ENTERED IN STATIONERS' HALL.

P. NEILL, Printer, Edinburgh.

4-26-41 NƯỚ

PREFACE.

Ir is
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 distance of two thousand years,
continue to form the most approved introduction
to the mathematical sciences. This remarkable
distinction the Greek Geometer owes not only to
the elegance and correctness of his demonstra-
tions, but to an arrangement most happily con-
trived for the purpose of instruction,-advantages
which, when they reach a certain eminence, se-
cure the works of an author against the injuries
of time, more effectually than even originality of
invention. The Elements of EUCLID, however, in
passing through the hands of the ancient editors,
during the decline of science, had suffered some
diminution of their excellence, and much skill and
learning have been employed by the modern ma-
thematicians to deliver them from blemishes,
which certainly did not enter into their original

composition. Of these mathematicians, Dr SIM. SON, as he may be accounted 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 suitable, the work now offered to the public will perhaps appear unnecessary. And indeed, if the geometer just 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 soon attempted. But his design was different: it was his object to re

store 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 ให their first 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 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 wherever it had been corrupted,

but in some cases removed imperfections that pro bably belonged to the original work; though his extreme partiality for his author never permitted him to suppose, that such honour could fall to the share either of himself, or of any other of the mo derns.

TOI

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, that would accommodate them better to a state of the mathematical sciences, so much more improved and extended than at the period when they were written. Accordingly, the object of the edition now offered to the public, is 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 made with this view, reespects 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, and the defects only accidental. To explain the nature of the former, requires a more minute examination bumping maod fui trendy auroof to toot

than is suited to this place, and must, therefore, be reserved for the Notes; but, in the mean time, it may be remarked, that no definition, except that of EUCLID, has ever been given, from which the properties of proportionals can be deduced by reasonings, which, at the same time that they are perfectly rigorous, are also simple and direct. As to the defects, the prolixness and obscurity that have so often been complained of in the fifth book, they seem to arise chiefly from the nature of the language employed, which being no other than that of ordinary discourse, cannot express, without much tediousness and circumlocution, the relations of mathematical quantities, when taken in their utmost generality, and when no assistance 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 simple form however, and without changing the nature of the reasoning, or departing in any thing from the rigour of geometrical demonstration. By this means, the steps of the reasoning which were before 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