Aristarchus’ seventh proposition is his most important, since there the essential numerical result is derived. The demonstration is reproduced here in brief. In the figure below, A represents the position of the sun, B the earth, C the moon when seen halved. Hence angle EBD = angle BAC = 3 degrees. Let the angle FBE (=45 degrees) be bisected by BG. Since the ratio of a great and a small tangent to a circle is greater than the ratio od the underlying arcs and angles, the ratio GE and HE will be greater than the ratio of 1/4 to 1/30 of a right angle, that is, greater than 15/2.
Further, FG:GE = BF:BE = the square root of 2, greater than 7/5; hence FE/GE is greater than 12/5. Combining it with the first inequality, we find the ratio of FE to HE greater than 15/2 x 12/5 = 18; and the ratio AB/BC which is equal to BH/HE, hence a little larger than BE/HE is certainly also greater than 18.
Applying, on the other hand, the proposition that the ratio of a great and a small chord is smaller than the ratio of the subtended arcs, upon DE subtending 6 degrees in the half-circle BDE, and the side of a regular hexagon, equal to the radius, subtending and arc of 60 degrees, we find the ratio of 1/2 BE to DE smaller than 10, hence the ratio of AB to BC smaller than 20.