China's knowledge of the Roman Empire was... really, really limited... for reasons already discussed well by /u/yolowolf... but I thought I'd chime in and say that, indeed, China had developed a rough-ish knowledge of the "Great [Western] Qin" well prior to 1 AD. It viewed it as its occidental analogy (hence naming it after its own great dynasty of the time). And that whole long, strange trip of discovery would open up an incredible new world of trade to both empires.
The Han Empire’s relationship with Rome must first begin with its tense, violent relationship to a far closer empire: the Xiongnu Confederacy of the Central Asian Steppes (think the Mongolians about 1400 years before they were Mongolians). The early reign of Emperor Wu of Han (one of its greats) had increased the heat and pressure along the two powers’ extensive borders to the boiling point. In the third year of his reign, 139 BCE, Wu had dispatched his envoy Zhang Qian to explore the unknown vastness of central Asia beyond the Han’s western border at the Jade Gate Pass, and establish relations with any peoples he might come across. This had long been a goal of the Chinese imperial court, but the Xiongnu Empire militarily controlling the only real passage westward had prevented it: then called Xiyu (literally, “Western Regions”), and today Xinjiang Autonomous Region that makes up China’s arid northwestern border. As China’s only reasonable land route westward, the Han already knew it would be an immensely valuable area to hold, and indeed it would eventually become a key portion of the Silk Road trade route… but first it would have to be taken.
Emperor Wu had chosen Zhang Qian because of his firsthand knowledge of the Xiongnu and their customs. He, along with a captured Xiongnu guide named Ganfu, and ninety-eight others, had been sent to both scout out the territory, as well as find a way through the Xiongnu-controlled Xiyu. On the far side, it was believed that a people called the Yuezhi (also, the Rouzhi and identified by the Greeks as Tókharoi), who had been expelled from their ancestral lands during the Xiongnu occupation existed. Emperor Wu wished to find and treat with the Yuezhi king, and convince his to aid the Han Empire in retaking Xiyu and opening up the western road to travel and trade.
Things didn’t start out so well for Zhang Qian, though. Virtually right out of the starting gate, he and his party were captured by Xiongnu patrols as they attempted to cross the world’s second largest shifting sand desert in the Tarim Basin – the Taklamakan, whose name either comes from Arabic, meaning “the abandoned place,” or Turki, meaning “the place of ruin.” Zhang Qian’s entire party was taken captive and enslaved by the Xiongnu (including the Xiongnu guide himself, Ganfu… welcome home). For a period of ten years, Zhang would work under the lash of his captors, but was able to build a rapport with the local tribal leader, and eventually married a Xiongnu woman who bore him a son.
Finally in 129 BCE, Zhang Qian was able to escape his captors, along with his wife, young son, and the guide Ganfu. Though the rest of his party had been lost, the four continued westward, around the northern edge of the Tarim Basin and the now dried-up salt lake Lop Nor, through the Dayuan Kingdom (whose markedly Caucasian-looking citizens were noted by Zhang, and were likely remnants of Alexander the Great of Greece’s conquest and colonization pf the area two centuries prior). It wasn’t until 126 BCE that Zhang Qian finally found the long-lost Yuezhi Kingdom, which by this time had settled in Bactria, centered around Samarkand (modern Uzbekistan, Kazakhstan, and northern Afghanistan). Zhang Qian was permitted an audience with the Yuezhi king, but was unable to convince him to return to the east to retake their lost homelands from the Xiongnu. The rather liked it where they were, thank you very much… plus, those horse-people were really scary. Regardless, Zhang Qian was welcomed within the Yuezhi Kingdom, and he stayed for a year, documenting their culture, lifestyle, and economy, as well as the myriad cultures surrounding the country that were wholly alien to the Chinese up until this point.
In late 126, Zhang, his family, and he guide Ganfu departed the Yuezhi Kingdom and began their return trip to Han China. This time they skirted the Tarim Basin to the south (both his north and south routes would ultimately make up the Silk Road’s path around the harsh desert basin), and were once again captured by the Xiongnu, but they released him – in spite of the hostilities which had been ongoing with the Han Empire for almost a decade at this point – because they respected his sense of duty and composure in the face of death. He was held for two years, before once again escaping when infighting broke out following the death of the tribal leader, and finally returned to Chang’an and the waiting Emperor Wu with his report.
Zhang Qian’s report described in great detail the kingdoms he had personally visited or traversed. I’ll relate some excerpts of Zhang’s report, as relayed through Sima Qian’s Shiji, itself translated by the esteemed Dr. Burton Watson: • the Dayuan/Ionian Kingdom, whom Zhang described as “southwest of the territory of the Xiongnu, some 10,000 li (5,000 kilometers) directly west of China. The people are settled on the land, plowing the fields and growing rice and wheat. They also make wine out of grapes. The people live in houses in fortified cities, there being some seventy or more cities of various sizes in the region. The population numbers several hundred thousand"
• the Yuezhi, described as “The Great Yuezhi live some 2,000 or 3,000 li (1,000 or 1,500 kilometers) west of Dayuan, north of the Gui (Oxus) river. They are bordered to the south by Daxia (Bactria), on the west by Anxi, and on the north by Kangju. They are a nation of nomads, moving place to place with their herds and their customs are like those of the Xiongnu. They have some 100,000 or 200,000 archer warriors."
• Daxia/Bactria, described as “situated over 2,000 li (1,000 kilometers) southwest of Dayuan, south of the Gui (Oxus) river. Its people cultivate the land, and have cities and houses. Their customs are like those of Dayuan. It has no great ruler but only a number of petty chiefs ruling the various cities. The people are poor in the use of arms and afraid of battle, but they are clever at commerce. After the Great Yuezhi moved west and attacked and conquered Daxia, the entire country came under their sway. The population of the country is large, numbering some 1,000,000 or more persons. The capital is Lanshi (Bactra) where all sorts of goods are bought and sold."
• the northern kingdom Zhang called the Kangju, which may have been of either Turkic, Iranian, or Tocharian origin, described as “situated some 2,000 li (1,000 kilometers) northwest of Dayuan. Its people are nomads and resemble the Yuezhi in their customs. They have 80,000 or 90,000 skilled archer fighters. The country is small, and borders Dayuan. It acknowledges sovereignty to the Yuezhi people in the South and the Xiongnu in the East."
He also made mention of many other peoples and kingdoms he had heard of or discussed while abroad, including
• Anxi (the Parthian Empire which was a kind of successor state to the ancient Persian Empire), "Anxi is situated several thousand li west of the region of the Great Yuezhi. The people are settled on the land, cultivating the fields and growing rice and wheat. They also make wine out of grapes. They have walled cities like the people of Dayuan, the region contains several hundred cities of various sizes. The coins of the country are made of silver and bear the face of the king. When the king dies, the currency is immediately changed and new coins issued with the face of his successor. The people keep records by writing on horizontal strips of leather. To the west lies Tiaozhi."
• Tiaozhi (which remains ill-defined, but might have been Mesopotamia), described as, “situated several thousand li west of Anxi and borders the Western Sea (probably either Persian Gulf or the Mediterranean). It is hot and damp, and the people live by cultivating the fields and planting rice... The people are very numerous and are ruled by many petty chiefs. The ruler of Anxi gives orders to these chiefs and regards them as vassals.
• Shendu (Northern India/Pakistan), saying, "Shendu, they told me, lies several thousand li southeast of Daxia (Bactria). The people cultivate the land and live much like the people of Daxia. The region is said to be hot and damp. The inhabitants ride elephants when they go in battle. The kingdom is situated on a great river (Indus)."
Zhang Qian was rewarded lavishly by Emperor Wu, and honored with a position as palace counselor. He would conduct a second and final journey to the west in his lifetime from 119-115 BCE as part of a trade mission to the Wusun people, which would result in a long-term link between Han China and the Parthinian/Persian Empire. Upon his triumphant return, Zhang was appointed the Imperial Grand Messenger, making him one of the nine most powerful ministers in government. He would die little more than a year later in the year 113, but had laid virtually all of the groundwork for the higly lucrative Silk Road trade network, both in terms of finding the literal western road out of China, and establishing connections with its distant neighbors.
Sima Qian. The Records of the Grand Historian (史記). http://ctext.org/shiji (Mandarin).
Watson (1993): Records of the Grand Historian' by Sima Qian, Han Dynasty II, (Revised Edition). Translated by Burton Watson, Columbia University Press, ISBN 0-231-08167-7.
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There is a lengthy description of Rome in the official history of the Eastern Han compiled by Fan Ye (c. 398 - 445), containing a mixture of fact and fiction. The same source also records that the Roman emperor "An-dun", presumed to be Marcus Aurelius Antonius Augustus (r. 161-181), sent diplomatic envoys the Han commandery in Vietnam, which was then under Chinese control. They reached the Han court in Luoyang, and offered elephant tusks, rhinoceros horn, and turtle shells to the Han emperor. It's not certain whether they were genuine envoys or Roman merchants mistaken as envoys. Even the Chinese historian chronicling Rome felt that the gifts, which were not considered rare, suggested that Chinese tales about Rome had been exaggerated.
Section 11 "The Kingdom of Da Qin" in Chapter 78 "Chronicle of the Western Regions" from the History of the Eastern Han Dynasty (Houhan Shu 後漢書) compiled by Fan Ye (范曄). The passage on Rome:
The kingdom of Da Qin is also called Lijiang. As it is found to the west of the sea, it is also called the kingdom of Haixi. Its territory extends for several thousands of li. It has more than four hundred walled towns. There are several tens of smaller dependent kingdoms. The walls of the towns are made of stone.
They have established postal relays at intervals, which are all plastered and whitewashed. There are pines and cypreses, as well as trees and plants of all kinds. The common people are farmers. They cultivate many type of trees, breed silkworms, and grow mulberries. They shave their heads, and their clothes are embroidered. They have screened coaches and small white-roofed one-horse carts. When carriages come and go, drums are beaten and flags and standards are raised.
The seat of government is more than a hundred li around. In this city are five palaces each ten li from the other. Moreover, in the rooms of the palace the pillars and the tableware are really made of crystal. The king does each day to one of the palaces to deal with business. After five days, he has visited them all. A porter with a sack has the job of always following the royal carriage. When somebody wants to discuss something with the king, he throws a note in the sack. When the king arrives at the palace, he opens the bag, examines the contents, and judges if the plaintiff is right of wrong. Each palace has officials in charge of the written documents.
A group of thirty-six leaders has been established to meet together to deliberate on affairs of state. Their kings are not permanent. They select and appoint the most worthy man. If they are unexpected calamaties, such as frequent extraodinary wind or rains, he is unceremoniously rejected and replaced. The one who has not been dismissed quietly accepts his demotion, and is not angry.
The people of this country are all tall and honest. They resemble the people of the Middle Kingdom and that is why this kingdom is called Da Qin.
Another passage describes the Roman envoys at the Eastern Han court that supposedly arrived through Vietnam:
The king of this country always wanted to send envoys to Han, but Anxi wishing to control the trade in multi-colored Chinese silks, blocked the route to prevent getting through.
In the ninth Yanxi year, during the reign of Emperor Huan, the king of Da Qin, Andun, sent envoys from beyond the frontiers through Rinan to offer elephant tusks, rhinoceros horn, and turtle shell. This was the very first time there was communication. The tribute brought was neither precious nor rare, therefore raising suspicious that the accounts might have been exaggerated.
The original translation is from John E. Hill's Through the Jade Gate to Rome: An Annotated Translation of the Chronicle on the "Western Regions" in the Hou Hanshu, cited in Li Feng's Early China: A Social and Cultural History.