been gotten with a little attention, "if we did not know that such simple relations were difficult to discover."

Though Wallis had obtained an entirely new expression for π, he was not satisfied with it; for instead of a finite number of terms yielding an absolute value, it contained merely an infinite number, approaching nearer and nearer to that value. He therefore induced his friend, Lord Brouncker (1620 ?-1684), the first president of the Royal Society, to investigate this subject. Of course Lord Brouncker did not find what they were after, but he obtained the following beautiful equality:

Continued fractions, both ascending and descending, appear to have been known already to the Greeks and Hindoos, though not in our present notation. Brouncker's expression gave birth to the theory of continued fractions.

Wallis' method of quadratures was diligently studied by his disciples. Lord Brouncker obtained the first infinite series for the area of an equilateral hyperbola between its asymptotes. Nicolaus Mercator of Holstein, who had settled in England, gave, in his Logarithmotechnia (London, 1668), a similar series. He started with the grand property of the equilateral hyperbola, discovered in 1647 by Gregory St. Vincent, which connected the hyperbolic space between the asymptotes with the natural logarithms and led to these logarithms being called hyperbolic. By it Mercator arrived at the logarithmic series, which Wallis had attempted but failed to obtain. He showed how the construction of logarith

mic tables could be reduced to the quadrature of hyperbolic spaces. Following up some suggestions of Wallis, William Neil succeeded in rectifying the cubical parabola, and Wren in rectifying any cycloidal arc.

A prominent English mathematician and contemporary of Wallis was Isaac Barrow (1630-1677). He was professor of mathematics in London, and then in Cambridge, but in 1669 he resigned his chair to his illustrious pupil, Isaac Newton, and renounced the study of mathematics for that of divinity. As a mathematician, he is most celebrated for his method of tangents. He simplified the method of Fermat by introducing two infinitesimals instead of one, and approximated to the course of reasoning afterwards followed by Newton in his doctrine on Ultimate Ratios.

He considered the infinitesimal right triangle ABB' having for its sides the difference between two successive ordinates, the distance between them, and the portion of the curve intercepted by them. This triangle is similar to BPT, formed by the ordinate, the tangent, and the sub-tangent. Hence, if we know the ratio of B'A to BA, then we know the ratio of the ordinate and the sub-tangent, and the tangent can be constructed at once. For any curve, say y2= px, the ratio of B'A

Since y2 = px, we get 2 ay + a2 = pe; neglecting higher powers of the infinitesimals, we have 2 ay=pe, which gives

a:e=p: 2y=p: 2√px.

But a: e = the ordinate: the sub-tangent; hence

p: 2 √px = √px: sub-tangent,

giving 2 x for the value of the sub-tangent. This method differs from that of the differential calculus only in notation.31

NEWTON TO EULER.

It has been seen that in France prodigious scientific progress was made during the beginning and middle of the seventeenth century. The toleration which marked the reign of Henry IV. and Louis XIII. was accompanied by intense intellectual activity. Extraordinary confidence came to be placed in the power of the human mind. The bold intellectual conquests of Descartes, Fermat, and Pascal enriched mathematics with imperishable treasures. During the early part of the reign of Louis XIV. we behold the sunset splendour of this glorious period. Then followed a night of mental effeminacy. This lack of great scientific thinkers during the reign of Louis XIV. may be due to the simple fact that no great minds were born; but, according to Buckle, it was due to the paternalism, to the spirit of dependence and subordination, and to the lack of toleration, which marked the policy of Louis XIV.

In the absence of great French thinkers, Louis XIV. surrounded himself by eminent foreigners. Römer from Denmark, Huygens from Holland, Dominic Cassini from Italy, were the mathematicians and astronomers adorning his court. They were in possession of a brilliant reputation before going to Paris. Simply because they performed scientific work in Paris, that work belongs no more to France than the discoveries of Descartes belong to Holland, or those of Lagrange to Germany, or those of Euler and Poncelet to Russia. We

must look to other countries than France for the great scientific men of the latter part of the seventeenth century.

About the time when Louis XIV. assumed the direction of the French government Charles II. became king of England. At this time England was extending her commerce and navigation, and advancing considerably in material prosperity. A strong intellectual movement took place, which was unwittingly supported by the king. The age of poetry was soon followed by an age of science and philosophy. In two successive centuries England produced Shakespeare and Newton!

Germany still continued in a state of national degradation. The Thirty Years' War had dismembered the empire and brutalised the people. Yet this darkest period of Germany's history produced Leibniz, one of the greatest geniuses of modern times.

There are certain focal points in history toward which the lines of past progress converge, and from which radiate the advances of the future. Such was the age of Newton and Leibniz in the history of mathematics. During fifty years preceding this era several of the brightest and acutest mathematicians bent the force of their genius in a direction which finally led to the discovery of the infinitesimal calculus by Newton and Leibniz. Cavalieri, Roberval, Fermat, Descartes, Wallis, and others had each contributed to the new geometry. So great was the advance made, and so near was their approach toward the invention of the infinitesimal analysis, that both Lagrange and Laplace pronounced their countryman, Fermat, to be the true inventor of it. The dif ferential calculus, therefore, was not so much an individual discovery as the grand result of a succession of discoveries by different minds. Indeed, no great discovery ever flashed upon the mind at once, and though those of Newton will

influence mankind to the end of the world, yet it must be admitted that Pope's lines are only a "poetic fancy":

"Nature and Nature's laws lay hid in night;

God said, 'Let Newton be,' and all was light."

33

Isaac Newton (1642-1727) was born at Woolsthorpe, in Lincolnshire, the same year in which Galileo died. At his birth he was so small and weak that his life was despaired of. His mother sent him at an early age to a village school, and in his twelfth year to the public school at Grantham. At first he seems to have been very inattentive to his studies and very low in the school; but when, one day, the little Isaac received a severe kick upon his stomach from a boy who was above him, he laboured hard till he ranked higher in school than his antagonist. From that time he continued to rise until he was the head boy. At Grantham, Isaac showed a decided taste for mechanical inventions. He constructed a water-clock, a wind-mill, a carriage moved by the person who sat in it, and other toys. When he had attained his fifteenth year his mother took him home to assist her in the management of the farm, but his great dislike for farmwork and his irresistible passion for study, induced her to send him back to Grantham, where he remained till his eighteenth year, when he entered Trinity College, Cambridge (1660). Cambridge was the real birthplace of Newton's genius. Some idea of his strong intuitive powers may be drawn from the fact that he regarded the theorems of ancient geometry as self-evident truths, and that, without any preliminary study, he made himself master of Descartes' Geometry. He afterwards regarded this neglect of elementary geometry a mistake in his mathematical studies, and he expressed to Dr. Pemberton his regret that "he had applied himself to the works of Descartes and other algebraic writers before he had