The Spanish Royal Academy of Sciences (The Royal Academy of Exact, Physical and Natural Sciences of Madrid) was founded in February, 1847. This was, however a reconstitution of the Academy, the origins of which go back to the reign of Alfonso el Sabio in the 13th century, to the Madrid Mathematical Academy of Phillip II, and also to the Academia Naturae curiosorum founded in 1657.
The Madrid Mathematical Academy (Academia de Matemáticas de Madrid) was founded in 1582 (some sources give the date 1575) by Phillip II of Spain. Its main work consisted of navigation, architecture and engineering, with experts in these areas working for the King. The Academy, which ceased to exist in 1625, is examined in detail in .
The Royal Academy of Exact, Physical and Natural Sciences of 1847 was largely due to Jorge Juan y Santacilia (1713-1773) who was an expert on navigation. He held weekly sessions in his home in Cádiz discussing mostly military and navigational affairs. It was Jorge Juan who drew up a detailed plan for the constitution and by-laws of the Spanish Royal Academy of Sciences (see  for details). The Royal decree, setting up the Academy, is dated 25 February 1847.
The foundation decree states:-
... the Exact, Physical and Natural Sciences so strongly influence the industry and prosperity of nations, but, unfortunately, they did not occupy the pre-eminent position in our old system of education that should be their right.
It explains that experts do not work most profitably in isolation but:-
... it is necessary that leading experts meet to confer with each other, to communicate their observations, to give mutual help and, finally, to establish extensive correspondence with other leading experts and major Academies in the world, in order that this immense range of ideas and discoveries spreads knowledge and increases the treasures of science.
There is then an interesting reference to the earlier Madrid Mathematics Academy that we mentioned above:-
Several times Spain tried to follow such praiseworthy examples, and our nation even went ahead of all the rest, since from the year 1580, that is to say, long before the famous Societies of Paris and London were established, already in Madrid there existed a Royal Academy of Sciences .... It had, nevertheless, a very ephemeral existence, being extinguished by the Austrian dynasty, and no memory of it has remained.
The Royal Academy of Exact, Physical and Natural Sciences was explicitly declared to be the equal of the other Spanish Academies of History and San Fernando. It also formally closed the Academy of Natural Sciences of Madrid. The Academy was set up with 36 members and given the power to elect future members. The newly formed Academy was asked to draw up its own statutes and it did so having them approved by Royal Order on 23 December 1847.
Today the Academy consists of 54 full members, 90 national members, and a number of foreign members. The Academy publishes a number of important journals, for example the mathematics journal Memorias de la Real Academia de Ciencias Exactas, Fisicas y Naturales de Madrid. Serie de Ciencias Exactas and Real Academia de Ciencias Exactas, F’sicas y Naturales de Madrid. Revista.
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