RAI
CpsO
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Sun-fhine, and viewing it in fuch a Pofture as that the Rays which come from the Globe to the Eye, may, with the Sun's Rays in- clude an Angle either of 42°, or 50 ; if, e. gr. the Angle be a- bout 42 ° j the Spectator, fuppofed at O, will Tee a full red Co- lour in that Side of the Globe oppofite to the Sun, as at F. And if that Angle be made a little lefs, fuppofe by depreffing the Glo- bule toE, the other Colours, Yellow, Blue, and Green, will ap- pear fuccefTively, in the fame Side of the Globe, alfo exceed- ingly bright.
But if the Angle be made about 50 , fuppofe by raifing the Globule G, there will appear a red Colour in that Side of the Globe towards the Sun- though that fomewhat faint; and if the Angle be made greater, fuppole by raiting the Globe to H ; the Red will change fuccefTively to the other Colours, Yellow^ Green, and Blue.
The fame thing is obfcrv'd in letting the Globe reft, and rai- fing, ordepreffingtheEye to make the Angle of a juft Magnitude.
Dimen/ions of the Rain-Bow.
Des Cartes firft derermind its Diameter by a tentative, and in- direct Method,- laying it down that the Magnitude of the Bow depends on the Degree of Refraction of the Fluid ; and alluming the Ratio of the Sine of Incidence to that of Refraction, to be in Water as 250 to 157. See Refraction. '
But, Dr. Halley has (ince, in the Philofoph. TravfaSl. given us a fimple, direct Method of determining the Diameter of the Rai.ibo'w from the Ratio of Retraction of the Fluid being given; or vice verfh the Rainbow being given, to determine the refra- ctive Power of the Fluid. The Praxis is as follows.
Firft, The Ratio of Refraftion being given, to find the Angles of fact/fence, and Refraction of a Ray -which becomes effetiual after a- xy given 'Number of Reflections. Suppofe any given Line as AC (Tab. Optick$> Fig. 49.J which divide in D ,• io, as that AC be to AD in the Ratio of Refraction ; and again divide it in E, fo as AC be to AE as the given Number of Reflexions increafed byUnity 3 istoUnity; with the Diameter CE defcribe aScmicir- cle CBE, and from the Centre A with the Radius AD defcribe an Arch DQ interfering the Semicircle in B. Then drawing AB, CBj ABC or its Complement to two right Angles, will be the Angle of Incidence 5 and AGO the Angle of Refraction required.
Secondly, The Ratio of Refratlion^ and any Angle of Incidence being given to find the Angle -which a Ray of Light emerging out of a refracting Sphere, after a given Number of R efleelions, makes with the Line of Afpecl, or an incident Ray j and confe^uently to find the Diameter of the Rainbow. The Angle of Incidence, and the Ra- tio of Refraction being given, the Angle of Refraction is given ; which Angle being multiplied by double the Number of Re- flexions increafed by 2, and double the Angle of Incidence fub- ftra&ed from the Product, the Angle remaining is the Angle fought.
Thus fuppofing, the Ratio of Refraction to be, as Sir Ifaac Newton has determined it, viz. as 108 to 81, in the red Rays, as 109 to 81 for the blue Rays, &c. the preceding Problem will give the Diftances of the Colours in the
I. Rain-Bow,
SRed 4 2 Blue 40
Red 41 I2i„ „ . , D , Blue 40 ,5^ The Spectators Back d „j . . o > beine turned to
9J
being turned to the Sun.
If the Angle made by a Ray after three or four Refkaions, were required, and therefore the Diameters of the third and fourth Rainbow, (which are fcarce ever ieen, by reafon of the great Diminution of the Rays, by fo many repeated Re- flexions) they will be found.
III. Rainbow, 5^^°
2 Blue 37
IV. Rainbow, S £; ed «
£Blue 49
3 VThe Spectator being / > turned towards the l\ Sun.
Hence, the Breadth of the Rainbow is ealily found : For the greater! Semidiameter of the firft Bow, i. e. from Red to Red being 42° 1', and the lead, viz. from Purple to Purple 40 16 ; the Breadth of the Vafcia. or Bow, meafured a-crofs from Red co Purple will be i» 45', and the greateft Diameter of the fe- condBow being 54° 9', and the lead 50 58', the Breadth of theFafcia will be 3° 10'. Andhence the Diftance between the two will be found 8° 15/.
In thefe Meafurcs the Sun is only efteem'd a Point; where- fore as his Diameter is really about 30' fo much mull be added to the Breadth of each Fa/da or Bow, from Red to Purple, and fo much be fubftrafied from the Diftance between them.
This will leave the Breadth of the primary Bow, 2." 15', that of the fico?idary Bow ]" 40', and the interval between the'liows 8° 25'; which Dimenfions deduced by Calculation, Sit Ifaac Newton allures us from his own Obfcrvations, agree very exactly with thofe found by actual Menfutation in the Heavens' '
Particular Fhanomena of the Rain-Bow, with the Caufes thereof.
From this Theory of the Rainfaw, all the particular Pheno-
mena are eafily deduced: Hence we fee why the his is always of the fame Bieadth; by reafon the intermediate Degrees of re- frangibility of the Rays between Red and Violet, which are its extreme Colours, are always the fame.
Secondly, Why it is more diittnflly terminated on the Side of the Red, than on that of the Violet ? There being no efficacious Rays in the Space adjoining to the red Drops, i. e. to the Space between the Bows ; whence it terminates abruptly ; whereas in the Space on the Side of the Violet ones there are fome Rays emit- ted to the Eye, which though too feeble to aftta it ftrongly, yet have this effed, that they foften the Violet Edge infallibly, fo that 'tis difficult to determine precifely where it terminates.
Thirdly, Why the Bow fhifts its Situation as the Eye does ; and, as the popular Phrafe has it, flies thofe who follow it, and fol- lows thofe that fly it ? The colour'd Drops being difpofed under a certain Angle about the Line of Afpeft, which is different in different Places : Whence, alfo, it follows that every different Spectator fees a different Bow.
Fourthly, Why the Bow is fometimes a larger Portion of a Circle, fometimes a lc-fs ? Its Magnitude depending on the grea- ter, or lels Part of the Surface of the Cone, above the Surface of the Earth at the Time of its appearance; and that Part being greater or lefs as the Line of Afped is more inclined or oblique to the Surface of the Earth ; which inclination, or obliquity, is greater as the Sun is higher : Whence, alfo, the higher the Sun, the lefs the Rainbow.
Fifthly, Why the Bow never appears when the Sun is above a certain Altitude? The Surface of rhe Cone wherein it fhould be feen, being loft in the Ground, at a little Diftance from theEye, When the bun is above 42 high.
Sixthly, Why the So™ never appears greater than a Semicircle, on a Plane? Since be rhe Sun never fo low, and even in the Ho- rizon ; the Centre of the Bow is ftill in the Line of Afped; which, in this Cafe, runs along the Earth, and is not all rais'd a* bove the Surface.
Indeed, if the Speftator be placed on a very confiderable Emi- nence, and the Sun in the Horizon ; die Line of Alpect wherein the Centre of the Bow is, will be notably rais'd above the Horizon, (confidering the Magnitude of 'the Circle whereof the Bow ufes to be a Part.; Nay, if the Eminence be very high, and the Rain near, 'tis poffible the Bow may be an entire Circle.
Seventhly, How the Bow may chance to appear inverted, >. e. the Concave Side be turn'd upwards? To wit, a Cloud happening to intercept the Rays, and prevent their Alining on the upper Pan of the Arch : In which Cafe only the lower Part appear- ing, the Bow will feem as if turn'd upfide down: Which proba- bly has been the Cafe in fereral Prodigies of this Kind, related by Authors.
Indeed the Bow may appear inverted from another Caufe- For, if, when the Sun is 41 » 46' high, his Rays fall upon the (mooch Surface of fome (pacious Lake, in the middle whereof a Spedator is plac'd; and if, at the fame time there be Rain fall- ing to which the Rays may be reflected from the Lake: 'Twilt be the fame as if the Sun fliou'd mine below the Horizon, and the Line of View be extended upwards: Thus the Surface of the Cone wherein the coloured Drops are to be placed, will be wholly above the Surface or the Earth.
But lince the upper Part will fall among the unbroken Clouds; and only the lower Part be found among the Drops of Rain, the Arch will appear inverted.
Eighthly, Why the Bow fometimes appears inclined ? The ac- curate rounduefs of the Bow depending on its great Diftance which prevents us from judging of it exactly; if the Rain which exhibits it, chance to be much nearer, we (hall fee its irregula- rities; and if the Wind in that Cafe drive the Rain fo as the higher Part be further from the Eye than the lower, the Bow will appear inclined.
Ninthly, Why the Legs of the Rainbow fometimes appear un- equally diftant ? If the Rain terminate on the Side of the Specta- tor, in a Plane fo inclined to the Line of Afpeft as to make an acute Angle on the left Hand, and an obtufe Angle on the rMir ; the Surface of the Cone which determines what Drops will" ap- pear, will fall upon them in fuch manner as that thofe on the left Hand, will appear further from the Eye than thofe on the Right. For che Line of Afped being Perpendicular co che Plane of che Bow, if you fuppole cwo rectangular Triangles a Right and Left, the Cathetus of each to be Line of View, and theBafe che Semidiamecer of the Bow, inclined as above: 'Tis evident fince thofe Angles of the Triangles, next the Eye, mult always be che fame, (viz. 43 « in che inner Bow) che Balis of che Right- hand Triangle will appear much longer than chat of the Left.
Lunar Rain-Bow.
The Moon, fomecimes, alfo, exhibits the Phenomenon of an Iris or Bow ; by the Refraction of her Rays in the Drops of Rain in the Night-time. See Moon.
Ariftotle fays, he was the firft that ever obferved it; and adds, that it never happens, i. e. is never vifible, buc ac che Time of che Full-Moon; her Light at other cifties being too faint to affea the Sight, after two RelraSions, and one Rtfkaioa
The
