20 REPORT—1883. 
simultaneously described, or which might be simultaneously described, by 
these points respectively. But whether the notion of time does or does 
not sooner enter into mathematics, we at any rate have the notion in 
Mechanics, and along with it several other new notions. 
Regarding Mechanics as divided into Statics and Dynamics, we 
have in dynamics the notion of time, and in connection with it that of 
velocity : we have in statics and dynamics the notion of force ; and also a 
notion which in its most general form I would call that of corpus: viz. 
this may be the material point or particle, the flexible inextensible string 
or surface, or the rigid body, of ordinary mechanics; the incompressible 
perfect fluid of hydrostatics and hydrodynamics ; the ether of any undu- 
latory theory; or any other imaginable corpus ; for instance, one really 
deserving of consideration in any general treatise of mechanics is a 
developable or skew surface with absolutely rigid generating lines, but 
which can be bent about these generating lines, so that the element of 
surface between two consecutive lines rotates as a whole about one of them. 
We have besides, in dynamics necessarily, the notion of mass or inertia. 
We seem to be thus passing out of pure mathematics into physical 
science; but it is difficult to draw the line of separation, or to say of 
large portions of the ‘ Principia,’ and the ‘ Mécanique céleste,’ or of the 
whole of the ‘ Mécanique analytique,’ that they are not pure mathematics. 
It may be contended that we first come to physics when we attempt to 
make out the character of the corpus as it exists in nature. I do not 
at present speak of any physical theories which cannot be brought under 
the foregoing conception of mechanics. 
I must return to the Theory of Numbers; the fundamental idea is 
here integer number: in the first instance positive integer number, but 
which may be extended to include negative integer number and zero. 
We have the notion of a product, and that of a prime number, which is 
not a product of other numbers; and thence also that of a number as 
the product of a determinate system of prime factors. We have here the . 
elements of a theory in many respects analogous to algebra: an equation 
is to be solved—that is, we have to find the integer values (if any) 
which satisfy the equation; and so in other cases: the congruence nota- 
tion, although of the very highest importance, does not affect the 
character of the theory. 
But as already noticed we have incommensurables, and the con- 
sideration of these gives rise to a new universe of theory. We may take. 
into consideration any surd number such as,/ 2, and so consider numbers 
of the form a + b,/2, (a and b any positive or negative integer numbers 
not excluding zero) ; calling these integer numbers, every problem which 
before presented itself in regard to integer numbers in the original and 
ordinary sense of the word presents itself equally in regard to integer 
numbers in this new sense of the word; of course all definitions must be 
altered accordingly : an ordinary integer, which is in the ordinary sense 
