Calculus Made Easy

Original Article ↗ Hacker News Discussion ↗

A century-old textbook, *Calculus Made Easy* by Silvanus P. Thompson, is earning renewed praise for its intuitive, non-rigorous introduction to derivatives and integrals, with many readers saying it succeeded where modern courses and texts failed. Commenters highlight how weak algebra and trig fundamentals—not the core ideas of calculus—are what usually hold learners back, and share alternative resources such as 3Blue1Brown, Khan Academy, Gilbert Strang’s linear algebra, and Tao’s *Analysis* for building deeper understanding. The exchange broadens into criticism of math pedagogy that emphasizes rote techniques over “why it works,” and into strategies adults are using—online courses, streaks, tutors, and boredom tolerance—to rebuild math skills later in life.

Reception of “Calculus Made Easy”

  • Many readers say this book finally made calculus click, sometimes more than multiple formal courses.
  • The geometric/infinitesimal style and step‑by‑step derivations are praised as intuitive and confidence‑building.
  • Others find the 1910 language formal, wordy, and culturally dated (e.g., old money units), and question whether it is actually “easy.”
  • Some feel that if you can follow its algebra (e.g., squaring x+dx), you could handle a modern, limit‑based intro just as well.
  • One commenter liked it only after first repairing their algebra with simpler “for dummies”–style books.

Infinitesimals vs Limits

  • Several people like the infinitesimal viewpoint and geometric arguments, seeing them as more intuitive than epsilon–delta limits.
  • Others note the historical criticism of informal infinitesimals and argue that without a rigorous framework they can mislead.
  • References are made to modern rigorous infinitesimal approaches, but these often require nonstandard or intuitionistic logic.

Foundations and Prerequisites

  • A major theme: calculus difficulty often comes from weak fluency in algebra, trig, and basic functions, not from core calculus ideas.
  • Commenters describe discovering gaps (e.g., simple root and fraction manipulations) while taking entrance or calculus courses.
  • There’s agreement that foundational practice and even rote memorization (identities, standard forms) can “free brain space” for higher concepts.
  • Others warn that conceptual understanding without enough exercise leads to a fragile, “expert beginner” level.

Teaching Quality and Pedagogy

  • Many criticize school and university math for:
    • Emphasizing symbol‑pushing and tricks over meaning and applications.
    • Presenting dot products, eigenvectors, triple integrals, etc., as procedures with no stated purpose.
    • Using calculus and linear algebra as “weeder” courses without explaining why they matter.
  • Some teachers and tutors in the thread argue that:
    • Most students lack the intuition needed to appreciate deep explanations early; they need mechanical practice first.
    • Often the “why” is mentioned, but students only notice years later when the ideas finally click.
  • Others strongly counter that good explanations and real‑world hooks can produce genuine “aha” moments without endless grinding.

Intuition, Motivation, and the “Why”

  • Multiple comments stress that integrals as “area under the curve” and derivatives as “rate of change” tied to speed/position are far more motivating than abstract limits.
  • Several people lament being taught math and physics as disconnected formulas, discovering much later how calculus underpins mechanics, probability, and engineering.
  • There is broad frustration with texts that never plainly say why the dot product, parametric curves, or higher‑dimensional calculus are useful.
  • Some frame this as a broader shift toward education as job‑training rather than cultivation of understanding.

Alternative Resources and Learning Strategies

  • Many recommendations surface:
    • Visual and intuition‑heavy video channels for calculus, linear algebra, and beyond.
    • Friendly analysis and “tour of calculus” books, and vector‑calculus texts that tie directly to electromagnetism.
    • Classic and modern linear algebra courses, including both computational and proof‑oriented tracks.
    • Online adaptive platforms that diagnose gaps and build a personalized path.
  • Several adults describe “starting over” with algebra playlists or high‑school level courses to prepare for machine learning, physics, or entrance exams.
  • Learners report success using: daily streaks, accountability partners, boredom/low‑distraction periods, and conversational help (including from AI) as lightweight tutoring.

Broader Reflections on Access and Era

  • Some note that highly readable, intuitive math texts have existed for over a century but remained obscure compared to standard textbooks.
  • There is debate over whether the internet truly fixes this: it increases access but also adds massive distraction and noise.
  • A recurring sentiment: with today’s resources, motivated learners can reach deep understanding—but they must still invest sustained, disciplined effort.