Page:Mathematical collections and translations, in two tomes - Salusbury (1661).djvu/298

because the angle BAD, conteined between the vertical lines, is equal to the difference of the Polar altitudes, it shall be 4gr. 40m. which I note here apart; and I finde the chord of it by the Table of Arches and Chords, and set it down neer unto it, which is 8142 parts, of which the semidiameter AB is 100000. Next, I finde the angle BDC with ease, for the half of the angle BAD, which is 2gr. 20 m. added to a right angle, giveth the angle BDF 92gr. 20 m. to which adding the angle CDF, which is the distance from the vertical point of the greatest altitude of the Star, which here is 62 gr. 15 m. it giveth us the quantity of the angle BDC, 154 grad. 45 min. the which I set down together with its Sine, taken out of the Table, which is 42657, and under this I note the angle of the Parallax BCD 0 gr. 2 m. with its Sine 58. And because in the Triangle BCD, the side DB is to the side BC; as the sine of the opposite angle BCD, to the sine of the opposite angle BDC: therefore, if the line BD were 58. BC would be 42657. And because the Chord DB is 8142. of those parts whereof the semidiameter BA is 100000. and we seek to know how many of those parts is BC; therefore we will say, by the Golden Rule, if when BD is 58. BG is 42657. in case the said DB were 8142. how much would BC be? I multiply the second term by the third, and the product is 347313294. which ought to be divided by the first, namely, by 58. and the quotient shall be the number of the parts of the line BC, whereof the semidiameter AB is 100000. And to know how many semidiameters BA, the said line BC doth contein, it will be necessary anew to divide the said quotient so found by 100000. and we shall have the number of semidiameters conteined in BC. Now the number 347313294. divided by 58. giveth 59881601/4. as here you may see.

59881601/4
58|3473132941/4
571794101/4
5430001/4

And this divided by 100000. the product is 59 88160/100000

1|00000|59|88160.

But we may much abbreviate the operation, dividing the first quotient found, that is, 347313294. by the product of the multiplication of the two numbers 58. and 100000. that is,59