Late Greek Math

Nichomachus and Elementary Number Theory

Nicomachus likely studied in Alexandria due to the Pythagorean influence in his work
Two surviving works: Introduction to Arithmetic and Introduction to Harmonics
Arithmetic: explains Pythagorean number philosophy (600 years after Pythagoras)
Book I: classified numbers
Euclidean algorithm given, examples of perfect numbers, no proofs
Book II: plane and solid numbers, proportion

Diophantus and Algebra

Alexandrian mathematician
Introduced symbolic abbreviations for terms
Aware of rules for multiplying with the minus (i.e., subtraction; no negative numbers)
Broke with Greek tradition and considered powers higher than 3
Diophantine equations common in intro number theory courses

Examples

Problem I-1: To divide a given number into two having a given difference.

Problem II-11: To add the same (required) number to two given numbers so as to make each of them a square.

Problem A-25: To find two numbers, one a square and the other a cube, such that the sum of their squares is a square.

Problem I-1: \(2x + b = a\), where \(a\) is the given number and \(b\) the given difference; the required numbers are \(\frac{1}{2} (a-b)\) and \(\frac{1}{2}(a+b)\)

Problem II-11: Diophantus took the given numbers as 2 and 3; if the required number is \(x\), he needs \(x+2 = u^2\) and \(x+3 = v^2\) to be squares. He solves it by taking the difference between the two expressions and resolving it into factors, then adding or subtracting the factors.

Problem A-25: That is, find \(x,y,z\) such that \((x^2)^2 + (y^3)^2 = z^2\). He set \(x = 2y\) (2 is arbitrary) and performed the exponentiation to conclude that \(16y^4 + y^6\) must be a square, which he took to be the square of \(ky^2\). So \(16y^4 + y^6 = k^2 y^4\), and \(y^6 = (k^2 - 16)y^4\), so \(y^2= k^2 - 16\). He chose \(k^2 = 25\), so that \(y = 3\). Thus the desired numbers are \(y^3 = 27\) and \((2y)^2 = 36\).

Can generalize the solution: Take \(x = ay\) for any positive \(a\). Then \(k\) and \(y\) must be found so that \(k^2 - a^4 = y^2\), or so that \(k^2 - y^2 = a^4\). He had already demonstrated in Problem II-10 that one can always find two squares whose difference is given.

Hypatia (355-415)

Christianity became the state religion of the Roman Empire by 397
Hypatia, daughter of Theon of Alexandria was a respected mathematician and Platonic philosopher
Maintained non-Christian religious beliefs
Wrote and commented on many mathematical works
Cyril, bishop of Alexandria, spread rumors that she practiced sorcery
Dragged from her chariot and murdered during Lent

The End of Greek Mathematics

Question for discussion: What does it take for mathematical study to flourish?
Babylonians invested in mathematics to help train the minds of future leaders
Greek systems of government provided philosophy and mathematics with encouragement and space to flourish
Even so, there weren’t that many who understood theoretical mathematics well (approximately 1000 over 1000 years)
Political strife around the first century destabilized the mathematical enterprise
Romans did not value mathematics in the ways the Greeks had
Continued in Egypt, particularly due to existence and influence of the Museum and Library of Alexandria until the late fourth century

An answer to the question: Curiosity (about matters practical or not), but also resources

At this point, mathematics in Europe goes fairly quiet for the next 600 years or so.

We’re going to head to Asia, first to China, then India, before circling back to Islamic mathematics, which incorporated elements of Indian math (particularly the Hindu numerals) which in turn fed into a reenergized European mathematics around 1000-1100.

But I want to point out that there’s a lot that we’re skipping. I want to emphasize that one could teach an entire course about non-Western mathematics. And if you’re looking for a topic for your historical term paper, understanding how different cultures approached, say, trig, etc., would be a pretty interesting one! I have resources.

Our lack of in-depth coverage of these topics does not mean I don’t think they’re important or valuable, but a focus of this course is understanding the history of Western school mathematics, and so our focus will tilt back toward the West pretty soon.