[14]

R - 1V ) R ( V + V/R, V/RR, &c.
VR- V
+ V
+ V/VR
+ V/R
+ VV/RRR
+ V/RR
&c.

[15]

m - 1 ) 1 ( 1/m + 1/mm + 1/m3 + &c.
1 - 1/m
+ 1/m
+ 1/m- 1/mm
+ 1/mm
+ 1/mm- 1/mmm
+ 1/mmm
&c.

[16]
1
1 1
1 1 1
1 1 1 1
&c.

[17]
1/m
1/m2 1/m
1/m3 1/m2 1/m
1/m4 1/m3 1/m2 1/m

An Instance of the Excellence of the Modern Algebra, in the Resolution of the Problem of finding the Foci of Optick Glasses Universally. By E. Halley, S. R. S.

THE Excellence of the Modern Geometry is in nothing more evident, than in those full and adequate Solutions it gives to Problems; representing all the possible Cases at one view, and in one general Theorem, many times comprehending whole Sciences; which deduced at length into Propositions, and demonstrated after the manner of the Ancients, might well become the Subjects of large Treatises: For whatsoever Theorem solves the most complicated Problem of the kind, does with a due Reduction reach all the subordinate Cases. Of this I now design to give a notable Instance in the Doctrine of Dioptricks.

This Dioptrick Problem is that of finding the Focus of any sort of Lens, exposed either to converging, diverging, or parallel Rays of Light, proceeding from, or tending to a given Point in the Axis of the Lens, be the Ratio of Refraction what it will, according to the Nature of the transparent Material whereof the Lens is formed, and also with allowance for the thickness of the Lens between the Vertices of the two Spherical Segments. This Problem being solved in one Case, mutatis mutandis, will exhibit Theorems for all the possible Cases, whether the Lens be Double-Convex or Double-Concave, Plano-Convex, or Plano-Concave, or Convexo-Concave, which sort are usually call'd Menisci. But this only to be understood of those Beams which are nearest to the Axis of the Lens, so as to occasion no sensible difference by their Inclination thereto; and the Focus here formed, is by Dioptrick Writers commonly call'd the principal Focus, being that of use in Telescopes and Microscopes.