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Ergo A1 =2AC = 1,70130161. Hence AC=0,85065080
O. E. D. From hence it will be easy to resolve the following Problem.
PROBLEM VIII. The Side of any regular Pentagon being givèn, to find its Area.
Example, Suppose the given Side to be 15 Inches long, then it will be, as I ; 1,53884176::15:22,0826264 the perpendicular Height; and by the general Rule 22,0826264x'<= 165,619698 the--Area requir’d.
Example. Let the given Side be 14 Inches long; then, as 1:1,53884176::14:21,543784 = the Radius of the inscrib'd Circle ; and 14 X 5 = 70 is half the Sum of its Sides. Lastly, 21,543784 X 70 = 1508,06488 the Area required.
Confeitary. Hence, if the side of any regular Dodecagon be given, the Radius of its infcrib'd Circle may be easily obtain’d, and thence the Area found; as in the last Problem.
The Work of the 'foregoing Polygons, being well consider’d, will help the voung Geometer to raise the like Proportions for others, if his Curiosity requires them : And not only so, but they will also help to form a truc Idea of a Circle's Periphery and Area, according to che Method which I Mall lay down in the next Chapter for finding them both. .
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 our 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 32 1.)
“'Tis true. Jaith he, we might in like Manner proceed by cone tinual Trifection, Quinquifection, or ocher Section, if we had “ for these as convenient Methods of Operation as we have for “ Bisection : But because Euclid shews how to bisect an Arch “ Geomatrically, but not to trisect, &c. and the one may be “ done (Algebraically) by resolving a Quadratick Æquation, but " not those other, without Æquations of a higher Compofition, I " therefore make Choice of a continual Bisection, &c.
And then he lays down these following Canons."
How tedious and troublesome the Work of these complicated Extractions is, I leave to the Confideration 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 Dostor inserts a particular Method propos’d by Libuities, publiñ'd in the Aeta Eruditorum at Leipfick, for the Month of February in 82, in order to find the Circle's Area, and consequently its Feriphery which is this: As I : to
it-it-istio, &c. infinitely : : so is the Square of the Diameter to the Circle's Area. But this convergeth so very slowly, 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