An Explanatory Approach to. Archimedes’s Quadrature of the Parabola. by. A. Kursat ERBAS. Have you ever been in a situation where you are trying to show the. Archimedes’ Quadrature of the Parabola is probably one of the earliest of Archimedes’ extant writings. In his writings, we find three quadratures of the parabola. Archimedes, Quadrature of the Parabola Prop. 18; translated by Henry Mendell ( Cal. State U., L.A.). Return to Vignettes of Ancient Mathematics ยท Return to.

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An Explanatory Approach to. Archimedes’s Quadrature of the Parabola. Have you ever been in a situation where you are trying to show the validity of something with a limited knowledge?

I hope you have not. Because, you just have to use your ingenuity.

Archimedes: “Quadrature of the parabola”

Think about a situation where you do not know “coordinate geometry”, “calculus in the modern sense differentiation, integration etc You just know the definition of a parabola.

You want to discover certain properties of the parabola, and solve a problem. History has lots of examples of this kind of situation. Wherever you go in the written history of human beings, you will find that civilizations built up with mathematics. With this respect, I think we must teach mathematics with a little bit history of mathematics.


We need to learn and teach to our kids how the concepts in mathematics are developed.

Quadrature of the Parabola | work by Archimedes |

And I think this will allow to pupils to get a broad vision of mathematics as well as a well built mathematical thinking. That is why I intended to write an essay on ” Quadrature of Parabola ” which is a famous work of Archimedes B.

The ” Quadrature of Parabola ” is one of his works besides crying “Eureka. The quadrature of the parabola investigates the ratio between the area of the parabolic section bounded by a parabola and a chord and the area of the triangle which has the vertex of the parabolic section and two points of intersection of the segment and the parabola as its vertices See Figure Quadrature of the Parabola.

Quadrature of the Parabola

The significance of the Archimedes’ solution to this problem is hidden in the fact that none of differention, integration, or coordinate geometry were known in his time. But, if you look at Greek mathematics carefully, they were using the idea of limit, or in other words “approximation”. Archimddes let’s start to Archimedes’ solution to Quadrature of Parabola. Let A be the midpoint of the segment SS’.


Click here to see on GSP.

If we consider Figure-3a and -3b which are extracted from Figure If the same argument applied to the left side of the Figure-2. Applying Claim-II each of them shows that area of the triangle VCS parablla four times the sum of the areas of the two blue triangles at right.

Similarly, the area of the triangle VC’S’ is four timesthe sum of the areas of the two blue riangles at left. Recalling that the light blue area in Figure-2 is.

The process forming triangles can be repeated again and again. Thus the sum the blue triangles approximate the area of the parabolic section See the Figure below. Click Here for a demonstration!

Click Here for a little example of “Quadrature of the Parabola” carried by Mapple