can be put in the places of the antecedents a, c; or, conversely, a and c in the places of b and d. 466. Also, from the same equality a-b-c-d, there will arise, by adding m-n to each of its sides, (a+m)-(b+n)=(c+m)-(d+n); where it appears that the proportion is not altered, by augmenting the antecedents a and c by the same quantity m, and the consequents b-d by another quantity n. In short, every operation by way of addition, subtraction, multiplication, and division, made upon each member of the equation, a→b=c -d, gives a new property of this kind of proportion, without changing its nature. 467. The same principles are also equally applicable to any continued set of equi-differences of the form a-bb-c-c-d-d-e, &c. which denote the relations of a series of terms in what has been usually called arithmetical progression. 468. But these relations will be more commodiously shown, by taking a, b, c, d, &c. so that each of them shall be greater or less than that which precedes it by some quantity d'; in which case the terms of the series will become a, a±d, a±2d', a±3d', a±4d', &c. Where, if be put for that term in the progression of which the rank is n, its value, according to the law here pointed out, will evidently be la±(n-1)d'; which expression is usually called the general term of the series; because, if 1, 2, 3, 4, &c. be successively substituted for n, the results will give the rest of the terms. Hence the last term of any arithmetical series is equal to the first term plus or minus, the product of the common difference, by the number of terms less one. 469. Also, if s be put equal to the sum of any number of terms of this progression, we shall have s=a+(a±d')+(a±2d') + .... +[a±(n−1)d']. And by reversing the order of the terms of the series, s=[a±(n−1)d']+[a±(n−2)d']+ ...(a±d)+a. Whence, by adding the corresponding terms of these two equations together, there will arise 28=[2a±(n−1)d']+[2a±(n−1)d'], &c. to n terms. And, consequently, as all the n terms of this series are equal to each other, we shall have 2s=n[2a=±(n−i)d'], or s=[2a±(n−1)d']..(1). 470. Or, by substituting for the last term a± (n-1)d, as found above, this expression (1) will become Hence, the sum of any series of quantities in arithmetical progression is equal to the sum of the two extremes multiplied by half the number of terms. It may be observed, that from equations (1) and (2), if any three of the five quantities, a, d', n, l, s, be given, the rest may be found. 471. Let l, as before, be the last term of an arithmetic series, whose first term is (a), common difference (d'), and number of terms (n); then l=a+(n−1)ď′; ..d': Now the intermediate terms between -α n- 1 the first and the last is n-2; let n-2=m, then n-1 l-a m+1' +1. Hence d': = which gives the following rule for finding any number of arithmetic means between two numbers. Divide the difference of the two numbers by the given number of means increased by unity, and the quotient will be the common difference. Having the common difference, the means themselves will be known. Example 1. Find the sum of the series 1, 3, 5, 7, 9, 11, &c. continued to 120 terms. Here a=1, d'=2, .*. (Art. 469), s=[2a+(n−1)d']";= the What are Ex. 2. The sum of an arithmetic series is 567, first term 7, and the common difference 2. the number of terms? Here s=567,).·.(Art. 469),2s=n[2a+(n−1)d']= n[14+ (n-1)2]=14n+2n2 - 2n 1134;.. n2+6n+9=576, and n=21. a=7, d=2; Ex. 3. The sum of an arithmetic series is 1455, the first term 5, and the number of terms 30. is the common difference? What Ans. 3. What Ex. 4. The sum of an arithmetic series is 1240, common difference-4, and number of terms 20. is the first term? Ans. 100. Ex. 5. Find the sum of 36 terms of the series, 40, 38, 36, 34, &c. Ex. 6. The sum of an arithmetic first term 3, and common difference 2. number of terms? Ans. 180. series is 440, What are the Ex. 7. A person bought 47 sheep, and gave 1 shilling for the first sheep, 3 for the second, 5 for the third, and so on. What did all the steep cost him? Ans. 1107. 9s. Ex. 8. Find six arithmetic means between 1 and Ans. 7, 13, 19, 25, 31, 37. 43. § II. GEOMETRICAL PROPORTION AND PROGRESSION. 472. GEOMETRICAL PROPORTION, is the relation which two numbers, or quantities, of the same kind, have to two others, when the antecedents or leading terms of each pair, are the same parts of their consequents, or the consequents of their antecedents. 473. And if two quantities only are to be compared together, the part, or parts, which the antecedent is of the consequent, or the consequent of the antecedent, is called the ratio; observing, in both cases, to follow the same method. 474. Direct proportion, is when the same relation subsists between the first of four quantities, and the second, as between the third and fourth. Thus, a, ar, b, br, as in direct proportion. 475. Inverse, or reciprocal proportion, is when the first and second of four quantities are directly proportional to the reciprocals of the third and fourth. Thus, a, ar, br, b, are inversely proportional; be 1 1 cause, a, ar, are directly proportional. br' b' 476. The same reason that induced the writers mentioned in (Art. 462), to give the name of equidifferences to arithmetical proportionals, also led them to apply that of equi-quotients to geometrical proportionals, and to express their relations in a similar way by means of equations. Thus, if there be taken any four proportionals, a, b, c, d, which it has been usual to express by means of points, as below, a: b::c: d This relation, according to the method above-men tioned, will be denoted by the equation a C ・b d' (Art. 24); where the equal ratios are represented by fractions, the numerators of which are the antecedents, and the denominators the consequents. Hence, (Art. 190), ad=bc. 477. And if the third term c, in this case, be the same as the second, or c=b, the proportion is said to be continued, and we have ad=b3, or b=√ud; where it is evident, that the product of the extremes of three proportionals, is equal to the square of the mean: or, that the mean is equal to the square root of the product of the two extremes. a C 478. Also, from the equality, there will re b ď : for, by adding or subtracting 1 from α each side of the equation ; then+1+1 a±b b Hence, when four quantities are proportionals, the sum or difference of the first and second is to the second as the sum or difference of the third and fourth, is to the fourth. 479. In like manner, if a: b::c:d; then, ma: C mb: Acid. For=; .. (Art. 118),· and, (Art. 478), ma mb :: c: c. : ma mb = Hence, when four quantities are proportionals, if the first and second be multiplied, or divided by any quantity, and also the second and fourth, the resulting quantities will still be proportionals. 480. Also, if a :: b::c:d; then, and (Art. 478), a* : b" : : c" ; d′′ ; where n may : be any number either integral or fractional. Hence, if four quantities be proportionals, any power or roots of those quantities will be proportionals. And, by proceeding in a similar manner, all the Properties and transformations of ratios and proportion, can be easily obtained from the equality b d or ad=bc. a 481. In addition to what is here said, it may be observed, that the ratio of two squares is frequently called duplicate ratio; of two square roots, subduplicate ratio; of two cubes, triplicate ratio; and of two cube roots, subtriplicate ratio. See the APPENDIX at the end of this Treatise, where the doctrine of ratios and proportion is fully explained and clearly illus trated. 482. GEOMETRICAL PROGRESSION, is when a series of numbers, or quantities, have the same constant ratio, or which increase, or decrease, by a common multiplier, or divisor. Thus, the numbers 1, 2, 4, 8, 16, &c. (which increase by the continual multiplication of 2), and the numbers 1, 3, 4, 7, &c. (which decrease |