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 Rome had become the (1927) . 79 88: "Oh izmeny intensivno,ti line y .spektrakh capital of' Italy 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 Accademia dei Lincei 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 `geodesics . 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 geodesic, 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-Euclidean geometry 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 differential geometry of curves . to it . Beltramml now offered a representation of , this After the establishment of the kingdom of Italy 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 tractrix 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 space, 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 manifolds 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 .