The geometry that was called " ancient mathematics" , after 17th century, has been developed as analytic geometry through the algebraic methods. The Inventor of the analytic geometry which can solve geometrical problems by way of algebraic equations, was Descartes.

In other words, he devised a systematic way of labeling each point on a flat plane by a pair of numbers. But, he did not have the concept of coordinate , only concentrated on the drawing problem ,and did not make the relation of x-y clear.

But Descartes did not use the term coordinate . "Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain straight lines is sufficient for its construction. Just as arithmetic consists of only four or five operations, namely, addition, subtraction, multiplication, division and the extraction of roots, which may be considered a kind of division, so in geometry, to find required lines it is merely necessary to add or subtract other lines .... Often it is not necessary thus to draw the lines on paper, but it is sufficient to designate each by a single letter."

At that time, x-y axis was only a concept of segment. His coordinate system was used as the instrument which draw the geometrical figure. If we take two straight lines which are perpendicular to each other, then the Cartesian coordinate is produced on the Euclid plane. The values of x and y at their intersections define the point. Also, to choose (take) the coordinate axis, that is, the choice of the coordinate axis is to mean a probability . In Euclidean space, to choose two perpendicular lines or three perpendicular lines is only a choice of two or three perpendicular lines among the countless intersectional lines. In physics, space is typically defined in "Cartesian coordinates"

To define intersection point as an origin(zero point) or to define the unit , are only a mathematical convenience. Because there is a necessary and sufficient condition . Now, it becomes a very universal and common concept. Also, to define a interval on the straight line is arbitrary. Among these intervals, if the small interval is dx ( dx is not zero, but dx is near zero) , the coordinate axis constitutes of the extension of the interval. The coordinate axis means a infinite straight line.

In order to make those intervals, it needs to at least three points. In other words, there must be an interval which the continuos three points exist for drawing the coordinate.

One point ( . ) is such a point defined by Euclid.

Two point (.. ) the interval dx between the continuous two points is not existing.

According to these definitions, if we draw the coordinate axis on the Euclid space is as follows.

If we inversely draw this coordinate axis, then we will obtain the coordinate axis as follows .

This inverse coordinate is not different from the Cartesian coordinate because all conditions is the same. Gauss's imaginary axis is only a same sort of expression. If we apply this coordinate axis to space, then we will obtain the 3-dimensional coordinates as follows. The comparison of the straight line and curved line in this inverse coordinate is more wider than the Cartesian coordinate .Also, this inverse coordinate make us know that three coordinate axes which has three right angles are difficult to easily draw. To draw three straight lines perpendicular to each other may be a great mathematical adventure. It is difficult to find the segment perpendicular to each other in space.

After the birth of analytic geometry, the rule of coordinate axis has been achieved naturally. Using the inverse coordinates, we can analyze the picture as follows. The Cartesian coordinates cannot analyze this picture.