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

SCHOLIƲM II.

HEnce also we have(α) 1.1 a new genesis of the Ellipse in Plano about the diameters given, from the speculations of Monsieur de Witt; viz. If about the rectilinear angle DCB (Fig. 126. n. 1. and 2.) consider'd as immove∣able, the rule NLK (which all of it will equal the greater semidiameter CB, and with the prominent part LK the lesser CD) be so moved that N going from C to D, and L from B to C may perpetually glide along the sides of the angle, the extreme point in the K in the mean while describing the curve BKE (and in a like application the other quadrants) and that this curve thus described will be the ellipsis of the ancients is hence manifest, because it has the second property of the ellipse just now described. For, 1. if the angle DCB or NCB be supposed to be a right one (as in Fig. 126. num. 1.) and the rule KN in the same station, it marked out the point K, and having apply'd the semiordinate KI, and drawn the perpendicular LM, from the square KL and the square CE (as being equal) substract mentally the equal squares LM and CI, and there will remain by virtue of the Pythagorick Theor. on the one hand □ KM and on the other by

Prop. 8. lib. 1. □ DIE equal among themselves. But now the square of KI is to the square of KM (i e. to the □ DIE) as the square of KN to the square of KL (i. e. as the square of CB to the square of CE) by reason of the similitude of the ▵ ▵ KLM and KNI; and since the same may be demonstra∣ted after the same manner of any other semiordinate Ki viz. that its □ Ki is to the □ DiE as the square CB to the square CE. It also follows, that the □ KI is to the □ DIE as □ Ki to the □ DiE, and alternatively, the square KI will be to the square Ki as the □ DIE to the □ DiE; which is the se∣cond property of the ellipse. 2. If the angle NCE be not a right one (as in Fig. 126. n. 2. and the like cases) having drawn NO and KP parallel to the rule nlB in the first station, [in which station the angle NCE, to which the flexible ruler is to be made, is determined, viz. by letting fall the perpendicular Bl from the extremity of one diameter upon the other, and moreover by adding or substracting the difference of the semi-diameters ln] having also drawn the Ordinate KIM, and PI parallel to CN; which being done the ▵ ▵ CBl and IKF, and also CBn and IKP will be similar. Where∣fore having joined NP, from the parallelism of the lines IP and NC and the similitude of the aforesoid ▵ ▵, as also of NCO and nCl, it will be easie to conclude that NCIP is a parallelogram. Wherefore since KN is = CE and □ KN = □ CE, having substracted the squares of the equal lines NP and CI, there will remain on the one hand □ KP on the other the □ DIE equal among themselves as above. Therefore the square of KI will be to the square of KP (i. e. to the □ DIE) as the square of BC to the square of Bn (i. e. to the square of CE) as in the former case: And since here also the same may be demonstrated after the same manner of any o∣ther semiordinate Ki; we may infer as above, that the □ □ KI and Ki are to one another as the rectangles DIE and DiE, &c.

But after what way the same ellipses may be described by these right lined angles without any of these rulers thro' infi∣nite points given, will be be manifest from the same figures to any attentive Person. For having once determined the angle NCE or nCD (num. 2. e. g.) if NL or nl be applyed where you please by help of a pair of compasses, and continued to

K, so that LK or lk shall be equal to lB, you will have eve∣ry where the point K, &c.

Notes

• (α) 1.1

  De Witt Elem. Curv. Lib. 1. Cap. 3. Prop. 13.