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

Proposition VII.

IN the hyperbola and ellipsis(α) 1.1 the Latus Rectum is to the Latus Transversum, 〈◊〉〈◊〉 the square of any semiordinate (e. g. IK in Fig. 118▪ and 120.) to the rectangle (DIE) con∣tained under the lines intercepted between it and the Vertex's of the Latus Transversum.

Demonstration.

For the Latus Rectum is on both sides 〈 math 〉〈 math 〉, the Latus Trans∣versum ob, &c. Wherefore if you make in the same series as the Latus R. to the Lat. Transv. so the □ IK to ▭ DIE

• 〈 math 〉〈 math 〉 in hyperb. 〈 math 〉〈 math 〉
• in ellips. 〈 math 〉〈 math 〉

The rectangles of the extremes and means will both be ooebcd±oeebcd, and so will prove the proportionality of the said quantities, by Prop. 19. lib. 1. Q. E. D.

CONSECTARY I.

HEnce having given in the ellipsis (see Fig. 124) the La∣tus Rectum and the transverse ax, you may easily obtain the second ax or diameter, if you make as the Lat. Transv. to the Lat. Rect. so the □ DCE to □AC 〈 math 〉〈 math 〉.

CONSECTARY II.

THerefore the □ of the whole AB will be = oocd=□ of the Latus Rectum into the Lat. Transv. (which Apllo∣nius calls the Figure) so that the second Ax (and any second Diameter) will be a mean proportional between the Latus Re∣ctum and the Latus Transversum. Hence in the hyperbola also the second or conjugate diameter may be called a mean proportional between the Latus Rectum and Transversum, i. e. √oocd or a line which is equal in power to the Figure, as A∣pollonius speaks.

SCHOLIƲM I.

HEnce may be derived another and more simple way of delineating organically the ellipsis in Plano about the given axes AB, DE (Fig 127.) which Schooten has given us▪ viz. by the help of two equal rulers CG and GK move∣able about the points G and G: If, viz. the portions CF and HK are equal to half the lesser ax AC, but taken with both the augments (viz. CF+FG+GH) may = ½ the greater ax CD or CI; and the point K moving along the produced line DE the point H may describe the curve EHAD. That this will be an ellipsis will be evident by vertue of this seventh Prop. from a property that agrees to this curve in all its points H. For having drawn circles about each diameter, and the lines IHN, FO perpendicular to CE; and having made the Latus Rectum EL, which is a third proportional to DE and AB by the second Consect. of this Prop. &c. by reason of the similitude of the triangles CFO, CIN, FO will be to FC as

IN to IC, and alternatively FO to IN as FC to IC i. e. as AC to CE or AB to DE. Therefore also the square of FO (or HN) will be to the square of IN, as the square of AB to the square of DE, by Prop. 22. lib. 1. i. e. as EL the Latus Rectum to ED the Latus Transversum, by Prop. 35. lib. 1. But the □ IN is = DNE from the proporty of the circle. Therefore □ FO (or of the semiordinate HN) is to the ▭ DNE as EL the Latus Rectum to ED the Latus Transv. therefore by ver∣tue of the pres. Prop. the point H is in the Ellipsis, and so any other, &c. Q. E. D.

CONSECTARY III.

NOW if in the ellipsis the □ of AC the second Ax (= 〈 math 〉〈 math 〉 by Consect. 1.) and □ CN the distance of the Fo∣cus from the centre (= 〈 math 〉〈 math 〉 by Censect 3. Prop. 5. the figure whereof you may see n. 124.) be joined in one sum; the □ AN will be = 〈 math 〉〈 math 〉, and so the line AN=〈 math 〉〈 math 〉 i. e. to half the Latus Transversum: So that hence having the axes given you may find the Foci, if from A at the interval CD you cut the transverse ax in N and N.

CONSECTARY IV.

NOW if, on the contrary, in an hyperbola (Fig. 123.) the □ AC or EF=〈 math 〉〈 math 〉 be substracted from the □ CF or CN=〈 math 〉〈 math 〉 by vertue of Consect. 2. Prop. 5. there will remain 〈 math 〉〈 math 〉 and its root 〈 math 〉〈 math 〉, i. e. half the Latus Trans∣versum CD: So that here also, the axes being given, you may find the Focus's, if from the vertex E you make EF a per∣pendicular to the ax = to the second Ax AC, and at the in∣terval CF from the centre C you cut the Latus Transversum continued in N and N.

SCHOLIƲM II.

BUT now, that the right lines KN and KN drawn from any other point (e. g. K) to the Foci, when taken toge∣ther in the ellipsis, but when substracted the one from the o∣ther in the hyperbola, are equal to the Latus Transversum DE, we will a little after demonstrate more universally, and also shew an easie and plain Praxis of delineating the ellipsis and hyperbola in Plano, having the axes and consequently the Fo∣ci given.

CONSECTARY V.

SInce we have before demonstrated Consect. 2. and 3. Prop. 5. that the □ DNE in the hyperbola and also in the el∣lipsis is = 〈 math 〉〈 math 〉; and here in Consect. 1. the □ of the second semi-diameter AC is also = 〈 math 〉〈 math 〉; it is evident that this □ AC is equal to the □ DNE.

CONSECTARY VI.

IT is hence moreover evident, if the square of half the trans∣verse diameter GE=〈 math 〉〈 math 〉 be compared with the square of half the second diameter AC or EF=〈 math 〉〈 math 〉, multiplying both sides by 4 and dividing by o; they will be to one another as obb to ocd i. e. further dividing both sides by b, as ob to 〈 math 〉〈 math 〉 the Latus Transversum to the Latus Rectum.

CONSECTARY VII.

BUT since also the □ DIE is to the □ IK as the Latus Transversum to the Latus Rectum, by vertue of the pre∣sent 7. Prop. the square of CE the transverse semidiam. will be

to the square of AC the second semidiam. (or by the 5th. Con∣sect. of this, to the □ DNE) as the □ DIE to the square of IK.

CONSECTARY VIII.

YOU may also now have the □ IK (which otherwise in the hyperb. is eocd+eecd, in the ellipse oecd−eecd, by vertue of Prop. 2. and 3.) in other terms, if you make as □ CE to the □ DNE so the □ DIE to a fourth; i. e.

as 〈 math 〉〈 math 〉 to 〈 math 〉〈 math 〉 by vertue of Consect. 4. Prop. 5.

(so oebb+eebb in the hyperb.

as 〈 math 〉〈 math 〉 to 〈 math 〉〈 math 〉 so oebb−eebb in the ellipsis.

For hence by the Golden Rule the square IK may be infer'd as a fourth proportional.

In the hyperbola 〈 math 〉〈 math 〉;

In the ellipsis 〈 math 〉〈 math 〉:

The use of which quantities will presently appear.

Notes

• (α) 1.1

  The 3. Property of the Hyperb. and El∣lips. Apollon. lib. I. Prop. 21. pars prior.