Dodwell: The Obliquity of the Ecliptic
ANCIENT GREEK OBSERVATIONS
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:
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:
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.”
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:
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:
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:
Excess of 1 over 2 = 8’ 27”
Eratosthenes 230 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.
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.”
Looking down into the Well
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)
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.
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.
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:
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:
Summing up, we have for the obliquity of the ecliptic, derived from the observations of Eratosthenes:
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.
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
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
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.