Chinese technology compared to Western technology
In the early middle ages China was clearly well ahead of the West, but by 1300 both places seem to have been roughly even. China would have had some things Europe didn't, but the reverse was also true, with Europe having things such as hourglasses, artesian wells, vertical windmills, spectacles, and more reliable mechanical clocks.
The Chinese built many impressive temples and pagodas, and medieval Europe had architectural examples such as Lincoln cathedral, Malbork castle, the Torrazzo of Cremona, and the medieval Louvre.
Looking at both places objectively, claims that China was "way more advanced" just don't seem to hold up. Indeed, the Chinese evidently held the western Jesuits of the 16thC in high regard for their knowledge of astronomy, geography, mathematics and hydraulics.
I'd say 1300 is too early for the catching up of Europe. 1500, perhaps. 1300 is more like the beginning of the catching-up, with the roots of the scientific and industrial revolutions. But in 1300, still only the roots, with not many fruits.

By 1300, there's still plenty of major science and engineers that China has and Europe hasn't. China was still well ahead in shipbuilding, chemistry, printing, a lot of basic sciences (like chemistry and physics) and mathematics, and a whole bunch of military technology, especially gunpowder technology (cannon, rockets, explosive mines (land and seas), handguns).

By 1500, A lot of that has been done in Europe.

In the mid/late 16th century, the Ming government realised that they'd fallen behind in a lot of important military technology, and worked to catch up. One reasons why the liked the Jesuits was their expertise in cannon casting. But they weren't very far behind, and the Qing government caught up.

By 1700, Europe is clearly ahead. By 1800, overwhelmingly ahead, due to the maturity of the Industrial Revolution.
The Chinese practiced alchemy, as did a lot of cultures, but chemistry as we think of it today didn't really exist anywhere until the 17thC.
With regards to printing, although the Chinese developed the first moveable type printing press, Europeans were the first to use the watermark (13thC) and water powered paper mills (14thC).
The Chinese also held to the idea of a flat earth until they were introduced to Western astronomy in the 17th century, while in Europe the round earth had been commonly accepted since the early middle ages.
Practical industrial chemistry, not scientific chemistry. Scientific chemistry as we know it is recent and European in origin (and postdates 1300 by long time). But practical applications can and do exist in the absence of what we'd call correct theory today.

The adoption of paper in Europe (from Islamic Spain(?), from the Middle East, from Central Asia, from China), and large-scale manufacture helped lower the price of books even before printing, and helped drive the beginnings of the scientific revolution. The printing press in the 15th century made a big impact; it's a major contributor to Europe being ahead of China by 1700 (or whenever we place the date for being ahead). But the printing press in the 15th century doesn't support a date of 1300 for equality. In 1300, China is still well ahead in printing technology (and quantity). (The Islamic world might have been ahead in book production in 1300?)

One needs to look at a broad range of technologies and sciences; anybody can argue either side by cherry-picking a few select inventions. For example, look at shipbuilding in general, not some specific inventions. Arguing based on the lack of some specific invention/discovery can be even more misleading.

One important European advance was the development of modern banking - without this, would we have seen the Industrial Revolution?
I think a lot of Chinese innovation was of the practical kind, while a lot of Western innovation had a somewhat more theoretical basis. Obviously there was still plenty of practical innovation in the West, but not quite to the same extent.

It would seem unfair to say that the West lacked the practical chemistry and alchemy that existed in China. Stained glass windows, forging metal alloys, clocks, calendars (they got better), telescopes etc are all practical chemistry/science, and knowledge (if not supported by the theory) would have been quite advanced in both cultures.

I'd argue that by 1520-30, Europe has at least matched, if not overtaken, China. Case in point, Pizarro and the conquest of South America. The technology of the Spanish which enabled/facilitated that was very important: guns, cannons, steel armour, ships, books and paper, complex and large central state that could fund the expeditions, and so on. While you could argue that China did send treasure fleets to Africa, indicating they had the technology, in the 15the Century, I think its a case of: Europe pressed its advantage, China did not.

Once Europe had started to catch up, and we can debate when that was, they kept gathering speed and were constantly using their improving technologies. Average income in Europe went up (not much, but a bit) when the GDP grew, whereas average income did not grow in China when the Chinese GDP grew. (Sources: Guns, Germs and Steel by J Diamond; The Soulful Science by Coyle).

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