Mazarin Cut Diamond
The Mazarin Cut Diamond is often considered to be the first true brilliant cut and the earliest predecessor of the modern Round Brilliant Cut diamond. The Mazarin Cut Diamond has 17 crown facets and 17 pavilion facets and has also been known as the "double cut brilliant", as it was the next stage in diamond cutting 'evolution' after the Single Cut. The Old Single Cut had 9 crown facets.
The Mazarin Cut was developed in the mid 17th century and is accredited to Cardinal Jules Mazarin, a powerful political figure in the court of France, who is said to have been the first to commission a diamond to be cut to this shape. However, the term "Mazarin Cut" is a modern one and does not appear in literature (no mention in Google Books search) prior to the 20th century; I have not so far been able to ascertain the source of the legend. The Mazarin Cut was also once referred to as the "Brilliant en seize" - "The Brilliant in sixteen." A further name for it is the Lisbon Cut - although from online sources it is not completely clear whether this is exactly the same cut or a variation.
As the Mazarin cut was an advance on the old single cut, so it too was superseded in time by a more complex cut: It was from the Mazarin Cut that diamond cutter Vincent Peruzzi of Venice developed the Peruzzi Cut, also known as the Triple-Cut Brilliant or Old Mine Cut. Each of these successive cuts significantly increased both the number of facets and the brilliance of the finished gems, triggering the "quest" to maximize brilliance and "sparkle" that led at last to the ideal cuts of today, with angles precisely optimized for light performance within the stone.
However the early 'proto-brilliants' such as the Mazarin Cut lacked the symmetry and precision of modern diamond cutting. They were also modified squares, rather than rounds; and this was because the modern bruting process had not yet been developed. Bruting is a process by which a diamond is set on a lathe and another is brought gradually into contact with it. The diamonds grind each other, resulting ultimately in a rounded / conical shape.  Modern diamond bruting is performed using computer-controlled machining, which controls the speed, pressure and angles with very high precision. 
The "Mazarin Diamonds"
Cardinal Mazarin was a famed, wealthy courtier of the 17th century. Mentioned by Alexandre Dumas in the famous "Three Musketeers" books, Mazarin was a contemporary and associate of Cardinal Richelieu, and something of a character: Despite being a Cardinal of the Holy Roman Catholic Church, he was heavily into gambling and was also said to have 'flirted' with the Queen of France, winning her favor at court through flamboyant gifts! He was also a serious aficionado and avid collector of diamonds. At his passing in 1661 (or 1662 according to some sources), he left a spectacular collection of 18 large diamonds to the French crown, with one condition: That these diamonds would forever be known as "The Mazarins". Eight of the diamonds in this collection were over 20 carats.
It's important to note that none of these diamonds was Mazarin cut - and to distinguish clearly between a "Mazarin Cut diamond" and "The Mazarin Diamonds". Of the 18 Mazarin Diamonds, it is interesting to note the prevalence of shapes as recorded in 1691: Eleven of the diamonds were square / table cut, two were "almond shaped", three were Heart Cut, one was pear cut and one was Marquise Cut. 
Many of the Mazarin Diamonds were stolen in the notorious robbery of the Garde Meuble (Public Treasury) of Paris in 1792. On the night of September 16th-17th 1792, thieves scaled the colonnade of the building facing the palace of Louis XV, climbed through a window and broke into cabinets containing the crown jewels, coronation regalia and other famous gems of incalculable value. Six of the Mazarin Diamonds were later recovered, but twelve were never found and have vanished into legend. It is most likely that they were recut into other shapes / smaller stones in order to re-enter circulation without detection. Every so often, a diamond surfaces with the claim that it was one of the Mazarins; however none has been verified and none is now likely to be, as accurate identification would probably be impossible.  
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Note - this website is intended for general informational and entertainment purposes, and should not be considered to be professional consultation. If you are considering purchasing precious stones, be sure to seek the advice of a qualified professional.