How Euler Did It, by Ed Sandifer

Euler’s Pi Formulas and BBP-Type Results

• A reader connects an Euler paper (1779) to modern BBP-type formulas for π that allow computation of arbitrary binary/hex digits without prior digits.
• It was previously believed such formulas began with work from the 1990s; the commenter wrote a paper arguing Euler had them first.
• Others add historical context: some arctangent-based formulas and series transformations involved were known earlier to other mathematicians, but Euler’s methods parallel modern BBP proofs.

Scale and Nature of Euler’s Output

• Multiple comments highlight Euler’s staggering productivity: thousands of pages, letters, and manuscripts, constituting a large fraction of 18th-century scientific output.
• Opera Omnia, the complete-works project, has taken over a century, suggesting much remains underappreciated.
• Anecdotes emphasize that his productivity increased after near-total blindness with the help of scribes; he reportedly joked that losing sight reduced distractions.

Mathematical Style and Intuition

• Commenters admire his intuitive handling of infinities and infinitesimals before rigorous analysis, noting examples like factoring trigonometric functions into infinite products in the Basel problem.
• His approach is described as almost “magical” but usually correct.

Education, Talent, and Nurturing “Whizzkids”

• The thread branches into criticism of modern education as ill-suited for extreme talent, more focused on conformity and box-checking than deep exploration.
• Some argue gifted students will self-educate; others counter that many potential geniuses are lost without support or inspiration.
• Competitive math/physics systems (e.g., Olympiads) are cited positively.
• There is debate over AI/LLMs as tutors: supporters expect personalized pacing, instant feedback, and gamification; skeptics warn about “number-crunching” without real understanding and hallucinations.

Modern Equivalents and “Low-Hanging Fruit”

• Discussion asks why there seems to be no contemporary figure with Euler’s breadth.
• One view: mathematics is now too large for one person to “know it all”; modern work is more specialized and collaborative.
• Another view challenges the “low-hanging fruit” narrative as ahistorical, arguing earlier breakthroughs depended on long cultural build-up.
• Several past and current mathematicians and scientists are mentioned as partial analogues, but consensus is that Euler remains uniquely prolific.

Nationality and Patronage

• Debate over whether to regard Euler as primarily Swiss or Russian: born in Switzerland, spent long periods working under Russian (and other) patronage, and died in St. Petersburg.
• Some argue birthplace and early formation dominate; others emphasize where he worked and was funded, concluding his identity is mixed and labels are somewhat arbitrary.

Euler as Educator and Writer

• Commenters praise his textbooks, especially early, accessible treatments of calculus and algebra that helped spread new ideas across Europe.
• His explanations, including informal use of infinitesimals in elementary contexts, are described as playful, intuitive, and unusually clear.

Miscellaneous

• Side notes include Euler’s role in music theory (e.g., structures like the tonnetz), and a brief technical digression on the linked site’s outdated/weak TLS certificate causing browser and corporate firewall warnings.