Opticks (1721, 3rd Edition) — part 4 of 18

Proportions obtain in all the refl. That it fhould be fo is very realbnable, Nature being ever conformable to her felf: but an experi- mental Proof, is defired. And fuch a Proof will be had if we can fliew that the Sines of Refraction of Rays differently refrangible are one to another in a given Proportion when their Sines of Incidence are equal. For if the Sines of Refraction of all the Rays are in given Proportions to the Sine of Refradion of a Ray which has a mean degree of Refrangibility, and this Sine is in a given Proportion to the equal Sines of Incidence, thofe other Sines of Refra- dion will alfo be in given Proportions to the equal Sines of Incidence. Now when the Sines of Incidence are equal; it will appear by the following Experiment that the Sines of Refra- dion are in a given Proportion to one ano- ther.

E^per. II). The Sun fhining into a dark Charnber through a little round hole in the Window-fliut, let S [in Fig. 26.] reprefent his round white Image painted on the oppofite Wall by his diredt Light, P T his oblong co- loured Image made by refrading that Light with a Prifm placed at the Window; and p t, orzpzt, or 3 Z' 3 t, his oblong colour'd Image made by refrading again the fame Light fide- ways with- a fecond Prifm placed immediately after the firft in a crofs pofition to it , as was explained in the fifth Experiment : that is to fay, / t when the Refradion of the fecond Prifm is fmall ^ z p 1 t when its Refraftion is ' greater, and ip i t when it is greateft. For

fuch

[ ^7 ]

fuch will be the diverfity of the Reflations if the refrading Angle of the fecond Prifm be of various magnitudes ; fuppofe of fifteen or twen- ty Degrees to make the Image/ t^ of thirty or forty to make the Image x/' 1 1, and of fixty to make the Image 3/ 3 /^. But for want of folid Glafs Prifms with Angles of convenient big- nefles, there may be VefTels made of poliilied Plates of Glafs cemented together in the form of Prifms and filled with Water. Thefe things being thus ordered, I obferved that all the fo- lar Images or coloured Spedrums P T, / ^ ^ / nt, 3 / 3 r did very nearly converge to the place S on which the dired: Light of the Sun fell and painted his white round Image when the Prifms were taken away. The Axis of the Speftrum P T, that is the Line drawn through the middle of it parallel to its re^ilinear Sides, did when produced pafs exadly through ths middle of that white round Image S. And when the Refraftion of the fecond Prifm was equal to the Refradion of the firft, the refrading An- gles of them both being about 60 Degrees, the I Axis of the Speftrum 3/ 3 ^ made by that Re- ! fraftion, did when produced pafs alfo through the middle of the fame white round Image S. '■ But when the Refraction of the fecond Prifm was lefs than that of the firft , the produced Axes of the Speftrums r/ ox 'l t x p made by that Refraftion did cut the produced Axis of the Speftrum T P in the points m and ;/, a lit- tle beyond the center of that white round I- mage S. Whence the proportion of the Line 3 ^ T to the Line 3/ P was a little greater than

F 2 • the

[ ^8 ]

the Proportion of x^ T to x/P, and this Pro- portion a Uttle greater than that of ^ T to /> P. Now when the Light of the Spedrum P T falls perpendicularly upon the Wall, thofe Lines 3 2f T, 3 Z' Pj and x /f T, 2 /> P and t T, p P, are the Tangents of the Refraftions, and therefore by this Experiment the Proportions of the Tan- gents of the Refra61ions are obtained, from whence the Proportions of the Sines being de- rived, they come out equal, fo far as by view- ing the Spedrums and ufing fome mathemati- cal Reafoning I could eitimate. For I did not make an accurate Computatioji. So then the Proportion holds true in every Ray apart, fo far as appears by Experiment. And that it is accurately true, may be demonftrated upon this Suppofition. That Bodies refratl Light hy a^l- ing upon its Rays in Lines perpendicular to their Surfaces. But in order to this Demon- flration, I mull diltinguifli the Motion of every Ray into two Motions, the one perpendicular to the refracting Surface, the other parallel to it , and concerning the perpendicular Motion lay down the following Propofition.

If any Motion or moving thing whatfoever be incident with any velocity on any broad and thin fpace terminated on both fides by two pa- rallel Planes, and in its paffage through that fpace be urged perpendiciilarly towards the far- ther Plane by any force which at given diltances from the Plane is of given quantities; the per- pendicular velocity of that Motion or Thing, at its emerging out of that fpace, fliali be al- ways equal to the fquare Root of the fum of

the.

[ 69 3

the fquare of the perpendicular velocity of that Motion or Thing at its Incidence on that fpace ; and of the fquare of the perpendicular velocity which that Motion or Thing would have at its Emergence, if at its Incidence its perpendicu- lar velocity was infinitely little.

And the fame Propofition holds true of any Motion or Thing perpendicularly retarded in its pafTage through that fpace, if inftead of the fum of the two Squares you take their diffe- rence. The demonitration Mathematicians will eafily find out, and therefore I fhall not trouble the Reader with it.

Suppofe now that a Ray coming mofl oblique- ly in the LineM C [in Fig. i.] be refracted at C by the Plane R S into the Line C M, and if it be. required to find the Line C E, into which any other Ray AC fliall be refra^led; let MC, AD, be the Sines of Incidence of the two Rays, and N G, E F, their Sines of Refradion, and let the equal Motions of the incident Rays bd reprefented by the equal Lines M C and AC, and the Motion M C being ccnfidered as paral- lel to the refrafting Plane, let the other Motion A C be diftinguilhed into two Motions A D and DC, one of which A D is parallel, and the other D C perpendicular to the refrading Sur- face. In Hke manner, let the Motions of the emerging Rays be diilinguifli'd into two, whereof

the perpendicular ones are ~ C G and ~ C F.

And if the force of the refrading Plane begins to aft upon the Rays either in that Plane or at a certain diitance from it on the one fide, and ends at a certain diftance from it on the other

F 3 iide

[ 70 ]

fide, and in all places between thofe two limits adts upon the Rays in Lines perpendicular to that refrading Plane, and the Adions upon the Rays at equal diflances from the refrafting Plane be equal, and at unequal ones either equal or unequal according to any rate whatever ; that Motion of the Ray which is parallel to the re- frafting Plane will fuffer no Alteration by that force ; and that Motion which is perpendicular to it will be altered according to the rule of the foregoing Proportion. If therefore for the per- pendicular velocity of the emerging Ray C N

you write — C G as above, then the perpendi- cular velocity of any other emerging Ray C E which was — r C F, will be equal to the fquare

Root of C D ^ 4- ^^^ C G f . And by fquaring

thefe Equals, and adding to them the Equals; A D ^ and M C ^ — C D f, and dividing the Sums by the Equals C F ^ -f EF ^ and C G ^ -^-j

NG^, you will have ^-~ equal to^^\ Whence

A D, the Sine of Incidence, is to E F the Sine of RefradHon, as MC to N G, that is, in a given ratio. And this Demonltration being general, without determining what Light is, or by what kind of force it is refrafted, or alTuming any thing farther than that the refrafting Body adls upon the Rays in Lines perpendicular to its Surface ; I take it to be a very convincing Argu- ment of the full truth of this Propofition.

So then, if the rafio of the Sines of Inci- dence and Refraction of any fort of Rays be

found

C 71 3

found in any one cafe,/tis given in all cafes; and this may be readily found by the method in the following Propoiition.

T R OT.yW, Theor.VI.

The TerfeBton of Telefeopes is impeded by the different Refrangibitity of the Rays of Light.

' I ^ H E Imperfeftion of Telefeopes is, vul- JL garly attributed to the fpherical Figures ot the Glafles, and therefore Mathematicians have propounded to figure them by the coni- cal Sedions. To fliew that they are miflaken, I have inferted this Propofition ; the truth of which will appear by the meafures of the Re- fraftions of the feveral forts of Rays ; and thefe meafures I thus determine.

In the third Experiment of the firft Book, where the refracting Angle of the Prifm was 61-;: Degrees, the half of that Angle 31 deg. 15' min. is the Angle of Incidence of the Rays at their going out of the Glafs into the Air ; and the Sine of this Angle is 5188, the Radius being 1 0000. When the Axis of this Prifm was pa- rallel to the Horizon, and the Refradion . of the Rays at their Incidence on this Prifm equal to that at their Emergence out of it, I obferved with a Quadrant the Angle which the mean re- frangible Rays (that is, thofe which went to the middle of the Sun's coloured Image) made with the Horizon, and by this Angle and the Sun's, altitude obferved at the fame time, I found the Angle which the eniergent Rays contained with

F 4 the

[72 ]

the incident to be 44 deg. and 40 min. and the half of this Angle added to the Angle of Inci- dence 31 deg. 15" min. makes the Angle of Re- fraftion, which is therefore 5-3 deg. 35- min. and its Sine 8047. Thefe are the Sines of Incidence and Refradion of the mean refrangible Rays, and their proportion in round numbers is zo to 31. This Glafs was of a colour inclining to green. The laft of the Prifms mentioned in the third Experiment was of clear white Glafs. Its refrading Angle 634 Degrees. The Angle which the emergent Rays contained, with the incident 45 deg. 50 min. The Sine of half the firll An- gle 5^6l. The Sine of half the fum of the An- gles 8157. And their proportion in rotmd num- bers 20 to 31, as before.

From the length of the Image, which was about 9-1 or 10 Inches, fubdud: its breadth, which was 24 Inches, and the remainder 7-I Inches would be the length of the Image were the Sun but a point, and therefore fubtends the Angle which the moft and leaft refrangible Rays, when incident on the Prifm in the fame Lines, do contain with one another after their Emergence, Whence this Angle is 2 deg. o'. 7'^. For the diltance between the Image and the Prifm where this Angle is made, was 184 Feet, and at that diftance the Chord 7^ Inches fubtends an Angle of x deg. o'. y^\ Now half this Angle is the Angle which thefe emergent Rays con- tain with the emergent mean refrangible Rays, and a quarter thereof, that is 30^. x^\ may be accounted the Angle which they would contain with the fame emergent mean refrangible Rays,

were

[73 ]

were they co-incident to them within the Glafs and fuffered no other Refraclion than that at their Emergence. For if two equal" Refradiions, the one at the Incidence of the Rays on the Prifm, the other at their Emergence, make half the Angle % deg. o^. 7'^. then one of thofe Re- fractions'will make about a quarter of that An- gle, and this quarter added to and fubdufted from the Angle of Refraftion of the mean re- frangible Rays, which was 5-3 deg. 35^ gives the Angles of Refraction of the molt and lead refrangible Rays 54 deg. 5^ -l^^ and 53 deg. 4^ 5-8^'', whofe Sines are 8099 and 7995", the common Angle of Incidence being 31 deg. if^ and its Sine 5-188; and thefe Sines in the leaft round numbers are in proportion to one another, as 78 and yy to 5-0.

Now if you fubdu(?c the common Sine of In- cidence 50 from the Sines of Refradion 77 and 78, the remainders ly and x8 ihew that in fmall Refractions the RefraCtion of the leaft refran- ^ gible Rays is to the RefraCtion of the moil re- frangible ones as 27 to 28 very nearly, and that the difference of the Refractions of the leaft re- frangible and moit refrangible Rays is about the Z74th part of the whole RefraCtion of the mean refrangible Rays.

Whence they that are skilled in Opticks will eafily underftand, that the breadth of the leall circular fpace into which ObjeCt glaffes of Te- lefcopes can colleCt all forts of parallel Rays, is about the 274th part of half the Aperture of the Glafs, or 5fth part of the whole Aperture ; and tjiat the Focus of the molt refrangible Rays is • ^ ■ ■ ' nearer

[74]

nearer to the Objeft-glafs than the Focus of the Jeaft refrangible ones, by about the 174th part of the diftance between the Objeft-'glafs and the Focus of the mean refrangible ones.

And if Rays of all forts, flowing from any one lucid point in the Axis of any convex Lens,- be made by the Refraction of the Lens to con- verge to points not too remote from the Lens, the Focus of the mod refrangible Rays iliall be nearer to the Lens than ihe Focus of the leafl refrangible ones, by a diftance which is to the xy-irth part of the diftance of the Focus of the mean refrangible Rays from the Lens as the di- ftance between that Focus and the lucid point from whence the Rays flow , is to the diftance between that lucid point and the Lens very nearly.

Now to examine whether the difference be- tween the Refradions which the moft refrangi- ble and the leaft refrangible Rays flowing from the. fame point fuffer in the Objeft-glalFes of Telefcopes and fuch like GlalTes, be fo great as is here described, I contrived the following Ex- periment.

Exfer. 16. The Lens which I ufed in the fe- cond and eighth Experiments, being placed fix Feet and an Inch diftant from any Objedt, col- lefted the Species of that Objed by the mean refrangible Rays at the diftance of fix Feet and an Inch from the Lens onthe other fide. And therefore by the foregoing Rule it ought to col- led the Species of that Objeft by the leaft re- frangible Rays at the diftance of fix Feet and 37 Inches from the Lens, and by the moft re- frangible

[ 75]

frangible ones at the diftance of five Feet and 104 Inches from it: So that between the two places where thefe lead and mdft refrangible Rays colleft the Species, there may be the di-- fiance of about $j Inches. For by that Rule, as fix Feet and an Inch (the diftance of the Lens from the lucid Objeft) is to twelve Feet and two Inches (the diftance of the lucid Objeft from the Focus of the mean refrangible Rays ) that is, as one is to two, fo is the zyith part 'of fix Feet and an Inch (the diftance between the Lens and the fame Focus) to the diftance between the Focus of the moft refrangible Rays and the Focus of the leaft refrangible ones, which is therefore 5-4.4 Inches, that is very near- ly 57 Inches. Now to know whether this mea- fure was true, I repeated the fecond and eighth Experiment with coloured Light, which was lefs compounded than that I there made ufe of : For I now feparated the heterogeneous Rays from one another by the method I de- fcribed in the eleventh Experiment , fo as to make a coloured Spedrum about twelve or fif- teen times longer than broad. This Speftrum I caft on a printed Book, and placing the above- mentioned Lens at the diftance of fix Feet and an Inch from this Speftrum to colled the Spe- cies of the illuminated Letters at the fame di- ftance on the other fide , I found that the Spe- cies of the Letters illuminated with blue were nearer to the Lens than thofe illuminated with deep red by about three Inches or three and a quarter : but the Species of the Letters illumi- nated with indigo and violet appeared fo con-

fufed

c in

fufed and indiflinft, that I could not read them : Whereupon viewing the Prifm, I found it was full)f Veins Tunning from one end of the Glafs to the other ; fo that the Refradion could not be regular. I took another Prifm therefore which was free from Veins, and inltead of the Letters I ufed two or three parallel black Lines a little broader than the flroaks of the Let- ters, and calling the Colours upon thefe Lines in fuch manner that the Lines ran along the. Colours from one end of the Spedriim to the other, I found that the Focus where the indigo, or confine of this Colour and violet cait the Species of the black Lines moil diftindly , to be about four Inches or 4^ nearer to the Lens than the Focus where the deepeft red call the Species of the fame black Lines moil di{lin6l- ]y. The violet was fo faint and dark, that I could not difcern the Species of the Lines di- Itindly by that Colour ; and therefore confi- dering that the Prifm was made of a dark co- loured Glafs inclining to green, I took another Prifm of clear white Glafs ; but the Spedrum of -Colours which this Prifm made had long white ftreams of faint Light lliooting out from both ends of the Colours, which made me con- clude that fomething was amifs ; and viewing the Prifm, I found two or three httle bubbles in the Glafs which refrafted the Light irregu- larly. Wherefore I covered that part of the Glafs with black Paper, and letting the Light' pafs through another part of it which was free from fuch bubbles, the Speftrum of Colours became free from thofe irregular Streanps of Light, and

was

[77]

Was now fuch as I defired. But ftill I found the violet fo dark and faint, that I could fcarce fee the Species of the Lines by the violet, and not at all by the deepeil part of it, which was next the end of the Speftrum. I fufpeded therefore that this faint and dark Colour might be allayed by that fcattering Light which was refraded , and reflefted irregularly, partly by fome very fmall bubbles in the Glalfes, and partly by the inequalities of their Polifli : which Light, tho' it was but little, yet it being of a white Colour, might fuffice to affed: the Senfe fo llrongly as to diiturb the Phaenomena of that weak and dark Colour the violet, and there- fore I tried, as in the 12th, 13th -and 14th Ex- periments, whether the Light of this Colour did not confid of a fenfible mixture of heteroge- neous Rays, but found it did not. Nor did the Refraftions caufe any other fenfible Colour than violet to emerge out of this Light , as they would have done out of white Light, and by confequence out of this violet Light had it been fenfibly compounded with white Light. And therefore I concluded , that the realbn why I could not fee the Species of the Lines diftinft- ly by this Colour, was only the darknefs of this Colour and thinnefs of its Light, and its di- Itance from the Axis of the Lens ; I divided therefore thofe parallel black Lines into equal parts, by which I might readily know the di- llances of the Colours in the Speftrum from one another, and noted the diltances of the Lens from the Foci of fuch Colours as caft the Species of the Lines diltindly, and then confi-

fidered

[ 78 ]

dered whether the difference of thofe diftances bear fuch proportion to si Inches, the greateft difference of the diftances which the Foci of the deepeit red and violet ought to have from the Lens, as the diftance of the obferved Go- lours from one another in the Spedrum bear to the greatefl diftance of the deepefl red and violet meafured in the reftilinear fides of the Spedrum, that is, to the length of thofe Sides or Excefs of the length of the Spedrum above its breadth. And my Obfervations were as fol- lows.

When I obferved and compared the deepeft fenfible red, and the Colour in the Confine of green and blue, which at the redilinear Sides of the Spedrum was diftant from it half the length of thofe Sides, the Focus where the Con- fine of green and blue caft the Species of the Lines diltindly on the Paper, was nearer to the Lens than the Focus where the red cafl thofe Lines ditiindly on it by about x^ or i4 Inches. For fometimes the Meafures were a little great- er," fometimes a little lefs, but feldom varied from one another above i of an Inch. For it was very difficult to define the places of the Foci, without fome little Errors. Now if the Colours diftant half the length of the Image, (meafured at its reftilinear Sides) give Zt or 2-| difference of the diftances of their Foci from the Lens, then the Colours diftant the whole length ought to give s or 5^ Inches difference of thofe diftances.

But here it's to be noted, that I could not fee the red to the full end of the Spedrum ,

but

[79]

but only to the center of the Semicircle which bounded that end , or a Uttle farther ; and therefore I compared this red not with that Colour which was exadly in the middle of the Speftrum, or Confine of green and blue, but with that which verged a Uttle more to the blue than to the' green: And as I reckoned the whole length of the Colours not to be the whole length of the Spedlrum, but the length of its rectilinear Sides, fo completing the femicircu- lar Ends into Circles, when either of the ob- ferved Colours fell within thofe Circles, I mea- fured the diftance of that Colour from the fe- micircular end of the Spectrum , and fubduft- ing half this diftance from the meafured di- ftance of the two Colours, I took the remain- der for their corre(5led diftance ; and in thefe Obfervations fet down this corrected diftance for the difference of the diftances of their Foci from the Lens. For as the length of the refti- linear Sides of the Spedrum would be the whole length of all the Colours, were the Circles of which (as we ihewed) that Spedrum confifts contraded and reduced to phylical Points, fo in that cafe this corrected diltance would be the real diftance of the two obferved Colours.

When therefore I farther obferved the deep- eft fenfible red, and that blue whofe correded diftance from it was -t parts of the length of the reftilinear Sides of the Spedrum, the dif- ference of the diftances of their Foci from the Lens was about 3-^ Inches, and as 7 to iz fo is 3 -T to 55.

When 3

[80 ]

When I obferved the deepeft fenfible red, and that indigo whofe correded diflance was TT or 4 of the length of the redilinear Sides of the Spedrum, the difference of the diltances of their Foci from the Lens, was about 34 Inches, and as ^ to 3 fo is 34 to 5-4.

When I obferved the deepefl fenfible red, and that deep indigo whofe correded diflance from one another was -% or -| of the length of the redilinear Sides of the Spedrum, the dif- ference of the diilances of their Foci from the Lens was about 4 Inches ; and as 3 to 4 fo is 4 to 5-4..

When I obferved the deepeft fenfible red, and that part of the violet next the indigo, whofe corredied diftan'ce from the red was 44 or 4 of the length of the reftilinear Sides of the Speftrum., the difference of the diftances of their Foci from the Lens was about 44 Inches, and as 5 to 6, fo is 44 to 5-4. For fometimes when the Lens was advantagioufly placed, fo that its Axis refpefted the blue, and all things elfe were well ordered, and the Sun fhone clear, and I held my Eye -very near to the Paper on which the Lens caft the Species of the Lines, I could fee pretty diftinflly the Species of thofe Lines by that part of the violet which was next the indigo ; and fometimes I could fee them by above half the violet. For in making thefe Experiments I had obferved , that the Species of thofe Colours only appear diftindt which were in or near the Axis of the Lens : So that if the blue or indigo were in the Axis, I could fee their Species diftindiy ; and then the red ap- peared

r 8i ]

peared much lefs dillinft than before. Where- fore I contrived to make the Spe(^trum of Co- lom's fhorter than before, fo that both its ends might be nearer to the Axis of the Lens. And now its length was about 24 Inches and breadth about 4 or 4 of an hich. Alfo inftead of the black Lines on which the SpeArum was caft, I made one black Line broader than thofe, that I might fee its Species more eafily ; and this Line I divided by Ihort crofs Lines into equal parts, for meafuring the diftances of the obfer- ved Colours. And now I could fometimes fee the Species of this Line with its divifions al- moft as far as the center of the femicircular violet end of the Spedrum, and made thefe farther Obfervations.

When I obferved the deepefl fenfible red, and that part of the violet whofe correfted diftance from it was about -I parts of the redilinear fides of the Spedrum the difference of the diftances of the Foci of thofe Colours from the Lens, was one time 44, another time 4-I, another time 4-r Inches, and as 8 to 9, fo are 47, 4-^-, 44, to 54, 544 5I4 refpedively.

When I obferved the deepeft fenfible red, land deepeft fenfible violet, (the correfted di- iftance of which Colours when all things were ■ordered to the beft advantage, and the Sun jflione very clear, was about 44 or 44 parts of Sthe length of the redilinear Sides of the co- iloured Speftrum ) I found the difference of the jdiftances of their Foci from the Lens fometimes 1 44 fometimes 54, and for the moft part 5- In-

• G ches

[ 82 ]

ches or thereabouts: and as ii to ix or 15- to 16, fo is f Inches to 54 or Si Inches.

And by this progreilion of Experiments I fa- tisiied my felf, that had the Light at the very ends of the Speflrum been lirong enough to make the Species of the black Lines appear plainly on the Paper, the Focus of the deepeft violet would have been found nearer to the Lens, than the Focus of the d^epefl red, by about si Inches at lead. And this is a farther evidence, that the Sines of Incidence and Re- fradion of the feveral forts of Rays, hold the fame proportion to one another in the fmalleft Refradions which they do in the greateft.

My progrefs in making this nice and trouble- fome Experiment I have fet down more at large, that they that Ihall try it after me may be aware of the circumfpedion requifite to make it fuc- ceed well. And if they cannot make it fuc- ceed fo well as I did, they may notwithftand- ing collect by the proportion of the diftance of the Colours of the Spedrum, to the difference pf the diflances of their Foci from the Lens, what would be the fuccefs in the more diitant Colours by a better trial. And yet if they ufe a broader Lens than I did, and tix it to a long Itraight Staff by means of which it may be rea- dily and truly dire61ed to the Colour whofe Fo- v cus is defired, I quellion not but the Experi- ment will fucceed better with them than it did i with me. For I direded the Axis as nearly as I could to the middle of the Colours, and then the faint ends of the Spedrum being remote from the Axis, call their Species lefs diflindly

• on

[83 ]

on the Paper than they would have done had the Axis been fucceflively dire6led to them.

Now by what has been faid, it's certain that the Rays which differ in RefrangibiUty do not converge to the fame Focus, but if they flow from a lucid point, as far from the Lens on one fide as their Foci are on the other, the Focus of the moft refrangible Rays iliall be nearer to the Lens than that of the leaft refrangible, by above the fourteenth part of the whole diltance : and if they flow from a lucid point, fo very re- mote from the Lens that before their Incidence they may be accounted parallel, the Focus of the moft refrangible Rays iliall be nearer to the Lens than the Focus of the lead refrangible, by about the x7th or i8th part of their whole diftance from it. And the diameter of the Cir- cle in the middle fpace between thofe two Fo- ci which they illuminate when they fall there on any Plane, perpendicular to the Axis (which Circle is the leaft into which they can all be ga- thered) is about the 5-5th part of the diameter of the Aperture of the Glafs. So that 'tis a won- der that Telefcopes reprefent Objefts fo diftind as they do. But were all the Rays of Light equally refrangible, the Error arifing only from the fphericalnefs of the Figures of Glaffes would be many hundred times lefs. For if the Obje^l- glafs of a Telefcope be Plano-convex, and the Plane fide be turned towards the Objeft, and the diameter- of the Sphere whereof this Glafs is a fegment be called D, and the femidiame- ter of the Aperture of the Glafs be called S, and the Sine of Incidence out of Glafs into Air,

G z be

[ 84 ]

be to the Sine of Refradion as I to R : the Rays which come parallel to the Axis of the Glafs, fliall in the place where the Image of the Objed is mofl diilinftly made, be fcattered all over a

little Circle whofe diameter is -j^ x 5-^ very

nearly, as I gather by computing the Errors of the Rays by the method of infinite Series, and ^ rejedting the Terms whofe Quantities are in- confiderable. As for inftance, if the Sine of Incidence I, be to the Sine of Refradion R, as 20 to 31, and if D the diameter of the Sphere to which the convex fide of the Glafs is ground, be 100 Feet or 1200 Inches, and S the femidia- meter of the Aperture be two Inches, the dia- meter of the little Circle (that is ^"^ \ "* ' will

be ^' * ^^ ''^' — Cor ' "^^^ parts of an Inch.

iO X lo X I ICO X 12,00 ^ 7ZOOOOUO *■

But the diameter of the little Circle through which thefe Rays are fcattered by unequal Re- frangibility, will be about the 5- 5th part of the Aperture of the Objed glafs which here is four Inches. And therefore the Error arifmg from the fpherical Figure of the Glafs, is to the Er- ror arifmg from the different Refrangibility of

the Rays, as to — that is as i to 5449 :

•' 7ioooooo 55: y I ly

and therefore being in comparifon fo very little, deferves not to be confidered.

But you will fay, if the Errors caufed by the

different Refrangibility be fo very great, how

comes it to pafs that Obje61s appear through

Telefcopes fo diilindt as they do ? I anfwer, 'tis

i becaufe

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becaufe the erring Rays are not fcattered uni- formly over all that circular fpace, but colled- ed infinitely more denfely in the center than in any other part of the Circle, and in the way from .the center to the circumference, grow continually rarer and rarer, fo as at the circum- ference to become infinitely rare ; and by rea- fon of their rarity are not ftrong enough to be vifible, unlefs in the center and very near it. Let A D E [in Fig. 27.] reprelent one of thofe Circles defcribed with the Center C and Semi- diameter A C, and let B F G be a fmaller Circle ^ concentrick to the former, cutting with its cir- ' ^ cumference the Diameter A C in B, and bifed ^ AC in N, and by my reckoning the Denfity of the Light in any place B will be to its Denfity in N, as AB to BC ; and the whole Light with- in the lefTer Circle BFG, will be to the whole Light within the greater A ED, as the Excefs of the Square of AC above the Square of AB, is to the Square of A C. As if BC be the fifth part of AC, the Light will be four times den- fer in B than in N, and the whole Light within the lefs Circle, will be to the whole Light with- in the greater, as nine to twenty five. Whence it's evident that the Light within the lefs Cir- cle, mull: ftrike the Senfe much more flrongly, than that faint and dilated Light round about between it and the circumference of the grea- ter.

But it's farther to be noted, that the mod lu- minous of the prifmatick Colours are the yel- low and orange. Thefe affedl the Senfes more ftrongly than all the reft together, and next to

G 3 thefe

I 8^ ]

thefe in ftrength are the red and green. The blue compared with thefe is a faint and dark Colour, and the indigo and violet are much darker and fainter, fo that thefe compared with the Itronger Colours are little to be regarded. The Images of Obje61s are therefore to be pla- ced, not in the Focus of the mean refrangible Rays which are in the confine of green and blue, but in the Focus of thofe Ra57s which are in the middle of the orange and yellow ; there where the Colour is moll luminous and fulgent, that is in the brighteit yellow, that yellow which inclines more to orange than to green. And by the Refradion of thefe Rays (whofe Sines of Incidence and Refradion in Glafs are as 17 and 11) the Refradion of Glafs and Cryftal for op- tical Ufes is to be meafured. Let us therefore place the Image of the Object in the Focus of thefe Rays, and all the yellow and orange will fall within a Circle, whofe diameter is about the 150th part of the diameter of the Aperture of the Glafs. And if you add the brighter half of the red, (that half which is next the orange) and the brighter half of the green, (that half which is next the yellow) about three fifth parts of the Light of thefe two Colours will fall within the fame Circle, and' two fifth parts will fall without it round about ; and that which falls without will be fpread through almoft as much more fpace as that which falls within, and fo in the grofs be almofl three times rarer. Of the other half of the red and green, (that is of the deep dark red and willow green) about one quarter will fall within this Circle, and

three

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