FIRST is to the SUM or DIFFERENCE of the first and second, as the THIRD to the SUM or LIFFERENCE of the third and fourth. : For (by THEOREM 6,) a + b b :: c+d: d, and alternately a + b c + db: d; but (by THEOREM 4,) b d :: a: c; hence, (by THEOREM 5,) a + b : c + d::a: c, and alternately a+b: a ::cd: c, .. inversely a: a+bcc+d. (115.) THEOREM 8. If four quantities be proportional, then the sum of the first and second is to their DIFFERENCE, as the SUM of the third and fourth is to THEIR difference. a+b c+d abcd therefore (by Art. 103), a+b: a-b:: c+d: c-d. or = C (116.) THEOREM 9. If four quantities be proportional, and any EQUIMULTIPLE or EQUAL PARTS whatever be taken of the first and second, and also of the third and fourth; then will the resulting quantities, taken in the same order, be still proportional. : For let a b; cd; then (by CASE I. Art. 104,) the ratio of ma mb is the same with the ratio of a b; and for the same reason, the ratio of nc: nd is the same with the ratio of cd; hence (Art. 101), ma mb :: nc: nd, where m and n may be any quantities whatever, either integral or fractional. : (117.) THEOREM 10. The same theorem is true if any EQU1MULTIPLE OF EQUAL PARTS whatever be taken of the FIRST and THIRD, and also of the SECOND and FOURTH; a C For since multiply each side of the equation by ma mc = nb nd' d' .. ma : nb:: mc: nd, where m and n may be any quan tities whatever, either integral or fractional. (118.) THEOREM 11. If four quantities be proportional, any POWERS or ROOTS of those quantities will also be proportional. a C For since we have :. a" : b′′ : c" : d", where n may be any number, either integral or fractional. (1.9.) THEOREM 12. If the corresponding terms of two sets proportionals be multiplied together, or divided by each other, the resulting quantities taken in order will still be proportional. ad : hence eh fg Again (by THEOREM 1), ad=bc, and eh=fg; :. Th (120.) THEOREM 13. If there be two rows of proportional quantities, whereof the SECOND and FOURTH of the first row are the same with the FIRST and THIRD of the second row, then will the remaining quantities, taken in order, be proportional; For, et ab: c: d and be :: d:f, then (by THEOREM 12), ab: be:: cd: df, or (reducing each ratio to its lowest terms) a: e: cf. This is Euclid's ex æquali proportion. : : :: : (121.) THEOREM 14. If there be a set of proportional quantities, a b c d e f g h, &c. &c. then will the FIRST be to the SECOND as the SUM OF ALL THE ANTECEDENTS to the SUM OF ALL THE CONSEQUENTS. Again, since a+c: b+d::a: b, or :: c d and c d e fi .. by THEOREM 5, a+c: b+d alternately, a+c: e By THEOREM 6, a+c+e: e :: e :: b+d :f, .. alternately, a+c+e: b+d+f :: e therefore a : But a b :: e :f. :f, ::a+c+e :b+&+f; And so on for any number of these proportions. (122.) THEOREM 15. If there be a set of quantities, a, b, c, d, e, `in CONTINUED proportion; then will ac: a: b, or in the duplicate ratio of a: b; a: d:: a3: b3, or in the triplicate ratio of a : ; : a: e :: a b', or in the quadruplicate ratio of a : b; &c. &c. &c. &c. For, by Art. 98, a: b :: b : c :: C d: de :: &c. &c. (123.) The following Examples are designed to illustrate the application of the foregoing Theorems. Er. 1. To divide the number 60 into two such parts, that the product shall be to the sum of the squares :: 2:5. First, let x= one part; then 60-x= the other part, (60—x) ×x=60x—x2= the product; and x2 + (60—x2)=2x2+3600—120x=sum of the squares. Hence, 60x-x2 : 2x2+3600−120x :: 2 : 5, by the question; consequently, by THEO. 1, (60x-xo) × 5=(2x2+3600-120x) x 2, or 300x-5x=4x2+7200-240x; whence by transposition and division, x2-60x=-800; therefore, x-60x+900=900−800=100, and x−30=±10; or x=30+10=40 or 20 the parts required. 2. The number 20 is divided into two parts, which are to each other in the duplicate ratio of 3: 1. It is required to find a mean proportional between those parts. First let x= greater part, then 20 x= lesser part; x: 32: 12:9: 1. Hence, by Theorem 1, x = 180 −9x, or 10x = 180; Therefore x = 18 greater part, and 20 lesser part. x = 20 - 18 2 Consequently by Theorem 3, a mean proportional between 18 and 2 is equal to 18 × 2 = √36 = 6 the number required. 3. If (a+x)2 : (a− x)2 :: x + y : x - y, shew that a : x :: √2a—y : Ny. Here by expansion, a2 + 2ax + x2 ; a2 2ax + x2 :: x+y: x—y But by Theorem 8, 2a2 + 2x: 4ax :: 2x: 2y. =2ax x x = 2a x x2. Therefore, by Theorem 1, (a2 + x2) × y and by Theorem 11, (n being ) a:x:: √2a—y: Ny. 4. If xy in the triplicate Ya+y, shew that dr = cy. Here since : y and by Theorem 11, a3: b3 ratio of a: b, and a: b :: Vc + x : :: :: c+x : d + y; therefore by Theorem 5, x: y :: c+x:d+y, or c+x: d+y :: x: Y, and by Theorem 4, c+x:x :: d+y: y; .. by Theorem 6, c: x :: d: y; and by Theorem 1, dr = cy. What 5. There are two numbers, the product of which is 24, and the difference of their cubes: cube of their difference :: 19: 1. are the numbers? Let x = greater number, and y = less number. Then, by the question, xy 24, and a3 — y3 : (x—y)3 :: 19 : 1. or 3xy (xy): (≈—y)3 :: 18: 1. Divide now by x- y, then will 3xy: (x—y)2 :: 18:1; but xy = 24; ..72 : (x—y)2 :: 18 : 1. Hence, by Theorem 1, 18 x (x—y)2 = = 72, or (x-y)2 = 4; ..x―y = 2. Again, x-y)2=x2—2xy+y2 Therefore 2+2xy + y2 3 = = 4. and 4xy 6. It is required to divide the number 24 into two such parts, that their product shall be to the sum of their squares :: 3 : 10. Ans. 18 and 6. If 6 7. There are two numbers which are to each other as 3 : 2. be added to the greater, and subtracted from the less, the sum and remainder will be to each other :: 3 : 1. What are those two numbers ? Ans. 24 and 16. 8. There are two numbers which are to each other in the duplicate ratio of 4: 3, and 24 is a mean proportional between them. What are Ans. 32 and 18. those two numbers ? 9. If xy:: 36: 25, and 2x + y : x + 2 in a ratio compounded of the ratios of 17: 2 and 2:7; numbers? what are the two Ans. 12 and 10. 11. If a + x : a − x :: 9 : 5, shew (by Theor. 8.) that a : r :: 7 : 2. 10. There are two numbers whose product is 135, and the difference of their squares is to the square of their difference :: 4 : 1. What are these numbers ? Ans. 15 and 9. PROGRESSION. (124.) When three or more quantities are in continued pro. portion, such that, as the first is to the second, so is the second to the third, so is the third to the fourth, so is the fourth to the fifth, &c. the series is called a progression, and is either an Arithmetical or a Geometrical Progression, according as the several terms of the series are in Arithmetical or Geometrical Proportion. And any progression is said to be ascending when the terms of the series increase, but to be descending when they decrease. Arithmetical Progression. (125.) The chief properties of an arithmetical series are the following: 1. The sum of the extreme terms is equal to the sum of any two means equidistant from them, or to twice the middle term when the number of terms is odd. 2. The difference of the extremes, divided by 1 less than the number of terms, is equal to the common difference. 3. The difference of the extremes is equal to the common difference multiplying 1 less than the number of terms. 4. The sum of the series is equal to half the number of terms multiplying the sum of the extremes. (126.) If, therefore, in any Arithmetical Series The first term be denoted by .. The last term by The number of terms by The common difference by a % n d ... |