4 ms·

The Birth of Calculus (1986) [video]

Strilanc 6y ago
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 6y ago
  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 https://en.wikipedia.org/wiki/Haversine_formula
29athrowaway 6y ago
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-researcher-discovers-integration-gets-75-citations/ https://fliptomato.wordpress.com/2007/03/19/medical-research...
- lqet 6y ago
  OP is not joking, the original paper is available here: https://math.berkeley.edu/~ehallman/math1B/TaisMethod.pdf 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 6y ago
    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 6y ago
    Is this not just the trapezoidal rule for numeric integration? https://en.wikipedia.org/wiki/Trapezoidal_rule 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 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.
acqq 6y ago
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 https://en.wikipedia.org/wiki/Stepped_reckoner
- beervirus 6y ago
  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.
z991 6y ago
Transcript: https://youtubetranscript.com/?v=ObPg3ki9GOI https://youtubetranscript.com/?v=ObPg3ki9GOI
melling 6y ago
Are the original notebooks online?
- laichzeit0 6y ago
  I've seen Newton's notebooks online before.
Vysero 6y ago
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.

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 6y ago
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 https://en.wikipedia.org/wiki/Haversine_formula

29athrowaway 6y ago

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-researcher-discovers-integration-gets-75-citations/ https://fliptomato.wordpress.com/2007/03/19/medical-research...

lqet 6y ago
OP is not joking, the original paper is available here: https://math.berkeley.edu/~ehallman/math1B/TaisMethod.pdf 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 6y ago
  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 6y ago
  Is this not just the trapezoidal rule for numeric integration? https://en.wikipedia.org/wiki/Trapezoidal_rule 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 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.

acqq 6y ago

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

beervirus 6y ago
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.

z991 6y ago

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

melling 6y ago

Are the original notebooks online?

laichzeit0 6y ago
I've seen Newton's notebooks online before.

Vysero 6y ago

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.