Also, in the right-angled triangle CGF, we find CG2+GF'= CF2; that is, sin. 'A+cos. 'A=R'; or, The square of the sine of an arc, together with the square of its cosine, is equal to the square of the radius. (29.) A table of natural sines, tangents, &c., is a table giving the lengths of those lines for different angles in a circle whose radius is unity. Thus, if we describe a circle with a radius of one inch, and divide the circumference into equal parts of ten degrees, we shall find the sine of 10° equals 0.174 inch; If we draw the tangents of the same arcs, we shall find the tangent of 10° equals 0.176 inch;. 70° In the accompanying table, pages 116-133, the sines, cosines, tangents, and cotangents are given for every minute of the quadrant to six places of figures. (30.) To find from the table the natural sine, cosine, &c., of an arc or angle. If a sine is required, look for the degrees at the top of the page, and for the minutes on the left; then, directly under the given number of degrees at the top of the page, and opposite to the minutes on the left, will be found the sine required. Since the radius of the circle is supposed to be unity, the sine of every arc below 90° is less than unity. The sines are expressed in decimal parts of radius; and, although the decimal point is not written in the table, it must always be prefixed. As the first two figures remain the same for a great many numbers in the table, they are only inserted for every ten minutes, and the vacant places must be supplied from the two leading figures next preceding. Thus, on page 120, the sine of 25° 11' is 0.425516; The tangents are found in a similar manner. the tangent of 31° 44' is 0.618417; Thus The same number in the table is both the sine of an arc and the cosine of its complement. The degrees for the cosines must be sought at the bottom of the page, and the minutes on the right. Thus, on page 130, the cosine of 16° 42' is 0.957822; 66 The cotangents are found in the same manner. the cotangent of 19° 16' is 2.86089; Thus It is not necessary to extend the tables beyond a quadrant, because the sine of an angle is equal to that of its supplement (Art. 27). Thus (31.) If a sine is required for an arc consisting of degrees, minutes, and seconds, we must make an allowance for the seconds in the same manner as was directed in the case of logarithms, Art. 7; for, within certain limits, the differences of the sines are proportional to the differences of the corresponding arcs. Thus the sine of 34° 25' is .565207; 66 34° 26' is .565447. The difference of the sines corresponding to one minute of arc, or 60 seconds, is .000240. The proportional part for 1' is found by dividing the tabular difference by 60, and the quotient, .000004, is placed at the bottom of page 122, in the column headed 34°. The correction for any number of seconds will be found by multiplying the proportional part for 1" by the number of seconds. Required the natural sine of 34° 25' 37". The proportional part for 1", being multiplied by 37, becomes 148, which is the correction for 37". Adding this to the sine of 34° 25', we find the sine of 34° 25' 37" is .565355. Since the proportional part for 1" is given to hundredths of a unit in the sixth place of figures, after we have multiplied by the given number of seconds, we must reject the last two figures of the product. In the same manner we find the cosine of 56° 34′ 28′′ is .550853. It will be observed, that since the cosines decrease while the arcs increase, the correction for the 28" is to be subtracted from the cosine of 56° 34'. (32.) To find the number of degrees, minutes, and seconds belonging to a given sine or tangent. If the given sine or tangent is found exactly in the table, the corresponding degrees will be found at the top of the page, and the minutes on the left hand. But when the given number is not found exactly in the table, look for the sine or tangent which is next less than the proposed one, and take out the corresponding degrees and minutes. Find, also, the difference between this tabular number and the number proposed, and divide it by the proportional part for 1" found at the bot tom of the page; the quotient will be the required number of seconds. Required the arc whose sine is .750000. The next less sine in the table is .749919, the arc corresponding to which is 48° 35'. The difference between this sine and that proposed is 81, which, divided by 3.21, gives 25. Hence the required arc is 48° 35′ 25′′. In the same manner we find the arc whose tangent is 2.00000 is 63° 26' 6". If a cosine or cotangent is required, we must look for the number in the table which is next greater than the one proposed, and then proceed as for a sine or tangent. Thus the arc whose cosine is .40000 is 66° 25' 18"; 66 66 66 cotangent is 1.99468 is 26° 37' 34". (33.) On pages 134-5 will be found a table of natural secants for every ten minutes of the quadrant, carried to seven places of figures. The degrees are arranged in order in the first vertical column on the left, and the minutes at the top of the page. Thus the secant of 21° 20' is 1.073561; If a secant is required for a number of minutes not given in the table, the correction for the odd minutes may be found by means of the last vertical column on the right, which shows the proportional part for one minute. Let it be required to find the secant of 30° 33′ The secant of 30° 30' is 1.160592. The correction for 1' is 198.9, which, multiplied by 3, be comes 597. Adding this to the number before found, we obtain 1.161189. For a cosecant, the degrees must be sought in the righthand vertical column, and the minutes at the bottom of the page. Thus the cosecant of 47° 40' is 1.352742; (34.) When the natural sines, tangents, &c., are used in proportions, it is necessary to perform the tedious operations of multiplication and division. It is, therefore, generally preferable to employ the logarithms of the sines; and, for convenience, these numbers are arranged in a separate table, called logarithmic sines, &c. Thus the natural sine of 14° 30' is 0.250380. Its logarithm, found from page 6, is 1.398600. The characteristic of the logarithm is negative, as must be the case with all the sines, since they are less than unity. To avoid the introduction of negative numbers in the table, we increase the characteristic by 10, making 9.398600, and this is the number found on page 38 for the logarithmic sine of 14° 30'. The radius of the table of logarithmic sines is therefore, properly, 10,000,000,000, whose logarithm is 10. (35.) The accompanying table contains the logarithmic sines and tangents for every ten seconds of the quadrant. The degrees and seconds are placed at the top of the page, and the minutes in the left vertical column. After the first two degrees, the three leading figures in the table of sines are only given in the column headed 0", and are to be prefixed to the numbers in the other columns, as in the table of logarithms of numbers. Also, where the leading figures change, this change is indicated by dots, as in the former table. The correction for any number of seconds less than 10 is given at the bottom of the page. (36.) To find the logarithmic sine or tangent of a given arc. Look for the degrees at the top of the page, the minutes on the left hand, and the next less number of seconds at the top; then, under the seconds, and opposite to the minutes, will be found four figures, to which the three leading figures are to be |