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It now appeared that Galileo had everything in his favour, a good permanent salary and abundant leisure for his favourite pursuits, with the support of a powerful prince. But the event proved that this was a very bad exchange for the Padua post, and it was not very long before the philosopher must have begun to regret leaving the one government in Italy where he would have been perfectly safe. Before Galileo entered its service, the Venetian republic, finding that the Jesuits were scheming to obtain control of education, starting with a school of their own, had decreed that no Jesuit should be allowed to teach in its territory. On the intervention of the Pope, who supported the Jesuit cause by laying Venice under an interdict, the senate retorted by expelling the Jesuits altogether. In no other part of Italy was absolute freedom of thought possible.

At first, however, all was plain sailing; the very first month of Galileo's return to Florence was marked by another important discovery. Once more he hid it in an anagram while repeating his observations to ensure himself against mistakes:—

"Haec immatura a me iam frustra leguntur o.y." The intention of this device, which was frequently employed, was to secure priority and prevent plagiarism. Many of Galileo's previous discoveries had been claimed by others. When he was quite sure he was right he sent the solution,

"Cynthiæ figuras aemulatur mater amorum,"

i.e. Venus imitates the phases of the moon. It had been regarded as an objection to the theory of Copernicus that Venus and Mercury ought to show phases but did not, and also that Venus ought to appear much larger when nearest the earth. Galileo's new discovery disposed of these objections, making it clear that Venus did show phases and that it did appear much larger when nearer the earth, but that as it was then nearly "new" most of its disc was not illuminated, so that it did not look so bright and consequently not so large to the naked eye.

Galileo found little trustworthy evidence of phases of Mars or Mercury, as the former does not reach crescent shape, and the latter is always close to the sun, but at any rate he found no reason to doubt that both these planets shared the motion of revolution round the sun. There seems to have been a suspicion in his mind that his credit might be assailed at Rome by the many enemies whom he had made, not only by his opposition to the Aristotelian tradition, but by his great skill in argument, his pungent satire, and his capacity for turning the opposition into ridicule after, adducing ingenious points in its favour. It was not altogether unlike the case of Socrates, whose system of debate must have been excessively annoying to anyone who dared seriously to maintain any proposition against him. Be that as it may, Galileo now considered it expedient to go to Rome to make His discoveries known at first hand to the leaders of the Church and forestall any possible accusation. The Grand Duke not only granted him leave of absence but provided for his journey and arranged for him to lodge at the Tuscan Embassy in Rome. Under such auspices and armed with many letters of introduction, including one to Cardinal Barberini (afterwards Pope Urban VIII.), he had no difficulty in obtaining a hearing and opportunity of demonstrating his discoveries, or "celestial novelties". Cardinal Bellarmine arranged for the appointment of four commissioners to examine and report. The commissioners, chosen for their scientific attainments from among the members of the Roman College, although previously prejudiced against the new discoveries, were constrained to admit the reality of the observations, and to this extent the Church's sanction was secured. The Pope, Paul V., received Galileo favourably and other high dignitaries followed his example, while the Academy "dei Lincei" (of the lynx-eyed ones), precursor of the present Italian scientific society of the same name, elected him a member. His friends were well satisfied with the result of his visit to Rome.

It was during this visit that Galileo announced another new discovery, which entailed to all appearances more important consequences than any that had gone before. In April, 1611, he declared the existence of spots on the sun, and showed them to several leading people. At first inclined to regard them as small planets revolving round the sun, he soon found that this would not explain the appearance, and in the following year he announced positively that they were actually on the sun's surface, and rotated with the sun in rather less than a lunar month, their shape being irregular and occasionally variable. Possibly a delay of a year in the making of this discovery might have changed the course of his subsequent career, for it so happened that during 1611 a Jesuit father, Christopher Scheiner, Professsor of Mathematics at Ingolstadt, also discovered sun-spots, or claimed to have done so. The weight of evidence is unmistakably in favour of Galileo, but the question of priority is really of very small consequence if different people make absolutely independent discoveries. It is surprising at the present time to note the careful computation of difference of longitude to settle rival claims to the discovery of a new comet which has been picked up on the same night by different comet seekers. The discovery of Neptune more than seventy years ago is a classical instance of the amount of attention this priority question continues to excite. We need not, therefore, wonder that three centuries ago this discovery of sun-spots caused acute controversy, especially as Scheiner's explanation of the phenomenon was that already discarded by Galileo, namely, that the spots were really small planets revolving close to the sun. Other observers about the same time made the same discovery, and one of them, Fabricius, is generally accepted as the first to publish his observations, so that in modern times he is usually credited with priority. The idea of change or imperfection in the sun was exceedingly repugnant to accepted tradition. Scheiner was only permitted to publish his discovery anonymously, but nevertheless he was supported against Galileo by the Jesuits, as a body, and we may regard this as the actual beginning of Galileo's serious troubles.

About the same time Galileo had been drawing his own conclusions from observations of the moon. Having determined that the moon always turned the same face towards the earth, as easily proved by the configuration of its surface, he saw at once that the lunar day must be of the same length as the lunar month, and became convinced that no plants or animals could exist under such conditions.

A controversy on a totally different subject was the cause of the publication of Galileo's discourse on Floating Bodies in 1612. Aristotle had stated that the chief cause of a body floating was its shape, so that ice floated not because it was lighter than water, but because it was flat. Presumably neither Aristotle nor any of his followers had ever tried whether a ball of ice would sink. The subject came up for discussion at one of the Grand Duke's scientific parties, and Galileo took up the cudgels against the opinion of Aristotle. The company was divided, and among those who supported Galileo's side was Cardinal Barberini, who afterwards became Pope Urban VIII. The chief point alleged by the Aristotelian side was that a thin slab of ebony will float while a ball of ebony sinks at once. Galileo pointed out that the thin slab would also sink if it was wetted all over, and that similarly a flat piece of lighter wood would not remain at the bottom but would rise in spite of its shape. He performed many ingenious experiments with bodies compounded of wax and lead so as to be nearly of the same specific gravity as water, but apparently failed to point out that the floating of heavy bodies involves not only the hydrostatic pressure but also capillary action which is a distinct phenomenon. His treatise contained many experiments and clever arguments but met with violent opposition. Galileo himself always seems to have arrived very quickly at his conclusions, but the great difficulty of convincing others led him to multiply his experiments far beyond what he considered necessary in his own case; and this naturally strengthened his convictions. In this way he justified the aphorism that Ignorance had been his best teacher.

Many attacks on his principles of hydrostatics as set forth in the "Discourse on Floating Bodies" were published, and produced scathing replies either from Galileo himself or from his loyal pupil Castelli. The most elaborate, attacks, by, Colombe and Grazia, called forth the most detailed refutation, an essay printed at Florence in 1615, ostensibly by Castelli but really written by Galileo himself. It is not to his credit that in this essay, after completely crushing the arguments of Colombe and Grazia, he suggests that as he is only the pupil, they would have been much more severely pulverised if Galileo had thought it worth while to tackle them himself. It is fairly certain that this rather cheap sneer would have been seized upon by his enemies as a special grievance if the true authorship had been discovered before his death. At the same time the plan of complete or partial anonymity has frequently found favour, especially in controversial questions.