By G. H. Hardy

There may be few textbooks of arithmetic as recognized as Hardy's natural arithmetic. for the reason that its booklet in 1908, it's been a vintage paintings to which successive generations of budding mathematicians have grew to become initially in their undergraduate classes. In its pages, Hardy combines the keenness of a missionary with the rigor of a purist in his exposition of the elemental rules of the differential and crucial calculus, of the homes of endless sequence and of different issues regarding the proposal of restrict.

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**Example text**

Let 1/1: V -+ ~2 be any other admissible chart such that the point e(t) E V for all t E [a, b]. Then the function y(t) = I/I[e(t)] is defined. We obviously have y(t) = q[x(t)], where q = qJ 0 1/1-1 is the transition function for the given charts qJ and 1/1. The function q has all the partial derivatives of any order, and these derivatives are continuous. Hence it follows that y(t) is a piecewise smooth path of class C. 1f x'(t)-# 0, then since the Jacobian of the function q is non-zero it follows from the equalities indicated that y'(t) -# o.

Corollary. The natural metric PM of the two-dimensional Riemannian manifold M is compatible with the natural topology of M. In order to prove the given assertion we first need to prove that any neighbourhood of an arbitrary point Po of the manifold M contains some disc BM(po, 8). The truth of this follows from the first inclusion of the lemma. Secondly, we need to prove that any disc, in the sense of the metric PM' with centre Po contains some neighbourhood of the point Po. This follows from the second inclusion, and thus the corollary is proved.

Then p-1(X) consists of two distinct elements x E L and x' E L', where the curves Land L' are identified according to the given pasting rule. G. Reshetnyak pasting together some neighbourhood V of the point x and some neighbourhood V' of the point x'. Here V and V' can be chosen so that each ofthem is homeomorphic to a half-disc and the pasting is carried out along the boundaries of these neighbourhoods. Finally we consider the case when some point x E p-l(X) is an end of one of the curves L E R.