298 Cambridge Course of Mathematics. 
is attached to this section, in which a few of the elementa- 
ry properties of isoperimetrical figures are demonstrated. 
Among the rest, it is shewn, that among polygons of the 
same perimeter and of the same number of sides, that is a 
maximum which has its sides equal; that of ail triangles” 
formed with two given sides making any angle at pleasure 
with each other, the maximum is that in which the two 
given sides make a right angle; that among polygons of the 
same perimeter and the same number of sides, the regular 
polygon is a maximum ; and that the circle is greater than 
The difficulty, however, which had arrested the progress 
of this part of geometry ever since the time of Euclid, has 
at length been surmounted by M. Gauss, a Professor at the 
university of Gottingen, and one of the greatest mathema- 
ticians of the present time. The work containing the ori- 
inal demonstration is entitled, ‘‘Disqwisitiones Arithmetice, 
ipsie, 1801,’ anda French translation of it was published 
by M. Delisle at Paris in 1807. In this demonstration. it is 
own, that the circumference of a circle may be divided 
into a number of equal parts designated by the formula 
2"+-1, when this is a prime number. Some of the numbers 
resulting from this formula are 17,257, 65537, &c. The 
circumstance that M. Gauss’ invention ‘is limited to the 
cases where the formula 2"+1 designates a prime number, 
greatly diminishes its value. No demonstration of this 
principle has, we believe, found its way into any elementa- 
ry treatise of geometry, and we are not sure, that it is 
cnnaie of a strictly geometrical elementary demonstra- 
ion. ; 
The first section of part II, contains the properties of 
planes and solid angles. This part'is intimately connect- 
ed with the demonstrations of the properties of solids, and 
figures in which different planes are considered. A com- 
plete underestanding of it is indispensable, also, in descrip- 
‘ive geometry, where the principal difficulty consists in 
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