The Development of Algebra in the Renaissance

Notation

The existing knowledge of both arithmetic and algebra came to Western Europe through the study of Arab mathematics. But not until the fifteenth century were symbols used, as Diophantus had done, for the commonest arithmetical operations. About that time, the symbols

and

for plus and minus were usual in Italy and France. They had been introduced by Lucia Pacioli (1445-1514) as abreviations for the words piu (more) and meno ( less). The symbols + and - occurred in Germany in 1480. These symbols were first to be printed in 1489 in a book by the Rechenmeister Johan Widmann.

The symbols and for multiplication and division do not appear until the 17th century. At this time, the sign for equality caught on, although it occurs earlier in an algebra textbook by the englishman Robert Recorde (1510-58), which appeared in 1557. Recorde introduced the sign with the justification that no two things can be more equal than a pair of parallel lines.

The use of letters for numbers is already found in the work of Jordanus Nemorarius (13th century). But his work lacks the combination of these letters with the operational symbols. This important step, which led to algebra as we know it today, is due to François Viète (1591). For unknown quantities Viète used capital vowels A, E, I, .... and for known quantities capital consonants B, C, D, .... It was René Descartes (1637) who introduced our current usage: small letters, of which the ones at the beginning of the alphabet denote known quantities, and the ones at the enbd unknown quantities.

For a long time there was much uncertainty in the use of negative numbers, as is apparent from the name. Moreover Stifel (c.1487 - 1567) called the numbers less than zero absurd numbers. Even Viète did not allow negative numbers as solutions of equations.

Albert Girard (1595-1632) seems to have been the first to give negative solutions full recognition. Also, the interpretation of negative numbers as line segments in the opposite direction was taken up again. However a precise foundation for the arithmetic of negative numbers had to wait until the beginning of the nineteenth century. Complex numbers were used from the 16th century, initially to aid in the solution of cubic equations, but these were viewed with even more scepticism.

Relationship of Algebra to Geometry

Prior to 1500, algebra had remained tied to geometry in the sense that all equations and formulas had to have some geometric interpretation. Thus for instance squared and cubed quantities were viewed as areas and volumes, but equations involving fourth and higher powers were considered nonsensical. Stifel wrote "Going beyond the cube just as if there were more than three dimensions..... is against nature".

In the 16th century it was recognised that mathematical results could be proved algebraically rather than geometrically, and that this would have to be the way forward in the future. This realisation provided the impetus for the development of coordinate geometry by Fermat and Descartes in the early 17th century, which in turn enabled the development of the function concept and the invention of the calculus.

With the freeing of algebra from the necessity of geometrical meaning, the way was clear for the development of algebra as an independent, abstract subject, and in particular for the study of various types of equation.

Solution of Higher Degree Equations

Up to the end of the 15th century cubic equations could be solved for special cases only. The University of Bologna, in particular, was a centre for attempts at solving the general cubic equation. It had been recognised that all cubic equations could be reduced to one of the following three cases:

x + px = q
x = px + q
x + q = px (p, q > 0)

Scipione del Ferro (c.1465 - 1526), one of the teachers at the University of Bologna, found an algorithm for the solution of the cubic equation sometime around 1500, but did not publish his method. In 1535 Niccolo Tartaglia (c.1500 - 57), a compatriot of del Ferro, found the solution independently. Mathematicians often preferred to keep their work secret in those days, and Tartaglia passed his solution on to Geronimo Cardano (1501-76) in the strictest confidence. But Cardano was a rather unprincipled character, and without permission from Tartaglia, published the solution in his own book, "Ars Magna", in 1545.

He gave the solution in case 1 as:

x =

This expression is still called Cardano's formula.

The "Ars Magna" also contained the general solution of an equation of the fourth degree. Cardano gave a method due to his pupil Ludovico Ferrari (1522-65), in which a solution is found by solving a related cubic equation.

Further significant contributions to the solution of the cubic equation were made by Raphael Bombelli in his "Algebra" (1572). He considered pure imaginary numbers, which he wrote in the form R [0m. 4], which in our notation means (0 - 4), i. e. 2i. He could thus solve those cases in which three real solutions appear as the sum of difference of complex numbers in Cardano's formula. Viète gave the solutions of these cases in trigonometric form so avoided the use of complex numbers.

The fact that an equation of fifth or higher degree does not, in general, have solutions which can be expressed by means of radicals (i. e. in terms of the coefficients, using the four rules of aritmetic and extraction of nth roots) was first recognised and proved by the Norwegian Niels Abel (1802-29). At first Abel thought that he had found a solution of the quintic. After he noticed his error, he showed in1824 that for equations of degree 5 or more no general solutions could be found.

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