Having in this way computed the sines and cosines, we may [see equations (B), §8] obtain the tangents by dividing the sines by their corresponding cosines; the cotangents will be obtained by dividing the cosines by the sines. The reciprocal of the cosines will give the secants, and the reciprocal of the sines will give the cosecants. The values of the sines, cosines, tangents, and cotangents are given in Table III. CHAPTER II. EXPLANATION OF TABLES. TABLE I. LOGARITHMS. $ 19. THIS table gives the logarithms of all numbers from 1 to 10000. For the method of calculating these logarithms, the student is referred to my "Treatise on Algebra," chap. IX. The common logarithm, or Briggean logarithm, as sometimes called, in honor of the inventor Briggs, has 10 for its base. Hence, for common logarithms, we have From the above, we see that the logarithm of any number between 1 and 10 is a proper fraction between 0 and 1. The logarithm of any number between 10 and 100 is 1 plus a proper fraction. The logarithm of any number between 100 and 1000 is 2 plus a fraction. We also see that the logarithm of any number between 1 and 0.1 is some number between 0 and -1; it may therefore be represented by -1 plus a fraction. For a similar reason, the logarithm of any number between 0-1 and 0.01 may be represented by -2 plus a fraction. The logarithm of any number between 0.01 and 0·001 is −3 plus a fraction. Hence, we see that logarithms of all numbers greater than 10 or less than 1 consist of an integral part, positive or negative, plus a proper fraction. The integral part is called the characteristic, and may readily be assigned by the following RULE. The characteristic of the logarithm of any number greater than 1 is one less than the number of places of figures which express the integral part of the given number. The characteristic of the logarithm of a decimal fraction is a negative number, and is equal to the number of places by which the first significant figure is removed from the units' place. The characteristic being so readily obtained by means of the above rule, has been omitted in the tables, except in the small table on page 1, where are found the logarithms of all numbers from 1 to 100, with their characteristics. But for all the other numbers of our table only the decimal part of the logarithms are given. The decimal part of all logarithms are regarded as positive. The characteristic may, however, be considered as negative. When this is the case, it is usual to place the negative sign immediately over the characteristic. Thus the logarithm of 0.75 is 1-875061, or 0.875061–1. § 20. To find, in the table, the logarithm of any number from 1 to 100. Seek for the given number in one of the columns headed N, of the first page of the tables of logarithms, and against it on the right in the next column, will be found the logarithm. We thus find log. 7=0·845098, §21. To find the logarithm of any number consisting of three places of figures. Seek the given number in one of the left-hand columns headed N, and against it in the column headed 0 will be found the decimal part of the logarithm, observing that if only four figures are given in column 0, they are the last four: the first two are to be taken from the number directly above which contains six places of figures. Thus, we find §22. To find the logarithm of a number consisting of four places of figures. Seek the first three figures in column headed N, then passing horizontally across the page until you reach the column headed with the fourth figure of the given number, and you will have the last four decimal figures of the logarithm. The first two figures are to be taken from the column headed 0; observing, if the four figures found, stand opposite to a row of six figures in the column 0, to use the first two. But if the four figures. found are opposite a line of only four figures, you are then to ascend the column till you come to a line of six figures, and then to use the two left-hand figures. The six figures thus obtained will be the decimal part of the logarithm. The characteristic is to be found by the Rule under § 19. Thus, we find log. 36573.563125, log. 7641 = 3.883150. In some portions of the table, in the columns headed 1, 2, 3, 4, &c., the character is to be found in place of a figure, which indicates that the two figures of the first column, which are to be prefixed, have changed, and are to be taken from the horizontal line directly below. This change of the two left-hand figures of the logarithm is indicated by an asterisk placed in the column 0. The place occupied by this character is to be supplied with The two figures from the column 0 must also be taken from the line below the asterisk, if the character has been passed over while crossing the page horizontally. Thus, we find a zero. log. 21383.330008, $23. To find the logarithm of a number consisting of five or more places of figures. For example, let us seek the logarithm of 734582. We readily find as follows: Diff. of numb. =100 Diff. of logs.=59 : The difference between the first number 734500 and the given number is 734582-734500 = 82. Hence, if we suppose the difference of logarithms to be to each other as the difference of their corresponding numbers, which supposition is very nearly correct, we shall have 100: 82: 59: 48-38, for the difference between the logarithm of 734500 and the logarithm of 734582. Hence, we have Had there been only five figures in the given number, that is, only one beyond the fourth place, the first term of the above proportion would have been 10 instead of 100; had there been three additional figures beyond the fourth place, the first term would have been 1000. The difference between the logarithms of consecutive numbers of four places of figures is given in the table in a column headed D. Thus by turning to the table, we find 59, our third term of the above proportion, immediately opposite the loga rithm of 7345. From what has been done, we see that we may find the logarithm of a number consisting of more than four places of figures by the following method: Consider all the figures after the fourth as zeros. Then find the decimal part of the logarithm of the number given by the first four figures, observing to give a characteristic for the whole number of figures by rule under § 19. Take from the column |