mathematicians, Dr Simson, as he may be ac. counted the last, has also been the most successful, and has left very little room for the ingenuity of future editors to be exercised in, either by amend. ing the text of Euclid, or þy improving the translations from it. CLID Such being the merits of Dr Simson's edition, and the reception it has met with having been every way suited to them, the work now offered to the public will perhaps appear unnecessary, ; And indeed, if that geometer, had written with à. view of accommodating the Elements of EU to the present state of the mathematical sciences, it is not likely that any thing new in Elementary Geometry would have been foon attempted. But his design was different; it was his object to restore the writings of EuCLID to their original perfection, and to give them to niodern Europe as nearly as possible in the ftate wherein they made their first appearance in ancient Greece. For this undertaking no body could be better qualified than Dr SIMSON ; who, to an accurate knowledge of the learned languages, and a most indefatigable spirit of research, added a profound skill in the ancient Geometry, and an admiration of it almost enthufiaftic. Accordingly, he not only restored the text of Euclid whereever it had been corrupted, but in some cases removed imperfections that probably belonged tą to the original work; though his extreme par. tiality for his author never permitted him to suppose, that this was an honour that could fall to the share either of himself, or of any other of the moderns. But, after all this was accomplished, something still remained to be done, fince, notwithstanding the acknowledged excellence of Euclid's Elements, it could not be doubted, that some alterations might be made upon them, that would accommodate them better to a state of the mathematical sciences, so much more improved and extended than at any former period. This accordingly is the object of the edition now offered to the public, which is intended not so much to give to the writings of Euclid the form which they originally had, as that which may at present render them most useful. One of the alterations that has been made with this view, respects the Doctrine of Proportion, the method of treating which, as it is laid down in the fifth of EUCLID, has great advantages, accompanied with considerable defects; of which, however, it must be observed, that the advantages are essential to it, and the defects only accidental. To explain the nature of the former, requires a more minute examination than is suited to this place, and which' must, therefore, be reserved for the a 3 the notes; and, in the mean time, it may be sufficient to remark, that no definition of proportionals, except that of Euclid, has ever been given, from which their properties can be deduced by reasonings, which, at the same time that they are perfectly rigorous, are also fimple and direct. As to the defects, on the other hand, the prolixness and obscurity, that have so often been complained of in this book, they feem to arise entirely from the nature of the language ; for, in mathematics, common language can seldom be applied, without much tedioufness and circumlocution, in reasoning about the relations of such things as cannot be represented by means of diagrams, which happens here, where the subject treated of is magnitude in general. It is plain, therefore, that the concise language of Algebra is directly calculated to remedy this inconvenience; and such a one I have, accordingly, endeavoured to introduce, in the simplest form, and without changing at all the nature of the reasoning, or departing in any thing from the rigour of geometrical demonstration. By this contrivance the steps of the reasoning which were before fo far separated, are brought near to one another, and the force of the whole is so clearly and directly perceived, that I am persuaded no more difficulty will be found in understanding the pro positions positions of the fifth Book, chan of any other of the Elements. A few changes have also been made in the enunciations of this book, chiefly in those of the subsidiary propofitions which Euclid introduced for the sake of the rest; they are exprefl'ed here in the manner that seemed best adapted to the new notation. The alterations above mentioned are the moft material that have been attempted on the books of EUCLID. There are, however, a few others, which, though less considerable, it is hoped, may in some degree facilitate the understanding of them. Such are those made on the definitions in the first Book, and particularly on that of a straight line. A new Axiom is also introduced in the room of the 12th, for the purpose of demonstrating more ealily some of the properties of parallel lines. In the third Book, the remarks concerning the angles made by a straight line, and the circumference of a circle, are left out, as tending to perplex one who has advanced no farther than the elements of the science. The 27th, 28th and 29th of the 6th are changed for easier and more fimple propositions, which do not materially differ from them, and which answer exactly the same purpose. Some propofitions also have been added ; but, for a fuller detail concerning these changes, 'Q 4 I I must refer to the notes, in which several of the more difficult, or more interesting subjects of Elementary Geometry are treated at confiderable length. Thus much for the part of the Elements that treats of Plane Figures. With respect to the Geometry of Solids, I have departed from Euclid altogether, with a view of rendering it both shorter and more comprehensive. This, however, is not attempted by introducing a mode of reasoning looser or less rigorous than that of the Greek geometer; for this would be to pay too dear even for the time that might thereby be faved; but it is done chiefly by laying aside a certain rule, which, though it be not essential to the accuracy of demonstration, Euclid has thought it proper; as much as poflible, to observe. The rule referred to, is one which regulates the arrangement of Euclid's propofitions through the whole of the Elements, viz. That in the demonftration of a theorem he never supposes any thing to be done, as any line to be drawn, or any figure to be constructed, the manner of doing which he has not previously explained. Now, the only use of this rule is to prevent the admilfion of imposible or contradictory suppositions, which no doubt might lead into error; and it is a rule well calculated to answer that end; as |