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various kinds may be turned about certain points and be what may be termed introgyrant, or turn upon their own ground without change of place. And on receipt of the remark mentioned above, this track was pursued with renewed vigour, and the results are presented in the sequel.
For the use of those who are not disposed to go into details, the substance may be stated as follows. A solid may be described, all the points in whose surface shall be equidistant from a given point within ; such a solid is called a sphere. A sphere may be turned in any manner whatsoever about its centre, without change of place. Consequences deducible from this are, that if two spheres touch one another externally, they touch only in a point ; and if they are turned as one body about the two centres which remain at rest, the point of contact remains unmoved. Hence if about two assigned points be described a succession of spheres touching one another, any number of intermediate points may be determined that shall be desired, which, on the whole being turned about the two centres, shall be without change of place; and if this be extended to imagining one sphere to increase continuously in magnitude and the other to decrease, the line described by their point of contact will be without change of place throughout; such a line is called a straight line. If two equal spheres be placed touching one another externally, and about the centre of each a sphere be described passing through the centre of the other, and a straight line of unlimited length be drawn from the point of contact of the two smaller spheres to any point in the intersection of the others; this straight line, on the whole being turned about the two centres of the spheres, will describe a surface in which any two points being taken, the straight line between them, with its prolongation either way, may be proved to lie wholly in that surface. A surface of this kind is called a plane. From these, all the relations of straight lines and planes may be inferred. If in this there is any novelty and truth, it is surprising that a property which was the foundation of the Platonic notion of the perfection of the sphere (Vide Plat. De Anim. Mund.) should not have been sooner carried into its consequences.
*These results, together with the proofs offered of what have usually been termed the Axioms and Postulates, have been formed into an Intercalary Book, with a view to facilitate their postponement in the case of students beginning geometry for the first time. And for the further convenience of this class, a Recapitulation of the principal contents of the Intercalary Book has been given at page 46 ; thereby placing the beginner in the same situation as by the ordinary proceeding. The best time for commencing the Intercalary Book would probably be after having gone once through as much of the Elements of Euclid as is usually read; not, of course, that the contents are in any degree dependent upon what follows, but to take advantage of the habits of reasoning that may be thus acquired, before attempting what must be characterized as, in some parts, at least equal in complexity to anything in the succeeding Books.
If this process is objected to as irregular, it may be a great irregularity that nature should not have framed the elements of geometry so as to present a concinnous whole with the easiest parts always foremost and the Planes in the Eleventh Book. But if she has not, or till somebody can establish that she has, there seems to be no cause why bad reasoning should be admitted for the sake of a conventional arrangement. If the sphere is the simplest of figures and the properties of all others are derivable from it, it is more reasonable to be thankful for the knowledge, than to quarrel with the dispensation. The same argument appears to apply to objections against the introduction of motion. If motion is introduced without utility, it is superfluous; and all superfluity is an evil. There may be places where it would be possible to evade the recognition of continuous motion, by a forced and affected substitution of a succession of positions instead, with the effect of greatly reducing the distinctness of the whole ; as for instance in the conclusions drawn with respect to the sphere and straight line from turning them about certain points. But there are others where the reasoning depends altogether on the supposition of a continuity of motion; as, notably, in the very pinch and nip of the argument on Parallel Lines in Prop. XXVIII D in the First Book, where the conclusion rests entirely on the impossibility of a certain line ceasing to cut a series of other lines during a continuous motion, in a way incapable of being supplied by any succession of insulated positions ; as likewise in part of Prop. XII Cor. 8 of the Intercalary Book, and elsewhere. On the whole therefore it may be an interesting question in what place a geometer would be warranted in saying, ' I could have proved these preliminaries, but it would have been necessary to disturb the order which directs that lines be treated of first and solids afterwards, and to introduce motion ; for which reason it was considered better that they should be adopted without proof.'
In labouring to get rid of Axioms, the object has been to assail the belief in the existence of such things as self-evident truths. Nothing is self-evident, except perhaps an identical proposition. There may be things of which the evidence is continually before the senses ; but these are not self-evident, but proved by the continual evidence of the senses. There may be things whose connexion with other things is so constantly impressed upon us by experience, that few people ever think of inquiring into the cause; but for that very reason there is often considerable difficulty in clearly explaining the cause, and among this class of things the admirers of axioms have found their greatest crop of self-evident truths. In arguments on the general affairs of life, the place where every man is most to be suspected, is in what he starts from as
what nobody can deny. It was therefore of evil example, that science of any kind should be supposed to be founded on axioms; and it is no answer to say, that in a particular case they were true. The Second Book of Euclid would be true, if the First existed only in the shape of the heads of the Propositions under the title of Axioms; but this would make a most lame and imperfect specimen of reasoning. The ways in which the Axioms and their kindred the Postulates have been disposed of, will be easily traced. Some have been demonstrated as Theorems, or executed as Problems; others (as the Axiom on coincidence) resolved into the mere declaration of the matter to which a certain Nomenclature is assigned; and others into Corollaries from the rest. The Axiom which declared the whole to be greater than its part, has been omitted as amounting, after the explanation of the terms 'greater' and 'less' introduced from Euclid's Book of Data, to no more than the proposition that 'the greatest is greatest.'
In the present Edition, the part relating to the disputed principle of Parallel Lines has been reduced in bulk to one half, and in substance to the following. If a tessera (or quadrilateral rectilinear plane figure of which two of the opposite sides are equal to one another and make equal interior angles with a side between them which shall be called the base] has the angles at the base less than right angles, the angles opposite to the base cannot be right angles. And this because, if a number of such figures are placed side by side, a straight line of unlimited length which shall travel continuously from one of the angular points in the series of bases, along the straight line formed by the junction of two sides, keeping ever at right angles to it, cannot cease to cut the bases and make angles at the point of section greater than given angles, before it has reached the series made by the sides opposite to the bases ; which is inconsistent with those sides forming one straight line as must be the case if the angles opposite to the bases in the tesseras were right angles. Whence may be inferred that the three angles of every triangle are not less than two right angles; and Euclid's Axion. The formula of generalization (see p. 5) has been extended and rigidly adhered to, for the express purpose of protesting against the unfortunate use which has been made of the term abstract as applied to what ought to have been called universal propositions; than which nothing has given a greater handle to the supporters of practical mal-reasoning. With a view to further resisting the notion of self-evident truths, no Corollary has been inserted without being followed by the reason ; a practice which, like the preceding, it may not be necessary to carry through the whole range of the sciences, but which will be found very useful at the threshold. Nomenclature has been substituted for Definitions, as being more closely accordant with the principle of LAVOISIER, de ne procéder jamais que du connu à l'inconnu. The difference may at first sight appear to be not much ; but if all men would agree to treat of nothing that had not previously been laid out for naming after the manner of the anatomists, the consequences in the aggregate would be considerable. The latest innovation has been the assertion, that an angle (or the thing spoken of under that term by geometers whether they knew it or not,) is a plane surface; an alteration which will probably be considered as among the most violent in the book, and for which reference must be made to the text (see p. 50 and elsewhere). No Proposition or Corollary has been admitted throughout, which is not subsequently cited; with the exception of the concluding one, and of a small number the application of which in the following Books of Euclid, or other reason for insertion, is pointed out in the Notes.
In all this the object has been to do something towards giving Geometry a right to the denomination of an exact science; a title, to which, after allowing for its superior capabilities, it has in fact had less claim than several other branches of knowledge. Its value, even as an imperfect model of reasoning, has been admitted by the fear at different times displayed of it by the allies of general obscuration. Not the least significant of the processes understood to have been in operation in France during the temporary restoration of the Bourbons, was the discouragement of geometrical lectures in the institutions under the influence of the government, and the attempt to substitute a series of philosophical recreations in the form of experiments. The hint is too good to be lost. Can nobody write a book of geometry that should be prohibited in Austria ?
T. Perronet Thompson,
Queen's Coll. Camb.
May 15, 1833.