28.
Cambridge.
December 10th 1672.
SrSir
My unwillingness to trouble you in the midst of yoryour late dbuisiness made me
suspend writing, though I stood obliged to thank you for the rest of Mr Horrox
works wchwhich I received some weeks since. But by yorsyours wchwhich I received two
days since I presume the most of that trouble is over. I am heartily
glatd at the acceptance wchwhich orour Reverend friend Dr Barrow's Lectures finds wthwith
forreign Mathematicians, & it pleased me not a little to
they are falln into the same method of drawing Tangents wthwith me. if
What I guess their method to be you will apprehend by this example.
Suppose applyed to in any given angle be terminated at any Curve line , and calling & let the relation between & be expressed by any æquation as whereby the curve is determined. To draw the tangent the Rule is this. Multiply the termes of the æquation by any arithmeticall progression according to the dimensions of suppose thus , also according to the dimensions of suppose thus . The first product shall be the Numerator, & the last divided by the Denominator of a fraction wchwhich expresseth the length of to whose end the tangent must be drawn. The length therefore is .
This SrSir is one particular, or rather a Corollary of a Generall Method wchwhich extends it selfe wthwithout any troublesome calculation, not onely to the drawing tangents to all curve lines whether Geometrick or mechanick or how ever related to streight lines or to other curve lines but also to the resolving other abstruser kinds of Problems about the crookedness, areas, lengths, centers of gravity of curves &c. Nor is it (as Huddens method de maximis et minimis & consequently Slusius his new method of Tangents as I presume) limited to æquations wchwhich are free from surd quantities. This method I have interwoven wthwith that other of working in æquations by reducing them to infinite series. I remember I once occasionally told Dr Barrow when he was about to publish his Lectures that I had such a method of drawing Tangents but some divertisment or other hindered me from descrdescribing it to him.
Of resolving by Cardans rules Equations that have 3 possible roots there may be examples framed at pleasure, but unlesse bBrasser show a direct method of performing it, wchwhich Ferguson doth not, it will not be allowed scientifick. How it is to be done directly I may possibly show upon occasion.
Mr Gregory is pleased to consider further the most advantageous construction of Catadioptricall Telescopes. And as his designe in his Optica promota excells that of M. Cassegrain (though they differ so sleightily that I thought it not worth the while to takeing notice of the difference) the advantage being that the little concave Ellipsis comes nearer to a sphericall figure then the convex Hyperbola; so I conceive his present designe proposall excells them both, of making that speculum plane. And this I conjecture is the way wchwhich SignōorSignor Salvetti one of yethe great Dukes Musitians mentioned in the last Transactions intends to make experiment of, excepting that instead of the convex eye glass he may probably substitute a concave one to erect the object. But yet I cannot think it the best it being liable to the 1st, 3d, & last of those difficulties wchwhich I urged against M. Cassegrain, & in my Judgment not wholly capable of the advantages wchwhich Mr Gregory propounds. The first disadvantage was that more light is lost in direct then oblique reflexion. I am convinced by severall observations that reflexion is not made by the solid parts of a body, as is commonly presumed, bput by the confine of the two Mediums whereof one is within, & the other wthwithout the body. And as stones are reflected by water when thrown obliquely wchwhich force their way into it when thrown directly downwards so light the rays of light (whether corporeall like stones or not) are most easily & copiously reflected when incident most obliquely. This you may observe in the passage of light out of glass into Air wchwhich is reflected more & more copiously as the obliquity is inclreased untill beyond a certain degree of obliquity it be wholly reflected. Also in the reflexion of light by an imperfectly polished plate of Brass or Silver or any other metall you may observe that the images of objects wchwhich by direct reflexion appear dull & confused, appiarappear by very oblique reflexion pretty distinct & vigorous. This advantage of oblique reflexion would be inconsiderable if metall reflected almost all the light directly incident on it, but so far as I can observe there is at least a third part if not the better half of the light lost & stifled in the metall at every reflexion ; & it is of some estimation if a third or 4th part of that can be redeemed by setting the flat speculum obliquely. As for Mr Gregory's insinuation that direct rays have the advantage of oblique because a direct ball is reflected more regularly from a rough wall then an oblique one, if he please to consider how different are the causes & circumstances of those reflexions, possibly upon second thoughts he may apprehend why the contrary ought to happen in light, at least the Experiment of the rudely polished plate of metall may perswade him.
The next disadvantage arising from the distance of the little speculum from the eye-glass being allowed I pass to yethe last wchwhich is to this effect, That if to diminish the magnifying virtue of the instrument the little speculum be made of a larger sphære, whose center is infinitely distant (as it is in Mr Gregories designe, a plane being equivalent to a sphære whose center is infinitely distant,) that would cause too many of the best rays to be intercepted. And though in his designe scarce a fourth part of the whole light be intercepted yet those rays seem to me of more value then twice their number next the circumference of yethe Tube, because they principally conduce to distinct vision. Their loss will be judged considerable by those that have thought the loss of scarce the fort fortieth part of the light in my way worthy of being objected by reason that they were the best of the rays.
There are yet other considerations by which Mr Gregory's Tube may perhaps be thought less advantageous, as that unless the speculum be made so broade as to intercept more then a quarter or perhaps then a third part of the whole light, it will be difficult to inlarge the aperture as is requisite for viewing dull & obscure objects. That the eye-glass if placed at the bottom will scarcely be well defended from the unusefull glareing light wchwhich in the day time comes from objects on all sides the flat speculum, at least not so well as by setting it at the side. And that an Artificer can scarcely polish the great concave so truly when perforated in the middle. For the metall neare that hole will be apt to weare away too fast as it doth neare the exterior limb. And though the hole may be made after tis polished, yet if the figure happen to be less true, or if afterwards the metall chance to tarnish, it must be polished again.
As for the advantages propounded by Mr Gregory I see not why the first should be recconed for one, viz: That the distance groweth almost the one half less & therefore the errors of the concave are also diminished upon the plane by one half. For how much those errors of the concave are increased or diminished is to be estimated by the prevarication of the rays not at the plane but at the Focus of that concave . And there the errors in both cases will be alike, provided the speculum be accurately plane, but if there be any irregularities in the figure of that speculum they will cause errors so much greater in one case then in the other, as that speculum is remoter from the eye-glass, wchwhich in large Telescopes may be more then 15 or 20 times.
The other advantage, viz: That his Tube will be little more then half the length of mine I should allow to be very considerable, if I thought that wthwith equall art in the mechanism it could be made to doe the same effect. The greatest difficulty is in forming the great Concave, wchwhich when once well done, perhaps it may be thought most advantageous to make the best use of it though with a longer Tube.
The propounded supposed advantage of Telescopes wthwith convex or concave speculums in that they may have any desirable charge by altering the distances of the eye-glass & specula, agrees more conveniently to my design of the instrument if that speculum be made use of wchwhich I described in a letter to Mr Oldenburg in answer to MonsrMonsieur Auzouts considerations on these instruments, wchwhich possibly you may have seen. For instance to double the charge the eye-glass in the other way must be drawn out almost as far behind the great concave as the little speculum is before it, whereby the length of the Tube will be almost doubled whereas in my way it need be drawn out no farther from the side of the Tube then a quarter of the Tubes diameter. The charge may be also conveniently varied by having two or three eye-glasses of severall depths set in a girdle, any of wchwhich may be adjusted to the metall by sliding that girdle about the Tube or by sliding the ring within the Tube to wchwhich that girdle metall is fastened.
That Telescopes by convex or concave speculums should be overcharged is not necessary but yet it is not avoydable without running upon one of the other two inconveniences described in the 7th particular of my considerations on M. Cassegrains Tube, as I there intimated.
29.
To diminish some of the afforesaid disadvantages there may be other still new variations or additions to these designes. As for instance by using two eye-glasses. Suppose represent the great Concave, the little speculum the eye-glass & another double-convex glass between & on both sides of wchwhich the rays cross. This way of redoubling these Tubes seems not inferior to the rest. For thus the object appeares erect, the speculum intercepts less light, & the charge may be varied at pleasure onely by changing the positions of & . But yet this is not wthwithout its imperfections, & particularly the (besides those common wthwith the other Designes) the glass will intercept many of the best rays in their passage from the concave to yethe little speculum , unless it be made less then is consistent wthwith some other conveniences. And by the iterated decussations of the rays objects will be rendered less distinct, as is manifest in Dioptrick Telescopes where two or three eye-glasses are applyed to erect the object.
As to the attempt in wchwhich Mr Rieve was imployed, I presumed it had been done wthwith much more accurateness then Mr Gregory now signifys, because Mr Hook, who you know is a curious & accurate experimenter, affirms in his considerations on my letter to Mr Oldenburg concerning refractions & colours published in the Transactions NumbNumber 80, that he made severall experiments wthwith that Instrument. And though he lays the blame on Mr Rieve's encheiria, yet he says not ytthat he blamed him then when the Experiment was made. His words are these
" I have made many tryalls both for Telescopes & Microscopes by reflexion wchwhich I have mentioned in my Micrographia, but deserted it as to Telescopes when I con sidered that the focus of a sphericall concave is not a point but a line, & that the rays are less true reflected to a point by a Concave then refracted by a Convex, wchwhich made me seek that by refraction wchwhich I found could not be ex pected from reflexion. Nor indeed could I find any effect of it by one of six foot radius wchwhich about 7 or 8 yeares since Mr Rieve made for Mr Gregory wthwith wchwhich I made severall tryalls; but it now appeares that it was for want of a good encheiria; From wchwhich cause many good experiments have been lost. Both wchwhich considerations discouraged me from attempting further that way; especially since I found the Parabola much more difficult to describe then the Hyperbola or Ellipsis.
From hence I might well infer that yethe want of a good encheiria appeared not till now. And that Mr Hook was discouraged from attempting further that way onely by these two or three considerations, That a Convex (as he presumes) refracts more truly then a concave reflects, that he found no effect by one of six foot radius, wchwhich till now he attributed to some other cause then the want of a good encheiria, as perhaps to the un namely to the supposedly less true reflexion of a sphæricall concave, & he supposed apprehended a greater difficulty of describing a Parabola then an Hyperbola or Ellipsis. Nor could I well interpret the cause from * * from wchwhich many good experiments have been lost to have been other then the want of a good encheiria wchwhich till afterwards appears not to have been wanting. I contend not that this was Mr Hooks meaning, but onely that his words seemed to import thus much, which gave me occasion to think there was no diligence wanting in making that experiment, especially since he expresseth that he made severall tryalls wthwith it.
And that you may not think I strained Mr Gregory's sense where he spake of Hyperbolick & Elliptick Glasses & Speculums attempted in vain; I would ask to what end those Speculums were attempted if not to compose optiqꝫue Instruments wchwhich is all I would infer from those words. For that those instruments if at all attempted were attempted in vain is evident by the want of success.
This SrSir I have said not that I desire to discourage the tryall of any practicable way or to contend wthwith Mr Gregory about so slender a subject difference. For I doubt not but when he wrote his Optica promota he could have described more fashions then one of these Telescopes & perhaps have run through all the possible cases of them if he had then thought it worth his paines. Because M. Cassegrain propounded his supposed invention pompously, as if the main buisiniess was in the contrivance of these instruments I thought fit to signify that that was none of his contrivance, nor so advantageous as he imagined. And I have now sent you these further considerations on on Mr Gregory's answer onely to let you see that I chose the most easy & practicable way to make the first tryalls. Others may try other ways. Nor doe I think it materiall wchwhich way these instruments are perfected so they be perfected.
You will pardon this long scribble in wchwhich I have been the more particular because Mr Gregory's discours looks as if intended for the Press. We are here very glad that we shall enjoy Dr Barrow again especially in the circumstances of a Master, nor doth any rejoyce at it more then SrSir
YorYour obliged humble Servant
Newton.
I suppose Slusius his method of TangntsTangents will shortly appear abroad, when it comes over I'le beg of you the trouble of transmitting a coppy to me if you will give me leave to be accountable to you for it.