PROPOSITION I THEOREM.

If four quantities are in proportion, the product of the means will be equal to the product of the extremes.

Assume the proportion,

A:

B

D

B C D; whence, A = a;

Cor. If B is equal to C, there will be but three proportional quantities; in this case, the square of the mean is equal to the product of the extremes.

PROPOSITION II. THEOREM.

If the product of two quantities is equal to the product of two other quantities, two of them may be made the means, and the other two the extremes of a proportion.

If we have,

AD = BC,

by changing the members of the equation, we have,

PROPOSITION III. THEOREM.

If four quantities are in proportion, they will be in proportion by alternation.

If one couplet in each of two proportions is the same, the other couplets will form a proportion.

Cor. If the antecedents, in two proportions, are the same the consequents will be proportional. For, the antecedents of the second couplets may be made the consequents of the first, by alternation (P. III.).

PROPOSITION V. THEOREM.

If four quantities are in proportion, they will be in proportion by inversion.

If we take the reciprocals of both members (A. 7), we have,

If four quantities are in proportion, they will be in pro portion by composition or division.

If we add 1 to both members, and subtract 1 from both

whence, by reducing to a common denominator, we have,

A: B+A :: C: D+C, and, A: B-A :: C: D-C

which was to be proved.

PROPOSITION VII. THEOREM.

Equimultiples of two quantities are proportional to the quan tities themselves.

Let A and B be any two quantities; then denote their ratio.

If we multiply both terms of this fraction by m, its value will not be changed; and we shall have,

If four quantities are in proportion, any equimultiples of the first couplet will be proportional to any equimultiples of the second couplet.

Assume the proportion,

B D

A B C : D; whence,

=
Α C

If we multiply both terms of the first member by m, and both terms of the second member by n, we shall have,

PROPOSITION IX. THEOREM.

If two quantities be increased or diminished by like part. of each, the results will be proportional to the quantities themselves.

If both terms of the first couplet of a proportion be increased or diminished by like parts of each; and if both terms of the second couplet be increased or diminished by any other like parts of each, the results will be in proportion.