Elements. 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 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 several of the more difficult, or more interesting subjects of Elementary Geometry are treated at considerable length. The SUPPLEMENT now added to the Six Books of Euclid is arranged differently from what it was in the first edition of these Elements. 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, especially 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 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. 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 proposition,-a problem which, though in itself extremely simple, has been omitted by Euclid, and hardly ever treated of, in an elementary form, by any geometer. With respect to the Geometry of Solids, in the third Book, 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 less rigorous than that of the Greek Geometer : for this would be to pay too dear even for the time that might thereby be saved; but it is done chiefly by laying aside à certain rule, which, though it be not essential to the accuracy of demonstration, Eucưid has thought it proper, as much as possible, to observe. The rule referred to, is one which influences the arrangement of his 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 figure 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 very far 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 in solid contents to any given solid, (because a solid, whatever be its base, as its height may be indefinitely varied, is capable of al degrees of magnitude, from nothing upwards), and yet it may in many cases be a problem of extreme difficulty to assign the height of such a solid, and actually to exhibit it. Now, this very supposition, that on a given base a solid of a given magnitude may be constituted, 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 in stance. In the Book, therefore, on the Properties of Solids, which I now offer to the public, I have not sought to subject the demonstrations to the law just mentioned, 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 I have been enabled to offer that very refined artifice 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 the method may, I think, be best learned when it is thus disengaged from every thing not essential. That it may be the better understood, and because the demonstrations which require exhaustions are, no doubt, the most difficult in the Elements, they are all conducted as nearly as possible in the same way, in the cases of the different solids, from the pyramid to the sphere. The comparison of this last solid with the cylinder, concludes the last Book of the Supplement, and is a proposition that may not improperly be considered as terminating the elementary part of Geo metry. The Book of the Data has been annexed to several editions of Euclid's Elements, and particularly to Dr. Simson's, but in this it is omitted altogether. It is omitted, however, not from any opinion of its being in itself useless, but because it does not belong to this place, and is not often read by beginners. It contains the rudiments of what is properly called the Geometrical Analysis, and has itself an analytical form ; and for these reasons, I would willingly reserve it, or rather a compend of it, for a separate work, intended as an introduction to the study of that analysis. In explaining the elements of Plane and Spherical Trigonometry, there is not much new that can be attempted, or that will be expected by the intelligent reader. Except, perhaps, some new demonstrations, and |