Mathesis enucleata, or, The elements of the mathematicks by J. Christ. Sturmius ; made English by J.R. and R.S.S.

Proposition XXI.

IF from any point of the other diameter in the generating cir∣cle e. g. from G (Fig. 111. n. 1.) you draw a perpendicu∣lar GE thro' the cissoid of Diocles. the lines CG, GE, GD, and GH will be continual proportionals.

Demonstration.

For since GE and IF, as right sines, and also GD and IC as versed sines of equal arches by the Hypoth. are equal; you'l have as ID to IF (i. e. CG to GE) so IF to IC (i. e. GE to GD) per n. 3. Schol. 2. Prop. 34. lib. 1. But GD is to GH as ID to IF (i. e. as GE to GD) by the forecited Prop. 34. lib. 1. Therefore CG to GE, GE to GD, and GD to GH, are all in the same continual proportion. Q. E. D.

CONSECTARY.

HEnce it was easie for Diocles to find two mean proportio∣nals x and y between two given right lines V and Z; (Fig. 146) for he made (having first described his curve DHB) as V to Z so CL to LK, and having drawn CKH to the curve, and thro' H the perpendicular GE, he had between CG and GH two mean proportionals GE and GD by vertue of the pre∣sent Prop. when in the mean while CG the first would be to GH the last, as CL to LK, i. e. as the first V to the last Z given by vertue of the Constr. Therefore nothing remain'd but to make, 1. as CG to GE so V to X; and lastly, as GE to GD so x to y.

SCHOLIƲM.

IT may not be amiss to mention here another way of find∣ing two mean proportionals between any two given lines by the help of two Parabola's, which Menechmus formerly made use of, viz. by joining at right angles the given lines AB and BC (Fig. 147.) and prolonging them as occasion shall require thro' E and D; and then describing a Parabola about BE as its axis, so made that BC shall be its Latus Rectum, and in like manner describing another Parabola about BD as its ax, that shall have AB for its Latus Rectum, and that shall cut the for∣mer in F: Which being done, the semiordinate FE (or BD which is equal to it) being drawn to the point of Intersection F, will be the two mean proportionals sought. For by ver∣tue of the fourth Consectary of Prop. 1. of this Book, DF or

BE is a mean proportional between AB and BD, and in like manner EF or BD is a mean proportional between BE and BC, and consequently as AB to BE so BE to BD, and as BE to BD so BD to BC; Q. E. D.

To this way of Menechmus that of Des Cartes is not unlike, which he gives us p. m. 91. except only that instead of two Parabola's, he makes use only of one and a circle in room of the other: In imitation of whom Renatus Franciscus Slusius has since shewn infinite methods of doing the same thing by help of a circle, and either infinite Ellipse's or Hyperbola's, in his ingenious Treatise which he thence names his Mesolabium.