In algebraic geometry , a lemniscate ( / l ɛ m ˈ n ɪ s k ɪ t / or / ˈ l ɛ m n ɪ s ˌ k eɪ t , - k ɪ t / ) is any of several figure-eight or ∞ -shaped curves . The word comes from the Latin lēmniscātus , meaning "decorated with ribbons", from the Greek λημνίσκος ( lēmnískos ), meaning "ribbon", or which alternatively may refer to the wool from which the ribbons were made.
29-428: Curves that have been called a lemniscate include three quartic plane curves : the hippopede or lemniscate of Booth , the lemniscate of Bernoulli , and the lemniscate of Gerono . The hippopede was studied by Proclus (5th century), but the term "lemniscate" was not used until the work of Jacob Bernoulli in the late 17th century. The consideration of curves with a figure-eight shape can be traced back to Proclus ,
58-421: A 2 − x 2 ) {\displaystyle y^{2}-x^{2}(a^{2}-x^{2})} . Viviani's curve , a three-dimensional curve formed by intersecting a sphere with a cylinder, also has a figure eight shape, and has the lemniscate of Gerono as its planar projection. Other figure-eight shaped algebraic curves include Quartic plane curve Too Many Requests If you report this error to
87-570: A horse fetter (a device for holding two feet of a horse together), or "hippopede" in Greek. The name "lemniscate of Booth" for this curve dates to its study by the 19th-century mathematician James Booth . The lemniscate may be defined as an algebraic curve , the zero set of the quartic polynomial ( x 2 + y 2 ) 2 − c x 2 − d y 2 {\displaystyle (x^{2}+y^{2})^{2}-cx^{2}-dy^{2}} when
116-415: A logarithmic spiral and the motto Eadem mutata resurgo ('Although changed, I rise again the same') engraved on his tombstone. He wrote that the self-similar spiral "may be used as a symbol, either of fortitude and constancy in adversity, or of the human body, which after all its changes, even after death, will be restored to its exact and perfect self". Bernoulli died in 1705, but an Archimedean spiral
145-420: A Greek Neoplatonist philosopher and mathematician who lived in the 5th century AD. Proclus considered the cross-sections of a torus by a plane parallel to the axis of the torus. As he observed, for most such sections the cross section consists of either one or two ovals; however, when the plane is tangent to the inner surface of the torus, the cross-section takes on a figure-eight shape, which Proclus called
174-454: A constant. Under very particular circumstances (when the half-distance between the points is equal to the square root of the constant) this gives rise to a lemniscate. In 1694, Johann Bernoulli studied the lemniscate case of the Cassini oval, now known as the lemniscate of Bernoulli (shown above), in connection with a problem of " isochrones " that had been posed earlier by Leibniz . Like
203-498: A finite limit less than 2. Euler was the first to find the limit of this series in 1737. Bernoulli also studied the exponential series which came out of examining compound interest. In May 1690, in a paper published in Acta Eruditorum , Jacob Bernoulli showed that the problem of determining the isochrone is equivalent to solving a first-order nonlinear differential equation. The isochrone, or curve of constant descent,
232-601: A limit (the force of interest ) for more and smaller compounding intervals. Compounding weekly yields $ 2.692597..., while compounding daily yields $ 2.714567..., just two cents more. Using n as the number of compounding intervals, with interest of 100% / n in each interval, the limit for large n is the number that Euler later named e ; with continuous compounding, the account value will reach $ 2.7182818.... More generally, an account that starts at $ 1, and yields (1+ R ) dollars at compound interest , will yield e dollars with continuous compounding. Bernoulli wanted
261-496: A pamphlet on the parallels of logic and algebra published in 1685, work on probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines. By 1689, he had published important work on infinite series and published his law of large numbers in probability theory. Jacob Bernoulli published five treatises on infinite series between 1682 and 1704. The first two of these contained many results, such as
290-515: Is in fact e ): One example is an account that starts with $ 1.00 and pays 100 percent interest per year. If the interest is credited once, at the end of the year, the value is $ 2.00; but if the interest is computed and added twice in the year, the $ 1 is multiplied by 1.5 twice, yielding $ 1.00×1.5 = $ 2.25. Compounding quarterly yields $ 1.00×1.25 = $ 2.4414..., and compounding monthly yields $ 1.00×(1.0833...) = $ 2.613035.... Bernoulli noticed that this sequence approaches
319-405: Is the curve along which a particle will descend under gravity from any point to the bottom in exactly the same time, no matter what the starting point. It had been studied by Huygens in 1687 and Leibniz in 1689. After finding the differential equation, Bernoulli then solved it by what we now call separation of variables . Jacob Bernoulli's paper of 1690 is important for the history of calculus, since
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#1732791631156348-467: The logarithmic spiral and epicycloids around 1692. The lemniscate of Bernoulli was first conceived by Jacob Bernoulli in 1694. In 1695, he investigated the drawbridge problem which seeks the curve required so that a weight sliding along the cable always keeps the drawbridge balanced. Bernoulli's most original work was Ars Conjectandi , published in Basel in 1713, eight years after his death. The work
377-607: The Wikimedia System Administrators, please include the details below. Request from 172.68.168.133 via cp1102 cp1102, Varnish XID 606643142 Upstream caches: cp1102 int Error: 429, Too Many Requests at Thu, 28 Nov 2024 11:00:31 GMT Jacob Bernoulli Jacob Bernoulli (also known as James in English or Jacques in French; 6 January 1655 [ O.S. 27 December 1654] – 16 August 1705)
406-450: The atmosphere of collaboration between the two brothers turned into rivalry as Johann's own mathematical genius began to mature, with both of them attacking each other in print, and posing difficult mathematical challenges to test each other's skills. By 1697, the relationship had completely broken down. The lunar crater Bernoulli is also named after him jointly with his brother Johann. Jacob Bernoulli's first important contributions were
435-571: The differential calculus in " Nova Methodus pro Maximis et Minimis " published in Acta Eruditorum . They also studied the publications of von Tschirnhaus . It must be understood that Leibniz's publications on the calculus were very obscure to mathematicians of that time and the Bernoullis were among the first to try to understand and apply Leibniz's theories. Jacob collaborated with his brother on various applications of calculus. However
464-482: The fundamental result that ∑ 1 n {\displaystyle \sum {\frac {1}{n}}} diverges, which Bernoulli believed were new but they had actually been proved by Pietro Mengoli 40 years earlier and was proved by Nicole Oresme in the 14th century already. Bernoulli could not find a closed form for ∑ 1 n 2 {\displaystyle \sum {\frac {1}{n^{2}}}} , but he did show that it converged to
493-408: The hippopede, it is an algebraic curve, the zero set of the polynomial ( x 2 + y 2 ) 2 − a 2 ( x 2 − y 2 ) {\displaystyle (x^{2}+y^{2})^{2}-a^{2}(x^{2}-y^{2})} . Bernoulli's brother Jacob Bernoulli also studied the same curve in the same year, and gave it its name,
522-406: The last part of the book, Bernoulli sketches many areas of mathematical probability , including probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; and
551-460: The latest discoveries in mathematics and the sciences under leading figures of the time. This included the work of Johannes Hudde , Robert Boyle , and Robert Hooke . During this time he also produced an incorrect theory of comets . Bernoulli returned to Switzerland, and began teaching mechanics at the University of Basel from 1683. His doctoral dissertation Solutionem tergemini problematis
580-417: The law of large numbers. Bernoulli was one of the most significant promoters of the formal methods of higher analysis. Astuteness and elegance are seldom found in his method of presentation and expression, but there is a maximum of integrity. In 1683, Bernoulli discovered the constant e by studying a question about compound interest which required him to find the value of the following expression (which
609-404: The lemniscate. It may also be defined geometrically as the locus of points whose product of distances from two foci equals the square of half the interfocal distance. It is a special case of the hippopede (lemniscate of Booth), with d = − c {\displaystyle d=-c} , and may be formed as a cross-section of a torus whose inner hole and circular cross-sections have
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#1732791631156638-453: The parameter d is negative (or zero for the special case where the lemniscate becomes a pair of externally tangent circles). For positive values of d one instead obtains the oval of Booth . In 1680, Cassini studied a family of curves, now called the Cassini oval , defined as follows: the locus of all points, the product of whose distances from two fixed points, the curves' foci , is
667-411: The same diameter as each other. The lemniscatic elliptic functions are analogues of trigonometric functions for the lemniscate of Bernoulli, and the lemniscate constants arise in evaluating the arc length of this lemniscate. Another lemniscate, the lemniscate of Gerono or lemniscate of Huygens, is the zero set of the quartic polynomial y 2 − x 2 (
696-461: The term integral appears for the first time with its integration meaning. In 1696, Bernoulli solved the equation, now called the Bernoulli differential equation , Jacob Bernoulli also discovered a general method to determine evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the parabola ,
725-439: Was between 1684 and 1689 that many of the results that were to make up Ars Conjectandi were discovered. People believe he was appointed professor of mathematics at the University of Basel in 1687, remaining in this position for the rest of his life. By that time, he had begun tutoring his brother Johann Bernoulli on mathematical topics. The two brothers began to study the calculus as presented by Leibniz in his 1684 paper on
754-561: Was in the field of probability , where he derived the first version of the law of large numbers in his work Ars Conjectandi . Jacob Bernoulli was born in Basel in the Old Swiss Confederacy . Following his father's wish, he studied theology and entered the ministry. But contrary to the desires of his parents, he also studied mathematics and astronomy . He traveled throughout Europe from 1676 to 1682, learning about
783-540: Was incomplete at the time of his death but it is still a work of the greatest significance in the theory of probability. The book also covers other related subjects, including a review of combinatorics , in particular the work of van Schooten, Leibniz, and Prestet, as well as the use of Bernoulli numbers in a discussion of the exponential series. Inspired by Huygens' work, Bernoulli also gives many examples on how much one would expect to win playing various games of chance. The term Bernoulli trial resulted from this work. In
812-583: Was one of the many prominent mathematicians in the Swiss Bernoulli family . He sided with Gottfried Wilhelm Leibniz during the Leibniz–Newton calculus controversy and was an early proponent of Leibnizian calculus , which he made numerous contributions to; along with his brother Johann , he was one of the founders of the calculus of variations . He also discovered the fundamental mathematical constant e . However, his most important contribution
841-665: Was submitted in 1684. It appeared in print in 1687. In 1684, Bernoulli married Judith Stupanus; they had two children. During this decade, he also began a fertile research career. His travels allowed him to establish correspondence with many leading mathematicians and scientists of his era, which he maintained throughout his life. During this time, he studied the new discoveries in mathematics, including Christiaan Huygens 's De ratiociniis in aleae ludo , Descartes ' La Géométrie and Frans van Schooten 's supplements of it. He also studied Isaac Barrow and John Wallis , leading to his interest in infinitesimal geometry. Apart from these, it
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