the second, the multiple of the third is also equal to that of the fourth; or, if the multiple of the first be greater than that of the second, the multiple of the third is also greater than that of the fourth. Conversely. If four magnitudes be proportional according to Euc. V., def. 5, their numerical values shall also be proportional according to Euc. VII., def. 20. Let a, b, c, d be four magnitudes which are proportional by Euc. V., def. 5, where any equimultiples whatever ma and me of a, e the first and third magnitudes being taken, and any equimultiples whatever nb and nd of b, d the second and fourth, such that if ma be greater than, equal to, or less than nb, then mc is greater than, equal to, or less than nd; then also a, b, c, d shall be proportional according to Euc. VII., def. 20; and if ma be greater than, equal to, or less than nb, then me is also greater than, equal to, or less than cd'; but these conditions are assumed of ma, nb, me, nd; ... nd' = nd and d' = d; therefore proportionals according to Euc. VII., def. 20. 22. In the proportion x: b:: y:d, or a с = b-d and a, b, c, d are if b, d remain constant while x and y change their values so that always then x is said to vary directly as y, and thus a proportion may be expressed by two terms. The mark is used for the words "varies as," and the proportion is written xxy, and is read "r varies as y.” if b and d remain constant In the proportion ≈ : b : 1 1 : or y X d by' constant while x, y, z change their values, then in ≈ = to vary jointly as x and y, or as the product of x and y. b ̈ày, x = bd.. bd., b and 3= dxy, It is obvious from the equations x= that variable quantities can always be connected by some constant multiplier, either integral or fractional; and that every variation is convertible into an equation, and conversely.* 23. Prop. If x vary as y, and y vary as z, then x varies as z. Since xxy and y xz, let x=py and y = qz, where p and q are some constants. Then x = pqx, and .. x × z. 24. Prop. If v vary as x and constant. Then vy=pqxz, .. vy x xz. y vary as z, then vy varies as xz. = px and y=qz, where Р and qare v = 25. Prop. If x vary as y and z vary as y, then both the sum and the difference of x and z, varies as y, and the product of x and z varies as y2, the square of y. For since xxy, .. x=py, and z = qy, where p and q are some constant quantities. Then x±z=py±qy=(p±q)y, and .. x±≈ œ y. Also xz=pqy3, and .'. xz ∞ y2. 26. Prop. If z vary as x when y is constant, and z vary as y when x is constant; then z varies as xy when all three are variable. Let ', ' be correspondent values of x and x when y is constant, then z:::x : x', and z' = zx But this value z' of z, will like z vary as y. Let z", y be correspondent values of z' and y when x is constant, yz" الح and z and %= is constant ..zx xy, or z. x'y' These three definitions of variation may be illustrated by considering that the area of a triangle is equal to half the product of the base and the perpendicular from the vertex on the base. If A, b, p denote the area, base, and perpendicular, then ▲-bp. (1) If b be constant, then A varies directly as p, and if p be constant, A varies. directly as b. (2) If A be constant, then p varies inversely as b, and b varies inversely as p. (3) If all three A, b, p vary, then A varies as bp, or the area varies as b and p jointly. EXERCISES. I. 1. EXPLAIN why in Algebra a ratio is treated as a fraction, when it is not so treated in Geometry? 2. Define the ratio of a to b; what it is; what it equals; and what is measured by it. 3. Determine the ratio of x to y from the equation 2x+5y=4x+3y. 4. Two quantities are in the ratio of m:n; what quantity must be added to each that they may be in the ratio of p : q? 5. What is the number, which, if added to each term of the ratio 5 3, makes it three fourths of what it would have become, if the same number had been subtracted from each term ? 6+1 12(a-1)=46+3; find the ratio of a to b. a+b2b-3c' and 6(a+b)=186-8c; find the ratios of a to to a and b; a being less than b. 9. If the ratio a: b be unchanged when c is added to the antecedent and 2c to the consequent; then tho value of the ratio is . 11. If x be very small compared with a, shew that the ratio of (a+x)3: a3, is very nearly that of a+3x: a, and the ratio of (a+x) at is very nearly that of a +x: a. Ex. Find the approximate ratio of (403): (402); and of (403)*: (402)*. 12. If x 5 y: 8, find the numerical ratio of x+5 to y+8. a a+c b 14. If and be two unequal ratios, then the ratio is greater b+d than one of them and less then the other. 15. A number of two figures is altered in the ratio of n to m, if its digits be interchanged; shew that the digits are to each other in the ratio of 10m-n to 10n-m. 17. Explain what are the necessary or sufficient tests of the ultimate ratio of two quantities which by continuous diminution become indefinitely small ultimately. 18. Shew that the ratio (1+3x)(x+1)* is equal to 1+5x+3x2-8x3 1+x+x2 nearly, when x is small; and equal to 9x-3 nearly, when x is large. II. 1. Find a third proportional to 999 and 33.3 2. Find a mean proportional between 016 and 2.704. 3. Find a fourth proportional to 0004, 14 and 02. : 4. If x2+y2: xy:: 34: 15, then x y: 53 and conversely. 5. Find the value of the common ratio when a, b, c, d are in continued proportion, and 2(a+d) = 3(b+c). proportion true if it involve concrete quantities? 7. Find the proportion deducible from the equation, a2+b2 = 2ac. 8. If a, b, c, d are proportionals; if a be prime to b, then c and d are equimultiples of a and b. 9. Can any number be added to each of four numbers in arithmetical progression, so that the sums are in geometrical proportion? 10. If a, b, c, d be proportionals, there is no number which can be added to, or subtracted from each, so that the four terms so increased or diminished shall be proportionals. 11. If a, b, c, d, are proportionals, deduce (a+b)(c+d) from each of the following expressions: (1) (a-d)2= (b-c)2+(c—a)2+(d—b)3. (2) (a+b)3+(d+c)2+(a−b-c+d)2 = 2(a+d)3. (3) (a−b+c—d)2 = (a—b)2+(c—d)2+2(b—c)2. 12. When are two quantities said to be commensurable or incommensurable? Find the approximate ratio of the diagonal to the side of a square, so that the error may be within one thousandth part of unity. 13. Shew that the geometrical definition of proportion (Euc. V., def. 5) is a consequence of the arithmetical definition (Euc. VII., def. 20), and the converse. III. Given a bed, prove the following proportions: 1. ab a±e: b+d; and conversely. 6. (a2+b2)2: (c2+d2)2 : : a1+b1 : c1+d1 :: ab3 ; cds. 7. (a3 + c3)2: (63 + d3)2 :: (a2+c2)3: (b2+d2)3. ́8. (a3 — b3)3 : (c3 — d3)3 :: (a3 — b3)5 : (c3 — d3)5. 1 ǝ. a+(a2+b3)*: a− (a2+b2)* : : c+(c2+d2)* : c−(c2 — ď2)3. IV. If a, b, c, d be proportionals; shew the truth of the following properties : 1. a(a+b+c+d)=(a+b)(a+c). 2. (a+b+c+d)(a−bc+d)=(a−b+c-d)(a+b-c-d). 3. (a2+c2)(b2+d2) = (ab+dc)2. 4. (a*+b*+c1+d1)(a+b+c+d)* 5. = {(a+b)*+(c+d)*}.{(a+c)*+(b+d)*}. = a2 c2 2ac 6. (a+d)−(b+c)=(a—b)(a–c), a being the greatest. a 7. (a+b)−(c+d) = (a+b)(b−d) (a−b)(a–c). a 9. (a2+3ab+b2) 2ab+3b2 = a1+b1+c1+d1 a+b+c+d→ 10. 11. =-, m, n, p, being any numbers whatever. ma+nb+pe a x2+y2+z2 (x+y+z)2 xy+xz+yz x2 a2+b3+c2 ̄ (a+b+c)2 ̄ ab+ac+bca2 2. |