The Birth of Calculus (1986) [video] | Hacker News Reader

The Birth of Calculus (1986) [video]

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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.

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

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

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.

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...

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

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.

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.

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.

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

Are the original notebooks online?

I've seen Newton's notebooks online before.

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