The Romans were famously enthusiastic about bathing. Their baths included rooms with water at different temperatures — the cold frigidarium, the warm tepidarium, and the hot calidarium.
If your calidarium is going to be worth its name, you need some system for heating a large volume of water. A miliarium is a Roman hot-water heater. It is a metal tank, typically with lead walls and a bronze bottom, placed above a furnace.
Wait, What's a Miliarium?
"Miliarium", variously spelled with one and two L's, commonly means a milestone marker along a Roman road. However, see this definition:
Palladius and his Opus Agriculturæ
In the 21st century you have the Toilet Guru to explain plumbing technology. In ancient times you had Palladius to explain everything.
Rutilius Taurus Æmilianus Palladius, usually known simply as Palladius, was a member of a prominent family in Roman Gaul. That is, today's France. In the late 4th century or the first half of the 5th century he wrote Opus Agriculturæ, sometimes known as De re Rustica.Opus Agriculturæ
The Opus Agriculturæ is a manual on farming as done by the Romans around the year 400 CE. It takes the form of 14 books. The first book is general, an introduction and overview. Books 2-13 then provide detailed instructions for each month of the year. Book 14, only rediscovered in the 20th century, describes the care for animals and veterinary science.On
Opus Agriculturæ was well-known in the Middle Ages. We have a Middle English verse translation from about 1420, On Husbondrie. Several works from the 13th century are based on Palladius' work, and the original was reproduced throughout the Middle Ages and into the Early Modern era.
Of The Baths
Section 40 of Book 1 is Of The Baths. Here is the 1807 translation by T. Owen of Queen's College at the University of Oxford.
It is not foreign to this business, if there is plenty of water, that the master should think of building a bath, which greatly conduces to pleasure as well as to health. Let us therefore build a bath in that situation where there is a degree of heat, the place being free from dampness, lest the moisture being near the furnaces render it cold. We are to give it light from the south and from the west in winter, that it may be benefited and enlightened all day by the sun. But you are to make the bathing-cells thus.
You first of all pave the area with tiles; but let the paving be inclined to the furnace, that, if you erect a buttress, it may not stand within, but recede towards the furnace: thus it will happen, that the flame, by rising high, may make the cells heat with greater efficiency.
Above this paving, let the buttresses be made of brick and well-wrought clay mixt with hair, a foot and a half distant from each other, two and a half in height.
Upon these buttresses, let two sequare tiles by placed, and let testaceous pavements be laid over these; and then, let marble be set on, if you can get it.
Let a miliarium of lead, which has a copper bottom, be fixt between the baths, on the outside, with a furnace underneath; to which miliarium let a pipe, for conveying cold water, be directed, and from this let a pipe of the same size proceed to the bath, which will convey as much hot water as the other pipe conveyed of that which was cold.
Let the cells be thus laid out, that they may be perfectly square; but, for example, that they may be fifteen feet in length and ten in breadth, for the vapour will make its way more effectually between the narrow passages.
Let the form of the baths themselves be according to every man's choice. Let the cold-water cells be enlightened from the north, in summer-baths; from the south, in winter-baths. If it may be done, let the baths be so built, that all the filth may run down through the gardens.
If the rooms in baths be made of Signine work, they are stronger; but those that are made of boards, are supported by transverse iron-rods and arches. But if you are unwilling to lay boards over the arches and rods, you may lay square tiles, fastened with iron crooks, held together by well-wrought hair mortar; and you then lay on a testaceous coat: you will then decorate it with white wash.
We may also, if we study our interest, build winter-edifices over the baths; then we have warmth under our habitation, and we gain the benefit of a foundation already laid.
In Section 41 he goes on to describe ways of sealing the cracks and mortar joints with pitch, wax and various water-proof mortar replacements.
The Hypocaustum and Miliarium
"On Ventilation and Warming of Buildings, and Heating of Baths, as practised by the Ancients"  describes Roman heating systems. The hypocaustum was the furnace itself. It was constructed below the lowest occupied level of the structure, so as to heat the floor. The chimney would go up through the center of a room, as chimney flues within walls were only introduced into Britain in the 11th or 12th century.
The miliarium was the water heater. This was a metal vessel. Water was put into the miliarium, and heat was transferred from the hypocaustum to the water. The water was then drawn off to the bath.
A more sophisticated system used a series of three water heating tanks. Cold water was supplied to the frigidarium tank from a spring, well, or stream. Water flowed from the frigidarium tank to the tepidarium tank, where it was heated further, and from there to the calidarium tank where it became even hotter. Cool, warm, and hot water was then drawn from those three tanks to rooms with the same names.
A simpler system heated the three rooms to varying temperatures based on their distance from the center of the hypocaustum, with one miliarium providing hot water used in varying proportions.
A different translation of Palladius describes the miliarium thus:
We place on the outside between the bathing vessels a hollow column of lead with a brazen flat bottom, and under it a furnace. Into this column the pipe of the cold water cistern is directly led, and from it another pipe of the same calibre proceeds to the bathing vessel, which conveys into it as much hot water as is brought in a cold state to the column by the pipe from the cold water cistern.