74 JEROME CARDAN 



be permitted to take a rapid survey of the condition 

 of Algebra at the time when Cardan sat down to write. 

 Up to the beginning of the sixteenth century the 

 knowledge of Algebra in Italy, originally derived from 

 Greek and Arabic sources, had made very little progress, 

 and the science had been developed no farther than to 

 provide for the solution of equations of the first or 

 second degree. 1 In the preface to the Liber Artis 

 Magnce Cardan writes : " This art takes its origin 

 from a certain Mahomet, the son of Moses, an Arabian, 

 a fact to which Leonard the Pisan bears ample testimony. 

 He left behind him four rules, with his demonstrations 

 of the same, which I duly ascribe to him in their proper 

 place. After a long interval of time, some student, 

 whose identity is uncertain, deduced from the original 

 four rules three others, which Luca Paciolus put with 

 the original ones into his book. Then three more were 

 discovered from the original rules, also by some one 

 unknown, but these attracted very little notice though 

 they were far more useful than the others, seeing that 

 they taught how to arrive at the value of the cubus and 

 the numerus and of the cubus quadratus? But in 

 recent times Scipio Ferreo of Bologna discovered 



1 Fra Luca's book, Summa de Arithmetica Geometria Propor- 

 tioni e" Proportionalita, extends as far as the solution of quadratic 

 equations, of which only the positive roots were used. At this 

 time letters were rarely used to express known quantities. 



2 The early writers on Algebra used numerus for the absolute or 

 known term, res or cosa for the first power, quadratum for the 

 second, and cubus for the third. The signs + and - first appear 

 in the work of Stifelius, a German writer, who published a book of 

 Arithmetic in 1544. Robert Recorde in his Whetstone of Wit 

 seems first to have used the sign of equality =. Vieta in France 

 first applied letters as general symbols of quantity, though the 

 earlier algebraists used them occasionally, chiefly as abbreviations. 

 Aristotle also used them in the Physics Libri. Hist, des Sciences 

 MathhnatiqueS) i. 104. 



