If two magnitudes are equimultiples of two magnitudes, and any magnitudes subtracted from them are equimultiples of the same, then the remainders also either equal the same or are equimultiples of them. If any magnitudes are proportional, then they are also proportional inversely. Ratios which are the same with the same ratio are also the same with one another. If any number of magnitudes are proportional, then one of the antecedents is to one of the consequents as the sum of the antecedents is to the sum of the consequents. If a first magnitude has to a second the same ratio as a third to a fourth, and the third has to the fourth a greater ratio than a fifth has to a sixth, then the first also has to the second a greater ratio than the fifth to the sixth.
Parts have the same ratio as their equimultiples.
If four magnitudes are proportional, then they are also proportional alternately. If magnitudes are proportional taken jointly, then they are also proportional taken separately. If magnitudes are proportional taken separately, then they are also proportional taken jointly. If a whole is to a whole as a part subtracted is to a part subtracted, then the remainder is also to the remainder as the whole is to the whole. If magnitudes are proportional taken jointly, then they are also proportional in conversion.