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I just last night watched a mini-documentary on the origins of calculus in the work of Newton and Leibnitz, focusing specifically on how they came to be interested in the idea and their very earliest thoughts on it, often quoting directly from their notebooks.

The doc is on Youtube here: https://youtu.be/ObPg3ki9GOI

It was very clear that Newton in particular was building incrementally on work that other mathematicians had done before him. Leibniz was also making incremental improvements, but was apparently somewhat more visionary in his work. The Youtube comments are also interesting, including anecdotes of ancient Egyptian land surveyors using the same "sum of lines" technique (what Leibniz generalized into our modern antiderivative) in order estimate the areas of unevenly-shaped land plots along waterways for taxation.

Leibniz in particular was fascinating because it shows the incrementality of science continuing after him as well. Apparently Leibniz had built several computing machines, 150 years before Babbage, and was deliberately trying to work towards general and abstract paradigms for solving mathematical problems, 300 years before Hilbert, Church, Gödel, and Turing!

Watch "The Birth of Calculus"[0]

[0]: https://www.youtube.com/watch?v=ObPg3ki9GOI

⬐ acqq

Wonderful. Some very interesting and very precise historical details.

I can only be sad to imagine how something on that subject would be produced today (with so much sound and visual effects and removing the substance to be practically unwatchable). They just don't make them like that anymore, sadly.

Also good to be remembered that a lot of work of Leibniz was in some way inspired or motivated by, or related to his work on his calculating machine:

https://en.wikipedia.org/wiki/Stepped_reckoner

⬐ beervirus

There's some math content on youtube that's great, and wouldn't have been possible 30-odd years ago. I'm mostly thinking of 3blue1brown, but I'm sure there are other examples.

⬐ 29athrowaway

People often know about Isaac Newton but not Isaac Barrow.

Or the greatest of all, the doctor that rediscovered integration in 1994. https://fliptomato.wordpress.com/2007/03/19/medical-research...

⬐ lqet

OP is not joking, the original paper is available here:

https://math.berkeley.edu/~ehallman/math1B/TaisMethod.pdf

> RESEARCH DESIGN AND METHODS— In Tai's Model, the total area under a curve is computed by dividing the area under the curve between two designated values on the X-axis (abscissas) into small segments (rectangles and triangles) whose areas can be accurately calculated from their respective geometrical formulas. The total sum of these individual areas thus represents the total area under the curve

⬐ 29athrowaway

Tai's "integration" method also has references to previous, similar work, and has 293 citations. You can buy the paper for $35.

Academia in a nutshell.

⬐ olooney

Is this not just the trapezoidal rule for numeric integration?

https://en.wikipedia.org/wiki/Trapezoidal_rule

It's not even a particular good choice for the specific problem (glucose curve) because the trapezoidal rule will systematically underestimate the true area when the curvature is always negative. Simpson's rule is almost always a better choice:

https://en.wikipedia.org/wiki/Simpson%27s_rule

Fun fact: although the method is attributed to the 18th century mathematician Simpson, Kepler is known to have used it in the 17th century.

⬐ Vysero

The difference between Newton and Leibniz seems remarkably similar to the difference between say: Einstein and Feynman. One seems to be discovering the maths while the other seems to be forging it. Personally, I prefer and understand the forged methodology better myself, but then again I have always been a tinkerer.

⬐ z991

Transcript: https://youtubetranscript.com/?v=ObPg3ki9GOI

⬐ melling

Are the original notebooks online?

⬐ laichzeit0

I've seen Newton's notebooks online before.

⬐ Strilanc

The really interesting thing about this video, to me, is the explanation of how tangents were calculated before calculus. You can see how awkward it would have been to do things that way, and how it would have been difficult to realize they was something far far easier.

⬐ 29athrowaway

Trigonometric functions have existed for a long time. And in the past, more trigonometric functions were used:

- versed sine (versin) and versed cosine (vercos)

- coversed sine (coversin) and coversed cosine (covercos)

- haversed sine (haversin) and haversed cosine (havercos)

- hacoversed sine (hacoversin) and hacoversed cosine (hacovercos)

- exsecant and excosecant

Perhaps I am forgetting some. Many alternative mnemonics exist for these too.

This formula was very important: https://en.wikipedia.org/wiki/Haversine_formula

For anyone interested, here's the BBC looking into Newton and Leibniz notebooks as they cook it up for the first time - https://www.youtube.com/watch?v=ObPg3ki9GOI

Also Steven Strogatz is an excellent writer. His other books Sync and Joy of X can be read by anyone.

I think that's this one https://www.youtube.com/watch?v=ObPg3ki9GOI

They don't review all of it, but you can get a few pages read

A very nice video about the work of Newton and Liebniz (with the actual papers on a table read by the host)

https://www.youtube.com/watch?v=ObPg3ki9GOI