Articles

Diophant


Diophant It is named after the city that was the largest center of mathematical activity in ancient Greece. Little is known about his life, ignorance even prevents us from securely fixing in which century he lived. Distant dates of a century have been suggested before or after the year 250 d. C. From verses found on his tomb written in the form of an enigmatic problem, it appears that he lived 84 years. Positively, such a problem should not be taken as the paradigm of the problems on which Diophantus was interested because he paid little attention to equations of the first degree.

Alexandria was always a very cosmopolitan center and the mathematics that originated in it was not all the same type. Heron's results were quite different from those of Euclid or Apollonius or Archimedes, and in Diophantus's work there is again an abrupt break from the classical Greek tradition. It is well known that the Greeks, in classical times, divided arithmetic into two branches: arithmetic itself as "theory of natural numbers". Often, it had more in common with Platonic and Pythagorean philosophy than with what is commonly thought of as mathematics, and logistics or practical calculus that laid down the practical rules of calculation that were useful to astronomy, mechanics, and so on.

The main treatise of Diophantus known, and that. Apparently, only partly reached us, it is the "Arithmetica". Only six of the original Greek books have survived, the total number (13) being a guess. It was a treatise characterized by a high degree of mathematical skill and ingenuity, and can therefore be compared to the great classics of the "Alexandrian First Age", that is, of the "golden age" of Greek mathematics, however, they have almost nothing to do with it. common with these or, indeed, with any traditional Greek math. It essentially represents a new branch and uses a different method, hence the time when Diophantus possibly lived was called the "second age Alexandrina," in turn known as the "silver age" of Greek mathematics.

Diophantus, more than a cultivator of arithmetic, and especially of geometry, as were the previous Greek mathematicians, must be considered a precursor of algebra, and in a sense more closely linked to the mathematics of the eastern peoples (Babylon, India, …) That with that of the Greeks. His "Arithmetica" resembles Babylonian algebra in many respects, but while Babylonian mathematicians were concerned primarily with "approximate" solutions of "determined" equations and above all "undetermined" equations of the 2nd and 3rd degrees of canonical forms, in current notation, Ax ^ 2 + Bx + C = y ^ 2 and Ax ^ 3 + Bx ^ 2 + Cx + D = y ^ 2, or sets (systems) of these equations. It is precisely for this reason - in honor of Diophantus - that this "indeterminate analysis" is called "diophantine analysis" or "diophantic analysis".

In the historical development of algebra it is generally considered that three stages can be recognized: the primitive or rhetorical, in which everything was completely written in words, an intermediate or syncopated, in which some abbreviations and conventions were adopted, and a final or symbolic, where only symbols are used. Diofanto's "Arithmetica" should be placed in the second stage; In his six books there is a systematic use of abbreviations for number powers and for relations and operations.

Source: Journal of Elementary Mathematica