24. The mistakes made by the early orientalists have naturally misled the historians of mathematics, and the opinions of Chasles, Wœpcke, Hankel and others founded upon such mistakes are now no longer authoritative. In spite, however, of the progress made in historical research there are still many errors current, of which, besides those already touched upon, the following may be cited as examples: (a) The proof by "casting out nines" is not of Indian origin and occurs in no Indian work before the 12th century; (b) The scheme of multiplication, of which the following is
| 1 | 3 | 5 | ||||
| 1 | ||||||
| 1 | 3 | 5 | ||||
| 2 | 1 | |||||
| 2 | 6 | 0 | ||||
| 1 | 6 | 2 | 0 | |||
an Indian example of the 16th century, was known much earlier to the Arabs and there is no evidence that it is of Indian origin; (c) The Regula duorum falsorum occurs in no Indian work; (d) The Indians were not the first to give double solutions of quadratic equations; Bhaskāra was not the discoverer of the "principle of the differential calculus," etc., etc.
