tion. Of these mathematicians, Dr SIMSON, as he may be accounted the laft, has also been the moft fuccefsful, and has left very little room for the ingenuity of future editors to be exercifed in, either by amending the text of EUCLID, or by improving the translations from it. Such being the merits of Dr SIMSON's edition, and the reception it has met with having been every way fuitable, the work now offered to the public will perhaps appear unneceffary. And indeed, if the geometer juft named, had written with a view of accommodating the Elements of EUCLID 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 defign was different: it was his object to restore the writings of EUCLID to their original perfection, and to give them to modern Europe as nearly as poffible in the ftate wherein they made their firft appearance in ancient Greece. For this undertaking, nobody could be better qualified than Dr SIMSON; who, to an accurate knowledge of the learned languages, and an indefatigable spirit of refearch, added a profound fkill in the ancient Geometry, and an admiration of it almoft enthufiaftic. Accordingly, he not only restored the text of EUCLID wherever it had been corrupted, but in fome cafes removed imperfections that probably belonged to the original work; though his extreme partiality partiality for his author never permitted him to fuppofe, that fuch honour could fall to the fhare either of himself, or of any other of the moderns. But, after all this was accomplished, fomething ftill remained to be done, fince, notwithstanding the acknowledged excellence of EUCLID's Elements, it could not be doubted, that fome alterations might be made upon them, that would accommodate them better to a ftate of the mathematical fciences, fo much more improved and ex tended than at the period when they were written. Accordingly, the object of the edition now offered to the public, is not fo much to give to the writings of EUCLID the form which they originally had, as that which may at prefent render them most useful. One of the alterations made with this view, refpects the Doctrine of Proportion, the method of treating which, as it is laid down in the fifth of EUCLID, has great advantages, accompanied with confiderable defects; of which, however, it must be obferved, that the advantages are effential to it, and the defects only accidental. To explain the nature of the former, requires a more minute examination than is fuited to this place, which muft, therefore, be referved for the Notes; and, in the mean time, it may be fufficient a 3 ficient to remark, that no definition of proportionals, except that of EUCLID, has ever been given, from which their properties can be deduced by reafonings, which, at the fame time that they are perfectly rigorous, are alfo fimple and direct. As to the defects, the prolixnefs and obfcurity, that have fo often been complained of in this book, they seem to arise entirely from the nature of the language, which being no other than that of ordinary difcourfe, cannot exprefs, without much tedioufnefs and circumlocution, the relations of mathematical quantities, when taken in their utmost generality, and when no affiftance can be received from diagrams. As it is plain, that the concise language of Algebra is directly calculated to remedy this inconvenience, I have endeavoured to introduce it here, in a very fimple form however, and without changing the nature of the reafoning, or departing in any thing from the rigour of geometrical demonftration. By this means, the steps of the reafoning which were before far feparated, are brought near to one another, and the force of the whole is fo clearly and directly perceived, that I am perfuaded no more difficulty will be found in understanding the propofitions of the fifth Book, than thofe of any other of the Elements. In the Second Book, alfo, fome Algebraic figns have been introduced, for the fake of representing more readily the addition and fubtraction of of the rectangles on which the demonftrations depend. The use of fuch fymbolical writing, in tranflating from an original, where no fymbols are used, cannot, I think, be regarded as an unwarrantable liberty; for, if by that means the tranflation is not made into English, it is made into that univerfal language fo much fought after in all the fciences, but deftined, it would seem, only for the mathematical. In The alterations above mentioned are the most material that have been attempted on the books of EUCLID. There are, however, a few others, which, though lefs confiderable, it is hoped, may in fome degree facilitate the ftudy of the Elements. Such are those made on the definitions in the first Book, and particularly on that of a ftraight line. A new Axiom is alfo introduced in the room of the 12th, for the purpofe of demonftrating more eafily fome of the properties of parallel lines. 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 fixth are changed for eafier and more fimple propofitions, which do not materially differ from them, and which anfwer exactly the fame purpofe. Some propofitions alfo have been added: but, for a fuller detail concerning thefe changes, I must refer to the Notes, in which feveral of the more difficult, or more interefting fubjects of Elementary Geometry are treated at confiderable length. The SUPPLEMENT now added to the Six Books of Euclid, is arranged differently from what it was in the first edition of these Elements. The Firft of the three books, into which it is divided, treats of the rectification and quadrature of the circle, fubjects that are often omitted altogether in works of this kind. They are omitted, however, as I conceive, without any good reafon, because, to measure the length of the fimplest of all the curves which Geometry treats of, and the space contained within it, are problems that certainly belong to the elements of the fcience, efpecially as they are not more difficult than other propofitions which are ufually admitted into them. When I fpeak of the rectification of the circle, or of measuring the length of the circumference, I must not be fuppofed to mean, that a ftraight line is to be made equal to the circumference exactly, a problem which, as is well known, Geometry has never been able to refolve: All that is propofed is, to determine two straight lines that fhall differ very little from one another, not more, for inftance, than the four hundred and ninety-feventh part of the diameter of the circle, and of |