One aspect central to the astronomical-cosmological discourse since antiquity was the actual size of the cosmos. This became particularly relevant to the astronomical system debate following Tycho’s star size argument. He argued given his failure to detect the stellar parallax, which should be observable in a heliocentric system, the stars must be so far away that the apparent size of the star discs would mean they must be quite literally unimaginably large and thus the system was not heliocentric. He also argued that under these circumstances there must also be an unimaginably vast distance between the orbit of Saturn and the sphere of the fixed stars. He thought it was ridiculous to suppose that there exists so much empty space, which for him also spoke against heliocentricity.

The earliest known serious attempt to determine the dimensions of the solar system was made by Aristarchus of Samos (c. 310–c. 230 BCE) infamous for proposing a heliocentric theory of the cosmos. We only have second-hand accounts of that system from Archimedes and Plutarch. However, the only manuscript attributed to him is *Peri megethon kai apostematon* (*On the Sizes and Distances (of the Sun and Moon)*). Aristarchus assumed that at half-moon the Earth, Moon and Sun form a right-angle triangle and that the angle between the Earth and the Moon is 87°.

From these assumptions he calculated that the ratio of the Earth/Sun distance to the Earth/Moon distance is approximately 1:19. In reality the ratio is approximately 1:400 because the angle is closer to 89.5° and is not differentiable by the human eye. Also, it is almost impossible to say exactly when half-moon is.

Aristarchus used a different geometrical construction based on the lunar eclipse to determine the actual sizes of the Earth, Moon and Sun.

It is possible to reconstruct Aristarchus’ values (Source: Wikimedia Commons

Relation |
Reconstruction |
Actual Values |

Sun’s radius in Earth radii (e.r.) |
6.7 |
109 |

Earth’s radius in Moon radii |
2.85 |
3.5 |

Earth/Moon distance in e.r.) |
20 |
60.32 |

Earth/Sun distance in e.r.) |
380 |
23,500 |

Hipparchus (c. 190 – c. 120 BCE) used a modified version of Aristarchus’ eclipse diagram, using a solar rather than a lunar eclipse, to make the same calculations arriving at a value of between 59 and c. 67 e.r. for the Moon’s distance and 490 e.r. for the Sun.

As with almost all of Hipparchus’ other writings, his work on this topic has been lost but we have his method and results from Ptolemaeus, who also used a modified version of the solar eclipse diagram to make the same calculations. Ptolemaeus got widely different values for the furthest c. 64 e.r. and nearest c. 34 e.r. distance of the Moon from the Earth. The first is almost the correct value the second wildly off. He determined the Sun to be 1,210 e.r. distant.

In the history of astronomy literature, particularly the older literature, it is often claimed that Copernicus’ heliocentric model leads automatically to a set of relative distances for all the known planets from the Sun, which is true, but there is no equivalent set of measures for a Ptolemaic geocentric system, which is false. It is the case that in his great astronomical work, the *Mathēmatikē Syntaxis* (*Almagest*), he gives detailed epicycle-deferent models for each of the then known seven planets–Moon, Mercury, Venus, Sun, Mars, Jupiter, Saturn–but does not deal with their distances from each other or from the Earth. However, he wrote another smaller work, his *Planetary Hypotheses*, and here he delivers those missing dimensions. For Ptolemaeus each planetary orbit is embedded in a crystalline sphere the dimensions of which are determined by the ecliptic-deferent model for the planet. How this works is nicely illustrated in Georg von Peuerbach’s (1423–1461) *Theoricae Novae Planetarum* (New Planetary Theory) published by Regiomontanus in Nürnberg in 1472.

It was long thought that Peuerbach’s was an original work but when in 1964 the first ever know manuscript in Arabic, and till today the only one, of Ptolemaeus *Planetary Hypotheses* was found it was realised that it was merely a modernised version of Ptolemaeus’ work.

Ptolemaeus’ model of the cosmos was quite literally spheres within spheres, a sort of babushka doll model of the solar system. The Moon’s sphere enclosed the Earth. Mercury’s sphere began where the Moon’s sphere stopped, Venus’ sphere began where Mercury’s stopped, the Sun’s sphere began where Venus’ stopped and so on till the outer surface of Saturn’s sphere. Using this model Ptolemaeus calculated the following values and a value of 20,000 e.r. for the distance from the Earth to the sphere of fixed star and c. 1,200 e.r. for the Earth/sun distance.

Ptolemaeus’ model and at least his basic dimensions–Earth/Moon, Earth/Sun and fixed star sphere distances–remained the astronomical/cosmological norm for nearly all astronomers in the Islamic and European Middle Ages and we first begin to see new developments in the sixteenth century and the so-called astronomical revolution.

In the geocentric model the order of the orbits of Mercury, Venus and the Sun moving away from the Earth and the Moon is purely arbitrary as they all have an orbital period of one year relative the Earth. Ptolemaeus’ order was, in antiquity, only one of several; in fact, he played with different possible orders himself. In a heliocentric system the correct order of the planets moving away from the Sun is given automatically by the length of their orbits. This is, of course, the basis of Kepler’s third law of planetary motion. The relative size of those orbits is also given with respect to the distance between the Earth and the Sun, the so-called astronomical unit. This gives a new incentive to trying to find the correct value for this distance, determine the one and you have determined them all.

Copernicus determined the distances between the Earth and the other planets using his epicycle models and Ptolemaeus’ data, which produced much smaller values for those distances that by Ptolemaeus. Although he appeared to calculate the astronomical unit for himself, however, he chose parameters that gave him approximately Ptolemaeus’ value of 1,200 e.r.

Tycho Brahe’s values were also smaller than those of Ptolemaeus, but he also chose a value for the astronomical unit that was in the same area of those of Ptolemaeus and Copernicus. Tycho’s failure to detect stellar parallax led him to argue that the parallax value for the fixed stars, if it exists, must have a maximum of one minute, i.e. one sixtieth of a degree, meaning that in a Copernican cosmos the fixed stars must have a minimum distance of approximately 7,850,000 e.r. Copernicans had no choice but to accept this, for the time, literally unbelievable distance. Tycho himself set the distance of the fixed stars in his system just beyond the orbit of Saturn at 14,000 e.r.

Up till now all of those distances had been calculated based on a combination series of dubious assumptions and rathe dodgy geometrical models, this would all change with the advent of Johannes Kepler in the game. Through out his career Kepler returned several times to the problem of the distance of the planets from the Sun expressed relative to the astronomical unit. By the time he wrote and published his *Harmonices Mundi* containing his all-important third law of planetary motion in 1619, the values that he had obtained were largely correct, but he still had no real measure for the astronomical unit or from the distance of the fixed stars. For his own estimate of the astronomical unit Kepler turned to a parallax argument. He argued that no solar parallax was visible, not even with the recently invented telescope, so the parallax could be, at the most, one minute i.e. one sixtieth of a degree. This would give him a minimum value for the astronomical unit of c. 3,500 e.r., three times as big as the Ptolemaic/Copernican value. As a convinced Copernican Kepler was more than prepared to accept Tycho’s argument for very distant fixed stars, his minimum value was 60,000,000 e.r.

Because the astronomical unit was essential for turning his relative values for the distances of the planets into absolute values, over the years he considered various methods for determining it. He even reconsidered Aristarchus’ half-moon method, hoping that the telescope would make it possible to accurately determine the time of half-moon and measure the angle. His own attempts failed and in his ephemeris for 1618 he appeals to Galileo and Simon Marius to make the necessary observations. However, even they would not have been able to oblige, as the telescopes were still too primitive for the task.

For once Galileo did not take part in the attempts to establish the dimensions of the solar system, accepting Copernicus’ values. He did make some measurements of the size of the planets, a parallel undertaking to determining the planetary distances. He never published a systematic list of those measurements preferring instead just to snipe at other astronomers, who published different values to his.

Kepler’s work was a major game changer in the attempts to calculate the size of the cosmos and its components. His solar system has very different dimensions to everything that preceded it and for those supporting his viewpoint it meant the necessity to find new improved ways to find a value for the astronomical unit.

* All diagrams and tables are taken from Albert van Helden, Measuring the Universe: Cosmic Dimensions from Aristarchus to Halley, University of Chicago Press, 1985, unless otherwise stated.

I like you blog and this series in particular very much and I have two questions for you:

1) Did you ever write a post on the discovery that the “fixed stars” are not actually fixed? If yes, can you link it to me? If not, do you think you’ll write that in the future (maybe even in this series)?

2) Which book would you recommend on the history of geometrical optic to someone interested in it?

The discovery that the sphere of the fixed stars is not fixed is not a topic that I have handled to date and I wasn’t intending to include it in this series, although it probably does belong here. Maybe I’ll rethink that.

Before the end of the year I intend to write a whole blog post entitled “Telescopes, Light and Optics” discussing bibliographical sources for the study of the history of the three topics.

Thanks for the infos, but i have a new question for you: in a previous post you said that the transits of Venus and Mercury were seen as a sort of indirect proof of Kepler’s sistem, but doesn’t the other sistems (mainly the Ursus one) predicts that the transits should happen?

“…but doesn’t the other sistems (mainly the Ursus one) predicts that the transits should happen?”

Yes, but throughout the history of astronomy astronomical tables played a very significant role in the acceptance of systems and/or an astronomer’s work and in this case it was Kepler’s Rudolphine Tables, which accurately predicted the transits that were pivotal. The Rudolphine Tables did more to cement Kepler’s reputation than any of his other works. The equivalent Tychonic tables produced by Longomontanus had far less impact.

Here is a good article on the measurement of a star’s diameter by Galileo

http://articles.adsabs.harvard.edu/full/seri/JBAA./0111//0000266.000.html

For the Aristarchus method of determining the Earth-Sun distance, 3º corresponds to ~0.77 arcminutes as seen by the observer on Earth (the deviation from true straightness of the terminator). An observer at a single point on the Earth’s surface would never be able to observe the moon around first quarter over the length of time required to interpolate between ±3º because this would take 12 hours and the maximum observing time would be no more than half of this (from moon on eastern horizon and sun at zenith to moon at zenith and sun on western horizon). Hence one could never correct for the systematic error of taking the first measurement where the terminator appeared straight. [1]

[1] In theory one could if the observer was at the North Pole and the measurement was made in mid-summer; however all these measurements were made at mid-latitudes.