fed to vanith. Hence it follows, that if a number of parallel rays, as AB, CD, EG, &c. fall upon a convex furface, (as fig. 1. Pl. CCLVI.) and if BA, DK, the reflected rays of the incident ones AB, CD, proceed as from the point F, thofe of the incident ones CD, EG, viz. DK, GL, will proceed as from N, thofe of the incident ones ED. HI, as from O, &c. because the farther the inciden: ones CD, EG, &c. are from AB, the nearer to the furface are the points F, f, f, in the line BF, from which they proceed after reflection; fo that properly the foci of the reflected rays BA, DK, GL, &c. are not in the line AB produced, but in a curve line paffing through the points F, N, O, &c. The fame is applicable to the cafe of parallel rays reflected from the concave furface, as expref. fed by the pricked lives on the other half of the figure, where PQ, RS, TV, are the incident rays; QF, Sf. Vf, the reflected ones, interfecting each other in the points X, Y, and F; fo that the foci of those rays are not in the line FB, but in a curve paffing through thofe points. Had the furface BH in 89, or 10, Pl. CCLV. been formed by the revolution of a parabola about its axis, having its focus in the point F, all the rays reflected from the convex surface would have proceeded as from the point F, and those reflected from the concave would have fallen upon it, however diftant their incident ones AB, DH, might have been from each other. For in the parabola, all lines, drawn parallel to the axis, make angles with the tangents to the points where they cut the parabola (that is, with the furface of the parabola) equal to thofe which are made with the fame tangents by lines drawn from thence to the focus; there fore, if the incident rays defcribe thofe parallel lines, the reflected ones will neceffarily defcribe thefe other, and fo will all proceed as from, or meet in, the same point. 383. PROP. III. Of the reflection of diverging and converging rays from a spherical furtace. "When rays fall upon any spherical furface, if they diverge, the diftance of the focus of the reflected rays from the furtace is to the diftance of the radiant point from the fame (or, if they converge, to that of the imaginary focus of the incident rays), as the diftance of the focus of the reflected rays from the centre is to the diftance of the radiant point (or imaginary focus of the incident rays) from the fame." This proposition admits of ten cafes. 384. CASE 1. Of diverging_rays falling upon a convex furface. DEM. Let RB, RD (Pl. CCLVI. fig. 2.) reprefent two diverging rays flowing from the point R, as from a radiant, and falling, the one perpendicularly, the other obliquely, on the convex furface BD, whose centre is C. Let DE be the reflected ray of the incident one RD, produce ED to F, and through R draw the line RH paralel to FE, till it meets CD produced in H. Then will the angle RHD be equal to EDH the angle of reflection, as being alterate to it, and therefore equal alfo to RDH which is the angle of incidence; wherefore the triangle DRH is ifofceles, and confequently DR is equal to RH. Now the lines FD and RH being parallel, the triangles FDC and RHC are fimilar, (or, to exprefs it in Euclid's way, the fides of the triangle RHC are cut pro portionably, 2 Elem. 6.): and therefore FD is to RH, or its equal RD, as CF to CR; but BD vanithing, FD and KD differ not from FB and RB : wheretore FB is to RB alfo, as CF to CR; that is, the diftance of the focus from the furface is to the dittance of the radiant point from the fame, as the diftance of the focus from the centre is to the diftance of the radiant from thence. Q. E. D. 385. CASE 2. Of converging rays falling upon a concave furface. DEM. Let KD and CB be the converging incident rays, having their imaginary focus in the point R, which was the radiant in the foregoing cale. Then as RD was in that cafe reflected into DE, KD will in this be reflected into DF; for, fince the angles of incidence in both cales are equal, as they are by being vertical, the angles of reflection will be fo too; fo that F will be the focus of the reflected rays: but it was there demonftrated, that FB is to RB as CF to CR; that 1s, the diftance of the focus from the furface is to the distance (in this Cafe) of the imaginary focus of the incident rays, as the distance of the imaginary focus of the incident rays from the fame. Q. E. D. 386. CASE 3. Of converging rays falling upon a convex iurface, and tending to a point between the focus of parallel rays and the centre. DEM. Le BD (P CCLI, fig. 12.) reprefent a convex furface whole centre is C, and whofe focus of parallel rays is P; and let AB, KD, be two converging rays incident upon it, and having their imaginary focus at R, a point between P and C. Now becauf KD tends to a point between the focus of parallel rays and the centre, the reflected ray DE will diverge from fome point on the other fide of the centre, suppose F; as explained above (§ 375.) under Prop. vii. Through D draw the perpendicular CD, and produce it to H; then will KDH and HDE be the angles of incidence and reflection, which being equal, their vertical ones RDC and CDF will be fo too, and therefore the vertex of the triangle RDF is bifected by the line DC: wherefore FD and DR, or, BD vanishing, FB and BR are to each other as FC to CR; that 18, the distance of the focus of the reflected rays is to that of the imaginary focus of the incident ones, as the diftance of the former from the centre is to the distance of the latter from the fame. E. D. 387. CASE 4. Of diverging rays falling upon a concave furface, and proceeding from a point between the focus of parallel rays and the centre. DEM. Let RB, RD, (Pl. CCLI, fig 12.) be the diverging rays incident upon the concave furface BD, having their radiant point in the point R, the imaginary focus of the incident rays in the foregoing cafe. Then as KD was in that cafe reflected into DE, RD will now be reflected into DF. But it was there demonftrated, that FB and RB are to each other as CF to CR; that is, the duftance of the focus is to that of the radiant as the distance of the former from the centre is to the diftance of the latter from the fame. Q. E. D. The angles of incidence and reflection being equal, it is evident, that if, in any cafe, the reflected ray be made the incident one, the incident will become the reflected one; and therefore the four following cafes may be confidered refpective ly ly as the converfe of the four foregoing; for in each of them the incident rays are fuppofed to coincide with the reflected ones in the other. Or they may be demonftrated independently of them as follows. 388. CASE 5. Of converging rays falling upon a convex furface, and tending to a point nearer the furface than the focus of parallel rays. DEM. Let ED, RB (Pl. CCLVI, fig. 2.) be the conver ging rays incident upon the convex furface BD, whofe centre is C, and focus of parallel rays is P; and let the imaginary focus of the incident rays be at F, a point between P and B; and let DR be the reflected ray. From C and R draw the lines CH, RH, the one paffing through D, the other parallel to FE. Then will the angle RHD be equal to HDE the angle of incidence, as alternate to it; and therefore equal to HDR, the angle of reflection; wherefore the triangle HDR is ifofceles, and confequently DR is equal to RH. Now the lines FD and RH being parallel, the triangles FDC and RHC are fimilar; and therefore RH, or RD, is to FD as CR to CF: but PD vanishing, RD and FD coincide with RB and FB; wherefore RB 18 to FB as CR to CF; that is, the diftance of the focus from the furface is to the diftance of the imaginary focus of the incident rays, as the distance of the focus from the centre is to the distance of the imaginary focus of the ir cident rays from the fame. Q. E. D. 389. CASE 6. Of diverging rays falling upon a concave furface, and proceeding from a point between the focus of parallel rays and the surface. DEM. Let FD and FB reprefent two diverging rays flowing from the point F as a radiant, which was the imaginary focus of the incident rays in the foregoing cafe. Then as ED was in that cafe reflected into DR, FD will be reflected into DK (for the reafon mentioned in Cafe 2.), fo that the reflected ray will proceed as from the point R: but it was demonftrated in the cafe immediately foregoing, that RB is to FB as CR to CF; that is, the distance of the focus from the furface is to that of the radiant from the fame, as the distance of the former from the centre is to that of the latter from the fame. Q. E. D. 390 CASE 7. Of converging rays falling upon a convex surface, and tending towards a point be. yond the centre. DEM. Let AB, ED (Pl. CCLI, fig. 12.) be the incident rays tending to F, a point beyond the centre C, and let DK be the reflected ray of the incident one ED. Then, because the incident ray ED tends to a point beyond the centre, the reflected ray DK will proceed as from one on the contrary fide, fuppofe R; as explained above under Prop. VII. Through D draw the perpendicular CD, and produce it to H. Then will EDH and HDK be the angles of incidence and reflection; which being equal, their vertical ones CDF and CDR will be fo too: confequently the vertex of the triangle FDR is bifected by the line CD: Wherefore, RD is to DF, or (3 Elem. 6.) BD vanishing, RB is to BF as RC to CF; that is, the diftance of the focus of the reflected rays is to that of the imaginary focus of the incident rays, as the diftance of the former from the centre is to the distance of the latter from the fame. Q. E. D. 391. CASE 8. Of diverging rays falling upon a concave surface, and proceeding from a point beyond the centre. DEM. Let FB, FD, be the incident rays having their radiant in F, the imaginary focus of the incident rays in the foregoing cafe. Then as ED was in that cafe reflected into DK, FD will now be reflected into DR; so that R will be the focus of the reflected rays. But it was demonftrated in the foregoing cafe, that RB is to FB as RC to CF; that is, the diftance of the focus of the reflected rays from the furface is to the diftance of the radiant from the fame, as the distance of the focus of the reflected rays from the centre is to the distance of the radiant from thence. Q. E. D. 392. The two remaining cafes may be confidered as the converfe of thofe under Prop. II. (§ 380. -382.) because the incident rays in these are the reflected ones in them; or they may be demonftrated in the fame manner with the foregoing, as follows: 393 CASE 9. Converging rays falling upon a convex surface, and tending to the focus of parallel rays, become paralle! after reflection. DEM. Let ED, RB (Plate CCLVI, fig. 2.) represent two converging rays incident on the convex furface BD, and tending towards F, which we will now fuppofe to be the focus of parallel rays; and let DR be the reflected ray, and C the centre of convexity of the reflecting furface. Through C draw the line CD, and produce it to H, drawing RH parallel to ED produced to F. Now it has been demonftrated (Cafe 5. where the incident rays are fuppofed to tend to the point F), that RB is to FB as RC to CF; but F in this Cafe being fuppofed to be the focus of parallel rays, it is the middle point between C and B (by Prop. II.), and therefore FB and FC are equal; and confequently the two other terms in the proportion, viz. RB and RC, muft be fo too; which can only be upon the fuppofition that R is at an infinite distance from B; that is, that the reflected rays BR and DR be parallel. Q. E. D. 394 CASE 10. Diverging rays falling upon a concave furface, and proceeding from the focus of parallel rays, become parallel after reflection, DEM. Let RD, RB (Pl. CCLI, fig. 12.) be two diverging rays incident upon the concave furface BD, as fuppofing in Case 4, where it was demonftrated that FB is to RB as CF to CR. But in the prefent cafe RB and CR are equal, because R is fuppofed to be the focus of parallel rays; therefore FB and FC are fo too; which cannot be, unlefs F be taken at an infinite diftance from B; that is, unless the reflected rays BF and DF be parallel. Q. E. D. OBS. In the case of diverging rays failing upon a convex surface (Pl. CCLVI, fig. 2.) the farther the point D is taken from B, the nearer the point F, the focus of the reflected rays, approaches to B, while the radiant R remains the fame. For it is evident from the curvature of a circle, that the point D (fig. 3.) may be taken fo far from B, that the reflected ray DE fhall proceed as from F, G, H, or even from B, or from any point between B and R; and the farther it is taken from B, the fafter the point from which it proceeds approaches towards R, as will eafily appear if we draw several incident rays with their refpective reflected ones, in fuch manner that the angles angles of reflection may be all equal to their refpective angles of incidence, as is done in the figure. The like is applicable to any of the other cafes of diverging or converging rays incident up. on a spherical surface. This is the reafon, that when rays are confidered as reflected from a fphe rical surface, the distance of the oblique rays from the perpendicular one is taken so small, that it may be supposed to vanish. From hence it follows, that if a number of diverging rays are incident upon the convex furface BD, at the several points B, D, D, &c. they fhail not proceed after reflection as from any point in the line RB produced, but as from a curve line paffing through the feveral points F, ff, &c. The fame is applicable in all the other cafes. Had the curvature BD (fig. 2. Pl. 256.) been hyperbolical, having its foci in R and F; then R being the radiant (or the imaginary focus of incident rays), F would have been the focus of the reflected ones, and vice versa, however diftant the points B and D might be taken from each other. In like manner, had the curve BD (fig. 12. Pl. 251.) been elliptical, having its foci in F and R, the one of thefe being made the radiant (or imaginary focus of incident rays), the other would have been the focus of reflected ones, and vice verfa. For both in the hyperbola and ellipfis, lines drawn from each of their foci through any point make equal angles with the tangent to that point. Therefore, if the incident rays proceed to or from one of their foci, the reflected ones will all proceed as from or to the other. So that, in order that diverging or converging rays may be accurately reflected to or from a point, the reflecting furface must be formed by the revolution of an hyperbola about its longer axis, when the incident rays are fuch, that their radiant or imaginary focus of incident rays fhall fall on one fide the furface, and the focus of the reflected ones on the other: when they are both to fall on the fame fide, it must be formed by the revolution of an ellipfis about its longer axis. However, upon account of the great facility with which spherical furfaces are formed, in comparison of that with which furfaces formed by the revolution of any of the conic sections about their axes are made, the latter are very rarely used. Add to this another inconvenience, viz. that the foci of these curves being mathematical points, it is but one point of the surface of an object that can be placed in any of them at a time; fo that it is only in the ory that surfaces formed by the revolution of these curves about their axes render reflection perfect. Now, because the focal diftance of rays reflected from a fpherical furface cannot be found by the analogy laid down in the third propofition, without making use of the quantity fought; we shall here give an inftance whereby the method of doing it in all others will readily appear. 395. PROB. Let it be required to find the focal distance of diverging rays incident upon a convex furface, whose radius of convexity is 5 parts, and the distance of the radiant from the furface is 20. SOL. Call the focal diftance fought; then will the distance of the focus from the centre be 5-x, and that of the radiant from the fame 25: therefore by Prop. 3. we have the following proportion, viz. x: 20 :: 5-x: 25; and multiplying extremes together and means together, we have 25 x=100-20 x; which, after due reduction gives, x. If in any cafe it fhould happen that the value of x fhould be a negative quantity, the focal point muft then be taken on the contrary fide of the furface to that on which it was fuppofed that it would fall in ftating the problem. If letters inftead of figures had been made use of in the foregoing folution, a general theorem might have been raised, to have determined the focal diftance of reflected rays in all cases whatever. (See Suppl. to Gregory's Opticks, ad edit. p. 112.) Becaufe it was, in the preceding fection, obferved, that different incident rays, though tending to or from one point, would, after refraction, proceed to or from different points, a method was there inferted of determining the diftinct point which each feparate ray entering a spherical furface converges to, or diverges from, after refraction: the fame has been observed here with regard to rays reflected from a spherical surface. (See Obf. in Cafe 2. and Cafe ro.) But the method of determining the diftinct point, to or from which any given incident ray proceeds after reflection, is much more fimple. It is only neceffary to draw the reflected ray such, that the angle of reflection may be equal to the angle of incidence, which will determine the point it proceeds to or from, in any cafe whatever. $3. Of the APPEARANCE of BODIES, feen by LIGHT reflected from PLANE and SPHERICAL SURFACES. 396. WHATEVER has been faid concerning the appearance of bodies feen by refracted light through lenfes, refpects alfo the appearance of bodies feen by reflection. But there is one thing peculiar to images by reflection, viz. that each point in the reprefentation of an object made by reflection appears fituated fomewhere in an infi nite right line that paffes through its correfpondent point in the object, and is perpendicular to the reflecting furface. The truth of this appears from the propofitions formerly laid down: in each of which, rays, flowing from any radiant point, are fhown to proceed after reflection to or from fome point in a line that paffes through the faid radiant, and is perpendicular to the reflecting furface. For inftance, (fig. 6. pl. CCLV.), rays flowing from Y are collected in X, a point in the perpendicular CD, which, being produced, paffes through Y: again, (fig. 7.) rays flowing from G, proceed, after reflection as from N, a point in the perpendicular CD, which, being produced, paffes through G; and fo of the reft. This obfervation, however, except where an object is seen by reflection from a plane furface, relates only to thofe cafes where the representation is made by fuch rays as fall upon the reflecting furface with a very small degree of obliquity; becaufe fuch as fall from a confiderable diftance from the perpendicular, proceed not after reflection as from any point in that perpendicular, but as from other points fituated in a certain curve, upon which account thefe rays are neglected, as making a confused and deformed reprefentation. And therefore, however the fituation of the eye with refpect to the object and reflecting furface may be reprefented in the following figures, it is to be supposed as situated in such |