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  "body": "\n## Summary\n\nThe lemniscate sine $\\operatorname{sl}$ and lemniscate cosine $\\operatorname{cl}$ are [[elliptic functions|FF9ZPCZ59K]].\nThey play for the lemniscate of Bernoulli the role that sine and cosine play for the circle.\nThe lemniscate sine is the inverse of the integral $\\int_0^x dt/\\sqrt{1 - t^4}$, which measures arc length on the lemniscate.\nTheir period lattice is the square lattice of Gaussian integers multiplied by $(1 + i)\\varpi$, with $\\varpi$ the [[lemniscate constant|V5A2S8QYQQ]].\nSo they have complex multiplication by $i$.\nThey were studied by Fagnano, Euler and Gauss, and Abel's work on dividing the lemniscate grew out of them.\n\n## Explanations\n\n### L1 human\n\nThe lemniscate of Bernoulli is the figure-eight curve $(x^2 + y^2)^2 = x^2 - y^2$.\nIf you walk along it from the centre, the arc length you have covered when you are at distance $r$ from the centre is $\\int_0^r dt/\\sqrt{1 - t^4}$.\nThe lemniscate sine reverses this: given the arc length, it returns the distance from the centre.\nThis mirrors the circle, where $\\arcsin x = \\int_0^x dt/\\sqrt{1 - t^2}$ and the sine reverses it.\nThe whole figure-eight has length $2\\varpi \\approx 5.244$.\n",
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