17thC math/science/natural-philosophy?
Jorn Barger
knowledge-representation (400k, 2000 links):
http://www.robotwisdom.com/ai/timeline/0000.html
But lately I've been zeroing in on the 17thC flowering
of natural philosophy:
http://www.robotwisdom.com/science/natphils/index.html
...trying to see exactly how the paradigms changed so
dramatically between (eg) Copernicus and Newton...
The telescope in 1609 seems to have been the real
turningpoint, allowing anyone with access to see wonders
unknown, that visibly overthrew both Aristotle and the
Bible.
But simultaneously mathematics came into fashion (perhaps
mainly due to Descartes' personal charisma?) where before
it had been mainly for engineers. And in England at least,
the group that founded the Royal Society managed to promote
the free exchange of ideas despite the many gigantic egos
pursuing personal glory...
There's a series of names that each contributed on multiple
levels, but the earliest (pre-Galileo) ones were obviously
floundering compared to Galileo-et-seq:
Copernicus - Paracelsus - Peter Ramus
John Dee - della Porta - Giordano Bruno
Thomas Harriot - Francis Bacon - [here the miracle occurs]
Galileo - Kepler - Hobbes
Mersenne - Gassendi - Descartes
Kircher - Wilkins - Wallis
Boyle - Huygens - Hooke - Newton
(The St-Andrews site has useful biographies of most of
these, but I tweaked my own version of their index-page:
http://www.robotwisdom.com/science/math.html )
One thing I'm unclear on is how much NON-applied math
was being explored pre-Descartes, and whether Euclid's
geometry was conventionally seen as superior to practical
calculations.
I've tried to find a timeline or history of *applied* math
that explains what problems were being attacked (pre-
Descartes, especially)-- eg when the Greeks explored conic
sections, were they trying to solve engineering problems,
or just enjoying the beauty of the puzzles?
[My experience on netnews is that many of the followups
will just say "Read this book" and I'm not _completely_
averse to those suggestions, but my library-access is
pretty limited so I'd appreciate hearing some detail
about what the books cover, so I can decide if they're
really worth the extra effort.]
George Cox
>
> ...
>
> One thing I'm unclear on is how much NON-applied math
> was being explored pre-Descartes, and whether Euclid's
> geometry was conventionally seen as superior to practical
> calculations.
Note that Euclid lived in Alexandria in Egypt. In Egypt geometry _was_
a practical matter: field boundaries were washed away by regular floods.
>
> I've tried to find a timeline or history of *applied* math
> that explains what problems were being attacked (pre-
> Descartes, especially)-- eg when the Greeks explored conic
> sections, were they trying to solve engineering problems,
> or just enjoying the beauty of the puzzles?
Interesting: "focus" is a Latin word not a Greek one. Focus = fireplace
= the hot spot when the suns rays fall on a parabola parallel to its
axis.
GC
George Cox
>
> ...
>
> One thing I'm unclear on is how much NON-applied math
Pre-Descartes was the applied/pure distiction made at all? Was it even
made for a long time after Descartes?
GC
George Cox
>
> Jorn Barger wrote:
> >
> > ...
> >
> > One thing I'm unclear on is how much NON-applied math
> > was being explored pre-Descartes, and whether Euclid's
> > geometry was conventionally seen as superior to practical
> > calculations.
>
> Note that Euclid lived in Alexandria in Egypt.
_If_ there was a Euclid, that is. He might have been the Bourbaki of
his day.
GC
Ken Moore
<jo...@enteract.com> writes
>One thing I'm unclear on is how much NON-applied math
>was being explored pre-Descartes, and whether Euclid's
>geometry was conventionally seen as superior to practical
>calculations.
Reading Galileo's Dialogue, I am struck by the way he could use
geometry, very ingeniously, where nowadays we would use algebra or
calculus, of which the first was primitive and the second non-existent
in his day. There is a particularly striking demonstration of the law
of the lever, which he demonstrates (it is probably just short of a
proof, by modern standards of rigour, by I will be interested in your
reactions) by a thought experiment. He considers a beam, supported by a
central pivot, holding up a rectangular prism of constant density.
Symbolic expressions indicate horizontal distances from the origin at
the left:
y
___________________________________________
|_____________________*_____________________| Beam
. | : | .
. | : | .
._______|______:_____________|______________.
| + | Prism
|___________________________________________|
0 x 2x 2x+(y-x) 2y
= 2y-(y-x)
= x + y
Initially, the prism is whole, and the beam supports it by wires which
are indicated by the dots. The colons indicate that the wires are
doubled. This system is in equilibrium, by symmetry (i.e. the beam
remains horizontal on its single pivot at y). The prism is cut through
at 2x, two wires supporting the left, two others the right portion. The
length of the latter is 2y - 2x. The cut being of infinitesimal
thickness, no masses move, and the equilibrium of the beam is
unaffected. Next the four wires (dots and colons) are replaced by two
(vertical bars), at x and x+y. The equilibrium of the parts is
unaffected, since they are now supported at their centres. With loads V
and W in the wires and linear density s of the prism, we have:
V = 2 * s * x, and W = 2 * s * (y - x), giving
W/V = (y-x)/x
On the beam, the left lever arm is y - x, and the right lever arm is
(x+y) - y = x, so the loads are inversely proportional to their lever
arms, which is the lever formula.
Galileo's demonstration of this is much more striking than mine, because
he uses geometric language where I revert to algebraic, but I found the
basic concept very interesting, since I had not seen it in several
years' study of mechanics.
--
Ken Moore
k...@mooremusic.org.uk
Web site: http://www.mooremusic.org.uk/
I reject emails > 300k automatically: warn me beforehand if you want to send one
Dan Drake
[It's a big and complicated question, and I don't know who's an expert in
it. Till someone stands up, here are a couple of things.]
> One thing I'm unclear on is how much NON-applied math
> was being explored pre-Descartes, and whether Euclid's
> geometry was conventionally seen as superior to practical
> calculations.
The latter is difficult to answer, because the practical people used
Euclid; see below. The universities had chairs in mathematics --
arithmetic and geometry were in the Quadrivium -- and these generally had
no connection with practical applications.
>
> I've tried to find a timeline or history of *applied* math
> that explains what problems were being attacked (pre-
> Descartes, especially)-- eg when the Greeks explored conic
> sections, were they trying to solve engineering problems,
> or just enjoying the beauty of the puzzles?
Even Euclid has a more complicated history in post-classic Europe than you
might think. Ken Moore has pointed out (in a post I haven't read
carefully yet) the remarkable things Galileo did with geometry and
proportions where we'd crank it out with algebra. Mostly, not always, the
old way is much harder, and it's serious work even to follow the arguments
based on the theory of proportion. Here's the odd part: Even those
arguments were not available 100 years before Galileo.
From the 12th to the 16th century there were two Latin editions of Euclid
that were pretty much standard. From internal evidence, they were both
taken from the same Arabic text (now unknown) and both had serious errors
in the chapter on continuous proportions, which rendered that theory
unusable. There was, in fact, an accurate translation from the Greek at
about the same time, but it remained obscure. Meanwhile, the kind of
arguments Galileo was to make were not possible.
In the 16th century new translations of Euclid appeared, from uncorrupted
Greek texts, and sometimes into vulgar languages like Italian; good
commentaries followed quickly. In the second half of the 16th century
there were many books published on science and math, and the use of the
"new" theory was widespread. If you wanted to go wild, you could argue
for these new translations of Euclid as essential to the scientific
revolution after 1600. Anyway, academics were not the main motive force
in getting out these corrected versions of Euclid.
One of the most famous of the people involved was Nicolo Tartaglia, who
also produced a solution of the cubic equation (called Cardan's solution,
but that's another story). His commentary made clear the difference
bewteen the correct and the spurious forms of the theory. Tartaglia was
mostly self-educated. "From youth he had been teaching mathematics to
practical men. Respect for academic tradition and authority did not cloud
his mathematical judgment." [1]
In fact, there is a book from around 1600, attributed to Galileo, which
takes the form of a dialogue between surveyors in which they mock
academics. Surveyors were practical men who used Euclid's geometry
routinely.
So there was practical math in active use before 1600, *and* it employed
Euclid. And the amount of contact at that time between the academics and
the practical sorts was about what you'd expect: very little indeed.
[1] "Euclid Book V from Eudoxus to Dedekind", Stillman Drake, reprinted in
_Essays on Galileo and the History of Science_ vol. 3 pp 61-75. See also
"Early Science and the Printed Book: The Spread of Science beyond the
Universities" in the same volume.
--
Dan Drake
d...@dandrake.com
http://www.dandrake.com
Amount that Congress proposes to provide this year
for reconstruction in Afghanistan: $435,000,000
Amount that the Bush government proposed: $0.00
Amount that Iran has pledged: $900,000,000
Dan Drake
> Reading Galileo's Dialogue, I am struck by the way he could use
> geometry, very ingeniously, where nowadays we would use algebra or
> calculus, of which the first was primitive and the second non-existent
> in his day. There is a particularly striking demonstration of the law
> of the lever, which he demonstrates (it is probably just short of a
> proof, by modern standards of rigour, by I will be interested in your
> reactions) by a thought experiment.
Clarification for anyone wanting to look it up: This is not in the
Dialogue, but in Two New Sciences, at the very beginning of the Second
Day. It's an ingenious demonstration, and also shows how much better off
we are thanks to Descartes and all.
Ken Moore
<d...@dandrake.com> writes
>Clarification for anyone wanting to look it up: This is not in the
>Dialogue, but in Two New Sciences, at the very beginning of the Second
>Day. It's an ingenious demonstration, and also shows how much better off
>we are thanks to Descartes and all.
Thanks for the correction.
Ken Moore
<d...@dandrake.com> writes
>The latter is difficult to answer, because the practical people used
>Euclid; see below. The universities had chairs in mathematics --
>arithmetic and geometry were in the Quadrivium -- and these generally had
>no connection with practical applications.
Was any of the Quadrivium of practical use? The music was theoretical
stuff about integer frequency ratios giving consonance, IIRC, and the
astronomy was presumably Ptolemaic, so it might give you a convincing
date for Easter, but not much more. Grammar, rhetoric and logic (in the
Trivium) look pretty much like the usual tools of non-scientific
intellectuals aiming to be convincing whether or not they are correct.
Jorn Barger
> [It's a big and complicated question
I appreciate that confirmation!
> Here's the odd part: Even those
> arguments were not available 100 years before Galileo.
I think this is a key: the 16thC showed gradual solid progress
in building the mathematician's toolkit, along with a gradually
increasing belief that careful observation and mathematical
reasoning could solve mysteries in astronomy, ballistics,
optics, hydrodynamics, acoustics, chemistry, medicine, etc.
The telescope, Galileo, Kepler, Gilbert, and Bacon emerged
together when the time was ripe, and Descartes tied most of
this progress up into a neat-but-mostly-false package that
gave the next generation a rallying point and target...
Dan Drake
> In article <vhIsdqY67dTD-pn2-2tr1FndPwBz9@localhost>, Dan Drake
> <d...@dandrake.com> writes
> >The latter is difficult to answer, because the practical people used
> >Euclid; see below. The universities had chairs in mathematics --
> >arithmetic and geometry were in the Quadrivium -- and these generally had
> >no connection with practical applications.
>
> Was any of the Quadrivium of practical use? The music was theoretical
> stuff about integer frequency ratios giving consonance, IIRC, and the
> astronomy was presumably Ptolemaic, so it might give you a convincing
> date for Easter, but not much more. Grammar, rhetoric and logic (in the
> Trivium) look pretty much like the usual tools of non-scientific
> intellectuals aiming to be convincing whether or not they are correct.
Well, I take a somewhat more charitable view of the Trivium: even
scientists have to communicate, and to think clearly, for that matter, as
the humanists keep telling us. If you seek negative examples, look about
you on the Internet. Taken in good faith, grammar and logic (even
rhetoric, in some sense) are prerequisite to the study of anything. Not
that the amount of good faith and good sense in teaching things was any
greater in 1550 than now, but perhaps it was not less.
For the Quadrivium I make no case. Remarkable, how radical it was when
some guy (Vincenzo Galilei, as it happens) noticed that a ratio of 5001 to
3001 was not a hideous dissonance because of being so far from a ratio of
small numbers, but a nice fifth.
Jorn Barger
> "Dan Drake" <d...@dandrake.com> wrote in message news:<vhIsdqY67dTD-pn2-2tr1FndPwBz9@localhost>...
> > arguments were not available 100 years before Galileo.
>
> I think this is a key: the 16thC showed gradual solid progress
> in building the mathematician's toolkit, along with a gradually
> increasing belief that careful observation and mathematical
> reasoning could solve mysteries in astronomy, ballistics,
> optics, hydrodynamics, acoustics, chemistry, medicine, etc.
Also, dissatisfaction with Aristotle's paradigms was widespread
during the 16thC, and Lucretius's atomism became available as
a clearly superior alternative.
Translations of Archimedes, Apollonius, and Euclid led to
better math, and Vieta built on Diophantus and the Arabic
algebraists. Increasingly precise astronomical observations
forced Kepler to the recognition of elliptical orbits,
which required much more sophisticated math to calculate.
Clavius raised the Jesuits' standards for math education,
with Descartes, Gassendi, and Mersenne as early beneficiaries.
These three became a nucleus for mathematically sophisticated
natural philosophy in Paris in the 1620s, with further
inspiration from Bacon and Galileo. By the 1640s, Wallis and
Wilkins had started a parallel group in London based on Bacon,
Galileo, Gassendi, Mersenne, and Descartes.
As techniques of measurement and calculation were successfully
applied to many fields, the fields where they _failed_ (like
astrology and alchemy) were simply dropped.
Robert Ehrlich
Just went to your web page. What an interesting and valuable resource! Thank you.
Bob Ehrlich
Jorn Barger
> Just went to your web page. What an interesting and valuable resource!
Thanks! I've been experimenting with showing intersecting lives
in parallel. Here's a very different example, for Dublin
literary gossip c1900 (Yeats, Maud Gonne, Joyce, Aleister
Crowley, Synge, Havelock Ellis, George 'AE' Russell, etc):
http://www.robotwisdom.com/jaj/gossip.html
The idea is to show how they viewed each other (and to liven
it up with as much salacious gossip as possible).