Final chapters of Greek mathematics

By Ptolemy's time

• the entire eastern Mediterranean was part of the Roman Empire
• the local rulers and romans left the local language and culture intact.

and the great Cicero admitted that the Romans were not that interest in math. 

The last of the Greek mathematians were

1. Nicomachus of Gerasa
2. Diophantus of Alexandria
3. Pappus
4. Hypatia

Not much is know about Nicomachus although it is speculated that he studied in Alexandria. He dealt with perfect primes where he calculated the first four: 6, 28, 496 and 8128. He, however did not present any proof. Besides perfect numbers, Nicomachus also worked with pentagonal, hexagonal and heptagonal numbers.

Pentagonal Numbers

Nicomachus used a proportion in a different sense form Euclid"s. For Nicomachus there was 3 types of proportions:

1. arithmetic proportion- consecutive pair of numbers differ by the same quantity (for example 1,5,9,13 ... etc differ by 4)
2. geometric proportion- "the greatest terms is to the next greatest as that one is to the next" (for example 3,9,27...etc)
3. Harmonic proportion-"greatest term is to the smallest as the difference between the great and mean terms is the the difference between the mean and the smallest terms" (for example: 3,4,6 are harmonic proportion since 6:3 = (6-4): (4-3)

Diophantus worte arithmetica which is divided into 13 books. He introduced symbolic abbreviations and dealt with powers higher then 3, a break frmo the traditional Greek usage.He also presented some intersting problems

1. To divide a give number into two having a given difference
2. To divide a given number into two number such that given fractions (not the same) of each number when added together produced a given number
3. To divide a given square into two squares

Pappus is best know for his Collections,  which are works on various topics in mathematics. Through one of his remarks we learned that women where involved in mathematics in Alexandria.

Hypatia(355-415) was the 1st women mathematician of whom any details are know and her violent death marked the effective end of the Greek mathematician tradition in Alexandria.

Hypatia

Hypatia wa a respected and eminent teach in Alexandria. She was given a thorough education in math and philosophy by her father, Theon of Alexandria. Despite having influential friends, when a rival of her friends spread rumors about her about practicing sorcery a group of citizens gruesomely ended her life.

Heron wrote a detailed work on indirect measurement. For instance, we have two points, A and B, on one inaccessible side of a river (where the observer is on the other side). We want to find the distance between A and B. Heron used similar triangles.

We can measure the distance CA. And the two similar trianlges are DAB and CED.
So we have

CD/AD = CE/AB

So now we can solve for AB

Unanswered questions:

1. How steady was the decline in Greek mathematics? Did it slowly start to fall apart due to political strife and war or was it a long drawn out process?

Evaluations
Interesting: 8
Complexity: 7
Quality:  8