You have to understand that even in an absolute monarchy, Kings and Queens who were underage nearly always had a regent, that is, someone who would run the Kingdom until they were an adult (Whatever age that may be in the culture/society they were in).

I have never heard of a single underaged King/Queen who've had an actual effect on the day-to-day running of a country, and to be fair, it's somewhat of a silly idea. People in the past were not unreasonable or fundamentally different from us now, if they had seen something like this, they probably would've had the same thoughts as us; that is, that it'd be unreasonable. Can you imagine someone taking a 6-year old heir to the throne who says 'make me ice cream for every meal' and being forced to do such a thing?

Hobbes did, in Chapter 19 of Leviathan. He didn't so much call it absurd, as discuss how the potential for an infant to succeed a monarch distributes the genuine power of sovereignty to the regent and/or advisors.

Keep in mind the context of his writing, also. In 1651, his native England had just been through a series of civil wars between Parliament and the monarchy. Hobbes was writing in France, since his association with royalists (one of his patrons was a staunchly royalist earl) came with significant danger.

Not sure I understand what you mean by "madness" or "appointing". Monarchs succeeded, they were normally not appointed. This was not considered mad or sane; it was simply the law of the land.

Very young heirs could be a problem though. Ecclesiastes 10:16 says: "Woe to thee, O land, when thy king is a child". This was often quoted by medieval commentators. The reason why this was problematic, was that it would often lead to infighting within the political elite over who would control government, i.e. act as regent(s).

Furthermore, the regents were only expected to act as caretakers, and not pursue active political agendas the way a ruler would. This could lead to stagnation in both domestic and international affairs.

A prime example of this would be Henry VI of England. Henry became king at nine months, and his long nonage has often been blamed for England's turn of fortune in the Hundred Years' War.

The real problem with Henry VI, however, was that his childhood never really ended, though he grew to be 49. But that's another story...

/u/Skyicewolf 's answer is correct - that when an underage monarch ascended to the throne, a regent or regency council would generally rule in their name. More generally, in the Middle Ages a country per se without a king was more or less conceptually impossible - kingship was thought to be the natural state of affairs, and kingless "nations" (in the Latin sense of the term) were considered sort of a logical paradox (eg, the case of "Commonwealth"-era Iceland.) Medieval corporatist theory (or assumptions, really, as thinkers didn't tend much towards introspection in this aspect of political thought) held that society was innately hierarchical, and that a king as "lex animata" (the animate law) was more or less a prerequisite for its functioning. So in response to your question, a medieval observer would likely have asked "How could we not have a king?"

(Certainly there were kingless societies - much of Italy, the later medieval Hanseatic free states - and to be honest I'm not really sure how their existence was justified. It's not something I've studied much.)

In Britain from 1751 to 1937, whenever a child was heir to the throne there was specific pre-emptive legislation outlining who would be regent in the particular circumstances. From 1937 general, rather than specific, rules deal with such cases.

The first act solely concerning arrangements for regency during a child's rule was the Minority of Successor to Crown Act 1751, which dealt with the 12-year-old Prince George (later George III), whose father Frederick (King George II's eldest son) had died while George II was still alive.

Once King George III had children, there was another act concerning what would happen if he were to die while they were still young: the Minority of Heir to the Crown Act 1765.

(The first acts solely about the regency of minority rulers concerned George III - before and during his reign - so it's interesting to note that he endured the only period of regency in modern British history at the end of his reign (1811 - 1820).)

The next potential child-ruler was Queen Victoria, and a Regency Act was passed in 1830 in anticipation. However, she turned 18 on 24 May 1837, one month she became queen on 20 June 1837, and regency was avoided.

The final case-specific Regency Act was passed in 1910 and concerned the possibility of George V being succeeded by the then 16-year-old Edward (later Edward VIII). George V died in 1936; thus regency was well avoided.

The Regency Act 1937 (passed in anticipation of the accession of the then under-aged Elizabeth) laid down general rules. This act (amended in 1943 and 1953) forms the basis of what would happen today if Queen Elizabeth II, Prince Charles (her eldest child), and Prince William (his eldest child) all lost their place in the line of succession before Prince George (William's son) came of age.

Thomas Paine gave a great breakdown of the arguments in favor of and opposing monarchy in Common Sense. The frequency of children inheriting the throne comes up. In fact it's one of his arguments for why a monarchical government is less likely to produce capable leaders than an elected one:

"Another evil which attends hereditary succession is, that the throne is subject to be possessed by a minor at any age; all which time the regency acting under the cover of a king have every opportunity and inducement to betray their trust. The same national misfortune happens when a king worn out with age and infirmity enters the last stage of human weakness. In both these cases the public becomes a prey to every miscreant who can tamper successfully with the follies either of age or infancy.

The most plausible plea which hath ever been offered in favor of hereditary succession is, that it preserves a nation from civil wars; and were this true, it would be weighty; whereas it is the most bare-faced falsity ever imposed upon mankind. The whole history of England disowns the fact. Thirty kings and two minors have reigned in that distracted kingdom since the conquest, in which time there has been (including the revolution) no less than eight civil wars and nineteen Rebellions. Wherefore instead of making for peace, it makes against it, and destroys the very foundation it seems to stand upon."

Given the popularity of Common Sense at the time of its publication, it's fair to assume that Americans by this time held similar views.

Weren't there cases of this in China? I recall there was a child emperor during the Mongol invasion however I don't know if they had any authority over their appointed ministers or not.

To tack another question onto this one. Was it common to question the idea of a monarchy full stop?