magic, that the degrees of grace of the Virgin Mary were in number the 256th power of 2, calculated that number. Whether or no his number correctly represented the result he announced, he certainly calculated it rightly, as we find by comparison with Mr. Shanks."
There is a point about Mr. Shanks's 608 figures of the value of π which attracts attention, perhaps without deserving it. It might be expected that, in so many figures, the nine digits and the cipher would occur each about the same number of times; that is, each about 61 times. But the fact stands thus: 3 occurs 68 times; 9 and 2 occur 67 times each; 4 occurs 64 times; 1 and 6 occur 62 times each; 0 occurs 60 times; 8 occurs 58 times; 5 occurs 56 times; and 7 occurs only 44 times. Now, if all the digits were equally likely, and 608 drawings were made, it is 45 to 1 against the number of sevens being as distant from the probable average (say 61) as 44 on one side or 78 on the other. There must be some reason why the number 7 is thus deprived of its fair share in the structure. Here is a field of speculation in which two branches of inquirers might unite. There is but one number which is treated with an unfairness which is incredible as an accident; and that number is the mystic number seven! If the cyclometers and the apocalyptics would lay their heads together until they come to a unanimous verdict on this phenomenon, and would publish nothing until they are of one mind, they would earn the gratitude of their race.—I was wrong: it is the Pyramid-speculator who should have been appealed to. A correspondent of my friend Prof. Piazzi Smyth[[139]] notices that 3 is the number of most frequency, and that 3-1/7 is the nearest approximation to it in simple digits. Professor Smyth himself, whose word on Egypt is paradox of a very high order, backed by a great quantity of useful labor, the results which will be made available by those who do not receive
the paradoxes, is inclined to see confirmation for some of his theory in these phenomena.
CURIOUS CALCULATIONS.
These paradoxes of calculation sometimes appear as illustrations of the value of a new method. In 1863, Mr. G. Suffield,[[140]] M.A., and Mr. J. R. Lunn,[[141]] M.A., of Clare College and of St. John's College, Cambridge, published the whole quotient of 10000 ... divided by 7699, throughout the whole of one of the recurring periods, having 7698 digits. This was done in illustration of Mr. Suffield's method of Synthetic division.
Another instance of computation carried to paradoxical length, in order to illustrate a method, is the solution of x3 - 2x = 5, the example given of Newton's method, on which all improvements have been tested. In 1831, Fourier's[[142]] posthumous work on equations showed 33 figures of solution, got with enormous labor. Thinking this a good opportunity to illustrate the superiority of the method of W. G. Horner,[[143]] not yet known in France, and not much known in
England, I proposed to one of my classes, in 1841, to beat Fourier on this point, as a Christmas exercise. I received several answers, agreeing with each other, to 50 places of decimals. In 1848, I repeated the proposal, requesting that 50 places might be exceeded: I obtained answers of 75, 65, 63, 58, 57, and 52 places. But one answer, by Mr. W. Harris Johnston,[[144]] of Dundalk, and of the Excise Office, went to 101 decimal places. To test the accuracy of this, I requested Mr. Johnston to undertake another equation, connected with the former one in a way which I did not explain. His solution verified the former one, but he was unable to see the connection, even when his result was obtained. My reader may be as much at a loss: the two solutions are:
2.0945514815423265...
9.0544851845767340...
The results are published in the Mathematician, Vol. III, p. 290. In 1851, another pupil of mine, Mr. J. Power Hicks,[[145]] carried the result to 152 decimal places, without knowing what Mr. Johnston had done. The result is in the English Cyclopædia, article Involution and Evolution.