but when the segments are incommensurable the ratio can only be symbolized, and cannot be expressed arithmetically except approximately. 189°. If we suppose CD to be capable of being stretched until it becomes equal in length to AB, the numerical factor which expresses or denotes the amount of stretching necessary may conveniently be called the tensor of AB with respect to CD. (Hamilton.) As far as two segments are concerned, the tensor, as a numerical quantity, is identical with the ratio of the segments, but it introduces a different idea. Hence in the case of commensurable segments the tensor is arithmetically expressible, but in the case of incommensurable ones the tensor may be symbolically denoted, but cannot be numerically expressed except approximately. Thus if AB is the diagonal of a square of which CD is the side, AB=CD√2 (180°); and the tensor of AB on CD, i.e., the measure of AB with CD as unit-length, is that numerical quantity which is symbolized by √2, and which can be expressed to any required degree of approximation by that arithmetical process known as "extracting the square root of 2." 190°. That the tensor symbolized by √2 cannot be expressed arithmetically is readily shown as follows :— If √2 can be expressed numerically it can be expressed as a fraction, which is in its lowest terms, and where accord m n ingly m and n are not both even. m If possible then let √2= n Then 2n2=m2. is odd. m2 But if m is even, is even, and n2 and n are both even. Therefore m2 and m are both even and n 2 But n cannot be both odd and even. Therefore √2 cannot be arithmetically expressed. A Illustration of an incommensurable tensor. Let BD be equal to AB, and let AC be equal to the diagonal of a square of which AB is the side. E' F' B Then some tensor will bring AB to AC. Let BD be divided into 10 equal parts whereof E and F are those numbered 4 and 5. Then the tensor 1.4 stretches AB to AE, and tensor 1.5 stretches AB to AF. But the first of these is too small and the second too great, and C lies between E and F. Now, let EF be divided into 10 equal parts whereof E', F' are those numbered 1 and 2. Then, tensor 1.41 brings AB to AE', and tensor 1.42 brings AB to AF'; the first being too small and the second too great. Similarly by dividing E'F' into 10 equal parts we obtain two points e, f, numbered 4 and 5, which lie upon opposite sides of C and adjacent to it. Thus, however far this process be carried, C will always lie between two adjacent ones of the points last obtained. But as every new division gives interspaces one-tenth of the length of the former ones, we may obtain a point of division lying as near C as we please. Now if AB be increased in length from AB to AD it must at some period of its increase be equal to AC. Therefore the tensor which brings AB to AC is a real tensor which is inexpressible, except approximately, by the symbols of Arithmetic. The preceding illustrates the difference between magnitude and number. The segment AB in changing to AD passes through every intermediate length. But the commensurable or numerically expressible quantities lying between 1 and 2 must proceed by some unit however small, and are therefore not continuous. Hence a magnitude is a variable which, in passing from one value to another, passes through every intermediate value. 191o. The tensor of the segment AB with respect to AC, or the tensor of AB on AC is the numerical factor which brings AC to AB. But according to the operative principles of Algebra, AB AC AB AC=AB, AC is the tensor which brings AC to AB. Hence the algebraic form of a fraction, when the parts denote segments, is interpreted geometrically by the tensor which brings the denominator to the numerator; or as the ratio of the numerator to the denominator. SECTION I. PROPORTION AMONGST LINE-SEGMENTS. 192°. Def.-Four line-segments taken in order form a proportion, or are in proportion, when the tensor of the first on the second the same as the tensor of the third on the fourth. This definition gives the relation a с = b ď ...(A) where a, b, c, and d denote the segments taken in order. The fractions expressing the proportion are subject to all the transformations of algebraic fractions (158°), and the result is geometrically true whenever it admits of a geometric interpretation. The statement of the proportion is also written a; b = c; d,......... ...(B) where the sign: indicates the division of the quantity denoted by the preceding symbol by the quantity denoted by the following symbol. In either form the proportion is read 193°. In the form (B) a and d are called the extremes, and b and c the means; and in both forms a and c are called antecedents and b and d consequents. In the form (A) a and d, as also b and c, stand opposite each other when written in a cross, as a/c and we shall accordingly call them the opposites of the proportion. 194°. 1. From form (A) we obtain by cross-multiplication ad=bc, which states geometrically that When four segments are in proportion the rectangle upon one pair of opposites is equal to that upon the other pair of opposites. Conversely, let ab and a'b' be equal rectangles having for adjacent sides a, b, and a', b' respectively. Then ab=a'b', and this equality can be expressed under any one of the following forms, or may be derived from any one of them, viz. : b b ba 2 " a b a in all of which the opposites remain the same. Therefore 2. Two equal rectangles have their sides in proportion, a pair of opposites of the proportion coming from the same rectangle. 3. A given proportion amongst four segments may be written in any order of sequence, provided the opposites remain the same. 195°. The following transformations are important. a C Let = then b ď I. a±b c±d b d' a C a+c a-c then a с e a+c+e+ etc. = b d f b+d+f+etc. etc., (ab for sign) (a>c for sign) = say, (194°, 3) 196°. Def.-1. Two triangles are similar when the angles of the one are respectively equal to the angles of the other. (77°, 4) 2. The sides opposite equal angles in the two triangles are corresponding or homologous sides. The symbol will be employed to denote similarity, and will be read "is similar to." |