query: part_of measure 
supports: Euclid ("Eukleids" ; fl. 300 BCE), sometimes called Euclid of Alexandria to distinguish him from Euclides of Megara, was a Greek mathematician, often referred to as the "father of geometry". He was active in Alexandria during the reign of Ptolemy I (323283 BCE). His "Elements" is one of the most influential works in the history of mathematics, serving as the main textbook for teaching mathematics (especially geometry) from the time of its publication until the late 19th or early 20th century. In the "Elements", Euclid deduced the principles of what is now called Euclidean geometry from a small set of axioms. Euclid also wrote works on perspective, conic sections, spherical geometry, number theory and rigor. Geometry (from the ; "geo-" "earth", "-metron" "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer. In mathematics, solid geometry is the traditional name for the geometry of three-dimensional Euclidean space.  The term proposition has a broad use in contemporary philosophy. It is used to refer to some or all of the following: the primary bearers of truth-value, the objects of belief and other "propositional attitudes" (i.e., what is believed, doubted, etc.), the referents of that-clauses, and the meanings of declarative sentences. Propositions are the sharable objects of attitudes and the primary bearers of truth and falsity. This stipulation rules out certain candidates for propositions, including thought- and utterance-tokens which are not sharable, and concrete events or facts, which cannot be false. A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, and plasma), and is the only state with a definite volume but no fixed shape. A liquid is made up of tiny vibrating particles of matter, such as atoms, held together by intermolecular bonds. Water is, by far, the most common liquid on Earth. Like a gas, a liquid is able to flow and take the shape of a container. Most liquids resist compression, although others can be compressed. Unlike a gas, a liquid does not disperse to fill every space of a container, and maintains a fairly constant density. A distinctive property of the liquid state is surface tension, leading to wetting phenomena. Paint is any liquid, liquefiable, or mastic composition that, after application to a substrate in a thin layer, converts to a solid film. It is most commonly used to protect, color, or provide texture to objects. Paint can be made or purchased in many colorsand in many different types, such as watercolor, synthetic, etc. Paint is typically stored, sold, and applied as a liquid, but most types dry into a solid. Mathematical analysis is the branch of mathematics dealing with limits
  and related theories, such as differentiation, integration, measure, infinite series, and analytic functions. In mathematics, an axiomatic system is any set of axioms from which some or all axioms can be used in conjunction to logically derive theorems. A theory consists of an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory typically means an axiomatic system, for example formulated within model theory. A formal proof is a complete rendition of a mathematical proof within a formal system. In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of "n"-dimensional Euclidean space. For "n" = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called n"-dimensional volume, n"-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue measurable; the measure of the Lebesgue measurable set "A" is denoted by ("A"). In geometric measurements, length is the most extended dimension of an object. In the International System of Quantities, length is any quantity with dimension distance. In other contexts "length" is the measured dimension of an object. For example, it is possible to cut a length of a wire which is shorter than wire thickness. A wire is a single, usually cylindrical, flexible strand or rod of metal. Wires are used to bear mechanical loads or electricity and telecommunications signals. Wire is commonly formed by drawing the metal through a hole in a die or draw plate. Wire gauges come in various standard sizes, as expressed in terms of a gauge number. The term "wire" is also used more loosely to refer to a bundle of such strands, as in "multistranded wire", which is more correctly termed a wire rope in mechanics, or a cable in electricity. Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.
  Volume is often quantified numerically using the SI derived unit, the cubic metre. The volume of a container is generally understood to be the capacity of the container, i. e. the amount of fluid (gas or liquid) that the container could hold, rather than the amount of space the container itself displaces. Real analysis (traditionally, the theory of functions of a real variable) is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions. Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics. The language of set theory can be used in the definitions of nearly all mathematical objects. The surface area of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with flat polygonal faces), for which the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration.  Quantity is a property that can exist as a magnitude or multitude. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value in terms of a unit of measurement. Quantity is among the basic classes of things along with quality, substance, change, and relation. Some quantities are such by their inner nature (as number), while others are functioning as states (properties, dimensions, attributes) of things such as heavy and light, long and short, broad and narrow, small and great, or much and little. A small quantity is sometimes referred to as a quantulum. Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma). A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or compound molecules made from a variety of atoms (e.g. carbon dioxide). A gas mixture would contain a variety of pure gases much like the air. What distinguishes a gas from liquids and solids is the vast separation of the individual gas particles. This separation usually makes a colorless gas invisible to the human observer. The interaction of gas particles in the presence of electric and gravitational fields are considered negligible as indicated by the constant velocity vectors in the image. One type of commonly known gas is steam. Algebra (from Arabic ""al-jabr"" meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is generally considered to be essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Greek "axíma" 'that which is thought worthy or fit' or 'that which commends itself as evident.'  Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although modern physicists usually consider it, with time, to be part of a boundless four-dimensional continuum known as spacetime. The concept of space is considered to be of fundamental importance to an understanding of the physical universe. However, disagreement continues between philosophers over whether it is itself an entity, a relationship between entities, or part of a conceptual framework. In mathematics, a real number is a value that represents a quantity along a line. The adjective "real" in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called "The Queen of Mathematics" because of its foundational place in the discipline. Number theorists study prime numbers as well as the properties of objects made out of integers (e.g., rational numbers) or defined as generalizations of the integers (e.g., algebraic integers). Solid is one of the four fundamental states of matter (the others being liquid, gas, and plasma). It is characterized by structural rigidity and resistance to changes of shape or volume. Unlike a liquid, a solid object does not flow to take on the shape of its container, nor does it expand to fill the entire volume available to it like a gas does. The atoms in a solid are tightly bound to each other, either in a regular geometric lattice (crystalline solids, which include metals and ordinary ice) or irregularly (an amorphous solid such as common window glass). The International System of Units (SI) specifies a set of seven base units from which all other SI units of measurement are derived. Each of these other units (SI derived units) is either dimensionless or can be expressed as a product of (positive or negative, but usually integral) powers of one or more of the base units.  In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces. It is named after the Ancient Greek mathematician Euclid of Alexandria. The term "Euclidean" distinguishes these spaces from other types of spaces considered in modern geometry. Euclidean spaces also generalize to higher dimensions. In mathematics, especially in set theory, a set "A" is a subset of a set "B", or equivalently "B" is a superset of "A", if "A" is "contained" inside "B", that is, all elements of "A" are also elements of "B". "A" and "B" may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment. The International System of Quantities (ISQ) is a system based on seven base quantities: length, mass, time, electric current, thermodynamic temperature, amount of substance, and luminous intensity. Other quantities such as area, pressure, and electrical resistance are derived from these base quantities by clear, non-contradictory equations. The ISQ defines the quantities that are measured with the SI units and also includes many other quantities in modern science and technology. The ISQ is defined in the international standard ISO/IEC 80000, and was finalised in 2009 with the publication of ISO 80000-1. René Descartes (Latinized: Renatus Cartesius; adjectival form: "Cartesian"; 31 March 159611 February 1650) was a French philosopher, mathematician, and scientist. Dubbed the father of modern western philosophy, much of subsequent Western philosophy is a response to his writings, which are studied closely to this day. He spent about 20 years of his life in the Dutch Republic. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface such as a plane or the surface of a cylinder or sphere has a dimension of two because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. The inside of a cube, a cylinder or a sphere is three-dimensional because three coordinates are needed to locate a point within these spaces. In mathematics, a polynomial is an expression consisting of variables (or indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate is . An example in three variables is . Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept). In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others. Functions of each type are infinitely differentiable, but complex analytic functions exhibit properties that do not hold generally for real analytic functions. A function is analytic if and only if its Taylor series about "x" converges to the function in some neighborhood for every "x" in its domain. In mathematical analysis , a measure on a set is a systematic way to assign a number to each suitable subset of that set , intuitively interpreted as its size . In this sense , a measure is a generalization of the concepts of length , area , and volume . A particularly important example is the Lebesgue measure on a Euclidean space , which assigns the conventional length , area , and volume of Euclidean geometry to suitable subsets of the n - dimensional Euclidean space Rn . For instance , the Lebesgue measure of the interval ( 0 , 1 ) in the real numbers is its length in the everyday sense of the word -- specifically , 1 . Technically , a measure is a function that assigns a non-negative real number or +  to ( certain ) subsets of a set X ( see Definition below ) . It must assign 0 to the empty set and be ( countably ) additive : the measure of a ' large ' subset that can be decomposed into a finite ( or countable ) number of ' smaller ' disjoint subsets , is the sum of the measures of the `` smaller '' subsets . In general , if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure , one only finds trivial examples like the counting measure . This problem was resolved by defining measure only on a sub-collection of all subsets ; the so - called measurable subsets , which are required to form a  - algebra . This means that countable unions , countable intersections and complements of measurable subsets are measurable . Non-measurable sets in a Euclidean space , on which the Lebesgue measure can not be defined consistently , are necessarily complicated in the sense of being badly mixed up with their complement . Indeed , their existence is a non-trivial consequence of the axiom of choice . Measure theory was developed in successive stages during the late 19th and early 20th centuries by Émile Borel , Henri Lebesgue , Johann Radon , and Maurice Fréchet , among others . The main applications of measures are in the foundations of the Lebesgue integral , in Andrey Kolmogorov 's axiomatisation of probability theory and in ergodic theory . In integration theory , specifying a measure allows one to define integrals on spaces more general than subsets of Euclidean space ; moreover , the integral with respect to the Lebesgue measure on Euclidean spaces is more general and has a richer theory than its predecessor , the Riemann integral . Probability theory considers measures that assign to the whole set the size 1 , and considers measurable subsets to be events whose probability is given by the measure . Ergodic theory considers measures that are invariant under , or arise naturally from , a dynamical system . Euclidean geometry is a mathematical system attributed to the Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the "Elements". Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The "Elements" begins with plane geometry, still taught in secondary school as the first axiomatic system and the first examples of formal proof. It goes on to the solid geometry of three dimensions. Much of the "Elements" states results of what are now called algebra and number theory, explained in geometrical language. A secondary school, often referred to as a high school or a senior high school, is a school which provides secondary education, between the ages of 11 and 19 depending on location, after primary school and before higher education. Alexandria (or ; Arabic: '; '; "") is the second largest city and a major economic centre in Egypt, extending about along the coast of the Mediterranean Sea in the north central part of the country. Its low elevation on the Nile delta makes it highly vulnerable to rising sea levels. Alexandria is Egypt's largest seaport, serving approximately 80% of Egypt's imports and exports. It is an important industrial center because of its natural gas and oil pipelines from Suez. Alexandria is also an important tourist destination. Mathematics (from Greek  "máthma", knowledge, study, learning) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics. In mathematics, a planar lamina is a closed set in a plane of mass formula_1 and surface density formula_2 such that: In mathematics, a theorem is a statement that has been proved on the basis of previously established statements, such as other theoremsand generally accepted statements, such as axioms. A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system. The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally "deductive", in contrast to the notion of a scientific law, which is "experimental". Henri Léon Lebesgue (June 28, 1875  July 26, 1941) was a French mathematician most famous for his theory of integration, which was a generalization of the 17th century concept of integrationsumming the area between an axis and the curve of a function defined for that axis. His theory was published originally in his dissertation "Intégrale, longueur, aire" ("Integral, length, area") at the University of Nancy during 1902.
candidates: 17th century 20th century abstract algebra algebra arabic atom center city country curve day drawing earth economics euclidean geometry february film french gas geometry higher education history human infinite international system of units july june law liquid march mass mathematical analysis mathematical logic mathematics mechanics mediterranean sea nature nile number theory part philosophy physics publication queen reasoning school sea set theory solid system teaching technology the following theory thought universe university word
answer: mathematics