Translation:Ayil Meshulash/Discourse 2
This chapter contains sections 29 to 47
A fraction is a number that describes a specific number's relationship to another number, such as a half, a third, or a fourth.
A ratio is a number that describes the generalized relationship between numbers, such as the ratio of 2 to 4 is a half, and the ratio of 2 to 6 is a third.
Two points regarding ratios:
1) Two ratios can be equal, for example 2 is to 4 is the same as 3 is to 6, for both are describing a ratio of a half, expressed as: A is to B as C is to D.
2) When two sets of numbers fall under one ratio, for example 2:4 and 5:10, B is to A as D is to C.
In the above example, the ratios of A to C and B to D are equal.
Likewise, the ratios of C to A and D to B are equal.
These 2 sets of numbers can be rearranged four ways while maintaining the correctness of the equalities above, as long as the two extremes are at the ends and the means are in the middle, or the two extremes are in the middle and the means at the ends.
Two sets of two ratios, such as 2:4=3:6 and 1:4=3:12, where the means of both sets are the same [4 and 3], have the property that when the extremes of each set are arranged so that the extremes of one set are taken as extremes and the extremes of the other set are taken as the means they still remain in proportion [for example, 1:2=6:12]. There are eight such possible arrangements, according to the methods outlined in the previous sections [section 31 to 34 and the four listed in section 35]. Similarly, if the extremes of one set are equal to the means of the other set they can also be rearranged as ratios of each other [4:1=12:3] as was explained in the previous sections.
When considering four numbers under proportion, if the 1st term is subtracted from the 2nd, and the 3rd term is subtracted from the 4th, the ratio of the remainder of the first subtraction compared to the 1st term is the same as the ratio of the remainder of the second subtraction to the 3rd term. For example, in the ratio 3:4=6:8, subtracting 3 from 4 yields 1 and 6 from 8 yields 2, resulting in the ratio 1:3 = 2:6.
The ratio that remains of the first subtraction to the second term is also in proportion with the ratio of the remainder of the second subtraction to the fourth term, which in the above example is 1:4=2:8.
If the first two terms are added together and the last two terms are added together, the ratio of the sum of the first addition to the first term is equivalent to the ratio of the sum of the second addition to the third term. For example, the ratios 3:4 = 6:8 become 7:3 = 14:6.
So to, the ratio of the sum of the first addition to the second term is equivalent to the ratio of the sum of the second addition to the fourth term, like so: 7:4=14:8.
Similarly, if you perform the same addition mentioned above along with the subtraction mentioned above [to two separate groups of these ratios], the ratio of the sum of the first addition to the remainder of the first subtraction is equivalent to the ratio of the second addition to the remainder of the second subtraction, like so: 7:1 = 14:2
If the first term is divided by the second term, it will be equal to the number returned by dividing the third term by the fourth, like so: 3/4 = 6/8.
If the first term is multiplied by the fourth term, it will be equal to the number returned by multiplying the second term by the third term, like so: 2:3 = 4:6 has extremes whose multiplication produce 12 as well as means whose multiplication produce 12.
From these methods it becomes clear that given two equal planes, if one places two of the edges of one plane as the extremes and two edges of the other plane as the means they will be in proportion to each other. As an example, take two planes, each with an area of 12, the first having two edges of 2 and 6, and the second having edges of 3 and 4, if placed as outlined they will be in proportion: 2:3 = 4:6.
If one of the four terms is unknown: If one of the extreme terms is unknown, multiple the two mean terms and divide the result by the known extreme term; If one of the means is unknown, multiply the two extreme terms and divide them by the known mean term.
If one subtracts the same number from two equivalent terms, or adds the same number, or multiplies or divides, the results are still equivalent to each to other. Similarly, if results are found to be equivalent, the original terms must have been equivalent as well.
If two terms are equal to a third then they are equal to each other as well.