outlay, an idea not indeed strange to some of the Greeks. The law of refraction of a ray of light he deals with correctly as a particular case of the principle of economy, a principle which exerted a potent influence in the scientific philosophy of the following century. Thus for example Euler says in 1744:

Since the organization of the world is the most excellent, nothing is found in it, out of which some sort of a maximum or minimum property does not shine forth. Therefore no doubt can exist, that all action in the world can be derived by the method of maxima and minima as well as from the actual operating causes.

Fermat's work in the theory of probability is fundamental. He discusses the case of two players, A and B, where A wants two points to win and B three points. Then the game will certainly be decided in the course of four trials. Take the letters a and b, and write down all the combinations that can be formed of four letters. These combinations are 16 in number, namely aaaa, aaab, aaba, aabb, abaa, abab, abba, abbb, baaa, baab, baba, babb, bbaa, bbab, bbba, bbbb. Now every combination in which a occurs twice or oftener represents a case favorable to A, and every combination in which b occurs three times or oftener represents a case favorable to B. Thus, on counting them, it will be found that there are 11 cases favorable to A, and 5 cases favorable to B; and, since these cases are all equally likely, A's chance of winning the game is to B's chance as 11 is to 5.

Like Descartes, Pascal (1623-1662) devoted but a fraction of his great talent to mathematical science.

I have spent much time in the study of the abstract sciences, — but the paucity of persons with whom you can communicate on such subjects gave me a distaste for them. When I began to study man, I saw that these abstract studies were not suited to him, and that in diving into them, I wandered farther from my real track than those who were ignorant of them, and I forgave men for not having attended to these things. But I thought at least I should find many companions in the study of mankind, which is the true and proper study of man. Again I was mistaken. There are yet fewer students of Man than of Geometry.

Learning geometry surreptitiously at 12 years, he had at 18 written an essay on conic sections and constructed the first computing machine. While most of his later life was devoted to religion, theology, and literature, he undertook a wide range of physical experimentation, and made important contributions to the then new theories of numbers and probability, besides a discussion of the cycloid. The juvenile essay on conic sections contains the beautiful theorem since named for him that the opposite sides of a hexagon inscribed in a conic section meet in a straight line. Of geometry and logic Pascal says:

Logic has borrowed the rules of geometry without understanding its power. . . . I am far from placing logicians by the side of geometers who teach the true way to guide the reason. ... The method of avoiding error is sought by every one. The logicians profess to lead the way, the geometers alone reach it, and aside from their science there is no true demonstration.

His work on probability connected itself with the problem of two players of equal skill wishing to close their play, of which Fermat's solution has been given above.

The following is my method for determining the share of each player when, for example, two players play a game of three points and each player has staked 32 pistoles.

Suppose that the first player has gained two points and the second player one point; they have now to play for a point on this condition, that if the first player gain, he takes all the money which is at stake, namely 64 pistoles; while if the second player gain, each player has two points, so that they are on terms of equality, and if they leave off playing, each ought to take 32 pistoles. Thus if the first player gain, then 64 pistoles belong to him, and if he lose, then 32 pistoles belong to him. If therefore the players do not wish to play this game, but separate without playing it, the first player would say to the second, 'I am certain of 32 pistoles, even if I lose this point, and as for the other 32 pistoles, perhaps I shall have them and perhaps you will have them; the chances are equal. Let us then divide these 32 pistoles equally, and give me also the 32 pistoles of which I am certain.’ Thus the first player will have 48 pistoles and the second 16 pistoles.

By similar reasoning he shows that if the first player has gained two points and the second none, the division should be 56 to 8; while if the first has gained one point, the second none, it should be 44 and 20.

The calculus of probabilities, when confined within just limits, ought to interest, in an equal degree, the mathematician, the experimentalist, and the statesman. From the time when Pascal and Fermat established its first principles, it has rendered, and continues daily to render, services of the most eminent kind. It is the calculus of probabilities, which, after having suggested the best arrangements of the tables of population and mortality, teaches us to deduce from those numbers, in general so erroneously interpreted, conclusions of a precise and useful character; it is the calculus of probabilities which alone can regulate justly the premiums to be paid for assurances; the reserve funds for the disbursements of pensions, annuities, discounts, etc. It is under its influence that lotteries and other shameful snares cunningly laid for avarice and ignorance have definitely disappeared. Arago.

With this work connected itself his arithmetical triangle in which successive diagonals contain the coefficients which occur in expansions by the binomial theorem, which Newton was soon to generalize.

1 1 1 1 1 1

1 2 3 4 5

1 3 6 10

1 4 10

15 1

He applies the method of indivisibles successfully to the cycloid (the curve generated by a point on the rim of a rolling wheel). Pascal invented in 1645 an arithmetical machine, writing the Chancellor in regard to it:

Sir: If the public receives any advantage from the invention which I have made to perform all sorts of rules of arithmetic in a manner as novel as it is convenient, it will be under greater obligation to your

Highness than to my small efforts, since I should only have been able to boast of having conceived it, while it owes its birth absolutely to the honor of your commands. The length and difficulty of the ordinary means in use have made me think on some help more prompt and easy to relieve me in the great calculations with which I have been occupied for several years in certain affairs which depend on the occupations with which it has pleased you to honor my father for the service of his Majesty in Normandy. I employed for this investigation all the knowledge which my inclination and the labor of my first studies in mathematics have gained for me, and after profound reflection, I recognized that this aid was not impossible to find.

MECHANICS AND OPTICS: HUYGENS. Most notable among the successors of Galileo in mechanics before we reach Newton was Huygens of Holland (1629-1695) who combined mathematical power with exceptional practical ingenuity. He first (in 1655) explained as a ring the excrescences of Saturn which had been misunderstood by Galileo and others, publishing his discovery in the occult form a'c5d1eg1h1ilm2n o1p2qlr2s1t5u5. (Annulo cingitur tenui, plano, nusquam cohærente ad eclipticam inclinato.) He also discovered Saturn's largest moon. About the same time he made his great invention of the pendulum clock. Accepting a call to Paris by Colbert at the founding of the French Academy, he remained there from 1666 to 1681.

In optics he developed and maintained even in opposition to the authority of Newton the undulatory or wave theory which only found general acceptance a century later. The velocity of light Galileo had failed to measure by means of signal lanterns, and Descartes had likewise been unable to ascertain it by comparing the observed and computed instants of a lunar eclipse. Huygens points out that even this latter test does not prove instantaneous transmission. Römer's conclusive report on observations of a satellite of Jupiter dates from 1675. On this basis Huygens estimated the velocity of light at 600,000 times that of sound, a result about one-third too small.

The medium in which light waves travel Huygens named the ether, attributing to its particles three properties in comparison