whether means might not be found for mentally extracting roots without limitation to exact cubes. This was an incomparably harder problem, nor did I arrive at its solution till a year or two later. Each process, as soon as discovered, I taught to my pupils, who in the easier task—all that Colbourn ever attempted—became more rapid and far more correct than Colbourn himself; for with him, in extracting a cube root expressed in three figures, it was a common incident to fail in the second, an error which my pupils learned for the most part to avoid. I may add that some of them became so quick and accurate in both processes, that when on a public occasion, viz., at Midsummer, 1822, printed tables of cubes and their roots had been placed in the hands of examiners, and questions asked therefrom ranging up to two thousand millions, and of course without any limitation to exact cubes, the answers—fractions, however being disregarded—were given so quickly as to lead some sceptics, little aware of the monstrous absurdity of the hypothesis, to declare that the whole must have been previously learned by rote. I reduced my discovery to writing, intending to publish it in a contemplated manual of mental arithmetic; but unfortunately this, with other papers, was lost in a manner never fully known, and to repeat the discovery I fear I should now find quite impracticable.[39] “While on the subject of mental arithmetic, I may mention that I brought the pupils in my class to perform mentally other difficult calculations with a facility that excited no small surprise. Thus they would readily find the moon’s age (approximately, by epacts) for any day of any year; also, the day of the week corresponding with any day of the month; and, by a combination of the two processes, ascertain the day of the month corresponding with Easter Sunday in any year.” It was with some reason that Mr. Sargant, in describing his old school, writes: “Our arithmetic was amazing, even excelling, by our laborious acquisition of mental arithmetic, the success of the present Privy Council Schools.”[40] In surveying, also, the young teacher’s pupils made almost as much progress as in mental arithmetic. He had undertaken to make a complete survey of Birmingham:— “I now made my first trigonometrical survey; taking my first stations on our own playground (which fortunately commanded a view of many of the principal objects in the town), and, as before, engaging my surveying class in the work, both for their instruction and my own assistance.