Nikolai Ivanovich Lobachevsky (Russian: Никола́й Ива́нович Лобаче́вский , IPA: [nʲɪkɐˈlaj ɪˈvanəvʲɪtɕ ləbɐˈtɕefskʲɪj] ; 1 December [ O.S. 20 November] 1792 – 24 February [ O.S. 12 February] 1856) was a Russian mathematician and geometer , known primarily for his work on hyperbolic geometry , otherwise known as Lobachevskian geometry , and also for his fundamental study on Dirichlet integrals , known as the Lobachevsky integral formula .
64-533: William Kingdon Clifford called Lobachevsky the " Copernicus of Geometry" due to the revolutionary character of his work. Nikolai Lobachevsky was born either in or near the city of Nizhny Novgorod in the Russian Empire (now in Nizhny Novgorod Oblast , Russia ) in 1792 to parents of Russian and Polish origin – Ivan Maksimovich Lobachevsky and Praskovia Alexandrovna Lobachevskaya. He
128-420: A hyperbolic triangle must be less than 180 degrees. Non-Euclidean geometry stimulated the development of differential geometry which has many applications. Hyperbolic geometry is frequently referred to as "Lobachevskian geometry" or "Bolyai–Lobachevskian geometry". Some mathematicians and historians have wrongly claimed that Lobachevsky in his studies in non-Euclidean geometry was influenced by Gauss, which
192-464: A collection of fairy stories, The Little People . In 1876, Clifford suffered a breakdown, probably brought on by overwork. He taught and administered by day, and wrote by night. A half-year holiday in Algeria and Spain allowed him to resume his duties for 18 months, after which he collapsed again. He went to the island of Madeira to recover, but died there of tuberculosis after a few months, leaving
256-463: A decimal number cannot be expressed in a finite number of binary digits. Related to approximation of functions is the asymptotic value of a function, i.e. the value as one or more of a function's parameters becomes arbitrarily large. For example, the sum k / 2 + k / 4 + k / 8 + ⋯ + k / 2 n {\displaystyle k/2+k/4+k/8+\cdots +k/2^{n}}
320-480: A former teacher and friend of the German mathematician Carl Friedrich Gauss (1777-1855). Lobachevsky received a Master of Science in physics and mathematics in 1811. In 1814 he became a lecturer at Kazan University, in 1816 he was promoted to associate professor. In 1822, at the age of 30, he became a full professor , teaching mathematics, physics, and astronomy. He served in many administrative positions and became
384-486: A four-dimensional space. Quaternion versors , which inhabit this 3-sphere, provide a representation of the rotation group SO(3) . Clifford noted that Hamilton's biquaternions were a tensor product H ⊗ C {\displaystyle H\otimes C} of known algebras, and proposed instead two other tensor products of H : Clifford argued that the "scalars" taken from the complex numbers C might instead be taken from split-complex numbers D or from
448-489: A hyperbola". He wrote an expression for a parametrized unit hyperbola , which other authors later used as a model for relativistic velocity. Elsewhere he states: This passage makes reference to biquaternions , though Clifford made these into split-biquaternions as his independent development. The book continues with a chapter "On the bending of space", the substance of general relativity . Clifford also discussed his views in On
512-410: A scientific theory can differ from actual measurements. This can be because there are factors in the real situation that are not included in the theory. For example, simple calculations may not include the effect of air resistance. Under these circumstances, the theory is an approximation to reality. Differences may also arise because of limitations in the measuring technique. In this case, the measurement
576-488: A similar meaning. It is often found abbreviated as approx. The term can be applied to various properties (e.g., value, quantity, image, description) that are nearly, but not exactly correct; similar, but not exactly the same (e.g., the approximate time was 10 o'clock). Although approximation is most often applied to numbers , it is also frequently applied to such things as mathematical functions , shapes , and physical laws . In science, approximation can refer to using
640-490: A simpler process or model when the correct model is difficult to use. An approximate model is used to make calculations easier. Approximations might also be used if incomplete information prevents use of exact representations. The type of approximation used depends on the available information , the degree of accuracy required , the sensitivity of the problem to this data, and the savings (usually in time and effort) that can be achieved by approximation. Approximation theory
704-409: A slur on [Lobachevsky's] character" and the name was chosen "solely for prosodic reasons". In Poul Anderson 's 1969 fantasy novella "Operation Changeling" – which was later expanded into the novel Operation Chaos (1971) – a group of sorcerers navigate a non-Euclidean universe with the assistance of the ghosts of Lobachevsky and Bolyai . Roger Zelazny 's science fiction novel Doorways in
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#1732790187424768-449: A small number of significant digits . Calculations are likely to involve rounding errors and other approximation errors . Log tables , slide rules and calculators produce approximate answers to all but the simplest calculations. The results of computer calculations are normally an approximation expressed in a limited number of significant digits, although they can be programmed to produce more precise results. Approximation can occur when
832-430: A small piece of mind-stuff. When molecules are so combined together as to form the film on the under side of a jelly-fish, the elements of mind-stuff which go along with them are so combined as to form the faint beginnings of Sentience. When the molecules are so combined as to form the brain and nervous system of a vertebrate, the corresponding elements of mind-stuff are so combined as to form some kind of consciousness; that
896-716: A widow with two children. Clifford and his wife are buried in London's Highgate Cemetery , near the graves of George Eliot and Herbert Spencer , just north of the grave of Karl Marx . The academic journal Advances in Applied Clifford Algebras publishes on Clifford's legacy in kinematics and abstract algebra . "Clifford was above all and before all a geometer." The discovery of non-Euclidean geometry opened new possibilities in geometry in Clifford's era. The field of intrinsic differential geometry
960-446: Is a branch of mathematics, and a quantitative part of functional analysis . Diophantine approximation deals with approximations of real numbers by rational numbers . Approximation usually occurs when an exact form or an exact numerical number is unknown or difficult to obtain. However some known form may exist and may be able to represent the real form so that no significant deviation can be found. For example, 1.5 × 10 means that
1024-562: Is a rule in Euclidean geometry which states (in John Playfair 's reformulation) that for any given line and point not on the line, there is only one line through the point not intersecting the given line. Lobachevsky would instead develop a geometry in which the fifth postulate was not true. This idea was first reported on February 23 (Feb. 11, O.S. ), 1826 to the session of the department of physics and mathematics, and this research
1088-419: Is also used to analyze the motion of several planets orbiting a star. This is extremely difficult due to the complex interactions of the planets' gravitational effects on each other. An approximate solution is effected by performing iterations . In the first iteration, the planets' gravitational interactions are ignored, and the star is assumed to be fixed. If a more precise solution is desired, another iteration
1152-538: Is an approximation to the actual value. The history of science shows that earlier theories and laws can be approximations to some deeper set of laws. Under the correspondence principle , a new scientific theory should reproduce the results of older, well-established, theories in those domains where the old theories work. The old theory becomes an approximation to the new theory. Some problems in physics are too complex to solve by direct analysis, or progress could be limited by available analytical tools. Thus, even when
1216-588: Is asymptotically equal to k . No consistent notation is used throughout mathematics and some texts use ≈ to mean approximately equal and ~ to mean asymptotically equal whereas other texts use the symbols the other way around. The approximately equals sign , ≈ , was introduced by British mathematician Alfred Greenhill . Symbols used in LaTeX markup. Symbols used to denote items that are approximately equal are wavy or dotted equals signs. Approximation arises naturally in scientific experiments . The predictions of
1280-551: Is being measured. Within the European Union (EU), "approximation" refers to a process through which EU legislation is implemented and incorporated within Member States ' national laws, despite variations in the existing legal framework in each country. Approximation is required as part of the pre-accession process for new member states, and as a continuing process when required by an EU Directive . Approximation
1344-442: Is chiefly associated with two phrases of his coining, mind-stuff and the tribal self . The former symbolizes his metaphysical conception, suggested to him by his reading of Baruch Spinoza , which Clifford (1878) defined as follows: That element of which, as we have seen, even the simplest feeling is a complex, I shall call Mind-stuff. A moving molecule of inorganic matter does not possess mind or consciousness; but it possesses
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#17327901874241408-533: Is now known as the Dandelin–Gräffe method , named after two other mathematicians who discovered it independently. In Russia, it is called the Lobachevsky method. Lobachevsky gave the definition of a function as a correspondence between two sets of real numbers ( Peter Gustav Lejeune Dirichlet gave the same definition independently soon after Lobachevsky). E. T. Bell wrote about Lobachevsky's influence on
1472-630: Is one long sin against mankind." "I was not, and was conceived. I loved and did a little work. I am not and grieve not." Approximation An approximation is anything that is intentionally similar but not exactly equal to something else. The word approximation is derived from Latin approximatus , from proximus meaning very near and the prefix ad- ( ad- before p becomes ap- by assimilation ) meaning to . Words like approximate , approximately and approximation are used especially in technical or scientific contexts. In everyday English, words such as roughly or around are used with
1536-411: Is referred to as hyperbolic geometry . Lobachevsky replaced Playfair's axiom with the statement that for any given point there exists more than one line that can be extended through that point and run parallel to another line of which that point is not part. He developed the angle of parallelism which depends on the distance the point is off the given line. In hyperbolic geometry the sum of angles in
1600-424: Is that mind is the one ultimate reality; not mind as we know it in the complex forms of conscious feeling and thought, but the simpler elements out of which thought and feeling are built up. The hypothetical ultimate element of mind, or atom of mind-stuff, precisely corresponds to the hypothetical atom of matter, being the ultimate fact of which the material atom is the phenomenon. Matter and the sensible universe are
1664-602: Is then performed, using the positions and motions of the planets as identified in the first iteration, but adding a first-order gravity interaction from each planet on the others. This process may be repeated until a satisfactorily precise solution is obtained. The use of perturbations to correct for the errors can yield more accurate solutions. Simulations of the motions of the planets and the star also yields more accurate solutions. The most common versions of philosophy of science accept that empirical measurements are always approximations — they do not perfectly represent what
1728-410: Is to say, changes in the complex which take place at the same time get so linked together that the repetition of one implies the repetition of the other. When matter takes the complex form of a living human brain, the corresponding mind-stuff takes the form of a human consciousness, having intelligence and volition. Regarding Clifford's concept, Sir Frederick Pollock wrote: Briefly put, the conception
1792-424: Is untrue. Gauss himself appreciated Lobachevsky's published works highly, but they never had personal correspondence between them prior to the publication. Although three people—Gauss, Lobachevsky and Bolyai—can be credited with discovery of hyperbolic geometry, Gauss never published his ideas, and Lobachevsky was the first to present his views to the world mathematical community. Lobachevsky's magnum opus Geometriya
1856-464: Is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." "If a man, holding a belief which he was taught in childhood or persuaded of afterwards, keeps down and pushes away any doubts which arise about it in his mind, purposely avoids the reading of books and the company of men that call in question or discuss it, and regards as impious those questions which cannot easily be asked without disturbing it—the life of that man
1920-403: The dual numbers N . In terms of tensor products, H ⊗ D {\displaystyle H\otimes D} produces split-biquaternions , while H ⊗ N {\displaystyle H\otimes N} forms dual quaternions . The algebra of dual quaternions is used to express screw displacement , a common mapping in kinematics. As a philosopher, Clifford's name
1984-466: The inner product and Grassmann's outer product. The geometric product was eventually formalized by the Hungarian mathematician Marcel Riesz . The inner product equips geometric algebra with a metric, fully incorporating distance and angle relationships for lines, planes, and volumes, while the outer product gives those planes and volumes vector-like properties, including a directional bias. Combining
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2048-408: The quaternions , developed by William Rowan Hamilton , with Grassmann's outer product (aka the exterior product ). He understood the geometric nature of Grassmann's creation, and that the quaternions fit cleanly into the algebra Grassmann had developed. The versors in quaternions facilitate representation of rotation. Clifford laid the foundation for a geometric product, composed of the sum of
2112-405: The rector of Kazan University in 1827. In 1832, he married Varvara Alexeyevna Moiseyeva. They had a large number of children (eighteen according to his son's memoirs, though only seven apparently survived into adulthood). He was dismissed from the university in 1846, ostensibly due to his deteriorating health: by the early 1850s, he was nearly blind and unable to walk. He died in poverty in 1856 and
2176-592: The Lobachevskian method of challenging axioms has probably yet to be felt. It is no exaggeration to call Lobachevsky the Copernicus of Geometry, for geometry is only a part of the vaster domain which he renovated; it might even be just to designate him as a Copernicus of all thought. A fictional mathematician named "Lobachevsky" is the subject of songwriter/mathematician Tom Lehrer 's humorous song " Lobachevsky " from his 1953 Songs by Tom Lehrer album. In
2240-717: The Sand contains a poem dedicated to Lobachevsky. In the sitcom 3rd Rock from the Sun , "Dick and the Single Girl" (season 2 episode 24) originally aired on May 11 1997, Sonja Umdahl ( Christine Baranski ), a forgotten colleague who is transferring to teach at another university, gives as the reason behind her departure that Columbia is the only holder of Nikolai Lobachevsky's manuscripts. Informational notes Citations William Kingdon Clifford William Kingdon Clifford FRS (4 May 1845 – 3 March 1879)
2304-676: The Space-Theory of Matter in 1876. In 1910, William Barrett Frankland quoted the Space-Theory of Matter in his book on parallelism: "The boldness of this speculation is surely unexcelled in the history of thought. Up to the present, however, it presents the appearance of an Icarian flight." Years later, after general relativity had been advanced by Albert Einstein , various authors noted that Clifford had anticipated Einstein. Hermann Weyl (1923), for instance, mentioned Clifford as one of those who, like Bernhard Riemann , anticipated
2368-413: The claims of sect above those of human society. The alarm was greater, as theology was still unreconciled with Darwinism ; and Clifford was regarded as a dangerous champion of the anti-spiritual tendencies then imputed to modern science. There has also been debate on the extent to which Clifford's doctrine of ' concomitance ' or ' psychophysical parallelism ' influenced John Hughlings Jackson 's model of
2432-469: The exact representation is known, an approximation may yield a sufficiently accurate solution while reducing the complexity of the problem significantly. Physicists often approximate the shape of the Earth as a sphere even though more accurate representations are possible, because many physical characteristics (e.g., gravity ) are much easier to calculate for a sphere than for other shapes. Approximation
2496-430: The following development of mathematics in his 1937 book Men of Mathematics : The boldness of his challenge and its successful outcome have inspired mathematicians and scientists in general to challenge other "axioms" or accepted "truths", for example the "law" of causality which, for centuries, have seemed as necessary to straight thinking as Euclid's postulate appeared until Lobachevsky discarded it. The full impact of
2560-409: The geometric ideas of relativity. In 1940, Eric Temple Bell published The Development of Mathematics , in which he discusses the prescience of Clifford on relativity: John Archibald Wheeler , during the 1960 International Congress for Logic, Methodology, and Philosophy of Science (CLMPS) at Stanford , introduced his geometrodynamics formulation of general relativity by crediting Clifford as
2624-414: The information available to him at the time. Clifford famously concludes with what has come to be known as Clifford's principle : "it is wrong always, everywhere, and for anyone, to believe anything upon insufficient evidence." As such, he is arguing in direct opposition to religious thinkers for whom 'blind faith' (i.e. belief in things in spite of the lack of evidence for them) was a virtue. This paper
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2688-537: The initiator. In The Natural Philosophy of Time (1961), Gerald James Whitrow recalls Clifford's prescience, quoting him in order to describe the Friedmann–Lemaître–Robertson–Walker metric in cosmology. Cornelius Lanczos (1970) summarizes Clifford's premonitions: Likewise, Banesh Hoffmann (1973) writes: In 1990, Ruth Farwell and Christopher Knee examined the record on acknowledgement of Clifford's foresight. They conclude that "it
2752-402: The moral law by the development in each individual of a 'self,' which prescribes the conduct conducive to the welfare of the 'tribe.' Much of Clifford's contemporary prominence was due to his attitude toward religion . Animated by an intense love of his conception of truth and devotion to public duty, he waged war on such ecclesiastical systems as seemed to him to favour obscurantism , and to put
2816-472: The nervous system and, through him, the work of Janet, Freud, Ribot, and Ey. In his 1877 essay, The Ethics of Belief , Clifford argues that it is immoral to believe things for which one lacks evidence. He describes a ship-owner who planned to send to sea an old and not well-built ship full of passengers. The ship-owner had doubts suggested to him that the ship might not be seaworthy: "These doubts preyed upon his mind, and made him unhappy." He considered having
2880-420: The night, which may have hastened his death. He published papers on a range of topics including algebraic forms and projective geometry and the textbook Elements of Dynamic . His application of graph theory to invariant theory was followed up by William Spottiswoode and Alfred Kempe . In 1878, Clifford published a seminal work, building on Grassmann's extensive algebra. He had succeeded in unifying
2944-446: The relations between particular organisms, that is, mind organized into consciousness , and the rest of the world. This leads to results which would in a loose and popular sense be called materialist . But the theory must, as a metaphysical theory, be reckoned on the idealist side. To speak technically, it is an idealist monism . Tribal self , on the other hand, gives the key to Clifford's ethical view, which explains conscience and
3008-437: The same symbolic form as they do in 2 or 3-dimensions. The importance of general Clifford algebras has grown over time, while their isomorphism classes - as real algebras - have been identified in other mathematical systems beyond simply the quaternions. The realms of real analysis and complex analysis have been expanded through the algebra H of quaternions, thanks to its notion of a three-dimensional sphere embedded in
3072-438: The ship refitted even though it would be expensive. At last, "he succeeded in overcoming these melancholy reflections." He watched the ship depart, "with a light heart…and he got his insurance money when she went down in mid-ocean and told no tales." Clifford argues that the ship-owner was guilty of the deaths of the passengers even though he sincerely believed the ship was sound: " [H]e had no right to believe on such evidence as
3136-401: The song, Lehrer portrays a Russian mathematician who sings about how Lobachevsky influenced him: "And who made me a big success / and brought me wealth and fame? / Nikolai Ivanovich Lobachevsky is his name." Lobachevsky's secret to mathematical success is given as " Plagiarize !", as long as one is always careful to "call it, please, research ". According to Lehrer, the song is "not intended as
3200-572: The theory of general relativity, the framework existed in which a geometrical perspective in physics could be developed and allowed the challenging geometrical conceptions of Riemann and Clifford to be rediscovered. "I…hold that in the physical world nothing else takes place but this variation [of the curvature of space]." "There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture—that it came to him from outside, and that he did not consciously create it from within." "It
3264-407: The true value of something being measured is 1,500,000 to the nearest hundred thousand (so the actual value is somewhere between 1,450,000 and 1,550,000); this is in contrast to the notation 1.500 × 10 , which means that the true value is 1,500,000 to the nearest thousand (implying that the true value is somewhere between 1,499,500 and 1,500,500). Numerical approximations sometimes result from using
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#17327901874243328-549: The two brought the operation of division into play. This greatly expanded our qualitative understanding of how objects interact in space. Crucially, it also provided the means for quantitatively calculating the spatial consequences of those interactions. The resulting geometric algebra, as he called it, eventually realized the long sought goal of creating an algebra that mirrors the movements and projections of objects in 3-dimensional space. Moreover, Clifford's algebraic schema extends to higher dimensions. The algebraic operations have
3392-418: Was Clifford, not Riemann, who anticipated some of the conceptual ideas of General Relativity." To explain the lack of recognition of Clifford's prescience, they point out that he was an expert in metric geometry, and "metric geometry was too challenging to orthodox epistemology to be pursued." In 1992, Farwell and Knee continued their study of Clifford and Riemann: [They] hold that once tensors had been used in
3456-602: Was a British mathematician and philosopher . Building on the work of Hermann Grassmann , he introduced what is now termed geometric algebra , a special case of the Clifford algebra named in his honour. The operations of geometric algebra have the effect of mirroring, rotating, translating, and mapping the geometric objects that are being modelled to new positions. Clifford algebras in general and geometric algebra in particular have been of ever increasing importance to mathematical physics , geometry , and computing . Clifford
3520-421: Was before him ." Moreover, he contends that even in the case where the ship successfully reaches the destination, the decision remains immoral, because the morality of the choice is defined forever once the choice is made, and actual outcome, defined by blind chance, doesn't matter. The ship-owner would be no less guilty: his wrongdoing would never be discovered, but he still had no right to make that decision given
3584-535: Was born, with the concept of curvature broadly applied to space itself as well as to curved lines and surfaces. Clifford was very much impressed by Bernhard Riemann ’s 1854 essay "On the hypotheses which lie at the bases of geometry". In 1870, he reported to the Cambridge Philosophical Society on the curved space concepts of Riemann, and included speculation on the bending of space by gravity. Clifford's translation of Riemann's paper
3648-557: Was buried in Arskoe Cemetery , Kazan. In 1811, in his student days, Lobachevsky was accused by a vengeful supervisor of atheism ( Russian : признаки безбожия , lit. 'signs of godlessness'). Lobachevsky's main achievement is the development (independently from János Bolyai ) of a non-Euclidean geometry , also referred to as Lobachevskian geometry. Before him, mathematicians were trying to deduce Euclid 's fifth postulate from other axioms . Euclid's fifth
3712-482: Was completed in 1823, but was not published in its exact original form until 1909, long after he had died. Lobachevsky was also the author of New Foundations of Geometry (1835–1838). He also wrote Geometrical Investigations on the Theory of Parallels (1840) and Pangeometry (1855). Another of Lobachevsky's achievements was developing a method for the approximation of the roots of algebraic equations . This method
3776-500: Was famously attacked by pragmatist philosopher William James in his " Will to Believe " lecture. Often these two works are read and published together as touchstones for the debate over evidentialism , faith , and overbelief . Though Clifford never constructed a full theory of spacetime and relativity , there are some remarkable observations he made in print that foreshadowed these modern concepts: In his book Elements of Dynamic (1878), he introduced "quasi-harmonic motion in
3840-490: Was one of three children. When he was seven, his father, a clerk in a land-surveying office, died, and Nikolai moved with his mother to Kazan . Nikolai Lobachevsky attended Kazan Gymnasium from 1802, graduating in 1807, and then received a scholarship to Kazan University , which had been founded just three years earlier in 1804. At Kazan University, Lobachevsky was influenced by professor Johann Christian Martin Bartels ,
3904-797: Was part of an expedition to Italy to observe the solar eclipse of 22 December 1870. During that voyage he survived a shipwreck along the Sicilian coast. In 1871, he was appointed professor of mathematics and mechanics at University College London , and in 1874 became a fellow of the Royal Society . He was also a member of the London Mathematical Society and the Metaphysical Society . Clifford married Lucy Lane on 7 April 1875, with whom he had two children. Clifford enjoyed entertaining children and wrote
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#17327901874243968-670: Was printed in the periodical 'Kazan University Course Notes' as On the Origin of Geometry ( О началах геометрии ) between 1829 and 1830. In 1829 Lobachevsky wrote a paper about his ideas called "A Concise Outline of the Foundations of Geometry" that was published by the Kazan Messenger but was rejected when it was submitted to the St. Petersburg Academy of Sciences for publication. The non-Euclidean geometry that Lobachevsky developed
4032-503: Was published in Nature in 1873. His report at Cambridge, " On the Space-Theory of Matter ", was published in 1876, anticipating Albert Einstein 's general relativity by 40 years. Clifford elaborated elliptic space geometry as a non-Euclidean metric space . Equidistant curves in elliptic space are now said to be Clifford parallels . Clifford's contemporaries considered him acute and original, witty and warm. He often worked late into
4096-555: Was the first to suggest that gravitation might be a manifestation of an underlying geometry. In his philosophical writings he coined the expression mind-stuff . Born in Exeter , William Clifford was educated at Doctor Templeton's Academy on Bedford Circus and showed great promise at school. He went on to King's College London (at age 15) and Trinity College, Cambridge , where he was elected fellow in 1868, after being Second Wrangler in 1867 and second Smith's prizeman. In 1870, he
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