On a Generalization of Poncelot's Theorems for the Linear Representation of Quadratic Radicals. By Professor SYLVESTER, M.A., F.R.S. The author explained the application of Poncelet's theorems, to practical questions of mechanics in the case of forces acting in a single plane as in the theory of bridges. He next referred to the mode of extension of this theorem, suggested by Poncelet, applicable to the case of forces in space, and pointed out its insufficiency, and, in a certain sense, its incorrectness. The essential preliminary question to be resolved in the first instance (after which the matter became one of easy calculation), was shown to be that of cutting off by a plane the smallest possible segment of a sphere that should contain the whole of a given set of points lying on the sphere's surface. Some years ago Prof. Sylvester had proposed in the Quarterly Mathematical Journal,' without any suspicion of its having any practical applications, the following question :-" Given a set of points in a plane to draw the smallest possible circle that should contain them all." By a singular coincidence, Professor Pierce, of Cambridge University, U.S., had studied this question and obtained a complete solution of it, which he had communicated to the author during the present meeting of the British Association. A slight consideration served to show that precisely the same solution as Professor Pierce had found for the problem of points in a plane was applicable with a merely nominal change to the sphere also; and thus the solution of a question set almost in sport was found to supply an essential link for the complete development of a method of considerable importance in practical mechanics. The author stated that it would be easy to draw up tables of the values of the constants appearing in the linear function, representing the resultant of three forces at right angles to one another, for the principal cases likely to occur in practice, the values of these constants depending solely upon the condition of relative magnitude to which the component forces are supposed to be subjected. LIGHT, HEAT. On the Influence of very small Apertures on Telescopic Vision. [The manuscript of this paper has been lost.] On some Optical Illusions connected with the Inversion of Perspective. By Sir DAVID BREWSTER, K.H., F.R.S. The term "Inversion of Perspective" has been applied to a class of optical illusions, well known and easily explained, in which depressions are turned into elevations, and elevations into depressions. One of the most remarkable cases of this kind, which has not yet been explained, presented itself to the late Lady Georgiana Wolf, and has been recorded by her husband Dr. Wolf. When she was riding on a sand-beach in Egypt, all the footprints of horses appeared as elevations, in place of depressions, in the sand. No particulars are mentioned, in reference to the place of the sun, or the nature of the surrounding objects, to enable us to form any conjecture respecting the cause of this phenomenon. Having often tried to see this illusion, I was some time ago so fortunate as not only to observe it myself, but to show it to others. In walking along the west sands of St. Andrews, the footprints, both of men and of horses, appeared as elevations. In a short time they sank into depressions, and subsequently rose into elevations. The sun was at this time not very far from the horizon, on the right hand; and on the left there were large waves of the sea breaking into very bright foam. The only explanation which occurred to me was, that the illusion appeared when the observer supposed that the footprints were illuminated with the light of the breakers, and not by the sun. Having, however, more recently observed the phenomenon, when the sun was very high on the right, and the breakers on the left very distant, and consequently very faint, I could not consider the preceding explanation as well-founded. Upon attending to the circumstances under which they were now seen, I observed that the human footprints were all covered with dry sand that had been blown into them, so that they were much brighter than the surrounding sand, and than the dark side of the impression next the sun; and hence it is probable that they appeared to be nearer the eye than the dark sand in which they were formed, and consequently elevations. After repeated examinations of them, I found the footprints appeared as elevations as far as the eye could see them; and they were equally visible with one or both eyes. But whenever the eye rested for a little while on the nearest footprint, it resumed its natural concavity. I have observed other illusions of this kind, which are more easily explained, though they differ from any hitherto described. In the Church of Saint Agostino in Rome, there is above each arch a painted festoon suspended on two short pillars; but instead of appearing in relief, as the painter intended, by shading the one side of them, they appeared concave, like an intaglio. In other positions in the church they rose into relief. Upon a subsequent visit to the church, I found that the festoon, or suspended wreath, was concave when it was illuminated, or rather when the observer saw that it was illuminated, by a window beneath it, and in relief when the eye saw that it was illuminated by a window above it, the object being similarly illuminated in both cases. In the common cases of inverted perspective, the eye is deceived by looking at the inversion of the shadow in the cameo or intaglio itself; but in the present case the eye is deceived by perceiving that the body painted, supposed to be in relief, is illuminated by a light either above or below it. An optical illusion of a different kind presented itself to me in the Church of Santa Giustina at Padua. Upon entering the church we see three cupolas. The one beneath which we stood appeared very shallow; the next appeared much deeper, and the third deeper still. They were all, however, of the same depth, as we ascertained by placing ourselves under each in succession, and observing that it was always the shallowest. On Microscopic Vision, and a New Form of Microscope. By Sir DAVID Brewster, K.H., F.R.S. In studying the influence of aperture on the images of bodies as formed in the camera, by lenses or mirrors, it occurred to me that in microscopic vision it might exercise a still more injurious influence. Opticians have recently exerted their skill in producing achromatic object-glasses for the microscope with large angles of aperture. In 1848 the late distinguished optician, Mr. Andrew Ross, asserted "that 135° was the largest angular pencil that could be passed through a microscopic object-glass," and yet in 1855 he had increased it to 170! while some observers speak of angular apertures of 175°. In considering the influence of aperture, we shall suppose that an achromatic object-glass with an angle of aperture of 170° is optically perfect, representing every object without colour and without spherical aberration. When the microscopic object is a cube, we shall see five of its faces; and when it is a sphere or a cylinder, we shall see nine-tenths or more of its circumference. How then does it happen that large apertures exhibit objects which are not seen when small apertures with the same focal length are employed? This superiority is particularly shown with test-objects marked with grooves or ridges, and obliquely illuminated. The marginal part of the lens will enlarge the grooves and ridges, and they will thus be rendered visible, not because they are seen more distinctly, but because they are expanded by the combination of their incoincident images. Hence we have an explanation of the fact-well known to all who use the microscope,-that objects are seen more distinctly with object-glasses of small angular aperture. In the one case we have, with the same magnifying power, not only an enlarged and indistinct image of objects, but a false representation of them, from which their true structure cannot be discovered; while in the other we have a smaller and distinct image, and a more correct representation of the object. But these are not the only objections to large angular apertures and short focal lengths. 1. In the first place, it is extremely difficult to illuminate objects when so close to the object-glass. 2. There is a great loss of light, from its oblique incidence on the surface of the first lens. 3. The surface of glass,-with the most perfect polish,-must be covered with minute pores, produced by the attrition of the polishing powder; and light, falling upon the sides of these pores with extreme obliquity, must not only suffer diffraction, but be refracted less perfectly than when incident at a less angle. 4. When the object is almost in contact with the anterior lens, the microscope is wholly unfit for researches in which mechanical or chemical operations are required, and also for the examination of objects enclosed in minerals or other transparent bodies. 5. In object-glasses now in use, the rays of light must pass through a great thickness of glass of doubtful homogeneity. It is a question yet to be solved whether or not a substance can be truly transparent, --in which the elements are not united in definite proportion,-in which the substances combined have very different refractive and dispersive powers; and in which the particles are so loosely united that they separate from one another, as in the various kinds of decomposition to which glass is liable. If the best microscopes are affected by these sources of error, every exertion should be made to diminish or remove them. 1. The first step, we conceive, is to abandon large angular apertures, and to use object-glasses of moderate focal length, effecting at the eye-glass any additional magnifying power that may be required. 2. In order to obtain a better illumination, either by light incident vertically or obliquely, a new form of the microscope would be advantageous. In place of directing the microscope to the object itself, placed as it now is almost touching the object-glass, let it be directed to an image of the object, formed by the thinnest achromatic lens, of such a focal length that the object may be an inch or more from the lens, and its image equal to, or greater, or less than the object. In this way the observer will be able to illuminate the object, whether opake or transparent, and may subject it to any experiments he may desire to make upon it. It may thus be studied without a covering of glass, and when its parts are developed by immersion in a fluid. 3. The sources of error arising from the want of perfect polish and perfect homogeneity of the glass of which the lenses are composed, are, to some extent, hypothetical; but there are reasons for believing, and these reasons corroborated by facts,-that a body whose ingredients are united by fusion, and kept in a state of constraint from which they are striving to get free, cannot possess that homogeneity of structure, or that perfection of polish, which will allow the rays of light to be refracted and transmitted without injurious modifications. If glass is to be used for the lenses of microscopes, long and careful annealing should be adopted, and the polishing process should be continued long after it appears perfect to the optician. We believe, however, that the time is not distant when transparent minerals, in which their elements are united in definite proportions, will be substituted for glass. Diamond, topaz, and rock-crystal are those which appear best suited for lenses. The white topaz of New Holland is particularly fitted for optical purposes, as its double refraction may be removed by cutting it in plates perpendicular to one of its optical axes. In rock-crystal the structure is, generally speaking, less perfect along the axis of double refraction than in any other direction, but this imperfection does not exist in topaz. On the decomposed Glass found at Nineveh and other places. The different kinds of glass which are in common use, consist of sand or silex combined by fusion with earths or alkalies, or metals which either act as fluxes, or communicate different colours or different degrees of lustre or refractive power to the combination. In quartz or rock-crystal, which is pure silex, and in other regularly crystallized bodies, the molecules or atoms unite in virtue of regular laws, the pole of one atom uniting with the pole of another. Such substances, therefore, do not decompose under the ordinary action of the elements. The lens of Rock-Crystal, for example, found by Mr. Layard at Nineveh, is as sound as it was many thousand years ago when in the form of a crystal. In the case of glass, however, the silex has been melted and forced into union with other bodies to which it has no natural affinity; and therefore its atoms, which have their similar poles lying in every possible direction, have a constant tendency to recover their crystalline position as when in a state of silex. For the same reason, the earths, alkalies and metals, with which the atoms of silex have been constrained by fusion to enter into union, all tend to resume their crystalline position and separate themselves from the silex. Owing to the manner in which melted glass is cooled and annealed, whether it is made by flashing or blowing, or moulding, the cohesion of its parts is not the same throughout the mass; and consequently its particles are held together with different degrees of force, varying in relation to points, lines, and surfaces. An atom of the flux, or other ingredient, may be less firmly united to an atom of silex in one place than in another, depending on the degree of heat by which they were combined, or upon the relative positions of the poles of the atoms themselves when combined. There are some remarkable cases where flint-glass without any rude exposure to the elements has become opake, and I have seen specimens in which the disintegration had commenced a few years after it was made. In general, however, the process is very slow, excepting in stables, where the prevalence of ammonia hastens the decomposition and produces all the beautiful colours of the soap-bubble. It is, however, from among the ruins of ancient buildings that glass is found in all the stages of decomposition; and there is perhaps no material body that ceases to exist with such grace and beauty, when it surrenders itself to time and not to disease. In damp localities, where acids and alkalies prevail in the soil, the glass rots as it were by a process which, owing to the opacity of the rotten part, it is difficult to study. It may be broken between the fingers of an infant; and we often find in the middle of the fragment a plate of the original glass which has not yielded to the process of decay. In dry localities, where Roman, Greek, and Assyrian glass has been found, the process of decomposition is exceedingly interesting, and its results singularly beautiful. At one or more points in the surface of the glass the decomposition begins. It extends round that point in spherical surfaces so that the first film is a minute hemispherical cup of exceeding thinness. Film after film is formed in a similar manner, till perhaps twenty or thirty are crowded into the 50th of an inch. They now resemble the section of a pearl or of an onion, and as the films are still glass, the colours of thin plates are seen when we look down through their edges which form the surface of the glass. These thin edges, however, being exposed to the elements, suffer decomposition. The particles of silex and the other ingredients now readily separate, and the decomposition goes on downwards in films parallel to the surface of the glass, the crystals of silex in one specimen forming a white ring, and the other ingredients rings of a different colour. (See the Figure.) Such is the process round one point, but the decomposition commences at many points, and generally these points lie in lines, so that the circles of decomposition meet one another and form sinuous lines. When there are only two points, these circles, when they meet, surround the two points of decomposition like the rings round two knots of wood; and in like manner, when there are many points, and these points near each other, the curves of decomposition unite as already mentioned, and form sinuous lines. When the decomposition is uniform and the little hemispheres have nearly the same depth, we can separate the upper film from the one below it, the convexities of the one falling into the concavities of the other. This general description was illustrated by drawings on the table, all of which were executed by Miss Mary King, of Ballylin, now the Hon. Mrs. Ward. But beautiful and correct as these drawings are, they convey a very imperfect idea of the brilliant colours and singular forms which characterize glass in a particular stage of its decomposition, and of the optical phenomena which it exhibits in common and polarized light. When the decomposition has gone regularly on round a single point, and there is no other change, a division of the glass into a number of hemispherical films within one another takes place, the group of films exhibiting in the microscope circular cavities, which under different circumstances become elliptical and polygonal. In salt water the decomposition of glass goes on more rapidly, as I have found in examining one of the bottles brought up in the wreck of the 'Royal George;' and the same effect may be produced by a quicker process. M. Brame, of Paris, having seen a notice of the decomposed glass from Nineveh which I read at the Association some years ago, succeeded in producing, in a very short time, regular and irregular circles of decomposition, in the centre of which there was always a small cavity or nucleus. This effect was obtained by immersing fragments of thick glass in a mixture of fluoride of calcium and concentrated sulphuric acid, or by exposing them to the action of the vapour of fluorhydrique acid. Such are some of the general phenomena of decomposed glass when seen by light reflected from its exposed surfaces; but when we separate the films and examine them in the microscope, either by common or polarized light, a series of phenomena are seen of the most beautiful kind,- -so various and so singular that it would be a vain attempt to describe them. A general idea of them, however, may be obtained from the drawings, and from a description of three varieties of these films. I. The first of these varieties has rough surfaces,-the roughness arising from an almost infinite number of hemispherical cavities on one side of the film, and hemispherical convexities on the other side. When these cavities are separated by flat portions of the film, they are perfectly circular; but when they are crowded together, they are irregularly polygonal, the sides of the polygons forming a sort of network, the cavities or convexities forming the meshes of the net. The convex and concave surfaces are not rough but specular, and reflect and transmit white light, exhibiting none of the colours of thin plates. In polarized light, each of the cavities, whether circular or polygonal, act as negative uniaxal crystals, exhibiting by the interference of the refracted and transmitted pencils the black cross, and the white of the first order in Newton's scale, rising sometimes to yellow or falling to the palest blue, or disappearing altogether, according to the number or curvature of the films which compose it. II. The second variety of these films has perfectly specular surfaces, in consequence of having almost no cavities. They exhibit in common light, and in a very beautiful manner, the colours of thin plates, the transmitted being complementary to the reflected light. This variety is exceedingly rare. In a specimen on the table the reflected light is blue and the transmitted yellow. In some of the fragments a few insulated circular cavities with the black cross occur, the tints which surround it being modified by the general tint of the film. III. The third variety of decomposed glass consists of films containing cavities of all sizes and forms, from the 30th of an inch to such a size that they are hardly visible by the microscope, giving to the film which they compose a sort of stippled appearance, or an imperfectly specular surface. These cavities or combinations of hemispherical films are circular, elliptical, or irregularly polygonal. The colours which they reflect and transmit are complementary, and the tints and rings which in polarized light surround the black cross are curiously modified by the general tint of the fragment, and the curvature of its component films,-the black cross itself varying its shape with the form of the cavities. When the cavities are flat, the black cross disappears as in thin slices of uniaxal crystals; but the tints reappear, rising to higher orders by inclining the plate. The cavities are often arranged in sinuous curves, and encroach upon one another, so that the polarized tints appear only at the margin of the line which they form. They frequently run in perfectly straight lines, and when they are very small and invisible as cavities, their margins form in polarized light brilliant lines, which are often grouped in bands like the stripes in a ribbon. Sometimes they are only a few thousands of an inch in diameter, and might be used as micrometers in the microscope, every trace of the cavities which form them having disappeared. These lines of polarized light all disappear when they lie in the plane of polarization of the incident light, or perpendicular to that plane. In some specimens a decomposition has taken place on several points of the convex or concave surfaces of the cavities, so as to form new cavities; and each of these minute cavities, often ten or twelve in number, exhibit the black cross with its tints, but disfiguring, of course, those of the cavity upon which they have encroached. In the three varieties of decomposed glass which I have described, the films are * Comptes Rendus, &c., Nov. 2, 1852. |