PROP. 3.-The angles of incidence and reflection are in the same plane, and they are equal to each other.

Let a ray of light AC, (fig. 2,) admitted through a small hole into a dark chamber, be incident upon the reflecting surface RS at C; and let CB be the reflected ray; draw CP perpendicular to the reflector. Then, if the plane surface of a board TS be made to coincide with CA and CP, the reflected ray CB is found also to coincide with the plane TS; or the angles of incidence and reflection are in the same plane.

Again, if from.C as a centre, with any radius ČA, the circle RPS be described, the arc AP is found to be equal to the arc PB; therefore, the angle of incidence, ACP, is equal to the angle of reflection, BCP.

The angles of incidence and reflection are also found to be equal when rays are reflected at a curve surface.

Cor. 1.-The angles ACR, BCS, which are the complements of the angles of incidence and reflection, are also equal.

Cor. 2.-If BC be the incident ray, CA will be the reflected ray. For, the angle PCA is equal to the angle PCB, and in the same plane; therefore CA is the reflected ray.

Cor. 3.-If the ray PC be incident perpendicularly upon the reflecting surface, it will be reflected in the perpendicular CP.

Cor. 4.-If AC be produced to E, the angle BCE, which measures the deviation of the ray AC from its original course, is 180° - ACB; or 180°-2 of incidence.

Cor. 5.-A ray of light will be reflected at a curve surface, in the same manner as at a plane which touches the curve at the point of incidence. For, the angle of incidence, and consequently the angle of reflection, is the same, whether we suppose the reflection to take place at the curve, or the plane.

PROP. 4.-The angles of incidence and refraction are in the same plane; and, whilst the mediums are the same, the sine of the angle of incidence is to the sine of the angle of refraction, in a given ratio.

Upon the surface of a board TV, (fig. 3,) with the centre C and any radius CA, describe a circle PRQ, draw the diameters RS, PQ, at right angles to each other, and immerse the board into a vessel of water, in such a manner that PQ may be perpendicular to, and RS coincide with, the surface of the water. Then, if a ray of light, admitted through a small hole into a dark chamber, be incident upon the surface RS in the direction AC, coincident with the plane of the board CB, the direction of the refracted ray is found to coincide with that piane; that is, the angles of incidence and refraction are in the same plane.

Iso, if A and B be drawn at right angles to PQ, they are the sines incidence and refraction to the radius CA; and it is found that AD bas to BF the same ratio, whatever be the inclination of the incident ray to the refracting surface. That is, if aC be any other incident ray, Cb the refracted ray, ad and bf the sines of incidence and refraction, then AD· BF:: ad: bf.

The ratio of the sines of incidence and refraction is the same, when the refracting surface is curved.

Cor. 1.-Hence, if the angles of incidence of two rays be equal, the angles of refraction are also equal.

Cor. 2.-As the angle of incidence increases, the angle of refraction increases.

Cor. 3.-A ray of light is refracted at a curve surface, in the same manner as at a plane which touches the curve at the point of incidence.

PROP. 5.-If a ray AC be refracted at the surface RS in the direction CB, then a ray BC, coming the contrary way, will be refracted in the direction CA.

The construction being made as before, let a small object be placed upon the board at B, (fig. 4;) and, when it is immersed perpendicularly in water, till RS coincides with the surface, the object B will be seen from A, in the direction AC; and, since the motion of light, in the same medium, is rectilinear, the ray, by which the object is seen, is incident at C, and refracted in the direction CA.

Cor. 1.-The angle of deviation of the ray AC, is equal to the angle of deviation of the ray BC, which is incident in the contrary direction.

Cor. 2.-When a ray of light passes out of air into water, the sine of incidence the sine of refraction :: 4:3; consequently, when a ray passes out of water into air, the sine of incidence: the sine of refraction :: 3 : 4.

In the same manner, out of air into glass, the sine of incidence: the sine of refraction :: 3 : 2; therefore, out of glass into air, the sine of inci dence the sine of refraction :: 2 : 3.

SCHOLIUM. The preceding propositions, which are usually called the Laws of Reflection and Refraction, are the principles upon which the theory of vision is founded. They were discovered, and their truth has been established by repeated experiments, made expressly for this purpose; and it is also confirmed by the constant agreement of the conclusions derived from them, with each other, and with experience.

When light is reflected or refracted at a polished surface, the motion of the general body of the rays is conformable to the laws above laid down: some are, indeed, thrown to the eye in whatever situation it is placed; and, consequently, a part of the light is dispersed, in all directions, by the irregularity of the medium upon which it is incident. This dispersion is however, much less than would necessarily be produced, were the rays reflected or refracted by the solid parts of bodies.

II. On the Reflection of Rays at Plane and Spherical Surfaces. Definitions.-1. By a pencil of rays, we understand a number of rays taken collectively, and distinct from the rest.

These pencils consist either of parallel, converging, or diverging

rays.

Converging rays are such as approach to each other in their progress, and, if not intercepted, at length meet.

Diverging rays are such as recede from each other, and whose directions meet if produced backwards.

2. The focus of a pencil of rays is that point towards which they converge, or from which they diverge.

If the rays in a pencil, after reflection, or refraction, do not meet exactly in the same point, the pencil must be diminished; and the focus is the limit of the intersections of the extreme rays, when they approach nearer and nearer to each other, and at length coincide. In this case, the focus is usually called the geometrical focus.

The focus is real, when the rays actually meet in that point; and imaginary, or virtual, when their directions must be produced to

meet.

3. The axis of a pencil is that ray which is incident perpendicularly upon the reflecting or refracting surface.

4. The principal focus of a reflector, or refractor, is the geometrical focus of parallel rays incident nearly perpendicularly upon it.

PROP. 6.-If a ray of light be reflected once by each of two plane surfaces, and in a plane which is perpendicular to their com mon intersection, the angle contained between the first and last directions of the ray, is equal to twice the angle at which the reflectors are inclined to each other.

Let AB, CD, (fig. 5,) be two plane reflectors, inclined at the angle AGD; SB, BD, DH, the course of a ray reflected by them. Produce HD to O, and SB till it meets DH in H. Then, because the HBG= the ABS the DBG, the whole angle DBH2/DBG. In the same manner, the DBO = 24 BDC. And since the BGD = the BDC-the DBG, we have 2 BGD = 2 ▲ BDC-2▲ DBG

the BDO- the DBH the BHD.* 'PROP. 7.-Parallel rays, reflected at a plane surface, continue parallel.

Case 1. When the angles of incidence are in the same plane. Let RS, fig. 6, be the reflecting surface; AB, CD the incident, BG, DH the reflected rays.

Then the ABR = the but, since AB and CD are therefore the GBS the 28. 1.)

GBS, and the ▲ CDR = the ≤ HDS; parallel, the ABR the CDR : HDS, and BG, DH are parallel (Euc

Euc. 32. 1.

Case 2. When the angles of incidence are in different planes.

Let AB, CD, fig 7, be the rays; BE, DF perpendiculars to the refleeting surface at the points of incidence; join BD, and let AB be reflected in the direction BG; also let DH be the intersection of the planes CDF, GBD.

Then, since BE, DF, which are perpendicular to the same plane, are parallel (Euc. 6. 11), and AB, CD, are parallel, by the supposition, the angles of incidence ABE, CDF are equal (Euc. 10. 11); therefore the angles of reflection are equal. Again, since EB and FD are parallel, as also AB and CD, the planes ABG, CDH, are parallel (Euc. 15. 11), and they are intersected by the plane GBDH: consequently DH is paralle! to BG (Euc. 16. 11); therefore the angles EBG, FDH are equal (Euc, 10. 11); but the angle EBG is the angle of reflection of the ray AB; therefore the angle FDH is equal to the angle of reflection of the ray CD; and since DH is in the plane CDF, CD is reflected in the direction DH (Art. 18), which has before been shown to be parallel to BG,

PROP. 8.-If diverging or converging rays be reflected at a plane surface, the foci of incident and reflected rays are on contrary sides of the reflector, and equally distant from it.

Let QAB be a pencil of rays diverging from Q, and incident upon the plane reflector ACB; draw QC perpendicular to the surface; then will QC be reflected in the direction CQ (fig. 8). Let QA be any

Fig. 8.

E

other ray, and since a perpendicular to the surface at A is in plane with QC and QA (Euc. 6. and 7. 11), QA will be reflecte plane. Produce CA to D, and make the angle DAO equal to the angle QAC, then will AO be the reflected ray. Produce OA, QC, till they meet in q. Then, since the qAC the OAD= the ZQAC, and also the CA the QCA, and the side CA is common to the two triangles QCA, CAq, the side QC is equal to Cq, In the same manner it may be shown, that every other reflected ray in the pencil, will, if produced backwards, meet the axis in g; that is the rays, after reflection, diverge from the focus q.

If QABE be a pencil of ravs converging to q, they will, after reflection

at the surface ACB, converge to Q; therefore, in this case also, the foct of incident and reflected rays are on contrary sides of the reflector, and equally distant from it.

Cor. 1. The divergency, or convergency of rays, is not altered by reflection at a plane surface.

Cor. 2. In the triangles QAC, CAq, Aq is equal to QA; if therefore any reflected ray AO be produced backwards to q, making Aq=AQ, ¶ is the focus of reflected rays.

Cor. 3. If the incident rays QA, Qa, be parallel, or the distance of Q from the reflector be increased without limit with respect to Aa, the distance of q is increased without limit, or the reflected rays are parallel.

PROP. 9.-If parallel rays be incident nearly perpendicularly upon a spherical reflector, the geometrical focus of reflected rays is the middle point in the axis between the surface and centre.

=

Let ACB be a spherical reflector, whose centre is E; DA, EC, two rays of a parallel pencil incident upon it, of which EC passes through the centre, and is therefore reflected in the direction CE; join EA, and in the plane DACE, make the angle EAq equal to the angle DAE, and DA will be reflected in the direction Aq (fig. 9); draw GAT in the same plane, touching the reflector in A, and let it meet EC produced in T. Then, since the EAg the DAE the AEq (Euc. 29. 1), Eq Aq; also, the qAT the DAG (Art. 19) = the ATq (Euc. 29. 1); therefore Aq=qT; consequently Eq=qT; that is, q bisects ET the sec nt of the arc AC. Now let DA approach to EC, and the arc AC will ecrease, and its secant, at length, become equal to the radius; consequently the limit of the intersections of Aq and CE is F. the middle point between E and C.

=

If the rays be incident upon the convex side of the reflector, the reflected rays must be produced backwards to meet the axis; and, in this case, F, the middle point between E and C, may be shown to be the limit of the intersections of CE and Aq, as before.

Cor. 1. As the arc AC decreases, Eq, or Fq, decreases. Thus, when AC is 60°, Eq=EC; and when AC is 45°, Aq is perpendicular to EC, and Eq: EC :: 1 : √2.

Cor. 2. If different pencils of parallel rays be respectively incident, nearly perpendicularly, upon the reflector, the foci of reflected rays will lie in the spherical surface SFV, whose centre is E and radius EF.Fig. 10.

Cor. 3. If the axes EA, EC, EB, of these pencils, lie in the same plane, the foci will lie in the circular arc SFV.

.

Cor 4. If any point S, in the arc SFV, whose radius EF is on half of EC, be the focus of a pencil of rays incident nearly perpendicularly upon the reflector, these rays will be reflected parallel to cach other, and to EA the axis of that pencil.