関孝和による『楊輝算法』の解法の訂正について

概要

In May 1673, Seki Takakzu completed a revised manuscript of “Yang Hui suanfa”. Seki corrected a proof in “Yang Hui suanfa”. Yang Hui claimed to have corrected the errors in three of the solutions of “Wucao suanjing”, but in the third and final solution, he had no choice but to show a non-mathematical solution: he drew a reduced drawing to find the required length and then measured it to find the actual length. Seki showed the correct length in his manuscript. Although Seki did not describe his argument for finding the length, his argument can be easily deduced. To do so, a simultaneous equation consisting of two unknowns had to be solved, and one of the unknowns had to be eliminated. This suggests that the “Yang Hui suanfa” problem was one of his first possible encounters with a problem that required many unknowns for its solution. In the manuscript, Seki gave a solution that is quite different from the solution given by Yang Hui. Again, he does not say how he obtained that solution. In this solution, it is not necessary to find the length of the line segment as Yang Hui did, but only to find the area from the three sides of a triangle. This is substantially described in Problem 19 of the “Daijutsu bengi no ho”. This formula for area is essentially the same as Heron's formula. However, it is doubtful that Seki found this solution immediately after solving the quadratic equation of the line segment numerically to find the true length. In this note, I will discuss another possibility for Seki's argument: the possibility that he used the root formula for a special type of quadratic equations. This formula could have been obtained by Seki combining Problem 14 in the second volume of “Tianmu bilei chengchu jiefa” of “Yang Hui suanfa” and Problem 8 in the final chapter of “Suanxue qimeng.”