144 quotes found
"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."
"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."
"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."
"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."
"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."
"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."
"Persisting in their original resolve to destroy me and everything mine by any means they can think of, these men are aware of my views in astronomy and philosophy. They know that as to the arrangement of the parts of the universe, I hold the sun to be situated motionless in the center of the revolution of the celestial orbs while the earth revolves about the sun. They know also that I support this position not only by refuting the arguments of Ptolemy and Aristotle, but by producing many counter-arguments; in particular, some which relate to physical effects whose causes can perhaps be assigned in no other way. In addition there are astronomical arguments derived from many things in my new celestial discoveries that plainly confute the Ptolemaic system while admirably agreeing with and confirming the contrary hypothesis."
"To this end they make a shield of their hypocritical zeal for religion. They go about invoking the Bible, which they would have minister to their deceitful purposes. Contrary to the sense of the Bible and the intention of the holy Fathers, if I am not mistaken, they would extend such authorities until even in purely physical matters — where faith is not involved — they would have us altogether abandon reason and the evidence of our senses in favor of some biblical passage, though under the surface meaning of its words this passage may contain a different sense."
"Copernicus never discusses matters of religion or faith, nor does he use argument that depend in any way upon the authority of sacred writings which he might have interpreted erroneously. ... He did not ignore the Bible, but he knew very well that if his doctrine were proved, then it could not contradict the Scriptures when they were rightly understood."
"Nature … is inexorable and immutable; she never transgresses the laws imposed upon her, or cares a whit whether her abstruse reasons and methods of operation are understandable to men. For that reason it appears that nothing physical which sense-experience sets before our eyes, or which necessary demonstrations prove to us, ought to be called in question (much less condemned) upon the testimony of biblical passages which may have some different meaning beneath their words. For the Bible is not chained in every expression to conditions as strict as those which govern all physical effects; nor is God any less excellently revealed in Nature's actions than in the sacred statements of the Bible."
"I do not feel obliged to believe that the same God who has endowed us with senses, reason, and intellect has intended us to forgo their use and by some other means to give us knowledge which we can attain by them."
"I would say here something that was heard from an ecclesiastic of the most eminent degree [probably Caesar Baronius]: "The intention of the Holy Ghost is to teach us how one goes to heaven, not how heaven goes.""
"Philosophy is written in this grand book, which stands continually open before our eyes (I say the 'Universe'), but can not be understood without first learning to comprehend the language and know the characters as it is written. It is written in mathematical language, and its characters are triangles, circles and other geometric figures, without which it is impossible to humanly understand a word; without these one is wandering in a dark labyrinth."
"Whence do you have it that the terrestrial globe is so heavy? For my part, either I do not know what heaviness is, or the terrestrial globe is neither heavy nor light, as likewise all other globes of the universe. Heaviness to me (and I believe to Nature) is that innate tendency by which a body resists being moved from its natural place and by which, when forcibly removed therefrom, it spontaneously returns there. Thus a bucketful of water raised on high and set free, returns to the sea; but who will say that the same water remains heavy in the sea, when being set free there, does not move?"
"I tell you that if natural bodies have it from Nature to be moved by any movement, this can only be circular motion, nor is it possible that Nature has given to any of its integral bodies a propensity to be moved by straight motion. I have many confirmations of this proposition, but for the present one alone suffices, which is this. I suppose the parts of the universe to be in the best arrangement, so that none is out of its place, which is to say that Nature and God have perfectly arranged their structure. This being so, it is impossible for those parts to have it from Nature to be moved in straight, or in other than circular motion, because what moves straight changes place, and if it changes place naturally, then it was at first in a place preternatural to it, which goes against the supposition. Therefore, if the parts of the world are well ordered, straight motion is superfluous and not natural, and they can only have it when some body is forcibly removed from its natural place, to which it would then return by a straight line, for thus it appears that a part of the earth does [move] when separated from its whole. I said "it appears to us," because I am not against thinking that not even for such an effect does Nature make use of straight line motion."
"It always seems to me extreme rashness on the part of some when they want to make human abilities the measure of what nature can do. On the contrary, there is not a single effect in nature, even the least that exists, such that the most ingenious theorists can arrive at a complete understanding of it. This vain presumption of understanding everything can have no other basis than never understanding anything. For anyone who had experienced just once the perfect understanding of one single thing, and had truly tasted how knowledge is accomplished, would recognize that of the infinity of other truths he understands nothing."
"To apply oneself to great inventions, starting from the smallest beginnings, is no task for ordinary minds; to divine that wonderful arts lie hid behind trivial and childish things is a conception for superhuman talents."
"I cannot without great astonishment — I might say without great insult to my intelligence — hear it attributed as a prime perfection and nobility of the natural and integral bodies of the universe that they are invariant, immutable, inalterable, etc., while on the other hand it is called a great imperfection to be alterable, generable, mutable, etc. For my part I consider the earth very noble and admirable precisely because of the diverse alterations, changes, generations, etc. that occur in it incessantly. If, not being subject to any changes, it were a vast desert of sand or a mountain of jasper, or if at the time of the flood the waters which covered it had frozen, and it had remained an enormous globe of ice where nothing was ever born or ever altered or changed, I should deem it a useless lump in the universe, devoid of activity and, in a word, superfluous and essentially non-existent. This is exactly the difference between a living animal and a dead one; and I say the same of the moon, of Jupiter, and of all other world globes. The deeper I go in considering the vanities of popular reasoning, the lighter and more foolish I find them. What greater stupidity can be imagined than that of calling jewels, silver, and gold "precious," and earth and soil "base"? People who do this ought to remember that if there were as great a scarcity of soil as of jewels or precious metals, there would not be a prince who would not spend a bushel of diamonds and rubies and a cartload of gold just to have enough earth to plant a jasmine in a little pot, or to sow an orange seed and watch it sprout, grow, and produce its handsome leaves, its fragrant flowers, and fine fruit. It is scarcity and plenty that make the vulgar take things to be precious or worthless; they call a diamond very beautiful because it is like pure water, and then would not exchange one for ten barrels of water. Those who so greatly exalt incorruptibility, inalterability, etc. are reduced to talking this way, I believe, by their great desire to go on living, and by the terror they have of death. They do not reflect that if men were immortal, they themselves would never have come into the world. Such men really deserve to encounter a Medusa's head which would transmute them into statues of jasper or of diamond, and thus make them more perfect than they are."
"If what we are discussing were a point of law or of the humanities, in which neither true nor false exists, one might trust in subtlety of mind and readiness of tongue and in the greater experience of the writers, and expect him who excelled in those things to make his reasoning most plausible, and one might judge it to be the best. But in the natural sciences, whose conclusions are true and necessary and have nothing to do with human will, one must take care not to place oneself in the defense of error; for here a thousand Demostheneses and a thousand Aristotles would be left in the lurch by every mediocre wit who happened to hit upon the truth for himself. Therefore, Simplicio, give up this idea and this hope of yours that there may be men so much more learned, erudite, and well-read than the rest of us as to be able to make that which is false become true in defiance of nature."
"If you could see the earth illuminated when you were in a place as dark as night, it would look to you more splendid than the moon."
"In the long run my observations have convinced me that some men, reasoning preposterously, first establish some conclusion in their minds which, either because of its being their own or because of their having received it from some person who has their entire confidence, impresses them so deeply that one finds it impossible ever to get it out of their heads. Such arguments in support of their fixed idea as they hit upon themselves or hear set forth by others, no matter how simple and stupid these may be, gain their instant acceptance and applause. On the other hand whatever is brought forward against it, however ingenious and conclusive, they receive with disdain or with hot rage — if indeed it does not make them ill. Beside themselves with passion, some of them would not be backward even about scheming to suppress and silence their adversaries."
"Among all the great men who have philosophized about this remarkable effect, I am more astonished at Kepler than at any other. Despite his open and acute mind, and though he has at his fingertips the motions attributed to the earth, he nevertheless lent his ear and his assent to the moon's dominion over the waters, to occult properties, and to such puerilities."
"The sun, with all those planets revolving around it and dependent on it, can still ripen a bunch of grapes as if it had nothing else in the universe to do."
"Of such are the mathematical sciences alone; that is, geometry and arithmetic, in which the Divine intellect indeed knows infinitely more propositions, since it knows all. But with regard to those few which the human intellect does understand, I believe its knowledge equals the Divine in objective certainty, for here it succeeds in understanding necessity, beyond which there can be no greater sureness."
"I cannot sufficiently admire the eminence of those men's wits, that have received and held it to be true, and with the sprightliness of their judgments offered such violence to their own senses, as that they have been able to prefer that which their reason dictated to them, to that which sensible experiments represented most manifestly to the contrary. ...I cannot find any bounds for my admiration, how that reason was able in Aristarchus and Copernicus, to commit such a rape on their senses, as in despite thereof to make herself mistress of their credulity."
"After the publication of my dialogues, I was summoned to Rome by the Congregation of the holy Office, where, being arrived on the 10th of February 1633, I was subjected to the infinite clemency of that tribunal, and of the Sovereign Pontiff, Urban the Eighth; who, notwithstanding, thought me deserving of his esteem."
"I am certainly interested in a tribunal in which, for having used my reason, I was deemed little less than a heretic. Who knows but men will reduce me from the profession of a philosopher to that of historian of the Inquisition! But they behave to me in order that I may become the ignoramus and the fool of Italy..."
"I was obliged to retract, like a good Catholic, this opinion of mine; and as a punishment my dialogue was prohibited; and after five months being dismissed from Rome (at the time that the city of Florence was infected with plague), the habitation which with generous pity was assigned to me, was that of the dearest friend I had in Siena, Monsignor the Archbishop Piccolomini, whose most agreeable conversation I enjoyed with such quite and satisfaction of mind, that having there resumed my studies, I discovered and demonstrated a great number on the mechanical conclusions on the resistance of solids … after about five months, the pestilence having ceased, the confinement of that house was changed by His Holiness for the freedom of the country so agreeable to me, whence I returned to the villa of Bellosguardo, and afterwards to Arcetri, where I still breathe salubrious air near my dear native-country Florence. Stay sane."
"Well, since paradoxes are at hand, let us see how it might be demonstrated that in a finite continuous extension it is not impossible for infinitely many voids to be found."
"I am quite convinced; and, believe me, if I were again beginning my studies, I should follow the advice of Plato and start with mathematics, a science which proceeds very cautiously and admits nothing as established until it has been rigidly demonstrated."
"My purpose is to set forth a very new science dealing with a very ancient subject. There is, in nature, perhaps nothing older than motion, concerning which the books written by philosophers are neither few nor small; nevertheless I have discovered by experiment some properties of it which are worth knowing and which have not hitherto been either observed or demonstrated. Some superficial observations have been made, as, for instance, that the free motion [naturalem motum] of a heavy falling body is continuously accelerated; but to just what extent this acceleration occurs has not yet been announced; for so far as I know, no one has yet pointed out that the distances traversed, during 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."
"It has been observed that missiles and projectiles describe a curved path of some sort; however no one has pointed out the fact that this path is a parabola. But this and other facts, not few in number or less worth knowing, I have succeeded in proving; and what I consider more important, there have been opened up to this vast and most excellent science, of which my work is merely the beginning, ways and means by which other minds more acute than mine will explore its remote corners."
"This [experimentation] is the custom—and properly so—in those sciences where mathematical demonstrations are applied to natural phenomena, as is seen in the case of perspective, astronomy, mechanics, music, and others where the principles, once established by well-chosen experiments, become the foundations of the entire superstructure."
"See now the power of truth; the same experiment which at first glance seemed to show one thing, when more carefully examined, assures us of the contrary."
"Indeed, I think we may concede to our Academician, without flattery, his claim that in the principle [principio, i. e., accelerated motion] laid down in this treatise he has established a new science dealing with a very old subject. Observing with what ease and clearness he deduces from a single principle the proofs of so many theorems, I wonder not a little how such a question escaped the attention of Archimedes, Apollonius, Euclid and so many other mathematicians and illustrious philosophers, especially since so many ponderous tomes have been devoted to the subject of motion. (Galileo referred to himself as the/our Academician in his dialogue)"
"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..."
"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."
"I have never met a man so ignorant that I could not learn something from him."
"Eppur si muove."
"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 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."
"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."
"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.""
"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."
"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."
"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."
"It is to the Italian astronomer, forced in old age by the Inquisition to turn aside from the more dangerous study of the machinery of the heavens, that we owe the first exposition of many of the problems of mechanics and statics, published... in 1638. Not only did Galileo put together whatever the sixteenth century had learned in the sciences affecting building construction, but from his study of the bending strength of a beam there dates a new branch of science—the theory of the strength of materials."
"Conclusions obtained by purely rational processes are, so far as Reality is concerned, entirely empty. It was because he recognized this, and especially because he impressed it upon the scientific world that Galileo became the father of modern physics and in fact of the whole of modern natural science."
"It has always hurt me to think that Galilei did not acknowledge the work of Kepler … That, alas, is vanity … You find it in so many scientists."
"Galileo was no idiot. Only an idiot could believe that science requires martyrdom — that may be necessary in religion, but in time a scientific result will establish itself."
"The beginning of astronomy, except observations, I think is not to be derived from farther time than from Nicolaus Copernicus; who in the age next preceding the present revived the opinion of Pythagoras, Aristarchus, and Philolaus. After him, the doctrine of the motion of the earth being now received, and a difficult question thereupon arising concerning the descent of heavy bodies, Galileus in our time, striving with that difficulty, was the first that opened to us the gate of natural philosophy universal, which is the knowledge of the nature of motion. So that neither can the age of natural philosophy be reckoned higher than to him."
"A light was kindled amongst the investigators of nature when Galilei let balls of a definite weight roll down the inclined plane. For they saw that they only understand what is produced according to a predetermined plan or hypothesis... for otherwise planless observations made according to no ideas could never be brought into the form of a law which reason demands and seeks. ...Thus physics was brought into the position of a certain science after groping about blindly for so many hundred years."
"Copernicus, Kepler and Galileo were ‘revisionists’ in rejecting the geocentric system of Ptolemy (which held sway for some 1500 years) and, against an oppressive and repressive mainstream opinion (and officialdom), reinstated—with improvements—the heliocentric system of Aristarchos of Samos (3rd cent BCE)."
"Galileo's program offers us a dead world: Out go sight, sound, taste, touch, and smell, and along with them have since gone esthetic and ethical sensibility, values, quality, soul, consciousness, spirit. Experience as such is cast out of the realm of scientific discourse. Hardly anything has changed our world more during the past four hundred years than Galileo's audacious program. We had to destroy the world in theory before we could destroy it in practice."
"Galileo deduced the laws of freely falling bodies and the parabolic paths of projectiles, initiating an era of applications of mathematics to physics. In his book Two New Sciences, he used indivisible methods to study the motion of a falling body, and he planned, but never wrote, an entire book on indivisibles."
"The first mathematician to consider the nature of the resistance of solids to rupture was Galileo. Although he treated solids as inelastic, not being in possession of any law connecting the displacements produced with the forces producing them, or of any physical hypothesis capable of yielding such a law, yet his enquiries gave the direction which was subsequently followed by many investigators."
"He endeavoured to determine the resistance of a beam, one end of which is built into a wall, when the tendency to break it arises from its own or an applied weight; and he concluded that the beam tends to turn about an axis perpendicular to its length, and in the plane of the wall. This problem, and, in particular, the determination of this axis is known as Galileo's problem."
"History can be helpful in making sense of the world we live in. It can also be fascinating, even fun. How can even the best novelist or playwright invent someone like Augustus Caesar or Catherine the Great, Galileo or Florence Nightingale? How can screenwriters create better action stories or human dramas than exist, thousand upon thousand, throughout the many centuries of recorded history? There is a thirst out there both for knowledge and to be entertained, and the market has responded with enthusiasm."
"Galileo's comprehension of the concept of acceleration, which he defined as a change of velocity either in magnitude or direction... was an abstract idea that no one seems to have thought much about before. And in using it to test the still accepted Aristotelian precept that a moving object requires a force to maintain it, Galileo easily demonstrated that it is not motion but rather acceleration which cannot occur without an external force. Deliberately rejecting common sense as a prejudiced witness, he let nature herself speak in the form of a "hard, smooth and very round ball" rolling down a "very straight" ideal groove lined with polished parchment, and then rolling up another groove, clocking each roll "hundreds or times"... he showed that, while downward motion (helped by gravity force) makes speed increase and upward motion (hindered by gravity force) makes speed decrease, there is always a "boundary case" in between... where speed remains constant (without any appreciable force)—and that, by reducing friction, this boundary case can be made to approach a horizontal level where gravity has no effect. Similarly testing... he also drafted a law of falling bodies: "that the distances traversed, during equal intervals of time... stand to one another in the same ratio as the odd numbers beginning with unity." And his beautiful analysis of a cannonball's trajectory into horizontal and vertical components... was one day to be of enormous help to Isaac Newton in solving the riddle of gravity."
"It was in Galileo's time that firearms were invented... What is the path of a cannonball? ...Characteristically, Galileo was engrossed with the problem; characteristically, he solved it. The outcome of his ingenuity we know today as the method of superposition."
"How do heavy bodies fall? ...Galileo's investigation of dynamics was physical; Aristotle's was metaphysical. But unlike Galileo, we have the additional convenience of algebraic notation. Had it been invented in his day he would certainly have known it; almost certainly he would have been able to push his development of dynamics much farther."
"Galileo's head was on the block... The crime was looking up for truth... And then you had to bring up reincarnation... How long 'til my soul gets it right... Can any human being ever reach that kind of light... I call on the resting soul of Galileo... King of night vision, king of insight... I'm not making a joke, you know me...I take everything so seriously... If we wait for the time 'til all souls get it right... Then at least I know there'll be no nuclear annihilation... Can any human being ever reach the highest light... Except for Galileo — God rest his soul... Except for the resting soul of Galileo... King of night vision, king of insight."
"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."
"The greatest advantage in gambling lies in not playing at all."
"Better it is to have the worst, than none at all. for example we see, that houses are nedefull, such as can not possese & stately pallaces of stone, do persuade themselves to dwell in houses of timber and clap, and wanting them, are contented to inhabite the simple cotage, yea rather than not to be housed at all refuse not the pore cabbon, and most beggerly cave. So necessarie is this gift of consolacion, as there livith no man, but that hathe cause to embrace it. for in these things better is it to have any than none at al."
"And wel we see ther is none alive that in every respect may be accompted happie, yea though mortall men were free from all calamities, yet the torments & feare of death should stil attend them But b:sides them, behold, what, and how manye evilles there bee, that unlesse the cloude of error bee removed, impossible it is to see the truth, or receive allay of our earthly woes."
"So shall we voyd of all craft and sail, with true reason declare how much each man erreth in life, judgement, opinion, and will. Some things there are that so wel do prove themselves, as besides nature nede no profe at all."
"Among other myseries what I pray you tá be greater than whē a man riseth frō bed in the morning, to be incertaine of his returne to rest againe. or being in bed, whether his life shall continue tyll he ryse. besydes that, what labour, what hazard & care, are men constrained to abyde with these our brittle bodies, our feeble force, and incertayne lyfe: so as no nacion I thinke a man better or more fitlye named than the Spaniard, who in their language do terme a man shadow. And sure ther is nothing to be found of lesse assurance or soner passed than the lyfe of man, no... may more rightly be resembled to a shadow."
"From these beginnings, as it were, have issued bitterness, contentious obstinancy, lack of amenity, hasty judgement, anger, and an intense desire for revenge—to say nothing of headstrong will; that which many damn, by word at least, was my delight."
"I am cold of heart, warm of brain, and given to never-ending meditation; I ponder over ideas, many and weighty, and even over things which can never come to pass."
"I am able to admit two distinct trains of thought to my mind at the same time."
"I have accustomed my features always to assume an expression quite contrary to my feelings; thus I am able to feign outwardly, yet within know nothing of dissumulation. This habit is easy if compared to the practice of hoping for nothing, which I have bent my efforts toward acquiring for fifteen successive years, and have at last succeeded."
"My personal affairs are not as highly esteemed as men commonly value their own interests—vain, empty affairs like those great clouds seen in the wake of the sunset which are meaningless and soon pass away."
"This I recognize as unique and outstanding among my faults—the habit...of preferring to say above all things what I know to be displeasing to the ears of my hearers. ...I keep it up wilfully, in no way ignorant of how many enemies it makes for me. ...Yet I avoid this practice in the presence of my benefactors and of my superiors. It is enough not to fawn upon these, or at least not to flatter them."
"What if one should address a word to the kings of the earth and say, "Not one of you but eats lice, flies, bugs, worms, fleas—nay the very filth of your servants! With what an attitude would they listen to such statements, though they be truths? What is this complacency then but an ignoring of conditions, a pretense of not being aware of what we know exists, or a will to set aside a fact by force? And so it is with everything else foul, vain, confused and untrue in our lives."
"I have not lost my faith; and this I must attribute more to a miracle than to my own wisdom; more to Divine Providence than to my own virtue. Steadfastly, in fact from my earliest childhood, I have made this my prayer, "Lord God... grant me long life, and wisdom, and health of mind and body.""
"My father, in my earliest childhood, taught me the rudiments of arithmetic, and about that time made me acquainted with the arcana; whence he had come by this learning I know not. This was about my ninth year. Shortly after, he instructed me in the elements of the astronomy of Arabia, meanwhile trying to instill in me some system of theory for memorizing, for I had been poorly endowed with the ability to remember. After I was twelve years old he taught me the first six books of Euclid, but in such a manner that he expended no effort on such parts as I was able to understand by myself. This is the knowledge I was able to acquire and learn without any elementary schooling..."
"Since this art surpasses all human subtelty and the perspecuity of mortal talent and is truly a celestial gift and a very clear test of the capacity of man's minds, whoever applies himself to it will believe that there is nothing that he cannot understand."
"Although a long series of rules might be added and a long discourse given about them, we conclude our detailed consideration with the cubic, others being merely mentioned, even if generally, in passing. For as positio refers to a line, quadratum to the surface, and cubum to a solid body, it would be very foolish for us to go beyond this point. Nature does no permit it."
"Cardano reasoned that the end of man is to know God and to mediate between the divine and the mortal. The is immortal and when permeated with is inseparable from God. True wisdom is gained from and by mathematics, as God has subjected the world to mathematical law."
"You troubled mindes with tormentes loste that sighes and sobs consumes: (Who breathes and puffes from burning breast, both smothring smoke and fumes.) Come reade this booke that freelye bringes, a boxe of balme full swete, An oyle to noynt the brused partes, of everye heavye spirete. ...The lame whose lack of legges is death, unto a loftye mynde, Wyll kiss his crotche and creepe on knees,Cardanus workes to fynde."
"It appears... from this short chapter [Ars Magna, lib x. ch. 1], that he had discovered most of the principal properties of the roots of equations, and could point out the number and nature of the roots, partly from the signs of the terms, and partly from the magnitude and relations of the co-efficients."
"Every medieval and Renaissance court had a royal astrologer who advised the duke or prince he served. ...Men such as Roger Bacon, who even in the thirteenth century was a clear and outspoken champion of the experimental method in science, and Jerome Cardan, one of the foremost mathematicians and physicians of the sixteenth century, subscribed to astrology."
"Jerome Cardan is... the founder of the higher algebra; for, whatever he may have borrowed from others, we derive the science from his Ars Magna, published in 1545. It contains many valuable discoveries; but that which has been most celebrated is the rule for the solution of cubic equations, generally known by Cardan's name, though he had obtained it from a man of equal genius in algebraic science, Nicolas Tartaglia. The original inventor appears to have been Scipio Ferreo, who, about 1505, by some unknown process, discovered the solution of a single case; that of x3 + px = q. Ferreo imparted the secret to one Fiore, or Floridus, who challenged Tartaglia to a public trial of skill, not unusual in that age. Before he heard of this, Tartaglia, as he assures us himself, had found out the solution of two other forms of cubic equation; x3 + px2 = q, and x3 - px2 = q. When the day of trial arrived, Tartaglia was able, not only to solve the problems offered by Fiore, but to baffle him entirely by others which resulted in the forms of equation, the solution of which had been discovered by himself. This was in 1535; and, four years afterwards, Cardan obtained the secret from Tartaglia under an oath of secrecy. In his Ars Magna, he did not hesitate to violate this engagement; and, though he gave Tartaglia the credit of the discovery, revealed the process to the world."
"Anticipations of Cardan are more truly wonderful when we consider that the symbolical language of algebra, that powerful instrument not only expediting the processes of thought, but in suggesting general truths to the mind, was nearly unknown in his age. Diophantus, Fra Luca, and Cardan make use occasionally of letters to express indefinite quantities besides the res or cosa, sometimes written shortly, for the assumed unknown number of an equation. But letters were not yet substituted for known quantities. Michael Stifel, in his Arithmetics Integra, Nuremberg, 1544, is said to have first used the signs + and -, and numeral exponents of powers. It is very singular that discoveries of the greatest convenience, and apparently, not above the ingenuity of a village schoolmaster, should have been overlooked by men of extraordinary acuteness like Tartaglia, Cardan, and Ferrari; and hardly less so, that by dint of this acuteness they dispensed with the aid of these contrivances, in which we suppose that so much of the utility of algebraic expression consists."
"His career is an account of the most extraordinary and inconsistent acts. A gambler, if not a murderer, he was an ardent student of science, solving problems which had long baffled all investigation; at one time in his life he was devoted to intrigues which were a scandal even in the sixteenth century, at another he did nothing but rave on astrology, and yet at another he declared that philosophy was the only subject worthy of a man's attention. His was the genius that was closely allied to madness."
"After spending a year or so in France, Scotland, and England, he returned to Milan as professor of science, and shortly afterward was elected to a chair at Pavia. Here he divided his time between debauchery, astrology, and mechanics. His two sons were as wicked and passionate as himself: the elder was in 1560 executed for poisoning his wife, and about the same time Cardan in a fit of rage cut off the ears of the younger who committed some offence; for this scandelous outrage he suffered no punishment, as Pope Gregory XIII granted him protection."
"In 1570 he was imprisoned for heresy on account of his having published the horoscope of Christ, and when released he found himself... generally detested..."
"Cardan was the most distinguished astrologer of his time, and when he settled in Rome he received a pension in order to secure his services as astrologer to the papal court. This proved fatal to him for, having foretold that he should die on a particular date, he felt obliged to commit suicide in order to keep up his reputation—so at least the story runs."
"The Ars Magna is a great advance on any algebra previously published. Hitherto algebraists had confined their attention to those roots of equations which were positive. Cardan discussed negative and even complex roots, and proved that the latter would always occur in pairs, though he declined to commit himself to any explanation as to the meaning of these "sophistic" quantities which he said were ingenious though useless."
"Most of his analysis of cubic equations seems to have been original; he shewed that if the three roots were real, Tartaglia's solution gave them in a form which involved imaginary quantities. Except for the somewhat similar researches of Bombelli a few years later, the theory of imaginary quantities received little further attention from mathematicians until John Bernoulli and Euler took up the matter after the lapse of nearly two centuries. Gauss first put the subject on a systematic and scientific basis, introduced the notation of complex variables, and used the symbol i, which had been introduced by Euler in 1777, to denote the square root of (-1): the modern theory is chiefly based on his researches."
"The first epoch-making algebra to appear in print was the Ars Magna of Cardan (1545). This was devoted primarily to the solution of algebraic equations. It contained the solution of the cubic and biquadratic equations, made use of complex numbers, and in general may be said to have been the first step toward modern algebra."
"[Zuanne de Tonini] da Coi... impuned Tartaglia to publish his method, but the latter declined to do so. In 1539 Cardan wrote to Tartaglia, and a meeting was arranged at which, Tartaglia says, having pledged Cardan to secrecy, he revealed the method in cryptic verse and later with a full explanation. Cardan admits that he received the solution from Tartaglia, but... without any explanation. At any rate, the two cubics x^3 + ax^2 = c and x^3 + bx = c could now be solved. The reduction of the general cubic x^3 + ax^2 + bx = c to the second of these forms does not seem to have been considered by Tartaglia at the time of the controversy. When Cardan published his Ars Magna however, he transformed the types x^3 = ax^2 + c and x^3 + ax^2 = c by substituting x = y + \frac{1}{3}a and x = y - \frac{1}{3}a respectively, and transformed the type x^3 + c = ax^2 by the substitution x = \sqrt[3]{c^2/y}, thus freeing the equations of the term x^2. This completed the general solution, and he applied the method to the complete cubic in his later problems."
"Cardan's originality in the matter seems to have been shown chiefly in four respects. First, he reduced the general equation to the type x^3 + bx = c; second, in a letter written August 4, 1539, he discussed the question of the irreducible case; third, he had the idea of the number of roots to be expected in the cubic; and, fourth, he made a beginning in the theory of symmetric functions. ...With respect to the irreducible case... we have the cube root of a complex number, thus reaching an expression that is irreducible even though all three values of x turn out to be real. With respect to the number of roots to be expected in the cubic... before this time only two roots were ever found, negative roots being rejected. As to the question of symmetric functions, he stated that the sum of the roots is minus the coefficient of x2"
"He states that the root of x^3 + 6x = 20 is{{center|1=x = \sqrt[3]{\sqrt{108} + 10} - \sqrt[3]{\sqrt{108} - 10}.}}"
"He... gave thirteen forms of the cubic which have positive roots, these having already been given by Omar Kayyam."
"The problem of the biquadratic equation was laid prominently before Italian mathematicians by Zuanne de Tonini da Coi, who in 1540 proposed the problem, "Divide 10 parts into three parts such that they shall be continued in proportion and that the product of the first two shall be 6." He gave this to Cardan with the statement that it could not be solved, but Cardan denied the assertion, although himself unable to solve it. He gave it to Ferrari, his pupil, and the latter, although then a mere youth, succeeded where the master had failed. ...This method soon became known to algebraists through Cardan's Ars Magna, and in 1567 we find it used by Nicolas Petri [of Deventer]."
"The law which asserts that the equation X = 0, complete or incomplete, can have no more real positive roots than it has changes of sign, and no more real negative roots than it has permanences of sign, was apparently known to Cardan; but a satisfactory statement is possibly due to Harriot (died 1621) and certainly to Descartes."
"The application of the theory [of probability] to mortality tables in any large way may be said to have started with John Graunt... The first tables of great importance, however, were those of Edmund Halley... however... Cardan seems to have been the first to have been the first to consider the problem in a printed work, although his treatment is very fanciful. He gives a brief table in his proposition "Spatium vitae naturalis per spatium vitae fortuitum declarare," this appearing in the De Proportionibus Libri V..."
"Cardano's entertaining books on science and curiosities were among the best read and most pirated works in the sixteenth century. ...his work on the "Great Art" has been characterized as the first that goes decisively beyond the attainments of classical Greek mathematics."
"Most important for the history of science is the fact that Liber de Ludo Aleae, "The Book of Games of Chance," contains the first study of the principles of probability. ...it would seem much more just to date the beginnings of probability theory from Cardano's treatise rather than the customary reckoning from Pascal's discussions with his friend de Méré and the ensuing correspondence with Fermat... at least a century after Cardano..."
"Cardano was a man of universal interests, and much of his ability must have been inherited from his father, Fazio Cardano... a lawyer... but also deeply steeped in the medical sciences, mathematics, and all kinds of occult lore... he had... a high reputation as a scholar in his native town; even Leonardo da Vinci notes several times that he consulted Messer Fazio on geometric questions... he was appointed as a public lecturer in geometry."
"When Cardano's Consolation or Comforte was translated into English in 1573... one of the readers is known to have been William Shakespeare. ...Hamlet's thoughts on death and slumber are believed to have been inspired by... passages in Comforte..."
"A great number of writers on the history of medicine have indicated important observations and suggestions which made their intitial appearance with Cardano."
"...they make no use of tables; but only of the addition, subtraction, multiplication, and division of certain numbers, of which we do not presently discern the ground, nor to what these numbers refer."