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 · Return to.

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Go to theorem If a straight line is drawn from the middle of the base in a segment which is enclosed by a straight line and a section of a right-angled cone, the point will be a vertex of the segment at which the line drawn parallel to the diameter cuts the section of the cone.

The main idea of the proof is the dissection of the parabolic segment into infinitely many triangles, as shown in the figure to the right.

That is why I intended to write an essay on ” Quadrature of Parabola ” which is a famous work of Archimedes B. Views Read Edit View history. This picture shows a unit square which has been dissected into an infinity of smaller squares. Go to theorem If magnitudes are placed successively in a ratio of four-times, all the magnitudes and yet the third part of the least composed into the same magnitude will be a third-again the largest.

I say that area Z is less than area L.

In other projects Wikimedia Commons. Here T represents the area of the large blue triangle, the second term represents the total area of the two green triangles, the third term represents the total qrchimedes of the four yellow triangles, and so forth. Go to theorem If a triangle is inscribed in a segment which is enclosed by a paravola line and a section of a right-angled and has the same base as the segment and the same height, the inscribed triangle will be more than half the segment.

I say that area Z is a third part of triangle BDG.

Archimedes: “Quadrature of the parabola”

Archimedean solid Archimedes’s cattle problem Archimedes’s principle Archimedes’s screw Claw of Archimedes. This assumes that there is only one vertex archimeds the section, something which we may want proved from fthe properties of cones.

But BD and BE are parallel, which is also impossible. Each successive purple square has one fourth the area of the previous square, with the total purple area being the sum. The formula above is a geometric series —each successive term is one fourth archiimedes the previous term. Think about a situation where you do not know “coordinate geometry”, “calculus in the modern sense differentiation, integration etc Archimedes’s Quadrature of the Parabola.


If the same argument applied to the left side of the Figure-2. This represents the most sophisticated use of the method of exhaustion in ancient mathematics, and remained unsurpassed until the development of integral calculus in the 17th century, being succeeded by Cavalieri’s quadrature formula. Go to a sample proof. The converse is easy to prove: And these have been proved in the Conic Elements.

Quadrature of the Parabola

Recalling that the light blue area in Figure-2 is. For they use this lemma itself to demonstrate that archimedss have to one another double ratio of the diameters, and that spheres have triple ratio to one another of the diameters, and further that every pyramid is a third part of the prism having the same base as the pyramid and equal height.

Click Here for a demonstration!

Similarly, the area of the triangle VC’S’ is four timesthe sum of the areas of the two blue riangles at left. Theorem 0 D with converse Case where BD is parallel to the diameter with converse.

Archimedes to Dositheus, greetings. Let A be the midpoint of the segment SS’.

The quadraturee 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. We need to learn and teach to our kids how the concepts in mathematics are developed. In a way similar to those earlier, area Z will be proved smaller than L similarly to those previously.

It is necessary, in fact, that either the line drawn from point B parallel to the diameter be on the same sides as the segment or that the line drawn from G make an obtuse angle with BG.

Wikimedia Commons has media related to Quadrature of the Parabola. Have you wrchimedes been in a situation where you are trying to show the validity of something with a limited knowledge?

Case where BD is the diameter: For always more than half being taken away, it is obvious, on account of this, that by repeatedly diminishing archmedes remaining segments we will make these smaller than any proposed area.

quxdrature Hence, there are two vertices of the segment, which is impossible as noted above, we may want to prove this from the properties of cones. First, let, in fact, BG be at right angles to the diameter, and let BD be drawn from point B parallel to quadrahure diameter, and let GD from G be a tangent to the section of the cone at G.


Earlier geometers have also used this lemma. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord. A parabolic segment is the region bounded by a parabola and line. And so, having written up the demonstrations of it we are sending first, how it was observed through mechanical means and afterwards how it is demonstrated through geometrical pf.

But we do not know of anyone previously who attempted to square the segment enclosed by the straight-line and right-angled section of a cone, which has now, in fact, been found by us.

Thus the sum the blue triangles approximate the area of the parabolic section See the Figure below. I hope you have not. Go to theorem Again let there be a segment BQG enclosed by a straight-line and section of a right-angled cone, and let BD be drawn through B parallel to the diameter, and arvhimedes GD be drawn from G touching the section of ;arabola cone at G, and let area Z be a third part of triangle BDG.

No proof is given.

Quadrature of the parabola, Introduction

If in fact some line parallel to AZ be drawn in triangle ZAG, the line drawn will be cut in the same ratio by the section of a right-angled cone as AG by the line drawn [proportionally], but the segment of AG at A will be homologous same parts of their ratios as the segment of the line drawn at A.

Applying Claim-II th of them shows that area of the triangle VCS is four times the sum of the areas of the two blue triangles at right. Theorem 0 B Case where BD is parallel to the diameter. It will be proved similarly to that earlier that Z is larger than L afchimedes smaller than M. The ” Quadrature of Parabola ” is one of his works besides crying “Eureka. Of the twenty-four propositions, the first three are quoted without proof from Euclid ‘s Elements of Conics a lost work by Euclid on conic sections.