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As in most such cases, nobody was "first". One can present already the Pythagorean proof of incommensurability of the side and the diagonal of a square as an "impossibility proof". It amounts to saying that it is impossible to split the side into a whole number of equal segments so that the diagonal can be assembled from a whole number of ...


3

Wells' book is available for download on PdfDrive. Puzzle #38 appears in the section of problems attributed to Abū al-Wafā and is preceeded by the comment that "he is best known for his study of geometrical dissections and of constructions with a rusty compass, meaning a compass which is so stiff that it can be used with only one opening". In the ...


3

Denote $\beta=\angle MBC,\;\alpha=\angle DMN,\;\gamma=\angle CDH$. We have: $$\tan\beta=1/2,\quad\cos\alpha=-\sin\beta=-1/\sqrt{5},\quad\sin\alpha=\cos\beta=2/\sqrt{5}.$$ Take $DM=1$, then $MN=\sqrt{5}-2$, and let $x=DN$. By the rule of cosines, $$x^2=1^2+(\sqrt{5}-2)^2+2(\sqrt{5}-2)/\sqrt{5}=12-24/\sqrt{5}.$$ Then by the rule of sines, $$\sin\gamma=\frac{(\...


3

Darrigol's History of Optics names Barrow (1669) and Huygens (1653) as giving semi-verbal case by case variants of the lens formula before Halley, with Molyneux being the first to publish in 1692. Halley (1693) is credited as the first who wrote the formula algebraically. Dijksterhuis in Lenses and Waves gives detailed description of Huygens's Dioptrica (...


3

It is not quite accurate to say that ancient astronomy made use of epicycloids. Epicycles, yes, but curiously enough, hypocycloids, epicycloids and cycloids were not studied as geometric curves (as far as we know), despite the use of epicycles in astronomy. It would have been possible to draw tangents to them by the same technique that Archimedes used for ...


1

The basic reason is that the straightline and circle are the most basic of shapes. Aristotle theorises about them in his Physics, for example. Its still the case today. Except of course the idea has ramified into many new and different ways. For example, when we think topologically, the straight-line is the standard example of a non-compact space and the ...


1

Virtually nobody reads Newton's Principia, yet virtually all physicists start off with Newton's laws. Likewise with geometry, we learn from Euclid even if his book is not right there before us. Even now. Take for example non-Euclidean geometry. One might think that this is far from Euclid geometry as one might imagine. Yet, it still conceived in exactly ...


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