Worked solution: Lindemann's transcendence proof of $\pi$ settles squaring the circle (1882)

Frequently asked questions

Was Ferdinand von Lindemann primarily known for this one result?

Largely yes, within mathematics — the transcendence of $\pi$ made him famous. He was also a professor at Königsberg and later Munich, and supervised the doctoral thesis of David Hilbert, who would go on to simplify Lindemann's own proof and pose the influential list of 23 problems that shaped 20th-century mathematics.

Source

Ferdinand von Lindemann, 1882

Über die Zahl π

Step 6 of 6: Legacy: the last classical problem falls, and the proof gets simpler

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In plain words

Lindemann's original 1882 proof was intricate and hard to follow even for professional mathematicians. Over the next decade, Karl Weierstrass (1885), and then David Hilbert, Adolf Hurwitz, and Paul Gordan independently (1893), found shorter, cleaner routes to the same conclusion — the streamlined version taught in courses today descends from their simplifications, not Lindemann's original argument.

With squaring the circle settled in 1882, all three of antiquity's great 'impossible' construction problems — trisecting an angle, doubling the cube, and squaring the circle — had finally been proven impossible, closing questions that professional and amateur mathematicians alike had attacked for over two thousand years.

\text{Hermite (1873)} \to \text{Lindemann (1882)} \to \text{Weierstrass, Hilbert, Hurwitz, Gordan (1885--1893)}

Detailed analysis

Lindemann's 1882 argument, while correct, was widely regarded as difficult; within a decade Karl Weierstrass (1885) gave a cleaner treatment, and Karl Weierstrass's version was further streamlined independently by David Hilbert, Adolf Hurwitz, and Paul Gordan in 1893, using more systematic symmetric-function bookkeeping. It is this simplified Hilbert–Hurwitz–Gordan line of argument, not Lindemann's original 1882 paper, that underlies most modern textbook presentations of the theorem (including the version summarized in Steps 2–4 here).

The result also had an unusually direct effect on a two-thousand-year-old open question outside pure mathematics: by 1882, Wantzel's 1837 impossibility proofs for angle trisection and doubling the cube, together with Lindemann's transcendence proof, meant all three of antiquity's classical straightedge-and-compass problems had been rigorously settled as impossible — squaring the circle being the hardest, since it required a genuinely new tool (transcendence theory) rather than the algebraic field-degree arguments that sufficed for the other two.

Lindemann's transcendence method also opened a durable research direction: it was extended in 1900 by Hilbert's seventh problem (are numbers like $2^{\sqrt2}$ transcendental?), finally resolved by Gelfond and Schneider in 1934, showing the reach of the auxiliary-function technique first used by Hermite in 1873 continues well beyond the original circle-squaring question.