Academics From Italy

"Quòd tertio loca à nobis fuit obſeruatum, eſt ipſiuſmet LACTEI Circuli eſſentia, ſeu materies, quam Perſpicilli beneficio adeò ad ſenſum licet intueri, vt & altercationes omnes, quæ per tot ſæcula Philoſophos excrucia runt ab oculata certitudine dirimantur, nosque à verboſis dſputationibus liberemur."

"The passage of time has revealed to everyone the truths that I previously set forth; and, together with the truth of the facts, there has come to light the great difference in attitude between those who simply and dispassionately refused to admit the discoveries to be true, and those who combined with their incredulity some reckless passion of their own. Men who were well grounded in astronomical and physical science were persuaded as soon as they received my first message. There were others who denied them or remained in doubt only because of their novel and unexpected character, and because they had not yet had the opportunity to see for themselves. These men have by degrees come to be satisfied. But some, besides allegiance to their original error, possess I know not what fanciful interest in remaining hostile not so much toward the things in question as toward their discoverer. No longer being able to deny them, these men now take refuge in obstinate silence, but being more than ever exasperated by that which has pacified and quieted other men, they divert their thoughts to other fancies and seek new ways to damage me."

"Some years ago, as Your Serene Highness well knows, I discovered in the heavens many things that had not been seen before our own age. The novelty of these things, as well as some consequences which followed from them in contradiction to the physical notions commonly held among academic philosophers, stirred up against me no small number of professors — as if I had placed these things in the sky with my own hands in order to upset nature and overturn the sciences. They seemed to forget that the increase of known truths stimulates the investigation, establishment, and growth of the arts; not their diminution or destruction."

"It seems to me that it was well said by Madama Serenissima, and insisted on by your reverence, that the Holy Scripture cannot err, and that the decrees therein contained are absolutely true and inviolable. But I should have in your place added that, though Scripture cannot err, its expounders and interpreters are liable to err in many ways; and one error in particular would be most grave and most frequent, if we always stopped short at the literal signification of the words."

"Surely it is a great thing to increase the numerous host of fixed stars previously visible to the unaided vision, adding countless more which have never before been seen, exposing these plainly to the eye in numbers ten times exceeding the old and familiar stars."

"Revealing great, unusual, and remarkable spectacles, opening these to the consideration of every man, and especially of philosophers and astronomers; as observed by Galileo Galilei, Gentleman of Florence, Professor of Mathematics in the University of Padua, with the aid of a spyglass lately invented by him, in the surface of the Moon, in innumerable fixed stars, in nebulae, and above all in four planets swiftly revolving about Jupiter at differing distances and periods, and known to no one before the author recently perceived them and decided they should be named the Medicean Stars"

"About ten months ago a report reached my ears that a certain Fleming had constructed a spyglass by means of which visible objects, though very distant from the eye of the observer, were distinctly seen as if nearby. Of the truly remarkable effect several experiences were related, to which some persons gave credence while others denied them. A few days later a report was confirmed to me in a letter from a noble Frenchman in Paris, Jacques Badovere, which caused me to apply myself wholeheartedly to inquire into means by which I might arrive at the invention of a similar instrument. This I did shortly afterwards, my basis being the theory of refraction. First I prepared a tube of lead, at the ends I fitted two glass lenses, both plane on one side while on the other side one was spherically convex and the other concave. Then placing my eye near the concave lens I perceived objects satisfactorily large and near, for they appeared three times closer and nine times larger than when seen with the naked eye alone. Next I constructed another one, more accurate, which represented objects as enlarged more than sixty times. Finally, sparing neither labor nor expense, I succeeded in constructing for myself so excellent an instrument that objects seen by means of it appeared nearly one thousand times larger and over thirty times closer than when regarded with our natural vision."

"Galileo was the first scientist to recognise clearly that the only way to further our understanding of the physical world was to resort to experiment. ...the Greeks, in spite of their proficiency in geometry, never seem to have realised the importance of experiment (Democritus and Archimedes excepted). ...an excuse ...can scarcely be put forward when the elementary nature of Galileo's experiments and observations is recalled. Watching a lamp oscillate in the cathedral of Pisa, dropping bodies from the leaning tower of Pisa, rolling balls down inclined planes, noticing the magnifying effect of water in a spherical glass vase... might just as well have been performed by the Greeks."

"For measurements of time he collected and weighed water flowing from a container at a constant rate of about three fluid ounces per second, He recorded weights of water in grains and, and defined his time unit, called a tempo, to be the time for 16 grains of water to flow, which was equivalent to 1/92 second. These units were small enough so Galileo's measurements of distance and time always resulted in large numbers. That was a necessity because decimal numbers were not part of his mathematical equipment; the only way he could add significant digits in his calculations was to make the numbers larger."

"Koyré's exaltation of the "Platonic and Pythagorean" elements of the Scientific Revolution... was based on a demonstrably false understanding of how Galileo reached his conclusions. Koyré asserted that Galileo merely used experiments as a check on the theories he devised by mathematical reasoning. But later research has definitively established that Galileo's experiments preceded his attempts to give a mathematical account of their results."

"In his founding treatise, the Dialogues Concerning Two New Sciences, Galileo boasted that he was setting forth "a very new science dealing with a very ancient subject." ...No one before him, he declared, had discovered that "the distance traversed, during [successive] equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the odd numbers beginning with unity." ...Galileo's rule can be expressed differently, that the total distance fallen is proportional to the square of the total elapsed time. ...he devised an experiment in which he "diluted" gravity, slowing down the motion of falling. For this purpose he used an inclined plane... He allowed a small metal ball to roll down the board at different inclinations, and recorded the distances and times. ...Galileo presented the numerical values that he found in his experiments as proof... Thus he could proudly boast of an agreement to within "one-tenth of a pulse beat.""

"The pre-Galilean thinkers were... concerned with motion in the sense used by Aristotle. For them, "motion" was any process in which there was transmission from any state or condition to another state. Thus the process of aging, the change in a person's degree of wisdom, or the growth in the weight of a boy could all be considered examples of motion. By contrast Galileo was concerned with physical motion, motion involving a change of place... One of the major kinds of motion that Galileo studied was the motion of free fall."

"His conflict with the Catholic Church arose because deep in his heart Galileo was a believer. There was for him no path of compromise, no way to have separate secular and theological cosmologies. If the Copernican system was true as he believed, what else could Galileo do but fight with every weapon he had in his arsenal... to make his Church accept a new system of the universe. ...In the contrast between Galileo's heroic stand when he tried to reform the cosmological basis of orthodox theology and his humbled, kneeling surrender when he disavowed his Copernicanism, we may sense the tremendous forces attendant on the birth of modern science."

"It is characteristic of Galileo as a scientist of the modern school that as soon as he found any kind of phenomenon, he wanted to measure it. It is all very well to be told that the telescope discloses that there are mountains on the moon, just as there are mountains on earth. But how much more extraordinary it is, and how much more convincing, to be told that there are mountains on the moon and that they are exactly four miles high! Galileo's determination of the height of the mountains on the moon has withstood the test of time..."

"It is impossible to exaggerate the effects of his telescopic discoveries on Galileo's life, so profound were they. Not only is it true of Galileo's personal life and thought, but it equally true of their influence on the history of scientific thought. Galileo had the experience of beholding the heavens as they actually are for perhaps the first time, and wherever he looked he found evidence to support the Copernican system against the Ptolemaic, or at least weaken the authority of the ancients. This shattering experience—of observing the depths of the universe, of being the first mortal to know what the heavens are actually like—made so deep an impression... that it is only by considering the events of 1609... that one can understand the subsequent direction of his life."

"Galileo is the founder of the science of dynamics. Among his contemporaries it was chiefly the novelties he detected in the sky that made him celebrated, but Lagrange claims that his astronomical discoveries required only a telescope and perseverance, while it took an extraordinary genius to discover laws from phenomena, which we see constantly and of which the true explanation escaped all earlier philosophers. The first contributor to the science of mechanics after Galileo was Descartes."

"While Stevin investigated , Galileo pursued principally dynamics. Galileo was the first to abandon the Aristotelian idea that bodies descend more quickly in proportion as they are heavier; he established the first law of motion; determined the laws of falling bodies; and, having obtained a clear notion of acceleration and of the independence of different motions, was able to prove that projectiles move in parabolic curves. Up to his time it was believed that a cannon-ball moved forward at first in a straight line and then suddenly fell vertically to the ground. Galileo had an understanding of s, and gave a correct definition of '. Though he formulated the fundamental principles of statics, known as the s, yet he did not fully recognise its scope. The principle of virtual velocities was partly conceived by Guido Ubaldo (died 1607), and afterwards more fully by Galileo."

"In Santa Croce's holy precincts lie Ashes which make it holier, dust which is Even in itself an immortality, Though there were nothing save the past, and this, The particle of those sublimities Which have relapsed to chaos: here repose Angelo's, Alfieri's bones, and his, The starry Galileo, with his woes; Here Machiavelli's earth returned to whence it rose. These are four minds, which, like the elements, Might furnish forth creation:—Italy! Time, which hath wronged thee with ten thousand rents Of thine imperial garment, shall deny, And hath denied, to every other sky, Spirits which soar from ruin: thy decay Is still impregnate with divinity, Which gilds it with revivifying ray; Such as the great of yore Canova is to-day."

"Copernicus had taken one course in treating the earth as virtually a celestial body in the Aristotelian sense—a perfect sphere governed by the laws which operated in the higher reaches of the skies. Galileo complemented this by taking now the opposite course—rather treating the heavenly bodies as terrestrial ones, regarding the planets as subject to the very laws which applied to balls sliding down inclined planes. There was something in all this which tended to the reduction of the whole universe to uniform physical laws, and it is clear that the world was coming to be more ready to admit such a view."

"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."

"[I]f Bacon had never lived, the student of nature would have found in the writings and labours of Galileo, not only the boasted principles of the inductive philosophy, but also their practical application to the highest efforts of invention and discovery."

"His brilliant discoveries the man of science regards as his peculiar property; the means by which they were made, and the development of his intellectual character, belong to the logician and to the philosopher; but the triumphs and the reverses of his eventful life must be claimed for our common nature, as a source of more than ordinary instruction."

"The credit of first using the telescope for astronomical purposes is almost invariably attributed to Galilei, though his first observations were in all probability slightly later in date than those of Harriot and Marius, is to a great extent justified by the persistent way in which he examined object after object, whenever there seemed any reasonable prospect of results following, by the energy and acuteness with which he followed up each clue, by the independence of mind with which he interpreted his observations, and above all by the insight with which he realised their astronomical importance."

"Galileo observed as early as 1638 that there are precisely as many squares 1, 4, 9, 16, 25,... as are positive integers all together. This is evident from the sequences1, 2, 3, 4, 5, 6, ... , n, ... 12, 22, 32, 42, 52, 62, ..., n, ... He thus recognized the fundamental distinction between finite and infinite classes that became current in the late nineteenth century. An infinite class is one in which there is a one-to-one correspondence between the whole class and a subclass of the whole. Or, what is equivalent, there are as many things in one part of an infinite class as there are in the whole class. ...A class whose elements can be put in a one-to-one correspondence with the integers 1, 2, 3, ... is said to be denumerable. All the points in any line segment, finite or infinite in length, form a non-denumerable set. A basic course in calculus starts from the theory of point sets. The distinction between denumerable and non-denumerable classes was not started by Galileo; it was observed about 1840 by Bolzano and in 1878 by Cantor. But Galileo's recognition of the cardinal property of all infinite classes makes him one of the genuine anticipators in the history of calculus. The other was Archimedes."

"[I]t was upon... inequality of motions in point of velocity that Galileo built his theory of flux and reflux of the sea; supposing that the earth revolved faster than the water could follow; and that the water was therefore first gathered in a heap and then fell down, as we see in a basin of water moved quickly. But this he devised upon an assumption which cannot be allowed, viz. that the earth moves; and also without being well informed as to the sexhorary motion of the tide."

"Measure what is measurable, and make measurable what is not so."

"Mathematics is the key and door to the sciences."

"It is only in order to shield your ignorance that you put the Lord at every turn to the refuge of a miracle."

"Eppur si muove."

"I have never met a man so ignorant that I could not learn something from him."

"All truths are easy to understand once they are discovered; the point is to discover them."

"Names and attributes must be accommodated to the essence of things, and not the essence to the names, since things come first and names afterwards."