Every mathematician knows that scores of methods, differing altogether from each other in process, all end in this mysterious 3.14159..., which insists on calling itself the circumference to a unit of diameter. A reader who is competent to follow processes of arithmetic may be easily satisfied that such methods do actually exist. I will give a sketch, carried out to a few figures, of three: the first two I never met with in my reading; the third is the old method of Vieta.[[354]] [I find that both the first and second methods are contained in a theorem of Euler.]
What Mr. James Smith says of these methods is worth noting. He says I have given three "fancy proofs" of the value of π: he evidently takes me to be offering demonstration. He proceeds thus:—
"His first proof is traceable to the diameter of a circle
of radius 1. His second, to the side of any inscribed equilateral triangle to a circle of radius 1. His third, to a radius of a circle of diameter 1. Now, it may be frankly admitted that we can arrive at the same result by many other modes of arithmetical calculation, all of which may be shown to have some sort of relation to a circle; but, after all, these results are mere exhibitions of the properties of numbers, and have no more to do with the ratio of diameter to circumference in a circle than the price of sugar with the mean height of spring tides. (Corr. Oct. 21, 1865)."
I quote this because it is one of the few cases—other than absolute assumption of the conclusion—in which Mr. Smith's conclusions would be true if his premise were true. Had I given what follows as proof, it would have been properly remarked, that I had only exhibited properties of numbers. But I took care to tell my reader that I was only going to show him methods which end in 3.14159.... The proofs that these methods establish the value of π are for those who will read and can understand.
| 200000000 | 31415 | 3799 |
| 66666667 | 2817 | |
| 26666667 | 1363 | |
| 11428571 | 661 | |
| 5079365 | 321 | |
| 2308802 | 156 | |
| 1065601 | 76 | |
| 497281 | 37 | |
| 234014 | 18 | |
| 110849 | 9 | |
| 52785 | 5 | |
| 25245 | 2 | |
| 12118 | 1 | |
| 5834 | ||
| ————— | ——— | —— |
| 314153799 | 31415 | 9265 |
1. Take any diameter, double it, take 1-3d of that double, 2-5ths of the last, 3-7ths of the last, 4-9ths of the last, 5-11ths of the last, and so on. The sum of all is the circumference of that diameter. The preceding is the process when the diameter is a hundred millions: the errors arising from rejection of fractions being lessened by proceeding on a thousand millions, and striking off one figure. Here 200 etc. is double of the diameter; 666 etc. is 1-3rd of 200 etc.; 266 etc. is 2-5ths of 666 etc.; 114 etc. is 3-7ths of 266 etc.; 507 etc. is 4-9ths of 114 etc.; and so on.
2. To the square root of 3 add its half. Take half the third part of this; half 2-5ths of the last; half 3-7ths of the last; and so on. The sum is the circumference to a unit of diameter.
| Square root of 3.... | 1.73205081 |
| .86602540 | |
| ————— | |
| 2.59807621 | |
| .43301270 | |
| .08660254 | |
| 1855768 | |
| 412393 | |
| 93726 | |
| 21629 | |
| 5047 | |
| 1188 | |
| 281 | |
| 67 | |
| 16 | |
| 4 | |
| 1 | |
| ————— | |
| 3.14159265 |