A few notes here from G.G.Joseph's excellent book, The Crest of the Peacock, specifically looking at evidence for transmission of mathematical knowledge and number systems.

Babylonian origins

See p.111 for first example of Diophantine technique used by Babylonians.

The second example is a cuneiform tablet, catalogued as Plimpton 322. It is not a trigonometrical table but is evidence for the methodical production of Pythagorean triples. We are all familiar with the (3,4,5) triangle and its properties, (that 3²+4²=5²), but not everybody will be aware that there are many more such combinations, infinitely many in fact. They are known as Pythagorean triples. Examples include (7,24,25) and (96,110,146).

With the help of some fairly elementary algebra, it is possible to generate triples using a formula.

This method of generating integral Pythagorean triples is usually attributed to Diophantus (c. AD 250) who, as we have seen, may be thought of as working in the Babylonian mathematical tradition and introducing it into Greek mathematics. p.117

Indian numerals

Indian numerals. The earliest trace of Brahmi numerals is from the third century BC, on the Asoka pillars scattered around India, though more detailed pieces of evidence are found elsewhere later. At the top of Nana Ghat near Poona in Central India is a cave, which must once have been a resting-place for travellers; inscribed on the cave walls are Brahmi numerals which date back to 150 BC. p.240

The early number systems had individual symbols for the digits 1 to 9, with further symbols specifically for 10 and its multiples up to 90, 100 and multiples up to 900. They were therefore not truly place-value systems to begin with, but later similar numerals (just the 1 to 9) were used in positional notation. Significantly also the novel concept of the digit zero was introduced, to make the decimal place-value system complete. This event is hard to date precisely.

GGJ notes an example in India (Gwalior) dated to 876 CE, and others from SE Asia (at the time under the cultural influence of India) at Palembang, Sumatra around 684, Sambor, Cambodia in 683, and Ponagar, Vietnam in 813 CE. The Bakshali manuscript, if an early date is accepted, may be considerably earlier than all of these. In it we see a fully developed decimal place-value system incorporating zero.

Bakshali Manuscript

p.257 Bakshali MS. Dating uncertain, but likely first few centuries of the Christian era.

The manuscript may therefore be the next substantial piece of evidence, after Jaina mathematics, to bridge the long gap between the Sulbasutras of the Vedic period and the mathematics of the Classical period, which began around AD 500.

It is also the earliest evidence we have of Indian mathematics free from any religious or metaphysical associations. Indeed, there is some resemblance between the Bakshali manuscript and the Chinese Chiu Chang from a few centuries earlier, which we examined in chapter 6, both in the topics discussed and in the style of presentation of results. It should, however, be added that the premier Chinese text is far more wide-ranging and ‘advanced’ than the Bakshali work.

pp344-348

Babylonian and Egyptian influence on Greek mathematics not felt until later Hellenistic period.

Euclid & Apollonius (Conics) were not trained in Babylonian methods

Babylonians had an analytic and algebraic approach to geometry

Source

• G. G. Joseph 2000, The Crest of the Peacock: Non-European roots of mathematics (Penguin)