Numbers and Counting
In the light of the achievements of the Greek geometers, we sometimes forget that the Greeks also devoted a great deal of study to numbers. This chapter illustrates their theoretical as well as their practical interest in numbers. The selection from Euclid has to do with number theory; the selection from Archimedes deals with the more mundane problem of how to name numbers.
We will begin with the brief selection from Euclid. His approach to arithmetic is very similar to his approach to geometry; in both sciences he begins with a long series of definitions which define both terms the reader is already familiar with and terms that are probably new to him. The early definitions, because they deal with the basic terms, present the greatest difficulty; see, for example, Euclid’s definitions of "unit" and "number."
The definitions are not followed by postulates or axioms. The absence of axioms can easily be explained: axioms are no different for arithmetic than for geometry. The lack of postulates is a different matter, however. It seems as though Euclid did not think that he needed to postulate anything here as he did in geometry. Yet this is clearly wrong. Just as there are geometrical constructions the possibility of which must be granted to Euclid in geometry, so there are a number of operations in arithmetic which must be granted to him if he is to prove anything here. For example, there should be a postulate which says: "Let it be granted that if a, b, and c are three numbers, then the sum of a and b added to c is the same as the number obtained by adding a to the sum of b and c." Or,"(a + b) + c = a + (b + c) ." There are several other, similar arithmetical postulates which are also omitted by Euclid. Euclid, who was so careful and precise in his formulation of the geometrical postulates, is apparently quite careless and happy-go-lucky here. In contrast to this, modern arithmetic and algebra pay much attention to the problem of tinding the right postulates.
Definition 11 defhres a prime number as one "which is measured by an unit alone." Another definition of a prime number is that it is not divisible by any number (except itself and unity). Examples of prime numbers are 2 (the only even prime number), 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, and so on. Even from these few examples it is obvious that prime numbers become more scarce as we count higher. Between 23 and 29 five not-prime numbers (composite numbers) intervene. These intervals become larger and larger; between 199 and 211, there are 11 composite numbers. This increasing rarity of prime numbers naturally leads to the question whether perhaps beyond a certain point in the number scale, there might be no more prime numbers at all. Is it possible that all numbers beyond a certain one (probably very large) are composite numbers? Or do prime numbers keep recurring, although less and less frequently?
The answer of this question is: The quantity of prime numbers is infinitely large. Euclid’s way of stating the proposition does not immediately reveal what he has in mind: "Prime numbers are more than any assigned multitude of prime numbers." This means the following: Suppose it is claimed that the number of prime numbers is finite, say equal to n. Then Euclid proves that there must be more than n prime numbers.
The last statement is a rather curious one. On the asumption that something is the case, namely, that there are just n prime numbers, the opposite is proved, namely, that there are more than n prime numbers. This oddity in the proof, together with the intrinsic interest in the statement of the proposition, constitutes the reason for our including this single proposition from the arithmetical books of Euclid’s Elements. The proof is also remarkable for the fact that it depends on nothing previously proved; it is an exercise in pure logic alone.
Instead of stating the proof in general terms, let US first exemplify it. Suppose that someone said: "The number of prime numbers is finite." We would then be justified in asking him: "How many prime numbers are there?’ His answer would have to be some number; let us assume that he answers: "There are just four prime numbers." Using Euclid’s method, we will now show that if there are four prime numbers, then there is at least another, a fifth prime number.
The four prime numbers claimed to be the only ones would have to be the fist four primes, of course; that is, they would have to be 2,3,5,7. Form the product of these four numbersnamely, 2 • 3 • 5 • 7 = 210. Add 1 to this product: 210 + 1 = 211. This new number is either a prime number or not. In this case, 211 is a prime number and, therefore, the proposition has been proved, for we have found a fifth prime number.
Suppose it had been claimed that there are just six prime numbers-namely, 2, 3, 5, 7, 11, 13. Form the product of these numbers. 2 • 3 • 5 • 7 • 11 • 13 = 30,030. Add 1 to this product: 30,030 + 1 = 30,031. Again we say that this number is either prime or not. In this case, it is a composite number and therefore divisible by some prime number. This prime number cannot be any of the original six, for if any of them is divided into 30,031, it leaves a remainder of 1. (This is the case because all of the original six prime numbers are divisible into 30,030.) Therefore, the proposition has again been proved, since a seventh prime number has been found. This seventh prime number is the one which is a factor of 30,031. In this example, the number would be 59, since 30,031 = 59 • 509. (509 is also prime, so that we have actually found not only a seventh but also an eighth prime number.)
Euclid’s proof is merely a generalization of this. If it is asserted that there are just n prime numbers, form the product of these n prime numbers. Add 1 to this product. This number -call it K-is itself either prime or not. If K is prime, the proposition has already been proved. If K is not a prime number, then it must be divisible by some prime number. This prime number is not one of the original n primes, for any of these n primes, if divided into K, leaves the remainder 1. Hence a new prime number has been found-namely, the one which is the factor of K.
What is the method of this proof? It somewhat resembles reduction to the absurd. We are to prove that the number of primes is larger than any given number, and so we begin by assuming the contradictory, namely that the number of primes is equal to a given number. But the conclusion which we come to is not in itself absurd; it merely contradicts the original assumption. From the assumption that there are just n prime numbers. we are able to demonstrate that there are. at least n + 1 prime numbers. We might call this .method "reduction to the opposite." Although this method is powerful, the number of instances where it can be. applied is small.
Now we turn our attention to Archimedes. There was probably no branch of mathematics known to him to which Archimedes did not make a valuable contribution. Living. in the third century B.C. (from aproximately 287 to 212 B.C.), Archimedes displayed a dazzling skill in geometry, in arithmetic, in the calculus, in the physics of the lever, and of floating bodies-a skill that was not matched until two thousand years later.
Archimedes lived in Syracuse in Sicily, though he had studied at Alexandria. The Sand -Reckoner is addressed to Gelon, the king of Syracuse; Archimedes was on friendly terms with both Gelon and his father, Hiero. On behalf of the kings of Syracuse, Archimedes constructed many clever mechanical devices, especially for repelling besieging armies. Archimedes attached little importance to these ingenious machines; he considered himself a mathematician and requested that on his tombstone there be displayed a sphere with a circumscribed cylinderthus commemorating what he considered to be his outstanding achievement, namely, the discovery of the relation of the volume of asphere and a cylinder.
Archimedes died when Syracuse was conquered 'by the Romans under the command of Marcehus in 212 B.C. Although Marcellus had given orders that Archimedes was not to be harmed, in the confusion of the battle Archimedes was slam. Marcellus was chagrined by the unfortunate event and gave Archimedes a decent burial. Much of our knowledge of Archimedes as a person stems from Plutarch’s Life of Marcellus. He is best seen, however, through his works, of which a great many have survived. The Sand-Reckoner, though it is a short work, displays his general scientific erudition as well as his skill as a mathematician.
All of us have at one time or. another encountered someone given to constant exaggeration. One of the most common exaggerations is the substitution of the word "infinite" for the phrase "very large." Many people say that something is "infinitely better than something else," or that "a modern ballistic missile is infinitely more complicated than the airplane of the brothers Wright," or that "the number of atoms in a given piece of matter is infinite." All of these expressions are not merely inaccurate, but wrong. Nothing on this earth is infinitely more complicated or infinitely better than anything else, and there is no number that is infinite. (Throughout this chapter, we use the word "number" to stand for "whole number" or "integer.") An infinite amount-leaving aside the question of whether or not there is such a thing-would mean an amount that cannot be counted, no matter how much time is taken to do it. An infinite quantity is not enumerable-it cannot be counted. And conversely, anything which can be counted-any quantity, no matter how large, to which a number can be assigned-is by that token not infinite. No number can ever be said to be infinite, for every number always has a next one; hence the former number cannot be called infinite, since there is at least one number greater than it. In fact, a good definition of intinity state; that infinity is larger than any number that you may name and that consequently, infinity itself is not a number.
King Gelon, to whom The Sand-Reckoner is addressed, was evidently a person foe. whom "very large" and "infinite" were synonymous, especially when "very large" means something of the order of ,millions or even more. One of the major tasks that Archimedes sets for himself in this little treatise is to show the king that "large"-no matter how large-is not infinite, but very definitely finite. Archimedes takes a quantity which seems to the uneducated to be so large as to be indistinguishable from infinity-the number of grains of sand in the universe - and counts it. At least, he shows that this quantity cannot possibly exceed a certain number which he names. And so, if the quantity can be numbered, it is not infinite.
In order to accomplish his purpose, Archimedes must first have some notion of the size of the universe. He must tell us what he means by "the universe," and how large he conceives it to be. He must also tell us how large i-e takes a grain of sand to be. Then Archimedes must find a way of naming very large numbers, so that he can tell us in a definite way the number of grains of sand in the universe. It will not do for him simply to say "it’s a very large number"; for nobody denies this. What is desired is a definite number to be assigned to the quantity of sand; this will show that the quantity is finite.
By "universe" Archimedes means the space enclosed by the sphere of the fixed stars. (In ancient astronomy, all fixed stars were thought to be attached or "fixed" to one celestial sphere.)
In defining what he means by "universe," Archimedes writes as follows (remember that the entire work is addressed to King Gelon):
Now you are aware that "universe" is the name given by most astronomers to the sphere whose centre is the centre of the earth and whose radius is equal to the straight line between the centre of the sun and the centre of the earth.
This view of the universe is based on the geocentric hypothesis: The earth is thought to be in the center of the universe, with sun, moon, planets, and the fixed stars all revolving around the earth. In this hypothesis, the fixed stars are usually considered to be farther out than any other heavenly body, but as Archimedes states the theory here, it appears that the sun is at the greatest distance from the earth.
Archimedes then reports that there is also another view of the universe:
Aristarchus of Samos brought out a book consisting of some hypotheses, in which the premises lead to the result that the universe is many times greater than that now so called. His hypotheses are that the fixed stars and the sun remain unmoved, that the earth revolves about the sun in the circumference of a circle, the sun lying in the middle of the orbit, and that the sphere of the fixed stars, situated about the same centre as the sun, is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the lixed stars as the centre of the sphere bears to its surface.
This is a heliocentric view: the sun is the center of the universe and the earth revolves around the sun. The fixed stars are truly fixed-that is, motionless-but appear to move because of the daily rotation of the earth. This is, of course, exactly the theory put forth by Copernicus some 1,700 years later. Aristarchus’ theory apparently could not hold its own against the rival geocentric theory and was not generally accepted. (We may surmise that the reason for Aristarchus’ failure lay in the apparent greater simplicity of the geocentric theory. In the course of time, however, the geocentric theory needed so many modifications and additions that, by the time of Copernicus, it was far more complicated than the rediscovered heliocentric theory.)
If the heliocentric theory is adopted, the fixed stars must be far more distant from the earth than they need be in the geocentric theory. Although the earth is sometimes closer to, and sometimes farther from, a given star (depending on where the earth is in the course of its annual revolution around the sun), the earth always seems to be exactly in the center of the universe. This can be the case only if the distance to the fixed stars is so great that in relation to it, the distance from the earth to the sun is so small as to be negligible. This is what Archimedes means when he writes that "the sphere of the fixed stars . . . is so great that the circle in which he supposes the earth to revolve bears such a proportion to the distance of the iixed stars as the centre of the sphere bears to its surface."
Now Archimedes begins to put down some hypothetical figures about the actual size of the universe. He is not so much concerned to give accurate figures for the astronomical distances as to be sure always to give a greater distance than anyone has proposed. In this way-if he succeeds in showing that the grains of sand in such a universe are enumerable-it will certainly be obvious that the quantity of sand in the actual universe, being smaller, must be also enumerable.
Archimedes begins by giving a value for the circumference of the earth. He assumes that it is- no larger than 3 million stadia. A stadium is a Greek unit of length; it was not everywhere the same length. (Just as "mile" can mean a statute mile or a nautical mile, and just as "gallon" designates a different volume in the United States and in Canada.) For our purposes we may say that a stadium is approximately 600 feet long. Consequently, as brief calculation will show, the figure of 3 million stadia is far too large jfor the circumference of the earth; in fact, 300,000 stadia, which, as Archimedes notes, some other astronomers proposed for the size of the earth, is much closer. But Archimedes is only interested in giving estimates that are not too small.
Further, Archimedes notes that the diameter of the sun is greater than the diameter of the earth, while the diameter of the earth is greater than that of the moon. In addition, Archimedes assumes that the diameter of the sun is about 30 times as great as the diameter of the moon, but not more than that. For this result he relies on experimental work by various astronomers; again, to be on the safe side he elects a value which makes the sun greater than any of the astronomers has found it to be.
So far all the assumptions have dealt with the diameters of three bodies: the earth, the sun, and the moon. Since Archimedes is interested in the size of the "universe," he must connect these diameters with the diameter or with the circumference of the universe. This he does in Assumption 4, in which he tells us that if a regular chiliagon (figure of a thousand sides) is inscribed in the "equator" of the universe, then the diameter of the sun is greater than the side of the chiliagon. Actually, Archimedes proves this statement _ by means of experimental evidence. Then he goes on:
the diameter of the sun is equal to or less than 30 diameters
of the moon
the diameter of the moon is less than the diameter of the
30 diameters of the moon are less than 30 diameters of
it follows that
the diameter of the sun is less than 30 diameters of the
Assumption 4 states that
the diameter of the sun is greater than the side of the
chiliagon inscribed in the universe.
1000 diameters of the sun are greater than 1000 sides
of the chiliagon
which means that
1000 diameters of the sun are greater than the circumference
of the chiliagon.
Turning this last inequality around, we have
the circumference of the chiliagon is less than 1000 diameters
of the sun
the circumference of the chiliagon is less than 30,000
diameters of the earth.
The circumference of a regular hexagon (six-sided figure) inscribed in a circle is three times the diameter of the circle. Any regular figure which has more than six sides has a circumference larger than that of the hexagon, but smaller than that of the circle. Consequently, the circumference of a regular chiliagon inscribed in the equator of the universe is greater than three times the diameter of the universe. Let us write this down:
The circumference of the chiliagon is greater than 3
diameters of the universe
or, turning this around,
3 diameters of the universe are less than the circumference
of the chiliagon.
Dividing by three, we have
the diameter of the universe is less than y3 of the circumference
of the chiliagon.
Reverting to the relation between the circumference of the chiliagon and the diameter of the earth, we have
the diameter of the universe is less than 1/3 of 30,000
diameters of the earth,
the diameter of the universe is less than 10,000 diameters
of the earth.
Since the circumference of the earth has been assumed to be at most 3 million stadia, the diameter of the earthsmust be less than 1 million stadia. (This is true because the diameter of a circle is multiplied by π, which is greater than 3, in order to obtain the circumference of a circle.)
the diameter of the earth is less than 1 million stadia,
it follows that
the diameter of the universe is less than 10,000 million
the diameter of the universe is less than 10 billion stadia.
Since, as we noted earlier, a stadium is about 600 feet or l/9 of a mile, the "universe" in this calculation turns out to have a diameter of about 1.1 billion miles. Imagine the vast quantity of sand, if this entire universe were filled with sand! Nevertheless, Archimedes proposes to tell us the number of grains of sand if this universe contained nothing but sand.
Let us simplify Archimedes' statements just a little. Let us say, for example, that he maintains that
1 stadium equals 10,000 fingerbreadths.
1 fingerbreadth equals 40 diameters of a poppy seed,
it follows that
1 stadium equals 400,000 diameters of a poppy seed.
Now the volumes of spheres are to each other as the cubes of their diameters. Hence we have
Volume of a sphere with the diameter of 1 stadium =
Volume of a sphere with the diameter of 1 poppy seed
(400,000)3/1 = (4 • 105)3/1 = 64 • 1015.
Since, according to Archimedes, a sphere with the diameter of one poppy seed contains 10,000 grams of sand it follows that a sphere with a diameter of 1 stadium contains 64 • 1015 • 10,000 grams of sand, or 64 • 1015 • 104 = 64 • 1019 grains of sand.
How many grains of sand are there in a sphere the size of the universe, or 10 billion stadia? Again we use the relation hetween volumes of spheres:
Volume of a sphere with a diameter of 1010 stadia =
Volume of a sphere with the diameter of 1 stadium
= (1010)3/1 = 1030.
Since the smaller sphere (with the diameter of 1 stadium) contains 64 • 1019 grains of sand, the larger sphere must contain 1030 times as many grains. Now 1019 • 1030 = 1049. Thus the number of grains of sand in the universe, using Archimedes’ assumptions, is 64 • 1049 (written as 64 followed by 49 zeros).
In making this calculation, we have employed the decimal system of numerical notation. This system is based on the powers of 10-10, 100, 1000, and so on. Each power of 10 gives its name to a whole series of numbers; there are units, tens, hundreds, thousands, and so on. However, we very quickly run out of names for the powers of ten, and in any case it becomes difficult to remember just what we mean, for example, by a quadrillion. For that reason, mathematicians do not even try to name very large numbers with. words. They merely write them as powers of ten. Thus 5 million is very often written as 5 • 106. For numbers larger than a million, this manner of notation is almost mandatory. The reader will have noticed that we employed this notation for the number of grains in the universe.
Let us now take a look at the system of naming numbers that Archimedes devised and see whether it is adequate to his purpose. That is, can numbers as large as 64 • 1049 (or even larger) be written in his notation? The Greeks, unlike us, had a single name for the number 10,000, namely "myriad." Thus they had distinct names up to the fourth power of ten, namely, ten, a hundred, a thousand, and a myriad. Apparently they. had no names for larger numbers; for instance, they had no name for a million. But given the’names they had, they could give distinct names, Archimedes notes, up to a myriad myriads. For example, there might be a number as follows:
4838 myriads, 659 thousands, 76 hundreds, 3 tens, 5.
This number means:
4838 • 10,000 plus 659 • 1000 plus 76 • 100 plus 3 • 10
plus 5 or
48,380,000 plus 659,000 plus 7600 plus 30 plus 5.
In the decimal system this number would be written as 49,046,635
Since a myriad myriads ( 100,000,000) is the last number that can be given a distinct name, Archimedes proposes that this number become the unit of a second group of numbers, which he calls numbers of the second order. (Numbers from 1 to 100,000,000 he calls numbers of the first order.) Numbers of the second order run from 100,000,000 to (100,000,000)2. This last number becomes the unit of numbers of the third order. In general, the numbers of the nth order are those beginning with ( 100,000,000)n - 1 and ending with ( 100,000,000)n. We can continue until we reach the 100,000,000th order of numbers, which, will end with the number (100,000,000)100,000,000. Archimedes calls this number P. In decimal notation, P would be written as (108)108 or 10(8 + 108).
Archimedes now calls the entire group of numbers from 1 to P the first period of numbers. Then he ,considers the number P as the unit of the first order of the second period. The first order of the second unit would go from P up to 100,000,000P. There is no need to describe the rest of the scheme, since Archimedes does it adequately. But what isof interest is this: although Archimedes at this point has barely begun to develop his scheme, we are already far past the number needed to express the number of grains of sand in the universe. As we saw, this number was approximately 64 • 1049, or less than 1052. Where does this number fail in Archimedes’ scheme?
The first order of numbers goes from 1 to 108 (1 to a myriad).
The second order of numbers goes from 108 to 1016 (a myriad to a myriad of myriads).
The third order of numbers goes from 1016 to 1024.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The seventh order of numbers goes from 1048 to 1056.
The number of grains of sand in the universe can be expressed, therefore, by a number of the seventh order. There is no need even to go to the-end of the first period of numbers!
To appreciate Archimedes :achievementr in developing such a scheme, remember .that in explaining it, we constantly had recourse to the decimal system We expressed all of Archimedes’ numbers in terms. of powers of ten: Archimedes, it must be remembered, did not possess the symbol '0’ for writing numbers. What seems easy to, us," therefore, required a tremendous effort of imgination and insight. Even without the symbol "0" Archimedes took the basic step in the writing of numbers: he uses each number that he can express as the unit for a new group of numbers: This is exactly what is done in the decimal system, or in any other system that writes its numbers by reference to the powers of some unit.