The Birth of Literal Algebra

Abstract

1. MATHEMATICS IN THE FIRST CENTURIES AD. DIOPHANTUS. The Babylonians developed a kind of numerical algebra. Then came Greek geometric algebra. The third-very important-stage of the development of algebra began in the first centuries AD and came to an end at the turn of the 17th century. Its beginning was marked by the introduction of literal symbolism by Diophantus of Alexandria and its end, by the creation of literal calculus in the works of Viete and Descartes. It was then that algebra acquired its own distinctive language, which we use today. The first century BC was a period of Roman conquests and of Roman civil wars. Both took place in the territories of the Hellenistic states and the Roman provinces and were accompanied by physical and economic devastation. One after another, these states lost their independence. The last to fall was Egypt (30 BC). The horrors of war and the loss of faith in a secure tomorrow promoted the spread of religious and mystical teachings and undermined interest in the exact sciences, and in abstract problems in mathematics and astronomy. In Cicero's dialogue On the state one of the participants proposes a discussion of why two Suns were seen in the sky. But the topic is rejected, for even if we acquired profound insight into this matter, we would not become better or happier. In the second half of the first century BC mathematical investigations came to a virtual halt and there was an interruption in the transmission of the scientific tradition. At the beginning of the new era, economic conditions in the Hellenistic countries, now turned Roman provinces, gradually improved, and there was a revival of literature, art, and science. In fact, the 2nd century came to be known as the Greek Renaissance. It was the age of writers such as Plutarch and Lucian and of scholars such as Claudius Ptolemy. Alexandria continued its role as the cultural and scientific center of antiquity and, in this respect, Rome was never its rival. Nor did it ever develop an interest in the depths of Hellenistic science. As noted by Cicero in his Tusculanae disputationes, the Romans, unlike the Greeks, did not appreciate geometry; just as in the case of arithmetic, they stopped at narrow, practical knowledge of this subject.