may be tilted through a finite angle without cutting the natural plane at a finite distance.
Then, I will state, what there is no difficulty in proving, that the delineation of the infinitely distant parts of the natural plane will occupy a space on the picture bounded by a conic section.
The picture looks exactly as before, only that the REAL DISTANCES of certain parts which on the first assumption were finite, now become infinite.
The following two figures show the straight line which on the first assumption alone represents parts really infinitely distant, as well as the conic which on the second assumption bounds the delineation of the really infinitely distant parts. With two lines parallel on the first assumption.
In this case, space is limited. But though limited, it is immeasurable. Imagine, for instance, that every kind of unit of linear measure should shrink up as it was removed from a fixed centre, and perhaps differently according as it was radially or peripherally placed, so that it never could in any finite number of repetitions get beyond a certain spherical surface. Then that surface would be at an infinite distance and no moving body could ever traverse it, for the distance moved over in a unit of time would be a unit of distance. But unless this linear unit were to shrink according to a peculiar law, the result would be that different parts of space would be unlike. That is to say, for example, if we were to draw the plane figure ABCD, by means of given lengths AB, AC, AD, BC, we should find the resulting length of CD to be different in different parts of space. Now, geometers assume, perhaps with little reason, that the fact that rigid bodies move about readily in space without change of proportions, shows that all the parts of space are alike.
With this condition, I will state, what is again readily proved, what the law of the representation of equal distances and equal angles is in this second or hyperbolic geometry. For the distances. Let the conic be the “absolute,” or the perspective representation boundary of infinitely distant regions. Let AB be a unit of distance. Through AB draw the line
IJ, let I′J′ be any other line. Draw straight lines through their intersections II′ and JJ′ with the absolute. Let O be the point where these lines intersect; from O draw rays to A and B; then the distance A′B′ cut off by these rays on the line I′J′ is equal to AB.
The rule for angles is this. Let the conic be the absolute. Let ACB be a given angle. From C draw CI and CJ tangent to the absolute. Take any other vertex C′ and draw tangents C′I and C′J to the absolute. Through I the intersection of CI and C′I and J the intersection of CJ and C′J draw the straight line IJ cutting AC and BC in A and B. Then AC′B = ACB.
The third kind of geometry, which denies the 1st natural assumption, supposes space to be unlimited but finite, that is measurable. It is as if the linear unit so expanded in departing from a centre as to enable us to pass through what is naturally supposed to be at an infinite distance.
1. Merely because so called by writers on Perspective. Nothing to do with the “natural assumptions” of page 25.
7
Review of Jevons’s Pure Logic
| 3 July 1890 | The Nation |
Pure Logic, and Other Minor Works. By W. Stanley Jevons. Edited by Robert Adamson and Harriet A. Jevons. Macmillan & Co. 1890.
Though called Minor, these are scientifically Jevons’s most important writings. As when they first appeared, they impress us by their clearness of thought, but not with any great power. The first piece, “Pure Logic
The substance of the second piece in this volume, “The Substitution of Similars,” is in its title. Cicero had a wart on his nose; so Burke would be expected to have something like it. This is Mill’s inference from particulars to particulars. As a matter of psychology, it is true the one statement suggests the other, but logical connection between them is wholly wanting. The substitution of similars might well be taken as the grand formula of bad reasoning.
Both these tracts warmly advocate the quantification of the predicate—that it is preferable in formal logic to take A = B as the fundamental form of proposition rather than “If A, then B,” or “A belongs among the Bs.” The question is not so important as Jevons thought it to be; but we give his three arguments with refutations. First, he says the copula of identity is logically simpler than the copula of inclusion. Not so, for the statement that “man = rational animal” is equivalent to a compound of two propositions with the copula of inclusion, namely, “If anything is a man, it is a rational animal,” and “If anything is a rational animal, it is a man.” True, Jevons replies that these propositions can be written with a copula of identity, A = AB. But A and B are not symmetrically situated here. They are not simply joined by a sign of equality. Second, Jevons says that logic takes a more unitary development with the proposition of identity than with that of inclusion. He thinks his doctrines of not quantified logic and the substitution of similars call for this copula, but this is quite an error. And then an inference supposes that if the premises are true, the conclusion is true. The relation of premises to conclusion is thus just that of the terms of the proposition of inclusion. Thus the illative “ergo” is really a copula of inclusion. Why have any other? Third, Jevons holds the proposition of identity to be the more natural. But, psychologically, propositions spring from association. The subject suggests the predicate. Now the difficulty of saying the words of any familiar thing backwards shows that the suggesting and suggested cannot immediately change places.
The third piece in the volume describes Jevons’s logical machine, in every respect inferior to that of Prof. Allan Marquand, and adequate only to inferences of childish simplicity. The higher kinds of reasoning concerning relative terms cannot (as far as we can yet see) be performed mechanically.
The fourth paper advocates the treatment of logic by means of arithmetic—without previous logical analysis of the conception of number, which would call for the logic of relatives. To exhibit the power of his method, Jevons shows that it draws at once such a difficult conclusion as this: “For every man in the house, there is a person who is aged; some of the men are not aged. It follows, that some of the persons in the house are not men.” Unfortunately, this is an exhibition not of the power of the method, but of its imbecility, since the reasoning is not good. For if we substitute for “person,” even number, for “man,” whole number, for “aged,” double of an integer, we get this wonderful reasoning: “Every whole number has its double; some whole numbers are not doubles of integers. Hence, some even numbers are not whole numbers.”
The remainder of the book is taken up with Jevons’s articles against Mill, which were interrupted by his death. The first relates to Mill’s theory of mathematical reasoning, which in its main features is correct. The only defect which Jevons brings out is, that no satisfactory mode of proving the approximate truth of the geometrical axioms is indicated. But this