OF SINES, TANGENTS, AND SECANTS. 21 between 18° and 15° be found*; from which the co-sine of 45' will be had, by two bisections only: whence the sines of all the arches in the progression 1° 30′, 2° 15′, 3° 00′, 3° 45', &c. may be determined (by Theor. 1. p. 17), and that to any assigned degree of exactness. The sines of all the terms of the progression 45', 1° 30', 2° 15', &c. up to 60°, being thus derived, the next thing is to find, by the help of these, the sines of all the intermediate arches, to every single minute. This, if you desire no more than the 4 or 5 first places of each (which is exact enough where nothing less than degrees and minutes is regarded), may be effected by barely taking the proportional parts of the differences. But if a greater degree of accuracy be insisted on, and you would have a table carried on to 7 or 8 places, each number (which is sufficient to give the value of an angle to seconds, and even to thirds, in most cases), then the operation may be as follows: 1o. Multiply the sum of the sines of any two adjacent terms of the progression 45', 1° 30′, 2° 15′, 3° 00′, 3° 45′, &c. (betwixt which you would find all the intermediate sines) by the fraction,0000000423, for a first product; and this, again, by 22, for a second product; to which last let of the difference of the two proposed sines (or extremes) be added, and the sum will be the excess of the first of the intermediate sines above the lesser extreme. *Note. The co-sine of the difference of two arches (suppos ng radius unity) is found by adding the product of their sines to that of their co-sines; as is hereafter demonstrated, 20. From this excess let the first product be continually subtracted; that is, first, from the excess itself; then from the remainder; then from the last remainder, and so on 44 times. 3°. To the lesser extreme add the forementioned excess; and to the sum add the first remainder; to this sum add the next remainder, and so on continually: then the several sums thus arising will respectively exhibit the sines of all the intermediate arches, to every single minute, exclusive of the last; which, if the work be right, will agree with the greater extreme itself, and therefore will be of use in proving the operation. But to illustrate the matter more clearly, let it be proposed to find the sines of all the intermediate arches between 3° 00' and 3° 45' to every single minute, those of the extremes being given, from the foregoing method, equal to ,05233595 and ,06540312 respectively. Here, the sum of the sines of the extremes being multiplied by ,0000000423, the first product will be ,00000000498, &c. or ,0000000050, nearly (which is sufficiently exact for the present purpose); and this, again, multiplied by 22, gives ,00000011 for a 2d product; which added to ,0002903815, part of the difference of the two given extremes, will be ,0002904915, the excess of the sine of 3° 01' above that of 3° 00'. From whence, by proceeding according to the 2d and 3d rules, the sines of all the other intermediate arches are had, by addition and subtraction only. See the operation. 1 ,0002904915 excess ,0000000050 OF SINES, TANGENTS, AND SECANTS. ,05233595 sine 3° 0' 23 4415 10th. rem. ,0552406400 sine 3° 10' &c. Again, as a second example, let it be required to find the sines of all the arches, to every minute, between 59° 15' and 60° 00'; those of the two extremes being first found, by the preceding method. In this case, the two extremes being ,85940641 and,86602540, their sum will be 1,72543, &c. and their difference =,00661899; whereof the former, multiplied by ,0000000423 (see the rule), gives ,00000007298, = &c. or ,0000000730, nearly, for the first product (which is exact enough for our purpose); therefore the 2d product, or ,0000000730 x 22, will be ,0000016060; which, added to of the difference, gives ,0001486947; from whence the operation will be as follows: 43 1 &c. ,8604457399 sine 59° 22' After the same manner the sines of all the intermediate arches between any other two proposed extremes may be derived, even up to 90 degrees; but those of above 60° are best found from those below, as has been shown elsewhere. The reasons upon which the foregoing operations are founded depend upon principles too foreign from the main OF SINES, TANGENTS, AND SECANTS. 25 design of this treatise to be explained here (even would room permit); however, as to the correctness and utility of the method itself, I will venture to affirm, that, whoever has the inclination, either to calculate new tables, or to examine those already extant, will not find one quarter of the trouble this way as he unavoidably must according to the common methods. |