The microscopical investigations of Malpighi, Kircher, Leeuwenhoek, Grew, and Hooke opened up an immense field for research. They developed microscopical chemistry and anatomy, and changed the prevailing ideas regarding animal and vegetable tissues. The sciences of mineralogy, botany and entomology were benefited and the medical sciences were practically revolutionized. The first publications of the Royal Society show the widespread attention microscopical and telescopic studies were then receiving. [Illustration: Copyright, Keystone View Co. WEATHER AND ASTRONOMICAL INSTRUMENTS ON THE ROOF OF GREENWICH OBSERVATORY, ENGLAND] [Illustration: Courtesy "Aeronautics," London A MOORING TOWER FOR AIRSHIPS, WITH THE R-24 FASTENED HEAD ON] Francis Bacon (1561-1626), René Descartes (1596-1650) and Gottfried Leibnitz (1646-1716), in England, France, and Germany, respectively, lent powerful aid to the advance of science at this time. Bacon's great learning enabled him effectively to describe scientific methods and to direct scientific criticism. He attracted general attention to scientific methods based on inductive processes. Descartes, seeing that the world's best intellects had long been exercised with philosophy and metaphysics, without discovering anything with certainty, resolved to accept no beliefs upon the authority of any name or reputation. He would reach his own conclusions based upon the scrupulous examination of data. He hoped to solve the mysteries of nature by the aid of mathematics and geometry, and developed the Cartesian philosophy. The mathematical works of Descartes are now better known than his general scientific ideas. He published in 1637 his "Discourses on Method" and on Geometry. In the last-named work, suggestions are given for the development of analytic methods. It has been said of his formulæ that they are even cleverer than himself. The general use of his analytic methods by other mathematicians resulted in the solution of many scientific problems that had been handed down for centuries as insoluble. Descartes also advanced algebra. The application of the doctrine of curved lines to algebra greatly enlarged the scope of its usefulness. In making these innovations, Descartes introduced the methods and symbols of modern exponential notation. The English mathematician Wallis was also an important agent in the development of mathematical notation. He based his work on the Greek notation and that of Nicolas Chuquet (1484), J. Bürgi, Thomas Harriot (1631), Johann Hudde (1659), and others. Descartes was familiar with the writings of these scholars and, undoubtedly, was influenced by them. Roberval, Fermat, and Pascal were contemporary mathematicians in France and left great names in the history of the mathematical sciences. They all made contributions which permanently enriched mathematics and made further progress in other sciences possible.