Opticks (1721, 3rd Edition) — part 7 of 18

hefs, or from white and black, that is, a grey, or dun, or rufTet brown, fuch as are the Co- lours of a Man's Nail, of a Moufej of Allies^ of ordinary Stones, of Mortar, of Dull and Dirt in High- ways, and the like. And fuch a dark white I have often produced by mixing colour'd Powders. For thus one part of red Lead, and five parts of V'tride JEar'tL, compo- fed a dun Colour like that of a Moufe. For thefe two Colours were feverally fo compound- ed of others, that in both together were mix*- ture of all Colours ; and there was lefs red Lead ufed than Viride Mris^ becaufe of the fulnefs of its Colour. Again, one part of red Lead, and four parts of blue Bife, compofed a dun Colour verging a little to purple, and by ad- ding to this a certain mixture of Orpiment and Vtr'tde Mrts in a due proportion , the mixture loft its purple tindure, and became perfeAly dun. But the Experiment fucceeded beft with- out Minium thus. To Orpiment I added by little and little a certain full bright purple > which Painters ufe until the Orpiment ceafed to be yellow, and became of a pale red. Then I diluted that red by adding a little Viride M- rif, and a little more blue Bife than Viride M- ris, until it became of fuch a grey or pale white, as verged to no one of the Colours more than to another. For thus it became of a Colour e- ' qual in whitenefs to that of Afhes or of Wood ' newly cut, or of a Man's Skin. The Orpiment ! refleded more Light than did any other of the Powders, and therefore conduced more to ihs whitenefs of the compounded Colour than they.

K * Tq

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To aflign the Proportions accurately may be difficult, by reafon of the different goodnefs of Powders of the fame kind. Accordingly as the Colour of any Powder is more or lefs full and luminous, it ought to be ufed in a lefs or greater proportion.

Now confidering that thefe grey and dun Co- lours may i)e alfo produced by mixing whites and blacks, and by confequence differ from perfe6l whites not in fpecies of Colours but on- ly in degree of Luminoufnefs , it is manifeft that there is nothing more requilite to make them perfedly white than to increafe their Light fufficiently; and, on the contrary, if by increa- fmg their Light they can be brought to perfed whitenefs, it will thence alfo follow, that they are of the fame fpecies of Colour with the belt whites, and differ from them only in the quan- tity of Light. And this I tried as follows. I took the third of the abovemention'd grey Mixtures (that which was compounded of Orpiment, Pur- ple, Bife, and FiridejEris) and rubbed it thick- ly upon the Floor of my Chamber, where the Sun flione upon it through the opened Cafe- ment ; and by it, in the fhadow, I laid a piece of white Paper of the fame bignefs. Then going from them to the dillance of ix or i8 Feet, fo that I could not difcern the uneavennefs of the Surface of the Powder, nor the little ihadows let fall from the gritty Particles thereof; the Powder appeared intenfely white, fo as to tran- fcend even the Paper it felf in whitenefs, efpe- cially if the Paper were a little iliaded from the Light of the Clouds, and then the Paper com- pared

[ 133 ]

pared with the Powder appeared of fuch a grey Colour as the Powder had done before. But by laying the Paper where the Sun (hines thro' the Glafs of the Window, or by fliutting the Window that the Sun might fhine through the Glafs upon the Powder, and by fuch oihcr fit means of increafing or decreafing the Lights wherewith the Powder and Paper were illumi- nated, the Light wherewith the Powder is illu- minated may be made ftronger in fuch a due proportion than the Light wherewith the Paper is illuminated, that they Ihall both appear ex- aftly alike in whitenefs. For when I was try- ing this, a Friend coming to vifit me, I llopp'd him at the Door, and before I told him what the Colours were, or what I was doing ; I ask- ed him, Which of the two Whites were the bell, and wherein they differed ? And after he had at that diftance viewed them well, he an- fwer'd, That they were both good Whites, and that he could not fay which was bell, nor wherein their Colours differed. Now if you confider, that this white of the Powder in the Sun- fhine was compounded of the Colours which the component Powders (Orpiment, Purple, Bife, and Viride Mr'ts) have in the fame San- Ihine, you mull acknowledge by this Experi- ment, as well as by the former, that perfed' whitenefs may be compounded of Colours.

From what has been faid it is alfo evident, that the whitenefs of the Sun's Light is com- pounded of all the Colours wherewith the fe- veral forts of Rays whereof that Light confifts, when by their feveral Refrangibilities they are

K 5 fepa'

[ 134- ]

feparated from one another, do tinge Paper or any other white Body whereon they fall. For thofe Colours by Trop./L. are unchangeable, and whenever all thofe Rays with thofe their Colours are mix'd again, they reproduce the fame white Light as before.

TROT. VI. Prob. IL

In a mixture of primary Colours, the quantity and quality of each being given^ to know the Colour of the Com_pound.

WI T H the Center O [in Fig. 1 1 .] and Ra- dius O D defcribe a Circle ADF, and dillinguifli its circumference into feven parts DE, EF, FG, GA, AB, BC, CD, propor- tional ,ta the feven mufical Tones or Intervals of the eight Sounds, Sol., la, fa, fol, la, mi, fa^ fol, contained in an eight, that is, proportiorKal to the number 4, tV, tV, 4, tV, tV, -i-. Let the firfl part D E reprefent a red Colour, the fe- cond E F orange, the third F G yellow, the fourth C A green, the fifth A B blue, the fixth BC indigo, and the feventh CD violet. And conceive that thefe are all the Colours of un- compounded Light gradually palling into one another, as they do when made byPrifms; the circumference DEFGABCD, reprefenting the whole feries of Colours from one end of the Sun's colour'd Image to the other, fo that from D to E be all degrees of red, at E t^e mean Colour between red and orange, from E

toF

to F all degrees of orange, at F the mean be- tween orange and yellow, from F to G all de- grees of yellow, and fo on. Let / be the cen- ter of gravity of the Arch D E, and q^ r, j-, t^ u, AT, the centers of gravity of the Arches E F, FG, GA, AB, BC and C D refpedively, and about thofe centers of gravity let Circles pro- portional to the number of Rays of each Co- lour in the given Mixture be defcrib'd ; that is, the Circle p proportional to the number of the red -making Rays in the Mixture, the Circle q proportional to the number of the orange-ma- Idng Rays in the Mixture, and fo of the reit. Find the common center of gravity of all thofe Circles />, ^, r, j, t, Uy x. Let that center be Z, and from the center of the Circle ADF, through Z to the circumference, drawing the right Line O Y, the place of the Point Y in the circumference lliall fliew the Colour ariling from the compofition of all the Colours in the given Mixture, and the Line OZ ihall be pro- portional to the fulnefs or intenfenefs of the Colour, that is, to its diitance from whitenefs. As if Y fall in the middle between F and G, the compounded Colour lliall be the bed yel- low ; if Y verge from the middle towards F or G, the compound Colour lliall accordingly be a yellow% verging towards orange or green. If Z fall upon the circumference the Colour fliall be intenfe and florid in the higheft degree ; if it fall in the mid way between the circum- ference and center, it Ihall be but half fo intenfe, that is, it Ihall be fuch a Colour as would be made by diluting the mtenfell yellow

K 4 with

C 13^ ]

with an equal quantity of whitenefs ; and if it fall uppn the center O, the Colour fhall have loft all its intenfenefs, and become a white. But it is to be noted, That if the point Z fall in or near the line O P, the main ingredients being the red and violet, the Colour compounded fliall not be any of the prifmatick Colours, but a purple, inclining to red or violet, according- ly as the point Z lieth on the fide of the Una t> O towards E or towards C , and in general the compounded violet is more bright and more fiery than the uncompounded. Alfo if only two of the primary Colours which in the circle are oppofite to one another be mixed in an equal proportion, the point Z lliall fall upon the cen- ter O, and yet the Colour compounded of thofe two iliall not be perfectly white, but fome faint anonymous Colour. For I could never yet by mixing only two primary Colours pro- duce a perfed white. Whether it may be com- pounded of a mixture of three taken at equal diftances in the circumference I do not know, but of four or five i do not much queftion but it may. But thefe are Curiofities ofUttle or no moment to the underftanding the Phaenomena of Nature. For in all whites produced by Na- ture, there ufes to be a mixture of all forts of Rays, and by confequence a compoiition of all Colours.

To give an inflance of this Rule ; fuppofe a Colour is compounded of thefe homogeneal Colours, of violet one part, of indigo one part, of blue two parts, of green three parts, of yel- low five pans, of orange fix parts, and of red

teu

C 137 ]

ten parts. Proportional to thefe parrs defcribe the Circles a;, v^ ^ j, r, q, /, refpecftively, that is, fp that if the Circle x be one, the Circle 'u may be one, the Circle t two, the Circle s three, and the Circles r, q and /, five, fix and ten. Then I find Z the common center of gravity of thefe Circles, and through Z drawing the Line

• O Y, the Point Y falls upon the circumference between E and F, fome thing nearer to E than to F, and thence I conclude, that the Colon? compounded of thefe Ingredients will be an o- range, verging a Uttle more to red than to yel-

^ low. Alfo I find that O Z is a little lefs than one half of O Y, and thence I conclude, that this orange hath a little lefs than half the ful- nefs or intenfenefs of an uncompounded o- range ; that is to fay, that it is fuch an orange as may be made by mixing an homogeneal o- range with a good white in the proportion of the Line O Z- to the Line Z Y, this Proportion being not of the quantities of mixed orange and white Powders , but of the quantities of the Lights reflefted from them.

This Rule I conceive accurate enough for

■ praftice , though not mathematically accurate ; and the truth of it may be fufficiently proved to Senfe, by flopping any of the Colours at the Lens in the tenth Experiment of this Book. For the reft of the Colours which are not ftopp'd, but pafs on to the Focus of the Lens, will there compound either accurately or very nearly fuch a Colour as by this Rule ought to refult from dieir mixture.

, TROT.

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T R OT.\ni. Theor. V.

^11 the Colours in the 1)niverfe which are made by Lights and depend not on the ^ower of /- magination, are either the Colours of homoge- neal Lights, or compounded of thefe, and that either accurately or very nearly^ according to the Rule of the foregoing Problem.

XT' OR it has been proved (in Trop. i. Tart.z.) P that the changes of Colours made by Re- fraftions do not arife from any new Modifica- tions of the Rays imprefs'd by thofe Refradions, and by the various Terminations of Light and Shadow, as has been the condant and general Opinion of Philofophers. It has alfo been pro- ved that the feveral Colours of the homogeneal Rays do conftantly anfwer to their degrees of Refrangibility, (Trop. i. Tart i. and ^rop. z, IPart Z.J and that their degrees of Refrangibi- lity cannot be changed by Refradions and Re- flexions, (Trop.x. Tarti.J and by confequence that thofe their Colours are likewife immuta- ble. It has alfo been proved direftly by refra- fting and refleding homogeneal Lights apart, that their Colours cannot be changed, (Trop. 2. Tart Z.J It has been proved alfo, that when the feveral forts of Rays are mixed, and in crof- fmg pafs through the fame fpace, they do not ad: on one another fo as to change each others colorific qualities. (Exper. 10. Tart zj but by mixing their Adions in the. Senforium beget a Senfation differing from what either would do apart, that is a Senfation of a mean Colour be- tween

[ ^39 I

tween their proper Colours ; and particularly when by the concourfe and mixtures of all forts of Rays, a white Colour is produced, the white is a mixture of all the Colours which the P.ays would have apart, (Trop. f. Tarn. J The Rays in that mixture do not lofe or alter their feveral colorific qualities, but by all their various kinds of Anions mix'd in the Senfori- urn, beget a Senfation of a middling Colour between all their Colours, which is whitenefs. For whitenefs is a mean between all Colours, having it felf indifferently to them all, fo as with equal facility to be tinged with any of them. A red Powder mixed with a little blue, or a blue with a little red, doth not prefently lofe its Colour, but a white Powder mix'd with any Colour is prefently tinged with that Colour , and is equally capable of being tinged with any Colour whatever. It has been fliewed alfo, that as the Sun*s Light is mix'd of all forts of Rays, {b its whitenefs is a mixture of the Colours of all forts of Rays ; thofe Rays having from the beginning their feveral colorific qua- lities as well as their feveral Refrangibilities, and retaining them perpetually unchanged not- withflanding any Refraftions or Reflexions they may at any time fuffer, and that whenever any fort of the Sun's Rays is by any means (as by Reflexion in Exper. 9. and 10. Tart 1. or by Refraftion as happens in all Refraftions) fepa- rated from the relf, they then manifelt their proper Colours. Thefe things have been prov'd, and the fum of all this amounts to the Propofi- tion here to be proved.. For if the Sun's Light

ifr^

[ HO 3

is mix'd of feveral forts of Rays, each of which have originally their feveral Refrangibilities and colorific Qualities, and notwithftanding their Refra(^tions and Reflexions , and their various Separations or Mixtures, keep thofe their ori- ginal Properties perpetually the fame without alteration; then all the Colours in the World muft be fuch as conflantly ought to arife from the original colorific qualities of the Rays where- of the Lights confift by which thofe Colours are feen. And therefore if the reafon of any Colour whatever be required, we have nothing elfe to do than to confider how the Rays in the Sun's Light have by Reflexions or Refradions, or other caufes been parted from one another, or mixed together ; or otherwife to find out what forts of Rnys are in the Light by which that Colour is made, and in what proportion ; and then by the lalt Problem to learn the Co- lour which ought to arife by mixing thofe Rays (or their Colours) in that proportion. I fpeak here of Colours fo far as they arife from Light. For they appear fometimes by other Caufes, as when by the power of Phantafy we fee Colours in a dream, or a mad IVlan fees things before him which arediot there; or when we fee Fire by llriking the Eye, or fee Colours like the Eye of a Peacock's Feather, by preffing our Eyes in either corner whild w^e look the other w^ay. Where thefe and fuch like Caufes interpofe not, the Colour always anfwers to the fort or forts of the Rays whereof the Light confifls, as I have conilantly found in whatever Phsenomena of Colours I have hitherto been able to exa- mine.

r HI 1

^' mine. I fliall in the following Propofitions give inftances of this in the Phaenomena of chiefell note.

TROT.Vm. Prob.III.

By the d'tfcovered Troferttes of Light to ex* flatn the Colours made by ^rijms,

LET A B C [in Fig. 12.] reprefent a Prifm refrafting the Light of the Sun, which comes into a dark Chamber through a hole F (p almolt as broad as the Prifm, and let M N re- prefent a white Paper on which the refrafted Light is cafl, and fuppofe the moll refrangible or deepeft violet -making Rays fall upon the Space P TT, the lead refrangible or deeped red- making Rays upon the fpace T7j the middle fort between the indigo making and blue- ma- king Rays upon the Space Q x^ the middle fort of the green-making Rays upon the fpace R ^, the middle fort between the yellow-making and orange- making Ra}js upon the fpace S o-, and o- ther intermediate forts upon intermediate Spa- ces. For fo the Spaces upon wnich the feveral forts adequately fall will by reafon of the dif- ferent Refrangibility of thofe forts be one lower than another. Now if the Paper MN be fo near the Prifm that the Spaces P T and cr7 do not interfere with one another, the diftance be- tween them T tt will be illuminated by all the forts of Rays in that proportion to one another which they have at their very firit coming out

of

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of the Prifm, and confequently be white. But the Spaces PT and ^7 on either hand, will not be illuminated by them all, and therefore will appear coloured. And particularly at P, where the outmoft violet-making Rays fall alone, the Colour mufl be the deepelt violet. At Q where the violet-making and indigo-making Rays are mixed, it mull be a violet incHning much to indigo. At R where the violet-making, indi- go-making, blue-making, and one half of the green- making Rays are mixed, their Colours mult (by the conttru^lion of the fecond Pro- blem ) compound a middle Colour between in- digo and blue. At S where all the Rays are mixed except the red-making and orange-mak- ing, their Coloitrs ought by the fame Rule to compound a faint blue, verging more to green than indigo. And in the progrefs from S to T, this blue will grow more and more faint and dilute, till at T, where all the Colours begin to be mixed, it ends in whitenefs.

So again, on the other fide of the white at r, where the lead refrangible or utmoll red-mak- ing Rays are alone, the Colour mult be the deepelt red. At <r the mixture of red and o- range will compound a red inclining to orange. At ^ the mixture of red, orange, yellow, and one half of the green muit compound a middle Colour between orange and yellow. At x the mixture of all Colours but violet and indigo will compound a faint yellow, verging more to green than to orange. And this yellow will grow more faint and dilute continually in its progrefs

from

C 143 3

from X to TT, where by a mixture of all forts of Rays it will become white.

Thefe Colours ought to appear were the Sun's Light perfeftly white : But becaufe it inclines to yellow, the Excefs of the yellow-making Rays whereby 'tis tinged with that Colour, be- ing mixed with the faint blue between S and T, will draw it to a faint green. And fo the Co» lours in order from P to t ought to be violet, indigo, blue, very faint green, white, faint yel- low, orange, red. Thus it is by the computa- tion : And they that pleafe to view the Colours made by a Prifm will find it fo in Nature.

Thefe are the Colours on both fides the white when the Paper is held between the Prifm, and the Point X where the Colours meet, and the interjacent white vanilhes. For if the Paper be held itill farther off from the Prifm, the moft refrangible and lealt refrangible Rays will be wanting in the middle of the Light, and the rell of the Rays which are found there, will by mixture produce a fuller green than before. Al- fo the yellow and blue will now become lefs compounded, and by confequence more intenfe than before. And this alfo agrees with expe- rience.

And if one look through a Prifm upon a white Objeft encompalTed with blacknefs or darknefs, the reafon of the Colours arifing on the edges is much the fame, as will appear to one that Ihall a little confider it. If a black Ob- jeft be encompalTed with a white one, the Co- lours which appear through the Prifm are to be derived from the Light of the white one, fpread-

4

C H4 ]

ing into the Regions of the black, and there- fore they appear in a contrary order to that; wheii a white Objeft is furrounded with black. And the fame is to be underftood when an Ob- jed: is viewed, whofe parts are fome of them lefs luminous than others. For in the borders of the more and lefs luminous parts, Colours ought always by the fame Principles to arife from the Excefs of the Light of the more lu- minous, and to be of the fame kind as if the darker parts were black, but yet to be more faint and dilute.

What is faid of Colours made by Prifnis may be eafily appHed to Colours made by the GlaiTes of Telefcopes or Microfcopes, or by the Hu- mours of the Eye, For if the Objed-glafs of a Telefcope be thicker on one fide than on the other, or if one half of the Glafs, or one half of the Pupil of the Eye be covered with any opake fubllance : the Objeft-glafs, or that part of it or of the Eye which is not cover'd, may be confider'd as a Wedge with crooked Sides^ and every Wedge of Glafs or other pellucid Subftance has the effeft of a Prifm in refrafting the Light which palTes through it.

How the Colours in the ninth and tenth Ex- periments of the firfl: Part arife from the diffe- rent Reflexibility of Light, is evident by what w^as there faid. But it is obfervable in the ninth Experiment, that whilfl the Sun's dired Light is yellow, the Excefs of the blue making Rays in the refleded beam of Light MN, fuffices only to bring that yellow to a pale white incli- ning to blue, and not to tinge it with a mani-

feftly

r H5 ij

feftly blue Colour. To obtain therefore a bet- ter blue , I ufed inllead of the yellow Light of the Sun the white Light of the Clouds, by vary- ing a little the Experiment, as follows.

Exper. i6. . Let HFG \x\ Fig. 13.] repre- fent a Prifm in the open Air, and S the Eye of the Spectator, viewing the Clouds by their Light coming into the Prifm at the' plane iide FIGK, and refleded in it by its bafe HEIG, and thence going out through its plane fide HEFK to the Eye. And when the Prifm and Eye are conveniently placed, fo that the Angles of Incidence and Reflexion at the Bafe may be about 40 Degrees, the Spedator will fee a Bow MN of a blue Colour, running from one end of the Bafe to the other, with the Concave fide towards him, and the part of the Bafe IMNG beyond this Bow will be brighter than the other part EMNH on the other fide of it. This blue Co- lour MN being made by nothing elfe than by re- flexion of a fpecular Superficies, feems fo odd a Phaenomenon, and fo difficult to be explain- ed by the vulgar Hypothefis of Philofophers, that I could not but think it deferved to be ta- ken notice of. Now for underflanding the rea- fon of it, fuppofe the Plane A B C to cut the plane Sides and Bafe of the Prifm perpendicu- larly. From the Eye to the Line B C, where- in that Plane cuts the Bafe, draw the Lines S/ and S ^ in the Angles S/ c 5-0 degr. ^, and '^t c 49 degr. 4^, and the Point p will be the Hmic beyond which none of the mod refrangible Rays can pafs through the Bafe of the Prifm^ and be refraded, whofe Incidence is fuch that

L they

[ H^ ]

they may be reflefted to the Eye ; and the Point t will be the like limit for the lead re- frangible Rays, that is, beyond which none of them can pafs through the Bafe, whofe Inci- dence is fuch that by Reflexion they may come to the Eye. And the Point r taken in the mid- dle way between / and t^ will be the like limit for the meanly refrangible Rays. And there- fore all the leait refrangible Rays which fall up- on the Bale beyond ^, that is, between t and B, and can come from thence to the Eye will be rcfleded thither : But on this fide r, that is, between t and f, many of thefe Rays will be tranfmitted through the Bafe. And all the moft refrangible Rays which fall upon the Bafe be- yond />, that is, between / and B, and can by reflexion come from thence to the Eye, will be reflerted thither, but every where between p and r, mnny of thefe Rays will get through the Bafe and be refraded ; and the fame is to be underftood of the meanly refrangible Rays on either iide of the Point r. Whence it follows, that the Bafe of the Prifm muft every where between t and B, by a total reflexion of all forts of Rays to the Eye , look white and bright. And every where between / and C, by reafon of the tranfmiflion of many Rays of every fort, look more pale , obfcure and dark. But at r, and in other places between / and t^ where all the more refrangible Rays are refleded to the Eye, and many of the lei's refrangible are tranf- mitted, the Excefs of the molt; refrangible in the refleded Light will tinge that Light with their Colour , which is violet and blue. And

;his

t 147 ]

this happens by taking the Line Cp r t E anf where between the ends of the Prifm H G and E I.

TROT. IX. Prob. IV.

By the difcovered Troperties of Light to explain the Colours of the Rain-bow,

HIS Bow never appears but where it

rains in the Sun-fhine, and may be made

artiricially by fpouting up Water which may break aloft, and fcatter into drops, and fall down like Rain. For the Sun lliining upon thefe drops certainly caufes the Bow to appear to a Spectator Handing in a due pofition to the Rain and Sun. And hence it is now agreed upon, that this Bow is made by refraftion of the Sun's Light in drops of falling Rain. This was uri- derilood by fome of the Ancients, and of late more fully difcover'd and explain'd by the fa- mous Antonius de T)om'mis Archbiihop of Spa- lato, in his Book De Radiis Visits & Lucis, pub- liihed by his Friend Bartolus at Venice^ in the Year 161 1, and written above 20 Years before. For he teaches there how the interior Bow is made in round drops of Rain by two Rerra- (Sions of the Sun's Light, and one Reflexion between them, and the exterior by two Refra- ^ions and two forts of Reflexions between them in each drop of Water, and proves his Explications by Experiments made with a Phial full of Water, and with Globes of Glafs. filled

L % witb

[148]

with Water, and placed in the Sun to make the Colours of the two Bows appear in them. The fame Explication T>es - Cartes hath purfued in his Meteors, and mended that of the exterior Bow. But whilft they underftood not the true origin of Colour's, it's necefTary^to purfue it here a little farther. For underllanding there- fore how the Bow is made, let a drop of Rain or any other fpherical tranfparent Body be repre- fented by the Sphere BNFG, [in Ftg, 14] de- fcribed with the center C, and femi-diameter C N. And let A N be one of the Sun's Rays incident upon it at N, and thence refraded to F, where let it either go out of the Sphere by Refraftion towards V, or be refleAed to G; and at G let it either go out by Refradion to R, or be refledted to H ; and at H let it go out by Refradion towards S, cutting the incident Ray in Y; produce AN and RG, till they meet in X, and upon AX and NF let fall the perpen- diculars C D and CE, and produce CD till it fall upon the circumference at L. Parallel to the incident Ray A N draw the diameter B Q, and let the Sine of Incidence out of Air into Water be to the Sine of Refraftion as I to R. Now if you fuppofe the Point of Incidence N to move from the Point B, continually till it come to L, the Arch QF will firll increafe and then decreafe, and fo will the Angle A X R which the Rays A N and G R contain ; and the Arch Q F and Angle A X R will be biggelt when N D is to C N asVji-RR to -/ 3 RR, in which cafe N E will be to N D as i R to I. Al- fo the Angle AYS which the Rays AN and HS

contain

[ H9 ]

contain will firft decreafe, and then increafe and grow lead when N D is to C N as Vii- kk to 1/ 8 RR, in which cafe NE will be to N D as 3 R to I. And fo the Angle which the next emergent Ray (that is, the emergent Ray after three Reflexions) contains with the incident Ray AN will come to its limit when ND is to CN as v'li-KK to V 15- RR, in which caJe NE will be to ND as 4 R to I. And the Angle which the Ray next after that emergent, that is, the Ray emergent after four Reflexions, contains with the incident will come to its limit, when N D is to C N as v/u -kk to 1/ 14 RR, in which cafe N E will be to N D as 5- R to I ; and fo on infinitely, the numbers 3, 8, 15-, 14, ^c. being gather'd by continual addition of the terms of the arithmetical Progreflion 3» f* 7j 9j ^^- The truth of all this Mathematicians will eafily ex- amine.

Now it is to be obferved, that as when the Sun comes to his Tropicks, Days increafe and decreafe but a very little for a great while to- gether ; fo when by increafing the diitance C D, thefe Angels come to their limits, they vary their quantity but very httle for fome time to- gether, and therefore a far greater number of the Rays which fall upon all the Points N in the Quadrant B L, ihall emerge in the Hmits of thefe Angles, than in any other IncUnations. And farther it is to be obferved, that the Rays which differ in Refrangibility will have diffe- rent limits of their Angles of Emergence, and by confequence according to their different de- gree? of Refrangibility emerge mofi: copioufly

L 3 in

[ i5o]

in different Angles, and being feparated froni one another appear each in their proper Co- lours. And what thofe Angles are may be ea- fily gather'd from the foregoing Theorem by computation.

For in the leait refrangible Rays the Sines I and R (as was found above) are io8 and 8i, and thence by computation the greateit Angle A X R will be found 4x Degrees and i Minutes and the lead Angle AY S, 5-0 Degrees and 57 Minutes. And in the moll; refrangible Rays the Sines I and R are 109 and 81, and thence by computation the greatell Angle A X R will be found 40 Degrees and 17 Minutes, and the leait Angle AYS 54 Degrees and 7 Minutes.

Suppofe now that O [in Fig. 15-.] is the Spe- iftator's Eye, and OP a Line drawn parallel to the Sun's Rays, and let P O E, P O F, P O G, POH, be Angles of 40 Degr. 17. Min. 4x. Degr. i Min. 5-0 Degr. 57 Min. and 5-4 Degr. 7 Min. refpedively, and thefe Angels turned about their common Side O P, fliall with their other Sides OE, OF ; OG, OH, defcribe the Verges of two Rain.bows AF BE and CH DG. For if E, F, G, H, be drops placed any where in the conical Superficies defcribed by OE, OF, OG, OH, and be illuminated by the Sun's Rays SE, SF, SG, SH; the Angle SEO being e- qual to the Angle POE or 40 Deg. 17 Min. lliall be the greateit Angle in which the moit refrangible Rays can after one Reflexion be re- fracted to the -Eye, and therefore all the drops in the Line OE ihall fend the moit refrangible

Rays moit copioufiy to the Eye, and thereby \ ' ftrike

r 151 3

ftrike the Senfes with the deepefl violet Colour in that Region. And in like manner the Angle SFO being equal to the Angle POF, or 41 Degr. 2, Min. Ihall be the greatell in which the lealt refrangible Rays after one Reflexion can emerge out of the drops, and therefore thofe Rays fliall come molt copioufly to the Eye from the drops in the Line O F, and ftrike the Senfes with the deepeft red Colour in that Region. And by the fame Argument, the Rays which have intermediate degrees of RefrangibUty iliall come mod copioufly from drops between E and F, and Itrike the Senfes with the intermediate Colours in the order which their degrees of RefrangibiUty require, that is in the progrefs from E to F, or from the infide of the Bow to the outfide in this order, violet, indigo, blue, green, yellow, orange, red. But the violet, by the mixture of the white Light of the Clouds, will appear faint and incline to purple.

Again, the Angle SGO being equal to the Angle POG, or 50 Gr. fi Min. fliall be the leafl Angle in which the lealt refrangible Rays can after two Reflexions emerge out of the drops, and therefore the lealt refrangible Rays fhail come moft copioufly to the Eye from the drops in the Line O G, and Itrike the Senfe with the deepeft red in that Region. And the Angle SHO being equal to the Angle POHor 5-4 Gr, 7 Min. fhall be the lealt Angle in which the moit refrangible Rays after two Reflexions can e- merge out of the drops, and therefore thofe Rays fliall come moft copioufly to the Eye from the drops in the Line OH, and ftrike the Senfes

. L 4 witfe

C ^52 ]

with the deepelt violet in that Region/ And by the fame Argument, the drops in the Re- gions between G and H fhall ftrike the Senfe with the intermediate Colours in the order which their degrees of Refrangibihty require, that is, in the progrefs from G to H, or from the infideofthe Bow to the outfide in this or- der, red, orange, yellow, green, blue, indigo, violet. And lince thefe four Lines OE, OF, OG, OH, may be fituatcd any where in the abovemention'd conical Superficies, what is faid of the Drops and Colours in thefe Lines is to be underflood of the Drops and Colours every where in thofe Superficies.

Thus ihall there be made two Bows of Co- lours, an interior and ilronger, by one Reflexion in the drops, and an exrerior and fainter by two ; for the Light becomes fainter by every Reflexion. And their Colours iliaii lie in a con- trary order to one another, the red of both Bovvs bordering upon the Space G F which is between the Bows. The breadth of the inte- rior Bow EOF meafured crofs the Colours Ihall be i Degr. 45- Min. and the breadth of the exterior GOH Ihall be 3 Degr. 10 Min. and the diftance between them GOF ihall be 8 Gr. 15: Min. the greatell Semi-diameter of the inner- moft, that is, the Angle P O F being 41 Gr. % Min. and the leafl: Semi- diameter of the outer- molt P O G, being 50 Gr, 5-7 Min. Thefe are the Meafures of the Bows, as they would be were the Sun but a point; for by the breadth pf his Body the breadth of the Bows will be in- ^reafed" and their diitance decreafed by half a

Degree,

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