here formed, is by Dioptrick Writers commonly call'd the principal Focus , being that of use in Telescopes and Microscopes .
Let then (in Fig. 7. Tab. 5.) BEβ be a double Convex Lens, C the Center of the Segment EB, and K the Center of the Segment Eβ, Bβ the thickness of the Lens, D a Point in the Axis of the Lens; and it is required to find the Point F, at which the Beams proceeding from the Point D, are collected therein, the Ratio of Refraction being as m to n. Let the distance of the Object DB = DA = d (the Point A being supposed the same with B, but taken at a distance therefrom, to prevent the coincidence of so many Lines) the Radius of the Segment towards the Object CB or CA = r, and the Radius of the Segment from the Object Kβ or K = ρ; and let Bβ the thickness of the Lens be = t, and then let the Sine of the Angle of Incidence DAG be to the Sine of the refracted Angle HAG or CAφ as m to n: And in very small Angles, the Angles themselves will be in the same proportion; whence it will follow that,
As d to r , so the Angle at C to the Angle at D, and d + r will be as the Angle of Incidence GAD; and again as m to n , so d + r to dn + rn / m , which will be as the Angle GAH = CAφ; This being taken from ACD which is as d , will leave m - nd - nr / m analogous to the Angle AφD; and the Sides being in this Case proportional to the Angles they subtend, it will follow, that as the Angle AφD is to the Angle ADφ, so is the Side AD or BD to Aφ or Bφ: That is, Bφ will be =