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the notes; and, in the mean time, it may be sufficient to remark, that no definition of proportionals, except that of Euclid, has ever been given, from which their properties can be deduced by reasonings, which, at the fame time that they are perfectly rigorous, are also simple and direct. As to the defects, on the other hand, the prolixness and obscurity, that have so often been complained of in this book, they seem to arise entirely from the nature of the language; for, in mathematics, common language can seldom be applied, without much tediousness and circumlocution, in reasoning about the relations of such things as cannot be represented by means of diagrams, which happens here, where the subject treated of is magnitude in general. It is plain, therefore, that the concise language of Algebra is directly calculated to remedy this inconvenience; and such a one I have, accordingly, endeavoured to introduce, in the simplest form, and without changing at all the nature of the reasoning, or departing in any thing from the rigour of geometrical demonstration. By this contrivance the steps of the reasoning which were before so 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
positions of the fifth Book, jhan of any other of the Elements.
A few changes have also been made in the enunciations of this book, chiefly in those of the subsidiary propositions which Euclid introduced for the fake of the rest; they are expressed here in the manner that seemed best adapted to the new notation.
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 understanding of them. Such are those made on the definitions in the first Book, and particularly on that of a straight line. A new Axiom is also introduced in the room of the 12th, for the purpose of demonstrating 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 tending to perplex one who has advanced no farther than the elements of the science. The 27th, 28th and 29th of the 6th 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, a 4 I
I must refer to the notes, in which several of the more difficult, or more interesting subjects of Elementary Geometry are treated a£ considerable length. ' . i •!'
Thus much for the part of the Elements that treats of Plane figures. With respect to the Geometry of Solids, I have departed from Euclid altogether, with a view of rendering it both shorter and more comprehensive. This, however, is not attempted by introducing a mode of reasoning looser or less rigorous than that of the Greek geometer; for this would be to pay too dear even for the time that might thereby be faved; but it is done chiefly by laying aside a certain rule, which, though it be not essential to the accuracy of demonstration, Euclid has thought it proper; as much as possible, to observe.
The rule referred to, is one which regulates the arrangement of Euclid's propositions through the whole of the Elements, viz. That in the demonstration of a theorem he never supposes any thing to be done, as any line to be drawn, or any sigure to be constructed, the manner of doing which he has not previously explained. Now, the only use of this rule is to prevent the admission of impossible or contradictory suppositions, which no doubt might lead into error; and it is a rule well calculated to answer that end; as
it does not allow the existence of any thing to be supposed, unless the thing itself be actually exhibited. But it is not always necessary to make use of this defence, for the existence of many things is obviously possible, and far enough from implying a contradiction, where the method of actually exhibiting them may be altogether unknown. Thus, it is plain, that on any given figure as a base, a solid may be constituted, or conceived to exist, equal to a given solid, (because a solid, whatever be its base, as its height may be indefinitely varied, is capable of all degrees of magnitude, from nothing upwards), and yet, it may in many cafes be a problem of extreme difficulty to assign the height of such a solid, and actually to exhibit it. Now, this very supposition is one of those, by the introduction of which, the Geometry of Solids is much shortened, while all the real accuracy of the demonstrations is preserved; and therefore, to follow, as Euclid has done, the rule that excludes this, and such like hypotheses, is to create artificial difficulties, and to embarrass geometrical investigation with more obstacles than the nature of things has thrown in its way. It is a rule, too, which cannot always be followed, and from which even Euclid himself has been forced to depart, in more than one instance.
In the two Books, therefore, on the Properties of Solids, that I now offer to the public, though I have followed Euclid very closely in the simpler parts, I have no where sought to subject the demonstrations to such a law as the foregoing, and have never hesitated to admit the existence of such solids,or such lines as are evidently possible, though the manner of actually describing them may not have been explained. In this way also, I have been enabled to offer that very resined artisice in geometrical reasoning, to which we give the name of the Method of Exhaustions, under a much simpler form than it appears in the 12th of Euclid; and the spirit of it may, I think, be best learned when it is disengaged from every thing not essential to it. That this method may be the better understood, and because the demonstrations that require it are, no doubt, the most difficult in the Elements, they are all conducted as nearly as possible in the fame way through the different solids, from the pyramid to the sphere. The comparison of this last solid with the cylinder concludes the eight Book, and is a proposition that may not improperly be considered as terminating the elementary part of Geometry.
In the beginning of the Book just mentioned, I have treated pretty fully of the rectisication 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