First, let us define monarchy. It is a system of government where the leader inherits his/her position and/or where the leader or the leader's family chooses his/her successor. Republicanism is where the leader is selected in some other way. In a democratic republic, the leader is chosen directly or indirectly by the citizens, while in an oligarchic republic, the leader is chosen by some high-level committee that is not a family. It's possible for a nation to have a mixture of the two systems, and some monarchies are essentially crowned republics, where the monarchs are figureheads, professional socialites and celebrities.
Monarchy has been around for at least 5000 years; most large-scale societies have entered recorded history as monarchies. This includes those with the longest recorded history, Egypt and southeast Iraq. Some monarchies have been VERY long-lived, even if rather discontinuous. The Pharaonic monarchy lasted for over 3000 years; the last Pharaoh was Cleopatra VII (the famous one), who ended the line by committing suicide in 30 BCE, avoiding being captured by the oncoming Roman armies. The Chinese monarchy lasted over 2500 years, ending in the abdication of Puyi in 1911 CE. The Roman Empire was a monarchy, of course, with the western half lasting 500 years and the eastern half lasting 1500 years.
But not all societies were monarchies during all this time. The classical-era Greek city-states had rather tumultuous politics, with the kings often getting sidelined or deposed, and various citizens' assemblies and committees and strongmen often ruling. It was there that the word "democracy" was invented, though it was often a put-down meaning roughly "mob rule". However, this political experimentation did not survive for long under Philip of Macedon and his successors. Rome became a republic around 500 BCE, replacing its king with two consuls elected for a year, but as Rome ruled more and more territory, this caused social stresses and strife and civil war. It gradually ended from 49 BCE, when Julius Caesar became strongman ruler, to 25 BCE, when the Roman Senate gave Octavian the title Augustus.
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But over the last few centuries, something remarkable happened. Many monarchies either fell or became crowned republics. Let's see what happened. For a long time, republics did not grow very large, and the longest-lived republic larger than a city-state has been Switzerland at 700 years. But in the 1770's and 1780's, some rebellious North American colonies created a republic that dwarfed Switzerland: the United States of America. Its first leader, George Washington, had no desire to make himself King George I.
This event was quickly followed by the French Revolution, but that was a poor advertisement for republicanism, and France alternated between monarchy and republicanism before settling down as a republic in 1870. Most 19th century European nation-builders wanted monarchs for their nations, all the way up to the Great War, as it was known back then. Belgium, Holland, Norway, Finland (almost), Italy, Serbia, Albania, Romania, Bulgaria, Greece.
But that war destroyed four great monarchies and deposed lots of princelings. The Hohenzollerns were gone from Germany, the Habsburgs from Austria and Hungary, the Romanovs from Russia, and the Osmans from Turkey. Their successors created republics, and those nations have been republics ever since. New nations were created, and they were almost all republics: Czechoslovakia, Poland, Lithuania, Latvia, Estonia. Serbia's monarch became Yugoslavia's, however. Later in the 20th cy., other monarchies would be deposed: Italy, Yugoslavia, Albania, Romania, Bulgaria, Greece.
Though Britain elevated various local princelings to monarchs in the Middle East and North Africa, some of those monarchies have also fallen: Egypt, Iraq, Yemen, Libya. Most of the rest of the former British Empire is either crowned republics (Canada, Australia, New Zealand, etc.), or pure republics (the US, South Africa, India, Kenya, ...). Most of post-colonial Latin America and Africa has been republics, though often not very democratic ones. In Asia, both Nationalist and Communist China are pure republics, Japan is a European-style crowned republic, and South Korea is a pure republic.
The only restored monarch I know of over the last century is Spain's. For all the rest, when a monarchy has been deposed, it has stayed deposed. However, some leaders have made themselves monarchs.
- North Korea has long been the world's only Communist monarchy (Kim Il Sung - Kim Jong Il - Kim Yong Un), though Cuba may now be competition (Fidel Castro - Raul Castro).
- Hafez Assad of Syria has been succeeded by his son Bashar.
Also some abortive ones:
- Saddam Hussein had wanted to be succeeded by either of his sons, Qusay or Uday, but they are all dead.
- Muammar Khadafy had likely wanted to be succeeded by his son Saif al Islam, but Muammar is dead, Saif al Islam is in hiding, and his other sons and his daughter are either dead or in exile.
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Is this a reasonable assessment? Has anyone tried to guess why monarchy has lasted so long or why it went out of style so rapidly? I wouldn't know where to look for discussions of such questions.