What Hacker News says

ewiethoffJan 2009

Article subtitle is "A long-lost text by the ancient Greek mathematician shows that he had begun to discover the principles of calculus." This has been known for several decades, ever since Heiberg transcribed some of the text (I don't know how much), per the article.

I refer you to The History of the Calculus and Its Conceptual Development, a 1949 book by Carl B. Boyer, reprinted by Dover Publications [ISBN 0486605094]. Boyer writes several pages about Archimedes' method of exhaustion, notions of the infinitesimal, and so on. The main footnote reads:

For the works of Archimedes in general, see Heiberg, Archimedes opera omnia and T. L. Heath, The Works of Archimedes. For Archimedes' Method, see T. L. Heath, The Method of Archimedes, Recently Discovered by Heiberg; Heiberg and Zeuthen, "Eine neue Schrift des Archimedes"; and Smith, "A Newly Discovered Treatise of Archimedes."

Boyer is clearly using Heiberg's work on the Archimedes Palimpsest.

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mindcrimeAug 2024

I've been going through that very video lecture series the last couple of weeks. Good stuff. And in the lectures he mentions a number of books. I looked a few up on Amazon, and then looked at the associated Amazon recommendations, and so far have this small list of books related to Maths history that look worth reading:

  Mathematics and Its History (Undergraduate Texts in Mathematics) 3rd ed. 2010 Edition by John Stillwell

  The History of the Calculus and Its Conceptual Development (Dover Books on Mathematics) by Carl B. Boyer

  A History of Mathematics by Carl B. Boyer

  A Concise History of Mathematics: Fourth Revised Edition (Dover Books on Mathematics)A Concise History of Mathematics: Fourth Revised Edition (Dover Books on Mathematics) by Dirk J. Struik

  Introduction to the Foundations of Mathematics: Second Edition (Dover Books on Mathematics) Second Edition by Raymond L. Wilder

  Mathematical Thought from Ancient to Modern Times by Morris Kline (3 volume set)

  The Calculus Wars: Newton, Leibniz, and the Greatest Mathematical Clash of All Time by Jason Socrates Bardi

There is also a "thing" in mathematics that is sometimes called the "genetic approach" where "genetic" is roughly equivalent to "historical" or maybe "developmental". IOW, a "genetic approach" book teaches a subject by tracing the development of the subject over its history. One popular book in this mold is:

  The Calculus: A Genetic Approach by Otto Toeplitz

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bikenagaJan 2024

Leibniz and the product/quotient rules: "Although at the outset Leibniz was uncertain about his method and hesitated as to whether or not d(x y) is the same as dx dy and whether d(x/y) is equal to dx/dy, he in the end answered these questions correctly, determining that d(x y) = x dy + y dx and d(x/y) = (y dx - x dy)/y^2. These values he found by allowing x and y to become x + dx and y + dy respectively. Upon subtracting the original value of the function from the new one and observing that dx dy is infinitely small in comparison with the terms x dy and y dx, the results are obtained." [1]

Newton's algebraic vs. geometric approaches to calculus: Boyer [1] notes that in Principia (1687) "Newton presented them [his propositions] in the form of synthetic geometrical demonstrations with an almost complete lack of analytical calculations" - hence the geometrical approach you point out. But he actually wrote up accounts of his calculus in three papers (De analysi ... (1669), Methodus fluxionum ... (1671), De quadratura ... (1676)), all of which were published after 1700. In those earlier papers, his approach is algebraic, often using the binomial theorem. In one incredible result from De analysi, he shows that the area under y = a x^(m/n) is given by z = (m/(m + n)) a x^((m + n)/n). To do this, he increments z and x:

  z + o y = (m/(m + n)) a (x + o)^((m + n)/n)

Expand the right side using the binomial theorem, subtract the z-expression from both sides, divide both sides by o, then drop any terms on the right containing o ... and you get y = a x^(m/n). That's pretty algebraic, right?

But the incredible thing is that he's basically showing that (roughly) "the rate of change of the area under the y-curve is y", which is the Fundamental Theorem of Calculus! He's figured out that derivatives (rates of change) and integrals (areas under curves) are somehow inverses (at least in this case). Newton was pretty amazing!

[1] Carl Boyer, "The History of the Calculus and Its Conceptual Development". Dover Publications, 1949.

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