As I understand it the steam engine created in the time of Ancient Greece as a toy. How come no thought to scale up this process and apply it to mill production or a magnet?
The first commercially successful steam engines - the Newcomen engine - were not very efficient, and only worked because the fuel was essentially free. The Newcomen engine saw use in coal mines, for pumping water. The fuel was the least saleable coal (which probably wouldn't have been sold, anyway). These engines only had an efficiency of 0.2%, i.e., for every 1000J of heat energy produced by burning the fuel, 2J of mechanical work was done. Watt's more efficient engine - delivering over 2% of the input thermal energy as useful work - proved efficient enough to be used when the owners had to buy fuel, and powered the Industrial Revolution. The ancient Roman/Greek steam engine, the aeolipile, is less efficient that the Newcomen engine. This measurement:
gave an efficiency of a mere 0.0128%, under 1/10 of that of the Newcomen engine. This efficiency includes the drive train used to raise the weight; an ancient version, with ancient bearings and gears would have been significantly less efficient.
Could an aeolipile be an economically-sound choice of power in the Roman Empire? According to the Price Edict of Diocletian, 1 day of unskilled/semi-skilled labour cost the same as 250 Roman pounds of firewood. This is about 80kg of firewood, with a total thermal energy content of about 1200MJ. Using the aeolipile efficiency above, this would yield about 150kJ of mechanical work. A shaduf (an ancient counter-weight manual water-lifting device) can provide about 50W of average water-lifting power. In one hour of operation, about 180kJ of water-lifting can be done. That is, a labourer operating a shaduf for 1 hour can lift more water than an aeolipile could pump using firewood worth 1 day's wages for the labourer. This is not including the labour cost for operating and maintaining the aeolipile.
Conclusion: hiring a labourer would be cheaper. Other options would be slave labour, and animal power. An ox can do over 6MJ of useful work in one day; assuming mechanical losses of 50% in a pumping or water-lifting mechanism, an ox would lift about 20 times as much as an aeolipile with 1 day's wages worth of firewood.
The Watt engine, about 200 times as efficient as an aeolipile, running on firewood at Roman prices, would do about 30MJ of water-lifting with 1 day's wages worth of firewood, which would require about 10 ox-days or 160 man-hours. This illustrates why the Watt engine was able to transform industry, while the aeolipile did not.
Reference:
On the shaduf:
On the power of an ox:
'Scaling up' in your sentence is a taller order than I think you realise. More can always be said on why the aeliopile was, as far as the Classical Period is concerned, a dead end, so if you'd like to write up a post of your own, please don't hesitate to do so! For the meantime, OP, here are some posts for your perusal: