***
Lewis Carroll may be entertaining, but he really does not get us to large numbers. Eighteen million liters is not such an overwhelming number. Even if we measured it as three and a half billion teaspoons we can deal with these numbers. There are about a billion grains of sand in a liter, so this beach may contain a few quadrillion grains of sand, and even that number we can handle. But thinking about the number of grains of sand has often been associated with a vast or even uncountable number. For example, in the Iliad, Iris in the voice of Polites warns the Trojans about the Greek army, “… I have been in many a battle, but never yet saw such a host as is now advancing. They are crossing the plain to attack the city as thick as leaves or as the sands of the sea …” (Book II). Also, in the Bible, sand and stars are often seen as being uncountable, as in “… so many as the stars of the sky in multitude, and as the sand which is by the sea shore innumerable …” (Hebrews 11:12).
“There are some, King Gelon, who think that the number of sand is infinite in multitude.” wrote Archimedes (287–212 BC) at the beginning of his essay The Sand Reckoner. I have always wondered what it was that prompted this treatise. I like to imagine that Archimedes and King Gelon, who apparently was interested in mathematics, were walking along a beach near Syracuse where they both lived. King Gelon may have said something as flippant as “… as countless or innumerable as the stars above or grains of sand on the beach …” and that this got Archimedes thinking “are they really countless?” I expect that Archimedes immediately recognized that the number of grains of sand was not infinite, but in some sense they were innumerable, because the number of grains of sand exceed a myriad-myriad, the largest number the Greeks could write in their number system.
Archimedes tells us in The Sand Reckoner that he has in fact devised a number system that could enumerate every grain of sand and he has already sent it off to Zeuxippus. But he also wanted to demonstrate this new system by calculating something really big, like the number of grains of sand on the beach.
There are some, King Gelon, who think that the number of the sand is infinite in multitude; and I mean by the sand not only that which exists about Syracuse and the rest of Sicily but also that which is found in every region whether inhabited or uninhabited. Again there are some who, without regarding it as infinite, yet think that no number has been named which is great enough to exceed its multitude.
So Archimedes sets himself a task, to calculate the amount of sand not only on all beaches, but if the whole universe was filled with sand. In fact he ups the challenge a bit; he did not want anyone in the future to say his number was too small, so at every stage he would overestimate the size of things to make sure his number was truly massive.
The reason we are going to spend a whole chapter with Archimedes is firstly he has an intriguing number system, and secondly because the way he estimates the size of the universe, even if it is really too small, parallels how modern astronomers work. But first, I will describe the Greek number system. When you learn a second language, it often leads you to better understand the structure of your first language; the same is true with number systems.
***
If you saw MDCLXXXVII you would immediately recognize it as a Roman number, and translate it into 1687, the year Isaac Newton published Principia. You can do that because you know the translation table (see Table 6.1).
The system we normally use is called the Hindu–Arabic numeral system or a positional number system with Arabic numerals. This system is a positionally based system, which the Roman system is not—or almost not. If you saw the first digit of Principia’s publication date, ‘M’, you would understand a value of 1000. In contrast, if you saw the first digit of 1687—the ‘1’—by itself you would think it meant “one.” It is its position as the fourth digit as “1 ---” that tells you that the meaning is “one thousand.” The Greek number system is more like the Roman, but with a lot more symbols. In fact the Greek system needed twenty-seven symbols and so used all the characters of their alphabet, plus others.
Table 6.1 The symbols and values of numerals in the Roman number system.
In the Greek system, if I wanted to write out Principia’s publication date I would write ͵αχπζʹ (see Table 6.2). The little mark in front of alpha is called a low keraia and is needed because we ran out of letters. α is 1, whereas ͵α is a thousand. The Greeks also needed a way to distinguish a word from a number and so they marked numbers with a ʹ: a keraia. For example ροζ is a word meaning “pink” (try pronouncing rho-omicron-zeta, and you are speaking Greek), whereas ροζʹ is the number 177. Actually most of the time in classical Greece they would mark a number with an over-line instead of a keraia. There are also three Greek letters rarely seen, (diagamma), (koppa) and (sampi). These letters had been dropped from use in words by the time of classical Greek, but were still needed to make the number system work.
Table 6.2 The symbols, values and names of numerals in the Greek number system.
With this system you can write any number from αʹ (1) to ͵ϑϑʹ (9999). The next larger number in this system has its own name, a myriad. A myriad specifically meant 10,000. Athens in the fifth century had about 40,000 citizens and probably a population of a few hundred thousand, and so a need to record larger numbers. For these purposes the Athenians would mark the digits above 9999 with a μ (mu) for myriad. So, for example in the number 12,345,678 they would first group as 1234,5678 and then write as:
With this system they could count up to 99,999,999 or a myriad-myriad, which was as high as most Greeks needed. But that number was just the starting point for Archimedes. However, before we delve into how Archimedes proposed to extend this number system, we will look at his astronomy.
***
To measure the universe, either as a part of modern astronomy or cosmology, or to help Archimedes fill it with sand, we must start with the shape and size of the Earth. Once we know that, we can estimate the distance to the Moon, Sun and stars.
One of the myths we were taught as children was that Columbus had to fight the flat-earthers of his time and the fear that if you sailed into the far western seas you would eventually reach the edge of the Earth and fall off. Apparently the story of the fear of the edge of the Earth among Columbus’s crew was primarily a construction of the storyteller Washington Irving. It makes a good tale but has little basis in fact: at the time of Columbus, most Europeans who thought about these things understood that the Earth was spherical.
Primitive societies, however, often started with a flat-Earth cosmology. When you look at the sea, large lakes, or great grassy plains or steppes they do look flat. Also, things fall off of globes but not table tops. A lot of people will tell you that they can see the curvature of the horizon while standing at the seaside, but that really is a tiny aberration, a barely measurable effect that is hardly noticeable unless you are certain it is there. A more dramatic effect is when you are sailing towards an island and you can see mountain tops while the shores of the island are still hidden below the horizon. Sailors speak of a ship being “hull-down,” which means that it is so far away that you may be able to see the mast, but the hull is below the horizon. Still, one could argue that this is an optical illusion, like a mirage caused by heat or humidity.
Pythagoras (c. 572–c. 495 BC) may have been among the first to recognize that the Earth was a sphere, and within a few centuries of his discovery most Greek astronomers had come to the same conclusion. Aristotle tells us that the most convincing piece of evidence to support the idea is that the shadow of the Earth that falls on the Moon during a lunar eclipse is round. In addition, a lunar eclipse will appear later in the night to observers in the east then in the west. Since the eclipse is an event that is observed by everyone simultaneously, sunsets, sunrises and noon must happen earlier in the east then the west. Finally, the southern constellations drop below the horizon as you travel north. Most Greek astronomers found the evidence when put together overwhelming.
Once
the shape of the Earth was established, the size of the Earth was a natural next question.
***
The most famous measurement of the Earth’s circumference, and the one Archimedes cited, was made by Eratosthenes of Cyrene (c. 276– c. 195 BC). Eratosthenes was the third head librarian of the great library of Alexandria. Before its burning, this library was probably the greatest repository of knowledge in the ancient world. Contemporaries of Eratosthenes referred to him as “Beta” because he was not the top intellect in any one field, but he was near the top in many fields. His original writings have been lost to us, but we do know that they are based on him considering the city of Syene, which we now call Aswan, to be on the Tropic of Cancer. It is not clear exactly how he knew this, but Aratus, who described Eratosthenes’ works, tells us that we will recognize the Tropic of Cancer because on the first day of summer, at noon, a gnome of a sundial, or any vertical column, will not have a shadow (see Figure 6.1). The Sun will be directly overhead in Syene (Aswan) while at the same time a gnome in Alexandria will have a shadow of just over 7°, or a fiftieth of a circle. Eratosthenes concluded that the distance between Syene and Alexandria was of the circumference of the Earth. The trek between the two cities had been measured at 5000 stade and so the circumference of the Earth, he concluded, must be 250,000 stade.
Figure 6.1 Eratosthenes’ method for measuring the size of the Earth. This involved comparing shadows in Alexandria and Syene (Aswan), and knowing the distance between.
A book that has been lost for over a thousand years leaves a lot of questions. Which stade did Eratosthenes use? And how did he decide that Syene was on the Tropic of Cancer? Legends have grown up about Eratosthenes actually visiting Syene and looking down into a deep well on the first day of summer and seeing the reflection of the Sun. In fact on the island of Elephantine in the Nile river is a well now called Eratosthenes’ well, a deep cool moist haven in the hot dry Egyptian desert. But we know of no ancient connection between Eratosthenes and this well.
A lot of people have argued about what that 5,000 stade from Alexandria to Syene means. As was mentioned in Chapter 1, there were many different stades in use in the ancient world. Was it the Olympic stade (176 m), the Roman stade (185 m), the itinerary stade (157 m) or some other unit? The 5000 stade number itself is probably pretty good. At the time there were professional “pacers” who measured long distances. But he may have also just asked the opinion of caravans who had trekked the distance. Most people who study this problem use the itinerary stade to convert Eratosthenes’ 250,000 stade into 39,000 km, which is just a bit less than modern measurements.
By modern measurements, Alexandria is at latitude 31°12′ N and Aswan is at 24°5′ N, so the difference is just over 7°, and a bit more than a fiftieth of a circle. The distance about 840 km, although not all of that is north–south. Dividing the 840 km by the 5000 stade Eratosthenes reported would suggest a stade of 168 meters, something in the middle of those cited above. I actually think the number 5000 tells us that Eratosthenes understood that this was not an exact measurement. It was rounded off and only a rough attempt to describe the size of the Earth.
Eratosthenes’ measurement is within about 10% of our modern one, but more important for this chapter, it forms the basis for measuring Archimedes’ cosmos. However, Archimedes wanted a large universe, so he rounded 250,000 up to 300,000 and then multiplied by 10 to make sure he did not underestimate.
***
The next step is to measure the distance to the Moon, and the Greeks did a good job on this too. However, measuring the distance to the Moon is a bit different than the distance around the Earth. You cannot pace off even part of the way.
When I was young I remembered seeing a chart in my classroom that showed the heights of different types of clouds. Cumulus were at less than 2000 m whereas cirrus were at a much dizzier height of 7000 m. I also learned that people knew this before airplanes or even hot air balloons and was left with the question of how they had measured these distances. Years later, on a beautiful summer day, I was sitting on a hilltop watching clouds when the secret of how to measure their heights came to me. First, I could figure out how big the clouds were by looking at their shadows: that big puffy one that reminds me of the face of a moose cast a shadow on that far hill, which goes from that big oak tree to the Holstein grazing in the western pasture. If that cow would remain still I could run over to the hill and pace off the distance between the tree and Holstein, and so measure the size of the cloud. Once I know its size, by measuring a few angles and using trigonometry I would be able to calculate its altitude, thickness and any other dimension. An earthbound observer could in fact measure the height of this non-terrestrial body.
In a similar way Aristarchus of Somos (c. 310–c. 230 BC) measured the size and then the distance to the Moon. He started out by watching a lunar eclipse and noted at what time the Moon started to enter the shadow of the Earth, at what time it was completely in the shadow, and at what time it emerged from the other side (see Figure 6.2). He reasoned that since the Sun was very far away, the distance across the shadow of the Earth is about the diameter of the Earth. Then the ratio of the time to enter the shadow to the time to cross the shadow was about equal to the ratio of the diameter of the Moon to the diameter of the Earth. From that he deduced that the diameter of the Moon was about a third that of the Earth.
Figure 6.2 Aristarchus’s method for measuring the size of the Moon.
Curiously enough, when Aristarchus proposed this technique he could not report the distance to the Moon in stade because Aristarchus preceded Eratosthenes and so he did not know the size of the Earth. Of course Archimedes followed both of them and so had access to both Aristarchus’s technique and Eratosthenes’ baseline: the diameter of the Earth.
Just now I introduced the term “baseline.” We are going to see a lot of baselines in this and later chapters as we survey space. So it is worth taking time now to try and understand what a baseline is. Some of the techniques used to map space are the same as the techniques used to map the surface of the Earth, especially when trying to measure the distance to a place you cannot touch. Let us imagine that you would like to measure the distance across the Grand Canyon. You are not going to stretch a tape measure from rim to rim. Also, hiking down the Bright Angel trail on one side and up the north Kaibab trail on the other side, with its switchbacks and serpentine route, is not going to tell you the straight-line distance. However, you can survey the distance, using geometry and trigonometry.
You can stand on Yaki point, near the park’s visitor center on the south rim and note that due north, on the other rim, is Widforss point (see Figure 6.3). Now you can hike west on the relatively straight and level trail for about 5.5 km to the Bright Angel trail’s south-rim trail-head. From there you can again sight Widforss point and see that it is now north-by-northeast of you, or 23° from north. So, if you sketch this out you have a triangle with Yaki point, the Bright Angel trailhead and Widforss point. Since you know two angles, 90° at Yaki Point and 67° at the trailhead, the last angle at Widforss must be 23°. You know everything about the shape of the triangle, but you do not know the size until you remembered the 5.5 km you just hiked. That one distance sets the scale of the whole triangle. This one distance that you measured directly is the baseline of the survey, and the distance across the Grand Canyon, or at least from Yaki to Widforss Point, can be determined at about 13 km.
The baseline to measure the Earth was the distance from Alexandria to Syene. The baseline to measure the distance to the Moon was the diameter of the Earth. The baseline to measure the distance to the Sun is the distance to the Moon. All measurements of great distance build upon previous, smaller measurements, something which is true even today when we measure the distances to galaxies.
Figure 6.3 Surveying the distance to unreachable points with angles and baselines.
Figure 6.4 Aristarchus’s method for measuring the size of and distance to the Sun.
The me
asurement of the distance to the Sun is hard, but again it was Aristarchus who provided us with a method. His technique was to look at the Moon and note the moment at which exactly half the Moon is illuminated (see Figure 6.4). He reasoned that this was the moment that the Earth–Moon line was at right angles to the Moon–Sun line, so the Earth–Moon–Sun angle was 90°. At that same moment he could measure the Moon–Earth–Sun angle as 87°. So he had a triangle, he knew two angles and a baseline, and so he could calculate the distance to the Sun.
Mathematically, this was a beautiful solution. Experimentally, it was very hard to carry out. First and foremost it is difficult to determine exactly when the Moon is half full. Aristarchus picked his moment, measured his angle, and found that the distance to the Sun was 190 times the diameter of the Earth. But he was a few hours too early. Ideally he would have measured 89.5° but as the angle approaches 90° a few minutes makes a huge difference. At 90° the mathematical distance is infinite.
In fairness to Aristarchus I have oversimplified his method. He recognized that because the Sun was not infinitely far away, the shadow of the Earth would taper and be smaller than the Earth when it reached the Moon’s orbit. To correct for this he combined the two measurements, and added one more observable, the angle size of the Moon and Sun. Aristarchus underestimated the size of the Sun by a factor of six, and the distance to the Sun by a factor of about fifteen.
How Big is Big and How Small is Small Page 10