This ratio, invariably denoted by 7r, is constant for all circles, but it does not admit of exact arithmetical expression, being of the nature of an incommensurable number.
Investigation of the writings of Indian mathematicians has exhibited a fundamental distinction between the Greek and Indian mind, the former being pre-eminently geometrical and speculative, the latter arithmetical and mainly practical.
Statistical inquiries as to the incidence of taxation or of particular taxes, though ideal or even approximate equality of a palpable arithmetical kind is practically unattainable by governments, are not altogether to be put aside.
Even when the formal evolution of the science was fairly complete, it was taken for granted that its symbols of quantity invariably stood for numbers, and that its symbols of operation were restricted to their ordinary arithmetical meanings.
The harmony between arithmetical and geometrical measurement, which was disturbed by the Greek geometers on the discovery of irrational numbers, is restored by an unlimited supply of the causes of disturbance.
That for the conversion of a fraction into decimals (giving the complete period for all the prime numbers up to 997) is a specimen of the extraordinary love which Gauss had for long arithmetical calculations; and the amount of work gone through in the construction of the table of the number of the classes of binary quadratic forms must also have been tremendous.
The association of algebra with arithmetic on the one hand, and with geometry on the other, presents difficulties, in that geometrical measurement is based essentially on the idea of continuity, while arithmetical measurement is based essentially on the idea of discontinuity; both ideas being equally matters of intuition.
The difficulty first arises in elementary mensuration, where it is partly met by associating arithmetical and geometrical measurement with the cardinal and the ordinal aspects of number respectively (see Arithmetic).
During the later years of his life he resided in London, devoting himself to the construction of machines capable of performing arithmetical and even algebraical calculations.
The development is based on the necessity of being able to represent geometrical magnitude by arithmetical magnitude; and it may be regarded as consisting of three stages.