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Equations with other Data in them; the which I shall here omit pursuing, and leave them for the Learner's Practice.

Se&t. 2. Of Duantities in Geometrical Proportion.,

GEOMETRICAL Propartion continued has been already U defined in Sect. 2. Chap. 6. Part 1. And what is there faid concerning Numbers in :: may easily be applied to any Sort of Homogeneal Quantities that are in .

The moft natural and simple Series of Geometrical Proportionals, is when it begins with Unity or 1.

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That is, when all the middle Terms betwixt the two Extreams are both Consequents and Antecedents, that Series is in Geome. trical Proportion continued. Therefore in every Series of Quantities in - all the Terms except the last are Antecedents; and all the Terms except the firft are Consequents. But universally putting a the first Term in the Series, and the Ratio, viz. the common Multiplier, or Divisor; then it will be

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Q.ac.all.alle.acere. abs.a66. &c. in : Dra. For 2:20::20:444 = 216, &c. Anda : I. In any of these Series it is evident, that if three Quantities are in ;, 'the Rectangle of the two Extrems will be equal to the Square of the Mean; as in these, aiae. ann, here a xarx Hexal, ADIR. &c.

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III. If never fo many Quantities are in :; it will be, as any one of the Antecedents is to it's Consequents; fo is the Sum of all the Antecedents, to the Sum of all the Consequents.

ra.ne.aceiaeee..aceio.al?, &c. increasing. Asin thefc. a. 8:9c:a factace tre' tae*:cetare tali tast tee

Orar::a+ + + + + + + ++), viz. axaltantae taet tei=4. xataetaeitap Fact

That is, the Rectangle of the Extreams is equal to the Rectangle of the Means; per Second of this Seet.

Note, The Ratio of any Series in ;: increasing, is found by die viding any of the Consequents by it's Antecedent.

Thus, a) a ele . Or a é) are (e, &c.'

But if the Series be decreasing, then the Ratio is found by die viding any of the Antecedents by it's Consequent.

Thus, ) ( 0) (s de

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Note, The ::: set in the Margin at the second Step, is instead. of ergo; and imports that the Rectangle of the two Extrcams in the firtt Step, is equal to the Rectangle of the Means. And so for any other Proportion.

Sect. 3: Of barmonical Proportion. HARMONICAL or Musical Proportion is, when of three

Quantities (or rather Numbers) the first hath the same Ratio to third, as the Difference between the first and second, hath to the Difference between the second and third. As in these fol.

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Ç H A P. VII. Of Proportior Disjund, and bow to turn Equations into

Analogies, &c. PROPORTION Disjundt, or the Rule of Three in 4 Numbers, is already explained in Chap. 7. Part 1. And what hath been there said, is applicable to all Homogeneous Quantities, viz. of Lines to Lines, &c.

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If four Quantities are Proportionals they will also be Proportionals in Alternation, Inversion, Composition, Division, Conversion, and Mixtly, Euclid s. Def. 12, 13, 14, 15, 16..

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