Chapter 5 THE SANDRECKONER.

The Sandreckoner deserves a place by itself. It is not mathematically very important; but it is an arithmetical curiosity which illustrates the versatility and genius of Archimedes, and it contains some precious details of the history of Greek astronomy which, coming from such a source and at first hand, possess unique authority. We will begin with the astronomical data. They are contained in the preface addressed to King Gelon of Syracuse, which begins as follows:-

"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 filled 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 size to the earth filled up in the way described, but also that of a mass equal in size to the universe.

"Now you are aware that 'universe' is the name given by most astronomers to the sphere the centre of which is the centre of the earth, while the 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 conclusion 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 centre 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 ratio to the distance of the fixed stars as the centre of the sphere bears to its surface."

Here then is absolute and practically contemporary evidence that the Greeks, in the person of Aristarchus of Samos (about 310-230 B.C.), had anticipated Copernicus.

By the last words quoted Aristarchus only meant to say that the size of the earth is negligible in comparison with the immensity of the universe. This, however, does not suit Archimedes's purpose, because he has to assume a definite size, however large, for the universe. Consequently he takes a liberty with Aristarchus. He says that the centre (a mathematical point) can have no ratio whatever to the surface of the sphere, and that we must therefore take Aristarchus to mean that the size of the earth is to that of the so-called "universe" as the size of the so-called "universe" is to that of the real universe in the new sense.

Next, he has to assume certain dimensions for the earth, the moon and the sun, and to estimate the angle subtended at the centre of the earth by the sun's diameter; and in each case he has to exaggerate the probable figures so as to be on the safe side. While therefore (he says) some have tried to prove that the perimeter of the earth is 300,000 stadia (Eratosthenes, his contemporary, made it 252,000 stadia, say 24,662 miles, giving a diameter of about 7,850 miles), he will assume it to be ten times as great or 3,000,000 stadia. The diameter of the earth, he continues, is greater than that of the moon and that of the sun is greater than that of the earth. Of the diameter of the sun he observes that Eudoxus had declared it to be nine times that of the moon, and his own father, Phidias, had made it twelve times, while Aristarchus had tried to prove that the diameter of the sun is greater than eighteen times but less than twenty times the diameter of the moon (this was in the treatise of Aristarchus On the Sizes and Distances of the Sun and Moon, which is still extant, and is an admirable piece of geometry, proving rigorously, on the basis of certain assumptions, the result stated). Archimedes again intends to be on the safe side, so he takes the diameter of the sun to be thirty times that of the moon and not greater. Lastly, he says that Aristarchus discovered that the diameter of the sun appeared to be about 1?720th part of the zodiac circle, i.e. to subtend an angle of about half a degree; and he describes a simple instrument by which he himself found that the angle subtended by the diameter of the sun at the time when it had just risen was less than 1?164th part and greater than 1?200th part of a right angle. Taking this as the size of the angle subtended at the eye of the observer on the surface of the earth, he works out, by an interesting geometrical proposition, the size of the angle subtended at the centre of the earth, which he finds to be > 1?203rd part of a right angle. Consequently the diameter of the sun is greater than the side of a regular polygon of 812 sides inscribed in a great circle of the so-called "universe," and a fortiori greater than the side of a regular chiliagon (polygon of 1000 sides) inscribed in that circle.

On these assumptions, and seeing that the perimeter of a regular chiliagon (as of any other regular polygon of more than six sides) inscribed in a circle is more than 3 times the length of the diameter of the circle, it easily follows that, while the diameter of the earth is less than 1,000,000 stadia, the diameter of the so-called "universe" is less than 10,000 times the diameter of the earth, and therefore less than 10,000,000,000 stadia.

Lastly, Archimedes assumes that a quantity of sand not greater than a poppy-seed contains not more than 10,000 grains, and that the diameter of a poppy-seed is not less than 1?40th of a dactylus (while a stadium is less than 10,000 dactyli).

Archimedes is now ready to work out his calculation, but for the inadequacy of the alphabetic system of numerals to express such large numbers as are required. He, therefore, develops his remarkable terminology for expressing large numbers.

The Greek has names for all numbers up to a myriad (10,000); there was, therefore, no difficulty in expressing with the ordinary numerals all numbers up to a myriad myriads (100,000,000). Let us, says Archimedes, call all these numbers numbers of the first order. Let the second order of numbers begin with 100,000,000, and end with 100,000,0002. Let 100,000,0002 be the first number of the third order, and let this extend to 100,000,0003; and so on, to the myriad-myriadth order, beginning with 100,000,00099,999,999 and ending with 100,000,000100,000,000, which for brevity we will call P. Let all the numbers of all the orders up to P form the first period, and let the first order of the second period begin with P and end with 100,000,000 P; let the second order begin with this, the third order with 100,000,0002 P, and so on up to the 100,000,000th order of the second period, ending with 1,000,000,000100,000,000 P or P2. The first order of the third period begins with P2, and the orders proceed as before. Continuing the series of periods and orders of each period, we finally arrive at the 100,000,000th period ending with P100,000,000. The prodigious extent of this scheme is seen when it is considered that the last number of the first period would now be represented by 1 followed by 800,000,000 ciphers, while the last number of the 100,000,000th period would require 100,000,000 times as many ciphers, i.e. 80,000 million million ciphers.

As a matter of fact, Archimedes does not need, in order to express the "number of the sand," to go beyond the eighth order of the first period. The orders of the first period begin respectively with 1, 108, 1016, 1024, ... (108)99,999,999; and we can express all the numbers required in powers of 10.

Since the diameter of a poppy-seed is not less than 1?40th of a dactylus, and spheres are to one another in the triplicate ratio of their diameters, a sphere of diameter 1 dactylus is not greater than 64,000 poppy-seeds, and, therefore, contains not more than 64,000 × 10,000 grains of sand, and a fortiori not more than 1,000,000,000, or 109 grains of sand. Archimedes multiplies the diameter of the sphere continually by 100, and states the corresponding number of grains of sand. A sphere of diameter 10,000 dactyli and a fortiori of one stadium contains less than 1021 grains; and proceeding in this way to spheres of diameter 100 stadia, 10,000 stadia and so on, he arrives at the number of grains of sand in a sphere of diameter 10,000,000,000 stadia, which is the size of the so-called universe; the corresponding number of grains of sand is 1051. The diameter of the real universe being 10,000 times that of the so-called universe, the final number of grains of sand in the real universe is found to be 1063, which in Archimedes's terminology is a myriad-myriad units of the eighth order of numbers.

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