The same reasoning is applicable, if any other term of the analogy be supposed the greatest.

Cor. Hence the mean term of three proportionals, is less than half the sum of both extremes,

PROP. XV. THEOR.

If two analogies have the same antecedents, another analogy may be formed, having the consequents of the one for its antecedents, and the consequents of the other for its consequents.

: ::

Let A B C D and A: E::C: F; then B: E:: D: F.

For, alternating the first analogy, A: C:: B: D, and alternating the second, A: C:: E: F; whence, by identity of ratios, B: D:: E: F. This inference is named a direct equality.

PROP. XVI. THEOR.

If the consequents of one analogy be antecedents in another, a third analogy will arise, having the same antecedents as the former, and the same consequents as the latter.

:

Let A B C : D, and B : E::D: F; then A : E :: C: F.

For, alternating both analogies, A: C:: B: D, and

B: D:: E: F; whence, by identity of ratios, A: C;; E: F. This conclusion is also named a direct equality,

PROP. XVII. THEOR.

If two analogies have the same means, the extremes of the one, with those of the other as the mean terms, will form a third analogy.

:

:

Let A B C : D, and E: B:: C: F; then A: E:: F: D.

For, since A: B:: C: D, AD=BC (V. 6.); and because E: B: C: F, EF BC. Whence AD=EF, and A:EF: D.

:

=

Cor. Hence the extreme and mean terms being interchangeable, it likewise follows, that, if A: B:: C: D, and A: EFD, then B: E:: F: C.

PROP. XVIII. THEOR.

If the extremes of one analogy are the mean terms in another, a third analogy will subsist, having the means of the former as its extremes, and the extremes of the latter as its means.

Let A: B::C: D, and E: A:: D: F; then B: E:: F: C.

For, from the first analogy AD=BC, and, from the se

cond, EF AD; whence BC=EF, and consequently B: E: F: C.

Cor. Hence also, if A: B :: C: D and B: E:: F:C; then E: A:: D: F. The principle of this and the preceding Proposition is named inverse, or perturbate, equality.

PROP. XIX. THEOR.

If there be any number of proportionals, as one antecedent is to its consequent, so is the sum of all the antecedents to the sum of all the consequents.

:

:

Let A B C:D::E: F::G: H; then A: B:: A+C+E+G:B+D+F+H.

Because A: B:: C: D, (V. 6.) AD=BC; and, since AB:: E: F, AFBE, and, for the same reason, AH =BG. Consequently, the aggregate products, AB+AD +AF+AH=BA+BC+BE+BG; and, by resolution, A(B+D+F+H)=B(A+C+E+G); whence A: B :: A+C+E+G: B+D+F+H.

Cor. 1. It is obvious, that the Proposition will extend likewise to the difference of the homologous terms, and may, therefore, be more generally expressed thus: A: B:: A+C+E+G: B÷D÷F÷H.

Cor. 2. Hence, in continued proportionals, as one antecedent is to its consequent, so is the sum or difference of the several antecedents to the corresponding sum or difference of the consequents. For, if A: B:: B:C::C:

3

then, by corollary 1, A : B :: A±B÷E: B±D±F, or (V. 8.) A: A÷C÷E:: B: BDF; wherefore (V. 11.) A: CE:: B: DF, and (V. 8.) A: B:: C+E DF.

PROP. XX. THEOR.

If two analogies have the same antecedents, another analogy may be formed of these antecedents, and the sum or difference of the consequents.

Let A: B::C:D, and A: E : : C: F; then A: B÷E :: CD F. For, by alternation, these analogies become A: C: B: D, and A: C:: E: F; whence (V. 19. cor. 2.) A: C:: B÷E: DF, and alternately, A: B÷E :: C: DF.

Cor. If A: B:: C: D, and E:B:: F: D; then A±E: B:: C÷±F: D. For, by alternating the analogies, A C: B : D, and E: F: B: D; whence (V. 19. cor. 2.) B : D :: A±E: C÷F, and, by alternation and inversion, A±E: B:: CF: D.

PROP. XXI. THEOR.

In continued proportionals, the difference between the first and second is to the first, as the difference between the first and last terms to the sum of all the terms excepting the last.

I

Let A B

:

B:C::C: D:: D: E; then if A➡B,

A-BA:: A-E: A+B+C+D.

For (V. 19.), A:B::A+B+C+D: B+C+D+E, and consequently (V. 11. cor.), A—B:A::(A+B+C+D) -(B+C+D+E) : A+B+C+D; that is, omitting B+C+Din the third term, A-B:A::A-E: A+B+C+D.

If AB, then B-A: A :: (B+C+D+E)—(A+B +C+D): A+B+C+D, that is, B-A: A :: E—A : A +B+C+D.

The same reasoning, it is evident, will hold for any number of terms.

Scholium. Hence the summation of continued progressions, whether ascending or descending, is easily derived.

PROP. XXII. THEOR.

The products of the like terms of any numerical proportions, are themselves proportional.

Let A: B::C: D

E: F:: G: H

I: K:: L: M;

then AEI: BFK :: CGL : DHM.

For (V. 6.), from the first analogy AD=BC, from the second analogy EH=FG, and from the third analogy IM =KL; whence the compound product AD.EH.IM= BC.FG.KL. But AD.EH.IM=AEI.DHM (V. 2.), and BC.FG.KL = BFK.CGL ; wherefore AEI.DHM = BFK.CGL, and consequently (V. 6.), AEI : BFK : : CGL: DHM.