Dodwell: The Obliquity of the Ecliptic





These are taken partly from Wendelin’s Memoir, (Solis Obliquitas), and partly from such sources as the Encyclopaedia Britannica, 9th Edition, article “Astronomy,”  Hutton’s Mathematical and Philosophical Dictionary, vol. 1, p. 455, article “Ecliptic,” list of Obliquities contained in Ball’s Astronomy, page 354, E. Bernard’s list of Obliquity determinations, previously mentioned, and other source to which reference will be made.


Early in the 11th century, the development of astronomical research spread from the East into Europe, and was particularly encouraged in Spain, both amongst the Moors and the Spanish Princes, particularly Alphonso X of Castile, who employed several notable astronomers at Toledo, where he prepared the Alphonsian Tables.  Amongst these astronomers was Al Zarkali (Arzachel), who determined the Obliquity of the Ecliptic at Toledo in 1070 A.D.  His solstitial observations, together with some others mentioned, according to Wendelin’s statement, are recorded as follows:

Arzachel the Moor, about our year 1070, noted that the declination of the sun did not exceed 23° 34’, and Almaeon, who followed him in the year 1140, observed 23° 33 ½ ‘, and then Prophatius Judaeus, in the year 1328, added to this 23° 32'….
Great support is given to the observations of these authors, when we compare the decrement, which is exceedingly constant at that time, and most especially because they used no reckoning of the parallaxes, but determined their obliquities from the maximum and minimum altitudes of the sun, so that they did not err beyond a total of 2’; but above all the very praiseworthy diligence of Almaeon is conspicuous in that he wished them to be more certain to the half of a minute.

Al Zarkali
When the observations of Al Zarkali are corrected for modern values of parallax and refraction, we have the following result:
Al Zarkali, 1070 A.D., Toledo, Latitude  39° 51’ N., Longitude 4° 0’ W.:  Corrected Obliquity 23° 34’ 42”

The observations of Almaeon similarly require the modern corrections for parallax and refraction.  The corrected result is as follows:
Almaeon 1140 A.D. Toledo, Latitude 39° 51’ N., Longitude 4° 0’ W.:  Corrected Obliquity 23° 34’ 12”

Ben Maimon
E Bernard, Philosophical Transactions 1684, says

The learned Jew, Rabbi Moyses Ben Maimon, A.D., 1174, found the Obliquity 23° 30’.  As this is one of the low values, it is probable, as pointed out earlier, that Ben Maimon applied corrections for the Ptolemaic value of solar parallax and semi-diameter.

Allowing for this, and using modern values for parallax, correction to sun’s centre, and refraction, we obtain
Ben Maimon, 1174 A.D., Cordova, Latitude 37° 52' N., Longitude 4° 48’ W.: Corrected Obliquity 23° 31’ 51”.

Alphonsian Tables, prepared from observations at Toledo
The value of Obliquity obtained from these observations is given in the Encyclopaedia of Islam (Houtsma, Vol III, page 503), namely 23° 32’ 29".  This value is derived from the observations made by the astronomers at Toledo, and a total correction of + 42" for parallax and refraction is to be applied, as in the case of Al Zarkali.  We then have
Alphonsian Tables, 1250 A.D. Latitude  39° 51’ N., Longitude 4° 0’ W.:  Corrected Obliquity 23° 33’ 11”

Nasir Al Din
The Obliquity found by Nasir al Din, in 1290 A.D., 23°  30’ is given in E. Bernard’s list, (Philosophical Transactions, 1684), also in Hutton’s Dictionary.  Correcting this value for Ptolemaic parallax and semi-diameter of the sun used by Nasir al Din, and applying modern corrections, including refraction, we obtain
Nasir Al Din, 1290 A.D. Maragha (Persia), Latitude 37° 21’N., Longitude 46° 17’E.:  Corrected Obliquity 23° 31’ 40".

Guilliame de St. Cloud
The Obliquity obtained from the observations of St. Cloud is given in Mediaeval Astronomy, by J.L.E Dreyer, in C. Singer’s Studies in the History and Method of Science, Vol II.  On page 117, referring to St. Cloud’s Almanac for 20 years from 1292 A.D., he says, “From observed Soltitial altitudes of the sun in 1290, the Obliquity of the Ecliptic is found to be 23° 34’, and the latitude of Paris, 48° 50’ N.” (Note.  The actual latitude of the Paris Observatory is 48° 50’ 11” N.)
Guilliame de St. Cloud, 1290 A.D., Paris,  Latitude 48° 50’N., Longitude 2° 20 E.:   Obliquity 23° 34’

Jean de Lignieres
The Obliquity obtained by de Lignieres is given in an account of his work by G. Bigourdan in Comptes Rendus, 1915, December 20th, pages 753-754.  With regard to his Obliquity, 23° 31’ 45”, Bigourdan gives the following quotation from a letter written by Wendelin to Gassendi, (Gassendi, Opus VI. 428):

…and remark with me that in 1332 the Obliquity of the ecliptic was 23° 31’ 45”.  This result could not have been obtained at Paris, where the winter refraction diminishes the distance of the two tropics.  May it be that the Provincial might have himself brought from the Province to Paris the result which he attributes to Lignieres?

Bigourdan says that there is no reason to believe that this was the case, but that this reported value of the Obliquity must be considered as having been obtained at Paris by Jean de Lignieres.  It appears from Wendelin’s statement that the Obliquity obtained by de Lignieres was not corrected for refraction.  Applying this correction, and the modern value of parallax, we obtain
Jean de Lignieres, 1332 A.D. Paris.  Latitude 48° 50’N., Longitude 2° 20’E.:  Corrected Obliquity 23° 33’ 31”.

Ibn al Shatir
The value of Obliquity found by Ibn al Shatir, 23° 31’, is given in the Encyclopaedia of Islam, Vol. III, page 503.  It is also given by E. Bernard, in Philosophical Transactions, 1684 A.D., as follows:

Ibn Shatir Damascenus, anno 1363, says that he corrected the Obliquity, making it 23° 31’, not neglecting the sun’s horizontal parallax, which he says he found of the extraordinary quantity 2° 59’. (This is obviously 2’ 59”, the same as was used 74 years later by Ulugh Beigh.)

Nallino, in his article on Arab Astronomy in Encyclopaedia of Islam, Vol 1, page 499, says, “At Cairo, Ibn al Shatir took good observations; his tables were renowned in Syria, Egypt, and the whole of North Africa.”   From Ibn al Shatir’s observations, applying modern corrections instead of those which he used, we have
Ibn al Shatir, 1363 A.D., Cairo, Latitude 32° 04’ N., Longitude 31° 17’ E.: Corrected Obliquity 23° 32’ 32”.

Ulugh Beigh
According to Hutton’s Mathematical Dictionary, Ulugh Beigh, grandson of Tamerlane, built very large instruments at his Observatory at Samarkand, in Turkestan, particularly a quadrant about 180 feet in height.  With this instrument, in 1437 A.D., he found the altitude of the sun’s centre, corrected for the value of parallax which he used (horizontal parallax 2’ 29”.4),  to be 73° 52’ 54” at the summer solstice, and 26° 52’ 20” at the winter solstice.  When the modern value of parallax is used, instead of that adopted by Ulugh Beigh, and with corrections also for refraction, we obtain
Ulugh Beigh, 1437 A.D., Samarkand, Latitude 39° 39’ 40” N., Longitude 66° 59’ E., altitude 2150 feet, Corrected Obliquity 23° 31’ 44”



Purbach and Regiomontanus
About 1460 A.D., the Obliquity of the Ecliptic was observed at Erfurt and Nürnberg (Nuremberg) by George Purbach, eminent astronomer and mathematician of Vienna, and by his pupil assistant, Johannes Muller, who afterwards became the most famous astronomer of his time in Europe, and was called Regiomontanus, from Mons Regius or Königsberg, where he was born in 1436 A.D.  According to Wendelin, Purbach and Regiomontanus made observations of the Obliquity about the year 1460 A.D., at Erphordia (Erfurt), latitude 51° 0’ N, longitude 11° 01’ E.,  in Middle Germany, and also at Nürnberg, latitude 49° 27’ 31” N., longitude 11° 01’ E.  Wendelin states that at the former place, “they found the maximum declination of the sun 23° 28’, and the interval of the tropics 46° 56’, but if the parallaxes subtracted by them are restored, then the interval is increased to 46° 57’ 55” nearly, and this was the distance seen by their own eyes.

Using the modern values of parallax and refraction, we have
Purbach and Regiomontanus, 1460 A.D., Erfurt, Latitude 51° 0’ N, Longitude 11° 01’ E.corrected Obliquity 23° 30’ 24”.

Purbach and Regiomontanus (continued) Observations at Nürnberg
Wendelin makes the further statement that Purbach and Regiomontanus found the observed Zenith distance of the sun at Nürnberg 24° 56 ½' .  Using the latitude of Nürnberg 49° 27’ 31”, and applying corrections for parallax and refraction, we obtain from this observation
Purbach and Regiomontanus, 1460 A.D., Nürnberg, Corrected Obliquity 23° 30’ 38”.

According  to Clavius in Opera Math. Tome III page 149, quoted in Ball’s Astronomy, page 354, Regiomontanus observed the Obliquity in the year 1460 A.D., at Vienna, with a quadrant.  As Purbach and Regiomontanus were attached to the Vienna University at this period, it has been suggested that the foregoing observations at Erfurt and Nürnberg may have been obtained by assistant observers, under their direction.

The Obliqity found at Vienna, and given in Ball’s Astronomy, is evidently the corrected value.  The mean Obliquity for the year 1460 is therefore as follows

Obliquity in 1460 A.D. at  
Erfurt 23° 30’ 24”
Nürnberg 23° 30’ 38”
Vienna 23° 30’ 49”
Mean Obliquity 23° 30’ 37”


Paolo Toscanelli is said to have constructed his famous gnomon in the Florence Cathedral with the object of demonstrating the variation of the Obliquity of the Ecliptic, which had long been suspected.  The method of measuring the sun’s altitude with the gnomon consisted in causing the sunlight to enter the South window of the lantern above the dome of the Cathedral, and to pass through an aperture 1.5 inches in diameter, in a bronze plate, let into the marble sill of this window, and then noting the spot where the long ray of light struck the floor of the North transept, about 350 feet below.

On the day of the summer solstice, the lantern was provided with a temporary flooring, allowing only a narrow ray to pass, while the transept was suitably darkened to enhance the effect.  The position of the solar image at the solstices was traced on solstitial slabs in the floor pavement of the transept.  With this method of observation, Toscanelli determined the Obliquity as 23° 30’ (see account given by Dr. Parr in the British Astronomical Journal, June 25, 1930, Vol. 40, No. 8, page 296).

Toscanelli used the Ptolemaic parallax, but allowing for this, and applying the modern correction for parallax and refraction, we obtain from Toscanelli’s observations
Toscanelli, 1468 A.D., Florence, Latitude 43° 46’ N., Longitude 11° 20’ E.: Corrected Obliquity 23° 31’ 47”.

Bernhard Walther of Nürnberg, friend and colleague of Regiomontanus from 1475 A.D. till his death in 1504 A.D., made 746 measurements of solar altitudes with a Dreistab and Regula Ptolemaica [a Triquetrum], besides a great many other astronomical observations.

Following the example of Regiomantanus, he used an instrument (a Dreistab) which consisted of three wooden rods.

The vertical rod was provided with a plumb-bob and two turning pivots for the two other rods, which were movable in altitude, and arranged for setting on the sun, with two sighting holes in order to determine its altitude.  One of these movable rods carried continuous scale divisions, marked in hundreds up to 140,000.  From the measured chords, the corresponding zenith distance of the Sun was obtained, and from this its altitude was calculated. (see E. Zinner, Frankische Sternkunde im bis 16 Jahrhundert).

Wendelin states that “Bernard Walther, the disciple of Regiomontanus, left many chords, published in writing, of solar distances from the zenith, in which for the year 1489, and likewise 1490, are those which descend to 44,895, the chord of the whole sine being 100,000.  Whence it emerges that the observed distance of the summer sun from the zenith was 25° 56’ 38”.

Zinner, however, in the work above mentioned, gives a comprehensive analysis of the numerous solar altitudes observed by Walther at the summer solstice between the years 1473 A.D., and 1503 A.D. These are divided into four groups, and applying modern corrections for parallax and refraction, and using the latitude of Nürnberg as given in the Berliner Jahrbuch 49° 27’ 31”, we obtain the obliquities for the four periods as follows

Year A.D.
23°  30’  56”
23°  30’  10”
23°  29’  52”
23°  31’  32”


Zinner points out that the mean error of the summer observation is 1’.02, and is reduced to 0’.65 if two discordant observations during the period 1487-1501 are excluded.  Giving weight 2 to the observations of 1473-1478 and 1503, and weight 1 to the observations 1487-1495, and 1497 - 1501, then the final reslt for the mean date 1491 is
Walther, 1491 A.D., Nürnberg, Latitude 49°  27’  31” N, Longitude 11°  02’ E.: Mean Corrected Obliquity 23°  30’  50”.

Domenico Maria Da Novarra (A.D. 1454-1504) was a notable astronomer at the Bologna University, Italy, with whom Copernicus was associated for about 3.5 years, from 1497 to 1500, as a student, but “rather as a friend and assistant than as a pupil.”  In connection with Novarra’s astronomical observations we are informed that in the year 1491 he determined the Obliquity of the Ecliptic equal to a trifle over 23° 29’. (see Dreyer’s Planetary Systems, page 307, and reference to a note in the original manuscript of the work De Revolutionibus of Copernicus, Edition 1873, pages 171, 172).   Novarra used the Ptolemaic parallax, and correcting this to the modern value of parallax, and applying also corrections for refraction, we obtain
Novarra, 1491 A.D., Bologna, Latitude 44°  29’ N, Longitutde 11°  21’ E.: Corrected Obliquity 23°  30’  56”

Johann Werner
With regard to the observations of Werner at Nürnberg, Wendelin says, “In our year 1514, Werner observed the declination 23°  28’  30” at Nürnberg, and by similar correction of parallax, and refraction added, there arises the distance of the tropics 47°  0’  20” and in the same way the true declination 23°  30’  10”.”  Applying modern corrections instead of those used by Wendelin, we obtain the following result from Werner’s observations
Johann Werner, 1514 A.D. Nürnberg, Latitude 49°  27’  31” N., Longitude 11°  02’ E.:  Corrected Obliquity 23°  30’  33”.

Copernicus was born at Thorn, in Poland, on February 19th, 1473 A.D.  He was admitted as a Canon of the Frauenberg Cathedral in 1498, but he was then studying at the Bologna University, in Italy, where he was associated with Novarra in practical astronomy.  From Bologna he proceeded to Rome but returned home to take possession of his Canonry at Frauenberg in 1501.  He then went back to Italy for further study, and continued there until 1506, and worked at Frauenberg until his death at the age of 70, in 1543 A.D.  Frauenberg is in the province of Ermland, in East Prussia, and is on the Southern Shore of the Frische Haff, Gulf of Danzig.

In his great work De Revolutionibus Orbium Ceolestium, published at Nürnberg in 1543, Copernicus says that during 30 years he had frequently determined the Obliquity of the Ecliptic.  For this purpose, as mentioned previously,  the instrument which he used was a Triquetrum, or “Ptolemy’s Rule” (Regula Ptolemaica).  This was eight feet in height.  It was made of pine wood, and was divided by ink marks, the two equal arms into 1000 parts, and the long rule into 1414 parts.  This instrument was provided with a large foresight, and the latitude obtained by Copernicus for Frauenberg, where his observations were made, indicates that these observations were made upon the shadow cast by the upper edge of the Sun.

The semi-diameter of the sun used by Copernicus was 16’  0” at the summer solstice, and 16’  16” at the winter solstice.  He also used a mean horizontal parallax of the sun, 3’  1”.  He obtained the latitude of Frauenberg 54°  19’  30”, and the Obliquity of the Ecliptic 23°  28’  0”. 

Wendelin says that the value of Obliquity obtained by Copernicus depended on observations at both summer and winter solstices, by which he obtained the total distance between the tropics exactly 46°  56’, “not having used the correct proportion of parallax, and much less that of refraction.”  On page 151 of his Memoir, Wendelin says that “Copernicus suspected nothing of winter refractions."

After applying modern corrections for refraction, parallax, and semi-diameter, we obtain from the observations of Copernicus
Copernicus, 1525 A.D., Frauenberg, East Prussia, Latitude 54°  20’ N.  Longitude 19°  41’ E.:  Corrected Obliquity 23°  30’  30”

(The corrected value of latitude of Frauenberg, resulting from these observations, made  by Copernicus, is 54°  21’  15” N.)

Hutton’s list of observations of the Obliquity of the Ecliptic gives three values determined in the year 1570 A.D., namely

23°  29’ by Ignatius Dante at Bologna
23°  31’ by the Prince of Hess at Cassel
23°  30’  20”  by Rothmann and Byrge at Cassel

            The mean of these three values is 23°  30’  07”

It is uncertain, however, to what extent corrections have been applied to these results, so that, pending further investigations, they are not used at present.

William, Prince of Hess, a Cassel benefactor of astronomy, is said to have erected the first observatory in regular operation in Europe, at Cassel on the river Fulda, in the year 1561.  He took a considerable part in the observations himself, and also employed as his assistants two well known astronomers of that period, Christopher Rothmann and Juste Byrge.  This observatory was furnished with good instruments, and the observations were published in Leyden in 1618 A.D. by Snell.

Tycho Brahe
The observations by Tycho Brahe were made at his famous Observatory at Uraniborg, on the island of Huen, latitude 55°  54’  25” N, longitude 12°  42’ E, near Copenhagen. 

Wendelin says, on page 110 of his Memoir, that Tycho noted the visible declination of the summer sun from the Equator 23°  29’  55” taking the latitude of his Observatory to be 55°  54’  55”,  and consequently the altitude of the Equator 34°  05’  15”, thus obtaining the sun’s observed summer altitude 57°  35’  10”.  To this Tycho added a parallax correction of 1’  35” in accordance with his table of parallaxes, in which the sun’s horizontal parallax was taken as 3’, practically the same as Ptolemy’s value.

Taking the Sun’s observed zenith distance in 1587 A.D., as 32°  24’  50”, and applying corrections for parallax, -5”, and refraction +36”, we obtain the corrected zenith distance of the Sun, 32°  25’  21”.  Subtracting this from the latitude (modern value 55°  54’  25”) we obtain the Obliquity, derived from Tycho Brahe’s summer solstitial observations in 1587 A.D., 23°  29’  04”.

Wendelin does not give the result of Tycho’s solstitial observations in the winter, and only refers to the summer observation in 1587 A.D.  As there would be other observations of the Obliquity made by Tycho Brahe, these have doubtless all been taken into account by later authorities. The values given by these authorities, e.g. Encyclopaedia Britannica (9th Edition, article “Astronomy”) 23°  29’  52”, and Ball’s Astronomy, page 354, 23°  29’  46”, are to be preferred.  The value in Ball’s Astronomy is that given by Bugge, Berliner Jahrbuch, 1794, page 100.
The mean of these two values, 23°  29’  49”  has been adopted in this investigation.

Edward Wright
Edward Wright was a distinguished English mathematician of Cambridge, who made valuable contributions to navigation and astronomy.  He made observations of the sun and stars with a six-foot quadrant in the years 1594 to 1597 A.D.  The Obliquity determined by him was 23°  30’ as given in Hutton’s Table of the Obliquity of the Ecliptic

Godifred Wendelin
Wendelin, famous mediaeval Belgian astronomer, made his observations of the Obliquity of the Ecliptic at the French town of Forcalquier, in the Basses Alpes.  He says on page 15 of his Memoir that his instrument was a quadrant, divided into single minutes in the quadrants.  It was made of wood, but so solid that both in the winter and in the summer it gave the same distances of the stars as were published by Tycho and Kepler. 

He also says on page 15, and again on page 110, that the apparent or visible declination of the sun at the summer solstice was 23°  30’, but at the winter solstice it was 23°  31’  15”, both being dependent on the latitude 43°  54’, which he found at Forcalquier from his observations of circumpolar stars.  That is to say, that his observed zenith distance of the Sun at the winter solstice was 67°  25’  15”, and at the summer solstice it was 20°  24’  0”.  Applying modern corrections for parallax and refraction to these observations, we obtain
Godifred Wendelin, 1609 A.D. Forcalquier, Latitude 43°  57’  34” N. Longitude 5°  47’  11” E., altitude 1804 feet, Corrected Obliquity 23°  31’  31”

The somewhat large difference between the observed and the true latitude of Forcalquier seems to indicate a combination of errors, possibly index and gradation errors of the quadrant, together with errors of level and verticality; with the possibility also that the site of the observations may not have been at Forcalqueir itself, but some distance away from that centre. 

The Obliquity result, however, would remain unaffected by this error in the adopted latitude.

Johann Kepler
The observations made by this famous astronomer were at Linz, on the River Danube, in Upper Austria.  The value of the Obliquity obtained from his observations (Tabulae Rudolphinae, page 116) as given in Ball’s Astronomy, is
Johann Kepler, 1627 A.D., Linz, Latitude 48°  17’ N. Longitude 14°  18’ E.:  Obliquity 23°  30’  30”

Nicolas Claude de Peiresc
Nocolas Claude de Peiresc was a noted astronomer of the early 17th century.  He undertook, in conjunction with Gassendi, to repeat the observations made by Pytheas at Marseilles in 326 B.C., of the altitude of the sun at mid-day on the day of the summer solstice, with a gnomon, in order to ascertain to what extend the Obliqity of the Ecliptic had altered since the time of Pytheas.

The following is a translation of an extract from an article “Sur les Travaux Astronomiques de Peiresc,” by G. Bigourdan, in Comptes Rendus, Academie de Sciences, Paris, 8th November 1915, pages 543-545, describing 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 latitude, as we have seen from some extracts of his letters.

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

(The attention of the reader is specially drawn to this statement by this eminent modern astronomer, in regard to the degree of accuracy of observations made with the gnomon.)

And he gave the example himself, with Gassendi, by repeating at Marseilles, 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, hardly touched upon then, of the variation 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 in diameter, of which the gnomon was more than 9 canes in height, and it was divided into more than 80,000 recognisable parts,, such that one could recognise  and determine the difference of that upon which the solar shadow fell precisely, exclusively from the other parts, either above or below. (This ‘cane’ was probably a local variant of the old French toise, as used at Marseilles, equivalent to 5 ½ feet in length.)

And this was done so skillfully, and with such little expense, that all those present were delighted. (This expense was borne by the town, which was thus a prelude to the sacrifices which it generally made afterwards to uphold the high renown of Marseillaise Astronomy.)

We had only to pierce the roof of a very high bilding, of three or four storeys, 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.

Pytheas had only observed with a gnomon divided into 600 parts.  The computation is found to agree with that of Pytheas with so little difference, gained successively by so many centuries, that it will be of great service to confirm the certainty of  the fundamental quantities, which are adopted to regulate all the celestial movements and all geography.

The observations were immediately communicated to Wendelin, but the publication did not take place till 1658, after the death of Gassendi; you can see why this is; Requier suspects indeed that they were not very satisfactory.

The height of the gnomon was 51 feet 8 inches, 4 lines, 0 parts, or 89,328 parts (twelfths of a line), and the shadow was found to be 31,750 parts, neglecting refraction, and suspecting the solar parallax to be equal to 20.  (A line is one-twelfth of an inch)

The solstitial altitude of the centre of the sun was thus found to be 70°  11’  15".

Taking 43° 19’  19” for the latitude, the Obliquity of the Ecliptic was taken as 23°  30’  24”.

The latitude adopted today (1915) for Marseilles (Old Observatory) is 43°  17’  52” (1).  We see that the suspicion of Requier was not justified, and that the observation is remarkably exact; even when ‘we compare the values obtained on this occasion for the latitude of Marseilles by meridian altitudes of the Sun, or Mars, or Saturn, and of various stars, we are surprised that a quadrant with pinnules could have given results so concordant.’

The Obliquity derived from the observations of Peiresc is as follows (the calculation being given in full on account of the interest which is has, from association with the ancient observation of Pytheas, 1963 years before Peiresc):

With Peiresc’s corrections
With modern corrections

L tan altitude of sun’s upper edge =
L 89328
L tan 70°  25’  59”


Observed altitudes of Sun’s upper edge  =  70°  25’  59”
Parallax  used by Peiresc  +20
Refraction used by Peiresc  0
Sun’s semi-diameter used by Peiresc  -15.04

Observed altitudes of Sun’s upper edge =  70°  25’  59”
Modern parallax correction  +3
Modern refraction correction   -19
Modern correction to sun’s centre  - 13’ 12”

(a) Corrected altitude of Sun’s Centre (Peiresc) = 70°  11’  15”

(b) Zenith distances of Sun’s Centre (Peiresc) = 19° 48’  45”

Latitude used by Peiresc = 43° 19’ 09”

Corrected altitude of Sun’s Centre = 70°  12’  31”

(a)Zenith distance of Sun’s Centre = 19° 47’ 29”

(b)Approx. latitude of site (Old Marseilles) = 43°  17’  52”

Difference (b) – (a) = Obliquity
= 23°  30’  24”
Difference (b) – (a) = Obliquity
= 23°  30’  23”


Peter Gassendi
Peter Gassendi was a celebrated French philosopher and astronomer.  In 1636 A.D. he was associated with Peiresc in the repetition of the measurement of the Obliquity of the Ecliptic made by Pytheas in 326 B.C. at Marseilles at the summer solstice.  Riccioli (Almagestum Novum, Vol. 1, liv. 3, Cap. 14) says “But in the same place Gassendi made use of a gnomon of 52 Royal Feet, and found at the summer solstice the meridional shadow 31,950 units, of which the gnomon was 90,000; whence he deduced that the Obliquity of the Ecliptic was the same as in the time of Alexander the Great.”(2)

There is also an inconsistency in this account, as 52 Royal Feet are an equivalent to 89,856 units (52 x 12 x 12 x 12), whereas 90,000 units represent 52 feet 1 inch.

Before 1812 A.D., when the metric foot was adopted in France, the Royal foot (pied de Roi) or Paris food (equivalent to 1.0658 English feet) duo-decimally divided into inches, lines and parts, or twelfths of a line, was in use.

If we take the statement that the gnomon used by Gassendi was 52 Royal feet, compared with Peiresc’s height of gnomon 51 feet 8.5 inches, then we get the value of the Obliquity almost exactly the same as that which was measured by Peiresc.

On the other hand, if we take the alternative statement that the height of the gonomon was 90,000 units, it gives the Obliquity 1’ 20” greater than Peiresc’s value, and one that is discordant from the true value.  Probably the statement that the gnomon was 52 Royal feet in height is correct. 

The observation was made independently by Gassendi, but evidently at a point in the building closely adjoining that used by Peiresc, the difference in height being exactly that of one brick plus layer of mortar, i.e. 3 2/3 inches. 

The length of the shadow measured by Gassendi is correspondingly longer (namely 31,950 – 31,750 = 200 parts, or 1 inch, 4 lines 8 parts, i.e. very slightly more than 1 1/3 inches) than Peiresc’s shadow-measurement.  This is what would be naturally expected from these respective heights of gnomon.

The calculation of the Obliquity determined by Gassendi, but with modern correction, is as follows:

Gassendi’s Observation of the Obliquity

L tan altitude of sun’s upper edge =

L 89856
L 31950




L tan 70° 25’ 34”

Observed altitude of Sun's upper edge
Modern correction for parallax
Modern correction for refraction
Modern Correction to Sun's Centre
+ 3"
- 19"
- 13'
Corrected altitude of Sun's Centre =
(a) Zenith distance of Sun's centre
(b) Latitude of Old Marseilles
Difference (b) - (a) = Obliquity =


Peter Gassendi, 1636 A.D. Marseilles, Latitude 43°  17’  52” N. Longitude 5°  24’ E.: the Mean Obliquity from the combined observations of Peiresc and Gassendi is 23°  30’ 10”

NOTE:  The Mean Obliquity obtained from the ancient observations at Old Marseilles in 326 B.C., is 23°  54’  20”, thus giving a diminution of 24’  10” during the intervening 1963 years to the time of Peiresc and Gassendi. 

Again, the latitude of Old Marseilles derived from the ancient observations is 43°  18’  25”, which is only 33” above the modern value.  This is a striking confirmation of the accuracy of those ancient observations.

J.B. Riccioli
Joannes-Baptiste Riccioli, a learned Italian astronomer, philosopher and mathematician, was the author of the Almagestum Novum, up to this time the greatest mediaeval work on Astronomy.  He made observations of the Obliquity of the Ecliptic, and two results are given, the first in 1636 A.D., 23°  30’  20”, quoted in the Encyclopaedia Britannica, 11th Edition, article on Astronomy; also in Hutton’s Mathematical and Philosophical Dictionary, Vol. 1, page 455, article “Ecliptic.”  Another value is given in the latter work for the year 1655 A.D., namely 23°  29’.
J.B. Riccioli, 1650 A.D. Bologna, Latitude 44°  30’ N., Longitude 11°  21’ E.:  the mean of these two, the Mean Obliquity, 23°  29’  40”

Johannes Hevelius
Johannes Havelius was a celebrated astronomer and Burgomaster of Danzig on the Baltic coast of East Prussia.  Among his numerous observations were those of the Obliquity of the Ecliptic, and
the values given for Danzig, Latitude 54°  22’ N., Longitude 18°  39’  E., are
1653 -- 23°  30’  20”  Hutton’s Mathematical Dictionary
1660 -- 23°  29’  10”  Encyclopaedia Britannica
1661 -- 23°  28”  53”  (corrected) Hutton’s Mathematical Dictionary

Jean Dominique Cassini
Jean Dominique, the first of four generations of Cassinis to occupy the position of Director of the Paris Observatory, from its commencement in 1671, was at first a professor of mathematics and astronomy at Bologna, Italy.  In 1653 A.D., he was authorised by the Senate of the Bologna University to correct and settle the meridian line, in the Church of St. Petronius at Bologna, which had been drawn by Toscanelli in 1575 A.D.

In the absence of the records of his observations at Paris, which are contained in the archives of that Observatory, it is only possible to state here the values of the Obliquity ascribed to him in 1655 at Bologna, 23°  29’  15” (Hutton) and in 1672 A.D., at Paris, 23°  29’  0” (Encyclopaedia Britannica, 11th Edition).  These are clearly the corrected values, and from them we obtain for the middle period
Jean Dominique Cassini, 1655 A.D., Bologna, Latitude 44°  30’ N., Longitude 11°  21’ E., and  1672 A.D., Paris, Latitude 48°  50’ N. Longitude 2°  20’ E.:  Mean Obliquity (1664 A.D.) 23°  29’  08”


From the epoch of the establishment of the Paris and Greenwich Observatories, in 1671 A.D. and 1676 A.D., respectively, the values of the Obliquity of the ecliptic given by various authorities are the corrected values according to the following list, with the exception of Wurtzelbaur, 1686 A.D., whose observed meridian altitudes of the sun at the summer and winter solstices of that year are given by Edmond Halley, the second Astronomer Royal of England, in a paper published in the Philosophical Transactions of the Royal Society of London, 1687 A.D., Vol 16, No. 190, page 403 (Vol III, page 407 in the Abridged Transactions).

In this paper, the observed Summer Zenith distance of the sun’s centre is 25°  57’  30”, and the observed winter zenith distance is 72°  52’  50”.  Using modern corrections for parallax and refraction, we obtain Corrected Obliquity 1686 A.D. 23°  28’  58”

The latitude of Nürnberg (where the observations were made) calculated from these observations, is 49°  26’  50”, which compares very favourably with the modern value for that city, 49°  27’  31”.


  1. Present Marseilles Observatory:  Latitude 43°  18’  16” N., Longitude 5°  23’  37” E.  Old Marseilles Observatory (former site of National Observatory at Accoules in the old town of Marseilles), Latitude 43°  17’  52” N.  This is near the harbour, and therefore is likely to be very close to the ancient site of the observations made by Pytheas. return to text
  2. This statement by riccioli is obviously incorrect. return to text


Obliquity Results from 1660 A.D. to 1868 A.D.

Date A.D. Observer Obliquity Difference (1) Authority (2)
1660 Montons 23° 29' 03"
1660 Hevelius 23° 29' 10"
Ency. Brit.
1661 Hevelius 23° 28' 52"
1672 Cassini 23° 29' 0"
Ency. Brit.
1672 Richer 23° 28' 52"
1686 Wurtzelbauer 23° 28' 58"
+ 10"
Halley (3)
1686 La Hire 23° 28' 52"
+ 4"
1689 Flamstead -5 (4) 23° 28' 56"
+ 9"
Ball's Astronomy
1690 Flamstead 23° 28' 48"
+ 1"
Ency. Brit
1690 Flamstead 23° 29' 0"
+ 13"
1703 Bianchini 23° 28' 35"
- 6"
Ency. Brit
1703 Bianchini 23° 28' 25"
- 16"
1706 Roemer 23° 28' 41"
+ 2"
1715 Louville 23° 28' 24"
- 11"
1730 Godin 23° 28' 20"
- 8"
1736 Condamine 23° 28' 24"
- 1"
Ency. Brit.
1743 Cassini de Thury 23° 28' 26"
+ 4"
Ency. Brit.
1750 Lacaille 23° 28' 19"
Ency. Brit.
1750 Bradley 23° 28' 18"
- 1"
1755 Bradley 23° 28' 15"
- 1"
Ency. Brit.
1756 Mayer 23° 28' 16"
Ency. Brit. & Hutton
1769 Maskelyne 23° 28' 10"
Ency. Brit. & Hutton
1772 Hornsby 23° 28' 08"
1779 Nautical Almanac 23° 28' 07"
+ 2"
quoted by Hutton
1780 Cassini 23° 27' 54"
- 10"
Ency. Brit.
1795 Maskelyne 23° 27' 58"
+ 1"
Ball's Astronomy
1800 Nautical Almanac 23° 27' 50"
- 5"
quoted by Hutton
1800 Maskelyn 23° 27' 57"
+ 2"
Ency. Brit. & Hutton
1800 Piazzi 23° 27' 56"
+ 1"
Ency. Brit. & Hutton
1800 Pond 23° 27' 56"
+ 1"
1800 Delambre (5) 23° 27' 57"
+ 2"
Ency. Brit. & Hutton
1800 Bessel 23° 27' 55"
Ball's Astronomy
1812 Eng. and Fr. astronomers 23° 27' 42"
1813 Pond 23° 27' 49"
Ency. Brit.
1813 Brinkley 23° 27' 50"
+ 1"
Ball's Astronomy
1813 Arago & Mathieu 23° 27' 49"
Grant (6)
1815 Bessel 23° 27' 48"
Ency. Brit.
1816 Brinkley 23° 27' 49"
+ 1"
Ency. Brit.
1825 Pearson 23° 27' 44"
+ 1"
Ency. Brit.
1835 Airy 23° 27' 40"
+ 1"
Ency. Brit.
1850 Leverrier 23° 27' 32"
Ency. Brit.
1868 Airy 23° 27' 22"
- 1"
Ball's Astronomy


  • Difference from Newcomb's Formula (+ = greater than; - = smaller than)
  • “Hutton” is Hutton's Philosophical and Mathematical Dictionary, 1815,
    “Ency. Brit.” is Encyclopaedia Britiannica, 9th Edition
  • From Hutton's Philosophical and Mathematical Dictionary, 1815
  • Five observations
  • Many hundreds of observations
  • History of Physical Astronomy

continue to Chapter 8