Few of the mathematicians and philosophers of the present age have acquired a greater or better founded reputation than the celebrated Abbé Boscovich; and there is none from whose writings I have received such variety of instruction and entertainment. His Theory of Natural Philosophy will ever be considered by impartial judges, not only as one of the boldest, but also as one of the most ingenious researches into the secrets of nature. There is hardly a branch of physico-mathematical philosophy which he has not cultivated with success; and in this cultivation he has exhibited the most acute penetration and the greatest address. In all his investigations too he has given the most beautiful specimens of geometrical invention and elegance, and greatly heightens the pleasure of his readers, by marking out distinctly the progress of his own mind in his researches.

That excellent Geometer the late Dr Matthew Stewart, as long ago as the year 1746, published his book of General Theorems, all of them, except the first five, without the demonstrations. As I do not find that any demonstration of them has ever been made public, I may perhaps flatter myself that what I now communicate to this Society will not be wholly unacceptable. The demonstrations given are of the propositions relating to the sums of the squares, and of the fourth powers of lines drawn in a certain manner, and are selecled from the rest, as most connected with one another. The theorems that respect the cubes and other higher powers, may afford materials for another paper, should this meet with the approbation of the Society.

Since the time when Astronomy emerged from the obscurity of ancient fable, nothing is better known than its progress through the different nations of the earth. With the era of Nabonassar, regular observations began to be made in Chaldea; the earliest which have merited the attention of succeeding ages. The curiosity of the Greeks was, soon after, directed to the same object; and that ingenious people was the first that endeavoured to explain, or connect by theory, the various phenomena of the heavens. This work was supposed to be so fully accomplished in the Syntaxis of Ptolemy, that his system, without opposition or improvement, continued, for more than five hundred years, to direct the Astronomers of Egypt, Italy and Greece. After the sciences were banished from Alexandria, his writings made their way into the east, where, under the Caliphs of Bagdat, Astronomy was cultivated with diligence and success. The Persian Princes followed the example of those of Bagdat, borrowing besides, from Trebisond, whatever mathematical knowledge was still preserved among the ruins of the Grecian empire.

It is a fundamental principle in Algebra, that a problem admits of solution, when the number of independent equations is equal to that of the unknown quantities. If simple expressions only occur, the answers will always be found in numbers, either whole or fractional. But if the higher functions be concerned, the values of the unknown quantities will commonly be involved in surds, which it is impossible to exhibit on any arithmetical scale, and to which we can only make a repeated approximation. Hence the origin of that branch of analysis which is employed in the investigation of those problems, where the number of unknown quantities exceeds that of the proposed equations, but where the values are required in whole or fractional numbers. The subject is not merely an object of curiosity; it can be applied with advantage to the higher calculus. Yet the doctrine of indeterminate equations has been seldom treated in a form equally systematic with the other parts of algebra. The solutions commonly given are devoid of uniformity, and often require a variety of assumptions. The object of this paper is to resolve the complicated expressions which we obtain in the solution of indeterminate problems, into simple equations, and to do so, without framing a number of assumptions, by help of a single principle, which, though extremely simple, admits of a very extensive application.

In a paper published in the second volume of the second decade of the Medical Commentaries of Edinburgh, I mentioned a design of endeavouring to trace the influence of a cold climate on the human body and its diseases, which should form a contrast with the many accounts published of late years relative to the effects of hot climates; and I likewise mentioned my having given a detached piece, some years ago, as a commencement of the subject, in the fixty-eighth volume of the Philosophical Transactions of London, which contains matter necessary to illustrate some parts of the following Dissertation.

In the year 1777, an account was published by Mr John Williams, mineral-engineer, of certain remains of ancient buildings on the summits of some of the hills in the Highlands of Scotland, which had hitherto escaped observation, and which to him afforded grounds for a very extraordinary supposition, That they had been cemented together by means of Fire. He mentioned several of those hills exhibiting remains of building, which he had visited and examined; particularly the hill of Knockfarril in Ross-shire, Craig-Phadrick near Inverness, Dun-Evan and Castle-Finlay in the county of Nairn, and the Castle-hill of Finhaven in the county of Angus. He described the vestiges of regular fortifications on the summits of those hills, of which the walls, remaining in some places of several feet in height, were evidently compacted together by the vitrification of the stones of which they were built; and he offered some ingenious conjectures with regard to the means employed in forming such extraordinary structures, and the purposes for which they might have been reared.

The poetical beauties of Virgil's sixth book are great and many; and a most agreeable task it would be to point them out: but that is not my present purpose. Nor do I intend to draw a comparison of the sentiments of our poet with those of Homer, concerning a future state. From Homer, no doubt, Virgil received the first hint of this episode ; but the evocation of the ghosts, in the eleventh book of the Odyssey, is not in any degree so striking, or so poetical, as Eneas's descent into the world of spirits. Nor does the former exhibit any distinct idea of retribution. In it all is dark and uncomfortable. “I would rather, says the ghost of “Achilles, be the slave of a poor peasant among the living, “than reign sole monarch of the dead:” a passage blamed, not without reason, by Plato, as unfriendly to virtue, and tending to debase the soul by an unmanly fear of death.

Man is formed to derive pleasure from a variety of different sources. Many of his pleasures are communicated by the channel of the external senses. Each of these has particular classes of objects that are suited to its gratification ; and these, being diffused in great abundance through the works of nature, or framed by the art and ingenuity of men, become to him a source of frequent and diversified enjoyment.

The power of pronouncing articulate sounds is one of the most obvious marks which distinguish man from the other animals. No philosophical investigation is necessary for pointing it out, and therefore it has not escaped the notice of the poets, the most ancient of all authors. In the works of Homer and Hesiod, we often meet with the expression μέϱοπες ἄνθϱωποι, men having an articulate voice; the word μέϱοψ being evidently compounded of μείϱω, to divide, and ὄψ, the voice.

No country perhaps affords a more interesting literary speculation at present than Germany. For researches in science and philosophy, for laborious investigations into the principles of public polity and law, she had long been conspicuous ; but, till very lately, she made scarce any pretension to fame in the other departments of literature, which usually precede those more abstract and laborious pursuits I have just mentioned.

In the prosecution of certain philological and philosophical speculations, very ample specimens of which have already been submitted to the consideration of the Royal Society of Edinburgh, I had occasion to consider more minutely than I believe had ever been done before, many particulars relating to the nature, the structure, and the import of Verbs.

The Character of Hamlet, has been variously judged of by critics, and what might be expected, it has been still more variously represented by performers upon the stage. Shakespeare himself seems to have apprehended that this would happen; and that injustice would be done to a hero, who probably, in his estimation, ranked higher than any other that he has brought into the drama.