English 233: Introduction to Western Humanities - Baroque and Enlightenment

Study Guide

to Jacob Bronowski's

The Ascent of Man:

"The Music of the Spheres"

Note:  "The Music of the Spheres" is Program 5 of 13 that comprise the entire series called The Ascent of Man. Each of these 13 programs is available at the Manhattan Public Library, at the corner of Poyntz and Juliette ("7th" St.). KSU students are eligible to obtain a library card there (upon showing of a student id), and with it you can take out 2 videos per day at $1.00 rental each. Note:  If you are going to print out this document from one of the KSU public computer labs, you will first need to go into your browser's File menu, choose the Page Setup option and click on Black Type. This will ensure that any colored text in the document as it appears in your browser will come through (as black) in the copy you print out.

The following questions are offered in order to prime your attention to the video. During and/or after your viewing the program, test yourself to see how well you can come up with answers to them. The basic material (and more) is also available in Chapter 5 of the companion volume:  The Ascent of Man (Boston: Little, Brown & Co., 1973), pp. 154-187). The call number is Q175.B7918/1974.  I have put the book on reserve at Farrell Library. (Go to the Reserve Desk on the 2nd floor.)

F Notice something about the striking opening images Bronowski chooses (from the environment on the island of Samos in the Aegean Sea) for this program as a whole.  Even before we hear Bronowski's voice, introducing us to Pythagoras, how many purely visual references to you detect to the idea of nature as a realm of flux, of motion?  (This will be something that gets focused on at the very end of the program.)
F What difference does Bronowski have in mind when he distinguishes what he calls "arithmetic" from what he calls "mathematics"? (When you reflect on the entire program after it is over, see if you can collect your thoughts around the following question:  how might this entire part of his series be understood as an attempt to explain the importance of this distinction?)

1. There are two main points Bronowski stresses in his discussion of Pythagoras (who lived on the island of Samos near Asia Minor around 580 BC).

A. One is Pythagoras' success in providing a foundation for the picture of the world according to which "nature is commanded by numbers." (An alternative image is that numbers are "the language of" nature.) This conception in turn suggested that it might be possible for the human mind to grasp a fundamental unity behind the multifarious variety of appearance encountered in our sensory apprehension of the world. Part of what impressed Pythagoras' followers is that he was able to present convincing demonstrations that this might be true for each of two distinct physical dimensions of the world.

i. What did this achievement consist in concerning the world of sound?

ii. What did it consist in concerning the world of vision, both in the world we experience and the world we construct?
B. The other is his demonstration of the nature and power of rational proof.
Bronowski illustrates the latter two points with an interesting reconstruction of Pythagoras' proof of what has ever since been known as "the Pythagorean theorem." The next several questions have to do with this segment of his exposition.
i. What are the two aspects of our experience of being in the world which, in Bronowski's view, fix the nature of the right triangle?

ii. If set-squares (the construction tool) had been known since at least 2000 BC, what was special about Pythagoras' achievement?

iii. Can you reconstruct Bronowski's demonstration for a friend or roommate?
iv. Why does Bronowski think that this is the single most important theorem in mathematics?

2. Bronowski's next stop is Alexandria, Egypt, a center of Greek culture later even than the era of classical Athens. (Recall that Socrates was put to death in 399 BC)

A. He locates Euclid there, around 300 BC What is Euclid's special achievement, such that his book comes to be the most translated, copied and printed book in all of history, with the single exception of the Bible?

B. The other major science practiced at Alexandria was astronomy. Bronowski locates Ptolemy there around 150 AD

i. What was Ptolemy's outstanding achievement?
ii. By what circuit did his text (the Almagest) eventually find its way into Europe?

iii. What geometric figure did he choose as the basic elements of his model, and why? And what combination of these figures did he construct in order to "account for appearances"?

3. The next phase of Bronowski's tale concerns the emergence and spread of Islam. In his account, we are indebted to this broad development for several essential transmissions of achievements from the ancient heritage that otherwise might have been lost to the West, for the transmission of a momentous mathematical invention from India, and for a major step forward of its own in "the ascent of man."

First, here are some basic dates it is useful to keep in mind in connection with Bronowski's story:
630:  Mohammed conquers Mecca.
730:  Islam has spread from the Arabian Penninsula to embrace not only Alexandria in Africa and Baghdad to the North, but as far as Córdoba (in Spain) in the West and to the edge of India (Isfahan in Persia) in the East.
(Later on, it will conquer India via the Moghuls, on its way to Indonesia and even to what will later be known as the Philippines. It will also absorb Asia Minor and [via the expansion of the Ottoman Empire] Greece, the Balkans and Hungary.)
1085:  Toledo (in Spain) plays a central role in the culture of larger Europe.
1492:  The last remaining Muslim stronghold, Granada, will fall to the combined forces of Ferdinand & Isabella (of the combined kingdoms of Aragon & Castille, respectively). (The date is fateful for other reasons, as well, right?)
A. What is the state of intellectual life in Europe, according to Bronowski, during the period of the triumphant spread of Islam?
B. What special attitude towards the intellectual achievements of "conquered" cultures distinguished the culture of Islam?

C. Bronowski stresses that Islam, in distinction to early and medieval Christianity, is not a "religion of miracles," but one rather of "contemplation and analysis." How does this connect, in his picture, with
i.  what is special about the Islamic conception of the deity?
ii.  with Islam's attitude towards definite images of the deity or of man?
iii.  with classical Islamic culture's interest in numbers and calculation, in the construction and solution of puzzles?
D. What is an astrolabe, and what's important about it as an intellectual device?
And what practical use was made of it, within Islamic culture?)
E. What, according to Bronowski, is "the single most important innovation in mathematics [that] the ... Arab scholars brought [West] from afar," around 750 AD, from India?
How long did it take the Christian West to adopt that system?)
F. After a brief glimpse from afar at Isfahan, Bronowski takes us inside the fortress-palace of the Alhambra in southern Spain (which held out against the Christians until 1492).
i.  What is the interior of the palace meant to symbolize?
ii.  How is the decoration of the harem connected with important points of theology?
iii.  Bronowski examines two sets of tile patterns that he says make the points that (1) "the artist and the mathematician in Arab civilization have become one" (in a quite literal way) and that (2) these designs "represent a high point of the Arab exploration of the subtleties and symmetries of space itself."
What does the mathematical operation of rotation have to do in turn with the idea of symmetry?
v. At this point, Bronowski poses the question "So what?" and undertakes to answer it by explaining how this sort of exploration is not a mere "elegant game."
(1)  What is the answer he gives?

(2)  How does he illustrate its force by pointing to several examples of crystals in nature?

(3)  What is the connection of this fact about crystals with the fundamental insights of Pythagoras some 20 centuries earlier?
G. What is the main point of contrast Bronowski draws between the culture he sees expressed in the Alhambra and the culture of medieval Christianity he points to in the village of Santillana, on the southern coast of Spain (never conquered by the Moors)?

What is the point of that contrast? (This may be easier to appreciate when you look back from the vantage point of the conclusion of the program.)
5. What was special about intellectual life in Toledo around 1085 AD? (There are several things you might want to note.)

A. Who was Gerard of Cremona, and what was important about his work?
B. How did the mathematician Alhazan, in his Optics, explain an important fact about perception of objects at various distances that was inexplicable on the theory put forward by the ancient Greeks?  (What concept did he devise that becomes the basis for the science of perspective?)

6. "The excitement of perspective passed into art in north Italy, in Florence and Venice, in the fifteenth century."

Bronowski maintains that, in turn, the exploration of perspective (by artists of the Italian Renaissance [and pupils like Albrecht Dürer who came to study from places like Germany]) in turn were a major force in revivifying mathematics. One eventual result was the development of what we now call (after Leibniz) the differential calculus (what Newton, who invented it independently, called "the method of fluxions"). The concluding segment of this episode of The Ascent of Man is devoted to explaining this intriguing connection.
A. He begins by contrasting Carpaccio's St. Ursula and Her Suitor taking leave of her parents (set, he says, "in a vaguely Venetian port"), painted in 1495, with a view in fresco of Florence, painted around 1350. In the course of his discussion, he puts the picture of Florence through a visual transformation into the mode of the "perspectivi" (as the practitioners of the new visual language were called). Can you see how this alteration supports the following two claims?
i.  The earlier painter "thought of himself as recording things, not as they look, but as they 'are': a God's-eye view, a map of eternal truth" (that is, a conception of what is represented that pretends to look at it in abstraction from its embedment in time).
ii.  "The perspective painter has a different intention. He deliberately makes us step away from any absolute and abstract view. Not so much a place as a moment is fixed for us, and a fleeting moment: a point of view in time more than in space."
B. Next he wants to make the point that this new intention "was achieved by exact and mathematical means." This he illustrates with Albrecht Dürer's engraving of himself at work drawing a nude model with the help of a vertical sighting point in combination with grid of wires stationed between himself and the object, "to hold the instant of vision."
i. Note how Dürer draws our attention to the implication of the idea of a "choice of a moment" by depicting from one point of view himself engaged in depicting the model from another point of view:  the viewer is implicitly invited to imagine himself walking around the room to behold the woman from the point of view of the painter at work. This walk transpires in time, and the artist is therefore confronted with the task, as an artist (if he chooses to work in this mode), of picking one moment out of the continuum of possible moments in which it is possible to contemplate the subject of the painting.
ii. Note how Bronowski takes up this invitation by taking us on a "wheeling tour" of the scene, via a moving camera. See how this technique supports his thesis?
C. He then puts before us Dürer's painting The Adoration of the Magi.  His claim is that "[a]ll the natural details in which Dürer delights are expressions of the dynamic of time:  the ox and the ass, the blush of youth on the cheek of the Virgin."
D. He points out that the chalice at the center of this painting became the focus of a standard assignment in teaching perspective, and shows us the studies made by the artist Uccello. Using a computer to turn the image, he stresses that the artist's eye "worked like a turntable to follow and explore its shifting shape, the elongation of the circles into ellipses, and to catch the moment of time as a trace in space."
He now clinches the basic theme of the last part of this program:  "Analyzing the changing movement of an object, as I can do on the computer, was quite foreign to Greek and to Islamic minds. They looked always for what was unchanging and static, a timeless world of perfect order. The most perfect shape to them was the circle. Motion must run smoothly and uniformly in circles; that was the harmony of the spheres. ¶ This is why the Ptolemaic system was built up of circles, along which time ran uniformly and imperturbably."

E. Bronowski's climax is so tightly composed that it warrants being quoted here in its entirety, so that you'll have it for your considered reflection. As you review it, try to recall the images with which he counterpoints it. He continues:

     "But movements in the real world are not uniform. They change direction and speed at every instant, and they cannot be analyzed until a mathematics is invented in which time is a variable. That is a theoretical problem in the heavens, but it is practical and immediate on earth -- in the flight of a projectile, in the spurting growth of a plant, in the single splash of a drop of liquid that goes through abrupt changes of shape and direction. The Renaissance did not have the technical equipment to stop the picture frame instant by instant.  But the Renaissance had the intellectual equipment: the inner eye of the painter, and the logic of the mathematician.
     "In this way Johannes Kepler after the year 1600 became convinced that the motion of a planet is not circular and not uniform. It is an ellipse along which the planet runs at varying speeds.  That means that the old mathematics of static patterns will no longer suffice, nor the mathematics of uniform motion.  You need a new mathematics to define and operate with instantaneous motion.
     "The mathematics of instantaneous motion was invented by two superb minds of the late seventeenth century -- Isaac Newton and Gottfried Wilhelm Leibnitz.  It is now so familiar to us that we think of time as a natural element in a description of nature; but that was not always so.  It was they who brought in the idea of a tangent, the idea of acceleration, the idea of slope, the idea of infinitesimal, of differential.  There is a word that has been forgotten but that is really the best name for that flux of time that Newton stopped like a shutter:  Fluxions was Newton's name for what is usually called (after Leibniz) the differential calculus.  To think of it merely as a more advanced technique is to miss its real content.  In it, mathematics becomes a dynamic mode of thought, and that is a major mental step in the ascent of man.  The technical concept that makes it work is, oddly enough, the concept of an infinitesimal step; and the intellectual breakthrough came in giving a rigorous meaning to that.  But we may leave the technical concept to the professionals, and be content to call it the mathematics of change.
     "The laws of nature had always been made of numbers since Pythagoras said that was the language of nature.  But now the language of nature had to include numbers which described time.  The laws of nature become laws of motion, and nature herself becomes not a series of static frames but a moving process."

  Go to Study Guide for Part 6 of Bronowski's The Ascent of Man ("The Starry Messenger").  This program covers developments in astronomy during the 16th and 17th Centuries (touched upon above), culminating in the trial of Galileo Galilei before the Inquisition in Rome.

  Suggestions are welcome.  Please send your comments to lyman@ksu.edu .

      Contents copyright © 1997 by Lyman A. Baker

Permission is granted for non-commercial educational use (except for the final quotation from Bronowski); all other rights reserved.

  This page last updated 20 January 1999.

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