logarithmic sines of 27° 34', and of 27° 35', are, respectThe difference between their ively, 9.665375 and 9.665617. mantissas is 242; this, divided by 60, the number of seconds in one minute, gives 4.03, which is the change in the mantissa for 1", between the limits 27° 34' and' 27° 35'. For the sine and cosine, there are separate columns of lifferences, which are written to the right of the respective columns; but for the tangent and cotangent, there is but a single column of differences, which is written between them. The logarithm of the tangent increases, just as fast as that of the cotangent decreases, and the reverse, their sum being always equal to 20. The reason of this is, that the product of the tangent and cotangent is always equal to the square of the radius; hence, the sum of their logarithms must always be equal to twice the logarithm of the radius, or 20. The angle obtained by taking the degrees from the top of the page, and the minutes from any line on the left hand of the page, is the complement of that obtained by taking the degrees from the bottom of the page, and the minutes from the same line on the right hand of the page. But, by definition, the cosine and the cotangent of an arc are, respectively, the sine and the tangent of the complement of that arc (Arts. 26 and 28): hence, the columns designated sine and tang, at the top of the page, are designated cosine and cotang at the bottom. USE OF THE TABLE. To find the logarithmic functions of an arc which is expressed in degrees and minutes. 34. If the arc is less than 45°, ook for the degrees at the top of the page, and for the minutes in the left hand column; then follow the corresponding horizontal line till you come to the column designated at the top by sine, cosine, tang, or cotang, as the case may be; the number there found is the logarithm required. log sin 19° 55' Thus, 45°, look for the degrees at for the minutes in the right corresponding horizontal line If the angle is greater than the bottom of the page, and hand column; then follow the backwards till you come to the column designated at the bottom by sine, cosine, tang, or cotang, as the case may be; the number there found is the logarithm required. Thus, To find the logarithmic functions of an arc which is ex pressed in degrees, minutes, and seconds. 35. Find the logarithm corresponding to the degrees and minutes as hefore; then multiply the corresponding number taken from the column headed "D," by the number of seconds, and add the product to the preceding result, for the sine or tangent, and subtract it therefrom for the cosine or cotangent. EXAMPLES. 1. Find the logarithmic sine of 40° 26' 28". The same rule is followed for decimal parts, as in Art. 12. 2. Find the logarithmic cosine of 53° 40′ 40′′. If the arc is greater than 90°, we find the required function of its supplement (Arts. 26 and 28). 3. Find the logarithmic tangent of 118° 18' 25". 4. Find the logarithmic sine of 32° 18′ 35′′. Ans. 9.727945. 5. Find the logarithmic cosine of 95° 18' 24". Ans. 8.966080. 6. Find the logarithmic cotangent of 125° 23′ 50′′. Ans. 9.851619. To find the arc corresponding to any logarithmic function. 36. This is done by reversing the preceding rule: Look in the proper column of the table for the given logarithm; if it is found there, the degrees are to be taken from the top or bottom, and the minutes from the left or right hand column, as the case may be. If the given logarithin is not found in the table, then find the next less logarithm, and take from the table the corresponding degrees and minutes, and set them aside. Subtract the logarithm found in the table, from the given logarithm, and divide the remainder by the corresponding tabular difference. The quo tient will be seconds, which must be added to the degrees and minutes set aside, in the case of a sine or tangert, and subtracted, in the case of a cosine or a cotangent. Hence, the required arc is 15° 19′ 51′′. Hence, the required arc is 74° 28′ 43′′. 37. In what follows, we shall designate the three angles of every triangle, by the capital letters A, B, and C, A denoting the right angle; and the sides lying opposite the angles, by the corresponding small letters a, b, and C. Since the order in which these letters are placed may be changed, it follows that whatever is proved with the letters placed in any given order, will be equally true when the letters are correspondingly placed in any other order. Let CAB represent any triangle, right-angled at A. With C as a centre, and a radius CD, equal to 1, describe the arc DG, and draw GF and DE perpendicular to CA: then B E G will FG be the sine of the angle C, CF will be its cosine, and DE its tangent. Since the three triangles CFG, CDE, similar (B. IV., P. XVIII.), we may write the propor |