REF
( P 8i )
REF
fame manner as if the Glafs were away ■ For the Rays being Perpendicular will pafs without Refraction. — If the Glafs be turn'd obliquely to the Sun, the Light after Refraction will be of the famelntcniityas before; thelntenfity depending on the Spifiitude orClofenefsof the Rays, and on the Angle wherein they ftrike the Object, or the Eye, both which are here unvaried. See Ray.
2. If two Rays CD and CP, (Fig. 50.) proceeding from the fame Radiant C, and tailing on a Plane Surface of a different Deniity, fo as the Points of Refraction D and P are equally di- ftant tiom the Cathetus of Incidence GK ; the refracted Rays DF and PQJiave the fame virtual Focus, or Point of Difper- (ion G. See Virtual.
Hence, i°. Since in Rays very near each other, the Diftance from the Cathetus is the fame as to Senfe ; very near Rays wilt diverge from the fame Point G, i. e. have the fame virtual Fo- cus G.— And hence, 2°. When refracted Rays falling on the Eye placed out of the Cathetus of Incidence, are either equally di- itent from the Cathetus, or very near each other ; they will flow upon the Eye, as if they came to it from the Point G ; confe- quently the Point C will be feen by the refracted Rays as in G.
- . If a Ray CD fall obliquely out of a a thinner into a
doter Medium, having a plane Surface; the Diftance of the Ra- diant Point CK has a lefs Ratio to the Point of Difperlion, or virtual Focus KG, than the Sine of the refracted Angle to the Sine of the Angle of Inclination.— But if the Diftance of the Point of Refraction from the Cathetus of Incidence KD be lefs than the eleventh or nineteenth Part of the Diftance of the radi- ant Point CK ; and if in the former Cafe the tenth, in the lat- ter the hundredth Part thereof be lo fm.ill that it can't be affign'd or need not be minded, then will CK be to KG, as to Senfe : in the Ratio of the Sine of the refracted Angle, to the Sine of the Angle of Inclination.
Hence, 1°. If the Refraction be out of Air into Glafs, the Diftance of the Point of Difperlion of Rays near the Cathetus, is Sefquialterate, of the radiant Point ; of more remote Rays, greater than Selquiakerate.
Hence, 2.". If the Eye be placed in a denfe Medium, Ob- jects in a Rarer will appear more remote than they are; and the Place of the Image in any given Cafe, may be determined from the Ratio of the Rfr action. — Thus to Fi/les fwimnjing under Water, Objects out of the Water m-ift app ear further dijlant than in reality thy arc.
4. If a Ray GD fall obliquely out of a denfer, into a rarer Medium AB; the Diftance of the radiant Poinr GK, hasa grea- ter Ratio to the Diftance of the Point of Difperlion KC, than the Sine of the refracted Angle has to the Sine of the Angle of Inclination,— In the other Cafe of the preceding Theorem, KG Will be to KC, as to Senfe, in the Ratio of the Sine of refract- ed Angle, to the Sine of the Angle of Inclination.
Hence, i". If the Refraction be out of Glafsinto Air, the Di- ftance of the Point of Difperlion of Rays near the Cathetus of Incidence, is Subfefquialterate of the Diftance of the radiant Point. That of the more remote Rays is lefs than the Subfef- quialterate.
But, 2 . if the Refraction be out of Water into Air; the Di- ftance of the Point of Difperlion of Rays near the Cathetus, is Subfefqukertian ; of thofe more remote, lefs than Subfefqui- tertian.
And 5'. The Eye therefore being in a tarer Medium, Objeas placed in a Denier appear nearer than they are ; and the Place of the Image may be determined in any given Cafe by the Ratio of Refraction — Hence the Bottom of a Veffel full of Water, is rais'd by Refraction, to a third Part of its Height, with refpefi to an Eye perpendicularly over the refracting Surface; and hence Fi/bes, md other Bodies under Water, appear nearer than they really are.
5. If the Eye be placed in a rarer Medium, an Object feen in a denier Medium, by a Ray refractedin a plane Surface, will ap- pear larger than it really is — If the Objeft be in a rarer, and the Eye in a denier Medium, the Object: will appear lefs than it is. —And in each Cafe the apparent Magnitude is to the real one in a Ratio compounded of the Diftance of the Point to which the Rays tend before Refraaion, from the refracting Surface FL (% 60 ) to the Diftance of the Eye GL, from the fame, and of the Diftance of the Object from the Eye GM, to its Diftance from a Point to ">hich the Rays FL tend before Refraaion.
Hence, i°. If the Object AB be vety remote; FM will be phylically equal to GM; and therefore the real Magnitude MB K> its apparent one MH, as GL to FL, or the Diftance of the Eye G from the refracting Plane to the Diftance of the Point of Convergence F from the fame Plane.
Hence, 2 . Objects under Water, to an Eye in the Air, appear lar ger than they are ; and to Fi/bes under Water, Objects in the Air Vfetr lefs than they are.
av " "f Refraction in Spherical Surfaces, both Concave and Convex.
I. A Ray f Light DE (Fig. 61 ) Parallel to the Axis of a
- „ C l e n phere ' aftera fingle Refraction in E, falls in with the Axis
m the Point F, beyond the Centre C.
For the Semidiameter CE dtawn to the Point of Refraction h, is perpendicular to the Surface KL, and is therefore the Ax- is of Refraction; bur a Ray out of a rarer into a denfer Medium, we have fhewn, mefrelbd towards the Perpendicular, or the Ax- is of Refractvi; therefore the RayDE will converge to the Ax- is of the Sphere AF; and will, thetefore at length concur wiih it; and that beyond the Centre C, in F; becaufe the Angle of Refraction ¥hH is lels than the Angle ot Inclination CEH.
2. If a Ray DE fall on a fpherically Convex Surface of a denfer Medium, parallel to its Axis AF; the Semidiameter CE will be to then/raSWRay EF in the Ratio of the Sine of the refr.:lhd Angle to the Sine of the Angle of Inclination : But the; Diftance of the Focus or Point of Concurrence to the refracted Ray FE is in the Ratio of the Sine of the refracted Angle to the Sine of the Angle of Inclination.
3. If a Ray DE fall on a denfer fpherically Convex Surface KL, Parallel to the Axis AF; the Diftance of the Focus from the refracting Surface FB, is to its Diftance from the Cen- tre FC ; in a Ratio greater than that of the Sine of the Angle of Inclination to the Sine of the refri.ctd Angle.— But if the Rays be very near the Axis, and the Angle of Inclination BCE cf a few Degrees; the Diftances of the Focus from the Surface, and the Centre FBand FC, will be, nearly, in the Ratio of the Sine of the Angle of Inclination, to the 'Sine of the refracted Angle.
Hence, 1 °. If the Refraction be out of Air into Glafs ; in the Cafe of Rays near the Axis, BF : FC : : 3 : 2. And in the Cafe of the Rays remoter from the Axis, BF : FC > 3 : 2. Confequently in the former Cafe, BC : BF • • 1 • 5 ; and in the latter BC : BF < 1 : 3.
And 2 . If the Refraction be out of Air into Water ; in the former Cafe, BF : FC : : 4 : 3 ; abd in the latter, BF : FC > 4 : 3- Confequently in the former BC : BF .• ; 1 : 4, and on the latter BC : BF < 1 : 4.
Hence, 3 . Since the Sun's Rays are Parallel, asto Senfe; if they fall on the Surface of a folid Glafs Sphere, or of a Sphere full of Water, they will nut concur with the Axis within the Sphere. So that Vitcllio was miflafcen when he imagined that the Sun's Rays falling on the Surface of a Cryftallin Sphere, were refracted to the Centre. See Focus.
4. It a Ray DE (Fig. 62 ) fall out of a denfer into a rarer fpherical Medium ; after Refraaion it will diverge from rhe Axis; and the Diftance of the Point of Difperlion, or the virtual Fo- cus from the Centre of the Sphete, FC, will be to its Semidk- metcr CE in the Ratio of the Sine of the refracted An$e to the Angle of Refraction; but to the Pottion of the refracted Ray drawn back, FE, in the Ratio of the Sine of the refract.'d Angle to the Sine of the Angle of Inclination.
5. If a Ray ED fall Parallel to the Axis AF on the fphetically Convex Surface KL, of a rarer Medium, out of a denfer; the Diftance of the Point of Difpcrfion from the Centre, FC, is to its Diftance from the Surface FB : In a Ratio grearer than that of the Sine of the refrttted Angle to the Sine of the Angle of In- clination—But if the Rays DE be very near the Axis FA, the Ratio will be very nearly the fame with that of the reft acted An- gle to the Sine of the Angle of Inclination.
Hence, 1°. If rhe Refraction be out of Glais into Air; in the Cafe of Rays near the Axis, FC : FB : : 3 : 2. Confequent- ly BC : FB : : 1 : 2. Therefore in the Cafe of Rays more remote from the Axis, BC : FB < 1 : 2.
2 . If the Refralliou be out of Water into Air; in the former Cafe FC : FB : : 4 : 3. Confequently BC : FB : : 1 : 3 ; in the latter Cafe, therefore, BC : FB : 1 < 1 : 3.
3 . Since then the Point of Difperlion F is more remote from the refracting Surface KL, if rhe Rays proceed out of Water, than out of Glafs, into Air ; Parallel Rays are lefs difpers'd in the former Cafe than in the latter.
6. If a Ray HE (Fig. cSt ) fall parallel to the Axis FA, out of a rarer, on the Surface ofa Ipherically concave denfer Medium ; the refracted Ray EN will be made to recede from the Point of the Axis F ; fo as FE will be to FC, in the Ratio of the Sine ot the Angle of Inclination, to the Sine of the refracted Angle.
7. If a Ray EH tall parallel to the Axis FB on the Concave Surface KL of a fpherical denfer Medium, from a rarer ; the Diftance of the Point of Difperfion from the refracting Surface FB ; is to Diftance from the Centre, FC, in a Ratio greater than that of the Sine of the Angle of Inclination, to the Sine of the refracted Angle. But if the Rays be very near the Axis, and the Angle BCE very fmall; BH will be to BC very nearly in the Ratio of the Sine of the Angle of Inclination; to the Sine of the refr-aited Angle.
Hence, t°. If the Refraction be out of Air into Glafs; in the Cafe of the Rays near the Axis, FB : FC : : 3 : 2; in the Cafe of Rays more remote from the Axis FB : FC : : > 3 : confequently in the former, BC 1 : FC : : 1 : 2 : And hence, in the latter, BC : FC < 1 : 2.
Hence alfo 2 . If the R fraction be cut of Air into Water; in Cafe of the Rays near the Axis; FB : FC : : 4 : 3. In the Cafe of Rays more remote from the Axis FB : FC > 4:3. Confequently in thefirft Cafe BC : FC : : I : 3. And hence, in the latter, BC ; FC > 1 : }.
And
