Thread by @wojtek_jk79848

thread on history of calculus, in this thread i will trace origin of calculus in most unbiased way possible

The fundamental ideas of calculus arose from the logical difficulties encountered by ancient Greek mathematicians in their efforts to relate continuous concepts, such as the ratios of lines, to discrete numbers

The recognition of the need for concepts to handle incommensurable magnitudes (like the diagonal and side of a square), stemming from the work of the Pythagoreans, is considered the earliest starting point (terminus a quo) in the development of the concepts of calculus. This discovery exposed the inadequacy of the Pythagorean view of identifying the realm of number with that of geometry or continuous magnitude, forcing them to abandon the attempt to unite the two fields.

The Greek investigation into the problem of the continuum was complicated by Zeno's paradoxes. These paradoxes (such as Achilles and the Tortoise) involved difficulties concerning continuous movement and led to Greek mathematicians banning the concepts of the infinite and infinitesimal from their reasoning for a time, in order to retain logical precision

Greek thinkers, including Plato, displayed a lively interest in resolving these foundational problems of the continuum, the infinite, and the infinitesimal, urging their study upon his associates. The Method of Exhaustion (Precursor to Integration)
The earliest major idea in calculus, accumulation (integration), is traceable in the historical progression back to at least the fourth century BCE in Greece. To circumvent the logical difficulties of the infinitesimal, Eudoxus of Cnidus (followed later by Euclid) developed the method of exhaustion. Eudoxus proved propositions concerning the volumes of pyramids and cones using this method.

Eudoxus formulated a new, more general definition of proportion (stated in Euclid's Elements, Book V, Definition 5) that allowed all four terms to be geometrical entities rather than numbers, thus avoiding the problem of incommensurability and expressing the ordinal idea involved in the modern conception of a real number

The Greek method of exhaustion was wholly geometrical, focused on continuous magnitude. It worked by comparing the figure to be measured with inscribed and circumscribed rectilinear figures, showing that the difference in area could be made smaller than any assignable magnitude . The method avoided passing to the numerical limit, as the Greeks did not possess the general concept of real numbers necessary to define the limit of an infinite sequence.

then comes the archimedes. probably one of the most important figure in hustory of calculus. field medalist david mumphord explicitly said his rigour in proto calculus wasn't matched till cauchy. lets take a look at what he did.

Archimedes is credited with anticipating the calculus of the infinite. He combined the rigorous deductive method of exhaustion with infinitesimal considerations in his investigative work (the "mechanical method" described in The Method of Mechanical Theorems), imagining geometric objects as built from infinitely thin slices.

Archimedes considered his greatest achievement to be the derivation of the formula for spherical volume, noting that the volume of a sphere is two-thirds that of the smallest cylinder containing it.He successfully applied the method of exhaustion to find the area of a parabolic segment by summing a geometrical progression, using a technique equivalent to evaluating the limit of a sum, although he intentionally stopped short of expressing it as an infinite sum to maintain rigour. He also developed calculations equivalent to performing the integration indicated by ∫xdx and ∫x^2dx geometrically. Archimedes also approached questions of tangency in his study of the spiral (defined by uniform radial and angular motion), one of the few considerations found in Greek geometry corresponding to the differential calculus. method shown below.

Crucially, Archimedes explicitly stated that his infinitesimal method only indicated the correct result and did not constitute valid mathematical demonstration. For this reason, he recast all his analysis by infinitesimals into the orthodox synthetic form using the method of exhaustion. he was praised by field medalist david mumphord for giving most rigorous version of proto calculus more than kerala school and even newton/leibiniz, that was not surpassed till cauchy .

Source

Carl C boyer (1949)
David Mumbphord (2010)

*until

Plus bonus, here is the link to my notebook llm. It has all the best books which talk about the history of calculus. If anyone is interested you can chat with it, it doesn't hallucinate and strictly only quotes from the source materials with link to the exact line and para its taking its reference from.notebooklm.google.com/notebook/23d62…

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