wherefore e: b=f: d, where e and f, the consequents of the one, are the antecedents, and b and d, the consequents of the other, are consequents. Q. E. D. 112. Prop. 11. If the consequents of one proportion be the antecedents in another, a third proportion will arise, having the same antecedents as the former, and the same consequents as the latter. Let a:b=c:d, and bed:f; then a: ec:f; for 113. Prop. 12. If there be any number of proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents. Let a: bc: d=e:f=g:h; then a: b=a+c+e+g:b+d+f+h. For ab-ba, and Prop. 1st ad=bc, af be, and ah=bg; therefore by adding equals to equals, we have ab+ad+af +ah ba+bc+be+bg; hence a(b+d+f+h)=b(a+c +e+g); and therefore by Prop. 2d we obtain a:b=a+c+e+g:b+d+f+h. Q. E. D. 114. DEFINITION. When any number of quantities is in continued proportion, the first is said to have to the third the duplicate ratio of the first to the second, and the first is said to have to the fourth the triplicate ratio that the first has to the second. 115. Prop. 13. The duplicate ratio is the same as the ratio of the squares of the terms expressing the simple ratio; and the triplicate ratio is the same as the ratio of the cubes of the terms expressing the simple ratio. Let a: bb:cc: d, then a : c=a2 : b2. 116. Prop. 14. The product of the like terms of any numerical proportions are themselves proportional. hence aei: bfk-cgl: dhm. m Q. E. D. 117. Prop. 15. If there be three magnitudes, a, b, c, and other three, d, e, f, such that a: b=d: e, and b: c=e: f, then a c=d: f. 118. INTEREST is the allowance given for the loan or forbearance of a sum of money, which is lent for, or becomes due at a certain time; this allowance being generally estimated at so much for the use of L.100 for a year. The money lent is called the principal, the sum paid for its use is called the interest, the sum of the principal and interest is called the amount, and the interest of L.100 for one year is called the rate per cent. Interest is either Simple or Compound. Simple interest is that which is allowed upon the original principal only, for the whole time or forbearance. 119. PROBLEM 1. To find the simple interest of any sum for any period, and at any given rate per cent. Let r the interest of one pound for a year, p=the principal or sum lent, t=the time of the loan in years, i= the interest of the given principal for the given time, and a=the amount of the given principal and its interest for the time t; then we will obviously have the following relations among the quantities: 1: ptr: i.. i=prt. (1.) a=p+prt=p(1+rt). (2.) and hence i 1+rt rt a-p 2 pr pr By means of the above five formulæ all the circumstances connected with the simple interest of money are readily determined. But as the rules for the calculation of simple interest are generally given in reference to the rate per cent., instead of the rate per pound, as above, the formulæ may be all changed into those relating to rate per cent., by making r represent the rate per cent, and substituting in 100 stead of r throughout all the formula; and the student is requested to write in words the rules which the five formulæ contain, by which he will be led to see the advantage of algebraic formula. 1. What is the interest of L.560 for 3 years, at 4 per cent.? Ans. L.75, 12s. 2. What is the amount of L.420 for 6 years, at 3 per cent.? Ans. L.495, 12s. 3. What principal laid out at interest for 5 years at 4 per cent. will gain L.60? Ans. L.300. 4. What principal laid at interest for 10 years at 3 per cent. will amount to L.607, 10s.? Ans. L.450. 5. In what time will L.500 amount to L.800 at 4 per cent. ? Ans. 15 years. 6. At what rate per cent. will L.200 amount to L.344 in 18 years? COMPOUND INTEREST. Ans. 4 per cent. 120. In compound interest, the interest is added to the principal at stated intervals or periods, and this amount is made the principal for the next period. Hence if R represent the amount of one pound at the end of one period, since this is the sum laid at interest during the next period, we will evidently have the following proportions to find the amount of L.1 at the end of any number of periods. 1: R=R: R2, amount at the 2d period. 1: R=R3 : R3, amount at the 3d period. 1: R=R3 : R4, amount at the 4th period. 1: R=R"-1 : R", amount at the nth period. From which it appears that the amount of one pound at the end of any number of periods is R raised to the power denoted by the number of periods, and it is plain that the amount of p pounds will be p times the amount of one pound; hence, representing the amount of p pounds by A, and the number of periods by t, we will have F 121. The interest is generally converted into principal yearly, but sometimes half-yearly, and sometimes even quarterly. If r represent the simple interest of L.1 for a year, and n the number of years for which the calculation is to be made, then when the interest is convertible into principal half-yearly, R and t will have the following values: R=1+1⁄2, t=2n; and when it is convertible quarterly, R=1+, t=4n; also R=1+, and t=mn, when the interest is convertible into principal m times per annum. 5 per EXERCISES. 1. What will be the amount of L.1000 in ten years, at cent. compound interest? Ans. L.1628, 17s. 91d. 2. What principal laid at compound interest will amount to L.700 in eleven years, at 4 per cent.? cent. compound interest? at 4 per Ans. L.454, 14s. 1d. 3. In what time will L.365 amount to L.400, Ans. 2 years 122 days. 4. At what rate per cent. compound interest will L.50 amount to L.63, 16s. 34d. in five years? Ans. 5 per cent. 5. In what time will a sum of money double itself, at Ans. 14.2 years. 5 per cent. compound interest? 6. In what time will a sum of money double itself, at Ans. 17.67 years. 4 per cent. compound interest. 7. In what time at compound interest, reckoning 5 per cent. per annum, will L.10 amount to L.100? Ans. 47-19 years. 8. What will be the compound interest of L.100 for twelve years at 4 per cent., if the interest be payable yearly? what if payable half-yearly? and what if payable quarterly? Ans. L.60, 2s. Oĝd., L.60, 16s. 101d., and L.61, 4s. 51d. ANNUITIES. 122. ANNUITIES signify any interest of money, rents, or pensions, payable from time to time, at particular periods. The most general division of annuities, is into annuities certain, and annuities contingent; the payment of the latter depending upon some contingency, such, in particular, as the continuance of a life. Annuities have also been divided into annuities in possession, and annuities in reversion, the former meaning such as have commenced, or are to commence immediately, and the latter such as will not commence till some particular future event has happened, or till some given period of time has expired. Annuities may be farther considered as being payable yearly, half-yearly, or quarterly. The present value of an annuity is that sum, which being improved at compound interest, will be sufficient to pay the annuity. The present value of an annuity certain, payable yearly, and the first payment of which is to be made at the end of a year, is computed as follows:— Let the annuity be supposed L.1; the present value of the first payment is that sum in hand, which being put to interest, will amount to L.1 in a year; in like manner, the present value of the second payment, or of L.1 to be received two years hence, is that sum, which being put to interest immediately, will amount to L.1 in two years, and so on for any number of years or payments; and the sum of the values of all the payments will be the present value of the annuity. 123. Let the interest of L.1 for one year be represented by r, then L.1 will amount to 1+r in a year, and the sum that will amount to one in one year, which call x, will evidently bear the same proportion to L.1, that L.1 bears to 1+r; hence we have the following proportion: In the same manner, that sum which in two years will amount to L.1, is evidently that sum which in one year 1 will amount to Stating the proportion so that the 1+r quantity sought may stand last, we have |