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termine 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. T'his, 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 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 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, (be
cause a solid, whatever be its base, as its height may be indefinitely varied, is capable of all degrees of inagnitude, from nothing upwards), and
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
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 geometri
cal 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 Geometry.
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 some changes in the arrangement, these two treatises have, accordingly, no novelty to boast of The Plane Trigonometry is so divided, that the part of it that is barely sufficient for the resolution of Triangles may be easily taught by itself. The method of constructing the Trigonometrical Tables is explained, and a demonstration is added of those properties of the sines and cosines of arches, which are the foundation of those applications of Trigonometry lately introduced, with so much advantage, into the higher Geometry.
In the Spherical Trigonometry, the rules for preventing the ambiguity of the solutions, wherever it can be prevented, have been particularly attended to; and I have availed myself as much as possible of that excellent abstract of the rules of this science, which Dr. MASKELYNE has prefixed to Taylor's Tables of Logarithms.
An explanation of NAPIER's very ingenious and useful rule of the Circular Parts is here added as an appendix to Spherical Trigonometry.