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BY JOHN PLAYFAIR, F. R. S. Edin.
Professor of Mathematics in the University of Edinburgh.

EDINBURGH:

Printed For Bell & Bradfute, And
G. G. & J. Robinson, London.

M,DcC,XcV.

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IT is a remarkable fact in the history of science, that the oldest book of Elementary Geometry is still considered as the best, and that the writings of Euclid, at the distanceof two thoufand years, continue to form the most approved introduction to the mathematical sciences. This peculiar distinction the Greek geometer owes to the elegance and correctness of his demonstrations, it added to an arrangement most happily contrived j for the purposes of instruction; advantages which, i when they reach a certain eminence, secure the $ works of an author from being forgotten, more effectually than even originality of invention. In pasting, however, through the hands of the ancient editors, during the decline of science, the excellence of his writings had been considerably obscured, 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 a 2 mathematicians,

mathematicians, Dr Simson, as he may be acr 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 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 suited to them, the work now offered to the public will perhaps appear unnecessary. And indeed, if that geometer, had written with a. view of accommodating tue 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 soon attempted. But his design was different j it was his object to restore the writings of EuClid to their original perfection, and to give them to modern Europe as nearly as possible in the state wherein they made their sirst appearance in ancient Greece. For this undertaking no body could be better qualisied 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 enthusiastic. Accordingly, he not only restored the text of Euclid whereever it had been corrupted, but in some cases 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 thai 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, since, 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 arid 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 sifth 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 iuited to this place, and which' must, therefore, be reserved for

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