Galileo (1564-1642) took up the work of Tycho Brahe and Kepler and carried it forward to new triumphs. He made the first telescope ever used for astronomical observation, and with it was able to discern that the Milky Way was composed of aggregations of innumerable stars; that the surface of the moon was covered with plains and mountains, that there were four moons revolving around the planet Jupiter, that the planet Venus showed phases like those of the moon as she moved around her orbit, and that there were black spots, at times, upon the sun, which revealed its rotation on its axis. Galileo did equally fundamental work in developing the laws of motion, and the principles of mechanism and physics.
The development of modern mathematics began with three intellectual feats—the invention of the Arabic notation, of decimal fractions, and of logarithms. The notation was derived by the Arabs from India about 700 A. D. They had used numerals long before, but the old system was crude like the systems employed by the Egyptians and Greeks. The Textbook on Mathematics by Mohammed ibn Musa, published at Bagdad about 825 A. D., contained the first notable exposition of modern numerals. This important work gave rise to many more Arabic treatises, some of which showed improved methods.
Decimal fractions were used by the early peoples of central Asia and were transmitted by them to the Babylonians. Their system was based, apparently, upon a sexagesimal scale. Simon Stevin (1548-1620), a Belgian, made great improvements in decimals. He adopted the plan of William Buckley, of England, and other mathematicians, and made the base 100,000, instead of 60.
John Napier (1550-1617), a Scottish nobleman, invented logarithms. The story of this great mathematician's work is one of the most interesting in the history of science. Napier's first table of logarithms was published in 1614. Henry Briggs (1556-1631), professor at Oxford, made suggestions for the improvement of the tables, and persuaded Napier to make the base 10, as is now done in tables of common logarithms. Briggs published tables in 1624 containing the logarithms to 14 places of decimals for the numbers between 1 and 20,000 and from 90,000 to 100,000. Adrian Vlacq (1600-1667), a Dutchman, computed the logarithms of the numbers running from 20,000 to 90,000, and thus completed the whole series of logarithms between 1 and 100,000. Edmund Gunter (1581-1626), of London, calculated the logarithmic sines and tangents of angles for every minute to seven places. He invented the terms cosine and cotangent and used them in a work published in 1620.
Another Englishman, William Oughtred (1574-1660), wrote textbooks on mathematics, and invented numerous mathematical symbols which are now in general use, as well as rectilinear and circular slide rules.
Bonaventura Cavalieri (1598-1647) made many improvements in mathematical formulæ and expounded a new method of indivisibles which solved some of the difficult astronomical problems raised by Kepler, and enabled Torricelli, Viviani, de Roberval, and others to solve abstruse problems relating to all types of curved figures.
Pierre de Fermat (1601-1665), one of the greatest of French mathematicians, developed rules for calculating maxima and minima. His functions in this type of equation closely approached those of the differential calculus. The calculus was developed from Fermat's work by Lagrange, Laplace, Fourier, and other Frenchmen.
Pascal and Fermat developed the theory of probability. Pascal worked out many useful methods for dealing with curves.
The intense mathematical activity in England and France resulting from the stimulation given by the invention of Napier, prepared the way for the discovery of the infinitesimal calculus by Newton and Leibnitz.