BE LTRAMI BELTRAMI

Spectrum of the Nova of 1918 [Preliminary Report]"), in tarv algebra and analytic geometry at Bologna, which I: rev!ira lmperaiorskoi akmlemii nauk ; Astrospe/ iroskopia he held from 1862 to 1864 : from 1864 to 1866 he (''A,trospectroscopy" : Petroerad_ 1921 ): "Comet, and held the chair of' geodesy in Pisa . where Enrico Betti Ionization ." in Ohserratorr, 46 ( 1923). 124 125 : ''I_lher die was his friend and colleague . From 1866 to 1873 he Intensitiitsveranderlichtkeit der Spektrallinien einiger was hack in Bologna, where he occupied the chair /_re .stira Pulkoro Ohs., 2nd ser., 2. no . 101 Cephciden ." in of rational mechanics. After had become the (1927) . 79 88: "Oh izmeny intensivno,ti line y .spektrakh capital of' in 1870, Beltrami became professor nekotorvkh tsefeid." ("On Chances in the Intensity of Lines in the Spectra of Certain Cepheids"), in /:reviii'a of' rational mechanics at the new University of Rome, Akademii nauk . 7th ser .. no. I (1928) : and ''Be,timmuna but served there only from 1873 to 1876, after which der Sonnenrotation auf spektroskopischen Wegc in den he held the chair of mathematical physics at Pavia, Jahren 1931 . 1932 and 1933 in Pulkovo " in teirvchrift where he also taught higher mechanics . In 1891 lie /fir .k trophr.vik, 7 ( 1933). 357-363. returned to Rome . where lie taught until his death . 11. St c c>yiARY Li 11 RA1 t Rt: . Works on Belopolsky are He became the president of the S. N . Blazhko. "A. A. Belopolsky ." in Bolshara sorervkava in 1898 and, the following year, a senator of the ent.viklopedia, IV ( 1950), 462-464 : V. G . Fcsenkoy, kingdom . A lover of music, Beltrami was interested ." in l.rmdi rtcs.vkor "Aristarkh Apollonovich Belopolsky in the relationship between mathematics and music . (''Pe(,ple of Russian Science" : Moscow . 1961). nauki Beltrami's works can he divided into two main pp. 185 192 : B . P. Gerasimovich . "A. A . Belopolsky . groups : those hefure ca. 1872, which deal with differ- 1854-1934.'' in , h1ronomi(he.vkii =hurnal ( .S' .5'SR), 2. pt . 3 ( 1934), 251 254 : 0. A. Melnikov, "Aristarkh Apollonovich ential geometry of curves and surfaces and were Belopolsky (1854-1934) . Nauchno-hiografiche,ky ocherk" influenced by Gauss . Lame, and Rieniann, and the ("Aristarkh Apollonovich Belopolsky 11854 19341 . A Sci- later ones. which are concerned with topics in applied entific-biographical Essay'') . in .f . .1 . Belopolvki . Asiro- mathematics that range from elasticity to elect ronmg- nomicheskie trm/r ("A . A . Belopolsky . Astronomical netics . His most lasting work belongs to this first Works" : Moscow, 1954). pp. 7-58 : Y. (i. Perel. I t- period, and the paper "Saggio di interpretazione della daviolic/i,esra russkie astronom_i (''Outstanding Russian geometria non-euclidea" (1868) stands out . In a paper Astronomers" : Moscow. 1951). pp. 85 107: K . 1) . Pok- of 1865 Btfltrami had shown that on surfaces of con- roy,ky. ''A . A. Belopolsky- (k 50-letiyu ego nauchnoy stant curvature, and only oil them . the line element devatelnosti 1877 1927)" ("A . A . Belopolsky [on the 50th 1_ 111 d.v c 2 + 2l-'dnch' + Gi/t 2 can be written in such Anniversary of His Scientific Career 1877 19271") . In ' - a form that the ` . and only these . are repre- Astrononticheskii kalendai' rill 1)'A' hod ("Astronomical Calendar for 1928" : Nizhni Nov`gorod. 1927). pp . 123-125 . sented by linear expressions in a and r. For positive with illustrations : and D. A . %hukov . ''Spisok nauchnvkh curvature R ' this form is

rahot akademika A . A . Belopol,kogo 1877 1934" ("A List (1 2) ( 1 ,21 clx- - R - [( 1'- + (1-)dti' - 2urc/uch © + (ti' + 1 of the Scientific Papers of Academician A . A . Belopolsky 2 X (ii' + 1 .' + 112 ) . 1877 1934") . in BI alleten' isnntis.vii po ivvledovaniI'll solnLV(l akademii mad, .S.SSR, no,. 1)) I I ( 1934). 7 20 . The geodesics in this case behave, locally speaking, like the great circle, on a sphere . It now occurred to 1). G . Kt i IK0y si ' Beltrami that . b y changing R to iR and ct to is (i - V - I ), the line element thus obtained,

BELTRAMI, EUGENIO (b. Cremona, Italy, 16 No- R21((12 cls' - + 2arcludr + (ti' + (I-)cli''[ vember 1835 ; d. Rome, Italy, 18 February 1900), X (a- - it 2 - r 2 ) - =, mathematics . Beltrami was born into an artistic family : his grand- which defines surfaces of constant curvature -R -. father. Giovanni . was an engraver of precious stones . others a new type of geometry for its , inside especially cameos : his father, Eugenio . Painted mini- the region ti' + r- < ci= . This geometry is exactly that atures . Young Eugenio studied mathematics from of the so-called non- of Lohachev- 1853 to 1856 at the University of Payia . where Fran- ski. if geodesics on such a surface are identified with cesco Brioschi was his teacher . Financial difficulties the "straight line," of non-Euclidean geometry . forced Beltrami to become secretarv to a railroad This geometry . developed between 1826 and 1832 . engineer. first in Verona and then in Milan . In Milan was known to Beltrami through some of Gauss's he continued his mathematical studies and in 1862 letters and some translations of the work of Lohachev- published his first mathematical papers, which deal ski. Few mathematicians, however, had paid attention with the of curves . to it . Beltramml now offered a representation of , this After the establishment of the in geometry in term, of the acceptable Euclidean geom- 1861 . Beltrami was offered the chair of complcmen- etry : "We have tried to find a real foundation [suh-

5 9 9 BELTRAMI BENEDEN strato] to this doctrine, instead of having to admit for dynamics, optics, and conduction of heat that led to it the necessity of a new order of entities and con- linear partial differential equations . Some papers deal cepts." He showed that all the concepts and formulas with Maxwell's theory and its mechanistic interpre- of Lobachevski's geometry are realized for geodesics tation, suggesting a start from d'Alembert's principle on surfaces of constant negative curvature and, in rather than from that of Hamilton (1889). particular, that there are rotation surfaces of this kind . The simplest of this kind of"pseudospherical" surface (Beltrami's term) is the surface of rotation of the BIBLIOGRAPHY about its asymptote, now usually called the I . ORIGINAL WORKS . Beltrami's works are collected in pseudosphere, which Beltrami analyzed more closely Opere matematiche, 4 vols . (Milan, 1902-1920) . Important in a paper of 1872 . individual works are "Saggio di interpretazione della geo- Thus Beltrami showed how possible contradictions metria non-euclidea," in Giornale di matematiche, 6 (1868) . in non-Euclidean geometry would reveal themselves 284-312, and Opere, 1, 374-405, also translated into French in the Euclidean geometry of surfaces ; and this re- in Annales scientifiques de l'Eco/e Norrnale Superieure, 6 moved for most, or probably all, mathematicians the (1869) . 251-288: "Richerche di analisi applicata alla geo- feeling that non-Euclidean geometry might be wrong . metria," in Giornale di matematiche, 2 (1864) and 3 (1865), Beltrami, by "mapping" one geometry upon another, also in Opere, I . 107-206: a paper on surfaces of constant made non-Euclidean geometry "respectable ." His curvature, in Opere, I, 262-280 : "Teoria fondamentale degli spazi di curvatura costante," in Annali di malematica, method was soon followed by others, including Felix ser. 2 (1868-1869), 232-255, and Opere, 1406-429 : "Richerche Klein, a development that opened entirely new fields sulle cinematica dei fluidi," in Opere, II, 202-379 ; a paper on of mathematical thinking . the pseudosphere . i n Opere, II, 394-409; a non-Euclidean Beltrami pointed out that his representation of approach to , in Opere, III, 383-407 ; "Sully teoria della non-Euclidean geometry was valid for two dimensions scala diatonica," in Opere, III, 408-412; an article on Sac- only. In his "Saggio" he was hesitant to claim the cheri, in Rendiconti del/a Reale Accademia dei Lincei, ser. possibility of a similar treatment of non-Euclidean 4, 5 (1889), 441-448, and Opere, IV, 348-355 ; and papers geometry of space . After he had studied Riemann's on Maxwell's theory, in Opere, IV, 356-361 . Uber die Hypothesen welche der Geometrie zu Grunde 11 . SECONDARY LITERATURE. There is a biographical liegen, just published by Dedekind, he had no scruples sketch by L. Cremona in Opere, I, ix-xxii . See also about extending his representation of non-Euclidean L. Bianchi, "Eugenio Beltrami," in Encic/opedia italiana, VI (1930), 581 ; G. H. Bryan, "Eugenio Beltrami," in Proceed- geometry to of n > 2 dimensions in "Teoria ings ofthe London Mathematical Society, 32 (1900), 436-439 ; fondamentale degli spazi di curvatura costante ." and G . Loria, "Eugenio Beltrami e Ie sue opere mate- In a contribution to the history of non-Euclidean matiche," in Bibliotheca mathematica, ser. 3, 2 (1901), geometry, Beltrami rescued from oblivion the Jesuit 392-440 . On Beltrami's contribution to non-Euclidean ge- mathematician and logician Giovanni Saccheri ometry, consult, among others, R . Bonola, Non-Euclidean (1667-1733), au'hor of Euclides ab omni naevo yin- Geometry (Chicago, 1912; New York, 1955), pp. 130-139. dicatus, which foreshadowed non-Euclidean geometry 234-236. but did not achieve it . D . J . STRUIK In his "Ricerche di analisi applicata alla geome- tria," Beltrami, following an idea of Lames, showed BENEDEN, EDOUARD VAN (b. Louvain, Belgium, the power of using so-called differential parameters 5 March 1846; d. Liege, Belgium, 28 April 1910), in surface theory . This can be considered the begin- zoology, embryology. ning of the use of invariant methods in differential Van Beneden's father, the zoologist P . J. Van geometry. Beneden, was professor at the Catholic University in Much of Beltrami's work in applied mathematics Louvain . Edouard was appointed professor at the shows his fundamental geometrical approach, even University of Liege in 1870 with the qualification of in his analytical investigations. This trait characterizes charge de cours and was promoted to ordinarius in the extensive "Richerche sulle cinematica dei fluidi" 1874. (1871-1874) and his papers on elasticity . In these he His scientific work is characterized by a great unity recognized how Lame's fundamental formulas de- resulting from the character of the problems to which pend on the Euclidean character of space . and he he devoted his attention . In 1872 Van Beneden went sketched a non-Euclidean approach (1880-1882) . He to Brazil, and from there he brought back a number studied potential theory, particularly that of ellipsoids of specimens, particularly Anne/ida, which were and cylindrical discs ; wave theory in connection with studied by Armauer Hansen . P. J. Van Beneden had Huygens' principle : and further problems in thermo- organized a modest laboratory on the Belgian sea-

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