In probability theory, the sample space of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is denoted using set notation, the possible ordered outcomes are listed as elements in the set, it is common to refer to a sample space by the labels S, Ω, or U. For example, if the experiment is tossing a coin, the sample space is the set. For tossing two coins, the corresponding sample space would be written. If the sample space is unordered, it becomes. For tossing a single six-sided die, the typical sample space is. A well-defined sample space is one of three basic elements in a probabilistic model. A set Ω with outcomes s 1, s 2, …, s n must meet some conditions in order to be a sample space: The outcomes must be mutually exclusive, i.e. if s j takes place no other s i will take place, ∀ i, j = 1, 2, …, n i ≠ j. The outcomes must be collectively exhaustive, i.e. on every experiment there will always take place some outcome s i ∈ Ω for i ∈.
The sample space must have the right granularity depending on. We must remove irrelevant information from the sample space. In other words, we must choose the right abstraction. For instance, in the trial of tossing a coin, we could have as a sample space Ω 1 =, where H stands for heads and T for tails. Another possible sample space could be Ω 2 =. Here, R stands for N R not rains. Ω 1 is a better choice than Ω 2 as we do not care about how the weather affects the tossing of a coin. For many experiments, there may be more than one plausible sample space available, depending on what result is of interest to the experimenter. For example, when drawing a card from a standard deck of fifty-two playing cards, one possibility for the sample space could be the various ranks, while another could be the suits. A more complete description of outcomes, could specify both the denomination and the suit, a sample space describing each individual card can be constructed as the Cartesian product of the two sample spaces noted above.
Still other sample spaces are possible, such as. Some treatments of probability assume that the various outcomes of an experiment are always defined so as to be likely. However, there are experiments that are not described by a sample space of likely outcomes—for example, if one were to toss a thumb tack many times and observe whether it landed with its point upward or downward, there is no symmetry to suggest that the two outcomes should be likely. Though most random phenomena do not have likely outcomes, it can be helpful to define a sample space in such a way that outcomes are at least equally since this condition simplifies the computation of probabilities for events within the sample space. If each individual outcome occurs with the same probability the probability of any event becomes simply: P = number of outcomes in event number of outcomes in sample space For example, if two dice are thrown to generate two uniformly distributed integers, D1 and D2, each in the range, the 36 ordered pairs constitute a sample space of likely events.
In this case, the above formula applies, such that the probability of a certain sum, say D1 + D2 = 5 is shown to be 4/36, since 4 of the 36 outcomes produc
Three-dimensional space is a geometric setting in which three values are required to determine the position of an element. This is the informal meaning of the term dimension. In physics and mathematics, a sequence of n numbers can be understood as a location in n-dimensional space; when n = 3, the set of all such locations is called three-dimensional Euclidean space. It is represented by the symbol ℝ3; this serves as a three-parameter model of the physical universe. However, this space is only one example of a large variety of spaces in three dimensions called 3-manifolds. In this classical example, when the three values refer to measurements in different directions, any three directions can be chosen, provided that vectors in these directions do not all lie in the same 2-space. Furthermore, in this case, these three values can be labeled by any combination of three chosen from the terms width, height and length. In mathematics, analytic geometry describes every point in three-dimensional space by means of three coordinates.
Three coordinate axes are given, each perpendicular to the other two at the origin, the point at which they cross. They are labeled x, y, z. Relative to these axes, the position of any point in three-dimensional space is given by an ordered triple of real numbers, each number giving the distance of that point from the origin measured along the given axis, equal to the distance of that point from the plane determined by the other two axes. Other popular methods of describing the location of a point in three-dimensional space include cylindrical coordinates and spherical coordinates, though there are an infinite number of possible methods. See Euclidean space. Below are images of the above-mentioned systems. Two distinct points always determine a line. Three distinct points determine a unique plane. Four distinct points can either coplanar or determine the entire space. Two distinct lines can either be parallel or be skew. Two parallel lines, or two intersecting lines, lie in a unique plane, so skew lines are lines that do not meet and do not lie in a common plane.
Two distinct planes are parallel. Three distinct planes, no pair of which are parallel, can either meet in a common line, meet in a unique common point or have no point in common. In the last case, the three lines of intersection of each pair of planes are mutually parallel. A line can intersect that plane in a unique point or be parallel to the plane. In the last case, there will be lines in the plane. A hyperplane is a subspace of one dimension less than the dimension of the full space; the hyperplanes of a three-dimensional space are the two-dimensional subspaces. In terms of cartesian coordinates, the points of a hyperplane satisfy a single linear equation, so planes in this 3-space are described by linear equations. A line can be described by a pair of independent linear equations, each representing a plane having this line as a common intersection. Varignon's theorem states that the midpoints of any quadrilateral in ℝ3 form a parallelogram, so, are coplanar. A sphere in 3-space consists of the set of all points in 3-space at a fixed distance r from a central point P.
The solid enclosed by the sphere is called a ball. The volume of the ball is given by V = 4 3 π r 3. Another type of sphere arises from a 4-ball, whose three-dimensional surface is the 3-sphere: points equidistant to the origin of the euclidean space ℝ4. If a point has coordinates, P x2 + y2 + z2 + w2 = 1 characterizes those points on the unit 3-sphere centered at the origin. In three dimensions, there are nine regular polytopes: the five convex Platonic solids and the four nonconvex Kepler-Poinsot polyhedra. A surface generated by revolving a plane curve about a fixed line in its plane as an axis is called a surface of revolution; the plane curve is called the generatrix of the surface. A section of the surface, made by intersecting the surface with a plane, perpendicular to the axis, is a circle. Simple examples occur. If the generatrix line intersects the axis line, the surface of revolution is a right circular cone with vertex the point of intersection. However, if the generatrix and axis are parallel, the surface of revolution is a circular cylinder.
In analogy with the conic sections, the set of points whose cartesian coordinates satisfy the general equation of the second degree, namely, A x 2 + B y 2 + C z 2 + F x y + G y z + H x z + J x + K y + L z + M = 0, where A, B, C, F, G, H, J, K, L and M are real numbers and not all of A, B, C, F, G and H are zero is called a quadric surface. There are six types of non-degenerate quadric surfaces: Ellipsoid Hyperboloid of one sheet Hyperboloid of two sheets Elliptic cone Elliptic paraboloid Hyperbolic paraboloidThe degenerate quadric surfaces are the empty set, a single point, a single li
A plot is a graphical technique for representing a data set as a graph showing the relationship between two or more variables. The plot can be drawn by a mechanical or electronic plotter. Graphs are a visual representation of the relationship between variables useful for humans who can derive an understanding which would not come from lists of values. Graphs can be used to read off the value of an unknown variable plotted as a function of a known one. Graphs of functions are used in mathematics, engineering, technology and other areas. Plots play an important role in data analysis; the procedures here can broadly be split into two parts: graphical. Quantitative techniques are the set of statistical procedures that yield tabular output. Examples of quantitative techniques include: hypothesis testing analysis of variance point estimates and confidence intervals least squares regressionThese and similar techniques are all valuable and are mainstream in terms of classical analysis. There are many statistical tools referred to as graphical techniques.
These include: scatter plots spectrum plots histograms probability plots residual plots box plots, block plotsGraphical procedures such as plots are a short path to gaining insight into a data set in terms of testing assumptions, model selection, model validation, estimator selection, relationship identification, factor effect determination, outlier detection. Statistical graphics give insight into aspects of the underlying structure of the data. Graphs can be used to solve some mathematical equations by finding where two plots intersect. Biplot: These are a type of graph used in statistics. A biplot allows information on both samples and variables of a data matrix to be displayed graphically. Samples are displayed as points while variables are displayed either as vectors, linear axes or nonlinear trajectories. In the case of categorical variables, category level points may be used to represent the levels of a categorical variable. A generalised biplot displays information on both categorical variables.
Bland-Altman plot: In analytical chemistry and biostatistics this plot is a method of data plotting used in analysing the agreement between two different assays. It is identical to a Tukey mean-difference plot, what it is still known as in other fields, but was popularised in medical statistics by Bland and Altman. Bode plots are used in control theory. Box plot: In descriptive statistics, a boxplot known as a box-and-whisker diagram or plot, is a convenient way of graphically depicting groups of numerical data through their five-number summaries. A boxplot may indicate which observations, if any, might be considered outliers. Carpet plot: A two-dimensional plot that illustrates the interaction between two and three independent variables and one to three dependent variables. Comet plot: A two- or three-dimensional animated plot in which the data points are traced on the screen. Contour plot: A two-dimensional plot which shows the one-dimensional curves, called contour lines on which the plotted quantity q is a constant.
Optionally, the plotted values can be color-coded. Dalitz plot: This a scatterplot used in particle physics to represent the relative frequency of various manners in which the products of certain three-body decays may move apart Funnel plot: This is a useful graph designed to check the existence of publication bias in meta-analyses. Funnel plots, introduced by Light and Pillemer in 1994 and discussed in detail by Egger and colleagues, are useful adjuncts to meta-analyses. A funnel plot is a scatterplot of treatment effect against a measure of study size, it is used as a visual aid to detecting bias or systematic heterogeneity. Dot plot: A dot chart or dot plot is a statistical chart consisting of group of data points plotted on a simple scale. Dot plots are used for continuous, univariate data. Data points may be labelled. Dot plots are one of the simplest plots available, are suitable for small to moderate sized data sets, they are useful for highlighting gaps, as well as outliers. Forest plot: is a graphical display that shows the strength of the evidence in quantitative scientific studies.
It was developed for use in medical research as a means of graphically representing a meta-analysis of the results of randomized controlled trials. In the last twenty years, similar meta-analytical techniques have been applied in observational studies and forest plots are used in presenting the results of such studies also. Galbraith plot: In statistics, a Galbraith plot, is one way of displaying several estimates of the same quantity that have different standard errors, it can be used to examine heterogeneity in a meta-analysis, as an alternative or supplement to a forest plot. Heat map Nichols plot: This is a graph used in signal processing in which the logarithm of the magnitude is plotted against the phase of a frequency response on orthogonal axes. Normal probability plot: The normal probability plot is a graphical technique for assessing whether or not a data set is normally distributed; the data are plotted against a theoretical normal distribution in such a way that the points should form an approximate straight line.
Departures from this straight line indicate departures from normality. The normal probability plot is a special case of the probability plot. Nyquist plot: Plot is used in automatic control and
Statistics is a branch of mathematics dealing with data collection, analysis and presentation. In applying statistics to, for example, a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of surveys and experiments. See glossary of probability and statistics; when census data cannot be collected, statisticians collect data by developing specific experiment designs and survey samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample to the population as a whole. An experimental study involves taking measurements of the system under study, manipulating the system, taking additional measurements using the same procedure to determine if the manipulation has modified the values of the measurements.
In contrast, an observational study does not involve experimental manipulation. Two main statistical methods are used in data analysis: descriptive statistics, which summarize data from a sample using indexes such as the mean or standard deviation, inferential statistics, which draw conclusions from data that are subject to random variation. Descriptive statistics are most concerned with two sets of properties of a distribution: central tendency seeks to characterize the distribution's central or typical value, while dispersion characterizes the extent to which members of the distribution depart from its center and each other. Inferences on mathematical statistics are made under the framework of probability theory, which deals with the analysis of random phenomena. A standard statistical procedure involves the test of the relationship between two statistical data sets, or a data set and synthetic data drawn from an idealized model. A hypothesis is proposed for the statistical relationship between the two data sets, this is compared as an alternative to an idealized null hypothesis of no relationship between two data sets.
Rejecting or disproving the null hypothesis is done using statistical tests that quantify the sense in which the null can be proven false, given the data that are used in the test. Working from a null hypothesis, two basic forms of error are recognized: Type I errors and Type II errors. Multiple problems have come to be associated with this framework: ranging from obtaining a sufficient sample size to specifying an adequate null hypothesis. Measurement processes that generate statistical data are subject to error. Many of these errors are classified as random or systematic, but other types of errors can be important; the presence of missing data or censoring may result in biased estimates and specific techniques have been developed to address these problems. Statistics can be said to have begun in ancient civilization, going back at least to the 5th century BC, but it was not until the 18th century that it started to draw more from calculus and probability theory. In more recent years statistics has relied more on statistical software to produce tests such as descriptive analysis.
Some definitions are: Merriam-Webster dictionary defines statistics as "a branch of mathematics dealing with the collection, analysis and presentation of masses of numerical data." Statistician Arthur Lyon Bowley defines statistics as "Numerical statements of facts in any department of inquiry placed in relation to each other."Statistics is a mathematical body of science that pertains to the collection, interpretation or explanation, presentation of data, or as a branch of mathematics. Some consider statistics to be a distinct mathematical science rather than a branch of mathematics. While many scientific investigations make use of data, statistics is concerned with the use of data in the context of uncertainty and decision making in the face of uncertainty. Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, measure-theoretic probability theory.
In applying statistics to a problem, it is common practice to start with a population or process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Ideally, statisticians compile data about the entire population; this may be organized by governmental statistical institutes. Descriptive statistics can be used to summarize the population data. Numerical descriptors include mean and standard deviation for continuous data types, while frequency and percentage are more useful in terms of describing categorical data; when a census is not feasible, a chosen subset of the population called. Once a sample, representative of the population is determined, data is collected for the sample members in an observational or experimental setting. Again, descriptive statistics can be used to summarize the sample data. However, the drawing of the sample has been subject to an element of randomness, hence the established numerical descriptors from the sample are due to uncertainty.
To still draw meaningful conclusions about the entire population, in
Model selection is the task of selecting a statistical model from a set of candidate models, given data. In the simplest cases, a pre-existing set of data is considered. However, the task can involve the design of experiments such that the data collected is well-suited to the problem of model selection. Given candidate models of similar predictive or explanatory power, the simplest model is most to be the best choice. Konishi & Kitagawa state, "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling". Relatedly, Cox has said, "How translation from subject-matter problem to statistical model is done is the most critical part of an analysis". Model selection may refer to the problem of selecting a few representative models from a large set of computational models for the purpose of decision making or optimization under uncertainty. In its most basic forms, model selection is one of the fundamental tasks of scientific inquiry. Determining the principle that explains a series of observations is linked directly to a mathematical model predicting those observations.
For example, when Galileo performed his inclined plane experiments, he demonstrated that the motion of the balls fitted the parabola predicted by his model. Of the countless number of possible mechanisms and processes that could have produced the data, how can one begin to choose the best model? The mathematical approach taken decides among a set of candidate models. Simple models such as polynomials are used, at least initially. Burnham & Anderson emphasize throughout their book the importance of choosing models based on sound scientific principles, such as understanding of the phenomenological processes or mechanisms underlying the data. Once the set of candidate models has been chosen, the statistical analysis allows us to select the best of these models. What is meant by best is controversial. A good model selection technique will balance goodness of fit with simplicity. More complex models will be better able to adapt their shape to fit the data, but the additional parameters may not represent anything useful.
Goodness of fit is determined using a likelihood ratio approach, or an approximation of this, leading to a chi-squared test. The complexity is measured by counting the number of parameters in the model. Model selection techniques can be considered as estimators of some physical quantity, such as the probability of the model producing the given data; the bias and variance are both important measures of the quality of this estimator. A standard example of model selection is that of curve fitting, given a set of points and other background knowledge, we must select a curve that describes the function that generated the points. Data transformation Exploratory data analysis Model specification Scientific method Below is a list of criteria for model selection; the most used criteria are the Akaike information criterion and the Bayes factor and/or the Bayesian information criterion. Akaike information criterion, a measure of the goodness fit of an estimated statistical model Bayes factor Bayesian information criterion known as the Schwarz information criterion, a statistical criterion for model selection Cross-validation Deviance information criterion, another Bayesian oriented model selection criterion False discovery rate Focused information criterion, a selection criterion sorting statistical models by their effectiveness for a given focus parameter Hannan–Quinn information criterion, an alternative to the Akaike and Bayesian criteria Likelihood-ratio test Mallows's Cp Minimum description length Minimum message length Structural risk minimization Stepwise regression Watanabe–Akaike information criterion called the applicable information criterionBurnham & Anderson say the following.
There is a variety of model selection methods. However, from the point of view of statistical performance of a method, intended context of its use, there are only two distinct classes of methods: These have been labeled efficient and consistent..... Under the frequentist paradigm for model selection one has three main approaches: optimization of some selection criteria, tests of hypotheses, ad hoc methods. Aho, K.. R. Model Based Inference in the Life Sciences, Springer Ando, T. Bayesian Model Selection and Statistical Modeling, CRC Press Breiman, L. "Statistical modeling: the two cultures", Statistical Science, 16: 199–231, doi:10.1214/ss/1009213726 Burnham, K. P.. R. Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, Springer-Verlag, ISBN 0-387-95364-7 Chamberlin, T. C. "The method of multiple working hypotheses", Science, 15: 93, Bibcode:1890Sci....15R..92. Doi
Julius Plücker was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led to the discovery of the electron, he vastly extended the study of Lamé curves. Plücker was born at Elberfeld. After being educated at Düsseldorf and at the universities of Bonn and Berlin he went to Paris in 1823, where he came under the influence of the great school of French geometers, whose founder, Gaspard Monge, had only died. In 1825 he returned to Bonn, in 1828 was made professor of mathematics. In the same year he published the first volume of his Analytisch-geometrische Entwicklungen, which introduced the method of abridged notation. In 1831 he published the second volume, in which he established on a firm and independent basis projective duality. In 1836, Plücker was made professor of physics at University of Bonn. In 1858, after a year of working with vacuum tubes of his Bonn colleague Heinrich Geißler, he published his first classical researches on the action of the magnet on the electric discharge in rarefied gases.
He found that the discharge caused a fluorescent glow to form on the glass walls of the vacuum tube, that the glow could be made to shift by applying an electromagnet to the tube, thus creating a magnetic field. It was shown that the glow was produced by cathode rays. Plücker, first by himself and afterwards in conjunction with Johann Hittorf, made many important discoveries in the spectroscopy of gases, he was the first to use the vacuum tube with the capillary part now called a Geissler tube, by means of which the luminous intensity of feeble electric discharges was raised sufficiently to allow of spectroscopic investigation. He anticipated Robert Wilhelm Bunsen and Gustav Kirchhoff in announcing that the lines of the spectrum were characteristic of the chemical substance which emitted them, in indicating the value of this discovery in chemical analysis. According to Hittorf, he was the first who saw the three lines of the hydrogen spectrum, which a few months after his death, were recognized in the spectrum of the solar protuberances.
In 1865, Plücker returned to the field of geometry and invented what was known as line geometry in the nineteenth century. In projective geometry, Plücker coordinates refer to a set of homogeneous co-ordinates introduced to embed the set of lines in three dimensions as a quadric in five dimensions; the construction uses 2×2 minor determinants, or equivalently the second exterior power of the underlying vector space of dimension 4. It is now part of the theory of Grassmannians. 1828: Analytisch-Geometrische Entwicklungen from Internet Archive 1835: Syteme der analytische Geometrie auf neue Betrachtungsweise gegrundet 1839: Theory of Algebraic Curves 1852: System der Geometrie des Raumes neue analytische Behandlungsweise 1865: On a New Geometry of Space Philosophical Transactions of the Royal Society 14: 53–8 Plücker was the recipient of the Copley Medal from the Royal Society in 1866. Plücker's conoid Plücker coordinates Plücker embedding Plücker formula Plücker surface Plücker matrix Timeline of low-temperature technology Born, Die Stadt Elberfeld.
Festschrift zur Dreihundert-Feier 1910. J. H. Born, Elberfeld 1910 Giermann, Stammfolge der Familie Plücker, in: Deutsches Geschlechterbuch, 217. Bd, A. Starke Verlag, Limburg a.d. L. 2004 Strutz, Die Ahnentafeln der Elberfelder Bürgermeister und Stadtrichter 1708-1808. 2. Auflage, Verlag Degener & Co. Neustadt an der Aisch 1963 ISBN 3-7686-4069-8 Gustav Karsten, "Plücker, Julius", Allgemeine Deutsche Biographie, 26, Leipzig: Duncker & Humblot, pp. 321–323 Julius Plücker at the Mathematics Genealogy Project The Cathode Ray Tube site Weisstein, Eric Wolfgang. "Plücker, Julius". ScienceWorld. O'Connor, John J.. Julius Plücker in the German National Library catalogue Julius Plücker in der philosophischen Fakultät der Universität Halle Julius Plücker und die Stammfolge der Familie Plücker, Deutsches Geschlechterbuch, 217. Bd. A. Starke Verlag, Limburg a.d. L. 2004 uni-bonn.de „Ein streitbarer Gelehrter im 19. Jahrhundert. Der Mathematiker Julius Plücker starb vor 140 Jahren.“ Pressemitteilung der Universität Bonn vom 21.
Mai 2008 "Discussion of the general form for light waves"
Geometry is a branch of mathematics concerned with questions of shape, relative position of figures, the properties of space. A mathematician who works in the field of geometry is called a geometer. Geometry arose independently in a number of early cultures as a practical way for dealing with lengths and volumes. Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid, whose treatment, Euclid's Elements, set a standard for many centuries to follow. Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC. Islamic scientists expanded on them during the Middle Ages. By the early 17th century, geometry had been put on a solid analytic footing by mathematicians such as René Descartes and Pierre de Fermat. Since and into modern times, geometry has expanded into non-Euclidean geometry and manifolds, describing spaces that lie beyond the normal range of human experience.
While geometry has evolved throughout the years, there are some general concepts that are more or less fundamental to geometry. These include the concepts of points, planes, surfaces and curves, as well as the more advanced notions of manifolds and topology or metric. Geometry has applications to many fields, including art, physics, as well as to other branches of mathematics. Contemporary geometry has many subfields: Euclidean geometry is geometry in its classical sense; the mandatory educational curriculum of the majority of nations includes the study of points, planes, triangles, similarity, solid figures and analytic geometry. Euclidean geometry has applications in computer science and various branches of modern mathematics. Differential geometry uses techniques of linear algebra to study problems in geometry, it has applications in physics, including in general relativity. Topology is the field concerned with the properties of geometric objects that are unchanged by continuous mappings. In practice, this means dealing with large-scale properties of spaces, such as connectedness and compactness.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues using techniques of real analysis. It has close connections to convex analysis and functional analysis and important applications in number theory. Algebraic geometry studies geometry through the use of multivariate polynomials and other algebraic techniques, it has applications including cryptography and string theory. Discrete geometry is concerned with questions of relative position of simple geometric objects, such as points and circles, it shares many principles with combinatorics. Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc; the earliest recorded beginnings of geometry can be traced to ancient Mesopotamia and Egypt in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles and volumes, which were developed to meet some practical need in surveying, construction and various crafts.
The earliest known texts on geometry are the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets such as Plimpton 322. For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum. Clay tablets demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space; these geometric procedures anticipated the Oxford Calculators, including the mean speed theorem, by 14 centuries. South of Egypt the ancient Nubians established a system of geometry including early versions of sun clocks. In the 7th century BC, the Greek mathematician Thales of Miletus used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore, he is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. Pythagoras established the Pythagorean School, credited with the first proof of the Pythagorean theorem, though the statement of the theorem has a long history.
Eudoxus developed the method of exhaustion, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose Elements considered the most successful and influential textbook of all time, introduced mathematical rigor through the axiomatic method and is the earliest example of the format still used in mathematics today, that of definition, axiom and proof. Although most of the contents of the Elements were known, Euclid arranged them into a single, coherent logical framework; the Elements was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes of Syracuse used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, gave remarkably accurate approximations of Pi.
He studied the sp