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but in some cases removed imperfections that probably 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.

Buts 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, respects the doctrine of Proportion, the of

merno 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

e nature of the former, requires a more minute examination syon moos bei 39,7333 for LOOT to trot

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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 Algebrà 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

understanding the propositions of the fifth Book, than those of any other of the Elements.

In the Second Book also, some algebraic signs have been introduced, for the sake of representing more readily the addition and subtraction of the rectangles on which the demonstrations depend. The use of such symbolical writing, in translating from an original, where no symbols are used, cannot, I think, be regarded as an unwarrantable liberty : for, if by that means the translation is not made into English, it is made into that universal language so much sought after in all the sciences, but destined, it would seem, to be enjoyed only by the mathematical.

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 less considerable, it is hoped may in some degree facilitate the study of the Elements. Such are those 'made on the definitions in the first Book, and particularly on the definition of a straight line. A new axiom is also introduced in the room of the 12th, for the purpose of demon strating more easily 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 tend

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ing to perplex one who has advanced no farther than the elements of the science. The 27th, 28th and 29th of the sixth are changed for easier and more simple propositions, which do not materially differ from them, and which answer exactly the same purpose. Some propositions also have been added; but, for a fuller detail concerning these changes, I must refer to the Notes, in which ses veral of the more difficult, or more interesting sub jects of Elementary Geometry are treated at. considerable length.

i : The SUPPLEMENT now added to the six Books of Euclid is arranged differently from what it was in the first Edition of these Elements.y03 Thorin

The First of the three Books, into which it is divided, treats of the rectification and quadrature: of the circle,--subjects that are often omitted altogether in works of this kind. They are omitted, however, as, I conceive, without any good reason, because, to measure the length of the simplest of all the curves which Geometry treats of, and the space contained within it, are problems that certainly belong to the elements of the science, espea cially as they are not more difficult than other propositions which are usually admitted into them. ! When I speak of the rectification of the circle, or of measuring the length of the circumference, I

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must not be supposed to mean, that a straight line is to be made equal to the circumference exactly, a problem which, as is well known, Geometry has never been able to resolve : All that is proposed is, to determine two straight lines that shall differ very little from one another, not more, for instance, than the four hundred and ninety-seventh part of the diameter of the circle, and of which the one shall be greater than the circumference of that circle, and the other less. In the same manner, the quadrature of the circle is performed only by approximation, or by finding two rectangles nearly equal to one another one of them greater, and another less than the

space contained within the circle. ei In the Second Book of the Supplement which treats of the intersection of Planes, I have departed as little as possible from EUCLID's method of considering the same subject in his eleventh Book. The demonstration of the fourth proposition is from LEGENDRE'S Elements of Geometry; that of the sixth is new, as far as I know; as is also the solution of the problem in the nineteenth propo sition;--a problem which, though in itself extreme ly simple, has been omitted by EUCLID, and hardly ever treated of, in an elementary form, by any geometer. 5 h: 35,

With respect to the Geometry of Solids, in the

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