instead of breaking the continuity of the text, by inserting it between two of the books. The chief new feature of the edition is the addition, to each book, of exercises adapted to the order of the theorems of the book. These exercises are such as have been thoroughly tested in the instruction of classes in Elementary Geometry. They serve the purpose of increasing the pupil's knowledge of geometrical truths as well as the more important one of giving him thorough instruction in the methods of Geometrical Analysis. They are numerous, in order that the teacher may have a wide range of selection. They have been chosen mainly from the work of Planche, alluded to above, but some of them from the collections of Guilmin, Amiot, Ritt, Rouchè and De Comberousse, and from Potts' Euclid. While most of the exercises relate to the methods of pure Geometry, a number of numerical problems have been added to some of the books. It is believed that these arithmetical applications will interest the pupil and help to fix his knowledge of the theorems on which they depend. The feature of Hints to the Solutions, which serve as a guide to the study of Geometrical Analysis, will be appreciated both by teachers and pupils. For, while it is most desirable to engage the attention of beginners with geometrical exercises, help enough should be given to prevent them from being disheartened. In conclusion, the editor cannot refrain from adverting to the excellence of the mechanical execution of the book and from expressing the hope that this will prove a practically useful edition of Elementary Geometry. CHAS. S. VENABLE. UNIVERSITY OF VIRGINIA, January, 1875. ELEMENTS OF GEOMETRY. INTRODUCTION. [For the sake of those who have not studied the Doctrine of Proportion, it is requisite to prefix to the treatise of Legendre a brief outline of its fundamental principles. After studying Book I. and Book II. to Prop. XVI., the beginner should study Sections II. and III. of this Introduction ; reserving, whenever it is thought advisable, the remaining sections for the review of the book.] I. PRELIMINARY NOTIONS. 1. Our first notion of whole numbers arises from considering distinct and similar objects. The measure of magnitudes brings us to a necessary extension of this first notion. 2. When a magnitude is the sum of 2, 3, 4 parts, equal to another magnitude of the same species, we say that the first is a multiple of the second, and that the second is an aliquot part of the first. 3. Two magnitudes are said to be commensurable with each other when they are both multiples of a third magnitude, which is called their common measure. When there is no third quantity of which they are both multiples, they are said to be incommensurable with each other. 4. To measure a magnitude, we seek a common measure between this magnitude and an arbitrary magnitude of the same species, which we call the unit of measure, or simply the unit. If this common measure is the unit itself, and the given magnitude contains it, for example, three times, we say that the magnitude is measured by the number 3. If the common measure is an aliquot part of the unit; for example, if, when the unit is divided into five equal parts, the given magnitude is the sum of three of these parts ; we say that this magnitude is three-fifths of the unit, and that it is measured by the fractional number . To sum up: to measure a magnitude commensurable with the unit, is to seek how often this magnitude contains the unit or aliquot parts of the unit. According as the magnitude is a multiple of the unit or a multiple of an aliquot part of the unit, the number which expresses its measure is entire or fractional. Conversely, every magnitude measured by an entire or fractional number is commensurable with the unit of measure. 5. Let us now consider a magnitude incommensurable with the chosen unit of measure. Here we shall find useful the following definition: The limit of a variable number is a number which that variable number may approach in value as near as we please, but which it can never reach. Now, conceive the unit to be divided into any number, n, of parts equal to one another and less than the magnitude, G, to be measured. Taking 1, 2, 3, 4 of these parts we shall form a series of magnitudes, A, A, A, A, : . .• All Ax+1 (1) measured respectively by the numbers 3 k k+1 I 2 4 By going far enough in the series (1) we shall find two consecutive magnitudes, Ax and Ax+1, between which G lies. Now G differs from Ax or Ax+1 by a quantity less than Ak+1 – Ak, and this difference, being the nth part of the unit, can be made as small as we please by making n sufficiently great. The magnitude G being then the common limit of the commen, surable numbers Ax and Ax+1, the number which measures it is, by k k+1 definition, the common limit of the numbers - and which meas n n ure Ax and Ax+1; for, as this measure differs from k or n by less n than we can make either of these numbers approach it as near as n we please by taking n sufficiently great. 6. A number is called commensurable or incommensurable according as the magnitude of which it expresses the measure, is commensurable or incommensurable with the unit adopted. The commensurable numbers are entire numbers or fractions. The result of operations to be performed on incommensurable numbers, is the limit of the results obtained by substituting for them commensurable numbers which approach them more and more nearly in value. |