CHAPTER III. 



RADICAL QUANTITIES AND IRRATIONAI, EXPRESSIONS. 



I. From the last chapter the student has learned that there are 

 two methods in use for indicating the root of a quantity, one by 

 the ordinary radical sign and the other by a fractional exponent. 

 Of course it is entirely unnecCvSsary to have two modes of writing 

 the same thing, and in this sense either one of the two ways may 

 be considered superfluous. But practically each method of nota- 

 tion has an advantage in special cases, and the student will feel 

 this as he proceeds. This fact that both methods are better than 

 either one, accounts for the retention of both in mathematics. 



2. Historic All Note— The introduction of the present symbols into alge- 

 bra was very gradual, and the use of a particular symbol did not generally 

 become common until some time after its suggestion. The signs -|- and — 

 were first used at the beginning of the 16th century in the works of Gramma- 

 teus, Rudolf and Stifel. Recarde (born about 1500) is said to have invented 

 the sign of equality about this time. Scheubet's work (1552) is the first one 

 containing the sign >/ . and Vieta (born 1540) first used the vinculum in con- 

 nection with it. Before this, root- extraction was indicated by a symbol some- 

 thing like R. /VStexln (born 1548) first used numbers to indicate powers of a 

 quantity, and lie'even suggested the use of fractional exponents, but not until 

 Descartes (born 1596) did exponents take the modern form of a superior figure. 



The development of the general notion of an exponent (negative, fractional, 

 incommensurable) first appears in a work of John Wallis (published in 1665) 

 in connection with the quadrature of plane curves. 



To show the appearance of mathematical works before the introduction cf 

 the common symbols, we give the following expression taken from Cardan's 

 works (1545) : 



55 V. cu. J]i 108 p7 10 I mj^ cu. J^ 108 w 10, 

 which is an abbreviation for "Radix universalis cubica radicis ex 108 plus 10, 

 minus radice universali cubica radicis ex 108 minus 10." Or, in modern sym- 

 bols, ■_ 



^x/l"o8-f IO-'^^/Io8-IO 

 Here is a sentence from Vieta's work (1615). 

 Et omnibus perE cubum ductis et ex arte concinnatus, 



E cubi quad. + Z solido 2 in E cubum, acquabitur B plani cubo. 

 This translated reads : Multiplying both members (" all ") by E"! .'iiul unit- 

 ing like terms, 



