Astronomers From Greece

"[[Physics|[P]hysicists]]... try to find a fundamental law of motion for matter from which all elementary particles and their properties can be derived mathematically. This fundamental equation... may refer either to waves of a known type... or to waves... which have nothing to do with any of the known waves or elementary particles. In the first case it would mean that all other elementary particles can be reduced in some way to a few sorts of "fundamental" elementary particles... In the second case all different elementary particles could be reduced to some universal substance... energy or matter, but none of the... particles could be preferred... as... more fundamental. The latter... corresponds to... Anaximander, and... this view is the correct one."

"Anaximander denied the fundamental substance to be... any of the known substances. He taught that the primary substance was infinite, eternal and ageless and... encompassed the world. This... is transformed into the various substances... Theophrastus quotes from Anaximander: "Into that from which things take their rise they pass away once more... for they make reparation and satisfaction to one another for their injustice according to the ordering of time." In this... the antithesis of Being and Becoming plays the fundamental role. The primary substance, infinite... ageless... undifferentiated Being, degenerates into... forms which lead to endless struggles. ...Becoming is ...a ...debasement of the infinite Being—a disintegration into the struggle ultimately expiated by a return into that ...without shape or character. The struggle ...is the opposition between hot and cold, fire and water, wet and dry, etc. ...[T]emporary victory ...is the injustice for which they ...make reparation in the ordering of time. ...[T]here is "eternal motion," the creation and passing away of worlds from infinity to infinity."

"The big bang and the steady state debate in some ways echoed that between the ideas of Anaximander and Anaxagoras from two and a half millennia earlier. Anaxagoras had envisaged that at one time "all things were together" and that the motive force for the universe originated at a single point... Anaximander on the other hand wanted a universe determined by "the infinite" and needed an "eternal motion" to explain the balancing process of things coming into being and passing away in an eternal universe... ancient philosophy was debating the alternatives of a creation event starting the universe from a single point versus a continuous creation in an eternal universe."

"Anaximander relies on the accuracy of geometry in matters beyond the range of any kind of verification—in its application to cosmic proportions—and also in contradiction to appearance, which suggests that the sun is about as large in diameter as the width of a human foot. The concept of geometrical similarity is also the precondition for Anaximander's attempt to construct a map of the world."

"The Earth is cylindrical, three times as wide as it is deep, and only the upper part is inhabited. But this Earth is isolated in space, and the sky is a complete sphere in the center of which is located, unsupported, our cylinder, the Earth, situated at an equal distance from all the points of the sky."

"With regard to the Pythagorean theorem my conjecture is that... it was known to Thales. ...the hypotenuse theorem is a direct consequence of the principle of similitude, and... Thales was fully conversant with the theory of similar triangles."

"It was not Zeno, the founder of the Stoics, alone, who taught that the Universe evolves, and its primary substance is transformed from the state of fire into that of air, then into that of water, etc. Heraclitus of Ephesus maintained that the one principle that underlies all phenomena in Nature is fire. The intelligence that moves the Universe is fire, and fire is intelligence. And while Anaximenes said the same of air, and Thales of Miletus (600 years b.c.) of water, the Esoteric Doctrine reconciles all these philosophers, by showing that though each was right, the system of none was complete."

"It has fallen to the lot of one people, the ancient Greeks, to endow human thought with two outlooks on the universe neither of which has blurred appreciably in more than two thousand years. ...The first was the explicit recognition that proof by deductive reasoning offers a foundation for the structure of number and form. The second was the daring conjecture that nature can be understood by human beings through mathematics, and that mathematics is the language most adequate for idealizing the complexity of nature into appreciable simplicity. Both are attributed by persistent Greek tradition to Pythagoras in the sixth century before Christ. ...there is an equally persistent tradition that it was Thales... who first proved a theorem in geometry. But there seems to be no claim that Thales... proposed the inerrant tactic of definitions, postulates, deductive proof, theorem as a universal method in mathematics. ...in attributing any specific advance to Pythagoras himself, it must be remembered that the Pythagorean brotherhood was one of the world's earliest unpriestly cooperative scientific societies, if not the first, and that its members assigned the common work of all by mutual consent to their master."

"It seems probable that the early Greeks were largely indebted to the Phoenicians for their knowledge of practical arithmetic or the art of calculation, and perhaps also learnt from them a few properties of numbers. It may be worthy of note that Pythagoras was a Phoenician; and according to Herodotus, but this is more doubtful, Thales was also of that race."

"Thales had a motto — sophotaton chronos aneuriskei gar panta — which means time is wisest because it discovers everything. We still live by that motto — we mark the time and aid the discoveries by keeping the soul lines intact."

"Thales asserted Water to be the principle of things. For he saw that matter was principally dispensed in moisture, and moisture in water; and it seemed proper to make that the principle of things, in which the virtues and powers of beings, and especially the elements of their generations and restorations, were chiefly found. He saw that the breeding of animals is in moisture ; that the seeds and kernels of plants (as long as they are productive and fresh), are likewise soft and tender; that metals also melt and become fluid, and are as it were concrete juices of the earth, or rather a kind of mineral waters; that the earth itself is fertilised and revived by showers or irrigation, and that earth and mud seem nothing else than the lees and sediment of water; that air most plainly is but the exhalation and expansion of water; nay, that even fire itself cannot be lighted, nor kept in and fed, except with moisture and by means of moisture. He saw, too, that the fatness which belongs to moisture, and which is the support and life of flame and fire, seems a kind of ripeness and concoction of the water."

"Nothing is more ancient than God, for He was never created; nothing more beautiful than the world, it is the work of that same God; nothing is more active than thought, for it flies over the whole universe; nothing is stronger than necessity, for all must submit to it."

"Do not ask who started it. Finish it"

"Placing your stick at the end of the shadow of the pyramid, you made by the sun's rays two triangles, and so proved that the pyramid [height] was to the stick [height] as the shadow of the pyramid to the shadow of the stick."

"Water is the first principle of everything."

"Hope is the only good that is common to all men; those who have nothing else possess hope still."

"According to tradition, Thales is the first to reveal the study of nature to the Greeks; although he had many predecessors, in Theopharastus' view, he so surpassed them as to eclipse everyone before him."

"In committing himself to a form of materialism, Thales rejects a picture of the universe found in the Homeric poems, one which posits, in addition to the natural world, a supernatural quadrant populated by beings who are not subject to such laws as may govern the interactions of all natural bodies. If all things are composed of matter, then it ought to be possible to explain all there is to explain about the universe in terms of material bodies and their law-governed interactions. This simple thought already stands in sharp contrast to a world supposed to be populated by supernatural immaterial beings whose actions may be capricious or deliberate, rational or irrational, welcome or unwelcome, but which as a matter of basic principle cannot be explicated in terms of the forms of regularity found in the natural world. In Thales’ naturalistic universe, it ought to be possible to uncover patterns and laws and to use such laws as the basis for stable predictions about the direction the universe is to take; to uncover causes and to use that knowledge to find cures for illnesses or to develop strategies for optimizing our well-being; and, less practically, to find broad-based explanations to fundamental questions which crop up in every organized society. Such questions persist: Where did the universe come from? What, ultimately, is its basic stuff?"

"[Thales] first went to Egypt and hence introduced this study [geometry] into Greece. He discovered many propositions himself, and instructed his successors in the principles underlying many others, his method of attack being in some cases more general [i.e. more theoretical or scientific], in others more empirical [...more in the nature of simple inspection or observation]."

"While [Thales] was studying the stars and looking upwards, he fell into a pit, and a neat, witty Thracian servant girl jeered at him, they say, because he was so eager to know the things in the sky that he could not see what was there before him at his very feet."

"It has been asserted that metaphysical speculation is a thing of the past and that physical science has extirpated it. The discussion of the categories of existence, however, does not appear to be in danger of coming to an end in our time, and the exercise of speculation continues as fascinating to every fresh mind as it was in the days of Thales."

"Since Alyattes would not give up the Scythians to Cyaxares at his demand, there was war [ Battle of Halys ] between the Lydians and the Medes for five years; each won many victories over the other, and once they fought a battle by night. They were still warring with equal success, when it happened, at an encounter which occurred in the sixth year, that during the battle the day was suddenly turned to night. Thales of Miletus had foretold this loss of daylight to the Ionians, fixing it within the year in which the change did indeed happen."

"Thales' statement was the first expression of... a fundamental substance, of which all other things were transient forms. The word "substance"... was... not interpreted in the purely material sense ...Aristotle ascribes to Thales also ...All things are full of gods. ...[We can] imagine ...Thales took his view ...from meteorological considerations. ...[W]ater can take the most various shapes... ice and snow... vapor, and... clouds. It seems to turn into earth where the rivers form their delta, and it can spring from the earth. Water is the condition for life. Therefore... [as] a fundamental substance, it was natural to think of water first."

"[W]ith Thales... Aristotle ascribes the statement: "Water is the material cause of all things." This... expresses, as Nietzsche... pointed out, three fundamental ideas of philosophy. First, ...the material cause of all things; second, ...that this ...be answered in conformity with reason, without ...myths or mysticism; third, ...that ...it must be possible to reduce everything to one principle."

"Thales the teacher produced the first geometers, even as Thales the thinker founded the first geometry worthy of the name."

"The more one studies the period of Thales—the more one compares the knowledge he bequeathed to prosterity with the one he had found when he began his work—the more does his mathematical stature grow, until one is impelled to range Thales with such figures as Archimedes, Fermat, Newton, Gauss and Poincaré."

"The length of lifetakes the leading place among inquiries about events following birth."

"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)."

"To give here an elaborate account of Pappus would be to create a false impression. His work is only the last convulsive effort of Greek geometry which was now nearly dead and was never effectually revived. It is not so with Ptolemy or Diophantus. The trigonometry of the former is the foundation of a new study which was handed on to other nations indeed but which has thenceforth a continuous history of progress."

"Claudius Ptolemy's great contribution to astronomy was his famous work the Almagest, which presented formally the astronomical theories of the day that had evolved from the great debates within the different Greek philosophical schools. Claudius Ptolemy freely admitted that he had contributed little original research to the treatise but rather had based his conclusions principally on the work of Hipparchus. ...Ptolemy did not claim that his cosmological model described the actual conditions. It simply reproduced geometrically the observed motions of the known heavenly bodies and enabled their positions to be easily predicted for any particular time. … Ironically, even when Copernicus' heliocentric theory had replaced the Ptolemaic system, many astronomers used Ptolemy's model to predict the motion of the planets, since its intricate calculations produced more accurate values."

"Ptolemy... against the champions of this or that cosmology of the heavens... had dared to claim that it is legitimate to interpret the facts of astronomy by the simplest geometrical scheme which will 'save the phenomena,' no matter whose metaphysics might be upset. His conception of the physical structure of the earth, however, prevented him from carrying through in earnest this principle of relativity, as his objections to the hypothesis that the earth moves amply show."

"He left in his Optics, the earliest surviving table of angles of refraction from air to water. ...This table, quoted and requoted until modern times, has been admired... A closer glance at it, however, suggests that there was less experimentation involved in it than originally was thought, for the values of the angles of refraction form an arithmetic progression of second order... As in other portions of Greek Science, confidence in mathematics was here greater than that in the evidence of the senses, although the value corresponding to 60° agrees remarkably well with experience."

"Ptolemy's Geography is the only book on cartography to have survived from the classical period and one of the most influential scientific works of all time."

"There are three classes of friendship and enmity, since men are so disposed to one another either by preference or by need or through pleasure and pain."

"As material fortune is associated with the properties of the body, so honor belongs to those of the soul."

"πᾶν μὲν τὸ δυσέφικτον παρὰ τοῖς πολλοῖς εὐδιάβλητον ἔχει φύσιν"

"We consider it a good principle to explain the phenomena by the simplest hypothesis possible."

"I know that I am mortal by nature and ephemeral, but when I trace at my pleasure the windings to and fro of the heavenly bodies, I no longer touch earth with my feet. I stand in the presence of Zeus himself and take my fill of ambrosia."

"How many theorems in geometry which have seemed at first impracticable are in time successfully worked out!"

"Tragically for all of us, he [Archimedes] was cut down by a Roman soldier because he refused to stop working. … If Archimedes hadn't been killed before his time, what could have he achieved? The industrial revolution could have happened two thousand years earlier. He might have kick-started the modern age."

"According to legend, nothing could get between him [Archimedes] and his work, and sometimes he would even forget to eat. Ideas would come to him at any moment, and he would scribble them on any available surface. Famously, he was in the bath when he discovered the laws of buoyancy, leading him to run naked through the streets shouting "Eureka!" … Eureka means "I have found it," and it could be argued that Archimedes found out more than anyone else before or since."

"Archimedes was a brilliant inventor and a mathematician. He says to the people around him, "Don't just live in the lap of the gods. Don't be dominated by Mother Nature. You, as a man, can take control of your own destiny.""

"Archimedes was the earliest thinker to develop the apparatus of an infinite series with a finite limit ...starting on the conceptual path toward calculus. Of the giants on whose shoulders Isaac Newton would eventually perch, Archimedes was the first."

"Using his masterful understanding of mechanics, equilibrium, and the principles of the lever, he weighed in his mind solids or figures whose volumes or areas he was attempting to find against ones he already knew. After determining in this way the answer...he found it much easier to prove geometrically... Consequently The Method starts with a number of statements on centers of gravity and only then proceeds to the geometrical propositions and their proofs. ...[He] essentially introduced the concept of a thought experiment into rigorous research. ...[He] freed mathematics from the somewhat artificial chains that Euclid and Plato had put on it. ...He did not hesitate to explore and exploit the connections between the abstract mathematical objects (the Platonic forms) and physical reality (actual solids and flat objects) to advance his mathematics."

"Almost all modern translations of Archimedes’ works stem from a single Greek manuscript that was copied from an earlier original at Constantinople in the ninth or tenth century, was translated into Latin in the thirteenth century, and eventually disappeared without a trace in the sixteenth century."

"In Euclidean geometry the infinitely small was rejected and in the classical treatises of Archimedes we have the finest example of mathematical rigour in antiquity. Notwithstanding, in the discovery method we find him manipulating line and surface indivisibles skilfully, imaginatively and non-rigorously"

"The estimations, which occur in the summing of infinite series and in limiting operations, the "epsilontics", as the calculation with an arbitrarily small ε is sometimes called, were for Archimedes an open book. In this respect, his thinking is entirely modern."

"To conceive of a parabolic segment or of a triangle as the sum of infinitely many line segments, is closely akin to the idea of Leibniz, who thought of the integral \int y~dx as the sum of infinitely many terms y~dx. But, in contrast to Leibniz, Archimedes is fully aware that this conception is... incorrect and that the derivation should be supplemented by a rigorous proof."

"Modern mathematics was born with Archimedes and died with him for all of two thousand years. It came to life again with Descartes and Newton."

"There is here, as in all great Greek mathematical masterpieces, no hint as to the kind of analysis by which the results were first arrived at; for it is clear that they were not discovered by the steps which led up to them in the finished treatise. If the geometrical treatises had stood alone, Archimedes might seem, as Wallis said, "as it were of set purpose to have covered up the traces of his investigations, as if he has grudged posterity the secret of his method of inquiry, while he wished to extort from them assent to his results.""