Euclid’s first proposition in Book I states that an equilateral triangle can be constructed from a segment AB only with a ruler and compass. Here is a poem by the British poet Samuel Taylor Coleridge on this proposition:<\/p>\n
<\/p>\n
A Mathematical Problem<\/b><\/p>\n
This is now–this was erst,
\nProposition the first–and Problem the first.<\/p>\n
I.
\nOn a given finite Line
\nWhich must no way incline;
\nTo describe an equi–
\n–lateral Tri–
\n–A, N, G, L, E.
\nNow let A. B.
\nBe the given line
\nWhich must no way incline;
\nThe great Mathematician
\nMakes this Requisition,
\nThat we describe an Equi–
\n–lateral Tri–
\n–angle on it:
\nAid us, Reason–aid us, Wit!<\/p>\n
II.
\nFrom the centre A. at the distance A. B.
\nDescribe the circle B. C. D.
\nAt the distance B. A. from B. the centre The round A. C. E. to describe boldly venture.
\n(Third Postulate see.)
\nAnd from the point C.
\nIn which the circles make a pother
\nCutting and slashing one another,
\nBid the straight lines a journeying go,
\nC. A., C. B. those lines will show.
\nTo the points, which by A. B. are reckon’d,
\nAnd postulate the second
\nFor Authority ye know.
\nA. B. C.
\nTriumphant shall be
\nAn Equilateral Triangle,
\nNot Peter Pindar carp, not Zoilus can wrangle.<\/p>\n
III.
\nBecause the point A. is the centre
\nOf the circular B. C. D.
\nAnd because the point B. is the centre
\nOf the circular A. C. E.
\nA. C. to A. B. and B. C. to B. A.
\nHarmoniously equal for ever must stay;
\nThen C. A. and B. C.
\nBoth extend the kind hand
\nTo the basis, A. B.
\nUnambitiously join’d in Equality’s Band.
\nBut to the same powers, when two powers are equal,
\nMy mind forbodes the sequel;
\nMy mind does some celestial impulse teach,
\nAnd equalises each to each.
\nThus C. A. with B. C. strikes the same sure alliance,
\nThat C. A. and B. C. had with A. B. before;
\nAnd in mutual affiance,
\nNone attempting to soar
\nAbove another,
\nThe unanimous three
\nC. A. and B. C. and A. B.
\nAll are equal, each to his brother,
\nPreserving the balance of power so true:
\nAh! the like would the proud Autocratorix do!
\nAt taxes impending not Britain would tremble,
\nNor Prussia struggle her fear to dissemble;
\nNor the Mah’met-sprung Wight,
\nThe great Mussulman
\nWould stain his Divan
\nWith Urine the soft-flowing daughter of Fright.<\/p>\n
IV.
\nBut rein your stallion in, too daring Nine!
\nShould Empires bloat the scientific line?
\nOr with dishevell’d hair all madly do ye run
\nFor transport that your task is done?
\nFor done it is–the cause is tried!
\nAnd Proposition, gentle Maid,
\nWho soothly ask’d stern Demonstration’s aid,
\nHas prov’d her right, and A. B. C.
\nOf Angles three
\nIs shown to be of equal side;
\nAnd now our weary steed to rest in fine,
\n‘Tis rais’d upon A. B. the straight, the given line.<\/p>\n
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