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Geometry
Oxyrhynchus papyrus with Euclid's Elements.jpg
Oxyrhynchus papyrus (P.Oxy. I 29) showing fragment of Euclid's Elements
History of geometry
Geometry (Ancient Greekγεωμετρία; geo- "earth", -metria "measurement") "Earth-Measuring" is a part of mathematics concerned with questions of size, shape, relative position of figures, and the properties of space. Geometry is one of the oldest sciences. Initially a body of practical knowledge concerning lengths, areas, and volumes, in the 3rd century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow. The field of astronomy, especially mapping the positions of the stars and planets on the celestial sphere, served as an important source of geometric problems during the next one and a half millennia. A mathematician who works in the field of geometry is called a geometer.
The introduction of coordinates by René Descartes and the concurrent development of algebra marked a new stage for geometry, since geometric figures, such as plane curves, could now be represented analytically, i.e., with functions and equations. This played a key role in the emergence of calculus in the 17th century. Furthermore, the theory of perspective showed that there is more to geometry than just the metric properties of figures. The subject of geometry was further enriched by the study of intrinsic structure of geometric objects that originated with Euler and Gauss and led to the creation of topology and differential geometry.
Since the 19th-century discovery of non-Euclidean geometry, the concept of space has undergone a radical transformation. Contemporary geometry considers manifolds, spaces that are considerably more abstract than the familiar Euclidean space, which they only approximately resemble at small scales. These spaces may be endowed with additional structure, allowing one to speak about length. Modern geometry has multiple strong bonds with physics, exemplified by the ties between Riemannian geometry and general relativity. One of the youngest physical theories, string theory, is also very geometric in flavor.
The visual nature of geometry makes it initially more accessible than other parts of mathematics, such as algebra or number theory. However, the geometric language is also used in contexts that are far removed from its traditional, Euclidean provenance, for example, in fractal geometry, and especially in algebraic geometry.[1]

Contents

Overview

Visual proof of the Pythagorean theorem for the (3, 4, 5) triangle as in the Chou Pei Suan Ching 500–200 BC.
Recorded development of geometry spans more than two millennia. It is hardly surprising that perceptions of what constituted geometry evolved throughout the ages.

Practical geometry

There is little doubt that geometry originated as a practical science, concerned with surveying, measurements, areas, and volumes. Among the notable accomplishments one finds formulas for lengths, areas and volumes, such as Pythagorean theorem, circumference and area of a circle, area of a triangle, volume of a cylinder, sphere, and a pyramid. Development of astronomy led to emergence of trigonometry and spherical trigonometry, together with the attendant computational techniques.

Axiomatic geometry

A method of computing certain inaccessible distances or heights based on similarity of geometric figures and attributed to Thales presaged more abstract approach to geometry taken by Euclid in his Elements, one of 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. In the 20th century, David Hilbert employed axiomatic reasoning in his attempt to update Euclid and provide modern foundations of geometry.

Geometric constructions

Ancient scientists paid special attention to constructing geometric objects that had been described in some other way. Classical instruments allowed in geometric constructions are those with compass and straightedge. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found. The approach to geometric problems with geometric or mechanical means is known as synthetic geometry.

Numbers in geometry

Already Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths, which contradicted their philosophical views, made them abandon (abstract) numbers in favor of (concrete) geometric quantities, such as length and area of figures. Numbers were reintroduced into geometry in the form of coordinates by Descartes, who realized that the study of geometric shapes can be facilitated by their algebraic representation. Analytic geometry applies methods of algebra to geometric questions, typically by relating geometric curves and algebraic equations. These ideas played a key role in the development of calculus in the 17th century and led to discovery of many new properties of plane curves. Modern algebraic geometry considers similar questions on a vastly more abstract level.

Geometry of position

Even in ancient times, geometers considered questions of relative position or spatial relationship of geometric figures and shapes. Some examples are given by inscribed and circumscribed circles of polygons, lines intersecting and tangent to conic sections, the Pappus and Menelaus configurations of points and lines. In the Middle Ages new and more complicated questions of this type were considered: What is the maximum number of spheres simultaneously touching a given sphere of the same radius (kissing number problem)? What is the densest packing of spheres of equal size in space (Kepler conjecture)? Most of these questions involved 'rigid' geometrical shapes, such as lines or spheres. Projective, convex and discrete geometry are three sub-disciplines within present day geometry that deal with these and related questions.
Leonhard Euler, in studying problems like the Seven Bridges of Königsberg, considered the most fundamental properties of geometric figures based solely on shape, independent of their metric properties. Euler called this new branch of geometry geometria situs (geometry of place), but it is now known as topology. Topology grew out of geometry, but turned into a large independent discipline. It does not differentiate between objects that can be continuously deformed into each other. The objects may nevertheless retain some geometry, as in the case of hyperbolic knots.

Geometry beyond Euclid

For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori.[2] This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), published only after his death. Riemann's new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.

Symmetry

The theme of symmetry in geometry is nearly as old as the science of geometry itself. The circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail by the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the bewildering graphics of M. C. Escher. Nonetheless, it was not until the second half of 19th century that the unifying role of symmetry in foundations of geometry had been recognized. 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 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' proved most influential. Both discrete and continuous symmetries play prominent role in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry.

Modern geometry

Modern geometry is the title of a popular textbook by Dubrovin, Novikov and Fomenko first published in 1979 (in Russian). At close to 1000 pages, the book has one major thread: geometric structures of various types on manifolds and their applications in contemporary theoretical physics. A quarter century after its publication, differential geometry, algebraic geometry, symplectic geometry and Lie theory presented in the book remain among the most visible areas of modern geometry, with multiple connections with other parts of mathematics and physics.

Contemporary geometers

The E8 Lie Group Petrie Projection
Some of the representative leading figures in modern geometry are Michael Atiyah, Mikhail Gromov and William Thurston. The common feature in their work is the use of smooth manifolds as the basic idea of space; they otherwise have rather different directions and interests. Geometry now is, in large part, the study of structures on manifolds that have a geometric meaning, in the sense of the principle of covariance that lies at the root of general relativity theory in theoretical physics. (See Category:Structures on manifolds for a survey.)
Much of this theory relates to the theory of continuous symmetry, or in other words Lie groups. From the foundational point of view, on manifolds and their geometrical structures, important is the concept of pseudogroup, defined formally by Shiing-shen Chern in pursuing ideas introduced by Élie Cartan. A pseudo-group can play the role of a Lie group of 'infinite' dimension.

Dimension

Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians have used higher dimensions for nearly two centuries. Dimension has gone through stages of being any natural number n, possibly infinite with the introduction of Hilbert space, and any positive real number in fractal geometry. Dimension theory is a technical area, initially within general topology, that discusses definitions; in common with most mathematical ideas, dimension is now defined rather than an intuition. Connected topological manifolds have a well-defined dimension; this is a theorem (invariance of domain) rather than anything a priori.
The issue of dimension still matters to geometry, in the absence of complete answers to classic questions. Dimensions 3 of space and 4 of space-time are special cases in geometric topology. Dimension 10 or 11 is a key number in string theory. Exactly why is something to which research may bring a satisfactory geometric answer.

Contemporary Euclidean geometry

The study of traditional Euclidean geometry is by no means dead. It is now typically presented as the geometry of Euclidean spaces of any dimension, and of the Euclidean group of rigid motions. The fundamental formulae of geometry, such as the Pythagorean theorem, can be presented in this way for a general inner product space.
Euclidean geometry has become closely connected with computational geometry, computer graphics, convex geometry, discrete geometry, and some areas of combinatorics. Momentum was given to further work on Euclidean geometry and the Euclidean groups by crystallography and the work of H. S. M. Coxeter, and can be seen in theories of Coxeter groups and polytopes. Geometric group theory is an expanding area of the theory of more general discrete groups, drawing on geometric models and algebraic techniques.

Algebraic geometry

The field of algebraic geometry is the modern incarnation of the Cartesian geometry of co-ordinates. After a turbulent period of axiomatization, its foundations are stable in the 21st century. Either one studies the "classical" case where the spaces are complex manifolds that can be described by algebraic equations; or the scheme theory provides a technically sophisticated theory based on general commutative rings.
The geometric style which was traditionally called the Italian school is now known as birational geometry. It has made progress in the fields of threefolds, singularity theory and moduli spaces, as well as recovering and correcting the bulk of the older results. Objects from algebraic geometry are now commonly applied in string theory, as well as diophantine geometry.
Methods of algebraic geometry rely heavily on sheaf theory and other parts of homological algebra. The Hodge conjecture is an open problem that has gradually taken its place as one of the major questions for mathematicians. For practical applications, Gröbner basis theory and real algebraic geometry are major subfields.

Differential geometry

Differential geometry, which in simple terms is the geometry of curvature, has been of increasing importance to mathematical physics since the suggestion that space is not flat space. Contemporary differential geometry is intrinsic, meaning that space is a manifold and structure is given by a Riemannian metric, or analogue, locally determining a geometry that is variable from point to point.
This approach contrasts with the extrinsic point of view, where curvature means the way a space bends within a larger space. The idea of 'larger' spaces is discarded, and instead manifolds carry vector bundles. Fundamental to this approach is the connection between curvature and characteristic classes, as exemplified by the generalized Gauss-Bonnet theorem.

Topology and geometry

A thickening of the trefoil knot
The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the dictum 'topology is rubber-sheet geometry'. Contemporary geometric topology and differential topology, and particular subfields such as Morse theory, would be counted by most mathematicians as part of geometry. Algebraic topology and general topology have gone their own ways.

Axiomatic and open development

The model of Euclid's Elements, a connected development of geometry as an axiomatic system, is in a tension with René Descartes's reduction of geometry to algebra by means of a coordinate system. There were many champions of synthetic geometry, Euclid-style development of projective geometry, in the 19th century, Jakob Steiner being a particularly brilliant figure. In contrast to such approaches to geometry as a closed system, culminating in Hilbert's axioms and regarded as of important pedagogic value, most contemporary geometry is a matter of style. Computational synthetic geometry is now a branch of computer algebra.
The Cartesian approach currently predominates, with geometric questions being tackled by tools from other parts of mathematics, and geometric theories being quite open and integrated. This is to be seen in the context of the axiomatization of the whole of pure mathematics, which went on in the period c.1900–c.1950: in principle all methods are on a common axiomatic footing. This reductive approach has had several effects. There is a taxonomic trend, which following Klein and his Erlangen program (a taxonomy based on the subgroup concept) arranges theories according to generalization and specialization. For example affine geometry is more general than Euclidean geometry, and more special than projective geometry. The whole theory of classical groups thereby becomes an aspect of geometry. Their invariant theory, at one point in the 19th century taken to be the prospective master geometric theory, is just one aspect of the general representation theory of algebraic groups and Lie groups. Using finite fields, the classical groups give rise to finite groups, intensively studied in relation to the finite simple groups; and associated finite geometry, which has both combinatorial (synthetic) and algebro-geometric (Cartesian) sides.
An example from recent decades is the twistor theory of Roger Penrose, initially an intuitive and synthetic theory, then subsequently shown to be an aspect of sheaf theory on complex manifolds. In contrast, the non-commutative geometry of Alain Connes is a conscious use of geometric language to express phenomena of the theory of von Neumann algebras, and to extend geometry into the domain of ring theory where the commutative law of multiplication is not assumed.
Another consequence of the contemporary approach, attributable in large measure to the Procrustean bed represented by Bourbakiste axiomatization trying to complete the work of David Hilbert, is to create winners and losers. The Ausdehnungslehre (calculus of extension) of Hermann Grassmann was for many years a mathematical backwater, competing in three dimensions against other popular theories in the area of mathematical physics such as those derived from quaternions. In the shape of general exterior algebra, it became a beneficiary of the Bourbaki presentation of multilinear algebra, and from 1950 onwards has been ubiquitous. In much the same way, Clifford algebra became popular, helped by a 1957 book Geometric Algebra by Emil Artin. The history of 'lost' geometric methods, for example infinitely near points, which were dropped since they did not well fit into the pure mathematical world post-Principia Mathematica, is yet unwritten. The situation is analogous to the expulsion of infinitesimals from differential calculus. As in that case, the concepts may be recovered by fresh approaches and definitions. Those may not be unique: synthetic differential geometry is an approach to infinitesimals from the side of categorical logic, as non-standard analysis is by means of model theory.

History of geometry

Woman teaching geometry. Illustration at the beginning of a medieval translation of Euclid's Elements, (c.1310)
The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia, Egypt, and the Indus Valley from around 3000 BCE. 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 the Egyptian Rhind Papyrus and Moscow Papyrus, the Babylonian clay tablets, and the Indian Shulba Sutras, while the Chinese had the work of Mozi, Zhang Heng, and the Nine Chapters on the Mathematical Art, edited by Liu Hui.
Euclid's Elements (c. 300 BCE) was one of the most important early texts on geometry, in which he presented geometry in an ideal axiomatic form, which came to be known as Euclidean geometry. The treatise is not, as is sometimes thought, a compendium of all that Hellenistic mathematicians knew about geometry at that time; rather, it is an elementary introduction to it;[3] Euclid himself wrote eight more advanced books on geometry. We know from other references that Euclid’s was not the first elementary geometry textbook, but the others fell into disuse and were lost.[citation needed]
In the Middle Ages, mathematics in medieval Islam contributed to the development of geometry, especially algebraic geometry[4][5] and geometric algebra.[6] Al-Mahani (b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra.[5] Thābit ibn Qurra (known as Thebit in Latin) (836-901) dealt with arithmetical operations applied to ratios of geometrical quantities, and contributed to the development of analytic geometry.[7] Omar Khayyám (1048-1131) found geometric solutions to cubic equations, and his extensive studies of the parallel postulate contributed to the development of non-Euclidian geometry.[8] The theorems of Ibn al-Haytham (Alhazen), Omar Khayyam and Nasir al-Din al-Tusi on quadrilaterals, including the Lambert quadrilateral and Saccheri quadrilateral, were the first theorems on elliptical geometry and hyperbolic geometry, and along with their alternative postulates, such as Playfair's axiom, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo, Levi ben Gerson, Alfonso, John Wallis, and Giovanni Girolamo Saccheri.[9]
In the early 17th century, there were two important developments in geometry. The first, and most important, 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 is the study of geometry without measurement, just the study of how points align with each other.
Two developments in geometry in the 19th century changed the way it had been studied previously. These were the discovery of non-Euclidean geometries by Lobachevsky, Bolyai and 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, working primarily with tools from mathematical analysis, and introducing 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 traditional type of geometry was recognized as that of homogeneous spaces, those spaces which have a sufficient supply of symmetry, so that from point to point they look just the same. Although various laws concerning lines and angles were known to the Egyptians and the Pythagoreans, the systematic treatment of geometry by the axiomatic method began with the Elements of Euclid. From a small number of explicit axioms, postulates, and definitions Euclid deduces theorems concerning the various figures of geometrical interest. Until the 19th century this work stood as a supreme example of the exercise of reason, which all other intellectual achievements ought to take as a model. With increasing standards of formal rigor it was recognized that Euclid does contain gaps, but fully formalized versions of his geometry have been provided. For example, in the axiomatization of David Hilbert, there are six primitive terms, in that of E. V. Huntington only two: ‘sphere’ and ‘includes’.
In the work of Kant, Euclidean geometry stands as the supreme example of a synthetic a priori construction, representing the way the mind has to think about space, because of the mind's own intrinsic structure. However, only shortly after Kant was writing non-Euclidean geometries were contemplated. They were foreshadowed by the mathematician K. F. Gauss (1777-1855), but the first serious non-Euclidean geometry is usually attributed to the Russian mathematician N. I. Lobachevsky, writing in the 1820s. Euclid's fifth axiom, the axiom of parallels, states that through any point not falling on a straight line, one straight line can be drawn that does not intersect the first. In Lobachevsky's geometry several such lines can exist. Later G. F. B. Riemann (1822-66) realized that the two-dimensional geometry that would be hit upon by persons confined to the surface of a sphere would be different from that of persons living on a plane: for example, π would be smaller, since the diameter of a circle, as drawn on a sphere, is relatively large compared to the circumference. In the figure, BCB, the circumference of the circle, is less than 2π AB, where AB is the radius. Generalizing, Riemann reached the idea of a geometry in which there are no straight lines that do not intersect a given straight line, just as on a sphere all great circles (the shortest distance between two points) intersect.
The way then lay open to separating the question of the mathematical nature of a purely formal geometry from the question of its physical application. In 1854 Riemann showed that space of any curvature could be described by a set of numbers known as its metric tensor. For example, ten numbers suffice to describe the point of any four-dimensional manifold. To apply a geometry means finding coordinative definitions correlating the notions of the geometry, notably those of a straight line and an equal distance, with physical phenomena such as the path of a light ray, or the size of a rod at different times and places. The status of these definitions has been controversial, with some such as Poincaré seeing them simply as conventions, and others seeing them as important empirical truths. With the general rise of holism in the philosophy of science the question of status has abated a little, it being recognized simply that the co-ordination plays a fundamental role in physical science. See also relativity theory, space-time

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  1. ^ It is quite common in algebraic geometry to speak about geometry of algebraic varieties over finite fields, possibly singular. From a naïve perspective, these objects are just finite sets of points, but by invoking powerful geometric imagery and using well developed geometric techniques, it is possible to find structure and establish properties that make them somewhat analogous to the ordinary spheres or cones.
  2. ^ Kline (1972) "Mathematical thought from ancient to modern times", Oxford University Press, p. 1032. Kant did not reject the logical (analytic a priori) possibility of non-Euclidean geometry, see Jeremy Gray, "Ideas of Space Euclidean, Non-Euclidean, and Relativistic", Oxford, 1989; p. 85. Some have implied that, in light of this, Kant had in fact predicted the development of non-Euclidean geometry, cf. Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164.
  3. ^ Boyer (1991). "Euclid of Alexandria". pp. 104. "The Elements was not, as is sometimes thought, a compendium of all geometric knowledge; it was instead an introductory textbook covering all elementary mathematics-" 
  4. ^ R. Rashed (1994), The development of Arabic mathematics: between arithmetic and algebra, London
  5. ^ a b O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html .
  6. ^ Boyer (1991). "The Arabic Hegemony". pp. 241–242. "Omar Khayyam (ca. 1050-1123), the "tent-maker," wrote an Algebra that went beyond that of al-Khwarizmi to include equations of third degree. Like his Arab predecessors, Omar Khayyam provided for quadratic equations both arithmetic and geometric solutions; for general cubic equations, he believed (mistakenly, as the 16th century later showed), arithmetic solutions were impossible; hence he gave only geometric solutions. The scheme of using intersecting conics to solve cubics had been used earlier by Menaechmus, Archimedes, and Alhazan, but Omar Khayyam took the praiseworthy step of generalizing the method to cover all third-degree equations (having positive roots). .. For equations of higher degree than three, Omar Khayyam evidently did not envision similar geometric methods, for space does not contain more than three dimensions, ... One of the most fruitful contributions of Arabic eclecticism was the tendency to close the gap between numerical and geometric algebra. The decisive step in this direction came much later with Descartes, but Omar Khayyam was moving in this direction when he wrote, "Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved."" 
  7. ^ O'Connor, John J.; Robertson, Edmund F., "Al-Sabi Thabit ibn Qurra al-Harrani", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Thabit.html .
  8. ^ O'Connor, John J.; Robertson, Edmund F., "Omar Khayyam", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Khayyam.html .
  9. ^ Boris A. Rosenfeld and Adolf P. Youschkevitch (1996), "Geometry", in Roshdi Rashed, ed., Encyclopedia of the History of Arabic Science, Vol. 2, p. 447-494 [470], Routledge, London and New York:
    "Three scientists, Ibn al-Haytham, Khayyam and al-Tusi, had made the most considerable contribution to this branch of geometry whose importance came to be completely recognized only in the 19th century. In essence their propositions concerning the properties of quadrangles which they considered assuming that some of the angles of these figures were acute of obtuse, embodied the first few theorems of the hyperbolic and the elliptic geometries. Their other proposals showed that various geometric statements were equivalent to the Euclidean postulate V. It is extremely important that these scholars established the mutual connection between tthis postulate and the sum of the angles of a triangle and a quadrangle. By their works on the theory of parallel lines Arab mathematicians directly influenced the relevant investiagtions of their European counterparts. The first European attempt to prove the postulate on parallel lines - made by Witelo, the Polish scientists of the 13th century, while revising Ibn al-Haytham's Book of Optics (Kitab al-Manazir) - was undoubtedly prompted by Arabic sources. The proofs put forward in the 14th century by the Jewish scholar Levi ben Gerson, who lived in southern France, and by the above-mentioned Alfonso from Spain directly border on Ibn al-Haytham's demonstration. Above, we have demonstrated that Pseudo-Tusi's Exposition of Euclid had stimulated both J. Wallis's and G. Saccheri's studies of the theory of parallel lines."

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1911 encyclopedia

Up to date as of January 14, 2010

From LoveToKnow 1911

.GEOMETRY, the general term for the branch of.^ GEOMETRY , the general term for the branch of.

^ Geometry is the Branch of math known for shapes (polygons), 3D figures, undefined terms, theorems, axioms, explanation of the universe, and pi.
  • WikiAnswers - Geometry Questions including "What is a tessellation" 16 January 2010 23:55 UTC wiki.answers.com [Source type: FILTERED WITH BAYES]

^ Geomtry Glossary General listing of common Geometry terms.
  • Geometry Dictionary, Glossary and Terms directory. 16 January 2010 23:55 UTC www.glossarist.com [Source type: Reference]

.mathematics which has for its province the study of the properties of space.^ He enlarged the world of geometries yet again in another major way, declaring that a geometry is the study of those properties which are preserved by a group of transformations, in any space, whether metric or not.
  • Some History of Geometry 16 January 2010 23:55 UTC www.math.wichita.edu [Source type: FILTERED WITH BAYES]

^ This part of mathematics implies questions of size, shape, and relative position of figures and with properties of space.
  • Geometry Assignment | Online Geometry Homework Help | Geometry Assistance 16 January 2010 23:55 UTC www.assignmentexpert.com [Source type: General]

^ Mathematics can be subdivided into the study of quantity, structure, space, and change.
  • Go Geometry Step by Step from the Land of the Incas, Cuzco, Machu Picchu. Elearning. 16 January 2010 23:55 UTC www.gogeometry.com [Source type: FILTERED WITH BAYES]

.From experience, or possibly intuitively, we characterize existent space by certain fundamental qualities, termed axioms, which are insusceptible of proof; and these axioms, in conjunction with the mathematical entities of the point, straight line, curve, surface and solid, appropriately defined, are the premises from which the geometer draws conclusions.^ These are called chief-tangent curves; on a quadric surface they are the above straight lines.

^ Draw the straight line a through point P at right angles with the segment P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ From experience, or possibly intuitively, we characterize existent space by certain fundamental qualities, termed axioms, which are insusceptible of proof; and these axioms, in conjunction with the mathematical entities of the point, straight line, curve , surface and solid, appropriately defined, are the premises from which the geometer draws conclusions.

.The geometrical axioms are merely conventions; on the one hand, the system may be based upon inductions from experience, in which case the deduced geometry may be regarded as a branch of physical science; or, on the other hand, the system may be formed by purely logical methods, in which case the geometry is a phase of pure mathematics.^ Music is, after all, based on geometry and mathematics.

^ Obviously, geometry and mathematics march hand in hand; one is implied in the other.
  • Earth/matriX:The Geometry of Ancient Sites 16 January 2010 23:55 UTC earthmatrix.com [Source type: FILTERED WITH BAYES]

^ Note : Plane geometry is one of the oldest branches of mathematics .
  • Plane geometry Definition | Definition of Plane geometry at Dictionary.com 16 January 2010 23:55 UTC dictionary.reference.com [Source type: General]

.Obviously the geometry with which we are most familiar is that of existent spacethe three-dimensional space of experience; this geometry may be termed Euclidean, after its most famous expositor.^ Obviously the geometry with which we are most familiar is that of existent spacethe three-dimensional space of experience; this geometry may be termed Euclidean, after its most famous expositor.

^ The universe can be described as a three dimensional space.

^ The book utilizes dynamic geometry software, specifically Geometer's Sketchpad ® , to explore the statements and proofs of many of the most interesting theorems in advanced Euclidean geometry.
  • FOUNDATIONS OF GEOMETRY 16 January 2010 23:55 UTC www.calvin.edu [Source type: General]

.But other geometries exist, for it is possible to frame systems of axioms which definitely characterize some other kind of space, and from these axioms to deduce a series of non-contradictory propositions; such geometries are called nori-Euclidean.^ Some history of the development of Euclidean Geometry.
  • College Geometry 16 January 2010 23:55 UTC www.westminster.edu [Source type: FILTERED WITH BAYES]

^ The meaning of Non-Euclidean Geometry and related axiomatic systems.
  • College Geometry 16 January 2010 23:55 UTC www.westminster.edu [Source type: FILTERED WITH BAYES]

^ Traditionally, only some of these metric geometries were studied: the so-called definite ones.
  • Some History of Geometry 16 January 2010 23:55 UTC www.math.wichita.edu [Source type: FILTERED WITH BAYES]

It is convenient to discuss the subject-matter of geometry under the following headings:
.I. Euclidean Geometry: a discussion of the axioms of existent space and of the geometrical entities, followed by a synoptical account of Euclids Elements.^ I. Euclidean Geometry: a discussion of the axioms of existent space and of the geometrical entities, followed by a synoptical account of Euclids Elements.

^ Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.
  • Geometry Tutor - Geometry Tutoring - High School Geometry Tutor 16 January 2010 23:55 UTC steppingstonetutors.com [Source type: FILTERED WITH BAYES]

^ His most famous work is the Elements, a book in which he deduces the properties of geometrical objects and integers from a set of axioms, thereby anticipating the axiomatic method of modern mathematics.
  • Awesome Library - Mathematics - Middle-High School Math - Geometry 16 January 2010 23:55 UTC www.awesomelibrary.org [Source type: Reference]

.II, Projective Geometry: primarily Euclidean, but differing from I. in employing the notion of geometrical continuity points and lines at infinity.^ In Euclidean geometry, lines are infinite in length.
  • Geometry In Space Activity 16 January 2010 23:55 UTC universe.sonoma.edu [Source type: Original source]

^ Projective Geometry: Lines (the real projective plane), and 4.
  • Read This: Geometry 16 January 2010 23:55 UTC www.maa.org [Source type: FILTERED WITH BAYES]

^ The Journal of Algebraic Geometry is devoted to research articles in algebraic geometry, singularity theory, and related subjects such as number theory, commutative algebra, projective geometry, complex geometry, and geometric topology.
  • Journal of Algebraic Geometry Online 16 January 2010 23:55 UTC www.ams.org [Source type: Academic]

.III. Descriptive Geometry: the methods for representing upon planes figures placed in space of three dimensions.^ III. Of three dimensions.

^ Types of Geometry - Types of Geometry Euclidean geometry, elementary geometry of two and three dimensions (plane and ...
  • Geometry — FactMonster.com 16 January 2010 23:55 UTC www.factmonster.com [Source type: Reference]

^ III. Descriptive Geometry: the methods for representing upon planes figures placed in space of three dimensions.

.IV. Analytical Geometry: the representation of geometrical figures and their relations by algebraic equations.^ Algebraic and analytic geometry .
  • analytic geometry Facts, information, pictures | Encyclopedia.com articles about analytic geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ How is analytic geometry related to algebraic geometry?
  • analytic geometry Facts, information, pictures | Encyclopedia.com articles about analytic geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ Informally, that means that we can do geometry by algebraic equations.

.V. Line Geometry: an analytical treatment of the line regarded as the space element.^ V. Line Geometry: an analytical treatment of the line regarded as the space element.

^ The non-Euclidean geometries are exactly analogous to the geometry of Euclid, except that Euclid's postulate regarding parallel lines is replaced and all theorems depending on this postulate are changed accordingly.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ In solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines, λ, μ, and ν, the cosines of the angles the line makes with the x-, y-, and z -axes, respectively; these satisfy the relationship λ 2 +μ 2 +ν 2 = 1.
  • analytic geometry Facts, information, pictures | Encyclopedia.com articles about analytic geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

.VT. Non-Euclidean Geometry: a discussion of geometries other than that of the space of experience.^ Non-Euclidean geometry .
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^ Geometry and the calculus - Axiomatic Euclidean and non-Euclidean...
  • geometry (mathematics) -- Britannica Online Encyclopedia 16 January 2010 23:55 UTC www.britannica.com [Source type: Reference]

^ VT. Non-Euclidean Geometry: a discussion of geometries other than that of the space of experience.

.VII. Axioms of Geometry: a critical analysis of the foundations of geometry.^ VII. Axioms of Geometry: a critical analysis of the foundations of geometry.

^ In Foundations of Geometry , that appeared in 1899, he listed 21 axioms and analyzed their significance.
  • What Is Geometry? from Interactive Mathematics Miscellany and Puzzles 16 January 2010 23:55 UTC www.cut-the-knot.org [Source type: FILTERED WITH BAYES]

^ But previously a description of the chief characteristic properties of elliptic and of hyperbolic geometries will be given, assuming the standpoint arrived at below under VII. Axioms of Geometry.

.Special subjects are treated under their own headings: e.g.^ Special subjects are treated under their own headings: e.g.

^ It is convenient to discuss the subject-matter of geometry under the following headings: .

.Projection, Perspective; Curve, Surface; Circle, Conic section; triangle, Polygon, Polyhedron; there are also articles on special curves and figures, e.g.^ We may also project points and curves on the surface.

^ Special lines and circles in triangles .
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^ Projection , Perspective ; Curve , Surface ; Circle , Conic section ; triangle , Polygon , Polyhedron ; there are also articles on special curves and figures, e.g.

.Ellipse, Parabola, hyperbola; Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron, Cardioid, Catenary, Cissoid, Cochleoid, Cycloid, Epicycloid, Lfnaox, Oval, Quadratrix, Spiral, &c.^ Similar calculations establish the number of faces, edges, and vertices on the tetrahedron, cube, and icosahedron.
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

^ These Polyhedrons include the Cube, Triangular Prism, Square Pyramid, Tetrahedron, Octahedron, Dodecahedron, Icosahedron and Polyhedra Names which will appear on the assorted worksheets .
  • Geometry Worksheets. Free Worksheets with Angles, Shapes, Polygons, Solids and More! | Geometry Worksheets Org 16 January 2010 23:55 UTC geometryworksheets.org [Source type: Reference]

^ From left to right we see: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron.
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

History

.The origin of geometry (Gr.^ The origin of geometry (Gr.

.1i~, earth, uiTPOP, a measure) is, according to Herodotus, to be found in the etymology of the word.^ In fact, the word geometry means “measurement of the Earth”, and the Earth is (more or less) a sphere.
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

^ It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in surveying, and its name is derived from Greek words meaning "Earth measurement."
  • Geometry Definition | Definition of Geometry at Dictionary.com 16 January 2010 23:55 UTC dictionary.reference.com [Source type: FILTERED WITH BAYES]

^ Current revision (unreviewed) Jump to: navigation , search The word geometry Originates from the Greek words ( geo meaning world, metri meaning measure) and means, literally, to measure the earth.
  • Geometry - Wikibooks, collection of open-content textbooks 16 January 2010 23:55 UTC en.wikibooks.org [Source type: Academic]

.Its birthplace was Egypt, and it arose from thi need of surveying the lands inundated by the Nile floods.^ Its birthplace was Egypt , and it arose from thi need of surveying the lands inundated by the Nile floods.

^ The granddaddy of all science s, geometry probably arose out of techniques developed in Ancient Egypt to re-survey land after the Nile River's annual flood.
  • geometry@Everything2.com 16 January 2010 23:55 UTC www.everything2.com [Source type: FILTERED WITH BAYES]

.Jr its infancy it therefore consisted of a few rules, very rough and approximate, for computing the areas of triangles and quadrilaterals; and, with the Egyptians, it proceeded no further, the geometrical entitiesthe point, line, surface and solidbeing only discussed in so far as they were involved in practical affairs, The point was realized as a mark or position, a straight line as a stretched string or the tracing of a pole, a surface as an area; but these units were not abstracted; and for the Egyptians geometry was only an artan auxiliary to surveying.1 The first step towards its elevation to the rank of a science was made by Thales of Miletus, who transplanted the elementary Egyptian mensuration to Greece.^ These are called chief-tangent curves; on a quadric surface they are the above straight lines.

^ They override this function to compute the area.

^ A ruled quadric surface contains two sets of straight lines.

.Thales clearly abstracted the notions of points and lines, founding the geometry of the latter unit, and discovering per saltum many propositions concerning areas, the circle, &c.^ In Euclidean geometry, a line is made up of infinitely many points, each of which has zero area.
  • circular geometry 16 January 2010 23:55 UTC www.flowresearch.com [Source type: FILTERED WITH BAYES]

^ Initially a body of practical knowledge concerning lengths, areas, and volumes, in the third century BC geometry was put into an axiomatic form by Euclid, whose treatment—Euclidean geometry—set a standard for many centuries to follow.
  • Geometry Tutor - Geometry Tutoring - High School Geometry Tutor 16 January 2010 23:55 UTC steppingstonetutors.com [Source type: FILTERED WITH BAYES]

^ In geometry, a chord is often used to describe a line segment joining two endpoints that lie on a circle.
  • Geometry and the Circle 16 January 2010 23:55 UTC www.mathgoodies.com [Source type: Original source]

.The empirical rules of the Egyptians were corrected and developed by the Ionic School which he founded, especially by Anaximander and Anaxagoras, and in the 6th century B.C. passed into the care of the Pythagoreans.^ The empirical rules of the Egyptians were corrected and developed by the Ionic School which he founded, especially by Anaximander and Anaxagoras , and in the 6th century B.C. passed into the care of the Pythagoreans.

.From this time geometry exercised a powerful influence on Greek thought.^ From this time geometry exercised a powerful influence on Greek thought.

^ At this time thinking was dominated by Kant who had stated that Euclidean geometry is the inevitable necessity of thought and Gauss disliked controversy.

^ Just as the Greeks based their art on tactile qualities, they didn't stray far from this way of thought in their geometry.

.Pythagoras, seeking the key of the universe in arithmetic and geometry, investigated logically the principles underlying the, known propositions; and this resulted in the formulation of definitions, axioms and postulates which, in addition to founding a science of geometry, permitted a crystallization, fractional, it is true, of the amorphous collection of material at hand.^ Like Euclids geometry, Circular Geometry can be formulated as a series of axioms and definitions.
  • circular geometry 16 January 2010 23:55 UTC www.flowresearch.com [Source type: FILTERED WITH BAYES]

^ Pythagoras , seeking the key of the universe in arithmetic and geometry, investigated logically the principles underlying the, known propositions; and this resulted in the formulation of definitions, axioms and postulates which, in addition to founding a science of geometry, permitted a crystallization , fractional, it is true, of the amorphous collection of material at hand.

^ Meanwhile, Schweikart had found a noneuclidean geometry by 1816 [ 16 ], but did not take the final step of observing that it could not be determined if the universe were Euclidean or not.
  • Some History of Geometry 16 January 2010 23:55 UTC www.math.wichita.edu [Source type: FILTERED WITH BAYES]

.Pythagorean geometry was essentially a geometry of areas and solids; its goal was the regular solids the tetrahedron, cube, octahedron, dodecahedron and icosahedronwhich symbolized the five elements of Greek cosmology.^ From left to right we see: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron.
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

^ Yet it contains the and geometry of the cube, tetrahedron, and octahedron.
  • The Dodecahedron 16 January 2010 23:55 UTC www.kjmaclean.com [Source type: FILTERED WITH BAYES]

^ Tetrahedron Hexahedron (cube) Octahedron Dodecahedron Icosahedron .
  • Sacred Geometry Home Page by Bruce Rawles 16 January 2010 23:55 UTC www.geometrycode.com [Source type: General]

.The geometry of the circle,, previously studied in Egypt and much more seriously by Tbales, was somewhat neglected, although this curve was regarded as the most perfect of all plane figures and the sphere the most perfect of all solids.^ The geometry of the circle,, previously studied in Egypt and much more seriously by Tbales, was somewhat neglected, although this curve was regarded as the most perfect of all plane figures and the sphere the most perfect of all solids.

^ Geometry has be-en i as the study of series of two or more dimensions.

^ The study of these is called solid geometry.
  • Undefined: Points, Lines, and Planes 16 January 2010 23:55 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

.The circle, however, was taken up by the Sophists, who made most of their discoveries in attempts to solve the classical problems of squaring the circle, doubling the cube and trisecting an angle.^ The three classical problems - - - Doubling the cube .
  • geometry (mathematics) -- Britannica Online Encyclopedia 16 January 2010 23:55 UTC www.britannica.com [Source type: Reference]

^ The circle, however, was taken up by the Sophists , who made most of their discoveries in attempts to solve the classical problems of squaring the circle, doubling the cube and trisecting an angle.

^ The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry for this purpose in the late 18th cent.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

These problems, besides stimulating pure geometry, i.e. the geometry of constructions made by the ruler and compasses, exercised considerable influence mother directions. The first problem led to the discovery of the method of exhaustion for determining areas. Antiphon. inscribed a square in a circle, and on each side an isosceles .triangle having its vertex on the circle; on the sides of the octagon so obtained, isosceles triangles were again constructed, the process leading to inscribed polygons of 8, 16 and 32 sides; and the areas of these polygons, which are easily determined, are, successive approximations to the area of the circle.^ We also obtain easily for the area of a triangle the formula R1(irABC).

^ A triangle with an inscribed circle .
  • BRUNNERMATH -> your math connection 16 January 2010 23:55 UTC www.brunnermath.com [Source type: FILTERED WITH BAYES]

^ Triangles have 3 sides, squares have 4, pentagons have 5 and octagons have 8.
  • Geometry 16 January 2010 23:55 UTC www.ndt-ed.org [Source type: Original source]

.Bryson of Heraclea took an important step when he circumscribed, in addition to inscribing, polygons to a circle, but he committed an error in treating the circle as the mean of the two polygons.^ Bryson of Heraclea took an important step when he circumscribed, in addition to inscribing, polygons to a circle, but he committed an error in treating the circle as the mean of the two polygons.

^ A polygon is inscribed in a circle if each vertex of the polygon is a point on the circle.
  • Beginning Algebra Tutorial on Basic Geometry 16 January 2010 23:55 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ This point is the centre of the circles circumscribed about and inscribed in the regular polygon.

.The method of Antiphon, in assuming that by continued division.^ The method of Antiphon , in assuming that by continued division.

a polygon can. be constructed coincident with the circle, demanded that magnitudes are not infinitely divisible. .Much controversy ranged about this point; Aristotle supported the doctrine of infinite divisibility; Zeno attempted to show its absurdity.^ Much controversy ranged about this point; Aristotle supported the doctrine of infinite divisibility; Zeno attempted to show its absurdity.

.The mechanical tracing of loci, a principle initiated by Archytas of Tarentum to solve the last two problems, was a frequent subject for study, and several mechanical curves were thus discovered at subsequent dates (cissoid, conchoid, quadratrix).^ Having solved two problems (Props.

^ The mechanical tracing of loci, a principle initiated by Archytas of Tarentum to solve the last two problems, was a frequent subject for study, and several mechanical curves were thus discovered at subsequent dates (cissoid, conchoid , quadratrix ).

^ The problem of representing three-dimensional objects on a two-dimensional surface was solved by Gaspard Monge, who invented descriptive geometry for this purpose in the late 18th cent.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

.Mention may be made of Hippocrates, who, besides developing the known methods, made a study of similar figures, and, as a consequence, of proportion.^ Mention may be made of Hippocrates , who, besides developing the known methods, made a study of similar figures, and, as a consequence, of proportion.

^ Corresponding sides of similar figures are in proportion to each other.
  • Beginning Algebra Tutorial on Basic Geometry 16 January 2010 23:55 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ If four straight lines be proportionals, the similar rectilineal figures similarly described on them shall also be pro portionals; and if the similar rectilineal figures similarly described on four straight lines be pro portionals, those straight lines shall be proportionals.

.This step is important as bringing into line discontinuous number and continuous magnitude.^ This step is important as bringing into line discontinuous number and continuous magnitude.

^ If we now define a quotient ~ of two lines as the number which multiplie4 into b gives a, so that we see that from the equality of two quotients follows, if we multiply both sides by lid, ~b.d=~d.b, ad = cb.

^ Number Line Assumption Every line is a set of points which can be put into a one-to-one correspondence with the real numbers.
  • Undefined: Points, Lines, and Planes 16 January 2010 23:55 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

~A fresh stimulus was given by, the succeeding .Platonists, who, accepting in part the Pythagorean.^ A fresh stimulus was given by, the succeeding Platonists, who, accepting in part the Pythagorean.

cosmology, made the study of geometry preliminary to that of philosophy. .The many discoveries made by this school were facilitated in no small measure by the clarification of the axioms and definitions, thc logical sequence of propositions which was adopted, and, mor especially, by the formulation of the analytic method, i,e.^ The many discoveries made by this school were facilitated in no small measure by the clarification of the axioms and definitions, thc logical sequence of propositions which was adopted, and, mor especially, by the formulation of the analytic method, i,e.

^ Understand the different roles played by axioms, definitions and theorems in the logical structure of mathematics, especially in geometry: .
  • Math Benchmarks - K | Achieve.org 16 January 2010 23:55 UTC www.achieve.org [Source type: Academic]

^ Pythagoras , seeking the key of the universe in arithmetic and geometry, investigated logically the principles underlying the, known propositions; and this resulted in the formulation of definitions, axioms and postulates which, in addition to founding a science of geometry, permitted a crystallization , fractional, it is true, of the amorphous collection of material at hand.

ol assuming the truth of a proposition and then reasoning to 1
i .For Egyptian geometry see EGYPT. Science and Matherna~ics.^ For Egyptian geometry see EGYPT. Science and Matherna~ics.

^ The granddaddy of all science s, geometry probably arose out of techniques developed in Ancient Egypt to re-survey land after the Nile River's annual flood.
  • geometry@Everything2.com 16 January 2010 23:55 UTC www.everything2.com [Source type: FILTERED WITH BAYES]

known truth. .The main strength of the Platonist geometers lies in stereometry or the geometry of solids.^ The main strength of the Platonist geometers lies in stereometry or the geometry of solids.

.The Pythagoreans had dealt with the sphere and regular solids, but the pyramid, prism, cone and cylinder were but little known until the Platonists took them in hand.^ Identify and classify prisms, pyramids, cylinders and cones based on the shape of their base(s).
  • Mathematics - Geometry Core 16 January 2010 23:55 UTC www.uen.org [Source type: Reference]

^ A solid is a three-dimensional figure such as a cube, cylinder, cone, prism, or pyramid.
  • Geometric Shapes and Figures 16 January 2010 23:55 UTC 42explore.com [Source type: FILTERED WITH BAYES]

^ The Pythagoreans had dealt with the sphere and regular solids, but the pyramid , prism , cone and cylinder were but little known until the Platonists took them in hand.

.Eudoxus established their mensuration, proving the pyramid and cone to have one-third the content of a prism and cylinder on the same base and of the same height, and was probably the discoverer of a proof that the volumes of spheres are as the cubes of their radii.^ Calculates and converts for area of triangles, trapezoids, circles and the surface area and volume of cubes, pyramids, cones, prisms, cylinders and spheres.

^ The height of this pyramid is in Phi ratio to its base.
  • About Sacred Geometry 16 January 2010 23:55 UTC www.spiraloflight.com [Source type: FILTERED WITH BAYES]

^ Every cone is the third part of a cylinder which has the same base, and is of an equal altitude with it.

.The discussion of sections of the cone and cylinder led to the discovery of the three curves named the parabola, ellipse and hyperbola (see COMc SEcTIoN); it is difficult to over-estimate the importance of this discovery; its investigation marks the crowning achievement of Greek geometry, and led in later years to the fundamental theorems and methods of modern geometry.^ The discussion of sections of the cone and cylinder led to the discovery of the three curves named the parabola , ellipse and hyperbola (see COMc SEcTIoN); it is difficult to over-estimate the importance of this discovery; its investigation marks the crowning achievement of Greek geometry, and led in later years to the fundamental theorems and methods of modern geometry.

^ (However, in his later years Carl J. Ellipse proved that it is possible by adding a computer and printer to the list of standard tools.
  • Geometry - Uncyclopedia, the content-free encyclopedia 16 January 2010 23:55 UTC uncyclopedia.wikia.com [Source type: FILTERED WITH BAYES]

^ (See section VI. Non-Euclidean Geometry.

.The presentation of the subject-matter of geometry as a connected and logical series of propositions, prefaced by ~pof or foundations, had been attempted by many; but it is to Euclid that we owe a complete exposition.^ The presentation of the subject-matter of geometry as a connected and logical series of propositions, prefaced by ~pof or foundations, had been attempted by many; but it is to Euclid that we owe a complete exposition.

^ The Axiomatic Approach to Geometry Euclid's Elements organized the geometry then known into a systematic presentation that is still used in many texts.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ This geometry was codified in Euclid’s Elements about 300 bce on the basis of 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic.
  • geometry (mathematics) -- Britannica Online Encyclopedia 16 January 2010 23:55 UTC www.britannica.com [Source type: Reference]

.Little indeed in the Elements is probably original except the arrangement; but in this Euclid surpassed such predecessors as Hippocrates, Leon, pupil of Neocleides, and Theudius of Magnesia, devising an apt logical model, although when scrutinized in the light of modern mathematical conceptions the proofs are riddled with fallacies.^ Little indeed in the Elements is probably original except the arrangement; but in this Euclid surpassed such predecessors as Hippocrates , Leon , pupil of Neocleides, and Theudius of Magnesia , devising an apt logical model, although when scrutinized in the light of modern mathematical conceptions the proofs are riddled with fallacies.

^ His most famous work is the Elements, a book in which he deduces the properties of geometrical objects and integers from a set of axioms, thereby anticipating the axiomatic method of modern mathematics.
  • Awesome Library - Mathematics - Middle-High School Math - Geometry 16 January 2010 23:55 UTC www.awesomelibrary.org [Source type: Reference]

^ This lets students become familiar with the concepts little by little, and constantly keep building upon previous knowledge, just as is normally done in mathematics.
  • A discovery-based high school geometry course; Geometry: A Guided Inquiry with Geometer's Sketchpad and Home Study Companion. Review by Maria Miller. 16 January 2010 23:55 UTC www.homeschoolmath.net [Source type: General]

.According to the commentator Proclus, the Elements were written with a twofold object, first, to introduce the novice to geometry, and secondly, to lead him to the regular solids; conic sections found no place therein.^ According to the commentator Proclus , the Elements were written with a twofold object, first, to introduce the novice to geometry, and secondly, to lead him to the regular solids; conic sections found no place therein.

^ The Greek mathematician Euclid composed the first axiomatic formulation of geometry in his famous 13-volume work entitled the Elements around 300 B .
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^ Common geometric formulas Geometric shapes First: parallel postulate Euclid's Elements Euclidean Geometry Euclidean Space Affine Geometry analytic geometry (needs to be filled!
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.What Euclid did for the line and circle, Apollonius did for the conic sections, but there we have a discoverer as well as editor.^ What Euclid did for the line and circle, Apollonius did for the conic sections, but there we have a discoverer as well as editor.

^ Between Euclid and Apollonius there flourished the illustrious Archimedes , whose geometrical discoveries are mainly concerned with the mensuration of the circle and conic sections, and of the sphere, cone and cylinder, and whose greatest contribution to geometrical method is the elevation of the method of exhaustion to the dignity of an instrument of research.

^ With a centre on the point where these second circles meet the horizontal line, a third, smaller, circle is drawn which intersects with the horizontal line as well as the extended lines of the larger squares.
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.These two works, which contain the greatest contributions to ancient geometry, are treated in detail in Section I. Euclidean Geometry and the articles EUCLID; CONIC SECTION; AP0LL0NIUs.^ These two works, which contain the greatest contributions to ancient geometry, are treated in detail in Section I. Euclidean Geometry and the articles EUCLID; CONIC SECTION ; AP0LL0NIUs.

^ The non-Euclidean geometries are exactly analogous to the geometry of Euclid, except that Euclid's postulate regarding parallel lines is replaced and all theorems depending on this postulate are changed accordingly.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ The Greek mathematician Euclid composed the first axiomatic formulation of geometry in his famous 13-volume work entitled the Elements around 300 B .
  • Geometry (ACP/CP) 16 January 2010 23:55 UTC www.nd.edu [Source type: FILTERED WITH BAYES]

.Between Euclid and Apollonius there flourished the illustrious Archimedes, whose geometrical discoveries are mainly concerned with the mensuration of the circle and conic sections, and of the sphere, cone and cylinder, and whose greatest contribution to geometrical method is the elevation of the method of exhaustion to the dignity of an instrument of research.^ What Euclid did for the line and circle, Apollonius did for the conic sections, but there we have a discoverer as well as editor.

^ Between Euclid and Apollonius there flourished the illustrious Archimedes , whose geometrical discoveries are mainly concerned with the mensuration of the circle and conic sections, and of the sphere, cone and cylinder, and whose greatest contribution to geometrical method is the elevation of the method of exhaustion to the dignity of an instrument of research.

^ The mensuration of the sphere, like that of the circle, the cylinder and the cone, had not been settled in the time of Euclid.

.Apollonius was followed by Nicomedes, the inventor of the conchoid; Diodes, the inventor of the cissoid; Zenodorus, the founder of the study of isoperimetrical figures; Hipparchus, the founder of trigonometry; and Heron the elder, who wrote after the manner of the Egyptians, and primarily directed attention to problems of practical surveying.^ Apollonius was followed by Nicomedes, the inventor of the conchoid ; Diodes, the inventor of the cissoid ; Zenodorus, the founder of the study of isoperimetrical figures; Hipparchus , the founder of trigonometry ; and Heron the elder, who wrote after the manner of the Egyptians, and primarily directed attention to problems of practical surveying.

^ The mechanical tracing of loci, a principle initiated by Archytas of Tarentum to solve the last two problems, was a frequent subject for study, and several mechanical curves were thus discovered at subsequent dates (cissoid, conchoid , quadratrix ).

^ Practice Problems 5a - 5d: Use the following figure to answer the questions .
  • Beginning Algebra Tutorial on Basic Geometry 16 January 2010 23:55 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

.Of the many isolated discoveries made by the later Alexandrian mathematicians, those of Menelaus are of importance.^ Of the many isolated discoveries made by the later Alexandrian mathematicians, those of Menelaus are of importance.

^ These propositions, and also those of Hipparchus , were utilized and developed by Ptolemy , the expositor of trigonometry and discoverer of many isolated propositions.

^ Many of the geometries I describe later relate to this concept of sacred geometry, particularly those relating to the Golden Section and Fibonacci.
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.He showed how to treat spherical triangles, establishing their properties and determining their congruence; his theorem on the products of the segments in which the sides of a triangle are cut by a, line was the foundation on which Carnot erected his theory of transversals.^ Determine congruence of line segments, angles, and polygons.

^ He showed how to treat spherical triangles, establishing their properties and determining their congruence; his theorem on the products of the segments in which the sides of a triangle are cut by a, line was the foundation on which Carnot erected his theory of transversals.

^ Each line segment is called the side .
  • Beginning Algebra Tutorial on Basic Geometry 16 January 2010 23:55 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

.These propositions, and also those of Hipparchus, were utilized and developed by Ptolemy, the expositor of trigonometry and discoverer of many isolated propositions.^ These propositions, and also those of Hipparchus , were utilized and developed by Ptolemy , the expositor of trigonometry and discoverer of many isolated propositions.

^ They appeared in many parts of the world, but it is likely that those originating in Mesopotamia were developed by many of the civilisations that followed in the region, spreading out from there with the advance of Islam.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

^ Of the many isolated discoveries made by the later Alexandrian mathematicians, those of Menelaus are of importance.

.Mention maybe made of the commentator Pappus, whose Mathematical Collections is valuable for its wealth of historical matter; of Theon, an editor of Euclids Elements and commentator of Ptolemys Almagest; of Proclus, a commentator of Euclid; and of Eutocius, a commentator of Apollonius and Archimedes.^ Mention maybe made of the commentator Pappus , whose Mathematical Collections is valuable for its wealth of historical matter; of Theon , an editor of Euclids Elements and commentator of Ptolemys Almagest; of Proclus , a commentator of Euclid; and of Eutocius, a commentator of Apollonius and Archimedes .

^ What Euclid did for the line and circle, Apollonius did for the conic sections, but there we have a discoverer as well as editor.

^ The Arabs more closely followed the Greeks, a plan adopted as a sequel to the translation of the works of Euclid, Apollonius, Archimedes and many others into Arabic.

.The Romans, essentially practical and having no inclination to study science qua science, only had a geometry which sufficed for surveying; and even here there were abundant inaccuracies, the empirical rules employed being akin to those of the Egyptians and Heron.^ The Romans , essentially practical and having no inclination to study science qua science, only had a geometry which sufficed for surveying; and even here there were abundant inaccuracies, the empirical rules employed being akin to those of the Egyptians and Heron .

^ There are many limitations on the forms of GML geometries supported by this parser, but they are too numerous to list here.
  • Geometry (GDAL/OGR 1.7.0 Java bindings API) 16 January 2010 23:55 UTC gdal.org [Source type: Reference]

^ It is one of the oldest branches of mathematics, having arisen in response to such practical problems as those found in surveying, and its name is derived from Greek words meaning “Earth measurement.” Eventually it was realized that geometry need not be limited to the study of flat surfaces ( plane geometry ) and rigid three-dimensional objects (solid geometry) but that even the most abstract thoughts and images might be represented and developed in geometric terms.
  • geometry (mathematics) -- Britannica Online Encyclopedia 16 January 2010 23:55 UTC www.britannica.com [Source type: Reference]

.The Hindus, likewise, gave more attention to computation, and their geometry was either of Greek origin oi~ in the form presented in trigonometry, more particularly connected with arithmetic.^ The Hindus, likewise, gave more attention to computation, and their geometry was either of Greek origin oi~ in the form presented in trigonometry, more particularly connected with arithmetic .

^ For an exhaustive explanation on why 6 Cervélo sizes fit more people than 12 “normal” sizes, check out our geometry tech presentation .
  • http://www.cervelo.com/bikes.aspx?bike=S22009 16 January 2010 23:55 UTC www.cervelo.com [Source type: General]

^ Greek geometry was founded upon the original discoveries of the Egyptians.
  • Foundations of Greek Geometry 16 January 2010 23:55 UTC www.perseus.tufts.edu [Source type: Original source]

.It had no logical foundations; each proposition stood alone; and the results were empirical.^ It had no logical foundations; each proposition stood alone; and the results were empirical.

^ Westphal and Hardy said that "no one has, so far as we know, proposed modelling propositions as vectors while retaining classical logic."
  • The Geometry of Logic 16 January 2010 23:55 UTC finitegeometry.org [Source type: FILTERED WITH BAYES]

^ The presentation of the subject-matter of geometry as a connected and logical series of propositions, prefaced by ~pof or foundations, had been attempted by many; but it is to Euclid that we owe a complete exposition.

.The Arabs more closely followed the Greeks, a plan adopted as a sequel to the translation of the works of Euclid, Apollonius, Archimedes and many others into Arabic.^ The Arabs more closely followed the Greeks, a plan adopted as a sequel to the translation of the works of Euclid, Apollonius, Archimedes and many others into Arabic.

^ The Greek mathematician Euclid composed the first axiomatic formulation of geometry in his famous 13-volume work entitled the Elements around 300 B .
  • Geometry (ACP/CP) 16 January 2010 23:55 UTC www.nd.edu [Source type: FILTERED WITH BAYES]

^ I must go back and look at more of the flowers to see how many other variations in petals there are.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

.Their chief contribution to geometry is exhibited in their solution of algebraic equations by intersecting conics, a step already taken by the Greeks in isolated cases, but only elevated into a method by Omar al Hayyami, who flourished in the IIth century.^ Informally, that means that we can do geometry by algebraic equations.

^ The conic sections are also studied in algebraic geometry .
  • 51: Geometry 16 January 2010 23:55 UTC www.math.niu.edu [Source type: Academic]

^ Their chief contribution to geometry is exhibited in their solution of algebraic equations by intersecting conics, a step already taken by the Greeks in isolated cases, but only elevated into a method by Omar al Hayyami, who flourished in the IIth century.

.During the middle ages little was added to Greek and Arabic geometry.^ During the middle ages little was added to Greek and Arabic geometry.

^ These first three photographs may appear to have little in common with the subject of Arabic geometry, but I am including a note on this type of design here for four reasons.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

^ BIBLIOGRAPHY.FOr an account of the investigations on the axioms of geometry during the Greek period, see M. Cantor, Vorbesungen uber die Geschichte der Mathematik, Bd.

.Leonardo of Pisa wrote a Practica geometriae (1220), wherein Euclidean methods are employed; but it was not until the 14th century that geometry, generally Euclids Elements, became an essential item in university curricula.^ The geometry of Euclid was known for many centuries as 'the' geometry, but is nowadays referred to as Euclidean geometry."
  • Awesome Library - Mathematics - Middle-High School Math - Geometry 16 January 2010 23:55 UTC www.awesomelibrary.org [Source type: Reference]

^ Leonardo of Pisa wrote a Practica geometriae (1220), wherein Euclidean methods are employed; but it was not until the 14th century that geometry, generally Euclids Elements, became an essential item in university curricula.

^ The Greek mathematician Euclid composed the first axiomatic formulation of geometry in his famous 13-volume work entitled the Elements around 300 B .
  • Geometry (ACP/CP) 16 January 2010 23:55 UTC www.nd.edu [Source type: FILTERED WITH BAYES]

.There was, however, no sign of original development, other branches of mathematics, mainly algebra and trigonometry, exercising a greater fascination until the 16th century, when the subject again came into favor.^ There was, however, no sign of original development, other branches of mathematics, mainly algebra and trigonometry, exercising a greater fascination until the 16th century, when the subject again came into favor.

^ While your child may not study algebra in depth until 8th or 9th grade, it's a good idea to introduce her to the subject early in her school career.
  • Geometry Worksheets and Printables | Education.com 16 January 2010 23:55 UTC www.education.com [Source type: General]

^ For the next two millennia, mathematics expanded and refined the foundational theorems of Euclidean geometry, and modern mathematics has made several advances into the development of non-Euclidean geometry.
  • Geometry (ACP/CP) 16 January 2010 23:55 UTC www.nd.edu [Source type: FILTERED WITH BAYES]

.The extraordinary mathematical talent which came into being in the 16th and 17th centuries reacted on geometry and gave rise to all those characters which distinguish modern from ancient geometry.^ Traditional geometry is the basis of all mathematics.
  • Geometry eccentric view - Rafiki 16 January 2010 23:55 UTC www.codefun.com [Source type: FILTERED WITH BAYES]

^ The extraordinary mathematical talent which came into being in the 16th and 17th centuries reacted on geometry and gave rise to all those characters which distinguish modern from ancient geometry.

^ For thousands of years, people have wondered if the basics of geometry came naturally to all humans or if they were something you had to learn through instruction or cultural experiences.
  • We’re hard-wired for basic geometry - Science Mysteries- msnbc.com 16 January 2010 23:55 UTC www.msnbc.msn.com [Source type: News]

.The first innovation of moment was the formulatkni of the principle of geometrical continuity by Kepler.^ The first innovation of moment was the formulatkni of the principle of geometrical continuity by Kepler.

^ In contrast, this paper geometrically derives discrete equations of motion for fluid dynamics from first principles in a purely Eulerian form.
  • Applied Geometry Lab Publications 16 January 2010 23:55 UTC geometry.caltech.edu [Source type: Academic]

^ Descartes in his Gomtrie (1637) was the first to systematize the application of this principle to the inherent first notions of geometry; and the methods which he instituted have become the most potent methods of all in geometrical research.

.The notion of infinity which it involved permitted generalizations and systematizations hitherto unthought of (see GEOMETRICAL CONTINUITY); and the method of indefinite division applied to rectification, and quadrature and cubature problems avoided the cumbrous method of exhaustion and provided more accurate results.^ See general method above.
  • Geometry 16 January 2010 23:55 UTC icee.usm.edu [Source type: Reference]

^ The method of Antiphon , in assuming that by continued division.

^ The notion of infinity which it involved permitted generalizations and systematizations hitherto unthought of (see GEOMETRICAL CONTINUITY); and the method of indefinite division applied to rectification, and quadrature and cubature problems avoided the cumbrous method of exhaustion and provided more accurate results.

.Further progress was made by Bonaventura Cavalieri, who, in his Geometria indivisibilibus continuorutn (1620), devised a method intermediate between that of exhaustion and the infinitesimal calculus of Leibnitz and Newton.^ Further progress was made by Bonaventura Cavalieri, who, in his Geometria indivisibilibus continuorutn (1620), devised a method intermediate between that of exhaustion and the infinitesimal calculus of Leibnitz and Newton .

^ A bar above AB to indicate line segment AB was used in 1647 by Bonaventura Cavalieri (1598-1647) in Geometria indivisibilibae and Exercitationes geometriae sex, according to Cajori.
  • Earliest Uses of Symbols from Geometry 16 January 2010 23:55 UTC jeff560.tripod.com [Source type: Academic]

^ By 1813 he had made little progress and wrote: In the theory of parallels we are even now not further than Euclid .

.The logical basis of his system was corrected by Roberval and Pascal; and their discoveries, taken in conjunction with those of Leibnitz, Newton, and many others in the fluxional calculus, culminated in the branch of our subject known as differential geometry (see INTINITESIMAL CALCULUS; CURVE; SURFACE).^ The logical basis of his system was corrected by Roberval and Pascal; and their discoveries, taken in conjunction with those of Leibnitz , Newton , and many others in the fluxional calculus, culminated in the branch of our subject known as differential geometry (see INTINITESIMAL CALCULUS; CURVE; SURFACE).

^ The Axiomatic Approach to Geometry Euclid's Elements organized the geometry then known into a systematic presentation that is still used in many texts.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ There is an elementary course aimed mainly at secondary education students and to be taken soon after calculus, but then we jump to a beginning course in differential geometry.
  • Read This: Geometry 16 January 2010 23:55 UTC www.maa.org [Source type: FILTERED WITH BAYES]

.A second important advance followed the recognition that conics could be regarded as projections of a circle, a conception which led at the hands of Desargues and Pascal to modern projective geometry and perspective.^ A second important advance followed the recognition that conics could be regarded as projections of a circle, a conception which led at the hands of Desargues and Pascal to modern projective geometry and perspective .

^ Projective geometry, on the other hand, is more general and includes the metric geometries as a special case; pure projective geometry makes no reference to lengths or angle measurements.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ For the next two millennia, mathematics expanded and refined the foundational theorems of Euclidean geometry, and modern mathematics has made several advances into the development of non-Euclidean geometry.
  • Geometry (ACP/CP) 16 January 2010 23:55 UTC www.nd.edu [Source type: FILTERED WITH BAYES]

.A third, and perhaps the most important, advance attended the application of algebra to geometry by Descartes, who thereby founded analytical geometry.^ Algebraic and analytic geometry .
  • analytic geometry Facts, information, pictures | Encyclopedia.com articles about analytic geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ That is geometrys demonstration of the most important truth.
  • Masonic Dictionary | Geometry | www.masonicdictionary.com 16 January 2010 23:55 UTC www.masonicdictionary.com [Source type: Original source]

^ What is the most common application for analytic geometry?
  • analytic geometry Facts, information, pictures | Encyclopedia.com articles about analytic geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

.The new fields thus opened up were diligently explored, but the calculus exercised the greatest attraction and relatively little progress was made in geometry until the beginning of the 19th century, when a new era opened.^ The new fields thus opened up were diligently explored, but the calculus exercised the greatest attraction and relatively little progress was made in geometry until the beginning of the 19th century, when a new era opened.

^ There is an elementary course aimed mainly at secondary education students and to be taken soon after calculus, but then we jump to a beginning course in differential geometry.
  • Read This: Geometry 16 January 2010 23:55 UTC www.maa.org [Source type: FILTERED WITH BAYES]

^ More abstraction occurred in the early 19th century with formulations of non-Euclidean geometry by Janos Bolyai and N. I. Lobachevsky, and differential geometry, based on the application of calculus.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

.Gaspard Monge was the first important contributor, stimulating analytical and differential geometry and founding descriptive geometry in a series of papers and especially in his lectures at the cole polytechnique.^ Who first introduced analytic geometry?
  • analytic geometry Facts, information, pictures | Encyclopedia.com articles about analytic geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ Gaspard Monge was the first important contributor, stimulating analytical and differential geometry and founding descriptive geometry in a series of papers and especially in his lectures at the cole polytechnique.

^ Choose what geometry to focus on: -- A single geometry (for your first time we suggest the first of the series: Torus, due to its power to center and focus ones consciousness.
  • About Sacred Geometry 16 January 2010 23:55 UTC www.spiraloflight.com [Source type: FILTERED WITH BAYES]

.Projective geometry, founded by Desargues, Pascal, Monge and L. N. M. Carnot, was crystallized by J. V. Poncelet, the creator of the modern methods.^ Projective methods had been employed by Desargues (b.
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Because this method requires internal creation and initialization of an CoordinateTransformation object it is significantly more expensive to use this method to transform many geometries than it is to create the CoordinateTransformation in advance, and call transform() with that transformation.
  • Geometry (GDAL/OGR 1.7.0 Java bindings API) 16 January 2010 23:55 UTC gdal.org [Source type: Reference]

^ This creates a new linked drawing, called Mexico Geometry Data in the project.
  • Geometry in Tables 16 January 2010 23:55 UTC demo.manifold.net [Source type: Reference]

.In his Trait des proprits des figures (1822) the line and circular points at infinity, imaginaries, polar reciprocation, homology, crossratio and projection are systematically employed.^ In his Trait des proprits des figures (1822) the line and circular points at infinity, imaginaries, polar reciprocation, homology, crossratio and projection are systematically employed.

^ The figure formed by the points where these lines meet H is the projection of Φ on H from P .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Parallel lines have parallel projections, because points at infinity are projected to infinity.

In Germany, A. F. Mobius, J. Plucker and J. Steiner were making far-reaching contributions. .Mbius, in his Barycentrische Calcul (1827), introduced homogeneous co-ordinates, and also the powerful notion of geometrical transformation, including the special cases of collineation and duality; Plucker, in his Analytischgeometrische Eniwickelungen (1828-1831), and his System der anal ytischen Geometrie (1835), introduced the abridged notation, line and plane co-ordinates, and the conception of generalized space elements; while Steiner, besides enriching geometry in numerous directions, was the first to systematically generate figures by projective pencils.^ A Plane is the path generated by moving a Line through Space.
  • circular geometry 16 January 2010 23:55 UTC www.flowresearch.com [Source type: FILTERED WITH BAYES]

^ The special case of geometry .
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Projective Geometry: Lines (the real projective plane), and 4.
  • Read This: Geometry 16 January 2010 23:55 UTC www.maa.org [Source type: FILTERED WITH BAYES]

.We may also notice M. Chasles, whose Aperu historique (1837) is a classic.^ We may also notice M. Chasles, whose Aperu historique (1837) is a classic.

.Synthetic geometry, characterized by its fruitfulness and beauty, attracted most attention, and it so happened that its originally weak logical foundations became replaced by a more substantial set of axioms.^ Synthetic geometry, characterized by its fruitfulness and beauty, attracted most attention, and it so happened that its originally weak logical foundations became replaced by a more substantial set of axioms.

^ His most famous work is the Elements, a book in which he deduces the properties of geometrical objects and integers from a set of axioms, thereby anticipating the axiomatic method of modern mathematics.
  • Awesome Library - Mathematics - Middle-High School Math - Geometry 16 January 2010 23:55 UTC www.awesomelibrary.org [Source type: Reference]

^ From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry.
  • Awesome Library - Mathematics - Middle-High School Math - Geometry 16 January 2010 23:55 UTC www.awesomelibrary.org [Source type: Reference]

.These were found in the anharmonic ratio, a device leading to the liberation of synthetic geometry from metrical relations, and in involution, which yielded rigorous definitions of imaginaries.^ These were found in the anharmonic ratio, a device leading to the liberation of synthetic geometry from metrical relations, and in involution , which yielded rigorous definitions of imaginaries.

^ Marvin J. Greenberg (Hardcover) Buy new : $97.31 33 used and new from $80.94 Are these related to Geometry?
  • Amazon.com: The Geometry Community 16 January 2010 23:55 UTC www.amazon.com [Source type: General]

^ Thus, if we choose to take any one of these three theorems as an axiom, all the theorems of projective geometry which do not require ordinal or metrical ideas for their enunciation can be proved.

.These innovations were made by K. J. C. von Staudt.^ These innovations were made by K. J. C. von Staudt.

.Analytical geometry was stimulated by the algebra of invariants, a subject much developed by A. Cayley, G. Salmon, S. H. Aronhold, L. 0. Hesse, and more particularly by R. F. A. Clebsch.^ Algebraic and analytic geometry .
  • analytic geometry Facts, information, pictures | Encyclopedia.com articles about analytic geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ This text provides Adefinitive source for the development of intrinsic geometry and is indispensable for graduate students who want a better understanding of this subject.
  • Geometry 16 January 2010 23:55 UTC www.wordtrade.com [Source type: FILTERED WITH BAYES]

^ Grade 12 Math Standards (NCTM) Includes Algebra, Trigonometry, Geometry, Estimation, Probability and Statistics, and more.
  • Awesome Library - Mathematics - Middle-High School Math - Geometry 16 January 2010 23:55 UTC www.awesomelibrary.org [Source type: Reference]

.The introduction of the line as a space element, initiated by H. Grassmann (1844) and Cayley (1859), yielded at the hands of Plucker a new geometry, termed line geometry, a subject developed more notably by F. Klein, Clebsch, C. T. Reye and F. 0. R. Sturm (see Section V., Line Geometry).^ V. Line Geometry: an analytical treatment of the line regarded as the space element.

^ The introduction of the line as a space element, initiated by H. Grassmann (1844) and Cayley (1859), yielded at the hands of Plucker a new geometry, termed line geometry, a subject developed more notably by F. Klein, Clebsch, C. T. Reye and F. 0.

^ Lines in spherical geometry are more subtle.
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

.N~n-euclidean geometries, having primarily their origin in the discussion of Euclidean parallels, and treated by Wallis, Saccheri and Lambert, have been especially developed during the f9th century.^ N~n-euclidean geometries, having primarily their origin in the discussion of Euclidean parallels , and treated by Wallis, Saccheri and Lambert , have been especially developed during the f9th century.

^ Develop an understanding of an axiomatic system through investigating and comparing Euclidean and Non-Euclidean geometries This site currently under construction to become 508 compliant.
  • Geometry In Space Activity 16 January 2010 23:55 UTC universe.sonoma.edu [Source type: Original source]

^ It was from endeavours to improve the theory of parallels that non-Euclidean geometry arose; and though it has now acquired a far wider scope, its historical origin remains instructive and interesting.

.Four lines of investigation may be distinguished: the naive-synthetic, associated with Lobatschewski, Bolyai, Gauss; the metric differential, studied by Riemann, Helmholtz, Beltrami; the projective, developed by Cayley, Klein, Clifford; and the critical-synthetic, promoted chiefly by the Italian mathematicians Peano, Veronese, Burali-Forte, Levi Civitt, and the Germans Pasch and Hilbert.^ Four lines of investigation may be distinguished: the naive-synthetic, associated with Lobatschewski, Bolyai, Gauss; the metric differential, studied by Riemann, Helmholtz, Beltrami; the projective, developed by Cayley, Klein, Clifford ; and the critical-synthetic, promoted chiefly by the Italian mathematicians Peano, Veronese, Burali-Forte, Levi Civitt, and the Germans Pasch and Hilbert.

^ The first assumption was investigated by German mathematician Carl Friedrich Gauss (1777-1855), Russian mathematician Nikolai Lobachevsky (1792-1856), and Hungarian mathematician Janos Bolyai (1802-1860).
  • Geometry In Space Activity 16 January 2010 23:55 UTC universe.sonoma.edu [Source type: Original source]

^ We may, however, ignore the accented letters in translating theorems, for a configuration of lines and its polar with respect to a linear complex have the same projective properties.

(C. E.*)
I. EUCLIDEAN GEOMETRY
.This branch of the science of geometry is so named since its methods and arrangement are those laid down in Euclids Elements.^ This branch of the science of geometry is so named since its methods and arrangement are those laid down in Euclids Elements.

^ One theory is that geometry originated when whiny man named Euclid attempted to accomplish something productive.
  • Geometry - Uncyclopedia, the content-free encyclopedia 16 January 2010 23:55 UTC uncyclopedia.wikia.com [Source type: FILTERED WITH BAYES]

^ The Axiomatic Approach to Geometry Euclid's Elements organized the geometry then known into a systematic presentation that is still used in many texts.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

.I. Axioms.The object of geometry is to investigate the properties of space.^ Geometry Duplicates the geometry properties to another geometry object.
  • away3d.core.base.Geometry 16 January 2010 23:55 UTC away3d.com [Source type: FILTERED WITH BAYES]
  • away3d.core.base.Geometry 16 January 2010 23:55 UTC www.away3d.com [Source type: Reference]

^ I. Axioms.The object of geometry is to investigate the properties of space.

^ ARTICLE from the Encyclopædia Britannica mathematics the branch of mathematics concerned with the shape of individual objects, spatial relationships among various objects, and the properties of surrounding space.
  • geometry (mathematics) -- Britannica Online Encyclopedia 16 January 2010 23:55 UTC www.britannica.com [Source type: Reference]

.The first step must consist in establishing those fundamental properties from which all others follow by processes of deductive reasoning.^ The first step must consist in establishing those fundamental properties from which all others follow by processes of deductive reasoning.

^ Standard 5 Spatial Sense and Geometric Concepts The student will investigate, model, and apply geometric properties and relationships and use indirect reasoning to make conjectures; deductive reasoning to draw conclusions; and both inductive and deductive reasoning to establish the truth of statements.
  • Sites to use to practice skills needed on the Geometry end of course assessment 16 January 2010 23:55 UTC www.internet4classrooms.com [Source type: Reference]

^ That is, it may also make sense to put it into the proof in an order other than the first successive steps of the proof.
  • Mastering the Formal Geometry Proof - For Dummies 16 January 2010 23:55 UTC www.dummies.com [Source type: General]

.They are laid down in the Axioms, and these ought to form such a system that nothing need be added to them in order fully to characterize space, and that nothing may be omitted without making the system incomplete.^ They are laid down in the Axioms, and these ought to form such a system that nothing need be added to them in order fully to characterize space, and that nothing may be omitted without making the system incomplete.

^ From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry.
  • Awesome Library - Mathematics - Middle-High School Math - Geometry 16 January 2010 23:55 UTC www.awesomelibrary.org [Source type: Reference]

^ The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.

.They must, in fact, completely define space.^ They must, in fact, completely define space.

.2. Definitions.The axioms of Euclidean Geometry are obtained from inspection of existent space and of solids in existent space,hence from experience.^ Definitions.The axioms of Euclidean Geometry are obtained from inspection of existent space and of solids in existent space,hence from experience.

^ This stands in sharp contrast to Euclidean space, which has only three, and to spherical geometry, where there are only five non-degenerate possibilities (corresponding to the Platonic solids).
  • Hyperbolic Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

^ Types of Geometry Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

.The same source gives us the notions of the geometrical entities to which the axioms relate, viz, solids, surfaces, lines or curves, and points.^ This gives two curves which lie in the same plane and whose intersections will give us points on both surfaces.

^ The same source gives us the notions of the geometrical entities to which the axioms relate, viz, solids, surfaces, lines or curves, and points.

^ We may also project points and curves on the surface.

.A solid is directly given by experience; we have only to abstract all material from it in order to gain the notion of a geometrical solid.^ A solid is directly given by experience; we have only to abstract all material from it in order to gain the notion of a geometrical solid.

^ Later on in this book we will discover a remarkable polyhedron that defines the relationship and provides the proper nesting for all 5 Platonic Solids, including the icosahedron, directly on its vertices.
  • The Dodecahedron 16 January 2010 23:55 UTC www.kjmaclean.com [Source type: FILTERED WITH BAYES]

^ These principles of ordered geometric flows (explained in Chapter 1 after two Sturm's Theorems) are the only machinery of intersection comparison Galaktionov is going to use here.
  • Geometry 16 January 2010 23:55 UTC www.wordtrade.com [Source type: FILTERED WITH BAYES]

.This has shape, size, position, and may be moved.^ This has shape, size, position, and may be moved.

^ The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.

^ I, and on a few other occasions, viz, that figures may be moved in space without change of shape or size.

Its boundary or boundaries are called surfaces. .They separate one part of space from another, and are said to have no thickness.^ They separate one part of space from another, and are said to have no thickness.

^ Few wish to admit the obvious; many cling to the fixed idea that the different ancient cultures were completely separated from one another by the distances separating them on this immense planet.
  • Earth/matriX:The Geometry of Ancient Sites 16 January 2010 23:55 UTC earthmatrix.com [Source type: FILTERED WITH BAYES]

^ For example be writes: ~ Space does not represent any quality of objects by themselves, or objects in their relation to one another, i.e.

.Their boundaries are curves or lines, and these have length only.^ The symbol , which includes the arrow heads at both ends indicates the whole line where , which does not have the arrow heads, indicates a line segment, which is finite in length (only the part of the line from A to B).
  • Beginning Algebra Tutorial on Basic Geometry 16 January 2010 23:55 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ Parallel and Perpendicular Lines not only explains these terms, but also studies angles formed when a line intersects two parallel lines.
  • Math Mammoth Geometry 1 workbook - learn elementary geometry by drawing. 16 January 2010 23:55 UTC www.mathmammoth.com [Source type: General]

^ Convexity requires the use of "lines" or "curves of shortest length" and the use of different coordinate systems may result in different versions of the convex hull of an object.

.Their boundaries, again, are points, which have no magnitude but only position.^ I, I. A point is that which has no parts, or which has no magnitude.

^ Their boundaries, again, are points, which have no magnitude but only position.

^ The same thing is intended to be expressed by saying that a solid has length, breadth and thickness, a surface length and breadth, a line length only, and a point no extension whatsoever.

.We thus come in three steps from solids to points which have no magnitude; in each step we lose one extension.^ The equals predicate has the following equivalent definitions: The two geometries have at least one point in common, and no point of either geometry lies in the exterior of the other geometry.

^ Objects can be drawn in one- two- or three-point perspective, depending on how many vanishing points are used.
  • Undefined: Points, Lines, and Planes 16 January 2010 23:55 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

^ If A, B, and C are three distinct points lying on the same line, then one and only one of the points is between the other two.
  • Undefined: Points, Lines, and Planes 16 January 2010 23:55 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

.Hence we say a solid has three dimensions, a surface two, a line one, and a point none.^ (Hence a line has two distinct points at infinity, and not one only as in ordinary geometry.

^ Through two points exactly one straight line can be drawn.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Any two distinct points lie in one and only one straight line.

.Space itself, of which a solid forms only a part, is also said to be of three dimensions.^ Space itself, of which a solid forms only a part, is also said to be of three dimensions.

^ This stands in sharp contrast to Euclidean space, which has only three, and to spherical geometry, where there are only five non-degenerate possibilities (corresponding to the Platonic solids).
  • Hyperbolic Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

^ Types of Geometry Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

.The same thing is intended to be expressed by saying that a solid has length, breadth and thickness, a surface length and breadth, a line length only, and a point no extension whatsoever.^ I. A superficies is that which has only length and breadth.

^ The same thing is intended to be expressed by saying that a solid has length, breadth and thickness, a surface length and breadth, a line length only, and a point no extension whatsoever.

^ Three or more points are collinear if and only if they are on the same line.
  • Undefined: Points, Lines, and Planes 16 January 2010 23:55 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

.Euclid gives the essence of these statements as definitions: Def.^ Euclid points this out in his definitions,Def.

^ Euclid gives the essence of these statements as definitions: Def.

^ These definitions have given rise to much discussion, The only definitions which are essential for the fifth book are Defs.

.I, I. A point is that which has no parts, or which has no magnitude.^ I, I. A point is that which has no parts, or which has no magnitude.

^ We thus come in three steps from solids to points which have no magnitude; in each step we lose one extension.

^ Their boundaries, again, are points, which have no magnitude but only position.

Def. .2, I. A line is length without breadth.^ I. A line is length without breadth.

^ The same thing is intended to be expressed by saying that a solid has length, breadth and thickness, a surface length and breadth, a line length only, and a point no extension whatsoever.

Def. 5, I. A superficies is that which has only length and breadth.
Def. .1, XI. A solid is that which has length, breadth and thickness.^ XI. A solid is that which has length, breadth and thickness.

^ His several para-graphs, here quoted in succession although separated in his Monitor, read as follows: Geometry treats of the powers and properties of magnitudes in general, where length, breadth and thickness are considered; from a point to a line, from a line to a superficies and from a superficies to a solid.
  • Masonic Dictionary | Geometry | www.masonicdictionary.com 16 January 2010 23:55 UTC www.masonicdictionary.com [Source type: Original source]

^ The same thing is intended to be expressed by saying that a solid has length, breadth and thickness, a surface length and breadth, a line length only, and a point no extension whatsoever.

.It is to be noted that the synthetic method is adopted by Euclid; the analytical derivation of the successive ideas of surface, line, and point from the experimental realization of a solid does not find a place in his system, although possessing more advantages.^ It is to be noted that the synthetic method is adopted by Euclid; the analytical derivation of the successive ideas of surface, line, and point from the experimental realization of a solid does not find a place in his system, although possessing more advantages.

^ Recursion of points produces more points, lines, planes and solids.
  • Geometry eccentric view - Rafiki 16 January 2010 23:55 UTC www.codefun.com [Source type: FILTERED WITH BAYES]

^ Returns: 0 for points, 1 for lines and 2 for surfaces.
  • Geometry (GDAL/OGR 1.7.0 Java bindings API) 16 January 2010 23:55 UTC gdal.org [Source type: Reference]

.If we allow motion in geometry, we may generate these entities by moving a point, a line, or a surface, thus: The path of a moving point is a line.^ In spherical geometry, the “points” are points on the surface of the sphere.
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

^ His several para-graphs, here quoted in succession although separated in his Monitor, read as follows: Geometry treats of the powers and properties of magnitudes in general, where length, breadth and thickness are considered; from a point to a line, from a line to a superficies and from a superficies to a solid.
  • Masonic Dictionary | Geometry | www.masonicdictionary.com 16 January 2010 23:55 UTC www.masonicdictionary.com [Source type: Original source]

^ In solid analytic geometry the orientation of a straight line is given not by one slope but by its direction cosines, λ, μ, and ν, the cosines of the angles the line makes with the x-, y-, and z -axes, respectively; these satisfy the relationship λ 2 +μ 2 +ν 2 = 1.
  • analytic geometry Facts, information, pictures | Encyclopedia.com articles about analytic geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

The path of a moving line is, iii general, a surface.
The path of a moving surface is, in general, a solid. .And we may then assume that the lines, surfaces and solids, as defined before, can all be generated in this manner.^ And we may then assume that the lines, surfaces and solids, as defined before, can all be generated in this manner.

^ The path of a moving surface is, in general, a solid.

^ Generally, there are at least three types of distances that may be defined between points (and therefore between geometric objects): map distance, geodesic distance, and terrain distance.

.From this generation of the entities it follows again that the boundaries the first and last position of the moving elementof a line are points, and so on; and thus we come back to the considerations with which we started.^ Compare the following: 2 points determine a line; and 2 lines determine a point.
  • Undefined: Points, Lines, and Planes 16 January 2010 23:55 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

^ From this generation of the entities it follows again that the boundaries the first and last position of the moving elementof a line are points, and so on; and thus we come back to the considerations with which we started.

^ A point that moves without turning will follow a straight line.
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

.Euclid points this out in his definitions,Def.^ Euclid points this out in his definitions,Def.

^ XI. He does not, however, show the connexior which these definitions have with those mentioned before When ,points and lines have been defined, a statement lik Def.

^ Euclid gives the essence of these statements as definitions: Def.

3, I., Def. 6, I., and Def. .2, XI. He does not, however, show the connexior which these definitions have with those mentioned before When ,points and lines have been defined, a statement lik Def.^ To these we add the definition of a line parallel to a plane as a line which does not meet the plane.

^ XI. He does not, however, show the connexior which these definitions have with those mentioned before When ,points and lines have been defined, a statement lik Def.

^ Euclid gives the essence of these statements as definitions: Def.

.3, I., The extremities of a line are points, is a proposition which either has to be proved, and then it is a theorem, or which has to be taken for granted, in which case it is an axiom.^ I., The extremities of a line are points, is a proposition which either has to be proved, and then it is a theorem, or which has to be taken for granted, in which case it is an axiom .

^ A line extends infinitely and is named by labeling two points on the line with capital letters or by putting a lower case letter near it.
  • Beginning Algebra Tutorial on Basic Geometry 16 January 2010 23:55 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ This geometry was codified in Euclid’s Elements about 300 bce on the basis of 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic.
  • geometry (mathematics) -- Britannica Online Encyclopedia 16 January 2010 23:55 UTC www.britannica.com [Source type: Reference]

And so with Def. 6, I., and Def. 2, XI.
.3. Euclids definitions mentioned above are attempts to describe, in a few words, notions which we have obtained by inspection of and abstraction from solids.^ Euclids definitions mentioned above are attempts to describe, in a few words, notions which we have obtained by inspection of and abstraction from solids.

^ Definitions.The axioms of Euclidean Geometry are obtained from inspection of existent space and of solids in existent space,hence from experience.

^ The definitions which have not been mentioned are all nominal definitions, that is to say, they fix a name for a thing described.

.A few more notions have to be added to these, principally those of the simplest linethe straight line, and of the simplest surfacethe flat surface or plane.^ A few more notions have to be added to these, principally those of the simplest linethe straight line, and of the simplest surfacethe flat surface or plane.

^ Of three or more points: to lie on the same straight line; to lie on the same plane.
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ The Straight Line.This is the simplest type of locus.

.These notions we possess, but to define them accurately is difficult.^ These notions we possess, but to define them accurately is difficult.

.Euclids Definition 4, I., A straight line is that which lies evenly between its extreme points, must be meaningless to any one who has not the notion of straightness in his mind.^ If a point lies in two lines its projections must lie in the projections of both.

^ Euclid points this out in his definitions,Def.

^ Through two points exactly one straight line can be drawn.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

.Neither does it slate a property of the straight line which can be used in any further investigation.^ To see this, we used properties of parallel lines.
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

^ Neither does it slate a property of the straight line which can be used in any further investigation.

^ Hence, a straight line which does not lie in the plane cannot have more than one point in common with the plane.

.Such a property is given in Axiom lo, I. It is really this axiom, together with Postulates 2 and 3, which characterizes the straight line.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ Such a property is given in Axiom lo, I. It is really this axiom, together with Postulates 2 and 3, which characterizes the straight line.

.Whilst for the straight line the verbal definition and axiom are kept apart, Euclid mixes them up in the case of the plane.^ Whilst for the straight line the verbal definition and axiom are kept apart, Euclid mixes them up in the case of the plane.

^ If the base BC of a triangle is kept fixed and the vertex A moves in the fixed plane ABC so that the area ABC is constant, then the locus of A is a line of equal distance from BC. This locus is not a straight line.

^ Also the previous definition of an angle can be adapted to this case, by making If and ti to be the tangent planes through the line pipf to the imaginary conic.

.Here the Definition 7, I., includes an axiom.^ Here the Definition 7, I., includes an axiom.

.It defines a plane as a surface which has the property that every straight line which joins any two points in it lies altogether in the surface.^ If a point lies in two lines its projections must lie in the projections of both.

^ Ore the point A, lies in the plane; otherwise not.

^ A ruled quadric surface contains two sets of straight lines.

.But if we take a straight line and a point in such a surface, and draw all straight lines which join the latter to all points in the first line, the surface will be fully determined.^ Draw the straight line a through point P at right angles with the segment P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ As an example, take straight lines.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ To draw a straight line from any point to any point.
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

.Thisconstruction is therefore sufficient as a definition.^ Thisconstruction is therefore sufficient as a definition.

.That every other straight line which joins any two points in this surface lies altogether in it is a further property, and to assume it gives another axiom.^ This means that between any two points is another point.
  • Undefined: Points, Lines, and Planes 16 January 2010 23:55 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

^ If a point lies in two lines its projections must lie in the projections of both.

^ A ruled quadric surface contains two sets of straight lines.

.Thus a number of Euclids axioms are hidden among his first definitions.^ Thus a number of Euclids axioms are hidden among his first definitions.

^ If we wish to collect the axioms used in Euclids Elements, we have therefore to take the three postulates, the last three axioms as generally given, a few axioms hidden in the definitions, and an axiom used by Euclid in the proof of Prop.

^ The difficulty which thus arises is overcome by Euclid assuming that the first question has to be answered in the affirmative.

.A still greater confusion exists in the present editions of Euclid between the postulates and axioms so called, but this is due to later editors and not to Euclid himself.^ Five of the assumptions were called common notions (axioms, or self-evident truths), and the other five were postulates (required conditions).
  • Mathematics Curriculum Framework - November 2000 - Massachusetts Department of Elementary and Secondary Education 16 January 2010 23:55 UTC www.doe.mass.edu [Source type: Reference]

^ The Axiomatic Approach to Geometry Euclid's Elements organized the geometry then known into a systematic presentation that is still used in many texts.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

^ This geometry was codified in Euclid’s Elements about 300 bce on the basis of 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic.
  • geometry (mathematics) -- Britannica Online Encyclopedia 16 January 2010 23:55 UTC www.britannica.com [Source type: Reference]

.The latter had the last three axioms put togetherwith the postulates (air*uara), so that these were meant to include all assumptions relating to space.^ The latter had the last three axioms put togetherwith the postulates (air*uara), so that these were meant to include all assumptions relating to space.

^ Five of the assumptions were called common notions (axioms, or self-evident truths), and the other five were postulates (required conditions).
  • Mathematics Curriculum Framework - November 2000 - Massachusetts Department of Elementary and Secondary Education 16 January 2010 23:55 UTC www.doe.mass.edu [Source type: Reference]

^ Thus, if we choose to take any one of these three theorems as an axiom, all the theorems of projective geometry which do not require ordinal or metrical ideas for their enunciation can be proved.

.The remaining assumptions, which relate to magnitudes in general, viz, the first eight axioms in modern editions, were called common notions (icou-ai ~vpoiai).^ The remaining assumptions, which relate to magnitudes in general, viz, the first eight axioms in modern editions, were called common notions (icou- ai ~vpoiai).

^ Five of the assumptions were called common notions (axioms, or self-evident truths), and the other five were postulates (required conditions).
  • Mathematics Curriculum Framework - November 2000 - Massachusetts Department of Elementary and Secondary Education 16 January 2010 23:55 UTC www.doe.mass.edu [Source type: Reference]

^ In this book figures -are considered which are not confined to a plane, viz, first relations between lines and planes in space, and af terwards properties of solids.

.Of the latter a few may be said to be definitions.^ Of the latter a few may be said to be definitions.

^ We may then state the definitions 3 or 4 thus: Definition.A polygon is said to be inscribed in a circle, and the circle is said to he circumscribed about the polygon, if the vertices of the polygon lie in the circumference of the ircle.

Thus the eighth might be taken as a definition of equal, and the seventh of halves. .If we wish to collect the axioms used in Euclids Elements, we have therefore to take the three postulates, the last three axioms as generally given, a few axioms hidden in the definitions, and an axiom used by Euclid in the proof of Prop.^ Thus a number of Euclids axioms are hidden among his first definitions.

^ If we wish to collect the axioms used in Euclids Elements, we have therefore to take the three postulates, the last three axioms as generally given, a few axioms hidden in the definitions, and an axiom used by Euclid in the proof of Prop.

^ This geometry was codified in Euclid’s Elements about 300 bce on the basis of 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic.
  • geometry (mathematics) -- Britannica Online Encyclopedia 16 January 2010 23:55 UTC www.britannica.com [Source type: Reference]

.4, I, and on a few other occasions, viz, that figures may be moved in space without change of shape or size.^ I, and on a few other occasions, viz, that figures may be moved in space without change of shape or size.

^ This assumes that we may move a length about without changing it.

^ Figures maybe freely moved in space without change of shape or size.

4. Postulates.The assumptions actually made by Euclid may be stated as follows:
.(I) Straight lines exist which have the property that any one of them may be produced both ways without limit, that through any two points in space such a line may be drawn, and that any two of them coincide throughout their indefinite extensions as soon as two points in the one coincide with two points in the other.^ Through any point in space,.

^ Draw the straight line a through point P at right angles with the segment P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ If a point lies in two lines its projections must lie in the projections of both.

.(This gives the contents of Def.^ (This gives the contents of Def.

4, part of Def. 35, the first two Postulates, and Axiom 10.)
.(2) Plane surfaces or planes exist having the property laid down in Def.^ Plane surfaces or planes exist having the property laid down in Def.

^ To do this, the plane is turned about one of its traces till it is laid down into that plane of projection to which the trace belongs.

^ It defines a plane as a surface which has the property that every straight line which joins any two points in it lies altogether in the surface.

.7, that every straight line joining any two points in such a surface lies altogether in it.^ If a point lies in two lines its projections must lie in the projections of both.

^ A ruled quadric surface contains two sets of straight lines.

^ Through two points exactly one straight line can be drawn.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

.(3) Right angles, as defined in Def.^ Right angles, as defined in Def.

.10, are possible, and all right angles are equal; that is to say, wherever in space we take a plane, and wherever in that plane we construct a right angle, all angles thus constructed will be equal, so that any one of them may be made to coincide with any other.^ Every other edge is equal to one of them.

^ The space of planesthat is, all planes in space.

^ If two angles in diffeicent planes have the two limits of the one parallel to those of the other, then the angles are equal.

(Axiom 11.)
.(4) The 12th Axiom of Euclid.^ The 12th Axiom of Euclid.

.This we shall not state now, but only introduce it when we cannot proceed any further without it.^ This we shall not state now, but only introduce it when we cannot proceed any further without it.

^ The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.

.(5) Figures maybe freely moved in space without change of shape or size.^ I, and on a few other occasions, viz, that figures may be moved in space without change of shape or size.

^ Figures maybe freely moved in space without change of shape or size.

^ This has shape, size, position, and may be moved.

.This is assumed by Euclid, but not stated as an axiom.^ This is assumed by Euclid, but not stated as an axiom.

.(6) In any plane a circle may be described, having any point in that plane as centre, and its distance from any other point in that plane as radius.^ (E ~ i) touches a circle with the fixed point for centre.

^ In any plane a circle may be described, having any point in that plane as centre, and its distance from any other point in that plane as radius .

^ This point is called the centre of the circle.

(Postulate 3.)
.The definitions which have not been mentioned are all nominal definitions, that is to say, they fix a name for a thing described.^ The definitions which have not been mentioned are all nominal definitions, that is to say, they fix a name for a thing described.

^ As already mentioned, at each line I of a complex there is an infinite number of tangent linear complexes, and they all contain the lines adjacent to 1.

^ There is no definition of a line in terms of the discrete geometry, and, above all, the projected width on the x -axis of a line L is calculated according to a Euclidean distance function that is not explicitly mentioned.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

.Many of them overdetermine a figure.^ Many of them overdetermine a figure.

.5. Euclids Elements (see EUCLID) are contained in thirteen books.^ Euclids Elements (see EUCLID) are contained in thirteen books.

^ T. L. Heath , The Thirteen Books of Euclids Elements, a New Translation from I/ic Greek, with Introductory Essays and Commentary, Historical, Critical, andExplanatory (Cambridge, 1908)this work is the standard source of information; W. B. Frankland, Euclid, Book I., with a Commentary (Cambridge, 1905)the commentary contains copious extracts from the ancient commentators.

^ Euclid (he continued) was the author of a book which contains many different figures leading to the knowledge of how things, both known and hidden, really are.
  • Islamic Art and the Argument from Academic Geometry 16 January 2010 23:55 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

.Of these the first four and the sixth are devoted to plane geometry, as the investigation of figures in a plane is generally called.^ These are called the four quadrants.

^ These are called the traces of the plane.

^ Of these the first four and the sixth are devoted to plane geometry, as the investigation of figures in a plane is generally called.

.The 5th book contains the theory of proportion which is used in Book VI. The 7th, 8th and oth books are purely arithmetical, whilst the 10th contains a most ingenious treatment of geometrical irrational quantities.^ The 5th book contains the theory of proportion which is used in Book VI. The 7th, 8th and oth books are purely arithmetical, whilst the 10th contains a most ingenious treatment of geometrical irrational quantities.

^ It contains the theory of ratios and proportion of quantities, in general.

^ Doing so allows use of linked drawings that are dynamically created from tables or queries containing geometric data.
  • Geometry in Tables 16 January 2010 23:55 UTC demo.manifold.net [Source type: Reference]

These four books will be excluded from our survey. .The remaining three books relate to figures in space, or, as it is generally called, to solid geometry.^ The study of these is called solid geometry.
  • Undefined: Points, Lines, and Planes 16 January 2010 23:55 UTC www.andrews.edu [Source type: FILTERED WITH BAYES]

^ While the notation is a bit unwieldly, the figures are nice, and the book serves as a nice collection for common types of solids.
  • Solid Geometry -- from Eric Weisstein's Encyclopedia of Scientific Books 16 January 2010 23:55 UTC www.ericweisstein.com [Source type: Academic]

^ Types of Geometry Euclidean geometry, elementary geometry of two and three dimensions (plane and solid geometry), is based largely on the Elements of the Greek mathematician Euclid (fl.
  • geometry Facts, information, pictures | Encyclopedia.com articles about geometry 16 January 2010 23:55 UTC www.encyclopedia.com [Source type: Academic]

.The 7th, 8th, 9th, 10th, i3th and part of the 11th and 12th books are now generally omitted from the school editions of the Elements.^ The 7th, 8th, 9th, 10th, i3th and part of the 11th and 12th books are now generally omitted from the school editions of the Elements.

^ The 5th book contains the theory of proportion which is used in Book VI. The 7th, 8th and oth books are purely arithmetical, whilst the 10th contains a most ingenious treatment of geometrical irrational quantities.

^ The Thirteen Books of Euclid's Elements , translated from the text of Heiberg with introduction and commentary, New York: Dover, 3 volumes, 2nd edition, revised with additions.
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

.In the first four and in the 6th book it is to be understood that all figures are drawn in a plane.^ In the first four and in the 6th book it is to be understood that all figures are drawn in a plane.

^ They all relate to figures in a plane.

^ In this book figures -are considered which are not confined to a plane, viz, first relations between lines and planes in space, and af terwards properties of solids.

BOOK I. OF EUcLIDs ELEMENTS.
.6. According to the third postulate it is possible to draw in any plane a circle which has its centre at any given point, and its radius equal to the distance of this point from any other point given in the plane.^ (E ~ i) touches a circle with the fixed point for centre.

^ To draw a circle with any center and any radius.
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ In any plane a circle may be described, having any point in that plane as centre, and its distance from any other point in that plane as radius .

.This makes it possible (Prop.^ This makes it possible (Prop.

.1) to construct on a given line AB an equilateral triangle, by drawing first a circle with A as centre and AB as radius, and then a circle with B as centre and BA as radius.^ To draw a circle with any center and any radius.
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ To circumscribe a circle about a given triangle.

^ Again, draw a straight line and, with the centre of your compasses on the line, draw a circle.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

.The point where these circles intersect that they intersect Euclid quietly assumesis the vertex of the required triangle.^ The point where these circles intersect that they intersect Euclid quietly assumesis the vertex of the required triangle.

^ A polygon is inscribed in a circle if each vertex of the polygon is a point on the circle.
  • Beginning Algebra Tutorial on Basic Geometry 16 January 2010 23:55 UTC www.wtamu.edu [Source type: FILTERED WITH BAYES]

^ Area of triangle formed by the intersection of common tangents to three circles.
  • BRUNNERMATH -> your math connection 16 January 2010 23:55 UTC www.brunnermath.com [Source type: FILTERED WITH BAYES]

.Euclid does not suppose, however, that a circle may be drawn which has its radius equal to the distance between any two points unless one of the points be the centre.^ (E ~ i) touches a circle with the fixed point for centre.

^ Through two points exactly one straight line can be drawn.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ With the compasses centred on one of the two points where the circle cuts the line, draw another circle of the same radius.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

.This implies also that we are not supposed to be able to make any straight line equal to any other straight line, or to carry a distance about in space.^ This implies also that we are not supposed to be able to make any straight line equal to any other straight line, or to carry a distance about in space.

^ There are exactly two straight lines through Q , coplanar with a , that make an angle of size μ with P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means~ and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines, are pro portionals.

.Euclid therefore next solves the problem: It is required along a given straight line from a point in it to set off a distance equal to the length of another straight line given anywhere in the plane.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ It is now possible to solve the following two problems: To draw a straight line perpendicular to ~a given plane frirn a given point which lies 1.

This is done in two steps. It is shown in Prop. .2 how a straight line may be drawn from a given point equal in length to another given straight line not drawn from that point.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ Through two points exactly one straight line can be drawn.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

.And then the problem itself is solved in Prop.^ After solving a few problems we come to Prop.

^ Having solved two problems (Props.

^ And then the problem itself is solved in Prop.

.3, by drawing first through the given point some straight line of the required length, and then about the same point as centre a circle having this length as radius.^ Draw the straight line a through point P at right angles with the segment P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ To divide a straight line in a given ratio.

^ (E ~ i) touches a circle with the fixed point for centre.

.This circle will cut off from the given straight line a length equal to the required i one.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ To divide a straight line into a given number of equal parts.

.Nowadays, instead of going through this long process, we take a pair of compasses and set off the given length by its aid.^ Nowadays, instead of going through this long process, we take a pair of compasses and set off the given length by its aid.

^ This circle will cut off from the given straight line a length equal to the required i one.

^ This property gives a simple means to construct, by aid of the straight edge only, in an involution of which two pairs of conjugate points are given, to any point its conjugate.

.This assumes that we may move a length about without changing it.^ This assumes that we may move a length about without changing it.

^ The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.

^ I, and on a few other occasions, viz, that figures may be moved in space without change of shape or size.

.But Euclid has not assumed it, and this proceeding wotild be fully justified by his desire not to take for granted more than was necessary, if he were not obliged at his very next step actually to make this assumption, though without stating it.^ This is assumed by Euclid, but not stated as an axiom.

^ But Euclid has not assumed it, and this proceeding wotild be fully justified by his desire not to take for granted more than was necessary, if he were not obliged at his very next step actually to make this assumption, though without stating it.

^ Postulates.The assumptions actually made by Euclid may be stated as follows: .

.7. \Ve now come (in Prop.^ Ve now come (in Prop.

4) to the first theorem. .It is the fundamental theorem of Euclids whole system, there being only a very few propositions (like Props.^ As consequences of this fundamental theorem we get Prop.

^ It is the fundamental theorem of Euclids whole system, there being only a very few propositions (like Props.

^ The third case of this theorem is Euclids Prop.

.13, 14, 15, I.), except those in the 5th hook and the first half of the 11th, which do not depend upon it.^ I.), except those in the 5th hook and the first half of the 11th, which do not depend upon it.

^ These four types of quantity depend upon the two first among them as fundamental.

It is stated very accurately, though somewhat clumsily, as follows: -
.If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal; and their other angles shall be equ at, each to each, namely, those to which the equal sides are opposite.^ If two triangles are equal in area, and have one angle in the one vertically opposite to one angle in the other, then the two straight lines which join the remaining two vertices of the one to those of the other triangle are parallel.

^ The other sides and base are legs.
  • WikiAnswers - Geometry Questions including "What is a tessellation" 16 January 2010 23:55 UTC wiki.answers.com [Source type: FILTERED WITH BAYES]

^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

.That is to say, the triangles are identically equal, and one may be considered as a copy of the other.^ Every other edge is equal to one of them.

^ That is to say, the triangles are identically equal, and one may be considered as a copy of the other.

^ Hence the triangles are equal, and the angles in the one are equal to those in the other, viz.

.The proof is very simple.^ The proof is very simple.

^ Pole and Polar belong to this picture, as well as a very simple proof of a famous theorem of Pascal.
  • Geometry 16 January 2010 23:55 UTC www.wordtrade.com [Source type: FILTERED WITH BAYES]

.The first triangle is taken up and placed on the second, so that the parts of the triangles which are known to be equal fall upon each other.^ Moreover, if a triangle is divided into smaller triangles, the defect of the whole equals the sum of the defects of the parts.
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ (In elliptic geometry every straight line meets every other, and the three internal angles of a triangle always add up to more than two right angles.
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ In fact, such constructions were very much part of the motivation for passing from elementary geometry to higher geometry in the first place.
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.It is then easily seen that also the remaining parts of one coincide with those of the other, and that they are therefore equal.^ The equals predicate has the following equivalent definitions: The two geometries have at least one point in common, and no point of either geometry lies in the exterior of the other geometry.

^ I have to admit that I have not seen a designer making one of these patterns other than artisan craftsmen in Iran and Qatar who were producing relatively simple designs, and who were able to explain something of what they believed governed their work.
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^ The three circles producing the basis for this panel can be seen to be part of a pattern of lines of circles regularly aligned at right angles to each other in two directions, as can be seen in the lowest of the three illustrations above.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

.This process of applying one figure to another Euclid scarcely uses again, though many proofs would be simplified by doing so.^ This process of applying one figure to another Euclid scarcely uses again, though many proofs would be simplified by doing so.

^ Figure 2 We have a right-angled triangle pqr such that for simplicity the right sides pq and qr are equal to one another and are aligned with the axes of the grid.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Sometimes, however, it is convenient to use axes which are oblique to one another, so that (as in fig.

.The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.^ The process introduces motion into geometry, and includes, as already stated, the axiom that figures may be moved without change of shape or size.

^ I, and on a few other occasions, viz, that figures may be moved in space without change of shape or size.

^ This assumes that we may move a length about without changing it.

.If the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop.^ If two angles ir~ a triangle are equal, then the sides oppbsite them are equali.e.

^ The remaining two theorems (Props.

^ If the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop.

.5), if two sides of a triangle are equal, then the angles opposite these sides are equal.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

^ If two sides in one are proportional to two sides in the other, and if the angles contained by these sides are equal; 4.

.Euclids proof is somewhat complicated, and a stumbling-block to many schoolboys.^ Euclids proof is somewhat complicated, and a stumbling-block to many schoolboys.

^ This process of applying one figure to another Euclid scarcely uses again, though many proofs would be simplified by doing so.

.The proof becomes much simpler if we consider the isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC, and once as a triangle CAB; and now remember that AB, AC in the first are equal respectively toAC, AB in the second, and the angles included by these sides are equal.^ From this it follows that if the included angles are not equal, the third sides are not equal; and conversely, that if the third sides are not equal, the included angles are not equal.

^ The proof becomes much simpler if we consider the isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC, and once as a triangle CAB ; and now remember that AB, AC in the first are equal respectively toAC, AB in the second, and the angles included by these sides are equal.

^ ABC in the first equals angle ACB in the second, as they are opposite the equal sides AC and AB in the two triangles.

.Hence the triangles are equal, and the angles in the one are equal to those in the other, viz.^ Figure 2 We have a right-angled triangle pqr such that for simplicity the right sides pq and qr are equal to one another and are aligned with the axes of the grid.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ The equals predicate has the following equivalent definitions: The two geometries have at least one point in common, and no point of either geometry lies in the exterior of the other geometry.

^ (Where ‘right angle’ means, as usual, an angle equal to its adjacent angle, and two angles in Int(κ) are said to be equal if one is the image of the other by a transformation of group G κ ).
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

those which are opposite equal sides, i.e. angle .ABC in the first equals angle ACB in the second, as they are opposite the equal sides AC and AB in the two triangles.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ ABC in the first equals angle ACB in the second, as they are opposite the equal sides AC and AB in the two triangles.

^ If two angles ir~ a triangle are equal, then the sides oppbsite them are equali.e.

.There follows the converse theorem (Prop.^ From this follows easily the important theorem Prop.

^ There follow several important theorems: .

^ From this follows at once the theorem contained in Prop.

6). .If two angles ir~ a triangle are equal, then the sides oppbsite them are equali.e.^ Figure 2 We have a right-angled triangle pqr such that for simplicity the right sides pq and qr are equal to one another and are aligned with the axes of the grid.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ (Where ‘right angle’ means, as usual, an angle equal to its adjacent angle, and two angles in Int(κ) are said to be equal if one is the image of the other by a transformation of group G κ ).
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

the triangle is isosceles. .The proof given consists in what is called a reductio ad absurdum, a kind of proof often used by Euclid, and principally in proving the converse of a previous theorem.^ Prove and apply the Pythagorean theorem and its converse .

^ The proof given consists in what is called a reductio ad absurdum, a kind of proof often used by Euclid, and principally in proving the converse of a previous theorem.

^ This geometry was codified in Euclid’s Elements about 300 bce on the basis of 10 axioms, or postulates, from which several hundred theorems were proved by deductive logic.
  • geometry (mathematics) -- Britannica Online Encyclopedia 16 January 2010 23:55 UTC www.britannica.com [Source type: Reference]

It assumes that the theorem to be proved is wrong, and then shows that this assumption leads to an absurdity, i.e. to a conclusion which is in contradiction to a proposition proved beforethat therefore the assumption made cannot be true, and hence that the theorem is true. It is often stated that Euclid invented this kind of ~g,roof, hut the method is most likely much older.
.8. It is next proved that two triangles which have the three sides of the one equal respectively to those of the other are identically equal, hence that the angles of the one are equal respectively to those of the other, those being equal which are opposite equal sides.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

^ If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal; and their other angles shall be equ at, each to each, namely, those to which the equal sides are opposite.

This is Prop. 8, Prop. 7 containing only a first step towards its proof.
.These theorems allow now of the solution of a number of problems, viz.: To bisect a given angle (Prop.^ The first of these theorems gives Prop.

^ Wecan bisect any given arc (Prop.

^ These theorems allow now of the solution of a number of problems, viz.: To bisect a given angle (Prop.

9).
.To bisect a given finite straight line (Prop.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ Wecan bisect any given arc (Prop.

10).
.To draw a straight line perpendicularly to a given straight line through a given point in it (PrOp.^ Draw the straight line a through point P at right angles with the segment P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

.If), and also through a given point not in it (Prop.^ We next draw through the given point A a plarn nsrslr,.r ~ ~ ~ ,.,.

^ One version of this postulate asserts that there is one and only one line parallel to a given line through a point outside it.
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^ For the conic through four points which touches a given line ha~ its point of contact at a focus of the involution determined by th four-point on the line.

12).
.The solutions all depend upon properties of isosceles triangles, 9. The next three theorems relate to angles only, and might have been proved before Prop.^ It therefore requires additional arguments to claim that among all possible order relations on a given set, one and only one has a special status.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Pythagorean Theorem works only with right triangles.
  • WikiAnswers - Geometry Questions including "What is a tessellation" 16 January 2010 23:55 UTC wiki.answers.com [Source type: FILTERED WITH BAYES]

^ (In elliptic geometry every straight line meets every other, and the three internal angles of a triangle always add up to more than two right angles.
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

4, or even at the very beginning. The first (Prop. .13) says, The angles which one straight line makes with another straight line on one side of it either are two right angles or are together equal to two right angles.^ There are exactly two straight lines through Q , coplanar with a , that make an angle of size μ with P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Draw the straight line a through point P at right angles with the segment P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Through two points exactly one straight line can be drawn.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

.This theorem would have been unncce~sary if Euclid had admitted the notion of an angle such that its two limits are in the same straight line, and had besides defined the sum of two angles.^ What are the two angles that together form a straight angle?
  • WikiAnswers - Geometry Questions including "What is a tessellation" 16 January 2010 23:55 UTC wiki.answers.com [Source type: FILTERED WITH BAYES]

^ There are exactly two straight lines through Q , coplanar with a , that make an angle of size μ with P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ The top of the two photographs is a straight elevation of half of the work, the lower was taken at an oblique angle.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

Its converse (Prop. .14) is of great use, inasmuch as it enables us in many cases to prove that two straight lines drawn from the same point are one the continuation of the othei.^ Through two points exactly one straight line can be drawn.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Any two distinct points lie in one and only one straight line.

^ If two straig lit lines be parallel, the straight line drawn from any point in one to any point in the other is in the same plane with the parallels.

So also is Prop. .15. If two straight lines cut one another, the vertical or opposite angles shall be equal.^ Through two points exactly one straight line can be drawn.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ If two planes cut one another, their common section is a straight line.

^ The angles in the same segment of a circle are equal to one another; Prop.

-
10. Euclid returns now to properties of triangles. .Of great importance for the next steps (though afterwards superseded by a more complete theorem) is Prop.^ Of great importance for the next steps (though afterwards superseded by a more complete theorem) is Prop.

^ The next theorem (Prop.

^ This is of great importance for its consequences, of which the two following are the principal: Prop.

.16. If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles.

^ At least one interior angle is greater than 180 degrees.
  • WikiAnswers - Geometry Questions including "What is a tessellation" 16 January 2010 23:55 UTC wiki.answers.com [Source type: FILTERED WITH BAYES]

Prop. .17. Any two angles of a triangle are together less than two right angles, is an immediate consequence of it.^ Each angle in a regular triangle equals two-thirds of one right angle.

^ Each will be fieth of a right angle, or lth of two right angles.

^ The sum of the three angles of a triangle is always less than two right angles.

.By the aid of these two, the following fundamental properties of triangles are easily proved: Prop.^ From this it follows easily that the triangles FIG. 35.

^ Or it may be considered proved by aid of Prop.

^ By aid of these two propositions the following two are proved.

.18. The greater side of every triangle has the greater angle opposite to it; Its converse, Prop.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ The greater side of every triangle has the greater angle opposite to it; Its converse, Prop.

^ Any two sides of a triangle are together greater than the third side; And also Prop.

.19. The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ The greater side of every triangle has the greater angle opposite to it; Its converse, Prop.

^ Any two sides of a triangle are together greater than the third side; And also Prop.

.20. Any two sides of a triangle are together greater than the third side; And also Prop.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ If a solid angle be contained by three plane angles, any two of them are together greater than the third.

^ How do you find the third side of a triangle when two sides are known?
  • WikiAnswers - Geometry Questions including "What is a tessellation" 16 January 2010 23:55 UTC wiki.answers.com [Source type: FILTERED WITH BAYES]

.21. If from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle, these shall be less than the other two sides of the triangle, but shall contain a greater angle.^ There are exactly two straight lines through Q , coplanar with a , that make an angle of size μ with P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ X second end point of line.
  • Geometry 16 January 2010 23:55 UTC icee.usm.edu [Source type: Reference]

^ Draw the straight line a through point P at right angles with the segment P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

.11. Having solved two problems (Props.^ After solving a few problems we come to Prop.

^ Having solved two problems (Props.

^ And then the problem itself is solved in Prop.

.22, 23), he returns to two triangles which have two sides of the one equal respectively to two sides of the other.^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

^ Returns: true if the two Geometry s are equal .

^ Every other edge is equal to one of them.

It is ,known (Prop. .4) that if the included angles are equal then the third sides are equal; and conversely (Prop.^ Figure 2 We have a right-angled triangle pqr such that for simplicity the right sides pq and qr are equal to one another and are aligned with the axes of the grid.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ There is a geometric way of establishing this proportion – a right angle triangle with adjacent sides equal, will have a hypoteneuse of √2 to the adjacent sides.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

^ Recall that a regular polygon is a polygon with all sides the same length and all angles equal.
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

.8), if the third sides are equal, then the angles included by the first sides are equal.^ From this it follows that if the included angles are not equal, the third sides are not equal; and conversely, that if the third sides are not equal, the included angles are not equal.

^ Figure 2 We have a right-angled triangle pqr such that for simplicity the right sides pq and qr are equal to one another and are aligned with the axes of the grid.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ The proof becomes much simpler if we consider the isosceles triangle ABC (AB = AC) twice over, once as a triangle BAC, and once as a triangle CAB ; and now remember that AB, AC in the first are equal respectively toAC, AB in the second, and the angles included by these sides are equal.

.From this it follows that if the included angles are not equal, the third sides are not equal; and conversely, that if the third sides are not equal, the included angles are not equal.^ From this it follows that if the included angles are not equal, the third sides are not equal; and conversely, that if the third sides are not equal, the included angles are not equal.

^ Figure 2 We have a right-angled triangle pqr such that for simplicity the right sides pq and qr are equal to one another and are aligned with the axes of the grid.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ If we now define a quotient ~ of two lines as the number which multiplie4 into b gives a, so that we see that from the equality of two quotients follows, if we multiply both sides by lid, ~b.d=~d.b, ad = cb.

.Euclid now completes this knowledge by proving, that if the included angles are not equal, then the third side in that triangle is the greater which contains the greater angle; and conversely, that if the third sides are unequal, that triangle contains the greater angle which contains the greater side.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ From this it follows that if the included angles are not equal, the third sides are not equal; and conversely, that if the third sides are not equal, the included angles are not equal.

^ Euclid returns now to properties of triangles.

These are Prop. 24 and Prop. 25.
.12. The next theorem (Prop.^ The next theorem (Prop.

^ The next three theorems relate to angles only, and might have been proved before Prop.

^ Of great importance for the next steps (though afterwards superseded by a more complete theorem) is Prop.

.26) says that if two triangles have one side and two angles of the one equal respectively to one side and two angles of the other, viz, in both triangles either the angles adjacent to the equal side, or one angle adjacent and one angle opposite it, then the two triangles are identically equal.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ Every other edge is equal to one of them.

^ If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal; and their other angles shall be equ at, each to each, namely, those to which the equal sides are opposite.

.This theorem belongs to a group with Prop.^ This theorem belongs to a group with Prop.

4 and Prop. .8. Its first case might have been given immediately after Prop.^ Its first case might have been given immediately after Prop.

^ Now the general equation of the first degree Ax+By+C =o may be written y= ~x~, Unless B=o, in which case it represents a line parallel to the axis of y; and A/B, C/B are values which can be given to m and b, so that every equation of the first degree represents a straight line.

^ Since a twisted cubic can be made to satisfy twelve conditions, it might be supposed that a finite number could be drawn to touch four given lines, but this is not the case.

.4, but the second case requires Prop.^ The second case (Prop.

16 for its proof.
13. We come now to the investigation of parallel straight lines, i.e. of straight lines which lie in the same plane, and cannot be made to meet however far they be produced either way. The investigation which starts from Prop. .16, will become clearer if a few names be explained which are not all used by Euclid.^ He uses the following names: A treble infinite number of lines, that is, all lines which satisfy one condition, are said to form a complex of lines; e.g.

^ If we wish to collect the axioms used in Euclids Elements, we have therefore to take the three postulates, the last three axioms as generally given, a few axioms hidden in the definitions, and an axiom used by Euclid in the proof of Prop.

^ For the sake of brevity we shall presuppose a knowledge of Euclids Elements, although we shall use only a few of his propositions.

.If two straight lines be cut by a third, the latter is now generally called a transversal of the figure.^ If two straight lines be cut by a third, the latter is now generally called a transversal of the figure.

^ II. To find a third proportional to two given straight lines.

^ If two straight lines which meet are cut by a transversal, their alternate angles are unequal.

.It forms at the two points where it cuts the given lines four angles with each.^ I in two conies which have the point S and the points where it cuts af and.

^ Two, four and eight point .
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

^ It forms at the two points where it cuts the given lines four angles with each.

.Those of the angles which lie between the given lines are called interior angles, and of these, again, any two which lie on opposite sides of the transversal but one at each of the two points are called alternate angles.^ If a point lies in two lines its projections must lie in the projections of both.

^ Antipodal points Two points which are opposite each other on the sphere are called antipodal points .
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

^ Through two points exactly one straight line can be drawn.
  • Finitism in Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

.We may now state Prop.^ We may now state Prop.

^ If we call that side in a right-angled triangle which is opposite the right angle the hypotenuse, we may state it as follows: Theorem of Pythagoras (Prop.

.16 thus:If two straight lines which meet are cut by a transversal, their alternate angles are unequal.^ Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane.

^ If two straight lines which are cut by a transversal make alternate angles equal, the lines cannot meet, however far they be produced, hence they are parallel.

^ That is to say,, If alternate angles are unequal, do the lines meet ?

.For the lines will form a triangle, and one of the alternate angles will be an exterior angle to the triangle, the other interior and opposite to it.^ If one side of a triangle be produced, the exterior angle shall be greater than either of the interior opposite angles.

^ For the lines will form a triangle, and one of the alternate angles will be an exterior angle to the triangle, the other interior and opposite to it.

^ Hence the triangles are equal, and the angles in the one are equal to those in the other, viz.

.From this follows at once the theorem contained in Prop.^ From this follows easily the important theorem Prop.

^ From this follows at once the theorem contained in Prop.

^ There follows the converse theorem (Prop.

.27. If two straight lines which are cut by a transversal make alternate angles equal, the lines cannot meet, however far they be produced, hence they are parallel.^ Parallel planes have parallel traces, because parallel planes are cut by any plane, hence also by in and by in2, in parallel lines.

^ The angles which one straight line makes with another straight line on one side of it either are two right angles or are together equal to two right angles.

^ Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane.

.This proves the existence of parallel lines.^ This proves the existence of parallel lines.

Prop. 28 states the same fact in different forms. .If a straight line, falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on the same side together equal to two right angles, the two straight tines ihall be parallel to one another.^ There are exactly two straight lines through Q , coplanar with a , that make an angle of size μ with P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Draw the straight line a through point P at right angles with the segment P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ Draw a quadrangle that has 2 pairs of parallel sides and no right angles?
  • WikiAnswers - Geometry Questions including "What is a tessellation" 16 January 2010 23:55 UTC wiki.answers.com [Source type: FILTERED WITH BAYES]

.Hence we know that, if two straight lines which are cut by a transversal meet, their alternate angles are not equal and hence that, if alternate angles are equal, then the lines are parallel.^ Parallel planes have parallel traces, because parallel planes are cut by any plane, hence also by in and by in2, in parallel lines.

^ Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane.

^ Note also that two parallel lines 1 and m are not coplanar.

.The question now arises, Are the propositions converse to these true or not ?^ Of each of these propositions, which will easily be seen to be true, the converse holds also.

^ The question now arises, Are the propositions converse to these true or not ?

.That is to say,, If alternate angles are unequal, do the lines meet ?^ If two straight lines which meet are cut by a transversal, their alternate angles are unequal.

^ That is to say,, If alternate angles are unequal, do the lines meet ?

^ In this case, b 1 and b 2 make a right angle at Q and we thus have two mutually perpendicular straight lines on the same plane as a , which fail to meet a .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

.And if the lines are parallel, are alternate angles necessarily equal ?^ And if the lines are parallel, are alternate angles necessarily equal ?

^ This angle has in modern times been called the angle between the given lines: By the angles between two not intersecting lines we understand the angles which two intersecting lines include that are parallel respectively to the two given lines.

^ If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior andopposite angle on the same side, and also the two interior angles on the same side together equal to two right angles.

.The answer to either of the~e two questions implies the answer to the other.^ The answer to either of the~e two questions implies the answer to the other.

^ The equals predicate has the following equivalent definitions: The two geometries have at least one point in common, and no point of either geometry lies in the exterior of the other geometry.

^ But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude.

.But it has been found impossible to prove that the negation or the affirmation of either is true.^ But it has been found impossible to prove that the negation or the affirmation of either is true.

.The difficulty which thus arises is overcome by Euclid assuming that the first question has to be answered in the affirmative.^ The difficulty which thus arises is overcome by Euclid assuming that the first question has to be answered in the affirmative.

^ Thus a number of Euclids axioms are hidden among his first definitions.

.This gives his last axiom (12), which we quote in his own words.^ This gives his last axiom (12), which we quote in his own words.

.Axiom Ia-If a straight line meet two straight lines, so as ,to make the Iwo interior angles ,on the same side of it taken together less than two right angles, these straight lines, being continually produced, shall at length meet on that side oil which are the angles which areless than two right angles.^ If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior andopposite angle on the same side, and also the two interior angles on the same side together equal to two right angles.

^ If a straight line, falling on two other straight lines, make the exterior angle equal to the interior and opposite angle on the same side of the line, or make the interior angles on the same side together equal to two right angles, the two straight tines ihall be parallel to one another.

^ The angles which one straight line makes with another straight line on one side of it either are two right angles or are together equal to two right angles.

.The answer to the second of the above questions follows from this, and gives the theorem Prop.^ The first of these theorems gives Prop.

^ From this follows easily the important theorem Prop.

^ From this follows at once the theorem contained in Prop.

.29~If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior andopposite angle on the same side, and also the two interior angles on the same side together equal to two right angles.^ Draw a quadrangle that has 2 pairs of parallel sides and no right angles?
  • WikiAnswers - Geometry Questions including "What is a tessellation" 16 January 2010 23:55 UTC wiki.answers.com [Source type: FILTERED WITH BAYES]

^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

^ If a straight line fall on two parallel straight lines, it makes the alternate angles equal to one another, and the exterior angle equal to the interior andopposite angle on the same side, and also the two interior angles on the same side together equal to two right angles.

.4- \Vith this a new part of elementary geometry begins.^ Vith this a new part of elementary geometry begins.

^ I use elementary calculus to compute lengths of curves in 3‑space and on spheres, a topic usually found at the beginning of elementary books on differential geometry.
  • Geometry 16 January 2010 23:55 UTC www.wordtrade.com [Source type: FILTERED WITH BAYES]

^ However, basic elementary set theory is required for the axiomatic geometry part.
  • Geometry 16 January 2010 23:55 UTC www.wordtrade.com [Source type: FILTERED WITH BAYES]

.The earlier propositions are independent of this axiom, and would be true even if a wrong assumption had been made in it.^ The earlier propositions are independent of this axiom, and would be true even if a wrong assumption had been made in it.

^ The axioms are propositional functions.7 When a set of axioms is given, we can ask (I) whether they are consistent, (2) whether their existence theorem is proved, (3) whether they are independent.

^ And a set of formal geometrical axioms cannot in themselves be true or false, since they are not determinate propositions, in that they do not refer to a determinate subject matter.

.They all relate to figures in a plane.^ They all relate to figures in a plane.

^ In this book figures -are considered which are not confined to a plane, viz, first relations between lines and planes in space, and af terwards properties of solids.

^ I! a point O on 1 is related to A by a prospectivity, then all prospectivities, which (1) have the same double point U, and (2) relate 0 to A, give the same correspondent (Q, in figure) to any point P on the line 1; in fact they are all the same prospectivity, however m, ii, S, and S may have been varied subject to these conditions.

.But a, plane is,only oneamong an infinite number of conceivable surfaces.^ Symmetry can be defined in terms of grid and detailed design, and although there are only seventeen basic pattern arrangements, there are an infinite number of patterns that can be developed from them.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

.We may draw figures on any one of them and study their prope.rties.^ We may draw figures on any one of them and study their prope.rties.

^ Or, one or both of them may fall beyond the drawing paper, so that they are practically non-existent for the construction.

^ This branch of geometry is concerned with the methods for representing solids and other figures in three dimensions by drawings in one plane.

.We may, for instance, take a sphere instead.of the plane, and obtain spherical in the place of plane geometry.^ We may, for instance, take a sphere instead.of the plane, and obtain spherical in the place of plane geometry.

^ Instead of a circle or sphere we may take any conic or quadric.

^ In spherical geometry, the “points” are points on the surface of the sphere.
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

.If on one of these surfaces lines and figures could be drawn, answering to all the definitions of our plane figures, and if the axioms with the exception of the last ~ll hold, then all propositions up to the 28th will be true for these figures.^ These theorems do not hold for spherical figures.

^ To these we add the definition of a line parallel to a plane as a line which does not meet the plane.

^ Of each of these propositions, which will easily be seen to be true, the converse holds also.

,This is the case in spherical geometry if. we substitute shortest line or .great circle for straight line, small circle for circle, and if, besides, we limit all figures to a part of the sphere which is less than a hemisphere, so that two, points on it cannot be opposite ends of a diameter, and therefore determine always one and only one great circle.^ (Hence a line has two distinct points at infinity, and not one only as in ordinary geometry.

^ Hence, a straight line which does not lie in the plane cannot have more than one point in common with the plane.

^ Antipodal points Two points which are opposite each other on the sphere are called antipodal points .
  • Spherical Geometry - EscherMath 16 January 2010 23:55 UTC euler.slu.edu [Source type: FILTERED WITH BAYES]

.For spherical triangles, therefore, all the important propositions 4, 8, 26; 5 and 6; and 18, 19 and 20 will hold good.^ For spherical triangles, therefore, all the important propositions 4, 8, 26; 5 and 6; and 18, 19 and 20 will hold good.

^ In Euclids Elements almost all propositions refer to the magnitude of lines, angles, areas or volumes, and therefore to measurement.

^ A good plan, therefore, is to use GeomW KB if one must have interoperability with existing applications that support it and use Geom for all other things.
  • Geometry in Tables 16 January 2010 23:55 UTC demo.manifold.net [Source type: Reference]

.This remark will be sufficientto show the impossibility of proving Euclids last axiom, which would mean proving that this axiom is a consequence of the others, and hence that the theory of parallels would hold on a spherical surtace, where the other axioms do hold, whilst parallels do not even exist.^ This remark will be sufficientto show the impossibility of proving Euclids last axiom, which would mean proving that this axiom is a consequence of the others, and hence that the theory of parallels would hold on a spherical surtace, where the other axioms do hold, whilst parallels do not even exist.

^ I intend to show students how theories that underlie other fields of mathematics can be used to better under­stand the concrete models for the axiom systems of Euclidean, spherical, and hyperbolic geometries and to better visualize the abstract theorems.
  • Geometry 16 January 2010 23:55 UTC www.wordtrade.com [Source type: FILTERED WITH BAYES]

^ But it requires Desarguess theorem, and hence axiom 6, to prove that Harm.

.It follows that the axjom ip question states an inherent difference between the plane and other, surfaces, and that the plane is only fully characterized wheti this axiom is added to the other assumptiop~.^ It follows that the axjom ip question states an inherent difference between the plane and other, surfaces, and that the plane is only fully characterized wheti this axiom is added to the other assumptiop~.

^ We also demonstrate that other surface properties such as friction and texture can be added elegantly.
  • Applied Geometry Lab Publications 16 January 2010 23:55 UTC geometry.caltech.edu [Source type: Academic]

^ The following propositions follow: A quadric surface has at every point a tangent plane.

.15. The introduction of, the new axiom and of parallel lines leads to a new class of propositions.^ The introduction of, the new axiom and of parallel lines leads to a new class of propositions.

^ I., The extremities of a line are points, is a proposition which either has to be proved, and then it is a theorem, or which has to be taken for granted, in which case it is an axiom .

^ For many centuries the speculations of mathematicians on the foundations of geometry were almost confined to hopeless attempts to prove the parallel axiom without the introduction of some equivalent axiom.1 .

After proving (Prop. .30) that two lines which are each parallel to a third are parallel to eec/i other, we obtain the new properties of triangles contained in Prop.^ Note also that two parallel lines 1 and m are not coplanar.

^ Any two sides of a triangle are together greater than the third side; And also Prop.

^ By two parallel lines (Def.

.32.01 these the second part is, the most important, viz, the theorem, The three interior angles of every triangle are together equal to two right angles.^ Each will be fieth of a right angle, or lth of two right angles.

^ The sum of the three angles of a triangle is always less than two right angles.

^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

.As easy deductions not given by Euclid but added by Simson follow the propositions about the angles in polygons; they are given in English editions as corollaries to Prop.^ The next proposition, together with one added by Simson as Prop.

^ As easy deductions not given by Euclid but added by Simson follow the propositions about the angles in polygons; they are given in English editions as corollaries to Prop.

^ We conclude with a few theorems about regular polygons which are not given by Euclid.

32.
.These theorems do not hold for spherical figures.^ These theorems do not hold for spherical figures.

.The sum of the interior angles of a spherical triangle is always greater than two right angles, and increases with the area.^ At least one interior angle is greater than 180 degrees.
  • WikiAnswers - Geometry Questions including "What is a tessellation" 16 January 2010 23:55 UTC wiki.answers.com [Source type: FILTERED WITH BAYES]

^ Each angle in a regular triangle equals two-thirds of one right angle.

^ Each will be fieth of a right angle, or lth of two right angles.

.16. The theory of parallels as such may be said to be finished with Props.^ The theory of parallels as such may be said to be finished with Props.

^ We may mention, however, that the theorems about triangles contained in the propositions of Book I., which do not depend upon the theory of parallels (that is all up to Prop.

33 and 34, which state properties of the parallelogram, i.e. of a quadrilateral, formed by two pairs of parallels. They arc Prop. .33. The straight lines which join the extremities of two equal and parallel straight lines towards life same parts are themselves equal and parallel; and Prop.^ The straight lines which join the extremities of two equal and parallel straight lines towards life same parts are themselves equal and parallel; and Prop.

^ Note also that two parallel lines 1 and m are not coplanar.

^ Equal triangles, on the same or on equal bases, in the same straight line, and on the same side of it, are between the same parallels.

.34. The opposite sides and angles of a parallelogram are equal to, one another, and the diameter (diagonal) bisects the parallelogram, that is, divides it into two equal parts.^ Parallelograms on equal bases and between the same parallels are equal to one another.

^ The angles in the same segment of a circle are equal to one another; Prop.

^ If two sides in one are proportional to two sides in the other, and if the angles contained by these sides are equal; 4.

.17. The rest of, the first boo~c relates to areas of figures.^ The rest of, the first boo~c relates to areas of figures.

^ In this book figures -are considered which are not confined to a plane, viz, first relations between lines and planes in space, and af terwards properties of solids.

^ The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares.

.The theory is made to depend upon the theorems Prop.^ The theory is made to depend upon the theorems Prop.

^ An important application of these theorems is at once made to a right-angled triangle, viz.: Prop.

^ The construction of a rectangle having the base u, and being equal in area to a given rectangle, depends upon Prop.

.35. Parallelograms on the same base and between the same parallels areequal to one another; and Prop.^ This theorem corresponds to the theorem (VI. I) that parallelograms between the silme parallelsare to one another as their bases.

^ Parallelograms on equal bases and between the same parallels are equal to one another.

^ The angles in the same segment of a circle are equal to one another; Prop.

.36. Parallelograms on equal bases and between the same parallels are equal to one another.^ With the compasses centred on one of the two points where the circle cuts the line, draw another circle of the same radius.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

^ ToSameClass (java.lang.Object o) Returns whether this Geometry is greater than, equal to, or less than another Geometry having the same class.

^ ToSameClass (java.lang.Object o, CoordinateSequenceComparator  comp) Returns whether this Geometry is greater than, equal to, or less than another Geometry of the same class.

.As each parallelogram is bisected by a diagonal, the last theorems hold also if the word parallelogram be replaced by triangle, as is done in Props.^ As each parallelogram is bisected by a diagonal , the last theorems hold also if the word parallelogram be replaced by triangle, as is done in Props.

^ These theorems allow now of the solution of a number of problems, viz.: To bisect a given angle (Prop.

^ In other words, every theorem of Lobachevskian geometry holds for suitable figures formed from points of Int(κ), if the distance between any two of these points is given by the function d κ .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

37 and 38.
.It is .to be remarked that Euclid proves these propositions only in the case when the parallelograms or triangles have their bases in the sanie straight line.^ These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

^ In this case, b 1 and b 2 make a right angle at Q and we thus have two mutually perpendicular straight lines on the same plane as a , which fail to meet a .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ But not all elements in nature, in this particular case, flowers, have geometries radiating from a single point in straight lines.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

.The theorems converse to the last form the contents of the next three prooositions.^ The theorems converse to the last form the contents of the next three prooositions.

^ These last three theorems are fundamental for the theory of the circle.

^ The next three theorems relate to angles only, and might have been proved before Prop.

viz.: Props. .40 and 41.Equal triangles, on the same or on equal bases, in the same straight line, and on the same side of it, are between the same parallels.^ Parallelograms on equal bases and between the same parallels are equal to one another.

^ Equal triangles, on the same or on equal bases, in the same straight line, and on the same side of it, are between the same parallels.

^ The distance between two parallel lines or between two parallel planes is the length of the common perpendicular between the lines or the planes.

.That the two cases here stated are given by Euclid in two separate propositions proved separately is characteristic of his method.^ That the two cases here stated are given by Euclid in two separate propositions proved separately is characteristic of his method.

^ That Euclid treats of two cases is characteristic of Greek mathematics.

^ The axioms are propositional functions.7 When a set of axioms is given, we can ask (I) whether they are consistent, (2) whether their existence theorem is proved, (3) whether they are independent.

.18. To compare areas of other figures, Euclid shows first, in Prop.^ He first shows (Prop.

^ To compare areas of other figures, Euclid shows first, in Prop.

^ The rest of, the first boo~c relates to areas of figures.

.42, how to draw a parallelogram which is equal in area to a given triangle, and has one of its angles equal to a given angle.^ S solved to construct a parallelogram on a given line, which is equal in area to a given triangle, and which has one angle equal to a given angle (generally a right angle).

^ Hence the triangles are equal, and the angles in the one are equal to those in the other, viz.

^ If the given angle is right, then the problem is solved to draw a rectangle equal in area to a given triangle.

.If the given angle is right, then the problem is solved to draw a rectangle equal in area to a given triangle.^ Each angle in a regular triangle equals two-thirds of one right angle.

^ S solved to construct a parallelogram on a given line, which is equal in area to a given triangle, and which has one angle equal to a given angle (generally a right angle).

^ The angle in the regular pentagon equals ~ of a right angle.

.Next this parallelogram is transformed into another parallelogram, which has one of its sn/es equal to a given straight line, whilst its angles remain unaltered.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ Parallelograms on equal bases and between the same parallels are equal to one another.

.This may be done by aid of the theorem in Prop.^ Or it may be considered proved by aid of Prop.

^ This may be done by aid of the theorem in Prop.

^ A segment of a circle being given to describe the circle of which it is a segment, may be solved much more easily by aid of the construction described in relation to Prop.

.43. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.^ Parallelograms on equal bases and between the same parallels are equal to one another.

^ Parallelograms about the diameter of any parallelogram are similar to the whole parallelogram and to one another; and its converse (Prop.

^ Equal parallelograms which have one angle of the one equal to one angle of the other have their sIdes about the equal angles reciprocally proportional; and parallelograms which have one - angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.

.Thus the problem (Prop.^ Thus the problem (Prop.

.44) iS solved to construct a parallelogram on a given line, which is equal in area to a given triangle, and which has one angle equal to a given angle (generally a right angle).^ This will intersect with the first line at right angles.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

^ Draw the straight line a through point P at right angles with the segment P Q .
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ The three circles producing the basis for this panel can be seen to be part of a pattern of lines of circles regularly aligned at right angles to each other in two directions, as can be seen in the lowest of the three illustrations above.
  • Arabic geometry – page 1/2 16 January 2010 23:55 UTC catnaps.org [Source type: FILTERED WITH BAYES]

.As every polygon can be divided into a number of triangles, we can now construct a parallelogram having a given angle, say a right angle, and being equal in area to a given polygon.^ Given the number of sides or angles in a polygon you are to find the name of the polygon .
  • BRUNNERMATH -> your math connection 16 January 2010 23:55 UTC www.brunnermath.com [Source type: FILTERED WITH BAYES]

^ To divide a straight line into a given number of equal parts.

^ In a polygon you are to identify the number of sides or the angles in a given polygon (interactive) .
  • BRUNNERMATH -> your math connection 16 January 2010 23:55 UTC www.brunnermath.com [Source type: FILTERED WITH BAYES]

.For each of the triangles into which the polygon has been divided,, a parallelogram may be constructed, having one side equal to a given straight line and one angle equal to a given angle.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ The opposite sides and angles of a parallelogram are equal to, one another, and the diameter (diagonal) bisects the parallelogram, that is, divides it into two equal parts.

.If these parallelograms be placed side by side, they may be added together to form a single parallelogram, having still one side of the given length.^ They may be stated together.

^ If these parallelograms be placed side by side, they may be added together to form a single parallelogram, having still one side of the given length.

^ Having now shown how to represent points, planes and lines, we have to state the conditions which must hold in order that these elements may lie one in the other, or else that the figure formed by them may possess certain metrical properties.

This is done in Prop. 45.
.Herewith a means is found to compare areas of different polygons.^ Herewith a means is found to compare areas of different polygons.

.We need only construct two rectangles equal in area to the given polygons, and having each one side of given length.^ We need only construct two rectangles equal in area to the given polygons, and having each one side of given length.

^ The construction of a rectangle having the base u, and being equal in area to a given rectangle, depends upon Prop.

^ For each of the triangles into which the polygon has been divided,, a parallelogram may be constructed, having one side equal to a given straight line and one angle equal to a given angle.

.By comparing the unequal sides we are enabled to judge whether the areas are equal, or which is the greater.^ ToSameClass (java.lang.Object o) Returns whether this Geometry is greater than, equal to, or less than another Geometry having the same class.

^ Parameters: o - a Geometry with which to compare this Geometry comp - a CoordinateSequenceComparator Returns: a positive number, 0, or a negative number, depending on whether this object is greater than, equal to, or less than o , as defined in "Normal Form For Geometry" in the JTS Technical Specifications .

^ Specified by: compareTo in interface java.lang.Comparable Parameters: o - a Geometry with which to compare this Geometry Returns: a positive number, 0, or a negative number, depending on whether this object is greater than, equal to, or less than o , as defined in "Normal Form For Geometry" in the JTS Technical Specifications .

.Euclid does not state thisconsequence, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon.^ Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides.

^ Euclid does not state thisconsequence, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon.

^ (A) on each side of the square [Carroll does not say whether to do this by measurement or by geometric construction].
  • Islamic Art and the Argument from Academic Geometry 16 January 2010 23:55 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

Prop. .46 is: To describe a square on a given straight line.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ To cut a given straight line in extreme and mean ratio - leads to the equation ax+x2=a2.

.19. The first book concludes with one of the most important theorems in the whole of geometry, and one which has been celebrated since the earliest times.^ The first book concludes with one of the most important theorems in the whole of geometry, and one which has been celebrated since the earliest times.

^ A most important theorem is Prop.

^ We begin by quoting those definitions at the beginning of Book V. -which are most important.

.It is stated, but on doubtful authority, that Pythagoras discovered it, and it has been called by his name.^ It is stated, but on doubtful authority, that Pythagoras discovered it, and it has been called by his name.

^ If we call that side in a right-angled triangle which is opposite the right angle the hypotenuse, we may state it as follows: Theorem of Pythagoras (Prop.

.If we call that side in a right-angled triangle which is opposite the right angle the hypotenuse, we may state it as follows: Theorem of Pythagoras (Prop.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ The theorem may also be stated thus: - .

^ From this follows easily the important theorem Prop.

.47).In every right-angled triangle the square on the hypotenuse is equal to the sum of the squares of the other sides.^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

^ Every other edge is equal to one of them.

And conversely Prop. .48. If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

^ In other words, the areas of similar triangles are to one another as the squares on homologous sides.

^ Every other edge is equal to one of them.

On this theorem (Prop. .47) almost all geometrical measurement depends, which cannot be directly obtained.^ In Euclids Elements almost all propositions refer to the magnitude of lines, angles, areas or volumes, and therefore to measurement.

^ It is hardly too much to say that, when known facts as to a geometrical figure have once been expressed in algebraical terms, all strictly consequential facts as to the figure can be deduced by almost mechanical processes.

^ A solid is directly given by experience; we have only to abstract all material from it in order to gain the notion of a geometrical solid.

Booa II.
.20. The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares.^ The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares.

^ The last proposition in the fifth book is of a different character.

^ They all relate to figures in a plane.

.Their true significance is best seen by stating them in an algebraic form.^ Their true significance is best seen by stating them in an algebraic form.

^ The form is best seen from fig.

.This is often done by expressing the lengths of lines by aid of numbers, which tell how many times a chosen unit is contained in the lines.^ Then a plane contains points and as many lines.

^ This is often done by expressing the lengths of lines by aid of numbers, which tell how many times a chosen unit is contained in the lines.

^ Though a length might be recognized as known when measurement certified that it was so many times a standard length, it was not every length which could be thus specified in terms of the same standard length, even by an arithmetic enriched with the notion of fractional number.

.If there is a unit to be found which is contained an exact number of times in each side of a rectangle, it is easily seen, and generally shown in the teaching of arithmetic, that the rectangle contains a number of unit squares equal to the product of the numbers which measure the sides, a unit square being.^ If there is a unit to be found which is contained an exact number of times in each side of a rectangle, it is easily seen, and generally shown in the teaching of arithmetic, that the rectangle contains a number of unit squares equal to the product of the numbers which measure the sides, a unit square being.

^ If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means~ and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines, are pro portionals.

^ If we now define a quotient ~ of two lines as the number which multiplie4 into b gives a, so that we see that from the equality of two quotients follows, if we multiply both sides by lid, ~b.d=~d.b, ad = cb.

the square on the unit line. .If, however, no such unit can be found, this process requires that connexiun between lines and numbers which is only established by aid of ratios of lines, and which is therefore at this stage altogether inadmissible.^ If, however, no such unit can be found, this process requires that connexiun between lines and numbers which is only established by aid of ratios of lines, and which is therefore at this stage altogether inadmissible.

^ Similarly, all conies touching four fixed lines form a system such that any fifth tangent determines one and only one conic.

^ The second case is easily reduced to the firstviz, if by aid of the first we have drawn any perpendicular to the plane from some point without it, we need only draw through the ~1ven point in the plane a line parallel to it, in order to have the required perpendicular given.

.But there exists another way of connecting these propositions with algebra, based on modern notions which seem destined greatly to change and to simplify mathematics.^ But there exists another way of connecting these propositions with algebra, based on modern notions which seem destined greatly to change and to simplify mathematics.

^ The choice of this particular group is motivated by its mathematical simplicity, but also by the fact that “there exist in nature some remarkable bodies which are called solids , and experience tells us that the different possible movements of these bodies are related to one another much in the same way as the different operations of the chosen group” (ibid.
  • Nineteenth Century Geometry (Stanford Encyclopedia of Philosophy) 16 January 2010 23:55 UTC plato.stanford.edu [Source type: Academic]

^ There are an infinite number of ways in which Islamic geometries can be organised to form patterns, as these notes may demonstrate.
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.We shall introduce here as much of it as is required for our present purpose.^ We shall introduce here as much of it as is required for our present purpose.

.At tije beginning of the second book we find a definition according to which a rectangle is said to be contained by the two sides which contain one of its right angles; in the text this phraseology is extended by speaking of rectangles contained by any two straight lines, meaning the rectangle which has two adjacent sides equal to the two straight lines.^ If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means~ and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines, are pro portionals.

^ Each will be fieth of a right angle, or lth of two right angles.

^ At tije beginning of the second book we find a definition according to which a rectangle is said to be contained by the two sides which contain one of its right angles; in the text this phraseology is extended by speaking of rectangles contained by any two straight lines, meaning the rectangle which has two adjacent sides equal to the two straight lines.

.We shall denote a finite straight line by a single small letter, a, b, c, - - - x, and the area of the rectangle contained by-two lines a and b by ab, and this we shall call the product of the two lines a and b.^ A ruled quadric surface contains two sets of straight lines.

^ The prod~cl of two lines a and b (in symbols, ab) means the area of the rectangle contained by the lines a and b.

^ We shall denote a finite straight line by a single small letter, a, b, c, - - - x, and the area of the rectangle contained by-two lines a and b by ab, and this we shall call the product of the two lines a and b.

.It will be understood that this definition has nothing to do with the definition of a product of numbers.^ It will be understood that this definition has nothing to do with the definition of a product of numbers.

.We define as follows: The sum of two straight lines a and b means a straight line c,which may be divided in two parts equal respectively to a and b.^ To divide a straight line in a given ratio.

^ To divide a straight line into a given number of equal parts.

^ We define as follows: The sum of two straight lines a and b means a straight line c,which may be divided in two parts equal respectively to a and b.

This sum is denoted by a+b.
The differehce of two lines a and b (in symbols, ab) means a line c which when added to b gives a; that is, ab=c if b+c=a.
.The prod~cl of two lines a and b (in symbols, ab) means the area of the rectangle contained by the lines a and b.^ The prod~cl of two lines a and b (in symbols, ab) means the area of the rectangle contained by the lines a and b.

^ If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means~ and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines, are pro portionals.

^ But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude.

For aa, which means the square on the line a, we write a2.
.21. The first ten of the fourteen propositions of the second book may then be written in the form of formulae as follows: Prop.^ The first ten of the fourteen propositions of the second book may then be written in the form of formulae as follows: Prop.

^ In the same manner every one of the first ten propositions is proved.

^ The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares.

I. a(b+c+d+.. .)ab+ac+ad+
2. ab+ac=a2ifb+c=a.
3. a(a+b) =a2+ab.
4. (a+b)1=a2-4-2ab-l-b2.
5. (a+b) (ab)-l-b2=a2.
6. (a+b) (ab)+bi=at.
7. a2+(ab)2=2a(ab)+b2.
8.4(a+h)a+b2= (2a+b)2.
9. (a-~-b)i-{-(a b)2=2a2+2b2.
10. (a+b)2+(a b)22a2+2b2.
.It will be seen that 5 and 6, and also 9 and To, ire id,~ntical.^ It will be seen that 5 and 6, and also 9 and To, ire id,~ntical.

.In Euclids statement they do not look the same, the figures being arranged differently.^ In Euclids statement they do not look the same, the figures being arranged differently.

^ Euclid (he continued) was the author of a book which contains many different figures leading to the knowledge of how things, both known and hidden, really are.
  • Islamic Art and the Argument from Academic Geometry 16 January 2010 23:55 UTC www.sonic.net [Source type: FILTERED WITH BAYES]

If the letters a, b, c,. .. denoted numbers, it follows from algebra that each of these formulae is true. .But this does not prove them in our case, where the letters denote lines, and their products areas without any reference to numbers.^ But this does not prove them in our case, where the letters denote lines, and their products areas without any reference to numbers.

^ I., The extremities of a line are points, is a proposition which either has to be proved, and then it is a theorem, or which has to be taken for granted, in which case it is an axiom .

^ It is .to be remarked that Euclid proves these propositions only in the case when the parallelograms or triangles have their bases in the sanie straight line.

To prove them we have to discover the laws which rule the operations introduced, viz, addition and multiplication of segments. .This we shall do now; and we shall find that these laws are the same with those which hold in algebraical addition and multiplication.^ This we shall do now; and we shall find that these laws are the same with those which hold in algebraical addition and multiplication.

^ These laws about addition are reducible to the two a+b=b+a.

^ Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.

.22. In a sum of numbers we may change the order in which the numbers are added, and we may also add the numbers together in groups and then add these groups.^ In a sum of numbers we may change the order in which the numbers are added, and we may also add the numbers together in groups and then add these groups.

^ There are an infinite number of ways in which Islamic geometries can be organised to form patterns, as these notes may demonstrate.
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^ They are laid down in the Axioms, and these ought to form such a system that nothing need be added to them in order fully to characterize space, and that nothing may be omitted without making the system incomplete.

.But this also holds for the sum of segments and for the sum of rectangles, a~ a little consideration shows.^ But this also holds for the sum of segments and for the sum of rectangles, a~ a little consideration shows.

^ The consideration that any polygonal prism may be divided into a number of triangular prisms, which have the same altitude and the sum of their bases equal to the base of the polygonal prism, shows further that the same holds for any prism whatever.

^ That this holds for our geometrical products is shown by Euclid in his first proposition of the second book, where he proves that the area of a rectangle whose base is the sum of a number of segments is equal to the sum of rectangles which have these segments separately as bases.

.That the sum of rectangles has always a meaning follows from the Props.^ That the sum of rectangles has always a meaning follows from the Props.

^ We define as follows: The sum of two straight lines a and b means a straight line c,which may be divided in two parts equal respectively to a and b.

43-45 in the first book. .These laws about addition are reducible to the two a+b=b+a.^ These laws about addition are reducible to the two a+b=b+a.

^ If the line q is made to describe a pencil about a point Q, then the line p will describe a pencil about P. These two pencils will be projective, for the line p passes through the pole of q, and whilst q describes the pencil Q, its pole describes a projective row, and this row is perspective to the pencil P. .

^ This we shall do now; and we shall find that these laws are the same with those which hold in algebraical addition and multiplication.

.. (I),
a+(b+c)sa+b+c. .. (2);
or, when expressed for rectangles, ab-fed=ed+ab. .. (3),
ab+(cd+ef)=ab+cd+ef. .. (4).
.The brackets mean that the terms in the bracket have been added together before they are added to another term.^ The brackets mean that the terms in the bracket have been added together before they are added to another term.

^ The second graphic above shows how this geometric pattern develops when they are added together following the basic rules created by the selection process.
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^ If these parallelograms be placed side by side, they may be added together to form a single parallelogram, having still one side of the given length.

.The more general cases for more terms may be deduced from the above.^ We may state it more generally, thus: .

^ The more general cases for more terms may be deduced from the above.

^ Although I may seem to be making a case for one form of geometry being more attractive than another, that is not the case.
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.For the product of two numbers we have the law that it remains unaltered if the factors be interchanged.^ For the product of two numbers we have the law that it remains unaltered if the factors be interchanged.

This also holds for our geometrical product. .For if ab denotes the area of the rectangle which has a as base and b as altitude, then ba will denote the area of the rectangle which has b as base and a as altitude.^ For if ab denotes the area of the rectangle which has a as base and b as altitude , then ba will denote the area of the rectangle which has b as base and a as altitude .

^ But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude.

^ This shows how to multiply quotients in our geometrical calculus Further, Two triangles have the ratios of their areas com~oundei of the ratios of their bases and their altitude.

.But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude.^ But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude.

^ Either face of it may be taken as base, and its distance from the opposite face as altitude.

^ If a, b, c be the three edges meeting at a point, then we may take the rectangle contained by two of them, lay by b and c, as base and the third as altitude.

.This gives ab=ba.^ This gives ab=ba.

... (5).
.In order further to multiply a sum by a number, we have in algebra the rule :Multiply each term of the sum, and add the products thus obtained.^ In order further to multiply a sum by a number, we have in algebra the rule :Multiply each term of the sum, and add the products thus obtained.

^ We may give concrete interpretation to an algebraical equation by allowing its terms all to mean numbers of times the same unit length, or the same unit area, or &c.

^ The consideration that any polygonal prism may be divided into a number of triangular prisms, which have the same altitude and the sum of their bases equal to the base of the polygonal prism, shows further that the same holds for any prism whatever.

.That this holds for our geometrical products is shown by Euclid in his first proposition of the second book, where he proves that the area of a rectangle whose base is the sum of a number of segments is equal to the sum of rectangles which have these segments separately as bases.^ This also holds for our geometrical product.

^ But this also holds for the sum of segments and for the sum of rectangles, a~ a little consideration shows.

^ That this holds for our geometrical products is shown by Euclid in his first proposition of the second book, where he proves that the area of a rectangle whose base is the sum of a number of segments is equal to the sum of rectangles which have these segments separately as bases.

In symbols this gives, in the simplest case, a(b+c)=ab-l-ac 6
and (b+c)a=ba+ca .
.To these laws, which have been investigated by Sir William Hamilton and by Hermann Grassmann, the former has given special names.^ To these laws, which have been investigated by Sir William Hamilton and by Hermann Grassmann, the former has given special names.

^ In place of a, /3, 7 it is lawful to use, as coordinates specifying the position of a point in the plane of a triangle of reference ABC, any given multiples of these.

.He calls the laws expressed in (I)and (~) the commutative law for addition; (5) ,, ,, multiplication; (2) and (4) the associative laws for addition; (6) the distributive law.^ He calls the laws expressed in (I)and (~) the commutative law for addition; (5) ,, ,, multiplication; (2) and (4) the associative laws for addition; (6) the distributive law.

^ This we shall do now; and we shall find that these laws are the same with those which hold in algebraical addition and multiplication.

^ To prove them we have to discover the laws which rule the operations introduced, viz, addition and multiplication of segments.

.23. Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.^ In the same manner every one of the first ten propositions is proved.

^ Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.

^ Having now shown how to represent points, planes and lines, we have to state the conditions which must hold in order that these elements may lie one in the other, or else that the figure formed by them may possess certain metrical properties.

.The first is proved geometrically, it being one of the fundamental laws.^ The first is proved geometrically, it being one of the fundamental laws.

^ In the same manner every one of the first ten propositions is proved.

^ Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.

.The next two propositions are only special cases pf the first.^ The next two propositions (7 and 8) again belong together.

^ Of these we coirsider only the first two cases.

^ The next two propositions are only special cases pf the first.

Of the others we shall prove one, viz. the fourth: (a-+b)=(a+b) (a+b)=(a+b)a+(a+b)b by (6).
But (a-l-b)aaa+ba by (6),
=aa+ab by (5);
and (a+b)b=ab+bb by (6).
Therefore (a+b)2 = a-a +ab+ (ab+bb))
aa+(ab+ab)+bb f by (4);-
=aa+2ab+bb)
This gives the theorem in question.
.In the same manner every one of the first ten propositions is proved.^ In the same manner every one of the first ten propositions is proved.

^ Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.

^ Each involution in an axial pencil contains in the same manner one pair of conjugate planes at right angles to one another.

.It will be seen that the operations performed arc exactly the same as if the letters denoted numbers.^ It will be seen that the operations performed arc exactly the same as if the letters denoted numbers.

^ However, it is obvious from some of the photos seen here that there can be a number of different divisions of the same type of plant, and that few plants with many divisions consistently show the same number.
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^ But this does not prove them in our case, where the letters denote lines, and their products areas without any reference to numbers.

Props. 5 and 6 may also be written thus (a+b) (ab)=aZb2.
Prop. .7, which is an easy consequence of Prop.^ An easy consequence of this is the following theorem, which is essentially the same as Prop.

4, may be transformed. If we denote by c the line a+b, so that c=a+b, acb, we get c2+(c b)2=2c(c b)+b2
=2ci 2bC+bi.
Subtracting c2 from both sides, and writing a for c, we get (a b)2 =a2 2ab+b2.
.In Euclids Elements this formS of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them.^ Euclid does not give the theorem in this form.

^ The proposition itself does not state this.

^ In Euclids Elements this formS of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them.

.24. The remaining two theorems (Props.^ The remaining two theorems (Props.

^ We get, for instance, from I. 4, the theorem, If two trihedral angles have the angles of two faces in the one equal to the angles of two faces in the other, and have likewise the angles included by these faces equal, then the angles in the remaining faces are equal, and the angles between the other faces are equal each to each, viz.

^ If the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop.

.12 and 13) connect the square on one side of a triangle with the sum of the squares on the other sides, in case that the angle between the latter is acute or obtuse.^ As the latter are perpendicular, they will bisect the angles between the other pair.

^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

^ In other words, the areas of similar triangles are to one another as the squares on homologous sides.

.They are important theorems in trigonometry, where it is possible to include them in a single theorem.^ They are important theorems in trigonometry, where it is possible to include them in a single theorem.

.25. There are in the second book two problems, Props.^ Having solved two problems (Props.

^ There are in the second book two problems, Props.

^ On the same straight line, and on the same side of it, there cannot be two similar segments of circles, not coinciding with one another; Prop.

II and 14.
.If written in the above symbolic language, the former requires to find a line x such that a(ax) =xf.^ If written in the above symbolic language, the former requires to find a line x such that a(ax) =xf.

^ To find the projections of a line which joins two points A, B given by their projections A1, A2 and B~, B2, we join A1, B, and A2, B2 these will be the projections required.

^ Now the general equation of the first degree Ax+By+C =o may be written y= ~x~, Unless B=o, in which case it represents a line parallel to the axis of y; and A/B, C/B are values which can be given to m and b, so that every equation of the first degree represents a straight line.

Prop. .II contains, therefore, the solution of a quadratic equation, which we may write x2+ax =a2. The solution is required later on in the construction of a regular decagon.^ The solution is required later on in the construction of a regular decagon.

^ The next three propositions contain problems which may be said to be solutions of quadratic equations.

^ II contains, therefore, the solution of a quadratic equation , which we may write x2+ax =a2.

.More important is the problem in the last proposition (Prop.^ More important is the problem in the last proposition (Prop.

^ If the last proposition be applied to an isosceles triangle, which has two sides equal, we obtain the theorem (Prop.

^ Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them: Prop.

14). .It requires the construction of a square equal in area to a given rectangle, hence a solution of the equation x1=ab.^ It requires the construction of a square equal in area to a given rectangle, hence a solution of the equation x1=ab.

^ The solution is required later on in the construction of a regular decagon.

^ The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares.

.In Book 1., 42-45, it has been shown howa rectangle may be constructed equal in area to a given figure bounded by straight lines.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ To divide a straight line into a given number of equal parts.

.By aid of the new proposition we may therefore now determine a line such that the square on that line is equal in area to any given rectilinear figure, or we can square any such figure.^ These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle.
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.As of two squares that is the greater which has the greater side, it follows that now the comparison of two areas has been reduced to the comparison of two lines.^ As of two squares that is the greater which has the greater side, it follows that now the comparison of two areas has been reduced to the comparison of two lines.

^ If we now define a quotient ~ of two lines as the number which multiplie4 into b gives a, so that we see that from the equality of two quotients follows, if we multiply both sides by lid, ~b.d=~d.b, ad = cb.

^ From the definition it follows that every focus lies on an axis, for the line joining a focus to the centre of the conic is a diameter to which the conjugate lines are perpendicular; and every line joining two foci is an axis, for the perpendiculars to this line through the foci are conjugate to it.

.The problemof reducing other areas to squares is frequently met with among Greek mathematicians.^ The problemof reducing other areas to squares is frequently met with among Greek mathematicians.

^ In other words, the areas of similar triangles are to one another as the squares on homologous sides.

^ As of two squares that is the greater which has the greater side, it follows that now the comparison of two areas has been reduced to the comparison of two lines.

.We need only mention the problem of squaring the circle (see CIRCLE).^ We need only mention the problem of squaring the circle (see CIRCLE).

^ The circle, however, was taken up by the Sophists , who made most of their discoveries in attempts to solve the classical problems of squaring the circle, doubling the cube and trisecting an angle.

^ The fourth book contains only problems, all relating to the construction of triangles and polygons inscribed in and circumscribed about circles, and of circles inscribed in or circumscribed about triangles and polygons.

.In the present day the comparison of areas is performed in a simpler way by reducing all areas to rectangles having a common base.^ In the present day the comparison of areas is performed in a simpler way by reducing all areas to rectangles having a common base.

^ The propositions in the second book are very different in character from those in the first; they all relate to areas of rectangles and squares.

^ Incidentally, all of these first three examples are based on eight point geometry, a common and relatively easy framework to establish.
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.Their altitudes give then a measure of their areas.^ Their altitudes give then a measure of their areas.

.The construction of a rectangle having the base u, and being equal in area to a given rectangle, depends upon Prop.^ While many regard Islamic designs as being based on strict geometrical constructions, there are also Islamic designs that are formed with floral devices and where the geometry might not be so obvious.
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.43, 1. This therefore gives a solution of the equation a-b =ux, where x denotes the unknown altitude.^ This therefore gives a solution of the equation a-b =ux, where x denotes the unknown altitude.

^ II contains, therefore, the solution of a quadratic equation , which we may write x2+ax =a2.

BooK III.
.26. The third book of the Elements relates exclusively to properties of the circle.^ The fifth book of the Elements is not exclusively geometrical.

^ The third book of the Elements relates exclusively to properties of the circle.

^ In this book figures -are considered which are not confined to a plane, viz, first relations between lines and planes in space, and af terwards properties of solids.

.A circle and its circumference have been defined in Book I., Def.^ A circle and its circumference have been defined in Book I., Def.

^ The rest of the book relates to angles connected with a circle, viz, angles which have the vertex either at the centre or on the circumference, and which are called respectively angles at the centre and angles at the circumference.

.15. We restate it here in slightly different words: Definition.The circumference of a circle is a plane curve such that all points in it have the same distance from a fixed point in the plane.^ (E ~ i) touches a circle with the fixed point for centre.

^ We restate it here in slightly different words: Definition.The circumference of a circle is a plane curve such that all points in it have the same distance from a fixed point in the plane.

^ The geometry of the circle,, previously studied in Egypt and much more seriously by Tbales, was somewhat neglected, although this curve was regarded as the most perfect of all plane figures and the sphere the most perfect of all solids.

.This point is called the centre of the circle.^ (E ~ i) touches a circle with the fixed point for centre.

^ This point is called the centre of the circle.

^ If now the plane a be turned about a the point P will describe a circle about Q as centre with radius QP=QR, in a plane perpendicular to the trace a.

Of the new definitions, of which eleven are given at the beginning of the third book, a few only require special mention. .The first,, which says that circles with equal radii are equal,is in part a theorem, but easily proved by applying the one circle to the other.^ Every other edge is equal to one of them.

^ With equal ease the following theorem is proved: .

^ We prove the first part only.

Or it may be considered proved by aid of Prop. 24, equal circles not being used till after this theorem.
.In the second definition is explained what is meant by a line which touches a circle.^ With a centre on the point where these second circles meet the horizontal line, a third, smaller, circle is drawn which intersects with the horizontal line as well as the extended lines of the larger squares.
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.Such a line is now generally called a tangent to the circle.^ Such a line is now generally called a tangent to the circle.

^ At a special point which is such that the chief tangents there run to the circular points at infinity in the tangent plane, the principal tangents are indeterminate; such a special point is called an umbilic of the surface.

^ These are called chief-tangent curves; on a quadric surface they are the above straight lines.

.The introduction of this name allows us to state many of Euclids propositions in a much shorter form.^ The introduction of this name allows us to state many of Euclids propositions in a much shorter form.

^ In Euclid each proposition stands by itself; its connection with others is never indicated; the leading ideas contained in its proof are not stated; general principles do not exist.

^ It is often stated that Euclid invented this kind of ~g,roof, hut the method is most likely much older.

.For the same reason we shall call a straight line joining two points on the circumference of a circle a chord.^ Any two distinct points lie in one and only one straight line.

^ But two circles which have a common centre, and whose circumferences have a point in common, coincide.

^ This point is called the centre of the circle.

.Definitions 4 and 5 may be replaced with a slight generalization by the following:- Definition.By the distance of a point from a line is meant the length of the perpendicular drawn from the point to the line.^ Definitions 4 and 5 may be replaced with a slight generalization by the following:- Definition.By the distance of a point from a line is meant the length of the perpendicular drawn from the point to the line.

^ The distance between two parallel lines or between two parallel planes is the length of the common perpendicular between the lines or the planes.

^ From this follows: If there is one point on a quad nc surface through which one, but only one, line can be drawn on the surface, then throurh every Point one line can be drawn, and all these lines meet in a point.

.27. From the definition of a circle it follows that every circle has a centre.^ From the definition of a circle it follows that every circle has a centre.

^ From the definition it follows that every focus lies on an axis, for the line joining a focus to the centre of the conic is a diameter to which the conjugate lines are perpendicular; and every line joining two foci is an axis, for the perpendiculars to this line through the foci are conjugate to it.

^ Conversely every equation of this form represents a circle: we have only to take A, B, Ai+Bi_C for h, k, pi respectively, to obtain its centre and radius.

Prop. 1 require~s to find it when the circle is given, i.e. when its circumference is drawn. -
.To solve this problem a chord is drawn (that is, any two points in the circumference are joined), and through the point where this is bisected a perpendicular to it is erected.^ The problem is not a determinate one, inasmuch as the chord may be drawn from any point in the circumference.

^ To solve this problem a chord is drawn (that is, any two points in the circumference are joined), and through the point where this is bisected a perpendicular to it is erected.

^ Having solved two problems (Props.

.Euclid then proves, first, that no point off this perpendicular can be the centre, hence that the centre must lie in this line; and, secondly, that of the points on the perpendicular one only can be the centre, viz, the one which bisects the parts of the perpendicular bounded by the circle.^ We prove the first part only.

^ (E ~ i) touches a circle with the fixed point for centre.

^ (Hence a line has two distinct points at infinity, and not one only as in ordinary geometry.

.In the second part Euclid silently assumes that the perpendicular there used does cut the circumference in two, and only in two points.^ In the latter there appear, on first consideration, to be very few possibilities for variety as there are only a limited number of two-dimensional geometries on which forms or patterns can be based.
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^ On the left every second point of intersection of the surrounding circles with the basic circle has been connected, creating two interlocking, regular isosceles triangles, creating a regular six-pointed star.
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^ With the compasses centred on the two points where the first circle cuts the line and the compasses set at a greater distance than the radius of the first circles, describe two arcs from each of these two centres above and below the line.
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.The proof therefore is incomplete.^ The proof therefore is incomplete.

The proof of the first part, however, rs exact. .By drawing two non-parallel chords, and the perpendiculars which bisect them, the centre will be found as the point where these perpendiculars intersect.^ By drawing two non-parallel chords, and the perpendiculars which bisect them, the centre will be found as the point where these perpendiculars intersect.

^ Where these arcs intersect, draw a line.
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^ If a straight line through the centre of a circle bisect a chord, then it is perpendicular to the chord, and if it be perpendicular to the chord it bisects it.

28. In Prop. 2 it is proved that a chord of. circle lies altogether within the circle.
.What we have called the first part of Euclids solution of Prop.^ What we have called the first part of Euclids solution of Prop.

^ The solution of the first part is of ipterest in itself.

^ If we express this again in symbols, calling the given base a, the one part x, and the altitude y, we have to determine x and y in the first case from the equations (ax)y = .

.I may be stated as a theorem: Every straight line which bisects a chord, and is at right angles to it, passes through the centre of the circle.^ This will intersect with the first line at right angles.
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^ Again, draw a straight line and, with the centre of your compasses on the line, draw a circle.
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^ Draw a straight line and, with the centre of your compasses on the line, draw a circle.
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.The converse to this gives Prop.^ Book III., gives the converse to Prop.

^ The converse to this gives Prop.

.3, which may be stated thus: If a straight line through the centre of a circle bisect a chord, then it is perpendicular to the chord, and if it be perpendicular to the chord it bisects it.^ The theorem may also be stated thus: - .

^ Thus Lobatchewskys parallels are represented by straight lines intersecting on the circle.

^ A line through the centre is called a diameter.

An easy consequence of this is the following theorem, which is essentially the same as Prop. .4: Two chords of a circle, of which neither passes through the centre, cannot bisect each other.^ Two chords of a circle, of which neither passes through the centre, cannot bisect each other.

^ If a straight line through the centre of a circle bisect a chord, then it is perpendicular to the chord, and if it be perpendicular to the chord it bisects it.

^ The result is that the problem has always a solution, viz, the centre of the circle is the point where the bisectors of two of the interior angles of the triangle, meet.

.These last three theorems are fundamental for the theory of the circle.^ These last three theorems are fundamental for the theory of the circle.

^ The theorems converse to the last form the contents of the next three prooositions.

^ Thus, if we choose to take any one of these three theorems as an axiom, all the theorems of projective geometry which do not require ordinal or metrical ideas for their enunciation can be proved.

.It is to be remarked that Euclid never proves that a straight line cannot have more than two points in common with a circumference.^ I have to admit that I’m not sure whether five or six divisions are the more common, but my impression is that it’s likely to be the six-pointed.
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^ This interleaving is shown to be more conservative in number of Steiner point insertions than refinement alone, and to produce higher quality meshes than optimization alone.
  • Applied Geometry Lab Publications 16 January 2010 23:55 UTC geometry.caltech.edu [Source type: Academic]

^ With the compasses centred on the two points where the first circle cuts the line and the compasses set at a greater distance than the radius of the first circles, describe two arcs from each of these two centres above and below the line.
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.29. The next two propositions (5 and 6) might be replaced by a single and a simpler theorem, viz: Two circles which have a common centre, and whose circumferences have one point in common, coincide.^ (E ~ i) touches a circle with the fixed point for centre.

^ As a consequence of this we get If the circumferences of the two circles have three points in common they coincide.

^ The next two propositions (7 and 8) again belong together.

.Or, more in agreement with Euclids form: Two different circles, whose circumferences have a point in common, cannot have the same centre.^ (E ~ i) touches a circle with the fixed point for centre.

^ The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.

^ As a consequence of this we get If the circumferences of the two circles have three points in common they coincide.

.That Euclid treats of two cases is characteristic of Greek mathematics.^ That Euclid treats of two cases is characteristic of Greek mathematics.

^ That the two cases here stated are given by Euclid in two separate propositions proved separately is characteristic of his method.

^ These two works, which contain the greatest contributions to ancient geometry, are treated in detail in Section I. Euclidean Geometry and the articles EUCLID; CONIC SECTION ; AP0LL0NIUs.

.The next two propositions (7 and 8) again belong together.^ The next two propositions (7 and 8) again belong together.

^ The next proposition, together with one added by Simson as Prop.

^ The next two propositions (5 and 6) might be replaced by a single and a simpler theorem, viz: Two circles which have a common centre, and whose circumferences have one point in common, coincide.

.They may be combined thus: If from a point in a plane of a circle, which is not the centre, straight lines be drawn to the different points of the circumference, then of all these lines one is the shortest, and one the longest, and these lie both in that straight line which joins the given point to the centre, Of all the remaining lines each is equal to one and only one other, and these equal lines lie on opposite sides of the shortest or longest, and make equal angles with them.^ To divide a straight line in a given ratio.

^ (E ~ i) touches a circle with the fixed point for centre.

^ Hence, a straight line which does not lie in the plane cannot have more than one point in common with the plane.

.Euclid distinguishes the two cases where the given point lies within or without the circle, omitting the case where it lies in the circumference.^ But two circles which have a common centre, and whose circumferences have a point in common, coincide.

^ Given a circle and a point 0 (fig.

^ Euclid distinguishes the two cases where the given point lies within or without the circle, omitting the case where it lies in the circumference.

.From the last proposition it follows that if from a point more than two equal straight lines can be drawn to the circumference, this point must be the centre.^ If a point lies in two lines its projections must lie in the projections of both.

^ Any two distinct points lie in one and only one straight line.

^ But two circles which have a common centre, and whose circumferences have a point in common, coincide.

This is Prop. 9.
.As a consequence of this we get If the circumferences of the two circles have three points in common they coincide.^ As a consequence of this we get If the circumferences of the two circles have three points in common they coincide.

^ But two circles which have a common centre, and whose circumferences have a point in common, coincide.

^ The result is that the problem has always a solution, viz, the centre of the circle is the point where the bisectors of two of the interior angles of the triangle, meet.

.For in this case the two circles havefl a common centre, because from the centre of the one three equal Lines can be drawn to points on the circumference of the other.^ (E ~ i) touches a circle with the fixed point for centre.

^ As a consequence of this we get If the circumferences of the two circles have three points in common they coincide.

^ Every other edge is equal to one of them.

.But two circles which have a common centre, and whose circumferences have a point in common, coincide.^ (E ~ i) touches a circle with the fixed point for centre.

^ As a consequence of this we get If the circumferences of the two circles have three points in common they coincide.

^ Or, more in agreement with Euclids form: Two different circles, whose circumferences have a point in common, cannot have the same centre.

(Compare above statement of Props. 5 and 6.)
.This theorem may also be stated thus: Through three points only one circumference may be drawn; or, Three points determine a circle.^ The theorem may also be stated thus: - .

^ Through every point in space not on the twisted cubic one and only one secant to the cubic can be drawn.

^ All points describe, accordingly, one of the three types of circles.

.Euclid does not give the theorem in this form.^ Euclid does not give the theorem in this form.

^ In Euclids Elements this formS of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them.

.He proves, however, that the two circles cannot cut another in more than two points (Prop.^ On the left every second point of intersection of the surrounding circles with the basic circle has been connected, creating two interlocking, regular isosceles triangles, creating a regular six-pointed star.
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^ This interleaving is shown to be more conservative in number of Steiner point insertions than refinement alone, and to produce higher quality meshes than optimization alone.
  • Applied Geometry Lab Publications 16 January 2010 23:55 UTC geometry.caltech.edu [Source type: Academic]

^ With the compasses centred on the two points where the first circle cuts the line and the compasses set at a greater distance than the radius of the first circles, describe two arcs from each of these two centres above and below the line.
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.10), and that two circles cannot touch one another in more points than one (ProD. 13).^ (E ~ i) touches a circle with the fixed point for centre.

^ He proves, however, that the two circles cannot cut another in more than two points (Prop.

^ There are two singly infinite systems of curves on a surface, a pair cutting one another at right angles through every point upon it, all tangents to which are principal tangents of the surface at their respective points of contact.

.30. Propositions II and 12 assert that if two circles touch, then the point of contact lies on the line joining their centres.^ (E ~ i) touches a circle with the fixed point for centre.

^ If a point lies in two lines its projections must lie in the projections of both.

^ But two circles which have a common centre, and whose circumferences have a point in common, coincide.

.This gives two propositions, because the circles may touch either internally or externally.^ This gives two propositions, because the circles may touch either internally or externally.

^ Propositions II and 12 assert that if two circles touch, then the point of contact lies on the line joining their centres.

^ But in a rectangle we may take either of the two lines which contain it as base, and then the other will be the altitude.

.31. Propositions 14 and 15 relate to the length of chords.^ Propositions 14 and 15 relate to the length of chords.

The first says that equal chords are equidistant from the centre, and that chords which are equidistant from the centre are equal; Whilst Prop. 15 compares unequal chords, viz. .Of all chords the diameter is the greatest, and of other chords that is the greater which is nearer to the centre; and conversely, the greater chord is nearer to the centre.^ Of all chords the diameter is the greatest, and of other chords that is the greater which is nearer to the centre; and conversely, the greater chord is nearer to the centre.

^ If four magnitudes of the same kind be proportional, the greatest and least of them together shall be greater than the other two together.

^ Two chords of a circle, of which neither passes through the centre, cannot bisect each other.

32. In Prop. 16 the tangent to a circle is for the first time introduced. .The proposition is meant to show that the straight line at the end point of the diameter and at right angles to it is a tangent.^ Cbrollary.The straight line at right angles to a diameter drawn through the end point of it touches the circle.

^ This will intersect with the first line at right angles.
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^ One line through (x, y, z) is at right angles to the tangent plane.

.The proposition itself does not state this.^ The proposition itself does not state this.

^ In Euclid each proposition stands by itself; its connection with others is never indicated; the leading ideas contained in its proof are not stated; general principles do not exist.

^ In Euclids Elements this formS of the theorem does not appear, all propositions being so stated that the notion of subtraction does not enter into them.

.It runs thus: Prop.^ It runs thus: Prop.

.16. The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line cau be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle.^ The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line cau be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle.

^ Cbrollary.The straight line at right angles to a diameter drawn through the end point of it touches the circle.

^ P comes to a point S on the line P,Q perpendicular to ac, so that QS=QP. But QP is the hypotenuse of a triangle PP1Q with a right angle P1.

.Cbrollary.The straight line at right angles to a diameter drawn through the end point of it touches the circle.^ (E ~ i) touches a circle with the fixed point for centre.

^ The straight line drawn at right angles to the diameter of a circle, from the extremity of it, falls without the circle; and no straight line cau be drawn from the extremity, between that straight line and the circumference, so as not to cut the circle.

^ Cbrollary.The straight line at right angles to a diameter drawn through the end point of it touches the circle.

.The statement of the proposition and its whole treatment show the difficulties which the tangents presented to Euclid.^ The statement of the proposition and its whole treatment show the difficulties which the tangents presented to Euclid.

^ It is the fundamental theorem of Euclids whole system, there being only a very few propositions (like Props.

^ The proposition is meant to show that the straight line at the end point of the diameter and at right angles to it is a tangent.

Prop. .17 solves the problem through a given point, either in the circumference or without it, to draw a tangent to a given circle.^ To draw a straight line perpendicularly to a given straight line through a given point in it (PrOp.

^ To draw a plane through a line and a point without the line, we join the given point to any point in the line and determine the plane through this and the given line.

^ Through a given point to draw a planeparalll to a given plane.

Closely connected with Prop. 16 are Props. 18 and 19, which state (Prop. i .8), that the line joining the centre of a circle to the point of contact of a tangent is perpendicular to the tangent; and conversely (Prop.^ With the compasses centred on the two points where the first circle cuts the line and the compasses set at a greater distance than the radius of the first circles, describe two arcs from each of these two centres above and below the line.
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^ With a centre on the point where these second circles meet the horizontal line, a third, smaller, circle is drawn which intersects with the horizontal line as well as the extended lines of the larger squares.
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^ These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle.
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.19), that the straight line through the point of contact of, and perpendicular to, a tangent to a circle passes through the centre of the circle.^ (E ~ i) touches a circle with the fixed point for centre.

^ A line through the centre is called a diameter.

^ This point is called the centre of the circle.

.33. The rest of the book relates to angles connected with a circle, viz, angles which have the vertex either at the centre or on the circumference, and which are called respectively angles at the centre and angles at the circumference.^ The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.

^ This point is called the centre of the circle.

^ The rest of the book relates to angles connected with a circle, viz, angles which have the vertex either at the centre or on the circumference, and which are called respectively angles at the centre and angles at the circumference.

.Between these two kinds of angles exists the important relation expressed as follows Prop.^ From this follows easily the important theorem Prop.

^ The angle between two geodesics u=const., v=const.

^ Between these two kinds of angles exists the important relation expressed as follows Prop.

.20. The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.^ The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.

^ In equal circles equal angles stand on equal arcs, whether they be at the centres or circumferences; Prop.

^ The result is that the problem has always a solution, viz, the centre of the circle is the point where the bisectors of two of the interior angles of the triangle, meet.

.This is of great importance for its consequences, of which the two following are the principal: Prop.^ This is of great importance for its consequences, of which the two following are the principal: Prop.

^ By the aid of these two, the following fundamental properties of triangles are easily proved: Prop.

^ An easy consequence of this is the following theorem, which is essentially the same as Prop.

.21. The angles in the same segment of a circle are equal to one another; Prop.^ With the compasses centred on one of the two points where the circle cuts the line, draw another circle of the same radius.
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.22. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.^ Each angle in a regular triangle equals two-thirds of one right angle.

^ Each will be fieth of a right angle, or lth of two right angles.

^ The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

.Further consequences are: Prop.^ Further consequences are: Prop.

.23. On the same straight line, and on the same side of it, there cannot be two similar segments of circles, not coinciding with one another; Prop.^ If two planes cut one another, their common section is a straight line.

^ The angles in the same segment of a circle are equal to one another; Prop.

^ Any two distinct points lie in one and only one straight line.

.24. Similar segments of circles on equal straight lines are equal to one another.^ The angles in the same segment of a circle are equal to one another; Prop.

^ On the same straight line, and on the same side of it, there cannot be two similar segments of circles, not coinciding with one another; Prop.

^ Similar segments of circles on equal straight lines are equal to one another.

The problem Prop. .25. A segment of a circle being given to describe the circle of which it is a segment, may be solved much more easily by aid of the construction described in relation to Prop.^ A segment of a circle being given to describe the circle of which it is a segment, may be solved much more easily by aid of the construction described in relation to Prop.

^ The regular hexagon is more easily constructed, as shown in Prop.

^ The angles in the same segment of a circle are equal to one another; Prop.

1, III., in 27.
.34. There follow four theorems connecting the angles at the centre, the arcs into which they divide the circumference, and the chords subtending these arcs.^ The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.

^ There follow several important theorems: .

^ There follow four theorems connecting the angles at the centre, the arcs into which they divide the circumference, and the chords subtending these arcs.

.They are expressed for angles, arcs and chords in equal circles, but they hold also for angles, arcs and chords in the same circle.^ The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.

^ The angles in the same segment of a circle are equal to one another; Prop.

^ They are expressed for angles, arcs and chords in equal circles, but they hold also for angles, arcs and chords in the same circle.

The theorems are: Prop. .26. In equal circles equal angles stand on equal arcs, whether they be at the centres or circumferences; Prop.^ The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.

^ In equal circles the angles which stand on equal arcs are equal to one another, whether they be at the centres or the circumferences; Prop.

^ The angles in the same segment of a circle are equal to one another; Prop.

27. (converse to Prop. 26). .In equal circles the angles which stand on equal arcs are equal to one another, whether they be at the centres or the circumferences; Prop.^ The angle at the centre of a circle is double of the angle at the circumference on the same base, that is, on the same arc.

^ In equal circles the angles which stand on equal arcs are equal to one another, whether they be at the centres or the circumferences; Prop.

^ The angles in the same segment of a circle are equal to one another; Prop.

.28. In equal circles equal straight lines (equal chords) cut off equal arcs, the greater equal to the greater, and the less equal to the less; Prop.^ Similar segments of circles on equal straight lines are equal to one another.

^ This circle will cut off from the given straight line a length equal to the required i one.

^ In equal circles equal arcs are subtended by equal straight lines.

29 (converse to Prop. 28). .In equal circles equal arcs are subtended by equal straight lines.^ Similar segments of circles on equal straight lines are equal to one another.

^ In equal circles equal arcs are subtended by equal straight lines.

^ If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means~ and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines, are pro portionals.

i5. .Other important consequences of Props.^ Other important consequences of Props.

20-22 are: Prop. .31. In a circle the angle in a semicircle is a right anile; but the angle in a segment greater than a semicircle is less than a right angle; and the angle in a segment less than a semicircle is greater than a right angle; Prop.^ The angles in the same segment of a circle are equal to one another; Prop.

^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ The three circles producing the basis for this panel can be seen to be part of a pattern of lines of circles regularly aligned at right angles to each other in two directions, as can be seen in the lowest of the three illustrations above.
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.32. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.^ (E ~ i) touches a circle with the fixed point for centre.

^ The angles in the same segment of a circle are equal to one another; Prop.

^ Cbrollary.The straight line at right angles to a diameter drawn through the end point of it touches the circle.

~6. .Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them: Prop.^ And then the problem itself is solved in Prop.

^ After solving a few problems we come to Prop.

^ Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them: Prop.

.30. To bisect a given arc, that is, to divide it into two equal parts; Prop.^ This illustration of a hendecagon shows it divided, on the left, into twenty-two parts by lines running through its centre and, on the right, with all its chords drawn, illustrating its potential for complexity.
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^ The upper of the two can be seen to have eleven divisions, whereas the lower is divided into twelve.
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^ The form of the decoration, in this case, can be divided into two forms: free-flowing, and tailored.
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.33. On a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle; Prop.^ The three circles producing the basis for this panel can be seen to be part of a pattern of lines of circles regularly aligned at right angles to each other in two directions, as can be seen in the lowest of the three illustrations above.
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^ With the compasses centred on the two points where the first circle cuts the line and the compasses set at a greater distance than the radius of the first circles, describe two arcs from each of these two centres above and below the line.
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^ These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle.
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.34. From a given circle to cut off a segment containing an angle equal to a given rectilineal angle.^ The angles in the same segment of a circle are equal to one another; Prop.

^ On a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle; Prop.

^ This circle will cut off from the given straight line a length equal to the required i one.

i~. .If we draw chords through a point A within a circle, they will each be divided by A into two segments.^ If we draw chords through a point A within a circle, they will each be divided by A into two segments.

^ Let any point B divide 1 into two half-lines if and if.

^ To do this we draw two planes cii and hi through Si, cutting the surface I in two conies .which we also denote by cii and ~3i.

.Between these segments the law holds that the rectangle contained by them has the same area on whatever chord through A the segments are taken.^ Between these segments the law holds that the rectangle contained by them has the same area on whatever chord through A the segments are taken.

^ Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.

^ In the same way, if in a perpendicular to in through A2 a point A be taken such that A2A= A0A1, then this will give the point A relative to the plane s-i.

The value of this rectangle changes, of course, with the position of A.
.A similar theorem holds if the point A be taken without the circle.^ In this first exercise I have taken the basic seven-circle rose from the six-point geometry construction illustrated above and shown how the basic pattern lines evolve.
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.On every straight line through A, which cuts the circle in two points B and C, we have two segments AB and AC, and the rectangles contained by them are again equal to one another, and equal to the square on a tangent drawn from A to the circle.^ To describe a square on a given straight line.

^ I in two conies which have the point S and the points where it cuts af and.

^ A ruled quadric surface contains two sets of straight lines.

.The first of these theorems gives Prop.^ The first of these theorems gives Prop.

^ These theorems allow now of the solution of a number of problems, viz.: To bisect a given angle (Prop.

^ An important application of these theorems is at once made to a right-angled triangle, viz.: Prop.

35, and the second Prop. 36, with its corollary, whilst Prop. 37, the last of Book III., gives the converse to Prop. 36. The first two theorems may be combined in one: If through a point A in the plane of a circle a straight line be drawn cutting the circle in B and C, then the rectangle AB.A C has a constant value so long as the point A be fixed; and if from A a tangent AD can be drawn to the circle, touching at D, then the above rectangle equals the square on AD.
Prop. 37 may be stated thus: If from a point A without a circle a line be drawn cutting the circle in B and C, and another line to a point D on the circle, and AB.A C= AD2, then the line AD touches the circle at D.
.It is not difficult to prove also the converse to the general proposition as above stated.^ It is not difficult to prove also the converse to the general proposition as above stated.

^ In Euclid each proposition stands by itself; its connection with others is never indicated; the leading ideas contained in its proof are not stated; general principles do not exist.

^ That the two cases here stated are given by Euclid in two separate propositions proved separately is characteristic of his method.

This proposition and its converse may be expressed as follows: If four points A BCD be taken on the circumference of a circle, and if the lines AB, CD, produced if necessary, meet at E, then EA.EB =EC.ED;
and conversely, if this relation holds then the four points lie on a circle, that is, the circle drawn through three of them passes through the fourth.
That a circle may always be drawn through three points, provided that they do not lie in a straight line, is proved only later on in Book IV.
BooK IV.
.38. The fourth book contains only problems, all relating to the construction of triangles and polygons inscribed in and circumscribed about circles, and of circles inscribed in or circumscribed about triangles and polygons.^ And definitions 5 and 6 thus: Definition.A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the.

^ In each of them a circle may be inscribed, and another may be circumscribed about it.

^ To circumscribe a circle about a given triangle.

.They are nearly all given for their own sake, and not for future use in the construction of figures, as are most of those in the former books.^ The former is constantly used in nearly all problems concerning surfaces.

^ They all relate to figures in a plane.

^ They are nearly all given for their own sake, and not for future use in the construction of figures, as are most of those in the former books.

.In seven definitions at the beginning of the book it is explained what is understood by figures inscribed in or described about other figures, with special reference to the case where one figure is a circle.^ In seven definitions at the beginning of the book it is explained what is understood by figures inscribed in or described about other figures, with special reference to the case where one figure is a circle.

^ We begin by quoting those definitions at the beginning of Book V. -which are most important.

^ And definitions 5 and 6 thus: Definition.A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the.

.Instead, however, of saying that one figure is described about another, it is now generally said that the one figure is circumscribed about the other.^ Instead, however, of saying that one figure is described about another, it is now generally said that the one figure is circumscribed about the other.

^ If now the plane a be turned about a the point P will describe a circle about Q as centre with radius QP=QR, in a plane perpendicular to the trace a.

^ Having now shown how to represent points, planes and lines, we have to state the conditions which must hold in order that these elements may lie one in the other, or else that the figure formed by them may possess certain metrical properties.

.We may then state the definitions 3 or 4 thus: Definition.A polygon is said to be inscribed in a circle, and the circle is said to he circumscribed about the polygon, if the vertices of the polygon lie in the circumference of the ircle.^ The theorem may also be stated thus: - .

^ And definitions 5 and 6 thus: Definition.A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the.

^ In each of them a circle may be inscribed, and another may be circumscribed about it.

.And definitions 5 and 6 thus: Definition.A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the.^ And definitions 5 and 6 thus: Definition.A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the.

^ In each of them a circle may be inscribed, and another may be circumscribed about it.

^ We may then state the definitions 3 or 4 thus: Definition.A polygon is said to be inscribed in a circle, and the circle is said to he circumscribed about the polygon, if the vertices of the polygon lie in the circumference of the ircle.

polygon are tangents to the circle.
,p~. .The first problem is merely constructive.^ The first problem is merely constructive.

.It requires to draw in a given circle a chord equal to a given straight tine, which is not greater than the diameter of the circle.^ With the compasses centred on the two points where the first circle cuts the line and the compasses set at a greater distance than the radius of the first circles, describe two arcs from each of these two centres above and below the line.
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^ These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle.
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^ Again, draw a straight line and, with the centre of your compasses on the line, draw a circle.
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.The problem is not a determinate one, inasmuch as the chord may be drawn from any point in the circumference.^ The problem is not a determinate one, inasmuch as the chord may be drawn from any point in the circumference.

^ In 8 it has been seen that two conditions determine the equation of a straight line, because in Ax+By+C=o one of the coefficients may be divided out, leaving only two parameters to be determined.

^ All points at infinity in space may be considered as lying in one ideal plane, which is called the plane at infinity.

.This may be said of almost all problems in this book, especially of the next two.^ This may be said of almost all problems in this book, especially of the next two.

^ The next three propositions contain problems which may be said to be solutions of quadratic equations.

^ A double infinite number of lines, that is, all lines which satisfy two conditions, or which are common to two complexes, are said to form a congruence of lines; e.g.

They are: Prop. 2. In a given circle to inscribe a triangle equiangular to a given triangle; -
Prop. .3. About a given circle to circumscribe a triangle equsangular to a given triangle.^ To circumscribe a circle about a given triangle.

^ About a given circle to circumscribe a triangle equsangular to a given triangle.

^ In a given circle to inscribe a triangle equiangular to a given triangle; - .

.40. Of somewhat greater interest are the next problems, where the triangles are given and the circles to be found.^ Of somewhat greater interest are the next problems, where the triangles are given and the circles to be found.

^ To circumscribe a circle about a given triangle.

^ In a given circle to inscribe a triangle equiangular to a given triangle; - .

Prop. .4. To inscribe a circle in a given triangle.^ To circumscribe a circle about a given triangle.

^ To inscribe a regular quindecagon in a given circle.

^ In a given circle to inscribe a triangle equiangular to a given triangle; - .

.The result is that the problem has always a solution, viz, the centre of the circle is the point where the bisectors of two of the interior angles of the triangle, meet.^ The three circles producing the basis for this panel can be seen to be part of a pattern of lines of circles regularly aligned at right angles to each other in two directions, as can be seen in the lowest of the three illustrations above.
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^ On the left every second point of intersection of the surrounding circles with the basic circle has been connected, creating two interlocking, regular isosceles triangles, creating a regular six-pointed star.
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^ With the compasses centred on the two points where the first circle cuts the line and the compasses set at a greater distance than the radius of the first circles, describe two arcs from each of these two centres above and below the line.
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.The solution shows, though Euclid does not state this, that the problem has but one solution; and also, The three bisectors of the interior angles of any triangle meet in a point, and this is the centre of the circle inscribed in the triangle.^ The result is that the problem has always a solution, viz, the centre of the circle is the point where the bisectors of two of the interior angles of the triangle, meet.

^ (E ~ i) touches a circle with the fixed point for centre.

^ All points describe, accordingly, one of the three types of circles.

.The solutions of most of the other problems contain also theorems.^ The solutions of most of the other problems contain also theorems.

^ The next three propositions contain problems which may be said to be solutions of quadratic equations.

^ These theorems allow now of the solution of a number of problems, viz.: To bisect a given angle (Prop.

.Of these we shall state those which are of special interest; Euclid does not state any one of them.^ Of these we shall state those which are of special interest; Euclid does not state any one of them.

^ Those interested in these aspects of geometry will again have to look elsewhere as these areas are complex and have little or nothing to do with the main subject of this site.
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^ If two triangles have two sides of the one equal to two sides of the other, each to each, and have also the angles contained by those sides equal to one another, they shall also have their bases or third sides equal; and the two triangles shall be equal; and their other angles shall be equ at, each to each, namely, those to which the equal sides are opposite.

41. Prop. .5. To circumscribe a circle about a given triangle.^ To circumscribe a circle about a given triangle.

^ About a given circle to circumscribe a triangle equsangular to a given triangle.

^ In a given circle to inscribe a triangle equiangular to a given triangle; - .

.The one solution which always exists contains the following: The three straight lines which bisect the sides of a triangle at right angles meet in a point, and this point is the centre of the circle circumscribed about the triangle.^ The result is that the problem has always a solution, viz, the centre of the circle is the point where the bisectors of two of the interior angles of the triangle, meet.

^ (E ~ i) touches a circle with the fixed point for centre.

^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

.Euclid adds in a corollary the following property: The centre of the circle circumscribed about a triangle lies within, on a side of, or without the triangle, according as the triangle is acute-angled, right-angled or obtuse-angled.^ The three circles producing the basis for this panel can be seen to be part of a pattern of lines of circles regularly aligned at right angles to each other in two directions, as can be seen in the lowest of the three illustrations above.
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^ There is a geometric way of establishing this proportion – a right angle triangle with adjacent sides equal, will have a hypoteneuse of √2 to the adjacent sides.
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^ Illustrated to the near right, the roundel is comprised of six concentric circles of unequal widths containing triangles and a small number of lozenges.
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.42. Whilst it is always possible to draw a circle which is inscribed in or circumscribed about a given triangle, this is not the case with quadrilaterals or polygons of more sides.^ And definitions 5 and 6 thus: Definition.A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the.

^ In each of them a circle may be inscribed, and another may be circumscribed about it.

^ To circumscribe a circle about a given triangle.

Of those for which this is possible the regular polygons, i.e. polygons which have all their sides and angles equal, are the most interesting. In each of them a circle may be inscribed, and another may be circumscribed about it.
.Euclid does not use the word regular, but he describes the polygons in question as equiangular and equilateral.^ Euclid does not use the word regular, but he describes the polygons in question as equiangular and equilateral.

^ We shall use the name regular polygon.

^ Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides.

.We shall use the name regular polygon.^ We shall use the name regular polygon.

^ Euclid does not use the word regular, but he describes the polygons in question as equiangular and equilateral.

.The regular triangle is equilateral, the regular quadrilateral is the square.^ The regular triangle is equilateral, the regular quadrilateral is the square.

^ The angle in a square (the regular quadrilateral) equals one right angle.

.Euclid considers the regular polygons of 4, 5, 6 and 15 sides.^ Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides.

^ Euclid considers the regular polygons of 4, 5, 6 and 15 sides.

^ Three regular polygons of six or more sides cannot form a solid angle.

.For each of the first three he solves the problems(1) to inscribe such a polygon in a given circle; (2) to circumscribe it about a given circle; (3) to inscribe a circle iii, and (4) to circumscribe a circle about, such a polygon.^ With the compasses centred on the two points where this vertical line intersects the first circle, draw two circles with the same radius as the previous three circles.
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For the regular triangle the problems are not repeated, because more general problems have been solved.
Props. .6, ~, 8 and 9 solve these problems for the square.^ The circle, however, was taken up by the Sophists , who made most of their discoveries in attempts to solve the classical problems of squaring the circle, doubling the cube and trisecting an angle.

.The general problem of inscribing in a given circle a regular polygon of n sides depends upon the problem of dividing the circumference of a circle into n equal parts, or what comes to the same thing, of drawing from the centre of the circle n radii such that the angles between consecutive radii are equal, that is, to divide the space about the centre into n equal angles.^ The three circles producing the basis for this panel can be seen to be part of a pattern of lines of circles regularly aligned at right angles to each other in two directions, as can be seen in the lowest of the three illustrations above.
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^ These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle.
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^ With the compasses centred on one of the two points where the circle cuts the line, draw another circle of the same radius.
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.Thus, if it is required to inscribe a square in a circle, we have to draw four lines from the centre, making the four angles equal.^ Again, draw a straight line and, with the centre of your compasses on the line, draw a circle.
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^ Thus, if it is required to inscribe a square in a circle, we have to draw four lines from the centre, making the four angles equal.

^ Draw a straight line and, with the centre of your compasses on the line, draw a circle.
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.This is done by drawing two diameters at right angles to one another.^ The three circles producing the basis for this panel can be seen to be part of a pattern of lines of circles regularly aligned at right angles to each other in two directions, as can be seen in the lowest of the three illustrations above.
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^ With the compasses centred on one of the two points where the circle cuts the line, draw another circle of the same radius.
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^ With the compasses centred at one of the two points where this circle cuts the first circle, draw the next circle, continuing this process until there are six circles mutually intersecting and centred on the original circle.
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.The ends of these diameters are the vertices of the required square.^ The ends of these diameters are the vertices of the required square.

^ These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle.
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^ If, on the other hand, tangents be drawn at these ends, we obtain a square circum~cribed about the circle.

.If, on the other hand, tangents be drawn at these ends, we obtain a square circum~cribed about the circle.^ If, on the other hand, tangents be drawn at these ends, we obtain a square circum~cribed about the circle.

^ In seven definitions at the beginning of the book it is explained what is understood by figures inscribed in or described about other figures, with special reference to the case where one figure is a circle.

^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

43. To construct a regular pentagon, we find it convenient first to construct a regular decagon. .This requires to divide the space about the centre into ten equal angles.^ Generally all these rosettes are constructed the same way: the circumference is divided into the required number of points and those points joined, each to all the others.
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^ It follows that it is also very common in nature to find plants which are divided into ten divisions.
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.Each will be fieth of a right angle, or lth of two right angles.^ The three circles producing the basis for this panel can be seen to be part of a pattern of lines of circles regularly aligned at right angles to each other in two directions, as can be seen in the lowest of the three illustrations above.
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.If we suppose the decagon constructed, and if we join the centre to the end of one side, we get an isosceles triangle, where the angle at the centre equals 1/2th of two right angles; her,ce each of the angles at the base will be iths of two right angles, as all three anglea together equal two right -angles.^ With the other point, make two additional marks (B), one toward each corner along the same side.
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^ His friend had not known how to use it, so Carroll provides two method of layout: one employing the square itself, the other based on a decimal ratio.
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^ For a [regular] octagon, says Carroll (I have not checked the assertion), half the length of one of its sides equals 0.2071 multiplied by the length of the side of a square enclosing the octagon and coinciding with four of its sides.
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.Thus we have to construct an isosceles triangle, having the angle at the vertex equal to half an angle at the base.^ There is a geometric way of establishing this proportion – a right angle triangle with adjacent sides equal, will have a hypoteneuse of √2 to the adjacent sides.
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^ The pavement is based on four point geometry and is constructed entirely of only three different tiles: a square, a lozenge and a triangle.
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This is solved in Prop. .10, by aid of the problem in Prop.^ Propositions 30, 33, 34, contain problems which are solved by aid of the propositions preceding them: Prop.

.II of the second book.^ II of the second book.

.If we make the sides of this triangle equal to the radius of the given circle, then the base will be the side, of the regular decagon inscribed in tile circle.^ The result is that the side of the regular hexagon inscribed in a circle is equal to the radius, of the circle.

^ To circumscribe a circle about a given triangle.

^ To inscribe a regular quindecagon in a given circle.

.This side being known the decagon can be constructed, and if the vertices are joined alternately, leaving out half their number, we obtain the regular pentagon.^ Generally all these rosettes are constructed the same way: the circumference is divided into the required number of points and those points joined, each to all the others.
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^ So, we have a pattern, difficult to set out, which appears to be based on a slight but poor mathematical or geometrical relationship, and which suggests that the relationship is accidental, or that the numbers are significant and relate to something symbolic, the geometry of construction being incidental to the meaning.
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^ For a [regular] octagon, says Carroll (I have not checked the assertion), half the length of one of its sides equals 0.2071 multiplied by the length of the side of a square enclosing the octagon and coinciding with four of its sides.
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(Prop. If.)
.Euclid does not proceed thus.^ Euclid does not proceed thus.

.He wants the pentagon before the decagon.^ He wants the pentagon before the decagon.

.This, however, does not change the real flature of his solution, nor does his solution become simpler by not mentioning the decagon.^ This, however, does not change the real flature of his solution, nor does his solution become simpler by not mentioning the decagon.

^ XI. He does not, however, show the connexior which these definitions have with those mentioned before When ,points and lines have been defined, a statement lik Def.

.Once the regular pentagon is inscribed, it is easy to circumscribe another by drawing tangents at the vertices of the inscribed pentagon.^ Once the regular pentagon is inscribed, it is easy to circumscribe another by drawing tangents at the vertices of the inscribed pentagon.

^ In each of them a circle may be inscribed, and another may be circumscribed about it.

^ This side being known the decagon can be constructed, and if the vertices are joined alternately, leaving out half their number, we obtain the regular pentagon.

This is shown in Prop. 12.
Props. .13 and 14 teach how a circle may be inscribed in or circumscribed about any given regular pentagon.^ In each of them a circle may be inscribed, and another may be circumscribed about it.

^ To circumscribe a circle about a given triangle.

^ About a given circle to circumscribe a triangle equsangular to a given triangle.

.44. The regular hexagon is more easily constructed, as shown in Prop.^ The regular hexagon is more easily constructed, as shown in Prop.

^ A segment of a circle being given to describe the circle of which it is a segment, may be solved much more easily by aid of the construction described in relation to Prop.

.15. The result is that the side of the regular hexagon inscribed in a circle is equal to the radius, of the circle.^ These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle.
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^ Essentially they derive from the central small square upon which a circle is drawn of radius equal to half the diagonal of the square.
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^ For a [regular] octagon, says Carroll (I have not checked the assertion), half the length of one of its sides equals 0.2071 multiplied by the length of the side of a square enclosing the octagon and coinciding with four of its sides.
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.For this polygon the other three problems mentioned are not solved.^ For this polygon the other three problems mentioned are not solved.

^ For each of the first three he solves the problems(1) to inscribe such a polygon in a given circle; (2) to circumscribe it about a given circle; (3) to inscribe a circle iii, and (4) to circumscribe a circle about, such a polygon.

.45. The book closes with Prop.^ The book closes with Prop.

.16. To inscribe a regular quindecagon in a given circle.^ To inscribe a regular quindecagon in a given circle.

^ In a given circle to inscribe a triangle equiangular to a given triangle; - .

^ To inscribe a circle in a given triangle.

.If we inscribe a regular pentagon and a regular hexagon in the circle, having one vertex in common, then the arc from the common vertex to the next vertex of the pentagon is 1/2th of the circumference, and to the next vertex of the hexagon is lth of the circumference.^ To inscribe a regular quindecagon in a given circle.

^ If we inscribe a regular pentagon and a regular hexagon in the circle, having one vertex in common, then the arc from the common vertex to the next vertex of the pentagon is 1/2th of the circumference, and to the next vertex of the hexagon is lth of the circumference.

^ As a consequence of this we get If the circumferences of the two circles have three points in common they coincide.

.The difference between these arcs is, therefore, 1/2, 11-sth of the circumference.^ The difference between these arcs is, therefore, 1/2, 11-sth of the circumference.

^ There follow four theorems connecting the angles at the centre, the arcs into which they divide the circumference, and the chords subtending these arcs.

^ The difference between this and its original drawing is dramatic and shows how easily these patterns can be developed.
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.The latter may, therefore, be divided into thirty, and hence also in fifteen equal parts, and the regular quindecagoa be described.^ The latter may, therefore, be divided into thirty, and hence also in fifteen equal parts, and the regular quindecagoa be described.

^ To divide a straight line into a given number of equal parts.

^ Hence we can divide a circumference into 2n equal parts as soon as it has been divided into n equal parts, or as soon as a regular polygon of n sides has been constructed.

.46. We conclude with a few theorems about regular polygons which are not given by Euclid.^ We conclude with a few theorems about regular polygons which are not given by Euclid.

^ Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides.

^ Euclid considers the regular polygons of 4, 5, 6 and 15 sides.

.The straight lines perpendicular to and bisecting the sides of any regular polygon meet in a point.^ With a centre on the point where these second circles meet the horizontal line, a third, smaller, circle is drawn which intersects with the horizontal line as well as the extended lines of the larger squares.
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^ These lines are extended to the point where they meet the two squares that are based on, and have sides equal to, the diameter of the circle.
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^ From the points of intersection of each of these extended lines outside the circle, new lines are drawn across the circle to meet each fifth point of intersection.
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.The straight lines bisecting, the angle,s in the regular polygon meet in the sense point.^ The straight lines bisecting, the angle,s in the regular polygon meet in the sense point.

^ The straight lines perpendicular to and bisecting the sides of any regular polygon meet in a point.

^ The result is that the problem has always a solution, viz, the centre of the circle is the point where the bisectors of two of the interior angles of the triangle, meet.

.This point is the centre of the circles circumscribed about and inscribed in the regular polygon.^ On the left every second point of intersection of the surrounding circles with the basic circle has been connected, creating two interlocking, regular isosceles triangles, creating a regular six-pointed star.
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^ With the compasses centred on the two points where the first circle cuts the line and the compasses set at a greater distance than the radius of the first circles, describe two arcs from each of these two centres above and below the line.
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^ Repeat this with the compasses centred at the other point where the first circle cuts the line.
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.Wecan bisect any given arc (Prop.^ Wecan bisect any given arc (Prop.

^ To bisect a given finite straight line (Prop.

^ These theorems allow now of the solution of a number of problems, viz.: To bisect a given angle (Prop.

30, III.). .Hence we can divide a circumference into 2n equal parts as soon as it has been divided into n equal parts, or as soon as a regular polygon of n sides has been constructed.^ Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides.

^ Hence If a regular polygon of n sides has been constructed, then a regular polygon of 2n sides, of 40, of 8n sides, &c., may also be constructed.

^ To divide a straight line into a given number of equal parts.

.Hence If a regular polygon of n sides has been constructed, then a regular polygon of 2n sides, of 40, of 8n sides, &c., may also be constructed.^ Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides.

^ Hence If a regular polygon of n sides has been constructed, then a regular polygon of 2n sides, of 40, of 8n sides, &c., may also be constructed.

^ Hence we can divide a circumference into 2n equal parts as soon as it has been divided into n equal parts, or as soon as a regular polygon of n sides has been constructed.

.Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides.^ Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides.

^ Euclid considers the regular polygons of 4, 5, 6 and 15 sides.

^ Euclid does not state thisconsequence, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon.

It follows that we can construct regular polygon.s of 3, 6, 12, 24.. - sides 4, 8, 16, 32...,,
5, 10, 20, 40...,,
15, 30, 60, 120
.The construction of anynew regular polygon not included in one of these series will give rise to a new series.^ One of these panels is unfinished and shows the net of construction lines used to lay out the pattern.
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^ Fernández-Puertas gives instructions for constructing various kinds of star-and-polygon design in this fashion; his fig.
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.Till the beginning of the 19th century nothing was added to the knowledge ofregular polygons as given by Euclid.^ Till the beginning of the 19th century nothing was added to the knowledge ofregular polygons as given by Euclid.

^ As easy deductions not given by Euclid but added by Simson follow the propositions about the angles in polygons; they are given in English editions as corollaries to Prop.

^ Euclid does not state thisconsequence, but the problem is taken up again at the end of the second book, where it is shown how to construct a square equal in area to a given polygon.

.Then Gauss, in his celebrated Arithmetic, proved thatevery regular polygon of 2+I sides may be cOnstructed if this number 2+f be prime, and that no others except those with 2(2+1) sides can be constructed by elementary methods.^ You may find the methods slightly different from the constructions shown further down, but they all produce the correct results.
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^ Clearly this intricate and regular design was constructed with the simplest of tools and the easiest methods of layout, although with a crew of at least six.
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^ Stepping the weaving is the only method available to create the lines of geometry on a kilim other than those on the horizontal and vertical.
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.This shows that regular polygons of 7, 9, 13 sides cannot thus be constructed, but that a regular polygon of 1~ sides ispossible; for 17=2i-i-I. The next polygon isone of 257 sides.^ Euclid shows how to construct regular polygons of 3, 4, 5 and 15 sides.

^ This shows that regular polygons of 7, 9, 13 sides cannot thus be constructed, but that a regular polygon of 1~ sides ispossible; for 17=2i-i-I. The next polygon isone of 257 sides.

^ And definitions 5 and 6 thus: Definition.A polygon is said to be circumscribed about a circle, and a circle is said to be inscribed in a polygon, if the sides of the.

.The construction becomes already rather complicated for I 7 sides.^ The construction becomes already rather complicated for I 7 sides.

BOOK V~
4~. .The fifth book of the Elements is not exclusively geometrical.^ The fifth book of the Elements is not exclusively geometrical.

^ Considered from this point of view, we may say the fifth book of the Elements shows that some of the simpler algebraical operations hold for incommensurable numbers.

^ The third book of the Elements relates exclusively to properties of the circle.

.It contains the theory of ratios and proportion of quantities, in general.^ It contains the theory of ratios and proportion of quantities, in general.

^ The 5th book contains the theory of proportion which is used in Book VI. The 7th, 8th and oth books are purely arithmetical, whilst the 10th contains a most ingenious treatment of geometrical irrational quantities.

^ The ratio of two lines (and of two like quantities in general) is equal to that of their numerical values.

.The treatment, as here given, is admirable, and in every respect superior to the algebraical method by which Euclids theory is now generally replaced.^ Not every flower is regular in this respect, but I have generally sought out plants which display a relatively simple two-dimensional geometrical form.
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.We shall treat the subject in1order to show why the usual algebraical treatment of proportion is not really sound.^ We shall treat the subject in1order to show why the usual algebraical treatment of proportion is not really sound .

^ In the ordinary algebraical treatment of numbers this proof is altogether omitted, or given by a process of limits which does not seem to be natural to the subject.

We begin by quoting those definitions at the beginning of Book V. -which are most important. .These definitions have given rise to much discussion, The only definitions which are essential for the fifth book are Defs.^ These definitions have given rise to much discussion, The only definitions which are essential for the fifth book are Defs.

^ XI. He does not, however, show the connexior which these definitions have with those mentioned before When ,points and lines have been defined, a statement lik Def.

^ Euclid gives the essence of these statements as definitions: Def.

I, 2, 4, 5, 6 and 7. Of the remainder 3, 8 and 9 are more than useless, and probably not Euclids, but additions of later editors, of whom Theon of Alexandria was the most prominent. Defs. 10 and II belong rather to the sixth book, whilst all the others are merely nominal. The really important ones are 4,5, 6 and 7.
.48. To define a magnitude is not attempted by Euclid.^ To define a magnitude is not attempted by Euclid.

The first two definitions state what is meant by a part, that is, a submultiple or measure, and by a ~ multiple. Of .a given magnitude. The meaning of Def. .4 is that two given quantities can have a ratio to one another only in case that they are comparable as to their magnitude, that is, if they are of the same kind.^ We have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude.

^ Thus in the case of the two planes one and only one of the two, Oia and ~n, is real.

^ (Hence a line has two distinct points at infinity, and not one only as in ordinary geometry.

Def. .3, which is probably due to Theon, professes to define a ratio, but is as meaningless as it is uncalled for, for all that is wanted is given in Defs.^ Theon, professes to define a ratio, but is as meaningless as it is uncalled for, for all that is wanted is given in Defs.

5 and 7.
In Def. 5 it is explained what is meant by saying that two magnitudes have the same ratio to one another as two other magnitudes, and it a>c, then d>f, but if a=c, then d=f, and if a
.By aid of these two propositions the following two are proved.^ By the aid of these two, the following fundamental properties of triangles are easily proved: Prop.

^ By aid of these two propositions the following two are proved.

^ Having proved that these six laws hold, we can at once prove every one of the above propositions in their algebraical form.

55. Prop. .22. If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last.^ If there be any number of magnitudes, and as many others, which have the same ratio, taken two and two in order, the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last.

^ I must go back and look at more of the flowers to see how many other variations in petals there are.
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^ In the latter there appear, on first consideration, to be very few possibilities for variety as there are only a limited number of two-dimensional geometries on which forms or patterns can be based.
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We may state it more generally, thus:
If a:b:c:d:e:...~a:b:c:d:e:...,
then not only have two consecutive, but any two magnitudes on the first side, the same ratio as the corresponding magnitudes on the other. For instance a: c=a: c; b: e=b: e, &c.
Prop. 23 we state only in symbols, viz.:
.1.. ..i _I,I I I I
a.v.c ~
then a: c=c: a, b: e=e: b, and so on.
Prop. 24 comes to this: If a: b=c: d and e: b=f: d, then a+e: b=c+f: d.
.Some of the proportions which are considered in the above propositions have special names.^ Some of the proportions which are considered in the above propositions have special names.

^ Before, however, we treat of this we consider some special cases.

^ Their existence theorem is the proof that they are true when the fundamental ideas are considered as denoting some determinate subject matter, so that the axioms are developed into determinate propositions.

.These we have omitted, as being of no use, since algebra has enabled us to bring the different operations contained in the propositions under a common point of view.^ These we have omitted, as being of no use, since algebra has enabled us to bring the different operations contained in the propositions under a common point of view.

^ Incidentally, all of these first three examples are based on eight point geometry, a common and relatively easy framework to establish.
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^ And this study, based on a panel of ceramic tilework, was undertaken as an examination of its underlying geometry, particularly from the point of view of determining the relationships between the circles containing the ten points.
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56. The last proposition in the fifth book is of a different character.
Prop. .25. If four magnitudes of the same kind be proportional, the greatest and least of them together shall be greater than the other two together.^ With the compasses centred on the two points where the first circle cuts the line and the compasses set at a greater distance than the radius of the first circles, describe two arcs from each of these two centres above and below the line.
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^ With the other point, make two additional marks (B), one toward each corner along the same side.
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.In symbols If a, b, c, d be magnitudes of the same kind, and if a: b=c: d, and if a is the greatest, hence d the least, then a+d> b+c.^ We have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude.

^ If four magnitudes of the same kind be proportional, the greatest and least of them together shall be greater than the other two together.

^ But, in giving interpretation to such an equation, we must of course refer to numbers Ax, By, C of unit magnitudes of the same kind, of units of counting for instance, or unit lengths or unit squares.

5~. .We return once again to the question, What is a ratio?^ We return once again to the question, What is a ratio?

.We have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude.^ We have seen that we may treat ratios as magnitudes, and that all ratios are magnitudes of the same kind, for we may compare any two as to their magnitude.

^ In 8 it has been seen that two conditions determine the equation of a straight line, because in Ax+By+C=o one of the coefficients may be divided out, leaving only two parameters to be determined.

^ F(O), which express the coordinates of any point of it as two functions of the same variable parameter 0 to which all values are open.

.It will presently be shown that ratios of lines may be considered as quotients of lines, so that a ratio appears as answer to the question, How often is one line contained in another?^ While simple geometry is used to establish the layout, you can see how different it appears from Islamic patterns in its loose form and, of course, the overlapping curved line.
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^ With the compasses centred on one of the two points where the circle cuts the line, draw another circle of the same radius.
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^ Although I may seem to be making a case for one form of geometry being more attractive than another, that is not the case.
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.But the answer to this question is given by a number, at least in some cases, and in all cases if we admit incommensurable numbers.^ All these four motives may have and probably did overlap with some of the others, not least in connection with actual translations or those who could explain them; in the case of artisans this would have been the geometers.
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^ These designs are some of the most basic forms and can be found in Islamic work all over the Arab world, fabricated in a number of different materials.
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.Considered from this point of view, we may say the fifth book of the Elements shows that some of the simpler algebraical operations hold for incommensurable numbers.^ The fifth book of the Elements is not exclusively geometrical.

^ Considered from this point of view, we may say the fifth book of the Elements shows that some of the simpler algebraical operations hold for incommensurable numbers.

^ Considering a surface element through the origin, we may choose our axes so that, for this element, 13 = Z4 ...

In the ordinary algebraical treatment of numbers this proof is altogether omitted, or given by a process of limits which does not seem to be natural to the subject.
BooK VI.
.58. The sixth book contains the theory of similar figures.^ The sixth book contains the theory of similar figures.

^ We may mention, however, that the theorems about triangles contained in the propositions of Book I., which do not depend upon the theory of parallels (that is all up to Prop.

^ The next four propositions Contain the theory of similar triangles, of which four cases are considered.

After a few definitions explaining terms, the first proposition gives the first application of the theory of proportion.
Prop. .I. Triangles and parallelograms of the same altitude are to cne another as their bases.^ Parallelograms on equal bases and between the same parallels are equal to one another.

^ For a triangle is equa in area to half a parallelogram which has the same base and th same altitude.

^ I. Triangles and parallelograms of the same altitude are to cne another as their bases.

The proof has already been considered in 49.
From this follows easily the important theorem Prop. .2. If a straight line be drawn parallel to one of the sides of a triangle it shall cut the other sides, or those sides produced, proportionally; and if the sides or the sides produced be cut proportionally, the sirai~ht Line which joins the points of section shall be parallel to the remaining side of the triangle.^ If two triangles are equal in area, and have one angle in the one vertically opposite to one angle in the other, then the two straight lines which join the remaining two vertices of the one to those of the other triangle are parallel.

^ Of the others we shall prove one, viz.

^ If two straight lines are parallel, the straight line whieh joins any point in one to any point in tile other is in the same plane as the parallels.

.59. The next proposition, together with one added by Simson as Prop.^ The next two propositions (7 and 8) again belong together.

^ The next proposition, together with one added by Simson as Prop.

^ As easy deductions not given by Euclid but added by Simson follow the propositions about the angles in polygons; they are given in English editions as corollaries to Prop.

A, may be expressed more conveniently if we introduce a modern phraseology, viz, if in a line AB we assume a point C between A and B, we shall say that C divides AB internally in the ratio AC: CB; but if C be taken in the line AB produced, we shall say that AB is divided externally in the ratio AC: CB.
The two propositions then come to this:
Prop. .3. The bisector of an angle in a triangle div-ides the opposite side internally in a ratio equal to the ratio of the two sides including that angle; and convetsely, if a line through the vertex of a triangle divide the base internally in the ratio of the two other sides, then that line bisects the angle at the vertex.^ To divide a straight line in a given ratio.

^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ From this it follows that if the included angles are not equal, the third sides are not equal; and conversely, that if the third sides are not equal, the included angles are not equal.

Simsons Prop. .A. The line which bisects an exterior angle of a triangle divides the opposite side externally in the ratio of the other sides; and conversely, if a line through the vertex of a triangle divide the base externally in the ratio of the sides, then it bisects an exterior angle at the vertex of the triangle.^ To divide a straight line in a given ratio.

^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ The bisector of an angle in a triangle div-ides the opposite side internally in a ratio equal to the ratio of the two sides including that angle; and convetsely, if a line through the vertex of a triangle divide the base internally in the ratio of the two other sides, then that line bisects the angle at the vertex.

-
.If we combine both we have The two lines which bisect the interior and exterior angles at one vertex of a triangle divide the opposite side internally and externally in the same ratio, viz, in the ratio of the other two sides.^ To divide a straight line in a given ratio.

^ The greater angle of every triangle is subtended by the greater side, or has the greater side opposite to it; Prop.

^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

60. The next four propositions Contain the theory of similar triangles, of which four cases are considered. .They may be stated together.^ They may be stated together.

^ If these parallelograms be placed side by side, they may be added together to form a single parallelogram, having still one side of the given length.

.Two triangles are similar, 1. (Prop.^ Two triangles are similar, 1.

^ On the same straight line, and on the same side of it, there cannot be two similar segments of circles, not coinciding with one another; Prop.

^ By the aid of these two, the following fundamental properties of triangles are easily proved: Prop.

4). If the triangles are equiangular:
2. (Prop. 5). If the sides of the one are proportional to those of the other; 3. (Prop. 6). .If two sides in one are proportional to two sides in the other, and if the angles contained by these sides are equal; 4. (Prop.^ If the square described on one of the sides of a triangle be equal to the squares described on the other sides, then the angle contained by these two sides is a right angle.

^ The angles in the same segment of a circle are equal to one another; Prop.

^ If two sides in one are proportional to two sides in the other, and if the angles contained by these sides are equal; 4.

7). If two sides in one are proportional to two sides in the other, if the angles opposite homologous sides are equal, and if the angles opposite the other homologous sides are both acute, both right or both obtuse, homologous sides being in each case those which are opposite equal angles.
.An important application of these theorems is at once made to a right-angled triangle, viz.: Prop.^ The first of these theorems gives Prop.

^ From this follows easily the important theorem Prop.

^ An important application of these theorems is at once made to a right-angled triangle, viz.: Prop.

.8. In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.^ In other words, the areas of similar triangles are to one another as the squares on homologous sides.

^ This is done by drawing two diameters at right angles to one another.

^ Similar triangles are to one another in the duplicate ratio of their homologous sides.

.Corollary.From this it is manifest that the perpendicular drawn from the right angle of a right-angled triangle to the base is a mean proportional between the segments of the base, and also that each of the sides is a mean proportional between the base and the segment of the base adjacent to that side.^ There is a geometric way of establishing this proportion – a right angle triangle with adjacent sides equal, will have a hypoteneuse of √2 to the adjacent sides.
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^ Corollary.From this it is manifest that the perpendicular drawn from the right angle of a right-angled triangle to the base is a mean proportional between the segments of the base, and also that each of the sides is a mean proportional between the base and the segment of the base adjacent to that side.

^ In a right-angled triangle, if a perpendicular be drawn from the right angle to the base, the triangles on each side of it are similar to the whole triangle, and to one another.

61. There follow four propositions containing problems, in language slightly different from Euclids, viz.: Prop. .9. To divide a straight line into a given number of equal parts.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ To divide a straight line into a given number of equal parts.

Prop. .10. To divide a straight line in a given ratio.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ To divide a straight line into a given number of equal parts.

Prop. .II. To find a third proportional to two given straight lines.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ To find a fourth proportional to three given straight lines.

Prop. .12. To find a fourth proportional to three given straight lines.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ To find a fourth proportional to three given straight lines.

Prop. .13. To find a mean proportional between two given straight lines.^ To divide a straight line in a given ratio.

^ To describe a square on a given straight line.

^ To find a fourth proportional to three given straight lines.

The last three may be written as equations with one unknown quantityviz, if we call the given straight lines a, b, c, and the required line x, we have to find a line x so that Prop. 1I a: bb,: x; Prop. 12. a: b=c: x; Prop.13. a:x=x:b.
.We shall see presently how these may be written without the signs of ratios.^ We shall see presently how these may be written without the signs of ratios.

^ It will presently be shown that ratios of lines may be considered as quotients of lines, so that a ratio appears as answer to the question, How often is one line contained in another?

^ A M B, P To see how the value of 9t r-m-rT this ratio changes with ~ct F1o.

.62. Euclid considers next proportions connected with parallelograms and triangles which are equal in area.^ Euclid considers next proportions connected with parallelograms and triangles which are equal in area.

^ If the base BC of a triangle is kept fixed and the vertex A moves in the fixed plane ABC so that the area ABC is constant, then the locus of A is a line of equal distance from BC. This locus is not a straight line.

^ In 1832, in reply to the receipt of Bolyais Appendix, he gives an elegant proof that the amount by which the sum of the angles of a triangle falls short of two right angles is proportional to the area of the triangle.

Prop. .14. Equal parallelograms which have one angle of the one equal to one angle of the other have their sIdes about the equal angles reciprocally proportional; and parallelograms which have one - angle of the one equal to one angle of the other, and their sides about the equal angles reciprocally proportional, are equal to one another.^ Parallelograms on equal bases and between the same parallels are equal to one another.

^ Parallelograms which are equiangular to one another, have to one another the ratio which is compounded of the ratios of thel, sides.

^ It is necessary to attend to signs; x has one sign or the other according as the point P is on one side or the other of the axis of y, and y one sign or the other according as P is on one side or the other of the axis of x.

Prop. .15. Equal triangles which have one angle of the one equal to one angle of the other, have their sides about the equal angles reci procally proportional; and triangles which have one angle of the one equal to one angle of the other, and their s-ides about the equal angles reciprocally pro portional, are equal to one another.^ In other words, the areas of similar triangles are to one another as the squares on homologous sides.

^ If two angles ir~ a triangle are equal, then the sides oppbsite them are equali.e.

^ Similar triangles are to one another in the duplicate ratio of their homologous sides.

The latter p?oposition is really the same as the former, for if, as in the accompanying diagram, in the figure belonging to the A former the two equal parallelograms AB and BC be bisected by the lines DF and EG, and if EF be drawn, we get the D -.. B figure belonging to the latter.
It is worth noticing that the lines FE and DG are parallel. We may state there- fore the theorem G
If two triangles are equal in area, and have one angle in the one vertically opposite to one angle in the other, then the two straight lines which join the remaining two vertices of the one to those of the other triangle are parallel.
63. A most important theorem is Prop. .16. If four straight lines be proportionals, the rectangle contained by the extremes is equal to the rectangle contained by the means~ and if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines, are pro portionals.^ If four straight lines be pro portionals, the similar solie parallelepipeds.

^ To find a mean proportional between two given straight lines.

^ If three straight lines are proportional the rectangle contained by the extremes is equal to the square on the