The Five Astronomical Canons

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the square root of the difference of R² The square of the Radius and the square of sine 60°. — The method by which the other sines required are found will be understood with the help of figure 1. Of the arc the sine of which is required — let us say ab — double is taken = ac. The sine of this double arc = cf and the sine of the complementary arc The arc required to complete a 90-degree quadrant (cd) = ef. By deducting ef from the Radius we get af and by drawing from h (i. e., the point marking the centre of the chord ac) the line hg parallel to cf we halve the line af and thus obtain the value of ag. The line hg again is half of cf, and ah i. e., the desired sine = $\sqrt{hg^2 + ag^2}$. — The sines of 60° and 30° being given, the above method can of course be employed for finding the sine of 15° and so on. — The cosine of the given arc — Stanza 4 goes on to say — is found by deducting the square of the sine from the square of the Radius and taking the square root of the remainder. The sine of 45° is equal to $\sqrt{\frac{R^2}{2}}$ (the square of the chord of 90° being 2R² and consequently the square of half that chord i. e., the sine of 45° being equal to $\frac{2R^2}{4} = \frac{R^2}{2}$).

The so-called different method described in stanza 5 is not essentially different from the method described before. We are directed to multiply the difference of the Radius and the sine of the complement of twice the given arc by 60, the product being equal to the square of the desired sine. Now in the above diagram

Adding the expressions thus found for ag² and hg² we obtain

6. The sines in AriesThe first sign of the Zodiac, representing the first 30 degrees of the ecliptic are 7 ; 15 ; 20 + 3 = 23 ; 20 + 11 = 31 ; 20 + 18 = 38 ; 45 ; 50 + 3 = 53 ; 60 minutes ;