The Sand Reckoner, by Archimedes
Chapter I
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 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 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 premisses 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 to be equal to what we call the "universe".
I say then that, even if a sphere were made up of 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:
* The perimeter of the earth is three hundred myriad stadia and no greater, though some have tried to show, as you know, that this length is thirty myriad stadia. But I, surpassing this number and setting the size of the earth as being ten times that evaluated by my predecessors, suppose that its perimeter is three hundred myriad stadia and not greater.
* Secondly, that the diameter of the earth is greater than the diameter of the moon and that the diameter of the sun is greater than the diameter of the earth. My hypothesis is in agreement with most earlier astronomers.
* Third hypothesis: the diameter of the sun is thirty times larger than that of the moon and not greater, even though among earlier astronomers Eudoxus tried to show it as nine times larger and Pheidias, my father, as twelve times larger, while Aristarchus tried to show that the diameter of the sun lies between a length of eighteen moon diameters and a length of twenty four moon diameters; but I, surpassing this number as well, suppose, so that my proposition may be established without dispute, that the diameter of the sun is equal to thirty moon diameters, and not more.
* Finally, we state that the diameter of the sun is greater than the side of the polygon of one thousand sides inscribed in the great circle of the universe. I make this hypothesis because Aristarchus found that the sun appears as the seven hundred and twentieth part of the circle of the zodiac. While examining this question I have, for my part tried in the following manner, to show with the aid of instruments, the angle subtended by the sun, having its vertex at the eye. Clearly, the exact evaluation of this angle is not easy since neither vision, hands, nor the instruments required to measure this angle are reliable enough to measure it precisely. But this does not seem to me to be the place to discuss this question at length, especially because observations of this type have often been reported.
For the purposes of my proposition, it suffices to find an angle that is not greater than the angle subtended at the sun with vertex at the eye and to then find another angle which is not less than the angle subtended by the sun with vertex at the eye. A long ruler having been placed on a vertical stand placed in the direction of where the rising sun could be seen, and a little cylinder was put vertically on the ruler immediately after sunrise. Then, the sun being at the horizon, and could be looked at directly, the ruler was oriented towards the sun and the eye at the extremity of the ruler. The cylinder being placed between the sun and the eye, occludes the sun. The cylinder is then moved further away from the eye and as soon as a small piece of the sun begins to show itself from each side of the cylinder, it is fixed.
If the eye were really to see from one point, tangents to the cylinder produced from the end of the ruler where the eye was placed would make an angle less than the angle subtended by the sun with vertex at the eye. But since the eyes do not see from a unique point, but from a certain size, one takes a certain size, of round shape, not smaller than the eye and one places it at the extremity of the ruler where the eye had been placed. If one produces tangents to this size and to the cylinder, the angle between these lines is smaller than the angle subtended by the sun with vertex at the eye. And here is the way one finds the size not smaller to the eye: one takes two small thin cylinders of the same width, one white, the other not, and one places them in front of the eye, the white one at some distance, and the other one which is not white as close to the eye as possible without touching the face. In this way, if the small cylinders chosen are smaller than the eye, the cylinder neighbouring the eye is encompassed in the visual field and the eye sees the white cylinder. If the cyclinders are much smaller, the white one is completely seen. If they are not much smaller, one sees parts of the white one and parts of the one neighboring the eye. But if one choose cylinders of appropiate width one of them occludes the other without covering a larger space. It is therefore clear that the width of cylinders producing this effect is not smaller than the dimensions of the eye.
As for the angle not smaller than the angle subtending the sun with vertex at the eye, it was taken as follows: The cylinder being placed on the ruler at a distance which blocks all of the sun, if one produces from the end of the ruler where the eye is placed tangent lines to the cylinder, the angle made by these lines is not smaller than the angle subtended by the sun with vertex at the eye. A right angle being measured by the angles taken in this way, the angle placed at the point is found to be the one hundred and sixty fourth part of a right angle, while the smallest angle is found to be greater to the two hundredth part of a right angle. It is therefore clear that the angle subtended by the sun with vertex at the eye is also smaller than the one hundred and sixty fourth part of a right angle, and greater than the two hundredth part of a right angle.
With these measurements completed, one shows that the diameter of the sun is greater than the side of the polygon with one thousand sides inscribed in the great circle of the universe.
Let us imagine then a plane passing through the centre of the sun, the centre of the earth and the eye at the instant when the sun finds itself a little above the horizon; that this plane cuts the universe at the circle ABG, the earth at the circle DEZ, and the sun at the circle SH. Let T be the centre of the earth, K the centre of the sun, and let D be the eye; we produce from D the tangents DL, DX to the circle SH with contact points N and T, and from T the tangents TM and TO with contact points X and P. Let A and B be the points of intersection of the circle ABG and the lines TM and TO. Thus TK is greater than DK from the hypothesis that the sun finds itself above the horizon. If follows that the angle contained between DL and DX is greater than the angle contained between TM and TO. But the angle contained between DL and DX is greater than the two hundredth part of a right angle since it is equal to the angle subtended by the sun with vertex at the eye; and consequently, the angle contained between TM and TO is less than the one hundred and sixty fourth part of a right angle and the segment of the line AB is less than the chord of the circular sector which is the six hundred and sixty fifth part of the circle ABG. But the perimeter of the polygon in question has with the radius of the circle ABG a ratio less than fourty four to seven because the ratio of the perimeter of every polygon inscribed in a circle to the radius of the circle is less than the ratio fourty four to seven.
You know, in fact, that I have shown that in every circle the perimeter is greater, by a quantity smaller than the seventh, than triple the diameter and that the perimeter of the inscribed polygon is smaller than this circumference. The ratio of BA to TK is thus less than the ratio of eleven to one thousand one hundred and fourty eight. It follows that BA is smaller than a hundredth TK. But the diameter of the circle SH is equal to BA since half of SH, the segment \PhiA, is equal to KP. The segments TK and TA are in fact equal and their endpoints perpendiculars are produced under the same angle. It is thus clear that the diameter of the circle SH is less than the hundredth part of TK. Moreover, the diameter ETU is less than the diameter of the circle SH since the circle DEZ is less than the circle SH. If follows that the sum of TU and KS is less than the hundredth part of TK so that the ratio of TK to UK is less than the ratio of one hundred to ninety nine. And as long as SU is less than DT, the ratio of TP to DT is less than the ratio of one hundred to ninety nine. But since in the right triangles TKP and DKT the sides KP and KT are equal and the sides TP and DT unequal, TP being larger, the ratio of the angle contained between the sides DT and DK to the angle contained between the TP and TK is greater than the ratio of TK to DK, but less than the ratio of TP to DT. For if in two right triangles two of the sides containing the right angle are equal and the two others unequal, then the larger angle opposite the unequal sides has to the smaller of these angles a ratio greater than the ratio of the greater hypotenuse to the smaller, but smaller than the ratio of the greater side to the right angle to the smaller. As a consequence, the ratio of the angle contained between DL and DX to the angle contained between TO and TM is less than the ratio of TP to DT which is itself less than the ratio of one hundred to ninety nine. It follows that the ratio of the angle contained between DL and DX is greater than the two hundredth part of a right angle, the angle contained between TM and TO is greater than ninety nine twenty thousandths of a right angle; and as a consequence, this angle is greater than one two hundred and third of a right angle. The segment BA is thus greater than the chord of the sector which is a eight hundred and twelvth part of the circle ABG. But it is to the line segment AB that the diameter of the circle is equal to. It is therefore clear that the diameter of the circle is greater than the side of the polygon of one thousand sides.
Chapter II
These relations being given, one can also show that the diameter of the universe is less than a line equal to a myriad diameters of the earth and that, moreover, the diameter of the universe is less than a line equal to one hundred myriad myriad stadia. As soon as one has accepted the fact that the diameter of the sun is not greater than thirty moon diameters and that the diameter of the earth is greater than the diameter of the moon, it is clear that the diameter of the sun is less than thirty diamters of the earth. As we have also shown that the diameter of the sun is greater than the side of the polygon of one thousand sides inscribed in the great circle of the universe, it is clear that the perimeter of the indicated polygon of one thousand sides is less than one thousand diameters of the sun. But the diameter of the sun is less than thirty earth diameters so it follows that the perimeter of the polygon of one thousand sides is less than thirty thousand earth diameters.
Given that the perimeter of the polygon of one thousand sides is less than thirty thousand earth diameters and greater than three diameters of the universe — we have shown in fact that in every circle the diameter is less than one third the perimeter of any regular polygon inscribed in the circle for which the number of sides is greater than that of the hexagon — the diameter of the universe is less than a myriad earth diameters. One has thus shown that the diameter of the universe is less than a myriad earth diameters; that the diameter of the universe is less than one hundred myriad myriad stadia, which comes out of the following argument; since, in fact, we have supposed that the perimeter of the earth is not greater than three hundred myriad stadia and that the perimeter of the earth is greater than triple the diameter because in every circle the circumference is greater than triple the diameter, it is clear that the diameter of the earth is less than one hundred myriad stadia.
Given that the diameter of the universe is less than a myriad earth diameters it is clear that the diameter of the world is less than one hundred myriad myriad stadia. These are my hypotheses regarding sizes and distances.
Here now is what I assume about the subject of sand: if one has a quantity of sand whose volume does not exceed that of a poppy-seed, the number of these grains of sand will not exceed a myriad and the diamter of the grains will not be less than a fourtieth of a finger-breadth. I make these hypotheses following these observations: poppy seeds having been placed on a polished ruler in a straight line in such a way that each touches the next, twenty five seeds occupied a space greater than one finger-breadth. I will suppose that the diameter of the grains is smaller and to be about a fourtieth of a finger-breadth for the purpose of removing any possiblity of criticizing the proof of my proposition.
Chapter III
These are thus my hypotheses; but I think it useful to explain myself about the naming of numbers so that those readers, not having been able to get hold of my book addressed to Zeuxippus, may not be thrown off by the absence in this book of any indication of the subject of this nomenclature.
It so happens that tradition has given to us the name of numbers up to a myriad and we distinguish enough numbers surpassing a myriad by enumerating the number of myriads until a myriad myriad. We will therefore call first numbers those which, after the current nomenclature, go up to a myriad myriad.
We will call units of second numbers the myriad myriad of first numbers and we will count among second number units and, starting with units, tens, hundreds, thousands, myriads, until a myriad myriad.
We will call once again call third numbers a myriad myriad of second numbers and we will count among third numbers, starting with units, tens, hundreds, thousands, myriads, until the myriad myriad.
In the same way we will call units of fourth numbers a myriad myriad of third numbers, units of fifth numbers a myriad myriad of fourth numbers, and continuing in this way the numbers will be distinguishable until the myriad myriad of of myriad myriad numbers.
Numbers named in this way could certainly suffice but it is possible to go still further. Let us in fact call numbers of the first period the numbers given up to this point and units of first numbers of the second period the last number of the first period. Furthermore, call the unit of second numbers of the second period the myriad myriad of first numbers of the second period. In the same way, the last of these numbers will be called the unit of third numbers of the second period, and continuing in this way, progressing through the numbers of the second period will have their names up to the myriad myriad numbers. [A] myriad myriad [of the] last number of the second period will be in turn called the unit of the first numbers of the third period, and so forth until the myriad myriad [of] numbers myriad myriad of the myriad myriad period.
These numbers having been named, given numbers ordered by size starting from unity and if the number closest to unity is the tens, the first eight of these including the unity will belong to the numbers called first numbers, the following eight numbers called second, and the others in the same way by the distance of their octad of numbers to the first octad of numbers. The eighth number of the first octad is thus one thousand myriads and the first number of the second octad, since it multiplies by ten the number preceding it, will be a myriad myriad and this number is the unit of the second numbers. The eighth number of the second octad is one thousand myriad of second numbers. The first number of the third octad will once again be, as it multiplies by ten the preceding number, a myriad myriad of second numbers, the unity of the third numbers. It is clear that the same will hold as indicated for any octad.
It is useful to know what follows. If numbers are in proportion starting from unity and some which are in the same proportion are multiplied to each other, then the product will be increased from the larger of the factors by as many numbers as the smaller number is in proportion to unity and it will be increased from unity by the sum minus one of the numbers away from unity. In fact, let A,B,G,E,Z,H,T,I,K,L in proportion starting from unity, and let A be unity. Multiply D by T and let X be the product. Let us take in the proportion L whose distance to T holds as many numbers as the distance from D to unity. It must be shown that X equals L. If, among the numbers in proportion, the distance from D to A counts as many numbers as that from L to T, the ratio of D to A equals the ratio of L to T. But D is the product of D by A from which it follows that L is the product of D by T, so L is equal to X. It is therefore clear that the product is in the proportion and that its distance to the largest factor counts as many numbers as the distance of the smaller factor to unity. But it is also clear that this product is increased, from unity, by the sum minus one, of the distances of the numbers D and T to unity; for A,B,G,D,E,Z,H,T are numbers among which T is increased from unity, and I,K,L are, up to one number, those from which D is increased from unity; adding T one has the sum of the distances.
Chapter IV
The preceding being in part assummed and in part proved, I will now prove my proposition. As we have assumed that the diameter of a poppy-seed is not smaller than a fourtieth of a finger-breadth, it is clear that the volume of the sphere having diameter one finger-breadth does not exceed that of sixty four thousand poppy-seeds; for this number indicates how many times it is the multiple of the sphere having as diameter one fourtieth of a finger-breadth; it has in fact been shown that spheres are related to each other as the cubes of their diameters. As we have also assumed that the number of grains of sand contained in one poppy-seed does not exceed a myriad, it is clear that, if the sphere having diameter one finger-breadth were filled with sand, the number of grains would not exceed sixty four thousand myriads. But this number represents six units of second numbers increased by four thousand myriad of first numbers, and is thus less than ten units of second numbers.
The sphere with diameter one hundred finger-breadths is equivalent to one hundred myriad spheres of diameter one finger-breath, since spheres are related to each other as the cubes of their diameters. If one now had a sphere filled with sand of the size of the sphere of diameter one hundred finger-breadths, it is clear that the number of grains of sand would be less than the product of ten myriad second numbers and one hundred myriads. But since ten units of second numbers make up the tenth number starting from unity in the proportional sequence of constant multiple ten, and the one hundred myriads of the seventh number starting from unity in the same proportional sequence, it is clear that the number obtained will be the sixteenth starting from unity in the same proportional sequence. For we have shown that the distance of this product to unity is equal to the sum of, minus one, of the distance from unity of its two factors. From these sixteen numbers the first eight are among, with unity, the numbers called first numbers, the following eight are part of the second numbers, and the last of these is one thousand myriad second numbers. It is now evident that the number of grains of sand whose volume is equal to one hundred finger-breadths is less than one thousand myriad second numbers.
Similarly, the volume of the sphere of diameter one myriad finger-breadths is one hundred myriad times the volume of the sphere of diameter one hundred finger-breadths. If one now had a sphere, filled with sand, of the size of the sphere with diameter a myriad finger breadths, it is clear that the number of grains of sand would be less than the product of one thousand myriads of second numbers and one hundred myriads. But since one thousand myriad second numbers are the sixteenth number starting from unity in the proportional sequence and that one hundred myriad are the seventh number starting from unity in the same proportional sequence, it is clear that the product will be the twenty second number starting from unity in the same proportional sequence. Of these twenty two numbers, the first eight, with unity, are among the numbers called first numbers, the following eight are among the numbers called second, and the six remaining numbers are called third numbers, the last of which being ten myriad third numbers. It is then clear that the number of grains of sand whose volume is equal to a sphere of diameter of a myriad finger-breadths is less than ten myriads of third numbers.
And since the sphere with diameter one stade is smaller than the sphere with diameter a myriad finger-breadths, it is also clear that the number of grains of sand contained in a volume equal to a sphere with diameter one stade is less than ten myriad third numbers.
Similarly, the volume of a sphere of diameter one hundred stadia is one hundred myriad times the volume of a sphere of diameter one stade. If one now had a sphere, filled with sand, of the size of the sphere with diameter one hundred stadia, it is evident that the number of grains of sand would be less than the product of ten myriad third numbers with one hundred myriad. And since ten myriad third numbers are the twenty second numbers, starting from unity, in the proportional sequence, and that one hundred myriad are the seventh number starting from unity in the same proportional sequence, it is clear that the product will be the twenty eighth number starting from unity in the proportional sequence. Of these twenty eight numbers, the first eight, with unity, are part of the numbers called first numbers, the following eight are second numbers, the following eight are third numbers, and the four remaining are called fourth, the last being one thousand units of fourth numbers. It is then evident that the number of grains of sand whose volume equals that of a sphere of diameter a hundred stadia is less than one thousand units of fourth numbers.
Similarly, the volume of a sphere of diameter a myriad stadia is one hundred myriad times the volume of a sphere having diameter one hundred stadia. If one then had a sphere, filled with sand, of the size of a sphere of diameter a myriad stadia, it is clear that the number of grains of sand would be less than the product of one thousand units of fourth numbers with one hundred myriad. Just as one thousand units of fourth numbers represent the twenty eighth number, starting from unity, in the proportional sequence, and one hundred myriad the seventh number in the proportional sequence, starting from unity, of the same proportional sequence, it is clear that their product will be, in the same proportional sequence, with unity, the thirty fourth number starting form unity. But of these thirty four numbers, the first eight, with unity, are among those numbers called first numbers, the following eight among second numbers, the following eight among third numbers, the following eight among fourth numbers, and the two remaining among fifth numbers, the last of these being ten units of fifth numbers. It is thus clear that the number of grains of sand whose volume is equal to that of a sphere having diameter a myriad stadia will be smaller than ten units of fifth numbers.
And similarly, the volume of a sphere of diameter one hundred myriad stadia is one hundred myriad times the volume of a sphere of diameter a myriad stadia. If one had then had a sphere, filled with sand, of the size of the sphere with diameter one hundred myriad stadia, it is clear that the number of grains of sand would be smaller than the product of ten units of fifth numbers and one hundred myriads. As the ten units of fifth numbers represent the thirty fourth number starting from unity in the proportional sequence, and one hundred myriads the seventh number starting from unity in the same proportional sequence, it is clear that the product will be, in the same proportional sequence, the fourtieth number starting from unity. But of these fourty numbers, the first eight, with unity, are among the numbers called first numbers, the eight following are second numbers, the eight following are third numbers, the eight following are fourth numbers, the eight following are fifth numbers, the last of these being one thousand myriad fifth numbers. It is therefore clear that the number of grains of sand whose volume is equal to that of a sphere of diameter one hundred myriad stadia is less than one thousand myriad fifth numbers.
But the volume of a sphere of diameter a myriad myriad stadia is one hundred myriad times the sphere of diameter one hundred myriad stadia. Thus, if one had a sphere, filled with sand, of the size of a sphere of diameter a myriad myriad stadia, it is clear that the number of grains of sand would be less than the product of one thousand myriad fifth numbers by one hundred myriads. However, since one thousand myriad fifth numbers represent the fourtieth number, starting from unity, of the proportional sequence, and one hundred myriad the seventh number starting from unity in the same proportional sequence, it is clear that the product will be the fourty sixth number starting from unity. Of these fourty six numbers, the first eight, with unity, are part of the numbers called first numbers, the eight following second numbers, the eight following third numbers, the eight following fourth numbers, the eight following fifth numbers, and the six left over are numbers called sixth, the last among being ten myriads of sixth numbers. It is thus clear that the number of grains of sand whose volume is equal to a sphere of diameter a myriad myriad stadia is smaller than ten myriad sixth numbers.
But the volume of a sphere of diameter one hundred myriad myriad stadia is one hundred myriad times the multiple of a sphere of diameter a myriad myriad stadia. Thus, if one had a sphere, filled with sand, of the size of a sphere of diameter one hundred myriad myriad stadia, it is clear that the number of grains of sand would be smaller than the product of ten myriad sixth numbers by one hundred myriad. But, since ten myriad sixth numbers represent the fourty sixth number, starting from unity, in the proportional sequence, and one hundred myriad the seventh number starting from unity in the same proportional sequence, it is clear that the product will be the fifty second number starting in the same proportional sequence. But of these fifty two numbers, the first fourty eight, with unity, belong to numbers called first numbers, second numbers, third, fourth, fifth, and sixth, and the the four remaining are among numbers called seventh numbers, the last of them being one thousand units of seventh numbers. It is thus clear that the number of grains of sand in a volume equal to a sphere whose volume is equal to that of a sphere of diameter one hundred myriad myriad stadia is smaller than one thousand units of seventh numbers.
As we [have] shown that the diameter of the universe is less than one hundred myriad myriad stadia, it is clear that the number of grains of sand filling a volume equal to that of the universe is itself less than one thousand units of seventh numbers. We have thus shown that the number of grains of sand filling a volume equal to that of the universe, as the majority of astronomers understand it, is one thousand units of seventh numbers; we will now show that even the number of grains of sand filling a volume equal to the sphere as large as Artistarchus proposed for the fixed stars, is smaller than one thousand myriad eighth numbers. As we have assumed, in fact, that the ratio of the earth to what we commonly call the universe is equal to the ratio of this universe to the sphere of fixed stars as proposed by Aristarchus, the two spheres have the same ratio to each other. But it has been shown that that the diameter of the universe is less than a length a myriad times the multiple of the diameter of the earth. It is thus clear that the diameter of the sphere of fixed stars is itself smaller to a length a myriad times the diameter of the universe.
But since the sphere have the ratio among themselves of their diameters, it is clear that the sphere of fixed stars, as Aristarchus proposes, is less than a volume a myriad myriad myriad times a multiple the volume of the universe. But we have shown that the number of grains of sand filling a volume equal to that of the world is less than a thousand units of seventh numbers; it is therefore evident that if a sphere, as large as Aristarchus supposes that of the fixed stars to be, were to be filled with sand, the number of grains of sand would be less than the product of one thousand units [of seventh numbers] by a myriad myriad myriad. And since one thousand units of seventh numbers represent the fifty second number in the reciprocal sequence starting from unity, and a myriad myriad myriads the thirteenth number starting from unity in the same proportional sequence, it is clear that the product will be the sixty fourth number starting from unity in the same proportional sequence; but this number is the eighth of the eight numbers, which is one thousand myriads of eight numbers.
It is therefore obvious that the number of grains of sand filling a sphere of the size that Aristarchus lends to the sphere of fixed stars is less than one thousand myriad myriad eighth numbers.
I conceive, King Gelon, that among men who do not have experience of mathematics, such a thing might appear incredible. On the other hand, those who know of such matters and have thought about the distances and sizes of the earth, the sun, the moon, and the universe in its entirety will accept them due to my argument, and that is why I believed that you might enjoy having brought it to your attention.
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