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The universe of Archimedes: Archimedes tackled the size of the universe with grains of sand, and in the process found an entirely new system for large numbers.

AROUND 216 BC, shortly before his death, the septuagenarian Archimedes presented an astronomical treatise to Gelon II, king of his native Syracuse on the island of Sicily. In this curious tract, titled The Sand-Reckoner, Archimedes demonstrates his facility with large numbers by computing how many sand grains it would take to fill the universe. Yes, the universe.


Although Archimedes is better known for his studies of buoyancy and levers, as well as for his inspirational cry "Eureka!" (I have found it!), he took more than a passing interest in the heavens. His father, Phidias, was an astronomer, and Archimedes is known to have measured the Sun's apparent diameter. He also wrote a treatise on the design and construction of tabletop planetariums.

Archimedes' opening pitch in The Sand-Reckoner has a definite ring of the carnival barker: "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 ... 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." Sounds impressive, even today. But just how big is this universe that Archimedes intends to fill?

To make a lasting impression on his royal patron, Archimedes seeks a universe vaster than the commonly accepted geocentric system of his day, with its tightly nested heavenly spheres arrayed concentrically around a stationary Earth. So he selects a competing world system put forward by his elder contemporary Aristarchus of Samos. Archimedes describes a book in which Aristarchus "hypothesizes that the fixed stars and the sun remain unmoved; that the earth is borne around the sun on the circumference of a circle ... ; and that the sphere of the fixed stars, situated about the same center as the sun is so great that the circle in which he hypothesizes that the earth revolves bears such a proportion to the distance of the fixed stars as the center of the sphere does to its surface."

Here is none other than the Sun-centered, or heliocentric, model of the cosmos, laid out some 1,800 years before its acclaimed inventor Copernicus (see page 10). (To be fair, Copernicus developed his model independently of Aristarchus and endowed it with a mathematical force that far surpasses its classical forerunner.) Archimedes does not reveal why Aristarchus adopted a Sun-centered cosmic plan; presumably, his reasons were those of Copernicus--the Sun is larger than Earth and is unique among the known celestial bodies in its brilliance and warmth. Nor does Archimedes endorse the heliocentric cosmos; he had previously built a mechanical planetarium whose ersatz planets whirled around a central Earth. In The Sand-Reckoner, Archimedes is interested in only one feature of Aristarchus's model: its vastness.


Let's begin by laying out the groundwork for what Aristarchus has to say about the extent of the universe Archimedes proposes to fill with sand. Aristarchus approaches the issue in an oblique and problematic fashion.

In general, there are several ways to effectively convey the size of an object. One is to state an object's dimensions outright: the house is 30 feet wide. Another method is to compare its size to that of another object or, equivalently, to form a size ratio between the objects: the shopping mall is twice the length of a football field; or the ratio between the lengths of the shopping mall and the football field is 2:1.


Yet a third method is to express the size ratio of two objects in terms of the size ratio of two other objects: the tree's height is to the man's height as the man's height is to the child's height. In this case, if the man is 6 feet tall and the child 3 feet tall, the ratio of their heights is 2:1; since the tree's height bears this same ratio to the man's height, the tree is 12 feet tall. It is the last of these methods that Aristarchus chooses to reveal the size of his universe. Here, in modernized form, is what Archimedes claims Aristarchus wrote:

The distance of the stars bears the same relation to the diameter of the Earth's orbit as the surface of a sphere bears to its center.

To the modern ear, Aristarchus's statement is reminiscent of a thorny SAT problem. Parsing it, we see that the statement contains two ratios: the first ratio is a purely physical one, between the distance of the stars and the diameter of the Earth's orbit; the second ratio is a purely mathematical one, between the surface of a sphere and the center of a sphere (more about this in a moment). These two ratios, Aristarchus claims, are equal, which we can express as a simple equation:

Distance of the stars: Diameter of Earth's orbit = Surface of a sphere: Center of a sphere

Compute the second ratio--the mathematical on--and you simultaneously learn the first ratio--the physical one. Here Aristarchus apparently offers Archimedes a way to compute the extent of the heliocentric universe, or distance of the stars, in terms of the diameter of Earth's orbit. Specifically, how many times bigger is the universe than Earth's orbit? That we can answer by computing the second ratio. Or can we?

The geometrically savvy reader will already have recognized that the second ratio, between the surface of a sphere and the center of a sphere, is patently absurd. Yet similar phrasing was often used by Greek astronomers when expressing the immensity of the heavens. (By surface of a sphere, we'll take Aristarchus's meaning to be the sphere's diameter.) The center of a sphere is a point, which geometrically speaking has no size at all. Thus a sphere's diameter is infinitely larger than its center, because any number divided by zero yields infinity. In other words, the second ratio turns out to be infinite. Taken literally, Aristarchus's statement implies that the stars are infinitely far away.

But Archimedes has no use for an infinite universe, just a very large one. In The Sand-Reckoner, he chooses a less-literal interpretation of Aristarchus's enigmatic phrase comparing the surface to the center of a sphere. According to Archimedes, Aristarchus must have meant something like this:

The distance of the stars bears the same relation to the diameter of Earth's orbit as the diameter of Earth's orbit bears to the diameter of Earth.

Or, in equation form:

Distance of the stars: Diameter of Earth's orbit = Diameter of Earth's orbit: Diameter of Earth

Voila! The second ratio is no longer infinite, but a straightforward comparison between two measurable quantities: the diameter of Earth's orbit and the diameter of Earth. These were numbers for which crude estimates already existed, from Aristotle, among others. Using generous assumptions, Archimedes reasons that the diameter of Earth's orbit does not exceed 10,000 Earth diameters. (In reality, he is still too low by a factor of two.) The previous equation now becomes:

Distance of the stars: 10,000 Earth diameters = 10,000 Earth diameters: 1 Earth diameter

Therefore, according to Archimedes, the stars on the celestial sphere are 100 million Earth-diameters away, which dwarf the estimate of 10,000 Earth diameters proposed by the geocentrists. (Yet it still falls far short of the true distance to even the nearest star, which is more than 3 billion Earth diameters.)

Next, Archimedes converts his cosmic distance to stadia, a terrestrial measure used by the ancients and roughly equivalent to a tenth of a mile. For this calculation, he again fudges his assumptions. Remember, Archimedes was trying to impress King Gelon by filling the biggest possible universe, so he unabashedly gooses up the numbers for maximum effect. He takes Earth's diameter to be nearly 1 million stadia, which he freely admits is tenfold larger than the commonly accepted value at the time. At last, Archimedes deduces that the universe is about 100 trillion stadia in radius, roughly 10 trillion miles. (The modern equivalent, around 1.7 light-years, is only some 40% of the distance to the nearest star.)


This is only the halfway point for Archimedes! He now has to count how many sand grains would be required to fill such an expanse. That, of course, depends on the minuteness of the grains. Archimedes had no microscope with which to view samples of sand, much less one fitted with a measuring stage. He approaches the issue by comparing sand to a mote he could more easily measure: a poppy seed. A line of 40 poppy seeds, he informs King Gelon, matches the width of his forefinger; thus, on average, a poppy seed is V40 of a finger-width across. The volume of one such seed, he speculates, would hold up to 10,000 sand grains--a myriad, in the ancient Greek number system.

Having sized up both the vast universe and the minuscule sand grain, Archimedes arrives at his conclusion: to fill space all the way out to the sphere of the stars requires one-thousand trillion trillion trillion trillion trillion sand grains. In modern powers-of-ten notation, the requisite number of sand grains would be written 1063, that is, the numeral 1 followed by 63 zeroes.

This number is so large that it could not be expressed in the ancient Greek system, which represented numbers as strings of letters and which petered out at a measly 100 million. In characteristic fashion, Archimedes invented an entirely new counting system, involving successive powers of 100 million, which handles numbers up to a dizzying [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], or the numeral 1 followed by eighty-thousand trillion zeroes.

How close did Archimedes come to the right answer? Of course, he had no way of knowing the true extent of the universe, which might indeed be infinite. But were we to fill up his assumed 10-trillion-mile cosmos with the finest sand grains--Vk millimeter across, according to the website of the International Sand Collector's Society ( arrive at the number 1061 which, considering the magnitudes involved, is in fair agreement with Archimedes's tally of [10.sup.63]. Archimedes's more diminutive sand grain is actually classified as silt.

If Archimedes were sand-reckoning today, he would no doubt compute how many grains it would take to fill the observable universe. The universe is approximately 13.7 billion years old. Were it not expanding, the farthest object we could see now would be 13.7 billion light-years distant. But the universe is expanding, and the space through which light travels grows. Given the observed rate of expansion, astronomers estimate that the radius of the observable universe is around three times greater than if the universe were stationary, or about 47 billion light-years. Archimedes would find that [10.sup.92] sand grains are needed to fill up the observable universe, a figure well within the limits of his counting system.

Archimedes concludes The Sand-Reckoner by telling King Gelon that "these things will appear incredible to the numerous persons who have not studied mathematics; but to those who are conversant therewith and have given thought to the distances and the sizes of the earth, the sun, and the moon, and of the whole of the cosmos, the proof will carry conviction. It is for this reason that I thought it would not displease you either to consider these things." Whether Gelon went to sleep that night inspired, or with a headache, history does not report.

Alan Hirshfeld is Professor of Physics at the University of Massachusetts, Dartmouth and an Associate of the Harvard College Observatory. His latest book is Eureka Man: The Life and Legacy of Archimedes, published by Walker & Co.

Archimedes 287-212 BC

The greatest scientist of antiquity, and one of history's all-time great mathematicians, Archimedes lived in the Greek city-state of Syracuse on the island of Sicily. Little is known of his life, and some of what has passed through the ages is in the form of apocryphal stories. Take, for example, Archimedes's death, the best accounts of which were written decades later. All agree that Archimedes was killed by a Roman soldier during the sack of Syracuse. And while some suggest that Archimedes was so absorbed in a mathematical problem that he didn't realize the danger at hand, others suggest that he recognized his fate and begged only that the soldier not disturb his calculations.

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Title Annotation:Astronomical Calculations
Author:Hirshfeld, Alan
Publication:Sky & Telescope
Date:Nov 1, 2010
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