G B Halsted reviews Cayley's papers

George Bruce Halsted reviewed The Collected Mathematical Papers of Arthur Cayley in two different journals, namely:

G B Halsted, Review: The Collected Mathematical Papers of Arthur Cayley, by A Cayley, Amer. Math. Monthly 6 (3) (1899), 59-65.

G B Halsted, Review: The Collected Mathematical Papers of Arthur Cayley, by A Cayley, Science, New Series 9 (211) (1899), 59-63.

Except for minor editorial changes, these two articles are identical. In this review he quotes extensively from Cayley himself. We give an extract from these papers in which Halsted is looking at Cayley's contributions to non-Euclidean geometry:

Twenty years ago, in my "Bibliography of Hyper-Space and Non-Euclidean Geometry," (American Journal of Mathematics, Vol. I., Nos. 2 and 3, 1878) I cited seven of Cayley's papers written before 1873:

  1. Chapters in the Analytical Geometry of (n) Dimensions. Camb. Math. Jour., Vol, IV., 1845, pp. 119-127.

  2. Sixth Memoir on Quantics. Phil. Trans., vol. 149. pp. 61-90, (1859).

  3. Note on Lobatchevsky's Imaginary Geometry. Phil. Mag. XXIX,. pp. 231-233, (1865).

  4. On the rational transformation between two spaces. Lond. Math. Soc. Proc. III., pp. 127-180, (1869-71).

  5. A Memoir on Abstract Geometry. Phil. Trans. CLX., pp. 51-63, (1870).

  6. On the superlines of a quadric surface in five dimensional space. Quarterly Journ., Vol. XlI., pp. 176-180, (1871-72).

  7. On the Non-Euclidean Geometry. Clebsch. Math. Ann. V., pp. 630- 634, (1872).

Four of these pertain to Hyper-Space, and in that Bibliography I quoted Cayley as to its geometry as follows:

"The science presents itself in two ways - as a legitimate extention of the ordinary two- and three-dimensional geometries; and as a need in these geometries and in analysis generally. In fact, whenever we are concerned with quantities connected together in any manner, and which are, or are considered as variable or determinable, then the nature of the relation between the quantities is frequently rendered more intelligible by regarding them (if only two or three in number) as coordinates of a point in a plane or in space: for more than three quantities there is, from the greater complexity of the case, the greater need of such a representation: but this can only be obtained by means of the notion of a space of the proper dimensionality: and to use such a representation, we require the geometry of such space.

An important instance in plane geometry has actually presented itself in the question of the determination of the number of the curves which satisfy given conditions: the conditions imply relations between the coefficients in the equation of the curve; and for the better understanding of these relations it was expedient to consider the coefficients as the coordinates of a point in a space of the proper dimensionality."

For a dozen years after it was written, the Sixth Memoir on Quantics would not have been enumerated in a Bibliography of non-Euclidean geometry, for its author did not see that it gave a generalization which was identifiable with that initiated by Bolyai and Lobachevski, though afterwards, in his Address to the British Association, 1883, he attributes the fundamental idea involved to Riemann, whose paper was written in 1854.

Says Cayley: "In regarding the physical space of our experience as possibly non-Euclidean, Riemann's idea seems to be that of modifying the notion of distance, not that of treating it as a locus in four dimensional space."

What the Sixth Memoir was meant to do was to base a generalized theory of metrical geometry on a generalized definition of distance.

As Cayley himself says: "... the theory in effect is, that the metrical properties of a figure are not the properties of the figure considered per se apart from everything else, but its properties when considered in connection with another figure, viz., the conic termed the absolute."

The fundamental idea that a metrical property could be looked at as a projective property of an extended system had occurred in the French school of geometers. Thus Laguerre (1853) so expresses an angle. Cayley generalized this French idea, expressing all metrical properties as projective relations to a fundamental configuration.

JOC/EFR November 2014

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