in viewed by polarized light. Why is it that they exhibit a black cross? The answer according to the old theory will be, that it is in consequence of the layers varying in density. But what answer will be given to the question, Why do the layers vary density? Without calling in aid a new hypothesis, that the layers are formed from within (?), this question cannot be answered by the mere formation of layers; whilst the rolling up and ultimate growing together of the band produces, as already observed, a condensation or pressure of the parts around the socalled kernel. Hence the amylum grain resembles an unannealed glass disc, and consequently in polarized light it must exhibit the coloured cross. 8. Many additional proofs of my theory might be advanced, but I have no wish to trespass on the patience of the reader, and therefore will leave the discovery of contradictions to my adversaries; for had I myself found the slightest inconsistency, in a physical point of view, I would at once have rejected the whole theory or admitted its weakness. In conclusion, I will merely observe that, according to my experience, this theory will apply to other kinds of amylum, all of which, as far as I have hitherto seen, produce after boiling the vesicle as described, and in their unfolding perfectly agree with their different kinds of curl. My sincere wish is, that my observations may soon be either fully confirmed or completely refuted; a proceeding, which cannot, however, be performed by mere arguments, but must be achieved by observation on the stage of the microscope itself. XL. On a Remarkable Property of the Diamond. By Sir DAVID BREWSTER, K.H., D.C.L., F.R.S., and V.P.R.S. Ed.* [With a Plate.] AVING had occasion, some years agot, to examine the structure of a diamond plano-convex lens which gave triple images of minute microscopic objects, I discovered, by a particular method of observation, that the whole of its plane surface was covered with hundreds of minute bands, some reflecting more and some less light; and I naturally drew the inference that this diamond consisted of a great number of layers of different reflective, and consequently refractive powers, from which arose all its imperfections as a single microscope. In this case the veins or layers lay parallel, or nearly so, to the axis of the lens, so as to produce the worst effect upon the refracted pencil; for if the axis of the lens had been perpendicular to the surfaces * From the Phil. Trans. 1841, pp. 41, 42. + This Journal, vol. vii. p. 245. of these veins, its performance as a microscope would scarcely have been injured by them. In repeating Mr. Airy's experiments on the action of the diamond in modifying Newton's rings near the polarizing angle, I was led to re-examine the flat surface of the diamond above mentioned; but though I found my former observations perfectly correct, yet I was induced to suspect the accuracy of the inference which I drew from them, and which I could not but draw in the circumstances under which the phænomenon was presented to me. In order that the Society may be able to judge of the new results at which I have arrived, I have given in Plate VI. fig. 4 as accurate a drawing as I am able to make of the appearance of the flat surface of the diamond under consideration, as seen by light incident upon it nearly perpendicularly. The flat surface of the diamond is 0.058, orth of an inch in diameter, and owing to the great convexity of its other surface, the light reflected by it does not interfere with the examination of the structure above mentioned. The appearance shown in the figure is that which I observed some years ago; but upon shifting the line of illumination, I was surprised to perceive that all the dark bands became light ones, and all the light bands became dark ones, a phænomenon which placed it beyond a doubt that all the bands were the edges of veins or lamina whose visible terminations were inclined at different angles, not exceeding two or three seconds to the general surface. Had this surface been an original face of the crystal there would have been nothing surprising in its structure, excepting the exceeding minuteness of the strata and the slight inclination of their terminal planes to each other; but being a surface ground and polished by art, the phænomenon which it presents is one extremely interesting. The mineralogist will have no hesitation in admitting that this diamond is part of a composite crystal consisting of a great number of individual crystals, like certain specimens of felspar, carbonate of lime, and other minerals; but it is more difficult to conceive that the terminal planes of these individual crystals should retain their relative inclination after undergoing the operations of grinding and polishing upon a lapidary's wheel. To many persons such a result may appear inadmissible; but there are several physical facts, which, when well considered, cannot fail to diminish its improbability. If we grind and polish a surface of mother-of-pearl obliquely to the strata of which it is composed, we shall find it impossible to produce a perfectly flat surface: even if we grind it on the finest and softest hone, and polish it with the smoothest powder, the termination of each stratum will remain; and while the general surface reflects a white image, the grooves or striæ will give rise to the beautiful prismatic images produced by interference*. Another analogous fact presented itself to me many years ago in examining calcareous spar. Having had occasion to form an artificial face upon one of the edges of the rhomb containing the obtuse angle, I used a coarse file without water, and found that it exposed faces of cleavage which had never been previously seen, and which were inclined to the general surface produced by the filet. In examining the optical figures produced by the disintegration of crystallized surfaces, I have found that by coarse sandstone, or the action of a rasp, or large-toothed file, we can expose surfaces of crystallization with their natural polish differently inclined to the general surface ‡. In all these cases the faces, exposed by the mechanical action of grinding or filing, preserve their natural surfaces and polish, and will preserve them more perfectly and readily if they are faces of easy cleavage. The facility of exposing such faces by the action of grinding must increase as the veins or strata become thinner, and it is probable that their exceeding minuteness in the diamond may have aided in the production of the structure which has been described. I have found it quite impossible to measure the inclination of any of the faces by the goniometer; but I have succeeded, though with some difficulty, in taking an impression of the grooved surface upon wax. This structure sufficiently explains the existence of three images when the lens was used as a microscope, without supposing that the veins had different refractive powers. Faces of different inclinations would, of course, converge the rays to different foci on the retina, as effectually as if there had been only a variation in their refractive indices. St. Leonard's College, St. Andrews, XLI. Geometry and Geometers. Collected by the late THOMAS STEPHENS DAVIES, F.R.S.L. & E. &c.§ No. IX. [Continued from vol. ii. p. 446.] THIS appears to be an appropriate occasion for offering a few suggestions for the consideration of geometers respecting the ancient geometry and its modern cultivators. *See Philosophical Transactions, 1814. † Edinburgh Journal of Science, Oct. 1828, vol. ix. p. Trans. Royal Soc. Edinb. vol. xiv. 312. Communicated by James Cockle, Esq., M.A., Barrister-at-Law, who adds the following note: ["Unlike the two papers of this series, which I have already forwarded The charge most commonly made against the science is, the total absence of general methods of research, both as respects construction and demonstration. It is alleged, that of two properties of a figure intimately related as to their subject matter, the demonstration of the one furnishes no clue to the demonstration of the other; and that the most elegant construction of a problem fails to facilitate the construction of one nearly kindred to it-often, indeed, of a converse problem. "All is isolated," it is said; "and it rather requires a certain kind of haphazard dexterity of mind than the application of general methods to make an able geometer. In the coordinate geometry, on the contrary, we can always depend upon obtaining a solution, since we can always reduce the conditions into the form of equations, which only require the ordinary resources of algebraic transformation to complete the inquiry." No doubt there is a certain degree of truth in this, but there is yet a greater degree of misapprehension. Still, the inference being made from the general writings of geometers, and that too from the survey which an unpractised mind is obliged to take, even the misapprehension is pardonable. The brevity with which geometers put down their steps (consisting only of what constructions they make in the individual case before them, and the relations which successively result amongst the parts of the figure), without the slightest reference as to why they adopted their special method, tends very much to justify the opinion to the mind of the uninitiated, that the ancient geometry is a system of special expedients, each adapted to the individual case, like the solution of an enigma, and the whole incapable of reduction to any general principles. There is also another very plausible ground for the inference. It cannot be denied that nearly all geometers, however much they may add to the details of the science in the way of theorems or problems, do yet pursue it merely as a technical system. Their only ambition is "to discover new truths," to make mere to the Philosophical Magazine, there is nothing on the face of the above autograph of Professor Davies to indicate with certainty, or to afford anything like a conclusive inference, that he intended it to occupy its present position. I am responsible for the title given to it. But, even if its original destination be doubtful, it may with great propriety form part of this set of articles. The manuscript now forwarded is a portion of a longer autograph of Mr. Davies, which I have divided into two portions, thinking that such a form would be more convenient for publication. When this has appeared in print, I shall forward the other part for insertion in this admirable Journal. "JAMES COCKLE. "2 Pump Court, Temple, December 20, 1851."] deductions from previously established properties. What relations these new truths have to any previously known, in respect of systematic classification, they know not, and they care not; the mission of these geometers is fulfilled in making the deduction, and upon this they rest their hopes of distinction as geometers. Yet in reality they are but "the hewers of wood and drawers of water" for geometry. They are analogous to the ingenious, but unreasoning, experimenters who abound in physical science; they are, to use the language of Hartsoker, the manouvrières of the philosopher, whether geometrical or physical. At the same time they are as necessary in all sciences as the "hod-man" is to the builder, or the "bellows-boy" to the organist; and happily they are found to exist in abundance, or unhappily in such superabundance as to create the desire for a large promotion of them into the order of actual philosophers. The consequence is, that there already exists such an immense mass of theorems and problems relating to the ancient geometry, scattered in the most sibylline confusion, and without the slightest indication of connection, that they may be deemed as useless as the unreduced accumulations of an observatory; or, indeed, worse than these, for observations are so kept together that they can be reduced, whilst the labour of the whole life of a geometer would not suffice to reduce into order (both as to subject and method) the accumulations of English geometry within the last hundred years. The mathematician who looks at these accumulations in their present unreduced state, and considers them to be the end at which geometry proposes to terminate, may well be excused for the opinion he forms unfavourable to this form of the science. Yet this is not an inherent vice of geometry; though it may and does result from the inherent mental indolence of the geometer himself. He satisfies himself with the deduction, and affects to' consider everything which relates to classification or method as "too speculative" for so able a geometer as he is! He "leaves it to the talkers who call themselves philosophers, but who cannot solve problems, to amuse themselves about such trivialities." The truth however is, in the language of an eminent living philosopher, the "contest of mind" which this requires is such as to transcend the powers of the great majority of men, even though their problem-solving powers may be altogether unquestioned. Nor, if the inquirer apply directly for information on the subject from those geometers who are the most adroit in this class of deductions and constructions, does he find much to enlighten him. Such a geometer will at once sit down and analyse a problem or a theorem proposed to him; and in most cases obtain a solution, sometimes of considerable elegance, however complex |