Ueber Descartes’ Urteilslehre

Hippokrates Ueber die Krankheiten III

irrelevant. Chemistry and the relations between science, technology, and industry are well surveyed by Crosland and Cardwell respectively. Forbes's essay on mathematical cosmography is not a comprehensive account but the story (albeit interesting) of a few German cartographers. Compared with other recent collections of essays in the history of science, this volume parades no party line: witness...

The Descartes Rule of Sweeps and the Descartes Signature

The Descartes Rule of Signs, which establishes a bound on the number of positive roots of a polynomial with real coefficients, is extended to polynomials with complex coefficients. The extension is modified to bound the number of complex roots in a given direction on the complex plane, giving rise to the Descartes Signature of a polynomial. The search for the roots of a polynomial is sometimes ...

Arithmetic Multivariate Descartes' Rule Arithmetic Multivariate Descartes' Rule

Let L be any number field or p-adic field and consider F := (f1, . . . , fk) where fi∈L[x ±1 1 , . . . , x ±1 n ]\{0} for all i and there are exactlym distinct exponent vectors appearing in f1, . . . , fk. We prove that F has no more than 1+ ( σm(m− 1)2n2 logm )n geometrically isolated roots in Ln, where σ is an explicit and effectively computable constant depending only on L. This gives a sign...

Ueber Das Verschwinden Der Θ-functionen

7 Descartes ’ Perfect Lens

We give a new, elementary, purely analytical development of Descartes’ theorem that a smooth connected surface is a perfect focusing lens if and only if it is a connected subset of the ovoid obtained by revolving a cartesian oval around its axis of symmetry.