Here we return to the transformation of number in modern mathematics, where first Descartes' arithmetic algebra and then Leibniz's analytic algebra as a middle term transform both the understanding of number and the understanding of figure. First, we"ll look at the way that mathematicians moved from the natural numbers, the integers, and the rationals, to the reals, the complex numbers, the p-adic numbers, and the transfinite ordinals and cardinals. Just as algebra allowed Leibniz to master the infinitesimal and broach the topic of the infinite, so here mathematicians finally begin to master the infinite, though unsurprisingly the infinite still proves elusive and mysterious.