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.