I mentally conceive of some moveable [sphere] projected on a horizontal plane, all impediments being put aside. Now it is evident... that equable motion on this plane would be perpetual if the plane were of infinite extent, but if we assume it to be ended, and [situated] on high, the movable, driven to the end of this plane and going on further, adds on to its previous equable and indelible motion, that downward tendency which it has from its heaviness. Thus, there emerges a certain motion, compounded...

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

Mathematics is the key and door to the sciences.

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

[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.

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.

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.

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.

[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.

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.

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.

In Santa Croce's holy precincts lieAshes which make it holier, dust which isEven in itself an immortality,Though there were nothing save the past, and this,The particle of those sublimitiesWhich have relapsed to chaos: here reposeAngelo'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 rentsOf thine imperial garment, shall deny,And hath denied, to every other sky,Spirits which soar from ruin: thy decayIs still impregnate with divinity,Which gilds it with revivifying ray;Such as the great of yore Canova is to-day.

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.

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.

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.

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...

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.

Eppur si muove.

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

Proposition I. Theorem I: When a projectile is carried in motion compounded from equable horizontal and from naturally accelerated downward [motions], it describes a semiparabolic line in its movement.

The speed of the ball—thanks to opposition from the air—will not go on increasing forever. Rather, what will happen is seen in bodies of very little weight falling through no great distance; I mean, a reduction to equable motion, which will occur also in a lead or iron ball after the descent of some thousands of braccia. This bounded terminal speed will be called the maximum that such a heavy body can naturally attain through the air...

It seems to me proper to adorn the Author's thought here with its conformity to a conception of Plato's regarding the determination of the various speeds of equable motion in the celestial motions of revolution. ...he said that God, after having created the movable celestial bodies, in order to assign to them those speeds with which they must be moved perpetually in equable circular motion, made them depart from rest and move through determinate spaces in that natural straight motion in which we sensibly see our moveables to be moved from the state of rest, successively accelerating. And he added that these having been made to gain that degree [of speed] which it pleased God that they should maintain forever, He turned their straight motion into circulation, the only kind [of motion] that is suitable to be conserved equably, turning always without retreat from or approach toward any pre-established goal desired by them. The conception is truly worthy of Plato, and it is to be more esteemed to the extent that its foundations, of which Plato remained silent, but which were discovered by our Author in removing their poetical mask or semblance, show it the guise of a true story.

It now remains that we find the amount of time of descent through the channel. This we shall obtain from the marvelous property of the pendulum, which is that it makes all its vibrations, large or small, in equal times. This requires, once and for all, that two or three or four patient and curious friends, having noted a fixed star that stands against some fixed marker, taking a pendulum of any length, shall go counting its vibrations during the whole time of return of the fixed star to its original point, and this will be the number of vibrations in 24 hours. From the number of these we can find the number of vibrations of any other pendulums, longer or shorter, at will, so that if for example those counted by us in 24 hours were 234,567, then taking another shorter pendulum with which one counts 800 vibrations while another counts 150 of the longer pendulum, we already have, by the golden rule, the number of vibrations for the whole time of 24 hours; and if we want to know the time of descent through the channel, we can easily find not only the minutes, seconds, and sixtieths of seconds, but beyond that as we please. It is true that we can pass a more exact measure by having observed the flow of water through a thin passage, for by collecting this and having weighed what passes in one minute, for example, then by weighing what passes in the time of descent through the channel we can find the most exact measure and quantity of this time, especially by making use of a balance so precise as to weigh one sixtieth of a grain.

If I shall have sufficient strength to improve and amplify what was written and published by me up to now about motion by adding some little speculations, and in particular those relating to the force of percussion, in the investigation of which I have consumed hundreds and thousands of hours, and finally reduced this to very easy explanation, so that people can understand it in less than half an hour of time.

I esteem myself happy to have as great an ally as you in my search for truth. I will read your work … all the more willingly because I have for many years been a partisan of the Copernican view because it reveals to me the causes of many natural phenomena that are entirely incomprehensible in the light of the generally accepted hypothesis. To refute the latter I have collected many proofs, but I do not publish them, because I am deterred by the fate of our teacher Copernicus who, although he had won immortal fame with a few, was ridiculed and condemned by countless people (for very great is the number of the stupid).

What has philosophy got to do with measuring anything? It's the mathematicians you have to trust, and they measure the skies like we measure a field.

My dear Kepler, what would you say of the learned here, who, replete with the pertinacity of the asp, have steadfastly refused to cast a glance through the telescope? What shall we make of this? Shall we laugh, or shall we cry?

sì perché l'autorità dell'opinione di mille nelle scienze non val per una scintilla di ragione di un solo, sì perché le presenti osservazioni spogliano d'autorità i decreti de' passati scrittori, i quali se vedute l'avessero, avrebbono diversamente determinato.

We seek not what God could have done but what He has done.… God could have caused birds to fly with bones of solid gold, with veins full of quicksilver, with flesh heavier than lead and very small and heavy wings, so as to better show His power … but He wanted to make their bones, flesh and feathers very light … to teach us that He likes simplicity and ease.

After an injunction had been judicially intimated to me by this Holy Office, to the effect that I must altogether abandon the false opinion that the sun is the center of the world and immovable, and that the earth is not the center of the world, and moves, and that I must not hold, defend, or teach in any way whatsoever, verbally or in writing, the said false doctrine, and after it had been notified to me that the said doctrine was contrary to Holy Scripture — I wrote and printed a book in which I discuss this new doctrine already condemned, and adduce arguments of great cogency in its favor, without presenting any solution of these, and for this reason I have been pronounced by the Holy Office to be vehemently suspected of heresy, that is to say, of having held and believed that the Sun is the center of the world and immovable, and that the earth is not the center and moves: Therefore, desiring to remove from the minds of your Eminences, and of all faithful Christians, this vehement suspicion, justly conceived against me, with sincere heart and unfeigned faith I abjure, curse, and detest the aforesaid errors and heresies, and generally every other error, heresy, and sect whatsoever contrary to the said Holy Church, and I swear that in the future I will never again say or assert, verbally or in writing, anything that might furnish occasion for a similar suspicion regarding me; but that should I know any heretic, or person suspected of heresy, I will denounce him to this Holy Office, or to the Inquisitor or Ordinary of the place where I may be. Further, I swear and promise to fulfill and observe in their integrity all penances that have been, or that shall be, imposed upon me by this Holy Office. And, in the event of my contravening, (which God forbid) any of these my promises and oaths, I submit myself to all the pains and penalties imposed and promulgated in the sacred canons and other constitutions, general and particular, against such delinquents. So help me God, and these His Holy Gospels, which I touch with my hands. I, the said Galileo Galilei, have abjured, sworn, promised, and bound myself as above; and in witness of the truth thereof I have with my own hand subscribed the present document of my abjuration, and recited it word for word at Rome, in the Convent of Minerva, this twenty-second day of June, 1633.

I have been in my bed for five weeks, oppressed with weakness and other infirmities from which my age, seventy four years, permits me not to hope release. Added to this (proh dolor! [O misery!]) the sight of my right eye — that eye whose labors (dare I say it) have had such glorious results — is for ever lost. That of the left, which was and is imperfect, is rendered null by continual weeping.

Alas! Your dear friend and servant Galileo has been for the last month hopelessly blind; so that this heaven, this earth, this universe, which I by my marvelous discoveries and clear demonstrations had enlarged a hundred thousand times beyond the belief of the wise men of bygone ages, henceforward for me is shrunk into such a small space as is filled by my own bodily sensations.

Wine is a mixture of moisture and light.

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.

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

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.

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."

If Galileo had been willing to face the idea of a plurality of worlds, instead of resting on that of the Sun as the natural "centre of things," he might have been impelled to develop his system in the Newtonian sense. ...The position of the satellites in the Copernican scheme proved the existence of a multiplicity of centers. But the way that led from there was fraught with danger.

The allusion to the "puzzling" problem of [the orbit of] Mars shows that Galileo ought not to have been unaware of the great work of Kepler published in 1609: Astronomia nova... in which the first two of Kepler's laws were formulated. Yet he does not mention here at all Kepler's success in solving the problem, nor his laws, nor his name even, which is brought up... only to criticize his belief in the Moon's attraction [effect upon tides], which is quite reasonably presented in the Astronomia nova and founded on astronomical reasons and not on mystical speculations.

To give us the science of motion God and Nature have joined hands and created the intellect of Galileo.

The old Greek philosophy, which in Europe in the later middle ages was synonymous with the works of Aristotle, considered motion as a thing for which a cause must be found: a velocity required a force to produce and to maintain it. The great discovery of Galileo was that not velocity, but acceleration requires a force. This is the law of inertia of which the real content is: the natural phenomena are described by differential equations of the second order.

[T]he mathematical habit of mind and the mathematical procedure... had to be generated; otherwise Newton could never have thought of a formula representing the force between any two masses at any distance. ...Throughout the middle ages, under the influence of Aristotle, the science was entirely misconceived. Newton had the advantage of coming after a series of great men, notably Galileo... who in the previous two centuries had reconstructed the science and had invented the right way of thinking about it. He completed their work.

The way in which the persecution of Galileo has been remembered is a tribute to the quiet commencement of the most intimate change in outlook which the human race had yet encountered. Since a babe was born in a manger, it may be doubted whether so great a thing has happened with so little stir.

The worst that happened to men of science was that Galileo suffered an honorable detention and a mild reproof, before dying peacefully in his bed.

In Galileo's time, professors of philosophy and theology—the subjects were inseparable—produced grand discourses on the nature of reality... all based on sophisticated metaphysical arguments. Meanwhile, Galileo measured how fast balls roll down inclined planes. How mundane! But the learned discourses, while grand, were vague. Galileo's investigations were clear and precise. The old metaphysics never progressed, while Galileo's work bore abundant, and at length spectacular, fruit. Galileo too cared about the big questions, but he realized that getting the genuine answers requires patience and humility before the facts.

How does the world recognizes England, the United Kingdom, as the country that gave birth to the modern age? It was not Newton but Galilei who opened the Moderna age.

As a believing scientist [...] it is my deep conviction that it is our task to search nature and the universe, as Galileo Galilei, the father of modern science, did, for the footprints of God.

[http://www.galilean-library.org/hps.html The Galileo Affair] by Paul Newall.

[http://www.cslewis.org/journal/?p=10 Conflict or Cooperation] by Robert C. Fay

[http://galileo.rice.edu/ The Galileo Project] at Rice University