By the rules of primogeniture as later observed in choosing monarchs in England and Britain, the successor to Henry I should have been his daughter Matilda or Maud (Matilda was her name in Latin, Maud in French). She was born in 1102 and married at the age of 12 to the Holy Roman Emperor, Henry V. After his death in 1125, she continued to be known as the Empress. Her brother having been killed in the disaster of the White Ship, her father called her back to England and had his barons swear an oath of loyalty that they would recognise her as queen after her father's death. In the meantime she married again, this time to Geoffrey IV, Duke of Anjou, in 1128. Unfortunately, the Normans and Angevins did not get on and so many of the Normans in England, who after all had not conquered England that long ago, were worried that Angevin influence would become paramount in England if she became queen.
When Henry I died in 1135, Matilda's cousin Stephen of Blois (the son of Henry I's sister) moved quickly to take the crown for himself, with the support of many in the nobility and the church. She launched an invasion with her half-brother, Robert of Gloucester in 1139 and civil war began. Stephen was captured by her forces at Lincoln in February 1141, and Henry of Blois, bishop of Winchester, changed sides to join Matilda. Matilda went to London but made herself so unpopular that she was chased out of the city. After Robert of Gloucester was captured at the battle of Winchester in September 1141, Matilda had to agree to an exchange of prisoners, swapping Stephen for Robert.
William of Newburgh's contemporary history is at the Medieval Sourcebook. (illustrations from wikipedia in the public domain)
More details of Stephen and Matilda's struggle for the throne are narrated in this, the first of two YouTube video clips on the subject.
Cambridge's Fitzwilliam Museum has an online gallery of coins issued by Stephen and Matilda
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