simply ratio are therefore directly applicable to all cases of geometrical magnitudes; while the remainder are rendered so by the considerations mentioned in Scholium 2, page 117. In the above m and n may represent any numbers integral, fractional, or mixed.

Proposition 2.

Theorem.—If the product of two quantities be equal to the product of two others, these four quantities may form a proportion, one set being taken for the extremes and the other for the

Scholium.-From this it is manifest that we can form a number of proportions from the equation A×D=B× C. The only limitation being, that if A be made a mean or an extreme, D must be the other mean or extreme. Thus we might have AC: B: D,

Let A and B be two quantities, and mA and mB equimultiples of them.

Corollary.-Equal measures of two quantities are proportional to the quantities.

This may be proved as the above by supposing m to be fractional.

Proposition 4.

Theorem.-If equimultiples be taken of the first and second terms of a proportion, and also of the third and fourth, the resulting terms are proportional.

Theorem.-If equimultiples be taken of the first and third terms of a proportion, and also of the second and fourth, the resulting terms are proportional.

Theorem.-If four quantities be in proportion, if the first be any multiple of the second, the third is the same multiple of the fourth.

Theorem.-If four quantities of the same kind be in proportion, they are in proportion when taken alternately.

Theorem.-If four quantities be in proportion, they are in proportion when taken inversely.

Theorem. If four quantities be in proportion, they are in proportion by composition.

If

then

and

A B C: D,

A+B : B :: C+D : D,

A+B A :: C+D: C.

Theorem. If four quantities be in proportion, they are in proportion by division.

Theorem.-If four quantities be in proportion, the sum of the first and second is to their difference, as the sum of the third and fourth is to their difference.

Dividing one equation by the other, and striking out (IV. 3) the common terms, B and D, and we have

Theorem.—The products of the corresponding terms of two numerical proportions are in proportion.

Corollary 1.—The continued products of the corresponding terms of any number of proportions are in proportion.

Corollary 2.-If four quantities be in proportion, like powers of them are in proportion.

For, if in last corollary all the corresponding terms be the same, the products would be like powers of the terms.