tance fallen through toward the center in the short time t. Since P and Q are on a circle, the lengths PT (y), TQ (x) are connected by the relation

J2 + (r−x)2=p2

or, what is the same thing, by the relation between y2 and r(2r−x) proved in Euclid's Elements (III, 36). The above relation gives, when we substitute for y and x in it from what precedes and neglect powers of t higher than the lowest,

v2 = ar.

From this we get Huygens's formula for the magnitude of the central force,

a=v2/r.

It may be mentioned in passing that there is some analogy between this investigation and Galileo's investigation of the path of a projectile. If the projectile starts from P with a velocity v along PT and travels a distance vt in the time t, and at the same time suffers an acceleration in the direction PS by which in the time t it describes a space gt2/2, it arrives, by the end of the time t, at the point Q, where

y2 = v2t2 = 2v2x/g.

The difference lies in the fact that in this case t need not be infinitesimal. We may say that, in the former case, the curve begins by being a parabola, but that the direction of the force only remains parallel to its original direction at an infinitely small distance along the curve from P, where begins the motion that we are considering.

When the empirical laws of Kepler became generally known, it seems that they were combined with Huygens's theorem by Halley and possibly others as well as by Newton, and this combination gave rise to speculation on the orbits of the planets. Indeed, it seems natural to consider Galileo's parabola of projection as passing over into an ellipse or a circle when the center of attraction is brought from infinity to a finite distance. It is probable that Hooke had no more grounds for his assertions that the force of gravitation varies inversely as the square of the distance and that this law proves that the planets move in ellipses, than this plausible analogy together with the discovery-made also by Newton in 1666 and Halley at a later date that Kepler's third law and Huygens's theorem between them imply that the force keeping a body revolving in a circle about a larger one, as is approximately the case with the planets and

the sun or the moon and the earth, is inversely as the square of the distance.

Another aspect of Galileo's works was of especial importance in influencing Newton probably through Barrow, and this was in a purely mathematical direction. Galileo, in his attempt to find the law according to which a body falls near the surface of the earth, neglecting the resistance of the air, assumed that the velocity acquired by the end of a certain time during which the body falls is proportional to the length of the time of falling. This assumption turned out to be correct, and a previous mistaken assumption that the velocity acquired is proportional to the space fallen through will be referred to below. Since it was easier to find out by experiment in what way the distance fallen through increased with the time rather than in what way the velocity increased with the time, Galileo deduced, from the assumption that v is proportional to t, the relation between s and t. It must be remembered that Galileo was perfectly familiar with the ideas which were expressed in the methods of indivisibles of Kepler and Cavalieri. Galileo considered, unlike his predecessors, velocities varying from point to point, and consequently saw that we could not define "the velocity at a point" by the ratio of the space passed over in a finite time to that time, for different lengths of time would give different results. When, however, we consider, round a certain point, a distance which is infinitely small and therefore very nearly a straight line, then for the infinitely small time in which this space is described, we may regard the increase or decrease of space as uniform. This new notion of "velocity" as the (unique) ratio of infinitesimals includes the old one as a particular case; for if the ratio of ds to dt, as we may write these infinitesimal increments in the notation subsequently introduced by Leibniz, is constant, then s is proportional to t.

Galileo represented the lengths of time by lengths on an axis of abscissæ measured from a fixed point on it, and the magnitudes of the corresponding velocities by ordinates. In this diagram, which is like the diagrams introduced into geometry by Descartes soon after Galileo's ideas were formed, except that r and y replaced t and v, Galileo's assumed proportionality of v to t is represented by a straight line through O. That is to say, if P is any point on the t-axis and PQ is the corresponding ordinate at right angles to OP, then, when P varies in position, the ends Q of all such ordinates lie on the above straight line. Now Galileo proved that the tri

angular area OPQ represents, in units of square measure, the space (s), in units of linear measure, fallen through. Just as the velocity when variable can be measured, at any point, by ds/dt, the slope of the curve at that point, on a diagram in which times are abscissæ and spaces ordinates, so on a diagram in which times are abscissæ and velocities ordinates, the acceleration is measured by dv/dt, the slope of the curve. It can hardly be doubted that Galileo, knowing how areas of curves are found by the method of indivisibles, saw that when the acceleration is variable, the area of the figure OPQ, which is no longer a triangle, still represents s. It is also quite possible that Galileo saw in this way the inverse relation of the problems of tangents and quadratures: the ordinate (v) of any point on the st-diagram is given by calculating the corresponding area on the vt-diagram, while the ordinates on the vt-diagram are determined by the slopes or tangents at the corresponding points of the st-curve. Here we may mention that, in Galileo's mistaken assumption referred to above of the proportionality of v to s, triangles like OPQ on a vs-diagram do not represent s. It will be remembered from Mach's Mechanics that Galileo rejected this mistaken assumption on grounds which were also mistaken. In fact, if v=ds/dt, the integral of v.dt is s, but the integral of v.ds is not s unless is always unity, so that a vs-diagram does not show that integration is the inverse of differentiation. The notions of time being the independent variable and geometrical curves being generated by motions were used by Isaac Barrow, who was certainly influenced by Galileo and possibly by Roberval. Barrow denoted the areas by "t" and "v," and in this notation and the ideas which it implied he was followed by his pupil Isaac Newton, who, in his "method of fluxions," greatly developed the suggestive ideas of Barrow, especially Barrow's clear perception of the truth that the problems of tangents and of quadratures were inverse problems.

FLEET, HANTS, ENGLAND.

PHILIP E. B. JOURDAIN.

CURRENT PERIODICALS.

We hope that the increasing bulk of Science Progress, this "Quarterly Review of Scientific Thought, Work and Affairs," is the outward and visible sign of an increasing recognition of its interest

and value. The articles in the April 1918 number are of the best and by the best. The "Essay-Reviews," the as-yet-unimitated-andown-peculiar creation of Science Progress (to use the hyphenated form which gave to De Morgan many a chuckle), which combine with effect the brevity of the ordinary review and the majestic longueurs of the quarterly, give their writers an opportunity of using the book under discussion as a peg upon which to hang a good deal that is both relevant and urgent. The section entitled "Popular Science" has not perhaps assumed its final form. If it is a matter of popularization, it seems perhaps somewhat out of place in a periodical written by men of science for their brethren and who can manage to assimilate more solid stuff than the name would imply. On the other hand, there is a place for it if it were made to serve the function of a gibbet, so that the world at large might know the kind of nonsense that is dignified in the daily press and the magazines with the honored name of science. The "Notes" have in general a peculiar fragrance of their own, and it is easy to see in them the imprint of the wayward genius-is not all genius wayward, or is the epithet "unscientific"?-who is poet, mathematician, and savant, and who has had every reason to be out in revolt against the malignant stupidity which has characterized the attitude of the governing classes in Britain to those who have been making life possible for the white man in vast territories of the Empire, and who have been doing the work that will count the most in aid of human effort in every field of nature during the next half century. The ordinary reviews are to be counted among the best of their class. All these sections are to the good, most of them are helpful, and if irony does not always bring about the revolution desired, it plays a useful part in titillating the jaded, and in giving voice to those who have been shamefully inarticulate. But of all the features in Science Progress the one that strikes us as the most valuable are the forty pages or so entitled "Recent Advances in Science." There is, as far as we can recollect, nothing comparable to this section in scientific literature. Other attempts to keep the student in touch with what is going on in the world of scientific research are rather in the nature of catalogues with, at most, the shortest of summaries. But the summary is really critical, that is not mechanical but selective, that, where need be, reviews past and present and casts a penetrating eye on the future-that is what Science Progress gives us, and is as far as we know without a

parallel. It is the exceptional value to the writer of this notice of the pages on "Recent Advances" in two of the branches dealt with month by month in this periodical, that leads him to believe that this section is of unique and permanent value.

We have left ourselves no room to deal in detail with the contents of the April number, and must content ourselves with observing that it is brimful of interest from cover to cover, and that "R. R." seems to be here, there, and everywhere, and racier than

ever.