[15] _m_ - 1) 1 (1/_m_ + 1/_mm_ + 1/_m_³ + &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 1 ----- --- _m_² _m_ 1 1 1 ----- ----- --- _m_³ _m_² _m_ 1 1 1 1 ----- ----- ----- --- _m_⁴ _m_³ _m_² _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_.