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nydus/A History of MathematicsPublic

This text examines the transition from the Middle Ages to the Modern era, highlighting how the fall of Constantinople and the invention of the printing press catalyzed a revival of classical learning. It traces the shift toward scientific inquiry through the rise of pure mathematics and astronomy, detailing the intellectual struggle against established scholastic and ecclesiastical authority.

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Table of Contents

Newton to Euler

of Descartes' rule of signs, now given in books. This skilful

geometer wrote in 1740 a work on analytical geometry, the

object of which was to show that most investigations on curves could be carried on with the analysis of Descartes quite as easily as with the calculus. He shows how to find the tangents, asymptotes, and various singular points of curves of all degrees, and proved by perspective that several of these points can be at infinity. A mathematician who clung to the methods of the ancients was Philippe de Lahire (1640–1718), a pupil of

Desargues. His work on conic sections is purely synthetic,

but differs from ancient treatises in deducing the properties of conics from those of the circle in the same manner as did Desargues and Pascal. His innovations stand in close relation

with modern synthetic geometry. He wrote on roulettes, on

graphical methods, epicycloids, conchoids, and on magic squares. Michel Rolle (1652–1719) is the author of a theorem

named after him.

Of Italian mathematicians, Riccati and Fagnano must not

remain unmentioned. Jacopo Francesco, Count Riccati (1676–1754) is best known in connection with his problem, called Riccati's equation, published in the Acta Eruditorum in 1724. He succeeded in integrating this differential equation for some special cases. A geometrician of remarkable power was Giulio Carlo, Count de Fagnano (1682–1766). He discovered the following formula, π=2ilog1−i1+i, in which he anticipated Euler

in the use of imaginary exponents and logarithms. His studies

on the rectification of the ellipse and hyperbola are the starting-points of the theory of elliptic functions. He showed, for

instance, that two arcs of an ellipse can be found in an indefinite number of ways, whose difference is expressible by a right line.

In Germany the only noted contemporary of Leibniz is

Ehrenfried Walter Tschirnhausen (1651–1708), who discovered the caustic of reflection, experimented on metallic reflectors and large burning-glasses, and gave us a method of transforming equations named after him. Believing that the most

simple methods (like those of the ancients) are the most correct, he concluded that in the researches relating to the properties of curves the calculus might as well be dispensed with.

After the death of Leibniz there was in Germany not a single mathematician of note. Christian Wolf (1679–1754),

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