History of the formation of geometry

The development of geometry has roughly gone through four basic stages.

1. Formation and development of experimental geometry

Geometry first arose from the observation of the shape and arrangement of the stars in the sky, and from the need for practical activities such as surveying land, measuring volume, making utensils and drawing graphs, etc. People accumulated rich geometric experience on the basis of observation, practice, and experimentation, and formed a number of rough concepts reflecting the connection between certain empirical facts. The connection between some of the empirical facts, the formation of experimental geometry.

The geometry studied in ancient China, ancient Egypt, ancient India, Babylon, is largely the content of experimental geometry.

For example, in ancient China, we discovered the theorem of hook and strand and simple measurement knowledge very early, the "Mojing" contains "Won (circle), a center with the same length," "Ping (parallel), the same height," and the ancient Indians believe that "The area of the circle is equal to the area of a rectangle, and the bottom of the rectangle is equal to half the circumference of the circle, the height of the rectangle is higher than the radius of the circle" and so on, all belong to the category of experimental geometry.

2. Formation and Development of Theoretical Geometry

With the trade and cultural exchanges between ancient Egypt and Greece, Egyptian geometric knowledge was gradually introduced into ancient Greece.

Many mathematicians in ancient Greece, such as Thales, Pythagoras, Plato, Euclid and others, made significant contributions to the study of geometry.

In particular, Plato introduced logic into geometry, establishing a rigorous definition and clear axioms as the basis of geometry, and then Euclid, on the basis of the existing geometric knowledge of his predecessors, wrote the 13 volumes of Geometry in accordance with a rigorous system of logic, laying the foundation for theoretical geometry (also known as deductive geometry, deduction geometry, axiomatic geometry, Euclid's geometry, and so on), which became the basis of A famous masterpiece in history.

The Original Geometry, despite the incompleteness of the axioms, sometimes resorting to intuition and other shortcomings of the argument, but it is a collection of ancient mathematical achievements, rigorous arguments, far-reaching, the use of axiomatic method of the development of mathematics pointed out the direction of the future, and even become the milestone in the history of the development of human civilization, the cultural heritage of all mankind in the treasures.

3. The emergence and development of analytic geometry

In the 3rd century AD, the appearance of the Principia Geometria laid the foundation for theoretical geometry.

At the same time, people also did some research on conic curves, and discovered many properties of conic curves.

But for a long time afterward, theology ruled in feudal society, and science was not given the attention it deserved.

Until the 15th and 16th centuries, European capitalism began to develop, with the actual needs of production, natural science has been rapidly developed.

France Descartes (Descartes) found in the study, Euclidean geometry is overly dependent on graphics, and the traditional algebra is completely subject to formulas, laws of constraints, they believe that the traditional study of conic curves, only pay attention to the geometric aspects of the neglect of the algebraic aspects, and advocate for the combination of geometric and algebraic complementary to the strengths and weaknesses of this is to promote the development of mathematics, a new way.

In this way, he emphasized only the geometric aspects and ignored the algebraic ones.

Under the guidance of such an idea, Descartes proposed the concept of a plane coordinate system, realized the correspondence between points and pairs of numbers, expressed the conic curve in terms of equations containing two-sided, three-sliced numbers, and formed a series of brand-new theories and methods, and analytic geometry arose as a result.

The emergence of analytic geometry greatly broadened the study of geometry and contributed to its further development.

In the 18th and 19th centuries, due to the needs of engineering, mechanics, and geodesy, further branches of geometry such as pictorial geometry, projective geometry, affine geometry, and differential geometry were developed.

4. The emergence and development of modern geometry

In the development of elementary geometry and analytic geometry, people constantly found that the original geometry is not logically rigorous enough, and constantly enrich some axioms, especially in trying to prove the fifth axiom with other axioms, axioms, and axiomatic proof of the fifth axiom, "When a straight line intersects with two other straight lines, and the sum of the interior angles on the same side is less than two right angles, this is the same as a straight angle, but it is the same as a straight angle, which is the same as a straight angle. In particular, the failure of the attempt to prove the fifth axiom "a straight line intersects two other straight lines, the sum of the interior angles of the same side of the line is less than the two right angles, these two straight lines will intersect on this side" has prompted people to re-examine the logical foundations of geometry, and achieved outstanding research results in two areas.

On the one hand, the change of the axiomatic system of geometry, i.e., the replacement of the fifth axiom of Euclidean geometry by a proposition that contradicts it, led to a fundamental breakthrough in the object of study of geometry.

The Russian mathematician Lobachevsky replaced the fifth axiom with "In the same plane, two straight lines can be made parallel to a known straight line at a point outside the line", which led to a series of new conclusions, such as "the sum of the interior angles of a triangle is less than two right angles",

The German mathematician Riemann from another perspective, "in the same plane, over any point outside the line does not exist a straight line parallel to the known line" instead of the fifth axiom, also led to a series of new theories, such as "triangle angle sum is greater than two right angles",

It is customary to refer to Roche and Ricardian geometry collectively as non-Euclidean geometry.

It is customary to refer to Euclidean geometry (also known as parabolic geometry), and the metric **** part of Roche geometry collectively as absolute geometry.

On the other hand, people in the rigorous analysis of the axiomatic system of Euclidean geometry, the formation of axiomatic method, and by the German mathematician Hilbert in his book Foundations of Geometry perfected a rigorous axiomatic system, commonly known as the Hilbert axiomatic system, the Hilbert axiomatic system is complete, that is, using purely logical reasoning, will be able to deduce a systematic and rigorous Euclidean geometry.

But to deduce from this axiomatic system, step by step, what is known in Euclidean geometry, is a rather tedious task.