I am sensible that these Examples are too much for the compleat Analyst, though I fear too little for the less Skilful; it being very hard, if possible, in such Matters, so to write, as to give satisfaction to both; or to please the one, and instruct the other. But this may suffice to shew the extent of our Theorem, and how easie a Reduction adapts any one case to all the rest.
Nor is this only useful to discover the Focus from the other proposed data, but from the Focus given, we may thereby determine the distance of the Object; or from the Focus and Distance given, we may find of what Sphere it is requisite to take another Segment, to make any given Segment of another Sphere cast the Beams from the distance d to the Focus f. As likewise from the Lens, Focus, and Distance given, to find the Ratio of Refraction, or of m to n, requisite to answer those Data. All which it is obvious, are fully determined from the Equation we have hitherto used, viz. pdρr = drf + dρf - prρf, for to find d the Theorem is prρf / rf + ρf - pρr = d, the distance of the Object.
For ρ the Rule is drf / pdr + df + prf = ρ.
But for p will be 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.
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.