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

For Henry Oldenburg Esqꝫ.

< insertion from the top of f 53r >An M^{r} *Newtons* Answer to ye precedent Letter,

sent to the Publisher.

The Copy of this is sent to Liege to M^{r} Lucas Aug. {23}. 1676.

S^{r} The things opposed by M^{r} *Line* being upon tryalls found true & granted me; I begin w^{th} y^{e} new question about y^{e} proportion of y^{e} length of y^{e} Image{illeg} to it's breadth. This I call a new one, for though M^{r} 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 y^{t} 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 y^{e} just quantity of y^{e} refracting angle w^{th} w^{ch} I would have y^{e} exp^{t} repeated {illeg}: w^{ch} would have been a necessary circumstance had y^{e} dispute been about y^{e} just proportion of y^{e} length to y^{e} breadth. Yet I added ^{*} |{*} In my 1st letter {in} Phil. Trans. {N.} 121. p. 500.| this note, that y^{e} bigger y^{e} angle of y^{e} Prism is, the greater will be y^{e} length in proportion to y^{e} breadth: not imagining but that when he had found in any Prism y^{e} length of y^{e} Image transvers to y^{e} breadth axis, he would easily thence conclude y^{t} a Prism w^{th} a greater angle would make y^{e} Image longer, & consequently that by using an angle great enough he might bring it to equal or exceed y^{e} 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 y^{e} length not only 5 but 6 or 8 times y^{e} breadth & more. No wonder therefore that M^{r} Lucas found y^{e} Image shorter then I did, seing he tryed y^{e} exp^{t} w^{th} a less angle. T

The angle indeed w^{ch} I used was but about 63^{degr.}, 12^{min}, & his is set down 60^{degr}: the difference of w^{ch} from mine being but 3^{degr.}, 12^{min}, is too little to reconcile us, but yet it will bring us considerably nearer together. And if his angle was not exactly measured, but y^{e} round number of 60^{degr.} set down by guess or by a less accurate measure (as I suspect by the conjectural {illeg}|me|asure of y^{e} refraction of his prism by y^{e} ratio of y^{e} sines 2 to 3 set down instead of {illeg} at y^{e} 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} the greatest part of y^{e} 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 y^{e} receipt of M^{r} Lucas's letter, & that w^{th}out any considerable difference of my observations either from one another or from what I wrote before. And that it might appear experimentally how y^{e} increase of y^{e} angle increases y^{e} length of y^{e} Image, & also that no body who has a mind to try y^{e} experiment exactly might be troubled to procure a Prism w^{ch} has an angle just of {illeg}|y|^{e} bigness assigned by me; I have tryed y^{e} experiment w^{th} divers angles; {illeg} & have have set down my trialls in y^{e} following table; where y^{e} first column expresses the six angles of two Prisms w^{ch} I used, w^{ch} were measured as exactly as I could by applying \them/ to y^{e} angle of a Sector; & y^{e} second column expresses in inches the length of y^{e} image made by each of those angles; its breadth being two inches: its distance from y^{e} hole in y^{e} window shut \Prism/ 18 feet & 4 inches, & y^{e} breadth of y^{e} hole in y^{e} window shut $\frac{1}{4}$ of an inch.

$\begin{array}{cc}\begin{array}{r}\text{The angles of}\\ \text{degr}\phantom{\rule{0.7em}{0ex}}\text{min}\end{array}& \begin{array}{c}\text{The lengths}\\ \text{of}\phantom{\rule{0.5em}{0ex}}{\text{y}}^{\text{e}}\phantom{\rule{0.5em}{0ex}}\text{Image}\end{array}\\ {\text{y}}^{\text{e}}\phantom{\rule{0.5em}{0ex}}{\text{1}}^{\text{st}}\phantom{\rule{0.5em}{0ex}}\text{Prism}\phantom{\rule{0.5em}{0ex}}\{\begin{array}{rr}56\phantom{\rule{0.2em}{0ex}}.& 10\phantom{\frac{0}{0}}\\ 60\phantom{\rule{0.2em}{0ex}}.& 24\phantom{\frac{0}{0}}\\ 63\phantom{\rule{0.2em}{0ex}}.& 26\phantom{\frac{0}{0}}\end{array}& \begin{array}{c}\phantom{0}7\frac{3}{4}\\ \phantom{0}9\frac{1}{2}\\ 10\frac{1}{3}\end{array}\\ {\text{y}}^{\text{e}}\phantom{\rule{0.5em}{0ex}}{\text{2}}^{\text{d}}\phantom{\rule{0.5em}{0ex}}\text{Prism}\phantom{\rule{0.5em}{0ex}}\{\begin{array}{rr}54\phantom{\rule{0.2em}{0ex}}.& \phantom{0}0\phantom{\frac{0}{0}}\\ 62\phantom{\rule{0.2em}{0ex}}.& 12\phantom{\frac{0}{0}}\\ 63\phantom{\rule{0.2em}{0ex}}.& 48\phantom{\frac{0}{0}}\end{array}& \begin{array}{c}\phantom{0}7\frac{1}{3}\\ 10\frac{1}{8}\\ 10\frac{3}{4}\end{array}\end{array}$^{[1]}

You may perceive that the lengths of y^{e} images in respect of y^{e} angles that made them, are something greater in the 2^{d} Prism then in y^{e} first: but that was because y^{e} glass of w^{ch} y^{e} second Prism was made, had y^{e} greater refractive power.

The days in w^{ch} I ma{illeg}|d|e these trialls were pretty clear but not so clear as I desired, & therefore afterward meeting w^{th} a day as clear as I desired, I rep{illeg}|e|ated y^{e} experiment w^{th} y^{e} second Prism, & found y^{e} lengths of y^{e} 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.

$\begin{array}{cc}\begin{array}{r}\text{The angles of}\\ \text{degr}\phantom{\rule{0.7em}{0ex}}\text{min}\end{array}& \begin{array}{c}\text{The lengths}\\ \text{of}\phantom{\rule{0.5em}{0ex}}{\text{y}}^{\text{e}}\phantom{\rule{0.5em}{0ex}}\text{Image}\end{array}\\ {\text{y}}^{\text{e}}\phantom{\rule{0.5em}{0ex}}{\text{2}}^{\text{d}}\phantom{\rule{0.5em}{0ex}}\text{Prism}\phantom{\rule{0.5em}{0ex}}\{\begin{array}{rr}54\phantom{\rule{0.2em}{0ex}}.& \phantom{0}0\phantom{\frac{0}{0}}\\ 62\phantom{\rule{0.2em}{0ex}}.& 12\phantom{\frac{0}{0}}\\ 63\phantom{\rule{0.2em}{0ex}}.& 48\phantom{\frac{0}{0}}\end{array}& \begin{array}{c}\phantom{0}7\frac{2}{3}\phantom{\rule{0.2em}{0ex}}\text{.}\\ 10\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\text{.}\\ 11\phantom{\rule{0.2em}{0ex}}\text{.}\phantom{\frac{0}{0}}\end{array}\end{array}$

The reason of this difference I apprehend was \{partly}/ that in y^{e} clearest days, the light of y^{e} \white/ skies w^{ch} dilutes & renders invisible the faintest colours at y^{e} ends of y^{e} Image is a little diminished in a clear day & so gives leave to y^{e} l|c|olours to appear to a greater length; the Sun's light at y^{e} same time becoming brisker & so strengthening y^{e} colours & making y^{e} faint ones at y^{e} two ends more conspicuous. For I have observed that in days something cloudy, whilst y^{e} Prism has stood unmoved at y^{e} window, the image would grow a {illeg}little longer or a litt{illeg}|l|e shorter accordingly as y^{e} Sun was more or less obscured by thin clouds w^{ch} passed over it; y^{e} image being shortest when y^{e} cloud was brightest & y^{e} suns light faintest. Whence it is easy to apprehend that if y^{e} light of y^{e} clouds could be quite taken away, so y^{t} y^{e} Sun might appear surrounded w^{th} 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 y^{e} breadth of y^{e} Image was just two inches. But observing that y^{e} sides of y^{e} two Prisms I used were {illeg}|n|o{illeg}|t| exactly pl{illeg}|a|in 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 y^{e} other were convex; & this {illeg} made y^{e} breadth of y^{e} Image to be two inches & a third part of an inch: the angles of y^{t} \this/ Prism & y^{e} lengths of this|e| Image made by each of those angles being those exprest in this Table

$\begin{array}{cc}\begin{array}{c}\text{The angles}\\ \text{of the Prism}\\ \text{degr.}\end{array}& \begin{array}{c}\text{The lengths of}\\ \text{of}\phantom{\rule{0.5em}{0ex}}{\text{y}}^{\text{e}}\phantom{\rule{0.5em}{0ex}}\text{Image in inches.}\\ \phantom{\text{degr.}}\end{array}\\ \begin{array}{c}58\phantom{\frac{0}{0}}\\ 59\frac{1}{2}\\ 62\frac{1}{2}\end{array}& \begin{array}{c}\phantom{0}8\frac{1}{2}\phantom{\rule{0.2em}{0ex}}\text{.}\\ \phantom{0}9\phantom{\rule{0.2em}{0ex}}\text{.}\phantom{\frac{1}{2}}\\ 10\frac{1}{3}\phantom{\rule{0.2em}{0ex}}\text{.}\end{array}\end{array}$

In this case you see y^{e} concave figure of y^{e} sides of y^{e} Prism by making y^{e} rays diverge a little, causes y^{e} breadth of y^{e} 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 M^{r} Lucas may examin whether his Prism have not this fault. If a Prism may be had w^{th} sides exactly plain, it may do well to try y^{e} experiment w^{th} that: but it's better if y^{e} sides be about so much convex as those of mine are, because y^{e} Image will thereby become much better defined. For this convexity of y^{e} sides does y^{e} same effect as if you should use a Prism w^{th} sides exactly plain, & between it & y^{e} hole in y^{e} window shut, place an object-glass of an 18 foot Telescope to make y^{e} round image of y^{e} sun appear distinctly defined on y^{e} wall when the Prism is taken away, & consequently the long image made by y^{e} Prism to be much more distinctly defined (especially at it's streight sides) then it would be otherwise.

One thing more I shall ad{illeg}|d|: Th{illeg}|at| y^{e}e utmost length of y^{e}e Image from y^{e} faintest red at one end to y^{e} faintest blew at y^{e} 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 {illeg}length y^{e} utmost length of y^{e} image; & I measured y^{e} utmost length, because I account all that length to be caused by y^{e} im{illeg}med{illeg}iat{illeg}e light of y^{e} sun, seing the colours (as I noted above) become visible to y^{e} greatest length in y^{e} clearest days, that is, when y^{e} light of y^{e} Sun transcends most y^{e} light of y^{e} clouds. Sometimes th{illeg}|e|re will happen to shoot out from both ends of y^{e} Image a glaring light {illeg} a good way beyond these colours, but this is not to be regarded, as not apperteining to y^{e} Image. If y^{e} measures be taken right y^{e} whole length will exceed y^{e} length of y^{e} streight sides by about y^{e} breadth of y^{e} Image.

By these things set down thus circumstantially, I presume M^{r} Lucas will be enabled to accord his trials of y^{e} Exp^{t} w^{th} 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 {illeg}|seei|ng there may be many various circumstances w^{ch} may conduce to it; such as are not only y^{e} different figures of {illeg} prisms, but also y^{e} different refractive power of glasses, y^{e} different diameters of y^{e} Sun at divers times of y^{e} year, & y^{e} little errors that may happen in measuring lines & angles, or in placing y^{e} Prism at y^{e} {illeg}d window: though for my part I took care to do these things as exactly as I could. However M^{r} Lucas may make sure to find y^{e} Image as long as|or| longer then I have set down, if he take a Prism whose sides are not hollow grownd, but plain, or (w^{ch} 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 or 67 degrees, or (if he will) a little greater.

Concerning M^{r} *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 y^{e} first y^{t} has sent me an experimental examination of them. By this I \may/ presume he really desires to know what truth{illeg} there is in these matters. {illeg}|B|ut yet it will conduce to his more speedy & full ^{[2]}^{[3]}satisfaction if he a little change y^{e} method w^{ch} he has propounded, & instead of a multitude of things try only the *Experimentum Crucis*. For it is not number of Exp^{ts}, but weight to be regarded; & where one will do, what need {illeg} 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 y^{e} principall of y^{e} experiments I had tryed; {illeg} amongst {the rest} \which/ there has happened to be the principal of those w^{ch} Experiments w^{ch} M^{r} Lucas has now sent me. And as for y^{e} Experiments set down in my first letter to you, they were only such as I thought convenient to sele{illeg}|c|t out of that Tractate.

But yet suppose those had been my whole store, yet M^{r} 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 *y ^{e} different refrangibility* of light. And this I demonstrated by y

^{e}

*e|E|xperimentu Crucis*. Now if this demonstration be good, there needs no further examination of y

^{e}thing; if not good y

^{e}fault of it is to be shewn: for y

^{e}only way to examin a demonstrated proposition is to examin y

^{e}demonstration. Let that exp

^{t}therefore be examined in y

^{e}first place, & that w

^{ch}it proves be acknowledged, & then if M

^{r}Lucas want my assistance to unfold y

^{e}difficulties w

^{ch}he fancies to be in y

^{e}experiments he has propounded, he shall freely have it; for then I presume a few words may make them pl{illeg}|a|in to him: whereas should I be drawn from a demonstrative experiment to begin with those, it might create us both y

^{e}trouble of a long dispute, & by y

^{e}multitude of words cloude rather then clear up y

^{e}truth. For if it has already cost us so much trouble to agree upon y

^{e}matter of fact in y

^{e}first & plainest experiment, & yet we are not fully agreed: what an endless trouble might it create us, if we should give o

^{r}selves up to dispute upon every argument that occurs, & what would become of truth in such a tedious dispute.

^{[4]}The way therefore that I propound being y

^{e}shortest & clearest (not to say y

^{e}only proper way,) I question not but M

^{r}Lucas will be glad that I have recommended it, seeing he professes that it is {illeg}|y

^{e}| knowledge of truth that he seeks after. And therefore at p

^{r}sent I shall say nothing in answer to M

^{r}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 experi{illeg}|m|ents there is one, w

^{ch}when duely tryed, is, next to y

^{e}

*Experimentum Crucis*, the most conspicuous Experiment I know for proving the different refrangibility of light, w

^{ch}he brings it to d{is}prove against.

By y^{e} Post-script of M^{r} *Lucas's* letter, one not acquainted w^{th} what has passed, might think, that he quotes y^{e} Observation of y^{e} *R. Society* against me; whereas the relation of their Observation, w^{ch} you sent to Leige, conteined nothing at all about the just proportion of y^{e} Length of y^{e} Image to it's {illeg}|B|readth according to y^{e} angle of y^{e} Prism, nor any thing more (so far as I can perceive by yo^{r} last) than what was pertinent to the things then in dispute, viz, that they found them succeed as I had affirmed. And therefore since M^{r} *Lucas* has found the same success, I suppose suppose {sic} y^{t} when he expresses|d| that he much rejoyced to see y^{e} trialls of y^{e} R. Society agree so exactly w^{th} his, he meant only so far as his agreed w^{th} mine.

And because I am again upon this first experiment, I {illeg} \shall/ desire that M^{r} *Lucas* will repeat it with all \y^{e}/ exactness & caution that may be, regard being had to y^{e} information about it, {illeg}|set| down in this letter; & then I desire to have the length & breadth of y^{e} Image w^{th} its distance from y^{e} hole in y^{e} 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 y^{e} sun's diameter. For I know, that M^{r} *Lucas's* Observation cannot hold where y^{e} refracting angle of y^{e} Prism is full 60^{degr:}, & the breadth of y^{e} Im \day is clear & the/ full length of y^{e} colours is measured, & y^{e} breadth of y^{e} Imag{illeg}|e| answers to y^{e} sun's diameter: & seeing I am well assured of y^{e} 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 y^{e} first place.

S^{r}

I am

Yo^{r} humble Servant

Is. Newton.

Cambridge

Aug {illeg}|18|. 1676

*Postscript.*

I had like to \have/ forgotten to advise y^{t} y^{e} *Experimentum Crucis*, & such others as shall

be made for knowing y^{e} nature of colours;

be made w^{th} Prisms which refract so much

as to make y^{e} length of y^{e} Image five times

it's breadth, & rather more then less; for,

otherwise Exp^{ts} will not succeed so plainly w^{th}

others as they have done w^{th} me.

^{[1]} Yyyy *6.*

^{[2]} Zzzz {7} 703

^{[3]} pray {ed}{2^{d}} turn {tꝫ} a {Leaf}

^{[4]} {4 Zz 704}