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 a new genesis of the parabola in Plan•• from the spculations of De Witte, viz. if the rectiline∣ar angle HBG (Fig. 115.) conceived to be moveable about the fixed point B be conceived so to move out of its first situ∣ation with its other leg BH along the immoveable rule EF, that it may at the same time move also the ruler HG, from its first situation DK, all along parallel to it self, and with the other leg BG let it all along cut the said ruler HG, and with this point of its intersection continually moving from B to∣wards G it will describe a curve. That this curve will be the parabola of the antients is hence manifest, because it will have this same first property of the parabola. For, 1. if the an∣gle HBG (n. 1.) be supposed to be a right one, and BD or HI=a, BI or KG=b (viz. in that station of the angle and rule HG by which they denote the point G in the inter∣section) you'l have by reason of the right angle at B, BI, i. e. b a mean proportional Between HI i. e. a, and IG or BK, and so this as an abscissa = 〈 math 〉〈 math 〉. Wherefore if BK i. e. 〈 math 〉〈 math 〉 be multiplyed by BD=a, the rectangle DBK will be = bb =□KG; which is the first property of the parabola: So that it follows, since the same inference may be made of any other point in this curve, that this curve will be the parabo∣la, BD or HI its Latus Rectum, KG a semiordinate, and BK its axis, &c. 2. If the angle HBG be an oblique one (num. 2) it may be easily shewn from what we have supposed that the ▵ ▵ DBH and BKG will be equiangular: Therefore as BD (i. e. a) to DH sc. BI (i. e. b) so KG sc BI (i. e. b) to BK (i. e. 〈 math 〉〈 math 〉) Therefore again the □ DBK=bb□KG. QED.

Consect. 3. It is also evident in this second case, that BK drawn parallel to the ax, but not thro' the middle of the pa∣rabola,

will be a diameter which will have for its vertex B, its Latus Rectum BD, and semiordinate GK, &c.

Consect. 4. Therefore you may find the Latus Rectum in a given parabola geometrically, if you draw any semiordinate whatsoever IK (Fig. 116.) and make the abscissa EF equal to it, and from F draw a parallel to the semiordinate IK, and from E draw the right line EK thro' K cutting off FH the Latus Rectum sought; since as EI to IK so is EF (i. e. IK) to FH by Prop 34. lib. 1. wherefore having the ab∣scissa and semiordinate given arithmetically, the Latus Re∣ctum will be a third proportional.

Consect. 5. Since therefore the Latus Rectum found above is 〈 math 〉〈 math 〉, if you conceive it to be applyed to the parabola in LM, so that N shall be that point which is called the Focus, LN will be 〈 math 〉〈 math 〉 and its square 〈 math 〉〈 math 〉 and this divided by the Latus Rectum 〈 math 〉〈 math 〉 will give occ〈 math 〉〈 math 〉 for the abscissa EN: So that the distance of the Focus from the Vertex will be ¼ of the Latus Re∣ctum.

Consect. 6. Since therefore EN is = 〈 math 〉〈 math 〉 if for EF you put ib, NF will be = 〈 math 〉〈 math 〉, whose square will be found to be 〈 math 〉〈 math 〉, to which if there be added □ GF=oicc, by Prop. 1. the square of NG will be = 〈 math 〉〈 math 〉 whose root (as the extraction of it and, without that, the ana∣logy of the square NF with the square NG manifestly shews) will be 〈 math 〉〈 math 〉; so that a right line drawn from the Focus to the end of the ordinate, will always be equal to the abscissa EF

+EN i. e. (if EO be made equal to EN) to the compounded line FO.