Miscellanies. 207 
must have place, but also, whether it is possible for the same pig 
ty to exist under any change of those conditions.” 
While the author designs to establish the doctrine of enensen’ 
proportions with the rigor of proof peculiar to Euclid, he has endeay- 
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 ares 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 cireum- 
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 
same, and as the circle is between the two, it cannot ditar 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 _ or forty figures farther 
by later mathematicians.” 
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