First there had to be scientific instruments. The eyes could not see clearly enough, far enough, minutely enough; the flesh could not feel with requisite accuracy the pressure, warmth, and weight of things; the mind could not measure space, time, quantity, quality, density without mingling its personal equation with the facts. Microscopes were needed, telescopes, thermometers, barometers, hydrometers, better watches, finer scales. One by one they came.

In his *Magia naturalis* (1589) Giambattista della Porta wrote, “With a concave lens things appear smaller but plainer; with a convex lens you see them larger but less distinct; if, however, you know how to combine the two sorts properly, you will see near and far both large and clear.”__ ^{40}__ Here was the principle of the microscope, the field glass, the opera glass, the telescope, a whole hatful of inventions, and all histology. The simple microscope, a single convex lens, had long been known. The invention that transformed biology was the compound microscope combining several converging lenses. The industry of grinding and polishing lenses was especially developed in the Netherlands—Spinoza lived and died by it. About 1590 Zacharias Janssen, a spectacle-maker of Middelburg, combined a double convex lens and a double concave lens to make the earliest known compound microscope. From that invention came modern biology and modern medicine.

A further application of these principles transformed astronomy. On October 2, 1608, another spectacle-maker of Middelburg, Hans Lippershey, presented to the States-General of the United Provinces (still at war with Spain) the description of an instrument for seeing objects at a distance. Lippershey had placed a double convex lens (the “object glass”) at the farther end of a tube, and a double concave lens (the “eyepiece”) at the nearer end. The legislators saw the military value of the invention and awarded Lippershey nine hundred florins. On October 17 another Dutchman, Jacobus Metius, stated that he had independently made a similar instrument. Hearing of these developments, Galileo made his own telescopes at Padua in 1609, which magnified to three diameters; these were the instruments with which he began to enlarge the world. In 1611 Kepler suggested that still better results could be obtained by reversing the Galilean position of the lenses, using the convex lens as the “eyepiece” and the concave lens as the object glass; and in 1613–17 the Jesuit Christoph Scheiner made an improved telescope on this plan.^{41}

Meanwhile, on principles known to Hero of Alexandria in or before the third century A.D., Galileo had invented a thermometer (c. 1603). Into a vessel of water he placed the open end of a glass tube whose other end was an empty glass bulb, which he warmed by the touch of his hand; when he withdrew his hand the bulb cooled and water rose in the tube. Galileo’s friend Giovanni Sagredo (1613) marked off the tube into a hundred degrees.

A pupil of Galileo, Evangelista Torricelli, closed a long glass tube at one end, filled it with mercury, and stood it with its open end submerged in a dish of mercury; the mercury in the tube did not flow down into the dish. Scholastic physics explained this as due to “Nature’s abhorrence of a vacuum”; Torricelli explained it as due to the pressure of the surrounding atmosphere upon the mercury in the dish. He reasoned that this outside pressure would raise the mercury in the vessel into an empty tube freed from air; experiment proved him right. He showed that variations in the height of the mercury in the tube could be used as a measure of variations in atmospheric pressure. So in 1643 he constructed the first barometer—still the basic instrument of meteorology.

Armed with these new tools, the sciences called to mathematicians for improved methods of calculation, measurement, and notation. Napier and Bürgi, as we have seen, responded with logarithms, Oughtred with the slide rule; but a greater boon came with the decimal system. Tentative suggestions, as usual, had prepared the way. Al-Kashi of Samarkand (d. 1436) had expressed the ratio of the circumference of a circle to the diameter as 3 1415926535898 732, which is a decimal using a space instead of a point. Francesco Pellos of Nice in 1492 used a point. Simon Stevinus expounded the new system in an epochal treatise, *The Decimal* (1585), in which he offered to “teach with unheard-of ease how to perform all calculations … by whole numbers without fractions.” The metric system in Continental Europe carried out his ideas in the measurement of lengths, volumes, and currencies; but the circle and the clock paid tribute to Babylonian mathematics by retaining a sexagesimal division.

Gérard Desargues published in 1639 a classic treatise on conic sections. François Viète of Paris revived the languishing study of algebra by using letters for known as well as unknown quantities, and he anticipated Descartes by applying algebra to geometry. Descartes established analytical geometry in a flash of inspiration when he proposed that numbers and equations can be represented by geometrical figures and vice versa (so the progressive depreciation of currency in a course of time can be shown as a statistical graph); and that from an algebraic equation representing a geometrical figure consequences can be algebraically drawn which will prove geometrically true; algebra could therefore be used to solve difficult geometrical problems. Descartes was so charmed with his discoveries that he thought his geometry as far superior to that of his predecessors as the eloquence of Cicero was above the A B C of children.__ ^{42}__ His analytical geometry, Cavalieri’s theory of indivisibles (1629), Kepler’s approximate squaring of the circle, and the squaring of the cycloid by Roberval, Torricelli, and Descartes all prepared Leibniz and Newton to discover calculus.

Mathematics was now the goal as well as the indispensable tool of all the sciences. Kepler observed that when the mind leaves the realm of quantity it wanders in darkness and doubt.__ ^{43}__ “Philosophy,” said Galileo, meaning “natural philosophy,” or science,

is written in this grand book of the universe, which stands continually open to our gaze. But the book cannot be understood unless we first learn to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics.^{44}

Descartes and Spinoza longed to reduce metaphysics itself to mathematical form.

Science now began to liberate itself from the placenta of its mother, philosophy. It shrugged Aristotle from its back, turned its face from metaphysics to Nature, developed its own distinctive methods, and looked to improve the life of man on the earth. This movement belonged to the heart of the Age of Reason, but it did not put its faith in “pure reason”—reason independent of experience and experiment. Too often such reasoning had woven mythical webs. Reason, as well as tradition and authority, was now to be checked by the study and record of lowly facts; and whatever “logic” might say, science would aspire to accept only what could be quantitatively measured, mathematically expressed, and experimentally proved.