Archimedes of Syracuse

Commentary on the text

1.  The text is a letter from Archimedes to Dositheus.  Notice the formal greeting that introduces the letter; this has the same form as more famous letters written by men of the ancient world.  Compare it, for instance, to the introduction to the first letter of St. Paul to St. Timothy in the Christian scriptures.

2.  Archimedes talks of discovering his results "by means of mechanics" and then demonstrating them "by geometry".  He had developed a technique for determining areas of figures and volumes of solids that was based on the law of the lever which he had formulated and put to good use in many of the inventions for which he was famous.  The law of the lever says that if two weights, w and W, suspended from either end of a straight lever balance when w is a distance l from the fulcrum of the lever and W is a distance L from this point, then lw = LW, or as Archimedes would say "the ratio of the weights in balance is the same as the inverse ratio of their lengths from the fulcrum" (w/W = L/l).  By mentally balancing figures with weights of various shape, he was able to produce many area and volume calculations.  However, Archimedes realized that this method, while reliable, was a heuristic; it did not provide logical demonstrations of these facts.  So he would then work geometrically to formulate proofs of his results.  For many centuries it was not known how Archimedes developed these mechanical calculations, until Heiberg in 1906 recognized the text of a lost work of Archimedes, the Mechanics, in the palimpsest that is now undergoing restoration at the Walters Art Gallery in Baltimore.

3.  The "section of the whole cone" is an ellipse (see the discussion on the section of a right-angled cone):

4.  It is the main result of the text, found in Proposition 24.

5.  The axis of a parabola is the unique line of mirror symmetry for the curve; a chord is any line segment joining two points of the curve.  As you can see, no proofs are given for the first few propositions stated here.  Archimedes mentions below that these are already found "in the elements of conics", probably a reference to a standard source for the geometry of conics, perhaps even the lost Conics of Euclid.  We, too, will skip the proof of Proposition 1 here, and defer it to the exercises.

6.  We will see this proposition again, with proof, in the next text of Apollonius.

7.  Like Proposition 1, we leave the proof to the exercises.

8.  In the intervening propositions, Archimedes quotes a few more fundamental facts about parabolas, then shows how he deduces the main result of this treatise by mechanical means (see the end of note 2).  In Proposition 18, he begins the proof of the result by geometric means.
Here in Proposition 20, he begins to collect the various facts needed to apply the method of exhaustion.  This explains his intention in stating the Corollary that follows this proposition.  You will find a statement like this (that the figure inscribed is more than half of the area it is inscribed into) in Eudoxus' proof of Elements xii.2.  Observe that Archimedes calls P the vertex of the parabolic segment; this is the point on the curve furthest from the base. By virtue of Proposition 1, the tangent at P must be parallel to the base Qq.

9.  Consider the triangle PQV.  Since RM is parallel to PV and bisects PQ, it must also bisect QV.

10.  Proposition 19 states that PV =  (4/3)RM.  Here is the proof.  Let the parallel to Qq through R cut PV in V'.  Then by Proposition 3 above,

so that PV' = (1/4)PV.  Therefore PV = PV' + V'V = (1/4)PV + RM, so RM = (3/4)PV and thus PV = (4/3)RM.

11.  Triangles PQV and YQM are similar and QV = 2QM.

12.  DPQM = DYQM + DPYM and  DPRQ = DYRQ + DPRY.  But DYQM has the same height (to Q) as DYRQ and twice the base (YM = 2RY), so  DYQM = 2 DYRQ; similarly, DPYM has the same height (to P) as DPRY and twice the base, so  DPYM = 2 DPRY.  Therefore,  DPQM = 2 DPRQ.

13.  DPQM= DPMV since they have the same height (to P) and equal bases (QM = MV).  So DPQV = DPQM + DPMV = 2 DPQM = 4 DPRQ.

14.  The term ordinate is used to denote a line segment drawn to an axis that is parallel to some fixed transversal line to the axis.  In this case, the axis PV is cut by transversal Qq, so RW is the ordinate from R to PV.

15.  RW = MV = Vm = rW.

16.  Here Archimedes is preparing for the exhaustion of the parabolic segment by successively smaller inscribed triangles.

17.  This is Archimedes' version of the geometric series formula

18.  Note the now familiar form of the double reductio argument.
 
 

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