Archimedes - Numbers and Counting
We have already mentioned that Euclid’s Elements contains arithmetical as well as geometrical material. In the present selection, we present two short excerpts from the three number books (Books VII-IX). First, there are all the definitions that Euclid puts down at the beginning of Book VII; and secondly, there is one very important proposition from Book IX. Of course, the number books contain a great deal more than this, but much of it is very ordinary arithmetical stuff and some of it is also not of much interest to us any more. But the two excerpts which we have selected retain their validity and their utility.
In addition to Euclid, we also make the acquaintance here of Archimedes, with the little treatise called The Sand-Reckoner. 'Archimedes wrote many treatises, and quite a few of them have come down to us. Many deal with problems of physics; others deal with pure mathematics. The Sand-Reckoner attacks a fairly simple problem, but one that is nevertheless important: how to count up to large numbers, and how to name large numbers in a consistent fashion.
Elements of Geometry
1. An unit is that by virtue of which each of the things that exist is called one.
2. A number is a multitude composed of units.
3. A number is a part of a number, the less of the greater, when it measures the greater;
4. but parts when it does not measure it.
5. The greater number is a multiple of the less when it is measured by the less.
6. An even number is that which is divisible into two equal parts.
7. An odd number is that which is not divisible into two equal parts, or that which differs by an unit from an even number.
8. An even-times even number is that which is measured by an even number according to an even number.
9. An even-times odd number is that which is measured by an even number according to an odd number.
10. An odd-times odd number is that which is measured by an odd number according to an odd number.
11. A prime number is that which is measured by an unit alone.
12. Numbers prime to one another are those which are measured by an unit alone as a common measure.
13. A composite number is that which is measured by some number.
14. Numbers composite to one another are those which are measured by some number as a common measure.
15. A number is said to multiply a number when that which is multiplied is added to itself as many times as there are units in the other, and thus some number is produced.
16. And, when two numbers having multiplied one another make some number, the number so produced is called plane, and its sides are the numbers which have multiplied one another.
17. And, when three numbers having multiplied one another make some number, the number so produced is solid, and its sides are the numbers which have multiplied one another.
18. A square number is equal multiplied by equal, or a number which is contained by two equal numbers.
19. And a cube is equal multiplied by equal and again by equal, or a number which is contained by three equal numbers.
20. Numbers are proportional when the fist is the same multiple, or the same part, or the same parts, of the second that the third is of the fourth.
21. Similar plane and solid numbers are those which have their sides proportional.
22. A perfect number is that which is equal to its own parts. . .
Prime numbers are more than any assigned multitude of prime numbers.
Let A, B, C be the assigned prime numbers; I say that there are more prime numbers than A, B, C.
For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE.
Then EF is either prime or not.
First, let it be prime; then the prime numbers A, B, C, EF have been found which are more than A, B, C.
Next, let EF not be prime; therefore it is measured by some prime number.
Let it be measured by the prime number G.
I say that G is not the same with any of the numbers A, B, C.
For, if possible, let it be so.
Now A, B, C measure DE; therefore G also will measure DE.
But it also measures EF.
Therefore G, being a number, will measure the remainder, the unit DF: which is absurd.
Therefore G is not the same with any one of the numbers A, B, C.
And by hypothesis it is prime.
Therefore the prime numbers A, B, C, G have been found which are more than the assigned multitude of A, B, C. Q.E.D.
Archimedes: The Sand Reckoner
"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. 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 f2led up to a height equal to that of the highest of the mountains, would be many times further still from recognising 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 a mass equal in magnitude to the universe. 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 is the common account as you have heard from astronomers. But 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 fixed stars as the centre of the sphere bears to its surface. Now it is easy to see that this is impossible; for, since the centre of the sphere has no magnitude, we cannot conceive it to bear any ratio whatever to the surface of the sphere. We must however take Aristarchus to mean this: since we conceive the earth to be, as it were, the centre of the universe, the ratio which the earth bears to what we describe as the 'universe’ is the same as the ratio which the sphere containing the circle in which he supposes the earth to revolve bears to the sphere of the fixed stars. For he adapts the proofs of his results to a hypothesis of this kind, and in particular he appears to suppose the magnitude of the sphere in which he represents the earth as moving tc be equal to what we call the 'universe.’
I say then that, even if a sphere were made up of the sand, as great as Aristarchus supposes the sphere of the fixed stars to be, I shall still prove that, of the numbers named in the Principles, some exceed in multitude the number of the sand which is equal in magnitude to the sphere referred to, provided that the following assumptions be made.
1. The perimeter of the earth is about 3,000,OOO stadia and not greater.
It is true that some have tried, as you are of course aware, to prove that the said perimeter is about 300,000 stadia. But I go further and, putting the magnitude of the earth at ten times the size that my predecessors thought it, I suppose its perimeter to be about 3,000,OOO stadia and not greater.
2. The diameter of the earth is greater than the diameter of the moon, and the diameter of the sun is greater than the diameter of the earth.
In this assumption I follow most of the earlier astronomers.
3. The diameter of the sun is about 30 times the diameter of the moon and not greater.
It is true that, of the earlier astronomers, Eudoxus declared it to be about nine times as great, and Pheidias my father twelve times, while Aristarchus tried to prove that the diameter of the sun is greater than 18 times but less than 20 times the diameter of the moon. But I go even further than Aristarchus in order that the truth of my proposition may be established beyond dispute, and I suppose the diameter of the sun to be about 30 times that of the moon and not greater.
4. The diameter of the sun is greater than the side of the chiliagon inscribed in the greatest circle in the (sphere of the) universe.
I make this assumption because Aristarchus discovered that the sun appeared to be about &th part of the circle of the zodiac, and I myself tried, by a method which I will now describe, to find experimentally the angle subtended by the sun and having its vertex at the eye.”
[Up to this point the treatise has been literally translated because of the historical interest attaching to the ipsissima verba of Archimedes on such a subject. The rest of the work can now be more freely reproduced, and, before proceeding to the mathematical contents of it, it is only necessary to remark that Archimedes next describes how he arrived at a higher and a lower limit for the angle subtended by the sun. This he did by taking a long rod or ruler, fastening on the end of it a small cylinder or disc, pointing the rod in the direction of the sun just after its rising (so that it was possible to look directly at it), then putting the cylinder at such a distance that it just concealed, and just failed to conceal, the sun, and lastly measuring the angles subtended by the cylinder. He explains also the correction which he thought it necessary to make because "the eye does not see from one point but from a certain area."]
The result of the experiment was to show that the angle subtended by the diameter of the sun was less than 1/164th part, and greater than &th part, of a right angle.
To prove that (on this assumption) the diameter of the sun is greater than the side of a chiliagon, or figure with 1000 equal sides, inscribed in a great circle of the 'universe.’
Suppose the plane of the paper to be the plane passing through the centre of the sun, the centre of the earth and the eye, at the time when the sun has just risen above the horizon. Let the plane cut the earth in the circle EHL and the sun in the circle FKG, the centres of the earth and sun being C, O respectively, and E being the position of the eye.
Further, let the plane cut the sphere of the 'universe’ (i.e. the sphere whose centre is C and radius CO) in the great circle AOB.
Draw from E two tangents to the circle FKG touching it at P, Q, and from C draw two other tangents to the same circle touching it in F, G respectively.
Let CO meet the sections of the earth and sun in H, K respectively; and let CF, CG produced meet the great circle AOB in A, B.
Join EO, OF, OG, OP, OQ, AB, and let AB meet CO in M.
Now CO > EO, since the sun is just above the horizon.
Therefore PEQ > FCG.
And PEQ>R/200,but PEQ <R/164, where R represents a right angle.
Thus FCG <R/164, a fortiori, and the chord AB subtends an arc of the great circle which is less than 1/656th of the circumference of that circle, i.e. AB < (side of 656-sided polygon inscribed in the circle).
Now the perimeter of any polygon inscribed in the great circle is less than 44CO/7. [Cf. Measurement of a circle, Prop. 3.1]
Therefore AB : CO < 11 : 1148,
and, a fortiori, AB < CO/100. . . . . . . . . . . .(a).
Again, since CA = CO, and AM is perpendicular to CO, while OF is perpendicular to CA, AM = OF.
Therefore AB = 2AM = (diameter of sun).
Thus (diameter of sun) < CO/100, by (a),
and, a fortiori
(diameter of earth) < CO/100. [Assumption 2]
Hence CH + OK<CO/100,
so that HK >99CO/100,
or CO : HK<100 : 99.
And CO > CF,
while HK < EQ.
Therefore CF : EQ < 100 : 99 . . . . . . . . . . . . (β).
Now in the right-angled triangles CFO, EQO, of the sides about the right angles,
OF = OQ, but EQ < CF (since EO < CO).
Therefore OEQ : OCF > CO : EO,
but OEQ : OCF < CF : EQ.
Doubling the angles,
OEQ : OCF < CF : EQ
and OEQ : OCF < 100 : 99, by (β) above.
But PEQ >R/200, by hypothesis.
Therefore ACB >99R/20000,
It follows that the arc AB is greater than 1/812th of the circumference of the great circle AOB.
Hence, a fortiori, AB > (side of chiliagon inscribed in great circle), and AB is equal to the diameter of the sun, as proved above.
The following results can now be proved:(diameter of 'universe’) < 10,000 (diameter of earth), and (diameter of 'universe’) < 10,000,000,000 stadia.
(1) Suppose, for brevity, that du, represents the diameter of the 'universe,’ ds, that the sun, de, that of the earth, and dm, that of the moon.
By hypothesis, ds 30dm, [Assumption 3]
and de > dm [Assumption 2]
therefore ds < 30de.
Now, by the last proposition,
ds > (side of chiliagon inscribed in great circle),
so that (perimeter of chiliagon) < 1000ds,
(perimeter of chiliagon) < 30,000de.
But the perimeter of any regular polygon with more sides than 6 inscribed in a circle is greater than that of the inscribed regular hexagon, and therefore greater than three times the diameter. Hence (perimeter of chiliagon) > 3 du.
It follows that du, < 10,000de.
(2) (Perimeter of earth) 3,000,000 stadia. [Assumption 1]
and (perimeter of earth) > 3de.
Therefore de, < 1,000,000 stadia,
whence du, < 10,000,000,000 stadia.
Suppose a quantity of sand taken not greater than a poppyseed, and suppose that it contains not more than 10,000 grains.
Next suppose the diameter of the poppy-seed to be not less than 1/40th of a finger-breadth.
Orders and periods of numbers.
I. We have traditional names for numbers up to a myriad (10,000) ; we can therefore express numbers up to a myriad myriads (100,000,000). Let these numbers be called numbers of the first order.
Suppose the 100,000,000 to be the unit of the second order, and let the second order consist of the numbers from that unit up to (100,000,000)2.
Let this again be the unit of the third order of numbers ending with (100,000,000)3; and so on, until we reach the 100,000,000th order of numbers ending with (100,000,000)100,000,000, which we will call P.
II. Suppose the numbers from 1 to P just described to form the first period.
Let P be the unit of the first order of the second period, and let this consist of the numbers from P up to 100,000,000 P.
Let the last number be the unit of the second order of the second period, and let this end with (100,000,000)2 P.
We can go on in this way till we reach the 100,000,000th order of the second period ending with ( 100,000,000)100,000,000 P, or P2.
III. Taking P2 as the unit of the first order of the third period, we proceed in the same way till we reach the 100,000,000th order of the third period ending with P3.
IV. Taking P3 as the unit of the first order of the fourth period, we continue the same process until we arrive at the 100,000,000th order of the 100,000,000th period ending with P100,000,000. This last number is expressed by Archimedes as "a myriad-myriad units of the myriad-myriad-th order of the myriad-myriad-th period, which is easily seen to be 100,000,000 times the product of ( 100,000,000)99,999,999 and P99,999,999, i.e. P100,000,000.
[The scheme of numbers thus described can be exhibited more clearly by means of indices as follows.
The prodigious extent of this scheme will be appreciated when it is considered that the last number in the first period would be represented now by 1 followed by 800,000,000 ciphers, while the last number of the (108) th period would require 100,000,000 times as many ciphers, i.e. 80,000 million millions of ciphers.]
Consider the series of terms in continued proportion of which the tirst is 1 and the second 10 [i.e. the geometrical progression 1, 101, 102, 103, . . . ]. The first octad of these terms [i.e. 1, 101, 102,. . . 107] fall accordingly under the first order of the first period above described, the second octad [i.e. 108, 109, . . . 1015] under the second order of the first period, the first term of the octad being the unit of the corresponding order in each case. Similarly for the third octad, and so on. We can, in the same way, place any number of octads.
If there be any number of terms of a series in continued proportion, say A1, A2, A3, . . . Am, . . . An, . . . Am + n - 1. . . of A1 = 1, A2 = 10 [so that the series forms the geometrical progression 1, 101, 102, . . . 10m - 1, . . . 10n - 1, . . .10m + n - 2, ...] and if any two terms as Am, An be taken and multiplied, the product Am.An will be a term in the same series and will be as many terms distant from An as Am is distant from A1; also it will be distant from A1 by a number of terms less by one than the sum of the numbers of terms by which Am and An respectively are distant from A1.
Take the term which is distant from An by the same number of terms as Am is distant from A1 . This number of terms is m (the first and last being both counted). Thus the term to be taken is m terms distant from An and is therefore the term Am + n - 1.
We have therefore to prove that
Am.An = Am + n - 1
Now terms equally distant from other terms in the continued proportion are proportional.
Thus Am/A1 = Am + n - 1/An .
But Am = Am.A1 , since A1 = 1.
Therefore Am + n - 1 = Am.An .................. (1).
The second result is now obvious, since Am is m terms distant from A1, An is n terms distant from A1 , and Am + n - 1 is (m + n - 1) terms distant from A1.
Application to the number of the sand.
By Assumption 5,
(diam. of poppy-seed) 1/40 (finger-breadth);
and, since spheres are to one another in the triplicate ratio of their diameters, it follows that sphere of diam. 1 tiger-breadth 64,000 poppy-seeds
and sphere of diam. 1 tiger-breadth 64,000 X 10,000 (1)
and sphere of diam. 1 tiger-breadth 640,000,000 (2)
and sphere of diam. 1 tiger-breadth 6 units of second order + 40,000,000 units of first order (3)
(a fortiori) < 10 units of second order of numbers. (4)
(1), (2), (3), (4) ⇒ grains of sand.
We now gradually increase the diameter of the supposed sphere, multiplying it by 100 each time. Thus, remembering that the sphere is thereby multiplied by 1003 or 100,000,000, the number of grains of sand which would be contained in a sphere with each successive diameter may be arrived at as follows.
But by the proposition above,
(diameter of 'universe’) < 10,000,000,000 stadia.
Hence the number of grains of sand which could be contained in a sphere of the size of our 'universe’ is less than 1,000 units of the seventh order of numbers [or 1051].
From this we can prove further that a sphere of the size attributed by Aristarchus to the sphere of the fixed stars would contain a number of grains of sand less than 10,000,000 units of the eighth order of numbers [or 1056 + 7 = 1063].
For, by hypothesis,
(earth) : ('universe’) = ('universe’) : (sphere of fixed stars).
(diameter of 'universe’) < 10,000(diam. of earth);
(diam. of sphere of fixed stars) < 10,000(diam. of 'universe’).
(sphere of fixed stars) < ( 10,000)3.('universe’).
It follows that the number of grains of sand which would be contained in a sphere equal to the sphere of the fixed stars:
<(10,000)3 X 1,000 units of seventh order
< (13th term of series) X (52nd term of series)
< 64th term of series [i.e. 1063]
< [ 107 or] 10,000,000 units of eighth order of numbers.
"I conceive that these things, king Gelon, will appear incredible to the great majority of people who have not studied mathematics, but that to those who are conversant therewith and have given thought to the question of the distances and sizes of the earth and the sun and moon and the whole universe the proof will carry conviction. And it was for this reason that I thought the subject would be not inappropriate for your consideration."