Preface to the First German Edition XI one hopes to derive mathematical knowledge and principles from the solution of the equation. Let us hypothesize, for example, based on numeric calculation, the following identities: ^ 2 - 1 = ^ ( ^ 3 - ^ 6 + ^ 1 2 ) , e 7rvT63 = 262537412640768744, and 2TT 2 cos 17 \ + \y/YI+\\lu-2y/YI + J y 17 + 3\/l7 - ^3 4 - 2v/17 - 2^34 + 2VT7. Without going into detail, it seems plausible that behind such identities, if indeed they are correct, lie some mathematical laws. A direct check to determine whether they are in fact correct or are merely the result of chance numeric approximation would be difficult.1 But back to Cardano. In addition to the solution for cubic equa- tions, Cardano published in his Ars Magna a general formula for quartic equations, that is, equations of the fourth degree, also known as biquadratic equations. Using such formulas, the equation x4 - Sx + 6 = 0 I will reveal that only the first and third identities are correct. The first was discovered by the Indian mathematician Ramanujan (1887-1920) and can be easily checked. The third, which will be discussed in Chapter 7, contains within it a proof that the regular heptadecagon (seventeen-sided polygon) can be constructed with straight- edge and compass. The second equation is not exact. The actual value of the right-hand side is 262537412640768743.9999999999992501 However, this approximate identity is more than mere chance. It is based on some deep number-theoretic relationships. For more on this, see Philip J. Davies, Are there coincidences in mathematics? American Mathematical Monthly 88 (1981), pp. 311— 320.
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