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Squaring the circle is the age-old attempt to construct a square the same size as a circle, using a compass and straightedge, in a finite number of steps. It was finally proven impossible in 1882. However, Menaechmus, in his study of the issue, devised the conic sections, beginning a whole branch of mathematics.

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Jeff Miller's site gives the first occurrence of $\Delta$ as The symbol $\Delta$ for the Laplacian operator (also represented by $\nabla^2$) was introduced by Robert Murphy in 1833 in Elementary Principles of the Theories of Electricity. (Kline, page 786) The first use of the term Laplace's Operator is given as The term LAPLACE’S OPERATOR (for the ...

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This episode is discussed in A Mathematical History of the Golden Number by Herz-Fischler. The 1597 letter was not from Mästlin to Kepler but from Kepler to Mästlin. In it Kepler described a construction of a right triangle whose sides are in geometric progression, now called the Kepler triangle, which he attributes to "Magirus", who "pleased ...

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One can argue that most parts of mainstream mathematics are borne "by accident". That is we start with a very particular but still important example, then generalizations are discovered and a large theory emerges completely unexpected by the person who started all that. For example Max Dehn studied tessellations of hyperbolic plane and discovered ...

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Here's a link to Jordan Bell's English translation of Euler's original paper, "Various observations on angles proceeding in geometric progression". Euler begins exactly as the OP outlines, starting from the double-angle formula for sine. By the second page he has given the following version of the OP's formula: Therefore the arc $s$ itself can be ...

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What follows is from A History of Vector Analysis by Michael John Crowe (1967; 1985 Dover corrected reprint). (from middle of p. 167) In Heaviside's later papers of 1883 and 1884 use was made of vectors, but no new principles were $\text{introduced.}^{35}$ (from middle of p. 180) ${}^{35}$This should perhaps be qualified by the statement that [the symbol] $... 2 Yes, differential geometry is older. Though he had predecessors, Gauss can be considered the founding father of differential geometry with his book General investigation of curved surfaces, 1827. This was long before the subject of topology was born as a subject (though some isolated results in both differential geometry and topology are much older). ... 2 Apparently ever since precision in astronomy became really important professionals did use ecliptic coordinates. Hipparchus had demonstrated the precession of the vernal point and its position against the stars as the Zero of ecliptic became problematic. Already Babylonians had two values for it, either as 10 or 8 of Aries; Eudoxus said it was 15 but ... 1 I have found a reference for the injectivity property of the$j$function when it acts on a fundamental domain, and this constitutes the first part of the "special property". However, regarding the surjectivity of$j(\tau)$, I'm still searching. In a short (unpublished) paragraph in p.386 of vol.3, Gauss introduces a mapping from the set of binary ... 1 I believe the notation$\nabla^2 f\$ comes from $$\nabla^2 f = \nabla\cdot\nabla f = \operatorname{div}(\operatorname{grad} f)$$ See here

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Perhaps something closer to the question. in the effort of proving it, ending up proving something intuitively different One can count here numerous attempts to prove well known open conjectures which resulted in proving weaker and seemingly different statements. For example, Tao's "approximation" of Collatz problem is interesting by itself but ...

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