The Berlin Academy of Science was founded at a time when Berlin prospered. The population of the city grew from 12,000 in 1670 to 61,000 in 1712, while the University of Halle was founded in 1694, the Academy of the Arts in 1696. Frederick III, elector of Brandenburg, signed the foundation deeds for the Berlin-Brandenburg Society of Scientists (as it was originally called) on his 43rd birthday, 11 July 1700. On 16 November of that year Austria and Prussia signed a secret treaty that permitted Frederick to crown himself king in Prussia, which he did in Königsberg on 18 January 1701. Frederick I of Prussia made Berlin the royal city of residence.
On 12 July 1700, one day after the Berlin-Brandenburg Society of Scientists was founded, Leibniz was named as President of the Society. Leibniz had accepted a position from the Duke of Hanover, Johann Friedrich, of librarian and of Court Councillor at Hanover which he had taken up in December 1676. At the time the Berlin Society was founded, Leibniz was still employed in Hanover, now by Georg Ludwig who had become elector in 1698. Leibniz, as President of the Berlin-Brandenburg Society, spent some time in Berlin but he retained his position in Hanover. He was President for ten terms over the years 1700 to 1711, and this covered a total of three years.
The new Society soon set about the task of having a journal in which to publish the work of the Academy. The first volume of the Miscellanea Berolinensia appeared in 1710 and of the 61 papers printed in this volume, twelve were written by Leibniz. Of course Leibniz was a broad scholar, and these publications were certainly not all in the area of mathematics, but three of them were important mathematical contributions.
Frederick I's son Frederick William I began his reign in 1713 and his son, Frederick II, succeeded him in 1740. Frederick II was a man interested and knowledgeable about philosophy and aimed to reorganise the Berlin-Brandenburg Society of Scientists in Berlin to rival the Academy in Paris. He began to invite top people to participate in his Academy and he approached both Voltaire and Maupertuis in 1740. It was Voltaire who recommended Maupertuis to Frederick for the position of President of the Berlin Academy. In 1741 Frederick II invited Euler to join in the reorganisation of the Society into an Academy. Euler accepted, arrived in Berlin on 25 July 1741, and became Head of the Observatory of the Academy later that year. In a letter to a friend Euler wrote:-
I can do just what I wish [in my research] ... The king calls me his professor, and I think I am the happiest man in the world.
By 1743 Frederick's reorganisation was complete and the Academy was given the title of Académie Royale des Sciences et Belles Lettres. The first session of the new Academy took place in January 1744 and, on 12 May 1746, Maupertuis was officially appointed as president of the Berlin Academy. Euler was appointed as Director of the Mathematics Section of the Academy and, after Maupertuis' death in 1759, Euler took charge of the Academy (although he was never given the title of President). Euler undertook an unbelievable amount of work for the Academy from the time of his initial appointment:-
... he supervised the observatory and the botanical gardens; selected the personnel; oversaw various financial matters; and, in particular, managed the publication of various calendars and geographical maps, the sale of which was a source of income for the Academy. The king also charged Euler with practical problems, such as the project in 1749 of correcting the level of the Finow Canal ... At that time he also supervised the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence.
The book  includes fascinating detail of Euler's day to day life at the Academy. Of the 3000 documents from the Academy containing reports, notes, minutes of meetings, etc. summaries are given of all that mention Euler:-
... these summaries enable one to follow Euler's scientific, organisational and administrative activities between 1741 and 1766 almost day by day. As the editor explains in his preface, new light is shed by some of the documents upon important events in Euler's life: the quarrel, for instance, about Maupertuis' "Principle of least action", or the affairs which resulted in Euler's decision to leave Berlin after 25 years.
Other mathematicians served the Academy over these years. Johann(III) Bernoulli was appointed to the Academy in 1764, at the age of 19, and Frederick II asked him to revive the astronomical observatory of the Academy. In the same year of 1764 Lambert was appointed to the Academy and two years later, on 6 November 1766, Lagrange succeeded Euler as Director of Mathematics at the Academy. Euler had proposed Lagrange for election to the Berlin Academy which had taken place on 2 September 1756. In 1765 Frederick II had arranged for Lagrange to be offered a position in the Berlin Academy but he had turned it down, writing:-
It seems to me that Berlin would not be at all suitable for me while M Euler is there.
However, once it was known that Euler was returning to St Petersburg, Frederick II wrote again in April making a generous offer, and Lagrange finally accepted.
After the University of Berlin was founded in founded in 1809 the Academy had a less important role in the development of mathematics, as the University became a centre of mathematical research. However the Academy continued to be important for mathematics, as well as the full range of other disciplines. For example in  Crelle's contribution to the Academy is detailed:-
Crelle has won a permanent place in the history of mathematics for his scientific and organizational ability. He is especially important for the history of the Berlin Academy of Sciences, to which he was elected in 1827 with the strong support of Alexander von Humboldt. ... This paper is devoted primarily to the study of the recommendations Crelle wrote for prospective members of the Academy. The mathematical papers he read there, as well as the prize-problems he proposed and evaluated for the Academy, are also discussed.
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