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111-464: The János Bolyai Mathematical Society (Bolyai János Matematikai Társulat, BJMT) is the Hungarian mathematical society, named after János Bolyai , a 19th-century Hungarian mathematician, a co-discoverer of non-Euclidean geometry . It is the professional society of the Hungarian mathematicians, applied mathematicians, and mathematics teachers. It was founded in 1947, as one of the two successor societies of

222-405: A plane is described with Cartesian coordinates : The points are sometimes identified with generalized complex numbers z = x + y ε where ε ∈ { –1, 0, 1}. The Euclidean plane corresponds to the case ε = −1 , an imaginary unit . Since the modulus of z is given by For instance, { z | z z * = 1} is the unit circle . For planar algebra, non-Euclidean geometry arises in

333-517: A will in which he asked to be buried next to his wife. Aristotle left his works to Theophrastus, his successor as the head of the Lyceum, who in turn passed them down to Neleus of Scepsis in Asia Minor. There, the papers remained hidden for protection until they were purchased by the collector Apellicon . In the meantime, many copies of Aristotle's major works had already begun to circulate and be used in

444-541: A "survival of the fittest" origin of living things and their organs, and ridiculed the idea that accidents could lead to orderly results. To put his views into modern terms, he nowhere says that different species can have a common ancestor , or that one kind can change into another , or that kinds can become extinct . Aristotle did not do experiments in the modern sense. He used the ancient Greek term pepeiramenoi to mean observations, or at most investigative procedures like dissection. In Generation of Animals , he finds

555-559: A broad range of subjects spanning the natural sciences , philosophy , linguistics , economics , politics , psychology , and the arts . As the founder of the Peripatetic school of philosophy in the Lyceum in Athens , he began the wider Aristotelian tradition that followed, which set the groundwork for the development of modern science . Little is known about Aristotle's life. He was born in

666-495: A building known as the Lyceum (named after the sacred grove of Apollo Lykeios ), in which he established his own school. The building included a gymnasium and a colonnade ( peripatos ), from which the school acquired the name Peripatetic . Aristotle conducted courses and research at the school for the next twelve years. He often lectured small groups of distinguished students and, along with some of them, such as Theophrastus , Eudemus , and Aristoxenus , Aristotle built

777-410: A complex number z . Hyperbolic geometry found an application in kinematics with the physical cosmology introduced by Hermann Minkowski in 1908. Minkowski introduced terms like worldline and proper time into mathematical physics . He realized that the submanifold , of events one moment of proper time into the future, could be considered a hyperbolic space of three dimensions. Already in

888-438: A fertilized hen's egg of a suitable stage and opens it to see the embryo's heart beating inside. Instead, he practiced a different style of science: systematically gathering data, discovering patterns common to whole groups of animals, and inferring possible causal explanations from these. This style is common in modern biology when large amounts of data become available in a new field, such as genomics . It does not result in

999-569: A few salient points. Aristotle was born in 384 BC in Stagira , Chalcidice , about 55 km (34 miles) east of modern-day Thessaloniki . He was the son of Nicomachus , the personal physician of King Amyntas of Macedon , and Phaestis, a woman with origins from Chalcis , Euboea . Nicomachus was said to have belonged to the medical guild of Asclepiadae and was likely responsible for Aristotle's early interest in biology and medicine. Ancient tradition held that Aristotle's family descended from

1110-497: A fluid such as air. In this system, heavy bodies in steady fall indeed travel faster than light ones (whether friction is ignored, or not ), and they do fall more slowly in a denser medium. Newton's "forced" motion corresponds to Aristotle's "violent" motion with its external agent, but Aristotle's assumption that the agent's effect stops immediately it stops acting (e.g., the ball leaves the thrower's hand) has awkward consequences: he has to suppose that surrounding fluid helps to push

1221-406: A form of an apple. In this distinction, there is a particular apple and a universal form of an apple. Moreover, one can place an apple next to a book, so that one can speak of both the book and apple as being next to each other. Plato argued that there are some universal forms that are not a part of particular things. For example, it is possible that there is no particular good in existence, but "good"

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1332-454: A geometry several years before, though he did not publish. While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter  k . Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of

1443-422: A given line l through a point that is not on l . In hyperbolic geometric models, by contrast, there are infinitely many lines through A parallel to l , and in elliptic geometric models, parallel lines do not exist. (See the entries on hyperbolic geometry and elliptic geometry for more information.) Euclidean geometry is modelled by our notion of a "flat plane ." The simplest model for elliptic geometry

1554-550: A large library which included manuscripts, maps, and museum objects. While in Athens, his wife Pythias died and Aristotle became involved with Herpyllis of Stagira. They had a son whom Aristotle named after his father, Nicomachus . This period in Athens, between 335 and 323 BC, is when Aristotle is believed to have composed many of his philosophical works. He wrote many dialogues, of which only fragments have survived. Those works that have survived are in treatise form and were not, for

1665-597: A linear scale, and noted various exceptions, such as that sharks had a placenta like the tetrapods. To a modern biologist, the explanation, not available to Aristotle, is convergent evolution . Philosophers of science have generally concluded that Aristotle was not interested in taxonomy, but zoologists who studied this question in the early 21st century think otherwise. He believed that purposive final causes guided all natural processes; this teleological view justified his observed data as an expression of formal design. Aristotle's psychology , given in his treatise On

1776-424: A manner that the interior angles on the same side are together less than two right angles, then the straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. Other mathematicians have devised simpler forms of this property. Regardless of the form of the postulate, however, it consistently appears more complicated than Euclid's other postulates : For at least

1887-506: A model of the acute case on a sphere of imaginary radius. He did not carry this idea any further. At this time it was widely believed that the universe worked according to the principles of Euclidean geometry. The beginning of the 19th century would finally witness decisive steps in the creation of non-Euclidean geometry. Circa 1813, Carl Friedrich Gauss and independently around 1818, the German professor of law Ferdinand Karl Schweikart had

1998-401: A new viable geometry, but did not realize it. In 1766 Johann Lambert wrote, but did not publish, Theorie der Parallellinien in which he attempted, as Saccheri did, to prove the fifth postulate. He worked with a figure now known as a Lambert quadrilateral , a quadrilateral with three right angles (can be considered half of a Saccheri quadrilateral). He quickly eliminated the possibility that

2109-480: A non-Euclidean geometry, the parallel postulate (or its equivalent) must be replaced by its negation . Negating the Playfair's axiom form, since it is a compound statement (... there exists one and only one ...), can be done in two ways: Models of non-Euclidean geometry are mathematical models of geometries which are non-Euclidean in the sense that it is not the case that exactly one line can be drawn parallel to

2220-489: A pair of similar but not congruent triangles." In any of these systems, removal of the one axiom equivalent to the parallel postulate, in whatever form it takes, and leaving all the other axioms intact, produces absolute geometry . As the first 28 propositions of Euclid (in The Elements ) do not require the use of the parallel postulate or anything equivalent to it, they are all true statements in absolute geometry. To obtain

2331-427: A plant in the soil is potentially ( dynamei ) a plant, and if it is not prevented by something, it will become a plant. Potentially, beings can either 'act' ( poiein ) or 'be acted upon' ( paschein ), which can be either innate or learned. For example, the eyes possess the potentiality of sight (innate – being acted upon), while the capability of playing the flute can be possessed by learning (exercise – acting). Actuality

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2442-481: A possibility (some others of Euclid's axioms must be modified for elliptic geometry to work) and set to work proving a great number of results in hyperbolic geometry. He finally reached a point where he believed that his results demonstrated the impossibility of hyperbolic geometry. His claim seems to have been based on Euclidean presuppositions, because no logical contradiction was present. In this attempt to prove Euclidean geometry he instead unintentionally discovered

2553-498: A priori principles. Aristotle's "natural philosophy" spans a wide range of natural phenomena including those now covered by physics, biology and other natural sciences. In Aristotle's terminology, "natural philosophy" is a branch of philosophy examining the phenomena of the natural world, and includes fields that would be regarded today as physics, biology and other natural sciences. Aristotle's work encompassed virtually all facets of intellectual inquiry. Aristotle makes philosophy in

2664-562: A researcher and lecturer, earning for himself the nickname "mind of the school" by his tutor Plato . In Athens, he probably experienced the Eleusinian Mysteries as he wrote when describing the sights one viewed at the Mysteries, "to experience is to learn" ( παθεĩν μαθεĩν ). Aristotle remained in Athens for nearly twenty years before leaving in 348/47 BC after Plato's death. The traditional story about his departure records that he

2775-625: A significant impact across the world, and remain a subject of contemporary philosophical discussion. Aristotle's views profoundly shaped medieval scholarship . The influence of his physical science extended from late antiquity and the Early Middle Ages into the Renaissance , and was not replaced systematically until the Enlightenment and theories such as classical mechanics were developed. He influenced Judeo-Islamic philosophies during

2886-411: A spacetime event one moment into the future of a frame of reference of rapidity a . Furthermore, multiplication by z amounts to a Lorentz boost mapping the frame with rapidity zero to that with rapidity a . Aristotle Aristotle ( Attic Greek : Ἀριστοτέλης , romanized:  Aristotélēs ; 384–322 BC) was an Ancient Greek philosopher and polymath . His writings cover

2997-412: A special role for geometry. It was his prime example of synthetic a priori knowledge; not derived from the senses nor deduced through logic — our knowledge of space was a truth that we were born with. Unfortunately for Kant, his concept of this unalterably true geometry was Euclidean. Theology was also affected by the change from absolute truth to relative truth in the way that mathematics is related to

3108-425: A term that generally fell out of use ). His influence has led to the current usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry. There are some mathematicians who would extend the list of geometries that should be called "non-Euclidean" in various ways. There are many kinds of geometry that are quite different from Euclidean geometry but are also not necessarily included in

3219-469: A thousand years, geometers were troubled by the disparate complexity of the fifth postulate, and believed it could be proved as a theorem from the other four. Many attempted to find a proof by contradiction , including Ibn al-Haytham (Alhazen, 11th century), Omar Khayyám (12th century), Nasīr al-Dīn al-Tūsī (13th century), and Giovanni Girolamo Saccheri (18th century). The theorems of Ibn al-Haytham, Khayyam and al-Tusi on quadrilaterals , including

3330-549: A thrown stone, in the Physics (254b10), and "natural motion", such as of a falling object, in On the Heavens (300a20). In violent motion, as soon as the agent stops causing it, the motion stops also: in other words, the natural state of an object is to be at rest, since Aristotle does not address friction . With this understanding, it can be observed that, as Aristotle stated, heavy objects (on

3441-494: A two-dimensional plane that are both perpendicular to a third line (in the same plane): Euclidean geometry , named after the Greek mathematician Euclid , includes some of the oldest known mathematics, and geometries that deviated from this were not widely accepted as legitimate until the 19th century. The debate that eventually led to the discovery of the non-Euclidean geometries began almost as soon as Euclid wrote Elements . In

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3552-448: A two-dimensional plane, for any given line l and a point A , which is not on l , there is exactly one line through A that does not intersect l . In hyperbolic geometry, by contrast, there are infinitely many lines through A not intersecting l , while in elliptic geometry, any line through A intersects l . Another way to describe the differences between these geometries is to consider two straight lines indefinitely extended in

3663-454: Is a historical accident: his works on botany have been lost, but two books on plants by his pupil Theophrastus have survived. Aristotle reports on the sea-life visible from observation on Lesbos and the catches of fishermen. He describes the catfish , electric ray , and frogfish in detail, as well as cephalopods such as the octopus and paper nautilus . His description of the hectocotyl arm of cephalopods, used in sexual reproduction,

3774-495: Is a sphere, where lines are " great circles " (such as the equator or the meridians on a globe ), and points opposite each other are identified (considered to be the same). The pseudosphere has the appropriate curvature to model hyperbolic geometry. The simplest model for elliptic geometry is a sphere, where lines are " great circles " (such as the equator or the meridians on a globe ), and points opposite each other (called antipodal points ) are identified (considered

3885-419: Is also a final cause or end. Then Aristotle proceeds and concludes that the actuality is prior to potentiality in formula, in time and in substantiality. With this definition of the particular substance (i.e., matter and form), Aristotle tries to solve the problem of the unity of the beings, for example, "what is it that makes a man one"? Since, according to Plato there are two Ideas: animal and biped, how then

3996-423: Is impossible for two convergent straight lines to diverge in the direction in which they converge." Khayyam then considered the three cases right, obtuse, and acute that the summit angles of a Saccheri quadrilateral can take and after proving a number of theorems about them, he correctly refuted the obtuse and acute cases based on his postulate and hence derived the classic postulate of Euclid, which he didn't realize

4107-541: Is man a unity? However, according to Aristotle, the potential being (matter) and the actual one (form) are one and the same. Aristotle's immanent realism means his epistemology is based on the study of things that exist or happen in the world, and rises to knowledge of the universal, whereas for Plato epistemology begins with knowledge of universal Forms (or ideas) and descends to knowledge of particular imitations of these. Aristotle uses induction from examples alongside deduction , whereas Plato relies on deduction from

4218-442: Is not on l , there are infinitely many lines through A that do not intersect l . In these models, the concepts of non-Euclidean geometries are represented by Euclidean objects in a Euclidean setting. This introduces a perceptual distortion wherein the straight lines of the non-Euclidean geometry are represented by Euclidean curves that visually bend. This "bending" is not a property of the non-Euclidean lines, only an artifice of

4329-406: Is predicated. So, according to Aristotle, the form of apple exists within each apple, rather than in the world of the forms. Concerning the nature of change ( kinesis ) and its causes, as he outlines in his Physics and On Generation and Corruption ( 319b–320a), he distinguishes coming-to-be ( genesis , also translated as 'generation') from: Coming-to-be is a change where the substrate of

4440-497: Is probably not in its original form, because it was most likely edited by students and later lecturers. The logical works of Aristotle were compiled into a set of six books called the Organon around 40 BC by Andronicus of Rhodes or others among his followers. The books are: The order of the books (or the teachings from which they are composed) is not certain, but this list was derived from analysis of Aristotle's writings. It goes from

4551-480: Is still a proper universal form. Aristotle disagreed with Plato on this point, arguing that all universals are instantiated at some period of time, and that there are no universals that are unattached to existing things. In addition, Aristotle disagreed with Plato about the location of universals. Where Plato spoke of the forms as existing separately from the things that participate in them, Aristotle maintained that universals exist within each thing on which each universal

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4662-413: Is the fulfilment of the end of the potentiality. Because the end ( telos ) is the principle of every change, and potentiality exists for the sake of the end, actuality, accordingly, is the end. Referring then to the previous example, it can be said that an actuality is when a plant does one of the activities that plants do. For that for the sake of which ( to hou heneka ) a thing is, is its principle, and

4773-539: The Iliad , which reportedly became one of Alexander's most prized possessions. Scholars speculate that two of Aristotle's now lost works, On kingship and On behalf of the Colonies , were composed by the philosopher for the young prince. After Philip II's assassination in 336 BC, Aristotle returned to Athens for the second and final time a year later. As a metic , Aristotle could not own property in Athens and thus rented

4884-457: The Cayley–Klein metric because Felix Klein exploited it to describe the non-Euclidean geometries in articles in 1871 and 1873 and later in book form. The Cayley–Klein metrics provided working models of hyperbolic and elliptic metric geometries, as well as Euclidean geometry. Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry parabolic ,

4995-634: The Early Modern period. John Philoponus (in Late antiquity ) and Galileo (in Early modern period ) are said to have shown by experiment that Aristotle's claim that a heavier object falls faster than a lighter object is incorrect. A contrary opinion is given by Carlo Rovelli , who argues that Aristotle's physics of motion is correct within its domain of validity, that of objects in the Earth 's gravitational field immersed in

5106-522: The Elements , Euclid begins with a limited number of assumptions (23 definitions, five common notions, and five postulates) and seeks to prove all the other results ( propositions ) in the work. The most notorious of the postulates is often referred to as "Euclid's Fifth Postulate", or simply the parallel postulate , which in Euclid's original formulation is: If a straight line falls on two straight lines in such

5217-515: The Great Comet of 371 BC . Aristotle was one of the first people to record any geological observations. He stated that geological change was too slow to be observed in one person's lifetime. The geologist Charles Lyell noted that Aristotle described such change, including "lakes that had dried up" and "deserts that had become watered by rivers", giving as examples the growth of the Nile delta since

5328-471: The History of Animals in a graded scale of perfection, a nonreligious version of the scala naturae , with man at the top. His system had eleven grades of animal, from highest potential to lowest, expressed in their form at birth: the highest gave live birth to hot and wet creatures, the lowest laid cold, dry mineral-like eggs. Animals came above plants , and these in turn were above minerals. He grouped what

5439-464: The Hungarian mathematician János Bolyai separately and independently published treatises on hyperbolic geometry. Consequently, hyperbolic geometry is called Lobachevskian or Bolyai-Lobachevskian geometry, as both mathematicians, independent of each other, are the basic authors of non-Euclidean geometry. Gauss mentioned to Bolyai's father, when shown the younger Bolyai's work, that he had developed such

5550-680: The Lambert quadrilateral and Saccheri quadrilateral , were "the first few theorems of the hyperbolic and the elliptic geometries ". These theorems along with their alternative postulates, such as Playfair's axiom , played an important role in the later development of non-Euclidean geometry. These early attempts at challenging the fifth postulate had a considerable influence on its development among later European geometers, including Witelo , Levi ben Gerson , Alfonso , John Wallis and Saccheri. All of these early attempts made at trying to formulate non-Euclidean geometry, however, provided flawed proofs of

5661-451: The Milky Way was made up of "those stars which are shaded by the earth from the sun's rays," pointing out partly correctly that if "the size of the sun is greater than that of the earth and the distance of the stars from the earth many times greater than that of the sun, then... the sun shines on all the stars and the earth screens none of them." He also wrote descriptions of comets, including

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5772-508: The Physics (215a25), Aristotle effectively states a quantitative law, that the speed, v, of a falling body is proportional (say, with constant c) to its weight, W, and inversely proportional to the density, ρ, of the fluid in which it is falling:; Aristotle implies that in a vacuum the speed of fall would become infinite, and concludes from this apparent absurdity that a vacuum is not possible. Opinions have varied on whether Aristotle intended to state quantitative laws. Henri Carteron held

5883-404: The mathematical model of space . Furthermore, since the substance of the subject in synthetic geometry was a chief exhibit of rationality, the Euclidean point of view represented absolute authority. The discovery of the non-Euclidean geometries had a ripple effect which went far beyond the boundaries of mathematics and science. The philosopher Immanuel Kant 's treatment of human knowledge had

5994-513: The "extreme view" that Aristotle's concept of force was basically qualitative, but other authors reject this. Archimedes corrected Aristotle's theory that bodies move towards their natural resting places; metal boats can float if they displace enough water ; floating depends in Archimedes' scheme on the mass and volume of the object, not, as Aristotle thought, its elementary composition. Aristotle's writings on motion remained influential until

6105-431: The 1890s Alexander Macfarlane was charting this submanifold through his Algebra of Physics and hyperbolic quaternions , though Macfarlane did not use cosmological language as Minkowski did in 1908. The relevant structure is now called the hyperboloid model of hyperbolic geometry. The non-Euclidean planar algebras support kinematic geometries in the plane. For instance, the split-complex number z = e can represent

6216-504: The Great beginning in 343 BC. He established a library in the Lyceum, which helped him to produce many of his hundreds of books on papyrus scrolls . Though Aristotle wrote many treatises and dialogues for publication, only around a third of his original output has survived , none of it intended for publication. Aristotle provided a complex synthesis of the various philosophies existing prior to him. His teachings and methods of inquiry have had

6327-550: The Lyceum of Athens, Alexandria , and later in Rome . With the Prior Analytics , Aristotle is credited with the earliest study of formal logic, and his conception of it was the dominant form of Western logic until 19th-century advances in mathematical logic . Kant stated in the Critique of Pure Reason that with Aristotle, logic reached its completion. Most of Aristotle's work

6438-575: The Mathematical and Physical Society (Matematikai és Fizikai Társulat) founded in 1891. It is a member-society of the European Mathematical Society . The society publishes the following periodicals. Non-Euclidean geometry The essential difference between the metric geometries is the nature of parallel lines. Euclid 's fifth postulate, the parallel postulate , is equivalent to Playfair's postulate , which states that, within

6549-621: The Middle Ages, as well as Christian theology , especially the Neoplatonism of the Early Church and the scholastic tradition of the Catholic Church . Aristotle was revered among medieval Muslim scholars as "The First Teacher", and among medieval Christians like Thomas Aquinas as simply "The Philosopher", while the poet Dante called him "the master of those who know". His works contain

6660-472: The approach of Euclid and provides the justification for all of Euclid's proofs. Other systems, using different sets of undefined terms obtain the same geometry by different paths. All approaches, however, have an axiom that is logically equivalent to Euclid's fifth postulate, the parallel postulate. Hilbert uses the Playfair axiom form, while Birkhoff , for instance, uses the axiom that says that, "There exists

6771-489: The ball along to make it continue to rise even though the hand is no longer acting on it, resulting in the Medieval theory of impetus . Aristotle suggested that the reason for anything coming about can be attributed to four different types of simultaneously active factors. His term aitia is traditionally translated as "cause", but it does not always refer to temporal sequence; it might be better translated as "explanation", but

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6882-617: The basics, the analysis of simple terms in the Categories, the analysis of propositions and their elementary relations in On Interpretation , to the study of more complex forms, namely, syllogisms (in the Analytics ) and dialectics (in the Topics and Sophistical Refutations ). The first three treatises form the core of the logical theory stricto sensu : the grammar of the language of logic and

6993-399: The becoming is for the sake of the end; and the actuality is the end, and it is for the sake of this that the potentiality is acquired. For animals do not see in order that they may have sight, but they have sight that they may see. In summary, the matter used to make a house has potentiality to be a house and both the activity of building and the form of the final house are actualities, which

7104-574: The broad sense coextensive with reasoning, which he also would describe as "science". However, his use of the term science carries a different meaning than that covered by the term "scientific method". For Aristotle, "all science ( dianoia ) is either practical, poetical or theoretical" ( Metaphysics 1025b25). His practical science includes ethics and politics; his poetical science means the study of fine arts including poetry; his theoretical science covers physics, mathematics and metaphysics. In his On Generation and Corruption , Aristotle related each of

7215-433: The cause of earthquakes was a gas or vapor ( anathymiaseis ) that was trapped inside the earth and trying to escape, following other Greek authors Anaxagoras , Empedocles and Democritus . Aristotle also made many observations about the hydrologic cycle. For example, he made some of the earliest observations about desalination: he observed early – and correctly – that when seawater is heated, freshwater evaporates and that

7326-486: The city of Stagira in northern Greece during the Classical period . His father, Nicomachus , died when Aristotle was a child, and he was brought up by a guardian. At around eighteen years old, he joined Plato 's Academy in Athens and remained there until the age of thirty seven ( c.  347 BC ). Shortly after Plato died, Aristotle left Athens and, at the request of Philip II of Macedon , tutored his son Alexander

7437-424: The conventional meaning of "non-Euclidean geometry", such as more general instances of Riemannian geometry . Euclidean geometry can be axiomatically described in several ways. However, Euclid's original system of five postulates (axioms) is not one of these, as his proofs relied on several unstated assumptions that should also have been taken as axioms. Hilbert's system consisting of 20 axioms most closely follows

7548-487: The correct rules of reasoning. The Rhetoric is not conventionally included, but it states that it relies on the Topics . What is today called Aristotelian logic with its types of syllogism (methods of logical argument), Aristotle himself would have labelled "analytics". The term "logic" he reserved to mean dialectics . The word "metaphysics" appears to have been coined by the first century AD editor who assembled various small selections of Aristotle's works to create

7659-399: The data to come to the theory of evolution . Aristotle's writings can seem to modern readers close to implying evolution, but while Aristotle was aware that new mutations or hybridizations could occur, he saw these as rare accidents. For Aristotle, accidents, like heat waves in winter, must be considered distinct from natural causes. He was thus critical of Empedocles's materialist theory of

7770-787: The death. Following Alexander's death, anti-Macedonian sentiment in Athens was rekindled. In 322 BC, Demophilus and Eurymedon the Hierophant reportedly denounced Aristotle for impiety, prompting him to flee to his mother's family estate in Chalcis, Euboea , at which occasion he was said to have stated "I will not allow the Athenians to sin twice against philosophy" – a reference to Athens's trial and execution of Socrates . He died in Chalcis, Euboea of natural causes later that same year, having named his student Antipater as his chief executor and leaving

7881-479: The earliest known formal study of logic, and were studied by medieval scholars such as Peter Abelard and Jean Buridan . Aristotle's influence on logic continued well into the 19th century. In addition, his ethics , although always influential, gained renewed interest with the modern advent of virtue ethics . In general, the details of Aristotle's life are not well-established. The biographies written in ancient times are often speculative and historians only agree on

7992-543: The end of his life, the two men became estranged having diverging opinions over issues, like the optimal administration of city-states, the treatment of conquered populations, such as the Persians, and philosophical questions, like the definition of braveness. A widespread speculation in antiquity suggested that Aristotle played a role in Alexander's death, but the only evidence of this is an unlikely claim made some six years after

8103-409: The entirety of hyperbolic space, and used this to show that Euclidean geometry and hyperbolic geometry were equiconsistent so that hyperbolic geometry was logically consistent if and only if Euclidean geometry was. (The reverse implication follows from the horosphere model of Euclidean geometry.) In the hyperbolic model, within a two-dimensional plane, for any given line l and a point A , which

8214-533: The following. Brood size decreases with (adult) body mass, so that an elephant has fewer young (usually just one) per brood than a mouse. Lifespan increases with gestation period , and also with body mass, so that elephants live longer than mice, have a longer period of gestation, and are heavier. As a final example, fecundity decreases with lifespan, so long-lived kinds like elephants have fewer young in total than short-lived kinds like mice. Aristotle distinguished about 500 species of animals , arranging these in

8325-403: The form of the substance is the actual house, namely 'covering for bodies and chattels' or any other differentia that let us define something as a house. The formula that gives the components is the account of the matter, and the formula that gives the differentia is the account of the form. Like his teacher Plato, Aristotle's philosophy aims at the universal . Aristotle's ontology places

8436-509: The four elements proposed earlier by Empedocles , earth , water , air , and fire , to two of the four sensible qualities, hot, cold, wet, and dry. In the Empedoclean scheme, all matter was made of the four elements, in differing proportions. Aristotle's scheme added the heavenly aether , the divine substance of the heavenly spheres , stars and planets. Aristotle describes two kinds of motion: "violent" or "unnatural motion", such as that of

8547-408: The fourth angle is obtuse, as had Saccheri and Khayyam, and then proceeded to prove many theorems under the assumption of an acute angle. Unlike Saccheri, he never felt that he had reached a contradiction with this assumption. He had proved the non-Euclidean result that the sum of the angles in a triangle increases as the area of the triangle decreases, and this led him to speculate on the possibility of

8658-490: The germinal ideas of non-Euclidean geometry worked out, but neither published any results. Schweikart's nephew Franz Taurinus did publish important results of hyperbolic trigonometry in two papers in 1825 and 1826, yet while admitting the internal consistency of hyperbolic geometry, he still believed in the special role of Euclidean geometry. Then, in 1829–1830 the Russian mathematician Nikolai Ivanovich Lobachevsky and in 1832

8769-428: The ground, say) require more force to make them move; and objects pushed with greater force move faster. This would imply the equation incorrect in modern physics. Natural motion depends on the element concerned: the aether naturally moves in a circle around the heavens, while the 4 Empedoclean elements move vertically up (like fire, as is observed) or down (like earth) towards their natural resting places. In

8880-440: The image. According to Aristotle, spontaneity and chance are causes of some things, distinguishable from other types of cause such as simple necessity. Chance as an incidental cause lies in the realm of accidental things , "from what is spontaneous". There is also more a specific kind of chance, which Aristotle names "luck", that only applies to people's moral choices. In astronomy , Aristotle refuted Democritus 's claim that

8991-513: The intellectual life of Victorian England in many ways and in particular was one of the leading factors that caused a re-examination of the teaching of geometry based on Euclid's Elements . This curriculum issue was hotly debated at the time and was even the subject of a book, Euclid and his Modern Rivals , written by Charles Lutwidge Dodgson (1832–1898) better known as Lewis Carroll , the author of Alice in Wonderland . In analytic geometry

9102-558: The legendary physician Asclepius and his son Machaon . Both of Aristotle's parents died when he was still at a young age and Proxenus of Atarneus became his guardian. Although little information about Aristotle's childhood has survived, he probably spent some time in the Macedonian capital, making his first connections with the Macedonian monarchy . At the age of seventeen or eighteen, Aristotle moved to Athens to continue his education at Plato's Academy . He became distinguished as

9213-468: The modern zoologist would call vertebrates as the hotter "animals with blood", and below them the colder invertebrates as "animals without blood". Those with blood were divided into the live-bearing ( mammals ), and the egg-laying ( birds , reptiles , fish ). Those without blood were insects, crustacea (non-shelled – cephalopods, and shelled ) and the hard-shelled molluscs ( bivalves and gastropods ). He recognised that animals did not exactly fit into

9324-528: The most part, intended for widespread publication; they are generally thought to be lecture aids for his students. His most important treatises include Physics , Metaphysics , Nicomachean Ethics , Politics , On the Soul and Poetics . Aristotle studied and made significant contributions to "logic, metaphysics, mathematics, physics, biology, botany, ethics, politics, agriculture, medicine, dance, and theatre." While Alexander deeply admired Aristotle, near

9435-494: The nature of parallelism. This commonality is the subject of absolute geometry (also called neutral geometry ). However, the properties that distinguish one geometry from others have historically received the most attention. Besides the behavior of lines with respect to a common perpendicular, mentioned in the introduction, we also have the following: Before the models of a non-Euclidean plane were presented by Beltrami, Klein, and Poincaré, Euclidean geometry stood unchallenged as

9546-498: The near-by island of Lesbos . During this time, Aristotle married Pythias , Hermias's adoptive daughter and niece, and had a daughter whom they also named Pythias. In 343/42 BC, Aristotle was invited to Pella by Philip II of Macedon in order to become the tutor to his thirteen-year-old son Alexander ; a choice perhaps influenced by the relationship of Aristotle's family with the Macedonian dynasty. Aristotle taught Alexander at

9657-404: The oceans are then replenished by the cycle of rainfall and river runoff ("I have proved by experiment that salt water evaporated forms fresh and the vapor does not when it condenses condense into sea water again.") Aristotle was the first person to study biology systematically, and biology forms a large part of his writings. He spent two years observing and describing the zoology of Lesbos and

9768-512: The other cases. When ε = +1 , a hyperbolic unit . Then z is a split-complex number and conventionally j replaces epsilon. Then and { z | z z * = 1} is the unit hyperbola . When ε = 0 , then z is a dual number . This approach to non-Euclidean geometry explains the non-Euclidean angles: the parameters of slope in the dual number plane and hyperbolic angle in the split-complex plane correspond to angle in Euclidean geometry. Indeed, they each arise in polar decomposition of

9879-478: The parallel postulate, depending on assumptions that are now recognized as essentially equivalent to the parallel postulate. These early attempts did, however, provide some early properties of the hyperbolic and elliptic geometries. Khayyam, for example, tried to derive it from an equivalent postulate he formulated from "the principles of the Philosopher" ( Aristotle ): "Two convergent straight lines intersect and it

9990-406: The physical universe is Euclidean or non-Euclidean; this is a task for the physical sciences. Bernhard Riemann , in a famous lecture in 1854, founded the field of Riemannian geometry , discussing in particular the ideas now called manifolds , Riemannian metric , and curvature . He constructed an infinite family of non-Euclidean geometries by giving a formula for a family of Riemannian metrics on

10101-437: The primary kind of knowledge; but if there is some motionless independent thing, the knowledge of this precedes it and is first philosophy, and it is universal in just this way , because it is first. And it belongs to this sort of philosophy to study being as being, both what it is and what belongs to it just by virtue of being. Aristotle examines the concepts of substance ( ousia ) and essence ( to ti ên einai , "the what it

10212-567: The private school of Mieza , in the gardens of the Nymphs , the royal estate near Pella. Alexander's education probably included a number of subjects, such as ethics and politics , as well as standard literary texts, like Euripides and Homer . It is likely that during Aristotle's time in the Macedonian court, other prominent nobles, like Ptolemy and Cassander , would have occasionally attended his lectures. Aristotle encouraged Alexander toward eastern conquest, and his own attitude towards Persia

10323-433: The same certainty as experimental science, but it sets out testable hypotheses and constructs a narrative explanation of what is observed. In this sense, Aristotle's biology is scientific. From the data he collected and documented, Aristotle inferred quite a number of rules relating the life-history features of the live-bearing tetrapods (terrestrial placental mammals) that he studied. Among these correct predictions are

10434-417: The same). This is also one of the standard models of the real projective plane . The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric. In the elliptic model, for any given line l and a point A , which is not on l , all lines through A will intersect l . Even after

10545-549: The surrounding seas, including in particular the Pyrrha lagoon in the centre of Lesbos. His data in History of Animals , Generation of Animals , Movement of Animals , and Parts of Animals are assembled from his own observations, statements given by people with specialized knowledge, such as beekeepers and fishermen, and less accurate accounts provided by travellers from overseas. His apparent emphasis on animals rather than plants

10656-402: The thing that has undergone the change has itself changed. In that particular change he introduces the concept of potentiality ( dynamis ) and actuality ( entelecheia ) in association with the matter and the form. Referring to potentiality, this is what a thing is capable of doing or being acted upon if the conditions are right and it is not prevented by something else. For example, the seed of

10767-470: The time of Homer , and "the upheaving of one of the Aeolian islands , previous to a volcanic eruption ."' Meteorologica lends its name to the modern study of meteorology, but its modern usage diverges from the content of Aristotle's ancient treatise on meteors . The ancient Greeks did use the term for a range of atmospheric phenomena, but also for earthquakes and volcanic eruptions. Aristotle proposed that

10878-473: The traditional rendering will be employed here. Aristotle describes experiments in optics using a camera obscura in Problems , book 15. The apparatus consisted of a dark chamber with a small aperture that let light in. With it, he saw that whatever shape he made the hole, the sun's image always remained circular. He also noted that increasing the distance between the aperture and the image surface magnified

10989-401: The treatise we know by the name Metaphysics . Aristotle called it "first philosophy", and distinguished it from mathematics and natural science (physics) as the contemplative ( theoretikē ) philosophy which is "theological" and studies the divine. He wrote in his Metaphysics (1026a16): If there were no other independent things besides the composite natural ones, the study of nature would be

11100-450: The unit ball in Euclidean space . The simplest of these is called elliptic geometry and it is considered a non-Euclidean geometry due to its lack of parallel lines. By formulating the geometry in terms of a curvature tensor , Riemann allowed non-Euclidean geometry to apply to higher dimensions. Beltrami (1868) was the first to apply Riemann's geometry to spaces of negative curvature. It

11211-492: The universal ( katholou ) in particulars ( kath' hekaston ), things in the world, whereas for Plato the universal is a separately existing form which actual things imitate. For Aristotle, "form" is still what phenomena are based on, but is "instantiated" in a particular substance. Plato argued that all things have a universal form , which could be either a property or a relation to other things. When one looks at an apple, for example, one sees an apple, and one can also analyse

11322-528: The way they are represented. In three dimensions, there are eight models of geometries. There are Euclidean, elliptic, and hyperbolic geometries, as in the two-dimensional case; mixed geometries that are partially Euclidean and partially hyperbolic or spherical; twisted versions of the mixed geometries; and one unusual geometry that is completely anisotropic (i.e. every direction behaves differently). Euclidean and non-Euclidean geometries naturally have many similar properties, namely those that do not depend upon

11433-459: The work of Lobachevsky, Gauss, and Bolyai, the question remained: "Does such a model exist for hyperbolic geometry ?". The model for hyperbolic geometry was answered by Eugenio Beltrami , in 1868, who first showed that a surface called the pseudosphere has the appropriate curvature to model a portion of hyperbolic space and in a second paper in the same year, defined the Klein model , which models

11544-414: The world around it, that was a result of this paradigm shift. Non-Euclidean geometry is an example of a scientific revolution in the history of science , in which mathematicians and scientists changed the way they viewed their subjects. Some geometers called Lobachevsky the " Copernicus of Geometry" due to the revolutionary character of his work. The existence of non-Euclidean geometries impacted

11655-449: Was Gauss who coined the term "non-Euclidean geometry". He was referring to his own work, which today we call hyperbolic geometry or Lobachevskian geometry . Several modern authors still use the generic term non-Euclidean geometry to mean hyperbolic geometry . Arthur Cayley noted that distance between points inside a conic could be defined in terms of logarithm and the projective cross-ratio function. The method has become called

11766-670: Was disappointed with the Academy's direction after control passed to Plato's nephew Speusippus , although it is possible that the anti-Macedonian sentiments in Athens could have also influenced his decision. Aristotle left with Xenocrates to Assos in Asia Minor , where he was invited by his former fellow student Hermias of Atarneus ; he stayed there for a few years and left around the time of Hermias' death. While at Assos, Aristotle and his colleague Theophrastus did extensive research in botany and marine biology , which they later continued at

11877-411: Was equivalent to his own postulate. Another example is al-Tusi's son, Sadr al-Din (sometimes known as "Pseudo-Tusi"), who wrote a book on the subject in 1298, based on al-Tusi's later thoughts, which presented another hypothesis equivalent to the parallel postulate. "He essentially revised both the Euclidean system of axioms and postulates and the proofs of many propositions from the Elements ." His work

11988-570: Was published in Rome in 1594 and was studied by European geometers, including Saccheri who criticised this work as well as that of Wallis. Giordano Vitale , in his book Euclide restituo (1680, 1686), used the Saccheri quadrilateral to prove that if three points are equidistant on the base AB and the summit CD, then AB and CD are everywhere equidistant. In a work titled Euclides ab Omni Naevo Vindicatus ( Euclid Freed from All Flaws ), published in 1733, Saccheri quickly discarded elliptic geometry as

12099-474: Was strongly ethnocentric . In one famous example, he counsels Alexander to be "a leader to the Greeks and a despot to the barbarians". Alexander's education under the guardianship of Aristotle likely lasted for only a few years, as at around the age of sixteen he returned to Pella and was appointed regent of Macedon by his father Philip. During this time, Aristotle is said to have gifted Alexander an annotated copy of

12210-421: Was to be") in his Metaphysics (Book VII), and he concludes that a particular substance is a combination of both matter and form, a philosophical theory called hylomorphism . In Book VIII, he distinguishes the matter of the substance as the substratum , or the stuff of which it is composed. For example, the matter of a house is the bricks, stones, timbers, etc., or whatever constitutes the potential house, while

12321-588: Was widely disbelieved until the 19th century. He gives accurate descriptions of the four-chambered fore-stomachs of ruminants , and of the ovoviviparous embryological development of the hound shark . He notes that an animal's structure is well matched to function so birds like the heron (which live in marshes with soft mud and live by catching fish) have a long neck, long legs, and a sharp spear-like beak, whereas ducks that swim have short legs and webbed feet. Darwin , too, noted these sorts of differences between similar kinds of animal, but unlike Aristotle used

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