It is hardly necessary for me to express here my indebtedness to the patience and learning displayed by Sir Thomas Heath in his edition of the Works of Archimedes (Cambridge, 1897); but for his statements and suggestions I should never have even known that the greatest master of Greek geometry had left so fascinating a problem for the speculations of later mathematicians. In what follows I can only claim to have tried to frame my conjectures on the hypothesis that so consummate a geometer was at least on the same intellectual plane as such men as Newton or Gauss, so far as a humble admirer can compare those who have towered as giants in the subject.

The following elementary way of obtaining this expansion is interesting in connection with my paper on the Gamma-function in the May issue of the Mathematical Gazette.

On previous occasions in the Gazette (and in other places) I have repeatedly expressed the opinion that the most natural line of approach to the logarithmic function is the historical one (followed both by Napier and by Newton) in which the logarithm is effectively defined as an integral. From conversation with boys (from eleven to fourteen) I am still convinced that the idea of an area is much easier to grapple with than the idea of a limit. In spite, then, of other suggestions (which appear in the Gazette three or four times a year on the average), I still accept this method as the best: more details will be found in Appendix II. of my book on Infinite Series (both editions).

The object of the first conversation (Math. Gazette, March 1928) was to suggest the advantages of using a definition of right angle different from that of Euclid for the study of “spaces” of which the dimensions can be distinguished from each other, a very common type of space. For brevity we will refer to this study as “Dimensional Geometry,” and call the right angles “dimensional right angles.” The second conversation is intended to suggest that “dimensional spaces” facilitate the study of Non-Euclidean Geometry.