may be bisected, we can thus divide a right angle geometrically into five equal parts.
It follows from what is given in the fourth Book of the Elements that the circumference of a circle can be divided into 3, 6, 12, 24, .... equal parts; and also into 4, 8, 16, 32, .... equal parts; and also into 5, 10, 20, 40, .... equal parts; and
also into 15, 30, 60, 120,........ equal parts. Hence also regular polygons having as many sides as any of these numbers may be inscribed in a circle, or described about a circle. This however does not enable us to describe a regular polygon of any assigned number of sides; for example, we do not know how to describe geometrically a regular polygon of 7 sides.
It was first demonstrated by Gauss in 1801, in his Disquisitiones Arithmeticae, that it is possible to describe geometrically a regular polygon of sides, provided be a prime number; the demonstration is not of an elementary character. As an example, it follows that a regular polygon of 17 sides can be described geometrically; this example is discussed in Catalan's Théorèmes et Problèmes de Géometrié Elémentaire.
For an approximate construction of a regular heptagon see the Philosophical Magazine for February and for April, 1864.
THE FIFTH BOOK.
The fifth Book of the Elements is on Proportion. Much has been written respecting Euclid's treatment of this subject; besides the Commentaries on the Elements to which we have already referred, the student may consult the articles Ratio and Proportion in the English Cyclopædia, and the tract on the Connexion of Number and Magnitude by Professor De Morgan.
The fifth Book relates not merely to length and space, but to any kind of magnitude of which we can form multiples.
V. Def. 1. The word part is used in two senses in Geometry. Sometimes the word denotes any magnitude which is less than another of the same kind, as in the axiom, the whole is greater than its part. In this sense the word has been used up to the present point, but in the fifth Book Euclid confines the word to a more restricted sense. This restricted sense agrees with that which is given in Arithmetic and Algebra to the term aliquot part, or to the term submultiple.
