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 QUADKATURE QUADRUMANA 123 made his calculations, extended the calculation to 36 figures, which are engraved upon his tomb- stone in Leyden. These are 3-1415926535897- 9323846264338327950289. The last figure is too large, and 8 would be too small. This was obtained by calculating the chords of suc- cessive arcs, each one being half of the pre- ceding ; for the above result this was carried out so far, that the last arc was one side of a polygon of 36,893,488,147,419,103,232 sides. The method of calculation was greatly simpli- fied by Snell, who carried the computation to 55 decimal places by means of a polygon of only 5,242,880 sides. By other mathematicians the computation was carried on, reaching suc- cessively during the last century 75, 100, 128, and 140 places of decimals ; and Montucla re- ceived from Baron Zach 154 figures, said to have been obtained from a manuscript in the Radcliffe library at Oxford, of the existence of which there is no other evidence. The figures, however, except the last two, have since been proved correct. (See Montucla, Histoire des recherches sur la quadrature du cercle, 1754.) Notwithstanding that Lambert in 1761, and still later Legendre in his Ele- ments de geometrie, proved that the ratio of the diameter to the circumference cannot be expressed by any numbers, the wish to satisfy those who still sought the exact expression of this ratio led other mathematicians to continue to add to these figures ; and some must have derived a singular gratification in the compu- tation itself and its never terminating result. In May, 1841, a paper was communicated to the royal society by Dr. Rutherford of Wool- wich, presenting 208 figures of decimals, of which however 56 were afterward proved to be wrong, so that the series was not really carried beyond the result obtained from the Oxford manuscript. In 1846 200 decimals were correctly made out by Mr. Base ; and the next year 250 by Dr. Clausen of Dorpat. In 1851 Mr. William Shanks of Durham cal- culated 315 decimals, which Dr. Rutherford verified and extended to 350. Mr. Shanks soon carried these to 527 decimals, of which 411 were confirmed by Dr. Rutherford. Fi- nally in 1853 Mr. Shanks reached the num- ber of 607 decimals, and gave the result in "his "Contributions to Mathematics" (London, 1853). When it was made evident that the arithmetical expression was impossible, it was still hoped by many that the ratio might be determined by geometrical construction ; and the bare possibility of this, which a few math- ematicians have admitted, has given encour- agement to some to seek the solution in this direction. But this, too, is now generally ad- mitted to be impracticable. Little benefit has resulted from the vast amount of time and labor that have been expended upon this fa- mous problem. Wallis, investigating it at a time when the nature of the subject was not so well understood, and the investigation was consequently a proper one, was led to the dis- covery of the binomial theorem ; but most of those who have since interested themselves in the question understood too little of the math- ematical sciences to avail themselves of any opportunity that might be presented of in- creasing the means of mathematical research. The academy of sciences at Paris in 1775, and soon after the royal society in London, to dis- courage this and other similarly futile research- es, declined to examine in future any paper pretending to the quadrature of the circle, the trisection of an angle, the duplication of the cube, or the discovery of perpetual motion. QUADROIAJVA (Lat., from quatuor, four, and manus, hand), a division of the mammalia em- bracing the lemurs and monkeys or apes, and forming the highest order of Owen's subclass gyrencephala, so called from the generally pre- hensile nature of their four extremities. Al- though, on anatomical grounds, the term quad- rumanous cannot be considered as strictly ap- plicable to the members of this extensive or- der, it is nevertheless retained by the majority of naturalists in contradistinction to bimanous (two-handed), as restricted to man alone. The restoration of the Linnrean term primates (limited so as to exclude the cheiroptera) has of late been advocated by Prof. Huxley, as more conformable to the true nature of struc- tural affinities, a view in which he has been sustained by St. George Mivart. This order, which has been conveniently divided into the three families of strepsirrliini, platyrrMni, and catarrhini, may be briefly defined as follows : Animals wkh a deciduate, discoidal placenta; clavicles complete ; orbital ring completely cir- cumscribed, and usually separated by an osse- ous septum from the temporal fossa; pollex (when present) often, and hallux generally op- posable, the latter provided with a flat nail (ex- cept in orang, in which the nail is often want- ing) ; cerebral hemispheres well developed and strongly convoluted, covering the cerebellum (except in mycetes and certain genera of the lemuridce, where the cerebellum is naked, and in the marmoset, where the external gyri and sulci are almost entirely wanting) ; stomach in most cases simple (complex in semnopithecus and cololivs) and furnished with csecal appen- dages ; teetli never in an unbroken series, but separated by a diastema. The strepsirrhini (lemurs, aye-ayes, loris, galagos, potos, and in- dris) constitute the lowest family of the order, and inhabit portions of Africa, Madagascar, and some of the Asiatic islands. They are characterized by the twisted nature of their nostrils, and by the presence of a claw on the second digit of the foot. The aye-ayes (cheiro- mys), which seem to connect the lemurs with the lower rodents, form an abnormal group by themselves, by reason of the true rodent type of their dentition, which is, incisors -fz-f, canines, premolars ^i, and molars fif = 18. The chisel-shaped incisors, moreover, agree with thoee of the rodents in growing from per- sistent pulps, but differ in being entirely in-