0420 Irrational Numbers 



Irrational numbers, theory. Dantscher von 

 Kollesberg, V. Innsb. Nt. Md. B. 17 (1887) 



, Weierstrass-Cantor theory. Illigens, E. 



Mth. A. 33 (1889) 155- . 



quantities, approximate determination, two 

 methods. Giinther, S. Erlang. Ps. Md. S. 

 Sb. 6 (1874) 82-. 



, equidifferent multiples. Sang,E. [1880] 



Edinb. E. S. P. 11 (1882) 2-. 

 Irrationality, arithmetical. Glaisher, J. W. L. 



Ph. Mg. 45 (1873) 191-. 



of cos , except when n=l, 2, 3, 4 or 6. 



Wendt, C. Mh. Mth. Ps. 10 (1899) 97-. 

 Irrationals, algebraic, theory. Capelli, A. 



Nap. Ed. 30 (1891) 61-. 

 Limit-value and irrational numbers. Jensen, 



J. L. W. V. Ts. Mth. 3 (1885) 33-; Fschr. 



Mth. (1885) 203. 

 Limits and irrational numbers, theory. Arzela, 



C. (xn) Ev. Sc.-Ind. 10 (1878) 77-, 144-, 



177-. 



Periods of decimals equivalent to 1/p, termina- 

 tor. Sardi, C. [1868] G. Mt. 7 (1869) 24-. 

 for Jv, il?, -&, etc., short process. 



Glaisher, J. W. L. Mess. Mth. 7 (1878) 



190-. 

 Pythagorean series for side and diagonal of 



square. Hultsch, F. Bb. Mth. 1 (1900) 8-. 

 Eadicals, calculation. Eggers, H. Des Moines 



Anal. 3 (1876) 100-. 

 , extraction, general method. Ruffini, P. 



[1812] Mod. S. It. Mm. 16 (1813) 373-; 



17 (1815) 1-. 

 , quadratic, approximate value. Poncelet, 



J. V. Crelle J. 13 (1835) 277-. 

 , , values. Horvath, . (xn) Mag. 



Tud. Ak. Ets. 1 (No. 8) (1867) 150- ; (x) Par. 



S. Phlm. Bll. 4 (1867) 247-. 

 , , approximation, Poncelet's theorem. 



Zeuner, G. Schlomilch Z. 3 (1858) 383-. 

 , , Poncelet's approximate linear valuation. 



Sylvester, J. J. Ph. Mg. 20 (1860) 203-. 

 , , geometric approximation. LeauU, 



H. Par. S. Mth. Bll. 8 (1880) 106-. 

 , , theorem, extension to trinomial. 



Eesal,H.A. (xi) Par. S. Mth. Bll. 1 (1873) 



, , theorems for linear representation 



generalised. Sylvester, J. J. B. A. Ep. 



(1860) (pt. 2) 7. 

 Eeal numbers, theory of, according to 



Dedekind's conception. Eicci, G. Ven. I. 



At. (1892-93) 233- ; G. Mt. 35 (1897) 22-. 

 Eoot extraction. West, B. Bost. Mm. Am. 



Ac. 1 (1785) 165-. 

 . Anon, (vi 782) Madrid Ev. 2 (1851) 



. Kinkelin, H. Grunert Arch. 26 (1856) 



361-. 

 . Evans, A. Des Moines Anal. 3 (1876) 



. Chokrobarthy,S.N. Edinb. Mth. S. P. 



16 (1898) 77. 

 , approximate. Dienger, J. Grunert 



Arch. 17 (1851) 421-. 



Irrational Numbers 0420 



Eoot extraction, approximate. Heaton, JEf. 

 A. Mth. 1 (1884-85) 14-. 



, cubic, &c. Young, M. Ir. Ac. T. 1 



(1787) 59-. 



of fractions, Ampere's method; applica- 

 tion to gamut. Vincent, A. J. H. N. A. 

 Mth. 5 (1846) 5-. 



of integers by binomial theorem. 

 Martin, Art. Des Moines Anal. 1 [1874] 117-. 



, nth degree. Eggers, H. Des Moines 



Anal. 2 (1875) 14-. 

 Eoots of all powers, extraction. Hoare, C. 



Franklin I. J. 39 (I860) 50-. 

 , approximate calculation, and applications. 



Thallmayer, V. Arch. Mth. Ps. 10 (1891) 



32-. 

 , formulae for calculation. Borgogelli, M. 



Em. N. Line. At. 45 (1892J 101-. 

 , expression by rational fractions. Thiele, 



T. N. N. Ts. Mth. 1 (B) (1890) 33- ; Fschr. 



Mth. (1890) 221-. 

 , higher, extraction. Greene, D. Camb. 



(M.^Mth. M. 1 (1859) 123-. 



'HjN, last n figures by division. Anne, J/. 

 N. A. Mth. 4 (1845) 561-. 



, numerical, extraction, abridged. Bobillier, 

 . Gergonne A. Mth. 20 (1829-30) 125-. 



and powers, calculation. Cordeiro-Feyo, J. 

 Lisb. Mm. Ac. Sc. 12 (1837) (pte. 1) 1-. 



, rapid method of approximation. Collins, 

 M. B. A. Ep. 42 (1872) (Sect.) 13-. 



Square and cube root extraction, abridged. 

 Finck, B. N. A. Mth. 5 (1846) 250-. 



, degree of approximation. 

 Verhulst, P. F. N. A. Mth. 7 (1848) 46-. 



roots. Fitremann, . N. A. Mth. 



10 (1871) 87-. 

 , rules for contracting processes, 



incorrectness of. Hill, M. J. M. L. Mth. 



S. P. 18 (1886-87) 171-. 



other roots, principal values. Tilly, 

 de. Mathesis 15 (1895) 177-, 217- ; 16 

 (1896) 5-. 



root extraction. Crosson, A. N. A. Mth. 5 

 (1846) 128- . 



. Alexeeff, N. Par. S. Mth. Bll. 7 



(1879) 167-. 

 . Darboux, G. Bll. Sc. Mth. 11 



(1887) 176-. 

 . Pavlov, A. P. Kazan S. Nt. (Ps.- 



Mth.) P. 6 (1888) 8. 



. Bougon, . Ev. Sc. 46 (1890) 283-. 



. Kleiber, J. Ev. Sc. 46 (1890) 507-. 



, abridged. Catalan, E. C. N. A. 



Mth. 17 (1858) 232-. 

 , . Hill, M. J. M. B. A. Ep. 



(1886) 538. 

 , . Raffalli, J. Ev. Sc. 47 (1891) 



90-. 

 , . Decamps, A. Ev. Sc. 50 (1892) 



764. 

 , , history. Cantor, M. N. A. Mth. 



20 (1861) 46-. 

 , methods. Grunert, J. A. Grunert 



Arch. 30 (1858) 275-. 

 , rapidly converging series for. Muir, 



T. Edinb. E. S. P. 17 (1891) 14-. 



48 



