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Partially correct, and partially wrong. The great patricians and senators will try to keep down a plebeian, but this is equally true of a fellow-patrician emerging as a dominant power.
But people could and did succeed in Rome: your point is an objection, not an insoluble problem.
I would point out that not getting assassinated - or coming up against a power bloc that can stop you - is a problem with all 'Become Emperor of Rome' scenarios: and it was solved by all emperors.
It boils down to politics. A strong point of the Joint-Stock plan is that you make a lot of people wealthy: a broad power base is a strong power base.
Consider this: many ancient and noble Patrician families are in a state best described as 'genteel poverty'. Give them enough of a share to regain a place in public life - but about the same, or slightly less, than the share you sell to merchants, so that these aristocrats cannot dominate the power base and - and you've got a very powerful influence in the senate.
The other place to secure influence is the Army. Families of successful (or influential) generals and mid-level commanders must get a share. At least one Legion, within reach of Rome, should start receiving mass-produced high-quality equipment ....Or mass-produced equipment as good as the standard issue, but at factory prices - much to the profit of the commanders and the procurator. (This is legal in Rome: cost-effective supply was properly incentivised).
"The Joachimsthaler Legion stands with Rationus' is a declaration that will put a stop to any drastic setback.
Incidentally, I wouldn't even start on trying to get the Praetorian Guard 'onside' - too obvious, certain to alarm the Emperor, and fatal.
Improving the water supply, temple donations, and economic mobility: that's a good route to popular support - and the 'Vox populi' was a matter of importance in Rome. Not that it was a democracy by that time, but the ruling classes took great care (and great expense) to maintain popularity. And there was a direct 'Will of the People' channel in Rome: the Tribunes.
Arguably, they are a point of failure: your public works, equalisation of wealth for plebeians, and temple donatuons would count in your favour - and should - but the Tribunes are a point of failure if your enemies seek influence there. Note, however, that the tribunes existed explicitly as a counterbalance to the great patricians and the emperor: that's the way you play against 'the people who are already on top'.
Finally, be sure that you and your supporters are liberal with donations to the Temples. And, indeed, to the causes and projects of the existing Emperor's supporters... Some of whom will be happy to be shareholders, discreetly, because they're not all rich - old families, again - and financial dependence on the Emperor is an uncomfortable thing.
In short, you'd play politics. But politics powered by an important invention of the modern age - Shareholders - with crude steam engines to prime that particular pump with money.
It's bad form to reply to my own post, but a strong cup of tea has invigorated me with some ideas that follow on from building the first working beam engine.
I'm introducing a second invention, halfway through, because you need something to pump industrial quantities of capital as well as water.
Firstly, you probably won't have the cashflow to purchase another mine and build another engine for at least a decade. Not unless you're mining silver, or there's a severe economic supply contraint on copper, tin, or whatever you're extracting, permitting you to make extraordinary profits.
You will have the cashflow to improve - and probably replace - your engine, your workshop, and your craftsmmen in two or three years.
You now have a business model. No, TWO business models.
Firstly, municipal water supply already exists in the Roman Empire: there's an existing demand for big beam-engine pumps, and you can demonstrate a profitable and reliable working model. Your craftsmen might defect and start up a rival business - if they can get capital (which they can't) or interest a Patrician - but you can let them go, secure in the knowledge that they can only ever copy what you've already done.
They cannot compete with your next technological improvement, or the one after: your emerging commercial rivals are no better than the industrial pioneers and inventors who took decades to develop things that you know completely, a century ahead of their best possible learning-curve.
Your second business model is that you know where the all good ore lodes are, and how to get at them - flooded or not. Even if you don't, mine-owners with a flooding problem are going to tell you where the few known ones are...
...But developing those resources still needs far more capital than you possess; and your objective is to become Emperor, not just a provider of steam pumps to grand patricians with an ore lode, and Proconsuls with a municipal supply problem.
No, you need a second invention: the Joint Stock Company. Lots of wealthy families and merchants would love to have something to invest their money in - Rome's a surprisingly rigid economy, above a certain level of family wealth - and you've got an idea they can see working.
So: get the articles drawn up - and sworn before a Quaestor (and a Vestalis if you want it to be a convincing demonstration of good faith that'll get people killed for suborning or betraying) - and open up another mine.
And another. And a bigger one, with a horse-drawn tramway to move the ore and the coal. Lucky you, knowing the Mass Haul calculation, and the principles of bridge design, soil mechanics, and a simple optical level...
So now you have cashflow, an expanding demand for steel and machine tools, bulk transportation, and coal. And capital.
Sounds like an industrial revolution, right there.
And you're always going to be first with the right technology, at the right time: whatever significant technology anyone else invents - or copies - you will always know its flaws and how they were overcome in subsequent improvement and developments - or how that particular technology was superseded by something far, far better when materials became available.
You also have a substantial political power base among your shareholders: merchants and mid-level patrician families who will, within a decade or two, be wealthier than the entire senatorial class of Rome.
Purple rather suits me, don't you think?
This is wrong, but less wrong than I thought:
" Steam engines, for example, weren't invented until the 1700's because metals were so bad at the time that pressure vessels were impossible to make safe "
Not quite: the development of high-pressure engines was delayed, but the initial deployment if low pressure engines was an immediate success.
The first steam engines were, of course, atmospheric engines: fill a large piston with low-pressure steam, squirt cold water, and - whoosh! - the steam condenses to water and a pretty good vacuum, leaving the piston to be forced in by atmospheric pressure.
Inefficient? Actually, no. Heavy, yes: this is a building with an engine in it, not a locomotive; and slow. But good for pumping and acceptably efficient at it; beam engines remained in use as municipal pumping stations well into the twentieth century.
They were two or more orders of magnitude more efficient than a horse, given access to tons (but not tens of tons) per day of coal. You simply could not link up enough capstans and horses to do what an early beam engine did.
Crucially for our purposes, a working beam engine can be constructed by blacksmiths and coopers, with a little bit of skilled brasswork and solder for the valves. This is feasible and affordable, in Rome, with our limited start-up capital.
What we need next is a profitable application, and we can copy de Savary's business model as well as his first crude invention: drain a once-lucrative mine that's failing due to flooding.
The Romans had mines - huge ones for copper, in North Wales - and probaby silver elsewhere. If they had mines, they had flooding.
The question is whether we can travel back in time with a well-memorised geological map that shows the proximity of such an opportunity to any small (or large) coal measure.
So we might have an opportunity to make a lot of money from an abandoned and unprofitable enterprise that the owner will be delighted to sell or rent to us.
Note, also, that we're travelling back in time with some seriously useful - and profitable! - ideas about mining engineering in general.
Armed with a lot of money, we might just have the cash flow to kickstart an industrial economy in the region, and start constructing a reverberating furnace and a real machine shop with some interesting uses for good steel. High-pressure boilers included.
Interesting stuff: but I'm going to throw in a few disappointments on technological triumphs, and propose a peaceful takeover by affordable glazed pottery.
Firstly, the printing press: movable type is a great idea but you also need paper. And you'll need the 'killer app' - or rather, the book with a massive pre-existing demand. In 1 AD, that's not the Bible! If you got the authorities interested in the promulgation of official edicts and shool texts, you might be able to scale up the business to kickstart a commercial printing economy - but scaling-up is not the same as starting up, and you'd need a lot of capital just to demonstrate the technology to skeptical officials.
Without official patronage, getting the paper-and-printing economy started will need more capital than you've been given: you'd need a runaway success in some other small startup business or the patronage of someone with a medium-to-large country estate.
Maybe the 'mysterious healing' of some aristocrat's favourite slave and a religious conversion isn't such a bad starting point, after all. It's no less plausible as a way of gaining access to resources to start building a technological power-base.
A guns-and-steel route to power is even harder: handmade guns that offer a significant advantage over a disciplined formation of legionaries supported by archers will be prohibitively expensive in the quantities required to equip a company of musketeers. And that's a dead end: there's only so much money available to you, and a huge entrenched power-base of landowners and spear-wielding legions.
Better, cheaper guns will need very good steel and that, in turn, will need investment in lime kilns, the first successful blast furnace, good-quality coal, coking ovens, and trained workers.
And once you've got the steel, you need to mark out the standard millimetre and weigh out the standard kilogram, and start building the first precision-casting production line; presses, milling machines, lathes, screw-cutting, bench drills...
It took a seventy-five years to move from good blacksmiths and the reverberating furnace (an adapted lime kiln) to a machine shop that could build precision tools, valve-gear for steam-engines, rifle barrels, or machine tools to equip a factory for the volume production of firearms.
Given the knowledge, you could cut that period of not-quite-from-boots 'bootstrapping' to twenty years; but it might take twice that long to bootstrap demand alongside the supply and create a functioning market for the products - and interim products at that! - and that market would be necessary (and, necessarily, damn' lucrative) in order to support the vast investment for a technological expansion that utilises all these shiny (but unfamiliar and unreliable) new tools profitably in order to support continued growth.
No aristocrat will keep pouring money into that forever: you have to achieve profitability and an upward trajectory of organic growth before you exhaust the limited resources of your patron's latifundi.
One thing that is in your favour is that the Romans in 1AD are ready for mass production: urbanised, with good regional and local trade, and already centralising some of their economy into large production facilities (for grain and - with less evidence -for arrows).
So your new world ruler might not need to be Abraham Darby, Joseph Whitworth and Samual Colt, all rolled into one near-superhuman genius of invention, development, process management, and leadership: he or she might become the wealthiest of all Romans with the more prosaic skills of Josiah Wedgewood.
However, I would note that Wedgewood kickstarted the industrial production of consumer durables with a substantial injection of capital from a distant relative. Nevertheless, industrialising artisanal pottery is an easier route to the economic power of a factory economy than emulating Richard Arkwright - he had a lot more to do, to scale up and power-up the existing barn-size 'manufactuaries' of spinners and weavers into the modern cotton mill.
I note that printers did not become amazingly wealthy in Europe: they changed the intellectual 'economy' but they did not start an industrial revolution.
However you do it - printing, pottery, or cloth - you'll need to expand to the scale of a factory-driven county or city-state economy as a prerequisite to equipping militarily meaningful formations with firearms.
But here's the joker: with the wealth to to so equip an invincible legion, you'd be the emperor anyway. That kind of money - Josiah Wedgewood's fortune turbocharged by first-mover advantage and rational training - would buy the patrician families, the senate, the generals and (probably) the incorruptible tribunes.
Which would, in turn, put you in a position to enact the reforms that will - you hope - prevent the later collapse of Rome; establish rational schools, technological colleges, and scientific universities; impose Rationalism as the State Religion (!) and send forth the legions, with firearms, on steam railways to conquer all of Europe, Asia, and Africa.