Quadrature of the parabola, Introduction

Archimedes, Quadrature of the Parabola: Introduction and Prop. 1-7
translated by Henry Mendell (Cal. State U., L.A.)
Thanks to Jonathan Beere, Jacob Rosen, Matthias Schemmel, John Schmid

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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 (or segment enclosed by a straight-line and a section of a right-angled cone), two here and one in the Method 1, probably one of his last works among extant texts. The two here take very different approaches, and yet more different from that in the Method. The issue here has nothing to do with the use of the principle of the balance that distinguishes the mechanical proofs, 6-15 from the rest of the book. The first quadrature, 16-17, takes a triangle circumscribing the segment and divides up the segment and the triangle into trapezoids and triangle constructed by lines parallel to the base of the segment and lines parallel to the diameter, which is also parallel to one side of the circumscribing triangle. These form different sets which compress the segment. Among these trapezoids and triangles is a set on the curve, which are used to fuel the reductio. The geometrical construction instead uses a series of inscribed triangles, which in the Equilibria of Planes II 2 is called familiarly inscribed and forms the foundation for the analysis of the centers of weight of parabolas and cross sections of parabolas in that book. These approximate the segment from within. The reductio is based on a summation of a series, a₁, ..., a_n where each a_i+1 = 1/4 a_i. The sum is then 4/3 a₁ + 1/3 a_n, which each a₁ will turn out to be the next approximation of the parabola.

Introduction

Preliminary theorems on orthotomes (props. 1-5)
Prop. 1
Prop. 2
Prop. 3
Small comment after 3
Prop. 4
Prop. 5

Quadrature by the mechanical means (props. 6-17)
Basic Lemmas on triangles and trapezoids
Prop. 6
Prop. 7
Prop. 8
Prop. 9
Prop. 10
Prop. 11
Prop. 12
Prop. 13
Approximations of the Parabola
Prop. 14
Prop. 15
Archimedean reductio
Prop. 16
Getting the theorem in the right form
Prop. 17 Note that with this theorems, Archimedes' presentation becomes less informal, i.e., the enunciations of the theorems do not require reference to a diagram.

Quadrature by geometrical means (props. 18-24).
Definition before Prop. 18
Prop. 18
Prop. 19
Prop. 20 and corollary
Prop. 21
Prop. 22
Prop. 23
Prop. 24

Introduction:

Archimedes to Dositheus, greetings. After I heard that Conon, who fell no way short in our friendship, had died and that you had become an acquaintance of Conon and were familiar with geometry, we were saddened on behalf of someone both dear as a man and admirable at mathematics, and we resolved to write and send to you, just as we had meant to write to Conon, one of the geometrical theorems that had not been observed earlier, but which now has been observed by us, it being earlier discovered through mechanical means, but then also proved through geometrical means. Some of those who worked earlier in geometry attempted to prove (draw) how it would be possible to find a rectilinear area equal to a circle that’s given or a segment of a circle that’s given, and afterwards they attempted to square the area enclosed by the section of the cone as a whole(*) by assuming some lemmas that were not readily admissible. For this reason, these were condemned by most people as not being discovered by them. 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. For it is proved that every segment enclosed by a straight-line and right-angled section of a cone is a third-again the triangle having its base as the same and height equal to the segment, i.e., when this lemma is assumed for its demonstration: that is possible for the excess of unequal areas by which the larger exceeds the smaller, by being added to itself, to exceed any proposed finite area. Earlier geometers have also used this lemma. For they use this lemma itself to demonstrate that circles 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. And because every cone is a third part of the cylinder having the same base as the cone and equal height, they would assume some lemma similar to the one mentioned to prove (draw) them . It follows that we have no less conviction in each of the mentioned theorems than those demonstrated without this lemma. It is adequate given that those presented by us have been raised to a conviction similar to these. 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 means. But first there are also proved (drawn) conic elements that are useful for the demonstrations. Farewell.

Does he mean the ellipse, as Heiberg (Archimedes II 263 n.) and Heath (Archimedes, 233 n.), or, as J. Beere suggested to me, “cone generally,” as opposed to any of the three classes, or, as Knorr suggests ("Archimedes and the Elements," AHES 19 (1978), 232 n. 44), perhaps, but not necessarily, with emendation ('plane' for 'straight-line') 'the area bounded by the section of the whole (right/isosceles) cone and/by a plane'?

1. If there is a section of a right-angled cone, on which is ABG, and BD is parallel to its diameter or itself a diameter, equal ordinates and AG is parallel to the line touching the section of the cone at B, then AD will be equal to DG; and if AD is equal to DG, then AG and the line touching the section of the cone at B will be parallel.

No proof appears in Quadrature of the Parabola. Here are proofs.
Case where BD is the diameter: Theorem 0 B
Case where BD is parallel to the diameter