edition of Rudolff's Coss in 1553. Thus, by slow degrees, their adoption became universal. There is another short-hand symbol of which we owe the origin to the Germans. In a manuscript published sometime in the fifteenth century, a dot placed before a number is made to signify the extraction of a root of that number. This dot is the embryo of our present symbol for the square root. Christoff Rudolff, in his algebra, remarks that "the radix quadrata is, for brevity, designated in his algorithm with the character , as ." Here the dot has grown into a symbol much like our own. This same symbol was used by Michael Stifel. Our sign of equality is due to Robert Recorde (1510–1558), the author of The Whetstone of
Witte (1557), which is the first English treatise on algebra. He selected this symbol because no two things could be more equal than two parallel lines . The sign for division was first used by Johann Heinrich Rahn, a Swiss, in 1659, and
was introduced in England by John Pell in 1668.
Michael Stifel (1486?–1567), the greatest German algebraist
of the sixteenth century, was born in Esslingen, and died in Jena. He was educated in the monastery of his native place, and afterwards became Protestant minister. The study of the significance of mystic numbers in Revelation and in Daniel drew him to mathematics. He studied German and Italian works, and published in 1544, in Latin, a book entitled Arithmetica integra. Melanchthon wrote a preface to it. Its three parts treat respectively of rational numbers, irrational numbers, and algebra. Stifel gives a table containing the numerical values of the binomial coefficients for powers below the 18th. He observes an advantage in letting a geometric progression
correspond to an arithmetical progression, and arrives at the designation of integral powers by numbers. Here are the germs of the theory of exponents. In 1545 Stifel published
an arithmetic in German. His edition of Rudolff's Coss contains
rules for solving cubic equations, derived from the
writings of Cardan.
We remarked above that Vieta discarded negative roots of
equations. Indeed, we find few algebraists before and during the Renaissance who understood the significance even of negative quantities. Fibonacci seldom uses them. Pacioli
states the rule that "minus times minus gives plus," but applies it really only to the development of the product of ; purely negative quantities do not appear in his work. The great German "Cossist" (algebraist), Michael Stifel, speaks as early as 1544 of numbers which are "absurd" or "fictitious below zero," and which arise when "real numbers above zero" are subtracted from zero. Cardan, at last, speaks of a "pure minus"; "but these ideas," says Hankel, "remained sparsely, and until the beginning of the seventeenth century, mathematicians dealt exclusively with absolute positive quantities." The first algebraist who occasionally places a purely negative quantity by itself on one side of an equation, is Harriot in England. As regards the recognition of negative
roots, Cardan and Bombelli were far in advance of all writers