+-------+-------+-------+------+-------+------+------+------+ | 1, 2 | 2, 3 | 3, 4 | 4, 5 | 5, 6 | 6, 7 | 7, 8 | 8, 9 | | | | | | | | | | | 1, 3 | 2, 5 | 3, 5 | 4, 7 | 5, 7 | | 7, 9 | | | | | | | | | | | | 1, 4 | 2, 7 | 3, 7 | 4, 9 | 5, 8 | | | | | | | | | | | | | | 1, 5 | 2, 9 | 3, 8 | | 5, 9 | | 7, 1 | | | | | | | | | | | | 1, 6 | | | | 5, 11 | | | | | | | | | | | | | | 1, 7 | | 3, 11 | | 5, 13 | | | | | | | | | | | | | | 1, 8 | | 3, 13 | | | | | | | | | | | | | | | | 1, 9 | | | | | | | | | | | | | | | | | | 1, 11 | | | | | | | | | | | | | | | | | | 1, 13 | | | | | | | | | | | | | | | | | | 1, 15 | | | | | | | | | | | | | | | | | | 1, 17 | | | | | | | | +-------+-------+-------+------+-------+------+------+------+ Primary triangles may be found from the _angle of the satellite_, but it is an exceedingly round-about way. I will, however, give an example. Let us construct a primary triangle from the satellite 4, 9. Rad. × 4 -------- = ·4444444 = Tangt. < 23° 57′ 45·041″ 9 ∠ 23° 57′ 45·041″ × 2 = 47° 55′ 30·083″. therefore the angles of the "primary" are 47° 55′ 30·083″. and 42° 4′ 29·917″. The natural sine of 42° 4′ 29·917″ = ·6701025. The natural co-sine 42° 4′ 29·917″ = ·7422684. The greatest common measure of these numbers is about 102717, therefore-- Radius 10000000 ÷ 102717 = 97 Co-sine 7422684 ÷ 102717 = 72 Sine 6701025 ÷ 102717 = 65 and 65, 72, 97 is the primary triangle to which the satellites are 4, 9, and 5, 13. (_See Fig_. 63.) The figures in the calculation do not balance exactly, in consequence of the insufficient delicacy of the tables or calculations. Fig. 63. The connection between primaries and satellites is shown by figure 64. Fig. 64. Let the triangle ADB be a satellite, 5, 2, which we will call BD 20, and AD 8. Let C be centre of semi-circle ABE. AD : DB :: DB : DE = 50 (_Euc. VI_. 8) AD + DE = AE = 58 = diameter AE ÷ 2 = AC = BC = 29 = radius