British colonization in India has always been fascinating to me for, if anything, the sheer logistics of dominating an entire subcontinent. In terms of tracing it from its origins in Masulipatnam/Surat all the way to the Battle of Plassey (and beyond), how did the East India Company actually go about conquering India? How did this compare with British colonial projects elsewhere and how did this go about evolving into the British Raj? What were their most formidable challenges (i.e. local rivals) and how did they succeed against them?
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I can answer this question with respect to India and how the East India Company went from a bunch of traders out to make a quick buck to controlling a vast land stretching all the way from Afghanistan to Burma, and the Himalayas to the Indian ocean on the deep south.
For my sources, I will primarily rely upon William Dalrymple's recent work - "The Anarchy", an account of how the EIC actually took over vast parts of India in the late 18th century. I will also rely upon other sources such as Khushwant Singh's "A history of the Sikhs".
I will also talk about how the EIC actually "conquered India" and not so much about how they "controlled India" as I believe that is the crux of the question. I will not talk much about British Colonialism elsewhere as I am not knowledgeable enough to do so. Also, it is no mean task to compress 150 years of history into a reddit comment, but it worth trying.
So, let's look at the political situation in India at the end of the 17th century. By 1700, the Mughal Empire was a vast realm that stretched from Kabul and Kandahar in the far northwest to the Burmese borderlands in the east, and from Kashmir and the mountains of the far North to the Tamil Country in the deep south of India. The present day nations of India, Pakistan, Bangladesh and most of Afghanistan came under this realm. It was prosperous, it was industrious and militarily, one of the Superpowers of the age. It was ruled by the Emperor Aurangzeb, who had been on the throne for the past 42 years, having ascended the throne by overthrowing his father, the emperor Shahjahan, in 1658. Aurangazeb was a competent and ruthless ruler and expanded the Empire to it's greatest extent by completing the conquest of the Deccan states and the far south, a campaign that had begun nearly a century before. He kept an iron grip on power and to all outward appearances, the Mughal Empire was as strong as it ever was. However, the Deccan wars had fatally overstretched the Empire, something that Aurangazeb himself seemed to be aware of.
When he finally died at the age of 88 in 1707, having reigned for nearly 50 years, Aurangazeb is said to have wept that all of his life's work had been left incomplete. Indeed, upon his death, the empire very quickly broke up into fiefdoms which only nominally paid tribute to the Mughal Emperor who sat on the Peacock Throne at Delhi. The invasion of Nadir Shah in 1739, and the sacking of Delhi, demonstrated to everyone that the Mughal Empire was simply a shadow of what it once was. After that, it was a free for all. Every Chieftain, minor and major, sought to carve out his own area of influence all over India. In the Deccan, it was the Nizam of Hyderabad. Across his borders, the decentralized Maratha Confederacy, controlled a vast portion of the Northern Deccan and Central India. In the Punjab, the Sikhs sought to carve out a state of their own. The Rajput Kings of Rajasthan, for so long the beating heart of the Mughal administration and the Military, went their own ways. Awadh, the Mughal heartland and Bengal, the richest province, too cast of the Mughal yoke and paid only lip service to Delhi. To summarize, the Mughal Emperor in the 18th century was merely a figurehead, and his writ was largely confined to the environs of Delhi. Chaos reigned in vast parts of the country and armed bands roved the countryside looting merchants and killing lone travelers.
At the time of Aurangazeb's death, the EIC was a motley collection of traders and merchants who were more interested in enriching their own pockets than ruling over a vast and ancient land. Disreputable men with shady antecedents, Soldiers of Fortune and ambitious second sons of the landed gentry in England, all made their way to India. Initially, the EIC contented itself with trade and setting up minor fortified settlements on the coasts of India. The main settlements were at Bombay on western coast, Madras in the south and Calcutta in the east. The EIC recruited small armies (mainly native soldiers officered by Europeans) at these forts to safeguard their investments and to fight off the French (who had their own outposts in India) during the War of the Austrian Succession and the Seven Years War. All that changed in 1756-57 when a political dispute in Bengal saw one of the parties solicit the intervention of the British Garrison at Calcutta. Now Bengal was, by far, the richest province of the Mughal Empire. The land was, and still is, among the most fertile in the Indian subcontinent. It had a rich tradition of textile manufacturing. It was in a state of proto-industrialisation and had been tended to carefully by the rulers. Robert Clive, who commanded a body of troops at Calcutta, was bribed to intervene in the dispute and was allowed to line his pockets in doing so. The native authorities had hoped to utilize the British for their own purposes and send them back to the barracks when they were done with them. But once the British had got their foot in the door, they kept it there and eventually the door was opened ajar. They intervened again and again in domestic politics in Bengal. In 1765, after a string of victories against Native princes and even the Mughal forces, the British were granted the Diwani of Bengal, or the right to collect taxes in the whole of the richest province of the country! Note that there was no "grand design" on the part of the British to subjugate and conquer India. All the time this was happening, the board of directors of the EIC in London, nervously bemoaned the danger to trade and profits that getting involved in local disputes entailed.
Once the British had established a foothold in Bengal, they utilized the resources of that province to build up their forces and mount a challenge to native authority in other areas of the country. It must again be mentioned that the British did not actively go out to conquer India. They got involved in local politics nearly everywhere and by the last decades of the 18th century were a serious player in nearly all parts of the country. In the South, they faced a stiff challenge from Tipu Sultan of Mysore. Between 1782-1799, they fought 4 bloody wars with him and eventually emerged the political masters of nearly all of what is now South Karnataka, Tamil Nadu and Andhra Pradesh. This is the Carnatic country and rivalled Bengal in wealth and industriousness. Awadh and Hyderabad were cowed into submission by British feats of arms and had subsidiary alliances thrust upon them. A body of British troops was maintained in these states at the expense of the local ruler, ostensibly to guarantee their realm from external aggression but mainly to ensure good behaviour.
The one Indian polity that had the military means and the pan-Indian reach to seriously challenge the British was the Maratha Confederacy. The Maratha Confederacy consisted of 5 quasi independent chiefs (The Peshwa at Pune, the Bhonsale at Nagpur, the Holkar at Indore, the Gaekwad at Baroda, and the Scindia at Gwalior) who often quarreled with each other but would unite to face a common threat. The Marathas faced down the British in three epic showdowns featuring several bloody battles between two equally matched armies over a period of ~40 years from 1775 to 1818. The Maratha chiefs were unable to put aside their differences and were picked off one by one by the British. The wealth of the Carnatic and Bengal ensured that the British had far superior resources at their disposal. Central and Western India, by contrast, are arid regions whose main export in the 17th and 18th centuries was military labour. By the time the dust settled in 1818, the British were the masters of a territory that stretched from Delhi in the North to the Tamil country in the south and Bengal in the east.
Greetings! u/conqueror_of_destiny has already done an excellent job giving an overview of the "main" question at hand regarding the rise of the British East India Company on the Indian subcontinent, so I shall be focusing on the more secondary questions that OP has put forward in the question. For some foundational knowledge, consider reading this thread I weighed in on about some patterns of British rule across the world. Further, as I elaborated on further in this thread, the British Raj would soon be transformed into the British Empire's geostrategic lynchpin for Asia, serving as a major military base as well as an economic asset whose value made it the "jewel in the crown" of the Empire. With those two preambles and shameless plugs out of the way, let's begin.
The British Bridgehead
Centuries before the Sepoy Mutiny of 1857 and Queen Victoria's ascension as Empress of India, the British presence on the massive sub-continent was negligible, dwarfed by all manner of princely factions, Mughal territories, and the formidable Maratha Confederacy. OP has already touched on the foundations of the British East India trading company at its factory and mercantile hub of Masulipatam by 1617. It was not until Robert Clive began his conquest of the Bengal region in the 1750s that the British hold over the continent formally expanded. Even sixty years after the bridgehead had been established, the British civilian population (out of a total of 300,000 people, many locals and other foreigners among them) numbered an estimated 114 (unless you count the garrison, which still made the total a minute 400 Britons). The Company could do little in this time to attempt to expand its holdings, knowing full well that any of the surrounding kingdoms likely had the power to expel them from the subcontinent (or destroy the Fort where it stood).
This thought almost became reality in 1746, when the French forces stationed at nearby Pondicherry (as part of the larger War of Austrian Succession) forced the British out of their holdings at Madras. The British would go on to recapture Madras in 1748, though they also temporarily lost hold of Calcutta at the hands of the Nawab of Bengal, where the infamous "Black Hole of Calcutta" incident occurred. Clive was not only well-versed as a military commander, but had the necessary political skill of finding and keeping Indian allies, among them the Nawab of Bengal's chief advisers; allowing his forces to triumph over the Bengali-French ones at Plassey in 1757. John Darwin on the ramifications of this victory:
"Within less than ten years of Clive's scattering of the Nawab's army at Plassey, the Company took control of the province's revenues. With this second grand bridgehead, the richest region of India fell into its hands. Forty years later, its men ruled in Delhi. The merchants' frontier, their precarious perch on the coast, had turned into an empire."^(1)
Conquest, Colonise, or Control?
However, it is necessary that we realise that this control was not as concrete (or even as colonial) as one might imagine. Firstly, a fair portion of the Company's territory was a feudal grant from the Mughal emperor, for which the British East India merchants paid tribute up until 1772. It was only after 1800 however, that the Company asserted the claim that it was sovereign over the territories under its rule. Knowing full well the banking power of the Company, those holding the reigns of power in London demanded that they share some control over the "men on the spot" of the Company. This resulted in the India Act of 1784, by which the following were to be supervised by a Board of Control and a cabinet minister:
Yet the implementation of British law here was always a game of touch-and-go. In many places where the Company flag flew, they relied heavily on local partnerships with Indian rulers. Upsetting the local religious balance by introducing Christian missionaries into the region was also a key issue for many governor-generals, as well as the question of respecting traditional Hindu/Muslim laws, or ignoring them completely for "English" ones. The patterns of rule across the Company's holdings remained so different that one Chief Justice of Bengal remarked in 1830 that:
"There is no uniform, no definite opinion, either as to the true character... of the sovereignty of the Crown, nor of the dependence of the laws on Parliament, nor as to rights either of political power or of property of the East India Company, nor even of the relations in which the many millions of natives stand to the political authorities by which they are entirely governed."^(2)
So until the British government formally stepped in and took over control of the East India's Company's flagging command of the sub-continent in 1857, India was a curious British holding unto its own. Too valuable to give up, but too politically complex and diverse to effectively implement direct rule from Whitehall (at least until crisis propelled the government under PM Henry John Temple to step in and do so). To quote Historian Goldwin Smith, writing in 1906 (a fittingly old source when the British Raj was still around):
"So long as the Company held real sway, the policy was strictly and narrowly commercial..."Send us dividends" was the one great commandment of the Company to its servants in India. Nothing could be further from the thoughts of Leadenhall [the street in London where the Company was headquartered] than the idea of civilizing and Anglicizing Hindostan."^(3)
A British Raj
As a small bit of context (and 'looking forwards' if you will) before moving onto a comparison between British Company rule in India and British imperial presence elsewhere in the world, once the Sepoy Mutiny of 1857 had been violently and brutally crushed by the British government, they set about structuring a Colonial office specifically devoted to running the day-to-day affairs in India. Here is a primary source from 1900 explaining the actual staff positions and hierarchy of the India office:
"The administration of the Empire [in India] is entrusted to the Secretary of State for India, assisted by a council of not less than ten members, nine of whom have served or resided ten years in India, and have not left India more than ten years previous to the date of appointment. The administrative machinery in India consists of a Viceroy, who in theory represents the Queen-Empress, but who in reality is subordinate to the Secretary of State, and two councils, which owe their existence to a series of acts of parliament. One of these councils is executive, and the other legislative."^(4)
Once the Sepoy Mutiny had been crushed, the British also were effective at maintaining control of the remaining local princely states. Their rulers, in many forms and areas, still existed, though practically with little to no power at all. Many jostled for prestige in the eyes of the British government in India, several seeking to secure the highly valuable titles which conveyed a sense (but never in actuality) of equality with the British civilian rulers.
So there's an explanation of the British East India Company's rule from its first minor fort on the coast to the "jewel in the crown" of the Empire, and what Viceroy of India **Lord Curzon (**holding that position from 1899 - 1905) would even describe as the "pivot and center of the British Empire". Next up, a larger look at how this unique pattern (or lack thereof) of rule compared to other British colonial projects.
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