vol 4: Glossary
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Euclidean space
[Hazewinkel, euclidean space] [Borowski, euclidean space]
Euclidean (or Cartesian) space is the mathematical abstraction and extension of the 'ordinary' three dimensional space of everyday life. The first mathematical treatment of this space is Euclid's Elements, composed about 300 bce. The Elements begins with a set of definitions and postulates from which it derives the theorems of geometry that have been used since time immemorial by surveyors, engineers and builders. Heath
Rene Descartes (1596-1650) connected geometry to arithmetic and algebra through his introduction of coordinate representation of Euclidean space. This opened the way to the consideration of Euclidean spaces of more than three dimensions. Descartes
From a modern perspective, the Euclidean axiomatisation of geometry lacks rigour. David Hilbert produced the first precise axiomatisation of Euclidean geometry. Hilbert
Euclidean space is 'flat'. This flatness is encoded in Euclid's definition 23 and postulate 5.
Definition 23: 'Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.' Heath I:190.
Postulate 5 (Parallel postulate): 'That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.'
In the nineteenth century it was realised that space is not necessarily flat. Spaces may be devised in which there are no lines fulfilling definition 23, or in which lines described by Postulate 5 never meet. Such spaces were developed by Riemann (1826-66) and Lobachevsky (1793-1856).
An important feature of Euclidean space is that Pythagoras' theorem ('the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the other two sides') holds within it.
Books
| Borowski, Ephraim J, Collins Dictionary of Mathematics, Harper Collins 1089 'It is the immodest hope of the authors that this dictionary will not only prove valuable as a reference book for students of mathematics at all levels from secondary schools to a master's degree, but also offer much to interest a more general readership. Amazon back |
| Descartes, Rene, Geometrie , Dover 1956 Jacket: ' ... With this volume, Descartes founded modern analytical geometry. Reducing geometry to algebra and analysis and, conversely, showing that analysis can be translated into geometry, it opened the way for modern mathematics. ... This edition contains the entire definitive Smith-Latham translation of Descartes three books: Problems the Construction of which requires Only Straight Lines and Circles; On the Nature of Curved Lines; On the Construction of Solid and Supersolid Problems. Interleaved page by page with the translation is a complete facsimile of the 1637 French text, together with Descartes' original illustrations. ...' Amazon back |
| Hazewinkel, Michiel, Encyclopaedia of Mathematics (6 volumes), Kluwer Academic and Toppan 1995 'The Encyclopaedia of mathematics aims to be a reference work for all parts of mathematics. It is a translation with updates and editorial comments of the Soviet Mathematical Encyclopaedia published by 'Soviet Encyclopaedia Publishing House' in five volumes in 1977-85.' Amazon back |
| Heath, Thomas L, Thirteen Books of Euclid's Elements (volume 1, I-II), Dover 1956 'This is the definitive edition of one of the very greatest classics of all time - the full Euclid, not an abridgement. Utilizing the text established by Heiberg, Sir Thomas Heath encompasses almost 2500 years of mathematical and historical study upon Euclid.' Amazon back |
| Hilbert, David, Foundations of Geometry (Grundlagen der Geometrie), Open Court 1999 Jacket: 'Along with the writings of Hilbert's friend and correspondent Frege, Hilbert's Grundlagen der Geometrie is the major prop that set the stage for Russell and Whitehead's Principa Mathematica. Hilbert presents a new axiomatization of geometry, the reduction of geometry to algebra, and introduces the distinction between mathematics and metamathematics, with a new theory of proof. This edition is translated from the tenth German edition, including all the improvements which Hilbert derived from his own reflections and the contributions of other writers. Amazon back |
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