The most influential mathematics textbook of ancient times is easily named, for the Elements of Euclid has set the pattern in elementary geometry ever since. The most effective textbook of the medieval age is less easily designated; but a good case can be made out for the Al-jabr of Al-Khwarizmi, from which algebra arose and took its name. Is it possible to indicate a modern textbook of comparable influence and prestige? Some would mention the Géométrie of Descartes or the Principia of Newton or the Disquisitiones of Gauss; but in pedagogical significance these classics fell short of a work by Euler titled Introductio in analysin infinitorum. Here in effect Euler accomplished for analysis what Euclid and Al-Khwarizmi had done for synthetic geometry and elementary algebra respectively. Coordinate geometry, the function concept, and the calculus had arisen by the seventeenth century; yet it was the Introductio which in 1748 fashioned these into the third member of the triumvirate - comprising geometry, algebra, and analysis.
Euler was not the first to use the word analysis, but he gave it, a new emphasis. Plato's analysis had reference to the logical order of steps in geometrical reasoning, and the analytic art of Viète was akin to algebra; but the first volume of the Introductio resembles analysis in the current orthodox sense - the study of functions by means of infinite processes.
Euler avoided the phrase analytic geometry, probably to obviate confusion with the older Platonic usage; yet the second volume of the Introductio has been referred to, appropriately, as the first textbook on the subject. It contains the earliest systematic graphical study of functions of one and two independent variables, including the recognition of the quadrics as constituting a single family of surfaces. The Introductio was first also in the algorithmic treatment of logarithms as exponents and in the analytic treatment of the trigonometric functions as numerical ratios.
The Introductio does not boast an impressive number of editions, yet its influence was pervasive. In originality and in the richness of its scope it ranks among the greatest of textbooks; but it is outstanding also for clarity of exposition. Published two hundred and two years ago, it nevertheless possesses a remarkable modernity of terminology and notation, as well as of viewpoint. Imitation is indeed the sincerest form of flattery.
Brooklyn College,
Brooklyn, N. Y., U.S.A.
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