D. 1 H. Cor. 1, 17, V. Then A: B' A: B,

B;

B,

B'A': A.

and the 1st A' is the same with the third A', .. A'~ B' = A.

but A' B', B'~ C', C'~ D'&c., form a progression

~

in which A' ~B': B' C'
i. e. A: B;

=

= A': B',

5 14, V. 6 Remk.

.. B' C' = B, C'

~ D' = C &c.

7 Conc.

But the progressions A, B, C, D &c. and
A'~ B', B'~ C', C' ~ D', D'.

E' &c.

have the same first terms and the same

common ratio.

..those progressions cannot but be identical. Geom. Pl. Sol. & Sph. p. 42, 53.

Alg. & Arith. Hyp. Let a 15, b = 10 be two Ms; k = 6 & y = 4 the respective parts.

Because of this identity of products the quantities must be in proportion;

Convertendo. If four magnitudes be proportionals, they are also proportionals by conversion: that is, the first is to its excess above the second, as the third to its excess above the fourth.

"If four magnitudes be proportional, the first being greater than the second, and the third than the fourth; then the first shall be to its excess above the second, as the third is to its excess above the fourth."-HOSE.

"The terms of an analogy are proportional by conversion."-BELL.

DEM. 17, V. dividendo; B, V. invertendo; 18, V. componendo.

Or,

If A: B = C: D, by conversion, A : A-B= C: C — D.

Alg. & Arith. Hyp. Let a 12 : 69 = c 8 : d 6, then a 12 : a + b 12 +9

USE & APP. Among other results, Prop. E., bk. V., leads to the following

1. If any number of Ms be in continued proportion, as A: B:C: DE, the difference between the first and second terms, A ~ B, is to the first A, as the difference between the first and last, A ~ E, is to the sum of all the terms, except the last, A + B + C + D.

A: B= =A+B+C+D:B + C + D + E. .. Conv. A: A~B: A+B+C+D: A~

=

B being > A and D > C;

E;

And. (A + B + C + D) (B + C + D + E) =A~E,

A~BAZA~E: A+B+C+D.

2. In a series of continued proportionals, A: B: CD: E, &c., the differences of the successive terms Á~B, B ~ C, C~ D, &c., are also in continued proportion, — A ~ B: B~C: C~D, and B~C: C~D = C~D:D~ E.

CASE 1. Let A > B,—the series is continually decreasing; then A - B : B-C B-C: C · D, and B-C:C-DC —Ď:D — E.

=

D. 1. H. 17, V.

21 16, V. 3 Sim.

4 H. 11, V.

5 Sim. 6 Rec.

''A: B = B: C, .. div. A-B: B=B-C: C; and altern. A−B:B-C=B: C,

So, . B: C = C:D, .. B-C : C-D = C : D. but B: CC: D, ... A B: B - C=B-C

: C-D.

So BC: C-DC-D:D-E;

.. A-B: B-C: C-D: DE, are continued proportionals.

CASE II. If A < B, the series is continually increasing.

3

By a like method this case is also proved.

In an infinitely decreasing series of Magnitudes in continued proportion, the first term, A, is a mean proportional between its excess above the second, B, and the sum of the series.

Let A, B, denote the 1st and 2nd terms,- Z the last term, and S the sum of the Series, then, as in the last example but one, Use 1, E, V., A−B : A=Z: S-Z. But the last term Z may be less than any magnitude or quantity we fix on, however small; and hence to the values of A-Z and S-Z there will be limits namely A and S.

If there be three magnitudes, and other three, which, taken two and two, have the same ratio; then if the first be greater than the third, the fourth shall be greater than the sixth, and if equal, equal; and ij less, less.

Cor. 13, V. If 1st

1st 2nd > 5th

::

2nd 3rd 4th, but 3rd 4th 5th 6th, the 6th.

10, V. The M with gr. ratio the gr. of two Ms.

7, V. Equal Ms. the same ratio to the same M &c.

11, V. Ratios the same to the same r. the same to one another.
Pr. B, V. Invertendo. 9, V. Ms. with the same ratio equal &c.

E. 1| Hyp. 1.

2 Hyp 2. 3 Conc.

Let there be three Ms. A, B, C, & three other Ms. D, E, F ;
let A: BD: E, & B: C=E: F;
if A> or < C, then D > = or < F.

CASE II. Let A = C, then D = F. Fig. 2.