P12_2.3.1 - Archimedes
Archimedes The Sand Reckoner
##############################################################
##############################################################
NON_PUBLIC
UNDER CONSTRUCTION 2013
##############################################################
##############################################################
Archimedes of Syracuse
Archimedes of Syracuse (287-212 B.C.) is often regarded as the greatest mathematician of antiquity. However during the time that he was alive he was probably best known for his work in physics, especially for its military applications. He purportedly designed machines capable of lifting ships out of the water and another that could set them ablaze using mirrors! If that isn't a testament to his ingenuity and genius I don't know what is. Besides this he also made advances in Physics. He developed the foundations of hydrostatics, and discovered the principle of the lever. In mathematics Archimedes was able to compute the area under parabolic curves using a summation of an infinite series. These were methods that would later form the foundation of Calculus, being applied more than a thousand years before its time! Archimedes was also able to derive many of the formulae for the surface areas and volumes of common 3-dimensional shapes, and his methods led to improved approximations of pi. All that being said perhaps his greatest discovery in mathematics, advocated as such by Archimedes himself, was that the volume of a sphere was 2/3 the volume of its circumscribing cylinder. If you need proof of just how significant this discovery was to Archimedes himself, consider that he requested that his tomb be surmounted by a sphere inscribed in a cylinder. If that was not already enough, Archimedes was also the large number champion of his day, inventing an ingenious number system that allowed him to transcend the known universe in scale and majesty... Multitudes beyond Number
It was a common bit of folk wisdom at the time of Archimedes that the grains of sand over all the earth were "infinite"; by which it was meant that the multitude was beyond "counting". References to this mindset can be found in the bible. For example, in genesis God promises Abraham:
"I shall surely multiply your seed like the stars of the heavens and like the grains of sand that are on the seashore" - (Genesis 22:17)
Given that ones descendents are potentially infinite, does this not suggest that to promise a multitude like the stars of the heavens or grains of sand on the seashore is essentially to promise descendents beyond number? Surely this passage mainly serves to overwhelm Abraham with the prospects of such a promise, as well as the reader. In this sense, "stars of the heavens" and "grains of sand that are on the seashore" are merely hyperbole. It is clear that this is merely God's way of telling Abraham its a very large number. In a later passage in Genesis Jacob comments upon the promise made to Abraham, saying:
" And you, you have said, ‘Unquestionably I shall deal well with you and I will constitute your seed like the grains of sand of the sea, which cannot be numbered for multitude.’ " - (Genesis 32:12)
There is some ambiguity in the phrase "cannot be numbered for multitude". To a modern reader it might sound like Jacob is saying there is no finite number to represent the multitude. This is how a modern mathematician would understand "cannot be numbered". But I don't think this is what Jacob meant, because I can't believe that people in antiquity really thought the sands of the sea were infinite; they were just too large to encompass. To say something cannot be numbered may simply have meant they couldn't literally be counted, one by one, with an exact value arrived at. By that reckoning however the smallest number which cannot be "numbered" would not be very large, perhaps only the count of a sizable pile of sand, and the claim that the sand of the sea cannot be counted would seem to be quite an understatement! The most likely meaning is that it's size is "inestimable"; that is, too large even to be estimated! Now that sounds much more impressive.
This may seem a strange claim for Jacob to make, but we of coarse make that judgement in hindsight. Recall that in antiquity, most notational systems for numbers did not extend beyond a million. The Hebrew system of numeration used most of its alphabet to write numbers up to 999, and could be extended into the thousands with some potential ambiguity; for example "aleph", the first letter of the hebrew alphabet could mean both 1 and 1000. The largest exact number described in the bible is the "myriad myriad", where a myriad is 10,000. Thus a myriad myriad, would be a hundred million. It is possible that people thought that no notational system could encompass numbers like "the stars of the heavens", or the "grains of sands in the sea", or at least no notational system conceivable by man, and this is what it meant to say something could not be numbered for multitude.
Archimedes however was able to show that this assumption is mathematically naive. In fact, he showed that not only can the grains of sand of the sea be "numbered", but man could imagine numbers so vast that everything known to exist would shrink before the minds eye to an infinitesimal dot!
Numbers to Rival The Dreams of Kings
Little is known about the surviving letter in which Archimedes explored the very large and what spurred its creation. It is commonly known as "The Sand Reckoner" and is a classic of the large number field, representing one of the earliest known expositions on large numbers for their own sake. It was written to King Gelon II of Syracuse, presumably because the King had especially requested it.
Archimedes had recently wrote a more extensive work, now lost to time, in which he described an extensive numeration system. Perhaps Gelon, aware of this, had come to wonder how close Archimedes numbers were to the number of grains of sand over all the earth, and requested a response from Archimedes himself. If this was so then Archimedes went well beyond Gelon's request and instead got to the heart of the matter: he showed there was nothing in the known world so large it could not find measure in the mind of man!
The opening paragraph of the treatise really speaks for itself:
"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 magnitude. And it is clear that they who hold this view, if they imagined a mass made up of sand in other respects as large as the mass of the Earth, including in it all the seas and the hollows of the Earth filled up to a height equal to that of the highest of the mountains, would be many times further still from recognizing that any number could be expressed which exceeded the multitude of the sand so taken. "But I will try to show you by means of geometrical proofs, which you will be able to follow, that, of the numbers named by me and given in the work which I sent to Zeuxippus, some exceed not only the number of the mass of sand equal in magnitude to the Earth filled up in the way described, but also that of the mass equal in magnitude to the universe. "
- Archimedes, The Sand Reckoner
The treatise goes on to show that even if the entire universe (as understood at the time) were filled up completely with sand, the count of grains still could not exceed 1064. Along the way the treatise explains Archimedes system of numeration and how to construct numbers MUCH MUCH larger than this! Thus Archimedes proved to his own satisfaction that there was no magnitude in the world that he could not exceed. Unfortunately we can never know what Gelon's reaction was, but it probably came as a completely counter-intuitive surprise. Mathematics, and man's imagination were much more powerful than anyone at the time had assumed!
Greek Numeration
The Greek numeration system was a somewhat awkward and difficult notational system. It made use of the entire Greek alphabet, each symbol representing a different value. The table below displays each of the symbols used, and their default value[1]:
The order of the letters in greek was the same as the order of ascending value. Thus the first 9 symbols were used for the ones position, the next 9 for the tens position, and the last 9 for the hundreds position. Thus 27 symbols were used to notate every number from 1 to 999. This is actually not a very efficient method of numeration, although it has certain advantages over roman numerals as the expressions don't get prohibitively long.
To write a number you would simply write symbols whose sum was the number in question. For example to write "564" you would write:
φξδ
The symbols were always written in descending order from left to right, same as with our decimals. In order to denote the thousands place, a ones place symbol would have an apostrophe added to the left. For example "3285" would be:
'γσπε
In this manner, one could provide a written form for every counting number up to 9999. 10,000 was known as the myriad, which means uncountable in greek[2]. This name is particularly poignant for our story since Archimedes was going to use the "uncountable" as the base unit of his system. To notate a myriad, a capital m was used:
M
One could continue by adding any number from 1 to 9999 to the myriad. For example:
M φξδ
would be 1564. To specify a multiple of a myriad any number could be raised next to M. Some examples:
Mδ 'γσπε = 43,285
Mξδ φξδ = 640,564
Mχα 'δσξε = 6,014,265
Using this system one could get up to the "myriad myriad" without too much trouble. This would be written as:
MM = 100,000,000
This is probably as far as the system was ever carried, although the logic can certainly be used to extend to the "myriad myriad myriad", the "myriad myriad myriad myriad", etc. simply by stacking Ms. Clearly this would get cumbersome pretty quickly. Could any stack of Ms exceed the number of grains of sand over all the earth? Not only was Archimedes able to show that it would only take a stack of 16 Ms to fill the entire universe with sand, but he also devised an improved system for expressing numbers of this size and much much greater.
The Greek Universe
"The Sand Reckoner" is conveniently broken up into four chapters[3][4]. In the first two chapters Archimedes begins by making generous assumptions about the scale of the universe. He weighs various scientific theories of the time about the scale of the universe, and then goes with the largest such model to avoid any shadow of doubt that he is generating an upper bound for the number of grains of sand that could fit into the volume of the known universe.
The Greeks were a people remarkably ahead of their time. Historians have treated them as the foundation for so called "western civilization". The Greeks founded both mathematics and science as we understand it today. That is, they were the first to devise mathematical proofs, and the first to base scientific models on accumulated observations. Being a curious and questioning people, it should come as no surprise that even the greeks did not agree amongst themselves precisely what the dimensions and arrangement of the universe was.
The consensus at the time was that the earth was the center of the universe, that the sun, moon, and planets revolved around it, and that this was all surrounded by a spherical surface on which the stars were imprinted, referred to by Archimedes as the "sphere of the fixed stars". This probably comes as no surprise, as this is all in line with the typical historical narrative. What is generally not appreciated is that not only did the greeks know that the earth was spherical, but one greek, Aristarchus[5], had proposed a model in which the earth and planets orbited the sun; this more than a thousand years before Copernicus!
Archimedes begins his treatise by first weighing the common theory and that of Aristarchus. Ultimately he goes with Aristarchus model for two reasons. Firstly, Aristarchus' universe was much much larger, and secondly Aristarchus had provided a clue for estimating the radius of the "sphere of fixed stars". Archimedes says that Aristarchus had suggested that the ratio of the sphere of stars to the orbit of the earth, was the same as the orbit of the earth to radius of the earth itself!
Armed only with this suggestion, Archimedes then begins to make an estimate of the size of the sphere of the universe. He begins by intentionally using an overestimate for the size of the earth, stating that the perimeter must be less than 300,000 stadia. The stadia was essentially the greek mile. It was equal to 600 feet, but since a foot was a measure that varied from state to state their is some ambiguity in this measure. Since the differences in the length of a foot were generally small, I will assume a "foot" to be the same as the modern measurement for convenience. Thus we have:
PerimeterEarth < 300,000 stadia * 600ft/stadia = 180,000,000 ft
Checking up against modern measurements, the diameter of the earth is estimated at 12,756 km. Let's assume the earths diameter is less than 12,760 km. Multiplying this by 3.142 as an overestimate for π we get a perimeter of about 40,092 km when we round up the result to the nearest integer. Assuming a km to be less than 0.63 miles, we can say that the circumference of the earth must be less than 25,258 mi when rounded up. A mile is equal to 5,280 feet. Thus we can say:
PerimeterEarth ≈ 133,362,240 ft < 180,000,000 ft
Thus Archimedes provides a remarkably accurate upper bound on the size of the earth. But Archimedes isn't taking any chances, and he goes instead with a figure 10 times this! Thus he says the earth MUST have a perimeter less than 3,000,000 stadia, which would be 1,800,000,000 ft. Archimedes goes on to say that he will assume the moon is smaller than the earth, and that the sun is larger. Again going through various theories about how many times larger the sun is than the moon, he goes with the largest accepted value and rounds it up to 30.
Since the moon is smaller than the earth it follows that if "30" is an overestimate then the sun must be smaller than 30 times the size of the earth. Checking against modern measurements however it turns out that the sun is in fact more like 400 times the size of the moon. So the greeks were unfortunately off by an order of magnitude, probably due to poor equipment. Interestingly however, by overestimating the earth by an order of magnitude the errors cancel and Archimedes model sun ends up being larger than the actual one.
Archimedes goes on to prove that the diameter of the sun is larger than the length of the side of the "chiliagon" inscribed with the sphere of fixed stars. Since the it's larger, it follows that by using a chiliagon whose side length is the diameter of the sun, the circumscribing sphere of such a chiliagon would be larger than the actual sphere of the fixed stars.
Works Cited
[1] http://www.math.wichita.edu/history/topics/num-sys.html#greek
[2] http://physics.weber.edu/carroll/archimedes/sand.htm
[3] http://mrob.com/pub/math/sandreckoner.html
[4] http://www.calstatela.edu/faculty/hmendel/Ancient%20Mathematics/Archimedes/SandReckoner/Ch.1/Ch1.html