Miscellanies. 207 



must have place, but also, whether it is possible for the same proper- 

 ty to exist under any change of those conditions." 



While the author designs to establish the doctrine of geometrical 

 proportions with the rigor of proof peculiar to Euclid, he has endeav- 

 ored to relieve the subject from the intricacy and subtlety of the ela- 

 borate reasonings of that great geometer, which opposed very serious 

 obstacles in the way of the student. 



The Editor has interspersed some important propositions, for which 

 he acknowledges himself indebted to Legendre, and Leslie, and to 

 Bland's geometrical problems, and he has also added some methods 

 on the rectification of the circle. 



Although the eminence of the French philosophers is generally 

 acknowledged in most branches of abstract science, yet they have 

 not succeeded in demonstrating the quadrature of the circle. Dr. 

 Young, in common with many others, deems it incapable of being 

 rigorously ascertained ;" although by inscribing and circumscribing 

 polygons, on Gregory's method, (which Dr. Young employs,) within 

 and without a circle, a coincidence with it may be so nearly ascertain- 

 ed that for all practical purposes, it is equivalent to perfect accuracy. 

 The seventh Book is devoted to the properties of polygons, and in 

 the tenth proposition it is shown " that the arcs which the sides of a 

 polygon subtend are bisected. The chords of the half arcs will be 

 the sides of a regular polygon having double the number of sides." 

 And in the scholium to the thirteenth proposition he says, that " Hav- 

 ing obtained numerical expressions for polygons of eight sides, by an 

 application of the same two proportions in a similar way, the surfaces 

 of sixteen sides may be determined, and thence of thirty-two sides, 

 and so on, till we arrive at an inscribed and circumscribed polygon 

 differing from the circle, and from each other so little, as to be unas- 

 signable by any numerical expression. The inscribed and circum- 

 scribed polygons of 32,768 sides, differ so little from each other that 

 the numerical value of each, as far as seven places of decimals is the 

 sarne, and as the circle is between the two, it cannot differ so much 

 from either as they do from each other. 



"The number 3.1415926 expresses correctly the area of a circle, 

 whose radius is one, as far as seven places of decimals," and if it were 

 necessary, the approximation might be continued to an almost endless 

 extent. " An infinite series was discovered by Machin, by which he 

 reached the quadrature as far as the hundredth place of decimals, 

 and even this number has been extended thirty or forty figures farther 

 by later mathematicians." 



