much more convenient constructions than had been handed down of old, if certain assumptions, which are called axioms, be granted me. . . .

Accorded then these premises, I shall attempt to show briefly how simply the uniformity of motion can be saved. . . .” 15

These passages show clearly that to Copernicus' mind the question was not one of truth or falsity, not, does the earth move? He simply included the earth in the question which Ptolemy had asked with reference to the celestial bodies alone; what motions should we attribute to the earth in order to obtain the simplest and most harmonious geometry of the heavens that will accord with the facts? That Copernicus was able to put the question in this form is ample proof of the continuity of his thought with the mathematical developments just recounted, and this is why he constantly appealed to mathematicians as those alone able to judge the new theory fairly. He was quite confident that they, at least, would appreciate and accept

his view.

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'Nor do I doubt that skilled and scholarly mathematicians will agree with me if, what philosophy requires from the beginning, they will examine and judge, not casually but deeply, what I have gathered together in this book to prove these things." Mathematics is written for mathematicians, to whom these my labours, if I am not mistaken, will appear to contribute something.... What... I may have achieved in this, I leave to the decision of your Holiness especially, and to all other learned mathematicians." 'If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgment as rash."16 And it is not surprising that for the sixty years that elapsed before Copernicus' theory was confirmed in more empirical fashion, practically all those who ventured to stand with him were accomplished mathematicians, whose thinking was thoroughly in line with the mathematical advances of the day.

15 Commentariolus, Fol. 1a, h, 2a.

16 These quotations are all from his Letter to Pope Paul III in the De Revolutionibus. Cf. also Bk. I, Chs. 7 and 10.

(C) Ultimate Implications of Copernicus' Step—Revival of Pythagoreanism.

But now, of course, the question which Copernicus has thus easily answered carries with it a tremendous metaphysical assumption. Nor were people slow to see it and bring it to the forefront of discussion. Is it legitimate to take any other point of reference in astronomy than the earth? Mathematicians who were themselves subject to all the influences working in Copernicus' mind, would, so he hoped, be apt to say yes. But of course the whole Aristotelian and empirical philosophy of the age rose up and said no. For the question went pretty deep, it meant not only, is the astronomical realm fundamentally geometrical, which almost any one would grant, but is the universe as a whole, including our earth, fundamentally mathematical in its structure? Just because this shift of the point of reference gives a simpler geometrical expression for the facts, is it legitimate to make it? To admit this point is to overthrow the whole Aristotelian physics and cosmology. Even many mathematicians and astronomers might not be willing to follow the tendencies of their science to this extreme; the current of their general thinking flowed on another bed. To follow Ptolemy in ancient times meant merely to reject the cumbrous crystalline spheres. To follow Copernicus was a far more radical step, it meant to reject the whole prevailing conception of the universe. That Copernicus himself and some others were able to answer this ultimate question with a confident affirmative suggests a fourth contributory feature of Copernicus' environment; it suggests that for many minds of the age at least, there was an alternative background besides Aristotelianism, in terms of which their metaphysical thinking might go on, and which was more favourable to this astonishing mathematical movement. As a matter of fact there was just such an alternative

background. All students of philosophy are aware that during the early Middle Ages the synthesis of Christian theology and Greek philosophy was accomplished with the latter in a predominantly Platonic, or rather Neo-Platonic cast. Now the Pythagorean element in Neo-Platonism was very strong. All the important thinkers of the school liked to express their favourite doctrines of emanation and evolution in terms of the number theory, following Plato's suggestion in the Parmenides that plurality unfolded itself from unity by a necessary mathematical process.

Now during this early period of medieval philosophy it is significant that the only original work of Plato in the hands of philosophers was the Timeus, which presents Plato more in the light of a Pythagorean than any other dialogue. It was largely because of this curious circumstance that the first return to a serious study of nature, under Pope Gerbert and his disciple Fulbert about 1000, was undertaken as a Platonic venture. Plato appeared to be the philosopher of nature; Aristotle, who was known only through his logic, seemed like a barren dialectician. It was no accident that Gerbert was an accomplished mathematician, and that William of Conches, a later member of the school, stressed a geometrical atomism which he had drawn from the Timæus.

When Aristotle captured medieval thought in the thirteenth century, Neo-Platonism was not by any means routed, but remained as a somewhat suppressed but still widely influential metaphysical current, to which dissenters from the orthodox Peripateticism were accustomed to appeal. The interest in mathematics evidenced by such freethinkers as Roger Bacon, Leonardo, Nicholas of Cusa, Bruno, and others, together with their insistence on its importance, was in large part supported by the existence and pervading influence of this Pythagorean stream. Nicholas of Cusa found in the theory of numbers the essential

element in the philosophy of Plato. The world is an infinite harmony, in which all things have their mathematical proportions. Hence knowledge is always measurement."" number is the first model of things in the mind of the Creator"; in a word, all certain knowledge that is possible for man must be mathemation knowledge. The same strain appears strongly in Bruno, though in him even more than in Cusa the mystico-transcendental aspect of the number theory was 1st to be uppermost.

It was natural then that in the fifteenth and sixteenth centuries mer men's minds had become thoroughly restless but before they were independent enough to break more definitely with ancient traditions, there was a strong revival of Flatonism in Southern Furore. da Aademy was founded in Florence under the patronage of the Matice family, and boasting as is scholarets such names is Flette. Bessarion, Marsus Fiers and Famiz La this Platonic revival it was again the Pragerean element that assumed prominence, coming to striking expression in the thoroug-going macterical incerpretation of the world offered or Jean Free of Miraccia. The work of these thinkers zenetined to some extent every important Ence of nougat south of the Ales, including the Universier of Belegna, where their most inporant represenatve vis Deminicus Maria de Nova previsser of mathemaces and astronomy. Now's was Copernicus end and teacher during gde si vars of is say ʼn laly, and among the import tics who we know about him is this, ghac he was a se ce ne tolemaic system of ASCIANKOPY, AL CY MOUSe of some coservations which e'd not give cosy enough we deductions from it, but move (stava AOS R vs trougly caught in pla Padre Notagorean current and felt that the WASC CUBAANS SESON voldtet me rosculate that the

astronomical universe is an orderly mathematical harmony, 18

This was in fact the greatest point of conflict between the dominant Aristotelianism of the later Middle Ages and this somewhat submerged but still pervasive Platonism. The latter regarded a universal mathematics of nature as legitimate (though, to be sure, just how this was to be applied was not yet solved); the universe is fundamentally geometrical; its ultimate constituents are nothing but limited portions of space; as a whole it presents a simple, beautiful, geometrical harmony. On the other hand the orthodox Aristotelian school minimized the importance of mathematics. Quantity was only one of the ten predicaments and not the most important. Mathematics was assigned an intermediate dignity between metaphysics and physics. Nature was fundamentally qualitative as well as quantitative; the key to the highest knowledge must, therefore, be logic rather than mathematics. With the mathematical sciences allotted this subordinate place in his philosophy, it could not but appear ridiculous to an Aristotelian for any one to suggest seriously that his whole view of nature be set aside in the interest of a simpler and more harmonious geometrical astronomy. Whereas for a Platonist (especially as Platonism was understood at the time) it would appear a most natural, though still radical step, involving as it did a homogeneity of substance throughout the whole visible cosmos. However, Copernicus could take the step because, in addition to the motive factors already discussed, he had definitely placed himself in this dissenting Platonic movement. Already before he went to Italy in 1496 he had felt its appeal, and, while there, he found ample reinforcement for his daring leap in the energetic Neo-Platonic environment south of the Alps, and particularly in his long

1 Dorothy Stimson, The Gradual Acceptance of the Copernican Theory of the Universe, New York, 1917, P. 25.