infinitely near the Truth, with much greater Ease and Expedition than either that of Bifection, or that of Libnitius, as above, or any other Method that I have yet feen; it being perform'd by refolving only one Equation, deduced by an eafy Procefs from the Property of a Circle, (known to every Cooper) which is this: = The Radius of every Circle is equal to the Chord of one fixth Part of its Periphery. That is, AD DH HG, the Chords of one third Part of the Semicircle, are each equal to AF its Radius. Then, if the Arch AD be trifected, it will be AB=BZ=ZD. H 4 X &c. A AFB, and A B Ae, are alike. 4 Rc-2 Ra=Ra ааа R I 53 Ra-aaa RRc. That is, 3a—aaa=1, 1'8 Next, To trifect the Arch AB, Let 13y3a the laft Chord. 2 273 2715+ 9y7 — y9= a3 I X 3 39y3y2 = 3a 13 3 za 49130y+27y5—9y7+y9=3a —a3 = 1 Here y the Chord of Part of the Circle, Again, To trifect the Arch whereof y is the Chord. Let13a-a3 = y 227a3-27a5 +9a7 — a9 = y3 324395-405a7+270a? —90a" †15a13 —a15=y3 I 2187a7 — 5103a? + 5103a" — 2835a 13 + 945a=y7 Į 19683a-59049a11 + 78732a13 — 561236asy9 627a9a3 = gy X 30 7 3 X 27 810a3-810a5+ 270a7— 30a9= 30y3 85656195 4 X 99 6 7 +8-9 ΙΟ - 10935a7+7290a9-2430a" + ¿ 40593 +27α = 27ys 72930a107406a" + 1+}= Here a = the Chord of Part of the Circle. Proceeding on in this Method of continually trifecting the Arch of every new Chord and fill connecting the produced Equations into one, as in the two laft Trifections, 'will not be difficult to obtain the Chord of any affign'd Arch, how fmail foever it be. Now, in order to facilitate the Work of raifing these Equations to any confiderable Height, 'twill be convenient to add a few useful Obfervations concerning their Nature, and of fuch Contractions as may be fafely made in them; which, being well understood, will render the Work very easy. 1. I have obferv'd, that every Trifection will gain or advance one Figure in the Circle's Periphery, but no more. Therefore fo many Places of Figures as are at firft defign'd to be perfect in the Periphery, Jo many Trifections must be repeated to raife an Equation that will produce a Chord answerable to that Defign. 2. I have also found, that all the fuperior Powers (of a) whofe Indices are greater than the Number of Trifections, (viz. whofe Indices are greater than the Number of defign'd Figures) may be wholly rejected as infignificant. 3. When once the Number of Trif ions and thence the highest Power (of a) is determin'd, the third Procefs (viz the third Trifection) may be made a fix'd or conftant Canon; for by it, and Multiplication only, all the fucceeding Trifections (how many foever they are) may be compleated without repeating the several Involuti ons. 4. In raifing and collecting the Co-efficients of the feveral Powers (of a) 'twill be fufficient to retain only fo many fignificant Figures (at a) as there is defign'd to be Places of Figures in the Periphery (or at most but two more) and every fucceeding fuperior Power may be allow'd to decrease two Places of fignificant Figures: But herein great Care must be taken to fupply the Places of thofe Figures that are omitted, with Cyphers, that fo the whole and exact Number of Places may be truly adjusted; otherwife all the Work will be erroneous. Now the Number of those fupplying Cyphers may be very conveniently denoted by Figures placed within a Parenthefis, thus: 576 (8) a3, may fignify 57600000200a3, as in the following Equations. The like may be done with Decimal Parts, thus: (7)658 may fignify ,0000000658 &c. which will be found very useful in the Solution of thefe and the like Equations. The aforefaid Contractions may be fafely made, because both the fuperior Powers of a, which are rejected; as alfo thofe Numbers that are omitted in the Co-efficients (and fupply'd with Cyphers) would produce Figures fo very remote from Unity, as that they would not affect the Chord defign'd; that is, they would not affect the Chord in that Place wherein the defign'd Periphery is concerned; as will in Part appear in the following Example. If thefe Directions be carefully minded, 'twill be eafy to raise an Equation that will produce the Side of a regular Polygon, whose Number of Sides fhall be vaftly numerous, confequently infinitely fmall: But, I prefume, 'twill be fufficient for an Example to find the Side of a Polygon confifting of 258280326 equal Sides; that is, if I find the Chord of 258290326 Part of the Circle's Periphery, and that requires but fixteen Trifections, which being order'd, as before directed, will produce this Equation. 430467219-332360179486968612(4)a3 +769837653199714(20)a-8491218532841(35a7 +5463333114350)a-230083348(66)a" ·+6830988 (79) a13 — 15072(94)a15 Here the Value of a will have 23 Places of Figures true; that is, the Sides of the inferib'd and circumfcrib'd Polygons will be exactly the fame to 23 Places of Decimal Parts, but no farther; all which may be easily obtain'd at two Operations. And for the first, 'twill be fufficient to take only three Terms of the Equation, which will admit of being yet farther contracted, thus: Let Let { 430467214-3323601794(12)a3 } = 1 And let rea; then rejecting all the Powers of e, that arise by Involution above eee, it will be r3+3rre + 3reeeeeaaa And r5e+ 10r3ee + 1orreee = a5 Then the first fingle Value of r may be thus found: This ,00000002r being duly involv'd, and its Powers multiply'd into their refpective Co efficients, will produce +86093441+43046721e 02658881-3988322e-199416(9)ce—-3324(18)eee} = i 61587e+ 6159(9)ee+ 308(18)eee. +,00024635+ viz.,83459196+39119986e—193257(9)ee—3016(18)eee =i Hence 39119986e — 193257 (9)ee —3016(18) ee=0,16540804 All the Terms of this laft Equation being divided by 193257(9) the Co-efficient of ee, it will then become 0000002024e-ee-156(5)ece=,0000000000000008558968=D Confequently, { Operation. D+156(5)eee =e ,0000002024- e ,0000002024),0000000000000008558968 (,000000004 = • ;0000000043:+,0000000000000000009984 <= 156 (5) ece Now, if this firft Value of a =,000000024327 were not continued to more Places of Figures by a fecond Operation, but only multiply'd into the Number of Chords. viz. ,000000024327 X 258280326 6, 8318539, &c. the Periphery of that Circle whofe Diameter is 23 nearer than either Archimedes, or Moetius's Proportion: For Ar Archimedes makes it 6,285714 &c. viz. As 7 to 22. And Moetius makes it 6,28318584 &c. viz. As 113 to 355. But if the whole Equation before propos'd be now taken, and we proceed to a fecond Operation, the Value of a may be encreas'd with twelve Places of Figures more, and thofe may be obtain'd by plain Divifion only. Thus, let rea, as before, and let all the Powers of e be now rejected as infignificant; The feveral Powers of r=,000000024327 being rais'd, and multiply'd into their respective Co efficients, will produce these following Numbers. Viz. +1,047197581767 +43046721e ,047849196598394865 5900751e +,000655906484595355 + 134810e ,000004281440413375 +,000000016302517863 ,000000000040631167 +,000000000000071388 + 1232e бе Oe Ое Qe 1,000000026474745106 +37279554e = Hence 37279554e ——,000000026474745106 =D: Or Operation. ,000000026474745106 = D 37279554),000000026474745106( (,15)710167967=e 260956878 37905730 62617660 |