In what year was the first solar eclipse?
Canon of the solar eclipses of 2501 BC Until 1000 AD
The historical chronology is based on a close interplay between theoretical astronomy and historical events. The use of astronomy for chronology is to fix astronomical phenomena, which are mentioned in historical sources, in absolute arithmetic. Particularly important are solar and lunar eclipses, new moon and old moon sightings, as well as observations of the planets Mercury, Venus, Mars, Jupiter and Saturn.
Because of their rare occurrence in a specific place on earth, solar eclipses have the greatest potential to be able to date a historical event to the exact day, if a total or substantial obscuration of the sun by the moon is recorded. Characteristics for a considerable or total eclipse of the sun are:
- During a total solar eclipse, the brightness of the sky changes drastically, but not evenly. The day can almost turn into night. The change in brightness is felt most strongly shortly before and after the maximum of the eclipse.
- In the vicinity of the eclipsed sun, the brightest fixed stars and planets can be seen.
- Shadows become more contoured. Images of the sun sickle can be observed on the ground thanks to the pinhole effect near trees and shrubs.
- The temperature drops during a total solar eclipse. The birds fall silent, flowers close their leaves, diurnal animals withdraw and nocturnal animals become active.
- At the beginning and at the end of the total phase, the so-called pearl effect can be observed for a short time: the last or the first rays of the sun shine through the valleys of the mountainous moon silhouette and create the impression of a pearl string.
- During the total coverage, the solar corona shines around the dark disc of the moon.
In the historical sources up to 1000 AD, only points 1) and 2) are mentioned. In 2) it should be pointed out at once that ancient authors often write that stars became visible, although today's calculations show that this was completely impossible. The indication that the day turned into night is therefore a more reliable indication that the solar eclipse described is a total at the specified location. The phrase about the visibility of stars during solar eclipses should rather be understood as a standard phrase to add to the drama. It is noticeable that solar and lunar eclipses are often mentioned at the same time as earthquakes and the destruction that occurred, comets and other events that were viewed as negative omens - even at times when the cause of the eclipses had long been known.
On average, a total solar eclipse occurs every 375 years at a certain place on earth; if you include the ring-shaped solar eclipses, you get an average of about every 140 years.
Geometry and parameters of a solar eclipse
A solar eclipse occurs when the sun is completely or partially covered by the moon as seen from the earth. This can only happen when the sun, earth and moon are in line.
Since the lunar orbit is inclined to the sun's orbit, this case can only occur when the moon is near the so-called lunar nodes. This is the case on average after every sixth new moon, sometimes even after the fifth. Since the distances between the celestial bodies fluctuate due to the elliptical orbits of the earth and the moon and therefore the apparent diameters of the moon and the sun also vary, different types of solar eclipses are possible. Eclipses in which the axis of the moon's shadow crosses the earth are known as central eclipses. There are three types of these:
- Total solar eclipse (T): The apparent diameter of the moon is larger than that of the sun and is therefore sufficient to completely cover the sun. The totality zone is relatively narrow, in the best case the umbra has a diameter of about 273 kilometers.
- Annular Solar Eclipse (R): If the apparent diameter of the sun is larger than that of the moon, the outer edge of the sun remains visible. The umbra of the moon is too short to reach the surface of the earth.
- Annular-total or hybrid solar eclipse (RT): In a ring-shaped total solar eclipse, the umbra of the moon is only too short to reach the surface of the earth at the beginning and at the end or sometimes only at one of the two. In the middle of its orbit, however, the umbra hits the earth's surface because of the spherical shape. Such a solar eclipse usually begins and ends as a ring-shaped eclipse. In between, she is total. The diameter of the umbra is very small (approx. 20 kilometers) and the totality only lasts a few seconds.
An eclipse in which the earth is only hit by the penumbra of the moon is one partial solar eclipse (P). Such eclipses mainly occur in the polar regions of the earth, but they can also lead to occlusions of 50%, sometimes up to 70%, in the Mediterranean area.
Since the totality zones are relatively narrow, only comparatively few people are able to observe the complete coverage of the sun during a solar eclipse. The so-called penumbra, which is several thousand kilometers wide and enables a partial eclipse of the sun to be observed from more than a quarter of the earth's surface, is much wider than the umbra of the moon. Unfortunately, this only partial eclipse is also called a partial solar eclipse when there is a central eclipse.
The most important parameter of a solar eclipse for ancient scientists is the degree of coverage or the size or magnitude, with which the extent of the coverage is described.
- The degree of coverage is the ratio between the area covered by the moon and the total area of the solar disk. In the case of total eclipse, the degree of coverage within the totality zone reaches the maximum of 100%; in the case of a ring-shaped eclipse, the value remains slightly less than 100%.
- The size or magnitude of a partial eclipse is the proportion of the sun's diameter covered by the moon. In a total or ring-shaped eclipse, the size is the ratio between the diameter of the moon and the diameter of the sun. The value is slightly greater than 1 for a total eclipse, and slightly less than 1 for a ring-shaped eclipse.
During the course of an eclipse, the degree of coverage and size slowly increase, reach maximum values and then decrease again.
Calculation of solar eclipses
The calculation when and where on earth solar eclipses are observable is done with the help of the so-called Bessel elements executed. The method was developed by Friedrich Wilhelm Bessel in 1842 and has since been refined numerous times . The basic idea is that the Bessel elements determine the movement of the moon's shadow on the appropriately chosen, so-called Fundamental level reproduce. The fundamental plane goes through the center of the earth and is perpendicular to the axis of the shadow cone. To describe the movement of the shadow in this plane, it is sufficient to specify a few geometric quantities; In a further step, the conditions on the earth's surface are calculated from this by projection.
In the system shown above, if x, y, and z are the coordinates of the center of the moon, then the coordinates of the center of the umbra in the fundamental plane are x and y. The Bessel elements are then the following quantities:
- X and Y: coordinates of the center of the umbra in the fundamental plane
- L1: radius of the penumbra cone in the fundamental plane
- L2: Radius of the Umbra cone in the fundamental plane
- F1: Angle between the penumbra cone and the shadow axis
- F2: Angle between umbra cone and the shadow axis
- D: Declination of the point on the celestial sphere to which the shadow axis points
- M: Hour angle of the point on the celestial sphere to which the shadow axis points
The equatorial radius of the earth is used as the unit of length.
The Bessel elements are time-dependent; they must therefore be specified for a period of several hours. There are several variants of the publication of the Bessel elements of a solar eclipse: either the values of all elements that are not to be regarded as constant are tabulated in ten-minute intervals for the entire course of the eclipse, so that intermediate values can be interpolated. Or the Bessel elements are specified for a reference time t0 and, in addition, the hourly changes in the elements. This enables the calculation of the values for other points in time of the course of the eclipse as a linear function of time. Instead of a linear form, a polynomial form can also be selected for greater accuracy. Terms up to 3rd order ensure sufficient accuracy for most purposes.
If one could follow a solar eclipse on the earth's surface from the sun, one would see the large penumbra of the moon - gray in the following figure - and the much smaller umbra - black - move across the earth for a few hours:
The time of sunrise has come for all points on the curve N-PN1-PS1-S on the globe. It is noon for all points on the N-S meridian. All points on the curve N-PN2-PS2-S have sunset. If the points of intersection are projected onto a geographic map of the earth, the regions described above have oval outlines: PN1-P2-PS1-P1 or PN2-P4-PS2-P3 in the figure on the right.
ΔT and its uncertainty
For many centuries, the Earth's period of rotation with respect to the Sun has been the fundamental unit of time. Universal Time (UT), also known as Greenwich Mean Time (GMT), is based on Greenwich Mean Solar Time. Unfortunately, UT is not a uniform time scale because the period of rotation of the earth slows down over time. Therefore, the calculation of the local circumstances of solar eclipses for distant times is burdened with certain uncertainties. The so-called tidal friction is very important for the slowing down of the earth's rotation. The tides are caused by the gravitational effect of the moon and sun on the water and air masses of the earth. The hiking of the flood mountain consumes - especially in the narrow sea streets - rotational energy of the earth. The angular momentum of the earth also decreases over time. This has an effect on the moon: The flood mountain is carried along by the rotation and in turn "accelerates" the moon. As a result, the distance of the moon grows and the period of rotation of the moon becomes longer. The total angular momentum of the earth - moon system is retained. Today atomic clocks are used to measure time. An atomic time scale is Terrestrial Dynamic Time (TDT). Solar eclipse calculations are based on TDT, but the location of the eclipse visibility area on earth depends on UT. Therefore the calculation results have to be converted from TDT to UT. To do this, you have to know the time difference between TDT and UT. This time difference ΔT called, which dates back to about 12 hours in the year 2000 BC. And the uncertainty of the same (2000 BC about 2 hours) must be included in the calculations. For more information about ΔT, I recommend Robert van Gent's website.
Calculations in this canon
For the calculation of the moon and sun positions in the period between 2500 BC. The DE406 long-term ephemeris of the Jet Propulsion Laboratory were chosen, which allow the positions of the sun, moon and all planets between 3001 BC to be determined. And 3000 AD . For a more detailed description of the ephemeris used, see here. Espenak's polynomial expressions were used for ΔT, Huber's formula for estimating the uncertainty of these values. All calculations were made for a value of the secular tidal acceleration of the moon of -25,826 arcseconds / century2 carried out, and the ΔT values, assuming a value of -26.0 arc seconds / century2 for the secular tidal acceleration have been adjusted accordingly using a small correction term.
|-3000||20H 31m||±2H 30m|
|-2500||16H 30m||±1H 42m|
|-2000||12H 54m||±1H 02m|
The calculation of the solar eclipses and the local circumstances for certain locations was carried out in three steps.
- First, the Bessel elements X0, Y0, M0, D0, L10, L20 and their hourly changes X1, Y1, M1, D1, L11, L21 as well as tanF1 and tanF2, and the time of maximum eclipse for all in the period between 2500 v. The solar eclipses that occurred in AD 1000 and 1000 AD are calculated and stored for three different ΔT values: for a mean ΔT, which can be calculated from the formulas, for a lower ΔT, which corresponds to the mean ΔT minus the uncertainty of the ΔT value , and for an upper ΔT that corresponds to the mean ΔT plus the uncertainty of the ΔT value.
mean ΔT: 12H 54m
lower ΔT: mean ΔT uncertainty = 12H 54m - 1H 02m = 11H 52m
upper ΔT: mean ΔT + uncertainty = 12H 54m + 1H 02m = 13H 56m
Since this solar eclipse canon is designed for historical eclipses, the Bessel elements are calculated for a reference time t0 and the hourly changes in the elements are given. This linear form of the calculation of the elements is justified because the uncertainty of the ΔT value in the relevant period is much greater than the differences caused by a third-order polynomial form compared to the linear form. Bessel elements in polynomial form for all eclipses between 2000 BC Chr. And 3000 AD calculated for the mean ΔT and older lunar and solar ephemeris can be found on the NASA Eclipse web site.
- In a second step, the locations Athens, Rome, Babylon, Memphis, Thebes, Alexandria and Knossos as well as for the locations with the geographic coordinates (20 ° N, 5 ° E), (20 ° N, 50 ° E), (50 ° N, 5 ° E) and (50 ° N, 50 ° E) all those solar eclipses were selected that reached a magnitude of greater than or equal to 0.5 for at least one of the three ΔT values. For each of these locations, the times of the beginning, the maximum and the end of the solar eclipse, the magnitude, the sunrise and sunset times, as well as the position angles and sun heights were calculated. A limit of 0.5 was chosen based on the information in Ginzel (1899), who writes that an unpredicted solar eclipse is only noticed when a magnitude of about 0.75 is reached when the sun is high in the sky, or when a magnitude of 0.5 is reached for a sun just above the horizon.
- In the third step, the visibility curves were calculated for the selected solar eclipses: the outer points of the eclipse, the rise and set curves and the curve "maximum of the eclipse in the horizon" were calculated using the information in Seidelmann calculated, the position of the central line, the totality zone and the curves of equal magnitude according to Mucke & Meeus.
The present canon contains all solar eclipses between 2500 BC. And 1000 AD, which are in the geographical area between (20 ° N, 5 ° E), (20 ° N, 50 ° E), (50 ° N, 5 ° E) and (50 ° N, 50 ° E) ° O) were potentially conspicuous and thus come into question for identifications for eclipse reports from this geographical area, even if they were not predicted. Exact eclipse forecasts are first written in writing from around 300 BC. Tangible in Babylon; from then on there are reports in which solar eclipses with magnitudes smaller than 0.5 are also reported. For this reason, the data of all those solar eclipses are listed in a separate table, which reached a magnitude of less than 0.5 in the geographical area under consideration, but are still documented in writing.
History of the eclipse calculations
In the 19th century, interest in astronomical issues arose in the field of ancient studies. Moon observations, planet observations and eclipse reports were found and had to be set chronologically. For such tasks, Ginzel published a canon of eclipses between 900 BC in 1899. BC and AD 600. His work remained and still is groundbreaking for all later works on eclipses, which were created with a focus on the chronological purposes of eclipses. Ginzel collected eclipse quotes from ancient authors in his canon and suggested identifications. His work contains maps for the areas of classical antiquity, in which for each century between 900 BC. And 600 AD the totality zones of solar eclipses are drawn.
In 1986, Stephenson & Houlden published a solar eclipse canon for East Asia for the period between 1500 BC. And published in 1900 AD . The J = 2 moon phemerids of the International Astronomical Union (1968) were used to calculate the moon position, but adjusted to a more recent value for the tidal acceleration of the moon of -26 "/ cy2. The atlas contains maps in which the totality zones are drawn in and the associated latitudes, local time and sun elevation are given for selected degrees of longitude.
The last canon of eclipses so far, which is specifically geared to the needs of ancient scholars, was published in 2002 by Salvo de Meis; his canon contains calculations and maps of the areas of visibility for solar and lunar eclipses, which have been handed down in source texts, beginning in 763 BC. And ending in 1740 AD. For each eclipse, the quotation in English translation, the times and the magnitude for the observation location (s) are given. The calculations in the de Meis canon are based on the VSOP87 (for the sun) and ELP2000 ephemeris (for the moon) of the Bureau des Longitudes in Paris.
The traditional Babylonian solar and lunar eclipse observations and calculations were published in 2004 by Huber & de Meis. The calculations of the ephemeris are based on programs which Tuckerman’s tables and goldstine use. However, Huber includes some additional perturbation terms. The maps with the areas of visibility for the eclipses were made by S. de Meis, based on the same ephemeris that he used in his canon has used. For each identifiable solar eclipse, Huber gives the Babylonian date, universal time, start time and associated solar height, time of the maximum eclipse and associated solar height, end time and solar height, the magnitude reached, the duration of a possible totality, and the times of sunrise and sunset .
Apart from Ginzel's canon, the works mentioned above only contain maps of eclipses that are documented in writing. The electronic canon of solar eclipses presented here, on the other hand, is based heavily on Ginzel's. The present canon contains maps of all solar eclipses that occurred in the period between 2500 BC. Were potentially conspicuous in the selected geographical area, i.e. which reached a magnitude of at least 0.5.
The following excerpt from the tables created is intended to illustrate the structure of the canon:
For each solar eclipse, the following is given:
- the date,
- the ΔT value in seconds,
- Start and end time and time of maximum eclipse.
For the locations Athens, Rome, Babylon, Memphis, Thebes, Alexandria and Knossos then follow:
- the maximum magnitude,
- Time of maximum eclipse at this location in true local time
- Entry angle,
- Exit angle.
If a solar eclipse is documented in writing, the source is indicated, which can be clicked directly with the quote in the original (as far as possible) and in an English translation, e.g .:
For the solar eclipses from around 100 AD each was a single Quote selected, namely the one that contains the most information about the eclipse, even if several sources mention the corresponding eclipse. If the quotations are not displayed correctly or the file is not opened in the right place (depending on the browser!), The two .pdf files with the quotations can be loaded and saved here: babfinst.pdf and eclipsecitations.pdf. Gorodezkii Michail Leonidovich has put together a catalog of eclipse quotes that takes into account different sources for individual eclipses.
For each solar eclipse there is a map for the geographic area between (20 ° N, 5 ° E), (20 ° N, 50 ° E), (50 ° N, 5 ° E) and (50 ° N, 50 ° E) available. If an eclipse is documented in writing, there are further detailed plots for the areas of Italy, Greece and Asia Minor, Egypt and the Middle East. The maps are described in detail in the subsection Visibility area of a solar eclipse.
In addition to the publications on solar eclipses mentioned here, whose main focus is on historical eclipses, there are numerous other purely astronomical publications whose aim is to list all solar eclipses that have been and will be visible anywhere on earth within a certain period of time. These include the publications of Oppolzer, Mucke & Meeus, and Espenak.
Data and their download
If you wish to download and use the following data, please quote the following article in the Klio magazine and the address of this website as a reference:
R. Gautschy, Solar eclipses and their chronological significance: A new solar eclipse canon for ancient scholars, Klio 94, Issue 1, 2012, pp. 7-17.
The following table contains lists for the locations Athens, Rome, Babylon, Memphis, Thebes, Alexandria and Knossos, in which all solar eclipses can be found, and for which the sun is 50%, 60%, 70%, 80% in this location. , 90% or 100% was covered by the moon. These lists contain the following information:
- In the first three columns the calendar date in the order year, month and day (which are years BC astronomic counted, i.e. -999 corresponds to the year 1000 BC. Chr.),
- Information about the type of solar eclipse,
- the value of ΔT used for the calculation in seconds (calculated for mean ΔT),
- the maximum magnitude that the eclipse reached at this location,
- the start time in true local time,
- Time of maximum coverage in true local time,
- Time of the end of the darkness in true local time,
- Time of sunrise in true local time,
- Time of sunset in true local time.
Visibility area of a solar eclipse
This figure shows the visibility area of the total annular solar eclipse on May 12, 361 BC. BC, which has been handed down in writing in Plutarch, Vita Dionis XIX.4. At point P1 marked in green, the moon's shadow hits the earth's surface for the first time, i.e. H. the solar eclipse begins. The two red points PN1 and PS1 indicate the northernmost and southernmost point at which the solar eclipse can no longer be observed. For all points on the blue curve between PN1, P1 and PS1, the solar eclipse begins exactly at sunrise, i.e. the entire eclipse can be observed. For all points on the yellow curve between PN1, C1 and PS1, the sun is maximally covered at sunrise, i.e. only the second half of the eclipse can be observed. For all points on the blue curve between PN1, P2 and PS1, the solar eclipse ends at sunrise, i.e. no part of the eclipse can be observed here. Starting from the yellow curve PN1-C1-PS1, the position of the totality zone of the solar eclipse is drawn in red and so-called isomagnitudes are drawn in yellow or in different shades of green. In the case of the totality zone, the northern limit, the central line and the southern limit are indicated. The green isomagnitudes show where the sun is eclipsed to 90%, 80%, 70%, 50% or in yellow to 0%. At point P4 marked in green, the moon's shadow leaves the earth's surface again, i.e. H. the eclipse ends. For all locations on the blue curve between PN2, P4 and PS2, the solar eclipse ends exactly at sunset. For all locations on the yellow line PN2, C2 and PS2, the sun is maximally covered at sunset, i.e. only the first half of the eclipse can be observed. For all locations on the blue line PN2, P3 and PS2, the solar eclipse begins at sunset, i.e. no part of the eclipse can be observed here.
The maps in this eclipse canon are limited to the geographic area between (20 ° N, 5 ° E), (20 ° N, 50 ° E), (50 ° N, 5 ° E) and (50 ° N, 50 ° E) ). This means that the entire visibility area of the solar eclipse is never as shown in the figure above:
The two red solid lines indicate the area of the totality zone of the solar eclipse if one calculates with an average ΔT value. The green isomagnitude lines, which indicate where 90%, 80%, 70% and 50% of the sun are covered, are normally calculated on the maps with the mean ΔT value. The same applies to the blue sunrise and sunset curves and the yellow line, which indicates the maximum of the eclipse in the horizon. That is why the two red solid lines end exactly on the yellow "maximum in the horizon" line, while the red dashed or red dot-dashed lines end a little before or behind the yellow line. If during a solar eclipse in the defined geographical area the limit magnitude of 0.5 is not exceeded when calculating with the mean ΔT value, but if this is the case with the lower or upper ΔT value, then the curves mentioned are drawn with dashed or dotted lines . The two red dashed lines show the position of the totality zone of this solar eclipse when calculating with a lower ΔT value, the red dot-dashed lines when calculating with an upper ΔT value. The deviations of the dashed or dot-dashed lines from the solid line thus indicate the uncertainty of the calculations that are caused by the uncertainty of the time difference ΔT. For the example here this means that one can say that the totalitäszone ran safely over Sicily and Crete, but whether in Sparta (No. 6) the sun was covered 100% or only approx. 96% cannot say for sure. The black line indicates the central line of the solar eclipse, calculated with a mean ΔT value. Black ticks on the central line indicate the local time every quarter of an hour (in this example 16.5, i.e. 16:30), these times are correct exclusively for the central line and they should serve as a rough temporal orientation.
If the solar eclipse is a partial eclipse, the isomagnitude lines, which indicate where 90%, 80%, 70% and 50% of the sun were covered, are shown in different shades of purple on the maps.
For every solar eclipse, numerous places that are important during this period are shown as small blue circles, the names and coordinates of which can be found in the location register. Ten places important in the respective period are each marked in red, provided with numbers, and the names of the respective cities are indicated on the right outside the map.
For those solar eclipses that have been recorded in writing, further detailed maps are available for the areas of Italy, Greece and Asia Minor, Egypt and Voderasia. The following figure shows an example for the respective regions:
The fine subdivision of the area in 1 ° steps should make it possible to easily determine the location of a place that is not entered in the location register. The numbered places vary according to the period and their names can be looked up in the place register.
Due to the size of the files, the canon in the following table is divided into centuries. The link in the first column leads to all in the selected area conspicuous Solar eclipses in the corresponding century; in the last column .zip files can be downloaded which contain the calculated data. The last line in the table refers to the list of those solar eclipses that are documented in writing, but which have not reached the specified limit magnitude of 0.5 in the selected area.
- 1 J. Meeus, Mathematical Astronomy Morsels, Willmann-Bell 1997, 88ff.
- 2 F. W. Bessel, Astronomical Investigations II, Königsberg 1842 (online).
- 3 E. M. Standish, JPL Planetary and Lunar Ephemerides, DE405 / LE405, Jet Propulsion Laboratory Interoffice Memorandum 312.F, 1998.
- 4 F. Espenak, Polynominal expressions for deltaT, http://eclipse.gsfc.nasa.gov/SEhelp/deltatpoly2004.html
- 5 P. J. Huber, Modeling the Length of Day and Extrapolating the Rotation of the Earth, Journal of Geodesy 80, 2006, 283-303.
- 6 F. K. Ginzel, On the slightest phase that can still be seen with the naked eye when observing solar eclipses, Astronomische Nachrichten 118, 1887, 119-122.
- 7 P. K. Seidelmann (Ed.), Explanatory Supplement to the Astronomical Almanac, Sausalito 1992, 421-471.
- 8 H. Mucke & J. Meeus, Canon of solar eclipses -2003 to +2526, Vienna 1992, XXXIII-LI.
- 10 F. K. Ginzel, special canon of solar and lunar eclipses for the country area of classical antiquity and the period from 900 BC. BC to AD 600, Berlin 1899.
- 11 F. R. Stephenson & M. A. Houlden, Atlas of historical eclipse maps. East Asia 1500 BC - AD 1900, Cambridge 1986.
- 12 Transactions of the International Astronomical Union 13B, 48, 1968.
- 13 S. de Meis, Eclipses. An astronomical introduction for humanists, Serie Orientale Roma XCVI, Rome 2002.
- 14 P. Bretagnon & G. Francou, Planetary theories in rectangular and spherical variables. VSOP87 solutions, Astronomy and Astrophysics 202, 1988, 309-315.
- 15 J. Chapront & G. Francou, Lunar Solution ELP version ELP / MPP02; available online here.
- 16 P. J. Huber & S. de Meis, Babylonian eclipse observations from 750 BC to 1 BC, Milan 2004.
- 17 B. Tuckerman, Planetary, Lunar and Solar Positions 601 B.C. to A.D. 1, Memoirs of the American Philosophical Society Vol. 56, Philadelphia 1962.
- 18 H. H. Goldstine, New and Full Moons 1001 B.C. to A.D. 1651, Memoirs of the American Philosophical Society Vol. 94, Philadelphia 1973.
- 19 th. V. Oppolzer, Canon der Eclipse, Vienna 1887 (reprint Dover 1960).
- 20 H. Mucke & J. Meeus, Canon of solar eclipses -2003 to +2526, Astronomical Office, Vienna, 2nd edition, 1999.
- 21 F. Espenak, Five Millennium Canon of Solar Eclipses: -1999 to +3000, 2006, available online here.
This work was funded by the Swiss National Science Foundation as part of a Marie Heim-Vögtlin grant.
I would like to thank Alfred Gautschy, Bill Gray, Peter J. Huber, Kurt Locher, Salvo de Meis and Robert Nufer, who made an important contribution with suggestions, comments and corrections in various phases of the development of this canon.
Created by Rita Gautschy, Version 2.0, January 2012
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