Letter to Oldenburg on the lengths and angles of prism images, dated 18 August 1676

53Newton 1253

~~For Henry Oldenburg Esqꝫ..~~

An Mr Newtons Answer to yethe precedent Letter, sent to the Publisher.

~~The Copy of this is sent to Liege to Mr Lucas Aug. 23. 1676.~~

SrSir The things opposed by Mr Line being upon tryalls found true & granted me; I begin wthwith yethe new question about yethe proportion of yethe length of yethe Image to it's breadth. This I call a new one, for though Mr Line in his last letter spake against so great a length as I assigne, yet, as it seems to me, it was not to grant any transvers length shorter then ytthat assigned by me (for in his first letter he absolutely denyed that there would be any such length;) but to lay the greater emphasis upon his discours whilst he in defence of common Optiques he was disputing in general against a transvers Image: & therefore in my answer I did not prescribe yethe just quantity of yethe refracting angle wthwith wchwhich I would have yethe exptexperiment repeated : wchwhich would have been a necessary circumstance had yethe dispute been about yethe just proportion of yethe length to yethe breadth. Yet I added * * In my 1st letter in Phil. Trans. N. 121. p. 500. this note, that yethe bigger yethe angle of yethe Prism is, the greater will be yethe length in proportion to yethe breadth: not imagining but that when he had found in any Prism yethe length of yethe Image transvers to yethe ~~breadth~~ axis, he would easily thence conclude ytthat a Prism wthwith a greater angle would make yethe Image longer, & consequently that by using an angle great enough he might bring it to equal or exceed yethe length assigned by me: as indeed he might, for by taking an angle of $70$ or $75$ degrees, or a little greater, he might have made yethe length not only 5 but 6 or 8 times yethe breadth & more. No wonder therefore that Mr Lucas found yethe Image shorter then I did, seing he tryed yethe exptexperiment wthwith a less angle. T

The angle indeed wchwhich I used was but about 63degr., 12min, & his is set down 60degr: the difference of wchwhich from mine being but 3degr., 12min, is too little to reconcile us, but yet it will bring us considerably nearer together. And if his angle was not exactly measured, but yethe round number of 60degr. set down by guess or by a less accurate measure (as I suspect by the conjectural measure of yethe refraction of his prism by yethe ratio of yethe sines

2

3

set down instead of at yethe same time instead of an experimental one,) then might it be two or three degrees less then

60

, if not still less: & all this, if it should be so, would take away away the greatest part of yethe difference between us. But however it be, I am well assured my own observation was exact enough. For I have repeated it divers times since yethe receipt of Mr Lucas's letter, & that wthwithout any considerable difference of my observations either from one another or from what I wrote before. And that it might appear experimentally how yethe increase of yethe angle increases yethe length of yethe Image, & also that no body who has a mind to try yethe experiment exactly might be troubled to procure a Prism wchwhich has an angle just of yethe bigness assigned by me; I have tryed yethe experiment wthwith divers angles; & have have set down my trialls in yethe following table; where yethe first column expresses the six angles of two Prisms wchwhich I used, wchwhich were measured as exactly as I could by applying them to yethe angle of a Sector; & yethe second column expresses in inches the length of yethe image made by each of those angles; its breadth being two inches: its distance from yethe hole in yethe window shut Prism

18

feet &

4

inches, & yethe breadth of yethe hole in yethe window shut

\frac{1}{4}

of an inch.

\begin{matrix} \begin{array}{r} The angles of \\ degr min \end{array} & \begin{matrix} The lengths \\ of y^{e} Image \end{matrix} \\ y^{e} 1^{st} Prism {\begin{array}{r} 56 . & 10 \\ 60 . & 24 \\ 63 . & 26 \end{array} & \begin{matrix} 7 \frac{3}{4} \\ 9 \frac{1}{2} \\ 10 \frac{1}{3} \end{matrix} \\ y^{e} 2^{d} Prism {\begin{array}{r} 54 . & 0 \\ 62 . & 12 \\ 63 . & 48 \end{array} & \begin{matrix} 7 \frac{1}{3} \\ 10 \frac{1}{8} \\ 10 \frac{3}{4} \end{matrix} \end{matrix}

Yyyy 6. You may perceive that the lengths of yethe images in respect of yethe angles that made them, are something greater in the 2dsecond Prism then in yethe first: but that was because yethe glass of wchwhich yethe second Prism was made, had yethe greater refractive power. The days in wchwhich I made these trialls were pretty clear but not so clear as I desired, & therefore afterward meeting wthwith a day as clear as I desired, I repeated yethe experiment wthwith yethe second Prism, & found yethe lengths of yethe image made by its severall angles to be those set down in this table about

\frac{1}{4}

of an inch greater then before, the measures being those set down in this table.

\begin{matrix} \begin{array}{r} The angles of \\ degr min \end{array} & \begin{matrix} The lengths \\ of y^{e} Image \end{matrix} \\ y^{e} 2^{d} Prism {\begin{array}{r} 54 . & 0 \\ 62 . & 12 \\ 63 . & 48 \end{array} & \begin{matrix} 7 \frac{2}{3} . \\ 10 \frac{1}{2} . \\ 11 . \end{matrix} \end{matrix}

The reason of this difference I apprehend was partly that in yethe clearest days, the light of yethe white skies wchwhich dilutes & renders invisible the faintest colours at yethe ends of yethe Image is a little diminished in a clear day & so gives leave to yethe lcolours to appear to a greater length; the Sun's light at yethe same time becoming brisker & so strengthening yethe colours & making yethe faint ones at yethe two ends more53– more conspicuous. For I have observed that in days something cloudy, whilst yethe Prism has stood unmoved at yethe window, the image would grow a little longer or a little shorter accordingly as yethe Sun was more or less obscured by thin clouds wchwhich passed over it; yethe image being shortest when yethe cloud was brightest & yethe suns light faintest. Whence it is easy to apprehend that if yethe light of yethe clouds could be quite taken away, so ytthat yethe Sun might appear surrounded wthwith darkness, or if the suns light was much more stronger then it is the colours would still appear to a greater length. In all these observations yethe breadth of yethe Image was just two inches. But observing that yethe sides of yethe two Prisms I used were not exactly plain but a little convex, (the convexity being about so much as that of a double convex glass of an sixteen or eighteen foot Telescope), I took a third Prism whose sides were as much concave as those of yethe other were convex; & this made yethe breadth of yethe Image to be two inches & a third part of an inch: the angles of ytthat this Prism & yethe lengths of thise Image made by each of those angles being those exprest in this Table

\begin{matrix} \begin{matrix} The angles \\ of the Prism \\ degr. \end{matrix} & \begin{matrix} The lengths of \\ of y^{e} Image in inches. \end{matrix} \\ \begin{matrix} 58 \\ 59 \frac{1}{2} \\ 62 \frac{1}{2} \end{matrix} & \begin{matrix} 8 \frac{1}{2} . \\ 9 . \\ 10 \frac{1}{3} . \end{matrix} \end{matrix}

In this case you see yethe concave figure of yethe sides of yethe Prism by making yethe rays diverge a little, causes yethe breadth of yethe Image to be greater in proportion to its length then it would be otherwise. And this I thought fit to give you notice of, that Mr Lucas may examin whether his Prism have not this fault. If a Prism may be had wthwith sides exactly plain, it may do well to try yethe experiment wthwith that: but it's better if yethe sides be about so much convex as those of mine are, because yethe Image will thereby become much better defined. For this convexity of yethe sides does yethe same effect as if you should use a Prism wthwith sides exactly plain, & between it & yethe hole in yethe window shut, place an object-glass of an

18

foot Telescope to make yethe round image of yethe sun appear distinctly defined on yethe wall when the Prism is taken away, & consequently the long image made by yethe Prism to be much more distinctly defined (especially at it's streight sides) then it would be otherwise. One thing more I shall add: That yethee utmost length of yethee Image from yethe faintest red at one end to yethe faintest blew at yethe other, must be measured. For in my first letter about colours where I set down the length to be five times the breadth, I called that length yethe utmost length of yethe image; & I measured yethe utmost length, because I account all that length to be caused by yethe immediate light of yethe sun, seing the colours (as I noted above) become visible to yethe greatest length in yethe clearest days, that is, when yethe light of yethe Sun transcends most yethe light of yethe clouds. Sometimes there will happen to shoot out from both ends of yethe Image a glaring light a good way beyond these colours, but this is not to be regarded, as not apperteining to yethe Image. If yethe measures be taken right yethe whole length will exceed yethe length of yethe streight sides by about yethe breadth of yethe Image. By these things set down thus circumstantially, I presume Mr Lucas will be enabled to accord his trials of yethe ExptExperiment wthwith mine; so nearly at least that there shall not remain any very considerable difference between us. For if some little difference should still remain, that need not trouble us any further seeing there may be many various circumstances wchwhich may conduce to it; such as are not only yethe different figures of prisms, but also yethe different refractive power of glasses, yethe different diameters of yethe Sun at divers times of yethe year, & yethe little errors that may happen in measuring lines & angles, or in placing yethe Prism at yethe d window: though for my part I took care to do these things as exactly as I could. However Mr Lucas may make sure to find yethe Image as long asor longer then I have set down, if he take a Prism whose sides are not hollow grownd, but plain, or (wchwhich is better) a very little convex, & whose refracting angle is as much greater then that I used as that he has hitherto tryed it with is less; that is, whose angle is about

66

67

degrees, or (if he will) a little greater. Concerning Mr Lucas's other experiments, I am much obliged to him that he would take these things so far into consideration, & be at so much pains for examining them; & I thank him so much the more because he is yethe first ytthat has sent me an experimental examination of them. By this I may presume he really desires to know what truth there is in these matters. But yet it will conduce to his more speedy & full Zzzz 7 703pray ed2d turn tꝫthis a Leafsatisfaction if he a little change yethe method wchwhich he has propounded, & instead of a multitude of things try only the Experimentum Crucis. For it is not number of ExptsExperiments, but weight to be regarded; & where one will do, what need many? Had I thought more requisite, I could have added more. For before I wrote my first letter to you about colours I had taken much pains in trying experiments about them & written a Tractate on that subject wherein I had set down at large & considered yethe principall of yethe experiments I had tryed; amongst the rest which there has happened to be the principal of those wchwhich Experiments wchwhich Mr Lucas has now sent me. And as for yethe Experiments set down in my first letter to you, they were only such as I thought convenient to select out of that Tractate. But53– But yet suppose those had been my whole store, yet Mr Lucas should not have grownded his discourse upon a supposition of my want of experiments till he had examined those few. For if any of those be demonstrative, they will need no assistants nor leave room for further disputing about what they demonstrate. The main thing he goes about to examin is yethe different refrangibility of light. And this I demonstrated by yethe eExperimentuum Crucis. Now if this demonstration be good, there needs no further examination of yethe thing; if not good yethe fault of it is to be shewn: for yethe only way to examin a demonstrated proposition is to examin yethe demonstration. Let that exptexperiment therefore be examined in yethe first place, & that wchwhich it proves be acknowledged, & then if Mr Lucas want my assistance to unfold yethe difficulties wchwhich he fancies to be in yethe experiments he has propounded, he shall freely have it; for then I presume a few words may make them plain to him: whereas should I be drawn from a demonstrative experiment to begin with those, it might create us both yethe trouble of a long dispute, & by yethe multitude of words cloude rather then clear up yethe truth. For if it has already cost us so much trouble to agree upon yethe matter of fact in yethe first & plainest experiment, & yet we are not fully agreed: what an endless trouble might it create us, if we should give orour selves up to dispute upon every argument that occurs, & what would become of truth in such a tedious dispute. 4 Zz 704The way therefore that I propound being yethe shortest & clearest (not to say yethe only proper way,) I question not but Mr Lucas will be glad that I have recommended it, seeing he professes that it is yethe knowledge of truth that he seeks after. And therefore at prsentpresent I shall say nothing in answer to Mr his experimental discourse but this in general: that it has proceeded partly from some misunderstanding of what he writes against, & partly from want of due caution in trying experiments; & that amongst his experiments there is one, wchwhich when duely tryed, is, next to yethe Experimentum Crucis, the most conspicuous Experiment I know for proving the different refrangibility of light, wchwhich he brings it to disprove against. By yethe Post-script of Mr Lucas's letter, one not acquainted wthwith what has passed, might think, that he quotes yethe Observation of yethe R. Society against me; whereas the relation of their Observation, wchwhich you sent to Leige, conteined nothing at all about the just proportion of yethe Length of yethe Image to it's Breadth according to yethe angle of yethe Prism, nor any thing more (so far as I can perceive by yoryour last) than what was pertinent to the things then in dispute, viz, that they found them succeed as I had affirmed. And therefore since Mr Lucas has found the same success, I suppose suppose ytthat when he expressesd that he much rejoyced to see yethe trialls of yethe R. Society agree so exactly wthwith his, he meant only so far as his agreed wthwith mine. And because I am again upon this first experiment, I shall desire that Mr Lucas will repeat it with all yethe exactness & caution that may be, regard being had to yethe information about it, set down in this letter; & then I desire to have the length & breadth of yethe Image wthwith its distance from yethe hole in yethe window shut, Prism set down exactly in feet & inches & parts of an inch, that I may have an opportunity to consider what relation it's length & breadth have to yethe sun's diameter. For I know, that Mr Lucas's Observation cannot hold where yethe refracting angle of yethe Prism is full 60degr:, & the breadth of yethe Im day is clear & the full length of yethe colours is measured, & yethe breadth of yethe Image answers to yethe sun's diameter: & seeing I am well assured of yethe truth & exactness of my own observation, I shall be unwilling to be diverted by any other experiments from having a fair end made of this in yethe first place. SrSir I am YorYour humble Servant Is. Newton. Cambridge Aug 18. 1676 Postscript. I had like to have forgotten to advise ytthat yethe Experimentum Crucis, & such others as shall be made for knowing yethe nature of colours; be made wthwith Prisms which refract so much as to make yethe length of yethe Image five times it's breadth, & rather more then less; for, otherwise ExptsExperiments will not succeed so plainly wthwith others as they have done wthwith me.