(_drf + dρf_)/(_dρr + fρr_) = _p_,
which latter determines the _Ratio_ of Refraction, _m_ being to _n_, as
1 + _p_ to _p_.
I shall not expatiate on these Particulars, but leave them for the
Exercise of those that are desirous to be informed in Optical Matters,
which I am bold to say are comprehended in these three Rules, as fully
as the most Inquisitive can desire them, and in all possible Cases;
regard being had to the Signs + and -, as in the former Cases of finding
the _Focus_. I shall only shew two considerable Uses of them; the one to
find the distance whereat an Object being plac'd, shall by a given
_Lens_ be represented in a _Species_ as large as the Object it self,
which may be of singular Use in drawing Faces and other things in their
true Magnitude, by transmitting the _Species_ by a Glass into a dark
Room, which will not only give the true Figure and Shades, but even the
Colours themselves, almost as vivid as the Life. In this Case _d_ is
equal to _f_, and substituting _d_ for _f_ in the Equation, we shall have
_pdrρ_ = _ddr + ddρ - dpρr_,
and dividing all by _dprρ_ = _dr + dρ - prρ_, that is, _2prρ_/(_r + ρ_)
= _d_; but if the two Convexities be of the same Sphere so as _r_ = ρ,
then will the distance be = _pr_; that is, if the _Lens_ be Glass =
_2r_, so that if an Object be placed at the Diameter of the Sphere
distant, in this Case the _Focus_ will be as far within as the Object is
without, and the _Species_ represented thereby will be as big as the
Life; but if it were a _Plano-Convex_, the same distance will be =
_2pr_, or in Glass to four times the _Radius_ of the Convexity; but of
this Method I may entertain the Curious at some other Time, and shew how
to magnifie or diminish an Object in any proportion assign'd, (which yet
will be obvious enough from what is here deliver'd) as likewise how to
erect the Object which in this Method is represented inverted.
A Second Use is to find what Convexity or Concavity is required, to make
a vastly distant Object be represented at a given _Focus_, after the one
Surface of the _Lens_ is formed; which is but a Corollary of our Theorem
for finding ρ, having _p_, _d_, _r_ and _f_ given; for _d_ being
infinite, that Rule becomes
_rf_/(_pr - f_) = ρ,
that is in Glass _rf_/(_2r - f_) = ρ, whence if _f_ be greater than
_2r_, ρ becomes Negative, and _rf_/(_f - 2r_) is the _Radius_ of the
Concave sought.
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.
[Illustration: _Tab. 5. pag. 359_]
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.