To Explain the World: The Discovery of Modern Science (7 page)

There was one scientific subject of practical importance to which Greek scientists returned again and again, even into the Roman period: the behavior of light. This concern dates to the beginning of the Hellenistic era, with the work of Euclid.

Little is known of the life of Euclid. He is believed to have lived in the time of Ptolemy I, and may have founded the study of mathematics at the Museum in Alexandria. His best-known work is the
Elements
,
⁷
which begins with a number of geometric definitions, axioms, and postulates, and moves on to more or less rigorous proofs of increasingly sophisticated theorems. But Euclid also wrote the
Optics
, which deals with perspective, and his name is associated with the
Catoptrics
, which studies reflection by mirrors, though modern historians do not believe that he was its author.

When one thinks of it, there is something peculiar about reflection. When you look at the reflection of a small object in a flat mirror, you see the image at a definite spot, not spread out over the mirror. Yet there are many paths one can draw from the object to various spots on the mirror and then to the eye.
*
Apparently there is just one path that is actually taken, so that the image appears at the one point where this path strikes the mirror. But what determines the location of that point on the mirror? In the
Catoptrics
there appears a fundamental principle that answers this question: the angles that a light ray makes with a flat mirror when it strikes the mirror and when it is reflected are equal. Only one light path can satisfy this condition.

We don’t know who in the Hellenistic era actually discovered this principle. We do know, though, that sometime around AD 60 Hero of Alexandria in his own
Catoptrics
gave a mathematical proof of the equal-angles rule, based on the assumption that the path taken by a light ray in going from the object to the mirror and then to the eye of the observer is the path of shortest length. (See
Technical Note 8
.) By way of justification for this principle, Hero was content to say only, “It is agreed that Nature does nothing in vain, nor exerts herself needlessly.”
⁸
Perhaps he was motivated by the teleology of Aristotle—everything happens for a purpose. But Hero was right; as we will see in
Chapter 14
, in the seventeenth century Huygens was able to deduce the principle of shortest distance (actually shortest time) from the wave nature of light. The same Hero who explored the fundamentals of optics used that knowledge to invent an instrument of practical surveying, the theodolite, and also explained the action of siphons and designed military catapults and a primitive steam engine.

The study of optics was carried further about AD 150 in
Alexandria by the great astronomer Claudius Ptolemy (no kin of the kings). His book
Optics
survives in a Latin translation of a lost Arabic version of the lost Greek original (or perhaps of a lost Syriac intermediary). In this book Ptolemy described measurements that verified the equal-angles rule of Euclid and Hero. He also applied this rule to reflection by curved mirrors, of the sort one finds today in amusement parks. He correctly understood that reflections in a curved mirror are just the same as if the mirror were a plane, tangent to the actual mirror at the point of reflection.

In the final book of
Optics
Ptolemy also studied refraction, the bending of light rays when they pass from one transparent medium such as air to another transparent medium such as water. He suspended a disk, marked with measures of angle around its edge, halfway in a vessel of water. By sighting a submerged object along a tube mounted on the disk, he could measure the angles that the incident and refracted rays make with the normal to the surface, a line perpendicular to the surface, with an accuracy ranging from a fraction of a degree to a few degrees.
⁹
As we will see in
Chapter 13
, the correct law relating these angles was worked out by Fermat in the seventeenth century by a simple extension of the principle that Hero had applied to reflection: in refraction, the path taken by a ray of light that goes from the object to the eye is not the shortest, but the one that takes the least time. The distinction between shortest distance and least time is irrelevant for reflection, where the reflected and incident ray are passing through the same medium, and distance is simply proportional to time; but it does matter for refraction, where the speed of light changes as the ray passes from one medium to another. This was not understood by Ptolemy; the correct law of refraction, known as Snell’s law (or in France, Descartes’ law), was not discovered experimentally until the early 1600s AD.

The most impressive of the scientist-technologists of the Hellenistic era (or perhaps any era) was Archimedes. Archimedes lived in the 200s BC in the Greek city of Syracuse in Sicily, but is believed to have made at least one visit to Alexandria. He is credited with inventing varieties of pulleys and screws, and various
instruments of war, such as a “claw,” based on his understanding of the lever, with which ships lying at anchor near shore could be seized and capsized. One invention used in agriculture for centuries was a large screw, by which water could be lifted from streams to irrigate fields. The story that Archimedes used curved mirrors that concentrated sunlight to defend Syracuse by setting Roman ships on fire is almost certainly a fable, but it illustrates his reputation for technological wizardry.

In
On the Equilibrium of Bodies
, Archimedes worked out the rule that governs balances: a bar with weights at both ends is in equilibrium if the distances from the fulcrum on which the bar rests to both ends are inversely proportional to the weights. For instance, a bar with five pounds at one end and one pound at the other end is in equilibrium if the distance from the fulcrum to the one-pound weight is five times larger than the distance from the fulcrum to the five-pound weight.

The greatest achievement of Archimedes in physics is contained in his book
On Floating Bodies.
¹⁰
Archimedes reasoned that if some part of a fluid was pressed down harder than another part by the weight of fluid or floating or submerged bodies above it, then the fluid would move until all parts were pressed down by the same weight. As he put it,

Let it be supposed that a fluid is of such a character that, the parts lying evenly and being continuous, that part which is thrust the less is driven along by that which is thrust the more; and that each of its parts is thrust by the fluid which is above it in a perpendicular direction if the fluid be sunk in anything and compressed by anything else.

From this Archimedes deduced that a floating body would sink to a level such that the weight of the water displaced would equal its own weight. (This is why the weight of a ship is called its “displacement.”) Also, a solid body that is too heavy to float and is immersed in the fluid, suspended by a cord from the arm of a balance, “will be lighter than its true weight by the weight
of the fluid displaced.” (See
Technical Note 9
.) The ratio of the true weight of a body and the decrease in its weight when it is suspended in water thus gives the body’s “specific gravity,” the ratio of its weight to the weight of the same volume of water. Each material has a characteristic specific gravity: for gold it is 19.32, for lead 11.34, and so on. This method, deduced from a systematic theoretical study of fluid statics, allowed Archimedes to tell whether a crown was made of pure gold or gold alloyed with cheaper metals. It is not clear that Archimedes ever put this method into practice, but it was used for centuries to judge the composition of objects.

Even more impressive were Archimedes’ achievements in mathematics. By a technique that anticipated the integral calculus, he was able to calculate the areas and volumes of various plane figures and solid bodies. For instance, the area of a circle is one-half the circumference times the radius (see
Technical Note 10
). Using geometric methods, he was able to show that what we call pi (Archimedes did not use this term), the ratio of the circumference of a circle to its diameter, is between 3
¹
/
₇
and 3
¹⁰
/
₇₁
. Cicero said that he had seen on the tombstone of Archimedes a cylinder circumscribed about a sphere, the surface of the sphere touching the sides and both bases of the cylinder, like a single tennis ball just fitting into a tin can. Apparently Archimedes was most proud of having proved that in this case the volume of the sphere is two-thirds the volume of the cylinder.

There is an anecdote about the death of Archimedes, related by the Roman historian Livy. Archimedes died in 212 BC during the sack of Syracuse by Roman soldiers under Marcus Claudius Marcellus. (Syracuse had been taken over by a pro-Carthaginian faction during the Second Punic War.) As Roman soldiers swarmed over Syracuse, Archimedes was supposedly found by the soldier who killed him, while he was working out a problem in geometry.

Aside from the incomparable Archimedes, the greatest Hellenistic mathematician was his younger contemporary Apollonius. Apollonius was born around 262 BC in Perga, a city on
the southeast coast of Asia Minor, then under the control of the rising kingdom of Pergamon, but he visited Alexandria in the times of both Ptolemy III and Ptolemy IV, who between them ruled from 247 to 203 BC. His great work was on conic sections: the ellipse, parabola, and hyperbola. These are curves that can be formed by a plane slicing through a cone at various angles. Much later, the theory of conic sections was crucially important to Kepler and Newton, but it found no physical applications in the ancient world.

Brilliant work, but with its emphasis on geometry, there were techniques missing from Greek mathematics that are essential in modern physical science. The Greeks never learned to write and manipulate algebraic formulas. Formulas like
E
=
mc
²
and
F
=
ma
are at the heart of modern physics. (Formulas were used in purely mathematical work by Diophantus, who flourished in Alexandria around AD 250, but the symbols in his equations were restricted to standing for whole or rational numbers, quite unlike the symbols in the formulas of physics.) Even where geometry is important, the modern physicist tends to derive what is needed by expressing geometric facts algebraically, using the techniques of analytic geometry invented in the seventeenth century by René Descartes and others, and described in
Chapter 13
. Perhaps because of the deserved prestige of Greek mathematics, the geometric style persisted until well into the scientific revolution of the seventeenth century. When Galileo in his 1623 book
The Assayer
wanted to sing the praises of mathematics, he spoke of geometry:
*
“Philosophy is written in this all-encompassing book that is constantly open to our eyes, that is the universe; but it cannot be understood unless one first learns to understand the language and knows the characters in which it is written. It is written in
mathematical language, and its characters are triangles, circles, and other geometrical figures; without these it is humanly impossible to understand a word of it, and one wanders in a dark labyrinth.” Galileo was somewhat behind the times in emphasizing geometry over algebra. His own writing uses some algebra, but is more geometric than that of some of his contemporaries, and far more geometric than what one finds today in physics journals.

In modern times a place has been made for pure science, science pursued for its own sake without regard to practical applications. In the ancient world, before scientists learned the necessity of verifying their theories, the technological applications of science had a special importance, for when one is going to use a scientific theory rather than just talk about it, there is a large premium on getting it right. If Archimedes by his measurements of specific gravity had identified a gilded lead crown as being made of solid gold, he would have become unpopular in Syracuse.

I don’t want to exaggerate the extent to which science-based technology was important in Hellenistic or Roman times. Many of the devices of Ctesibius and Hero seem to have been no more than toys, or theatrical props. Historians have speculated that in an economy based on slavery there was no demand for laborsaving devices, such as might have been developed from Hero’s toy steam engine. Military and civil engineering
were
important in the ancient world, and the kings in Alexandria supported the study of catapults and other artillery, perhaps at the Museum, but this work does not seem to have gained much from the science of the time.

The one area of Greek science that did have great practical value was also the one that was most highly developed. It was astronomy, to which we will turn in Part II.

There is a large exception to the remark above that the existence of practical applications of science provided a strong incentive to get the science right. It is the practice of medicine. Until modern times the most highly regarded physicians persisted in practices, like bleeding, whose value had never been established
experimentally, and that in fact did more harm than good. When in the nineteenth century the really useful technique of antisepsis was introduced, a technique for which there
was
a scientific basis, it was at first actively resisted by most physicians. Not until well into the twentieth century were clinical trials required before medicines could be approved for use. Physicians did learn early on to recognize various diseases, and for some they had effective remedies, such as Peruvian bark—which contains quinine—for malaria. They knew how to prepare analgesics, opiates, emetics, laxatives, soporifics, and poisons. But it is often remarked that until sometime around the beginning of the twentieth century the average sick person would do better avoiding the care of physicians.