it becomes for a short space sensibly uniform, and at length de creases very slowly, according to the common constitution of the atmosphere. If we suppose an observer placed in a stratum of mean density, and looking at a distant object situated also in this stratum, he will be able to see it in two ways; directly through the uniform stratum which intervenes, and indirectly by the rays reflected from the inferior stratum. These rays at first directed from the object towards the terrestrial surface, with a certain inclination, enter the strata of less density, are refracted there, taking a direction more nearly approaching to a horizontal line, whence they rise, and traversing the superior and denser strata which attract them, pass on to the eye of the observer. There will then be two images of the object, the one upright, by direct vision, the other inverted, by reflection. If the object be insulated on the dark ground of the sky, an inverted image of the sky will surround also the reflected image of the object exactly as when objects are represented by reflection on the surface of water. 87. We are hence able to explain a very curious phenomenon, known to French mariners under the name of mirage,† and which the French army had many opportunities of observing in their expedition to Egypt. The land of Lower Egypt is a vast horizontal plain, the uniformity of which is interrupted only by some small eminences which serve for the sites of villages, and which are thus rescued from the inundations of the Nile. Morning and evening, the appearance of the country agrees with the actual disposition and distance of the objects; but when the surface of the soil is heated by the presence of the sun, the ground seems to terminate at a certain distance by a general inundation, and the villages beyond resemble islands situated in the midst of a large lake. Beneath each village is seen its inverted image exactly as it appears in water. As we advance, the limits of this inundation appear to recede; the imaginary lake which seems to surround the village retires; and at length entirely vanishes, while the appearance is reproduced with respect to another village more distant. Thus, according to Monge, from whom this account is borrowed, every thing conspires to complete an illusion which is sometimes painful, especially in the The English word looming is sometimes used to denote this appearance, generally however it is employed in a more restricted sense. desert, since it tantalizes one with the show of water at the very time when he has the greatest need of it. A similar appearance is exhibited at sea in very calm weather. A ship seen at a distance and in the horizon, sometimes presents two images, one direct, the other inverted. The second answers exactly to the first, and is equally distinct; indeed the effect is precisely that of a reflection from a mirror. Hence the name, mirage, which mariners have given to this phenomenon. As it is produced by the difference of heat between the water and air, it is commonly seen after sudden changes of temperature; since the density of the water at the surface of the sea does not admit of its taking these variations so rapidly as the air. But on the other hand, the temperature of the water, and the evaporation which is constantly going on at its surface, prevent it from taking so high a temperature as the sandy surface of an arid soil. For these reasons, the phenomenon of double images occurs rarely at sea, and continues only for a short time; whereas in Egypt and on some sandy plains in which the same circumstances present themselves at the same height of the sun, it occurs every clear day. 88. M. Mathieu and myself observed at Dunkirk, on the seashore, many phenomena of this kind, and I have given the mathematical, theory of them.* I have proved that the consecutive paths or trajectories, traced from the eye of the observer, interest in their second branches, in such a manner as to form a caustic, below which no point can be seen. The curve LT. represents this caustic, and DMS is the limiting trajectory, drawn from the eye of the observer, a tangent to the ground. I call it limiting trajectory, because it limits the height where the inversion takes place. In this figure all the points situated above the trajectory can send only one image to the eye of the observer. Those situated in the space SLT send two, the superior being erect, and the inferior inverted. Lastly, those situated below the caustic in the space MLT, not being capable of sending any, are invisible; so that a moving object, a man, for example, withdrawing successively to different distances, will present the successive appearances represented in figure 73. Theory and experiment agree in proving that no very considerable difference of temperature is necessary in order that these *Memoirs of the Institute for 1809. Fig. 72. appearances may be produced. Two or three degrees of Fahrenheit's thermometer are sufficient, when the place of observation is a level and extended plain, which allows the luminous rays to be prolonged without obstacle, and thus to manifest the cur vature of the trajectories described by them. Of this kind was the station selected at Dunkirk, it being a sandy beach situated in the downs, near fort Risban; and the observations were also favored by the existence of a great number of distant objects, such as steeples, trees, and cottages, which, rising like so many signals above the dry soil, manifested the course of the rays by the appearances they presented. Moreover, the phenomenon of the doubling and inverting of the images, presented itself almost every day, arising from differences of temperature not exceeding three degrees of Fahrenheit. 89. If we suppose a like difference of temperature to exist between two strata of air, not placed vertically one above the other, but laterally contiguous, the phenomenon will still present itself; only it will take place in a horizontal direction perpendicular to the common surface of the two strata. The late M. Jurine and M. Sorret observed an appearance of this kind on the lake of Geneva, at a place rendered very narrow by the approach of the opposite shores. Now upon examining the configuration of these shores, of which the one situated on the south side is bordered by high mountains, and the other, situated on the north side, is exposed to the rays of the sun, it will be easily perceived that such a difference of aspect, in calm weather, or even when a moderate wind is blowing parallel to the direction of the shores, may occasion for a moment, in the mass of air incumbent on the lake, a lateral inequality of temperature, and consequently of density; and this may produce inflexions in the luminous rays which traverse the mass of air in a longitudinal direction sufficiently near the limit where the lateral variation of density takes place. 90. It sometimes happens also, that distant objects seem suspended in the air; the image in this case is single, erect, and apparently not accompanied with any inverted figure.* *This phenomenon is called by the French suspension, and by the English looming. It has been shown in the memoir above cited, that when this phenomenon occurs, the second inverted image exists, but is infinitely reduced, so that we see only the erect image which is detached, on the inverted image of the sky. When vision thus takes place by trajectories convex towards the earth or sea, the reflection is negative; the apparent horizon is much lower than it ought to be, for the height where the observation is made. Mariners ought, therefore, to be on their guard with respect to this phenomenon which tends to occasion considerable errors in latitude. It is found, indeed, by experiment, that these errors often amount to four or five minutes. The apparent horizon will therefore be too low when the sea is warmer than the air. If, on the contrary, its temperature is lower, its density is increased, but by a much more rapid law than ordinary; and the apparent horizon is sometimes raised very considerably. These errors may be avoided by observing the height of the stars, not above the horizon of the sea, but above an artificial horizon, placed without the inferior strata, where the extraordinary variation of density always takes place. But this expedient is not always easy even on land; and on board of vessels it is altogether impracticable, on account of the motion of the sea. In this case the error may be corrected, by taking, if it be possible, the distance of the two opposite horizons of the sea. The excess of the sum above two right angles, will give double the apparent depression of the horizon which it is necessary to employ in the calculation. Thus we shall know this depression by taking half the result. Unfortunately this observation of the two horizons, proposed by Dr Wollaston, appears very difficult to be taken with accuracy; and besides, we can seldom expect to find the differences of temperature between the air and water to agree sufficiently to render the two depressions equal. If it is not in our power to correct the error to which we are liable under these circumstances, it is at least well to be apprized of its existence and tendency, in order that we may not be misled by it. Double Refraction. 91. It has already been stated, that a beam of light in traversing crystallized bodies, is generally divided into two portions of which the one, called the ordinary ray, follows the ordinary law of refraction assigned by Descartes; while the other, called the extraordinary ray, is subject to very different laws. According to a very ingenious remark, first made by Dufay and confirmed by all succeeding observations, this phenomenon takes place in all crystals, except those whose primitive form is the cube or some of its geometrical derivatives. The separation of the refracted rays, proceeding from an incident beam, is more or less considerable, according to the nature of the crystal, and the direction in which the light traverses it; but this last variation is subject to the same law for all substances. In general, there are two directions, and only two, in which the separation of the two rays is infinitely small or nothing, and their velocity the same. These two directions are called the axes of the crystal. In proportion as the refracted rays deviate from the axes, their velocities of transmission become different; and this diversity is manifested, in general, by the separation or divergence which takes place in the rays, when the obliquity of incidence or emergence is the same. The inequality of the velocities increases according as the refracted rays form greater angles with the two axes; and it is the greatest possible when these angles are both right angles, that is, when the refracted rays become perpendicular to the plane of the two axes. Among the different kinds of crystals there are some in which the angle formed by the two axes is absolutely nothing; these may be regarded therefore as having but one axis. It is thus indeed that we did at first consider them; but it is better to imagine two axes, coinciding with each other, in order to preserve the analogy, and for the purpose of comprehending all crystals under one law. This case, however, is evidently more simple than the general one, because the inequality of the velocities depends entirely upon a single angle; on this account it will be examined first; and to make the process still more easy, we shall take, for an example, the rhomboidal carbonate of lime, commonly called Iceland spar; which, besides being remarkable for a double refraction extremely energetic, has likewise the advantage of being frequently met with in collections of minerals. 92. The crystals of this substance present cleavages in three directions, remarkable for their evenness and the facility with which they are effected. They are hence often separated of themselves into an infinity of small rhomboidal solids, having each six faces, parallel two and two, and formed of similar par |