event in the development of the metaphysics of modern science. Aristotelianism had won out in the long preceding period of human thought because it seemed to make intelligible and rational the world of commonsense experience. Kepler early realized that the admission of validity to the Copernican world-scheme involved a radically different cosmology, a cosmology which could rest upon the revived Neo-Platonism for its general background, would find its historical justification in the remarkable developments in the sciences of mathematics and astronomy, and which could lay bare a marvellous significance and a new beauty in the observed events of the natural cosmos by regarding them as exemplifications of simple, underlying numerical relations. The task involved revising to this end the traditional ideas of causality, hypothesis, reality, and knowledge; hence Kepler offers us the fundamentals of a metaphysic based in outline upon the early Pythagorean speculations, but carefully accommodated to the new ideal and method. Fortunate indeed it was for Kepler's historical importance that his venture proved pragmatically successful. The acquisition of further empirical facts in astronomy by Galileo and his successors showed that the astronomical and physical universe was enough like what Copernicus and Kepler had dared to believe, for them to become established as fathers of the outstanding movement of human thought in modern times, instead of being consigned to oblivion as a pair of wild-minded apriorists. In particular, Kepler's method had just enough in common with the successful procedure of later science, so that out of a vast mass of painfully and laboriously won geometrisms in nature, three chanced to become fruitful foundations for the later stupendous scientific achievements of Newton. But only those who fix their attention on these three, forgetting the arduous amassing of quite useless numerical curiosities which to him were quite as significant, could

make, without qualification, such claims for him as are made by Eucken and Apelt :

Kepler is the first who ventured here an exactmathematical treatment of the problems (of astronomical science), the first to establish natural laws in the specific sense of the new science."47 Kepler was the first to discover the art of successfully inquiring her laws of nature, since his predecessors merely constructed explanatory concepts which they endeavoured to apply to the course of nature."48

Such laudations, while not wholly false, obscure our genuine debt to Kepler. His solid and forwardlooking achievement as a philosopher of science, is his insistence that valid mathematical hypotheses must be exactly verifiable in the observed world. He is entirely convinced on a priori grounds that the universe is basically mathematical, and that all genuine knowledge must be mathematical, but he makes it plain that the laws of thought innate in us as a divine gift, cannot come to any knowledge of themselves; there must be the perceived motions which furnish the material for their exact exemplification49. For this side of his thought we have to thank his training in mathematics and in particular his association with that giant of careful star-observation, Tycho Brahe. It is this, together with his reinterpretation, in terms set by the situation of his day, of such notions as causality, hypothesis, reality, and the like, that constitute the constructive portion of his philosophy. But his outlook and method were as fully dominated by an æsthetic as by a purely theoretic interest, and the whole of his work was overlaid and confused by crude inherited superstitions which the most enlightened people of his time had already discarded.

47 R. Eucken, Kepler als Philosoph (Philosophische Monatshefte, 1878, p. 42, ff.). 48 E. F. Apelt, Epochen der Geschichte der Menscheit, Vol. I, p. 243.

49 Opera, V, 229.

CHAPTER III

GALILEO

GALILEO was a contemporary of Kepler, his life overlapping that of the great German astronomer at both ends. After the two became acquainted through the publication of the Mysterium Cosmographicum in 1597, they remained firm friends and carried on a considerable and interesting correspondence, but it cannot be said that either influenced the philosophy of the other to any important extent. Each, of course, made use of the other's positive and fruitful scientific discoveries, but the metaphysics of each was conditioned primarily by general environmental influences and by intensive reflection on the ultimate bearings of his own achievements.

(A) The Science of Local Motion'

Galileo's father had destined his son for the study of medicine, but at the early age of seventeen the latter acquired a consuming interest in mathematics, and after securing his sire's reluctant consent, proceeded during the next few years to make himself master of the subject. Were it not for his more stupendous achievements he, like Kepler, would have won brilliant fame as a mathematician. He invented a geometrical calculus for the reduction of complex to simple figures, and wrote an essay on continuous. quantity. The latter was never published, but such was his mathematical name that Cavalieri did not publish his own treatise on the Method of Indivisibles, as long as he hoped to see Galileo's essay printed. At the youthful age of twenty-five he was appointed

professor of mathematics at the University of Pisa, largely because of the fame won by some papers on the hydrostatic balance, the properties of the cycloid, and the centre of gravity in solids. The direction of

his early mathematical studies is sufficiently indicated by these works; it was the mechanical branch that absorbed his attention and interest from the very beginning. The famous event in the Cathedral of Pisa, when he observed that the swings of the great hanging lamp were apparently isochronous, had just preceded, and in part inspired, his first interest in mathematics, hence the mathematical study of mechanical motions became quite naturally the focus of his work. Furthermore, as soon as he became competent in this new field he eagerly embraced the Copernican system (though continuing for many years to teach Ptolemaism to his classes out of deference to popular feeling), and the Copernican attribution of motion to the earth gave him a powerful impetus to study more closely, i.e., mathematically, such motions of small parts of the earth as occur in every-day experience, as we learn on the authority of his great English disciple, Hobbes1. Hence the birth of a new science, terrestrial dynamics, which presented itself to Galileo as a simple and natural extension of the exact mathematical method to a field of somewhat more difficult mechanical relations. Others before him had asked why heavy bodies fall; now, the homogeneity of the earth with the heavenly bodies having suggested that terrestrial motion is a proper subject for exact mathematical study, we have the further question raised how do they fall? with the expectation that the answer will be given in mathematical terms.

As Galileo notes in the introduction to his science of dynamics or local motion,' many philosophers had

2

Epistle Dedicatory to the Elements of Philosophy Concerning Body, Works, Molesworth edition, London, 1839, Vol. I (English), p. viii.

* Dialogues and Mathematical Demonstrations Concerning Two New Sciences, by Galileo Galilei (Crew and De Salvio translation), New York, 1914, p. 153. ff.

written on motion, " nevertheless I have discovered by experiments some properties of it which are worth knowing and which have not hitherto been either observed or demonstrated." Some, too, had observed that the motion of a falling body was one of acceleration, "but to just what extent this acceleration occurs has not yet been announced." The same thought is again expressed with reference to the motion of projectiles-others had observed that a projectile followed a curved path, but none had demonstrated that the path must be a parabola. It was this reduction of terrestrial motions to terms of exact mathematics which, fully as much as the significant astronomical discoveries that empirically confirmed Copernicanism, measured his import to those of his contemporaries who were fitted to appreciate this stupendous advance in human knowledge. His friend and admirer Fra Paolo Sarpi reflected the opinion of such minds when he exclaimed, "To give us the science of motion God and Nature have joined hands and created the intellect of Galileo." Galileo's practical mechanical inventions are themselves sufficiently remarkable. In his early years he invented a pulsimeter, operating by means of a small pendulum, and also a contrivance for measuring time by the uniform flow of water. Later he became the inventor of the first crude thermometer, and in the last year of his life sketched out complete plans for a pendulum clock. His achievements in the early development of the telescope are known to all students.

Now what are the main metaphysical conclusions that Galileo found implied in his work? Let us first consider briefly those in which his agreement with Kepler is most complete, passing then to a fuller treatment of his more novel suggestions. Our expectation that the reduction of the motions of bodies to exact mathematics must carry large metaphysical bearings to Galileo's mind will not be disappointed.

Two New Sciences, Editor's Preface.