y = BC, then AC-a-y. The velocity of the point C is

It is evident from this formula that Napier's logarithms are not the same as the natural logarithms. Napier's logarithms increase as the number itself decreases. He took the logarithm of sin 900; i.e. the logarithm of 100. The logarithm of sin a increased from zero as a decreased from 90°. Napier's genesis of logarithms from the conception of two flowing points reminds us of Newton's doctrine of fluxions. The relation between geometric and arithmetical progressions, so skilfully utilised by Napier, had been observed by Archimedes, Stifel, and others. Napier did not determine the base to his system of logarithms. The notion of a "base" in fact never suggested itself to him. The one demanded by his reasoning is the reciprocal of that of the natural system, but such a base would not reproduce accurately all of Napier's figures, owing to slight inaccuracies in the calculation of the tables. Napier's great invention was given to the world in 1614 in a work entitled Mirifici logarithmorum canonis descriptio. In it he explained the nature of his logarithms, and gave a logarithmic table of the natural sines of a quadrant from minute to minute.

Henry Briggs (1556-1631), in Napier's time professor of geometry at Gresham College, London, and afterwards. professor at Oxford, was SO struck with admiration of Napier's book, that he left his studies in London to do

homage to the Scottish philosopher. Briggs was delayed in his journey, and Napier complained to a common friend, "Ah, John, Mr. Briggs will not come." At that very moment knocks were heard at the gate, and Briggs was brought into the lord's chamber. Almost one-quarter of an hour was spent, each beholding the other without speaking a word. last Briggs began: "My lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help in astronomy, viz. the logarithms; but, my lord, being by you found out, I wonder nobody found it out before, when now known it is so easy." 28 Briggs suggested to Napier the advantage that would result from retaining zero for the logarithm of the whole sine, but choosing 10,000,000,000 for the logarithm of the 10th part of that same sine, i.e. of 5° 44' 22". Napier said that he had already thought of the change, and he pointed out a slight improvement on Briggs' idea; viz. that zero should be the logarithm of 1, and 10,000,000,000 that of the whole sine, thereby making the characteristic of numbers greater than unity positive and not negative, as suggested by Briggs. Briggs admitted this. to be more convenient. The invention of "Briggian logarithms" occurred, therefore, to Briggs and Napier independently. The great practical advantage of the new system was that its fundamental progression was accommodated to the base, 10, of our numerical scale. Briggs devoted all his energies to the construction of tables upon the new plan. Napier died in 1617, with the satisfaction of having found in Briggs an able friend to bring to completion his unfinished plans. In 1624 Briggs published his Arithmetica logarithmica, containing the logarithms to 14 places of numbers, from 1 to 20,000 and from 90,000 to 100,000. The gap from 20,000 to 90,000 was filled up by that illustrious successor of

Napier and Briggs, Adrian Vlacq of Gouda in Holland. He published in 1628 a table of logarithms from 1 to 100,000, of which 70,000 were calculated by himself. The first publication of Briggian logarithms of trigonometric functions was made in 1620 by Gunter, a colleague of Briggs, who found the logarithmic sines and tangents for every minute to seven places. Gunter was the inventor of the words cosine and cotangent. Briggs devoted the last years of his life to calculating more. extensive Briggian logarithms of trigonometric functions, but he died in 1631, leaving his work unfinished. It was carried on by the English Henry Gellibrand, and then published by Vlacq at his own expense. Briggs divided a degree into 100 parts, but owing to the publication by Vlacq of trigonometrical tables constructed on the old sexagesimal division, Briggs' innovation remained unrecognised. Briggs and Vlacq published four fundamental works, the results of which "have never been superseded by any subsequent calculations."

The first logarithms upon the natural base e were published by John Speidell in his New Logarithmes (London, 1619), which contains the natural logarithms of sines, tangents, and secants.

The only possible rival of John Napier in the invention of logarithms was the Swiss Justus Byrgius (Joost Bürgi). He published a rude table of logarithms six years after the appearance of the Canon Mirificus, but it appears that he conceived the idea and constructed that table as early, if not earlier, than Napier did his. But he neglected to have the results published until Napier's logarithms were known and admired throughout Europe.

Among the various inventions of Napier to assist the memory of the student or calculator, is "Napier's rule of circular parts" for the solution of spherical right triangles. It is, perhaps, "the happiest example of artificial memory that is known."

The most brilliant conquest in algebra during the sixteenth century had been the solution of cubic and bi-quadratic equations. All attempts at solving algebraically equations of higher degrees remaining fruitless, a new line of inquiry the properties of equations and their roots was gradually opened up. We have seen that Vieta had attained a partial knowledge of the relations between roots and coefficients. Peletarius, a Frenchman, had observed as early as 1558, that the root of an equation is a divisor of the last term. One who extended the theory of equations somewhat further than Vieta, was Albert Girard (1590-1634), a Flemish mathematician. Like Vieta, this ingenious author applied, algebra to geometry, and was the first who understood the use of negative roots in the solution of geometric problems. He spoke of imaginary quantities; inferred by induction that every equation has as many roots as there are units in the number expressing its degree; and first showed how to express the sums of their powers in terms of the coefficients. Another algebraist of considerable power was the English Thomas Harriot (1560-1621). He accompanied the first colony sent out by Sir Walter Raleigh to Virginia. After having surveyed that country he returned. to England. As a mathematician, he was the boast of his country. He brought the theory of equations under one comprehensive point of view by grasping that truth in its full extent to which Vieta and Girard only approximated; viz. that in an equation in its simplest form, the coefficient of the second term with its sign changed is equal to the sum of the roots; the coefficient of the third is equal to the sum of the products of every two of the roots; etc. He was the first to decompose equations into their simple factors; but, since he failed to recognise imaginary and even negative roots, he failed also to prove that every equation could be thus decomposed. Harriot made some changes in algebraic nota

tion, adopting small letters of the alphabet in place of the capitals used by Vieta. The symbols of inequality > and < were introduced by him. Harriot's work, Artis Analytica praxis, was published in 1631, ten years after his death. William Oughtred (1574-1660) contributed vastly to the propagation of mathematical knowledge in England by his treatises, which were long used in the universities. He introduced x as symbol of multiplication, and as that of proportion. By him ratio was expressed by only one dot. In the eighteenth century Christian Wolf secured the general adoption of the dot as a symbol of multiplication, and the sign for ratio was thereupon changed to two dots. Oughtred's ministerial duties left him but little time for the pursuit of mathematics during daytime, and evenings his economical wife denied him the use of a light.

Algebra was now in a state of sufficient perfection to enable Descartes to take that important step which forms one of the grand epochs in the history of mathematics, - the application of algebraic analysis to define the nature and investigate the properties of algebraic curves.

In geometry, the determination of the areas of curvilinear figures was diligently studied at this period. Paul Guldin (1577-1643), a Swiss mathematician of considerable note, rediscovered the following theorem, published in his Centrobaryca, which has been named after him, though first found in the Mathematical Collections of Pappus: The volume of a solid of revolution is equal to the area of the generating figure, multiplied by the circumference described by the centre of gravity. We shall see that this method excels that of Kepler and Cavalieri in following a more exact and natural course; but it has the disadvantage of necessitating the determination of the centre of gravity, which in itself may be a more difficult problem than the original one of finding the