TRIGONOMETRY. CHAPTER I. LOGARITHMS. 1. Logarithms are exponents of the powers of some number which is taken as a base. In the tables of Logarithms in common use, the number 10 is taken as the base, and all numbers are considered as powers of 10. And, since 10° 1, that is, the Logarithm of 1 is 0, the Logarithm of any number between 1 and 10 is between O and 1, i. e. is a fraction; the Logarithm of any number between 10 and 100 is between 1 and 2, i. e. is one plus a fraction; and the Logarithm of any number between 100 and 1000 is 2 plus a fraction; and so on. And, as 100.1, that is, the Logarithm of 0.1 66 66 10-30.001, &c., 66 0.01 the Logarithm of any number between 1 and 0.1 is between 0 and -1, i. e. is —1 plus a fraction; the Logarithm of any number between 0.1 and 0.01 is between —1 and —2, i. e. is -2 plus a fraction; and so on. The Logarithms of most numbers, therefore, consist of an integer, either positive or negative, and a fraction, which is always positive. The representation of the Logarithms of all numbers less than a unit by a negative integer and a positive fraction is merely a matter of convenience. 2. The integral part of a Logarithm is called the characteristic, and is not generally written in the tables, but can be found by the following RULE. The characteristic of the Logarithm of any number is equal to the number of places by which its first significant figure on the left is removed from units' place, the characteristic being positive when this figure is to the left, and negative when it is to the right of units' place. E. g. The Logarithm of 59 is 1 plus a fraction; i. e. the characteristic of the Logarithm of 59 is 1. The Logarithm of 5417.7 is 3 plus a fraction; i. e. the characteristic of the Logarithm of 5417.7 is 3. The Logarithm of 0.3 1 plus a fraction; i. e. the characteristic of the Logarithm of 0.3 is -1. The Logarithm of 0.00017 is -4 plus a fraction; i. e. the characteristic of the Logarithm of 0.00017 is -4. is 3. Since the base of this system of Logarithms is 10, if any number be multiplied by 10, its Logarithm will be increased by a unit; if divided by 10, diminished by a unit. Hence, the decimal part of the Logarithm of any set of figures is the same, wherever the decimal point of the set may be. The minus sign is generally written over the characteristic, as the characteristic only is negative. TABLE OF LOGARITHMS. 4. In the tables of Logarithms, generally, the decimal only is given, and the characteristic must be supplied, according to the Rule in Article 2. To find the Logarithm of any number of three or less figures: RULE. Find the given number in the column marked N., and directly opposite, in the column marked 0, is the decimal part of the Logarithm, to which must be prefixed the characteristic, according to the Rule in Article 2. The first two figures of the decimal, remaining the same for several successive numbers, are not repeated, but are left to be supplied. Thus the Log. of 839 is 2.923762. As, according to Article 3, multiplying a number by 10 increases its Logarithm by a unit, therefore, to find the Logarithm of any number containing only three significant figures with one or more ciphers annexed, we use the same rule as in the last case. E. g. The Log. of 8320 is 3.920123 68 "" 756000 5.878522 To find the Logarithm of any number consisting of four figures: RULE. Look for the first three figures in the column marked N., and for the fourth figure at the top of one of the columns. the fourth figure, will be the last four figures of the decimal part of the Logarithm, to which the first two figures in the column marked 0 are to be prefixed, and the characteristic, according to the Rule in Article 2. In some of the columns marked 1, 2, 3, &c., dots will be found. This shows that the two figures which are to be prefixed from the column marked 0 have changed to the next larger number, and are to be found in the horizontal line directly below. The dots are used to avoid any mistake, and their place is to be supplied with ciphers. As in Article 3, annexing ciphers to the number only increases the characteristic as many units as there are ciphers annexed. To find the Logarithm of any number consisting of more than four figures: RULE. Find the Logarithm of the first four figures as before; multiply by the remaining figures the number standing opposite, in the column marked D, reject from the right as many figures as you multiply by, and add what is left to the Logarithm previously found. E. g. Required, the Log. of 609946. The Log. of 609900 is Under D opposite is 71, wh. multiplied by 46 gives Therefore, the Log. of 609946 is Required, the Log. of 84997. Log. of 84990 is 5.785259 32.66 5.785292 * 4.929368 35.7 4.929404 Under D opposite is 51, which multiplied by 7 gives The column marked D contains the average difference of the ten Logarithms against which it stands. The reason * Wherever the fractional part omitted is larger than half the unit in the for rejecting from the product as many figures as you multiply by is, that these figures are just so many places farther to the right than the figures whose Logarithm has already been found. This process of finding the Logarithms of large numbers supposes that the Logarithms vary as the numbers, which is not strictly true, though sufficiently so for ordinary use. If the number whose Logarithm is sought contains decimal figures, the decimal part of the Logarithm, according to Article 3, is the same as though there were no decimal point; but the characteristic varies according to the Rule in Article 2. The Logarithm of a vulgar fraction may best be found by reducing the fraction to a decimal, and then proceeding as above. 5. To find the Natural Number corresponding to a given Logarithm: RULE. Neglecting the characteristic, find, if possible, in the table the Logarithm given. The three figures opposite in the column N., with the number at the head of the column in which the Logarithm is found, affixed, and the decimal point so placed as to make the number of integral figures one more than the characteristic of the given Logarithm, will be the number required. E. g. The Natural Number corresponding to Log. 5.531862 is 340300. The Natural Number corresponding to Log. 1.605951 is 40.36. If the decimal part of the Logarithm cannot be exactly found, take the Natural Number corresponding to the next less Logarithm, as before; then find the difference between this and the given Logarithm; divide this difference by the tabular difference in the column opposite, under D, annex |