Prehistoric civilizations possessed astonishing mathematical knowledge, usually accompanied and driven by a manic preoccupation with the measurement of time and the survey of the objects visible in the sky. In some cases, these historic calendars measured the lengths of the year and of the month to an accuracy that was only achieved again in the 19th century. There is no way around the conclusion that a civilisation with outstanding knowledge was lost in an interim period. Globally congruent – or at least very similar in content – myths detail such a civilization’s collapse. Accordingly, it was not so much the consequence of wars, but high-cultures perished as a result of environmental disaster.
A great deal of what was rediscovered during the renaissance had previously been lost from the conscious canon of knowledge in an alarmingly short time. Beyond the known loss we can hardly imagine what had irretrievably disappeared. The loss must have been immense. We know definitely that within a few hundred years the Old World managed to completely lose the level of education and knowledge attained by ancient civilizations and fell back to the level of a ‘disc world’ full of magic and superstition. Without monasteries, the Byzantium and knowledge-preserving Arabs the loss of knowledge would have been greater and probably final. Thus, only a little was saved and mostly by mere luck. Historically the religious zealotry of the monotheistic religions acted (and still act) as highly efficient purgers of the past or rewrites it according to their teachings.1 Berossos (contemporary of Alexander the Great) asserts that at his time in Babylon reports dating back as far as 150,000 years were still preserved.2 This time scale contradicts all dating of official early history. About Berossos and his work, we know little. Fragments of his books are only preserved in quotations, mainly in the writings of Eusebius of Caesarea (500 years later).3 Other writers from the pre-Christian era reported that Egyptian archives also dated back into a similarly far past.
It is not only in Mesopotamia and Egypt, where we find the remnants of lost knowledge, the situation in other places looks similarly disastrous. We know for example that the Chinese emperor Tsin-chi-hoang-ti (246-209 BC) excelled in the destruction of ancient knowledge. He not only built the Great Wall of China but also let all ancient books burn – except those for medicine and moral teachings.4
When we talk about antiquity, we usually refer to civilisations that date back no more than 3500 BC at maximum. Surprisingly, high cultures arose simultaneously in distant regions. This temporal coincidence of the flowering of these – to us – ‘new’ historic civilizations after the first and prehistoric dark age goes beyond coincidence. To many, this contemporaneity awakening of ancient civilizations in different regions suggests a global power acting in the background.
To become concrete in this assessment, the 13th August, 3114 BC is the Mayan calendar’s zero hour. On this day their famous Long Count of their new era began. Only slightly different to the Mayan calendar, the Indian book ‘Surya Siddhanta’ designates the hour zero at midnight on the 17th to 18th February 3102 BC. On this night, for the Hindu, the current Kali-Yuga age began. According to C. W. Beyer’s contribution 5 in S. J. Baumgarten’s book, the beginning of the Egyptian calendar falls on March 22nd of the year 3069 B.C. which is close to the two dates above.6
Additionally, incredible parallels can be drawn between the beginning of the Mayan Calendar and the emergence of calendars in other regions across the ancient world whose people also chose this century as the beginning of their epoch. Persian and Chinese myths report that four bright stars acted as the guardians of the four particular directions (equinox and solstice) in the nightly sky. If we assume the sunrise at the date of the equinoxes and the solstices as the time meant we detect a remarkable accord. 5,000 years ago, four stars of the 1st magnitude (Aldebaran (14.), Antares (16.), Regulus (21.) and Fomalhaut (18.)) were seen in the morning sky in such a way that they could be considered to act as the guardian of sunrise. (Note: The brackets behind each star indicates its rank in the list of the brightest stars at the firmament.)
Not only is the close accordance of the date on which the calendars began of historian interest, but also their unbelievable accuracy conveys a clear message about scientific and mathematical knowledge possessed at this time. The Mayan calendar even surpasses the precision of the Gregorian calendar in use today. Not only did the Mayan calendar measure the duration of the Earth’s orbit around the sun more accurately than our current calendar, but the Maya gave an even more precise value for the average duration of the Moon’s orbit around Earth. The precision achieved is all the more remarkable as the Moon in deserts or regions with clear skies may have played an important role in everyday life as a nightly source of light. But of what use is its dull light in the rainforest or cloudy regions of the world?
Compared to today’s most exact value of the length of a month (29.5305883 days), the Maya attributed a period of 29.5302 days to the synodic lunar month, which is 0.0013% shorter than today’s most precise current value. As accurate as this value may appear, in fact, it was even more precise. Since the distance between Moon and Earth increases by 3.8 cm every year, the orbital period increases by 9.4 s in 100 years. Given the length of a Mayan lunar month (source Codex Dresdensis) at 357 years ago the distance was 13.6 m narrower than today. At that time the Mayan value was without any deviation from the best modern value. The accuracy achieved is also unbelievable because the duration of the Moon’s orbit is subject to numerous fluctuations. The lunation – determining the variation of the duration of a lunar orbit – is one of the most difficult celestial-mechanical calculations. Without support by mathematical calculations, centuries of observations are necessary to average these fluctuations and achieve the above accuracy. Allegedly the Maya (and their earlier cultures) proceeded in such a way. Really?
Again, how exactly the Maya measured time illustrates their value for the length of the year. The deviation from the current best value (365.24219878173 days) occurs at the 7th digit. Is this still inaccurate or was the value 365.242… factually used deliberately broken off? Good reason exists. In fact, increased accuracy becomes absurd when one must wait many thousands of years before a leap day has been inserted or omitted. The difference in numerical values between the Mayan calendar and the best-known current value – not relative to the currently used calendar – is 0.002%. The error of 26 seconds, by which the Gregorian calendar indicates the year too long, is reduced in the Mayan calendar to 7 seconds too short.
To speak of an error is also misleading because the Earth slows down in its rotation due to tidal friction. In the accuracy, range quoted the prolongation of the period by 0.0016 s per century must no longer be neglected. 4,500 years ago, no deviation existed between the exact year length and the ancient Mayan value. The Mayan calendar was then – within the scope of today’s measuring accuracy – exact. If it was actually introduced in 3100 BC, the deviation between the civil and astronomical calendars would have been unmeasurable even by today’s methods and instruments.
How was it possible for a people who technologically barely got beyond the Neolithic Age to determine the length of the year to the millionth of a millionth? In the literature, a solution is circulated, which allegedly allows for this accuracy using the simplest measurement method. The solution and a – supposedly manageable – remaining challenge lies in a sufficiently long measurement period.
The Maya – and their predecessor cultures – actually erected buildings that functioned as chronometers. Only at a certain hour on a single day in the year did sunlight pass through a fissure or an aisle. If the days between two unique events were counted correctly, over time a constant refinement was achieved. According to this explanation, although the result looks sophisticated it was determined by mere brute force.
So far, so easy? Rather not. According to this theory, the Maya had to record and keep their books flawlessly during the whole counting period. After finishing the counting, they adjusted their calendar in a highly complex way. Allegedly, they completely relied on integers, their calendar inserted 13 days every 52 years and subtracted 25 days every 3,172 years. Since their calendar begins in the year 3114 BC, they made the last correction only one time before the conquistadors annihilated their culture. However, this explanation of the long counting regarded as the solution to the problem propagated by archaeology has a major drawback. What civilization keeps records for 3,000 years with such accuracy? Transferred to our time. If we want to establish a calendar by this method, already the pharaohs must have started counting and since then not a single day was allowed to slip in erroneously. How great this challenge is, becomes evident if we consider that the early Christians were not even able to determine the year of the birth of Christ correctly. Not to mention the day. And this event had to be traced back solely a few centuries rather than a few millennia.
Anyone who believes in this long-term counting theory is at least an optimist. The idea of a long counting, where at some time a shaman started to record the days and months as a tally sheet on a palm leaf in order to develop a complicated calendar from it, is not even absurd anymore, it is just ridiculous.
Why the Maya incorporated the orbital period of Venus into their calendar is yet another mystery.
The Mayans determined the Venus year to comprise 584 days. The remaining deviation from the exact value of 583.92 days probably doesn’t result from a measuring error, but was due to the necessity to establish a connection to the other two calendars used (the ritual calendar of 260 days length and solar calendar of 365 days plus corrections). Since the ratio of the orbital times of Venus to Earth is close to, but not exactly 13 : 8 and thus not commensurable, the mathematicians of the Stone Age had no choice but to conceal the dilemma of adaptation by means of inaccuracy.
The Maya pushed the calendar cult to its extremes. However, they were not the only ones who strived for uncanny accuracy in calendar matters. In the Indian Surya Siromani, the length of a sidereal year comprises 365 days, 6 hours, 12 minutes and 9 seconds. In decimal notation, this corresponds to the length of the year being 365.258438 days. Compared to the true value, this is 23.4 minutes too long. This error is far too large given the digits indicated and contradicts the accuracy with which star constellations and positions were calculated. If one dares to correct a possible sign error and use minus 12 minutes instead of plus 12 minutes in their calculation, this change of sign reduces the deviation from the true value to 37 seconds.
In another classic Indian book reporting on prehistoric astronomy, the Surya Siddhanta, the year comprises 365 days, 6 hours, 12 minutes and 36.56 seconds. This specification becomes exact when we assume the same sign error again. Doing so, this admittedly audacious sign correction reduces the deviation from the current best value to 9.4 seconds compared to the exact value.
Despite some doubt, the proposed corrections seem justified, especially since these two Indian books repeatedly show inconsistencies in the diagrams and explanations. We can assume that older and now lost books were revised during compilation by the copyists into problem-simplifying compendia. Most likely, the authors of Surya Siddhanta consulted and copied excerpts of sources of varying quality and only partly understood the details of the originals. Errors crept in when copying or taking over extracts. One even gets the impression that they corrected suspected errors to suit their own understanding when the issues became too incomprehensibly complex. Sometimes, it appears to be a lack of diligence. This fundamental lack of knowledge or carelessness becomes apparent in the case of obvious mix-ups. The Surya Siddhanta quotes the length of the sidereal day to comprise 86,400 seconds, although this is the length of the synodic day. A mistake, which no professional would have made and is best explained by carelessness.
Sometimes lost knowledge becomes recognizable only at the second view. Without mentioning a calendar background, Abydenus 7 says in his ‘Ancient Egyptian Chronicles’, 8 retained in part only, that 30 dynasties ruled Egypt for 36,525 years. The sequence of digits is certainly no coincidence. For in this number, all the digits of the number of days in the Julian calendar are reproduced. Although during Abydenus’ lifetime, the Julian calendar will have been in use, reproducing this string is nevertheless extremely astonishing. What to us appears to be a trivial representation of digits was foreign to the peoples of the ancient Egyptian and European cultures. The Greeks and Romans did not know how to write numbers in decimal notation, as it was simply impossible when using their number system. In any case, because of the notation in use, the representation of non-integer values in the form of decimal places was unknown.
In fact, the use of decimal fractions was not introduced until the 17th century. Thus, and perhaps surprisingly, on closer inspection even the simplest news become inexplicable.
The different quality of the sources which the two ancient Indian compendia consulted and the incorrect revision on copying is also strikingly illustrated by the data regarding the number Pi. The data on the relation between the diameter and circumference of the Earth suggests that the first authors knew the number Pi to at least five decimal digits. The Surya Siddhanta as well as the Surya Siromani measure lengths in units of yojana. If the original unit of length has been preserved over time until today, then 1 yojana equals 14.56 km.9 In the Surya Siromani the diameter of Earth is indicated to be 1,581 (1/24) yojana, and the circumference of Earth amounts to 4,967 yojana. Calculated on basis of these two figures, Pi is 3.141599684…, 10 which is actually remarkably close to the first digits of Pi (3.141592654…). Note: No matter how well we know the actual length of a yojana, this does not change the digits of Pi.
The ancient Indians not only knew the value of Pi they obviously also knew the formula for calculating the surface of a sphere from the radius:
The stated ratio of Earth’s circumference and diameter hints at an impressively precise knowledge of the value of Pi. This accuracy was not a coincidence. The same accuracy for Pi follows from the data about Earth’s surface, which is given as 7,853,034 square yojanas. The back-calculation of Pi from the surface indication and Earth’s diameter deviates by 2 millionths (deviation in the 6th digit) from the true value of Pi.
The formula for calculating the surface of a sphere was probably already known to the Egyptians. 11 Both the Indians and Egyptians were several thousand years ahead of the officially listed discoverer of the formula, the Greek scholar Archimedes. He nested Pi in the interval between greater than 3 + 10/71 and smaller than 3 + 10/70. Thus, he approximated the value of Pi to be 3.1428571 which is about 100 times less precise than the value given in ancient India.12
Earth and Space
The books Surya Siddhanta and Siddhanta-Siromani impress not only with mathematical knowledge and – to a certain extent – accurate information about the size of Earth, but also in stating further quantitative geophysical and astronomical data. In the Siddhanta-Siromani the height of the Earth’s atmosphere is specified as 175 km. This is twice the height of the stratification we call stratosphere. At this altitude shooting stars start to glow, radio waves are reflected and northern lights appear. In an extraordinary contradiction to the statement of the limited height of the atmosphere, Siddhanta-Siromani states that above the atmosphere winds blow which push the planets.13 The concept of assuming winds for the movement is one of the incomprehensible inconsistencies and fits the confusing mixture of right and wrong ideas.
The knowledge of the Indians about the solar system was fairly correct, as the distance of the planets was estimated to the correct order of magnitude. Incredibly, however, they are completely mistaken in the distance of the Moon from Earth. For this distance, the Surya Siddhanta indicates 324.000 yojana. This figure corresponds to 4,720,000 km and thus overestimates the average lunar distance by more than ten times. The reason for this overestimation probably derives from the wrong model of wind as the drive for the planet motion, since the authors assumed that the wind, which was concurrently responsible for pushing planets and the Moon, blew at any distance at the same speed. From this assumption, an unsolvable dilemma arises. At least as long as this wind theory is applied, only an overestimated Moon distance brings the planetary distances halfway in line with the true values.
With the wrongly estimated lunar distance and the evenly blowing winds as a basis, the ancient Indians calculated the distance between the planets (and the Sun) in a geocentric model from the ratio of the orbital times:
Using the lunar orbital period of the Moon equal to 27.322 days as a basis, the formula leads to the data of Diagram 1.
This irresolvable issue arises because the actual periods of orbital cycle times are proportional to:
(3. Kepler’s Law) and not as the Indians assumed:
T is equal to the orbital period and the parameter ‘a’ is the major semi-axis of the orbital ellipse. As an explanation for the extraordinary mistake in the indication of the Moon’s distance, one can only assume that a helpless copyist realized the topic and tried to save the unsavable.
Although the values given are systematically wrong according to the Kepler factor, they are of the correct order of magnitude. For Jupiter, the distance value of Surya Siddhanta even agrees rather well with the true distance. The distance stated for Mercury is too small by a factor of 3, whilst for Saturn, it is 30% too large. Because of the wrong formula used the distances are systematically wrong, but in the used model the calculations are astonishingly exact.
The per thousandth accuracy of the incorrectly but consistently calculated planet distances from the Sun applies to all planets except for Venus. Its distance is quoted by 1.3 % too small, clearly deviating from the calculation formula. But, Venus as will be shown in a subsequent paper is a special case anyhow.
As the values for the orbital radii of the planets already show, the ancient Indians had no fear of large numbers. The planetary distances were by no means the upper limit of their imagination. In the Siddhanta Siromani, the radius of the solar surrounding region is specified to 18,712,080,864,000,000 yojana (~28,000 light-years). A proper value for our galaxy, which measures 100,000 light-years in diameter and on average of 10,000 light-years in thickness. Thus, the mean radius (= region) is determined to be 30,000 light-years in what the Indians called the solar neighborhood. Even the early twentieth century did not know any value for the size of this extended ‘neighborhood’.
Also, other data given are of a corresponding and correct order of magnitude. The existence of the world is quoted in the Surya Siddhanta at 4,300,560,000 years. Surprisingly – or maybe not – this value corresponds quite closely with the age of Earth (today’s estimated value: 4.5 billion years). According to Hinduistic belief, 4.32 billion years correspond to one day in the life of Brahma. Another example of how religion and astronomy are interwoven?
These giant numbers are astonishing when we consider that the early biblical exegetes assumed the creation of the world to be 7,000 years ago. To further respect the value of the ancient Indians contribution in this area, it is interesting to note that a top physicist of the 19th century, Lord Kelvin, estimated the age of Earth to be about 24 million years maximum.
These two books represent convincing proof of lost knowledge. The correct data about Earth, about astronomical distances and the figure regarding the age of Earth, are too precise and fall far off the possible knowledge of a Stone Age culture to have been accidentally estimated or guessed correctly.
The two ancient books were probably written about 3000 years ago. The time of origin can be narrowed down because the Siddhanta Surya puts the vernal equinox in the constellation of Aries so that its origin can be dated to around 1000 BC. The epoch of Aries comprises about 2000 BC until the beginning of our era.
It is not only in ancient America and India that we find unexpectedly sophisticated prehistoric knowledge but also in Mesopotamia corresponding reports have been handed down. As can be seen from the records found in the library of the Assyrian king Ashurbanipal (668 to 627 BC), the enumerative system of calculation was based on the number 12960000. 14 The same system is found on the plates of the ancient Sumerian town of Nippur and the Babylonian settlement of Sippar.15 Is it really just a coincidence when 2 x 12,960 years correspond to a platonic year (a 360° rotation of the Earth’s axis against the starry sky)? If the choice of this basic unit was not a coincidence, Sumerians and Babylonians already knew about the precession of the Earth axis with remarkable precision.
The boundaries between esotericism and facts blur when connections are made between prehistoric buildings and the scale of the Moon and the Earth and other astronomical data. Both the Pyramid of Khufu and the dimensions of Stonehenge can be interpreted to be related to the diameter of Earth and Moon or even to the distance to the Sun.16 The assumed relations may be surprising, but they are not entirely suitable as proof of prehistoric knowledge.
With some goodwill, sophisticatedly coded data of the planetary system can be derived from the dimensions and masses of the three pyramids. 17 By no means absurdly constructed relationships lead to deviations of nearly zero. For example, the relation of the volumes:
holds to < 0,1%. The deviation is within the error with which the volumes of the pyramids are known. To assign all this to coincidence strains this overused counterargument to its limits. If it is not a coincidence, then the planetary system and its objects were well known to the ancients.
Somewhat more questionably, besides masses, diameters, and the planetary orbits, when even fundamental natural numbers, such as the speed of light, are derived from found relations:
To call this number, the speed of light requires an identical definition of the unit second – considering that since time immemorial, the practised division of the day into 24 hours and then the further subdivision into sixtieths of units, this may be borderline but not impossible. The deviation to the actual number is solely 0,03%. Given the uncertainty in the pyramid data, this is again within the measuring error. In any case, the simplicity to get this result is surprising.
The knowledge of the classical ancient advanced civilizations – such as the Maya, Sumer, Indians and Egyptians – is partially preserved in writing. Such evidence is missing for megalithic Europe, whose culture dates back to 6,000 BC. If we do not understand the means by which the giant stones (menhirs) were moved to their sites, we are also surprised by the in-depth mathematical knowledge that was applied in the construction of these prehistoric structures. For example, the rational aspect ratios of right-angled triangles can be found both in large stone settings and small carvings. As A. Thom explains, 18 the early Northern European people knew Pythagoras’ theorem not only in the widely known and simply rational 3, 4, 5 ratio:
32 + 42 = 52
but also, in less obvious relations such as 5, 12, 13 and even 12, 35, 37. Alexander Thom and others19 consider the stone circles and stone marked streets to represent observatories, which at least in some cases served to the prediction of ebb and flow. Anyway, it remains a mystery how the Stone Age came to such knowledge, and why and when it was lost. 20
Moving from prehistory to antique we must state that the Greeks had a reasonably correct understanding regarding the structure of the planetary system far in advance of Kepler and Copernicus. Not only did Democritus deduce the existence of atoms by plain logic, but also the infinity of the universe.21 He guessed rightly that the Milky Way represents a collection of stars. Hipparchus concluded that the different length of the seasons stemmed from the elliptical shape of Earth’s orbit.22 By means of measurements, the Greeks, who initially copied from the Babylonians, developed a vague idea about the distance of the Moon and the Sun to the Earth. Aristarch’s approach, 23 to determine the relationship of distances from the position of the Moon in quadrature, basically repeats the method of Eratosthenes who determined the circumference of the globe by comparing the shadowing in Syene and Alexandria.24 This approach represents an intellectual masterpiece that ingeniously combines geometry and astronomy.25 From the missing movement of the stars during the orbit of Earth around the Sun, Aristarchus concluded correctly by assuming a heliocentric world view that Earth’s orbit must be tiny relative to the distance of the stars. In their astronomic model, the Greeks were 2,000 years ahead of Copernicus. Little known is that Copernicus knew about some of these theses of the antique scientists.26 Was he a bit of a plagiarist? To indeed be the first is not easy.
Undoubtedly, the Greeks were building upon the astronomy of the Chaldeans. But these teachers of the Greeks used a purely algebraic, rather than a geometric, description of the planetary movements. The planetary theory of the Kidinnu (Babylonian astronomer who lived around 400 BC) is only preserved in fragments, but they are sufficient to astonish.27 The accuracy of his tables and calculations was not achieved again until the 19th century.
Our compilation of ancient knowledge is more than incomplete and yet the evidence is that the officially taught story of early history not only paints an incomplete picture but to a great part is simply a misconception. A fatal shortcoming, since as weak as the clues may be, they are sufficient to question the learned worldview.
Alternatively, it could be argued that there was a slump in human culture because of a disastrous event, where maybe the knowledge lost was not of indigenous origin but, dwindled away with the disappearance of the teachers, who were considered to have been extraterrestrials. This topic was addressed in a previous paper regarding the Deluge and we will address it again in subsequent papers on this website.
As a matter of fact, in scholarly history, the existence of a lost prehistoric culture is completely hidden and not even addressed as an option. The fact is, no matter how hard we try, what has been completely erased and lost no historian can reconstruct. On the other hand, already knowing that humanity once fell back from cultural heights into deep barbarism is both instructive and a warning. Perhaps, there is no remedy as already Cicero noted, civilization-erasing global conflagrations are nothing extraordinary, but part of our existence.
1 Catherine Nixey, “Heiliger Zorn: Wie die frühen Christen die Antike zerstörten‘; Deutsche Verlags-Anstalt (2019)
3 http://prajnaquest.fr/blog/wp-content/uploads/Berossos-English-Verbrugghe-and-Wickersham-1996.pdf; http://prajnaquest.fr/blog/wp-content/uploads/Babylonaica-of-Berossus.pdf
4 M. Magold, “Lehrbuch der Chronologie“, https://opacplus.bsb-muenchen.de/Vta2/bsb10392469/bsb:BV001449944?page=5
5 C. W. Beyers,“Grundsätze der egyptischen Zeitrechnung“;in Siegmund Jacob Baumgarten, Samlung von Erleuterungen und Zusätzen zur algemeinen Welthistorie, Halle (1748)
6 The Jewish (biblical) calendar lacks such a comparable historical beginning. The calendar and its division of time go back to a Sumerian, Babylonian system and have been iteratively improved over centuries. The year zero was calculated back from biblical data. According to these considerations, God created the world in 3761 B.C. The Jewish year lasts 365.2468 days by inserting leap months to adapt it to the lunar year. In the Book of Enoch the year is still given as 364 days.
7 Lived around 200 BC
8 From ‘The ancient fragments’:… containing texts of Sanchoniatho, Berossus, Abydenus, Megasthenes, and Manetho. Also, the Hermetic Creed, the old (Egyptian) Chronicle, the ‘Laterculus’ of Eratosthenes, the ‘Tyrian Annals’, the ‘Oracles of Zoroaster’, and the ‘Periplus’ of Hanno, see: https://reader.digitale-sammlungen.de/de/fs1/object/display/bsb10239301_00077.html
The better known Manetho allows only the first Pharaoh (Hephaistos) an unusually long reign of 734.5 years. In total the gods reigned in Egypt for 3984 years.
10 Siddhanta Siromani, Chapter II, 53, 122 in translation by Lancelot Wilkinson, revised by Pundit Bapu Deva Sastri. Calcutta (1861)
11 Otto Neugebauer: Lectures on the History of Ancient Mathematical Sciences: Pre-Greek Mathematics; Springer-Verlag Berlin Heidelberg (1969)
13 The concept of pushing winds may be wrong, but personally I like it better than the idea propagated in the Middle Ages, which introduced angels as planet pusher.
14 J. C. Fricke, in: https://www.academia.edu/2393846/The_Babylonian_Texts_of_Nineveh._Report_on_the_British_Museums_Ashurbanipal_Library_Project D
15 J. Campbell, Oriental Mythology: Masks of God Digital Edition (2014)
18 A. Thom, ’Megalithic Lunar Observatories‘; Oxford Uni. Press, Oxford (1971)
21 * 460 or 459 BC in Abdera, Trace; † 370 BC
22 * around 190 BC; † 120 BC
23 * 310 BC; † c. 230 BC
24 * 275 BC; † around 194 BC
25 M. Hoskin, Storia dell’ Astronomia, BUR Scienza (2017)
26 The culture of the present; Third part, third volume: Astronomy, E. Lecher Ed., Teubner Verlag, Leipzig (1921), https://archive.org/stream/astronomie00hartuoft/astronomie00hartuoft_djvu.txt
27 A. Pannekoek, Journal: Popular Astronomy, Vol. 55, (1947); http://articles.adsabs.harvard.edu//full/1947PA…..55..422P/0000422.000.html