First, the new international sharpened its instruments. Pascal, Hooke, and Guericke developed the barometer; Guericke’s air pump explored the possibility of a vacuum; Gregory, Newton, and others made better telescopes than those of Kepler and Galileo; Newton invented the sextant; Hooke improved the compound microscope, which transformed the study of the cell; the thermometer became more reliable and accurate under Guericke and Amontons, and in 1714 Fahrenheit gave it its English-American form by using mercury instead of alcohol as the expanding medium, and dividing its scale at zero, 32 degrees, and 96 degrees (which he assumed to be the normal temperature of the human body).

The greatest instrument of all was mathematics, for this gave experience a quantitative and measured form, and in a thousand ways enabled it to predict, even to control, the future. “Nature plays the mathematician,” said Boyle; and Leibniz added, “Natural science is naught but applied mathematics.” __ ^{12}__ Historians of mathematics acclaim the seventeenth century as especially fruitful in their field, for it was the century of Descartes, Napier, Cavalieri, Fermat, Pascal, Newton, Leibniz, and Desargues. Ladies perfumed with pedigree attended lectures on mathematics; some of them, joked the

Pierre de Fermat fathered the modern theory of numbers (the study of their classes, characteristics, and relationships), conceived analytical geometry independently of—perhaps before—Descartes, invented the calculus of probabilities independently of Pascal, and anticipated the differential calculus of Newton and Leibniz. Yet he lived in comparative obscurity as a counselor of the Parlement of Toulouse, and formulated his contributions to mathematics only in letters to his friends—which were not published till 1679, fourteen years after his death. We catch the mathematical ecstasy in one of these letters: “I have found a very great number of exceedingly beautiful theorems.” __ ^{14}__ He was delighted by every new trick or surprising regularity in numbers. He challenged the mathematicians of the world “to separate a cube into two cubes, a fourth power into two fourth powers,” etc.; “I have discovered,” he wrote, “a truly marvelous demonstration” of this, now known as “Fermat’s last theorem”; but neither his nor any conclusive proof of it has yet been found. A German professor in 1908 left 100,000 marks to the first person who should prove Fermat’s proposition; no one has yet claimed the reward, perhaps discouraged by the depreciation of marks.

Christian Huygens, barring only one, was the outstanding scientist of this age—*facile secundus* to Newton. His father, Constantijn Huygens, was one of the most distinguished of Holland’s poets and statesmen. Born at The Hague in 1629, Christiaan (as the Dutch spelled him) began at the age of twenty-two to publish mathematical treatises. His discoveries in astronomy and physics soon won him a European renown; he was elected a fellow of the Royal Society in London in 1663, and in 1665 he was invited by Colbert to join the Académie des Sciences in Paris. He moved to the French capital, received a liberal pension, and remained there till 1681; then, uncomfortable under a King turned persecutor of Protestants, he returned to Holland. His correspondence in six languages with Descartes, Roberval, Mersenne, Fermat, Pascal, Newton, Boyle, and many others illustrated the growing unity of the scientific fraternity. “The world is my country,” he said, and “to promote science is my religion.” __ ^{15}__ His

Newton, of course, was now the central luminary in the galaxy of British science; he deserves a separate chapter; but there were satellites to his star. His friend John Wallis, an Anglican priest, became Savilian professor of geometry at Oxford in 1649 at the age of thirty-three, and held that chair for fifty-four years. Grammar, logic, and theology diverted his pen from science; nevertheless he wrote effectively on mathematics, mechanics, acoustics, astronomy, tides, botany, physiology, geology, and music; he lacked only some amours and wars to make him a full man. His *De Algebra Tractatus Historicus et Practicus* (1673) not only contributed original ideas to that science, but was the first serious attempt in England to write the history of mathematics. His contemporaries were delighted by his prolonged controversy with Hobbes over the quadrature of the circle; Wallis scored his point, but the old philosopher fought on to the end of his ninety-first year. History remembers Wallis chiefly for his *Arithmetica Infinitorum* (1655), which applied Cavalieri’s method of indivisibles to the quadrature of curves, and so prepared for infinitesimal calculus.

*Calculus* meant originally a small stone used by the ancient Romans in calculating; but only the devotees of calculus can now define their science properly.__*__ Archimedes had glimpsed it, Kepler had approached it, Fermat had discovered it but had not published his findings; Cavalieri and Torricelli in Italy, Pascal and Roberval in France, John Wallis and Isaac Barrow in England, James and David Gregory in Scotland, had all carried bricks to the building in this astonishing co-operation of a continent. Newton and Leibniz brought the work to fulfillment.

The term *calculus* was suggested to Leibniz by Johann Bernoulli, member of a family as remarkable as the Bachs, the Brueghels, and the Couperins for the social heredity of genius. Nikolaus Bernoulli (1623–1708), like his ancestors, was a merchant. In his son Jakob Bernoulli I (1654–1705) mercantile accounting passed into higher forms of reckoning. Taking as his motto *Invito patre sidera verso*—“Against my father’s will I study the stars”—Jakob dabbled in astronomy, contributed to analytical geometry, advanced the calculus of variations, and became professor of mathematics at the University of Basel. His studies of catenary curves (curves described by a uniform chain suspended between two points) came to later fruition in the designing of suspension bridges and high-voltage transmission lines. His brother Johann (1667–1748), also against paternal plans, took up medicine, then mathematics, and succeeded Jakob as professor at Basel; he contributed to physics, optics, chemistry, astronomy, the theory of tides, and the mathematics of sails; he invented exponential calculus, constructed the first system of integral calculus, and introduced the use of the word *integral* in this sense. Another brother, Nikolaus I (1662–1716), took his doctor’s degree in philosophy at sixteen, in law at twenty, taught law at Bern and mathematics in St. Petersburg. We shall find six more Bernoulli mathematicians in the eighteenth century, and there were two in the nineteenth. By that time the Bernoulli battery had run down.

The establishment of statistics as almost a science was among the achievements of this age. John Graunt, a haberdasher, amused himself by collecting and studying the burial records of London parishes. Usually these records stated the reported cause of death, including “dead in the street and starved,” “executed and prest to death,” “King’s evil,” “starved at nurse,” and “made away themselves.” __ ^{16}__ In 1662 Graunt published his

Statistics were applied to politics by Graunt’s friend Sir William Petty. Again exemplifying a versatility impossible today, Petty, after studying at Caen, Utrecht, Leiden, Amsterdam, and Paris, taught anatomy at Oxford and music at Gresham College, London, and won fortune and knighthood as physician to the royal army in Ireland.__*__ In 1676 he wrote the second classic in English statistics, *Political Arithmetic*. Politics, Petty held, could approach to a science only by basing its conclusions on quantitative measurements. Therefore he pleaded for a periodical census that would record the birth, sex, marital condition, titles, occupation, religion, etc., of every inhabitant of England. On the basis of mortality bills, number of houses, and annual excess of births over deaths, he estimated the population of London at 696,000 in 1682; of Paris, 488,000; Amsterdam, 187,000; Rome, 125,000. Like Giovanni Botero in 1589 and Thomas Malthus in 1798, Petty thought that population tends to increase faster than the means of subsistence, that this leads to war, and that by the year 3682 the habitable earth would be dangerously overcrowded, with one person to every two acres of land. ^{20}

Insurance companies used statistics to turn their business into an art and science that took account of everything except inflation. From the mortality reports of Breslau Edmund Halley drew up (1693) a table of expectable deaths for all years between one and eighty-four; on its basis he calculated the odds against persons of a given age dying during the calendar year, and deduced the logical price of a policy. The first life insurance companies established in London in the eighteenth century made use of Halley’s tables, and turned mathematics into gold.