Translating Newton’s Principia
principia.blog
principia.blog
As a woman, she didn't have any university education, but that wouldn't have included calculus anyway. Her problem was the modern concepts, not the old "geometric" way of reasoning about quantities. So she had to write a bulky Commentaire that really was her magnum opus
As a woman, she was completely accepted as an equal by scientists of her time, but treated rather dismissively by later historians of science. Clearly, history doesn't obey Newtons's first law: it doesn't progress in a straight line.
He has many lectures about it, I like this one: https://www.youtube.com/watch?v=D5in5EdjhD0, one of the best lectures ever.
(I also despise his continual use of the rhetoric device during his lecture of claiming that a particular philosophical debate has been settled and the "evident" correct answer is his own position on the subject. He keeps using that turn of phrase where his opinion is presented as the only tenable one by an intelligent thinker. He always does that. I think it's a weak trick so silence critics.)
It's not three pages long in Clarke (nor in Heath), and either way, the argument is only 10 statements long.
> But the reader who takes the trouble to decode the proposition will see that it is trivial primary school arithmetic.
This is also not super obvious from Clarke or Heath. I'd love to know how the author would render this proposition trivial to a fourth grader.
To be sure, the mathematics of Euclid is a very foreign country from what's commonly taught in schools today. But Newton's thought is firmly rooted in that foreign country.
Looking at the diagram in your link, and labelling AE x, EB y, CF a and FD b, "Let AE, EB, CF, and FD be magnitudes proportional taken separately, so that AE is to EB as CF is to FD. I say that they are also proportional taken jointly, that is, AB is to BE as CD is to FD.", means :
If x/y = a/b, then also (x+y) / y = (a+b) / b.
To prove this is true, operate on the second equation to make it the same as the first:
1. Expand: x/y + y/y = a/b + b/b
2. Subtract 1 from both sides: x/y = a/b. QED.
Notation really makes all the difference!
It's not, like, rocket science, but it's not trivial either.
Edit: Or I suppose "tabooing"[0] the word trivial would have helped – it was the trouble-maker.
[0] https://www.lesswrong.com/posts/WBdvyyHLdxZSAMmoz/taboo-your...
This is from the Todhunter edition of Euclid's elements. the proof occupies precisely three pages. in post-Euclidean notation, if a/b = c/d then (a+b)/b = (c+d)/d. For young children replace the letters by small positive integers.
To convert Euclid's formulation to the symbolic formulation requires turning a magnitude into a real number, and defining the division of two real numbers. Euclid's proof avoids these difficulties. His magnitudes are the lengths straight line sections. He can decode a/b = m/n where m and n are positive integers as meaning na = mb. And similarly he can define a/b > m/n and m/n > a/b. So now the statement a/b > c/d can be decoded as the existence of positive integers m and n such that a/b > m/n > c/d. Finally the statement a/b = c/d is decoded as asserting that both a/b > c/d and c/d > a/b are false.
In modern jargon, Euclid is using the Dedekind cut definition of a real number. Using this definition, and working from first principles, the proof unsurprisingly requires three pages.
He knows strictly no Polish, but ... he has a Polish dictionary.
Not a Polish to English dictionary, mind you ... a Polish to Polish dictionary.
Historical, religious, scientific, political context carries a ton of unspoken information, and when you try to read a text without the context ...