BY JOHN PLAYFAIR, F. R. S. EDIN. EDINBURGH: PRINTED FOR BELL & BRADFUTE, AND G. G. & J. ROBINSON, LONDON. M,DCC,XCV. Hodgson 7-5-39 38677 PREFACE. Τ IT is a remarkable fact in the history of science, that the oldest book of Elementary Geometry is ftill confidered as the beft, and that the writings of EUCLID, at the diftance of two thousand years, continue to form the most approved introduction to the mathematical fciences. This peculiar diftinction the Greek geometer owes to the elegance and correctness of his demonstrations, added to an arrangement most happily contrived for the purpofes of inftruction; advantages which, when they reach a certain eminence, fecure the works of an author from being forgotten, more effectually than even originality of invention. In paffing, however, through the hands of the ancient editors, during the decline of fcience, the excellence of his writings had been confiderably obfcured, and much skill and learning have been employed by the modern mathematicians to deliver them from blemishes, with which, it is certain, that they were not originally marked. Of these mathematicians, a 2 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 tranflations from it, Such being the merits of Dr SIMSON's edition, and the reception it has met with having been every way fuited to them, the work now offered to the public will perhaps appear unneceffary, And indeed, if that geometer, had written with à view of accommodating the Elements of EuCLID to the present state of the mathematical fciences, 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 no body could be better qualified than Dr SIMSON; who, to an accurate knowledge of the learned languages, and a moft 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 fome cafes removed imperfections that probably belonged to to the original work; though his extreme partiality for his author never permitted him to suppose, that this was an honour that 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 sciences, fo 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 fo much to give to the writings of Euclid the form which they originally had, as that which may at present render them moft 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 confiderable defects; of which, however, it must be observed, 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 iuited to this place, and which muft, therefore, be referved for a 3 the |