Salut mes amants des maths!
The class is once again not lined up with the tentative schedule, so we'll be covering some topics again and looking at some details that are different from the ones previously seen. According to historical evidence we have in relation to the Greek, there is an obvious shift in the focus and objectives of Greek mathematicians; they were beginning to incline more towards the idea of proofs and moving away from numerical calculation. This shift is further exemplified in the mid-fifth century attempts to solve two geometric problems, problems that went to on to occupy Greek mathematicians for centuries: the squaring of the circle (already attempted in Egypt) and the duplication of the cube (as noted in the oracle). The numerous attempts to solve these problems as well as the additional one of trisecting an angle support the belief that Greek mathematics was great focused on geometrical problem solving, which was relevant with respect to the various theorems found in the major extant works of Greek mathematics.
Hippocrates of Chios (mid-fifth centure BCE) was the first to attempt both geometrical problems. It is thought that Hippocrates realized the similarity between doubling the cube (picture above-left) and the simpler problem of doubling a square of side a. So, he was the first to reduce the problem of doubling the cube of side a to trying to find two mean proportionals b, c, between a and 2a. Since, if a : b = b : c = c : 2a, then a^3 : b^3 = (a : b)^3 = (a : b)(b : c)(c : 2a) = a : 2a = 1 :2 and b^3 = 2a^3. However, it was not possible to construct the two mean proportionals with only a straightedge and compass.
Similarly, Hipprocrates made progress with respect to the problem of squaring a circle, basically, by showing that certain lunes (picture above-right) could be "squared," that is to say, their areas could be shown equal to certain regions bounded by straight lines.
In addition to these geometrical problems, lecture covered more of the lives of two very well known mathematicians: Plato and Aristotle.
Plato (429-347):
- From his youth aspired to participate in public affairs
- Had a good education
- Met Socrates at age 20 and studied under him for 8 years
- Once Socrates was condemned to death in 399 BC, Plato and company retired briefly to Megara to pursue the Truth and master the "art of argument"
- He traveled about Greece on to Egypt, Southern Italy, and Sicily.
- He also studied under Archytas, improving his knowledge about Pythagoras and his findings.
- He returned to Athens in 388 BC.
- Upheld the philosphies "Let no man enter who is ignorant of Geometry" & "Everything is in flux, nothing is stable"
Aristotle (384-322):
- He was Plato's most famous student
- In 335 BC Alexander put Athens under Macedonia
- Created his own school "Lyceum" where he lectured for 12 years and introduced the concept of written exams.
Well, that's all for now! Until next time, mes amis!
Bisous!
-MB
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