In this chapter we give a detailed examination of the substantial treatise Leibniz composed in Paris in 1675-76, the De Quadratura Arithmetica, showing how the techniques that he developed in Proposition 6 of that treatise went beyond the traditional methods of quadrature using exhaustion or indivisibles, and relies upon what we have termed the “Principle of Unassignable Difference”. Although this is a “direct” proof, it does not depend on infinitesimals and infinites, which Leibniz does not introduce until Proposition 8. We show that in the intermediate Proposition 7, Leibniz provided an indirect proof by reductio of the legitimacy of the procedure adopted in Proposition 6. In Proposition 8 he shows how a “direct” proof relying on a continuity argument together with the fiction of infinities and the infinitely small can also be proved to be equivalent to the reductio proof of Proposition 7, thus establishing the inter-translatability between such proofs and direct proofs using the fiction. We then provide convincing evidence that Leibiz continued to value the methods of the DQA after developing his differential calculus, and show the intimate relationship between the two methods.