In mathematics, the trigonometric functions are functions of an angle. They relate the angles of a triangle to the lengths of its sides. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications; the most familiar trigonometric functions are the sine and tangent. In the context of the standard unit circle, where a triangle is formed by a ray starting at the origin and making some angle with the x-axis, the sine of the angle gives the y-component of the triangle, the cosine gives the x-component, the tangent function gives the slope. For angles less than a right angle, trigonometric functions are defined as ratios of two sides of a right triangle containing the angle, their values can be found in the lengths of various line segments around a unit circle. Modern definitions express trigonometric functions as infinite series or as solutions of certain differential equations, allowing the extension of the arguments to the whole number line and to the complex numbers.
Trigonometric functions have a wide range of uses including computing unknown lengths and angles in triangles. In this use, trigonometric functions are used, for instance, in navigation and physics. A common use in elementary physics is resolving a vector into Cartesian coordinates; the sine and cosine functions are commonly used to model periodic function phenomena such as sound and light waves, the position and velocity of harmonic oscillators, sunlight intensity and day length, average temperature variations through the year. In modern usage, there are six basic trigonometric functions, tabulated here with equations that relate them to one another. With the last four, these relations are taken as the definitions of those functions, but one can define them well geometrically, or by other means, derive these relations; the notion that there should be some standard correspondence between the lengths of the sides of a triangle and the angles of the triangle comes as soon as one recognizes that similar triangles maintain the same ratios between their sides.
That is, for any similar triangle the ratio of the hypotenuse and another of the sides remains the same. If the hypotenuse is twice as long, so are the sides, it is these ratios. To define the trigonometric functions for the angle A, start with any right triangle that contains the angle A; the three sides of the triangle are named as follows: The hypotenuse is the side opposite the right angle, in this case side h. The hypotenuse is always the longest side of a right-angled triangle; the opposite side is the side opposite in this case side a. The adjacent side is the side having both the angles in this case side b. In ordinary Euclidean geometry, according to the triangle postulate, the inside angles of every triangle total 180°. Therefore, in a right-angled triangle, the two non-right angles total 90°, so each of these angles must be in the range of as expressed in interval notation; the following definitions apply to angles in this range. They can be extended to the full set of real arguments by using the unit circle, or by requiring certain symmetries and that they be periodic functions.
For example, the figure shows sin for angles θ, π − θ, π + θ, 2π − θ depicted on the unit circle and as a graph. The value of the sine repeats itself apart from sign in all four quadrants, if the range of θ is extended to additional rotations, this behavior repeats periodically with a period 2π; the trigonometric functions are summarized in the following table and described in more detail below. The angle θ is the angle between the hypotenuse and the adjacent line – the angle at A in the accompanying diagram; the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. The word comes from the Latin sinus for gulf or bay, given a unit circle, it is the side of the triangle on which the angle opens. In that case: sin A = opposite hypotenuse The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse, so called because it is the sine of the complementary or co-angle, the other non-right angle; because the angle sum of a triangle is π radians, the co-angle B is equal to π/2 − A.
In that case: cos A = adjacent hypotenuse The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side, so called because it can be represented as a line segment tangent to the circle, i.e. the line that touches the circle, from Latin linea tangens or touching line. In our case: tan A = opposite adjacent Tangent may be represented in terms of sine and cosine; that is: tan A = sin A cos A = opposite
In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is related to the calculus of residues, a method of complex analysis. One use for contour integrals is the evaluation of integrals along the real line that are not found by using only real variable methods. Contour integration methods include direct integration of a complex-valued function along a curve in the complex plane application of the Cauchy integral formula application of the residue theoremOne method can be used, or a combination of these methods, or various limiting processes, for the purpose of finding these integrals or sums. In complex analysis a contour is a type of curve in the complex plane. In contour integration, contours provide a precise definition of the curves on which an integral may be suitably defined. A curve in the complex plane is defined as a continuous function from a closed interval of the real line to the complex plane: z: → C.
This definition of a curve coincides with the intuitive notion of a curve, but includes a parametrization by a continuous function from a closed interval. This more precise definition allows us to consider what properties a curve must have for it to be useful for integration. In the following subsections we narrow down the set of curves that we can integrate to only include ones that can be built up out of a finite number of continuous curves that can be given a direction. Moreover, we will restrict the "pieces" from crossing over themselves, we require that each piece have a finite continuous derivative; these requirements correspond to requiring that we consider only curves that can be traced, such as by a pen, in a sequence of steady strokes, which only stop to start a new piece of the curve, all without picking up the pen. Contours are defined in terms of directed smooth curves; these provide a precise definition of a "piece" of a smooth curve. A smooth curve is a curve z: → C with a non-vanishing, continuous derivative such that each point is traversed only once, with the possible exception of a curve such that the endpoints match.
In the case where the endpoints match the curve is called closed, the function is required to be one-to-one everywhere else and the derivative must be continuous at the identified point. A smooth curve, not closed is referred to as a smooth arc; the parametrization of a curve provides a natural ordering of points on the curve: z comes before z if x < y. This leads to the notion of a directed smooth curve, it is most useful to consider curves independent of the specific parametrization. This can be done by considering equivalence classes of smooth curves with the same direction. A directed smooth curve can be defined as an ordered set of points in the complex plane, the image of some smooth curve in their natural order. Note that not all orderings of the points are the natural ordering of a smooth curve. In fact, a given smooth curve has only two such orderings. A single closed curve can have any point as its endpoint, while a smooth arc has only two choices for its endpoints. Contours are the class of curves.
A contour is a directed curve, made up of a finite sequence of directed smooth curves whose endpoints are matched to give a single direction. This requires that the sequence of curves γ1,…,γn be such that the terminal point of γi coincides with the initial point of γi+1, ∀ i, 1 ≤ i < n. This includes. A single point in the complex plane is considered a contour; the symbol + is used to denote the piecing of curves together to form a new curve. Thus we could write a contour Γ, made up of n curves as Γ = γ 1 + γ 2 + ⋯ + γ n; the contour integral of a complex function f: C → C is a generalization of the integral for real-valued functions. For continuous functions in the complex plane, the contour integral can be defined in analogy to the line integral by first defining the integral along a directed smooth curve in terms of an integral over a real valued parameter. A more general definition can be given in terms of partitions of the contour in analogy with the partition of an interval and the Riemann integral.
In both cases the integral over a contour is defined as the sum of the integrals over the directed smooth curves that make up the contour. To define the contour integral in this way one must first consider the integral, over a real variable, of a complex-valued function. Let f: R → C be a complex-valued function of a real variable, t; the real and imaginary parts of f are denoted as u and v so that f = u + i v. The integral of the complex-valued function f over the interval is given by ∫ a b f d t = ∫ a b d t
In the mathematical field of complex analysis, a meromorphic function on an open subset D of the complex plane is a function, holomorphic on all of D except for a discrete set of isolated points, which are poles of the function. This terminology comes from the Ancient Greek meros, meaning "part," as opposed to holos, meaning "whole." Every meromorphic function on D can be expressed as the ratio between two holomorphic functions defined on D: any pole must coincide with a zero of the denominator. Intuitively, a meromorphic function is a ratio of two well-behaved functions; such a function will still be well-behaved, except at the points where the denominator of the fraction is zero. If the denominator has a zero at z and the numerator does not the value of the function will approach infinity. From an algebraic point of view, if D is connected the set of meromorphic functions is the field of fractions of the integral domain of the set of holomorphic functions; this is analogous to the relationship between the integers.
In the 1930s, in group theory, a meromorphic function was a function from a group G into itself that preserved the product on the group. The image of this function was called an automorphism of G. Similarly, a homomorphic function was a function between groups that preserved the product, while a homomorphism was the image of a homomorph; this terminology is now obsolete. The term endomorphism is now used for the function itself, with no special name given to the image of the function; the term meromorph is no longer used in group theory. Since the poles of a meromorphic function are isolated, there are at most countably many; the set of poles can be infinite, as exemplified by the function f = csc z = 1 sin z. By using analytic continuation to eliminate removable singularities, meromorphic functions can be added, subtracted and the quotient f / g can be formed unless g = 0 on a connected component of D. Thus, if D is connected, the meromorphic functions form a field, in fact a field extension of the complex numbers.
In several complex variables, a meromorphic function is defined to be locally a quotient of two holomorphic functions. For example, f = z 1 / z 2 is a meromorphic function on the two-dimensional complex affine space. Here it is no longer true that every meromorphic function can be regarded as holomorphic function with values in the Riemann sphere: There is a set of "indeterminacy" of codimension two. Unlike in dimension one, in higher dimensions there do exist complex manifolds on which there are no non-constant meromorphic functions, for example, most complex tori. All rational functions, for example f = z 3 − 2 z + 10 z 5 + 3 z − 1, are meromorphic on the whole complex plane; the functions f = e z z and f = sin z 2 as well as the gamma function and the Riemann zeta function are meromorphic on the whole complex plane. The function f = e 1 z is defined in the whole complex plane except for the origin, 0. However, 0 is not a pole of this function, rather an essential singularity. Thus, this function is not meromorphic in the whole complex plane.
However, it is meromorphic on C ∖. The complex logarithm function f = ln is not meromorphic on the whole complex plane, as it cannot be defined on the whole complex plane while only excluding a set of isolated points; the function f = csc 1 z = 1 sin is not meromorphic in the whole plane, since the point z = 0 is an accumulation point of poles and is thus not an isolated singularity. The function f = sin 1
In complex analysis, a discipline within mathematics, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves. It generalizes Cauchy's integral formula. From a geometrical perspective, it is a special case of the generalized Stokes' theorem; the statement is as follows: Let U be a connected open subset of the complex plane containing a finite list of points a1... an, f a function defined and holomorphic on U \. Let γ be a closed rectifiable curve in U which does not meet any of the ak, denote the winding number of γ around ak by I; the line integral of f around γ is equal to 2πi times the sum of residues of f at the points, each counted as many times as γ winds around the point: ∮ γ f d z = 2 π i ∑ k = 1 n I Res . If γ is a positively oriented simple closed curve, I = 1 if ak is in the interior of γ, 0 if not, so ∮ γ f d z = 2 π i ∑ Res with the sum over those ak inside γ; the relationship of the residue theorem to Stokes' theorem is given by the Jordan curve theorem.
The general plane curve γ must first be reduced to a set of simple closed curves whose total is equivalent to γ for integration purposes. The requirement that f be holomorphic on U0 = U \ is equivalent to the statement that the exterior derivative d = 0 on U0, thus if two planar regions V and W of U enclose the same subset of, the regions V \ W and W \ V lie in U0, hence ∫ V ∖ W d − ∫ W ∖ V d is well-defined and equal to zero. The contour integral of f dz along γj = ∂V is equal to the sum of a set of integrals along paths λj, each enclosing an arbitrarily small region around a single aj — the residues of f at. Summing over, we recover the final expression of the contour integral in terms of the winding numbers. In order to evaluate real integrals, the residue theorem is used in the following manner: the integrand is extended to the complex plane and its residues are computed, a part of the real axis is extended to a closed curve by attaching a half-circle in the upper or lower half-plane, forming a semicircle.
The integral over this curve can be computed using the residue theorem. The half-circle part of the integral will tend towards zero as the radius of the half-circle grows, leaving only the real-axis part of the integral, the one we were interested in; the integral ∫ − ∞ ∞ e i t x x 2 + 1 d x arises in probability theory when calculating the characteristic function of the Cauchy distribution. It resists the techniques of elementary calculus but can be evaluated by expressing it as a limit of contour integrals. Suppose t > 0 and define the contour C that goes along the real line from −a to a and counterclockwise along a semicircle centered at 0 from a to −a. Take a to be greater than 1, so that the imaginary unit i is enclosed within the curve. Now consider the contour integral ∫ C f d z = ∫ C e i t z z 2 + 1 d z. Since eitz is an entire function, this function has singularities only where the denominator z2 + 1 is zero. Since z2 + 1 =, that happens only where z = i or z = −i. Only one of those points is in the region bounded by this contour.
Because f is e i t z z 2 + 1 = e i t z 2 i ( 1 z − i − 1 z +
Niels Henrik Abel
Niels Henrik Abel was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solving the general quintic equation in radicals; this question was one of the outstanding open problems of his day, had been unresolved for over 350 years. He was an innovator in the field of elliptic functions, discoverer of Abelian functions, he made his discoveries while died at the age of 26 from tuberculosis. Most of his work was done in seven years of his working life. Regarding Abel, the French mathematician Charles Hermite said: "Abel has left mathematicians enough to keep them busy for five hundred years." Another French mathematician, Adrien-Marie Legendre, said: "quelle tête celle du jeune Norvégien!". The Abel Prize in mathematics proposed in 1899 to complement the Nobel Prizes, is named in his honour. Niels Henrik Abel was born in Nedstrand, Norway, as the second child of the pastor Søren Georg Abel and Anne Marie Simonsen.
When Niels Henrik Abel was born, the family was living at a rectory on Finnøy. Much suggests that Niels Henrik was born in the neighboring parish, as his parents were guests of the bailiff in Nedstrand in July / August of his year of birth. Niels Henrik Abel's father, Søren Georg Abel, had a degree in theology and philosophy and served as pastor at Finnøy. Søren's father, Niels's grandfather, Hans Mathias Abel, was a pastor, at Gjerstad Church near the town of Risør. Søren had spent his childhood at Gjerstad, had served as chaplain there; the Abel family came to Norway in the 17th century. Anne Marie Simonsen was from Risør. Anne Marie had grown up with two stepmothers, in luxurious surroundings. At Gjerstad rectory, she enjoyed arranging social gatherings. Much suggests she was early on an alcoholic and took little interest in the upbringing of the children. Niels Henrik and his brothers were given their schooling by their father, with handwritten books to read. An addition table in a book of mathematics reads: 1+0=0.
With Norwegian independence and the first election held in Norway, in 1814, Søren Abel was elected as a representative to the Storting. Meetings of the Storting were held until 1866 in the main hall of the Cathedral School in Christiania; this is how he came into contact with the school, he decided that his eldest son, Hans Mathias, should start there the following year. However, when the time for his departure approached, Hans was so saddened and depressed over having to leave home that his father did not dare send him away, he decided to send Niels instead. In 1815, Niels Abel entered the Cathedral School at the age of 13, his elder brother Hans joined him there a year later. They had classes together. Hans got better grades than Niels, he gave the students mathematical tasks to do at home. He saw Niels Henrik's talent in mathematics, encouraged him to study the subject to an advanced level, he gave Niels private lessons after school. In 1818, Søren Abel had a public theological argument with the theologian Stener Johannes Stenersen regarding his catechism from 1806.
The argument was well covered in the press. Søren was given the nickname "Abel Treating". Niels' reaction to the quarrel was said to have been "excessive gaiety". At the same time, Søren almost faced impeachment after insulting Carsten Anker, the host of the Norwegian Constituent Assembly, he began drinking and died only two years in 1820, aged 48. Bernt Michael Holmboe supported Niels Henrik Abel with a scholarship to remain at the school and raised money from his friends to enable him to study at the Royal Frederick University; when Abel entered the university in 1821, he was the most knowledgeable mathematician in Norway. Holmboe had nothing more he could teach him and Abel had studied all the latest mathematical literature in the university library. During that time, Abel started working on the quintic equation in radicals. Mathematicians had been looking for a solution to this problem for over 250 years. In 1821, Abel thought; the two professors of mathematics in Christiania, Søren Rasmussen and Christopher Hansteen, found no errors in Abel's formulas, sent the work on to the leading mathematician in the Nordic countries, Carl Ferdinand Degen in Copenhagen.
He too found no faults but still doubted that the solution, which so many outstanding mathematicians had sought for so long, could have been found by an unknown student in far-off Christiania. Degen noted, Abel's unusually sharp mind, believed that such a talented young man should not waste his abilities on such a "sterile object" as the fifth degree equation, but rather on elliptic functions and transcendence. Degen asked Abel to give a numerical example of his method. While trying to provide an example, Abel found a mistake in his paper; this led to a discovery in 1823 that a solution to a fifth- or higher-degree equation was impossible. Abel graduated in 1822, his performance was exceptionally high in average in other matters. A
Leonhard Euler was a Swiss mathematician, astronomer and engineer, who made important and influential discoveries in many branches of mathematics, such as infinitesimal calculus and graph theory, while making pioneering contributions to several branches such as topology and analytic number theory. He introduced much of the modern mathematical terminology and notation for mathematical analysis, such as the notion of a mathematical function, he is known for his work in mechanics, fluid dynamics, optics and music theory. Euler was one of the most eminent mathematicians of the 18th century and is held to be one of the greatest in history, he is widely considered to be the most prolific mathematician of all time. His collected works fill more than anybody in the field, he spent most of his adult life in Saint Petersburg, in Berlin the capital of Prussia. A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is the master of us all." Leonhard Euler was born on 15 April 1707, in Basel, Switzerland to Paul III Euler, a pastor of the Reformed Church, Marguerite née Brucker, a pastor's daughter.
He had two younger sisters: Anna Maria and Maria Magdalena, a younger brother Johann Heinrich. Soon after the birth of Leonhard, the Eulers moved from Basel to the town of Riehen, where Euler spent most of his childhood. Paul Euler was a friend of the Bernoulli family. Euler's formal education started in Basel. In 1720, aged thirteen, he enrolled at the University of Basel, in 1723, he received a Master of Philosophy with a dissertation that compared the philosophies of Descartes and Newton. During that time, he was receiving Saturday afternoon lessons from Johann Bernoulli, who discovered his new pupil's incredible talent for mathematics. At that time Euler's main studies included theology and Hebrew at his father's urging in order to become a pastor, but Bernoulli convinced his father that Leonhard was destined to become a great mathematician. In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono. At that time, he was unsuccessfully attempting to obtain a position at the University of Basel.
In 1727, he first entered the Paris Academy Prize Problem competition. Pierre Bouguer, who became known as "the father of naval architecture", won and Euler took second place. Euler won this annual prize twelve times. Around this time Johann Bernoulli's two sons and Nicolaus, were working at the Imperial Russian Academy of Sciences in Saint Petersburg. On 31 July 1726, Nicolaus died of appendicitis after spending less than a year in Russia, when Daniel assumed his brother's position in the mathematics/physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler. In November 1726 Euler eagerly accepted the offer, but delayed making the trip to Saint Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel. Euler arrived in Saint Petersburg on 17 May 1727, he was promoted from his junior post in the medical department of the academy to a position in the mathematics department. He lodged with Daniel Bernoulli with whom he worked in close collaboration.
Euler settled into life in Saint Petersburg. He took on an additional job as a medic in the Russian Navy; the Academy at Saint Petersburg, established by Peter the Great, was intended to improve education in Russia and to close the scientific gap with Western Europe. As a result, it was made attractive to foreign scholars like Euler; the academy possessed ample financial resources and a comprehensive library drawn from the private libraries of Peter himself and of the nobility. Few students were enrolled in the academy in order to lessen the faculty's teaching burden, the academy emphasized research and offered to its faculty both the time and the freedom to pursue scientific questions; the Academy's benefactress, Catherine I, who had continued the progressive policies of her late husband, died on the day of Euler's arrival. The Russian nobility gained power upon the ascension of the twelve-year-old Peter II; the nobility was suspicious of the academy's foreign scientists, thus cut funding and caused other difficulties for Euler and his colleagues.
Conditions improved after the death of Peter II, Euler swiftly rose through the ranks in the academy and was made a professor of physics in 1731. Two years Daniel Bernoulli, fed up with the censorship and hostility he faced at Saint Petersburg, left for Basel. Euler succeeded him as the head of the mathematics department. On 7 January 1734, he married Katharina Gsell, a daughter of Georg Gsell, a painter from the Academy Gymnasium; the young couple bought a house by the Neva River. Of their thirteen children, only five survived childhood. Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia, he lived for 25 years in Berlin. In Berlin, he published the two works for which he would become most renowned: the Introductio in analysin infinitorum, a text on functions published in 1748, the Institutiones calculi differentialis, published in 1755 on differential calculus.
Adrien-Marie Legendre was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him. Adrien-Marie Legendre was born in Paris on 18 September 1752 to a wealthy family, he received his education at the Collège Mazarin in Paris, defended his thesis in physics and mathematics in 1770. He taught at the École Militaire in Paris from 1775 to 1780 and at the École Normale from 1795. At the same time, he was associated with the Bureau des Longitudes. In 1782, the Berlin Academy awarded Legendre a prize for his treatise on projectiles in resistant media; this treatise brought him to the attention of Lagrange. The Académie des sciences made Legendre an adjoint member in 1783 and an associate in 1785. In 1789, he was elected a Fellow of the Royal Society, he assisted with the Anglo-French Survey to calculate the precise distance between the Paris Observatory and the Royal Greenwich Observatory by means of trigonometry.
To this end in 1787 he visited Dover and London together with Dominique, comte de Cassini and Pierre Méchain. The three visited William Herschel, the discoverer of the planet Uranus. Legendre lost his private fortune in 1793 during the French Revolution; that year, he married Marguerite-Claudine Couhin, who helped him put his affairs in order. In 1795, Legendre became one of six members of the mathematics section of the reconstituted Académie des Sciences, renamed the Institut National des Sciences et des Arts. In 1803, Napoleon reorganized the Institut National, Legendre became a member of the Geometry section. From 1799 to 1812, Legendre served as mathematics examiner for graduating artillery students at the École Militaire and from 1799 to 1815 he served as permanent mathematics examiner for the École Polytechnique. In 1824, Legendre's pension from the École Militaire was stopped because he refused to vote for the government candidate at the Institut National, his pension was reinstated with the change in government in 1828.
In 1831, he was made an officer of the Légion d'Honneur. Legendre died in Paris on 10 January 1833, after a long and painful illness, Legendre's widow preserved his belongings to memorialize him. Upon her death in 1856, she was buried next to her husband in the village of Auteuil, where the couple had lived, left their last country house to the village. Legendre's name is one of the 72 names inscribed on the Eiffel Tower. Abel's work on elliptic functions was built on Legendre's and some of Gauss' work in statistics and number theory completed that of Legendre, he developed the least squares method and firstly communicated it to his contemporaries before Gauss, which has broad application in linear regression, signal processing and curve fitting. Today, the term "least squares method" is used as a direct translation from the French "méthode des moindres carrés", his major work is Exercices de Calcul Intégral, published in three volumes in 1811, 1817 and 1819. In the first volume he introduced the basic properties of elliptic integrals, beta functions and gamma functions, introducing the symbol Γ normalizing it to Γ = n!.
Further results on the beta and gamma functions along with their applications to mechanics - such as the rotation of the earth, the attraction of ellipsoids, appeared in the second volume. In 1830, he gave a proof of Fermat's last theorem for exponent n = 5, proven by Lejeune Dirichlet in 1828. In number theory, he conjectured the quadratic reciprocity law, subsequently proved by Gauss, he did pioneering work on the distribution of primes, on the application of analysis to number theory. His 1798 conjecture of the prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1896. Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely, he is known for the Legendre transformation, used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics. In thermodynamics it is used to obtain the enthalpy and the Helmholtz and Gibbs energies from the internal energy.
He is the namesake of the Legendre polynomials, solutions to Legendre's differential equation, which occur in physics and engineering applications, e.g. electrostatics. Legendre is best known as the author of Éléments de géométrie, published in 1794 and was the leading elementary text on the topic for around 100 years; this text rearranged and simplified many of the propositions from Euclid's Elements to create a more effective textbook. Foreign Honorary Member of the American Academy of Arts and Sciences The Moon crater Legendre is named after him. Main-belt asteroid. Legendre is one of the 72 prominent French scientists who were commemorated on plaques at the first stage of the Eiffel Tower when it first opened. Essays1782 Recherches sur la trajectoire des projectiles dans les milieux résistants BooksEléments de géométrie, textbook 1794 Essai sur la Théorie des Nombres 1797-8, 2nd ed. 1808, 3rd ed. in 2 vol. 1830 Nouvelles Méthodes pour la Détermination des Orbites des Comètes, 1805 Exercices de Calcul Intégral, book in three volumes 1811, 1817, 1819 Traité des Fonctions Elliptiques, book in three volumes 1825, 1826, 1830Memoires in Histoire de l'Académie Royale des Scien