Peter Pesic

The formula for the solution of the general equation of the third degree by the italians Del Ferro, Tartaglia, and Cardano, in the 16th Century, was one of the triumphs of Renaissance mathematics, and one that was a clear improvement upon the achievements of the Ancients. Soon thereafter Cardano's student Ludovico Ferrari obtained the solution of the quartic equation. And at this point matters rested, with repeatedly failed attempts to get a formula for the solutions of the general equations of the fifth and higher degrees, until the 1824 paper by the norwegian mathematician Niels Henrik Abel, then twenty one years old, settled the matter for good, by proving such a formulae cannot exist. This great little book tells the story of this intelectual quest from the very beginning, with the Pythagoreans schoking discovery of the irrationals, and proceeding with the work of the Arabic and Italian algebraists on the solution of equations and other algebraic problems, not least among them the introduction of appropriate notation. The slow but steady development of ideas, with contributions by Viéte, Descartes, Newton, Gauss, Lagrange, and Ruffini, among others, resulted in the brilliant result by Abel (a translation of which is printed in Appendix A.) Also covered in the text is the aftermath of Abel's work, in particular his 1828 paper on the relation between solvability and noncommutativity, and the immense extension of this idea by Galois, with the development of the concept of group and the explanation of the solvability of algebraic equations in terms of the commutativity properties of certain quocient groups derived from the equation. The centrality of certain abstract algebraic notions subsequently introduced by Hamilton, Grassman, Gibbs, Sylvester, Cayley or Boole is also touched upon in this book. In short: this is a remarkable work that, although written for the educated lay person, is not shy to present and comment upon "real" mathematics (mainly in the boxes scattered throughout the text and in the appendices) and could very well serve as the backbone of an advanced course in the History of Algebra, guiding the study from the earlier examples of Babylonian mathematics to the development of several concepts of number (integers, rationals, reals, complex, quaternions,...), the notion of unsolvability, the development of mathematical notation, the gradual creation of the objects and concepts of present day linear and abstract Algebras. All of these themes could be introduced by starting with an appropriate part of this little gem of a book, and then take off from there, exploiting exciting events in this part of the intelectual history of mankind, and then get back again to this great little book to gain context and take off again a little later, and a little wiser...… (more)