Return to Vignettes of Ancient Mathematics

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Note: the numbering is from Heiberg's text (revised Stamatis), on which the translation is based.

[1] Some people believe, King Gelon, that the number of sand is infinite in multitude. I mean not only of the sand in Syracuse and the rest of Sicily, but also of the sand in the whole inhabited land as well as the uninhabited. There are some who do not suppose that it is infinite, and yet that there is no number that has been named which is so large as to exceed its multitude.

Note: Archimedes speaks of the number of the sand and not of the grains of sand. He does not use a word meaning 'grain of sand'. In deference to this, I shall treat 'sand' as a mass term (some sand), but allow that one can speak of the number of sand, meaning, of course, the number of the grains of sand.

[2] It is clear that if those who hold this opinion should conceive of a volume composed of the sand as large as would be the volume of the earth when all the seas in it and hollows of the earth were filled up in height equal to the highest mountains, they would not know, many times over, any number that can be expressed exceeding the number of it.

[3] I will attempt to prove to you through geometrical demonstrations, which you will follow, that some of the numbers named by us and published in the writings addressed to Zeuxippus exceed not only the number of sand having a magnitude equal to the earth filled up, just as we said, but also the number of the sand having magnitude equal to the world.

Note: the book to Zeuxippus (lost) would have been the formal presentation of the system, while the Sand-Reckoner is the popularization.

[4] You grasp that the world is called by most astronomers the sphere whose center is the center of the earth and whose line from the center is equal to the straight-line between the center of the sun and the center of the earth, since you have heard these things in the proofs written by the astronomers. But Aristarchus of Samos produced writings of certain hypotheses in which it follows from the suppositions that the world is many times what is now claimed.

Note: this claim is very odd and has not been adequately noticed by commentators. One would think that the whole world is the sphere of the fixed stars and everything within and that the sun is lower than the fixed stars, as Aristotle argues, and not the cosmology of Anaximander, who does place the sun as the outermost object. Instead, Archimedes seems to place the sun as the outermost, since 'world' (kosmos) should encompass everything, and he is aiming to give as large a universe as possible on each of the two rival theories. The issue is complicated by the fact that Hippolytus (3rd. cent. C.E.) preserves two versions of Archimedes' own dimensions of the universe:

  earth to sun earth to outermost in list
Version 1 5581,6195 stadia 2,4826,4780 stadia to zodiac
Version 2 1,2160,4451 2,2269,2711 stadia to Saturn
Sand Reckoner < 100,0000,0000 < 100,0000,0000,0000 (stadia to zodiac on Aristarchus' theory)

Source: O. Neugebauer, Hist. Anc. Math. Astron., 648-9.

Hence, Archimedes does not consider the sun to be at the edge of the world. Nonetheless, his value for the distance of the sun is much larger than the transmitted value for the size of the world. (For the placement of comma in numbers, cf. notes to Ch. 4).

[5] For he supposes that the fixed stars and the sun remain motionless, while the earth revolves about the sun on the circumference of a circle which is placed on the middle road, but that the sphere of the fixed stars, which is placed about the same center as the sun, is so large in magnitude that the circle on which he supposes the earth to revolve has the sort of proportion to the distance of the fixed stars that the center of the sphere has to the surface.

This is our principal source for this view and is one of the grounds for modern interest in the treatise. Aristarchus only extant treatise, On the Sizes and the Distances of the Sun and the Moon, gives not a hint of such an hypothesis.

[6] This is trivially impossible, since the center of the sphere has no magnitude. One must suppose that it doesn't have any ratio either to the surface of the sphere. We must understand this such that Aristarchus means this: since we suppose the earth is just like the center of the world, the ratio which the earth has to the world described by us is the same as the ratio that the sphere on which the circle is on which he supposes the earth to revolve has to the sphere of the fixed stars. For he applies the demonstrations of the phainomena to what is supposed here, and the magnitude of the sphere on which he makes the earth move appears above all to be supposed equal to the world described by us.

In order to have a ratio that he can use, Archimedes is being blatantly persnickety. Cf. Aristotle, Meteorology A 3.340a6-8, "For the volume of the earth, in which is taken all of it together, even the amount of water, is, so to speak, not even a portion in relation to the surrounding magnitude." Note that Aristotle thinks that the earth and the universe are finite. Aristarchus makes this claim also in his extant book, On the Sizes and the Distances of the Sun and the Moon, hypoth. 2.

[7] In fact we say that even if a sphere of sand were to become as large in magnitude as Aristarchus supposes the sphere of the fixed stars to be, we will also prove that some of the initial numbers having an expression (or: "numbers named in the Principles," cf. Heath, Archimedes, 222, and Dijksterhuis, Archimedes, 363) exceed in multitude the number of sand having a magnitude equal to the mentioned sphere, when the following are supposed.

[8] First that the perimeter of the earth is about 300,0000 stadia and not larger, although some have attempted to demonstrate it , as you too follow them, as being about 30,0000. But since I am exceeding this and posit the magnitude of the earth as ten-times what was believed by the earlier astronomers, I suppose the perimeter of it to be about 300,0000 and not larger.

Aristotle reports 40,0000 stadia, and Eratosthenes (contemporary of Archimedes) calculated 25,0000 or 25,2000 stadia. It is an old problem that the length of the stadium varies in different locals, so that it is a separate problem to know what these values are in actual distance.

After this, I suppose that the diameter of the earth is larger than the diameter of the moon and that the diameter of the sun is larger than the diameter of the earth, and assume the same things in like manner as most of the earlier astronomers.

[9] After these, I suppose that the diameter of the sun is about thirty-times the diameter of the moon and not larger, although of earlier astronomers Eudoxus declared about nine-times, Pheidias, my father, about twelve-times, and Aristarchus has attempted to prove that the diameter of the sun is more than eighteen-times the diameter of the moon and smaller than twenty-times. But I will also exceed this amount, so that what is proposed be indisputably proved, and suppose that the diameter of the sun is about thirty-times the diameter of the moon and not larger.

For Aristarchus, cf. his On the Sizes and the Distances of the Sun and the Moon.

[10] In addition to these, I suppose that the diameter of the sun is larger than the side of the chiliagon inscribed in the largest circle of those in the world. I suppose this given that Aristarchus has found the sun appears about one seven hundred and twentieth of the circle of the zodia, but having examined it in the following manner I attempted with instruments to get the angle into which the sun fits and which has its vertex at the eye. [11] And so it is not easy to get precision since neither the eye nor the hands nor the instruments through which we must get it are trustworthy at declaring precision. For the present it is not timely to lengthen our discussion about these things, especially since these sorts of things have been explained many times. For the demonstration of the proposed claim, it suffices for me to get an angle which is no larger than the angle into which the sun fits and which has its vertex at the eye, and again to get another angle which is not smaller than the angle into which the sun fits and which has its vertex at the eye.

[12] (diagram 1=initial diagram) And so a long ruler was placed on a straight footing lying in a place from where the sun would be seen rising, and a small bored cylinder was placed on the ruler upright straightaway after the rising of the sun, and then when it was on the horizon and could be looked at right on, the ruler was turned around into the sun, and the eye was positioned at the end of the ruler. Lying between the sun and the eye, the cylinder blocked the view of the sun. And so with the cylinder separated from the eye, the cylinder was positioned in the spot where a little bit of the sun would begin to appear on each side of the cylinder.

[13] And so if it had been the case that the eye sees from one point, with straight-lines drawn from the end of the ruler in the place where the eye was positioned and tangent to the cylinder, then the angle enclosed by the lines drawn would have been smaller than the angle into which the sun fits and which has its vertex at the eye, since a bit of the sun was glimpsed on each side of the cylinder. (diagram 2) Since eyes do not see from one point, but from a magnitude, a round magnitude was taken not smaller than an eye, and with the magnitude placed on the end of the ruler in the place where the eye was positioned, after straight lines tangent to the magnitude and the cylinder were drawn, (diagram 3) the angle enclosed by the drawn lines was thus smaller than the angle into which the sun fits and which has its vertex at the eye.

[14] The magnitude no smaller than the eye is found in this way. Two narrow cylinders equal to one another in thickness are taken, one white, one not, and they are placed before the eye, with the white set apart from it and the non-white as near as possible to the eye, so that it is even touching the face. And so, if the cylinders taken are narrower than the eye, the cylinder nearby will be encompassed by the eye and the white will be seen by it, and if they are much narrower, the whole will be seen, while if they are not much narrower, certain parts of the white will be seen on each side of the cylinder near the eye. But when these cylinders are somehow taken suitably for their thickness, one of them blocks the view of the other and not a larger place. Now such a magnitude which is the thickness of the cylinders which produce this effect is above all, I suppose, not smaller than the eye.

[15] (diagram 1=initial diagram) The angle which is not smaller than the angle into which the sun fits and which has its vertex at the eye was taken in this way. With the cylinder positioned on the ruler away from the eye in such a way that the cylinder blocked the view of the whole sun and (diagram 2) with straight-lines drawn from the end of the ruler in the place where the eye was positioned and tangent to the cylinder, (diagram 3) the angle enclosed by the straight-lines drawn becomes no smaller than the angle into which the sun fits and which has its vertex at the eye.

[16] When a right angle is measured out by angles taken in this way, with the right angle divided into 164 parts the angle at the point comes to be smaller than one of these parts, and, with the right angle divided into 200 parts, the smaller angle comes to be larger than one of these parts. And so it is clear that the angle into which the sun fits and which has its vertex at the eye is also smaller than, with the right angle divided into 164, one of these parts, but larger than, with the right angle divided into 200, one of these parts.

[17] When we put our trust in these things, the diameter of the sun is proved to be larger than the side of the chiliagon inscribed in the largest circle of those in the world.

(general diagram)

(diagram 1) For let there be conceived a plane extended through the center of the sun and the center of the earth and through the eye, with the sun a little above the horizon. Let the extended plane cut the world along circle ABG, the earth along circle DEZ, the sun along circle SH, and let there be a center of the earth, Q, a center of the sun, K, and let there be an eye, D. (diagram 2) And let straight-lines tangent to circle SH be drawn from D, namely DL, DX, with tangents at N and T, (diagram 3) but from Q, QM, QO, with tangent at C and R, and let QM, QO cut circle ABG at A and B. [18] (diagram 4) In fact QK is larger than DK, since the sun is supposed to be above the horizon. (diagram 5) Thus the angle enclosed by DL, DX is larger than the angle enclosed by QM, QO. (diagram 6) But the angle enclosed by DL DX is larger than a two hundredth part of a right angle and smaller than, with the right angle divided into 164 parts, one of these parts. For it is equal to the angle into which the sun fits and which has its vertex at the eye. (diagram 7) Thus the angle enclosed by QM, QO is smaller than, with the right angle divided into 164, one of these parts, (diagram 8) but straight-line AB is smaller than the line subtending one segment of the circumference of circle ABG divided into 656. [= 164*4] [19] (diagram 9) But the perimeter of the mentioned polygon to the line from the center of circle ABG has a smaller ratio than 44 to 7 since the perimeter of every polygon inscribed in a circle to the line from the center has a smaller ratio than 44 to 7. For you know what was proved by us, that the circumference of every circle is larger than three-times the diameter by smaller than a seventh part, but the perimeter of the inscribed polygon is smaller than this. (diagram 10) And so BA to QK has a ratio smaller than 11 to 1148. (diagram 11) Thus BA is smaller than a hundredth part of QK. [20] (diagram 12) But the diameter of circle SH is equal to BA, since its half FA is also equal to KR. (diagram 13) For, since QK, QA are equal, perpendiculars from their end-points are joined under the same angle. (diagram 14) And so it is clear that the diameter of circle SH is smaller than a hundredth part of QK.

(general diagram)

(diagram 15) And diameter EQU is smaller than diameter SH, since circle DEZ is smaller than circle SH. Therefore, both QU, KS are smaller than a hundredth part of QK. [1/2EU + 1/2SH] (diagram 16) Thus QK to US has a ratio smaller than 100 to 99. [US = QK – (KS + QU, while a < b/n b-a : b > n-1 : n b : (b-a) < n : (n-1))] (diagram 17) And since QK is not smaller than QR, but SU is smaller than DT, therefore QR to DT would have a ratio smaller than 100 to 99. [a c & b < d a : b > c : d] [21] (diagram 18) Since, given that QKR, DKT are right-angled triangles, sides KR, KT are equal, while QR, DT are unequal with QR larger, the angle enclosed by DT, DK to the angle enclosed by QR, QK has a ratio larger than QK to DK, but smaller than QR to DT. For if in two right-angled triangles one pair of sides about the right angle are equal and the others are unequal, the larger of the angles at the unequal sides to the smaller has a ratio larger than the larger of the lines subtending the right angle to the smaller, but smaller than the larger of the lines at the right angle to the smaller. [22] (diagram 19) Thus the angle enclosed by DL, DX to the angle enclosed by QO, QM has a ratio smaller than QR to DT, which has a ratio smaller than 100 to 99. Thus too the angle enclosed by DL, DX to the angle enclosed by QM, QO has a ratio smaller than 100 to 99.

(diagram 20) And since the angle enclosed by DL, DX is larger than a two hundredth part of a right angle, the angle enclosed by QM, QO would be larger than 99 of these parts of the right angle divided into 2,0000. [since LDX > /200 & LDX : MQO < 100 : 99] Thus it is larger than, with the right angle divided into 200 and 3, one of these parts. [since /203 < 99*/2,0000] Therefore, BA is larger than the line subtending one segment and dividing the circle into 812 [= 4 * 203]. But the diameter of the sun is equal to AB. And so it is clear that the diameter of the sun is larger than the side of the chiliagon.

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