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Bisection, repeating their Operations until they had approach'd to the Chord design'd.
And this Method is made choice of by the learned Dr. Wallis in his Treatise of Algebra; wherein, after he hath given us a large Account of the different Enquiries made by several (very eminent in Mathematical Sciences) in order to find out some easier and more expeditious Way of approaching to the Circle's Periphery, as in Chap. 82, 84, 85, 86, and several other Places, he comes to this Result, (Page 321.)
“ 'Tis true, faith he, we might in like Manner proceed by con“ tinual Trisećtion, Quinquisection, or other Section, if we had “ for these as convenient Methods of Operation as we have for “ Bisection : But because Euclid Thews how to bisect an Arch
Geomatrically, but not to trifect, &c. and the one may be "done (Algebraically) by resolving a Quadratick Æquation, but “ not those o:her, without Æquations of a higher Compofition, I " therefore make Choice of a continual Bisection, &c.
And then he lays down these following Canons.”
of 1 /
The Subtense of
V:2-V 3into 12 of 1
✓:2-V:27v 3&c. 24 V:2-V:2+v:2-V3 48
V:2-V:2+v:2+:2+V3 96 &c. V:2-V:2+v:2+v:2+v:2 tv3 192
✓:2-V:2+v:2+v:2+v:2+v:2+V 3 384 V:2-1:2+v:27v:2+v:2+v:2+v:27v3 768 &c.
How tedious and troublesome the Work of these complicated Extractions is, I leave to the Consideration of those, who either have had Experience therein, or out of Curiosity will give themselves the Trouble of making Trial.
Again, in Page 347, the Doctor inferts a particular Method propos’d by Libnitius, publish'd in the Aeta Eruditorum ac Leipfick, for the Month of February 1682, in order to find the Circle's Area, and consequently its Periphery, which is this:
As I : to_--+--itis-stin-1', &c. infinitely :: fo is the Square of the Diameter to the Circle's Area. But this convergeth fo very flowly, that it is not worth the Time to pursue it.
I shall here propose a new Method of my own, whereby the Circle's Periphery, and consequently its Area, may be obtain'd
infinitely near the Truth, with much greater Ease and Expedition than either that of Bilection, or that of Libritius, as above, or any other Method that I have yet seen; it being perform'd by refolving only one Æquation, deduced by an easy Process 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 A Fits Radius. Then, if the Arch A D be trifected, it will be AB-BZ=ZD.
: 62a R
And 2 R:a::R
And AB= de = Dx, &c.
4. Rc— 2 Ra=Ra-R
Here a=the Chord of is Part of the Circle.
Next, To trifeet the Arch AB.
2 27y} – 2795 +9y7 -- y = a
Here y = the Chord of 14 Part of the Circle,
Again, To trifeet the Arch whereof y is the Chord.
21870 - 5103a9 + 51032"' 2835a '3 + 43
19683a — 590492" + 787322'3—
3 6561a5 - 1093597 + 729099 - 24304" + 3 X 27
24052'3 + 27015 = 27y5
19683a1 - 45927a9 + 459270" — 4 x 919
255159'3 + 8505a'5 = 9y? 6 7
27a-81993 + 7371a) — 30888a7+ +8-9
729301 - 1074062" + +5
10465243 — 69768a'5 Here a = the Chord of oz Part of the Circle.
Proceeding on in this Method of continually trisecting the Arch of every new Chord and still connecting the produced Æquations into one, as in the two laft Trifections, 'twill not be difficuli to obtain the Chord of any assign'd Arch, how small foever it be.
Now, in order to facilitate the work of raising these Equations to any confiderable Height, 'twill be convenient to add a few useful Observations concerning their Nature, and of such Contractions as may be safely made in them ; which, being well understood, will render the Work very easy.
1. I have observ'd, that every Trifection will gain or advance one Figure in the Circle's Periphery, but no more. Therefore so many Places of Figures as are at first defign'd to be perfeet in the Periphery, so many Trifections must be repeated to raise an Æquation that will produce a Chord answerable to that Design,
2. I have also found, that all the superior Powers (of a ) whose Indices are greater than the Number of Trifections, (viz. whole İndices are greater than the Number of design’d Figures) may be wholly rejecte ed as insignificant.
-3. When once the Number of Trife&tions and thance the higher Power (of a) is determin'd, the third Process (viz the third Trifection) may be made a fix'd or constant Canon ; for by it, and Multiplication only, all the succeeding Trile&tions (how many foever they are) may be compleated without repeating the several Involuti
4. In raising and collecting the Co-efficients of the several Powers (of a) 'twill be sufficient to retain only so many fignificant Figures (at a) as there is design’d to be Places of Figures in the Periphery (or at móf but two more) and every succeeding superior Power may be allow'd to decrease two Places of significant Figures: But herein great Care mult be taken to supply the Places of those Figures that are omitted, with Cyphers, that so the whole and exact Number of Places may be truly adjusted; otherwise all the Work will be erroneous.
Now the Number of those supplying Cyphers may be very conveniently denoted by Figures placed within a Parenthesis, thus : 576 (8) as, may fignify 57400000 ooa, as in the following Æquations. The like may be done with Decimal Parts, thus: (47)658 may fignify ,0000000558 &c. which will be found very useful in the Solution of these and the like Æquations.
The aforesaid Contractions may be safely made, because both the superior Powers of a, which are rejected; as also those Numbers that are omitted in the Co-efficients (and supply'd with Cyphers) would produce Figures so very remote from Unity, as that they would not affect the Chord design'd; that is, they would not affect the Chard in that place wherein the design's Periphery is concerned; as will in Part appear in the following Example.
If there Directions be carefully minded, 't will be easy to raise an Æquation that will produce the side of a regular Polygon, whose Number of Sides (hall be vastly numerous, consequently infinitely small: But, I presume, 'cwill be sufficient for an Example to find the Side of a Polygon consisting of 258280326 equal Sides; that is, if I find the Chard of 178230376 Part of the Circle's Periphery, and that requires but fixteen Trifections, which being order'd, as before directed, will produce this Æquation.
+369837653199714(20)as-8491218532841(35a7 +54633331143(50)a!——230083348(66)a" +6830988(79)a'} – 15072(94)a's
Here the Value of a will have 23 Places of Figures true; that is, the sides of the inscribd and circumscribd Polygons will be exactly the same to 23 Places of Decimal Parts, but no farther; all which mav be easily obtain'd at two Operations. And for the first, 'twill be sufficient to take only three Terms of the Æquation, which will admit of being yet farther contracted, thus:
430467218-3323601794(12)a} And let te=a; then rejecting all the Powers of e, that arife by Involution above ere, it will be g3 + 3rre + 3ree to eee = aaa
And go5 et 5rtet 10rice + Torreee = as
43046721) 1,00000000 (,000000023, This ,00000002=r being duly involv'd, and its Powers multiply'd into their respective Co-efficients, will produce +,86093441+430457211 -,02658881– 39883220–199416(9)e~-3324(18)eee >=i +,00024635+ 61587et 6159(9)eet 308(18)eee ) viz.,83459196+391199861–193257(9)ee-3016(18)eee = Hence 39119986 — 193257 (9)ee~ 3016(18) ee =0,16540804
All the Terms of this last Æquation being divided by 193257(9) the Co-efficient of ee, it will then become 200000020246-ee- 156(5)ece=,0000000000000008558968=D
,0000002024-0 Operation. ,0000002024) ,0000000000000008558968 (,000000004 =. ,0000000043:+,000nco0000000000009984 = 156 (5) ece
Now, if this first Value of a =,000000024327 were not continued 10 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 whose Diameter is 23 nearer than either Archimedes, or Mætius's Proportion: For