II.6 Idea of Proof Development, Islamic and Renaissance Mathematics, Seventeenth-Century Mathematics: Pages 133 – 135.

Khwarizmi in the eighth century included algebraization of mathematical thinking and Euclidean geometric proof to legitimize his mathematics.  His Islamic book in translation entitled “The Compendious Book on Calculation by Completion and Balancing” is an example of these ideas.  He includes variables along with numbers in his problems and solutions.  The quadratic equation is broken up into six cases and is not yet treated as a singular entity.   On translation an example of this is the problem “What is the square which combined with ten of its roots will give a sum total of 39?”  This may now be written as: x2 + 10x = 39.  The solution is written out more in recipe form.  A justification is given by a geometric argument. 

Cardano in the 1545 book “Ars Magna” presented a complete treatment of 3rd and 4th degree polynomials.  Even though his reasoning is more abstract than that of Khwarizmi he still continued to utilize diagrams of Euclid-like geometric arguments.

In the seventeenth century Newton and Leibniz created and utilized infinitesimal calculus in their proofs.  Fermat and Descartes further utilized these tools.  More progressive ideas than geometrical proof were also utilized by Cavalieri, Roberval and Torricelli.  In 1643 Torricelli used an example of computing the volume of a rotating hyperbola.

In 1637 Descarte in “La Geometrie” algebratized geometry by making an analogy between operations in arithmetic with lines and line segments in geometry.  He included a unit length and was able to remove the dimensionality of values and thus do away with the requirement of homogeneity of variables.

It became possible to prove geometric facts with algebraic procedures.  This idea reached full maturity by Viete in 1591.  Newton (1707) was not a proponent of algebraic thinking.  In his “Principia” he used his own calculus sparingly and only where necessary and barred algebra from his treatise entirely.

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