What happened between the Spanish arabs and Europe was before what happened

to(mostly) Italy and the fall of the Byzantine empire. The dates are obvious;

the Arabs more or less fell around 1000 A.D.; the Byzantine empire fell around

1500 A.D.

The interaction between the Spanish Arabs and the Europeans

caused a pre-renaissance; it caused the crusades; it caused the Inquisition; the

knowledge that came out of Arab spain could only be copied so far; meaning, only

so many people could learn of all this knowledge; it was more for geeks like

Adelard who went out of their way to go into spain; real knowledge seekers; not

just guys who get hit on their head when some cultural force finds them. But,

this initial renaisance also was more about translation and commentary such as

Francis Bacon(or Roger bacon; i still get those two confused); they learned

about Aristotle and Plato; they certainly saw Euclid's elements; but . . . have

you looked at Euclid's elements? If you were even an Arab back then much less

some biblical copyist who first opens up a copy of Euclid's "Elements", you'd

probably burn it, throw it away, or loose it on the shelves(they didn't have any

form of order in the libraries of the christian monastaries back then; books

just got lost!); i mean, this thing starts out with a list of axioms, then

definitions; at which point, your like 'so what!?' Then, it goes straight into

theorem proving! If you don't have a clue about what deductive logic means and

how it works, you are lost by theorem one! Your only clue is Aristotle and

Plato; and, if you havn't gone through their mass of works and found the clues,

and gone through a period of reflection to note, dah! deduction! That must be

what all this theorem proving is in Euclid's "Elements", then, once again, that

literature was completelly meaningless to the majority of Europe . . .

anybody!

And then, the black plague destroyed any hope of this initial

renaissance to capitalize on this knowledge from Arab Spain. Then came a second

influx from the Byzantine empire!

What happened when the Byzantine empire

fell was that the printing press happened about the same time, the scholars from

the Byzantine empire knew what deductive logic meant in the first place. And, in

the end, Galileo was influenced to do his studies by all the scholars that came

before him about Aristotles physics. If it hadn't been for all that time and

effort to figure out what all that Greek knowledge meant, Galileo would have had

to spend all that time figuring out what all that meant.

This is why the

killing of Archimedes was such a tragedy; the Romans did not learn from him when

they could have. This is why the destruction of Athens was also a tragedy;

because, you can't just expect some culture that hasn't figured these things out

to figure it all out; it has to go through periods of development to even begin

on knew research, or to even understand something like the mathematics the

Greeks invented(a deductive approach to all the Babylonian mathematics; and

then, the Greeks went a little bit farther; although, from the Babylonian

viewpoint . . . except for the algebra of the Babylonians . . . was far beyond

anything they could ever imagine; trig, number theory . . . the infinitude of

prime numbers . . . conic sections, irrational numbers, Archimedes proof and

calculation as oppossed to mere measurement of Pi, some intial work on

integrations and differentiations, the balancing point of irregular shaped

surfaces, the use of higher curves to solve the three Greek classical problems;

the trisection of angles, the doubling of cubes, the squaring of the circle).

This is the tragedy of the dark ages; this is why the dark ages were the dark

ages!

I thought I'd put a bit of my review of Van Der Waerden's "Science

Awakening" in here to update my account of Archemedes!

'Most people have

heard of Archemedes; but, do they have any idea of why he is so highly regarded?

Highly douteful! For one thing(and this goes for all Greek mathematicians back

then), all mathematics was done in terms of an awkward geometrized

algebra(Eudoxes was highly regarded by Greek mathematicians like Plato because

he perfected it and made is fairly workable). So, when Archemedes proves

deductively how to calculate Pie(a double reductio absurdium proof!), he then

calculates it with this geometrized algebra! It's hard to appreciate the

difficulties that are brought in here that leaves those who look at this

absolutely intellectualy drunk; it's like when you show the trigonometry(and

Archemedes practicaly shows the way towards developing a trigonometry in his

calculation of pie) that comes from an isosceles triange; you find that one of

the sides requires a radical expression; now imagine having to do this with

geometrical algebra, and you should be feeling just . . . ; I've yet to explain

what makes this all amazing actually. The Babylonians at least observed that two

different sized circles have the same ratio of diameter to circumference. This

was proved in Euclid's Elements. C=(pie)D. But, what is pie(meaning the

numerical value)? As I've stated, Archemedes relates the area to the legs of a

right angle triangle. Basicaly, he's related the two dimensional property of

area to that of the one dimensinal constant of the circumference. Now, because

the area of a triangle is 1/2hb, we get (pie)r(squared). Archemedes goes on to

calculate the circumference of the circle as already described. But, then

Archemedes goes on to use much of the same strategy he used to show the way

towards calculating pie to calculating the area and volume of a sphere; he

relates the second and thired dimensions together! Euclid's "Elements" shows the

plane and solid geometry of his day before(and much else like number theory);

but, the solid geometry is flawed in areas; Archemdes comes up with this

dimensional analyses solution to the theory of solid geometry! That alone puts

him above most!

But! Archemedes goes on to do primitive calculus(using

geometrical algebra), he calculates the center of balance of odd shaped

figures(like obtuse and scalene triangles), he uses arithmetic and geometric

progressions to handle large numbers(and to deal with much of his irregular

surface results); he solved the area of triangles which is normaly attributed to

Heron(trigonomery students should know what I'm talking about). Some stuff that

was knew to me from Van Der Waerden that I didn't know before are Archemedes

construction of the hexagon which turns out to not be constructable by

straightedge and compass(a Plato restriction which isn't mathematical valid but

does have some mathematicaly interesting things; more on this later); Archemedes

constructs it with conics. Van Der Waerden relates that Pappas notes that

Archemedes constructed and explored much semi-regular solids(you can't be a

mathematician or consider yourself a mathematica enthusiasts if you don't know

what I'm talking about here). By this time, you should be putting Archemedes up

there with Gauss and many others!'

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