ELEMENTS OF GEOMETRY. BOOK V. " IN the demonstrations of this book there are certain Book V. signs or characters which it has been found convenient to employ. 1. The letters A, B, C, &c. are used to denote magnitudes of any kind. The letters m, n, p, q are used to denote numbers only. 2. The sign + (plus), written between two letters, that denote magnitudes or numbers, signifies the sum of those magnitudes or numbers. Thus, A+B is the sum of the two magnitudes denoted by the letters A and B; m+n is the sum of the numbers denoted by m and n. 3. The sign (minus), written between two letters, sige nifies the excess of the magnitude denoted by the first of these letters, which is supposed the greatest, above that which is denoted by the other. Thus, A-B signifies the excess of the magnitude A above the magnitude B. 4. When a number, or a letter denoting a number, is written close to another letter denoting a magnitude of any kind, R Book V. it signifies that the magnitude is multiplied by the number. Thus, 3A signifies three times A; mB, m times B, or a multiple of B by m. When the number is intended to multiply two or more magnitudes that follow, it is written thus, m.A+B, which signifies the sum of A and B taken m times ; ñ.A-B is m times the excess of A above B. Also, when two letters that denote numbers are written close to one another, they denote the product of those numbers when multiplied into one another. Thus, mn is the product of m into n; and mnA is A multiplied by the product of m into n. 5. The sign = signifies the equality of the magnitudes denoted by the letters that stand on the opposite sides of it. Thus, A=B signifies that A is equal to B; A+B=C—D signifies that the sum of A and B is equal to the excess of C above D. 6. The sign > is used to signify the inequality of the magnitudes between which it is placed, and that the magnitude to which the opening of the lines is turned is greater than the other. Thus, AB signifies that A is greater than B; and A<B signifies that A is less than B.” DEFINITIONS. I. nitude, when the less measures the greater, that is, when II. the greater is measured by the less, that is, when the greater III. Ratio is a mutual relation of two magnitudes, of the same kind, to one another, in respect of quantity, Book V. IV. be multiplied so as to exceed the greater; and it is only such V. soever be taken of the first and third, and any equimultiples VI. i usual to say that A is to B as C to D, and to write them VII. the fifth definition) the multiple of the first is greater than VIII. of which the first has to the second the same ratio that the Book V. IX. cond is said to be a mean proportional between the other X. the first is said to have to the last of them the ratio com- last magnitude. For example, if A, B, C, D be four magnitudes of the same kind, the first A is said to have to the last D the ratio compounded of the ratio of A to B, and of the ratio of B to C, and of the ratio of C to D; or the ratio of A to D is said to be compounded of the ratios of A to B, B to C, and C to D. And if A:B::E:F, and B: C::G:H, and C:D::K:L, then, since by this definition A has to D the ratio com- K to L. the same ratio which A has to D, then, for shortness sake, XI. the first to the third is said to be duplicate of the ratio of the first to the second. " Thus, if A be to B as B to C, the ratio of A to C is said to be duplicate of the ratio of A to B. Hence, since by the last definition the ratio of A to C is compounded of the ratios of A to B, and B to C, a ratio, which is compounded of two equal ratios, is duplicate of either of these ratios, Book V. XII. the first to the fourth is said to be triplicate of the ratio of third, &c. “ So also, if there are five continual proportionals, the ratio of the first to the fifth is called quadruplicate of the ratio of the first to the second; and so on, according to the number of ratios. Hence, a ratio compounded of three equal ratios is triplicate of any one of those ratios; a ratio compounded of four equal ratios quadruplicate,” &c. XIII. to one another, as also the consequents to one another. Geometers make use of the following technical words to signify certain ways of changing either the order or magnitude of proportionals, so as that they continue still to be proportionals. XIV. this word is used when there are four proportionals, and it XV. and it is inferred, that the second is to the first as the fourth XVI. tionals, and it is inferred, that the first, together with the |