Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/135

Lemma XIII.

The latus rectum of a parabola belonging to any vertex is quadruple to the diſŧance of that vertex from the focus of the figure.

This is demonſtrated by the writers on the conic ſections.

Lemma XIV.

The perpendicular let fall from the focus of a parabola on its tangent, if a mean proportional between the diſlances of the focus from the point of contact, and from the principal vertex of the figure. Pl. 5. Fig. 2.

Plate 5, Figure 2

For, let AP be the parabola, S its focus, A its principal vertex, P the point of contact, PO an ordinate to the principal diameter, PM the tangent meeting the principal diameter in M and SN the perpendicular from the focus on the tangent. join AM and becauſe of the equal lines MS and SP, MN and NP, MA and AO; the right lines AN, OP, will be parallel; and thence the triangle SAN will be right angled at A, and ſimilar to the equal triangles SNM, SNP: therefore PS is to SN as SN to SA. Q. E. D.

Cor 1. $\scriptstyle PS^{2}$ is to $\scriptstyle SN^{2}$ as PS to SA.