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 ) ,
e7rvT63
=
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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