I'm always surprised when reading the notes of scientists and mathematicians working in previous centuries to see just how steeped they were in synthetic geometry. This was taken to an extreme in the case of the Principia, but one can't read Gibbs or Maxwell either without realizing that they felt Euclid in their bones in a way that few people do today, with possible exceptions for mathematicians trained under the Soviet system.
I don't know-- usually the answer to such questions about academic priorities is "because it was cheaper", but they just seemed to emphasize geometry much more at the K-12 level. It's a generalization, of course, but a pretty robust one. One of my professors came from Kazakhstan, and once casually remarked that a certain problem on a homework set was "impossible unless you were Russian", since the proof was easy if you knew a certain proposition from Euclid, but extremely tedious without it.
EDIT: this interview with Izaac Wirzsup comparing the Soviet and US systems confirms my prejudice:
Another extremely harmful feature
of [the US] school mathematics programs
is that only about half of our students
take geometry, and for only one
year, generally in a concentrated high
school course. Students cannot be
expected to master the material taught
in this way. Moreover, they are not
being taught solid geometry, and they
rarely have a workable perception of
three-dimensional space, which is so
essential for studying science,
technical drawing, or engineering.
Soviet children study geometry
extensively for ten years, including
two years of solid geometry.
Jut from my own common sense it seems like geometry is important for understanding the relationship between abstract things and concrete things. It's easily understandable that shapes are described by geometry and it seems obviously useful. The square footage of a house. The volume of a bath.
If you try to describe what Calculus is or does, it's abstractions of abstractions. Rates of change or 'angle of a curve for a certain values. I think it's hard for students to see this as something useful or even see how it's a description of the world that opens up ways of understanding it.
I don't see calculus as an abstraction of abstractions. The fundamental idea is completely geometric: "break the domain of a problem into a bunch of pieces that can be easily described and related (e.g. by physics) then put the pieces back together." Time is a first-class dimension. Abstractions only enter the picture when you want to separate the problem of picking a mesh from the problem of representing mesh elements.
Differential operators perform the task of "breaking into pieces" in a mesh-invariant way. Differential forms are mesh-invariant pieces. Integration is the mesh-invariant description of putting the pieces back together.
It's convenient that differential forms can be interpreted physically (by normalizing, associating with geometric elements, etc) but I'd hesitate to associate them with any single physical interpretation (e.g. rates of change) because doing so de-emphasizes the generality of the approach; you can have a rate with respect to distance, area, or volume just as easily as a rate with respect to time.
Leibniz notation makes the hop from the geometric approach to the "operator that maps a function to a function" approach seamless, and since the latter description isn't nearly so intuitive, I've always suspected that the geometric approach could profitably be taught first.
> One of my professors came from Kazakhstan, and once casually remarked that a certain problem on a homework set was "impossible unless you were Russian", since the proof was easy if you knew a certain proposition from Euclid, but extremely tedious without it.
Do you remember what was the problem, by any chance?
I don't remember-- it must have been either differential geometry or topology, but I think the theorem in question was the inscribed angle theorem: http://www.proofwiki.org/wiki/Inscribed_Angle_Theorem. Not a difficult theorem, but you had to know it well to be able to see the application immediately.
There is an excellent chapter called 'on teaching of geometry in Russia' [1] discussing how Geometry stayed important even after west moved away from euclid.
The page uses SeaDragon (Ajax edition) from Microsoft former "Live Labs". It's the same infinite zoom technology that powers also Photosynth and DeepZoom both from Microsoft.
Reminds me of an incident (I forget where I read about it) where Dirac is giving a lecture in Europe, someone asks a question & Dirac has to start working it out on the board - and Ehrenfest turns around and yells: Kids, now we can see how he does these things!, or something like that.
I wished they would show notebooks like these early in school: "Look, this is the pinnacle of human thought and it's full of corrected mistakes, scribbles, attempts and mnemonics".
Thanks to Cambridge for making this available to the general public, under a CC 3.0 license! The images are crystal clear, and easy to read. Props to Newton, as well :)
Considering the original works are well outside of copyright, applying CC 3.0 to the images themselves really isn't especially grandiose of them. Though I appreciate their not trying to assert control.
IIRC getting unencumbered high-resolution images of, for example, famous paintings in public galleries is still far from assured, so (grading on the curve) it's not to be underestimated.
This is truly fascinating - a peek into the earliest thinking of arguably one of the greatest scientific minds who ever lived. It never ceases to amaze me that Newton, almost single-handedly and working all by himself pretty much laid the foundations for much of science for the next several hundred years. And in all likelihood, it all started with the thoughts he formulated while writing in this little notebook!
This is simply completely untrue. Newton synthesized a lot of earlier results in physics and mathematics into a unified framework, and he deserves a lot of credit for that obviously, but nevertheless it was a synthesis and not something done "single-handedly and all by himself", in fact you are contradicting Newtons famous own words that he was "standing at the shoulders of giants".
Isaac Barrow, Newtons teacher, had already discovered the rudiments of calculus:
And so on and so forth, lookup Kepler, Galileo, Huygens, Hooke, Barrow, Descartes, Fermat and so on in the Wikipedia, or better read any serious scholarly history of physics about this period, this is only scratching the surface of people whose work Newton very directly built upon. There is no synthesis to be done without a period of establishing a great many of particular results.
Easy to dismiss genius as 'obvious', don't err too far on that side! Newton was certainly the greatest Scientist in a couple of centuries, instrumental in advancing three areas of Science. Not single-handedly certainly but nothing we ever do works that way.
Hooke, btw, was an enemy of Newton. Hooke would publish books of drivel about 'what if' physics worked by a certain equation; Newton derived why physics Must work a certain way. 'Hookes Law' of springs should be another of Newton's laws, but Hooke had actually written down that spring law with no proof and no motivation, but nevertheless published first. Newton argued (correctly) why springs worked that way. So a basic rule of physics has the name of someone who did no physics.
I do not deny his genius, but popular accounts of scientific work certainly tend to overemphasize individual contributions and downplay the overall incremental progress a generation of scientists of various statue can make. It has the practical effect of discouraging young people from doing actual scientific work on small, particular results, instead they feel they should immediately have an idea for a grand theory of everything.
Of course, there is no denying that Newton built upon much of the work of his predecessors - notably Galileo's early experiments in motion, as well Kepler, Huygens, Barrow and others.
But there were a significant number of original ideas, research, analysis, and experiments that added to, refined and extended their work, and most importantly formalized it all into a coherent, unified corpus in the form of the Principia. And that work was done single-handedly, all by himself, working alone, and is considered seminal in laying the foundations of science for the next three hundred years. To call it just a grand synthesis, severely discounts the magnitude and impact of that achievement.
I've always found it strange that Newton is known for saying something so humble, yet when it came to calculus, he acted like a raging socio-path, abusing all of his political might in a successful effort to discredit Leibniz.
I recall reading a paper once that made the case that Newton was speaking sarcastically, mocking Hook, if I remember correctly. Certainly that interpretation seems a better fit for his character.
I recently read a quote on "Quicksilver" (Stephenson's novel) about his imaginary character "Daniel Waterhouse":
"Daniel was angry with God. God had implanted on him a passion for natural philosophy. He wanted to be one of the greats. But God brought him on earth the same era with individuals like Hooke, Leibniz and Newton. What where the chances?"
At this point in the novel, only Daniel has a clear view on Newton's genius, Hooke was renowned and Leibniz was not into mathematics (he studied to lawyer first, then turned into mathematics according to the novel, but knowing Stephenson I think it's true).
With all due respect to those comments, would it really be a loss? I really, really don't want this comment end up taken as a personal attack on those commenters, that's not the point or intent. Rather, the point is let's rationally, without emotions about our egos, question what is the point of those comments. There is none. If you like something, you upvote, and iff you have any new information to add, add it. It's bearable in a submission with a low number of comments such as this, but in large ones, there tends to be also a large amount of such redundant comments. There is simply no point in posting such comments---they don't get you any karma (I hope!), they don't add any information useful to anyone, and they make it harder to go through the whole comment set.
The page reads (from the manuscript image, interpreting the abbreviations) "[...] an arithmetic progression increasing from an unite by 1 composeth triangles by 2, composes squares by 3, composes pentangles by 4, hexangles &c. as 1.2.3.4.5.6. composes the triangles [...]".
In the "Transcription (normalised)" this is
"[...] an arithmet: progres: increasing from an unite by b=2 formula composeth triangles. by a=5/3, composes squares. by y=22/61, composes pentangles. by x=33/61, hexang: &c as 1 compose the triangles 2 &c likewise 3 compose 4 &c So 1.2.3.4.5.6. compose the quintangles [...].
It appears nearly all the 'MathML formulas' are wrong? There is also a textual transcription error "quintangles" which should read "triangles".
This is the only page I looked at. The "Transcription (diplomatic)" appears to bear the same errors. If this page is typical I hate to think how the hard to read or complex mathematical pages have been rendered in transcription.
Edit: I've just noticed that the erroneous MathML formulas are correct renderings of other expressions on the same page, this is probably a coding/markup error?
Thank you for spotting this. The transcriptions for the Newton papers are supplied by a third party (The Newton Project, http://www.newtonproject.sussex.ac.uk) so we will pass this onto them.
At a quick glance at a few pages it seems browsers are rendering various sections differently which may account for some errors.
Thanks again,
Cambridge Digital Library
Initially tried using WinHTTrack on the document folder but DL rate was only 25kB/s. So if mentally you resort to sublimated-OCD mode the whole notebook can be downloaded, one page at a time, then combined and cropped in Acrobat for offline reading in about 1/2 - 3/4 hour. Final size = 80MB (and a very nice .pdf it is to have in your archives!)
Curious to see the use of Y^e and Y^t for the words 'the' and 'that' in the mid 1600s. Figured they would have reverted to the 'thorn' letter rather than the earlier French printer's substitute.
A casual but chronological search of the material suggests Y^e was colloquial but in the process of fizzling out. Boyle, unlike the young Isaac, also made extensive use of the descending 's' (looked like an 'f' without the horizontal bar). As a kid, after seeing all these 'f-s' in ancient books you probably wondered whether people back then collectively spoke with a lisp. [fun diversion]
