44.
Extract of Mr Newtons Letter
to M. Collins. Dec. 10. 1672.
I am heartily glad at yethe acceptance, wchwhich D. Barrow's Lectures finds
wthwith forrein Mathematicians; and it pleas'ded me not a little to undderstand
ytthat they are falne into yethe same method of drawging Tangents wthwith me.
What I guess their method to be, you will apprehend by this example;
Suppose applied to in any given angle be terminated at any Curve line , and calling and , let the relaontion betwt yethe between and be exprest by any æquaontion; as , whereby the Curve is determin'ded. To draw the Tangent , yethe Rule is this. Multiply yethe termes of yethe æquation by any Arithmetical progression accordging to yethe dimensions of , suppose thus ; also accordging to yethe dimensions of suppose thus . The first product shall be yethe Numerator, and yethe last divided by yethe Denominator of a fraction wchwhich expresses yethe length of , to whose end yethe tangent must be drawn. The length therefore is .
This is one particular, or rather a Corollary of a general Method wchwhich extends itself, wthwithout any troublesome calculaontion, not only to yethe drawing tangents to all Curve lines, whether Geometric, or Mechanic, or however related to streight lines or to other Curve lines, but also to yethe resolvging other abstruser kinds of problems about yethe crookedness, areas, lengths, centers of gravity of curves &c. Nor is it (as Huddens method de maximis et minimis, and conseqnquently Slusius his new method of Tangents, as I presume,) limited to æquaonstions wchwhich are free from surd quantities. This method I have interwoven wthwith ytthat other of working in æquaonstions by reducing ymthem to infinit series. I remembber, I once occasionally told Dr Barrow, when he was about to publish his Lectures, ytthat I had such a method of drawging Tangents, but some divertisemt or other hinder'ded me from describging it to him.
Of resolvging by Cardans rules Æquaonstions ytthat have 3 possible roots, there may be examples fram'ded at pleasure; but unless Brasser show a direct method of performing it, wchwhich Ferguson dos not, it will not be allow'ded scientific. How it is to be done directly, I may possibly show upon occaonsion.
Mr Gregory is pleas'ded to c4344? Oldenb Copy43.
Ex ead.eadem epta Newtoni ad Colliniuum d. 10. Dec. 1672.
Mr Gregory is pleased to consider further yethe most advantageous construction of Cata-dioptrical Telescopes. And as his dessein in his Opt.Optica promota excels ytthat of M. Cassegrain (though they differ so slightly, ytthat I thought it not worth yethe while to take notice of yethe difference,) yethe advantage being, ytthat yethe litlelittle concave Ellipsis comscomes nearer to a Spherical figure, than yethe cConvex Hypperbola; so I conceive his present proposall excells ymthem both, of making ytthat speculuum plane. And this I conjecture is yethe way, wchwhich SignrSignor Salvetti, one of yethe G.Great Dukes Musicians, mention'ded in yethe last Transactions, intends to make Exptxperiment of, excepting ytthat instead of the Convex Ey-glas glass he may probably substitut a Concave one to erect yethe object. But yet I cannot think it yethe best, it being liable to the first, 3dthird and last of those difficulties, I urg'ded agstagainst M. Cassegrain, and in my JudgemtJudgement not wholly capable of yethe advantages, wchwhich M. Gregory propounds. The first disadvantage was, that more light is lost in direct than oblique reflections. I am convinc'ded by several ObservaonstionsObservations, ytthat reflexion is not made by yethe solid parts of a body, (as is commonly presum'ded,) but by yethe confine of yethe two mediums, whereof one is wthwithin, and yethe other wthwithout yethe body. And as stones are reflected by water, when thrown obliquely, wchwhich force their way into it when thrown directly downwards; so yethe rays of Light (whether Corporeal like stones, or not,) are most easily and copiously reflected when incident most obliquely. This you may observe in yethe passage of Light out of Glass into Air, wchwhich is reflected more and more copiously, as yethe obliquity is increas'ded, untill beyond a certain degree of obliquity it be wholly reflected. Also in yethe reflexion of Light by an imperfectly polish'tet plate of Brass or Silver or any other metall, you may observe ytthat yethe Images of objects, wchwhich by direct reflexion appear dull and confus'ded, appear by very oblique reflexion pretty distinct and vigorous. This advantage of oblique reflexion would be inconsiderable, if metall reflected almost all yethe light directly incident on it, but so far as I can observe, there is at least a 3dthird part, if not yethe better half, of yethe light lost and stifled in yethe metal at every reflexion; and it is of some estimaontion if a 3dthird or 4thfourth part of ytthat can be redeem'ded by setting yethe flat speculuum obliquely. As for Mr Gregory's insinuaontion, ytthat direct rays have yethe 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 circumstances of those reflexions, possibly upon second thoughts he may apprehend, why yethe contrary ought to happen in Light, at least yethe ExperimtExperiment of the rudely polish'tet plate of metall may persuade him.
The next dis-advantage arising from yethe distance of yethe litle speculuum from yethe Ey-glass, being allow'ded, I pass to yethe last, wchwhich is to this effect; That; if to diminish yethe magnifying virtue of yethe instrumtinstrument the litle speculuum 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,) ytthat would cause too many of yethe best rays to be intercepted. And tho in his designe scarce a forth part of yethe whole light be intercepted, yet those rays seem to me of more value than twice their number next yethe circumference of the; Tube, because they principally conduce to distinct vision. Their loss will be judged considerable by those, ytthat have thought yethe loss of scarce the 40thfortieth part of the Light in my way worthy of being objected by reason ytthat they were yethe best of yethe rays.
There are yet other Consideraonstions, by wchwhich Mr Gregory's Tube may perhaps be thought less advantagious, as, ytthat unles yethe speculuum 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 yethe Ey-glas, if placed at yethe bottom, will scarcely be well defended from yethe unusefull glaring light wchwhich in yethe day-time comes from objects on all sides yethe flat speculuum, at least not so well as by setting it at yethe side: And ytthat an Artificer can scarcely polish yethe great Concave so truly when perforated in yethe midle; for yethe metal near ytthat hole will be apt to weare away too fast, as it doth near yethe exterior limb. And tho yethe hole may be made after 'tisit is polish'tet, yet if yethe figure happen to be less true, or if afterwards yethe metal chance to tarnish, it must be polish'tet again.
As for the Advantages propounded by M. Gregory, I see not, why yethe first should be reckon'ded for one, viz. That yethe distance grows almost yethe one half less, and therefore yethe Errors of yethe Concave are also diminish'tet upon yethe plane by one half. For, how much those Errors of yethe Concave are increas'ded or diminish'ded is to be estimated by yethe prevarication of yethe rays not at yethe plane , but at yethe focus of ytthat concave . And there yethe Errors in both cases will be alike, provided yethe speculuum be accurately plane; but if there be any irregularities in yethe figure of ytthat Speculuum , they will cause Errors so much greater in one case than in yethe other, as ytthat speculuum is remoter from yethe Ey-glass; wchwhich in large Telescopes may be more than or times.
The other Advantage, viz. That his Tube will be litle more than half yethe length of mine, I should allow to be very considerable, if I thought, ytthat wthwith equall art in yethe mechanisme it could be made to doe yethe same effect. The greatest difficulty is in forming yethe great Concave, wchwhich when once well done, perhaps it may be thought most advantagious, to make yethe best use of it wthwith a longer Tube.
The suppos'ded Advantage of Telescopes wthwith Convex or Concave speculums in that they may have any desirable charge by altering yethe distances of the Ey-glass and specula, agrees more conveniontly wthwith my42? Oldenburg Copy42. dessein of yethe InstrumtInstrument if ytthat speculuum be made use of, wchwhich I described in a letter to M. Oldenburg in answer to M. Auzouts Considerations on these Instrum'tsents, wchwhich possibly you may have seen. For instance, to double yethe charge, yethe Ey-glass in yethe other way must be drawn out almost as far behind yethe great concave as yethe litle speculuum is before it, whereby yethe length of yethe Tube will be almost doubled; whereas in my way it need be drawn out no farther from yethe side of yethe Tube than a quarter of yethe 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 wchwhich may be adjusted to yethe metal , by sliding ytthat girdle about yethe Tube or by sliding yethe ring wthwithin yethe Tube, to wchwhich ytthat metal is fastned.
That Telescopes by Convex or concave speculums should be overcharg'ded is not necessary; but yet it is not avoidable wthwithout running upon one of yethe other two inconveniences, described in the 7thseventh particular of my consideraonstions on M. Cassegrains Tube, as I there intimated.
To diminish some of the aforesdaforesaid disadvantages, there may be still new variaonstions or additions to these designes. As, for instance, by using two Ey-glasses. Suppose represent yethe great Concave, yethe litle Speculuum, yethe Ey-glas and another double Convex-glas between and between and , on both sides of wchwhich yethe rays crosse. This way of redoubling yethe these Tubes seems not inferior to yethe rest: for, thus yethe object appears erect, yethe speculuum intercepts less light, and yethe charge may be varied at pleasure, only by changing yethe positions of and . But yet this is not wthwithout its imperfections, and particularly (besides those common wthwith yethe other designs,) yethe glass will intercept many of yethe best rays in their passage from yethe Concave to yethe litle speculuum , unless it be made less than is consistent wthwith some other conveniences. And by yethe iterated decussations of yethe rays, objects will be rendred less distinct, as is manifest in Dioptric Telescopes, where 2 or 3 Ey-glasses are applyed to erect yethe object.
As to yethe attempt in wchwhich Mr Reeves was imployed, I presum'ded, it had been done wthwith much more accuratness than Mr Gregory now signifyes, because Mr Hook, who you know is a curious and accurate Expperimenter, affirms in his consideraonstions on my letter to M. Oldenburg concerning refractions & colors, pu publish'tet in yethe Transactions No 80, ytthat he made several ExppertsExperiments wthwith ytthat Instrument. And though he lays yethe blame on M. Reeve's Encheiria, yet he says not, ytthat he blam'ded him ynthen; when the ExppertExperiment was made. His words are these;
"I have made many tryals both for Telescopes and Microscopes by reflexion, wchwhich I have mention'ded in my Micrography, but deserted it as to Telescopes, when I considered, ytthat ynthen focus of a spherical Concave is not a point but a line, and ytthat ynthen rays are lesse true reflected to a point by a Concave, than refracted by a Convex; wchwhich made me seek ytthat by refraction, wchwhich I found could not be expected by reflexion. Nor indeed could I find any effect of it by one of six foot radius wchwhich about 7 or 8 years since Mr Reeve made for M. Gregory, wthwith wchwhich I made severall tryals; but it now appears, ytthat it was for want of a good encheiria; from wchwhich cause many good ExptsExperiments have been lost. Both wchwhich consideraonstions discourag'ded me from attempting further ytthat way, especially since I found yethe Parabola much more difficult to describe, than the Hypperbola or Ellipsis."
From hence I might well infer ytthat yethe want of a good Encheiria appear'ded not till now: And ytthat Mr Hook was discouraged from attempting further ytthat way only by these 2 or 3 consideraonstions; That a Convex (as he presumes) refracts more truly, than a concave reflects; ytthat he found no effect by one of foot radius, wchwhich till now he attributed to some other cause then yethe want of a good encheiria, namly to yethe supposedly less true reflexion of a spherical concave; and ytthat he apprehended a greater difficulty of describing a parabola than an Hyperbola or Ellipsis. Nor could I well interpret yethe cause, from wchwhich many good ExptsExperiments have been lost, to have been other than yethe want of a good Encheiria, wchwhich till afterwards appears not to have been wanting. I contend not, ytthat this was M. Hooks meaning, but only ytthat his words seem'ded to import thus much: wchwhich gave me occasion to think, there was no diligence wanting in making that ExptExperiment, especially since he expresseth, ytthat he made severall tryals wthwith it.
Newton? Sept. 23. 72.
And ytthat you may not think I strain'ded Mr 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 InstrumtsInstruments; wchwhich is all I would inferr from those words. For, ytthat these InstrumtsInstruments, if at all attempted, were attempted in vain, is evident by yethe want of success.
This, SrSir, I have said, not ytthat I desire to discourage yethe tryall of any practicable way, or to contend wthwith Mr 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 yethe possible cases of ymthem, if he had thought it worth his pains. Because M. Cassegrain propounded his suppos'ded Invention pompously, as if yethe main busines was in yethe contrivance of these InstrumtsInstruments, I thought fit to signify, ytthat ytthat was none of his contrivance, nor so advantageous as he imagin'ded. And I have now sent you these further Consideraonstions on M. Gregory's Answer, only to let you see, ytthat I chose yethe most easy and practicable way to make yethe first Tryals. Others may try other ways. Nor doe I think it material, wchwhich way these InstrumtsInstruments are perfected, so they be perfected.