b d have still a proportion. For and are in the same proportion with b and d (Art. 236). n n 252. In two ranks of proportionals, if the corresponding terms be multiplied together, the products will be proportionals. that is, ae bf :: cg: dh. This is called compounding the proportions. с g d h bf dh The proposition is true if applied to any number of propor tions. 253. If four quantities be proportionals, the like powers, or roots, of these quantities, will be proportionals. 256. If four quantities are in continued proportion, that is, a b b : c :: cd, then a d :: a3: b3. a a b с (EUCLID, B. v. Def. x1.) a b For d - X X and C 257. The Definition of Proportion here used will not serve for a definition in Geometry, because there is no Geometrical method of representing the quotient of ab, a and b being any Geometrical magnitudes whatever of the same kind. But such magnitudes may always be multiplied geometrically; that is, a line may be produced till it becomes n times its original length-an area, or a solid, may be doubled, trebled, &c.— geometrically. Hence the strictness of Geometry requires such a definition as that which is the foundation of Euclid's 5th Book, and which may easily be shewn to follow from the Algebraic Definition. For suppose a, b, c, d to represent four quantities in proportion, according to the Algebraical definition; then from which it follows, by the nature of fractions, that if ma > nb, then mc > nd; if ma = nb, mc = nd; if ma < nb, mcnd: and ma, mc, are any equimultiples whatever of the 1st and 3rd quantities, nb, nd any equimultiples whatever of the 2nd and 4th. Therefore a, b, c, d are propor tional also according to the Geometrical Definition. 258. If two numbers, a and b, be prime to each other, they are the least in that proportion. If possible, let a b = с d' where a and b are prime to each other, and respectively greater than c and d. If the latter numbers be not prime to each other, divide them by their greatest common measure. Then divide a by b, and c by d, as in Art. 103; thus, b) a (m = C the first quotients m, m, are equal; again, d because b is greater than d, x is greater than r. b d or = In the same and y is greater than s; and so on, if there be r manner, = y S more remainders. Thus the remainder in the latter division will become 1 sooner than the remainder in the former. Let s = y 1; then 2 ; and y, which is greater than 1, will be a common measure of a and b (Art. 105), which is contrary to the supposition. 259. If a and b be each of them prime to c, ab is prime to c. = If not, let ab mr, and c = ms; then since a and b are prime to c, they are respectively prime to (Art. 258, Cor.), which is absurd, since it was before shewn to be prime to m. COR. 1. measure; or If b be equal to a, then a2 and c have no common COR. 2. In the same manner, &c. are fractions in their lowest terms. COR. 3. If a, b, and c, be each of them prime to d, e, and ƒ, abc is prime to def. For, if a be prime to d and e, it is prime to de, and if it be prime to de and f, it is prime to def. In the same manner, b and c are prime to def; consequently, abc is prime to def. COR. 4. If a be prime to b, a2 is prime to b2, and a3 to b3, &c. SCHOLIUM. 260. In the definition of Proportion it is supposed that one quantity is some determinate multiple, part, or parts of another; or that the fraction arising from the division of one by the other, (which expresses the multiple, part, or parts, that the former is of the latter,) is a determinate fraction. This will be the case, whenever the two quantities have any common measure whatever. Let a be a common measure of a and b, and let a = mx, b = n x; But it sometimes happens that the quantities are incommensurable, that is, admit of no common measure whatever, as when one represents the circumference of a circle and the other its diameter; in such cases the value of m a cannot be exactly expressed by any fraction, whose numerator and denominator are whole numbers; n yet a fraction of this kind may be found, which will express value to any required degree of accuracy. its Suppose to be a measure of b, and let b = nx; also let a be α greater than ma but less than (m + 1) x ; then is greater than b a m n and is less than b 1 1 ; and as a is diminished, since na b, n is increased, and nx = n diminished; therefore by diminishing, the difference between α and may be made less than n m n If a and b as well as c and d be incommensurable, and if when ever the magnitudes m and n are increased, then For, if they are not equal, they must have some m b n is equal to assignable dif ference; and because each of them lies between and 1 n ; but since n may, by the supposition, be difference is less than have no assignable difference; that is, is equal to ; and all b d the preceding propositions, respecting proportionals, are true of the four magnitudes, a, b, c, d. VARIATION. 261. In the investigation of the relation which varying and dependent quantities bear to each other the conclusions are more readily obtained by expressing only two terms in each proportion, than by retaining the four. |