In the present state of mathematical science, cultivated as it has been, with assiduity, during the two preceding centuries, it can hardly be expected that any considerable improvement remains to be made in Plane Trigonometry, one of its most elementary theories. There is, however, one case in the resolution of oblique-angled triangles, which appears to me to admit of a solution somewhat more simple and convenient than those which are commonly known; it is that in which two sides and the included angle are given to find the third side.

In the course of some inquiries into the affinity and structure of the Greek and Latin languages, I was led to analyse the superlative degree of both, and to trace, as I thought, some connection between it and the word Sublimis. While engaged in the investigation, I was naturally led to examine the common theories respecting the etymology of this remarkable word, and, in particular, the origin assigned to it by the late Dr Parr, and to weigh, with more attention than I had previously done, the arguments and proofs he had advanced in support of his opinions. All that I knew of them, till lately, was by verbal report, as I had not seen the abridged statement of them in an Appendix to the Notes of the 2d edition of Mr Stewart's Essays.

In the year 1820, I had occasion to suggest to the Royal Society the propriety of establishing Registers of the Thermometer in various parts of Scotland.

In a country embracing so many varieties of soil, climate and elevation, and extending over nearly six degrees of latitude, it was an object worthy of a public body to determine the Law of the Distribution of Temperature, even if such a subject had not possessed a separate interest in relation to the Horticulture and Agriculture of the Country. The Society did not hesitate in adopting this suggestion, and many intelligent individuals were found, who undertook to observe the thermometer twice a-day, and to measure occasionally the temperature of Springs and Wells, During the first year, viz. 1821, nearly sixty Meteorological Journals were regularly kept in different parts of Scotland. The number diminished considerably in subsequent years; but, notwithstanding this diminution, there is now in our possession a rich series of observations during five complete years, the results of which are nearly ready to be submitted to the Society.

As it is my intention soon to publish, in various works on Natural History, the observations on the Botany of India which I made during my residence there, I wish to place on record an account of the opportunities which I enjoyed of making such observations, with the view of explaining to the Botanist where he may find the various collections which I made in different parts. I also wish to explain the geographical terms that I shall employ, in giving an account of the places where I found each species. For this purpose I prefer using the ancient Sanscrita names, both as being more scientific, and as being more likely to remain permanent; for, after a lapse of many ages, they continue to be known to all Hindus of learning, while each new invasion or revolution sinks into immediate oblivion the mushroom appellations imposed by modern rulers, whether Muhammedans or Christians.

The Mineral called Analcime or Cubizite has been ranked by Haüy among those crystals which have the Cube for their primitive form; an opinion which has been adopted by all succeeding mineralogists. No distinct cleavage-planes, however, so far as I can learn, have been observed in it. Crystallographers presumed that such planes must exist, and, allowing conjecture to supply the place of observation, they considered Analcime as differing in no respect from other crystals of the same series. This opinion was first rendered doubtful by the observation, that at thicknesses of of an inch, it displayed a considerable action upon polarised light; but though, from the tessular form of the mineral, this fact indicated something singular in its organization, yet, owing to the great difficulty of obtaining proper specimens, I have been baffled in repeated attempts to investigate its structure.

A history of the revival of literature in Europe, after its extinction in the middle ages, has been long a desideratum in the annals of human knowledge; and from the wide, and almost untravelled field, which such a history would embrace, and the recondite sources of information which must be consulted, it will perhaps be long before any individual is found with sufficient learning to estimate its difficulties, and yet hardy enough to attempt to overcome them.

In the Paper which I had the honour of submitting to the Society, on the Two New Fluids in mineral bodies, I have given the index of refraction for the most expansible of the two, as it exists in the cavities of Amethyst; but as I had not then ascertained the refractive power of the second fluid, and as the principal phenomena of the two fluids, especially those which related to their properties when taken out of the cavities, were observed in specimens of Topaz, it became desirable to have an approximate measure of the refractive power of both of them, as they exist in that mineral. As the fluid in Amethyst had never been examined in the open air, its identity with that in Topaz was inferred solely from the equality of their expansion by heat, so that the determination of the refractive power of the latter was necessary to establish either a difference between these two substances, or their perfect identity.

The experiments which I now submit to the Royal Society are repetitions of those I made many months ago, for the purpose of ascertaining the Specific Heats of the Gases. The importance of the subject so impressed my mind, that I determined to spare no pains in the prosecution of the inquiry, and therefore I willingly withheld my first experiments from the public eye, until, by a fresh series, I might present them with the greater confidence. The apparatus employed in these experiments was calculated to operate upon greater quantities of the Gases than the former one, and as every precaution which had been suggested was adopted, they have, perhaps, given even more decisive results than the last. The results themselves, however, are in every important particular exactly the same. It is also but justice to myself to state, that the conclusions which the former experiments led to, were exactly the reverse of what I had anticipated, and that they seemed at the time totally opposed to the doctrines of Black and Crawford, which I am still disposed to credit to a limited degree.

The Natural History of the Pholades, so far as regards their mode of burrowing in wood and stone, seems yet to be but imperfectly understood, though the Pholas was known to the ancients, and Pliny notices its phosphorescent quality. Rondeletius, Johnston, and Rumphius have figured several species; Lister, among others, gives representations of three British species, the Pholas dactylus, Candida, and crispata; and Sir Robert Sibbald, in his Prodromus, has three rude figures of the dactylus or crispata, as Scottish shells. None of these authors, however, attempted to explain how the Pholades excavated their habitations in the rock, or perforated the submerged wood in which they seek protection. Bonanni, so far as I know, was the first who turned his attention particularly to this inquiry.

The history of the progress of discovery in the Natural Sciences, records innumerable instances, where the different ways of considering the same subject have created apparent contradictions, which have subsequently been reconciled by more accurate examination. The first observation of facts is very often far from being accurate; and, in most cases, this inaccuracy cannot be corrected, till the science itself has attained a higher degree of perfection. The optical and crystallographic inquiries into the nature of mineral substances, refer equally to the regular forms which these bodies present. The object of Crystallography is to ascertain this regular form from direct observations; and that of the department of Optics, which relates to this subject, is to determine the action of regularly crystallised bodies upon light. According to certain general laws, it has been found possible to argue from the optical phenomena to the external forms of minerals, and inversely from these forms to the actions dependent upon them, which affect light in its passage through crystallised substances.

On a former occasion, I had the honour of describing and exhibiting to this Society a new Registering Thermometer, by means of which the atmospheric temperature may be ascertained at any given instant during absence. In the construction of the instrument then described, a sliding index within the tube is indispensable; and, whenever such an index is employed, the diameter of the tube, and consequently that of its bulb, must be such as to render the instrument defective, when great accuracy and attention to minute fractions is requisite, such as in barometrical measurements, and various delicate experiments. Besides, though an instrument made with such an index may be so constructed as to perform with great accuracy, still, inasmuch as it is a complication, it is defective, and is more or less liable to error, as its construction may have been moire or less perfect.

The celebrated Bouguer was the first who discovered the important fact, that the eye can detect a very small difference between two similar illuminated surfaces, when viewed at the same moment,—the only principle which has yet been applied with any degree of success, in determining the relative illuminating powers of artificial flames. The following is perhaps the most commodious application of this principle, to determine the relative illuminating powers of different artificial lights, particularly of coal and oil gas. The instrument, or photometer, which I employ for this purpose, is extremely simple. It consists of a rectangular box, about an inch and a half, or two inches square, open at both ends, and blackened within for the purpose of absorbing the stray-light.