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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 a certain rule, which, though it be not essential to the accuracy of demonstration, EUCLID has thought it proper; as much as possible, to observė.
The rule referred to, is one which influences the arrangement of his propositions through the whole of the Elements, viz. That in the demons stration 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 all 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 instance.
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 this way,
existence of such solids, of such lines as are ever dently possible, though the manner of actually do scribing them may not have been explained. In
swav. I have been enabled to offer that very refined artifice in geometrical Teasoning, 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 máy, 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 conducta ed 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.
4:03) vivision to be Yü The Book of the Data has been annexed to several editions
of Euclid's Elements, and particularly to Dr SIMSON's, buit in this it is omitted alto gether. It is omitted, however, not from any opinion of its being in itself useless, but because it
es 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 in itself an analytical form, and for these rear sons 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, - fingir à 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.de monstrations, 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 80 much advantage, into the higher Geometry. : -Je In the Spherical Trigonometry, the rules for preventing the ambiguity of the solutions, whereever 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 MașKELYNE has prefixed to Taylor's Tables of Logarithms. Tie tip
An explanation of NAPIER's very ingenious and useful rule of the Circular Parts is here added as an appendix to Spherical Trigonometry.
It has been objected to many of the writers on Elementary Geometry, and particularly to EUCLID, that they have been at great pains to prove the truth of many simple propositions, which every body is ready to admit, without any demonstration, and that they thus take up the time, and fatigue the attention of the student, to no purpose. To this objection, if there be any force in it, the present treatise is certainly as much exposed as any other ; for no attempt is here made to abridge the Elements, by considering as self-evident any thing that admits of being proved. Indeed those who make the objection just stated, do not seem to have reflected sufficiently on the end of Mathematical Demonstration, which is not only to prove the truth of a certain proposition, but to shew its necessary connection with other propositions, and its dependence on them. The truths of Geometry are all necessarily connected with one another, and the system of such truths can never be rightly explained, unless that connection be accurately traced, wherever it exists. this that the beauty and peculiar excellence of the mathematical sciences depend : it is this, which, by
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