 and for any real \(x\), \(3xh+h^2\) is infinitesimal whenever \(h\) is infinitesimal.) For any particular real \(\epsilon\), this shows that there is some hyperreal \(\delta\) that works (namely, any infinitesimal), and by the transfer principle we can conclude that for this real \(\epsilon\) there is some real \(\delta\) that works, and we no longer need to worry about the details of how to find it. Thus, we can validate the reasoning of Newton and Leibniz that allows them to treat infinitesimals as non-zero for calculations until we get to the final result, and then treat them as zero at the end. They really work like the “ghosts of departed quantities” that Berkeley satirized! (The extent to which Robinson’s system is a vindication of Leibniz and Newton was the subject of extended discussion in articles by Robinson and others. See Bos (1974) for a classic source of this debate.)