tirely overcome. Now, the theorem just mention. ed would be easily demonstrated, if we had proved, when two angles of one triangle are equal to two angles of another, that the third angles are also equal, whatever may be the inequality of the basis, or of the triangles themselves. Of this proposition, Legendre gives the following demonstration. If the third angle of a triangle depend not on the other two angles alone, but on these angles and also on the base, then is there some function of these angles, and of the base, to which the third angle is equal. But, if this is true, an equation exists between the angles of a triangle and one of its sides; and, if so, a value of that side may be found in terms of the three angles; that is, the side has a given ratio to the angles; which is impossible; for they are quantities of different kinds, and can have no ratio to one another. Whenever, therefore, two angles of one triangle are equal to two of another, their third angles are also equal, whatever their basis may be. This reasoning appears to us extremely ingenious and satisfactory. It takes for granted nothing but that an angle and a line are magnitudes which admit of no comparison; a proposition, of which no one who understands the terms can entertain the smallest doubt. The reasoning, however, will not be readily followed by those who are unacquainted with algebra, or to whom the nature of functions and equations is not tolerably fa miliar. It is curious, that a principle which all the world knew, and which was received into geometry so long ago as the days of Plato, was never made subservient to the purposes of reasoning, till in the instance just mentioned, where it is found actually to involve in it the solution of a great difficulty. We must, however, take leave of Legendre's treatise, which we cannot sufficiently recommend. The Elements of Lacroix are also extremely valuable, though not marked, so strongly as the preceding, with the characters of originality and invention. Delambre goes on to remark, that the fine collection of the Greek mathematicians was completed in 1791, by the Archimedes of Torelli. We suppose that he has here in view the splendid edition of Torelli's Archimedes, printed at Oxford, not indeed in 1791, but in the year following. He makes further mention of a translation of the same into French by M. Peyrard, with a memoir by Delambre himself on the Arithmetic of the Greeks. "Before this memoir," he adds, " of which your Majesty yourself condescended to furnish the subject, it was difficult to conceive how the Greeks, with a notation so imperfect in comparison of ours, could possibly execute the arithmetical operations indicated by Archimedes and Ptolemy." This translation of Archimedes, so far as we know, has not yet reached England. The memoir of Delambre must be peculiarly interesting to mathematicians. On the subject of the ancient geometers and their writings, we must be indulged in a few more remarks. What the collection of the Greek geometers is to which Delambre refers, we do not perfectly understand; but of one thing we are certain, that that collection can never be considered as complete, while the Collections of Pappus, one of the most valuable remains of ancient science, are known only by a very imperfect translation, and while the original continues shut up in great libraries with other unpublished manuscripts. The most perfect MS. of Pappus, we believe is at Oxford, and is particularly described by Dr Horsley, in his restoration of the Inclinations of Apollonius. The late Professor Simson of Glasgow was the man of all others who had studied Pappus with the greatest care, as well as the greatest intelligence; and all the commentaries on that author which his papers afforded, were deposited in the Bodleian Library; so that the University of Oxford is certainly in possession of the best materials that the world affords, for a correct edition of this ancient author. We would willingly look to the learning of that celebrated university for a publication which will be most thankfully received by the whole mathematical world. * * Though the MSS. of Pappus, we believe, are but few, there are some now and then to be met with, which an editor would no doubt think it his duty to consult. One is now in the possession of the Advocates' Library, which was purchased a few years ago. It is very beautifully written; but is probably of no great antiquity. A circumstance that adds to its value is, that the name of Ortous de Mairan is inscribed on it; so that it probably was once the property of that learned and ingenious academician. Before we take leave of that part of the report which relates to the ancient geometry, we must observe, that the most interesting part of it, the geometrical analysis, has not, in later times, been cultivated in France; and very little, as far as we know, in any part of Europe, except Italy and Great Britain. This is so true, that the article of geometrical analysis is not to be found in that great work, which the French philosophers and mathematicians intended as a complete description of the science of the eighteenth century. The neglect, among these philosophers, of a branch of geometry that deserves so well to be cultivated, and is, in fact, one of the most beautiful and elegant inventions in the whole circle of the sciences, is the more wonderful, that the first of the moderns who understood this subject, and who, though destitute of many of the aids which have since been derived from a more complete knowledge of the ancient remains, became a great master of it, was a French geometer. Fermat flourished about the middle of the seventeenth century; and, in his Opera Varia, has resorted, or re-invented, some of the finest works of the ancient analysis, and has approached, at the same time, very near to several of the greatest discoveries of the modern. In the former, however, his course was not followed by the mathematicians of his own country; and the man who most nearly trode in the steps of Fermat, was Dr Edmund Halley, in the end of the same century, who, possessing great learning as well as genius, applied the former very successfully to the improvement of science. He was followed by the late Dr Simson of Glasgow, and Dr Matthew Stewart of Edinburgh, who cultivated the ancient analysis with singular assiduity; the former, restoring several of the most valuable works of the ancients; and the latter, introducing the geometrical analysis into those branches of physical science, which hitherto had been treated, either in the algebraical manner, or by synthetical demonstration. The late Dr Horsley was a proficient in the ancient analysis; and we might add some others of this country, who have cultivated the same subject with success, and whose writings fall within the period to which the report of the Institute is limited. In Italy, the ancient analysis has found several followers; among the Memoirs of the Società Italiana, many problems are found resolved by it; but, on the same subject, we have met with nothing in the Transactions of the other societies of Europe. There must |