Propositions

Proposition 1.: If any number of magnitudes are each the same multiple of the same number of other magnitudes, then the sum is that multiple of the sum.
Proposition 2.: If a first magnitude is the same multiple of a second that a third is of a fourth, and a fifth also is the same multiple of the second that a sixth is of the fourth, then the sum of the first and fifth also is the same multiple of the second that the sum of the third and sixth is of the fourth.
Proposition 3.: If a first magnitude is the same multiple of a second that a third is of a fourth, and if equimultiples are taken of the first and third, then the magnitudes taken also are equimultiples respectively, the one of the second and the other of the fourth.
Proposition 4.: If a first magnitude has to a second the same ratio as a third to a fourth, then any equimultiples whatever of the first and third also have the same ratio to any equimultiples whatever of the second and fourth respectively, taken in corresponding order.
Proposition 5.: If a magnitude is the same multiple of a magnitude that a subtracted part is of a subtracted part, then the remainder also is the same multiple of the remainder that the whole is of the whole.
Proposition 6.: If two magnitudes are equimultiples of two magnitudes, and any magnitudes subtracted from them are equimultiples of the same, then the remainders either equal the same or are equimultiples of them.
Proposition 7.: Equal magnitudes have to the same the same ratio; and the same has to equal magnitudes the same ratio.
Corollary. if any magnitudes are proportional, then they are also proportional inversely.
Proposition 8.: Of unequal magnitudes, the greater has to the same a greater ratio than the less has; and the same has to the less a greater ratio than it has to the greater.
Proposition 9.: Magnitudes which have the same ratio to the same equal one another; and magnitudes to which the same has the same ratio are equal.
Proposition 10.: Of magnitudes which have a ratio to the same, that which has a greater ratio is greater; and that to which the same has a greater ratio is less.
Proposition 11.: Ratios which are the same with the same ratio are also the same with one another.
Proposition 12.: 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.
Proposition 13.: 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.
Proposition 14.: If a first magnitude has to a second the same ratio as a third has to a fourth, and the first is greater than the third, then the second is also greater than the fourth; if equal, equal; and if less, less.
Proposition 15.: Parts have the same ratio as their equimultiples.
Proposition 16.: If four magnitudes are proportional, then they are also proportional alternately.
Proposition 17.: If magnitudes are proportional taken jointly, then they are also proportional taken separately.
Proposition 18.: If magnitudes are proportional taken separately, then they are also proportional taken jointly.
Proposition 19.: 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.
Corollary. if magnitudes are proportional taken jointly, then they are also proportional in conversion.
Proposition 20.: If there are three magnitudes, and others equal to them in multitude, which taken two and two are in the same ratio, and if ex aequali the first is greater than the third, then the fourth is also greater than the sixth; if equal, equal, and; if less, less.
Proposition 21.: If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them is perturbed, then, if ex aequali the first magnitude is greater than the third, then the fourth is also greater than the sixth; if equal, equal; and if less, less.
Proposition 22.: If there are any number of magnitudes whatever, and others equal to them in multitude, which taken two and two together are in the same ratio, then they are also in the same ratio ex aequali.
Proposition 23.: If there are three magnitudes, and others equal to them in multitude, which taken two and two together are in the same ratio, and the proportion of them be perturbed, then they are also in the same ratio ex aequali.
Proposition 24.: If a first magnitude has to a second the same ratio as a third has to a fourth, and also a fifth has to the second the same ratio as a sixth to the fourth, then the sum of the first and fifth has to the second the same ratio as the sum of the third and sixth has to the fourth.
Proposition 25.: If four magnitudes are proportional, then the sum of the greatest and the least is greater than the sum of the remaining two.