According to the rule for subtraction in arithmetic, any number is subtracted from 10, 100, 1000, &c. by beginning on the right hand, and taking each figure from 10, after increasing all except the first, by carrying 1. The difference, or arith'l compl't is 2.36875, which is obtained by taking 5 from 10, 3 from 10, 2 from 10, 4 from 10, 7 from 10, and 8 from 10. But, instead of taking each figure, increased by 1 from 10; we may take it without being increased, from 9. Thus, 2 from 9 is the same as 3 from 10, 3 from 9 the same as 4 from 10, &c. Hence, 55. To obtain the ARITHMETICAL COMPLEMENT of a number, subtract the right hand significant figure from 10, and cach of the other figures from 9. If, however, there are ciphers on the right hand of all the significant figures, they are to be set down without alteration. In taking the arithmetical complement of a logarithm, if the index is negative, it must be added to 9; for adding a negative quantity is the same as subtracting a positive one. (Alg. 81.) The difference between 3 and +9, is not 6, but 12. 56. The principal use of the arithmetical complement, is in working proportions by logarithms; where some of the terms are to be added, and one or more to be subtracted. In the Rule of Three or simple proportion, two terms are to be added, and from the sum, the first term is to be subtracted. But if, instead of the logarithm of the first term, we substitute its arithmetical complement, this may be added to the sum of the other two, or more simply all three may be added together, by one operation. After the index is diminished by 10, the result will be the same as by the common method. For subtracting a number is the same, as adding its arithmetical complement, and then rejecting 10, 100, or 1000, from the sum. (Art. 53.) It will generally be expedient, to place the terms in the same order, in which they are arranged in the statement of the proportion. 57. In compound, as in single proportion, the term required may be found by logarithms, if we substitute addition for multiplication, and subtraction for division. Ex. 1. If the interest of $365, for 3 years and 9 months, be $82.13; what will be the interest of $8940, for 2 years and 6 months? In common arithmetic, the statement of the question is made in this manner. And the method of calculation is, to divide the product of the third, fourth, and fifth terms, by the product of the first two.* This, if logarithms are used, will be to subtract the sum of the logarithms of the first two terms, from the sum of the logarithms of the other three. Sum of the logs. of the 3rd, 4th, and 5th, 6.26378 Do. Term required 1st and 2nd, 3.13632 1341 3.12746 58. The calculation will be more simple, if, instead of subtracting the logarithms of the first two terms, we odd their arithmetical complements. But, it must be observed, that each arithmetical complement increases the index of the logarithm by 10. If the arithmetical complement be introduced into two of the terms, the index of the sum of the logarithms will be 20 too great; if it be in three terms, the index will be 30 too great, &c. The result is the same as before, except that the index of the logarithm is 20 too great. See Arithmetic. Ex. 2. If the wages of 53 men for 42 days be 2200 dollars; what will be the wages of 87 men for 34 days? 59. In the same manner, if the product of any number of quantities, is to be divided, by the product of several others; we may add together the logarithms of the quantities to be divided, and the arithmetical complements of the logarithms of the divisors. Ex. If 29.67x346.2 be divided by 69.24X7.862X497; what will be the quotient? In this way, the calculations in Conjoined Proportion may be expeditiously performed. COMPOUND INTEREST. 60. In calculating compound interest, the amount for the first year, is made the principal for the second year; the amount for the second year, the principal for the third year, &c. Now the amount at the end of each year, must be proportioned to the principal at the beginning of the year. If the principal for the first year be 1 dollar, and if the amount of 1 dollar for 1 year-a; then, (Alg. 341.) 1:a:: a a2=the amount for the 2d year, or the principal for the 3d; a: a=the amount for the third year, or the principal for the 4th; a3: a=the amount for the 4th year, or the principal for the 5th. That is, the amount of 1 dollar for any number of years is obtained by finding the amount for 1 year, and involving this to a power whose index is equal to the number of years. And the amount of any other principal, for the given time, is found by multiplying the amount of 1 dollar, into the number of dollars, or the fractional part of a dollar. If logarithms are used, the multiplication required here may be performed by addition; and the involution by multiplication. (Art. 45.) Hence, 61. To calculate Compound Interest, Find the amount of 1 dollar for 1 year; multiply its logarithm by the number of years; and to the product, add the logarithm of the principal. The sum will be the logarithm of the amount for the given time. From the amount subtract the principal, and the remainder will be the interest. If the interest becomes due half yearly or quarterly; find the amount of one dollar, for the half year or quarter, and multiply the logarithm by the number of half years or quarters in the given time. If P the principal, a=the amount of 1 dollar for 1 year, n=any number of years, and A=the amount of the given principal for n years; then, |