Algebra at the more advanced level is often described as modern or abstract algebra. In fact, bothof these descriptions are partly misleading. Some of the great discoveries in the upper reaches of presentday algebra (for example, the so-called Galois theory) were known many years before the American Civil War ; and the broad aims of algebra today were clearly stated by Leibniz in the seventeenth century. Thus, “modern” algebra is not so very modern, after all! To what extent is it abstract? Well, abstraction is allrelative; one person’s abstraction is another person’s bread and butter. The abstract tendency inmathematics is a little like the situation of changing moral codes, or changing tastes in music: What shocksone generation becomes the norm in the next. This has been true throughout the history of mathematics.For example, 1000 years ago negative numbers were considered to be an outrageous idea. After all,it was said, numbers are for counting: we may have one orange, or two oranges, or no oranges at all; buthow can we have minus an orange? The logisticians, or professional calculators, of those days usednegative numbers as an aid in their computations; they considered these numbers to be a useful fiction, forif you believe in them then every linear equation ax + b = 0 has a solution (namely x = −b/a, provided a ≠0). Even the great Diophantus once described the solution of 4x + 6 = 2 as an absurd number. The idea ofa system of numeration which included negative numbers was far too abstract for many of the learnedheads of the tenth century!