OCTOHER 2, 1891.] 



SCIENCE. 



18= 



is true likewise of his Arabian successors, who, thoug-h they 

 advanced so far as to obtain the general solution of a cubic 

 equation, and to state such a proposition in integers of the 

 equation x^ -j- y^ ^ z^ is feasible, yet always adhered to 

 the rhetorical method, and made scarcely any progress in 

 general algebraic science. Indeed such progress was hardly 

 possible until the introduction of symbolic methods. 



The first decided steps in the direction of symbolism since 

 the work of Diophantos were taken by a mathematician of 

 India named Bhaskara in the twelfth century. He used ab- 

 breviations and initials to denote the unknown, a dot for 

 minus, and juxtaposition to indicate addition. A product is 

 denoted by the first syllable of the word for multiplication 

 subjoined to the factors, division by the divisor being written 

 beneath the dividend without a line between as our custom 

 is now. The two sides of an equation are written one under 

 the other, and explanatory records are introduced whenever 

 it is necessary to prevent misunderstanding. Occasionally 

 symbols are used for given as well as unknown quantities. 

 Square, cube, and square root are denoted by the initial 

 letters of the corresponding words. Using the Arabic, or 

 decimal, notation, he has a character for zero, which enables 

 him to write all his equations with all the powers of the un- 

 known arranged in regular order on each side of the equa- 

 tion, certain of them being multiplied by the factor zero. 

 This method of writing equations maintained itself till long 

 afterwai'ds. We have in this author a distinct advance over 

 Diophantos and the Arabians in the introduction of various 

 symbols for the unknown, so that several might be used in 

 the same problem, as well as in the use of zero. 



We have now to consider a new phase of algebraic prog- 

 ress arising from the introduction into western Europe of 

 the works of the Arabian mathematicians. This took place 

 through the Moors of Spam. The Greek and Arabic works 

 were studied at the Moorish universities of Granada, Cor- 

 dova, and Seville, but all knowledge of them was jealously 

 kept from the outside world until the twelfth century, during 

 which copies came into the possession of Christians. Up to 

 this time Christian Europe had been almost a mathematical 

 blank. The simple arithmetical operations they were able 

 to perform were accomplished by the aid of the abacus, and 

 they possessed some knowledge of astronomy and geometry, 

 but made no progress until they were able to avail them- 

 selves of the previous labors of Greek, Hindu, and Arab, 

 under the stimulus of which a career of advancement began 

 which has continued to the present time. This career, how- 

 ever, did not begin immediately; it took several centuries to 

 assimilate the material received from these sources, and thus 

 to lay the foundations on which subsequent progress should 

 rest. 



During this period the rhetorical method was used in all 

 algebraic processes, and it was not until the sixteenth cen- 

 tury that syncopated methods were introduced, preparing 

 the way for the symbolic methods that soon followed. Latin 

 being the language in use, the word res, or radix, was em- 

 ployed for the unknown quantity, the square being called 

 census, and the cube cuhus. These words were at first writ- 

 ten out in full and afterwards represented by R or Rj, Z or 

 C, and C or K respectively. 



The signs -|- and — are first found in a mercantile arith- 

 metic by Johaun Widmann, published in 1489, though they 

 did not come into general use by mathematicians till a hun- 

 dred years or more afterward. The most probable supposi- 

 tion as to their origin is that they were at first warehouse 

 marks indicating an excess or deficiency in the contents of a 



package wliich was supposed to contain a certain definite 

 amount. Widmann uses tliem purely as abbreviations, not 

 as symbols of operation. 



The first mathematical work ever printed was by Pacioli, 

 upon arithmetic, algebra, and geometry, and marks the begin- 

 ningof the syncopated stageof developmentin western Europe. 

 This book appeared in 1494, just before the beginning of the 

 sixteenth century, during which this method was in vogue. 

 Pacioli uses initials as abbreviations for the unknown, its 

 square and cube, and for the words "plus" and "equal," 

 al,~.o occasionally de for dernptiis, instead of minus. 



The sign now used for equality was introduced by Eecorde 

 in an arithmetic published in 1540. He uses also the present 

 signs -|- and — . At about the same time our present sym 

 bol for square root was introduced by Stifel, and Nicholas 

 Tartaglia discovered the solution of the cubic equation 

 x' -\- 23X = g, which is generally attributed to Cardan, and 

 goes by his name. Cardan obtained the solution from Tar- 

 taglia under promise of strict secrecy, and then published it 

 in his work " Ars Magna." Considerable advance is made 

 in this work over anything done by his predecessors. Neg- 

 ative and eve^ imaginary roots of equations are discussed, 

 and the latter are shown to always occur in pairs, though 

 no interpretation of them is attempted. Cardan shows that 

 when the roots of the cubic are all real. Tartaglia's solution 

 appears in an imaginary form. This is the first notice we 

 find of imaginaries, and, with the exception of a similar- 

 treatment by Bombelli a few years later, and a suggestion as 

 to their interpretation by Wallis in 1685, they were discussed 

 by no subsequent mathematician until Euler investigated 

 them nearly two hundred years afterward. Cardan also 

 discovered the relations between the roots and coefficients of 

 an algebraic equation, and the underlying principle of Des- 

 cartes' rule of signs. It is to be noted that his solutions 

 both of quadratics and cubics are geometrical. 



In 1572 Bombelli published an algebra in which the same 

 subjects discussed by Cardan are treated in about the same 

 way, but in which a marked advance is made in notation, 

 viz., the employment for the unknown of the symbol 1, 

 while its powers are denoted by 2^ 3, etc. Thus he would 

 write X' -\- 5x — 4 as l^p. 5 1?«. 4, p. and m. standing for 

 plus and minus. Other writers of the same period would 

 have written the expression thus, 



IZp. 5Rm. 4, or IQ -\- 5N — 4. 

 Up to this time in the development of algebraic notation, 

 whatever may have been the forms or symbols used, they 

 were regarded simply as abbreviations for the words neces- 

 sary to express the idea to be conveyed. But now the con- 

 ception of pure symbolism begins to appear. Vieta, who 

 lived in the last half of the sixteenth century, denoted 

 known quantities by consonants and unknown by vowels, 

 while powers were indicated by initials or abbreviations of 

 the words quadratus and cuhus. He was thus enabled to 

 deal with several unknowns in the same problem, together 

 with their powers. The following is a specimen of his nota- 

 tion. The equation 'iBA' — DA -\- A^ = Z he writes as 

 B 3 in A quad. — D piano in A -\- A cubo equatur 

 Z solido. 

 (It ma> be noted that he makes his equations homogeneous, 

 and lays stress on the desirability of so doing.) This and 

 the other examples that have been given above illustrate the 

 great variety of notations in use during this period, no con- 

 ventional system having yet been adopted to be adhered to 

 in the main by all mathematical writers. This is, of course, 

 an inevitable accompaniment of the formative stage of any 



