By Robert Trout

“But Aristarchus of Samos brought out a book consisting of certain 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 motionless, 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 center of 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 center of the sphere bears to its surface.”

**Archimedes, in “Sand-reckoner”**

In 1543, Copernicus published his book, “On the Revolutions of the Heavenly Spheres,” which located the sun and not the earth at the center of the solar system. He was severely censured for this by Aristotelian circles. In his handwritten and autographed copy, Copernicus had written “… Philolaus perceived the mobility of the earth, which also some say was the opinion of Aristarchus of Samos….” This sentence was not included in the printed version. Perhaps, he judged the publication of it simply too risky, because it would had exposed how the system of Ptolemy, which had been imposed on the West for 1300 years, had been a fraud.

Aristarchus (ca 310-230 B.C.) is credited by his younger contemporary Archimedes, and many other commentators, with establishing that the earth rotates around the sun and not the other way around. In his one still extant treatise, “On the Sizes and Distances of the Sun and Moon,” he demonstrated his method for calculating the sizes and distances of the sun and moon, which dramatically changed man’s estimate of the size of the solar system. We will examine how Aristarchus’s discoveries are a culmination of a project launched by Plato to explain the apparently erratic cycles of the universe with “uniform and ordered motions.” Aristarchus’s created a paradox, and then resolved it by a creating a Platonic idea which explained the most basic cycles of the earth’s relationship with the sun, moon and the Universe.

Aristarchus begins his treatise by demonstrating that an observer on the earth can determine when the sun, moon, and earth are situated, so that their relationship to each other is described by a right triangle, and to measure the angles of that right triangle. From this, he was able to calculate an estimate of the ratio of the distance of the earth to the sun, relative to the distance of the earth to the moon.

Aristarchus demonstrated that the phases of the moon are caused by the sun shining on the moon from different directions. Aristarchus knew the basic principles of eclipses from Anaxagoras. He also knew, probably from the previous work of Pythagoras and Anaxagoras, that the moon is a sphere. Only the side of the moon, which is facing the sun, is illuminated, and is visible to an observer on earth. When the moon appears to be near the sun in the sky, the sun is actually further away and behind the moon. The sunlight, then, falls on the back side of the moon, so an observer on earth can see only a small sliver of the moon, if anything. When the sun and the moon opposite each other in the heavens, then the sun lights up the side of the sphere which faces the observer on earth. The moon then appears full. You can demonstrate this by shining a flashlight on a small ball from different directions, and observing how the lit portion of the ball appears.

He demonstrated that when the moon appeared to be exactly half full, the angle from the sun to the moon to the earth, was then very close to a right angle. (This requires that the distance between the sun and moon is large, relative to the diameter of the sun, as it is.)

Make a drawing to demonstrate what Aristarchus was doing. (Figure 1) Near the edge of a sheet of paper, draw a small circle, representing the moon, and label its center, M. Below M, draw a second circle, representing the earth. Draw a line through the centers of the two circles. Label the point where this line intersects the top of the circle representing the earth, E, which represents the position of the observer on the earth. Finally, draw a line, which is perpendicular to the first line, through the point M. The sun will be represented by a third circle, whose center is labelled S. If the center of the sun, S, is on this perpendicular line, Angle SME will be 90 degrees, and the moon will appear, to the observer on the earth at point E, to be almost exactly half full.

Now place the sun, with it’s center, S, on the perpendicular line, at different distances from the moon, and measure the angle which an observer, standing at point E, will see between the sun and the moon (angle SEM). When the sun is close to the moon, Angle SEM will be small. As you move the sun further away from the moon, Angle SEM will become larger.

Aristarchus, in studying the relationship between the actual sun, moon and earth, calculated that when the moon was half full, indicating that Angle SME was approximately a right angle, Angle SEM was 87 degrees. He now had calculated two of the angles of the triangle SEM. From this he was able to use the knowledge of geometry, which the Greeks had discovered at that time, to calculate SE/ME, or the ratio of (the distance from the earth to the sun)/(the distance from the earth to the moon).

There are a number of ways to do this. Neither trigonometry nor pocket calculators had yet been developed. Instead, Aristarchus solved the problem by using the knowledge which the Greeks had developed of relationships between triangles, to come up with an approximation for the ratio of the two distances. It can be seen from the diagram, that when Angle SEM approaches a right angle, the ratio of the sides, (SE/ME) becomes large. He calculated that the distance from the earth to the sun was approximately 18 to 20 times the distance from the earth to the moon.

Since Aristarchus estimated that the sun was 18-20 times further away than the moon, and they appeared to be around same size in the sky, as is demonstrated by eclipses of the sun, he used the principles of similar triangles, which had been developed by Thales, to conclude that the diameter of the sun was approximately 18-20 times the diameter of the moon.

The actual value of the ratio SE/ME is around 389, which is around 20 times greater than Aristarchus’ estimate of approximately 18-20. The actual value for the angle SEM is around 89.9 degrees. The error in his measurement of this angle probably resulted from the difficulty of determining when the moon appears exactly half full, and not his inability to accurately measure angles. Try reproducing his experiment. You will see for yourself that the angle SEM is clearly near 90 degrees, although it is difficult to determine this angle with greater precision than Aristarchus did.

Aristarchus also measured the distance to the moon, using the moon’s diameter as his unit of measurement. An observer on the earth will see the moon as a small disk in the sky. Thales is reported to have measured the angular size of the moon at 1/2 degree, approximately 300 years earlier.

To demonstrate Aristarchus’ method, draw a circle approximately 2 inches in diameter representing the earth and label a point on this circle, P, representing the position of an observer on the earth. (Figure 2) Approximately 5-6 inches away from point P draw a circle of around 1/2 inch in diameter, representing the moon. (This is not a scale model.) Now draw lines from the observer’s point, P, tangent to the two sides of the circle representing the moon. Draw a line connecting the two points of tangency which are on the opposite sides of the circle representing the moon. (Aristarchus demonstrated that the length of this line is very close to the diameter of the circle representing the moon.) Since the two lines from the observer at point, P to the two points of tangency are the same length, this creates a long slender isosceles triangle.

Using principles of geometry, which were then known to the Greeks, Aristarchus was able to calculate the ratio of the length of one of the long sides of the triangle, to the length of the short side. This ratio represented the earth-moon distance, measured in moon diameters. Aristarchus calculated that the distance to the moon was approximately 26 times the diameter of the moon. Strangely, Aristarchus used an angular displacement for the sun and moon of 2 degrees in this treatise. This is 4 times larger than the 1/2 degree which was reported by Archimedes to be known to Aristarchus. By using 2 degrees as the size of the angle at P, he decreased his estimate of the distance to the moon to approximately 1/4 of what he would have calculated, had he used 1/2 degree.

Aristarchus then combined these discoveries with another piece of experimental evidence, which was known from studying eclipses of the moon, to calculate the ratio of the size of the earth to that of the sun and moon. The Greeks had estimated, by measuring the amount of time that it took the moon to travel across the earth’s shadow during an eclipse of the moon, that the shadow, which the earth made on the moon, was approximately twice the size of the moon. Knowing this, he was able to use the geometrical relationships which existed between the sun, earth, and moon, during an eclipse of the moon, to calculate the size of the earth relative to the sun and moon.

Make a drawing which is a simplified version of Aristarchus’ calculations in his treatise. (Figure 3) This drawing will not be to the correct scale, to represent the sizes and distances in either the actual solar system or Aristarchus’ estimates of them. To do that, you need to know the answers that you are seeking. (It would also require either a very long piece of paper, or else drawing the sun, moon and earth so small that you would not be able to see the geometrical relations clearly.) Near the right edge of the paper, draw a circle, with a radius about 1 1/2 inches, representing the sun. Label it’s center S. Draw a circle to represent the earth, with its center, E, 5 inches to the left of S. Make the radius of this circle about 1/2 inch. Draw a line through the points S and E, and extend it 3 to 4 inches to the left of E. Next, draw a line, tangent to the top of the two circles, and a line, tangent to the bottom of the two circles. These lines will intersect at a point which is located on the first line, if your drawing is reasonably accurate. Label that point, A. Also, label, as U, the point where the upper tangent line intersects the circle representing the sun, and label, as F, the point where the upper tangent line intersects the circle representing the earth. Draw lines connecting U to S and F to E.

The earth’s shadow forms a cone on the side away from the sun. When the moon travels through this cone, there is an eclipse of the moon. The two lines that are tangent to the sun and earth represent the boundary of this shadow cone. The Greeks had estimated that, at the distance where the moon traveled through this shadow cone, the radius of the shadow cone was twice the radius of the moon.

Finally, draw a point M on the line AS approximately 1 1/2 inches to the left of E. This will represent the center of the moon, during an eclipse. At point, M, draw a line, perpendicular to the line AS, so that it intersects the line UA. Label the point where it intersects the line UA, N. The line NM represents the radius of the shadow cone of the earth at the distance where the moon travels during an eclipse of the moon.

From these relationships, Aristarchus constructed 3 similar triangles, AMN, AFE and AUS. He goal is to find the ratio of the earth’s radius relative to the sun and moon. He has at his disposal the following. He knows that the radius of the earth’s shadow on the moon, NM is approximately twice the length of the radius of the moon. If MN is twice the length of the radius of the moon, and the radius of the sun is 18-20 times the radius of the moon, this establishes a ratio of US/MN at between 9 and 10. He has estimated the distance from the sun to the earth, ES, at 18-20 times the distance of the earth to the moon, EM. From knowing these ratios, he is able to use the relations between the three similar triangles to calculate an estimate of the ratio of the earth’s radius to the radius of the sun and moon.

I will not go through Aristarchus’ calculations, which are available in his treatise. They are made even more complicated by the relatively unadvanced state of Greek mathematics. Archimedes estimation of the value for pi, with only the mathematics available at the time, was once described as the equivalent of running the hurdles while wearing weights. Aristarchus calculated that the diameter of the sun was approximately 6.8 times that of the diameter of the earth. He also calculated that the diameter of the moon was around .36 the diameter of the earth. He now had established a value for the diameter of the sun and moon, using the diameter of the earth as his measuring stick.

Aristarchus now had values for the distances to the sun and moon using the earth’s diameter as his measuring stick. Since he had calculated the distance from the earth to the moon, EM, at 26 moon diameters, and the distance from the earth to the sun, ES, at 18 to 20 times EM, he arrived at values of approximately 9.5 earth diameters for EM and 180 earth diameters for ES.

Eratosthenes of Cyrene (ca 276-194 B.C.) was, like Archimedes, from the generation after Aristarchus. He made a remarkably accurate estimate of the diameter of the earth. With Aristarchus’s and, then, Eratosthenes’s discovery, the Greeks had demonstrated that the sun and moon could be measured with the same units that were used to measure the earth.

Aristarchus’s estimate of, especially, the distance to the sun dramatically expanded the size of the universe over previous conceptions. For example, Anaximander, a younger friend of Thales, had estimated the distance to the moon at 19 earth diameters and the distance to the sun at 28 earth diameters. Anaximander had also thought that the planets and fixed stars were closer than the sun and moon.

Although there was a large error in Aristarchus’s measurements, his discoveries were a crucial experiment which demonstrated that the sun, which appeared to sense certainty to be only a relatively small disk in the sky, was dramatically larger than the earth. Aristarchus’s had created a paradox which he could resolve, only by overthrowing the earth centered conception of the universe that was accepted at the time, as we shall see in part II.

How Aristarchus Measured the Universe, Part II

By Robert Trout

In Part I, we saw that Aristarchus discovered how to use the geometrical relationship, that exists between the sun, earth, and moon, when the moon is half full, to calculate the relative distances from the earth to the sun and moon. He then calculated the ratio of the distance to the moon relative to the diameter of the moon, using his knowledge of the relationships contained in an isosceles triangle. Finally, he used the geometrical relationships that exist between the sun, earth, and moon during an eclipse of the moon, to calculate the size of the sun and moon relative to the earth, showing that the sun was dramatically larger than the earth or moon. He was then able to measure the sizes of the sun and moon, and their distances from the earth using the earth’s diameter as his measuring stick.

Aristarchus’ methods of measurement were crude. However, his experiment demonstrated that the sun was not just larger than the earth and the moon, but very much larger. The ratio of the volume of 2 spheres is based on the cube of their diameters. He calculated that the volume of the sun was around 6860 times larger than the volume of the moon. Likewise, he calculated that the volume of the sun was around 315 times larger than the volume of the earth.

Aristarchus did not state his hypothesis that the earth orbits the sun in his treatise, “On the Sizes and Distances of the Sun and Moon.” The treatise, that Archimedes quoted, in which Aristarchus stated this hypothesis, has been lost. Its disappearance is undoubtedly a result of the suppression of true science, that occurred with the imposition of the hoax of Ptolemy. However, we can reconstruct how Aristarchus’s discovery of the relative sizes of the sun, earth and moon represented a paradox, which could only be resolved by developing a new hypothesis of the universe, with the earth orbiting the sun and not the other way around.

To sense certainty, the earth appears to be 99% of what one can see. All around, one can see the earth. The sun and moon appear to be very small discs in the sky with an angular diameter of 1/2 degree, that rise, cross the sky and set. The stars are only tiny specks in the night sky. The earth feels rather solid and unmoved, with only earthquakes a rare exception.

A good example of a cosmology, which was based simply on assertions of sense certainty is the work of Ptolemy. Approximately 400 years after Aristarchus’s work, (and after the Roman Empire had driven the region into a dark age) Ptolemy established a fraudulent cosmology, with the earth again at the center of the universe. Ptolemy stated that “the fact that the earth occupies the middle place in the universe, and that all weights move towards it, is made so patent by the observed phenomena themselves.” His “proof” that the earth was the center of the universe consisted in arguing that all objects “which have weight” fall towards the earth. It is a dramatic demonstration of the dark age into which Europe had descended, that the ideology of Ptolemy, who argued that the nature of the Universe could be determined, by what he “saw” objects doing, within a few hundred feet from the earth, was enforced on Europe as the only acceptable view of cosmology, or the study of the heavens, for 1300 years. Nicholas of Cusa, with his “On Learned Ignorance,” again overthrew the earth centered system of cosmology.

Aristarchus’ experiments has rudely overthrown sense certainty. Aristarchus had constructed, in his mind, geometrical relationships which allowed him to determine the relationship between the sun, moon, and earth which his eyes were unable to see. Aristarchus had demonstrated through reason that the sun, that small disc in the sky which appeared to circle the earth, was far larger than the earth, estimating its size to be around 315 times the size of the earth. The idea that the earth was the center of the universe, while the sun, which orbited it, dwarfed the earth in size, created a paradox. Aristarchus, whose method was based on rejecting the fetters of sense certainty, was able to construct a new hypothesis, which placed the larger body in the center. His new hypothesis also solved a host of other paradoxes, which were inherent in an earth centered conception of the Universe.

Aristarchus had as a precedent, the discovery of Heraclides of Pontus, who had demonstrated, less than 100 years earlier, that two planets orbited the sun. His hypothesis that Mercury and Venus orbited the sun explained their seemingly highly erratic movements with a “uniform and ordered movement” of circular motion. Aristarchus’ hypothesis, that the earth was also revolving around the sun, would resolve the paradox which he had created, and explain, what appeared to be otherwise erratic and arbitrary motions of the outer planets and stars, and the cycles of the earth, with “uniform and ordered movements.”

This was, of course, a tremendous leap at that time. Plutarch described Aristarchus’s hypothesis as, “Only do not, my good fellow, enter an action against me for impiety in the style of Cleanthes, who thought it was the duty of the Greeks to indict Aristarchus of Samos on the charge of impiety for putting in motion the Hearth of the Universe, this being the effect of his attempt to save the phenomena by supposing the heaven to remain at rest, and the earth to revolve in an oblique circle, while it rotates, at the same time, about its own axis.”

Make a diagram to represent Aristarchus’s hypothesis. (Figure 4) Draw 2 concentric circles. The center of the two circles represents the sun. The inner circle represents the path of the earth’s orbit. (That the orbits or the planets are actually ellipses was, of course, a discovery by Kepler not to be examined here.) According to the view of the universe, prevalent at that time, the stars were located on a celestial sphere. (Many of the leading thinkers of the time rejected the idea that the stars were located on a physical sphere, including undoubtedly Aristarchus.) The outer circle represents the celestial sphere. You can draw little constellation around it, if you like.

Now, let’s look at how Aristarchus’s hypothesis corresponds to Plato’s research project of finding “what are the uniform and ordered movements by the assumption of which the apparent movements of the planets can be accounted for.” His hypothesis of the motion of the earth combined 2 rotations. First, the rotation of earth about its axis would explain the daily cycle of the sun and the nightly rotation of the fixed stars. Second, the yearly orbit of the earth around the sun, explains the apparent shifting of the “fixed stars” from night to night. Each night, the position of the “fixed stars” appears to have rotated slightly less than one degree, making almost one complete rotation in a year. (Although, not exactly a complete rotation. Hipparchus discovered, a century later, that these two cycles were subsumed by another much longer cycle.) It should be clear from studying this diagram, that as the earth travels around its yearly orbit about the sun, the view that an observer, located on the earth, will have of the celestial sphere, will shift from night to night.

The path of the sun across the sky also varies on a yearly cycle, corresponding to the yearly cycle of the seasons. Aristarchus’s hypothesis in which the earth revolves around the sun “in an oblique circle,” or slanting or sloping circle, can, potentially, explain this.

Conceptualize Aristarchus’s hypothesis that the earth’s orbit is an “oblique circle.” The earth is travelling, yearly, around the sun in a circular orbit, while rotating, daily, on its axis. Imagine the northern direction of the earth’s axis as pointing “straight up,” while imagining that the plane of the earth’s orbit around the sun is sloped at an angle relative to the earth’s axis. (Although, globes are constructed to have the north pole pointing upward, there is no basis in the physical universe for this assumption. Since you are assuming that the north pole is pointed upward, this makes the direction of the rotation of the earth on its axis, and its orbit around the sun both counterclockwise.) During the part of the earth’s orbit around the sun, that you are imagining to be the “higher part,” the southern hemisphere is more directly exposed to the sun. During the part of the earth’s orbit, that you are imagining to be the “lower part,” the northern hemisphere is more directly exposed to the sun. This explains the yearly variation in the path, that the sun makes across the sky each day, as seen by an observer on the earth, and the resulting variations in the seasons.

Eratosthenes also studied the question of the tilt of the earth’s axis relative to the plane of the earth’s orbit around the sun, approximately 25 to 50 years later. Eratosthenes measured the angle between the earth’s axis and the plane of its orbit around the sun with a remarkable accuracy, being in error by only around 0.1 degree from the currently accepted value.

Finally, Heraclides of Pontus had demonstrated that the apparently erratic motion of the inner planets, Mercury and Venus could be explained as circular motion around the sun. Aristarchus’s hypothesis would allow one to comprehend that the apparently erratic motion of the outer planets also corresponded closely to rotation around the sun, as seen from an earth, which is also rotating around the sun.

As a result of Aristarchus’s hypothesis, numerous movements in the universe, including a number of the most important conditions regarding human existence, such as the ordering of the seasons, which according to sense certainty, just are that way, could be determined as the consequence of “the uniform and ordered movements by the assumption of which the apparent movements of the planets can be accounted for.”

In closing, let’s examine the last part of Archimedes statement that Aristarchus hypothesized, “that the sphere of the fixed stars, situated about the center of 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 center of the sphere bears to its surface.” To restate this, Archimedes attributed to Aristarchus the idea that the ratio (circle of the earth’s orbit)/(distance from the earth to the fixed stars) equals the ratio (the center of the sphere)/(its surface). Archimedes criticized Aristarchus for saying this stating, “Now it is obvious that this is impossible; for since the centre of a sphere has no magnitude, it cannot be conceived to bear any ratio to the surface of the sphere.” However, Aristarchus was probably developing a metaphor to illustrate that the distance to the stars was such a large distance, that the circle of the earth’s orbit around the sun would appear to be point in comparison.

Since this work, which Archimedes quoted, has been lost, we must reconstruct how he could have estimated the distance to the fixed stars. The geometrical methods, which he used to measure the distance to the moon and sun, would work well to solve this problem.

Aristarchus has now established the hypothesis that the earth is orbiting the sun following a circular orbit, with a diameter estimated at 180 times the earth’s diameter. Aristarchus had already expanded the size of the universe tremendously over the prevailing view.

Go back to your drawing of the two concentric circles representing the earth’s orbit and the celestial sphere. As the earth travels around it’s orbit during the year, the earth will be closer to the part of the celestial sphere which is directly overhead at midnight. A star will be closer to the earth when it is directly overhead at midnight, and approximately the diameter of the earth’s orbit around the sun further away from the earth when it is very near the sun in the sky.

The angle between two adjacent stars should be larger when the earth is close to them. Aristarchus, who was very adept at this type of measurement, could have calculated the size of the celestial sphere, based on measuring the change in the size of the angle between two stars, when the earth is near them, versus when the earth is on the opposite side of its orbit from them.

Draw two dots, representing stars, on the celestial sphere, maybe 10 degrees apart. Pick a point on the circle of the earth’s orbit which is near these two dots, and draw 2 lines from this point to the two stars. The angle between these two lines represents the angle, that an observer on the earth, would see between the two stars. Next pick a point on the opposite side of the earth’s orbit, and draw 2 lines from this point to the two stars. (Don’t pick a point directly across, or the sun will block the view of the stars.) The angle between those two lines will be less than the first angle that you constructed, at the the other point on the earth’s orbit, which was nearer to the two stars.

The size of the celestial sphere’s diameter relative to the size of the diameter of the earth’s orbit, can be estimated by comparing the difference in the size of these two angles. If the celestial sphere is only a little larger than the circle of the earth’s orbit, the difference between the 2 angles will be large. If the celestial sphere is much larger than the circle of the earth’s orbit, then the 2 angles will be closer to the same size.

However, Aristarchus must have found that the angle, between two adjacent stars, did not appear to change, regardless of whether the earth was near to or far from those stars. He could only conclude that the distance to the stars must be so large, that the distance to them could not be measured using this method. There was probably no other method that could have measured the distance to the stars, with the means available at that time. This would have led him try to communicate his discovery of the immense size of the universe, by describing the distance to the stars relative to the diameter of the earth’s orbit, by comparing it to a “ratio between a sphere and a point.” Aristarchus had expanded man’s conception of the Universe beyond the wildest imagination of the majority of men who still had their minds stuck down in the mud of sense certainty!