3. Archimedes’s Equilibrium of Planes

The names of many ancient mathematicians and scientists are quite unfamiliar to most of us. But there is one shining exception‒Archimedes of Syracuse. His name usually brings to mind a man sitting in a bathtub with the water running over, then running out into the streets unclad, crying “Eureka (I have found it),” the “it” being the answer ro a scientific problem. We like to think of him as typical of the man of science who is so immersed in his studies that he pays no attention to his surroundings. We also remember Archimedes’ famous boast: “Give me a fulcrum (a place to rest a lever on), and I will move the earth.” (Mortimer J. Adler and Peter Wolff, Foundations of Science and Mathematics, volume 3 of The Great Ideas Program, Encyclopedia Britannica, 1960, page 33)

In my rereading of selections from Great Books of the Western World guided by The Great Ideas Program, I’ve reached the third reading in the third volume of The Great Ideas Program, Foundations of Science and Mathematics by Mortimer J. Adler and Peter Wolff (Encyclopedia Britannica, 1960), Archimedes’s On the Equilibrium of Planes. Adler and Wolff consider the first seven propositions in Book One of On the Equilibrium of Planes and divide their study of them into these sections:
I ‒ a consideration of what the subject of the book is; after observing that it is neither a purely mathematics treatise nor an introduction to experimental natural science, they describe as an example of mathematical physics.
II ‒ a discussion of how a book about physics can get along without any experiments; they conclude that they aren’t necessary because “The Equilibrium of Planes takes existing experimental and observational knowledge and, with the help of mathematics tools, refines it and recasts it in different language” (Adler and Wolff, page 37).
III ‒ a close look at Propositions 6 and 7, which together constitute the Law of the Lever. They show how Propositions 1-5 prepare for Propositions 6-7; then they assign numerical values to the weights and distances involved in Proposition 6 and go through Archimedes’s proof of it (I got stuck going through Archimedes’s proof but was able to follow Adler and Wolff’s explanation of it); and finally they use Proposition 6 to prove Proposition 7.
IV – a presentation and discussion of three questions on the reading; see below.

Here I’ll identify Archimedes, present the questions which Adler and Wolff present on On the Equilibrium of Planes, and indicate how Adler and Wolff answer the questions.

Archimedes

Archimedes was born in the Greek city-state of Syracuse around 287 B.C. As a young man he spent some time in Egypt, where he may have studied with the pupils of Euclid in Alexandria and where he invented the water-screw as a means of drawing water out of the Nile for irrigating the fields. His mechanical inventions won great fame for him and figure largely in the traditions about him, such as those referred to in the passage from Adler and Wolff with which I introduced this article. However, except for a lost work On Sphere-Making, he wrote only on strictly mathematics subjects. His absorption in his mathematical investigations that he forgot his food and neglected his person. It may even have caused his death. “In the general massacre which followed the capture of Syracuse by Marcellus in 212 B.C., Archimedes was so intent upon a mathematical diagram that he took no notice, and when ordered by a soldier to attend the victorious general, he refused until he should have solved his problem, whereupon he was slain by the enraged soldier” (Great Books of the Western World, Encyclopedia Britannica, 1952, volume 11, page 400).

Questions

– What are the assumptions underlying Archimedes’ proof of Proposition 6?
Archimedes doesn’t define “centre of gravity” but seems to use two different meanings for it. “The first meaning, used in Propositions 4 and 5, is apparently that of a point of balance, or the point at which a fulcrum should be located in order to balance a system of weights.… The second meaning [which is used in Proposition 6] is a point at which we imagine the entire weight of a body or system of bodies concentrated” (Adler and Wolff, page 41). In Proposition 6 it has to be assumed that both meanings are the same.
– Why must there be two proofs of the Law of the Lever, one for commensurable and one for incommensurable magnitudes?
Adler and Wolff give a lengthy explanation, which includes: “Archimedes’ proof of Proposition 6 depends on finding a common measure of weights A and B and dividing each of them into parts equal to the common measure. These equal parts are then strung out along the line LK. It is obvious that this method of proof will not work for incommensurable magnitudes, since by definition they have no common measure.” (Adler and Wolff, pages 43-44)
– What is the Law of the Lever for magnitudes that do not balance?
Adler and Wolff show that Archimedes seems to know it in proving Proposition 7 and express it in modern terms as “The side with the smaller moment will rise, while the side with the larger moment will be depressed,” where the moment (of force) is “the product of the weight and the distance from the fulcrum at which that weight is applied.” (Adler and Wolff, page 45)

1. Aeschylus’s Prometheus Bound

Divine power has not always been regarded as just and beneficent. The tension between the divine and the human is a perennial problem. Aeschylus, the Greek tragedian, has dramatized this tension by staging the myth of Prometheus, the Greek hero, or demigod, who was cruelly punished by Zeus for bringing culture to mankind. This myth of the benefactor of man, chained to a rock and tortured for countless ages but always maintaining his defiance of the supreme power, has stirred the imagination of readers for thousands of years. (Mortimer J. Adler and Seymour Cain, Religion and Theology, volume 4 of The Great Ideas Program, Encyclopedia Britannica, 1961, page 1)

In my rereading of selections from Great Books of the Western World guided by The Great Ideas Program, I’ve reached Aeschylus’s Prometheus Bound. It is the first reading in the fourth volume of The Great Ideas Program, Religion and Theology by Mortimer J. Adler and Seymour Cain (Encyclopedia Britannica, 1961).

Adler and Cain introduce their study of Prometheus Bound with the above quotation and the presentation of some basic religious problems raised by Prometheus Bound. They go on to: (I) describe the mythical background of Prometheus Bound, (II) outline Prometheus Bound, (III) discuss what Prometheus Bound says about the nature of the ultimate power in the universe and of man’s relation to that power, (IV) compare Prometheus Bound and the book of Job, and (V) pose and discuss three questions. Here I’ll identify Aeschylus and share briefly from Adler and Cain’s study.

Aeschylus

Aeschylus was a Greek poet who lived in Athens around 525-456 B.C. He write more than eighty plays, of which seven (including Prometheus Bound) survive. He is regarded as the founder of Greek tragedy because he added a second actor to the single actor and chorus previously employed. He won the prize at the annual contest in tragedy at the festival of the City Dionysia at least twelve times. He is noted for the religious element in his tragedies.

Mythical Background of Prometheus Bound

Essential parts of the mythical background of Prometheus Bound are that Zeus is the supreme ruler of the gods; that Prometheus, one of the gods under Zeus, angered him by stealing fire from heaven and bringing it to men and by teaching men all the useful arts for maintaining themselves on earth; and that Zeus punished Prometheus by having him chained to a rock on Mt. Caucasus, where an eagle ate his liver every day and it was restored at night.

Plot of Prometheus Bound

The play opens with Kratos and Bia bringing in Prometheus and holding him while Hephaestus, the divine smith, shackles him to a rock. It closes with Prometheus (and, because of their loyalty to him, the daughters of Oceanus) sinking into the abyss. Others appearing are Oceanus, the god of the water which surround the earth, who offers to intercede with Zeus on Prometheus’s behalf; Io, a girl with whom Zeus had fallen in love and turned into a heifer to protect her from the wrath of his wife Hera and who is now being driven by a gadfly sent by Hera to wander over the face of the earth; and Hermes, the messenger of Zeus, who threatens Prometheus with horrible punishment unless he reveals a secret which he knows about Zeus’s future.

What Prometheus Bound says about the nature of the ultimate power in the universe and of man’s relation to that power

Adler and Cain discuss whether Zeus is the ultimate power. They observe that in the play he is “overpowering force, not only omnipotent, but tyrannical, merciless, and unjust” (Adler and Cain, page 6). However they also present evidence to show that he is bound by Fate or Necessity, that his power is not eternal, and that he is not omniscient.

Adler and Cain also discuss the characteristics of Prometheus and why he protests against the divine power. They note these interpretations: he is “the benevolent enlightener of mankind and the defiant protagonist of spiritual liberty against a divine tyrant”; he is “a tragic hero of noble character who falls through the defect of self-willed pride”; or he is “a heavenly being who tries to ursurp the supreme power, in this case for the good of mankind” (Adler and Cain, page 10). They conclude that perhaps none of the interpretations is quite true.

Prometheus Unbound and the Book of Job

The Great Ideas Program considered the Book of Job in an earlier volume. See https://opentheism.wordpress.com/2017/09/01/7-the-bibles-book-of-job/. Here Adler and Cain discuss how the Book of Job deals with the problem of the suffering of the righteous and the prosperity of the wicked in a world ruled over by a Good of righteousness and compare Prometheus and Job,

In the latter Adler and Cain begin their comparison of Prometheus and Job by observing that there is one basic similarity between them‒both question the sufferings they are forced to endure. Then Adler and Cain show how Prometheus and Job have different attitudes: Prometheus complains about the injustice of his punishment but doesn’t expect justice from Zeus, whom he views as the Enemy, but Job’s resistance is that of a man of faith who can’t understand why he is suffering so when God is the Friend. Finally Adler and Cain show the different ends for Job and Prometheus, Job’s accepting (“I despise myself, and repent in dust and ashes,” Job 42:6, ESV) and being rewarded with earthly happiness and Prometheus’s remaining intransigent and being punished with endless torment.

Questions

Is it right or wrong to rebel against divine power?

\What is the religious evaluation of man’s acquisition of the arts and sciences?

Is there a nonrational, nonethical element in the divine?

Adler and Cain discuss what both Prometheus Bound and the Book of Job say about each question.

1-2. Euclid’s Elements

Euclid’s Elements…is the classic textbook of Greek geometry, which has served as the basis of study for over twenty centuries, It is a model of clear and orderly presentation.… It has the classic simplicity and order that so often characterizes a great work which summarizes generations or centuries of study. (Mortimer J. Adler and Peter Wolff, Foundations of Science and Mathematics, volume 3 of The Great Ideas Program, Encyclopedia Britannica, 1960, pages 1-2)

It is sometimes said that next to the Bible, the Elements may be the most translated, published, and studied of all the books produced in the Western world (“Euclid,” The New Encyclopedia Britannica, Encyclopedia Britannica, 1974, volume 6, page 1019).

In my rereading of selections from Great Books of the Western World guided by The Great Ideas Program, I’ve reached Euclid’s Elements. Its Book I constitutes the first and second readings in the third volume of The Great Ideas Program, Foundations of Science and Mathematics by Mortimer J. Adler and Peter Wolff (Encyclopedia Britannica, 1960). The first reading considers Definitions, Postulates, Common Notions, and Propositions 1-26, which deal with triangles, and the second reading considers Propositions 27-48, which deal with parallel lines (27-32) and parallelograms (33-48).

Adler and Wolff divide their guide to the readings into these sections:
First Reading
I ‒ the structure and unity of Book I
II ‒ some difficulties with Definitions
III ‒ the function of Postulates and Common Notions
IV ‒ questions on problems in Propositions 1-26 and discussion of them
Second Reading
I ‒ the order of the Propositions in Book I
II ‒ Proposition 47, the Pythagorean Theorem
III ‒ the kinds of proof used by Euclid in Elements
IV ‒ questions on problems in Propositions 27-48 and discussion of them

Here I’ll identify Euclid, present the questions which Adler and Wolff present on problems in the readings, and indicate how they answer the questions.

Euclid
Euclid flourished about 300 B. C. About all that is known of his life is that he taught at a school at Alexandria in Egypt. His great work was the thirteen books of the Elements, which he compiled from a number of writings of earlier men and which became a classic soon after publication.

Questions on the First Reading
– Why does Euclid begin Book I with Propositions 1-3?
Noting that Propositions 1-3 are construction propositions rather than theorems, Adler and Wolff answer that constructions are needed in the proofs of theorems and that, like postulates, they show that certain geometric operations can be performed.
– Is there any need for Proposition 2?
Proposition 2 involves placing at a given point (A) a straight line equal to a given line \(BC). Adler and Wolff show how Proposition 2 adds to Postulate 3, to describe a circle with any center and distance, by showing that, given a point and a distance, a circle can be drawn around the point with the distance as its radius without the distance having to start at the center of the circle.
– What are the various parts of a Euclidean proof?
Noting that Euclidean theorems can be restated as if-then statements, Adler and Wolff explain how their proofs consist in going from the “if” clause or hypothesis to the “then” clause or conclusion by a series of steps using postulates, common notions, and previously proved propositions.

Questions on the Second Reading
– What is the role of diagrams in Euclid’s proofs?
After observing that diagrams help us follow Euclidean proofs, Adler and Wolff discuss whether they must be present or are just a convenience. They present the case for them thus: “Geometry is about figures. Figures can be drawn and seen. Hence a geometrical proof should begin with a diagram. It shows us the very thing being talked about.” (Adler and Wolff, Foundations of Science and Mathematics, page 28) They then present objections to the need for diagrams and answers to them.
– Does the way in which Euclid presents his propositions indicate the way in which they were discovered?
Adler and Wolff explain why they answer negatively..
– Does Euclid tell us how to measure the size of a triangle?
Adler and Wolff observe that Euclid tells us in Proposition 41 that a triangle is half the size of a parallelogram with the same base and within the same parallels but that he doesn’t deal with the area of a parallelogram until Book VI of Elements.