 Now consider the space \(\Delta^{\Delta}\) of maps from \(\Delta\) to itself. It follows from the microaffineness principle that the subspace \((\Delta^{\Delta})_0\) of \(\Delta^{\Delta}\) consisting of maps vanishing at 0 is isomorphic to  \(\bR\).[48]  The space \(\Delta^{\Delta}\) is a  monoid[49]  under composition which may be regarded as acting on \(\Delta\) by evaluation: for \(f \in \Delta^{\Delta},\) \(f \cdot \varepsilon = f (\varepsilon)\). Its subspace \((\Delta^{\Delta})_0\) is a submonoid naturally identified as the space of ratios of microquantities. The isomorphism between \((\Delta^{\Delta})_0\) and \(\bR\) noted above is easily seen to be an isomorphism of monoids (where \(\bR\) is considered a monoid under its usual multiplication). It follows that \(\bR\) itself may be regarded as the space of ratios of microquantities. This was essentially the view of Euler, who regarded (real) numbers as representing the possible results of calculating the ratio 0/0. For this reason Lawvere has suggested that \(\bR\) be called the space of Euler reals.