Beyond this point, analytical methods must be adopted, and the student passes to trigonometry and the infinitesimal calculus.

During this period logarithms were invented, trigonometry and algebra developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation.

He published, among other mathematical works, Clavis Mathematica, in 1631, in which he introduced new signs for certain mathematical operations (see Algebra); a treatise on navigation entitled Circles of Proportion, in 1632; works on trigonometry and dialling, and his Opuscula Mathematica, published posthumously in 1676.

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The introduction of hyperbolic functions into trigonometry was also due to him.

Cantor attributes to him (in the use of his prosthaphaeresis) the first introduction of a subsidiary angle into trigonometry (vol.