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

SCHOLIƲM.

BY means of this Consectary Nicomedes solves that noble Problem of finding two mean proportionals, after this way, which we will here shew from Eutocius, but drawn into a compendium, and somewhat changed as to the order. Let two given lines AB and BC (Fig. 144.) between which you are to find two mean proportionals, be joined together at right angles, and divide both into two parts in D and E, and ha∣ving compleated the rectangle ABCL, from L thro' D draw LG to BC prolonged; so that after this way GB may become = AL or BC: Having let fall a perpendicular from E cut off from C at the interval CF=AD the part EF, and having drawn FG make CH parallel to it; and lastly thro' the legs of the angle KCH draw the right line FHK, so that the part HK shall be equal to the line CF, by the preceding Consect. and also draw the right line KM from K thro' L to the con∣tinued line BA: All which being done, CK and AM will be two mean proportionals between AB and BC; which after our way we thus demonstrate: By reason of the similitude of the ▵ ▵ MAL and LCK

MA is to LC or AB as AL or BC to CK

b−eb−c−ec and moreover,

as MA to AD so GC to CK i. e. FH to HK

b−½eb−2c−ec by reason of GF and CH be∣ing parallel, by Consect. 4. Prop. 34. lib. 1. therefore since HK is = AD=½eb, FH will be = A=b, and consequent∣ly MD=FK, viz. both b+½eb, and the square of both =bb+eb+¼eebb=□EF+EK by vertue of the Pythag. Theor. Now if to these equal quantities you add the equal □□ DX and EC=¼cc, their sum, viz. □MD+□DX i. e. □ MX will be bb+ebb+¼eebb+¼cc, equal to the sum of these, viz. □EF+□EC i. e. □ CF (by the Pythag. Theor. or EX by Construct.) +□KX; whence these two things now follow: 1. That the lines MX and KX are equal. 2. If from those equal sums you take away the common quan∣tities

¼eebb+¼cc, the remainders will be equal, viz. bb+ebb =ecc+eecc; and (since the part taken away, viz. bb is ma∣nifestly to the other part taken away, viz. ecc as the remain∣der ebb to the remainder eecc, and the whole with the parts taken away and the remainders are in the same proportion by Prop. 26. lib. 1.) separately also bb will = ecc and ebb=eecc. But from the latter equation it follows that

as eb to ec so ec to b by vertue of the 19. Prop. lib. 1. AB to CK so CK to MA

and by the same reason it follows from the former Equation as ec to b so b to c

CK to MA so MA to BC i. e. CK and MA are two mean Proportionals between AB and BC. Q. E. D.

From which deduction you have also manifest the foun∣dation of that mechanical way, which Hiero Alexandrinus makes use of in Eutocius, lib. 2. of the Sphere and Cylinder, and which Swenterus has put into his practical Geometry lib. 1. Tract. 1. Prop. 23. when, viz having joined in the form of a rectangle the given right lines AB and BC (Fig. 145.) and continued them at the other ends, he so long moves the ruler in L, having a moveable center, backwards and forwards, 'till XK and XM by help of a pair of compasses are found equal. To which, another way of Philo's is not unlike, and flows from the same fountain, wherein, having made on AC a semicircle, the moveable ruler in L is so long moved back∣warks and forwards, until LM and NK are found equal: Which seems to Eutocius to be more accommodated to practice, and easier to be perform'd by help of a ruler divided into small equal particles.