Overview of three core areas of Mathematics: geometry

Mathematics has developed to the present , It has become owned in the scientific world 100 A large number of major sub disciplines “ republic ”. Generally speaking, mathematics has three core areas ：

1. The part of mathematics that studies numbers belongs to the category of algebra ;

2. Study the part of form , It belongs to the fan Chou of geometry ;

3. The part that communicates form and number and involves limit operation , It belongs to the scope of analysis .

These three kinds of mathematics constitute the ontology and core of the whole mathematics . Around this core , Because mathematics passes the two concepts of number and shape , Permeate with other sciences , And there are many marginal and interdisciplinary disciplines . This paper briefly introduces the historical development of more than ten main branches of mathematics in the three core fields .

1. Elementary Geometry

In Greek ,“ geometry ” By “ The earth ” And “ measurement ” It came from the merger , It originally has the meaning of measuring land , Paraphrase is “ Geodesy ”.“ geometry ” The term , It was translated by mathematicians in Ming Dynasty according to pronunciation , Still in use today .

Now elementary geometry mainly refers to Euclidean geometry , It is about graphics ( spot 、 Line 、 Noodles 、 horn 、 Circle, etc. ) Science of unchanging nature under motion . for example , The distance between two points in Euclidean geometry , The intersection angle of two straight lines , Radius is r The area of a circle of is some motion invariants .

Elementary geometry as a course , Arranged after elementary algebra ; But in history , The development of geometry once took precedence over algebra , It is mainly considered to be the contribution of the ancient Greeks .

Geometry discards all other properties of matter , Only the spatial form and relationship are reserved as the objects of my own research , So it is abstract . This abstraction determines the thinking method of geometry , It is necessary to use the method of reasoning , Derive some new conclusions from some conclusions . The theorem is proved by deduction , The representative work of this argumentation geometry , It is Euclid in the third century BC 《 Original 》, It starts from definition and axiom , Deduce various geometric theorems .

Now middle school 《 Plane triangle 》 The theory of trigonometric function in is 15 Developed and perfected in the 21st century , But some of its most basic concepts , However, it has been formed as early as the ancient study of right triangle . therefore , Trigonometry can be classified under the heading of elementary geometry .

Ancient Egypt 、 Babylon 、 China 、 Greece has studied the knowledge of spherical trigonometry . B.c. 2 century , Hippachas made a string watch , It can be said to be the founder of triangle . Later, Indians made sine tables ; Arab al · Batani calculated sinθ Value method to solve the equation , He is also with arbour · Wofa jointly derived the tangent 、 Cotangent 、 Secant 、 The concept of cosecant ; Ricius made a more accurate sine table , And connect trigonometric functions with arcs .

Because a right triangle is the simplest straight line , It also has very important practical value , Therefore, all ancient civilizations attach great importance to its research . In our country 《 Zhou Bi Suan Jing 》 The early years of the Zhou Dynasty were recorded at the beginning ( B.c. 1100 About years ago ) The dialogue between Duke Zhou and scholar Shang Gao , Among them “ Hook three strands, four strings and five ”, That is, the special form of Pythagorean theorem ; It also records Chen Zi after the Duke of Zhou , I used Pythagorean theorem and the proportional relationship of similar figures , Calculated the distance between the earth and the sun and the diameter of the sun , At the same time, there are dozens of illustrations for Pythagorean theorem . Beyond seas , Traditionally, Pythagorean theorem is called Pythagorean theorem , It is believed that the first proof of consistency comes from the Pythagorean School ( B.c. 6 century ), Although the Babylonians used to 1000 This theorem has been discovered for many years . Until now, people have at least provided the Pythagorean theorem 370 Kind of proof .

19 Since the 20th century , People are interested in elementary comprehensive geometry about triangles and circles , And conducted in-depth research . So far, this research field has not come to an end , Many data have been extended to tetrahedron and accompanying points 、 Line 、 Noodles 、 The ball .

2. Projective geometry

Projective geometry is a study of projecting points onto a line or plane , A geometry of the invariant properties of graphs . Points on the slide 、 Line , After the illumination projection of the slide projector , There are corresponding dots and lines in the pictures on the screen , After a finite number of perspectives, such a group of figures , Become another set of graphics , This is called projective correspondence in Mathematics . Projective geometry in aviation 、 Photography and surveying are widely used .

Projective geometry is where deshag and Pascal are 1639 Opened in . Deshag published — This very original pamphlet about circular dimensional curves , Start with Kepler's continuity principle , We have derived a lot about involution 、 Harmonic range 、 transmission 、 Polar axis 、 The pole and the basic principle of perspective , These topics are familiar to those who study projective geometry today . Years old 16 Pascal, 19, came up with some new 、 A profound theorem , And in 9 Years later, he wrote a manuscript with rich content .18 In the late 20th century , Mengri proposed a method to express three-dimensional objects by appropriate projection on two-dimensional plane , Therefore, the position of artillery positions can be quickly calculated from the data provided , Avoid lengthy 、 Troublesome arithmetic operation .

The truly independent study of projective geometry was initiated by penseller .1822 year , He published 《 On the projective properties of figures 》 One article , It will greatly promote the research in this field . Many of his concepts were further developed by Steiner .1847 year , Stout published 《 Positional geometry 》 A Book , Finally, projective geometry is liberated from the basis of measurement .

Later it turned out that , Adopt a projective definition with appropriate metrics , Be able to study metric geometry within the scope of projective geometry . Add an invariant conic to projective geometry on a plane , You can get the traditional non Euclidean geometry . stay 19 The late 19th century and 20 At the beginning of the century , Projective geometry has been treated with various postulates , And finite projective geometry has also been found . The fact proved that , Gradually add and change public facilities , You can transition from projective geometry to Euclidean geometry , During this period, I experienced many other important geometries .

3. Analytic geometry

Analytic geometry is coordinate geometry , It includes plane analytic geometry and solid analytic geometry . Analytic geometry passes through plane rectangular coordinate system and space rectangular coordinate system , Establish a one-to-one correspondence between points and real pairs , Thus, a one-to-one correspondence between the curve or surface and the equation is established , Therefore, we can study geometric problems with algebraic methods , Or use geometric methods to study algebraic problems .

In Elementary Mathematics , Geometry and algebra are two independent branches ; On the way , They are also basically irrelevant . Establishment of analytic geometry , Not only because of the introduction of variable research in the content, but also created variable Mathematics , Moreover, it also combines geometric methods with algebraic methods .

While dishag and Pascal developed projective geometry , Descartes and Fermat began to conceive the concept of modern analytic geometry . There is a fundamental difference between the two studies ： The former is a branch of geometry , The latter is a method of geometry .

1637 year , Descartes published 《 methodology 》 And its three appendices , His contribution to analytic geometry , In the third appendix 《 geometry 》 in , He proposed several new curves generated by mechanical motion . stay 《 Introduction to plane and solid trajectories 》 in , Fermat analytically defines many new curves . For the most part , Descartes starts from the track , Then find its equation ; Fermat starts from the equation , Then study the trajectory . These are exactly two opposite aspects of the basic principles of analytic geometry ,“ Analytic geometry ” The name of is decided later .

This course reaches the familiar form in current textbooks , yes 100 Years later . Use coordinates like today 、 Abscissa 、 The terms of ordinate , It was Leibniz in 1692 Put forward in .1733 year , Years old 18 The year-old crallo published 《 Research on double curvature curve 》 A Book , This is the earliest work on analytic geometry of space .1748 year , Written by Euler 《 Summary of infinite analysis 》, It can be said to be the first course of analytic geometry in line with modern significance .1788 year , Lagrange began to study the theory of directed line segments .1844 year , Glassman put forward the concept of multidimensional space , And introduce the notation of vector . So multidimensional analytic geometry appeared .

The development of analytic geometry in modern times , It produces some branches of infinite dimensional analytic geometry and algebraic geometry . Ordinary analytic geometry is just a part of algebraic geometry , The development of algebraic geometry is closely related to abstract algebra .

4. Non Euclidean geometry

Non Euclidean geometry has three different meanings ： The narrow sense , One finger Roche ( Robachevsky ) The geometric ; The generalized , It refers to everything and Euclidean ( Euclid ) Geometrically different geometries ; In the usual sense , Refers to Roche geometry and Riemann geometry .

Euclid's first 5 Public establishment ( Parallel public establishment ) It occupies a special position in the history of Mathematics , It's the same as before 4 Compared with the postulate , The nature is too complicated . It's in 《 Original 》 The first application of is to prove that 29 When there is a theorem , And it always seems to try to avoid using it . Therefore, people doubt the axiomatic status of the fifth postulate , And explore other axioms to prove it , To make it a theorem . For more than 3000 years , More than 2000 people have made such explorations and have records , Including many famous mathematicians , But they all failed .

Robachevsky in 1826 year , Bowyer in 1832 In, he published epoch-making research results , Created non Euclidean geometry . In this geometry , They assume that “ Pass a point that is not on a known straight line , At least two lines can be drawn parallel to the known line ”, To replace the fifth postulate , At the same time, other postulates of Euclidean geometry are preserved .

1854 year , Riemann introduced another kind of non Euclidean geometry . In this geometry , He assumes that “ Pass a point beyond a known straight line , There is no straight line parallel to the known straight line to lead ”, Used instead of 5 Public establishment , At the same time, other postulates of Euclidean geometry are preserved .1871 year , Klein put this 3 Kind of geometry ： Robachevsky — Bowyer's 、 Euclidean and Riemannian are respectively named hyperbolic geometry 、 Parabolic geometry and elliptic geometry .

The discovery of non Euclidean geometry not only finally solves the problem of parallel postulates —— The parallel postulate is proved to be independent of other postulates of Euclidean geometry , And liberate geometry from its traditional model , The path of geometry that created many different systems was opened .

1854 year , Riemann published “ A lecture on assumptions as the basis of geometry ”. He pointed out that ： Each of them is different ( Two infinitely close points ) The distance formula determines the properties of the resulting space and geometry .1872 year , Klein established a classification method of various geometric systems according to different transformation group invariants .

19 After the century , Another direction of the development of the concept of geometric space , It is a classification according to the differential geometric principles of the manifold studied , Every geometry corresponds to a system of theorems .1899 year , Hilbert published 《 Geometric basis 》 A Book , A complete system of geometric axioms is proposed , The rigorous foundation of Euclidean geometry is established , The compatibility of an axiom system is proved ( No contradiction )、 The universal principles of independence and completeness . According to his point of view , Different geometric spaces are sets of elements that are subordinate to the requirements of different geometric axioms . Euclidean geometry and non Euclidean geometry , In a large number of geometric systems , It's just an extremely special situation .

5. Topology

1736 year , Euler published a paper , Discuss the problem of the Seven Bridges in gunnysburg . He also proposed that spherical triangles divide graph vertices 、 edge 、 Euler formula of the relationship between faces , This can be said to be the beginning of topology .

Poincare in 1895～1904 Topology was established in , Topological properties are studied by algebraic combination . He extended Euler's formula to Euler — Poincare formula , The theories related to this are now called homology theory and homotopy theory . Later topology developed mainly according to Poincare's ideas .

Topology began as a branch of geometry , In the 20th century, it has been greatly promoted .1906 year , Frecher published his doctoral thesis , Take the function as a “ spot ” Look at , Describe the function convergence as the convergence of points , This connects Cantor's point set theory with the abstraction of analysis . He introduced the concept of distance into the set of functions , Form a distance space , Expand the theory of linear distance space . On this basis , Produces point set topology . In Hausdorf's 《 Outline of point set theory 》 In a Book , The complete idea of more general point set topology has emerged . After the Second World War , Introduce analysis into topology , Developed differential topology .

Now topology can be roughly defined as the mathematical study of continuity . The set of everything can form a topological space in a sense , The concepts and theories of topology have been basically completed and become one of the basic theories of Mathematics , Infiltrate into all branches , And it has been successfully applied to the research of electromagnetism and Physics .