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

Proposition III.

IN the Ellipsis (KDEK, Fig. 120.) the(α) 1.1 square of the semiordinate (IK) is equal to the rectangle (IL) of the Latus Rectum (EL) and the abscissa (EI) (less or) taking first out another rectangle (LS) of the same abscissa (EI or LR) and RS a fourth proportional to (DE) the Latus Transversum (EL) the Latus Rectum and (EI) the abscissa.

Demonstration.

Suppose the side of the cone to be AB here also = a and BM parallel to the section = b and the intercepted AM=c, and EI=eb; and NI will be again = ec, all as in the hyperbola. And mak••ng also here as in the hyper∣bola MC=d, and the Latus Transversum DE=ob, so that DI will be ob−eb; then will (by reason of the simili∣tude of the ▵ ▵ BMC, DEP and DIO) EP be=od, and IO=od−ed. Therefore ▭ of NIO will be = oecd−eecd=□IK. But this square divided by the abscissa EI=eb gives 〈 math 〉〈 math 〉 or 〈 math 〉〈 math 〉 for that line IS which with the abscissa would make the rectangle ES= to the said square IK. Now therefore if we call the Latus Rectum a right line found after the same way as in the parabola, by making ac∣cording to Cons. 1. Prop. 1.

as b to c− so od− to a fourth 〈 math 〉〈 math 〉 i. e. as the line pa∣rallel to the section — to the intercepted diameter — so the Latus Primarium, &c. It is manifest that the Latus Re∣ctum is one part of the line just now found; and the other part 〈 math 〉〈 math 〉 is a fourth proportional to b, c and ed, or (to speak with Apollonius as we have done in the Prop.) to 〈 math 〉〈 math 〉 and eb (for there will come out the same quantity 〈 math 〉〈 math 〉) where∣fore now it is evident that the □ of the semiordinate IK is e∣qual to the ▭ IL (of the Latus Rectum 〈 math 〉〈 math 〉 and the ab∣scissa eb=oecd) having first taken out thence the ▭ LS, or eecd out of that fourth proportional 〈 math 〉〈 math 〉 by the same abscissa eb; which was to be found and demonstrated.

CONSECTARYS.

I. HEnce yov have first of all the reason of the name of the Ellipse, which Apollonius gave to this section; viz. because the square of the semiordinate IK is defective of, or less than the rectangle of the Latus Rectum and the abscis∣sa.

II. Since therefore the Latus Rectum here also as well as in the parabola and hyperbola, is found by making as b to c so od to 〈 math 〉〈 math 〉 (i. e. as BM parallel to the section is to the inter∣cept. diam. AM so the Latus Primarium EP to a fourth EL) now if any one had rather express this Latus Rectum after A∣pollonius's way, he will easily see that the quantity above found being multiplyed both Numerator and Denominator by b, that there will come out an equivalent one 〈 math 〉〈 math 〉, which instead of the former proportion will give this other,

as

• bb− to cd− so ob to a fourth;
• BM−▭AMC—Latus Transvers. to a fourth;

which is the same with that we have also found in the hyper∣bola, and which also Apollonius has Prop. 13. Lib. 1.

III. This Latus Rectum may also be had geometrically, if you find, 1. in the hyperbola a third proportional FH to the abscissa EI (Fig. 121.) and semiordinate IK (= EF.) 2. But EL a fourth proportional to DI (the difference of the Latus Transversum and the abscissa) and the found FH, or IS equal to it, and the Latus Transversum DE, is the Latus Rectum sought.

SCHOLIƲM.

FRom this third Consect. we may reciprocally, having the Latus Rectum and Transversum given, apply as many semiordinates to the ax as you please, and so draw the ellipsis thro' as many points given as you please, viz. if, taking any ab∣scissa EI, you make as DE to EL so DI to a fourth IS; then between this IS and the abscissa EI find a mean proportional IK, and that will be the semiordinate sought: And this Prax∣is also and the third Consect. may be abundantly proved by making use of a literal Calculus. For e. g. here a fourth proportional to ob, 〈 math 〉〈 math 〉 and ob−eb will be 〈 math 〉〈 math 〉 and a mean proportional between this fourth and eb will be 〈 math 〉〈 math 〉, &c.

Notes

• (α) 1.1

  I. Pro∣pert. of the El∣lipse of Apoll. Prop. 13. Lib. 1.