Dodwell: The Obliquity of the Ecliptic





These observations are the work of a galaxy of famous astronomers of ancient times, viz., Thales, Pytheas, Hipparchus, Eratosthenes and Ptolemy.


The earliest recorded Greek observation of the obliquity of the ecliptic is by Thales of Miletus in 558 B.C.  Miletus was an important city on the S.W. coast of Asia Minor, 53 miles south of Smyrna.  Thales was “the first of Seven Sages of Greece.”  He was self-taught until he traveled to Egypt, where he interviewed the priest-astronomers of Egypt and, it is said, learnt the geometry of the Egyptians.

Plutarch describes how this philosopher  was able to measure the height of the pyramids by using only his cane.  “Having no need of any instrument, but lifting your cane at the extremity of the shadow produced by the pyramid, you formed two triangles at the point where the solar rays met, showing thus the proportion between the two shadows and between the pyramid and the cane.”  Other authors say that this measurement was made when the sun was at an altitude of 45 degrees.

Thales is said to have predicted the total solar eclipse of 584 B.C., which has on that account come down in history as the “Eclipse of Thales.”  The Lydians and the Medes, who were at war with one another and were engaged in battle, according to the account given by Herodotus,  “when they saw night coming on, instead of day, ceased from battle and both parties were more eager to make peace with each other.”

For the observation of the obliquity of the ecliptic, it is evident that Thales used a gnomon, as this instrument had been in use among the Babylonians long before his time.  Anaximander, who was a contemporary of Thales, and was also a philosopher of the Ionian School, is credited with having introduced the use of the gnomon among the Greeks.

Wendelin, in his  Solis Obliquitas, says of this observation:

 In the times of Kings Cyrus and Croesus, when Thales the Milesian flourished and imparted the first rudiments of astronomy to the Greeks, he defined the interval between the two tropics as 8 parts out of 60 of the whole circle.  From this we find the interval 48°, as we divide the circle into 360°, so that the maximum obliquity of the sun was 24 whole degrees.

Messala, in The Astrolabe, discussing the change in the obliquity of the sun at various times, states that this same value of 24° was formerly given by the Hindus in their writings.  The time when Thales flourished was in the 55th Olympiad, that is, 558 before our era.

We have seen the confirmation of this remark concerning the Hindus in the result obtained from the Surya Siddhanta for 510 B.C.  When the observation of Thales is corrected for solar parallax and refraction in the latitude of Miletus, we have for the obliquity in the time of Thales 24° 00’ 56”, which is 15’ 07” larger than that which is calculated from Newcomb’s Formula.

With regard to this value of 24° about the time of Thales in 558 B.C., and of the ancient Hindus in 510 B.C., it is of interest to note the opinion of Tannery, the eminent French authority on ancient astronomy, concerning the value of the ancient Greek observations.  He shows that the accurate measurement of the circumference of the earth by Eratosthenes in 230 B.C., was “by no means the result of a happy accident,” and in L’Astronomie Ancienne p. 120, he makes the following remarks concerning Eratosthenes’ measurement of the obliquity of the ecliptic:

 Before the measurement of Eratosthenes, the obliquity was valued as 1/15 of the circle, or 24°.  This determination, indicated by Eudemus on the report of Dercyllides  (Theon of Smyrna), was already incontestably known by Eudosius, and perhaps it was anterior to him, going back to the school of Pythagoras.  It is evidently tied to the solution, given by Euclid, of the problem of inscription, in a circle, of the regular pentagon, this solution having for its object the tracing of the mean circle of the zodiac on the celestial sphere.  The process to be followed for measuring the obliquity of the ecliptic was then known from the beginning of Greek astronomy.  The part taken by Eratosthenes was of doubting the accuracy of the simple relationship universally admitted before him, and of undertaking a measurement which he knew how to make so exact as to cause his successors to despair of doing better.

Tannery also points out that Eratosthenes “naturally held to the process, already familiar in all the Greek cities, of observing the shadows of the gnomon.”

With regard to Tannery’s suggestion that the determination of 24° for the obliquity of the ecliptic goes back to the school of Pythagoras, we shall see that this suggestion is correct.

Pythagoras commenced to travel in the pursuit of knowledge at the age of 18.  After visiting Phoenicia and Syria, he went to Egypt, where he spent 22 years in his studies, which were terminated there by the conquest of Egypt by Cambyses, King of Persia, in 525 B.C.  He was taken to Babylon as a captive, but continued his studies amongst the Chaldean and the Persian Magoi.  After that he visited India, Celtica, Iberia, and Sicily, and then settled at Croton in Southern Italy.

We may therefore conclude that the School of Pythagoras, where he carried out astronomical observations, was established about the year 515 B.C.  From the new curve and formula the obliquity of the ecliptic at this period was 24° 00’ 05”.  Hence Tannery was right in attributing the value 24° to a date corresponding with the School of Pythagoras.


Pytheas of Massilia (the ancient  Marseilles, which was then an important Greek colony), is celebrated as a great navigator and geographer, about the time of Alexander the Great, in 325 B.C.  He was also an astronomer, and was one of the first to make observations of latitude, amongst others, that of his native city Massilia, which he fixed with remarkable accuracy, so that his result was adopted by Ptolemy, and became the basis of his map of the Western Mediterranean.

He was also the first among the Greeks to arrive at a correct idea of the tides.  He indicated their connection with the moon, and pointed out their periodical changes in accordance with the moon’s phases.

In his famous voyage of exploration, Pytheas visited Britain  and explored a large part of it, adding an account of Thule, which may have been the Orkney and Shetland Islands, or even Iceland, and then visited the whole of the coasts of Europe, including probably the Baltic as far as the mouth of the Vistula.

Concerning this voyage, R.T. Gunther, in his description of ancient surveying instruments, says:

About 326 B.C., in preparation for a voyage of discovery, which ended in the finding of our island of Britain, Pytheas sailed from Phocoea to Marseilles.  There he erected a large gnomon divided into 120 parts, and fixed its latitude with a result that seems almost  incredibly accurate, for, applying the sun’s semi-diameter, which he omitted, the latitude which he obtained differs not more than one minute from the true latitude of Marseilles  Observatory. (1)

The figures for the observations of Pytheas at the summer solstice at Marseilles are given by Strabo and Ptolemy.  Using the latitude of the old Observatory at Marseilles, which is near the harbour, and applying corrections for solar parallax, semi-diameter, and refraction, we obtain the obliquity of the ecliptic from the observation by Pytheas in 326 B.C.:   23° 53’ 46”.  This is 9’ 38” greater than the value calculated from Newcomb’s formula.  As we carry the comparison forward into the Christian era, we shall see this difference becoming less and less.  Conversely, tracing it back into B.C. times, the failure of the formula to account for the abnormal change in the obliquity, which is revealed by the ancient observations, is increasingly evident.

It is interesting to compare the value of obliquity obtained from the observations made by Pytheas in 326 B.C. with observations made at Marseilles, nearly 2000 years later, viz. by the French astronomers Nicholas Claude de Peiresc and Peter Gassendi at the summer solstice in 1636 A.D.  The following is taken from an account given by G. Bigourdan of the operations of Peiresc:

While paying great attention to the question of longitudes, of which the errors were enormous, even at our doors, Peiresc did not neglect the question of latitudes .......

He recommended above all that they be determined by meridian altitudes of the sun, particularly at the summer solstices, by employing the gnomon, which could then compete advantageously as regards precision with the best instrument of  the epoch.

And he gave the example himself with Gassendi by repeating at Marseilles, at the summer solstice of 1636, the memorable observation of Pytheas.  Already in the preceding year he had been requested by Wendelin to make this observation, in order to elucidate the question of variability of the obliquity of the ecliptic.

Various accidental circumstances having delayed this observation, it was made in 1636.  The choice of a convenient place presented some difficulties.  Finally it was decided to use the Church of the College of the Oratory, then in construction, where, with the agreement of the religious  and consular authorities, Peiresc caused some sections of the wall to be knocked down, the roof to be pierced, and a plank to be displaced.  This is what he wrote to D’Arcos on the 20th July, 1636:

'We caused to be prepared one morning a machine 18 canes (a) in diameter, of which the gnomon was more than 9 canes in height, and it was divided into more than 80,000 recognizable parts, such that one could recognize and determine the difference of that upon which the solar shadow fell precisely, exclusively from the other parts, either above or below.  All this was done so skillfully and with such little expense that all those present were delighted. (b) We had only to pierce the roof of a very high building, of three or four stories, and receive the ray from the sun at the bottom, having well adjusted with the plumb-bob at right angles to the meridian line, which was drawn below, and having taken away some bricks at a height of 9 canes to measure more exactly the distance between the hole in the roof and the base of the lower angle of the meridian line.' ..........

The height of the gnomon was 51 feet 8 inches 4 lines( c) 0 parts, or 89,328 parts, (twelfths of a line), and the shadow was found to be 31,750 parts.

Explanatory notes from Dodwell:

a.   Modern value of 18 canes would be about 100 feet.
b.  “This expense was borne by the town, which was thus a prelude of the sacrifices which it generally made afterwards to uphold the high renown of Marseillian astronomy.”
c.  A line is one twelfth of an inch.

Using the latitude of the old Marseilles Observatory, near the harbour, the obliquity calculated from Peiresc’s observation is 23° 30’ 23”.  This is remarkably exact, being only 1’ 11” different from the value derived from Newcomb’s Formula, and still less, 35” from the New Curve.

Besides demonstrating the accuracy obtainable with the gnomon, Peiresc’s observation shows that the difference between his result and that of Pytheas in 326 B.C. confirms our belief that in past ages there has been a marked change in the obliquity of the ecliptic, over and above the normal change corresponding to Newcomb’s Formula.

This is seen in the following comparison:

  1. Observed Values of the Obliquity
  1. Newcomb’s Calculated Values of the Obliquity

Pytheas, 326 B.C. (2)   23°  53’  46”

Peiresc, 1636 A.D     23°  30’  23”

Difference                         23’  23”

326 B.C.       23°  44’  08”

1636 A.D       23°  29’  12”

Difference           14’  56”

Excess of 1 over 2 = 8’ 27”

We shall now see how this conclusion is in line with the results obtained by three other famous Greek astronomers of early times, Eratosthenes, Hipparchus and Ptolemy.

Eratosthenes 230 B.C.

Eratosthenes was born at Cyrene, chief city of the ancient Greek colony of Cyrenaica, in Libya, North Africa, in 276 B.C.  After studying at Alexandria and Athens, he was appointed, at the age of 41, by Ptolemy Euergetes, to be the Superintendent of the famous Library Museum and Observatory of Alexandria.  He continued there till his death, at the age of 82, in 194 B.C.

Eratosthenes has been called the Cosmographer, or Surveyor, of the Universe, from his celebrated measurement of the earth’s circumference about 230 B.C.  He accepted the Pythagorean doctrine that the earth is a sphere, rotating upon its axis.  From this, he concluded that it would be possible to measure the earth’s circumference by a method, simple in principle, but requiring a work of considerable magnitude, to carry it into effect.

This method was to measure by astronomical observations, the arc in the sky between the summer solstitial position of the sun when it was overhead at Syene [the modern Assuan or Aswan, near the site of the great Assuan (Aswan) Dam], and its position at the same time at Alexandria.  The terrestrial counterpart of this was the distance between Syene and Alexandria, to be measured on the earth’s surface and then reduced by calculation to the true distance between the parallels of latitude passing through Syene and Alexandria.  The circumference of the earth could then be determined by the simple proportion of these celestial and terrestrial arcs to the complete circle of 360°.

The reader’s attention is specially drawn to this work of Eratosthenes, because it is certain that he took great care with these historical observations.     They give us a crucial test of the discrepancy between the ancient observations of the obliquity of the ecliptic and modern theory.

The first measurement of the earth’s circumference, 2180 years ago, was carried out in an effective and excellent manner.  In addition to the astronomical observations at Syene and Alexandria, a complete measurement was made of the distance on the land between those two places.  Egyptian surveyors, called Bematistoi (professional pacers) were employed for this purpose.  They measured the great distance of 640 miles along the winding track, following the course of the river Nile, from Syene to Alexandria.  Strabo informs us that allowances were cut off for irregular winding of roads, (3) and gives the distance found by Eratosthenes, from the small cataract of the Nile, near Syene, to the sea, as 5300 stades (518 ¾  miles).  The first cataract is 2 ½  miles south of Syene, and the ancient Library and Observatory of Alexandria was about ¼ mile from the sea.

The route from Alexandria to Syene does not go due south, but is at first south east for 125 miles (Alexandria to Cairo);  and from Cairo onwards there is a prevailing SSE to SE trend over much of the way, which includes the Great Bend of 80 miles between Hammadt and Erment, where there is a great deviation from the meridian line.

Eratosthenes consequently reduced the distance 5300 stades (518 ¾ miles) to 5000 stades (489.33 miles) to give the meridional distance, i.e. the corrected distance between the latitude parallels passing through Syene and Alexandria.  The stade used by Eratosthenes was 516.73 feet in length. (4)  The true distance between these parallels from modern calculations is 489 ¾ miles, almost precisely the same as that which was estimated by Eratosthenes.

With regard to the celestial arc between the overhead position of the sun at the summer solstices at Syene and Alexandria, Kleomedes gives it in round numbers as one fiftieth of the circle of 360°, i.e. 7°.2 .  Combining this with the terrestrial arc of 5000 stades, the approximate circumference of the earth was given in round numbers as 250,000 stades, equal to 24,466 miles.  The final result given by Eratosthenes was 252,000 stades, which is equal to 24,661 miles.  Modern measurements give the polar circumference of the earth as 24,860 miles, and the equatorial circumference as 24,902 miles.  Eratosthenes’ measurement of the polar circumference was thus only 199 miles less than the modern value.

Let us now turn to the astronomical measurements.  The precise value of the obliquity of the ecliptic recorded by Eratosthenes is involved in this historic measurement of the world.  It does not rest on this operation alone, however, but is the result of observations of the solsitial shadows over a period of years, both at Alexandria and Syene, as well as at other places.  This is evident from the information which has come down to us through Strabo, Kleomedes, and Pliny; although unfortunately, Eratosthenes’ own special book describing the measurement of the earth, and referred to by Macrobius as "Libri dimensionum," has been lost.

diagram of angles


At Syene was the famous “Well of Eratosthenes,” where in his day the sun at midday on the day of the summer solstice was seen “to light up the well right down to the water, and to cast no shadow on the sides.”


The Well of Eratosthenes at Syene

well exterior


Looking down into the Well

well interior

Prior to the solstice, the northern side of the Well was fully illuminated at midday, but the southern side was in the shadow.  As the solstice approached the shadow retreated towards the lowest edge of the southern side, until on the day of the solstice no shadow at all could be perceived on that side.  The northern edge of the sun was then exactly and vertically above the Well.  This fact is also established by confirmatory statements concerning the shadows of the gnomon at Syene.

According to a statement attributed to Eratosthenes,  “at Syene the gnomon threw no shadow on the day of the summer solstice.”  This is verified by Strabo, who says (Geography, Book 11, Ch. 5), “for indeed the summer tropic passed through Syene, because there at the time of the summer solstice the gnomon does not cast a shadow at noon.”  Plutarch also mentions “the gnomons of Syene, which appear free from shadow at the summer solstice.”(5)

It must be clearly borne in mind that these shadows, whether at the Well or cast by the gnomon, correspond not to the center of the sun, but to its northern edge.  In order, therefore, to reduce the observations to the centre of the sun, for comparison with modern observations, we must apply the necessary corrections not only for parallax and refraction, omitted by the ancient observers, but also the much larger one from the sun’s edge to its centre.  When this is done, we find that the observations made by Eratosthenes were remarkably accurate.  Thus, in the case of Alexandria, the latitude used by Eratosthenes was 30° 58’ N., determined from his observations of the sun’s northern edge.  The total corrections, as above, amounting to 13’ 52”, must be added to Eratosthenes’ latitude of Alexandria, 30° 58’, giving the latitude of the Museum of Alexandria 31° 11’ 52” N.  This ancient site is at the corner of the present Sesostris Street and Coptic Church Street; and the Survey General of Egypt gives its true latitude as 31° 11’ 50” N., only 2” less than the value found from Eratosthenes’ observations.  We see then that the observations made by Eratosthenes appear in a very favorable light in regard to their accuracy.

Turning to the latitude of Syene, the famous Well of Erastosthenes at this place is in latitude 24° 05’ 06” N., according to modern measurements supplied by the Surveyor General of Egypt.  Now Eratosthenes took for the latitude of this spot 23° 51’ 15”, determined as before, from the sun’s northern edge.  This was also the same as the obliquity of the ecliptic, which he obtained from his observations with the gnomon at Syene, when the northern edge of the sun was vertically overhead at the time of the summer solstice, and the gnomon consequently cast no shadow at noon.

When, as on this occasion, the sun is in the zenith, there is no parallax or refraction, and the only correction necessary to get the true latitude is the correction from the sun’s edge to its centre, viz., 13’ 13”.  When this is added to Eratosthenes’ altitude of Syene, 23° 51’ 15”, we obtain 24° 04’ 28” N.  This result is only 38” less than the true latitude 24° 05’ 06” N., and is a remarkable confirmation, not only of the accuracy of Eratosthenes’ observations at Syene, but also of the obliquity of the ecliptic 23° 51’ 15” at the date of his observations, viz. 230 B.C.

The relationship between the latitude of Syene and the obliquity of the ecliptic will perhaps be made clearer by consideration of the following diagram.


Syene diagram


SNE is  “sun’s northern edge.”

(1)        Eq.C represents a place on the earth’s equator where the centre of  the sun is vertically overhead at midday on the day of the equinox.  This is the equator according to modern definition.

(2)        Eq.N represents a place about 14 miles north of Eq.C, where the northern edge of the sun is vertically overhead at mid-day on the day of the equinox.  At that time a gnomon at any place north of Eq.N. would cast a shadow on its northern side, but at Eq.N it would cast no shadow. Eq.N was therefore the equator as Eratosthenes defined it.

(3)       Tr.C represents a position on the Tropic of Cancer, where the centre of the sun was vertically overhead at mid-day on the day of the summer solstice.

(4)      Tr.N represents the position of Syene, about 14 miles north of  Tr.C, where the northern edge of the sun was overhead at mid-day on the day of the summer solstice.

This was Eratosthenes’ “Tropic of Cancer,” where the sun then cast no shadow at the Well of Eratosthenes, and the gnomons also were free  from shadow.

From this diagram the reader will see that the obliquity of the ecliptic, as defined in the modern system, Eq.C - Tr.C, (i.e. the maximum distance of the sun from the equator), is exactly equal to the obliquity in the system adopted by Eratosthenes, viz. Eq.N - Tr.N., and found by him to be 23° 51’ 15”.  It will also be seen that this is proved to be the true value, because, by applying the correction TrC - TrN (centre of sun to its edge), 13’ 51”, according to Eratosthenes’ determination, we get the true latitude of Syene, viz. 24° 05’ 06”.

Another point that comes out from this analysis is that Eratosthenes found from his observations at Alexandria and Syene that the latitude of the former place, according to his system of measurement referred to the observation of the sun’s northern edge.  By this measurement, the latitude of Alexandria was 30° 58’ N. and Syene was 23° 51’ N. (to the nearest minute of arc).

The difference of latitude was therefore 7° 07’, which Kleomedes gives in “round numbers” as 1/50 of the whole circle of 360°.  The fraction is more exactly 1/50.5.  Combining this latter fraction, which Eratosthenes finally used, with 500 stades for the distance on the earth’s surface between the two parallels of latitude gives for the whole circumference of the earth 252,500 stades, for which Eratosthenes in his final result gave 252,000 stades, evidently to the nearest 1000 stades.  This was equivalent, as we have already pointed out, to 24,661 miles, within 199 miles of the best modern value.

Tannery, the eminent French writer on ancient astronomy, who thoroughly studies the work of Eratosthenes, says:

 “Is it necessary then to conclude that the accuracy of the result obtained by Eratosthenes was due to a happy accident?  I do not think so by any means, and I believe it is necessary to make a radical distinction between the methods which the learned Cyrenean could employ to control his observations, and the mode of exposition which he adopted in publishing his measurement of the earth’s circumference.”

Special attention has been given to this measurement of the earth’s circumference by Eratosthenes in order to point out the accuracy of his work, and of his determination of the obliquity of the ecliptic.  The latter rests on three different series of observations, agreeing with one another within very small limits:           

  • Observations at Alexandria 

These were made with the gnomon as well as with circular instruments.

Ptolemy tells us that the double obliquity angle observed by Eratosthenes and Hipparchus was less than 47° 45’  (maximum value) and greater than 47° 40’ (minimum value). 

The mean of these two is  47° 42’ 30”, and half of this gives
us 23° 51’ 15” for the obliquity found by Eratosthenes. (6) This does not include corrections for parallax  and refraction, and the difference between the apparent semi-diameters of the sun in summer and winter.  When these corrections are applied, the final value for the obliquity of the ecliptic from Eratosthenes' observations at Alexandria is 23° 52’ 04”.

  •  Observations at Syene

    Observations with the gnomon

    At the summer solstice:   it was observed that the gnomon at Syene cast no shadow at mid-day.  The sun’s northern edge was exactly in the zenith.  There was therefore no parallax or refraction, and the only correction is the reduction from the sun’s shadow-casting edge to its centre.  Using the true latitude, which is known, and applying this correction we obtain for the obliquity at the summer solstice   23° 51’ 53.

    At the winter solstice:   it is stated by Kleomedes that observations of the shadow of the gnomon made by Eratosthenes at the winter solstice at Syene and Alexandria gave the same result for the difference of latitude between the two places as at the summer solstice.   We therefore take the same double angle of obliquity for Syene as was found at Alexandria.  Applying the necessary corrections as before, we find for the obliquity at Syene for both solstices, 23°  54'  02".

    Observations at the Well of Eratosthenes

    According to the description given by Howard Payn (Observatory, 1914, p.287),      the well is now about 25 feet deep, and has spiral steps leading down to the water.  This may be part of a later reconstruction.  It is referred to by Strabo (XVII, Ch. 1. p. 48) as follows:

    But in Syene is also the Well that marks the summer tropic for the reason that this region lies under the tropic circle and causes the gnomon to cast no shadow at mid-day; for if from our region, I mean that of Greece, we proceed towards the south it is at Syene that the sun first goes over our heads and causes the gnomon to cast no shadow at mid-day; and necessarily when the sun goes over our heads it also casts its rays into wells as far as the water even if they are very deep; for we ourselves stand perpendicular to the earth and wells are dug perpendicular to the surface.  And here are stationed three cohorts (7) as a guard.

    The statement that Syene is the first place where this happens is of importance, and the Well gives confirmatory evidence of the obliquity found for the summer solstitial observation with the gnomon of 23° 51’ 53”.

    Observations at the shadowless area

      “Posidonius knew on the authority of Eratosthenes that at Syene, under the Tropic of Cancer at the time when the sun is in the constellation of Cancer, no shadows are seen at noon within an area 300 stadia in diameter.(8)

    Strabo refers to this in his Geography Book II, Chapter 5, paragraphs 36, 37:

    At Syene, at Berenice on the Arabian Gulf, and in the country of the Troglodytes, the sun stands in the zenith at the time of the summer solstice .... in all the regions that lie between the tropic and the equator the shadows fall in both directions, i.e. towards the north and towards the south, but in those from Syene and away from the summer tropic, the shadows at noon fall towards the north; and the inhabitants of the former regions are called Amphiscians (both shadows), and of the latter Heteroscians (other shadows), all whose shadows either always fall towards the north as is the case with us, or always towards the south as is the case with the inhabitants of the other temperate zone.

    (Strabo here is speaking of people in Africa living in the temperate zone south of the Tropic of Capricorn, where the solar shadows at noon always fall towards the south).

    From the statements of Strabo, Syene, in latitude 24° 05’ 06” N, was exactly on  the northern limit of the shadowless area, which extended Southward for 300 stadia, equivalent to 29.36 miles or 25’ 36” of latitude, i.e. to latitude 23° 39’ 30”.

    The exact centre of the shadowless area, i.e. where the centre of the sun was vertically overhead at mid-day on the day of the summer solstice, was therefore, in the time of Eratosthenes, midway between latitudes 24° 05’  06” and 23°  39’ 30”, i.e. in latitude 23° 52’  18”.  This was therefore the maximum distance of the centre of the sun from the celestial equator at the time of the summer solstice;  i.e. the obliquity from this observation was 23°  52’  18”.

The Shadowless Area:   Tropic of Cancer


shadowless area


SN  =  300 stadia  =  29.36 miles  =  25'  36"  of latitude

C  =  Centre of Shadowless Area  (Tropic of Cancer)
(Sun's centre overhead at mid-day, summer solstice)

N =  Northern limit of Shadowless Area
(site of Syene and Well of Eratosthenes)

S  =  Southern limit of Shadowless Area

N' =  Position north of Shadowless Area
Northern side of gnomon in shadow
Southern side of gnomon illuminated

S'  =  Position south of Shadowless Area
Southern side of gnomon in shadow
Northern side of gnomon illuminated

The diameter of the Shadowless Area noted by Eratosthenes and Posidonius, (300 stadia  = 25'  36"  of latitude), is very close to the theoretical value,  26'  26" of latitude, derived by calculation.


Summing up, we have for the obliquity of the ecliptic, derived from the observations of Eratosthenes:

Observations made at Alexandria
23°  52’  04” at both solstices
Observations at Syene
23°  52’  04” at both solstices
Observations at Syene at summer solstice with gnomon, and at Well of Eratosthenes
23°  51’  53”
Observations at shadowless area  
23°  52'  18"
23°  52’  05”

We see how accordant these results are, and the analysis given above indicates a great measure of reliability.  When we compare the mean result, 23°  52’  05”, with the result obtained by calculation for the same date,  230 B.C., from Newcomb’s Formula, we find a large difference.

Eratosthenes    230 B.C.          23°  52’  05”
Newcomb       230 B.C.          23°  43’  47”

Difference   8’  18”

This strikingly confirms the difference between Newcomb’s Formula and the observations made by Pytheas (9’  38” in 326 B.C.), Thales ( 15’  07” in 558 B.C.) and the Ancient Hindus  (15’  16” about 510 B.C.).

Comparing also with earlier Chinese and Hindu results, we see how the difference increases backwards to the most ancient times

It is clear that these ancient observations inescapably reveal the fact that there has been a movement of the earth in the past which is not accounted for by modern theory represented in Newcomb’s Formula.

Hipparchus,  135  B.C.

Hipparchus was born about 190 B.C., either at Rhodes or at Nicasa, an important city -- or Metropolis -- of Bithynia, Asia Minor.  He died about 120 B.C.  Hipparchus carried out most of his astronomical work at Rhodes, the capital city of the island of Rhodes, in the Aegean Sea, famed for its beautiful climate; where it is said that there is hardly a day in the year in which the sun is not visible.

Rhodes was also celebrated as the site of the Colossus of Rhodes, one of the seven wonders of the ancient world.  This was a colossal bronze statue of the Greek sun-god Helios, 70 cubits (106 feet high) at the entrance of the harbour.  It was made by the sculptor Chares, but was destroyed by an earthquake in 224 B.C.,  56 years after its erection.

Hipparchus built an observatory at Rhodes, where he made his principal observations and calculations.  He also visited Alexandria, and while there made use of the astronomical instruments at the Museum of Alexandria.  Hipparchus is generally acknowledged as the greatest astronomer of the ancient world, and ranks among the greatest astronomers of all time.  He was greatly admired by Ptolemy, who praised him as “a labor-loving and truth-loving man.”

Hipparchus was the first man to number the stars, considered in early times a stupendous if not even presumptuous undertaking, “rem etiam Deo improbam” (Pliny).  He invented the method of describing the positions of the stars by celestial longitudes and latitudes, a system afterwards adopted by geographers for mapping the earth.  His original catalogue of 1080 stars, giving their positions and magnitudes, arranged according to their constellations, for the year 129 B.C., is unfortunately lost.  But a copy, including 1028 stars, revised to the date 137 A.D., has come down to us in the 7th and 8th books of Ptolemy’s Almagest.

Hipparchus is thus regarded as the founder of observational astronomy.  He made the great discovery of the precession of the equinoxes, and invented trigonometry, both plane and spherical.  He calculated solar and lunar tables for predicting the positions of the sun and moon, and their eclipses, and redetermined the lengths of the year and the month.   His observations were very extensive, and he invented new instruments to increase their accuracy.

In particular, as bearing upon the subject of this study, he checked the obliquity of the ecliptic obtained by Eratosthenes in 230 B.C., 65 years before his own time, and found precisely the same result.   Concerning this, Wendelin says that Hipparchus

from his own observations stated that the distance between the tropics was in proportion to the whole circle as 11 is to 83, exactly the same as Eratosthenes, and found the maximum obliquity 23°  51’  20”.

He also used this value, as Ptolemy witnesses.  Ptolemy also agrees as to the year in which this observation was made, because in the year 43 of the 3rd Callipic period he commenced writing most vigorously, after having returned seven years earlier from Alexandria to his native city of Rhodes, where he took up a peaceful life.  I do not think this value of the obliquity ought to be referred to any other year; this year was 135 before our own era.

Hipparchus, like the other ancient astronomers, applied no corrections for parallax, refraction, or the sun’s semi-diameter, and his observations at Rhodes were made with the vertical gnomon.  Applying these necessary corrections, we obtain for the obliquity of the ecliptic, resulting from the observations of Hipparchus at Rhodes  23°  52’  16”.  The value calculated by Newcomb’s formula for the same date is 23° 42’ 43”.

Here again we see a strikingly large difference viz. 9’  33”, between the ancient observation as made by Hipparchus and the calculated value. Hipparchus was far too good an observer to have made so large an error as 9.5 minutes in his observations, and we may compare this with Kepler’s comment on the observations of Tycho Brahe in the 16th century.  After every conceivable effort to make astronomical theory of his day agree with Tycho’s observations of the planets, Kepler still found errors amounting to eight minutes of a degree.  He then unhesitatingly declared his belief in the accuracy of Tycho’s observations, and claimed that he was so good an observer that it was impossible for his observations to be in error to the extent of 8 minutes.  He added,  “Out of these eight minutes we will construct a new theory that will explain the motions of all planets.”  This he did by abandoning the Platonic idea that the planets moved in perfect circles, and by substituting elliptic orbits for all the planets, having the sun in the focus of the elliptical orbit of each planet.

In a similar way we may claim, with the support of all the ancient observations thus far examined, that the 9.5 minutes of Hipparchus led to a new theory, which, combined with the existing theory, explains the complete movement of the earth in the variation of its axial inclination during the last 4000 years.

Ptolemy,  126  A.D.

The famous ancient astronomer Claudius Ptolemy was born about 70 A.D. in Egypt, at Pelusium, an ancient city at the most easterly mouth of the Nile;  he died about 147 A.D.  Ptolemy is celebrated as the founder of the Ptolemaic System of the universe, which held sway for more than 1500 years, when it was finally replaced by the Copernican system.

Ptolemy was a great mathematician, geographer, and astronomer, and was regarded by the ancients as the “Prince of Astronomers.”  His  Megale Syntaxis, afterwards called  Ptolemy's Almagest, in 13 books, has preserved and handed down to us the observations and discoveries of the ancient astronomers, and forms a most remarkable and complete account of the astronomy of his time. It was the standard text book of astronomy for 1500 years.

Ptolemy gives the obliquity of the ecliptic for his time as 23°  51’  20”.  This is obviously taken from the observations of Hipparchus.  It is probable that Ptolemy did not make the observations of the obliquity of the ecliptic himself, but simply took this value from Hipparchus, on account of his high opinion of the accuracy of the observations made by that great astronomer, and their agreement with the obliquity found by Eratosthenes.

Nevertheless, we are able to get an independent value from Ptolemy’s own numerous observations of the moon at the time when it was at its least distance from the zenith of Alexandria.  The mediaeval Belgian astronomer Wendelin drew attention to this in his memoir The Obliquity of the Sun  (Solis Obliquitas).  In this work, he pointed out in regard to Ptolemy’s observations that

Ptolemy himself records numerous observations of the moon in quadrature, (i.e. at the moon’s first quarter phase when it is exactly 90 degrees east of the sun), distant from the zenith of Alexandria 2 1/8 degrees, i.e. 2°  7’  30”.

When this observed distance is corrected for refraction and lunar parallax, it is reduced to 2°  5’  26”.  Now this observation, as Wendelin explains, refers to the zenith distance of the moon at a period in its 19-year cycle when, at the sun’s summer solstice, the moon at the same moment was at its maximum distance -- 5°  18’  0” north of the ecliptic.  This occurred in the time of Ptolemy in the year 126 A.D.

We have then only to add together the two quantities 2°  5’  26”  and 5°  18’  0”, obtaining 7°  23’  26”, which was the actual distance of the ecliptic, at its crossing point with the celestial equator, from the zenith of Alexandria.  We now subtract this quantity from the latitude of the Museum of Alexandria, and thereby obtain the obliquity of the ecliptic   23°  48’  24” in the year 126 A.D., from Ptolemy’s own observations of the moon.

It is of great interest to notice that these observations were made with a special instrument invented by Ptolemy for his lunar observations, called the “Organon Parallaktikon” or “Regula Ptolemaica,” afterwards called Ptolemy’s Rules or the Triquetrum.  The foresight was made large enough to cover the entire disc of the moon, and a small back sight was used.  It was capable of giving accurate results, and 1400 years later Copernicus, using an instrument of the very same kind, “made those measurements with which he overthrew the Ptolemaic System and gave us a new idea of the universe.”

When we compare the obliquity thus obtained from Ptolemy’s observations, 23°  48’  24”, with that which is calculated by Newcomb’s Formula, viz. 23°  40’  47” we find a difference of 7’  37”.  This is again a further confirmation of the previous results, showing also the progressive diminution of the difference as we now advance towards the modern era.

So many results all tending in the same direction, from different countries, ages, and observers, are sufficient for the student of this work to see that modern astronomical theory does not disclose the whole story of the change in the inclination of the earth’s axis during past ages.  We see, on the contrary, that these ancient observations point to a change of great and far-reaching importance in the earth’s history.


  1. R.T. Gunther, Early Science in Oxford, 1923, Vol. 1. p. 330 "Ancient Surveying Instruments" return to text
  2. See also Table 25 [to be added later] return to text
  3. Strabo.  Geography,  Vol. 1. Book 11, Ch. 4, par. 4 return to text
  4. J.L.E. dreyer, “Planetary Systems,” p. 175 return to text
  5. Plutarch. Opera Moralia on the Cessation of Oracles  1V return to text
  6. Ptolemy.  Syntaxis II.  Chapter 12 return to text
  7. A cohort was a tenth of a legion, and comprised between 500 and 600 men return to text
  8. J.L.E. Dreyer. Planetary Systems p.185.  300 stadia are equivalent to 29 1/2 miles return to text


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