(XII). If, in any proportion, the antecedents are alike or equal, the consequents must be equal; also, if the consequents are alike or equal, the antecedents must be equal. The reason is plain; for equal fractions having equal numerators, must have equal denominators; and equal fractions having equal denominators, must have equal numerators. Moreover, it is evident that, If, in any proportion, the second term is greater than the first, the fourth must be greater than the third, and conversely; and if the first two terms are equal, the last two must also be equal. (XII). Suppose we have a series of equal ratios, as a:bc:d=e:f=g: h, or Let q represent the value of each of these fractions. Removing the denominators, a=bq, c=dq, e=fq, g=hq. Adding these equations, a+c+e+g=bq+dq+fq+hq, or Dividing by b+d+f+h, a+c+e+g:b+d+f+h=a:b=c:d=e:f=g:h. Now, the first term of this proportion is the sum of the antecedents, and the second is the sum of the consequents, of the given ratios. Hence, In any series of equal ratios, the sum of the antecedents is to the sum of the consequents, as any one of the antecedents is to its consequent. (XIV). Suppose we have the two proportions By multiplying together the corresponding members of these two equations, we obtain This proportion is the same as we should have obtained from multiplying together the corresponding terms of the two given proportions in their first form. This is called multiplying the proportions in order; and it is evident that any number of proportions might be combined in the same way. Hence, If two or more proportions are multiplied in order, the result will form a proportion. Since division is the reverse of multiplication, it follows that, If proportions are divided in order, the result will form a proportion. (xv). Given a : bc: d. Putting this proportion in the form of raising both members to any power, the degree of which is denoted by m, we have Similar powers of proportional quantities form a proportion. Since extracting roots is the reverse of finding powers, it follows that Similar roots of proportional quantities will form a proportion. ART. 133. The following exercises are designed to exemplify the foregoing principles of proportions. The correctness of any proportion may be verified by ascertaining that the product of the means is equal to that of the extremes. 1. Illustrate (1) by the proportion 7: 10=21: 30. 2. Illustrate (11) by putting 12.8=32.3 into a proportion; also by forming a proportion from mn=xy. 3. Illustrate (III) by finding the value of x in each of the following proportions.. x:7=9:21; 10:x=5:15; 7:4=x:20; 4. According to (II), what is to be inferred respecting x and y in the proportions make all the changes authorized by (v). 7. Illustrate (vi) by the proportion 10:15 30: 45. 8 Illustrate (VII) by the proportion 30:4960: 98. 9 Illustrate (vi) by the two proportions 7:9 14:18. 10. Illustrate (1x) by the two sets of proportions (10:7=30: 21, 10:530: 15; ( 8:5= 16: 10, 12:524: 10. 11. Illustrate (x) by the proportion 3:79:21. 12. Illustrate (x1) by the proportion 12:860: 40. 13. According to (XII), what is to be inferred from the proportion 9:x=9:3? also, from the proportion y:7=5:7? 14. Also, according to (x1), what is to be inferred with regard to x in each of the proportions 4:10=12: x, 6: 6=20: x? 15. Illustrate (XIII) by the equal ratios = =4:8 =9:18 12:24. 1:2 3:6: = SECTION XLII. PROGRESSION BY DIFFERENCE. ART. 134. A progression by difference, or an arithmetical progression, is a series of quantities constantly increasing or constantly diminishing by a common difference; and these successive quantities are called the terms of the progression. Thus, 1, 2, 3, 4, 5, &c., is a progression by difference, the common difference being 1; also, 3, 5, 7, 9, 11, &c., the common difference being 2. A progression is called increasing, when the terms. increase from left to right; and it is called decreasing, when the terms decrease in the same direction. Thus, 8, 11, 14, 17, &c., is an increasing, but 25, 20, 15, 10, &c., is a decreasing progression. ART. 135. To exhibit a progression by difference in its most general form, let a be the first term, and d the common difference. Then, if the progression is increasing, 1st 2d за 4th 5th a, (a+d), (a+2d), (a+3 d), (a+4 d), &c., will be the successive terms at the commencement of the series. But if the progression is decreasing, 1st a, (a—d), (a—2 d), (a—3 d), (a—4 d,) &c., will be the initial terms. If we examine either of these series, we shall perceive that the coefficient of d in the second term is 1, in the third term it is 2, in the fourth, 3, in the fifth, 4, &c.; |