Determining Earth’s Size, Lunar Distance, and Solar Distance through Ancient Greek Methods

December 10, 2023

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Abstract

How did the ancient Greeks figure out important astronomical values, and how did they pave the way for modern astronomy? In the past, many archaeoastronomers have discussed the methods and calculations of the Greeks in depth. Many papers have also been written about the growth of astronomy and where it could possibly be headed towards in the near future. This paper combines both of those by doing the calculations of the ancient Greeks and then comparing them to modern techniques. The calculated astronomical numbers included in this work are the radius of the Earth, the distance to the sun, and the distance to the moon. The paper will also discuss how ancient Greek techniques have affected broader astronomy and what methods have survived throughout the years and are still being used.

Introduction

Over two millennia ago, the Greeks were setting up forms of government, creating architectural marvels, and mapping out the sky. All three of these have had profound effects on society. The principles behind the system the Greeks created are still used in modern government structures. Ancient Greek architecture still has a heavy influence on the buildings of post-renaissance society. The Greeks have arguably affected the fields of mathematics and astronomy the most¹⁵. They are responsible for geometry and trigonometry. In the field of astronomy, the ancient Greeks were ahead of their time. They could precisely calculate the length of a lunar month down to the second, among many other amazing feats⁴. Without the Ancient Greek astronomers, our understanding of the celestial bodies around us would not be as comprehensive as it is today. The Greeks paved the way for modern astronomy. It is widely accepted that ancient Greek astronomers laid the basis for modern-day astronomy. Some of their theories and philosophies have stood the test of time and survived for more than two millennia. The work of Greek astronomers before the 4th century is very incomplete, as there are only a few writings available. However, from the writings that are available, it can be said that they knew that the earth was a sphere and not flat¹⁹. There was also an effort to learn more about the stars and celestial bodies. At the same time, the Egyptians and Babylonians, the Greek neighbours, also engaged in astronomy; however, their motivations were different. For example, in Egypt, they used the stars to make a calendar to help them predict the flooding of the Nile river¹³. On the other hand, the Babylonians looked for signs or omens in the sky¹³. The first Greek astronomer to figure out that the earth was round and not flat was Pythagoras¹². Using information such as the fact that the light on the moon was a reflection of the sun from the earth and how eclipses worked, Pythagoras concluded that the earth was spherical and not flat. “On the Heavens” written by Aristotle, contains many popular astronomical beliefs of that time. Xenophanes of Colophon believed that the earth below them was infinite, while others, such as Thales, believed that the earth was on top of water⁵. Greek astronomers after Aristotle had different ideas. Plato, for example, believed that celestial bodies were not governed by natural laws. He used the unpredictable motions of certain celestial bodies to prove his point that natural laws could not predict changes in nature. Plato’s thinking was heavily grounded in myths, as he once described the universe as the Spindle of Necessity, attended by the sirens and turned by the three fates¹³. It was widely accepted at the time that the universe was earth-centred. These claims lacked scientific evidence; however, many Greek astronomers, such as Aristotle, had strong observational evidence of a spherical earth and eventually the circumference of the earth. He took the position of the polar star between Greece and Egypt and then estimated the size of the planet to be around 400,000 stadia The conversion of 1 stadia to modern measurements is still unknown, but it is estimated that 400,000 stadia is equal to around 64,000 km¹³. Even though the figure is much larger than the modern calculated number of 40,000 km. While the calculation is incorrect, the theory behind the calculation is very accurate. If Aristotle had obtained more accurate numbers, he could have reached a more accurate figure. Eratosthenes (276 BCE–195 BCE) would later calculate a more accurate value for the size of the earth. He compared the shadows cast at Alexandria and Syene at the same time. Using the arc length formula, he raised it to a value of 250,000 stadia. 250,000 stadia is estimated to be 40,000 km, which is much closer than Aristotle’s calculations. Eratosthenes’s calculation would remain unparalleled until modern times¹⁴. While the earth-centric model was widely accepted, there were many indications that hinted at its inaccuracy. Many signs, such as the brightness of planets, gave hints to Greek astronomers that an earth-centred model was incorrect. Aristarchus of Samos (310 BCE–290 BCE) was an astronomer and mathematician. Twenty centuries before Copernicus and Galileo, he claimed that our solar system was revolving around the sun and not the earth. He also claimed that the distant stars were like the sun and did not move. He also concluded that the sun was larger than the earth and that smaller celestial bodies orbit larger celestial bodies. Aristarchus’ model was not accepted, and it was considered blasphemy by many important astronomers at the time¹². Another respected and talented astronomer was Hipparchus of Nicea (190 BCE–120 BCE). He managed to calculate the lunar month down to the second and the solar year with an error of 6 minutes. He created a sky database that included the positions of 1080 stars as well as their exact celestial latitude and longitude¹⁷. A star chart, drawn by Timocharis 166 years before Hipparchus, helped Hipparchus determine that the stars had moved two degrees in apparent position and therefore identified and measured the Equinoctial Precession. He predicted the precession to be 36 seconds per year, which is a little less than modern calculations, which put it at 50 seconds per year. He was also responsible for the majority of the calculations found in the Ptolemy’s Almagest, a huge astronomical book written in the second century CE that remained the standard reference for researchers until the time of the Renaissance. Hipparchus refuted Aristarchus’ idea, claiming that the geocentric model explained the observations better than Aristarchus’ model. As a result, he is frequently criticised for reversing astronomical progress by favouring the incorrect earth-entered viewpoint¹⁷. There is a three-century gap between Hipparchus and Ptolemy’s Almagest. Some researchers have labelled this time period as the “dark age” for Greek astronomy, while others feel that the Almagest’s victory wiped out all earlier astronomical works. This is a pointless debate because the significance of a scientific contribution is frequently determined by the number of prior studies it renders obsolete. The Almagest is a massive astronomical work. It incorporates geometrical models linked to tables that might be used to calculate the movements of celestial bodies endlessly. The Almagest compiles all of the Greco-Babylonian astronomical achievements. It has a catalogue with about 1,000 fixed stars in it. For the next 14 centuries, the Almagest’s cosmology would dominate western astronomy^8. Even if it was not flawless, it was accurate enough to be accepted until the Renaissance. The instruments used by Hipparchus and Apollonius provided adequate observational accuracy, allowing the geocentric model to progress, although complete success was never reached. Ptolemy added yet another element to the model to “preserve the appearances”: the equant point. The equant was the point symmetrically opposite the eccentric earth, and the planet had to travel in its orbit in such a way that it appeared to be moving uniformly across the sky from the equant’s perspective. Planets had to alter their speed to meet this need because the equant was offset from the orbit’s centre. In brief, because some of the cosmological model’s essential assumptions were incorrect (e.g., the earth-centred idea, perfect circular orbits), it was necessary to introduce dubious and complicated techniques (e.g., eccentric circles, epicycles, and equant points) to avoid or at least reduce discrepancies. In the end, the Ptolemaic model failed not just due to its errors but also due to the fact that it lacked simplicity⁸. Greek astronomers have had a huge impact on border astronomy. They not only developed excellent scientific expertise, but they also sucessfully used and merged astronomical data obtained from Egyptian, Babylonian, and Chaldean astronomy with their own. Even when they made a mistake, they used their ingenuity to come up with devices to correct their errors¹². The world would not see philosophers with sufficient astronomical expertise to dispute the assumptions of ancient Greek astronomy until the Renaissance, during the emergence of modern science. Complementing their astronomical expertise, the ancient Greek philosophers also developed and improved upon trigonometry as well as Euclidean geometry. Many of these mathematical concepts are still widely used today in a multitude of fields and subjects. The calculations conducted in this paper also tap into these areas of math, which were heavily explored in ancient Greece. Ever since the Renaissance, science has evolved rapidly. Many important discoveries have been made to advance the field of astronomy. In 2019, the first ever picture of a black hole was taken at the heart of the galaxy Messier 87¹. In 2017, astronomers found a system of seven earth-sized planets around 40 light years away (Mann et al., 2017). There have been many new methods to calculate the distance to celestial bodies such as the moon. One interesting method to calculate the distance to the moon was the use of laser ranging in the Lunar Laser Ranging (LLR) project. The premise behind this project is to setup retro reflectors on the moon and then measure the round-trip time to the moon and back. With the time taken, they can use a formula to calculate the distance.

The formula to calculate the distance to the moon: distance = $\frac{speed of light duration of delay}{2}$

While this formula may seem very simple, a lot of factors that lead to the complication of the real formula used. Some of these factors include the atmosphere, the speed at which light travels through different parts of the air, the weather, the lunar phase, and lunar libration. Since the moon’s orbit around the earth is elliptical, the distance to the moon varies but averages at 385,000.6 km from the centre of the earth to the moon. The distance to the Moon may now be measured to millimetre precision as of 2009². In a relative sense, this is one of the most accurate distance measurements ever made, comparable to estimating the distance between Los Angeles and New York to within a hair’s width. Calculations for the size of the earth have also become extremely accurate. While there are a plethora of options available to calculate the size of the earth, the easiest method is to use satellites to map the earth and calculate distances. Given the vast technological achievements of the modern day, the vast astronomical knowledge of the ancient Greeks as well as other ancient civilizations’ previously mentioned knowledge can be tested for accuracy by combining their ancient methods with more accurate modern astrological values. Using modern values, the extreme estimation and variation in the values calculated by the natural philosophers can be replaced by the most up-to-date values. This can isolate the theory from the experimental values that the aforementioned ancient Greek philosophers have performed. Using modern values, the true accuracy of these conceptual methods described and conceptualised by the Greek scholars can be tested and determined. Through these calculations using a fusion of modern and ancient techniques, hopefully more light can be shed on the ingenious solutions of the Greek philosophers.

Methodology

To make an analytical comparison between modern techniques and techniques used by Greek astronomers, we had to try and replicate the calculations of the Greek astronomers. This paper calculates three important astronomical values: The radius of the earth, the distance to the moon, and the distance to the sun. For each of these calculations, two trials were completed to try and eliminate possible errors. To get important numbers and information, such as where the sun is at a certain point in time, stellarium and Google Maps were used. Stellarium helped in the finding of important astronomical values. With the aid of stellarium, precise altitudes were able to be obtained, right ascension, and declination values. These precise values helped my calculations be closer to the true values and remove possible errors in measuring. To measure the distance between the two points on the globe, the measure distance function on Google Maps was utilised. With the aid of Google Maps, distances could be measured with accuracies of up to four significant figures. These two tools were extremely important in my calculations, and were used for all of my calculations.

The radius of the earth

Eratosthenes’s technique was used to derive the radius of the earth. Using two distinct locations on earth, the sun was observed at the same time on the vernal equinox (March 20, 2021). To get the angle to the sun, the altitude function on Stellarium was used. To get the distance between the two points on the ground, Google Maps’ distance measurement feature was used. Once all the information was found, the following equation was used in order the get the radius of the earth: $Radius = \frac{360 * Distance}{2\pi * Angle of shadow}$ . The angle of the shadow being as show in the figures below. The position of both person A and person B is depicted in figure 1a and a better view of the angles regarding the shadow and person B is seen in figure 1b. Plugging all this information into the equation, we are able to estimate the radius of the Earth.

Figure 1b: Up close of the person whose shadows will be measured. Over here is the same as the angle in-between person A and Person B.

Distance to the moon

A method called parallax was used to calculate the distance to the moon. Parallax is the displacement of an object when viewed from two different places. A triangle was used, as shown in Figure 2a, where is the angular difference between the moon when viewed from two different points. Stellarium was used to look at the moon from two places at the same time. Then, using the celestial coordinates, which include right ascension and declination, to measure the angle between them. RA (right ascension) and dec (declination) are essentially the equivalent of longitude and latitude on the celestial sphere, respectively. To get the angle between the two points, the Pythagorean theorem was employed with the triangle shown in Figure 2b. Once the angle of the moon was obtained, when viewed from two different locations, the parallax technique can be used to solve for the distance to the moon (R). To use trigonometry, a right triangle was needed. To do this, the triangle was split in half. Once split, trigonometric functions could be used. Pythagorean was used to get the hypotenuse’s length, which was $\sqrt{r^{2} + (\frac{d}{2})^{2}}$ . Using this hypotenuse length equation, we can use basic trigonometry to solve for r: the distance to the moon. Using a sin function, we can create an equation that can help solve for the distance to the moon.

Figure 2b: The triangle we use to estimate the angle of the moon from the two observers.

Equation used to solve for r: $\sin \frac{\theta}{2}$ = $\frac{\frac{d}{2}}{\sqrt{r^{2} + (\frac{d}{2})^{2}}}$

Equation used to solve for $\theta$ : $\theta = \sqrt{(PersonA_{RA}-PersonB_{RA})^2+(PersonA_{dec}-PersonB_{Dec})^2}$

Distance to the sun

and the moon, is the angle between the sun and the moon during the first quarter.

Calculating the distance to the sun was trickier than the two previous calculations since the distance is much larger. The lunar phases to helped me calculate the distance. In total, there are eight lunar phases. For my calculations, two of them were used: the third quarter moon and the first quarter moon. These two phases are specifically used because when the moon is in these phases, a 90-degree triangle is formed, as shown in Figure 3. The calculations were performed twice. For the first calculation, the first quarter moon in the month of July 2021 was used, and for the second calculator, the third quarter moon in the month of July 2021 was used. Once the dates were obtained, the next step was to find the angular distance $\theta$ between the sun and moon. Since the distance was so large in the sky, using the normal Pythagorean theorem will not work. The reason for this is that the equation to find the distance on a sphere is different from that on a flat surface. In the two previous calculations, Pythagorean could be used as the distance was small. Since the distance was so large in this particular calculation, my mentor, Dr. Taweewat, helped me calculate the distance between the two points using spherical trigonometry from a Python package. Once the angle was obtained, basic geometry could be used to find $\beta$ , the angle in between the moon and earth. To do this 180 needed to be subtracted from $\theat$ and $90:180 - (90 + \theta)$ . Once that was done, trigonometric functions were employed to solve for $d_{S}$ . For the purpose of accuracy, the distance to the moon used in this calculation is the known distance (384400 km), not the distance previously calculated. Using the steps provided above, the numbers generated can be plugged into the trigonometric equation below to compute the distance to the sun.

Equation to find distance to the sun from $\beta:Sin(\beta) = \frac{d_{m}}{d_{s}}$

Equation for $\beta$ = $180 - (90 + \theta)$

Results

1) radius of the earth calculations:

Equation to get radius r: $\frac{distance \cdot 360}{2 \pi \theta}$

r = \frac{3335.86 \cdot 360 }{2\pi \cdot 35.041}

2) distance to the moon calculations:

|\frac{V_{observed} - V_{true}}{V_{true}}|

3) distance to the sun calculations:

	Calculation 1	Calculation 2
Location	Singapore(1°21’50.04” , 103° 50’ 13.19”)	Singapore(1°17’34.44”, 103° 51’ 16.91”)
Date	July 17th 2021 (first quarter)	31 July 2021(third quarter)
Time (SGT)	18:10 SGT	21:15 SGT
Moon (Ra,Dec)	(13h 39m 05.17s, -6° 25’ 26.3”)	(2h 30m 91.6s ,11° 51’ 16.91″)
Sun (Ra,Dec)	(7h 48m 0.3s , 21° 07’ 09.8”)	(8h 43m 53.56s, 18° 07’ 22.1”)
$\theta$	89.92169°	89.5725727°
$\beta$	180 – (90+ 89.92169) = 0.07830°	180 – (90 +89.5725727 ) = 0.42742°
Distance to the Sun	$d_{s} = 2.81 \cdot 10^8$ km	$d_{s} = 5.52 \cdot 10^8$ km
Average Distance to the Sun	$Average d_{s} = \frac{2.81\cdot 10^8 + 5.52 \cdot 10^7}{2}$ = 168100000
True Distance to the sun	151800000 km
Percent Error	$\|\frac{V_{observed} - V_{true}}{V_{true}}\|$ = $\|\frac{ 168100000 - 151800000 }{151800000}\|$ = 10.72%

Since the percent error for all three calculations computed above was under 15%, these calculations are all relatively accurate and close to the true values. While a number like 15% would generally be considered inaccurate, given the simplicity of the methods the Greek natural philosophers used and the complexity of the astronomical value it is attempting to find, under 15% can be considered accurate in the context of these values. Using simple cues such as shadows and basic high school trignometry, it is very impressive that the Greek astronomers were able to get this close to the real values. The calculation that was the closest to the true astronomical value out of the three was the radius of the earth. The reason for this is that the radius of the earth calculation has the fewest variables involved in the calculation. It also had the fewest steps. All that was needed for the calculation was the altitude of the sun and the distance between the two points. Once those two values had been obtained, the calculation was extremely simple. As explained in the methodology section, the sun was the hardest to calculate as the distance was extremely large, and the moon was relatively easy when compared to the sun. Refer to the methodology section for more detailed explanations.

Discussion

Before each of the calculations is discussed, the tool used to obtain the astronomical data must be introduced. Stellarium is free, open-source planetarium software. This powerful tool can be used to access astronomical data at almost any point in time, both in the future and in the past, as well as at any position on Earth. For example, as seen on the right, Stellarium is providing the astronomical values for the sun on September 8, 2040. Among these values, we can see RA, which stands for right ascension, and DEC, which stands for declination. Values such as those computed by Stellarium can be considered accurate enough to try and emulate the Greek techniques. After doing the calculations, some limitations of the greek methods became apparent. There are three main limitations to the Greek methods: measurement error, systematic bias, and uncertainty. Furthermore, for all these calculations, the ancient Greek philosophers assumed ideal conditions, leading to one possible reason for the slight difference between the calculated and theoretical values. To reduce the statistical error, two trials were conducted and the average was used for the final result. The following paragraphs will now go over the limitations of each of the calculations.

1) Radius of the earth

Most of the limitations regarding data and numbers with the use of Google Maps and Stellarium.With these two, precise numbers were available, which makes my calculations more accurate. In this context we can define accuracy as our being calculation being close to the true value of an astronomical constant. So the closer our calculation for the radius of the earth is to the real radius, the more accurate our calculation is. This definition of accuracy in regards to this paper can be carried forward to other uses of this term. Google Maps was used to calculate the distance between the two points. The distance was very accurate, and it went down to thousands of metres. For the astronomical values, such as the altitude to the sun, stellarium was used. With these two values, which were fairly precise, the radius of the earth was calculated. However, Greek astronomers did not have the same technology and equipment that modern-day researchers, such as myself, have access to. They had to estimate the altitude of the sun by hand, and they had to physically travel to measure the distance between the two locations. There are also some flaws in the technique itself. The farther the distance between the two points, the more accurate the number is. For my first calculation, the distance was 5454.49 km, and for my second calculation, the distance was 1000 kilometres larger at 6266.80 km. My second calculation was much closer to the actual radius of the earth: 6,371 km³.

2) Distance to the moon

For the distance to the moon, the parallax method was used. There is a lot of systematic bias when using parallax; however, it did not affect me, as those effects only have a noticeable effect when the object you are measuring is extremely far away. However, there was another factor that made parallax very impractical. If you plan to use trigonometric functions on parallax triangles, it will be very hard, as the angles are extremely small¹⁰. This is evident in my calculations. This would have been a barrier for the Greek astronomers, as they did not have the equipment to calculate such small values, such as modern-day calculators. They would also have had trouble measuring the angle in the parallax triangle. To calculate the angle, we must get the location of the moon at the same time from two different locations. Without modern communication devices, it is difficult to imagine how they would be able to measure such a value. Other than these limitations, this method works quite well to find the distance to objects which have parallax angles greater than 0.01 arcsec.

3) Distance to the sun

For this calculation, lunar phases were used to calculate the distance to the sun. One of the biggest limitations of this method is calculating the distance between the sun and moon. In this calculation, the conventional Pythagorean theorem will not work to calculate the angle between the sun and moon. The reason for this is that you can only do Pythagorean on a flat surface, not a sphere. For the other two calculations, using the traditional Pythagorean theorem was good enough as the error was negligible; however, the distance between the moon and the sun is quite large. When regular Pythagorean was used, the angle between the sun and the moon was greater than 90°, which should not be possible. Once the new method (spherical trigonometry) was used, the number was much more realistic. That was the only systematic bias in this system. The Greek astronomer estimated the angles for this calculation. Aristarchus used 87 degrees for the calculations, which is very far away from the actual value, which is much closer to 90°¹⁶. While these techniques had their limitations, most of the major problems the Greeks faced were due to a lack of technology to determine accurate astronomical data. With the accurate data from Stellarium, the calculations were significantly closer to the actual value. With more accurate values, the Greek methods have the potential to estimate a much better result with smaller errors. One benefit of the Greek methods is that they are relatively simple, making it easy to understand what we are calculating. In most cases, they simply required basic geometry and trigonometry. It was quite interesting to see how I could duplicate their methods with relative ease using just the open sky without any satellites or advanced equipment that we are using today to measure these values. With these calculations, the Greeks paved the way for modern astronomers and, arguably, even space travel.

Conclusion

In summary, our study emphasises the significant contributions of ancient Greek astronomers to the foundations of modern astronomy, as elucidated through the methodology section. We can deem the methods created by the Greeks accurate. All three calculations performed underscore the remarkable astronomical methods and theories given their technological limitations: All three calculations have a percent error of less than 15%. The ancient Greeks used countless techniques to map out the sky and calculate many important astronomical values. From this work, we have shown that their techniques were not only simple but also extremely effective. Scholars like Eratosthenes were able to calculate values with surprisingly extreme precision. Eratosthenes’ method to calculate the radius of the earth was by far the most superior method until the Renaissance period¹⁴. Even though the calculations are considered accurate by the definitions previously mentioned, the actual calculations done by the Greek scholars are severely limited by their access to reliable data and other benefits that modern technology offers. Since they did not have access to this, it impacted the values they found, resulting in them being different from the true astronomical values. While this paper follows mainly Greek astronomers, other ancient civilisations have also heavily impacted and shaped modern science and astronomy. Some of these civilisations could include the Indus Valley civilisation, the ancient Arabs, the Babylonians, and the Egyptians. Understanding and studying how other ancient civilisations, such as the ones mentioned and others that have not been mentioned, shed light and can uncover a broader theme, which can help researchers understand how post-renaissance science developed from these ancient civilisations and its impact on today’s science.

	Calculation 1	Calculation 2
Point A(Equator)	(0,104) (South China sea)	(0,104) (South China sea)
Point B	(30,104) (Sichuan, china)	(70,104)(Krasnoyarsk Krai, Russia)
Alt of sun at Point B	54.959°	18.836°
$\theta$	35.041°	71.164°
Distance	3335.86 km	7783.66 km
Radius	$r = \frac{3335.86 \cdot 360 }{2\pi \cdot 35.041}$ = 5454.49km	$r = \frac{7783.66 \cdot 360}{2 \pi \cdot 71.164}$ =6266.80km
Average Radius	5860.645 km
True Earth Radius	6371 km
Percent Error	$\|\frac{V_{observed} - V_{true}}{V_{true}}\|$ = $\|\frac{5860.645 - 6371 }{5860.645}\|$ = 8.010%

	Calculation 1	Calculation 2
Person A	(0,103)(South China sea)	(0,103)(South China sea
Person B	(20,103)(Luang Prabuang, Laos)	(70,103) (Krasnoyarsk, Russia)
Distance	2223.90 km	7792.4 km
Time & Date (SGT)	24 July 00:00:00	24 July 00:00:00
Moon (RA,Dec) Person A in deg	(281.6917, -26.0803)	(50.4875,17.6539)
Person B (RA,Dec) in deg	(281.6958, -26.3625)	(50.0917,16.6969)
?	0.2822°	1.03562°
R (distance to the moon)	451,523 km	431,108 km
Percent Error	17.4618%	12.1509%
Average Distance to the Moon	$\frac{451523+431108}{2}$ = 441315.5
True Distance to Moon	384400 km
Percent Error	$\|\frac{V_{observed} - V_{true}}{V_{true}}\|$ = $\|\frac{ 441315.5 - 384400 }{384400}\|$ = 14.8063%

Determining Earth’s Size, Lunar Distance, and Solar Distance through Ancient Greek Methods

Abstract

Introduction