Proof that the diameter of the sun has to the diameter of the earth a ratio greater than 19 to 3, but less than 43 to 6:

Let A be the center of the sun, B the center of the earth, C the center of the moon when the eclipse is total, so as to secure that A, B, C may be in a straight line.

Let a plane be carried through the axis, and let it cut the sun in the circle DEF, the earth in GHK, the shadow in the circumference NO, and the cone in the straight lines DM, FM.

Join NO, and from A let PAQ be drawn at right angles to AM.

Then, since by proposition 13, (link to prop list) NO is less than 1/9th part of the diameter of the sun, the ratio PQ to NO is much greater than 9 to 1.

Therefore the ratio AM to MR is also greater than 9 to 1; and, conversely, the ratio MA to AR is less than 9 to 8.

Again, since AB is greater than 18 times BC by proposition 7, (link to prop 7) it is therefore much greater than 18 times BR; therefore AB has to BR a ratio greater than 18 to 1; therefore, inversely, the ratio BR to BA is less than 1 to 18; similarly, the ratio RA to AB is less than 19 to 18.

But it was proved that the ratio MA to AR is less than 9 to 8; therefore, by equal measures, the ratio MA to AB will be less than 171 to 144, and therefore less than 19 to 16: for parts have the same ratio as the same multiples of them: therefore the ratio AM to BM is greater than 19 to 3.

But, as AM is to MB, so is the diameter of the circle DEF to that of circle GHK; therefore the diameter of the sun has to the diameter of the earth a ratio greater than 19 to 3.

Again, I say that it has to it a ratio less than 43 to 6.

For, since the ratio BC to CR is greater than 675 to 1 by proposition 14 (link to list), the ratio CB to BR is less than 675 to 674.

But the ratio AB to BC is also less than 20 to 1; therefore by equal measures, the ratio AB to BR will be less than 13500 to 674, that is, less than 6750 to 337; therefore, inversely, RA to AB is greater than 7087 to 6750.

Now, since NO to PQ by proposition 13 (link to list) is greater than 979 to 10125, inversely, PQ to NO is less than 10125 to 979.

And, as PQ is to NO, so is AM to MR; therefore AM also has to MR a ratio less than 10125 to 979; therefore the ratio MA to AR is greater than 10125 to 9146.

But the ratio RA to AB is greater than 7087 to 6750; therefore, by equal measures, MA to AB has a ratio greater than the product of 10125 and 7087 has to the product of 9146 and 6750, that is, 71755875 to 61735500.

But 71755875 / 61735500 is greater than 43 / 37; therefore the ratio MA to AB is greater than 43 to 37; therefore, conversely, the ratio AM to MB is less than 43 to 6.

But, as AM is to BM, so is the diameter of the sun to the diameter of the earth; therefore the diameter of the sun has to the diameter of the earth a ratio less than 43 to 6.

And it was before proved that it has to it a ratio greater than 19 to 3.