The Genius of René Descartes – Part 1

Descartes-300René Descartes (1596 – 1650), a French philosopher, mathematician and writer, discovered a way to synthesize geometry and algebra that resulted in a revolution in mathematics and science. Without Descartes’s brilliant insight, it would not have been possible to develop differential calculus, integral calculus, and many other branches of mathematics. What was revolutionary to Descartes’s contemporaries, now seems natural and almost intuitively obvious, a part of our culture. (Before Isaac Newton, the concept of gravity was unknown, and now all adults and most children know something about gravity.)

So what was Descartes’s world changing discovery all about? He first invented a right angle based coordinate system in which every point in the Euclidean plane is assigned a unique ordered pair of numbers, which represents the point’s location, denoted by (x, y) where both x and y are real numbers. He then demonstrated how to create algebraic equations or formulas to calculate the distance between two points, midpoint of a line segment, and the slope of a line. With these basics established, he showed how to find an x-y variable equation that describes the relationship between the x-coordinate and y-coordinate for every point on a curve and only those points on the curve. Once the equation of a curve is known, the equation can be algebraically manipulated to reveal important properties of the curve and solve a wide variety of application problems.

The diagrams below illustrates how Descartes’s great discovery is used to calculate the distance between points A and B, and the slope of the line that contains points A and B. The distance calculation is, of course, a direct application of the theorem of Pythagoras. If we let AB equal the distance from point A to point B and let m equal the slope of the line that contains points A and B, then AB = √( 82 + (-6)2 ) = √(100) = 10 units, and m = Δy / Δx = -6/8 = -3/4 or -0.75.

The Genius of RenŽ Descartes Blog

From the definition of a conic section and the theorem of Pythagoras, we can derive an x-y variable equation that describes the relationship between x and y for every point (x, y) on the curve. Study the graph of the circle and its equation. If you listen carefully, you will hear Pythagoras whisper from his grave, “x squared plus y squared equals 4 squared for every point (x, y) on the circle.” The graphs below are the graphs of various conic curves and a line. All equations are special cases of the general conic equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 where A, B, C, D, E, and F are real number constants.

Mathblog13-Edit1The above graphics, created with the program Basic Trig Functions, is offered by Math Teacher’s Resource. Except for exponents, all equations are entered as indicated to the right of the graphic. Example: The inequality x2 + 4y2 ≤ 64 is entered as x^2 + 4y^2 ≤ 64. Relationships can be implicitly or explicitly defined. The program automatically figures out how to treat an equation or inequality, and shading of all inequality relations is automatic. Users can specify whether to shade the intersection or union of a system of inequalities. The user interface provides numerous sample equations along with comments and suggestions for setting screen parameters in order to achieve best results.

The user interface for all program modules is simple and intuitive. After an equation is graphed, users can plot a point on a graph near the mouse cursor and view the x-y coordinates of the plotted point. In addition to plotting points, relative minimum points, relative maximum points, x-intercepts and intersection points can be found with simple mouse control clicks. A Help menu gives a quick summary of all the magical mouse control clicks. Go to to view multiple screen shots of the program’s modules. Click the ‘learn more’ button in the TRIGONOMETRIC FUNCTIONS section (or click here). Teachers will find useful comments at the bottom of each screen shot.

Theorem of Pythagoras and “The Ascent of Man”

Kapitolinischer PythagorasEveryone who has studied mathematics is aware of the theorem of Pythagoras. However, relatively few people are aware of the history of the theorem and how the theorem reveals some of the secrets of the universe. In the early 1970s, The Ascent of Man, a thirteen-part BBC series written and hosted by Dr. Jacob Bronowski, appeared on public television. (Both the DVD and book version can be purchased online.) Even to this day, I periodically read or view parts of the series in order to enjoy, ponder, and savor special moments that were epiphanies for me. In this blog, I would like to discuss some of the insights about the theorem of Pythagoras that Dr. Bronowski shared with his viewers.

Dr. Bronowski tells his viewers, “To this day, the theorem of Pythagoras remains the most important single theorem in the whole of mathematics.” To hear this statement from a tier-one mathematician and scientist is astonishing. I was awe struck by the profound insights he revealed as he explained history of mathematics and the proof of the theorem. As a high school geometry teacher, I was well aware of the importance of the theorem of Pythagoras, but did not understand or appreciate that Pythagoras established a fundamental characteristic of the space in which we move. His theorem describes the relationship between the lengths of the sides of a right triangle, and this relationship is true if and only if the triangle is a right triangle. The sum of the squares of the legs of a right triangle equals the square of the hypotenuse.

Theorem of Pythagoras Blog, Edit 2 The Egyptians used a set square with sides of 3 units, 4 units and 5 units to build the pyramids, and the Babylonians used set squares, beside the 3-4-5 set square, to build the Hanging Gardens. By 2000 BC, the Babylonians knew hundreds of Pythagorean triples. The fact that the Babylonians knew 3,367-3,456-4,825 is a Pythagorean triple is testimony that the Babylonians were very good at arithmetic. Listed below are all of the Pythagorean triples with sides less than 100 units and the lengths of sides are relatively prime. Many beginning geometry students are surprised when they learn that any multiple of a Pythagorean triple is another Pythagorean triple. Example: Since 3-4-5 is a Pythagorean triple, the triples 6-8-10, 9-12-15, 12-16-20, 15-20-25, and 120.75-161.00-201.25 are Pythagorean triples.

3-4-5,           5-12-13,     8-15-17,     7-24-25,     9-40-41,     11-60-61,   12-35-37,   13-84-85,   16-13-65,          20-21-29,   28-45-53,   33-56-65,   36-77-85,   39-80-89,   48-55-73,   65-72-97

About 550 BC, Pythagoras proved why the relationship for the sides of Egyptian and Babylonian set squares is true, and this relationship is true for any right triangle, not just the set squares of the ancient builders. Dr. Bronowski shows his viewers how Pythagoras probably proved the great theorem to his followers. Pythagoras first created the square pattern shown below and to the left. The area of this square = c2. Pythagoras then created the pattern to the right by rearranging the pattern on the left. The area of the square pattern on the left must be equal to area of the pattern on the right. Furthermore, the area of the pattern on the right equals the sum of the areas of two squares which equals a2 + b2. Therefore a2 + b2 = c2.

Theorem of Pythagoras Blog, Edit 2There are hundreds of proofs of the theorem of Pythagoras, but none are as elegant as the proof shown above. I can only begin to imagine how Pythagoras must have felt after he completed his magnificent proof. It is said that he offered a hundred oxen to the Muses in thanks for the great inspiration. Book 1, Proposition 47 of Euclid’s Elements, written in about 300 BC, gives a proof of the theorem of Pythagoras. In 1876, when James Garfield, the 20th President of the United States, was serving in the United States Congress as Representative of Ohio’s 19th District, constructed a proof of the great theorem.

Even if you don’t consider yourself to be a math or science type, I think you’ll find The Ascent of Man to be fascinating. Bronowski has a gift for explaining fundamental discoveries in a wide variety of human enterprises in a manner that makes perfect sense to the thoughtful reader or viewer. Musicians will learn that Pythagoras found a basic relationship between musical harmony and mathematics. Artists will learn about the geometric designs created by Arab artists-mathematicians which led to a complete understanding of the symmetries of space, which in turn explains why molecular structures can only have certain shapes.

Basic Trig Functions, offered by Math Teacher’s Resource, has a module that enables teachers to create a wide variety of trig-circle diagrams in which the sides of the right triangles can be multiples of any of the 16 Pythagorean triples listed above. These diagrams can be used to create handouts, homework assignments and test questions. Teachers can go to and download the free handouts Trig Exercises # 1 and Trig Exercises # 2. These handouts will give teachers a good idea of the kinds of course materials they can create. Teachers may also want to download the free handout Basic Math Facts which is a compilation of some of the basic math facts that I want my beginning algebra students to understand and be able to apply when they finish my course. I even give this handout to my college algebra and technical math students. You can use our software to create all types of course materials, which I invite you to post and share on our web site. Besides handouts, teachers can use our software to create dynamic classroom presentations. Go to to view numerous screen shots of different program modules.


Photo of Pythagorus by Galilea at de.wikipedia [GFDL ( or CC-BY-SA-3.0 (], from Wikimedia Commons