130

PROJECT FOR A NEW

TRACT 29.

where b, c, d, 8cc, denote the preceding co-efficients.And hence, with the help of the table of the first tenpowers of the first 100 numbers, in p. 101 of my tables ofpowers, published by order of the Board of Longitude,may be easily found the sines, &c, of all arcs up to 100,by only dividing those powers by their respective co-effi-cients, as also of all multiples of these arcs by 10, 100, &c,by only varying the decimal points in the several terms, asthe figures will be all the same: and thus a number of pri-mary sines, Stc, may be found, to check or verify the same,when computed by other methods. By this means will befound the sines, &c, of the arcs

1 , 10 , 100 , 1000 , 10000 , 100000 ;

2 , 20 , 200 , 2000 , 20000 ;

3, 30, 300, 3000, 30000;

4, 40, 400, 4000, 40000;

&c, till

99, 990, 9900, 99000, 990000.

13. Again, it is evident, that, of the terms in the seriesfor the sine, the first term a alone will give the sine true tothe nearest unit in the 9th place, in the first 144 sines, orthe arc and sine will be the same for nine places, as far asthe arc 144; but they will agree to the nearest unit in the'7th place, as far as the arc 669 ; after which, the secondterm of the series must be included.

14. When the second term is taken in, these two terms,a — \a?, will give the sines true to the nearest unit in the 9thplace, till the arc becomes 3500. Now the numbers in mytable of cubes, published by order of the Board of Longi-tude, extend to 10000, and therefore all the above cubes arefound in it; consequently, taking the 6th part of thosecubes, and subtracting it from the corresponding arcs, theremainders will be the sines of those arcs, as far as till thearc be 3500: alter which the third term of the series maybe taken in, or other methods may be used.

15. But since, for any arc a, this is a general theorem,