# Natures of Curved Lines in the Early Modern Period and the Emergence of the Transcendental

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## Abstract

The transition from the geometric study to the algebraic study of curved lines changed how early modern mathematicians understood the origins of these mathematical objects. To know a mathematical object was to know its nature. To know its nature was to know its origin. The emergence of the transcendental classification was a consequence of a shift in how mathematicians understood the natures of mathematical objects because they were classified according to their natures. This nature was useful to determine which objects were appropriate for geometrical study, especially when applied to curves. The development of calculus provided the tools necessary for algebraic analysis to uncouple the study of curves and geometry, greatly increasing the number of curves which could be made known. The geometrical motivation for classifying curves was rendered obsolete and was replaced by focusing on functional relationships between variables. The nature of mathematical objects inherited the nature of the algebraic expression used to represent them.

## Keywords

Mathematical Object Curve Line Classification Rule Geometric Construction Conic Section## Notes

### Acknowledgements

The author would like to thank the members of the CSHPM, members of his dissertation committee, and the reviewers for their encouraging comments and helpful criticisms.

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