TRACT 21.

LOGARITHMS.

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tion ; and if the difference of the two least terms be made themeasure of their proportion, then the differences of the restwill be greater than the measure of the proportion betweentheir terms.

Corol. If the measure of the proportion between the greatestexceed their difference, then‘the proportion of this measureto the said difference, will be less than that of a followingmeasure to the difference of its terms. Because proportionalshave the same ratio.

13 Prop. If three quantities follow one another in the orderof magnitude, the proportion of the two least will be con-tained in the proportion of the extremes, a less number oftimes than the difference of the two least is contained in thedifference of the extremes: And, on the contrary, the pro-portion of the two greatest will be contained in the proportionof the extremes, oftener than the difference of the former iscontained in that of the latter.

Corol. Hence, if the difference of the two greater be equalto the difference of the two less terms, the proportion betweenthe two greater will be less than the proportion between thetwo less.

14 Prop. Of three equidifferent quantities, taken in ordet,the proportion between the extremes is more than double theproportion between the two greater terms.

Corol. Hence it follows, that half the proportion of theextremes is greater than the proportion of the two greatestterms, but less than the proportion of the two least.

15 Prop. If two quantities constitute a proportion, andeach quantity be lessened by half the greater, the remainders"'ill constitute a proportion greater than double the former.

16 Prop. The aliquot parts of incommensurable proportionsare incommensurable to each other.

1”! Prop. If one thousand numbers follow one another inthe natural order, beginning at 1000, and differing all byunity, viz. 1000, .999, 998, 997, &c; and the proportion be-tween the two greatest 1000, 999, by continual bisection, becut into parts that are smaller than the excess of the propor-

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