The invention and cultivation of the differential calculus led to many new views and new methods, into which it is not our purpose to enter, as we intend the present article not for mathematicians, but for those who have just enough of the science to think it possible that the solution of the problem is reserved for them. The continued product of Wallis, the continued fraction of Brounker, the series of Mercator, Gregory, Newton, &c., were so many new algebraical expres sions of a result which, one imagine, would be considered as carried far enough by the arithmetician. Nevertheless the ratio was consecutively carried to 75 places by Abraham Sharp, to 100 by Machin, and to 123 places by De Lagny, and at the end of the last century to 140 places by Vega. And Baron Zach gave Montucla the copy of a manuscript in the Radcliffe Library at Oxford, in which it was carried to 154 places.
Vega's result, which, so far as they go, is confirmed by those of Machin and De Lagny, is wrong only in the last four figures. The Oxford manuscript is wrong only in the last two.
In the year (1841) in which this article was first published, Dr. Rutherford communicated to the Royal Society 208 places of decimals. Of these, however, it Was afterwards found that the last 56 were incor rect; and so, it was found, were the last two figures of the Oxford result : that is, the Oxford manuscript and Dr. Rutherford were cor rect just as far as they agreed. About 1346, Mr. Dase [TABLES], a very powerful mental calculator, calculated 200 decimals; and in 1847, Dr. Clausen, of Dorpat, calculated 250 decimals, by two methods. (` Astr. Nachr,' No. 439, according to Mr. Shanks.) In 1851, without being aware of what Dr. Clausen had done, Mr. William Shanks [TABLES], of Houghton-le-Spring, Durham, calculated 315 decimals, which Dr. Rutherford verified, and extended to 350 decimals. In 1851-52, Mr. Shanks extended his calculation to 527 decimals, while Dr. Rutherford independently calculated 441. In March and April, 1853, while Mr. Shanks was passing a very full account of his 527 figures through the press, he extended it to 607 decimals, and gave the result in ' Contributions to Mathematics,' London, 1853, 8vo. Accord ingly, this famous result is now certainly obtained to 441 decimals, and with high probability to 607.
We give Mr. Shanks's result, and also his values of the base of Napier 's logarithms, and of the modulus of Briggs's system ; three numbers which are often wanted together. The figures between the rules, and those in the lowest row, are to enable the reader to detect and correct any error of printing, or to decide upon any defaced figure. The number between the rules is the sum of the ten preceding digits; and the number in the lowest column is the unit's figure of the sum of the whole column. Thus 1415926535 has digits which sum into
41; and 1534447730952 has digits which sum into 59, of which the 9 is written below.
These excesses of calculation are useful in showing the way, and in destroying that belief in the impracticability of extending what has been done which has retarded the progress of many subjects. The 607 decimal places give no sufficient notion of their amount of accu racy; or rather, give no sufficient notion of the impossibility of placing the amount of accuracy before the imagination. But the following illustration has a tendency to do what is wanted.
The blood-globules of some nnimalculee aro a millionth of an inch in diameter. Let there be an inhabited globe so large that our great glebe is but fit to be a blood-globule in the body of one of its animal cules; and call this the first globe abort us. Let the second globe above us be so large that the globe is but as one of the blood globules in it; and so on to the twentieth globe above us. Next, let one of the blood.globules on our globe be an inhabited globe, with everything In proportion; and let this be the first globe below us. Let the second globe below us be but a blood-globule in an animalcule of the first ; and so on to the twentieth globe below us. Then if the inhabitants of the twentieth globe above us were to calculate the circumference of their globe from its diameter by help of the 607 decimals, the inhabitants of the twentieth globe below us could not detect the error with their best microscopes, unless their Tulleys and Mosses were much greater masters of their art than ours.
The newest attempt at quadrature of our century is that of Mr. R. Ambrose Smith, of Aberdeen. It was undertaken as n test of the theory of probabilities. If a thin rod, not so long as the breadth of a plank, be thrown at hazard upon a planked floor, the chance which the rod has of intersecting a amens between two planks is the fraction which the length of the rod is of the quadrant having the breadth of a plank for its radius. lu 3204 tosses with a rod three-fifths of the breadth of plank, the experimeutor found 11 mere contacts, and 1213 decided intersections. He counted the contacts as intersections, and, from the principle [PROBABILITY] that the result of a large number of trials is nearly that of the long run, ho presumed that 3 is to the quad rant of radius 5 nearly as 1224 to 3204. This gives 3.1412 for the ratio of the circumference to the diameter ; which comes nearer than Archi medes. If, which perhaps ought to have been done, the contacts had been equally divided between intersections and non-intersections, the result would have been 3'1553: and even this is a remarkable approach, the nature of the method being considered.