Astronomers From Italy

"Analytic mechanics... was brought to the highest degree of perfection by Lagrange. Lagrange's aim is ('... 1788) to dispose, once and for all, of the reasoning necessary to resolve mechanical problems, by embodying as much as possible of it in a single formula. This he did. Every case... can now be dealt with by a very simple, highly symmetrical and perspicuous schema; and whatever reasoning is left is performed by purely mechanical methods. The mechanics of Lagrange is a stupendous contribution to the economy of thought."

"The questions here dealt with have occupied me since my earliest youth, when my interest for them was powerfully stimulated by the beautiful introductions of Lagrange to the chapters of his Analytic Mechanics..."

"Who has studied the works of such men as Euler, Lagrange, Cauchy, Riemann, , and Weierstrass, can doubt that a great mathematician is a great artist? The faculties possessed by such men, varying greatly in kind and degree with the individual, are analogous with those requisite for constructive art. Not every mathematician possesses in a specially high degree that critical faculty which finds its employment in the perfection of form, hi conformity with the ideal of logical completeness; but every great mathematician possesses the rarer faculty of constructive imagination."

"Lagrange, struck with the circumstance that the calculus had never given any inequalities but such as were periodical, applied himself to the investigation of a general question, from which he found by a method peculiar to himself and independent of any approximation, that the inequalities produced by the mutual action of the planets must in effect be all periodical; that the periodical changes are confined within narrow limits; that none of the planets ever has been or ever can be a comet moving in a very eccentric orbit; but that the planetary system oscillates as it were round a medium state from which it never deviates far: that amid all the changes which arise from the mutual actions of the planets, two things remain perpetually the same, viz. the length of the greater axis of the ellipse which the planet describes, and its periodical time round the sun; or, which is the same thing, the mean distance of each planet from the sun and its mean motion remain constant. The plane of the orbit varies, the species of the ellipse and its eccentricity change, but never, by any means whatever, the greater axis of the ellipse, or the time of the entire revolution of the planet. The discovery of this great principle, which we may consider as the bulwark that secures the stability of our system, and excludes all access to confusion and disorder, must render the name of Lagrange for ever memorable in science, and ever revered by those who delight in the contemplation of whatever is excellent and sublime. After Newton's discovery of the elliptic orbits of the planets from gravitation, Langrange's discovery of their periodical inequalities is, without doubt, the noblest truth in physical astronomy, and in respect of the doctrine of final causes, it may truly be regarded as the greatest of all."

"The mathematician is perfect only in so far as he is a perfect being, in so far as he perceives the beauty of truth; only then will his work be thorough, transparent, comprehensive, pure, clear, attractive and even elegant. All this is necessary to resemble Lagrange."

"Lagrange, in one of the later years of his life, imagined that he had overcome the difficulty [of the parallel axiom]. He went so far as to write a paper, which he took with him to the Institute, and began to read it. But in the first paragraph something struck him which he had not observed : he muttered II faut que fy songe encore, and put the paper in his pocket."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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."

"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?"

"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."

"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.""

"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."

"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."

"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."

"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."

"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."

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

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

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

"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."

"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."