** Introduction **

The circular functions arise from ratios of lengths in a circle. In a similar manner, the elliptic functions can be defined by means of ratios of lengths in an ellipse. Many of the key properties of the elliptic functions follow from simple geometric properties of the ellipse.

Originally, Carl Gustav Jacobi defined the elliptic functions , , using the integral

He called the *amplitude* and wrote . It can be difficult to understand what motivated his definitions. We will define the elliptic functions , , in a more intuitive way, as simple ratios associated with an ellipse.

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