AC-AD = DC = 21 = co-sine and DB = 20 = sine From the preceding it is manifest that-- sine² ----- + ver-s = dia. ver-s The formula to find the "primary triangle" to any satellite is as follows:-- Let the long ratio line of the satellite or sine be called _a_, and the short ratio line or versed-sine be called _b_. Then-- (1) a = sine. a² + b² (2) ------- = radius. 2b a² - b² (3) ------- = co-sine. 2b Therefore various primary triangles can be constructed on a side DB (_Fig_. 64) as sine, by taking different measures for AD as versed-sine. For example-- } 5 = sine = 5 } } 5² + 1² From } ------- = radius = 13 Satellite } 2 × 1 5, 1. } } 5² - 1² } ------- = co-s. = 12 } 2 × 1 * * * * * } 5 = sine = 5 } {20 } } { } 5² + 2² } { From } ------- = radius = 7¼ } × 4 {29 Satellite} 2 × 2 } { 5, 2. } } { } 5² + 2² } { } ------- = co-s. = 5¼ } {21 } 2 × 2 } { * * * * * Finally arises the following simple rule for the construction of "primaries" to contain any angle--_Decide upon a satellite which shall contain half the angle_--say, 5, 1. Call the first figure _a_, the second _b_, then-- a² + b² = hypotenuse. a² - b = perpendicular. a × 2b = base. "PRIMARY" LOWEST RATIO. Thus-- | 5² + 1² = 26 = 13 Satellite 5,1 | 5² - 1² = 24 = 12 | 5 × 2 × 1 = 10 = 5 --------------- |---------------------------- and-- | 5² + 2² = 29 = 29 Satellite 5,2 | 5² - 2² = 21 = 21 | 5 × 2 × 2 = 20 = 20 Having found the lowest ratio of the three sides of a "primary" triangle, the lowest whole numbers for tangent, secant, co-secant, and co-tangent, if required, are obtained in the following manner. Take for example the 20, 21, 29 triangle, now 20 × 21 = 420, and 29 × 420 = 12180, a new radius instead of 29 from which with the sine 20, and co-sine 21, increased in the same ratio, the whole canon of the 20, 21, 29 triangle will come out in whole numbers. Similarly in the triangle 48, 55, 73, radius 73 × 13200 (the product of 48 × 55) makes radius in whole numbers 963600, for an even canon without fractions. This is because sine and co-sine are the two denominators in the fractional parts of the other lines when worked out at the lowest ratio of sine, co-sine, and radius.