objects. If the tube be placed in a horizontal position, and the plate of ground glass be illuminated with a powerful light, a six-rayed star will be seen upon the opposite screen, decorated with the richest ornamentation.* This is the reflexion in the mirror of the fragments of glass which are combined to form this regular mosaic. If the cap be turned, the pieces of glass constantly form new combinations, and thus an inexhaustible succession of the most delicate forms are obtained which the liveliest fancy could scarcely invent. What may in this way be represented for a large number of persons, as if it were an object on the screen, can also of course be seen separately by every one who looks into the tube for himself. 21. Not only this ingenious plaything, but an in strument of high practical value, is founded on the mutual action of two mirrors placed at an angle to one another. In fig. 23, A and B are two small mirrors, In this experiment a lens of short focus is placed at the front end of the kaleidoscope. the reflecting surfaces of which are turned towards each other. If two objects are placed at L and R, of which the former is visible to an observer at 0, above the edge of the mirror B, in the direction OB, the mirror A may have such a position given to it that the light coming from R reaches the eye after double reflexion in the direction RA B O, and consequently two objects are seen in the same direction, O B, the one direct, the other reflected. Thus it results from the law of reflexion that the angle a which is included by the visual lines extending from the eye to L and R, is exactly twice as large as the angle B which the two planes of the mirrors form with one another. In order to measure FIG. 24. the angle ẞ conveniently, the mirror A is made to rotate around the axis of a divided arc, M N, and is connected with an indicator, AZ. The mirror B is permanently fixed on the plane of the arc parallel to the radius AM which goes to the zero of the division. If any Mirror or reflecting sextant. object, L, be now looked at in the direction OL through a telescope attached to the instrument (fig. 24) and the indicator, and with it the mirror, be rotated until the image of R is seen in this direction, twice the angle read off by the If the perpendiculars A E and B D, which if sufficiently prolonged cut one another in the point D at an angle B are erected upon the mirror planes, it follows if and indicate the angle of incidence of the rays RA and A B on the mirrors A and B from the consideration of the triangle A B D Φ B=4-4 and from the consideration of the triangle AO B a=24-24 from whence it immediately follows that a= indicator immediately gives the angle which the visual lines directed towards L and R form with each other. This ingenious angle measurer, conceived by Newton and constructed by Hadley, is termed the reflecting sextant. It is superior to other instruments made for this purpose because it needs no support, but during the act of measuring the angle can be held freely in the hand. Hence for nautical purposes it is the only available angle-measuring instrument. By means of the reflecting sextant the seafarer makes those measurements by which he determines the latitude and longitude of his ship. The two almost invisible mirrors enable him to pursue his predetermined course through the pathless waste of waters. FIG. 25. CHAPTER IV. SPHERICAL MIRRORS. 22. A spherical shell, the inner surface of which is highly polished, is called a spherical concave mirror. It may be regarded as a portion of a hollow sphere cut off by a plane M M', fig. 25. A perpendicular, cd, let fall from the centre, c, of the sphere of which the mirror is a segment upon this plane, will strike the middle point of the mirror, and is termed its principal axis. The angle Mc M', which the lines Mc and M'c, drawn from two diametrically opposite points of the periphery of the mirror to the centre of the sphere form with one another, is called the aperture of the mirror. In practice, only mirrors of small aperture are in use, in which this angle amounts at most to six or eight degrees, and the remarks here made will only have reference to these. Concave mirror. If a beam of parallel solar rays, thrown horizontally into the chamber by means of the Heliostat, be allowed to fall upon a concave mirror with small aperture parallel to its axis (fig. 26), it will be seen-for the path of the rays can be distinctly followed by the illu FIG. 26. mination of the particles of dust always present in the air of a room-that it is reflected in the form of a cone of light, the apex of which, F, lies in front of the mirror in this axis. This point, F, through which all the rays falling on the mirror parallel to its axis pass after reflexion, is called the focus. It becomes brilliantly luminous if I throw some dust into the air in its vicinity. It appears as a white spot of dazzling brilliancy when Focus. a white sheet of paper is held in it, and the wreaths of smoke that are now rising from it show you that the paper has caught fire in the intense heat of the rays collected at this point, and that it has consequently been appropriately named the focus or burning-point (Brennpunkt). The space intervening between the focus and the mirror-the focal distance-can easily be measured, and is found to be equal to half the radius of curvature of the mirror, or in other words, the focus lies midway between the mirror and the centre of the circle of which it is a segment. 23. The reflexion of a ray of light from a curved surface follows the same law as from a plane surface; the portion of the curve which immediately surrounds the minute point of incidence on that each ray of light impinges can alone be considered to act as a reflector. The smaller we admit the superficial area of this part to be-and we may conceive it to be as small as we please -so much the more accurately can we regard it as a small plane mirror, and the perpendicular erected upon this is then the axis of incidence in regard to which the incident and the reflected ray behave as has been |