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

Proposition XI.

THE Elliptical Space(α) 1.1 comprehended by the Ellipsis DAEB (Fig. 127.) is to a circle described on the transverse ax DE, as the Axis Rectus or conjugate diameter AB to the transverse ax DE.

Demonstration.

THis is in the first place evident from the genesis of the ellipse we deduced in Schol. 1. Prop. 7. for in that de∣duction we shewed that FO, i. e. HN was to NI as AB to DE: Which since it is true of all the other indivisibles or or∣dinates HN and IN ad infinitum; it is manifest that the planes themselves constituted of these indivisibles will have the same reason among themselves, as the Axis Rectus AB to the trans∣verse DE. Q. E. D.

CONSECTARY I.

THerefore the quadrature of the ellipse will be evident, if that of the circle be demonstrated.

CONSECTARY II.

SInce a circle described on the least diameter AB will be to one described on the greater diameter DE, as AB to a third proportional by Prop. 35. lib. 1. it follows by vertue of the present Prop. that the ellipse is a mean proportional between the greater and lesser circle, i e. as the ellipse is to the great∣er circle so is the lesser circle to it, viz. as AB to DE.

CONSECTARY III.

HEnce you may have a double method of determining the area of an ellipse. 1. If having found the area of the greater circle, you should infer, as the greater diameter of the ellipsis to the less, so the area of the circle found to the area of the ellipse sought. 2. If having also found the area of the les∣ser circle, you find a mean proportional between that and the area of the greater.

SCHOLIƲM.

WE may also shew the last part of the second Consect. thus, 1. If having described the circle EadbE (Fig. 134.) about the least axis of the ellipse we conceive a regular hexagon to be inscribed, and an ellipse coinciding with one end E of its transverse ax, and with the other or opposite one D to be so elevated, that with the point d it may perpendicu∣larly hang over the circle, and further from all the angles of the figure inscribed in the circle you erect▪ the perpendiculars gG, bB, &c. it is certain that the sides ED and Ed of the triangle DEd will be cut by the parallel planes FGgf, &c. into proportional parts, and that those by reason of the simi∣litude of the ▵ ▵ FDG and fdg, and so also the other rectan∣gles will be among themselves as the intercepted parts of the lines ID and id, CI and ci, and in infinitum, (viz. of how many sides soever the inscribed figure consists:) Wherefore also all the parts of the ellipse taken together will be to all the parts of the circle taken together, i. e. the whole ellipse to the whole

[illustration]

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circle as all the parts of the diameter ED or ab, i. e. as DE it self to AB. Q. E D.

CONSECTARY IV.

IT is also evident that both these demonstrations of the pre∣sent Prop. will be also the same in any segments of the el∣lipsis or circle.

Notes

• (α) 1.1

  Archim. lib. de Conoid. &c. Prop. 5.