Comparing the Figures spectively, it follows that

166, 167, 168 with the Figures 175, 176, 177 re

[rA, sB]≈ [rX, sY].

ENUNCIATION. If K, L, M, P be four straight lines in proportion, if the lengths of L and M be fixed, if the length of K can be made smaller than that of any line however small, to show that the length of P can be made greater than that of any line Q, however great Q may be.

By the Axiom of Art. 23, it is always possible to find an integer r such that

rM > Q.

Now divide L into r equal parts, and take K smaller than one of these equal parts.

If K, L, M, P be four straight lines in proportion, if the lengths of L, M be fixed, and if the length of K can be made greater than that of any line however great, show that the length of P can be made smaller than that of any line Q however small.

SECTION XI.

OTHER PROPOSITIONS IN THE THEORY OF SCALES AND OF RATIO.

Props. 64, 65.

Art. 199. PROPOSITION LXIV. (Euc. V. 19.)

ENUNCIATION 1. If A, B, C, D are magnitudes of the same kind, and A: B= C: D,

ENUNCIATION 2. If A, B, C, D are magnitudes of the same kind, and

If X, A, S, A' are four harmonic points, A and A' being conjugate, and if C be the middle point of AA', prove that

SA: AX CS: CA = CA : CX.

Art. 201. PROPOSITION LXV. (Euc. V. 25.)

ENUNCIATION 1. If four magnitudes of the same kind are proportional, then the greatest and least of them together are greater than the sum of the other two.

if

ENUNCIATION 2. If A, B, C, D are four magnitudes of the same kind, and

[A, B]~ [C, D],

then the sum of the greatest and least is greater than the sum of the other two.

If three quantities be in proportion, show that the sum of the extremes will exceed double the mean.

NOTES.

Art. 203. NOTE 1. ON PROPS. 1-5, 6, 9, 11.

Props. 1-5 relate to certain simple cases of the application of the Commutative, Associative and Distributive Laws, with which the reader who has commenced elementary Algebra is already familiar.

it is seen that the multiplicand is divided (or distributed) into its parts A, B. Prop. II.

and

(a + b) R= aR + bR.

Treating a + b as the multiplier,

R as the multiplicand,

it is seen that the multiplier is distributed into its parts a, b.

Here the multiplicand A - B is distributed into its parts A, B.
Prop. IV. If

a> b,

(a - b) RaR-bR.

Here the multiplier ab is distributed into its parts a, b.

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