they may be reflected to the Eye; and the Point t will be the like Limit for the least refrangible Rays, that is, beyond which none of them can pass through the Base, whose Incidence is such that by Reflexion they may come to the Eye. And the Point r taken in the middle Way between p and t, will be the like Limit for the meanly refrangible Rays. And therefore all the least refrangible Rays which fall upon the Base beyond t, that is, between t and B, and can come from thence to the Eye, will be reflected thither: But on this side t, that is, between t and c, many of these Rays will be transmitted through the Base, And all the most refrangible Rays which fall upon the Base beyond p, that is, between p and B, and can by Reflexion come from thence to the Eye, will be reflected thither, but every where between p and c, many of these Rays will get through the Base, and be refracted; and the same is to be understood of the meanly refrangible Rays on either side of the Point r. Whence it follows, that the Base of the Prism must every where between 't and B, by a total Reflexion of all sorts of Rays to the Eye, look white and bright. And every where between p and C, by reason of the Transmission of many Rays of every fort, look more pale, obscure, and dark. But at r, and in other Places between p and t, where all the more refrangible Rays are reflected to the Eye, and many of the less refrangible are tranfmitted, the Excess of the most refrangible in the reflected Light will tinge that Light with their Colour, which is violet and blue. And

this happens by taking the Line Cprt B any where between the Ends of the Prism HG and EI.

PROP. IX. PROB. IV.

By the discovered Properties of Light to explain the Colours of the Rain-bow.

TH

HIS Bow never appears, but where it rains in the Sun-shine, and may be made aruficially by spouting up Water which may break aloft, and scatter into Drops, and fall down like Rain. For the Sun shining upon these Drops certainly causes the Bow to appear to a spectator standing in a due Position to the Rain and Sun. And hence it is now agreed upon, that this Bow is made by Refraction of the Sun's Light in Drops of falling Rain. This was understood by some of the Antients, and of late more fully discover'd and explain'd by the famous Antonius de Dominis Archbishop of Spalato, in his Book De Radiis Visûs & Lucis, published by his Friend Bartolus at Venice, in the Year 1611, and written above 20 Years before. For he teaches there how the interior Bow is made in round Drops of Rain by two Refractions of the Sun's Light, and one Reflexion between them, and the exterior by two Refractions, 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 Glass filled

L 2

with

1

with Water, and placed in the Sun to make the Colours of the two Bows appear in them. The same Explication Des-Cartes hath pursued in his Meteors, and mended that of the exterior Bow. But whilst they understood not the true Origin of Colours, it's necessary to pursue it here a little farther. For understanding therefore how the Bow is made, let a Drop of Rain, or any other spherical transparent Body be represented by the Sphere BNFG, [in Fig. 14.] described with the Center C, and Semi-diameter CN. And let AN be one of the Sun's Rays incident upon it at N, and thence refracted to F, where let it either go out of the Sphere by Refraction towards V, or be reflected to G; and at G let it either go out by Refraction to R, or be reflected to H; and at H let it go out by Refraction 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 Perpendiculars CD and CE, and produce CD till it fall upon the Circumference at L. Parallel to the incident Ray AN draw the Diameter B Q and let the Sine of Incidence but of Air into Water be to the Sine of Refraction as I to R. Now, if you suppose the Point of incidence N to move from the Point B, continually till it come to L, the Arch QF will first increase and then decrease, and so will the Angle A XR which the Rays AN and GR contain; and the Arch QF and Angle AXR will be biggest when ND is to CN as VII-RR to 3 RR, in which case NE will be to ND as 2 R to I. Also the Angle AYS, which the Rays AN and HS

contain

3

contain will first decrease, and then increase and grow least when ND is to CN as II-RR to ✓ 8 RR, in which case NE will be to ND, as R to I. And so 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 II-RR to 15 RR, in which cafe 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 ND is to CN as II-KR to 24 RR, in which cafe NE will be to ND as 5 R to I; and so on infinitely, the Numbers 3, 8, 15, 24, &c. being gather'd by continual Addition of the Terms of the arithmetical progression 3, 5, 7, 9, &c. The Truth of all this Mathematicians will easily examine.*

Now it is to be observed, that as when the Sun comes to his Tropicks, Days increase and decrease but a very little for a great while together; so when by increasing the distance CD, these Angles come to their Limits, they vary their quantity but very little for some time together, and therefore a far greater Number of the Rays which fall upon all the Points N in the Quadrant BL, fhail emerge in the Limits of these Angles, than in any other Inclinations. And farther it is to be observed, that the Rays which differ in Refrangibility will have different Limits of their Angles of Emergence, and by consequence according to their different Degrees of Refrangibility emerge most copiously *This is demonftrated in our Author's Lect. Optic. Part I. Sea. IV. Prop. 35 and 36.

in different Angles, and being separated from one another appear each in their proper Colours. And what those Angles are may be easily gather'd from the foregoing Theorem by Computation.

For in the least refrangible Rays the Sines I and R (as was found above) are 108 and 81, and thence by Computation the greatest Angle. AXR will be found 42 Degrees and 2 Minutes, and the least Angle AYS, 50 Degrees and 57 Minutes. And in the most refrangible Rays the Sines I and R are 109 and 81, and thence by Computation the greatest Angle AXR will be found 40 Degrees and 17 Minutes, and the least Angle AYS 54 Degrees and 7 Minutes.

For

Suppose now that O [in Fig. 15.] is the SpeCtator's Eye, and OP a Line drawn parallel to the Sun's Rays, and let POE, POF, POG, POH, be Angles of 40 Degr. 17 Min. 42 Degr. 2 Min. 50 Degr. 57 Min. and 54 Degr. 7 Min. respectively, and these Angles turned about their common Side OP, shall with their other Sides OE, OF; OG, OH, describe the Verges of two Rain-bows AF BE and CHDG. if E, F, G, H, be drops placed any where in the conical Superficies described by OE, OF, OG, OH, and be illuminated by the Sun's Rays SE, SF, SG, SH; the Angle SEO being equal to the Angle POE, or 40 Degr. 17 Min. shall be the greatest Angle in which the most refrangible Rays can after one Reflexion be refracted to the Eye, and therefore all the Drops in the Line OE shall send the most refrangible Rays most copiously to the Eye, and thereby