 the chord of the primary arc of 45 degrees is the sine of 
 th9 duplicated arc of 22 J degrees. The chord of the arc of 
 22J degrees is the sine of the duplicated arc of 11 J degrees, 
 and so on. It is to be also observed that the Bq;uare of 
 the chord length, by means of which the quantitive value 
 of the versed sine to each successive arc is obtained, is con- 
 tij^ually flirnished by the computation itself. 
 
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