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A-s n\ Tlio triantflo, of which tlio huso is (F.) F., bisoctod by tho lino X. X., is an oquiiatoral triangle, aro likowisc cquilaturul) * And tho trianglo C. F. /., in also equilateral ; and tho inve See Analysis of Fig. 31, pag •Becuuse tl.e sine o( the arc of 30^ B. «., equals halCthe radios, the triangle, of whi . X., is an oqui lateral triangle, (and thoroforo the aoveral trianf^los, which are parts thereof, I also equilateral ; and the inverted triangle e. C. A., is equilateral, &c., &c. See Analysis of Fig. 31, page 11. half the radius, the triangle, of which the base is (e.) e., is manifestly equilateral. Entered according to Act of Parliament in the year one thousand eight hundred and seventy-four, by Joen Uabkis, In the office of the Minister of Agriculture and Statistics it Ottawa. c^> MoirrHEAL.-^OHM LorsLL, Pbinteb. red ure Supplementary Illustrations. Analytical Examination by Fig. 23.-(R.) The Arc of 30 degrees (^). The Octuple-arc B.Q* (or B.K. of Fig. 24.) And the line B,b., which is a production of the churd of the octuple arc. The arc of 30 degrees B.e., t or D.e., hag its sine equal to 5-0000. If the Octuple arc B.Q. be bisected, and the secant to the half-arc be drawn, the line A.O.K.' so drawn shall intersect the line B.b. in the point bisecting the chord of the Octuple arc, and the line A.O.K.' shall be (manifestly) caual and similar to the line B.b. (Because B.b. bisects C.D.,and A.O.K. bi.sccts B.D.) The secant A.O.K.' to the half-octuple arc, bisects the tangential line B.D., and consequently the tangent to the half-octuple arc is also equal to 5-0000. And equal to the sine of the arc of 30 degrees. We have therefore the arc S.e., equal to half of the arc B.e. which is one third of the quadrant B.C. ; and also the sine of B.e. equal to one half of B.D. the tangent to the half-quadrant, and equal to the tangent of half the octuple arc. The Cosecant of the arc of 30 degrees equals 20-0000 (i.e., twice the radius) ; The Tangent = 5-7735... " Cosine = 8-66025... " Cotangent = 17-3205... (i.e., twice the Cosine.) • We have adopted this term for the moment to distinguish this arc of which the sine is to the tangent of the hah-quadrant in the ratio of 8 to 10, of -which the chord is to the tangent of the half- quadrant as 'the square-root of 80' to 10, and of which the tangent is to the tangent of the half-quadrant as 8 to 6. It is to be noted that the same (octuple) arc is the particular subject of the theorem and prop, belonging to Fig. 24, page 10, Part Third, and is further consi- dered in the following supplementary examination by Fig. 24. (R.) t B.e. of Fig. 31.— in which B.e. = V.e. = e.d. = C.d. SnPPLEMENTABY ILLUSTRATIONS. A particular arithmetical relationship is at once appa- rent on contrasting these tlirce quantities with each other... viz, the least is one third of the greatest, and the interme- diate term is the one half of the greatest, moreover the least and the greatest of these numbers form with the number 10 a proportion of wbich 10 is the reciprocal or intermediate term. Now, 10 is the tangent to the half-quadrant, and the greatest of these numbers (17o2051) is the tangent to the arc of sixty degrees. £o that, herein wo have a remarkable relation between a magnitudinal and arithmetical proportion. The inter-relation between the arcs contrasted with t' j inter-relation between the tangents is remarkable in respect to the dissimilarity in the correspondence (so to speak) ; for, in the arcs, the addition of one-half the first term gives the second, and the third term is twice the first (30°, 45° and 60°) ; but in the case of the tangent, cosine and cotangent, the third term is twice the second and three times the first. 5-7735, 8-66025, and 17-3205 ; and in the tangents, the second term is greater than the first in the same ratio that the third is greater than the second, 5-7735: 10 :: 10 : 17-3205 Scholium. — In consideiing the relation of the arc of 30" C C / — \ to the half-quadrant (-—), it should be noted that, if the chord of the arc of 30° be bisected, the one-half is the chord of an arc of 30° belonging to a circle of half the magni- tude, (i.e., drawn on one-half the scale,) and is also the sine of the arc of 15° ( -ht) cut off from the greater arc by bisec- tion. Since the arc of 15° is one-third of the half-quadrant this relationship is of especial interest. The definite divisions of the lines obtained by this con- struction are: — the line B.D. into three equal parts by perpendiculars drawn from the points z. and y. respectively. (Note. — If the tangent to the octuple arc be drawn, the dif- ference, by which this tangent is greater than the tangent to the half-quadrant, is equal to one of the three equal divi- sions of B.D.) The line B.D. divided into five equal parts by perpendi- culars drawn from the points Q'.o.t.Q. respectively. ANALYTICAL ILLUSTRATIONS. STTPPLFMENTARY ILLrbTRATIONS. The lino B.b. into fifths of v/hich b.Q. is one fifth, Q.O. is two-fifths, and B.O. is two-fifths. The secant A.D. into throe equal parts hj- the points z. and y. Note. — To assist the s iidcnt in distinguishing the trigono- metrical relation and values of those divisions, the following may bo found useful. Figs. 1. 2. 3. — In oacn of those figures the lino B.b. bisects the perpendicular a.C. ; which lino (a.C.) divides the triangle A.D.B. into two similar and equal triangles. Fig. 3., (in conn ctionwithFig. 23. (R.), indicates the harmony of the cycloinotrical and trigonometrical inter- relation of the lin-*8 and their equal divisions. Sine =.8-0000. Co-sinc =60000. F\g. 24. {R)—The Octuple-Arc B.Q. | B.K.', the tangent to the half-octup'e-arc... B.K.' : B.D.:: 5 : 10. B.Q. the chord of the octuple-arc... B.Q. : B.D. :; V80 : lO.' B.Y. equals the sine of the octuple-arc... B.Y. : B.D. :: 8 : 10. B.E. the tangent of the octuple-arc... B.E. : B.D. 8:6. (Because B.E. = B.D. + B.H.') B.E. = 13-3333, &c.— A.P. the cosine of theoctuplo-arc... A.P. ; B.D. :: 6 : 10. A.E. the secant of the octuple arc... A.E. : B.D. ;: 10 : 6 . And A.E. --^ 100 Since B.Q.'z.k. bisects A.C. and B.O.y.b. bisects CD., the part Q.'Z. of tho octuple-arc D.Q.', is equal to the part O.Z. of tho same arc. Now tho point O. is the point of bisec- tion of tho chord of tho ^respondent) octuple-arc B.Q., and therefore if the octuple-arc B.Q. be described of one-half tho magnitude, B.O. will bo tho chord thereof. But D.O. is the sextuple arc of the quadrant D.A. and responds to the • Hence the sine is to the chord of the octuple arc : : 8 : V'sIT Also — the sine of the octuple arc is to the sine of the half-quadrant ;; 8 : V 50. An d th e chord of the octuple arc is to the chord of the half- quadrant : : V 80 : the square-root of ' twice the versed sine of the half-quadrant multiplied by 10.' (i.e. V 58-5786.) 6 SUPPLEMENTARY ILLUSTRATIONS. sextuple-arc B.t. of the quadrant B.C. ; consequently the point at the extremity of the octuple-arc of half-magnitude coincides with the point at the extremity of the responding sex tuple-arc. Examination by Figs. 23. (R.), and 24. (R) The Sextuple-Arc— B.t. Sine = 6-0000. The close relation of this arc to the octuple-arc of which the sine = 80000, will bo made at once apparent by consi- dering CD. as the tangent and by taking the point C. for the original instead of the terminal extremity of the quadrant, in which case the point t. becomes the extremity of the octuple arc C.t. * The following are the comparative elements of the twa aics. The radius equalling 10. Octuple-arc. Sextuple-arc The Sine = 8-00000. . 6-00000 . " Chord t = 8-9-1427. . 6-32455 " Tangent = 13-3333, &c. 7-50000 , " Secant = 16-6666, &c. 12-50000 '« Co-sine = 6-00000 . . 8-00000 These two arcs (the Octuple and Sextuple) are also so related to the half-quadrant B.S. that the one is less by the same quantity by which the other is greater than the half- quadrant, i.e., — the point at the terminal extremity of the half-quadrant is at an equal distance between the points at the terminal extremities of the two arcs. The relationship, therefore, is : — B.S. -I- half thediff. ofB.Q. and B.t. = B.Q. B.S. - half thediff. of B.Q. and B.t. = B.t. B-Q. + B.t. = 2 B.S. . , . . = B.C. From this relation the quantitive values of the two arcs are readily obtainable and we find accordingly : — The Octuple-arc contains 534 degrees \ ar y, 2 The Sextuple-arc contains 36f degrees j Together equalling 90 degrees. »» B.C. • Evidently therefore the sextuple and tlie octuple arcs are so related that the one is the complement of the other ; it is none the less important to clearly distinguish between them. t Hence the chord of the Octuple arc is to the chord of the Sextu- ple arc : : V"P : V407 J-.. itly the gnitude ponding >f which »y couM- point C. y of the :ti*emity the two arc also so i by the :he half- Y of the oints at iionship, two arcs C8 are so none the he Sextu- Cj ^«5 ■\ / ■^ i> 'H ^ f > ^ SO hi \ -\ -:^ V^' \ v \ oo '-\ — 4"tS ^^ '^O ^ \ \ ^VJ \ ?s) ■fe/ \ v> f? !P^ o '^0 SO ::5^ ^^ ?:J cj ^ k ,* .' ". SUPPLEMENTARY ILLUSTRATIONS. Note. — For tho purpose of assisting the examination and appreciation of tho linear divisions, we furnish the follow- ing : Figs. 4. 5. 6. — In Fig. 4, tho arc B.g. is deseri' .'d with tho radius A.B., and the arc B.p. with the radius C.B. which is tho one-half of A. B. Tho tangent B.D. of the arc B.g. is divided into four equal parts in the points a.b.d. and the linos drawn from these points respectively to the central ])oint A. divide each of the arcs B.g. and B.p. into four (unequal) parts. It will be found that tho ratios of tho divisions of these arcs to each other (i.e. of each to the next) are proportionals ; thus — 4'.f. : fe. ::f.e. : e.c. :: e.c. : c.B. Consequently, fc. - g.f. = c.B. (or f.c. - c.B. = g.f.), and g.f, + c.B. = f c. That is tho arc-length of g.f, together with the arc-length of c.B. equals half the length of tho arc B.g. Now if the radius be reduced to the one- half and the arc B.p. described therewith be examined, tho ratios of the arcs will be found proportionals as before and we have p.o. : o.n. :: o.n. : n.m. ;: n.m. : m.B. ; there- fore the arc-lengths p. o. -f m.B. = o.m., and o.m. equals the half of B.p. Tho Figs. 5. & 6. are, in the first place, for the ]nirpose of showing that the same inter-relation of the divi- sions of an arc divided in the same manner — to wit, by division of the tangent into equal parts - holds good whether tho length of the radius be increased and the extent of tho arc be diminished, or whether the length of tho radius be diminished and the extent of the arc bo increased. The same figures servo to illustrate the trigonometrical relation of the radial lines drawn from A, which intersect the tan- gential line B.D. The perpendiculars at the points d.b.a. divide the line A.D. into four equal parts ; and the line A.d. into three equal parts, and the line A.b. into two equal parts. The perpendiculars themselves are in arithmetical progi-es- sion and are successive multiples of tho least. Consequently d. H- B. = b. -f - Figs. 7. and 8. nru to illustrate tho conditions which neces- sitate tho proportional inter-relations of the divisional arcs; also, in Fig. 7., we have a series of similar triangles, of which the ratios are proportionals A.c.d. : A.d.B. :; A.d.B. : A.B.D. :: A.B.D. : A.D.f. ; the bases of these triangles are therefore similarly projiortional, each to tho next. Tho two sides D.e. and e.B. of tho triangle D.e.B. aro '^■ "H QQ / / y / (I V t- i \ \ V, -if". -V X 11 "NO O so ^■>: /' \, -V v> +S0 V \ ^, -v -^^ \ \ \ \ \' \ I I C"%. y \ \^ \ V^^ \ \ ^i \ \ 0, \ / ^ \ vn ^ a I ■i- / / J5 I 8 SUPPLEMENTARY ILLUSTRATIONS. together, a mean proportional between D.B. and g.f., and, therefore, g.f. is greater than D.f. in the same proportion that D.e.B. is greater than D.B., &c. Fig. 31. — Examination of the figure. Of the quadrant B.C.... Q.C. is the complement of the octuple-arc B.Q.... B.d. (two thirds of the quadrant) » C.e.; ... C.d. = d.e. = e.B.; and C.d. + d.e. + e.B. = C.d. x 3 = C.B.... S.d. = S.e., and S.Q. = S.P. ; therefore Q.d. = ".e. Of the similar quadrant respondent through S.d, to wit, the quadrant (C.)(B.) the similar divisions are equal; thus(C.)(d.) + (d.)(e.) - C.d. + d.e.; also V.(o.) = B.e. z= (e.)(d.) = (d.)(C.), &c. Of the similar quadrant respondent through c., to wit, the quadrant A.D., the similar divisions are equal ; thus Q.'Z. is equal to Q.S., Q.'e. is equal to Q.e., and the lino K.K.' bisects the line joining Q.Q'. Of the arcs shown in this Fig. the arc of curvature des- cribed with radius h.X. intersects the central point K, of the square, which is also intersected by the arc of the sino- Icngth described with radius B.W. We note, in the first place, the remarkable relation between the square and the quadrant herein exhibited, and which may be thus stated as a theorem... that, if a quadrant be described in a square, and if the two adjacent sides of the square next the quadrant be bisected and a square half the magnitude of the first be formed by joining the points of bisection and the central point of the greater square, then will the quadrant be intersected by the two sides of the lesser square, and the two points of intersection shall divide the quadrant into three equal parts. This theorem may be said to be demonstrated by inspection of the figure — that is, in other words, it becomes manifest — but, moreover, wo have the central part of the three into which the quadrant is divided bisected by the line K.S.D. (half the secant of the half quadrant) and we are familiar with the quantitire values of the lines K.S. and S.D. of which K.S. equals the versed sine of the half-quadrant = 2'92893,.. and S.D., the difference of the radius and secant of the half-quadrant, = 4'14214... (i.e., taking the side of the greater square equal to 10-0000.) SUPPLEMENTARY ILLUSTRATIONS. "Wo may therefore, by taking the distance K.d. as a radius, describe a quadrant terminated by the points d. and e. which terminate the arc d.e. ; or, by taking the line K.S. as a radius we may describe an arc touching the arc cut oft" from the greater quadrant, at the central point thereof S. ; and the proportion of these arcs each to each and their quan- titive values may be very readily determined. Now if a fourth quadrant intermediate between the two last be described by making the radius equal to one-third of the radius A.B., this last quadrant Avill intersect the cen- tral part of the primary quadrant in two points, and if the point (K.), belonging to the primary responding quad- rant (C.)(B.), be taken as a centre and with the same radius — to wit, equalling one-third of A.B. — another second- ary quadrant be described, we then obtain Fig. 32. Fig. 32. — In this figure the line XX. occupies the half distance between the perpendicular diameters of the two (greater) circles, and since the radius of each equals the one- third of ten (3333...) those circles intersect each other on the line X.X' Setting aside for the moment the preceding demonstrations of the locality and characteristics of the lino X.X; it is at once apparent that the relative place of that line as shown in the figure necessarily belongs to the struc- tural plan of the circle, because K.S. equalf- S.T. or (S.W.), and the distance S-X- is included in the distance S.T. and the distance, also, between the lines (T.) V. and K.k'. is in- cluded in the side of the right-angled triangle of which K.S. is the base, therefore if S. be taken as the centre of a circle described with the radius S.K., that circle will intercept the point T., and if the point (S.), responding to S., be taken as the centre and a circle be described with the radius (S.)(K.), that circle will intercept the point (T.) The distance be- tween the contres K. and (K.) of the two (greater) circles is therefore necessitated and determined by the actual rela- tion of the lines. Now taking our demonstration of the quantitive value of E.X., as the arc-length of the half quadrant — to wit, 7-85674...— we have XT. = E.T. - KX- = 2-14326... But S.X., by that demonstration, equals the one-tenth of E.X i.e., -785674..., and S. K., by the figui-e, equals S.T., and also equals S.W. the versed eine of the half quadrant and of which the magnitudinal value is therefore 10 SUPPLEMENTARY ILLUSTRATIONS. 2-92S93... Ld the fact be particularly noted that 2-14326 •»- •T85674 = 2'92893. It may bo observed that, since the chord of the arc of sixty dei^rces equals the radius, several of the distances between the points indicated in this figure are necessarily equal each to each. Fig. 33. — If the responding quadrant (B.) (C.) bo located at a distance from the centre Iv. of the primary circle, a little greater than its true place as/ielermined by the arc- length of the half quadrant — to wit, by making the distance Il..9i\ equal 7-86565. . instead of 7-85674... the perpendicular W.S. produced through S. will intercept the responding quadrant (B.) (C.) in the point d' (on the line J.K.), and in that case d.'K. will be equal to d'.S., and d.d.' equal to (S.) S. And also, if from K. a peiq)endicular be drawn through the line R.T., at E., to the point e. in the quadrant B.C., the perpendicular K.o. is manifestly equal to K.d., and K.E. equal to K.d'. Hence wo obtain the formation of four squares equal each to each, and of eight parallelograms C'jual and similar each to each, as shown in Fig. 33. Of these squares, each of the sides is equal to K.d'. and the diagonal equal to K.S.... Now the quantitive value of K.S. is 2-92893... and K.d.' : K.S. :: 10 : 14-14214...; therefore K.d.' = 2-07107; but in place of a diagonal each of the parallelograms, of which the two sides are each equal to K.d.' and the other two sides each equal to d.' d., or S. (S.), contains a fraction of the quadrant B.C., and which fraction is the one-sixth part of the quadrant, (an arc therefore con- taining 15 degrees described with radius equal to 10.) Analytical Examination by Fig. 31. The arc of lb degrees ( — ) ; and the line J.K.N. The arc of 30° B.e. is bisected in the point m. ; B.m. is therefore an arc of 15". (B.m. = m.e. = e.S. B.m. + m.e. + e.S. = B.S. S.d. is also an arc of 15° and equals S.o. = B.m.) Since the line A.M.m. bisects the angle B.A.f., the lino C.d.g. similarly bisects the responding angle D.C.F. and the point M. manifestly responds to the point d., therefore the line C.d. bisects an arc of 30° and J.d., on the line J.K., is »V SUPPLEMENTARY ILLUSTRATIONS. II the tangent to an arc of 15° described with the radius C.J. which equals one half the radius A.B., and J.d. therefore eijuals one-half of B.n. the tangent to the arc B.m. But the lines A.d. and A.o., are likewise respondent each to the other, and, therefore, the point d., at the terminal extremity of the arc C.d., responds to the point c. at the terminal extremity of the arc B.C. Xow the hino of the arc B.e. = (the half of B.D.) = 5. And conse- (|uently the cosine of the arc B.e. equals 8*66025. There- fore, N.d., which responds to the cosine of the arc B.C., also equals 8-66025... and J.d. = (10 — 8-66025) = 1-33975... .Vgain N.M., it the opposite extremity of the line, responds to J.d. and also equals 1'33975. . . ; and N.L. : the sine of B.e. : : 2-88675 : 5.* Therefore, M.L., which is the distance lietwecn the points in which the line is intersected by the arc and by the chord of the arc respectively, equals (2-88675 - 1-33975) = 1-54700. The magnitudinal values of these distances having thus been with certainty ascertained, the entire line J.K.N, may bo analytically examined ; commencing from the extremity N. wo have : — N.M. = 1-33975... M.L. = l-54t00... L.K. = 211325... N.K. - K.d.' d.'S $y. y. X. K.X. x.r ($)d. d.J. 5-00000 )323... ■) L784... j2'07107 205323. •01784. •04218... •) •74349.. J 0-78567 7-85674 •74349... •04218... •01784... 1-33975... N.K.J. 214326.. 10-00000 Because N.L. is the sine of the same angle, and 288675 : 5 866025. 12 SUPPLEMENTARY ILLUSTRATIONS. Or again, by taking the (central) point X, wo have m definite parts of the line J.K. : — X.y. and X.y.' each of which equals -74349... X6. and X (8).' each of which = -785674... Xd. and Xd.' each of which = -80351 . . and finally XT. = 2-14326, and XK. = 2-85674 The definite division of the line J.K.N., thus determined evidently furnishes the means of again testing the correct- ness of that quantity of length which has been hitherto sup- posed to measure the half-quadrant, namely, -78539... It ap])ears almost needless to show here by the addition and subtraction of the figures that such quantity cannot be made to harmonize with those definite divisions of the line J.K.N., which directly result from trigonometrical measure- ment, and that, furthermore, no quantity, as representing the arc-length of the half-quadrant, can bo made to har- monize therewith other than the quantity 7-85674... As an example, let us first take this demonstrated quan- tity ; — to wit, N.X - 7-85674:— We have:- J.K. = 5.00000 K.X = 2-85674) Xd. = -80351 j = 3-66025 J.d. = 1-33975 And 2 J.d. = 2-67950, the tangent to the arc of 15° (B.n.) But assume N.X = 7-8539... Then:- J.K. = 5-0000 K.X = 2-8539... ■) X.d. = (X.$ + $d) -- •8007...} = 3-6546 J.d. 1-3454 And 2 J.d. = (2-6908) ; as the tangent to ( -g^") the arc B.n. (Note. — We have already shown that assigning 7-8539... as the arc-length of the half-quadrant, is in fact attributing two different lengths to the same line. See Part Second.) Noteworthy in Fig. 31, is the foursided figure d.e. M.H. of which each of the sides is an arc of 30°. containing the H SUPPLEMENTARY ILLUSTRATIONS. Jg central one-third of a quadrant described with radius = 10. The two longer diameters H.e. and M.d. each equals 7'3205... ; of the two shorter diameters S.U. and z.u. each equals 5-85786... Fig. 34. — If the lines A.H. and B.o. be produced through the points H. and e. respectively, until they meet each other, they will meet at a distance from the point d. equal to the radius A.B. *(or from the point J. equal to the distance N.d.) and if from the point where the two lines so meet as a centre an arc A.B. be described, the length of the radius = 18-6025... and the chord of the arc = lO.f 7 THE ARC OF 18 jg DEGREES. {The half-sextuph arc.) Fig. 34. — Bisect the arc B.P. (i.e., the sextuple arc) ; and through the point of bisection c. draw the line A.Q.m. 7 The arc B.C. is therefore an arc containing 18 vg degrees, J its linear elements are, The Sine = 3-16227... '• Co-sine = 9-48685... " Tangent... =-- 3-33333... &c. «' Co-tangent.. = 30-0000... " Secant = 10.5409... " Co-secant. . = 31-6227... (^ote.) — These figures may he compared with the elements of the are of 18°, The Sine = 3-09017... " Co-sine = 9-51057... " Tangent. .. = 3-24920... " Co-tangent. . = 30-7768... " Secant = 10-51462... " Co-secant... ,^ 32-36068... • (d.A. also equals A.B.. t Having regard to the relations of the triangle thus formed the base of which (i.e. the chord of A.B.) equals 10, it is probable that tlie characteristics of this triangle will render it of much utility if applied in the art of computation. t The sextuple arc has-been shown to contain 36 ^degrees. (Page 6.) 14 SUPPLEMENTARY ILLC8TRATI0NS. Since the secant of the nrc B.C. intersects the point Q., nt the extremity of the qimdrupio arc, D.Q., the quantitivo magnitiulinal values of the lines belonging to the arc may he very readily computed and verified ; the one-third of ten as the tangent to the arc presents itseif at once... thus : — the triangle A.Q.O. is a part of the similar greater tri- angle A.m.B. ; Now wo have A.O. = 6, and Q.O. = 2; * (As A.O. : Q.O. :: A.B. : m.B.) -TT-which represents m.B. the Therefore as C ; 2 tangent to the arc. "Wo have confined 10 these illustrations thus far to tlie lines and divisional arcs belonging to a circle described with a radius equal to 10. It is evident that by taking any one of these primary or more important lines as a radius, a series of quantitive magnitudes with similar inter-relations will be obtained, and which will have, through the radius common to all of them, a definite and known relationsliip to the lines and divisional parts of the primary cii'cle. Since some of these secondary lines may absolutely agree in mag- iiitiide with some of the primary linos or may have some very simple (qmntitive and numerical) relationship to them, it is very desirable that the relations should be investigated and classified. As a brief example we will take the line 5'85786... which equals twice the versed sine of the half, quadrant and has been now shown to be one of the most important of the primary linos belonging to the half- quadrant. If from the point B., Fig. 34. on the line B.A., a centre a. be taken at a distance = 5-85786... and from the centre a. with radius a.B. a quadrant B.c. be described, then if the quadrant be bisected in the point h., B.h. shall be the half- quadrant ; and if one-third of the quadrant be divided off at the point d. the remainder — to wit, the arc B.d. - shall be an arc containing 60°. and the versed sine of this secondary * Geometrical denionetration that the eine of the quadruple arc is to tlic tangent of the half-quadrant as 8 : 10 will be found at page 10, Part third. Fig. 24. t*> SUPPLEMENTARY ILLUSTRATIONS. 15 nrc of fiO". (closcribod with D.V.S. rndluM^.B. = 5-857?fi), shall c(iual tho vcrsctl sine of tho lialf-(jimdrnnt Itolon^ing to the j)riinnry circle, (becnutio the vornccl bine of the nrcofGO", eqimls one-half its radius). Taking, therefore, as the example, the half-quadrant B.h. we have : — The decimal Circle of the D.V.S. (^duplicated versed sine.) Radius = 5-8578G. Tho hnlf-quadrnnt. (-^) The Sine = 4-142134... " Tangent. = 5-85786... " Secant.. . = 8 -28420... Herein we have an agreement (coincidence) between cor. tain of the principal lines belonging to one civcle and cer- tain of tho principal lines belonging to another circle dif- fering from the first in magnitude ; the lines l)elonging to tho second being dissimilar * from the lines belonging to tho first, with which they agree in length. The decimal Circle of the D.V.S.f The arc of sixty degrees- ( -p ) t Tho Co-sine = 2-92893 " Co-tangent. = 3-38204 " Co-secant.. = 676408 The sine = 5-0730. " Tangent =10-1461. " Secant.. =11-7157. " Chord.. = 5-8578. (Note.) — The chc 'ds of tho secondary arcs coincide with and form a part of the chord of the primary arc if the arcs be so described that each commences from the same original point (B.) as that of the primary and agrees with the primary arc in position, which is the method adopted in the foregoing illustrations : Since the primary radius includes the radius of each of the circles, the extremity of tho sine may bo made the point of coincidence, as in some of the analy- • That is, they are constructively different in relationship. t That is, the duplicated versed sine of the primary half-quadrant. X Sine 5-073056670 taking V 50 = 7-07107. 16 SUPPLEMENTARY ILLUSTRATIONS. tical figures of oui* 'part second,' (See Fig. 12.); Or, again, the same point may be taken thi'oughout for the centres of the circles — that is, the arcs muy bo described all from the one centre — in which case the secant will be the line of coincidence. In systematic analysis it will evidently be convenient to adhere to one uniform method throughout the whole or a part of the figures be- longing to the series. Analysis by Decimal Circles. Illustrated in Figs. 35 (a.) and 35 (b.) In which Figures these different methods are applied to illustrate the system of decimal cyclometry. In Fig. 35 (a.) the tangent is made the lino of coincidence, in Fig. 35 (b.) the sine is the lino of coincidence with refer- ence to the radius A.B., and also the secant is indicated as the line of coincidence with reference to the radius A.C and illustrated by the arcs intersecting that line. The decimal system of cyclometry is illust'ated in these two figures by the three decimal circles, namely, the circle of the sine ; the circle of the duplicated versed sine ; and the circle of the secant. These lines may be termed * Capitals ' of the system ; they are all secondary to the radius A.B. which equals 10 and is the 'primary' of the system; but each of them is ' primary ' to the lines and divisional arcs belonging to the circle of which it is the radius, for instance, in the example which has just been given, the circle of the D.V.S. has for its 'primary' the line A.B. which is the radius of the primary quadrant, but the line a. B. equalling twice the versed sine of the primary quadrant) is the pri* mary to the half-quadrant, to the arc of 60°, and to all other divisional arcs, belonging to its own circle, and also primary to the lines belonging to those respective arcs. In con- structing a cyclometrical table it is evident that this system may be pursued by sub-divisions as far as may be found desirable. The immediate gain to the science of quantity and number from such a tabulated analysis of the circle would bo the increased knowledge obtained of the inter- relations of the subjects of that division of science ; for example, by taking a primary radius equal to 10, we find SUPPLEMENTARY ILLUSTRATIONS. 17 that the number 5 represents the sine of the arc of 30 de- grees; the number 8 represents the sine of the octuple arc ; and the number 6 tbo sir.n, of the sextuple arc ; wo become thereby aware of auu are able to appreciate a particular magnitudinal relationship between these numbers, and not only of each to each of these but also of each and of all of these to many other (magnitudinal quantities) numbers. Or, as another example, we may take again the number 5, which appears as the sine of the arc of 30° and the cosine of the arc of 60°, both belonging to the primary circle, and also appears as the sine and cosine of the half-quadrant belong- ing to the decimal-circle of the sine — namely, that which has the sine-length V50 as its (radius) * Capital.' Examination of these two plates, 35 (a.) and 35 (I).), together with the annexed table wtll suffice to render the cyclometrical decimal system clearly understood. Note. — In Fig. 35 (b.), wherein the line of coincidence includes the sines of all the arcs, the division of the primary radius A.B. should be noted. The entire part divided off above the line of coincidence is equal to the sine-length of the prim- ary (t'OtlOT) ; from centre A., at the upper extremity of A.B. , a part is divided off by the centre of the dec-circle J). V.S. equal in length to the versed sine of the primary ; and, ad- joining the line of coincidence — above that lino — a jiart is divided off by the centre of the circle of the secant, also equal in length to the versed sine of the primary half- quadrant. The method of analysis to which these figures belong will be better appreciated after consideration of Table II., which we furnish as an appendix, and in which the arc-lengths are included as elements of the circles. From that table it will immediately appear, for example, that the arc-length of the arc of nine degrees belonging to the circle of the sine equals the radius A.B. divided by nine ; and that the arc-longi h of the half-quadrant belonging to the same circle equals five times that quantity. il 18 SrPPLE.MENTARY ILLUSTRATIONS. In concliuling those illustrations we beg to state that they may be consiilered generally as contributions and sugges- tions towards an analytical investigation of the circle; their immediate pm-pose however is to illustrate the proposition thai the circle itself is not only a reality but is one of the primary facts of Creation, constituting, indeed, a central primary or great fundamental fact in or upon which very many of the secondary facts belonging to abstract science j have their common hosis. \ Abstract-Science nuiy be flelined as tluit division of Science whicli treats of tiie inter-relation of like subjects of Science, in respect to those general properties (number, quantity, condition, form, magnitmle,) \vliicn belonj^ to those sulrjects as existences (thinjrs). CYCLOMETHTCAL TABLE I. Decimal Series. Primary radius A.B. — 10. (1) Divisional Arcs I Half-quadrant. Arc of GO degreea. oi the prinaary. ) * ° Sine 7-07107 8-6G025 Tangent ...10-00000 17-32051 Secant 14-14214 20-00000 Co-sine 7-07107 5-00000 Chord 7-653GG 10-00000 Co-tangent 10-00000 5-77350 Co-secant 1414214 11-54701 Arc of 30°. Arc of 15 = . Sine 5-00000 2-58819 Tangent 5-77350 2-67949 Secant 11-54701 10-35275 Co-sine 8-G()025 9-G5926 Chord 5-17638 2-61052 Co-tangent 17-32051 37-32050 Co-secant 20-00000 38-63702 Octuple Arc. Sextuple Arc. ' Sine.... 8-00000 6-0000 Tangent 13-33333 7-5000 Secant 16-66666 12-5000 Co-sine 6-00000 80000 Co-secant 12-50000 1G-666G Co-tangent 750000 13-3333 Cliord 8-94427 G-32455 (2) Decimal Circle ] of tlie Sine. [■ Half-quadrant. Arc of 60 degrees. Radius = 7-07107. J Sine 5-00000 G-12372 Tangent 7-07107 12-24744 Secant 10-00000 1414213 Co-sine 5-00000 3-53553 Chord 5 41195 70710G Co-tangent 7-07107 4-08248 Co-secant 10-00000 8-16496 20 CYCLOMETRICAL TABLE I. Arc of 30'. Sine 353553 Tangent 4-08248 Secant 8-16497 Co-sine 6-123'72 Chord 3-66025 Co-tangont 12-24745 Co-secant 1414213 Octuple Arc. Sine 5-656856 Tangent 9-428093 Secant 11-785116 Co-bine 4242642 Co-secant 8-838837 Co-tangeut 5-303302 Chord 6-324555 (!?) DeciiDsil Circle of) tlieD.V.S:' [ Half-quadrant Radius = 5-8r)786. j Sine 4-14213 Tangent 5-85786 Secant 8-28426 Co-sine 4-142134 Chord 4-490456 Co-tangent 5-857864 Cosecant 8-284268 Arc of 30°. Sine 2-928932 Tangent 3382040 Secant 6-764080 Co-sine 5-573060 Chord 3032252 Co-tangent 11-146120 Co-secant 11-715728 Arc of 15' '1-830127 1-894687 7.320506 6-839313 1-845919 26-389589 27-321252 Sextuple Arc. 4-242642 5-303302 8-838837 5-656856 11-785116 9-428093 4-472133 Arc of 60". 5073053 10-146106 11-715728 2-928932 5-857864 3 382040 6-764080 Arc of 15°. 1-511121 1-569609 6-044482 5-659561 1-529207 21-794185 22-638244 That is, of the duplicated versed-sine. CTCLOMETRICAL TABLE I. 21 Octuple Arc. Sextuple Arc. Sine 4-68628 3-514716 Tangent •7-81048 4-393395 Secant 9-76310 7322325 Co-sine 3-51471 4-686288 Chord 5'23942 3-70841... Cu-tangent 4-39339 7-81048... Co-secant 7-32232 9-76313... (4) Decimal Circle ©f"| the Secant* J- Half-quadrant. Arc of GO'. Radius = 414214. J • Sine 2-92893 3-587196 Tangent 4-14214 7-174392 Secant 5-85787 8-284274 Co-sine 2-92893 t71069 Chord 3-17025 4142138 Co-tangent 4-14214 2-391466 Co-sccant 5-85787 4-782932 Arcof30'. Arc of 15". Sine 2-071068 1-096936 Tangent 2-391466 1109883 Secant 4-782932 4-439532 Co-sine 3-587199 4-001002 Chord 2-144129 1-081315 Co-tangent...., 7-174398 15-458674 Co-secant 8-284272 16-004008 Octuple Arc. Sextuple Arc. Sine 3-313712 2485284 Tangent 5-522853 3-106605 Secant 6-903566 5177675 Cosine ; 2-485284 3-313712 ChoiJ 3-704841 2-61971 Co-tangent 3-106605 552285 Co-secant 5-177675 6-90356 * We thus apply the term ' Secaut ' to the ditTerence of the Secant and Radius of half-quadraut. Since the Secant is twice the sine there is no danger of misapprehension from such application ; some other term might, however, be preferable. 22 CYCLOMETRICAL TABLE II. Appendix. We have endeavoured as much as po8sible to avoid the xmnecessary use of strange and unusual terms. It might be preferable instead of the expressions ' decimal system' and ' decimal circles ' to substitute the word ' decui)lo ' for ' decimal ' in order to express the relationship, and to writ ) * decuple circles ' belonging to the ' eyclometrieal decuple sys- tem.' A decimal sj-stem in accordance with the more custom- ary sense of the exjiression would be formed by making the successive ' capitals ' of the system the numbers 9, 8, 7, G, 5, 4, 3, 2, 1 ; that is, the radius of the first circle, to be nine-tenths the length of the ' primary,' of the second circle eight-tenths of the primarj', and so on. A quadratic decimal system might also be formed by atlopting the same 'primary' as the square root of 100, the first capital would then bo the square root of 90, the second cajjital the square root of 80, and so on. In a system so framed the ' circle of the sine ' would find its place with its cajiital as the square root of 50. For the present, however, it appears more desirable to extend the table now given by addition of other important divisional and related sub-circles ; as a first addition wo may take the half-octuple, the half-sextuple, the half-octant * and the arc of nine degrees ; we have accord- ingly :— Cyclometrical Table II. The Primary. — Radius = \■ ) Aic- ;]-27932 2-276234 277777 l-llllll Sine 31G227 2236068 2'70r)97 1-106155 Tnngont 3-5:}553 2-37023 2-92893 1-119944 Chord 3-24920 2 265379 2-75898 1-109576 Sociuit 7-91468 7-453565 765368 7-260210 Co-sine 6-32455 • 6708206 6-53282 6-984010 Co-ttuigont.. 14-14213 21-21321 17-07107 44-64487 Co-sccant.... 15-81140 22-36068 18-49467 45-20146 Dec-Circle of D. V.S.—Uudinx - 5-85 Ilalf-Octiiplc. lliilf-Sextuple. Ai-e 2-716676 1C85689 Sine 2-6197lf 1852417 Tangent 2-928930 1-952621 Chord 2-691728 1-876683 Secant 6-549278 6-174666 C.,-sine 5-239428 5-557212 Co-tangent... 11715720 17573580 Cosecant 13-078561 18-524182 7854... Arc-length of Octant = 4-G013:57... Ilalf-OcUint. C f Arc \ "40 \0{[) \ 2.301180 0-920492 2-241703 0-916368 2-426407 0-927781 2-285613 0-919198 6-340512 5-930816 5-411961 5-785738 14122213 36-985063 15 321473 37-446076 Dec-Circle of Secant. — liadius — 4 142U.. Arc-Uncith of Octant = 3-254371... Half-Uctuple. Half-Sextuple. Half-Octant. P f Arc ) 40 ( of 9'' j Arc 1-920983 1-333388 1-627185 0-650874 Sine' 1-852418 1-309856 1-585164 0-647971 Tangent 2-071068 11380713 1-715732 0-656048 Chord 1-903342 1-327029 1-616176 0-649975 Secant 4-631049 4366201 4483427 4-193751 Co-sine 3-704842 3-929579 3-826844 4-091143 Co-tangont. 8-284276 12-426414 40-000000 26-152436 Co-eecant... 9-262098 13-098599 10-823942 26-478422 i> 2i CYCLOMETRICAL TABLE 11. With respect to the two tables here given we have not the slightest doubt whatever as to their substantial cor- roctnessj; it is, however, quite possible that some arith- metical mistakes in computation may be found in them, and as to the last decimal places the figures of many of the quantities cannot bo strictly accurate because those quanti- ties are based upon the assumption that '7*0710t, is the square root of 50, and such is not exactly the square root which is 70710678... It is quite proper that the correction should be made and the figures bo furnished with strict accuracy to the last decimal place, but the published tables of natural sines, &c., now authorized, or, at least, those most in use for general reference, are based on this assumption, and it is desirable i;i the first instance to show wherein our results are in perfe(-t agreement with the results of recog- nized trigonometrical processes. The corrections may bo very easily made and more complete tables furnished ; as an example of the necessity of strict accuracy, worthy of note, we will specify the first of the elements of the half-sextuple arc belonging to the dec-circle of the secant. Table II., to wit, the arc-length thereof — which appears as 1-333388... It is most probable that strict accuracy will give the figure 3 as an interminable decimal... that is, will give the actual quantity as 10 + -3-. (Ry correcting the equivalent of the capital (radius) in the fifth decimal place the figure 8 becomes reduced to 5 in the fifth decimal place of th ex- ample, and, moreover, the half-sextuple element of the pri- mary, from which the corresponding element of the dec-cir- cle is derived, is very slightly in excess from the same cause). Fig. 36. — Is to some extent a development of Fig. 31. ; it has, moreover, for its object to bring prominently under observxtion certain triangles and squares which, in their relation each to each and to the divisional arcs and elements of the circle, may be found to possess much utility. * k CYCLOMETRICAL TABLE II. 25 Note. — The arc-lengths belonging to the first cyclomctri- cal table arc as follows : — <1) The Primary. Half-quadrant Arc of 60 degress - a u 30o . - u u 150 .. . Octuple arc -531" Sextuple arc - 36^° (2) Doc-Circle of Sine. Half-quadrant - ■ Arc of 60 degrees u u 30° - - - u <( 150 - . - Octuple arc - 53^" Sextuple arc - 36^° (3) Dec-Circle of D.V.S. Half-quadrant - Arc of 60 degrees " " 30° - - « » 150 . - . Octuple arc-53J° Sextuple arc - 36^° (4) Dec-Circle of Secant. Half-quadrant - Arc of 60 degrees " " 30° - - - " « 150 - . . Octuple arc-53|^"' Sectuple arc - 361° 7-85674 10-4756(} 5-23783 2-61891 - 9 27532 6-^i3RI6 5 05555... 7-407407... 3-703703... 1-851851... - 6-558642 4-552468 - 4-60237 6-13650 ■ 306825 1-53412 ■ 5-43337 3-77137 3-25437 4-33916 ■ 2-16958 1-08479 3-84197 2-66677 END OP APPENDIX. .i\ r * # « 1^ 1» '/ A / / / / ^k y J ;t s. i I f I \ \ / y X y y y / / / M y / "7 \ / \ / "-x-. J- / / / / / / D z^ 1+-^^ +^ J /tf' 7i 6 T -I I f^2h (R) I 1 -j j /> 6 ■ .3 20 1: 1 J # .'^ I: S>(- t' :^^^f / / Y // V- lyl ' /l I ftl // r^f,'/ f f f /,■ ■/ / /. (/■r f'f a . /r,,./'// f: ' / "rii / I Ih' f'/' /■///•/■/■< ////•/■' ,t/ . tM \'u H f, \ \ B k o> fr ; 1 ''' i / 1 / 1 ;' n, ! I / j:c / >''• oj' /a /<' '/r// / / r // v.'/ ///• ri/ '(■/ rf //',,/-///, f ,. .'/, rii'i'i-rii , /.' /■ 'y '/''///' /■'. /// f/f" fi 6 !>. If w m .■*.< (K. fn ^CJ FIG.$'^ / '\. J R / / / . / / w / v ./ / I '^ / j / X \. \ \ / /«" >'-^' \ / -^---> (t) JP- ' m ir' / r h' ^,\ . f \ -\ , i \ i ht \ I I 1 IB >yv x \ /^ \ / \ / v / \ X I r 1/ / ^ w 'fWU ! \ ^, I \ F fC. .j.) 'a J I \ S I » I ■, A /^ J' ri ■ F re. .J.) fnj Snu I » K "1 .J S, I-,, >/ ' \ ' /(> li TV F* I "7\ The^ Prirrvary. ~'r / V- / \ / / \ / )^ ->^ \ / \ -,^- h 7* 07/07 Sine- s' 857 e I) . v: s'. •i- V -■^A B ^4r^c of Sec a?/ f .,^?-o ofD.VS. ^rc of Sz-rte. B DecirTT^aL CiYcies