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Greek Science after Aristotle
Michael Fowler UVa Physics
Strato
As we mentioned before, Aristotle’s analysis of motion was criticized
by Strato (who died around 268 B.C., he is sometimes called Straton),
known as “the Physicist” who was the third director of the Lyceum
after Aristotle (the founder) and Theophrastus, who was mainly a botanist.
Strato’s career was curiously parallel to Aristotle’s. Recall
Aristotle spent twenty years at Plato’s academy before going to Macedonia
to be tutor to Alexander, after which Aristotle came back to Athens to found
his own “university”, the Lyceum. A few years later, Alexander
conquered most of the known world, dividing it into regions with his old
friends in charge. In particular, he had his boyhood friend Ptolemy in charge
of Egypt, where Alexander founded the new city of Alexandria. Now Strato, after
a period of study at the Lyceum, was hired by Ptolemy to tutor his son Ptolemy
II Philadelphus (as he became known) in Alexandria. Subsequently Strato
returned to Athens where he was in charge of the Lyceum for almost twenty
years, until his death.
Strato, like Aristotle, believed in close observation of natural phenomena,
but in our particular field of interest here, the study of motion, he observed
much more carefully than Aristotle, and realized that falling bodies usually
accelerate. He made two important points: rainwater pouring off a corner of a
roof is clearly moving faster when it hits the ground than it was when it left
the roof, because a continuous stream can be seen to break into drops which
then become spread further apart as they fall towards the ground. His second
point was that if you drop something to the ground, it lands with a bigger thud
if you drop it from a greater height: compare, say, a three foot drop with a
one inch drop. One is forced to conclude that falling objects do not
usually reach some final speed in a very short time and then fall steadily,
which was Aristotle’s picture. Had this line of investigation been
pursued further at the Lyceum, we might have saved a thousand years or more,
but after Strato the Lyceum concentrated its efforts on literary criticism.
Aristarchus
Strato did, however, have one very famous pupil, Aristarchus of Samos
(310 - 230 B.C.). Aristarchus claimed that the earth rotated on its axis every
twenty-four hours and also went round the sun once a year, and that the other
planets all move in orbits around the sun. In other words, he anticipated
Copernicus in all essentials. In fact, Copernicus at first acknowledged
Aristarchus, but later didn’t mention him (see Penguin Dictionary of
Ancient History). Aristarchus’ claims were not generally accepted,
and in fact some thought he should be indicted on a charge of impiety for
suggesting that the earth, thought to be the fixed center of the universe, was
in motion (Bertrand Russell, quoting Plutarch about Cleanthes). The other
astronomers didn’t believe Aristarchus’ theory for different
reasons. It was known that the distance to the sun was in excess of one million
miles (Aristarchus himself estimated one and a half million miles, which is far
too low) and they thought that if the earth is going around in a circle that
big, the pattern of stars in the sky would vary noticeably throughout the year,
because the closer ones would appear to move to some extent against the
background of the ones further away. Aristarchus responded that they are all so
far away that a million miles or two difference in the point of observation is
negligible. This implied, though, the universe was really huge—at
least billions of miles across—which few were ready to believe.
Euclid
Although the Ptolemies were not exactly nice people, they did a great deal
of good for Greek civilization, especially the sciences and mathematics. In
their anxiety to prove how cultured and powerful they were, they had
constructed a massive museum and a library at Alexandria, a city which grew to
half a million people by 200 B.C. It was here that Erastosthenes (275 - 195
B.C.) was librarian, but somewhat earlier Euclid taught mathematics
there, about 295 B.C. during the reign of Ptolemy I. His great work is his Elements,
setting out all of Greek geometry as a logical development from basic axioms in
twelve volumes. This is certainly one of the greatest books ever written, but
not an easy read.
In fact, Ptolemy I, realizing that geometry was an important part of Greek
thought, suggested to Euclid that he would like to get up to speed in the
subject, but, being a king, could not put in a great deal of effort. Euclid responded: “There is no Royal Road to geometry.”
Euclid shared Plato’s contempt for the practical. When one of his
pupils asked what was in it for him to learn geometry, Euclid called a slave
and said “Give this young man fifty cents, since he must needs make a
gain out of what he learns.”
The Romans, who took over later on didn’t appreciate Euclid. There is
no record of a translation of the Elements into Latin until 480 A.D. But
the Arabs were more perceptive. A copy was given to the Caliph by the Byzantine
emperor in A.D. 760, and the first Latin translation that still survives was
actually made from the Arabic in Bath, England, in 1120. From that point on,
the study of geometry grew again in the West, thanks to the Arabs.
Plato, Aristotle and Christianity
It is interesting to note that it was in Alexandria that the first crucial
connection between classical Greek philosophy and Christian thought was made.
As we have just seen, Alexandria was a major center of Greek thought, and also
had a very large Jewish community, which had self-governing privileges. Many
Jews never returned to Palestine after the Babylonian captivity, but became
traders in the cities around the eastern Mediterranean, and Alexandria was a
center of this trade. Thus Alexandria was a melting-pot of ideas and
philosophies from these different sources. In particular, St. Clement (A.D.
150-215) and Origen were Greek Christians living in Alexandria who helped
develop Christian theology and incorporated many of the ideas of Plato and
Aristotle.
(Actually, this St. Clement was demoted from the Roman martyrology in the
ninth century for supposed hereticism (but Isaac Newton admired him!). There is a
St. Clement of Rome, who lived in the first century.) Recall that St. Paul himself was a Greek speaking
Jew, and his epistles were written in Greek to Greek cities, like Ephesus near Miletus, Phillipi and Thessalonica on the Aegean, and Corinth between Athens
and Sparta. After St. Paul, then, many of the early Christian fathers were
Greek, and it is hardly surprising that as the faith developed in Alexandria and elsewhere it included Greek ideas. This Greek influence had of course been
long forgotten in the middle ages. Consequently, when monks began to look at
the works of Plato and Aristotle at the dawn of the Renaissance, they were
amazed to find how these pre-Christian heathens had anticipated so many of the
ideas found in Christian theology. (A History of Science, W. C. Dampier,
end of Chapter 1.)
The most famous Alexandrian astronomer, Ptolemy, lived from about 100 AD to
170 AD. He is not to be confused with all the Ptolemies who were the
rulers! We will discuss Ptolemy later, in comparing his scheme for the solar
system with that of Copernicus.
There were two other great mathematicians of this period that we must
mention: Archimedes and Apollonius.
Archimedes
Archimedes, 287 - 212 B.C., lived at Syracuse in Sicily, but also studied in
Alexandria. He contributed many new results to mathematics, including
successfully computing areas and volumes of two and three dimensional figures
with techniques that amounted to calculus for the cases he studied. He
calculated pi by finding the perimeter of a sequence of regular polygons
inscribed and escribed about a circle.
Two of his major contributions to physics are his understanding of the
principle of buoyancy, and his analysis of the lever. He also invented many
ingenious technological devices, many for war, but also the Archimedean screw,
a pumping device for irrigation systems.
Archimedes’ Principle
We turn now to Syracuse, Sicily, 2200 years ago, with Archimedes and his
friend king Heiro. The following is quoted from Vitruvius, a Roman historian
writing just before the time of Christ:
Heiro, after gaining the royal power in Syracuse, resolved, as a
consequence of his successful exploits, to place in a certain temple a golden
crown which he had vowed to the immortal gods. He contracted for its making at
a fixed price and weighed out a precise amount of gold to the contractor. At
the appointed time the latter delivered to the king’s satisfaction an
exquisitely finished piece of handiwork, and it appeared that in weight the
crown corresponded precisely to what the gold had weighed.
But afterwards a charge was made that gold had been abstracted and an
equivalent weight of silver had been added in the manufacture of the crown.
Heiro, thinking it an outrage that he had been tricked, and yet not knowing how
to detect the theft, requested Archimedes to consider the matter. The latter,
while the case was still on his mind, happened to go to the bath, and on
getting into a tub observed that the more his body sank into it the more water
ran out over the tub. As this pointed out the way to explain the case in question,
without a moments delay and transported with joy, he jumped out of the tub and
rushed home naked, crying in a loud voice that he had found what he was
seeking; for as he ran he shouted repeatedly in Greek, “Eureka, Eureka.”
Taking this as the beginning of his discovery, it is said that he made two
masses of the same weight as the crown, one of gold and the other of silver.
After making them, he filled a large vessel with water to the very brim and
dropped the mass of silver into it. As much water ran out as was equal in bulk
to that of the silver sunk in the vessel. Then, taking out the mass, he poured
back the lost quantity of water, using a pint measure, until it was level with
the brim as it had been before. Thus he found the weight of silver corresponding
to a definite quantity of water.
After this experiment, he likewise dropped the mass of gold into the full
vessel and, on taking it out and measuring as before, found that not so much
water was lost, but a smaller quantity: namely, as much less as a mass of gold
lacks in bulk compared to a mass of silver of the same weight. Finally, filling
the vessel again and dropping the crown itself into the same quantity of water,
he found that more water ran over for the crown than for the mass of gold of
the same weight. Hence, reasoning from the fact that more water was lost in the
case of the crown than in that of the mass, he detected the mixing of silver
with the gold and made the theft of the contractor perfectly clear.
What is going on here is simply a measurement of the density—the mass
per unit volume—of silver, gold and the crown. To measure the masses some
kind of scale is used, note that at the beginning a precise amount of gold is
weighed out to the contractor. Of course, if you had a nice rectangular brick
of gold, and knew its weight, you wouldn’t need to mess with water to
determine its density, you could just figure out its volume by multiplying
together length, breadth and height, and divide the mass, or weight, by the
volume to find the density in, say, pounds per cubic foot or whatever units are
convenient. (Actually, the units most often used are the metric ones, grams per
cubic centimeter. These have the nice feature that water has a density of 1,
because that’s how the gram was defined. In these units, silver has a
density of 10.5, and gold of 19.3. To go from these units to pounds per cubic
foot, we would multiply by the weight in pounds of a cubic foot of water, which
is 62.)
The problem with just trying to find the density by figuring out the volume
of the crown is that it is a very complicated shape, and although one could no
doubt find its volume by measuring each tiny piece and calculating a lot of
small volumes which are then added together, it would take a long time and be
hard to be sure of the accuracy, whereas lowering the crown into a filled
bucket of water and measuring how much water overflows is obviously a pretty
simple procedure. (You do have to allow for the volume of the string!). Anyway,
the bottom line is that if the crown displaces more water than a block of gold
of the same weight, the crown isn’t pure gold.
Actually, there is one slightly surprising aspect of the story as recounted
above by Vitruvius. Note that they had a weighing scale available, and a bucket
suitable for immersing the crown. Given these, there was really no need to
measure the amount of water slopping over. All that was necessary was first, to
weigh the crown when it was fully immersed in the water, then, second, to dry
it off and weigh it out of the water. The difference in these two weighings is
just the buoyancy support force from the water. Archimedes’
Principle states that the buoyancy support force is exactly equal to
the weight of the water displaced by the crown, that is, it is equal to the
weight of a volume of water equal to the volume of the crown.
This is definitely a less messy procedure—there is no need to fill the
bucket to the brim in the first place, all that is necessary is to be sure that
the crown is fully immersed, and not resting on the bottom or caught on the
side of the bucket, during the weighing.
Of course, maybe Archimedes had not figured out his Principle when the king
began to worry about the crown, perhaps the above experiment led him to it.
There seems to be some confusion on this point of history.
Archimedes and Leverage
Although we know that leverage had been used to move heavy objects since
prehistoric times, it appears that Archimedes was the first person to
appreciate just how much weight could be shifted by one person using
appropriate leverage.
Archimedes illustrated the principle of the lever very graphically to his
friend the king, by declaring that if there were another world, and he could go
to it, he could move this one. To quote from Plutarch,
Heiro was astonished, and begged him to put his proposition into
execution, and show him some great weight moved by a slight force. Archimedes
therefore fixed upon a three-masted merchantman of the royal fleet, which had
been dragged ashore by the great labours of many men, and after putting on
board many passengers and the customary freight, he seated himself at some
distance from her, and without any great effort, but quietly setting in motion
a system of compound pulleys, drew her towards him smoothly and evenly, as
though she were gliding through the water.
Just in case you thought kings might have been different 2200 years ago,
read on:
Amazed at this, then, and comprehending the power of his art, the king
persuaded Archimedes to prepare for him offensive and defensive weapons to be
used in every kind of siege warfare.
This turned out to be a very smart move on the king’s part, since some
time later, in 215 B.C., the Romans attacked Syracuse. To quote from Plutarch’s
Life of Marcellus (the Roman general):
When, therefore, the Romans assaulted them by sea and land, the
Syracusans were stricken dumb with terror; they thought that nothing could
withstand so furious an onslaught by such forces. But Archimedes began to ply
his engines, and shot against the land forces of the assailants all sorts of
missiles and immense masses of stones, which came down with incredible din and
speed; nothing whatever could ward off their weight, but they knocked down in
heaps those who stood in their way, and threw their ranks into confusion. At
the same time huge beams were suddenly projected over the ships from the walls,
which sank some of them with great weights plunging down from on high; others
were seized at the prow by iron claws, or beaks like the beaks of cranes, drawn
straight up into the air, and then plunged stern foremost into the depths, or
were turned round and round by means of enginery within the city, and dashed
upon the steep cliffs that jutted out beneath the wall of the city, with great
destruction of the fighting men on board, who perished in the wrecks.
Frequently, too, a ship would be lifted out of the water into mid-air, whirled
hither and thither as it hung there, a dreadful spectacle, until its crew had
been thrown out and hurled in all directions, when it would fall empty upon the
walls, or slip away from the clutch that had held it... .
Then, in a council of war, it was decided to come up under the walls
while it was still night, if they could; for the ropes which Archimedes used in
his engines, since they imported great impetus to the missiles cast, would,
they thought, send them flying over their heads, but would be ineffective at
close quarters, since there was no space for the cast. Archimedes, however, as
it seemed, had long before prepared for such an emergency engines with a range
adapted to any interval and missiles of short flight, and, through many small
and contiguous openings in the wall, short-range engines called “scorpions”
could be brought to bear on objects close at hand without being seen by the
enemy.
When, therefore, the Romans came up under the walls, thinking themselves
unnoticed, once more they encountered a great storm of missiles; huge stones
came tumbling down upon them almost perpendicularly, and the wall shot out
arrows at them from every point; they therefore retired.... . At last, the
Romans became so fearful that, whenever they saw a bit of rope or a stick of
timber projecting a little over the wall, “There it is,” they
cried, “Archimedes is training some engine upon us,” and turned
their backs and fled. Seeing this, Marcellus desisted from all fighting and
assault, and thenceforth depended on a long siege.
It is sad to report that the long siege was successful and a Roman soldier
killed Archimedes as he was drawing geometric figures in the sand, in 212 B.C.
Marcellus had given orders that Archimedes was not to be killed, but somehow
the orders didn’t get through.
Apollonius
Apollonius probably did most of his work at Alexandria, and lived around 220
B.C., but his exact dates have been lost. He greatly extended the study of
conic sections, the ellipse, parabola and hyperbola.
As we shall find later in the course, the conic sections play a central role
in our understanding of everything from projectiles to planets, and both Galileo
and Newton, among many others, acknowledge the importance of Apollonius’
work. This is not, however, a geometry course, so we will not survey his
results here, but, following Galileo, rederive the few we need when we need
them.
Hypatia
The last really good astronomer and mathematician in Greek Alexandria was a
woman, Hypatia,
born in 370 AD the daughter of an astronomer and mathematician Theon, who
worked at the museum. She wrote a popularization of Apollonius’ work on
conics. She became enmeshed in politics, and, as a pagan who lectured on
neoplatonism to pagans, Jews and Christians (who by now had separate schools)
she was well known. In 412 Cyril became patriarch. He was a fanatical
Christian, and became hostile to Orestes, the Roman prefect of Egypt,
a former student and a friend of Hypatia. In March 415, Hypatia was killed by a
mob of fanatical Christian monks in particularly horrible fashion. The details
can be found in the book Hypatia’s Heritage (see below).
Books I used in preparing this lecture:
Greek Science after Aristotle, G. E. R. Lloyd, Norton, N.Y.,
1973
A Source Book in Greek Science, M. R. Cohen and I. E. Drabkin,
Harvard, 1966
Hypatia’s Heritage: A History of Women in Science, Margaret
Alic, The Women’s Press, London 1986
A History of Science, W. C. Dampier, Cambridge, 1929
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