Gatoptrics sophical studies, and partly in forensic discussions. Desirous of honestly qualifying himself for the questorship, he commenced to study all the financial questions connected with it. Immediately after his election, he introduced, in spite of violent opposition from those interested, a rigorous reform into the treasury offices. He quitted the quæstorship at the appointed time amid general applause. In 63 B.C., he was elected tribune, and also delivered his famous speech on the Catiline conspiracy, in which he denounced Cæsar as an accomplice of that political desperado, and determined the sentence of the senate. Strongly dreading the influence of unbridled greatness, and not discerning that an imperial genius-like that of Cæsar-was the only thing that could remedy the evils of that overgrown monster, the Roman Republic, he commenced a career of what seems to us blind pragmatical opposition to the three most powerful men in Rome-Crassus, Pompey, and Cæsar. C. was a noble but strait-laced theorist, who lacked the intuition into circumstances which belongs to men like Cæsar and Cromwell. His first opposition to Pompey was successful; but his opposition to Cæsar's consulate for the year 59 not only failed, but even served to hasten the formation of the first triumvirate between Cæsar, Pompey, and Crassus. He was afterwards forced to side with Pompey, who had resiled from his connection with Cæsar, and become reconciled to the aristocracy. After the battle of Pharsalia (48 B.C.), C. intended to join Pompey, but hearing the news of his death, escaped into Africa, where he was elected commander by the partisans of Pompey, but resigned the post in favor of Metellus Scipio, and undertook the defence of Utica. Here, when he had tidings of Cæsar's decisive victory over Scipio at Thapsus (April 6, 46 B.C.), C., finding that his troops were wholly intimidated, advised the Roman senators and knights to escape from Utica, and make terms with the victor, but prohibited all intercessions in his own favor. He resolved to die rather than surrender, and, after spending the night in reading Plato's "Phædo," committed suicide by stabbing himself in the breast. CATO, Dionysius, is the name prefixed to a little volume of moral precepts in verse, which was a great favorite during the middle ages. Whether or not such a person ever existed, is a point of the greatest uncertainty. The title which the book itself commonly bears, is "Dionysii Catonis Disticha de Moribus ad Filium." Its contents have been differently estimated: some scholars have considered the precepts admirable; others, weak and vapid: some have found indications of a superior scriptural knowledge; others, of a deep-rooted paganism. The style has been pronounced the purest Latin and the most corrupt jargon. The truth would seem to be, that on a groundwork of excellent Latin of the Silver Age, the illiterate monks of a latter period have, as it were, inwoven a multitude of their own barbaric errors, which preclude us from determining precisely the period when the volume was composed. It begins with a preface addressed by the supposed author to his son, after which come 56 injunctions of rather a simple character, such as 66 parentem ama." This is followed by the substance and main portion of the bookviz., 144 moral precepts, each of which is expressed in two dactylic hexameters. During the middle ages, the "Disticha " was used as a text-book for young scholars. In the 15th c., more than 30 editions were printed. The best edition, however, is that published at Amsterdam in 1754 by Otto Arntzenius. Caxton translated it into English. CA'TODON and Catodo'ntida. See CACHOLOT. CATO PTRICS. The divisions of the science of optics are laid out and explained in the article OPTICS (q. v.). C. is that subdivision of geometrical optics which treats of the phenomena of light incident upon the surfaces of bodies, and reflected therefrom. All bodies reflect more or less light, even those through which it is most readily transmissible; light falling on such media, for instance, at a certain angle is totally reflected. Rough surfaces scatter or disperse (see DISPERSION OF LIGHT) a large portion of what falls on them, through which it is that their peculiarities of figure, color, &c., are seen by eyes in a variety of positions; they are not said to reflect light, but there is no doubt they do, though in such a way, owing to their inequalities, as never to present the proper phenomena of reflection. The surfaces with which C. accordingly deals, are the smooth and polished. It tracks the course Catoptrics of rays and pencils of light after reflection from such surfaces, and determines the positions, and traces the forms, of images of objects as seen in mirrors of different kinds. A ray of light is the smallest conceivable portion of a stream of light, and is represented by the line of its path, which is always a straight line. A pencil of light is an assemblage of rays constituting either a cylindrical or conical stream. A stream of light is called a converging pencil when the rays converge to the vertex of the cone, called a focus; and a diverging pencil, when they diverge from the vertex. The axis of the cone in each case, is called the axis of the pencil. When the stream con. sists of parallel rays, the pencil is called cylindrical, and the axis of the cylinder is the axis of the pencil. In nature, all pencils of light are primarily diverging-every point of a luminous body throwing off light in a conical stream; converging rays, however, are continually produced in optical instruments, and when light diverges from a very distant body, such as a fixed star, the rays from it falling on any small body, such as a reflector in a telescope, may, without error, be regarded as forming a cylindrical pencil. When a ray falls upon any surface, the angle which it makes with the normal to the surface at the point of incidence is called the angle of incidence; and that which the reflected ray makes with the normal, is called the angle of reflection. Two facts of observation form the groundwork of catoptrics. They are expressed in what are called the laws of reflection of light: 1. In the reflection of light, the incident ray, the normal to the surface at the point of incidence, and the reflected ray, lie all in one plane. 2. The angle of reflection is equal to the angle of incidence. These laws are simple facts of observation and experiment, and they are easily verified experimentally. Rays of all colors and qualities follow these laws, so that white light, after reflection, remains undecomposed. The laws, too, hold, whatever be the nature, geometrically, of the surface. If the surface be a plane, the normal is the perpendicular to the plane at the point of incidence; if it be curved, then the normal is the perpendicular to the tangent plane at that point. From these laws and geometrical considerations may be deduced all the propositions of catoptrics. In the present work, only those can be noticed whose truth can in a manner be exhibited to the eye, without any rigid mathematical proof. They are arranged under the heads Plane Surfaces and Curve Surfaces. Plane Surfaces.-1. When a pencil of parallel rays falls upon a plane mirror, the reflected pencil consists of parallel rays. A glance at the annexed figure (fig. 1), where PA and QB are two of the incident rays, and are reflected in the directions AR and BS respectively, will make the truth of this pretty clear to the eye. The proposition, however, may be rigidly demonstrated by aid of Euclid, book xi., with which, however, we shall not presume the reader to be acquainted. The reader may satisfy himself of Fig 2. its truth practically R MN P -2. If a diverging or converging pencil is incident on a plane mirror, the focus of the reflected pencil is situated on the opposite side of the mirror to that of the incident pencil, and at an equal distance from it. Suppose the pencil to be diverging from the focus Q (fig. 2), on the mirror of the surface of which CB is a section. Draw QNq perpendicular to CB, and make 9N-QN, then q is the focus of the reflected rays. For let QA, QB, QC be any of the incident N M 'rays in the plane of the figure; draw the line AM perpendicular to CB, and draw AR, making the angle MAR equal to the angle of incidence, MAQ. Then AR is the reflected ray. Join qA. Now it can be proved geometrically, and indeed is apparent at a glance, that qA and AR are in the same straight line; in other words, the reflected ray AR proceeds as if from q. In the same way, it may be shewn that the direction of any other reflected ray, as BS, is as if it proceeded from q; in other words, q is the focus of reflected rays; it is, however, only their virtual focus. See Focus. If a pencil of rays converged to q, it is evident that they would be reflected to Q as their real focus, so that a separate proof for the case of a converging pencil is unnecessary. The reader who has followed the above, will have no difficulty in understanding how the position and form of the image of an object placed before a plane mirror-as in fig. 3, where the object is the arrow AB, in the plane of the paper, to which the plane of the a mirror is perpendicular-should be of the same form and magnitude as the object (as ab in the fig.), and at an equal distance from the mirror, on the opposite side of it, but with its different parts inverted with regard to a given direction. The highest point a, for instance, in the image corresponds with the lowest point, A, in the object. He will also understand how, in the ordinary use of a looking-glass, the right hand of the image corresponds to the left hand of the object. n mb When two plane mirrors are placed with their reflecting surfaces towards each other, and parallel, they form the experiment called the Endless Gallery. Let (in fig. 4) the arrow Q, be placed vertically between the parallel mirrors, CD, BA, with their silvered faces turned to one another, Q will produce in the mirror CD the image 1. This image will act as a new object to produce with the mirror BA the image q2, which, again, will produce with the mirror CD another image, and so on. Another series of images, such as q, q', &c., will similarly be produced at the same. time, the first of the series being q', the image of Q in the mirror BA. By an eye placed between the mirrors, the succession of images will be seen as described; and if the mirrors were perfectly plane and parallel, and reflected all the light incident, on them, the number of the images of both series would be infinite. If, instead of being paralled, the mirrors are inclined at an angle, the form and position of the image of an object may be found in precisely the same way as in the former case, the image formed with the first mirror being regarded as a new (virtual) object, whose image with regard to the second has to be determined. For a curious application of two plane mirrors meeting and inclined at an angle an aliquot part of 180°, see art. KALEIDOSCOPE-3. The two propositions already established are of extensive application, as his partly been shewn. They include the explanation of all phenomena of light related to plane mirrors. The E B Fig 5. third proposition is one also of considerable utility, though not fundamental. It is: When a ray of light has been reflected at each of two mirrors inclined at a given angle to each other, in a plane perpendicular to their intersection, the reflected ray will deviate from its original course by an angle double the angle of inclination of the mirrors. Let A and B (fig 5) be sections of the mirrors in a plane perpendicular to their intersection, and let their directions be produced till they meet in C. Let SA, in the plane of A and B. be the ray incident on the first inirror at A, and let AB be the line in which it is thence reflected to B. After reflection at B, it will pass in the line BD, meeting SA, its original path produced in D. The angle ADB evidently measures its deviation from its original course, and this angle is readily shewn to be double of the angle at C, which is that of the inclination of the mirrors. It is on this proposition that the important mathematical instruments called the Quadrant and Sextant (q. v.) depend. Curved Surfaces.-As when a pencil of light is reflected by a curved mirror, each ray follows the ordinary law of reflection, in every case in which we can draw the normals for the different points of the surface, we can determine the direction in which the various rays of the pencil are reflected, as in the case of plane mirrors. It so happens that normals can be easily drawn only in the case of the sphere, and of a few "surfaces of revolution," as they are called. These are the paraboloid, the ellipsoid, and the hyperboloid of revolution. The paraboloid of revolution is of importance in optics, as it is used in some specula for telescopes. See arts. SPECULUM and TELESCOPE. The three surfaces last named are, however, all of them interesting, as being for pencils of light incident in certain ways what are called surfaces of accurate reflection-i. e., they reflect all the rays of the incident pencil to a single point or focus. We shall explain to what this property is owing in the case of the parabolic reflector, and state generally the facts regarding the other two. A P F G 1. The concave parabolic reflector is a surface of accurate reflection for pencils of rays parallel to the axis or central line of figure of the paraboloid. This results from the property of the surface, that the normal at any point of it passes through the axis, and bisects the angle between a line through that point, parallel to the axis, and a line joining the point to the focus of the generating parabola. Referring to fig. 6, suppose a ray incident on the surface at P, in the line SP. parallel to the axis AFG. Then if F be the focus of the generating parabola, join PF. PF is the direction of the reflected ray. For PG, the normal at P, by the property of the surface, bisects the angle FPS, and therefore (angle) FPG= /GPS. But SPG is the angle of incidence, aud SP, PG, and FP are in one plane, and therefore, by the laws of reflection, FP is the reflected ray. In the same way, all rays whatever parallel to the axis, must pass through F after reflection. If F were a luminous point, the rays from it, after reflection on the mirror, would all proceed in a cylindrical pencil parallel to the axis. This reflector, with a bright light in its focus, is accordingly of common use in light-houses. 2. In the concave ellipsoid mirror there are two points-viz, the foci of the generating ellipse, such that rays diverging from either will be accurately reflected to the other. This results from the property of the figure, that the normal at any point bisects the angle included between lines drawn to that point from the foci. Fig 6 Cat's-eye 3. Owing to a property of the surface similar to that of the ellipsord, a pencil of rays converging to the exterior focus of a hyperbolic reflector, will be accurately reflected to the focus of the generating hyperbola. The converse of the above three propositions holds in the case of the mirrors being convex. Though the sphere is not a surface of accurate reflection, except for rays diverging from the centre, and which on reflection are returned thereto, the spherical reflector is of great practical importance, because it can be made with greater facility and at less expense than the parabolic reflector. See art. TELESCOPE. It is necessary, then, to investigate the phenomena of light reflected from it. Spherical Mirrors.-It is usual to treat of two cases, the one the more frequent in practice, the other the more general and comprehensive in theory. First, then, to P A B F B S find the focus of reflected rays when a small pencil of parallel rays is incident directly on a concave spherical mirror. Let BAB' (fig. 7) be a section of the mirror, O its centre of curvature, and A the centre of its aperture. AO is the axis of the mirror, and therefore of the incident pencil, because it is incident directly on the mirror; a pencil being called oblique when its axis is at an angle to the axis of the mirror. As the ray incident in the line OA will be reflected back in the same line-OA being the normal at A-the focus of reflected rays must be in OA. Let SP be one of the rays; it will be reflected so that /qPO = L SPO. But Poq LOPS by parallel lines. Therefore, qPO LqOP, and Pq and Oq are equal. If, now, the incident pencil be very small-i. e., if P be very near A-then the line Pq will very nearly coincide with the line OA, and Pq and Og will each of them become very nearly the half of OA. Let F be the middle point of OA-the point, namely, to which ૧ tends as the pencil diminishes. Then F is called the principal focus of the mirror, and AF the principal focal length, which is thus radius of the mirror. It will be observed that when AP is not small, q lies between A and F. Fq is called the aberration of the ray. When AP is large, the reflected rays will continually intersect, and form a luminous curve with a cusp at F. This curve is called the caustic (q. v.). We shall now proceed to the more general case of a small pencil of diverging rays, incident directly on a concave spherical mirror. Let PAP' (fig. 8) be a section of A { Fig 7. Fig 8 tion is deduced the = the mirror, A the centre of its aperture, O of its QA formula qA very nearly. From this equa qA QAX AF QA-AF which enables us to find qA, when QA and AF are known. Thus, let the radius of curvature be 12 inches, and the distance of the source of the rays, or QA, 30 inches, the focal length qA 30 X 6 7 inches. If the rays had diverged from, q, it is clear they 30-6 |