Abstract. The normal form x 2 +y 2 = a 2 +a 2 x 2 y 2 for elliptic curves simplifies formulas in the theory of elliptic curves and functions. Its principal advantage is that it allows the addition law, the group law on the elliptic curve, to be stated explicitlyThe j-invariant of an elliptic curve determines 24 values of a for which the curve is equivalent to x 2 + y 2 = a 2 + a 2 x 2 y 2 , namely, the roots of (The symmetry in x and y implies that the two transcendental functions x(t) and y(t) that parameterize x 2 + y 2 = a 2 + a 2 x 2 y 2 in a natural way are essentially the same function, just as the parameterizing functions sin t and cos t of the circle are essentially the same function. Such a parameterizing function is given explicitly by a quotient of two simple theta series depending on a parameter τ in the upper half plane.Part I. The Addition Formula Why elliptic functions?The double periodicity of elliptic functions, the property by which they are often defined today, is not what attracted attention to them in the first place. Abel, whose own contribution to the study of elliptic functions was enormous, began his long memoir [1] about them with the observation that "the first idea of [elliptic]

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