By Morris Kline

This complete historical past strains the advance of mathematical principles and the careers of the boys accountable for them. quantity 1 appears on the discipline's origins in Babylon and Egypt, the production of geometry and trigonometry via the Greeks, and the function of arithmetic within the medieval and early smooth sessions. quantity 2 specializes in calculus, the increase of study within the nineteenth century, and the quantity theories of Dedekind and Dirichlet. The concluding quantity covers the revival of projective geometry, the emergence of summary algebra, the beginnings of topology, and the effect of Godel on fresh mathematical learn.

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Michel Chasle s (1793-1880) wa s another great proponent of geometrical 834 THE RENEWA L O F INTEREST I N GEOMETR Y 83 5 methods. I n hi s Aperfu historique sur I'origine e t le developpement de s methodes e n geometric (1837), a historical study in which Chasles admitted tha t he ignore d the German writer s because he did not know the language, h e states that th e mathematicians o f hi s tim e an d earlie r ha d declare d geometr y a dea d language which in the future would be of no use and influence .

I f , F an d F' ar e sai d t o b e properl y equivalent , and i f , then F and F' ar e sai d t o b e improperly equivalent . Gauss proved a numbe r of theorems on the equivalenc e o f forms. Thu s if F is equivalent t o F' an d F' t o F" then F is equivalent t o F". If F is equivalent t o F', any numbe r M representabl e b y F is representable by F' an d i n as many ways by one as by the other. H e then shows, if FandF' are equivalent, how to find all the transformations from Finto F'. He also finds all the representations o f a given number M b y a form F , provide d th e value s of x and y are relativel y prime.

Dirichlet's analytical proof was long and complicated. Specifically it use d wha t ar e no w calle d th e Dirichle t series, , wherei n th e < and z ar e complex . Dirichlet als o prove d tha t th e su m o f the reciprocal s of the prime s in th e sequenc e diverges . This extend s a resul t o f Euler on th e usua l primes (se e below) . I n 1841 36 Dirichle t proved a theore m o n the prime s i n progression s of complex number s a + hi. The chie f problem involvin g the introductio n of analysis concerned th e function tr(x) whic h represents the numbe r o f primes not exceedin g x .