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 reafonings, which, at the fame time that they are perfectly rigorous, are alfo fimple and direct. As to the defects, on the other hand, the prolixnefs and obfcurity, that have so often been complained of in this book, they feem to arife entirely from the nature of the language; for, in mathematics, common language can feldom be applied, without much tedioufnefs and circumlocution, in reasoning about the relations of fuch things as cannot be reprefented by means of diagrams, which happens here, where the fubject treated of is magnitude in general. It is plain, therefore, that the concife language of Algebra is directly calculated to remedy this inconvenience; and fuch a one I have, accordingly, endeavoured to introduce, in the fimpleft form, and without changing at all the nature of the reafoning, or departing in any thing from the rigour of geometrical demonftration. By this contrivance the steps of the reasoning which were before fo far feparated, are brought near to one another, and the force of the whole is fo clearly and directly perceived, that I am perfuaded no more difficulty will be found in understanding the pro pofitions pofitions of the fifth Book, than of any other of the Elements. A few changes have alfo been made in the enunciations of this book, chiefly in those of the fubfidiary propofitions which EUCLID introduced for the fake of the reft; they are expreffed here in the manner that feemed beft adapted to the new notation. The alterations above mentioned are the moft material that have been attempted on the books of EUCLID. There are, however, a few others, which, though less confiderable, it is hoped, may in fome degree facilitate the understanding of them. Such are thofe made on the definitions in the first Book, and particularly on that of a ftraight line. A new Axiom is alfo introduced in the room of the 12th, for the purpose of demonftrating more easily fome of the properties of parallel lines. In the third Book, the remarks concerning the angles made by a ftraight 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 eafier and more fimple propofitions, which do not materially differ from them, and which anfwer exactly the fame purpofe. Some propofitions also have been added; but, for a fuller detail concerning thefe changes, I must refer to the notes, in which several of the more difficult, or more interesting subjects of Elementary Geometry are treated at confiderable length. 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 comprehenfive. This, however, is not attempted by introducing a mode of reasoning loofer or lefs 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 afide a certain rule, which, though it be not effential to the accuracy of demonftration, EUCLID has thought it proper; as much as poffible, to observe. The rule referred to, is one which regulates the arrangement of EUCLID's propofitions through the whole of the Elements, viz. That in the demonftration of a theorem he never fuppofes any thing to be done, as any line to be drawn, or any figure to be conftructed, the manner of doing which he has not previously explained. Now, the only use of this rule is to prevent the admiffion of impoffible or contradictory fuppofitions, which no doubt might lead into error; and it is a rule well calculated to answer that end; as it does not allow the exiftence of any thing to be fuppofed, unless the thing itself be actually exhibited. But it is not always neceffary to make ufe of this defence, for the existence of many things is obviously poffible, and far enough from implying a contradiction, where the method of actually exhibiting them may be altogether un-: known. Thus, it is plain, that on any given figure as a base, a folid may be conftituted, or conceived to exist, equal to a given solid, (because a folid, whatever be its bafe, 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 affign the height of fuch a folid, and actually to exhibit it. Now, this very fuppofition is one of thofe, by the introduction of which, the Geometry of Solids is much fhortened, while all the real accuracy of the demonftrations is preserved; and therefore, to follow, as EUCLID has done, the rule that excludes this, and fuch like hypothefes, is to create artificial difficulties, and to embarrass geometrical inveftigation with more obftacles 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 clofely in the fimpler parts, I have no where fought to fubject the demonftrations to fuch a law as the foregoing, and have never hefitated to admit the existence of fuch folids, or fuch lines as are evidently poffible, though the manner of actually defcribing them may not have been explained. In this way alfo, I have been enabled to offer that very refined artifice in geometrical reasoning, to which we give the name of the Method of Exhauftions, under a much fimpler form than it appears in the 12th of EUCLID; and the spirit of it may, I think, be beft learned when it is disengaged from every thing not ef fential to it. That this method may be the better understood, and because the demonftrations that require it are, no doubt, the most difficult in the Elements, they are all conducted as nearly as poffible in the fame way through the different folids, from the pyramid to the sphere. The comparison of this laft folid with the cylinder concludes the eight Book, and is a propofition that may not improperly be confidered as terminating the elementary part of Geometry, In the beginning of the Book juft mentioned, I have treated pretty fully of the rectification and quadrature of the Circle, fubjects that are often omitted altogether in works of this kind. They are omitted, however, as I conceive, without any good reason, becaufe, to measure the length of the fimpleft of all the curves which Geometry treats |