QUADRATURE OF THE CIRCLE. 



QUADRATURE OF THE CIRCLE. 



870 



until it can be shown to be given in allowing the simple postulates 

 above mentioned. 



Aristophanes introduces into his comedy of the Birds a geometer 

 who is going to make a square circle. Plutarch asserts that Anaxagoras 

 employed himself upon this problem in prison. Hippocrates of Chios 

 actually found the way to make a rectilinear space equal to certain 

 circular spaces, and is reported to have attempted the general problem. 

 There is evidence enough that it acquired an early celebrity, and it 

 may be doubted whether the researches of Euclid in incommensurables 

 [IRRATIONAL QUANTITIES] had not some reference to a supposition 

 that the circle and its diameter might possibly be discovered to belong 

 to a particular class of these quantities. Archimedes, in his book on 

 the mensuration of the circle, is the first who made any approach even 

 to a practical determination of the question. By inscribing and circum- 

 scribing polygons of 96 sides in and about the circle, he demonstrates 

 that the excess of the circumference over three times the diameter 

 must be less than 10-70ths of the diameter, and greater than 10-71st 

 parts. Hia limits are perfectly correct, and even tolerably close. 

 According to him, a circle of 4970 feet diameter would have a circum- 

 ference lying between 15,810 and 15,620 feet, the truth being that 

 such a circle would have a circumference of 15,613| feet very nearly. 

 This measure of Archimedes gives 3-14286 for the approximate value 

 of *, the ratio of the circumference to the diameter ; several of the 

 Greeks are said to have made further approximation, but their results 

 are not preserved. 



Among the Hindus [ViCA GAXITA, in Bioo. Drv.] are found the 

 ratios of 3927 to 1250, and also that of the square root of 10 to 1. 

 The first gives = 3-1416 exactly, and is considerably more correct 

 than that of Archimedes; the second gives 3-1623, and is much less 

 exact. The date of the first result is not known ; but all agree that 

 the writings in which it is found are anterior to any European improve- 

 ment on the measure of Archimedes. The ratio given by Ptolemy, in 

 the Syntaxis, is 3'141552, not quite so correct as 3-1416, but so near 

 to it that those who doubt of the antiquity of Hindu science will 

 probably suppose the 3"1416 above mentioned to be a version of 

 Ptolemy's measure. 



This subject began to be reconsidered in the 16th century, in the 

 middle of which were calculated the tables [TABLES] of Rheticus, the 

 celebrated Copernican, from which the value of T might easily have 

 been calculated to eight decimals, but it does not appear that this was 

 done. Purbach used the ratio of 377 to 120, or 3-141667, not so exact as 

 : ly's. Regiomontanus slightly corrected the limits of Archimedes, 

 but Peter Metius, father of Adrian (to whom it is often attributed, 

 merely because Adrian records and delivers it), and of James (to 

 whom the invention of the telescope has been given), made a decided 

 improvement. He gives the ratio of 355 to 113, or 3'14159292, which 

 is correct to the sixth decimal inclusive. Nothing more precise could 

 be desired for practical purposes, insomuch that a circle of 113 in 

 diameter may be reckoned as one of 355 in circumference, which, 

 though a little too great, does not rive the circumference wrongly by 

 so much as one foot in 1900 miles. On looking, however, at the 

 account which Adrian Metius gives of his father's investigation, we 

 find that the extreme closeness of the approximation is only a piece of 

 good fortune. As it is very curious that this should never have been 

 noticed, we quote the whole passage : " Parens mens Illustrium U.D. 

 Oplinum ConfnHlaratarum Belgisc Provinciarum Geometra, in libello 

 quern conscripsit adversnm quadraturam circuit Simonis a Quercu 

 demonstravit proportionem peripherito ad snam diametrum esse mino- 

 rem qiiam 3^, hoc est {JJ, majorem vero quam 3^, hoc est ${, quarum 

 proportionum intermedia existit 3 r 'ft, sive {fj. Quse quidem intermedia 

 -tio aliquantulum existit major, quam ea, quam invenit Mr. Lu- 

 dolph a Collen, cujus tainen differentia est minus quam ronfe^." The 

 book nf Peter Metius was never published, that we can find; and we 

 doubt if Adrian Metius would have published his father's result, if it 

 had not come so very near to the then recent result of Van CVnlrn. 

 It seems that the elder Metius, finding ^ too great, and ^ too small, 

 took the mean of the numerators for a numerator, and the mean of 

 the denominators for a denominator, and presumed that the result 

 was nearer the truth than either limit : a presumption to which he 

 had no right whatever, except that of trusting to the chapter of 

 accidents. 



Metius lived m the latter half of the 16th century, as also did Vieta, 

 who gave a still more accurate though not so elegant a measure. He 

 was the first who exhibited a series of arithmetical operations by 

 which a mere calculator might carry on the process to any extent, and 

 gave the following "result : The circle whose diameter is ten thousand 

 million of parts, has a circumference greater than 31,415,928,535 of 

 those parts, and less than 31,416,926,537. Other approximations 

 rapidly followed : Adrianus Romanus calculated the perimeter of an 

 inscribed polygon of 1073741824 sides, by means of which he found 

 for the ratio 3-141592853589793; but his contemporary Ludolph van 

 Ceulen, by calculating the chords of successive arcs, each of which 

 is the half of the preceding, found the perimeter of a polygon of 

 368834881 47 II l']''S-2:i-2 sides, and obtained 86 figures of the ratio 

 8-14159, ftc,, presently given to a still greater length. So far the 

 method of calculating by means of inscribed polygons, though Vieta 

 bad reduced it to routine, had received no material simplification. 

 Thi* was given by Snell, who found some propositions (afterwards 



demonstrated by Huyghens) which very much abridge the labour. 

 He found a result as correct as that of Archimedes, by means of a 

 simple hexagon; making the 96-sided polygon of Archimedes give 

 seven decimals correctly, instead of three. He also calculated the ratio 

 to 55 decimal places, and by means of a polygon of only 5242880 sides. 

 Huyghens introduced some new theorems of the same species as those 

 of Snell. 



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 tnose 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 might imagine, would be considered as 

 carried far enough by the arithmetician. Nevertheless the ratio was 

 consecutively carried to 75 places by Abraham Sharp, to 1 00 by Machin, 

 and to 128 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 1846, 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. 489, according to Mr. Shanks.) In 1851, without 

 being aware of what Dr. Clausen had done, Mr. William Shanks 

 [TABLES], of Houghton-le-Spriug, 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 1584447730952 has digits which sum into 59, of which the 9 

 is written below. 



* On making inquiry at the Radcliffe Library, we find that no such thing is 

 known to be there, nor has been known for the laet forty years at least. The 

 inquiry has often been made, and every search has been unproductive. Baron 

 Zach must certainly have seen uch a manuscript somewhere : for, though fond 

 of a hoax among friend*, it is incredible that he should have correctly calculated 

 the result sixteen places farther than had been done, merely to make Montucla 

 believe In an Oxford manuscript. Montucla does not ay that Zach saw a 

 manuscript upon the subject : but only that he saw the result in a manuscript. 

 It may be that the figures were only jotted down on a fly-leaf, or something of 

 that sort : if not, Zach must have failed to remember where he saw the figures; 

 a thing in itself very unlikely. 



