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Definition and Origins

by Cristian Violatti
published on 13 June 2013
Fragment of Euclid's Elements (Jitse Niesen)
By careful thinking based upon observation, some ancient Greeks realized that it was possible to find regularities and patterns hidden in nature and that those regularities were the key to unlocking the secrets of the universe. It became evident that even nature had to obey certain rules and by knowing those rules one could predict the behaviour of nature.
Observation was eventually undervalued by the Greeks in favour of the deductive process, where knowledge is built by means of pure thought. This method is key in mathematics and the Greeks put such an emphasis on it that they falsely believed that deduction was the way to obtain the highest knowledge.


During the 26th Dynasty of Egypt (c. 685–525 BCE), the ports of the Nile were opened for the first time to Greek trade.Important Greek figures such as Thales and Pythagoras visited Egypt, and brought with them new skills and knowledge. Ionia, in addition to Egyptian influence, was exposed to the culture and ideas of Mesopotamia through its neighbour, the kingdom of Lydia.
According to Greek tradition, the process of replacing the notion of supernatural explanation with the concept of a universe that is governed by laws of nature begins in Ionia. Thales of Miletus, about 600 BCE first developed the idea that the world can be explained without resorting to supernatural explanations. It is high likely that the astronomical knowledge that Thales got from Egyptian and Babylonian astronomy allowed him to predict a solar eclipse which took place in May 28th 585 BCE.
Anaximander, another Ionian, argued that since human infants are helpless at birth, if the first human had somehow appeared on earth as an infant, it would not have survived. Anaximander reasoned that people must, therefore, have evolved from other animals whose young are hardier. It was Empedocles who first taught an early form of evolution and survival of the fittest. He believed that originally “countless tribes of mortal creatures were scattered abroad endowed with all manner of forms, a wonder to behold”, but in the end, only certain forms were able to survive.


The Greek achievements in mathematics and astronomy were one of the finest in antiquity. Mathematics developed first, aided by the influence of Egyptian mathematics; astronomy flourished later during the Hellenistic age, after Alexander the Greatconquered the East, aided by the influence of Babylon.


A powerful aspect of science is that it aims to detach itself from notions with specific use and looks for general principles with broad applications. The more general science becomes the more abstract it is and has more applications. What the Greeks derived from Egyptian mathematics were mainly rules of thumbs with specific applications. Egyptians knew, for example, that a triangle whose sides are in a 3:4:5 ratio is a right triangle. Pythagoras took this concept and stretched it to its limit by deducting a mathematical theorem that bears his name: that, in a right triangle, the square on the opposite side of the right angle (the hypotenuse) is equal to the sum of the squares on the other two sides. This was true not only for the 3:4:5 triangle, but it was a principle applicable to any other right triangle, regardless of its dimensions.
Pythagoras was the founder and leader of a sect where philosophy, religion, art and mysticism were all fused together. In ancient times, Greeks did not make a clear distinction between science and non-scientific disciplines. There is a widespread argument which states that the coexistence of philosophy, art, mysticism, and other non-scientific disciplines interacting together with science has interfered with the development of scientific ideas. This seems to show a misconception of how the human spirit works. It is true that in the past moral and mystic bias has either delayed or led some knowledge up a blind alley and that the sharp limits of scientific knowledge were not clear. However, it is equally true that non-scientific disciplines have enhanced the imagination of the human mind, provided inspiration to approach problems that seemed impossible to solve and triggered human creativity to consider counter-intuitive possibilities (such as a spherical earth in motion) that time proved to be true. The human spirit has found plenty of motivation for scientific progress in non-scientific disciplines and it is likely that without the driving force of art, mysticism and philosophy, scientific progress would have lacked much of its impetus.
Bust of Pythagoras

Bust of Pythagoras


By discovering mathematical theorems, the Greeks came across the art of deductive reasoning. In order to build their mathematical knowledge they came to conclusions by reasoning deductively from what appeared to be self-evident. This approach proved to be powerful and its success in mathematics encouraged its application in many other disciplines. The Greeks eventually came to believe that the only acceptable way of obtaining knowledge was the use of deduction.
However, this way of doing science had serious limitations when it was applied to other areas of knowledge, but from the standpoint of the Greeks it was hard to notice. In antiquity, the starting point to discover principles was always an idea in the mind of the philosopher: sometimes observations were undervalued and some other times the Greeks were not able to make a sharp distinction between empirical observations and logical arguments. Modern scientific method no longer relies on this technique; today science seeks to discover principles based on observations as a starting point. Likewise, the logical method of science today favours induction over deduction: instead of building conclusions on an assumed set of self-evident generalizations, induction starts with observations of particular facts and derives generalizations from them.
Deduction did not work for some kind of knowledge. “What is the distance from Athens to Chios?” In this case, the answer cannot be derived from abstract principles; we have to actually measure it. The Greeks, when necessary, looked at nature to get the answers they were looking for, but they still considered that the highest type of knowledge was the one derived directly from the intellect. It is interesting to note that when observations were taken in consideration, it tended to be subordinated to the theoretical knowledge. An example of this could be one of the surviving works of Archimedes, The Method, which explains how mechanical experiments can help the understanding of geometry. In general, ancient science used experimentation to help theoretical understanding while modern science uses theory to pursue practical results.
The undervaluing of empirical observation and the emphasis on pure thought as a reliable starting point for building knowledge can also be reflected in the famous account (in all probability apocryphal) of the Greek philosopher Democritus who removed his own eyes so the sight would not distract him from his speculations. There is also a story about a student of Plato who asked with irritation during a mathematics class “But what is the use of all this?” Plato called a slave, ordered him to give the student a coin, and said, “Now you need not feel your instruction has been entirely to no purpose” With these words, the student was expelled.



Aristotle was the first philosopher who developed a systematic study of logic. His framework would become an authority in deductive reasoning for over two thousand years. Although he repeatedly admitted the importance of induction, he prioritized the use of deduction to build knowledge. It eventually turned out that his influence strengthened the over-estimation of deduction in science and of syllogisms in logic.
The doctrine of syllogism is his most influential contribution to logic. He defined the syllogism as a “discourse in which certain things having been stated, something else follows of necesity from their being so”. A well known example is:
All men are mortal. (major premise)
Socrates is a man. (minor premise)
Socrates is mortal. (conclusion)
This argument cannot be logically challenged, nor can we challenge its conclusion. However, this way of doing science has, at least, two failures. In the first place, the way the major premise works. Why should we accept the major premise without question? The only way that a major premise can be accepted is to present an obvious statement, such as “all men are mortal”, which is considered self-evident. This means that the conclusion of this argument is not a new insight but rather, something that was already implied either directly or indirectly within the major premise. Secondly, it does not seem to be an actual need to go through all this argumentation in order to prove logically that Socrates is mortal.
Another problem of this way of building knowledge is that if we want to deal with areas of knowledge beyond the ordinary everyday life, there is a great risk of choosing wrong self-evident generalizations as a starting point of reasoning. An example could be two of the axioms upon which all Greek astronomy was built:
(1) The earth is resting motionless at the centre of the universe.
(2) The earth is corrupt and imperfect, while the heavens are eternal, changeless, and perfect.
These two axioms appear to be self-evident and they are supported by our intuitive experience. However, scientific ideas can be counter-intuitive. Today we know that intuition alone should never be the guide for knowledge and that all intuition should be sceptically tested. The errors in the way of reasoning are sometimes hard to detect and the Greeks were not able to notice anything wrong with their way of doing science. There is a very lucid example of this by Isaac Asimov:
...if brandy and water, whiskey and water, vodka and water, and rum and water are all intoxicating beverages, one may jump to the conclusion that the intoxicating factor must be the ingredient these drinks hold in common-namely, water. There is something wrong with this reasoning, but the fault in the logic is not immediately obvious;and in more subtle cases, the error may be hard indeed to discover. (Asimov, 7)
Aristotle's logic system was recorded in five treatises known as the Organon, and although it does not exhaust all logic, it was a pioneering one, revered for centuries and regarded as the ultimate solution to logic and reference for science.


Aristotle's contribution in logic and science became an authority and remained unchallenged as late as the modern age. It took many centuries to notice the flaws of Aristotle's approach to science. Platonic influence also contributed to undervalue inference and experimentation: Plato's philosophy considered the world to be only an imperfect representation of the ideal truth sitting in the world of ideas.
Another obstacle for Greek science was the notion of an “ultimate truth”. After the Greeks worked out all the implications of their axioms, further progress seemed impossible. Some aspects of knowledge seemed to them “complete” and some of their notions were turned into dogmas not open to further analysis. Today we understand that there are never enough observations that could turn a notion into “ultimate”. No amount of inductive testing can tell us that a generalization is completely and absolutely valid. A single observation that contradicts a theory forces the theory to be reviewed.
Many important scholars have blamed Plato and Aristotle for delaying scientific progress, since their ideas were turned into dogmas and, especially during medieval times, nobody could challenge their work while keeping their reputation intact. It is highly likely that science would have reached its modern state a lot earlier if these ideas had been open to review, but this by no means questions the genius of these two talented Greeks. The mistakes of a gifted mind can appear to be legitimate and remain accepted for centuries. The errors of a fool become evident sooner rather than later.

Greek Astronomy › Ancient History

Definition and Origins

by Cristian Violatti
published on 23 July 2013
Almagest (Joonasl)
Astronomy is an area where the Greeks displayed a remarkable talent. Observational astronomy, which was the main form of astronomy elsewhere, was taken a step further in Greece : they attempted to build a model of the universe that could account for the observations. They explored all imaginable alternatives, they considered many different solutions for the various astronomical problems they came across. They not only anticipated many ideas of modern astronomy but also some of their ideas endured for around two millennia. Even at the time of Isaac Newton, some aspects of Aristotelian cosmology were still taught at the University of Cambridge.
Our knowledge of Greek astronomy before the 4th century BCE is very incomplete. We have just a few surviving writings, and most of what we know are references and comments form Aristotle, mostly opinions he is about to criticize. What is clear is that the earth was believed to be a sphere, and that there was an increasing effort to understand nature in purely natural term, without recourse to supernatural explanations.
The Greeks' neighbours, Egyptians and Babylonians, had highly developed astronomies, but the forces driving them were different. Egyptian administration relied on well-established calendars to anticipate the flooding of the Nile ; rituals were required to be able to tell the time during the night, and the orientation of monuments in the cardinal directions was also important. Babylonians believed in the reading of omens in the sky as a mean to secure the state. These were all important stimuli to develop a fine astronomy.
Pythagoras is credited as the first Greek to think the earth spherical, but this idea was probably founded on mystic reasons rather than scientific. The Pythagoreans found conclusive evidence in favour of a spherical earth after it was discovered that the moon shines by reflecting light, and the right explanation for eclipses was found. The earth's shadow on the moon's surface suggested that the shape of our planet was spherical.
Aristotle's book "On the Heavens" summarizes some astronomical notions before his time. He says, for example, that Xenophanes of Colophon claimed the earth below us is infinite, that it has “pushed its roots to infinity”; others believed the earth rested upon water, a claim whose original author seems to be Thales (according to Aristotle); Anaximenes, Anaxagoras, and Democritus believed the earth was flat which “covers like a lid, the earth beneath it”.



Apart for a few exceptions, the general consensus among Greek astronomers was that the universe was earth-centred. During the 4th century BCE, Plato and Aristotle agreed on a geocentric model but both thinkers did so based on mystical arguments: The stars and planets were carried around the earth on spheres, arranged in a concentric fashion. Plato even described the universe as the Spindle of Necessity, attended by the Sirens and turned by the three Fates. Plato discarded the idea of a universe governed by natural laws, since he rejected any form of determinism. In fact, the unpredictable motions of some planets (especially Mars ), were seen by Plato as proof that natural laws could not account for all the changes in nature.Eudoxus, a student of Plato, challenged the views of his teacher by working on a more myth-free mathematical model, but the idea of concentric spheres and circular planetary motion still persisted.
While Aristotle's justifications for an earth-centred universe lack scientific support, he offers some compelling observational evidence to justify a spherical earth, the most important being the difference in the position of the polar star as one changes latitude, an observation that offered a way to measure the earth's circumference.
Indeed there are some stars seen in Egypt and in the neighbourhood of Cyprus which are not seen in the northerly regions; and stars, which in the north are never beyond the range of observation, in those regions rise and set. All of which goes to show not only that the earth is circular in shape, but also that it is a sphere of no great size: for otherwise the effect of so slight a change of place would not be quickly apparent.
(Aristotle: Book 2, Chapter 14, p. 75)
Aristotle, based on the position of the polar star between Greece and Egypt, estimated the size of the planet as 400,000 stadia. We do not know exactly about the conversion of stadia into modern measures, but the general consensus is that 400,000 stadia would be around 64,000 kilometres. This figure is much higher than modern calculations, but what is interesting is that from a theoretical perspective, the calculation is a valid method to calculate the size of our planet; it is the inaccuracy of the figures Aristotle dealt with that prevents him arriving at an acceptable conclusion.


A more accurate figure for the size of our planet would appear later with Eratosthenes (276-195 BCE) who compared the shadows cast by the sun at two different latitudes ( Alexandria and Syene) at the exact same time. By simple geometry he then calculated the earth's circumference to be 250,000 stadia, which is about 40,000 kilometres. Eratosthenes' calculation is about 15% too high, but the accuracy of his figure would not be equalled until modern times.
The fairly good observations of Aristotelian cosmology coexisted with a number of mystic and aesthetic prejudices. It was believed, for example, that the heavenly bodies were "unregenerate and indestructible" and also "unalterable". All bodies which existed above our planet were considered flawless and eternal, an idea that endured long after Aristotle: even during the Renaissance, when Galileo claimed that the surface of the moon was as imperfect as our planet and filled with mountains and craters, it caused nothing but scandal among Aristotelian scholars who still dominated European thought.
Despite the general consensus on the Earth-centred model, there were a number of reasons that suggested the model was not fully accurate and needed corrections. For example, it was not possible for the geocentric model to explain either the changes in the brightness of the planets or their retrograde motions. Aristarchus of Samos (310 BCE - 290 BCE) was an ancient Greek mathematician and astronomer who came up with an alternative astronomical hypothesis that could address some of these concerns. Anticipating Copernicus and Galileo by almost 20 centuries, he claimed the sun, not the earth, was the fixed centre of the universe, and that the earth, along with the rest of the planets, revolved around the sun. He also said that the stars were distant suns that remained unmoved, and that the size of the universe was much larger than his contemporaries believed. Using careful geometrical analysis based on the size of the earth's shadow on the moon during a lunar eclipse, Aristarchus knew that the sun was much larger than the earth. It is possible that the idea that tiny objects ought to orbit large ones and not the other way around motivated his revolutionary ideas.
Aristarchus works where the heliocentric model is presented are lost, and we know of them by piecing together later works and references. One of the most important and clear is the one mentioned by Archimedes in his book “The Sand Reckoner”:
[...] But Aristarchus of Samos brought out a book consisting of certain hypotheses, in which the premises lead to the result that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the Sun remain unmoved, that the Earth revolves about the Sun in the circumference of a circle, the Sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the Sun, is so great that the circle in which he supposes the Earth to revolve bears such a proportion to the distance of the fixed stars as the centre of the sphere bears to its surface.
(Archimedes, 1-2)
Aristarchus' model was a good idea during a bad time, since all Greek astronomers in antiquity took for granted that the orbit of all heavenly bodies had to be circular. The problem was that Aristarchus' theory could not be reconciled with the supposedly circular movements of the heavenly bodies. In reality planets' orbits are elliptical, not circular: elliptical orbits or any other non-circular orbit could not be accepted; it was almost a blasphemy from the viewpoint of Greek astronomers.
Hipparchus of Nicea by Raphael

Hipparchus of Nicea by Raphael

Hipparchus of Nicea (190 BCE - 120 BCE), the most respected and talented Greek astronomer in antiquity, calculated the length of the lunar month with an error of less than one second and estimated the solar year with an error of six minutes. He made a catalogue of the sky providing the positions of 1080 stars by stating their precise celestial latitude and longitude.Timocharis, 166 years before Hipparchus, had also made a chart. Comparing both charts, Hipparchus calculated that the stars had shifted their apparent position by around two degrees, and thus he discovered and measured the Equinoctial Precession.He calculated the precession to be 36 seconds per year, an estimation a little too short according to modern calculations, which is 50. He also provided most of the calculations that are the backbone of Ptolemy's work Almagest, a massive astronomical essay completed during the 2nd century CE which remained the standard reference for scholars and unchallenged until the Renaissance.
Hipparchus put an end to Aristarchus' theory by saying that the geocentric model better explained the observations than did the model of Aristarchus. As a result of this, he is often blamed for bringing astronomical progress backwards by favouring the mistaken earth-centred view. However, this is a risk that surrounds every genius, two sides of the same coin : when they are right they can trigger a revolution of knowledge, and when they are wrong they can freeze knowledge for centuries.
The Aristotelian model was “rescued” by introducing two geometrical tools created by Apollonius of Perga around 200 BCE and perfected by Hipparchus. The conventional circles were replaced by eccentric circles. In an eccentric circle the planets moved as usual in a uniform circular motion around the earth, but our planet was not the centre of the circle, rather, offset the centre. This way, the planet's speed changes could be accounted for and also the changes in brightness: planets would appear to move faster, and also brighter, when they were nearer the earth, and slower, and also dimmer, when they were away on the far side of their orbit. Apollonius came up with an additional tool, the epicycle, an orbit within an orbit (the moon revolves around the earth and the earth orbits the sun or, in other words, the moon moves around the sun in an epicycle). This device could also account for changes in brightness and speed, and it could also account for the retrograde motions of the planets which had puzzled most Greek astronomers.
Antikythera Mechanism

Antikythera Mechanism


Between Hipparchus and Ptolemy's Almagest we have a three century gap. Some scholars have suggested that this period was some sort of “dark age” for Greek astronomy, while other scholars believe that the Almagest 's triumph wiped out all previous astronomical works. This is a superfluous debate since the importance of a scientific work is often measured by the number of previous works it makes redundant.
The Almagest is a colossal work on astronomy. It contains geometrical models linked to tables by which the movements of the celestial bodies could be calculated indefinitely. All Greco-Babylonian astronomical achievements are summarized in this work.It includes a catalogue containing over 1,000 fixed stars. The cosmology of the Almagest would dominate western astronomy for the 14 centuries to come. Although not perfect, it had sufficient accuracy to remain accepted until the Renaissance.
Ironically, Ptolemy was more of an astrologer than astronomer: during his time, there was no sharp distinction between the obscure business of astrology and the science of astronomy. Astronomical observations were merely a side effect of the desire of Ptolemy as an astrologer to be able to tell and anticipate the positions of the planets at all times. Furthermore, Ptolemy was also the author of a work named Tetrabiblos, a classic work on astrology.
The tools devised by Hipparchus and Apollonius allowed sufficient observational accuracy, encouraging the progress of the geocentric model, but total success could never be achieved. Ptolemy added still another device to “save the appearances” of the model: the equant point. The equant was the point symmetrically opposite the eccentric earth, and the planet was required to move in its orbit in such a way that from the perspective of the equant, it would appear to be moving uniformly across the sky. Since the equant was offset from the centre of the orbit, planets had to vary their speed in order to fulfil this requisite. In short, because some basic assumptions of the cosmological model were wrong (the earth centred notion, the perfect circular orbits, etc.), there was the need to add questionable and complicated devices (eccentric circles, epicycles, equants, etc.) to prevent inconsistencies or, at least, try to minimize them. In the end, the Ptolemaic model collapsed not only because of its inaccuracies but mainly because it lacked simplicity. When the sun-centred hypothesis of Copernicus was published in the 16th century CE, it gained popularity not because it was more accurate, but because it was much simpler and it did not have the need of all the overly complex devices that Ptolemy had to use.


The Greek achievements in art, politics, and even in philosophy may be judged according to personal taste, but what they achieved in astronomy is totally beyond question. They not only developed a fine astronomical knowledge, but they also successfully exploited astronomical data that they got from Egyptian, Babylonian, and Chaldean astronomy and managed to merge it with their own knowledge. Even when they made a wrong assumption, they showed a unique creativity to come up with devices to save their mistakes. During the rise of modern science, not until the Renaissance would the world see thinkers with sufficient astronomical competence to challenge the notions of ancient Greek astronomy.


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with permission from the Website Ancient History Encyclopedia
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