Luigi Cremona. 279 



space, but it has not been worked out in detail. Cremona's theory of 

 transformation interested him all his life, and his last published paper 

 (Tr. R. Irish Acad., 1884) is on a transformation of the fourth order 

 in space of three dimensions, the inverse transformation being of the 

 sixth order. His method attracted universal attention, and has 

 proved to be of the highest importance, not only in geometry, but in 

 the analytical theory of algebraic functions and integrals. It is, in fact, 

 an algebraic theory, though it is convenient to state it in a geometrical 

 form, and its author preferred to regard it from a geometrical point 

 of view. 



Nearly connected with the foregoing is the representation of surfaces 

 on a plane. Map-drawing is a familiar example, which engaged the 

 attention of Hooke and possibly Mercator, as well as of Lagrange, 

 Gauss, &c. For a one-one correspondence the surface must be 

 unicursal, and this is sufficient ; Cremona is associated with Cayley, 

 Clebsch, Nother and others, in the development of the theory. He also 

 applied it to cubic surfaces, Steiner's surface, various scrolls, and some 

 singular surfaces with cuspidal curves. 



Among his minor papers may be noticed those on twisted cubics, 

 unicursal twisted quartics, and the three-cusp hypocycloid. Like all 

 his work they are written in the most clear and attractive style : in 

 fact, he may be compared with Dirichlet in his power of simplifying 

 and illuminating everything on which he wrote, even when the subject- 

 was not altogether new. 



Mathematical students all over the world are indebted to Cremona 

 for his truly admirable treatises : Introduzione ad una teoria delle curve 

 piane (" Mem. Ace. Bologna," 1862); Preliminari ad una teoria geometrica 

 delle superficie (ibid., 1866); Elementi di Geometria proiettiva (Rome, 

 1873) : Elementi di calcolo grafico (Turin, 1874); and Le figure reciproche 

 della statica grafica (3rd ed., Milan, 1879). These, and the translations 

 of them into foreign languages, must have greatly helped the progress 

 of the subjects with which they deal; they are lucid, elegant, and 

 stimulating, and are not surpassed in merit by any text-books of 

 their kind. 



Cremona was a member of all the leading academies of the world, 

 and received, in May, 1903, from the German Emperor, the order 

 Pour le mtrite. He was elected Foreign Member of the Royal Society 

 in 1879, but did not make any communications to the "Transactions " 

 or " Proceedings." 



Obituary notices of some length by D'Ovidio and Veronese have 

 appeared in the Rendiconti of the Academy of Turin and of the 

 Accademia dei Lincei. From the second of these assistance has been 

 derived in the preparation of the present notice ; it concludes with a 



list of Cremona's scientific publications. 



G. B. M. 



