131-450: Geometry (from Ancient Greek γεωμετρία ( geōmetría ) 'land measurement'; from γῆ ( gê ) 'earth, land' and μέτρον ( métron ) 'a measure') is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic , one of the oldest branches of mathematics. A mathematician who works in
262-412: A Euclidean space . This implies that surfaces can be studied intrinsically , that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry . Later in the 19th century, it appeared that geometries without the parallel postulate ( non-Euclidean geometries ) can be developed without introducing any contradiction. The geometry that underlies general relativity
393-516: A geodesic is a generalization of the notion of a line to curved spaces . In Euclidean geometry a plane is a flat, two-dimensional surface that extends infinitely; the definitions for other types of geometries are generalizations of that. Planes are used in many areas of geometry. For instance, planes can be studied as a topological surface without reference to distances or angles; it can be studied as an affine space , where collinearity and ratios can be studied but not distances; it can be studied as
524-543: A pitch accent . In Modern Greek, all vowels and consonants are short. Many vowels and diphthongs once pronounced distinctly are pronounced as /i/ ( iotacism ). Some of the stops and glides in diphthongs have become fricatives , and the pitch accent has changed to a stress accent . Many of the changes took place in the Koine Greek period. The writing system of Modern Greek, however, does not reflect all pronunciation changes. The examples below represent Attic Greek in
655-495: A plane table in 1551, but it is thought that the instrument was in use earlier as his description is of a developed instrument. Gunter's chain was introduced in 1620 by English mathematician Edmund Gunter . It enabled plots of land to be accurately surveyed and plotted for legal and commercial purposes. Leonard Digges described a theodolite that measured horizontal angles in his book A geometric practice named Pantometria (1571). Joshua Habermel ( Erasmus Habermehl ) created
786-417: A vector space and its dual space . Euclidean geometry is geometry in its classical sense. As it models the space of the physical world, it is used in many scientific areas, such as mechanics , astronomy , crystallography , and many technical fields, such as engineering , architecture , geodesy , aerodynamics , and navigation . The mandatory educational curriculum of the majority of nations includes
917-445: A GPS on large scale surveys makes them popular for major infrastructure or data gathering projects. One-person robotic-guided total stations allow surveyors to measure without extra workers to aim the telescope or record data. A fast but expensive way to measure large areas is with a helicopter, using a GPS to record the location of the helicopter and a laser scanner to measure the ground. To increase precision, surveyors place beacons on
1048-402: A common endpoint, called the vertex of the angle. The size of an angle is formalized as an angular measure . In Euclidean geometry , angles are used to study polygons and triangles , as well as forming an object of study in their own right. The study of the angles of a triangle or of angles in a unit circle forms the basis of trigonometry . In differential geometry and calculus ,
1179-477: A fixed base station and a second roving antenna. The position of the roving antenna can be tracked. The theodolite , total station and RTK GPS survey remain the primary methods in use. Remote sensing and satellite imagery continue to improve and become cheaper, allowing more commonplace use. Prominent new technologies include three-dimensional (3D) scanning and lidar -based topographical surveys. UAV technology along with photogrammetric image processing
1310-423: A great step forward in the instrument's accuracy. William Gascoigne invented an instrument that used a telescope with an installed crosshair as a target device, in 1640. James Watt developed an optical meter for the measuring of distance in 1771; it measured the parallactic angle from which the distance to a point could be deduced. Dutch mathematician Willebrord Snellius (a.k.a. Snel van Royen) introduced
1441-502: A height above sea level. As the surveying profession grew it created Cartesian coordinate systems to simplify the mathematics for surveys over small parts of the Earth. The simplest coordinate systems assume that the Earth is flat and measure from an arbitrary point, known as a 'datum' (singular form of data). The coordinate system allows easy calculation of the distances and direction between objects over small areas. Large areas distort due to
SECTION 10
#17327767250931572-433: A known size. It was sometimes used before to the invention of EDM where rough ground made chain measurement impractical. Historically, horizontal angles were measured by using a compass to provide a magnetic bearing or azimuth. Later, more precise scribed discs improved angular resolution. Mounting telescopes with reticles atop the disc allowed more precise sighting (see theodolite ). Levels and calibrated circles allowed
1703-477: A lack of contemporaneous evidence. Several theories exist about what Hellenic dialect groups may have existed between the divergence of early Greek-like speech from the common Proto-Indo-European language and the Classical period. They have the same general outline but differ in some of the detail. The only attested dialect from this period is Mycenaean Greek , but its relationship to the historical dialects and
1834-419: A lesser degree. Pamphylian Greek , spoken in a small area on the southwestern coast of Anatolia and little preserved in inscriptions, may be either a fifth major dialect group, or it is Mycenaean Greek overlaid by Doric, with a non-Greek native influence. Regarding the speech of the ancient Macedonians diverse theories have been put forward, but the epigraphic activity and the archaeological discoveries in
1965-434: A loop pattern or link between two prior reference marks so the surveyor can check their measurements. Many surveys do not calculate positions on the surface of the Earth, but instead, measure the relative positions of objects. However, often the surveyed items need to be compared to outside data, such as boundary lines or previous survey's objects. The oldest way of describing a position is via latitude and longitude, and often
2096-439: A more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms . Congruence and similarity are generalized in transformation geometry , which studies the properties of geometric objects that are preserved by different kinds of transformations. Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically,
2227-618: A multi frequency phase shift of light waves to find a distance. These instruments eliminated the need for days or weeks of chain measurement by measuring between points kilometers apart in one go. Advances in electronics allowed miniaturization of EDM. In the 1970s the first instruments combining angle and distance measurement appeared, becoming known as total stations . Manufacturers added more equipment by degrees, bringing improvements in accuracy and speed of measurement. Major advances include tilt compensators, data recorders and on-board calculation programs. The first satellite positioning system
2358-426: A multitude of forms, including the graphics of Leonardo da Vinci , M. C. Escher , and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein 's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group , determines what geometry is . Symmetry in classical Euclidean geometry
2489-447: A number of apparently different definitions, which are all equivalent in the most common cases. The theme of symmetry in geometry is nearly as old as the science of geometry itself. Symmetric shapes such as the circle , regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in
2620-442: A physical system, which has a dimension equal to the system's degrees of freedom . For instance, the configuration of a screw can be described by five coordinates. In general topology , the concept of dimension has been extended from natural numbers , to infinite dimension ( Hilbert spaces , for example) and positive real numbers (in fractal geometry ). In algebraic geometry , the dimension of an algebraic variety has received
2751-411: A plan or map, and the points at the ends of the offset lines could be joined to show the feature. Traversing is a common method of surveying smaller areas. The surveyor starts from an old reference mark or known position and places a network of reference marks covering the survey area. They then measure bearings and distances between the reference marks, and to the target features. Most traverses form
SECTION 20
#17327767250932882-525: A plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus , area and volume can be defined in terms of integrals , such as the Riemann integral or the Lebesgue integral . Other geometrical measures include the curvature and compactness . The concept of length or distance can be generalized, leading to
3013-406: A point inside a triangle using the angles cast between the vertices at the unknown point. These could be measured more accurately than bearings of the vertices, which depended on a compass. His work established the idea of surveying a primary network of control points, and locating subsidiary points inside the primary network later. Between 1733 and 1740, Jacques Cassini and his son César undertook
3144-550: A prefix /e-/, called the augment . This was probably originally a separate word, meaning something like "then", added because tenses in PIE had primarily aspectual meaning. The augment is added to the indicative of the aorist, imperfect, and pluperfect, but not to any of the other forms of the aorist (no other forms of the imperfect and pluperfect exist). The two kinds of augment in Greek are syllabic and quantitative. The syllabic augment
3275-474: A problem that was stated in terms of elementary arithmetic , and remained unsolved for several centuries. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss 's Theorema Egregium ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in
3406-478: A profession. They established the basic measurements under which the Roman Empire was divided, such as a tax register of conquered lands (300 AD). Roman surveyors were known as Gromatici . In medieval Europe, beating the bounds maintained the boundaries of a village or parish. This was the practice of gathering a group of residents and walking around the parish or village to establish a communal memory of
3537-598: A purely algebraic context. Scheme theory allowed to solve many difficult problems not only in geometry, but also in number theory . Wiles' proof of Fermat's Last Theorem is a famous example of a long-standing problem of number theory whose solution uses scheme theory and its extensions such as stack theory . One of seven Millennium Prize problems , the Hodge conjecture , is a question in algebraic geometry. Algebraic geometry has applications in many areas, including cryptography and string theory . Complex geometry studies
3668-428: A reflector or prism to return the light pulses used for distance measurements. They are fully robotic, and can even e-mail point data to a remote computer and connect to satellite positioning systems , such as Global Positioning System . Real Time Kinematic GPS systems have significantly increased the speed of surveying, and they are now horizontally accurate to within 1 cm ± 1 ppm in real-time, while vertically it
3799-608: A separate historical stage, though its earliest form closely resembles Attic Greek , and its latest form approaches Medieval Greek . There were several regional dialects of Ancient Greek; Attic Greek developed into Koine. Ancient Greek was a pluricentric language , divided into many dialects. The main dialect groups are Attic and Ionic , Aeolic , Arcadocypriot , and Doric , many of them with several subdivisions. Some dialects are found in standardized literary forms in literature , while others are attested only in inscriptions. There are also several historical forms. Homeric Greek
3930-505: 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 ( c. 287–212 BC ) of Syracuse, Italy used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series , and gave remarkably accurate approximations of pi . He also studied
4061-424: A size or measure to sets , where the measures follow rules similar to those of classical area and volume. Congruence and similarity are concepts that describe when two shapes have similar characteristics. In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert , in his work on creating
Geometry - Misplaced Pages Continue
4192-630: A standard subject of study in educational institutions of the Western world since the Renaissance . This article primarily contains information about the Epic and Classical periods of the language, which are the best-attested periods and considered most typical of Ancient Greek. From the Hellenistic period ( c. 300 BC ), Ancient Greek was followed by Koine Greek , which is regarded as
4323-476: A star is determined, the bearing can be transferred to a reference point on Earth. The point can then be used as a base for further observations. Survey-accurate astronomic positions were difficult to observe and calculate and so tended to be a base off which many other measurements were made. Since the advent of the GPS system, astronomic observations are rare as GPS allows adequate positions to be determined over most of
4454-495: A theodolite with a compass and tripod in 1576. Johnathon Sission was the first to incorporate a telescope on a theodolite in 1725. In the 18th century, modern techniques and instruments for surveying began to be used. Jesse Ramsden introduced the first precision theodolite in 1787. It was an instrument for measuring angles in the horizontal and vertical planes. He created his great theodolite using an accurate dividing engine of his own design. Ramsden's theodolite represented
4585-517: A theorem called Hilbert's Nullstellensatz that establishes a strong correspondence between algebraic sets and ideals of polynomial rings . This led to a parallel development of algebraic geometry, and its algebraic counterpart, called commutative algebra . From the late 1950s through the mid-1970s algebraic geometry had undergone major foundational development, with the introduction by Alexander Grothendieck of scheme theory , which allows using topological methods , including cohomology theories in
4716-593: A time component. Before EDM (Electronic Distance Measurement) laser devices, distances were measured using a variety of means. In pre-colonial America Natives would use the "bow shot" as a distance reference ("as far as an arrow can slung out of a bow", or "flights of a Cherokee long bow"). Europeans used chains with links of a known length such as a Gunter's chain , or measuring tapes made of steel or invar . To measure horizontal distances, these chains or tapes were pulled taut to reduce sagging and slack. The distance had to be adjusted for heat expansion. Attempts to hold
4847-579: A type of transformation geometry , in which transformations are homeomorphisms . This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology , differential topology , algebraic topology and general topology . Algebraic geometry is fundamentally the study by means of algebraic methods of some geometrical shapes, called algebraic sets , and defined as common zeros of multivariate polynomials . Algebraic geometry became an autonomous subfield of geometry c. 1900 , with
4978-510: A vowel or /n s r/ ; final stops were lost, as in γάλα "milk", compared with γάλακτος "of milk" (genitive). Ancient Greek of the classical period also differed in both the inventory and distribution of original PIE phonemes due to numerous sound changes, notably the following: The pronunciation of Ancient Greek was very different from that of Modern Greek . Ancient Greek had long and short vowels ; many diphthongs ; double and single consonants; voiced, voiceless, and aspirated stops ; and
5109-408: Is diffeomorphic to Euclidean space. Manifolds are used extensively in physics, including in general relativity and string theory . Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. In modern terms, an angle is the figure formed by two rays , called the sides of the angle, sharing
5240-563: Is a famous application of non-Euclidean geometry. Since the late 19th century, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods— differential geometry , algebraic geometry , computational geometry , algebraic topology , discrete geometry (also known as combinatorial geometry ), etc.—or on the properties of Euclidean spaces that are disregarded— projective geometry that consider only alignment of points but not distance and parallelism, affine geometry that omits
5371-570: Is a literary form of Archaic Greek (derived primarily from Ionic and Aeolic) used in the epic poems , the Iliad and the Odyssey , and in later poems by other authors. Homeric Greek had significant differences in grammar and pronunciation from Classical Attic and other Classical-era dialects. The origins, early form and development of the Hellenic language family are not well understood because of
Geometry - Misplaced Pages Continue
5502-414: Is a part of some ambient flat Euclidean space). Topology is the field concerned with the properties of continuous mappings , and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness . The field of topology, which saw massive development in the 20th century, is in a technical sense
5633-474: Is a staple of contemporary land surveying. Typically, much if not all of the drafting and some of the designing for plans and plats of the surveyed property is done by the surveyor, and nearly everyone working in the area of drafting today (2021) utilizes CAD software and hardware both on PC, and more and more in newer generation data collectors in the field as well. Other computer platforms and tools commonly used today by surveyors are offered online by
5764-399: Is a term used when referring to moving the level to take an elevation shot from a different location. To "turn" the level, one must first take a reading and record the elevation of the point the rod is located on. While the rod is being kept in exactly the same location, the level is moved to a new location where the rod is still visible. A reading is taken from the new location of the level and
5895-412: Is a three-dimensional object bounded by a closed surface; for example, a ball is the volume bounded by a sphere. A manifold is a generalization of the concepts of curve and surface. In topology , a manifold is a topological space where every point has a neighborhood that is homeomorphic to Euclidean space. In differential geometry , a differentiable manifold is a space where each neighborhood
6026-418: Is added to stems beginning with consonants, and simply prefixes e (stems beginning with r , however, add er ). The quantitative augment is added to stems beginning with vowels, and involves lengthening the vowel: Some verbs augment irregularly; the most common variation is e → ei . The irregularity can be explained diachronically by the loss of s between vowels, or that of the letter w , which affected
6157-400: Is also appearing. The main surveying instruments in use around the world are the theodolite , measuring tape , total station , 3D scanners , GPS / GNSS , level and rod . Most instruments screw onto a tripod when in use. Tape measures are often used for measurement of smaller distances. 3D scanners and various forms of aerial imagery are also used. The theodolite is an instrument for
6288-412: Is an alternate method of determining the position of objects, and was often used to measure imprecise features such as riverbanks. The surveyor would mark and measure two known positions on the ground roughly parallel to the feature, and mark out a baseline between them. At regular intervals, a distance was measured at right angles from the first line to the feature. The measurements could then be plotted on
6419-505: Is because divergent conditions further away from the base reduce accuracy. Surveying instruments have characteristics that make them suitable for certain uses. Theodolites and levels are often used by constructors rather than surveyors in first world countries. The constructor can perform simple survey tasks using a relatively cheap instrument. Total stations are workhorses for many professional surveyors because they are versatile and reliable in all conditions. The productivity improvements from
6550-666: Is called 'East Greek'. Arcadocypriot apparently descended more closely from the Mycenaean Greek of the Bronze Age. Boeotian Greek had come under a strong Northwest Greek influence, and can in some respects be considered a transitional dialect, as exemplified in the poems of the Boeotian poet Pindar who wrote in Doric with a small Aeolic admixture. Thessalian likewise had come under Northwest Greek influence, though to
6681-518: Is called a land surveyor . Surveyors work with elements of geodesy , geometry , trigonometry , regression analysis , physics , engineering, metrology , programming languages , and the law. They use equipment, such as total stations , robotic total stations, theodolites , GNSS receivers, retroreflectors , 3D scanners , lidar sensors, radios, inclinometer , handheld tablets, optical and digital levels , subsurface locators, drones, GIS , and surveying software. Surveying has been an element in
SECTION 50
#17327767250936812-448: Is considered by some linguists to have been closely related to Greek . Among Indo-European branches with living descendants, Greek is often argued to have the closest genetic ties with Armenian (see also Graeco-Armenian ) and Indo-Iranian languages (see Graeco-Aryan ). Ancient Greek differs from Proto-Indo-European (PIE) and other Indo-European languages in certain ways. In phonotactics , ancient Greek words could end only in
6943-585: Is currently about half of that to within 2 cm ± 2 ppm. GPS surveying differs from other GPS uses in the equipment and methods used. Static GPS uses two receivers placed in position for a considerable length of time. The long span of time lets the receiver compare measurements as the satellites orbit. The changes as the satellites orbit also provide the measurement network with well conditioned geometry. This produces an accurate baseline that can be over 20 km long. RTK surveying uses one static antenna and one roving antenna. The static antenna tracks changes in
7074-407: Is defined. 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, areas, and volumes, which were developed to meet some practical need in surveying , construction , astronomy , and various crafts. The earliest known texts on geometry are
7205-434: Is not viewed as the set of the points through which it passes. However, there are modern geometries in which points are not primitive objects, or even without points. One of the oldest such geometries is Whitehead's point-free geometry , formulated by Alfred North Whitehead in 1919–1920. Euclid described a line as "breadthless length" which "lies equally with respect to the points on itself". In modern mathematics, given
7336-414: Is of importance to mathematical physics due to Albert Einstein 's general relativity postulation that the universe is curved . Differential geometry can either be intrinsic (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric , which determines how distances are measured near each point) or extrinsic (where the object under study
7467-480: Is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations , geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group ' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry,
7598-528: Is used, but the defining function is required to be differentiable. Algebraic geometry studies algebraic curves , which are defined as algebraic varieties of dimension one. A surface is a two-dimensional object, such as a sphere or paraboloid. In differential geometry and topology , surfaces are described by two-dimensional 'patches' (or neighborhoods ) that are assembled by diffeomorphisms or homeomorphisms , respectively. In algebraic geometry, surfaces are described by polynomial equations . A solid
7729-432: Is with an altimeter using air pressure to find the height. When more precise measurements are needed, means like precise levels (also known as differential leveling) are used. When precise leveling, a series of measurements between two points are taken using an instrument and a measuring rod. Differences in height between the measurements are added and subtracted in a series to get the net difference in elevation between
7860-689: The Egyptian Rhind Papyrus (2000–1800 BC) and Moscow Papyrus ( c. 1890 BC ), and the Babylonian clay tablets , such as Plimpton 322 (1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum . Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid procedures for computing Jupiter's position and motion within time-velocity space. These geometric procedures anticipated
7991-713: The Great Pyramid of Giza , built c. 2700 BC , affirm the Egyptians' command of surveying. The groma instrument may have originated in Mesopotamia (early 1st millennium BC). The prehistoric monument at Stonehenge ( c. 2500 BC ) was set out by prehistoric surveyors using peg and rope geometry. The mathematician Liu Hui described ways of measuring distant objects in his work Haidao Suanjing or The Sea Island Mathematical Manual , published in 263 AD. The Romans recognized land surveying as
SECTION 60
#17327767250938122-759: The Greek region of Macedonia during the last decades has brought to light documents, among which the first texts written in Macedonian , such as the Pella curse tablet , as Hatzopoulos and other scholars note. Based on the conclusions drawn by several studies and findings such as Pella curse tablet , Emilio Crespo and other scholars suggest that ancient Macedonian was a Northwest Doric dialect , which shares isoglosses with its neighboring Thessalian dialects spoken in northeastern Thessaly . Some have also suggested an Aeolic Greek classification. The Lesbian dialect
8253-522: The Lambert quadrilateral and Saccheri quadrilateral , were part of a line of research on the parallel postulate continued by later European geometers, including Vitello ( c. 1230 – c. 1314 ), Gersonides (1288–1344), Alfonso, John Wallis , and Giovanni Girolamo Saccheri , that by the 19th century led to the discovery of hyperbolic geometry . In the early 17th century, there were two important developments in geometry. The first
8384-599: 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's theorem . Pythagoras established
8515-509: The Principal Triangulation of Britain . The first Ramsden theodolite was built for this survey. The survey was finally completed in 1853. The Great Trigonometric Survey of India began in 1801. The Indian survey had an enormous scientific impact. It was responsible for one of the first accurate measurements of a section of an arc of longitude, and for measurements of the geodesic anomaly. It named and mapped Mount Everest and
8646-605: The Pythagorean School , which is credited with the first proof of the Pythagorean theorem , though the statement of the theorem has a long history. Eudoxus (408– c. 355 BC ) 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
8777-505: The Riemann surface , and Henri Poincaré , the founder of algebraic topology and the geometric theory of dynamical systems . As a consequence of these major changes in the conception of geometry, the concept of " space " became something rich and varied, and the natural background for theories as different as complex analysis and classical mechanics . The following are some of the most important concepts in geometry. Euclid took an abstract approach to geometry in his Elements , one of
8908-621: The Torrens system in South Australia in 1858. Torrens intended to simplify land transactions and provide reliable titles via a centralized register of land. The Torrens system was adopted in several other nations of the English-speaking world. Surveying became increasingly important with the arrival of railroads in the 1800s. Surveying was necessary so that railroads could plan technologically and financially viable routes. At
9039-799: The U.S. Federal Government and other governments' survey agencies, such as the National Geodetic Survey and the CORS network, to get automated corrections and conversions for collected GPS data, and the data coordinate systems themselves. Surveyors determine the position of objects by measuring angles and distances. The factors that can affect the accuracy of their observations are also measured. They then use this data to create vectors, bearings, coordinates, elevations, areas, volumes, plans and maps. Measurements are often split into horizontal and vertical components to simplify calculation. GPS and astronomic measurements also need measurement of
9170-416: The complex plane using techniques of complex analysis ; and so on. A curve is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curves and those in 3-dimensional space are called space curves . In topology, a curve is defined by a function from an interval of the real numbers to another space. In differential geometry, the same definition
9301-419: The distances and angles between them. These points are usually on the surface of the Earth, and they are often used to establish maps and boundaries for ownership , locations, such as the designated positions of structural components for construction or the surface location of subsurface features, or other purposes required by government or civil law, such as property sales. A professional in land surveying
9432-501: The present , future , and imperfect are imperfective in aspect; the aorist , present perfect , pluperfect and future perfect are perfective in aspect. Most tenses display all four moods and three voices, although there is no future subjunctive or imperative. Also, there is no imperfect subjunctive, optative or imperative. The infinitives and participles correspond to the finite combinations of tense, aspect, and voice. The indicative of past tenses adds (conceptually, at least)
9563-485: The spiral bearing his name and obtained formulas for the volumes of surfaces of revolution . Indian mathematicians also made many important contributions in geometry. The Shatapatha Brahmana (3rd century BC) contains rules for ritual geometric constructions that are similar to the Sulba Sutras . According to ( Hayashi 2005 , p. 363), the Śulba Sūtras contain "the earliest extant verbal expression of
9694-630: The 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry as the central consideration in the Erlangen programme of Felix Klein (which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann (1826–1866), working primarily with tools from mathematical analysis , and introducing
9825-472: The 19th century, the discovery of non-Euclidean geometries by Nikolai Ivanovich Lobachevsky (1792–1856), János Bolyai (1802–1860), Carl Friedrich Gauss (1777–1855) and others led to a revival of interest in this discipline, and in the 20th century, David Hilbert (1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry. Points are generally considered fundamental objects for building geometry. They may be defined by
9956-1031: The 5th century BC. Ancient pronunciation cannot be reconstructed with certainty, but Greek from the period is well documented, and there is little disagreement among linguists as to the general nature of the sounds that the letters represent. /oː/ raised to [uː] , probably by the 4th century BC. Greek, like all of the older Indo-European languages , is highly inflected. It is highly archaic in its preservation of Proto-Indo-European forms. In ancient Greek, nouns (including proper nouns) have five cases ( nominative , genitive , dative , accusative , and vocative ), three genders ( masculine , feminine , and neuter ), and three numbers (singular, dual , and plural ). Verbs have four moods ( indicative , imperative , subjunctive , and optative ) and three voices (active, middle, and passive ), as well as three persons (first, second, and third) and various other forms. Verbs are conjugated through seven combinations of tenses and aspect (generally simply called "tenses"):
10087-495: The Archaic period of ancient Greek (see Homeric Greek for more details): Μῆνιν ἄειδε, θεά, Πηληϊάδεω Ἀχιλῆος οὐλομένην, ἣ μυρί' Ἀχαιοῖς ἄλγε' ἔθηκε, πολλὰς δ' ἰφθίμους ψυχὰς Ἄϊδι προΐαψεν ἡρώων, αὐτοὺς δὲ ἑλώρια τεῦχε κύνεσσιν οἰωνοῖσί τε πᾶσι· Διὸς δ' ἐτελείετο βουλή· ἐξ οὗ δὴ τὰ πρῶτα διαστήτην ἐρίσαντε Ἀτρεΐδης τε ἄναξ ἀνδρῶν καὶ δῖος Ἀχιλλεύς. The beginning of Apology by Plato exemplifies Attic Greek from
10218-613: The Classical period of ancient Greek. (The second line is the IPA , the third is transliterated into the Latin alphabet using a modern version of the Erasmian scheme .) Ὅτι [hóti Hóti μὲν men mèn ὑμεῖς, hyːmêːs hūmeîs, Surveying Surveying or land surveying is the technique, profession, art, and science of determining the terrestrial two-dimensional or three-dimensional positions of points and
10349-652: The Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples , which are particular cases of Diophantine equations . In the Bakhshali manuscript , there are a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero." Aryabhata 's Aryabhatiya (499) includes
10480-527: The angles between plane curves or space curves or surfaces can be calculated using the derivative . Length , area , and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively. In Euclidean geometry and analytic geometry , the length of a line segment can often be calculated by the Pythagorean theorem . Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in
10611-550: The aorist. Following Homer 's practice, the augment is sometimes not made in poetry , especially epic poetry. The augment sometimes substitutes for reduplication; see below. Almost all forms of the perfect, pluperfect, and future perfect reduplicate the initial syllable of the verb stem. (A few irregular forms of perfect do not reduplicate, whereas a handful of irregular aorists reduplicate.) The three types of reduplication are: Irregular duplication can be understood diachronically. For example, lambanō (root lab ) has
10742-419: The augment when it was word-initial. In verbs with a preposition as a prefix, the augment is placed not at the start of the word, but between the preposition and the original verb. For example, προσ(-)βάλλω (I attack) goes to προσ έ βαλoν in the aorist. However compound verbs consisting of a prefix that is not a preposition retain the augment at the start of the word: αὐτο(-)μολῶ goes to ηὐ τομόλησα in
10873-438: The basis for dividing the western territories into sections to allow the sale of land. The PLSS divided states into township grids which were further divided into sections and fractions of sections. Napoleon Bonaparte founded continental Europe 's first cadastre in 1808. This gathered data on the number of parcels of land, their value, land usage, and names. This system soon spread around Europe. Robert Torrens introduced
11004-457: The bearing from every vertex in a figure, a surveyor can measure around the figure. The final observation will be between the two points first observed, except with a 180° difference. This is called a close . If the first and last bearings are different, this shows the error in the survey, called the angular misclose . The surveyor can use this information to prove that the work meets the expected standards. The simplest method for measuring height
11135-468: The beginning of the century, surveyors had improved the older chains and ropes, but they still faced the problem of accurate measurement of long distances. Trevor Lloyd Wadley developed the Tellurometer during the 1950s. It measures long distances using two microwave transmitter/receivers. During the late 1950s Geodimeter introduced electronic distance measurement (EDM) equipment. EDM units use
11266-528: The boundaries. Young boys were included to ensure the memory lasted as long as possible. In England, William the Conqueror commissioned the Domesday Book in 1086. It recorded the names of all the land owners, the area of land they owned, the quality of the land, and specific information of the area's content and inhabitants. It did not include maps showing exact locations. Abel Foullon described
11397-438: The center of Greek scholarship, this division of people and language is quite similar to the results of modern archaeological-linguistic investigation. One standard formulation for the dialects is: West vs. non-West Greek is the strongest-marked and earliest division, with non-West in subsets of Ionic-Attic (or Attic-Ionic) and Aeolic vs. Arcadocypriot, or Aeolic and Arcado-Cypriot vs. Ionic-Attic. Often non-West
11528-425: The computation of areas and volumes. Brahmagupta wrote his astronomical work Brāhmasphuṭasiddhānta in 628. Chapter 12, containing 66 Sanskrit verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain). In
11659-412: The concept of angle and distance, finite geometry that omits continuity , and others. This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space , or simply a space is a mathematical structure on which some geometry
11790-405: The development of the human environment since the beginning of recorded history . It is used in the planning and execution of most forms of construction . It is also used in transportation, communications, mapping, and the definition of legal boundaries for land ownership. It is an important tool for research in many other scientific disciplines. The International Federation of Surveyors defines
11921-615: The dialect of Sparta ), and Northern Peloponnesus Doric (including Corinthian ). All the groups were represented by colonies beyond Greece proper as well, and these colonies generally developed local characteristics, often under the influence of settlers or neighbors speaking different Greek dialects. After the conquests of Alexander the Great in the late 4th century BC, a new international dialect known as Koine or Common Greek developed, largely based on Attic Greek , but with influence from other dialects. This dialect slowly replaced most of
12052-562: The early days of surveying, this was the primary method of determining accurate positions of objects for topographic maps of large areas. A surveyor first needs to know the horizontal distance between two of the objects, known as the baseline . Then the heights, distances and angular position of other objects can be derived, as long as they are visible from one of the original objects. High-accuracy transits or theodolites were used, and angle measurements were repeated for increased accuracy. See also Triangulation in three dimensions . Offsetting
12183-691: The field of geometry is called a geometer . Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry , which includes the notions of point , line , plane , distance , angle , surface , and curve , as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture , and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem ,
12314-544: The first prototype satellites of the Global Positioning System (GPS) in 1978. GPS used a larger constellation of satellites and improved signal transmission, thus improving accuracy. Early GPS observations required several hours of observations by a static receiver to reach survey accuracy requirements. Later improvements to both satellites and receivers allowed for Real Time Kinematic (RTK) surveying. RTK surveys provide high-accuracy measurements by using
12445-467: The first triangulation of France. They included a re-surveying of the meridian arc , leading to the publication in 1745 of the first map of France constructed on rigorous principles. By this time triangulation methods were well established for local map-making. It was only towards the end of the 18th century that detailed triangulation network surveys mapped whole countries. In 1784, a team from General William Roy 's Ordnance Survey of Great Britain began
12576-521: The former in topology and geometric group theory , the latter in Lie theory and Riemannian geometry . A different type of symmetry is the principle of duality in projective geometry , among other fields. This meta-phenomenon can roughly be described as follows: in any theorem , exchange point with plane , join with meet , lies in with contains , and the result is an equally true theorem. A similar and closely related form of duality exists between
12707-673: The forms of the Greek language used in ancient Greece and the ancient world from around 1500 BC to 300 BC. It is often roughly divided into the following periods: Mycenaean Greek ( c. 1400–1200 BC ), Dark Ages ( c. 1200–800 BC ), the Archaic or Epic period ( c. 800–500 BC ), and the Classical period ( c. 500–300 BC ). Ancient Greek was the language of Homer and of fifth-century Athenian historians, playwrights, and philosophers . It has contributed many words to English vocabulary and has been
12838-515: The function of surveying as follows: A surveyor is a professional person with the academic qualifications and technical expertise to conduct one, or more, of the following activities; Surveying has occurred since humans built the first large structures. In ancient Egypt , a rope stretcher would use simple geometry to re-establish boundaries after the annual floods of the Nile River . The almost perfect squareness and north–south orientation of
12969-748: The ground (about 20 km (12 mi) apart). This method reaches precisions between 5–40 cm (depending on flight height). Surveyors use ancillary equipment such as tripods and instrument stands; staves and beacons used for sighting purposes; PPE ; vegetation clearing equipment; digging implements for finding survey markers buried over time; hammers for placements of markers in various surfaces and structures; and portable radios for communication over long lines of sight. Land surveyors, construction professionals, geomatics engineers and civil engineers using total station , GPS , 3D scanners, and other collector data use land surveying software to increase efficiency, accuracy, and productivity. Land Surveying Software
13100-509: The ground to large beacons that can be seen from long distances. The surveyors can set up their instruments in this position and measure to nearby objects. Sometimes a tall, distinctive feature such as a steeple or radio aerial has its position calculated as a reference point that angles can be measured against. Triangulation is a method of horizontal location favoured in the days before EDM and GPS measurement. It can determine distances, elevations and directions between distant objects. Since
13231-400: The height difference is used to find the new elevation of the level gun, which is why this method is referred to as differential levelling . This is repeated until the series of measurements is completed. The level must be horizontal to get a valid measurement. Because of this, if the horizontal crosshair of the instrument is lower than the base of the rod, the surveyor will not be able to sight
13362-561: The historical Dorians . The invasion is known to have displaced population to the later Attic-Ionic regions, who regarded themselves as descendants of the population displaced by or contending with the Dorians. The Greeks of this period believed there were three major divisions of all Greek people – Dorians, Aeolians, and Ionians (including Athenians), each with their own defining and distinctive dialects. Allowing for their oversight of Arcadian, an obscure mountain dialect, and Cypriot, far from
13493-476: The historical circumstances of the times imply that the overall groups already existed in some form. Scholars assume that major Ancient Greek period dialect groups developed not later than 1120 BC, at the time of the Dorian invasions —and that their first appearances as precise alphabetic writing began in the 8th century BC. The invasion would not be "Dorian" unless the invaders had some cultural relationship to
13624-596: The idea of metrics . For instance, the Euclidean metric measures the distance between points in the Euclidean plane , while the hyperbolic metric measures the distance in the hyperbolic plane . Other important examples of metrics include the Lorentz metric of special relativity and the semi- Riemannian metrics of general relativity . In a different direction, the concepts of length, area and volume are extended by measure theory , which studies methods of assigning
13755-534: The idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra (known as Thebit in Latin ) (836–901) dealt with arithmetic operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry . Omar Khayyam (1048–1131) found geometric solutions to cubic equations . The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals , including
13886-549: The latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral . Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula ), as well as a complete description of rational triangles ( i.e. triangles with rational sides and rational areas). In the Middle Ages , mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry . Al-Mahani (b. 853) conceived
14017-464: The measurement of angles. It uses two separate circles , protractors or alidades to measure angles in the horizontal and the vertical plane. A telescope mounted on trunnions is aligned vertically with the target object. The whole upper section rotates for horizontal alignment. The vertical circle measures the angle that the telescope makes against the vertical, known as the zenith angle. The horizontal circle uses an upper and lower plate. When beginning
14148-409: The measurement of vertical angles. Verniers allowed measurement to a fraction of a degree, such as with a turn-of-the-century transit . The plane table provided a graphical method of recording and measuring angles, which reduced the amount of mathematics required. In 1829 Francis Ronalds invented a reflecting instrument for recording angles graphically by modifying the octant . By observing
14279-424: The measuring instrument level would also be made. When measuring up a slope, the surveyor might have to "break" (break chain) the measurement- use an increment less than the total length of the chain. Perambulators , or measuring wheels, were used to measure longer distances but not to a high level of accuracy. Tacheometry is the science of measuring distances by measuring the angle between two ends of an object with
14410-411: The modern systematic use of triangulation . In 1615 he surveyed the distance from Alkmaar to Breda , approximately 72 miles (116 km). He underestimated this distance by 3.5%. The survey was a chain of quadrangles containing 33 triangles in all. Snell showed how planar formulae could be corrected to allow for the curvature of the Earth . He also showed how to resect , or calculate, the position of
14541-408: The most influential books ever written. Euclid introduced certain axioms , or postulates , expressing primary or self-evident properties of points, lines, and planes. He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as axiomatic or synthetic geometry. At the start of
14672-427: The multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry , a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation , but in a more abstract setting, such as incidence geometry , a line may be an independent object, distinct from the set of points which lie on it. In differential geometry,
14803-408: The nature of geometric structures modelled on, or arising out of, the complex plane . Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables , and has found applications to string theory and mirror symmetry . Ancient Greek language Ancient Greek ( Ἑλληνῐκή , Hellēnikḗ ; [hellɛːnikɛ́ː] ) includes
14934-508: The older dialects, although the Doric dialect has survived in the Tsakonian language , which is spoken in the region of modern Sparta. Doric has also passed down its aorist terminations into most verbs of Demotic Greek . By about the 6th century AD, the Koine had slowly metamorphosed into Medieval Greek . Phrygian is an extinct Indo-European language of West and Central Anatolia , which
15065-440: The only instruments used in most geometric constructions are the compass and straightedge . Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using neusis , parabolas and other curves, or mechanical devices, were found. The geometrical concepts of rotation and orientation define part of
15196-672: The other Himalayan peaks. Surveying became a professional occupation in high demand at the turn of the 19th century with the onset of the Industrial Revolution . The profession developed more accurate instruments to aid its work. Industrial infrastructure projects used surveyors to lay out canals , roads and rail. In the US, the Land Ordinance of 1785 created the Public Land Survey System . It formed
15327-487: The perfect stem eilēpha (not * lelēpha ) because it was originally slambanō , with perfect seslēpha , becoming eilēpha through compensatory lengthening. Reduplication is also visible in the present tense stems of certain verbs. These stems add a syllable consisting of the root's initial consonant followed by i . A nasal stop appears after the reduplication in some verbs. The earliest extant examples of ancient Greek writing ( c. 1450 BC ) are in
15458-406: The placement of objects embedded in the plane or in space. Traditional geometry allowed dimensions 1 (a line or curve), 2 (a plane or surface), and 3 (our ambient world conceived of as three-dimensional space ). Furthermore, mathematicians and physicists have used higher dimensions for nearly two centuries. One example of a mathematical use for higher dimensions is the configuration space of
15589-481: The properties that they must have, as in Euclid's definition as "that which has no part", or in synthetic geometry . In modern mathematics, they are generally defined as elements of a set called space , which is itself axiomatically defined. With these modern definitions, every geometric shape is defined as a set of points; this is not the case in synthetic geometry, where a line is another fundamental object that
15720-477: The rod and get a reading. The rod can usually be raised up to 25 feet (7.6 m) high, allowing the level to be set much higher than the base of the rod. The primary way of determining one's position on the Earth's surface when no known positions are nearby is by astronomic observations. Observations to the Sun, Moon and stars could all be made using navigational techniques. Once the instrument's position and bearing to
15851-422: The satellite positions and atmospheric conditions. The surveyor uses the roving antenna to measure the points needed for the survey. The two antennas use a radio link that allows the static antenna to send corrections to the roving antenna. The roving antenna then applies those corrections to the GPS signals it is receiving to calculate its own position. RTK surveying covers smaller distances than static methods. This
15982-584: The study of Euclidean concepts such as points , lines , planes , angles , triangles , congruence , similarity , solid figures , circles , and analytic geometry . Euclidean vectors are used for a myriad of applications in physics and engineering, such as position , displacement , deformation , velocity , acceleration , force , etc. Differential geometry uses techniques of calculus and linear algebra to study problems in geometry. It has applications in physics , econometrics , and bioinformatics , among others. In particular, differential geometry
16113-423: The surface of the Earth. Few survey positions are derived from the first principles. Instead, most surveys points are measured relative to previously measured points. This forms a reference or control network where each point can be used by a surveyor to determine their own position when beginning a new survey. Survey points are usually marked on the earth's surface by objects ranging from small nails driven into
16244-406: The survey, the surveyor points the instrument in a known direction (bearing), and clamps the lower plate in place. The instrument can then rotate to measure the bearing to other objects. If no bearing is known or direct angle measurement is wanted, the instrument can be set to zero during the initial sight. It will then read the angle between the initial object, the theodolite itself, and the item that
16375-517: The syllabic script Linear B . Beginning in the 8th century BC, however, the Greek alphabet became standard, albeit with some variation among dialects. Early texts are written in boustrophedon style, but left-to-right became standard during the classic period. Modern editions of ancient Greek texts are usually written with accents and breathing marks , interword spacing , modern punctuation , and sometimes mixed case , but these were all introduced later. The beginning of Homer 's Iliad exemplifies
16506-536: The telescope aligns with. The gyrotheodolite is a form of theodolite that uses a gyroscope to orient itself in the absence of reference marks. It is used in underground applications. The total station is a development of the theodolite with an electronic distance measurement device (EDM). A total station can be used for leveling when set to the horizontal plane. Since their introduction, total stations have shifted from optical-mechanical to fully electronic devices. Modern top-of-the-line total stations no longer need
16637-459: The two endpoints. With the Global Positioning System (GPS), elevation can be measured with satellite receivers. Usually, GPS is somewhat less accurate than traditional precise leveling, but may be similar over long distances. When using an optical level, the endpoint may be out of the effective range of the instrument. There may be obstructions or large changes of elevation between the endpoints. In these situations, extra setups are needed. Turning
16768-480: Was Aeolic. For example, fragments of the works of the poet Sappho from the island of Lesbos are in Aeolian. Most of the dialect sub-groups listed above had further subdivisions, generally equivalent to a city-state and its surrounding territory, or to an island. Doric notably had several intermediate divisions as well, into Island Doric (including Cretan Doric ), Southern Peloponnesus Doric (including Laconian ,
16899-459: Was revolutionized by Euclid, whose Elements , widely 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, theorem, and proof. Although most of the contents of the Elements were already known, Euclid arranged them into
17030-504: Was the US Navy TRANSIT system . The first successful launch took place in 1960. The system's main purpose was to provide position information to Polaris missile submarines. Surveyors found they could use field receivers to determine the location of a point. Sparse satellite cover and large equipment made observations laborious and inaccurate. The main use was establishing benchmarks in remote locations. The US Air Force launched
17161-592: Was the creation of analytic geometry, or geometry with coordinates and equations , by René Descartes (1596–1650) and Pierre de Fermat (1601–1665). This was a necessary precursor to the development of calculus and a precise quantitative science of physics . The second geometric development of this period was the systematic study of projective geometry by Girard Desargues (1591–1661). Projective geometry studies properties of shapes which are unchanged under projections and sections , especially as they relate to artistic perspective . Two developments in geometry in
#92907