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1. Denomination. Make BD or DC=a, BN or FE=c, BF=y, and DG=x; the Perpendicular EH will be = a, and EG=BF, viz y (because the ▵ EHG is similar to ▵ BDF, by n. 3 Schol 2 Prop. 34. Lib. 1. Math••s. Enucl. and BD in the one = to EH in the other) and BG=a+x, BE=y+c; and BH will have its Denomination, if you m••ke (by reason of the similarity of the ▵ ▵ BFD and ••EH)
as BF to BD so BE to BH 〈 math 〉〈 math 〉. and you'l have also HG = 〈 math 〉〈 math 〉, i. e. having reduc'd them all to the same Denomination, 〈 math 〉〈 math 〉, i. e. 〈 math 〉〈 math 〉. Having therefore na∣med all the lines you have occasion for, you must find two E∣quations, because there are assumed two unknown quantities, viz. x and y.
2. For the first Equation and its Reduction. By reason of the similiarity of the ▵ ▵ BGE and BEH, as BG to GE so BE to EH 〈 math 〉〈 math 〉: Therefore the Rectan∣gle of the Extremes will be = to the Rectangle of the means, i. e. 〈 math 〉〈 math 〉; and taking from both sides 〈 math 〉〈 math 〉.
3. For the second Equation and its Reduction. Since BH, HE and HG 〈 math 〉〈 math 〉 are continual proportionals, the Rectangle of the extreams are equal to the square of the mean, i. e. 〈 math 〉〈 math 〉; and multiplying both sides by yy, and dividing by 〈 math 〉〈 math 〉, and taking away ayy and transposing the rest, 〈 math 〉〈 math 〉; and dividing by 〈 math 〉〈 math 〉; i. e. dividing actually as far as may be by 〈 math 〉〈 math 〉.
4. The comparison of these two Equations thus reduc'd▪ gives a third new one, in which there will be only one un∣known quantity, viz. 〈 math 〉〈 math 〉; and adding to both sides cy,
〈 math 〉〈 math 〉; and multiplying by 〈 math 〉〈 math 〉; i. e. 〈 math 〉〈 math 〉 and dividing both sides by 〈 math 〉〈 math 〉; and adding 〈 math 〉〈 math 〉. Therefore 〈 math 〉〈 math 〉.
5. The Geometrical Construction, which is the same Pap•• prescribes in Cartes, viz. having prolong'd the side of 〈◊〉〈◊〉 square BA to N, so that BN shall be = to a given right li•• since BA is = a, and BN=c, the Hypothenusa DN will 〈 math 〉〈 math 〉. Having therefore made DG=DN, a•• describ'd a semi-circle upon the whole line BG, if AC be pr••¦longed until it occur to the Periphery in E, you'l have do that which was requir'd.