By Girolamo Cardano
CARDANO, G.: ARS MAGNA OR the foundations OF ALGEBRA. TRANSLATED by means of T. R. WITMER [1968, REPRINT]. manhattan, new york, 1993, xxiv 267 p. figuras.Encuadernacion unique. Nuevo.
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Additional resources for Ars magna, or, The rules of algebra
Sin q’e qL sin = e cos + qm cos = , e sin . T h e choice of a specific value of is, therefore, a matter of conven› tion, not t h e result of observation. Similarly a (classical) magnetic pole of conventional strength qm = g (and qe = 0) can be descri› bed with equal right as a body endowed with an electric as well as magnetic charge given by ql = (456) gsmip, qm = g cos ł . F r o m t h e experimental observation t h a t no force is acting between t h e electron and t h e pole w h e n b o t h are at rest, it follows that one should have qlql + qlql = eg cos (ł ) = which is satisfied by (46) ł- = – 90 .
52a) cannot hold everywhere; at t h e origin, where t h e pole is located, eq. (52a) is contradictory with eq. (51). T h e problem t o be solved can be divided into two p a r t s : t h e first one consists i n determining an explicit expression for t h e vector potential A, which satisfies eq. (52) and corresponds as close as possible t o t h e field (50). T h e second part concerns t h e derivation of a very important property of A, which will b e obtained by utilizing t h e formal analogy between t h e two equa› tions (52) a n d t h e equations used in elementary electromagnetism for deriving t h e magnetic field generated by an assigned distribution of electric currents.
V Since V is completely arbitrary, eq. (82) can only be satisfied if glx is due to one or more pointlike sources, each with a magnetic charge multiple of g=2n\e. Therefore eq. (82) is t h e relativistic generalization of Dirac’s valid t h e o r e m (5) (written in rationalized units with h=c \)y not only w h e n gIA is a classical (^-number) source, b u t also w h e n it is a q u a n t u m operator, all eigenvalues Qv of which should be multiples of g=2n\e. ) is obtained by defining t h e gauge-invariant operator which d e › scribes a scalar pole g and imposing t h a t its change, due t o a change of the p a t h from L to L’, would be i n d e p e n d e n t of t h e choice of t h e two-dimensional surface S of c o u n t o u r L’ L.
Ars magna, or, The rules of algebra by Girolamo Cardano