translation in 1736. Newton's next mathematical work was Tractatus de Quadratura Curvarum, which he wrote in 1693 but it wasn't published until 1704 when he published it as an Appendix to his Optics. This work contains another approach, which involves taking limits. Newton says: "In the time in which x by flowing becomes x+o, the quantity x becomes (x+o) i.e. by the method of infinite series, x + nox + (nn-n)/2 oox +...... At the end he lets the increment o vanish by 'taking limits'."A well known mathematician Leibniz, learned much on a European tour, which led him to meet Huygens in Paris in 1672. He also met Hooke and Boyle in London in 1673 where he bought several mathematics books, including Barrow's works. Leibniz had a lengthy correspondence with Barrow. On returning to Paris Leibniz did some very fine work on the calculus, thinking of the foundations very differently from Newton. Newton considered variables changing with time. Leibniz thought of variables x, y as ranging over sequences of infinitely close values. He introduced dx and dy as differences between successive values of these sequences. Leibniz knew that dy/dx gives the tangent but he did not use it as a defining property. For Newton integration consisted of finding fluents for a given fluxion so the fact that integration and differentiation were inverses was implied. Leibniz used integration as a sum. He was also happy to use 'infinitesimal' dx and dy where Newton used x' and y' which were finite velocities. Of course neither Leibniz nor Newton thought in terms of functions, however, but both always thought in terms of graphs. For Newton the calculus was geometrical while Leibniz took it towards analysis. Leibniz was very conscious that finding a good notation was of fundamental importance and thought a lot about it. Newton, on the other hand, wrote more for himself and, as a consequence, tended to use whatever notation he thought of on the day. Leibniz's notatio...
