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SHORT ANSWER

In your case there was a man William who was William III, Prince of Orange, Stadtholder of the Netherlands, and King William III of England (1650-1702) and there was another man known as William III, King of the Netherlands (1817-1890). They were two different members of the same dynasty who were both the third William to hold their position but those were different positions.

LONG ANSWER

To make a long story short, in many cases regnal numbers are all messed up.

As far as I know the practice of giving rulers regnal numbers originated in western Europe, and the practice of giving those regnal numbers in the form of Roman numerals clearly originated in western Europe and spread around the world.

And for most rulers in history the regnal numbers were assigned by historians centuries or millennia after those rulers lived. Various European kings began to use official regnal numbers about 1500 to 1700.

In the middle ages it was very rare for a western European king to use an official regnal number.

Karl Knutsson Bonde was a Swedish noble who became King of Sweden three times (1448-57, 1464-65, & 1467-70) and King of Norway 1449-50. He was the first King of Sweden to have a regnal number, being described as Charles II on his wife's tombstone in 1451. And he actually was the second Swedish king named Karl or Charles.

But in 1554 the Historia Omnibus Gothorum Sueorumque Regibus (History of all Kings of Goths and Swedes) by Johannes Magnus was published with thousands of years of fictional history of fictional kings of Sweden. So King Charles Vasa (r. 1604-1611), accepting that false history, called himself Charles IX instead of Charles III, making Charles II Charles VIII, and so on.

Charles IX's brother Erick (r. 1560-68) had set an example by calling himself Erik XIV instead of Erik VII. So the Regnal numbers of all Swedish kings named Erik or Charles are historically inaccurate.

Who was king of Sicily before Roger II? Roger I? Nope. Count Roger II of Sicily made himself the first King of Sicily but he is not listed as King Roger I because that might get him confused with His father Count Roger I of Sicily.

Frederick, king of Sicily from 1295-1337, called himself Frederick III. But there was only one previous king Frederick of Sicily, who reigned from 1198-1250. Since that previous King Fredrick was better known as Emperor Frederick II, Sicilian King Frederick II called himself Frederick III. The next King Frederick of Sicily ruled from 1355 to 1377 and never used a regnal number, but is known to modern historians as Frederick III.

Edward Longshanks, king of England, reigned 1272-1307, his son Edward of Caernarfon r. 1307-1327, and Edward of Windsor r. 1327-1377. Edward of Windsor's oldest son was Edward Prince of Wales (1330-76), whose oldest son was Edward (1364-1371) - the last two Edwards never ruled but they could have been expected to. So during the reign of Edward of Windsor from 1327-77 it became the practice to informally call Edward Longshanks Edward the First, Edward of Caernarfon Edward the Second, and Edward of Windsor Edward the Third whenever they needed to be distinguished.

This informal numbering became official in 1547-53, during the reign of Edward, son of Henry VIII, whose title was:

Edwarde the Sixth, by the Grace of God, King of Englande, France, and Irelande,
Defender of the Faith, and
of the Church of Englande, and also of Irelande, in Earth the Supreme Head

http://eurulers.altervista.org/england.html1.

And that ignored the previous reigns of Edward the Elder, King of the Anglo-Saxons (r.899-924), Edward the Martyr, King of England (r.975-978), and Edward the Confessor, King of England (r.1042-1066).

As this extremely brief discussion shows, the regnal numbers of rulers are often weird and mixed up, so expecting them to always make sense is not realistic.

Added 05-18-2019

The first western European rulers to use official regnal numbers were the Holy Roman emperors and the popes.

The numbering of the popes is rather confusing and messed up because there were a number of times when two or more persons claimed to be pope at the same time. So the medieval papal curia eventually began began making official lists of all the men in the past it considered to be popes. And papal claimants not considered to be rightful popes were considered antipopes.

But over time various persons have been moved from the list of popes to the list of antipopes, or from the list of antipopes to the list of popes. And there are questions whether someone elected pope but never consecrated counts as a pope or not. And there are controversies whether some alleged popes ever actually existed. There was the famous legend of the female Pope Joan believed for many centuries, for example.

The numbers of popes named John are so messed up there is an article in Wikipedia about "Pope John numbering". To make a long story short, when Pedro Juliani became pope in 1277 and chose John as his throne name, he should have used the number XIX, but chose XXI by mistake. Thus there has never been a pope who called himself john XX. https://en.wikipedia.org/wiki/Pope_John_numbering#John_XX1

In James Branch Caball's fantasy novel Jurgen: a Comedy of Justice (1919) there is a scene when Jurgen tries to enter Heaven, claiming to be Pope John XX. When told there wasn't a John XX, he asks how there could have been a John XXI without a John XX.

I note that both Antipope Baldassare Cossa in 1410-15 and Pope Angelo Guisseppi Roncalli in 1958-63 used the throne name John XXIII.

The official ideology of the Holy Roman Empire was that the emperors were the successors of the Roman emperors from 27 BC to AD 395, of the eastern Roman or "Byzantine" emperors from AD 395 to 797, and then of Charlemagne crowned in 800.

Modern rulers are listed by their personal names followed by the regnal number. But that isn't the case with the classical roman emperors.

For example Imperator Caesar Marcus Aurelius Valerius Claudius Augustus (r. 268-270) is often listed as Claudius II Gothicus, though his personal name was Marcus. Claudius I would presumably be Tiberius Claudius Caesar Augustus Germanicus (r. 41-54) whose personal name seems to be Tiberius.

Imperator Caesar Flavius Claudius Constantinus Augustus (r.407-411) was an usurper proclaimed in Britain in 407 who was recognized as a legitimate co-emperor by western emperor Honorius in 409. He is usually listed as Constantine III. His son and co-emperor Imperator Caesar Flavius Constans Augustus is usually listed as Constans II.

Eastern Roman emperor Heraclius Novus Constantinus Augustus (r. 641) is usually lasted as Constantine III, thus making two Roman Emperors Constantine III. Imperator Caesar Flavius Constantinus Augustus (r. 641-668) is usually listed as Constans II, thus making two emperors Constans II. Constans II is sometimes counted as Constantine III.

The last emperor in Constantinople is usually listed as Constantine XI (r. 1449-1453). But he is often called Constantine XII, counting Constantine Laskaris who may have been emperor for a day in 1204. Counting all co-emperors named Constantine, Constantine XI could be numbered as Constantine XVIII, counting all Constantines who were supreme rulers, Constantine XI could be numbered as Constantine XIV. https://en.wikipedia.org/wiki/Constantine_XI_Palaiologos2

Imperator Caesar Marcus Julius Philippus Augustus (r. 244-249) is usually called Philip the Arab, and his son and co-emperor Imperator Caesar Marcus Julius Philippus Severus Augustus is usually called Philip II, even though their personal names were both Marcus.

King of the Romans Philip (r. 1198-1208) is sometimes listed as Philip I but called himself Philip the Second at least once:

Philippus secundus divina favente clementia Romanorum rex et semper augustus

http://eurulers.altervista.org/emperors.html3

So presumably he counted Philip the Arab as Philip I.

From 962 the emperor of the Romans was also the king of Germany and of Italy or Burgundy, and from 1032 also the king of Arles or Burgundy. Since the three kingdoms had different rulers before becoming united with the emperorship it is possible for an emperor of the Romans to have different numbers as emperor, and as king of the three different kingdoms.

Because there were kings of Germany named Conrad and Henry before 962, German historians listed holy Roman Emperors named Conrad and Henry with their regnal numbers as kings of Germany, one digit higher than their true ordinal numbers as emperors, and those are the numbers those emperors are usually known as. You should subtract one from the regnal numbers of emperors named Conrad or Henry.

Someone elected emperor would take the title of Rex Romanorum et semper Augustus King of the Romans and always emperor/imperial, until he was crowned emperor in Rome by the pope, when he would take the title of Imperator Romanorum et semper Augustus Emperor of the Romans and always emperor/imperial.

If someone became king of the Romans when there was no emperor, he immediately gained all the powers of the emperor. If someone was elected king of the Romans during the lifetime of his father the emperor he didn't gain any power at the time but gained the full powers of the emperor as soon as the emperor died.

In the time of Emperor Henry IV, who really was Henry III, his son Conrad was elected king of the Romans, and later deposed for rebellion against his father. Then in 1138 Conrad of Hohenstaufen became King of the Romans, usually listed as Conrad III, though he could also be counted as Conrad II or Conrad IV. When Conrad went on crusade in 1147 he had his son Henry Berenger elected Co-king.

The later King of the Romans Conrad IV could also be counted as Conrad III or Conrad V.

There were six Holy Roman Emperors named Henry: Henry II (I) (r.1014-24), Henry III (II) (r. 1046-56), Henry IV (III) (r. 1084-1105), Henry V (IV) (r. 1111-1125), Henry VI (V)(r. 1191-1197), and Henry VII (VI) (r. 1312-13).

Henry, son of emperor Frederick II, was elected King of the Romans in 1220 and deposed in favor of Conrad IV in 1235, and is sometimes called Henry VII or Henry (VII).

So depending on whether King Henry I of Germany, Henry Berenger, and Henry (VII) are counted, the Emperor usually called Henry VII could be counted as Henry VI, Henry VII, Henry VIII, or Henry IX.

There were only three emperors named Frederick: Fredrick I Barbarossa (r. 1155-1190), Frederick II Stupor Mundi (r. 1220-1250) and Frederick III (r. 1453-1493). But in 1314 Duke Fredrick I of Austria was elected rival king of the Romans and is sometimes listed as Frederick III. In 1400 nobles met at Frankfurt, complaining about King of the Romans Wenceslaus who seemed like "Bad King Wenceslaus" to them. Duke Frederick I of Brunswick-Luneburg was murdered when returning from that meeting, and there is a story that he was elected King of the Romans there. So possibly Frederick III should be listed as Frederick IV or Frederick V.

Students of German history also have a problem with the German emperors. Wilhelm (William) I (1871-1888) & Wilhelm II (1888-1918) were the only two German emperors and the only two kings of Prussia named Wilhelm. But Count William of Holland was elected King of the Romans in opposition to Emperor Frederick II in 11247. So the German emperors could have been called Wilhelm II & Wilhelm III.

Frederick, son of Wilhelm I and father of Wilhelm II, is also a problem, being the third king of Prussia and first German emperor named Frederick. If he is called Frederick I he can be confused with Fredrick I Barbarossa (r. 1155-1190), and if he is called Frederick III, as is the usual rule, he can be confused with Frederick III (r. 1453-1493).

So this brief discussion should show that not only is the numeration of many monarchs rather messed up, in some cases it would be nearly impossible to replace the present numeration with a totally correct numeration.

Your example seems incorrect. William of Orange was William III of England, whereas the other was William III of the Netherlands.

William of Orange was also William III in the Netherlands, but as Stadtholderate under the House of Orange-Nassau, while the other William III you found was King of the Netherlands.

Lastly, as point out by LangLangC in the comments, the III part that's appended to the name is used by others to distinguish that person from others who had an identical name.

I'd stick my neck out and suggest that there are no examples of Regnal numbers that match two rulers unless you look at different polities (kingdoms, duchies, etc.) -- in which case there are plenty. In the specific case of the Netherlands, where they reset the numbering as the government form changed shape, there were also two William I and William II.

You might want to consider Denmark and the House of Estridsen:

• 
  Valdemar III the Young (co-ruled 1215 - 31)


• 
  Valdemar III Eriksøn von Schleswig (ruled 1326 - 29)

1. Valdemar the Young or III was co-ruler with his father Valdemar II from 1215 to 1231. Valdemar the Young

is sometimes referred to as Valdemar III for example his tombstone
reads in Latin: Waldemarus Tertius Rex Daniae, Filius Waldemari
Secundi "Valdemar the Third, King of Denmark, son of Valdemar the
Second". Although Valdemar III is more commonly used to denote a later
king, Valdemar of Schleswig.

Source: fynhistorie.dk

The popular Valdemar died when he was 21 or 22 after being shot in a hunting accident. He was succeeded by his brother Eric IV. He, in turn, was succeeded by another brother, Abel, from whom Valdemar III Eriksøn was descended (see below).

2. Valdemar III Eriksøn (who was also Duke Valdemar V of Schleswig) was King of Denmark as a minor from 1326 to 1329 with his uncle, Gerhard III of Holstein-Rendsburg, as regent. He replaced the unpopular Christoffer II who had been ousted by Gerhard III and other powerful nobles in 1326, but further rebellions led to the removal of Valdemar III and the reinstatement of Christoffer II in 1329.

Valdemar III Eriksøn, who reigned until the age of 15, was a descendant of King Abel of Denmark (reigned 1232 to 1252). Abel was a son of Valdemar II and half-brother of Valdemar the Young or III (see above).

There were two Henry VII rulers as kings of Germany, the first of whom co-ruled with and pre-deceased his father. He is often referred to as Henry (VII) to distinguish him from the later Henry VII, much admired by Dante.

Some kingdoms do not number crowned kings who co-ruled with their fathers but never in their own right. Thus, Henry II of England's son Henry is never referred to as Henry III or Henry (III), but only as Henry the Young King. (Perhaps it helps that "young king" rhymes with "atheling".)

As another example, popes sometimes intentionally duplicate a supposed predecessor's number to indicate they don't consider that person actually to have been pope. The most recent example of this is John XXIII, whose very choice of name clarified the official position on the Pisan line of supposed popes during the Great Western Schism over half a millennium earlier.

Going back to Germany, there were three monarchs of Germany named Frederick III. The first was a 14th century king who co-ruled with and predeceased his cousin, and his number was then ignored by the 15th century Frederick III. And finally in the 19th century we have another Frederick III of Germany because this Germany traced its political roots and king numbering not to medieval Germany but to Prussia.

Within the Roman/Byzantine empire there were:

• Constantine III (Western Roman Emperor)


• Constantine III (Byzantine emperor)

Disputable because one may claim that these were two distinct political entities, or that the Western Roman Emperor was a usurper. Also Roman/Byzantine imperial names are a complicated matter - e.g. Caligula would have been known as "Gaius Julius Caesar Germanicus" during his reign.