English 233: Introduction to Western
Humanities - Baroque and Enlightenment
to Jacob Bronowski's
The Ascent of Man:
"The Music of the
"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.
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
- 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.)
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
- 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
ii. If set-squares (the
construction tool) had been known
since at least 2000 BC, what was
special about Pythagoras'
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
- 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
Mohammed conquers Mecca.
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.)
Toledo (in Spain) plays a central
role in the culture of larger Europe.
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,
- 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
- 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
- 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
- (1) What is the answer he
(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
- 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
- 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
- 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
- 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
- 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
- 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
- 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:
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
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 email@example.com .
Contents copyright © 1997 by Lyman A.
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.