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Pythagorean theorem (for instance, Proposition 31 of Book VI of Euclid’s
Elements
), or much more powerful and useful than the Pythagorean theorem, without these attracting anywhere near the same degree of attention. A striking example of the latter is the law of cosines –
c
2 =
a
2 +
b
2 – 2
ab
cos θ – which covers all triangles, not just right-angled triangles, and relates the lengths of the sides to the cosine of one of the angles; the Pythagorean theorem is but a special case of the law of cosines. Yet this law communicates no special magic – partly because one has to know trigonometry to prove it – and one can hardly imagine a Hobbes becoming as transformed.
    The full answer as to why the Pythagorean theorem seems magical is threefold: the visibility of the hypotenuse rule’s applications, the accessibility of the proof, and the way that actually proving the theorem seems to elevate us to contemplate higher truths and thus acquaint us with the joy of knowing.
    First, the theorem characterizes the space around us, and we thusencounter it not only in carpentry and architecture, physics and astronomy, but in nearly every application and profession. The Pythagorean rule for a distance in three-dimensional space – the diagonal of a shoebox, say – is the square root of
x
2 +
y
2 +
z
2 ; for four-dimensional Euclidean space it’s the square root of
x
2 +
y
2 +
z
2 +
w
2 ; in Minkowski’s interpretation of Einstein’s special theory of relativity, the four-dimensional space-time version is
x
2 +
y
2 +
z
2 1 (
ct
) 2 , where
c
is the speed of light. Suitably adapted, this formula enters into the equations of thermodynamics, in describing the three-dimensional motions of masses of molecules. It also enters into both the special and the general theory of relativity. (In the former, it is used to describe the path of light moving in one reference frame from the point of view of another, while in the latter, in a still more complex extension, it is used to describe the motion of light in curved, four-dimensional space-time.) And it is generalized still further in higher mathematics. In
The Pythagorean Theorem: A 4,000-Year History
, Eli Maor calls the Pythagorean theorem ‘the most frequently used theorem in all of mathematics.’ 17 This is not only because of its direct use but also due to what Maor calls ‘ghosts of the Pythagorean theorem’ – the host of other expressions that derive, directly or indirectly, from it. An example is Fermat’s famous ‘last theorem’, finally proven in 1994, which asserts that no integers satisfy the equation
a n
=
b n
+
c n
(all variables stand for positive integers) for any
n
greater than two. Though, being the denial rather than the assertion of an equality, Fermat’s last theorem cannot be put in the form of an equation.
    Second, as Hobbes’s experience indicates, even though the Pythagorean theorem involves a bit of knowledge whose proof seems implausible at the beginning, it can be proved simply and convincingly even without mathematical training. This is one reason why philosophers and scientists from Plato onward use it as an emblematic demonstration of reasoning itself. In
On the World Systems
, Galileo cited Pythagoras’s experience proving the theorem to illustrate the distinction between certainty and proof – what we now call the context of discovery and the context of justification. 18 In
The Rulesfor the Direction of the Human Mind
(Rule XVI), French philosopher and scientist René Descartes used the Pythagorean theorem to show the virtues of symbolic notation, which he was introducing into mathematics. G.W.F. Hegel viewed the proof as ‘superior to all others’ in the way it illustrates what it means for geometry to proceed scientifically, which for him meant showing how an identity contains differences. 19 German philosopher Arthur Schopenhauer, one of the few critics of the way Euclid proved the Pythagorean theorem, viewed that proof as emblematic