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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 chitfly 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, (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
pre. served; 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 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 12 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,
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 admity without any demonstration, and that thus they 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 show its necessary connection with other propositions, and its dependance 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. It is upon this that the beauty and peculiar excellence of the mathematical sciences depend : it is this, which by preventing any one truth from being single and insulated, connects the different parts so firmly, that they must all stand, or all fall together. The demonstration, therefore, even of an obvious proposition, answers the purpose of connecting that 'proposition with others, and ascertaining its place in the general system of mathematical truth. If, for example, it be alleged, that it is needless to demonstrate that any two sides of a triangle are greater than the third; it may be replied, that this is no doubt a truth which, without proof, most men will be inclined to admit; but are we for that reason to account it of no consequence to know what the propositions are, which
must cease to be true if this proposition were supposed to be false? Is it not useful to know, that unless it be true, that any two sides of a triangle are greater than the third, neither could it be true, that the greater side of every triangle is opposite to the greater angle, nor that the equal sides are opposite to equal angles, nor, lastly, that things equal to the same thing are equal to one another? By a scientific mind this information will not be thought lightly of; and it is exactly that which we receive from Euclid's demonstration.
To all this it may be added, that the mind, especially when beginning to study the art of reasoning, cannot be employed to greater advantage than in analysing those judgments, which, though they appear simple, are in reality complex, and capable of being distinguished into parts. No progress in ascending higher can be expected, till a regular habit of demonstration is thus acquired ; it is much to be feared, that he who has declined the trouble of tracing the connexion between the proposition already quoted, and those that are more simple, will not be very expert in tracing its connection with those that are more complex; and that, as he has not been careful in laying the foundation, he will never be successful in raising the superstructure. COLLEGE OF EDINBURGH,
Dec. 1, 1813.