Since the time of the ancient Greeks until only recently, geometry was studied and developed as a separate science. Today, we take for granted the many connections between mathematics and Euclidean geometry, but these connections were not so easily apparent to early thinkers. The quantity pi, by which the diameter of a circle is related to its circumference, is one of the few examples from antiquity in which some indication of the relationship of numbers to geometrical shapes was hinted at. But a more significant understanding of these relationships was not finally achieved until the seventeenth century.
In 1619, the French philosopher and mathematician René Descartes had a dream in which a fly flitted about the air of his bedroom. Descartes realized that the changing position of the fly in space could be described very accurately with respect to its distance from the walls, floor, and ceiling of the room. This vision was the seed from which sprang the Cartesian coordinate system and the branch of mathematics called analytic geometry.
The link between Euclidean geometry and arithmetic is achieved in the Cartesian system. The link is made if, instead of a fly buzzing around a room, you imagine a point floating in space. Just as the fly 's position in the room can be determined by noting its distance from three perpendicular surfaces, so can an abstract point in Euclidean geometry be assigned an "address," or coordinate-position, using three numbers. The three numbers can be found to lie along lines juxtaposed at right angles to each other forming three axes, given by Descartes the conventional names x, y and z.
The place where the axes intersect is called the origin. The entire abstract space generated by the x, y and z axes is called the Cartesian coordinate system or grid. Using this scheme, it is easy to see how geometric forms can be generated using numbers. One classic Euclidean shape, for instance, is the line. If a line is composed of a series of points, then each point on the line must have a unique address or position on a Cartesian grid. Each point has an x, y, and z coordinate which is unique to that point. As an example, a line drawn diagonally on a piece of paper can be described as having x and y coordinates on a two-dimensional Cartesian grid. If the address of each point of the diagonal line is such that the x value is the same as the y value, the line will form an exact diagonal angle of 45 degrees with respect to the x and y axis. A simpler way of stating this property is to say that the line is generated by the equation y=x. Thus, an abstract mathematical equation has a corresponding shape in the Euclidean, geometric world. Similarly, the equation y=x2, when graphed, will form a parabola. In fact, any equation can be graphed in this way. Conversely, any shape can be analyzed to determine what unique equation, or equations, can generate it.
The linking of algebraic equations to geometry made it possible to visualize mathematical concepts. This became important as the science of physics presented to mathematicians the challenge of reducing into numerical form the exceedingly complex phenomena involving motions and forces in the real world. The system of analytical geometry developed simultaneously by Descartes and Pierre de Fermat was later expanded upon and became the basis for the calculus of Isaac Newton and Gottfried Leibniz.
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