The Book of Number

Chapter 6: Ratios

There are three types of ratios:

1. Ratios that follow an arithmetical order, e.g., 1, 2, 3 (as a ratio cannot be established by fewer than three numbers); or 2, 4, 6; or 3, 6, 9. Here the ratio between all the numbers is equal, as the difference between 4 and 2 is the same as the difference between 6 and 4.

2. Proportional ratios, such as 4, 6, 9, where the ratio between 4 and 6 is the same as that between 6 and 9. Likewise, the multiplication of the smallest and largest numbers is equal to the middle number multiplied by itself, i.e., squared. You should know that these three numbers are effectively four, as the middle one can be considered to be like an additional number. Accordingly, for any four numbers, such that the ratio between the first and the second numbers is equal to the ratio between the third and fourth numbers, if you add together the squares of all four of them, find the sum, then add the first and the fourth numbers and take the square of the result, and add to it the square of the difference between the second and third numbers, it will equal the result of the first sum. Similarly, if you add together the second and third numbers and take the square of the sum, and then add to it the square of the difference between the first and fourth numbers, you will find that the total is equal to the previous sum.

Example: The ratio between 4 and 6 is equal to the ratio between 8 and 12, and the sum of their squares is 260. The sum of the first and fourth numbers is 16, whose square is 256. Now the difference between 6 and 8 is 2, whose square is 4, and thus the two sums are the same. Likewise, the sum of the second and third numbers is 14, whose square is 196, while the difference between the first and fourth numbers is 8, whose square is 64, and once again the sum of these two numbers is the same.

Know that most astrological wisdom, and the discovery of the placement of the constellations, depends on proportional ratios, and the same is true of the majority of mathematical questions.

3. The ratios of musicology, which is an exceptional field, as the ratio between the difference of the first and middle numbers and the difference of the middle and last numbers will always be equal to the ratio between the first and the last numbers.

Example: 2, 3, 6. The difference between 2 and 3 is 1, while the difference between 3 and 6 is 3, which is the ratio between 2 and 6. [ . . . ]

Example: Let us [take a circle and] say that a chord [of that circle] is 6, and let us take the square of its half, which is 9. Subtract that from 25, which is the square of half the diameter, and that leaves 16, whose square root is 4. Subtract this from half the diameter, which is 5, and you are left with 1, which is the “arrow” [i.e., the part of the diameter intersected by the chord].

To find the circumference from the diameter, mathematical scholars have stated that the circumference is three times the size of the diameter plus 1/7, which is 22 divided by 7. If you multiply the diameter by 3 and 1/7, the sum will be the circumference. Alternatively, you can find the circumference by multiplying the diameter by 22 and then dividing the result by 7. In the opposite case, when you know the circumference and you want to find out the diameter, multiply the circumference by 7 and divide the result by 22, and you will get the diameter. Based on the above, if the diameter is 1, the circumference will be a full 3 plus 8 of the first type of division, 34 of the second, and 17 of the third.¹

Now the wise Archimedes provided a proof that the correct figure is actually lower than this, arguing that the additional part was less than 1/7. He also offered proof that the additional part is more than 1 divided by 7.5. Thus, the additional part over and above the 3 is 8 of the first type of division, 24 of the second, and 35 of the third.² He then gave proof that it should be slightly larger than this number. Ptolemy stated an intermediate figure, according to which the additional part was 8 of the first type of division and 30 of the second.³ The sages of India maintain that if the diameter is 20,000 then the circumference will be 62,838. When you examine this, you will find that it is close to Ptolemy’s opinion—there is only a difference of 3 of the third type of division.⁴