Tendency Of The Algorithm Of The Sixteenth And Seventeenth Centuries The Development Of Mathematics

The rapid development of computer techniques is opening up a new period of algorithmic research, and it is therefore an important task for historians of mathematics to probe systematically the algorithmic tendencies in the development of mathematics. By three elaborate case studies(Cardano's Ars Magna, Descartes' Geometry and Wallis' interpolation), joined with the general investigation into the historical backgrounds, this paper supplies strong support and new vision for a profound understanding of the algorithmic character of mathematical development in the 16th and 17th century. Some traditional viewpoints are argued and new conclusions are drawn by the author.Descartes presented in his Geometry a standard construction for solving algebraic equations. Though Descartes declared that his program fits equations of any degree, he examined equations of no more than six degree. In this paper, an example of equations of seven and eight degree is contributed for the first time with help of computer, and the algorithmic character of Descartes' construction is then discussed. The author also makes a distinction between the role of algebra in analytical geometry and that in Descartes' Geometry.The paper observes different viewpoints on the history of cubic equation at the algorithmic angle, investigates the algorithms for computing cubic and quartic equation in Cardano's Ars Magna, and analyses their origins. Moreover, by discussing the role of geometrical proof in Ars Magna and Cardano's attitude on irrational number, negative number and imaginary number, the paper points out that the role of geometry in Renaissance algebra had been tending weaker.Interpolation was a kind of skillful numerical method for calendar-making in ancient and medieval China, India and Arab. Comparatively, the method had not been developed as much in Europe till the Renaissance. In the 16th and 17th century, however, it was advanced greatly by European scholars. The turn indicated that Europeans started to pay more attention to algorithm. Having discussed Wallis' interpolation in great detail, the paper shows that Wallis mainly relied on incomplete induction and analogical inference to obtain his results, an approach which was key to the creation of many important algorithms during the considered period. A comparison between Wallis' method and previous interpolations is then made and the influence of Wallis' thinking style embodied in his interpolation on Newton's invention of binomial theorem is clearly revealed.Finally, some open problems for further research are proposed.