Archimedes and The Student

Bruce Director

To Archimedes came a youth desirous of knowledge.

“Tutor me,” spake he to him, “in the most godly of arts, 
Which such glorious fruit to the land of our father hath yielded 
And the walls of the town from the Sambuca preserv’d!” 
“Godly nam’st thou the art?” She is’t, “responded the wise one; 
“But she was that, my dear son, ere she the state ever serv’d. 
Wouldst thou but fruits from her, there too can the mortal engender; 
What the Goddess doth woo, seek no the woman in her.”

This poem by Friedrich Schiller (translated here by Will Wertz) was cited by Carl F. Gauss in his famous introductory lecture on astronomy. In attacking the pragmatic thinking and the “indifference and insensibility to the great and that which honors humanity,” Gauss told the audience of faculty and students at Goettingen University, the real life implications of this way of thinking. “Unfortunately one cannot conceal the fact that one finds such a mode of thinking very prevalent in our age, and it is probably quite certain that this attitude is very closely connected with the ill fortune which of late has struck so many states. Understand me correctly, I am not speaking of the very frequent lack of feeling for the sciences themselves, but of the source from which this flows, of the tendency everywhere to ask first about the advantage and to relate everything to physical well-being, of the indifference to great ideas, of the aversion to effort due merely to pure enthusiasm for the thing in itself. I mean that such characteristics if they are predominating, can have given a strong decision in the catastrophes which we have experienced.”

He then harkened back to one of his Greek predecessors, “The great happy minds who have created and extended astronomy as well as the other more beautiful parts of mathematics, were certainly not fired by the prospect of future utility; they sought truth for its own sake and found in the success of their efforts alone their reward and their good fortune. I cannot avoid reminding you here of Archimedes, who was admired most by his contemporaries only on account of his ingenious machines, on account of their apparent magic effects; he however valued all this so slightly in comparison with his glorious discoveries in the field of pure mathematics, which at that time mostly had no visible utility in themselves according to the usual sense of the word, that he wrote nothing about the former for posterity, while he lovingly developed the latter in his immortal works. You certainly all know the beautiful poem by Schiller, {Archimedes and the Student}….

Is Gauss asking you to choose between “pure mathematics” and pragmatism? If Gauss’ words seem a bit politically incorrect to you, you have succeeded in identifying a pragmatic demon in your own mind.

Take a look at Archimedes correspondence with Eratosthenes, entitled, “The Method of Treating Mechanical Problems.” In that work, Archimedes poses the problem of determining the relationship between incommensurable solids, such as a sphere, cone and cylinder, or a spheroid and a cylinder. Gauss would later identify that these solids are characterized by different curvatures. The cone and cylinder are generated by surfaces of zero curvature; the sphere by a surface of constant curvature; the spheroid by non-constant curvature.

The problem Archimedes posed to Eratosthenes was: “Seeing moreover in you, as I say, an earnest student, a man of considerable eminence in philosophy, and an admirer [of mathematical inquiry], I thought fit to write out for you and explain in detail in the same book the peculiarity of a certain method, by which it will be possible for you to get a start to enable you to investigate some of the problems in mathematics by means of mechanics. This procedure is, I am persuaded, no less useful even for the proof of the theorems themselves; for certain things first became clear to me by a mechanical method, although they had to be demonstrated by geometry afterwards because their investigation by the said method did not furnish an actual demonstration. But it is of course easier, when we have previously acquired, by the method, some knowledge of the questions, to supply the proof than it is to find it without any previous knowledge….”

An example of this method is the determination that the volume of a sphere is four times the cone with base equal to a great circle of the sphere and height equal to its radius; and the volume of a cylinder with base equal to the great circle of the sphere and height equal to the diameter is 1.5 times the sphere.

(The second part of this proposition, was depicted on Archimedes tombstone.)

The difficulty in proving this proposition, lay in the differing curvatures of the volumes measured. To overcome this obstacle, Archimedes investigated the interaction of these volumes in a physical process, the pull of the Earth’s gravity.

To construct the experiment, think of the sphere cone and cylinder, all nested together in the following way. Think of a sphere, then, think of a cone whose base is formed by the equator of the sphere, and whose apex is at the north pole. Now think of that whole thing embedded in a cylinder whose bases touch the north and south pole of the sphere. The sphere will be tangent to the cylinder at its equator.

(You can draw a cross section of this arrangement, by drawing a circle with two perpendicular diameters. This represents a cross section of the sphere. Label the intersections of these diameters counter-clockwise A,B,C,D. Then draw a triangle with vertices A,B,D. This represents a cross section of the cone. Then draw a square around the circle such that A,B,C,D intersect the midpoints of the sides of the square. This represents the cross-section of the cylinder. Label the corners of the square, V,X,W,Y clock-wise from the upper left. And label the center of the sphere K.)

Now, extend the sides of the cone (A-B and A-D in the cross section drawing) and the bottom side of the square (W-Y in the drawing). Label the intersections, E and F. Now we can think of an enlarged cone, A, E, F. Also construct, an enlarged cylinder, with base E-F and height, A-C.

Then, Archimedes imagines that this entire complex of solids is resting on a balance, whose bar is twice the length of A-C, with A at the midpoint. (To depict this in our cross-section drawing, extend double line A-C and label the new endpoint H.)

Finally, draw a line perpendicular to C-A-H and parallel to B-D. This line will intersect the cross sections of all the figures previously imagined. Archimedes, using the Pythagorean theorem, and the principles of Euclidean geometry, determines the proportional relationships existing among these cross-sections.

To determine the relationship of the volumes of the sphere, cone and cylinder, Archimedes investigates under what conditions the various cross-sections of the cone, cylinder and sphere are balanced. Using the proportions he just calculated, he is then able to determine that the volume of the sphere is 4 times the cone and the volume of the cylinder is 1.5 times the sphere.

(The actual calculation is not difficult, but it would be too cumbersome to describe in this format. If you don’t have a copy of Archimedes piece, send an e-mail to BMD, and I will supply a copy of this proposition.)

Is Archimedes procedure a physical demonstration, or a mathematical one. Isn’t he investigating geometrical objects, which have no physical existence, with respect to a physical process, the pull of Earth’s gravity? As he said to Eratosthenes, this procedure makes the proposition clear, but he still requires a geometrical proof, or, as Kepler would later state with respect to the divisions of the circle, “knowability”?

Forty years after Gauss identified the consequences of pragmatism on the political condition of Europe, then former U.S. President John Quincy Adams gave a speech in Cincinnati, Ohio on the occasion of the laying of the cornerstone of the U.S.’s first astronomical observatory. After an extensive discussion of the history of astronomy, Adams ended his speech:

“But when our fathers abjured the name of Britons, and `assumed among the powers of the earth, the separate and equal station, to which the laws of Nature, and of Natures, God entitles them,’ they tacitly contracted the engagement for themselves, and above all, for their posterity, to contribute, in their corporate and national capacity, their full share; aye, and more than their full share, of the virtues, that elevate, and of the graces that adorn the character of civilized man….

“… We have been sensible of our obligation to maintain the character of a civilized, intellectual, and spirited nation. We have been, perhaps, over boastful of our freedom, and over sensitive to the censure of our neighbors. The arts and sciences, which we have pursued with most intense interest, and persevering energy, have been those most adapted to our own condition. We have explored the seas, and fathomed the depths of the ocean, and we have fertilized the face of the land. We–you- -you, have converted the wilderness into a garden, and opened a paradise upon the wild. But have not the labors of our hands, and the aspirations of our hearts, been so absorbed in toils upon this terraqueous globe, as to overlook its indissoluble connection even physical, with the firmament above? Have we been of that family of the wise man, who, when asked where his country lies, points like Anaxagoras, with his finger to the heavens.

“Suffer me to leave these questions unanswered. For, however chargeable we may have been, with inattention or indifference, to the science of Astronomy, heretofore — you, fellow citizens, of Cincinnati — you, members of the Astronomical Society, of this spontaneous city of the West, will wipe that reproach upon us, away….”