They include the real numbers alongside many types of infinities and infinitesimals. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Such numbers are infinite, and their reciprocals are infinitesimals. Augustin-Louis Cauchy (1823) defined the differential without appeal to the atomism of Leibniz's infinitesimals. In the context of Lawvere's approach to infinitesimals, let "dx" be a nilsquare infinitesimal. The latter theory is a rigorous formalisation of calculations with infinitesimals. This is another reformulation of the calculus in terms of infinitesimals. Archimedes used them, even though he didn't believe that arguments involving infinitesimals were rigorous. Non-standard analysis provides a number system with a full array of infinitesimals (and their inverses). Such zeroes can be seen as an approximation to infinitesimals. Several technical issues must be addressed to develop a calculus of infinitesimals. Infinitely many infinitesimals are summed to produce an integral. From this point of view, calculus is a collection of techniques for manipulating infinitesimals. That is, mathematicians felt that the concept of infinitesimals contained logical contradictions in its development. Greek mathematicians are also credited with a significant use of infinitesimals.