# Copy of an extract from Newton to John Collins, dated 10 December 1672

## Extract of M^{r} Newtons Letter

to M. Collins. Dec. 10. 1672.

I am heartily glad at y^{e} acceptance, w^{ch} D. Barrow's Lectures finds w^{th} forrein Mathematicians; and it pleas'd me not a little to undstand y^{t} they are falne into y^{e} same method of drawg Tangents w^{th} me. What I guess their method to be, you will apprehend by this example;

Suppose CB applied to AB in any given angle be terminated at any Curve line AC, and calling AB x and BC y, let the relaon betw{illeg}t y^{e} between x and y be exprest by any æquaon; as ${\mathrm{x}}^{3}-2\mathrm{x}\mathrm{x}\mathrm{y}+\mathrm{b}\mathrm{x}\mathrm{x}-\mathrm{b}\mathrm{b}\mathrm{x}+\mathrm{b}\mathrm{y}\mathrm{y}-{\mathrm{y}}^{3}=0$, whereby the Curve is determin'd. To draw the Tangent CD, y^{e} Rule is this. Multiply y^{e} termes of y^{e} æquation by any Arithmetical progression accordg to y^{e} dimensions of y, suppose thus $\begin{array}{ccccccccccc}{\mathrm{x}}^{3}& -& 2\mathrm{x}\mathrm{x}\mathrm{y}& +& \mathrm{b}\mathrm{x}\mathrm{x}& -& \mathrm{b}\mathrm{b}\mathrm{x}& +& \mathrm{b}\mathrm{y}\mathrm{y}& -& {\mathrm{y}}^{3}\\ 0& & 1& & 0& & 0& & 2& & 3\end{array}$; also accordg to y^{e} dimensions of x sup\po/se thus $\begin{array}{ccccccccccc}{\mathrm{x}}^{3}& -& 2\mathrm{x}\mathrm{x}\mathrm{y}& +& \mathrm{b}\mathrm{x}\mathrm{x}& -& \mathrm{b}\mathrm{b}\mathrm{x}& +& \mathrm{b}\mathrm{y}\mathrm{y}& -& {\mathrm{y}}^{3}\\ 3& & 2& & 2& & 1& & 0& & 0\end{array}$. The first product shall be y^{e} Numerator, and y^{e} last divided by x y^{e} Denominator of a fraction w^{ch} expresses y^{e} length of BD, to whose end D y^{e} tangent CD must be drawn. The length BC therefore is $\frac{-2\mathrm{x}\mathrm{x}\mathrm{y}+2\mathrm{b}\mathrm{y}\mathrm{y}-3{\mathrm{y}}^{3}}{3\mathrm{x}\mathrm{x}-4\mathrm{x}\mathrm{y}+2\mathrm{b}\mathrm{x}-\mathrm{b}\mathrm{b}}$.

This is one particular, or rather a Corollary of a general Method w^{ch} extends itself, w^{th}out any troublesome calculaon, not only to y^{e} drawing tangents to all Curve lines, whether Geometric, or Mechanic, or however related to streight lines or to other Curve lines, but also to y^{e} resolvg other abstruser kinds of problems abou{illeg}|t| y^{e} crookedness, areas, lengths, centers of gravity of curves &c. Nor is it (as Huddens method de maximis et minimis, and conseqntly Slusius his new method of Tangents, as I presume,) limited to æquaons w^{ch} are free from surd quantities. This method I have interwoven w^{th} y^{t} other of working in æquaons by reducing y^{m} to infinit series. I rememb, I once occasionally told D^{r} Barrow, when he was about to publish his Lectures, y^{t} I had such a method of drawg Tangents, but some divertisemt or other hinder'd me from describg it to him.

Of resolvg by Cardans rules Æquaons y^{t} have 3 possible roots, there may be examples fram'd at pleasure; but unless Brasser show a direct method of performing it, w^{ch} Ferguson dos not, it will not be allow'd scientific. How it is to be done directly, I may possibly show upon occaon.

## Ex ead. {epta} Newtoni ad Colliniu

d. 10. Dec. 1672.

M^{r} Gregory is pleased to consider further y^{e} most advantageous construction of Cata-dioptrical Telescopes. And as his dessein in his Opt. promota excels y^{t} of M. Cassegrain (though they differ so slightly, y^{t} I thought it not worth y^{e} while to take notice of y^{e} difference,) y^{e} advantage being, y^{t} y^{e} litle {sic} concave Ellipsis coms {sic} nearer to a Spherical figure, than y^{e} c|C|onvex Hy*p*bola; so I conceive his present proposall excells y^{m} both, of making y^{t} speculu plane. And this I conjecture is y^{e} way, w^{ch} Sign^{r} Salvetti, one of y^{e} G. Dukes Musicians, mention'd in y^{e} last Transactions, intends to make Expt of, excepting y^{t} instead of the Convex Ey-glas glass he may probably substitut a Concave one to erect y^{e} object. But yet I cannot think it y^{e} best, it being liable to the first, 3^{d} and last of those difficulties, I urg'd ag^{st} M. Cassegrain, and in my Judgem^{t} not wholly capable of y^{e} advantages, w^{ch} M. Gregory propounds. The first disadvantage was, that more light is lost in direct than oblique ref{illeg}|le|ctions. I am convinc'd by several ObservaonsObservations, y^{t} reflexion is not made by y^{e} solid parts of a body, (as is commonly presum'd,) but by y^{e} confine of y^{e} two mediums, whereof one is w^{th}in, and y^{e} other w^{th}out y^{e} body. And as stones are reflected by water, when thrown obliquely, w^{ch} force their way into it when thrown directly downwards; so y^{e} rays of Light (whether Corporeal like stones, or not,) are most easily and copiously reflected when incident most obliquely. This you may observe in y^{e} passage of Light out of Glass into Air, w^{ch} is reflected more and more copiously, as y^{e} obliquity is increas'd, untill beyond a certain degree of obliquity it be wholly reflected. Also in y^{e} reflexion of Light by an imperfectly polish't plate of Brass or Silver or any other metall, you may observe y^{t} y^{e} Images of objects, w^{ch} by direct reflexion appear dull and confus'd, appear by very oblique reflexion pretty distinct and vigorous. This advantage of oblique reflexion would be inconsiderable, if metall reflected almost all y^{e} light directly incident on it, but so far as I can observe, there is at least a 3^{d} part, if not y^{e} better half, of y^{e} light lost and stifled in y^{e} metal at every reflexion; and it is of some estimaon if a 3^{d} or 4^{th} part of y^{t} can be redeem'd by setting y^{e} flat speculu obliquely. As for M^{r} Gregory's insinuaon, y^{t} direct rays have y^{e} advantage of oblique, because a direct ball is reflected more regularly from a rough wall, than an oblique one; if he please to consider, how different are the causes and circu{illeg}|m|stances of those reflexions, possibly upon second thoughts he may apprehend, why y^{e} contrary ought to happen in Light, at least y^{e} Experim^{t} of the rudely polish't plate of metall may persuade him.

The next dis-advantage arising from y^{e} di{illeg}|s|tance of y^{e} litle speculu from y^{e} Ey-glass, being allow'd, I pass to y^{e} last, w^{ch} is to this effect; That; if to diminish y^{e} magnifying virtue of y^{e} instrum^{t} the litle speculu be made of a larger sphere, (as it is in M. Gregory's dessein, a plane being equivalent to a sphere whose center is indefinitly distant,) y^{t} would cause too many of y^{e} best rays to be intercepted. And tho in his designe scarce a forth part of y^{e} whole light be intercepted, yet those rays seem to me of more value than twice their number next y^{e} circumference of the; Tube, because they principally conduce to distinct vision. Their loss will be judged considerable by those, y^{t} have thought y^{e} loss of scarce the 40^{th} part of the Light in my way worthy of being objected by reason y^{t} they were y^{e} best of y^{e} rays.

There are yet other Consideraons, by w^{ch} M^{r} Gregory's Tube may perhaps be thought less advantagious, as, y^{t} unles y^{e} speculu F be made so broad as to intercept more than a quarter, or perhaps than a third part of the whole Light, it will be difficult to enlarge the aperture as is requisite for viewing dull and obscure objects. That y^{e} Ey-glas, if placed at y^{e} bottom, will scarcely be well defended from y^{e} unusefull glaring light w^{ch} in y^{e} day-time comes from objects on all sides y^{e} flat speculu, at least not so well as by setting it at y^{e} side: And y^{t} an Artificer can \scarcely/ polish y^{e} great Concave so truly when perforated in y^{e} midle; for y^{e} metal near y^{t} hole will be apt to weare away too fast, as it doth near y^{e} exterior limb. And tho y^{e} hole may be made after 'tis polish't, yet if y^{e} figure happen to be less true, or if afterwards y^{e} metal chance to tarnish, it must be polish't again.

As for the Advantages propounded by M. Gregory, I see not, why y^{e} first {illeg} should be reckon'd for one, viz. That y^{e} distance EF grows almost y^{e} one half less, and therefore y^{e} Errors of y^{e} Concave CD are also diminish't upon y^{e} plane F by one half. For, how much those Errors of y^{e} Concave CD are increas'd or diminish'd is to be estimated by y^{e} prevarication of y^{e} rays not at y^{e} plane F, but at y^{e} focus of y^{t} concave CD. And there y^{e} Errors in both cases will be alike, provided y^{e} speculu F be accurately plane; but if there be any irregularities in y^{e} figure of y^{t} Speculu F, they will cause Errors so much greater in one case than in y^{e} other, as y^{t} speculu is remoter from y^{e} Ey-glass; w^{ch} in large Telescopes may be more than 15 or 20 times.

The other Advantage, viz. That his Tube will be litle more than half y^{e} length of mine, I should allow to be very considerable, i{f} {I} thought, y^{t} w^{th} equall art in y^{e} mechanisme it could be made to doe y^{e} same effect. The greatest difficulty is in forming y^{e} great Concave, w^{ch} when once well done, perhaps it may be thought most advantagious, to make y^{e} best use of it w^{th} a longer Tube.

The suppos'd Advantage of Telescopes w^{th} Convex or Concave speculums i{illeg}|n| that they may have any desirable charge by altering y^{e} distances of the Ey-glass and specula, agrees more conveniontly dessein of y^{e} Instrum^{t} if y^{t} speculu be made use of, w^{ch} I described in a letter to M. Oldenburg in answer to M. Auzouts Considerations on these Instrum'ts, w^{ch} possibly you may have seen. For instance, to double y^{e} charge, y^{e} Ey-glass in y^{e} other way must be drawn out almost as far behind y^{e} great concave as y^{e} litle speculu is before it, whereby y^{e} length of y^{e} Tube will be almost doubled; whereas in my way it need be drawn out no f{illeg}|a|rther from y^{e} side of y^{e} Tube than a quarter of y^{e} Tube's diameter. The charge may be also conveniently varied by having 2 or 3 Ey-glasses of severall depths set in a girdle; any of w^{ch} may be adjusted to y^{e} metal F, by sliding y^{t} girdle about y^{e} Tube or by sliding y^{e} ring w^{th}in y^{e} Tube, to w^{ch} y^{t} metal F is fastned.

That Telescopes by Convex or concave speculums should be overcharg'd is not necessary; but yet it is not avoidable w^{th}out running upon one of y^{e} other two inconveniences, described in the 7^{th} particular of my consideraons on M. Cassegrains Tube, as I there intimated.

To diminish some of the afores^{d} disadvantages, there may be still new variaons or additions to these designes. As, for instance, by using two Ey-glasses. Suppos{illeg}|e|{illeg} CD represent y^{e} great Concave, F y^{e} litle Speculu, E y^{e} Ey-glas and G another double Convex-glas between E and F between E and F, {sic} on both sides of w^{ch} y^{e} rays crosse. This way of redoubling y^{e} these Tubes seems not inferior to y^{e} rest: for, thus y^{e} object appears erect, y^{e} speculu F intercepts less light, and y^{e} charge may be varied at pleasure, only by changing y^{e} positions of G and F. But yet this is not w^{th}out its imperfections, and particularly (besides those common w^{th} y^{e} other designs,) y^{e} glass G will intercept many of y^{e} best rays in their passage from y^{e} Concave CD to y^{e} litle speculu F, unless it be made less than is consistent w^{th} some other conveniences. And by y^{e} iterated decussations of y^{e} rays, objects will be rendred less distinct, as i{illeg}|s| manifest in Dioptric Telescopes, where 2 or 3 Ey-glasses are applyed to erect y^{e} object.

As to y^{e} attempt in w^{ch} M^{r} Reeves was imployed, I presum'd, it had been done w^{th} much more accuratness than M^{r} Gregory now signifyes, because M^{r} Hook, who you know is a curious and accurate Ex*p*imenter, affirms in his consideraons on my letter to M. Oldenburg concerning refractions & colors, pu publish't in y^{e} Transactions N^{o} 80, y^{t} he made several Ex*p*ts w^{th} y^{t} Instrument. And though he lays y^{e} blame on M. Reeve's Encheiria, yet he says not, y^{t} he blam'd him y^{n}; when the Ex*p*t was made. His words are these;

"I have made many tryals both for Telescopes and Microscopes by reflexion, w^{ch} I have mention'd in my Micrography, but deserted it as to Telescopes, when I considered, y^{t} y^{n} focus of a spherical Concave is not a point but a line, and y^{t} y^{n} rays are lesse true reflected to a point by a Concave, than refracted by a Convex; w^{ch} made me seek y^{t} by refraction, w^{ch} I found could not be expected by reflexion. Nor indeed could I find any effect of it by one of six foot radius w^{ch} about 7 or {8} years since M^{r} Reeve made for M. Gregory, w^{th} w^{ch} I made severall tryals; but it *now appears*, y^{t} it was for want of a good encheiria; from w^{ch} cause many good Expts have been lost. Both w^{ch} consideraons discourag'd me from attempting further y^{t} way, especially since I found y^{e} Parabola much more difficult to describe, than the Hy*p*bola or Ellipsis.{"}

From hence I might well infer y^{t} y^{e} want of a good Encheiria appear'd not till now: And y^{t} M^{r} Hook was discouraged from attempting further y^{t} way only by these 2 or 3 consideraons; That a Convex (as he presumes) refracts more truly, th{illeg}|a|n a concave reflects; y^{t} he found no effect by one of 6 foot radius, w^{ch} till now he attributed to some other cause then y^{e} want of a good encheiria, namly to y^{e} supposedly less true reflexion of a spherical concave; and y^{t} he apprehended a greater difficulty of describing a parabola than an Hyperbola or Ellipsis. Nor could I well interpr{illeg}|et| y^{e} cause, from w^{ch} many good Expts have been lost, to have been other than y^{e} want of a good Encheiria, w^{ch} till afterwards appears not to have been wanting. I contend not, y^{t} this was M. Hooks meaning, but only y^{t} his words seem'd to import thus much: w^{ch} gave me occasion to think, there was no diligence wanting in making that Exp^{t}, especially since he expresseth, y^{t} he made severall tr{illeg}|y|als w^{th} it.

Newton? Sept. 23. 72.

< text from f 42v resumes >And y^{t} you may not think I strain'd M^{r} Gregorys sense, where he spake of Hyperbolic and Elliptic Glasses and Speculums attempted in vain; I would aske, to what end those Speculums were attempted in va if not to compose optic Instrumts; w^{ch} is all I would inferr from those words. For, y^{t} these Instrumts, if at all attempted, were attempted in vain, is evident by y^{e} want of success.

This, S^{r}, I have said, not y^{t} I desire to discourage y^{e} tryall of any practicable way, or to contend w^{th} M^{r} Gregory about so slender a difference. For, I doubt not but when he wrote his Optica promota, he could have described more fashions than one of these Telescopes and perhaps have run through all y^{e} possible cases of y^{m}, if he had thought it worth his pains. Because M. Cassegrain propounded his suppos'd Invention pompously, as if y^{e} main busines was in y^{e} contrivance of these Instrumts, I thought fit to signify, y^{t} y^{t} was none of his contrivance, nor so advantageous as he imagin'd. And I have now sent you these further Consideraons on M. Gregory's {illeg}|A|nswer, only to let you see, y^{t} I chose y^{e} most easy and practicable way to make y^{e} first Tryals. Others may try other ways. Nor doe I think it material, w^{ch} way these Instrumts are perfected, so they be perfected.