Greek mathematics refers to mathematics texts and advances written in Greek, developed from the 7th century BC to the 4th century AD around the shores of the Eastern Mediterranean. Greek mathematicians lived in cities spread over the entire Eastern Mediterranean from Italy to North Africa but were united by culture and language. Greek mathematics of the period following Alexander the Great is sometimes called Hellenistic mathematics; the word "mathematics" itself derives from the Ancient Greek: μάθημα, translit. Máthēma Attic Greek: Koine Greek:, meaning "subject of instruction"; the study of mathematics for its own sake and the use of generalized mathematical theories and proofs is the key difference between Greek mathematics and those of preceding civilizations. The origin of Greek mathematics is not well documented; the earliest advanced civilizations in Greece and in Europe were the Minoan and Mycenaean civilizations, both of which flourished during the 2nd millennium BC. While these civilizations possessed writing and were capable of advanced engineering, including four-story palaces with drainage and beehive tombs, they left behind no mathematical documents.
Though no direct evidence is available, it is thought that the neighboring Babylonian and Egyptian civilizations had an influence on the younger Greek tradition. Between 800 BC and 600 BC, Greek mathematics lagged behind Greek literature, there is little known about Greek mathematics from this period—nearly all of, passed down through authors, beginning in the mid-4th century BC. Historians traditionally place the beginning of Greek mathematics proper to the age of Thales of Miletus. Little is known about the life and work of Thales, so little indeed that his date of birth and death are estimated from the eclipse of 585 BC, which occurred while he was in his prime. Despite this, it is agreed that Thales is the first of the seven wise men of Greece; the two earliest mathematical theorems, Thales' theorem and Intercept theorem are attributed to Thales. The former, which states that an angle inscribed in a semicircle is a right angle, may have been learned by Thales while in Babylon but tradition attributes to Thales a demonstration of the theorem.
It is for this reason that Thales is hailed as the father of the deductive organization of mathematics and as the first true mathematician. Thales is thought to be the earliest known man in history to whom specific mathematical discoveries have been attributed. Although it is not known whether or not Thales was the one who introduced into mathematics the logical structure, so ubiquitous today, it is known that within two hundred years of Thales the Greeks had introduced logical structure and the idea of proof into mathematics. Another important figure in the development of Greek mathematics is Pythagoras of Samos. Like Thales, Pythagoras traveled to Egypt and Babylon under the rule of Nebuchadnezzar, but settled in Croton, Magna Graecia. Pythagoras established an order called the Pythagoreans, which held knowledge and property in common and hence all of the discoveries by individual Pythagoreans were attributed to the order, and since in antiquity it was customary to give all credit to the master, Pythagoras himself was given credit for the discoveries made by his order.
Aristotle for one refused to attribute anything to Pythagoras as an individual and only discussed the work of the Pythagoreans as a group. One of the most important characteristics of the Pythagorean order was that it maintained that the pursuit of philosophical and mathematical studies was a moral basis for the conduct of life. Indeed, the words philosophy and mathematics are said to have been coined by Pythagoras. From this love of knowledge came many achievements, it has been customarily said that the Pythagoreans discovered most of the material in the first two books of Euclid's Elements. Distinguishing the work of Thales and Pythagoras from that of and earlier mathematicians is difficult since none of their original works survive, except for the surviving "Thales-fragments", which are of disputed reliability; however many historians, such as Hans-Joachim Waschkies and Carl Boyer, have argued that much of the mathematical knowledge ascribed to Thales was developed particularly the aspects that rely on the concept of angles, while the use of general statements may have appeared earlier, such as those found on Greek legal texts inscribed on slabs.
The reason it is not clear what either Thales or Pythagoras did is that no contemporary documentation has survived. The only evidence comes from traditions recorded in works such as Proclus’ commentary on Euclid written centuries later; some of these works, such as Aristotle’s commentary on the Pythagoreans, are themselves only known from a few surviving fragments. Thales is supposed to have used geometry to solve problems such as calculating the height of pyramids based on the length of shadows, the distance of ships from the shore, he is credited by tradition with having made the first proof of two geometric theorems—the "Theorem of Thales" and the "Intercept theorem" described above. Pythagoras is credited with recognizing the mathematical basis of musical harmony and, according to Proclus' commentary on Euclid, he discovered the theory of proportionals and constructed regular solids; some modern historians have questioned whether he constructed all five regular solids, suggesting instead that it is more reasonable to assume that he constructed just three of them.
Some ancient sources attribute the discovery of the Pythagorean theorem to Pythagoras, whereas others claim it was a proof fo
Hipparchus of Nicaea was a Greek astronomer and mathematician. He is considered the founder of trigonometry but is most famous for his incidental discovery of precession of the equinoxes. Hipparchus was born in Nicaea and died on the island of Rhodes, Greece, he is known to have been a working astronomer at least from 162 to 127 BC. Hipparchus is considered the greatest ancient astronomical observer and, by some, the greatest overall astronomer of antiquity, he was the first whose accurate models for the motion of the Sun and Moon survive. For this he made use of the observations and the mathematical techniques accumulated over centuries by the Babylonians and by Meton of Athens, Aristyllus, Aristarchus of Samos and Eratosthenes, among others, he developed trigonometry and constructed trigonometric tables, he solved several problems of spherical trigonometry. With his solar and lunar theories and his trigonometry, he may have been the first to develop a reliable method to predict solar eclipses.
His other reputed achievements include the discovery and measurement of Earth's precession, the compilation of the first comprehensive star catalog of the western world, the invention of the astrolabe of the armillary sphere, which he used during the creation of much of the star catalogue. There is a strong tradition that Hipparchus was born in Nicaea, in the ancient district of Bithynia, in what today is the country Turkey; the exact dates of his life are not known, but Ptolemy attributes astronomical observations to him in the period from 147–127 BC, some of these are stated as made in Rhodes. His birth date was calculated by Delambre based on clues in his work. Hipparchus must have lived some time after 127 BC because he analyzed and published his observations from that year. Hipparchus obtained information from Alexandria as well as Babylon, but it is not known when or if he visited these places, he is believed to have died on the island of Rhodes, where he seems to have spent most of his life.
It is not known what Hipparchus's economic means were nor how he supported his scientific activities. His appearance is unknown: there are no contemporary portraits. In the 2nd and 3rd centuries coins were made in his honour in Bithynia that bear his name and show him with a globe. Little of Hipparchus's direct work survives into modern times. Although he wrote at least fourteen books, only his commentary on the popular astronomical poem by Aratus was preserved by copyists. Most of what is known about Hipparchus comes from Strabo's Geography and Pliny's Natural History in the 1st century. Hipparchus was amongst the first to calculate a heliocentric system, but he abandoned his work because the calculations showed the orbits were not circular as believed to be mandatory by the science of the time. Although a contemporary of Hipparchus', Seleucus of Seleucia, remained a proponent of the heliocentric model, Hipparchus' rejection of heliocentrism, supported by ideas from Aristotle, remained dominant for nearly 2000 years until Copernican heliocentrism turned the tide of the debate.
Hipparchus's only preserved work is Τῶν Ἀράτου καὶ Εὐδόξου φαινομένων ἐξήγησις. This is a critical commentary in the form of two books on a popular poem by Aratus based on the work by Eudoxus. Hipparchus made a list of his major works, which mentioned about fourteen books, but, only known from references by authors, his famous star catalog was incorporated into the one by Ptolemy, may be perfectly reconstructed by subtraction of two and two thirds degrees from the longitudes of Ptolemy's stars. The first trigonometric table was compiled by Hipparchus, now known as "the father of trigonometry". Hipparchus was in the international news in 2005, when it was again proposed that the data on the celestial globe of Hipparchus or in his star catalog may have been preserved in the only surviving large ancient celestial globe which depicts the constellations with moderate accuracy, the globe carried by the Farnese Atlas. There are a variety of mis-steps in the more ambitious 2005 paper, thus no specialists in the area accept its publicized speculation.
Lucio Russo has said that Plutarch, in his work On the Face in the Moon, was reporting some physical theories that we consider to be Newtonian and that these may have come from Hipparchus. According to one book review, both of these claims have been rejected by other scholars. A line in Plutarch's Table Talk states that Hipparchus counted 103049 compound propositions that can be formed from ten simple propositions. 103049 is the tenth Schröder–Hipparchus number, which counts the number of ways of adding one or more pairs of parentheses around consecutive subsequences of two or more items in any sequence of ten symbols. This has led to speculation that Hipparchus knew about enumerative combinatorics, a field of mathematics that developed independently in modern mathematics. Earlier Greek astronomers and mathematicians were influenced by Babylonian astronomy to some extent, for instance the period relations of the Metonic cycle and Saros cycle may have come from Babylonian sources (see "Babylonian astron
Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz was a German physician and physicist who made significant contributions in several scientific fields. The largest German association of research institutions, the Helmholtz Association, is named after him. In physiology and psychology, he is known for his mathematics of the eye, theories of vision, ideas on the visual perception of space, color vision research, on the sensation of tone, perception of sound, empiricism in the physiology of perception. In physics, he is known for his theories on the conservation of energy, work in electrodynamics, chemical thermodynamics, on a mechanical foundation of thermodynamics; as a philosopher, he is known for his philosophy of science, ideas on the relation between the laws of perception and the laws of nature, the science of aesthetics, ideas on the civilizing power of science. Helmholtz was born in Potsdam the son of the local Gymnasium headmaster, Ferdinand Helmholtz, who had studied classical philology and philosophy, and, a close friend of the publisher and philosopher Immanuel Hermann Fichte.
Helmholtz's work was influenced by the philosophy of Johann Gottlieb Immanuel Kant. He tried to trace their theories in empirical matters like physiology; as a young man, Helmholtz was interested in natural science, but his father wanted him to study medicine at the Charité because there was financial support for medical students. Trained in physiology, Helmholtz wrote on many other topics, ranging from theoretical physics, to the age of the Earth, to the origin of the Solar System. Helmholtz's first academic position was as a teacher of Anatomy at the Academy of Arts in Berlin in 1848, he moved to take a post of associate professor of physiology at the Prussian University of Königsberg, where he was appointed in 1849. In 1855 he accepted a full professorship of physiology at the University of Bonn, he was not happy in Bonn and three years he transferred to the University of Heidelberg, in Baden, where he served as professor of physiology. In 1871 he accepted his final university position, as professor of physics at the Humboldt University in Berlin.
His first important scientific achievement, an 1847 treatise on the conservation of energy, was written in the context of his medical studies and philosophical background. His work on energy conservation came about while studying muscle metabolism, he tried to demonstrate that no energy is lost in muscle movement, motivated by the implication that there were no vital forces necessary to move a muscle. This was a rejection of the speculative tradition of Naturphilosophie, at that time a dominant philosophical paradigm in German physiology. Drawing on the earlier work of Sadi Carnot, Benoît Paul Émile Clapeyron and James Prescott Joule, he postulated a relationship between mechanics, light and magnetism by treating them all as manifestations of a single force, or energy in today's terminology, he published his theories in his book Über die Erhaltung der Kraft. In the 1850s and 60s, building on the publications of William Thomson and William Rankine popularized the idea of the heat death of the universe.
In fluid dynamics, Helmholtz made several contributions, including Helmholtz's theorems for vortex dynamics in inviscid fluids. Helmholtz was a pioneer in the scientific study of human audition. Inspired by psychophysics, he was interested in the relationships between measurable physical stimuli and their correspondent human perceptions. For example, the amplitude of a sound wave can be varied, causing the sound to appear louder or softer, but a linear step in sound pressure amplitude does not result in a linear step in perceived loudness; the physical sound needs to be increased exponentially in order for equal steps to seem linear, a fact, used in current electronic devices to control volume. Helmholtz paved the way in experimental studies on the relationship between the physical energy and its appreciation, with the goal in mind to develop "psychophysical laws." The sensory physiology of Helmholtz was the basis of the work of Wilhelm Wundt, a student of Helmholtz, considered one of the founders of experimental psychology.
More explicitly than Helmholtz, Wundt described his research as a form of empirical philosophy and as a study of the mind as something separate. Helmholtz had, in his early repudiation of Naturphilosophie, stressed the importance of materialism, was focusing more on the unity of "mind" and body. In 1851, Helmholtz revolutionized the field of ophthalmology with the invention of the ophthalmoscope; this made. Helmholtz's interests at that time were focused on the physiology of the senses, his main publication, titled Handbuch der Physiologischen Optik, provided empirical theories on depth perception, color vision, motion perception, became the fundamental reference work in his field during the second half of the nineteenth century. In the third and final volume, published in 1867, Helmholtz described the importance of unconscious inferences for perception; the Handbuch was first translated into English under the editorship of James P. C. Southall on behalf of the Optical Society of America in 1924-5.
His theory of accommodation went unchallenged until the final decade of the 20th century. Helmholtz continued to work for several decades on several editions of the handbook updating his work because of his dispute with Ewald Hering who held opposite views on spatial and color vision; this dispute divided the discipline
Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, more as "the analysis of perceptual processes by studying the effect on a subject's experience or behaviour of systematically varying the properties of a stimulus along one or more physical dimensions". Psychophysics refers to a general class of methods that can be applied to study a perceptual system. Modern applications rely on threshold measurement, ideal observer analysis, signal detection theory. Psychophysics has important practical applications. For example, in the study of digital signal processing, psychophysics has informed the development of models and methods of lossy compression; these models explain why humans perceive little loss of signal quality when audio and video signals are formatted using lossy compression. Many of the classical techniques and theories of psychophysics were formulated in 1860 when Gustav Theodor Fechner in Leipzig published Elemente der Psychophysik.
He coined the term "psychophysics", describing research intended to relate physical stimuli to the contents of consciousness such as sensations. As a physicist and philosopher, Fechner aimed at developing a method that relates matter to the mind, connecting the publicly observable world and a person's experienced impression of it, his ideas were inspired by experimental results on the sense of touch and light obtained in the early 1830s by the German physiologist Ernst Heinrich Weber in Leipzig, most notably those on the minimum discernible difference in intensity of stimuli of moderate strength which Weber had shown to be a constant fraction of the reference intensity, which Fechner referred to as Weber's law. From this, Fechner derived his well-known logarithmic scale, now known as Fechner scale. Weber's and Fechner's work formed one of the bases of psychology as a science, with Wilhelm Wundt founding the first laboratory for psychological research in Leipzig. Fechner's work systematised the introspectionist approach, that had to contend with the Behaviorist approach in which verbal responses are as physical as the stimuli.
During the 1930s, when psychological research in Nazi Germany came to a halt, both approaches began to be replaced by use of stimulus-response relationships as evidence for conscious or unconscious processing in the mind. Fechner's work was studied and extended by Charles S. Peirce, aided by his student Joseph Jastrow, who soon became a distinguished experimental psychologist in his own right. Peirce and Jastrow confirmed Fechner's empirical findings, but not all. In particular, a classic experiment of Peirce and Jastrow rejected Fechner's estimation of a threshold of perception of weights, as being far too high. In their experiment and Jastrow in fact invented randomized experiments: They randomly assigned volunteers to a blinded, repeated-measures design to evaluate their ability to discriminate weights. Peirce's experiment inspired other researchers in psychology and education, which developed a research tradition of randomized experiments in laboratories and specialized textbooks in the 1900s.
The Peirce–Jastrow experiments were conducted as part of Peirce's application of his pragmaticism program to human perception. Jastrow wrote the following summary: "Mr. Peirce’s courses in logic gave me my first real experience of intellectual muscle. Though I promptly took to the laboratory of psychology when, established by Stanley Hall, it was Peirce who gave me my first training in the handling of a psychological problem, at the same time stimulated my self-esteem by entrusting me fairly innocent of any laboratory habits, with a real bit of research, he borrowed the apparatus for me, which I took to my room, installed at my window, with which, when conditions of illumination were right, I took the observations. The results were published over our joint names in the Proceedings of the National Academy of Sciences; the demonstration that traces of sensory effect too slight to make any registry in consciousness could none the less influence judgment, may itself have been a persistent motive that induced me years to undertake a book on The Subconscious."
This work distinguishes observable cognitive performance from the expression of consciousness. Modern approaches to sensory perception, such as research on vision, hearing, or touch, measure what the perceiver's judgment extracts from the stimulus putting aside the question what sensations are being experienced. One leading method is based on signal detection theory, developed for cases of weak stimuli. However, the subjectivist approach persists among those in the tradition of Stanley Smith Stevens. Stevens revived the idea of a power law suggested by 19th century researchers, in contrast with Fechner's log-linear function, he advocated the assignment of numbers in ratio to the strengths of stimuli, called magnitude estimation. Stevens added techniques such as cross-modality matching, he opposed the assignment of stimulus strengths to points on a line that are labeled in order of strength. That sort of response has remained popular in applied psychophysics; such multiple-category layouts are misnamed Likert scaling after the question items used by Likert to create multi-item psychometric scales, e.g. seven phrases from "strongly
Stellar parallax is the apparent shift of position of any nearby star against the background of distant objects. Created by the different orbital positions of Earth, the small observed shift is largest at time intervals of about six months, when Earth arrives at opposite sides of the Sun in its orbit, giving a baseline distance of about two astronomical units between observations; the parallax itself is considered to be half of this maximum, about equivalent to the observational shift that would occur due to the different positions of Earth and the Sun, a baseline of one astronomical unit. Stellar parallax is so difficult to detect that its existence was the subject of much debate in astronomy for hundreds of years, it was first observed in 1806 by Giuseppe Calandrelli who reported parallax in α-Lyrae in his work "Osservazione e riflessione sulla parallasse annua dall’alfa della Lira". In 1838 Friedrich Bessel made the first successful parallax measurement, for the star 61 Cygni, using a Fraunhofer heliometer at Königsberg Observatory.
Once a star's parallax is known, its distance from Earth can be computed trigonometrically. But the more distant an object is, the smaller its parallax. With 21st-century techniques in astrometry, the limits of accurate measurement make distances farther away than about 100 parsecs too approximate to be useful when obtained by this technique; this limits the applicability of parallax as a measurement of distance to objects that are close on a galactic scale. Other techniques, such as spectral red-shift, are required to measure the distance of more remote objects. Stellar parallax measures are given in the tiny units of arcseconds, or in thousandths of arcseconds; the distance unit parsec is defined as the length of the leg of a right triangle adjacent to the angle of one arcsecond at one vertex, where the other leg is 1 AU long. Because stellar parallaxes and distances all involve such skinny right triangles, a convenient trigonometric approximation can be used to convert parallaxes to distance.
The approximate distance is the reciprocal of the parallax: d ≃ 1 / p. For example, Proxima Centauri, whose parallax is 0.7687, is 1 / 0.7687 parsecs = 1.3009 parsecs distant. Stellar parallax is so small that its apparent absence was used as a scientific argument against heliocentrism during the early modern age, it is clear from Euclid's geometry that the effect would be undetectable if the stars were far enough away, but for various reasons such gigantic distances involved seemed implausible: it was one of Tycho Brahe's principal objections to Copernican heliocentrism that in order for it to be compatible with the lack of observable stellar parallax, there would have to be an enormous and unlikely void between the orbit of Saturn and the eighth sphere. James Bradley first tried to measure stellar parallaxes in 1729; the stellar movement proved too insignificant for his telescope, but he instead discovered the aberration of light and the nutation of Earth's axis, catalogued 3222 stars. Stellar parallax is most measured using annual parallax, defined as the difference in position of a star as seen from Earth and Sun, i.e. the angle subtended at a star by the mean radius of Earth's orbit around the Sun.
The parsec is defined as the distance. Annual parallax is measured by observing the position of a star at different times of the year as Earth moves through its orbit. Measurement of annual parallax was the first reliable way to determine the distances to the closest stars; the first successful measurements of stellar parallax were made by Friedrich Bessel in 1838 for the star 61 Cygni using a heliometer. Being difficult to measure, only about 60 stellar parallaxes had been obtained by the end of the 19th century by use of the filar micrometer. Astrographs using astronomical photographic plates sped the process in the early 20th century. Automated plate-measuring machines and more sophisticated computer technology of the 1960s allowed more efficient compilation of star catalogues. In the 1980s, charge-coupled devices replaced photographic plates and reduced optical uncertainties to one milliarcsecond. Stellar parallax remains the standard for calibrating other measurement methods. Accurate calculations of distance based on stellar parallax require a measurement of the distance from Earth to the Sun, now known to exquisite accuracy based on radar reflection off the surfaces of planets.
The angles involved in these calculations are small and thus difficult to measure. The nearest star to the Sun, Proxima Centauri, has a parallax of 0.7687 ± 0.0003 arcsec. This angle is that subtended by an object 2 centimeters in diameter located 5.3 kilometers away. In 1989 the satellite Hipparcos was launched for obtaining parallaxes and proper motions of nearby stars, increasing the number of stellar parallaxes measured to milliarcsecond accuracy a thousandfold. So, Hipparcos is only able to measure parallax angles for stars up to about 1,600 light-years away, a little more than one percent of the diameter of the Milky Way Galaxy; the Hubble telescope WFC3 now has a precision of 20 to 40 microarcseconds, enabling reliable distance measurements u
Syracuse is a historic city on the island of Sicily, the capital of the Italian province of Syracuse. The city is notable for its rich Greek history, amphitheatres, as the birthplace of the preeminent mathematician and engineer Archimedes; this 2,700-year-old city played a key role in ancient times, when it was one of the major powers of the Mediterranean world. Syracuse is located in the southeast corner of the island of Sicily, next to the Gulf of Syracuse beside the Ionian Sea; the city was founded by Ancient Greek Corinthians and Teneans and became a powerful city-state. Syracuse was allied with Sparta and Corinth and exerted influence over the entirety of Magna Graecia, of which it was the most important city. Described by Cicero as "the greatest Greek city and the most beautiful of them all", it equaled Athens in size during the fifth century BC, it became part of the Roman Republic and the Byzantine Empire. Under Emperor Constans II, it served as the capital of the Byzantine Empire. After this Palermo overtook it as the capital of the Kingdom of Sicily.
The kingdom would be united with the Kingdom of Naples to form the Two Sicilies until the Italian unification of 1860. In the modern day, the city is listed by UNESCO as a World Heritage Site along with the Necropolis of Pantalica. In the central area, the city itself has a population of around 125,000 people. Syracuse is mentioned in the Bible in the Acts of the Apostles book at 28:12; the patron saint of the city is Saint Lucy. Syracuse and its surrounding area have been inhabited since ancient times, as shown by the findings in the villages of Stentinello, Plemmirio, Cozzo Pantano and Thapsos, which had a relationship with Mycenaean Greece. Syracuse was founded in 734 or 733 BC by Greek settlers from Corinth and Tenea, led by the oecist Archias. There are many attested variants of the name of the city including Συράκουσαι Syrakousai, Συράκοσαι Syrakosai and Συρακώ Syrakō. A possible origin of the city's name was given by Vibius Sequester citing first Stephanus Byzantius in that there was a Syracusian marsh called Syrako and secondly Marcian's Periegesis wherein Archias gave the city the name of a nearby marsh.
The settlement of Syracuse was a planned event, as a strong central leader, Arkhias the aristocrat, laid out how property would be divided up for the settlers, as well as plans for how the streets of the settlement should be arranged, how wide they should be. The nucleus of the ancient city was the small island of Ortygia; the settlers found the land fertile and the native tribes to be reasonably well-disposed to their presence. The city grew and prospered, for some time stood as the most powerful Greek city anywhere in the Mediterranean. Colonies were founded at Akrai, Akrillai and Kamarina; the descendants of the first colonists, called Gamoroi, held power until they were expelled by the Killichiroi, the lower class of the city. The former, returned to power in 485 BC, thanks to the help of Gelo, ruler of Gela. Gelo himself became the despot of the city, moved many inhabitants of Gela and Megara to Syracuse, building the new quarters of Tyche and Neapolis outside the walls, his program of new constructions included a new theatre, designed by Damocopos, which gave the city a flourishing cultural life: this in turn attracted personalities as Aeschylus, Ario of Methymna and Eumelos of Corinth.
The enlarged power of Syracuse made unavoidable the clash against the Carthaginians, who ruled western Sicily. In the Battle of Himera, who had allied with Theron of Agrigento, decisively defeated the African force led by Hamilcar. A temple dedicated to Athena, was erected in the city to commemorate the event. Syracuse grew during this time, its walls encircled 120 hectares in the fifth century, but as early as the 470's BC the inhabitants started building outside the walls. The complete population of its territory numbered 250,000 in 415 BC and the population size of the city itself was similar to Athens. Gelo was succeeded by his brother Hiero, who fought against the Etruscans at Cumae in 474 BC, his rule was eulogized by poets like Simonides of Ceos and Pindar, who visited his court. A democratic regime was introduced by Thrasybulos; the city continued to expand in Sicily, fighting against the rebellious Siculi, on the Tyrrhenian Sea, making expeditions up to Corsica and Elba. In the late 5th century BC, Syracuse found itself at war with Athens, which sought more resources to fight the Peloponnesian War.
The Syracusans enlisted the aid of a general from Sparta, Athens' foe in the war, to defeat the Athenians, destroy their ships, leave them to starve on the island. In 401 BC, Syracuse contributed a force of 300 hoplites and a general to Cyrus the Younger's Army of the Ten Thousand. In the early 4th century BC, the tyrant Dionysius the Elder was again at war against Carthage and, although losing Gela and Camarina, kept that power from capturing the whole of Sicily. After the end of the conflict Dionysius built a massive fortress on Ortygia and 22 km-long walls around all of Syracuse. Another period of expansion saw the destruction of
Ctesibius or Ktesibios or Tesibius was a Greek inventor and mathematician in Alexandria, Ptolemaic Egypt. He wrote the first treatises on the science of its uses in pumps. This, in combination with his work on the elasticity of air On pneumatics, earned him the title of "father of pneumatics." None of his written work has survived, including his Memorabilia, a compilation of his research, cited by Athenaeus. Ctesibius' most known invention today is a pipe organ, on which the invention of the piano was based. Ctesibius was the first head of the Museum of Alexandria. Little is known of his life, but his inventions were well known, it is said. During his time as a barber, he invented a counterweight-adjustable mirror, his other inventions include the hydraulis, a water organ, considered the precursor of the modern pipe organ, improved the water clock or clepsydra. For more than 1,800 years the clepsydra was the most accurate clock constructed, until the Dutch physicist Christiaan Huygens' invention of the pendulum clock in 1656.
Ctesibius described one of the first force pumps for producing a jet of water, or for lifting water from wells. Examples have been found at various Roman sites, such as at Silchester in Britain; the principle of the siphon has been attributed to him. According to Diogenes Laërtius, Ctesibius was miserably poor. Laërtius details this by recounting the following concerning the philosopher Arcesilaus: When he had gone to visit Ctesibius, ill, seeing him in great distress from want, he secretly slipped his purse under his pillow. Ctesibius's work is chronicled by Vitruvius, Pliny the Elder, Philo of Byzantium who mention him, adding that the first mechanicians such as Ctesibius had the advantage of being under kings who loved fame and supported the arts. Proclus and Hero of Alexandria mention him. Landels, J. G.. Engineering in the ancient world. Berkeley: Univ. of California Press. ISBN 0-520-03429-5. Lloyd, G. E. R.. Greek science after Aristotle. New York: Norton. ISBN 0-393-04371-1. Vitruvius; the Ten Books on Architecture.
Cambridge: Harvard University Press