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 all triangles 

 formed with two given sides making any angle at pleasure 

 with each other, the maximurn'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 

 any polygon of the same perimeter. This is a very beautiful, 

 interesting and useful addition to elenn^ntary geometry. 



Until a short time since, it was supposed, that no regular 

 polygons, except those treated of B. IV. of Euclid, and 

 the several series depending on them, could be inscribed 

 in, or circumscribed about, a circle by geometrical means. 

 The diftjculty, 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- 

 ginal demonstration is entitled, '"''Disquisitiones Arithmeticcc, 

 Lipsi(s, 1801," and a French translation of it was published 

 by M. Delisle at Paris in 1807. In this demonstration^ it is 

 shown, that the circumference of a circle may be divided 

 into a number of equal parts designated by the formula 

 2"+l, when this is a prime number. Some of the numbers 

 resultiiig from this formula are 17,257,05537, &.c. The 

 circumstance that M. Gauss' invention is limited to the 

 cases where the formula 2"-f-l 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 

 capable of a strictly geometrical elementary demonstra- 

 tion. 



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- 

 tive geometry, where the principal difficulty consists in 



