CodalSearch this book — or all of Codal…⌘K
Projects
Sign inStart a book
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.

Page 123 of 219
Table of Contents

Analytic Geometry

Gauss, Lagrange. Its importance in the construction of maps

is obvious. Gauss was the first to represent a surface upon another with a view of more easily arriving at its properties. Plücker, Chasles, Cayley, thus represented on a plane the

geometry of quadric surfaces; Clebsch and Cremona, that of

cubic surfaces. Other surfaces have been studied in the same way by recent writers, particularly M. Nöther of Erlangen,

Armenante, Felix Klein, Korndörfer, Caporali, H. G. Zeuthen

of Copenhagen. A fundamental question which has as yet received only a partial answer is this: What surfaces can be represented by a (1,1) correspondence upon a given surface? This and the analogous question for curves was studied by Clebsch. Higher correspondences between surfaces have been investigated by Cayley and Nöther. The theory of surfaces has been studied also by Joseph Alfred Serret (1819–1885), professor

at the Sorbonne in Paris, Jean Gaston Darboux of Paris,

John Casey of Dublin (died 1891), W. R. W. Roberts of Dublin,

H. Schröter (1829–1892) of Breslau. Surfaces of the

fourth order were investigated by Kummer, and Fresnel's

wave-surface, studied by Hamilton, is a particular case of

Kummer's quartic surface, with sixteen canonical points and sixteen singular tangent planes.56

The infinitesimal calculus was first applied to the determination of the measure of curvature of surfaces by Lagrange,

Euler, and Meusnier (1754–1793) of Paris. Then followed the

researches of Monge and Dupin, but they were eclipsed by

the work of Gauss, who disposed of this difficult subject in a

way that opened new vistas to geometricians. His treatment is embodied in the Disquisitiones generales circa superficies curvas (1827) and Untersuchungen über gegenstände der höheren Geodäsie of 1843 and 1846. He defined the measure of curvature at a point to be the reciprocal of the product of the two principal radii of curvature at that point. From this flows the theorem of Johann August Grunert (1797–1872;

professor in Greifswald), that the arithmetical mean of the radii of curvature of all normal sections through a point is the radius of a sphere which has the same measure of curvature as has the surface at that point. Gauss's deduction of the formula of curvature was simplified through the use of determinants by Heinrich Richard Baltzer (1818–1887) of Giessen.69

123