486 MATHEMATICS 



We wish finally to mention a line of investigation which makes 

 use of the infinitesimal calculus arid even the theory of functions. 

 Here belong the brilliant researches of Dirichlet relating to the num- 

 ber of classes of binary forms for a given determinant, the number 

 of primes in a given arithmetic progression; and Riemann's remark- 

 able memoir on the number of primes in a given interval. 



In this analytical side of the theory of numbers we notice also the 

 researches of Mertens. Weber, and Hadamard. 



Protective Geometry 



The tendencies of the eighteenth century were predominantly 

 analytical. Mathematicians were absorbed for the most part in 

 developing the wonderful instrument of the calculus with its countless 

 applications. Geometry made relatively little progress. A new era 

 begins with Monge. His numerous and valuable contributions to 

 analytical descriptive and differential geometry, and especially his 

 brilliant and inspiring lectures at the Ecole Polytechnique (1795, 

 1809), put fresh life into geometry and prepared it for a new and 

 glorious development in the nineteenth century. 



When one passes in review the great achievements which have 

 made the nineteenth century memorable in the annals of our science, 

 certainly protective geometry will occupy a foremost place. Pascal, 

 De la Hire, Monge, and Carnot are forerunners, but Poncelet, a pupil 

 of Monge, is its real creator. The appearance of his Traite des pro- 

 prieties projcctives des figures, in 1822, gives modern geometry its 

 birth. In it we find the line at infinity, the introduction of imagin- 

 aries, the circular points at infinity, polar reciprocation, a discus- 

 sion of homology, the systematic use of projection, section, and 

 anharmonic ratio. 



While the countrymen of Poncelet, especially Chasles, do not fail 

 to make numerous and valuable contributions to the new geometry, 

 the next great steps in advance are made on German soil. In 1827 

 Mobius publishes the Barycentrische Calcul; Pliicker's Analytisch- 

 geometrische Entwickelungen appears in 182831 and Steiner's Ent- 

 loickelung der Abhdngigkeit geometrischer Gestalten von einander in 

 1832. In the ten years which embrace the publication of these 

 immortal works of Poncelet, Pliicker, and Steiner, geometry has 

 made more real progress than in the two thousand years which had 

 elapsed since the time of Appolonius. The ideas which had been 

 slowly taking shape since the time of Descartes suddenly crystallized 

 and almost overwhelmed geometry with an abundance of new ideas 

 and principles. 



To Mobius we owe the introduction of homogeneous coordinates, 

 and the far-reaching conception of geometric transformation, includ- 

 ing collineation and duality as special cases. To Pliicker we owe the 



