Tag Archives: cartesian plane

Euclid’s five postulates in Descartes’ Coordinate System

  1. Introduction

As we learned about the Euclidean geometry and its five basic axioms in class, some terms like “straight line”, “circle”, and “right angle” kept jumping in my mind. I thought I had a picture of them. for example, a straight line is as straight as the rope with a ball attached and hang in the air, and a right angle is shown like a corner of a rectangular table. However, as a math major student, such a simple cognition of them is not enough, I hope to have some more mathematical concept to express them.

  1. The Cartesian coordinate system

2.1 The invention of Cartesian coordinates

In the 17th century, René Descartes (Latinized name: Cartesius), a well-known mathematician and philosopher to today’s people all around the world, published his work La Géométrie , in which he made a breakthrough. More concretely, Descartes uses two straight lines that are perpendicular to each other as axes x, y, and uses these axes to measure the positions of any points in a plane.

2.2 The rule of representing a point in Cartesian coordinates

One point in Cartesian coordinates has two parameters: one is the x parameter, the other is the y parameter. To measure the x parameter, we need to draw a straight line y’ parallel to the y axis(we will discuss the definition of parallel in Cartesian coordinates later) that through the point, and then set the x parameter of that point as the number of the intersection of y’ and x axis, for its y parameter, draw a line x’ parallel to the x axis through the point and take the number on the intersection of x’ and y-axis as this point’s y parameter.

2.3 To express a straight line in Cartesian coordinates

A straight line in Euclidean geometry is a straight object with negligible width and depth. So, it is an idealization of such objects in Euclidean geometry. However, in Cartesian coordinates, a line has a strict definition, a straight line is the set of points that satisfies a certain equation. And the line equation usually can be written as:

A*x + B*y + C = 0,

The A, B, and C are the coefficients of x, y, and constant. Moreover, the -A/B is the slope of the straight line, -C/B is the y-intercept of this line, which means the intersection of the y-axes and the line.

So, all above is how we express a straight line in Cartesian coordinates.

  1. To express the five postulates in the Cartesian coordinate system

1.”To draw a straight line from any point to any point.”

In Cartesian coordinates, to express a line we only need one point and a direction. Suppose we have two points a=(A, C) and b=(B, D). By doing a subtraction of the two points, we can get a vector (B – A, D – C). We only need this vector to provides a direction, which is (B – A)/(D – C). So this unique straight line can be expressed as

(x – A)*(B – A)/(D – C) = y – C;

2.”To produce [extend] a finite straight line continuously in a straight line.”

Any line in Cartesian coordinates is a straight line(infinite). It can be limited by a range for x or y, such as

A*x + B*y + C = 0,( a < x < b or c < y < d)

  • “To describe a circlewith any centerand distance [radius].”

To describe a circle in the Cartesian system, we only need the center’s x and y coordinates (x0, y0), and a distance as the radius r, it is

(x – x0)2 + (y – y0)2 = r2,

So, above is a typical circle in the Cartesian coordinates.

A right angle in the Cartesian system is always equal to the angle between x and y axes, for x, y axes in the Cartesian system is perpendicular to each others.

  • “That, if a straight line falling on two straight lines make 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 the two right angles.”

For the parallel postulate, it is far more easier to be expressed in Cartesian coordinates, suppose we already know a line as

A*x + B*y + C = 0,

And we have a point (x1, y1) out of the line, have the point and a direction of the other line, the slope is – A/B, and the other line can be described as

-A/B*(x – x1) = y – y1;

And it is easy to know these two lines are parallel, because they have same slope and do not share one point. And by the property of Cartesian coordinates, this is the only line that parallel with the first one.

  1. Conclusion

In Euclidean geometry, some concept are hard to imagine or describe, while Cartesian coordinate make it possible and easy to express, such a great combination of geometry and algebra!

René Descartes, the Philosophy of Truth, and Algebraic Geometry

Portrait of René Descartes. Painting: Frans Hals, via Wikimedia Commons.

“I think, therefore I am,” one of the most famous philosophical proclamations ever written. The man who wrote these words? René Descartes. Born in 1596 he spent much of his life in the Dutch Republic. His unconventional philosophical ideas eventually shaped western thought. So much so that he is considered the father of modern philosophy. He reflected in his self-evaluation and autobiography Discourse On Method, published in 1637, that “I had always a most earnest desire to know how to distinguish the truth from the false, in order that I might be able clearly to discriminate the right path of life.” His basis of truth came from the idea that if he could think, he must exist; if he could doubt himself, it only strengthened his first principle of philosophy: “Cogito ergo sum,” the translation of which heads this story. Descartes was a methodological skeptic; he shunned anything that could be doubted with a belief that anything unsure is not a foundation for the studies of philosophy.

Descartes had four principles of logic that led him to an understanding of the subject, as laid out in Discourse On Method section II. The first is that he never accepted anything as truth which could not be known immediately. The second, break down the problem into smaller difficulties. The third principle is to start with the easiest idea and work towards more difficult concepts. And finally, make the proof general such that there is no confusion. These axioms gave way to testing thought processes in a rigorous manner that gave undeniable truth if they could be proven. He believed with these logics “that there is nothing so far removed from us as to be beyond our reach, or so hidden that we cannot discover it” as long as truths are used to discover new truths.

Descartes’ philosophy appears dense at times, but like many thinkers of the mid-second millennium, his philosophical logics went hand in hand with mathematics. In Discourse on Method, he claims to be “delighted with the mathematics, on account of the certitude and evidence of their reasoning.” To him, there was something profound about the power of numbers, that they fit nicely with his tenets of belief. Geometric problems especially could be solved using his four principles of logic.

Descartes’ most important contribution to mathematics was “La Géométre,” an appendix to his Discourse on Method. This particular work focused on the “merging” of algebra and geometry. What ideas we now take for granted in seventh grade algebra did not exist, or have any real structure, before this publication. Descartes introduced the reader to the common notations of algebraic expression, that is x represents a variable value and is mapped to y with a function, and in 3-dimensions, there is also a z that can be dependent upon the x and y values. The publication also set notation for constants, using a,b, and c to represent multiplicative and additive properties to functions. The variables x and y played an important role in the development of a coordinate system that would visualize algebra.

The x-axis and the y-axis would be drawn by two lines, perpendicular to each other, and intersecting at a point labeled as the origin, where it was decided that x and y are 0. the y-axis would stand vertically, and the x-axis horizontally. Any 2-dimensional equation could be mapped on the plane created by the two axes. The “Cartesian” coordinate system, named after Descartes himself, revolutionized algebra, such that it could be solved visually, that is geometrically, rather than analytically. He devised the four quadrants, and labeled them I for both x,y positive, II for x negative and y positive, III for x,y negative, and IV for x positive and y negative.

Cartesian Coordinates with the origin of (0,0) at the center and each of the quadrants labeled. A point P(3,5) demonstrates how to map on the plane. Image: Gustavb, via Wikimedia Commons.

René Descartes also gave a standard notation for the superscripts to denote powers, that is he was the first to denote the variable x-squared as x2. His analytic geometry was the basis for Newtonian calculus, and he developed early concepts of the law of conservation of momentum. He set the path for other mathematicians to create standard analytical notation that we use today for both algebra and logic.