_Lastly_, As to the Effect of turning the two sides of a _Lens_ towards an Object; it is evident, that if the thickness of the _Lens_ be very small, so as that you neglect it, or account _t_ = 0; then in all Cases the _Focus_ of the same _Lens_, to whatsoever Beams, will be the same, without any difference upon the turning the _Lens_: But if you are so curious as to consider the thickness, (which is seldom worth accounting for) in the Case of parallel Rays falling on a _Plano-Convex_ of Glass, if the plain side be towards the Object, _t_ does occasion no difference, but the focal distance _f_ = 2_r_. But when the Convex-side is towards the Object, it is contracted to _2r - ⅔t_, so that the _Focus_ is nearer by ⅔_t_. If the _Lens_ be double Convex, the difference is less; if a _Meniscus_, greater. If the Convexity on both sides be equal, the focal length is about ⅙_t_ shorter than when _t_ = 0. In a _Meniscus_ the Concave-side towards the Object increases the focal Length, but the Convex towards the Object diminishes it. A General Rule for the difference arising on turning the _Lens_, where the _Focus_ is Affirmative, is this (_2rt - 2ρt_)/(_3r + 3ρ - t_), for double Convexes of differing Spheres. But for _Menisci_ the same difference becomes (_2rt + 2ρt_)/(_3r - 3ρ + t_); of which I need give no other Demonstration, but that by a due Reduction it will so follow from what is premised, as will the Theorems for all sorts of Problems relating to the _Foci_ of Optick-Glasses. APPENDIX. _An Analytical Resolution of certain Equations of the Third, Fifth, Seventh, Ninth Powers, and so on _ad Infinitum_, in finite Terms, after the manner of _Cardan_'s Rules for Cubicks. By Mr. _A. Moivre_, Transact. _Nᵒ 309_._ Let (_n_) be any Number, (_y_) an unknown Quantity, or Root of the Equation, (_a_) a Quantity altogether known, or what they call _Homogeneum Comparationis_: And let the Relation of these Quantities to each other be exprest by the Equation. _ny_ + ((_nn_ - 1)/(2 × 3))_ny_³ + ((_nn_ - 1)/(2 × 3)) × ((_nn_ - 9)/(4 × 5))_ny_⁵ + ((_nn_ - 1)/(2 × 3)) × ((_nn_ - 9)/(4 × 5)) × ((_nn_ - 25)/(6 × 7))_ny_⁷, _&c._ = _a_. Its plain from the Nature of this Series, that if _n_ be any odd Number (that is an Integer, it matters not whether Affirmative or Negative) then the Series will Terminate, and the Equation arising will be one of the above defin'd, whose Root is (1) _y_ = ½[ⁿ√](√(1 + _aa_) + _a_) - ½/[ⁿ√](√(1 + _aa_) + _a_) or, (2) _y_ = ½[ⁿ√](√(1 + _aa_) + _a_) - ½[ⁿ√](√(1 + _aa_) - _a_) or, (3) _y_ = ½/[ⁿ√](√(1 + _aa_) - _a_) - ½[ⁿ√](√(1 + _aa_) - _a_) or, (4) _y_ = ½/[ⁿ√](√(1 + _aa_) - _a_) - ½/[ⁿ√](√(1 + _aa_) + _a_) For Example, Let the Root of this Equation of the Fifth Power be required 5_y_ + 20_y_³ + 16_y_⁵ = 4 in which case _n_ is = 5, and _a_ = 4, and the Root, according to the first Form, is _y_ = ½[⁵√](√(17 + 4)) - ½/[⁵√](√(17 + 4))