But after Alexander’s conquests, the anti-mechanical taboo was lifted. A new atmosphere prevailed in the Hellenistic or part-Greek monarchies of his successors. World trade, improved navigation and better farming methods were the order of the day, and mathematicians turned their attention to practical matters. They made water clocks, irrigation devices, cogs for launching ships and various sorts of tackles and gears.
Archimedes
The most illustrious mathematician of this later age was also the mechanical wizard of antiquity, Archimedes of Syracuse. Among his many inventions was the so-called ‘screw of Archimedes’, a device for pumping water from the holds of leaky ships or draining flooded mines.
So vast was his knowledge of leverage that he once said, ‘Give me a place to stand on, and I will move the earth!’
But Archimedes’ most fabulous accomplishments were in connection with the defence of Syracuse, where he was attached to the court of King Hiero II. That rich and beautiful city was a tempting prize to Marcellus, who sailed the Mediterranean as head of the Roman fleet. Time after time, he directed attacks on Syracuse, but each assault was repulsed by the ingenious machines of Archimedes.
On one occasion the Roman ships were burned by fireballs hurled from a catapult behind the city walls.
Another time, a large claw manoeuvred by levers and pulleys actually grabbed the prows of the Roman ships, raised them in the air and flung them back into the sea.
In another attack, a huge mirror concentrated the sun’s rays and focused them on the Roman fleet, setting ships on fire.
Naturally, at each suggestion of a return to Syracuse, the unhappy Roman sailors would cry, ‘Oh no! Not again!’ Marcellus himself used to call Archimedes ‘that geometrical hundred-armed giant’.
Archimedes’ many machines proved the value of science in war and peace even at that early date. Yet he did much more important and far-reaching work in pure mathematics. He and the other Alexandrian geometers were every bit as devoted to abstract thought as their predecessors. In fact, from the Golden Age through the Age of Plato and right into the Hellenistic Age, they were all absorbed in a set of fascinating abstract questions.
The three geometric puzzles
These were the three famous puzzles – constructions that had to be performed using only straight-edge and string.
One was called ‘squaring the circle’: how could you construct a square with the same area as a circle?
Another was ‘trisecting an angle’: how could you divide an angle into three equal parts?
Another was ‘doubling the cube’: how could you construct a cube whose volume would be double that of another cube?
About the origin of the cube puzzle, a curious story was told. It seems that a great plague ravaged Athens in 430 BC, and the citizens appealed to the oracle at Delos for help. The oracle replied that the plague would be stopped if the Athenians would double in size the altar of Apollo without changing its shape. The altar was a cube.
Historians do not think this tale is true. Rather, they believe, it was made up later to hide the fact that the ‘three geometric puzzles’ were really useless problems. But working on what may seem useless has frequently been the task of mathematicians, and such tasks, pursued with care, patience and persistence, have led to most useful results. A whole book could be written about useful results from useless problems!
In the case of the three geometric puzzles, they were not only useless but quite impossible with only those tools. These constructions simply cannot be made with string and straight-edge alone! But more than two thousand years elapsed before that was definitely proved.
So from the fifth to the third centuries BC, many geometers worked in vain on the three puzzles. And attempts to solve them led to the invention of new curves that broke the rules – they were made by a variety of mechanical or three-dimensional means.
Conic sections
Menaechmus himself, struggling with the problem of doubling the cube, dreamed up the idea of cutting a cone with a plane. The sections he cut were an epochal discovery, and later on Archimedes worked on them himself.
They were in the shape of three new curves, the ellipse, the parabola and the hyperbola – the Greek names of that age are still in use today and we still call them conic sections, because they can be cut from a cone.
If you cut a cone of your own, you can see these curves for yourself. Take an ordinary ice-cream cone – not a real one, as it would crumble if you cut it – but the shape of one. Point the vertex up; this is a right-angled circular cone. If you cut straight across it, parallel to the base, you will get a circle which is also a conic section. Slice clear across it at an angle, and you get an ellipse. Now, just slice off one side of the cone, cutting parallel to the slanting outside of the cone itself (the line of generation), and you get a parabola. Finally, slice straight down parallel to the vertical axis of the cone – and stand another cone upside down on top of the first one and slice it in the same way – and you get the two curves of a hyperbola.
Picture the sides of your cone extending endlessly, and you will see the open arms of the parabola and the hyperbola reaching out toward infinity. These three curves – the ellipse, the parabola and the hyperbola – are a perfect example of remarkable results from a mathematical puzzle.
The use of conic sections today
In the next century, the conic sections were studied by Archimedes, the famous Apollonius of Perga and others, mainly out of sheer interest in geometry.
But it took two thousand years for the conics to be recognised for what they were: the paths of motion for all bodies, celestial and earthly alike.
It wasn’t until the seventeenth century that the great astronomer Kepler discovered his famous law that the path of each planet around the sun is actually an ellipse. In the same century, Galileo proved that a cannon ball or any other missile shot into the air will travel a path that is a parabola. Less than a century later, Newton developed his universal laws of motion and his great law of gravitation. Without the Conica of Apollonius, which he knew thoroughly, it is unthinkable that Newton could have formulated these basic laws of modern astronomy and physics.
And today, twenty-three centuries after Archimedes studied them, you can see the conics everywhere, for they have countless uses in science and industry.
So the conic sections – found by Menaechmus from a ‘useless’ puzzle – have proved tremendously useful to later science and industry. And so have the great practical achievements of the Alexandrian geometers.
Archimedes’ work on levers and floating bodies was the beginning of the science of mechanics. And his method of determining curved areas and volumes was the forerunner of Newton’s calculus. In fact, both the theoretical and the applied mathematics of the Hellenistic Age prepared the way for the massive achievements of Sir Isaac Newton, who laid the foundations of the modern exact sciences. As Newton himself said, ‘If I have seen further it is by standing on the shoulders of giants.’
Uses of parabolas
Every time you throw a ball, the ball’s path traces a parabola. When a jet of water rises from a fountain, it describes a parabola as it falls back into the pool beneath. And this curve has characteristics that make it valuable for reflecting light and sound waves. That is why parabolic reflectors are used in searchlights and car headlights, in radar antennas, radio telescopes and satellite dishes.
Uses of hyperbolas
The path that the sun’s shadow of an object traces on the ground in the course of a year is a hyperbola, and its applications are very useful. This curve was used in Long Range Navigation in and after the Second World War, a radar-like system that enabled a pilot to set and hold a course in any weather. This is done by means of radio signals and a map with Loran ‘lines of position’, hyperbolic curves.
17. The Whole, Round Earth
Our story of ancient geometry reaches its finale at the start of the third century BC, for it was then that a famous geometer wrote down t
he whole theoretical subject in the best-selling mathematics text of all time, and soon after, another geometer performed the most spectacular practical feat. He used a shadow to measure not just a pyramid, but the whole round earth!
These two events took place in the new Greek capital of the land of Egypt. Founded by Alexander the Great and named after him, Alexandria had become the leading metropolis of the ancient world. By now, it was a flourishing royal city, a beehive of commerce and the most important seaport on the whole Mediterranean. And Alexandria was also the world centre for ideas.
This sumptuous cosmopolitan city was the gathering place for the best scholars and scientists of the age. Savants from many lands made their discoveries in the ‘Museum’ – a graduate school that carried out studies in literature, medicine, astronomy and mathematics. The accumulated learning of the past was stored in the great annexed Library, with nearly a million books on scrolls.
Eratosthenes and the size of the earth
The librarian was a Greek named Eratosthenes. A universal mind, he was a mathematician, a specialist in history, an astronomer and a poet besides. And around 250 BC he did something almost incredible in those times. Eratosthenes measured accurately the girth of the planet he lived on!
Strange as it seems, he had a practical purpose in mind. As a great geographer, he understood that the earth was round and he was mapping the known parts of it. On his map of the world, Eratosthenes put all the data and distances he could get. The project was typical of that era, when the Mediterranean was becoming ‘one world’ for the first time. It was one world of scientists; astronomers, mathematicians and geographers in many lands were pooling their knowledge. And it was one world of trade, of ships and sailors, who needed maps. To make his map more accurate and useful, Eratosthenes wanted to determine the width of a degree of latitude. But for that, he had to know the circumference of the earth. How was he to measure it?
The inspiration came to him one day as he was travelling up the Nile on a summer study trip. He noticed with excitement that on the longest day of the year the noonday sun shone straight down a well at Syene, a town about 5000 stadia up the river from Alexandria. He could see the shape of the sun reflected on the surface of the water at the bottom of the well. But from there northward to Alexandria where he lived, the sun never got directly overhead.
So Syene was on the Tropic of Cancer! To a geographer that was most important and he explored the region in order to draw the Tropic on his map.
But the sight of the sun in the well fired his imagination even more. Just that single observation, plus his knowledge of geometry and his own active brain, told him how to determine the distance around the earth. He did it by means of a shadow and some remarkably shrewd deductions. Eratosthenes simply took the known distance between Syene and Alexandria, due north – as reported by camel caravans and professional ‘step-counters’ – and then measured a single angle at the right place and the right time.
He made his historic measurement at Alexandria, at noontime on the longest day of the year. At that moment, he knew, the sun was shining straight down the well at Syene, 5000 stadia away, casting no shadow. But at Alexandria, where he stood, an upright post was casting a shadow.
So Eratosthenes stretched a string from the top of the post to the tip of its shadow. Then he measured the angle between the post (at its top) and the string. The string represented the sun’s ray that was casting the shadow. He had measured the angle at which the post met the sun’s ray: It was a 7° 12’ angle.
Now look at the picture on page 142, where we have cut away a piece of the earth and you can see the brilliant deduction that Eratosthenes made from this angle.
He assumed an imaginary line from the sun rays at Syene continuing straight down through the well to the centre of the earth, and an imaginary line from the post at Alexandria continuing down to the centre of the earth, too. Of course, he assumed the sun’s rays were parallel. So at the centre of the earth, the post line met the parallel ray line at the same angle that the post met the ray at Alexandria. (When a transversal cuts two parallel lines, the alternate interior angles are equal.) So Eratosthenes made the alternate angle at the centre of the earth 7° 12’. The rest was easy.
This angle goes into 360° just 50 times. (There are 60 minutes to the degree.) Then, since the length of the arc made by 7° 12’ was about 5000 stadia, the distance around the whole earth must be 50 times that, or about 250,000 stadia.
In Egypt there were about 10 stadia to a mile, so his measurement was around 25,000 miles – very close to the actual circumference of the earth, as it was measured in later centuries.
Eratosthenes’ estimate was the most accurate in ancient times, and the climactic feat of the ancient practical art of geometry, or earth measurement. Yet his contemporaries thought of him as a ‘second-string man’, for around the same time and in the same city of Alexandria, there lived another geometer whose name is more widely known than any mathematician’s in history.
In the above illustration you can see, as Eratosthenes did, that the string- and post-lines at Alexandria formed the same angle as the alternate angle formed by the post- and sunray-lines at the centre of the earth. Both represent angles of 7° 12’. With this fact and simple arithmetic Eratosthenes was able to calculate the approximate circumference of the earth.
Euclid’s Elements
The other man was. of course Euclid. His masterpiece, the Elements, became famous in his own lifetime and nowadays it is still just as famous. For more than two thousand years, ever since he wrote it, students have been studying elementary geometry from his great work. The chances are that it is the basis of your geometry book today.
The incidents of Euclid’s life are unknown. But we can infer the traits of painstaking accuracy, imagination, dogged determination and above all logical thinking, that led him to assemble and organise everything that had been accomplished in geometry up to his time.
Of course, there had been previous attempts by other writers. But it was he who finally arranged the whole subject in the complete and orderly outline that was desperately needed.
Earlier Greeks had struggled to find, in simple geometric forms, some basic rules of line and construction and relationship for a clear and accurate description of nature. Afterward, proofs and ideas grew like mushrooms, with more and more theorems established and problems solved. By this time, there was a pressing need for one summary that could be accepted and used by Greek geometers and their students from Asia Minor to Sicily.
Euclid of Alexandria wrote that texbook, but it was far more than a mere textbook. The Elements was also a work of art. Out of the pieces of a mathematical jigsaw puzzle, he created a clear and beautiful picture. He traced geometric facts, logically and clearly, from the very first principles that Thales had found. He built these facts, one upon the next, into a truly magnificent edifice. We have spoken often of mathematics in the arts. But the Elements shows us the reverse, art in a work of mathematics.
The Elements told the history of an age. For in making his compilation, Euclid had carefully collected, put together and reported the work of the preceding centuries. His book contained most of the important discoveries of the Greek masters of the classical period. Here was an indispensable collection of beautiful and useful definitions and theorems. But the arrangement surpassed even the content.
From start to finish, Euclid’s Elements was held together by rigorous deductive reasoning. From a few wisely chosen axioms (agreed-upon common notions), he went on, step by step, to the most advanced proofs. Perhaps no other human creation has shown how so much knowledge can be derived from reasoning alone. So through the ages, his book has been used as a training in logical thought, not just for mathematicians, but for philosophers, theologians, logicians, lawyers and political leaders. Seekers of truth in all fields have studied and imitated its form and procedure.
But these are just highlights of Euclid’s unique accomplishment. By writing the Elements, by putt
ing everything together is lasting logical form, he transmitted intact to posterity almost the whole of Greek geometry.
With Euclid’s immortal Elements we close our tale of geometry in the ancient world. That story has brought us all the way from primitive hunters to scholars at the Alexandria University.
We have seen ancient people, using the early practical art, gradually learn to tell time and direction, to lay out their fields and dig irrigation ditches, to design and decorate their dwellings and temples and tombs, to record the movements of the sun, moon and planets. We have watched the merchant Thales working out the first abstract rules, and the mystical Pythagorean Order developing these rules, studying shapes and trying to link geometry with numbers. Finally, we have seen the later Greek geometers influencing philosophy and art of the Golden Age, and laying the foundations of Hellenistic and modern science. From start to finish, it was a great adventure of the human mind. And all of it, from the Stone Age to Eratosthenes and Euclid was achieved with just the string, the straight-edge and the shadow.
Copyright
Illustrated by Corydon Bell
First published in the United States in 1965 by Viking Press, Inc.
Republished in 2012 by Jamie York Press, USA
This edition first published in 2018 by Floris Books
String, Straightedge, and Shadow the Story of Geometry Page 9