Chapter XIII: Appendix (12)

Those that are wholly to begin with this Dioptrical Science, cannot do better than to read with Attention a late Treatise of Dioptricks, published by _W. Molineux_, Esq, R. S. S. who has at large shewn the Nature of Optick Glasses, and the Construction and Use of Microscopes and Telescopes; and though some nicely Critical have endeavour'd to spy Faults, and to traduce the Book; yet having long since examin'd it with Care, I affirm, that if I can judge, it hath but two things that with any Colour may be call'd Faults; the one, an over-careful acknowledgment of every Trifle the Author had receiv'd from others; and the other that he labours to make easie this curious Subject, so little understood by most, in a manner perhaps too familiar for the _Learned Critick_, and which demonstrates that it was writ _cum animo docendi_, both which require but very little Friendship or good Nature in the Reader, to pass for Vertues in an Author.

But to return to our first Theorem, which accounting for the thickness of the _Lens_, we will here again resume, _viz._

(_mpdrρ - ndρt + nprρt_)/(_mdr + mdρ - mprρ - m - ndt + nrt_) = _f_.

And let it be required to find the _Focus_ where a whole Sphere will collect the Beams proceeding from an Object at the distance _d_: Here _t_ is equal to _2r_, and _r_ equal to ρ. And after due Reduction, the Theorem will stand thus,

(_mpdr - 2ndr + 2nprr_)/(_2nd + 2nr - mpr_) = _f_;

but if _d_ be Infinite, it is contracted to

_mpr_/_2n_ - _r_ = ((_2n - m_)/(_2m - 2n_))_r_ = _f_,

wherefore a Sphere of Glass collects the Sun-Beams at half the Semi-diameter of the Sphere without it, and a Sphere of Water at a whole Semi-diameter. But if the _Ratio_ of Refraction _m_ to _n_ be as 2 to 1, the _Focus_ falls on the opposite Surface of the Sphere; but if it be of greater Inequality it falls within.

Another Example shall be when a Hemisphere is exposed to parallel Rays, that is, _d_ and ρ being infinite, and _t_ = _r_, and after due Reduction the Theorem results

(_nn_/(_mm - mn_))_r_ = _f_.

That is, in Glass it is at 4/3_r_, in Water at 9/4_r_; but if the Hemisphere were Diamant, it would collect the Beams at 1-4/15 of the _Radius_ beyond the Center.

_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.