Abstract
The article discusses the construction, function, and origin of the solar dial in the Olsztyn Castle, traditionally attributed to Copernicus. The dial, preserved partially on the wall of the cloister and presumably designed to determine the time of equinoxes, served as an astronomical instrument mapping the daily paths of the sun in the sky. The article provides a comprehensive mathematical model of the instrument, taking into account the astronomical and architectural factors which affected its functioning. The analysis allows to alienate the essential properties of the dial as an observational instrument and to contend that measurements were recorded indelibly on the wall and averaged by interpolation. Furthermore, it reconsiders the arguments which support the hypothesis ascribing the construction of the Olsztyn instrument to Copernicus. (Some mathematical appendices appear only in the online issue of the journal.)
Introduction
One of the last largely unsolved mysteries of Copernicus’ astronomy is the way he observed celestial phenomena. 1 This refers both to the number of his observations and to the methods used by Copernicus. 2 Based on Copernicus’ references in De revolutionibus, one can assume that he made many such observations, 3 and yet there are only approximately 30 observations quoted by him in his work and the other 30 evidenced by some marginal notes in the books from his library. In total, there are about 60 known astronomical observations made by Copernicus. However, it is also the report of Georg Joachim Rheticus in Narratio prima which suggests that Copernicus brought together an impressive number of astronomical observations: “… my teacher always has before his eyes the observations of all ages together with his own, assembled in order as in catalogues.” 4
In the light of De revolutionibus, Copernicus appears to employ a rather conservative observational technique. There are references to exclusively three astronomical instruments, all deriving from ancient astronomy, i.e., the solar quadrant (the plinth; II,2 and 14), the armillary astrolabe (II,14), and the parallactic instrument (IV,15). Their descriptions appear to paraphrase Ptolemy’s Almagest (I,12, V,1, and V,12, respectively). Copernicus mentions also a Hipparchan dioptra (IV,2 and 18) as an instrument for measuring the moon’s or the sun’s diameter, but similarly as Ptolemy (Almagest, V,14) refrains from describing it. 5 There are no suggestions that he had such instrument.
Copernicus’ observations of the sun appear quite puzzling in this context. Let us consider first his observations in Frombork (Frauenburg) of the autumnal equinox on 14 September 1515 and the vernal equinox on 11 March 1516 (De revolutionibus, III,13) and, presumably, the vernal equinox on 1515 (III,16). 6 Copernicus did not write how he made these observations. Ptolemy while presenting (far more numerous than in De revolutionibus) observations of the times of the equinoxes made by Hipparchus and himself describes yet another ancient instrument, i.e., the equatorial ring (Almagest III,1), 7 which is missing from De revolutionibus. The list of the observations compiled by Ptolemy is so vague that one can only suspect that he observed at least some equinoxes on equatorial rings. It is also possible that Hipparchus determined the times of the equinoxes from direct observations of solar declinations rather than using an equatorial ring. Hence, Hipparchus could use a meridian ring which is described by Ptolemy along with the solar quadrant in Almagest (I,12). 8
Ptolemy discussed in Almagest (III,1) the inaccuracy which can arise while determining the time of equinox due to an observational error of 6′ in declination. This discussion appears to pertain to the proper alignment of the equatorial ring. A similar discussion (though referring to the observations of the solar altitude in the meridian, both near the time of equinox and solstice) is offered by Johannes Regiomontanus in Epitome of the Almagest (III,1). Copernicus did not go into such details in De revolutionibus, although he refers in passing to the general opinion of Ptolemy about the observation of solstices: “Claudius Ptolemy, however, being aware that pinpointing of a solstice is difficult and uncertain …” (III,13).
9
However, Ptolemy’s reservation is discussed more extensively in Narratio prima: Nevertheless, it is impossible, as Ptolemy states, by means of instruments to determine with precision the times of the solstices. For a single minute of declination, which of course easily escapes the eye, may deceive us in this matter by about 4°, to which four days correspond.
10
And yet Copernicus did not offer any account of his observations of solstices, and he published only two (or three) of the aforementioned observations of equinoxes. Without any direct hints in the sources, one can only assume that Copernicus may have used in these observations the measurements of the altitude of the sun at meridian transits, using his solar quadrant, and interpolated between observations of transits before and after the equinox. 11
However, Narratio prima offers yet another interesting hint. Rheticus’ account is followed by the chapter titled In Praise of Prussia which describes his stay in the Warmia. Here, we learn that Copernicus’ closest friend, Bishop Tiedemann Giese, had in his residence in Lubawa (Löbau; i.e., in the place where in the summer 1539 the final decision about the publication of De revolutionibus was made) a bronze equatorial ring. 12 Unfortunately, we know nothing about the origin and use of this instrument. Needless to say, an equatorial ring is not a portable device. It needs to be aligned towards the horizon in a way corresponding to the geographical latitude of a given location. Accordingly, if it had been used by Copernicus earlier, it could be treated as a mere historical exhibit in Lubawa. 13 (Giese moved to the Bishops’ Palace in Lubawa in the first half of 1538, after he received the papal appointment.) 14 Whatever was the case, the presence of this instrument in the residence of Giese who was Copernicus friend since at least 1516 15 seems to testify to the vivid interest in various techniques of the observation of equinoxes in Copernicus’ nearest circle.
An interesting light on Copernicus’ readiness to experiment with new observational methods is cast by his observations of the eclipses of the sun. The results of these observations in the years 1530–1541 are extant in the form of Copernicus’ marginal notes in his copy of Calendarium Romanum magnum by Johann Stoeffler (Oppenheim, 1518). Based on some indirect evidence, we can assume that in the relevant period, Copernicus used a pinhole camera (camera obscura) to determine the eclipse magnitude. 16 Rheticus placed Copernicus’ observations of the eclipses in the broader temporal perspective: “For nearly 40 years in Italy and here in Frauenburg, he observed eclipses and the motion of the sun.” 17 Apart from the aforementioned observations of the equinoxes and later observations of the eclipses, one of the direct accounts of Copernicus’ studies of the motions of the sun is his observation of the sun at Scorpio 15o published in De revolutionibus (III,16). 18 It was used to derive the elements of the solar orbits. Further on, however, Copernicus added a reservation: “… even in putting the apogee at 6 2/3° within the Crab, I was not satisfied to trust the time-measuring instruments, unless my results were also confirmed by solar and lunar eclipses” (III,20). 19 And yet Copernicus did not use any of his observations of the eclipses of the sun in De revolutionibus, and he never described how he made such observations.
One of the greatest puzzles as regards Copernicus’ observational practice pertains to the solar dial whose remains survived on the plastered wall of the cloister in the northern part of the Olsztyn Castle. There are at least two reasons to investigate the dial. First, Copernicus has been traditionally considered to be the originator of the dial even though also, in this case, there are no references to such an instrument in any of his extant texts. The significance of such attributions is more than obvious as this would be the only instrument built by Copernicus which has survived till this day. 20 Second, whoever constructed the solar dial, it is a fascinating object. Despite all our efforts, we still hardly know how it was drawn on the wall and for what purpose.
The solar dial: a historical overview
The first known description of the solar dial in the Olsztyn Castle is a note by Heinrich Reinhold Hein, the first parish priest (1783–1797) of the Evangelical Parish in Olsztyn (Allenstein). In the introduction to his report, he wrote, In the room, whose two windows face west, he [Copernicus] made the sundial on the wall, which at the time when the clock was to show hours, namely in the mornings, could not be directly illuminated by the sun …
21
He was also first to attribute the design of the instrument to Copernicus. The next information about the instrument we owe to Jan Śniadecki (1756–1830), a mathematician and astronomer, a professor in the Cracow Academy. He wrote, for the contest announced in 1802 by the Towarzystwo Warszawskie Przyjaciół Nauk (Warsaw Scientific Society), a treatise that discusses in a concise way the scientific achievements of Nicolaus Copernicus. In the final part of this modest little book, we find information about the solar dial in Olsztyn: It was probably an astronomical gnomon, which Copernicus himself drew up in his apartment, to measure the time of the height of the sun at noon, the Solstitiorum et Aequinoctiorum observation and investigation of the slope of the ecliptic.
22
Śniadecki acquired this information from a report on the interviews conducted by Tadeusz Czacki and Marcin Molski with Pastor Christian Leopold Stuber, who lived in the castle at that time (1797–1806). 23
The construction of the instrument is characterized by substantial originality. Due to its location, on the north-eastern wall of the cloister of the castle, its author could not copy the well-known sundial patterns but was forced to use a novel method employing reflection of the rays of the sun. It is likely that, as noted by Tadeusz Przypkowski, this application of the sun’s reflection was the first in the history of gnomonics. 24 The astronomical dial measures 7.05 m by 1.4 m, and it was imprinted onto a wet fresco white wash, used probably around 1517. 25 In several sections of the preserved plaster, oblique, almost parallel, straight lines, painted in red, can be seen (Figure 1). These day lines correspond to the daily trajectories of the sun’s rays reflected from the fixed mirror, and they were drawn on the wall at intervals of several days. At the lower edge of the dial, in red, are painted Arabic numerals which are descriptions of some of the day lines. The line corresponding to the equinoxes is distinguished by the preserved remnants of a cobalt blue pigment (smalt) and by the description, “T I C,” in uppercase, located above it. The whole system of days is intersected by straight black lines which converge downwards. Some of them are described by Roman numerals, most likely indicating the hours of the day.
The remains of the solar dial in the cloister of the north-eastern part of the Olsztyn Castle.
In some places, the red lines exhibit visible scratches and small holes. These may be related to the process of construction of the instrument or they may be the result of subsequent maintenance and repainting. To draw a day line, several observations were needed. The observations would have served as basic nodes, enabling construction of a day line by graphical interpolation across some time intervals. However, based purely on visual inspection, it is hard to state quite how the instrument was created.
The solar dial retained its original form, but could only fulfil its function until 1761, when the gallery was rebuilt and divided into smaller rooms. In the drawing by Ferdinand von Quast, completed in 1852, there can be seen bricked-up arcades, behind which there was an apartment for the Pastor (Figure 2).
26
The small hole in the third arcade from the left was possibly the opening which Marcin Molski reported to Jan Śniadecki in his letter from Königsberg, dated 12 August 1802: Above the door was formerly a carved hole, through which the sun’s rays were admitted to the points marked in the second chamber, but six years before, as the current resident bricked-up the empty space with a dozen bricks.
27
View of the north side of the castle courtyard in 1852. The sketch is by Ferdinand von Quast, the first Prussian conservator of historical monuments. The residues of the solar dial are in the passage of the second-floor cloister with bricked-up arches.
The present view of the instrument, after preservation works carried out in 1955 and 1956 by a team of art conservators led by Bohdan Marconi, 28 is shown in Figure 1.
To commemorate the approaching 400th anniversary of the death of Copernicus, Ernst Zinner published a detailed description of the Olsztyn instrument. 29 This publication proved to be an important asset to any subsequent examination of the dial. Feliks Przypkowski, analysing the drawing of the instrument published by Zinner, and taking into account the information of Pastor Heinrich Hein, 30 pointed out that the hour lines converge down towards the bottom of the graph (see Figure 1), in contrast to the classical case of a vertical sundial. This was a significant contribution towards understanding the function of the instrument. As an explanation, he proposed the idea that these are traces of the solar beam reflected by a horizontal mirror. 31
The method of direct projection of solar rays onto the floor was in use in Copernicus’ time. A ray of sunshine would pass through a small hole in the roof in the south aisle of the room, projecting a small image of the sun onto the floor. By observing the day-to-day movement of this solar spot, it was possible to determine, e.g., the time of solstice. In Italy, c. 1475, Paolo Toscanelli drew such meridian line which can be seen in the cathedral of Santa Maria del Fiore in Florence till today. 32 We do not know whether Copernicus saw Toscanelli’s instrument during his stay in Italy or heard about it. 33 Interestingly enough, however, Regiomontanus noted that it was built to check whether the inclination of the earth’s axis changes over time. 34 As it is known, Copernicus was vitally interested in the variability of the obliquity of the ecliptic, described as a change of the inclination of the earth’s axis (towards the axis of ecliptic), when he was developing the fundamental principles of his system. 35
Whatever was Copernicus’ involvement in the construction of the instrument, there is no information that in the sixteenth century or previously, anyone had ever used a reflection method for this purpose. In the critical literature on the European history of sundials, the theory and design of reflective (anacamptic) sundials are usually associated with the Jesuit scholars Athanasius Kircher (1635 and 1646) 36 and Emmanuel Maignan (1648). 37
The solar dial v. Copernicus’ appointment in Olsztyn
If we accept the relatively late tradition which ascribes the authorship of the solar dial in Olsztyn to Copernicus, we have to decide when during his lifetime he could construct such instrument. Knowing that, we could consider the usefulness of the instrument from the point of view of Copernicus’ astronomical pursuits.
Following his studies in Italy, Copernicus returned to Warmia in the second half of 1503 and instantly entered into service of Lucas Watzenrode (1447–1512), the Bishop of Warmia and his uncle.
38
If he did not travel with the Bishop, he would probably spend most of his time in that period at the episcopal palace in Lidzbark Warmiński (Heilsberg). Around 1510, in 1512 at the latest, Copernicus left the Bishop’s service and moved to Frombork (Frauenburg). Here, in June 1512, he received from the Cathedral Council a house on a hill slope, outside the walls of the Cathedral. It was there that he probably built his observational platform (pavimentum) and drew the local meridian as he describes in De revolutionibus (II,2). We can assume that his pavimentum was built with 800 bricks and one barrel of chlorinated lime which he bought in spring 1513. At that time, the first description of Copernicus’ heliocentric theory, his Commentariolus, was already circulating in the form of a short manuscript. In the middle of 1513, Copernicus was invited by Paul of Middelburg (1446–1534), the Bishop of Fossombrone, to contribute to the discussion on the reform of calendar conducted under the patronage of Pope Leo X. Copernicus’ letter with some remarks in this regard reached Paul of Middelburg before June 1516. Unfortunately, the content of the letter is unknown. However, a reference to this engagement can be found in the Preface to De revolutionibus addressed to Pope Paul III: For not so long ago under Leo X the Lateran Council considered the problem of reforming the ecclesiastical calendar. The issue remained undecided then only because the lengths of the year and month and the motions of the sun and moon were regarded as not yet adequately measured. From that time on, at the suggestion of that most distinguished man, Paul, bishop of Fossombrone, who was then in charge of this matter, I have directed my attention to a more precise study of this topics.
39
It must be emphasized that the observations of the sun which Copernicus mentions in De revolutionibus and which led to some important conclusions as regards his solar theory were made in the years 1515–1516. Referring to these observations, Copernicus contends, “For during the ten or more years since I have devoted my attention to investigating these topics, and in particular in 1515
Bearing in mind all these reservations, let us review the purpose of the instrument based on the contemporary analysis of its surviving fragments.
The calendar lines
The day lines on the dial were created by projecting the beam of sunlight reflected by a mirror (which was probably the size of a thaler), placed in a hole in the windowsill of the cloister arcade, according to Pastor Hein’s account. 42 This role could have been fulfilled by a vessel filled with mercury (or red wine), which provides an exact horizontal mirror surface.
For analysis of the dial as a gnomonic instrument, it is convenient to adopt a geocentric reference frame. The mirror is assumed to be in the centre of the celestial sphere in the plane of the horizon. The axis of the celestial sphere is inclined according to the latitude of the place of observation. The great circle of the local meridian is perpendicular to it and passes through the celestial poles, the zenith, and the nadir. The meridian crosses the circular section of the horizon at north and south points. The great circle of the equator is determined by the intersection of the celestial sphere by the earth’s equator plane. The ecliptic is a great circle resulting from the intersection of the celestial sphere with the plane of the earth’s orbit. It is inclined towards the plane of the celestial equator at an angle of 23.5°, intersecting it at two points, which correspond, respectively, to the vernal equinox (the point of Aries) and the autumnal equinox (the point of Libra). The sun, as viewed from earth, apparently moves along the ecliptic eastward at an average speed of 0.986° per day. The irregularity of this motion throughout the year, which has been known since ancient times, has no effect on the observed track drawn by the reflected sunlight, but only on its speed along this line.
From the point of view of an observer located in the surface of earth, the sun traces every day its circular path in the sky. The mirror illuminated during a day by sunlight is the apex of the cone, and the reflected sunbeam corresponds to this cone-forming surface. This defined surface cuts the plane of the solar dial, resulting in a conic curve, which is the path of the reflected sunbeam. As a result, this curve maps the daily path of the sun above the horizon. The apparent annual motion of the sun across the sky, which is a result of the orbital motion of the earth, means that the route of the sun above the horizon varies from day to day throughout the year. These changes are reproduced on the solar dial by the position and shape of the curves mapping the sun’s movement.
The form of the conic sections discussed here depends on the latitude of the observation site. If the angle between the plane of projection and the axis of the cone is smaller than the angle between the cone axis and the cone generatrix, the intersection of the edge of the cone is a hyperbola. This occurs in the temperate climate zone. The hyperbola of greatest curvature is observed on the summer and winter solstices. These curves form a figure resembling a double-edged axe.
The Olsztyn Castle is located at latitude ϕ = 53°46′36″. 43 The ground plan of the castle, based on a drawing by Ferdinand von Quast, 44 with the location and description of the architectural elements essential for the operation of the instrument, is shown in Figure 3. The plan provides the most important architectural parameters used in the calculations, and further analysis, including the chosen coordinate system, the location of the solar dial, the position of the mirror, and the location of the castle tower.
Plan of the first floor of the Olsztyn Castle at a scale of 1:320. It provides the architectural parameters essential for the analysis: the chosen coordinate system, the location of the solar dial, the position of the mirror, and the location of the castle tower.
The geometry of the observational system is shown in Figure 4. The mirror is located in the horizontal plane at the height of the parapet wall of the arcade at a distance d from the solar dial. The position of the dial, in the plane of the horizon, determines the azimuth of the wall on which it resides: AT = 123°33′36″. 45 The orthogonal projection of the centre of the mirror onto the plane of the instrument determines the origin of the Cartesian coordinate system. The x-axis of this system is oriented along the plane of the instrument and the x-coordinate increases to the right, while the z-axis is the vertical axis in the plane of the instrument increasing upwards. The (x, y) pair of coordinates describes the position of the point in the plane of the instrument. The y-axis is perpendicular to the plate and increases towards the mirror.
Geometry of setup for the observation of the beam of sunlight projected onto vertical dial. The instantaneous position of the sun determines the coordinates of the system associated with the plane of the horizon and the local meridian: the azimuth of the sun A measured from 0° to 360° starting from the south S in a clockwise direction, and the height of the sun h, which is the angle between the horizontal plane and the direction pointing towards the sun. The lower panel presents the horizontal plane (x, y) of the adopted coordinate system, while the upper panel shows the plane p – p containing the solar ray that defines the z-coordinate of the position of the light spot.
It is convenient to describe the geometry and working principle of the solar dial in the coordinates of the horizon (A, h) defined by the place and time of observation. The instantaneous position of the sun in the plane of the horizon determines its time-dependent azimuth A(t). Assuming that the sun is effectively a point source of light, the ray of the sun falling on the mirror and the reflected ray are in the same vertical plane, the plane which is perpendicular to that of the horizon. The angle of incidence of the solar ray on the mirror is equal to the current zenith distance of the sun z = (90° – h), where h is the instant height of the sun above the horizon. This is also the angle of the beam r reflected from the mirror and entering the plane of the instrument at the point (x, z). The coordinates of this point are a function of the pairs of angles (A, h), which correspond to the current position of the sun on the celestial sphere in the horizontal reference frame.
The projection of the reflected ray r on the plane of the horizon may be denoted as
where, to simplify the notation, the symbol θ = AT – 90° was introduced. The instantaneous coordinates of the light spot in the horizontal reference frame can be expressed as follows
Some trigonometric identities were employed to separate the angles θ and A from their difference θ – A. This enables the effective use of trigonometric relationships involving the angles and sides in a parallactic triangle.
However, the coordinates (A, h) of the sun change rapidly in the horizontal frame, and moreover are both complex functions of time. Therefore, for the analysis of day lines, it is more convenient to use the equatorial coordinate system with coordinates of declination δ and hour angle t. This pair (δ, t) applied to the position of the sun reflects the direction of the rotation axis of the earth in space and its current state of rotational motion. 46 The transformation between the two coordinate systems (A, h) and (δ, t) is defined by trigonometric relationships in the parallactic triangle (see Appendix A, available in the online edition of the journal).
For the analysis of day lines, it is sufficient to know the daily path plotted by the reflective nodus. This simplification, eliminating the hour angle from the system of equation (A2), greatly facilitates the study of the day lines, which are the trace of reflected sunlight in the vertical (x, z) plane of the instrument. The study shows that for the location of the solar dial in moderate climatic zone, the trace corresponds to the hyperbola. In the particular case of the orientation of the dial perpendicular to the local meridian (θ = 0°), equation (A5) simplifies considerably and the trace takes a regular form of the equilateral opening N-S hyperbola, whose branches are symmetrical with respect to the horizontal symmetry axis (Figure 5).
Hyperbolic paths of the spot of the sunlight for orientation of the dial perpendicular to the local meridian (θ = 0°), for declinations of the sun −20° δ 20°. The light spot moves from west to east. The horizontal straight line corresponds to the path on the equinox when δ = 0°. It divides the chart into two parts: curves below it corresponds to observations made before; above it, those made after the vernal equinox.
The branches of hyperbolae observed prior to the vernal equinox are open downwards, whereas, following it, they are open upwards, while the horizontal line that divides these mirror images corresponds to the equinoctial observation. The opposite arms of each of the branches exhibit limiting linear asymptotes which intersect at the centre of symmetry of the hyperbola. The horizontal straight line corresponds to the sunlight trace observed in the equinox day.
To plot the real path of the sunspot in the plane of the instrument (θ ≠ 0°), the x-coordinate was assumed to be an independent variable, of which the coordinate z = z0 + z(ϕ, θ, d, δ, x) becomes a function. The parameters ϕ, θ, δ, and d define the actual shape of the hyperbola plotted in the plane of the instrument. Similarly, as shown earlier in the specific case θ = 0°, there are two solutions for z corresponding to the two branches of the hyperbola: one for the days before (lower) and the other for days after (upper) the vernal equinox, and vice versa for the autumnal equinox. However, the azimuth of the solar dial, θ ≠ 0°, causes that the collection of light tracks plotted in Figure 7 for dates around the equinox are distorted.
Using the function z = z0 + z(ϕ, θ, d, δ, x), an attempt can be made to fit theoretically determined curves to the original lines plotted on the solar dial. For that purpose, the coordinates of some points on each day line were digitized 47 using a high-resolution photogrammetric image of the dial. 48 Then, the functions given in equation (A5) that are parameterized by the day-dependent declination of the sun, δ, were fitted to these digitized points. The fitting of this non-linear function to the data set was accomplished using a least-squares Levenberg–Marquardt algorithm. 49 The calculations were implemented in the Maple 12 mathematical package, 50 using a freely available software procedure implemented by D. E. Holmgren, J. F. Ogilvie, and M. Monagan. 51 The following values of parameters characterizing the location of the instrument have been assumed: ϕ = 53°46′36″ and θ = 33°33′36″ in order to retrieve the declination of the sun, δ, related to each line plotted on the wall. The standard deviation obtained from the fit to each line was adopted as the criterion of the quality of the fit.
The choice of origin of the Cartesian reference frame and position of the mirror associated with it are of crucial importance in the fitting procedure. To give these locations the most credibility, the report of Pastor Hein, 52 the findings by previous investigators of the solar dial (Zinner, 53 Przypkowski 54 ), and the results of other surveys of the instrument and its proximity (Miałdun 55 ) were considered. On this basis, the initial assumption was made that the mirror was placed on the windowsill of the third cloister arcade, at a distance d = 3.99 m from the plane of the instrument and below its lower edge at z0 = –2.16 m. As displayed in Figure 5, the theoretical considerations (equation (A6)) show that the centres of curvature of the set of hyperbolas are located in the area of the dial situated opposite the mirror. It should be noted that there is a deep hole in the wall, whose relationship with the operation of the solar dial remains unknown. It may be suspected that it housed a mounting bracket for hanging the mirror.
The values of the declination of the sun for individual day lines resulting from numerical analysis are listed in Table 1. The values of declination δ of the sun for consecutive day lines were obtained by the least-squares fitting method for the mirror stationed at the offset points z0 below the lower edge of the solar dial, where MSE is the residual mean squared error. The uncertainty of the estimated value of Δδ was calculated by error propagation. 56
The values of declination δ of the sun and corresponding longitudes λ for each day line.
MSE: mean squared error.
The calendar lines were numbered from the left side of the solar dial (see Figure 7).
Figure 6 reveals that the declination of the sun increases almost linearly in the period that was covered by the observations. For the fixed value of the parameter z0 = –2.16 m, the minimum value of the declination recovered from the results for all the lines was δ = –0.280°. It corresponds to calendar day line No. 8, but, unfortunately, this is not the equinoctial line. Moreover, for line No. 8, a significant discrepancy between the theoretical curve and the points measured on the solar dial is also evident. The MSE in this case is 0.079 and is one of the highest among all the lines (Figure 6, □). It should be noted that neither the quality of fit nor the resulting value of the declination were found to depend on the choice of the initial value taken for δ in the analysis.
The resulting values for daily declination of the sun for each calendar line on the solar dial (see Table 1). The line with points marked □ corresponds to the mirror being positioned at z0 = –2.16 m and that marked with ❍ corresponds to the mirror being positioned at z0 = –1.86 m. Error bars for the MSE of δ are presented 20-fold enlarged in order to emphasize the quality of the fit for each of calendar line.
Close inspection of Table 1 reveals the best quality fit, MSE = 0.013, to be for line No. 9 (Figure 6, □). This is the straight line in the solar dial, indicated by the inscription “T I C,” that corresponds to the equinox. A very good quality of fit is not surprising in this case, because the hyperbola resulting from the theory deviates only very little from a straight line. There still remains, however, the problematic non-zero declination δ = 2.810° assigned to this line by the fitting procedure.
It follows from equation (A5) that the only factors which may affect the results of the analysis are the design parameters of the solar dial and, to more precision, the coordinates of the position of the mirror. On releasing the parameter z0, corresponding to the offset of the mirror, to be freely found in the analysis for the equinox line No. 9, an offset value for z0 = –1.86 m results in the best fit, with a virtually zero declination, δ = –0.001°, and an improved quality of fit: σ = 0.0078 (Figure 7, ○). On this basis, it can be concluded that the mirror was originally about 30 cm higher than the assumed initial value of the parameter z0.
The theoretical hyperbolic tracks of the sun’s light spot matched to points on the preserved fragments of the day lines. The labels present the value of the sun’s declination found in the fitting procedure. The results were obtained for the following parameters: ϕ = 53°46′36″, θ = 33°33′36″, and offset z0 = –1.86 m.
Jerzy Sikorski, based on the analysis of archival records and architectural inventory of the Olsztyn castle, determined that during the renovation in years 1865–1866, the walls dividing the cloister into smaller chambers were removed, leaving only the remaining sections of the instrument. 57 The layout and height of the arcade were also changed and the windowsill was lowered by the removal of three layers of bricks (see Figure 2). This last fact may explain why the value of the z0 offset resulting from the procedure of curve-fitting after releasing the parameter z0 was smaller by 30 cm.
Figure 7 shows the hyperbolic curves defined by equation (A5) for individual values of declination of the sun obtained for offset z0 = –1.86 m. The declination values listed in Table 1 show that the red lines in the solar dial correspond to daily observations of the sun, over a period of 67 days in early 1517, from 28 January (line No. 1) to 3 April (line No. 14) according to the Julian calendar or, correspondingly, from 7 February till 13 April according to the modern calendar. The calendar lines were numbered from the left side of the panel. The average value of the differences in declination between adjacent lines, Δδ, is 1.89° ± 0.07°, which corresponds to an average spacing of time, ΔT, of 5.07d ± 0.16d.
Fryderyk Koebke and Tadeusz Przypkowski, studying the solar dial in the mid-1950s, calculated the slope of the equinox line in the accepted reference system, which, as Przypkowski writes, “… closely corresponds to the current slope of the graph on the wall.” 58 Unfortunately, he does not delineate these calculations nor explicitly specify the value obtained. The slope of the equinox line (No. 9, δ = −0.001 in Figure 7) measured on the solar dial by matching the arbitrary linear function is α = –21°14′7″ ± 4″. It is worth comparing this result with the theoretical value. The directional coefficient of the straight line given by equation (A7) depends on the geographical latitude ϕ and azimuth of the instrument θ = AT – 90° only. The resulting slope of the line is α = –21°38′41.07″. The two, theoretically calculated and measured in situ, values of the slope of the equinox line thus differ by 24′3″.
As shown herein, the individual lines are uniquely defined by the daily declination of the sun. Alternatively, the ecliptic coordinates, longitude λ and latitude β, are used to define the positions of the sun and other objects in the solar system. According to the contemporary definition of the longitude, the angle increases along the ecliptic in the direction of the annual movement of the sun, from zero at the point of Aries to 360°. The amplitude, β, of the monthly deviations of the solar disc from the line of the ecliptic does not exceed 1.2″. For this reason, in order to assign individual lines corresponding to daily values of λ, the latitude β = 0° can be applied with negligible error in the transformation between the equatorial and ecliptic systems, leading to a simple relationship between these coordinates sinλ = sinδ/sinε, where ε is the angle of inclination of the equator to the plane of the ecliptic. 59 On a scale of hundreds of years, this slope has slowly changed in the range from 22.1° to 24.5°. The current value is ε = 23.439°, but in the time of Copernicus, it was a little bit larger, with a value ε = 23.505°. The deviation in the course of the function z = z0 + z(ϕ, θ, d, δ, x) that results from this variation does not exceed the width of lines drawn on the wall. Table 1 also shows, as well as the values of declination δ, the corresponding values of longitudes, λ.
The range of the longitudes given in Table 1 indicates that the range of λ covered by the solar dial is 65.539°. This corresponds to two and nearly a quarter signs of the zodiac (Figure 8). The end portion λ⊙ < 330° of the sign of Aquarius appears in the left part of the solar dial, while in the right part, there is a fragment of the sign of Aries, ranging from λ⊙ = 0° to λ⊙ = 23.4°. The central section of the solar dial includes the sign of Pisces, in 330° < λ⊙ < 360°. The average difference of longitude Δλ between two adjacent calendar lines is 5.00° ± 0.16°. On this basis, one can identify the meaning of the Arabic numerals that describe the day lines at the bottom margin of the solar dial. 60 The numeral “10” (left-hand red circle in Figure 8) below line No. 1 (λ⊙ = 316.88°) means that there remain 10° to the end of the sign of Aquarius, indicated by line No. 3 (λ⊙ = 327.99). Strictly speaking, the day corresponding to the entrance of the sun in the sign of Pisces is a little earlier (λ⊙ = 330°) than the subsequent observation recorded on the solar dial. Figure 8 is marked with a black dashed line. At the next line, No. 4 (λ⊙ = 333.03°), one can recognize (in the middle of the red circle) the old form of the numeral “2,” resembling the letter “Z,” used in the fourteenth and fifteenth centuries. 61 The plaster to the right of that digit is damaged. Thus, the question arises as to what this figure could mean, situated in this place on the dial? Presumably, it could constitute the remnant of the number “25” defining the number of degrees left to the end of Pisces, and thereby indicate the time remaining until the equinox. In the middle of the sign of Pisces is line No. 6 (λ⊙ = 342.76°), marked, as would be consistent with the earlier presumptive statement, with the numeral “15” (right-hand red circle). The subsequent line, No. 7 (λ⊙ = 348.69°), is again described with the numeral “10,” representing the number of degrees that are left until the line of equinox when the sun enters the sign of Aries is reached. This reference number is, in contrast to the numbers described previously, located not at the bottom margin of the solar dial but painted on line No. 7, just above the black Roman numeral II. It is also slightly smaller than the other numbers. A closer look at this inscription reveals the symbol “X” carved into the plaster inside the digit “0.” One would assume that this symbol represents the Roman numeral 10, and that it would be an original indication of the number of days remaining until the equinox, made by the author of the instrument. The last line plotted in the upper right corner of the solar dial is line No. 14, corresponding to the longitude λ⊙ = 23.37°. 62
View of the solar dial. The day line numbering convention used is from left to right. The red circles indicate the markings of the lines that are relevant for explanation of the operation of the instrument. Enlarged fragments of these inscriptions are shown below. A detailed description is presented in the text.
Line No. 9 corresponding to the vernal equinox occupies a central position on the solar dial (see Figure 9). The remnants of blue colour and the inscription “T I C,” which appears just above this line distinguish it from the other red lines. Regardless of whether the purpose of the solar dial was strictly to do with the calendar and/or had a research function, it is worth considering the significance of this inscription. In the traditional research carried out on the solar dial over the years, several suggestions as to the meaning of this signature have been made. According to the proposal of Jan Śniadecki, these three letters were regarded as a remnant of the term ECLIPTICA.
63
The problem is that all apparent annual paths of the sun coincide with the ecliptic. The equinox would not, therefore, constitute an especially highlighted day in this case. This fact attracted the attention of Tadeusz Przypkowski who pointed out that a more appropriate suggestion would be
The central part of the solar dial. The inscription “T I C” appears just above the day line corresponding to the vernal equinox. A detailed description is presented in the text.
The lines of hours
The day lines discussed so far are crossed out by a bunch of black lines that, most likely, indicated the consecutive hours of true solar time. Two external black lines are labelled at the bottom of the solar dial by the Roman numerals XI and II (Figure 8). Descriptions of the other two straight lines located between them have not survived. In this work, they shall be referred to as numbers 2 and 3 from left to right, respectively. In the history of research into the instrument, the interpretation of these black lines has been the most difficult task and the obtained results were rather inaccurate and speculative. 67 Adequate analysis of the forms of black lines is necessary for an understanding of the geometric principles that underlie their construction. In contrast to the classical sundial, in which the current time is shown by the shadow of the gnomon falling on the appropriate hour line at the given moment, individual points of the hour line in the case of the reflective clock are defined by the instantaneous position of the spot of sunlight on the dial. To draw an hour line, at least two such points, corresponding to the same time on different days, are needed. This requirement represents a significant impediment to the practical plotting of hour lines.
The manifold of hour lines may be derived from equation (A2). Each hour line in the plane is the mapping of a big wheel corresponding to the fixed values of the hour angle t for values of the declination δ changing in the range of −90° to +90°. The projection lines corresponding to different values of the hour angle t diverge. This feature is used in graphical methods of constructing sundials. 68
From a formal point of view, the reflected beam of light determines a point on the plane of the instrument which is the gnomonic mapping of the instantaneous position of the sun on the celestial sphere. Thus, the straight black lines, corresponding to the hours, are resulting from the gnomonic projection of the individual hour wheels of the equatorial coordinate system, on the plane tangential to the sphere. However, in the case of the reflective clock, the geometry of the classical gnomonic projection is modified by the presence of the horizontally oriented mirror. According to the law of reflection, the mirror constitutes a symmetrical reflection point which is the gnomonically mapped position of the sun relative to the plane of the horizon (see Appendix B, available in the online edition of the journal). 69
The straight lines z = z0 + z(ϕ, θ, d, x, t) defined by equation (B1) were fitted, using the previously described method of least squares, to the residues of hour lines preserved in the dial. As shown in Figure 4, the solar dial is in the plane (x, z) with its lower edge at z = 0. The mirror is located below the solar dial at a point (0, d, –z0). Figure 10 shows graphically the result of matching when the hour angle t was the only parameter allowed to be free in the analysis. It is easy to see from Figure 10 that none of the straight-line fits is even remotely satisfactory.
The linear function z = z0 + z(ϕ, θ, d, x, t) have been fitted to the digitized coordinates of the points (symbol o) lying on the hour lines identified in the order from right to left as XI, 2, 3, and II. The dashed line corresponds to a currently non-existent line for hour 12. The fixed parameters are ϕ = 53°46′36″, θ = 33°33′36″, z0 = −1.86 m, and d = 3.99 m. The only free parameter in the fitting is the hour angle t.
If the projection plane is not perpendicular to the local meridian (θ ≠ 0°), which is what occurs in the case of the solar dial, the bunch of hour lines is asymmetric with respect to the origin of the coordinate system and lines intersect at a point x ≠ 0. It should be mentioned here that the point of hour lines intersection is a mapping of the axis of the world that connects poles of the celestial sphere δ = ±90° on the plane of the instrument (see Appendix B in the online edition of the journal).
A further attempt was carried out to fit the z = z0 + z(ϕ, θ, d, x + x0, t) function to the same data but releasing also the parameter x0 which defines the offset of the mirror along the edge of the solar dial. The results are shown in Figure 11 and summarized in Table 2, where corresponding values of the hour angle t, x0, and the MSE obtained in this example are presented and demonstrate a vastly improved fit.
The linear function z = z0 + z(ϕ, θ, d, x + x0, t) has been fitted to the digitized coordinates of the points (symbol o) lying on the hour lines identified in the order from right to left as XI, 1, 2, 3, and II. The dashed line (No. 1) corresponds to a currently non-existent line for hour 12h00m00s. The free parameters are the hour angle t and offset x0.
Hour angle values obtained by fitting the z = z0 + z(ϕ, θ, t, d, x + x0) function to the hour lines, with free parameters t and x0, as well as fixed parameters z0 = –1.86 m and d = 3.99 m.
LHA: Local Hour Angle; MSE: mean squared error.
The line No. 1 does not exist anymore.
Among the hour lines surviving on the solar dial lines, Nos. 2, 3, and II intersect each other in a limited area and their offset values, x0, resulting from a fitting procedure are similar. Only the line indicated by XI shows an extremely large deviation from the direction resulting from the theory of gnomonic projection. This raises the essential question of why straight lines for the hour No. XI and the group of lines, consisting of two unmarked lines and lines marked as II, do not intersect at a single point? Is this a result of inaccuracy in the drawing? Where should we look for an explanation for the non-compliant slope of the hour lines? It should be emphasized that the determination of the hour line by the method of reflective gnomonics is not straightforward. Each point of the hour line lies on a different calendar line. Then, to effectively draw an hour line, one would need to specify at least two points corresponding to a given time for a relatively well-separated pair of dates and apply an interpolation. The line No. XI is located on the left edge of the solar dial, farthest away from the mirror. Discussion of factors that determine the accuracy of the instrument carried out in the section below leads to the conclusion that the lines in this area have been plotted with the least accuracy. The sunlight projected into this area has the largest spot size and is characterized by the lowest contrast and the greatest elliptical deformation (see Figure 16).
This allows the supposition that the lower absolute value of the slope of the line for hour XI may be the result of a lower precision of its determination. This seems to be the most likely explanation for the inconsistency of the course of the clock lines. Such an explanation is also supported by the fact that, in its lower part, this line crosses the day line for 15 February (according to the Gregorian calendar) and, in its upper part, reaches at its maximum the day line for 11 March, so the hour line No. XI was indicated by the light spot on the solar dial only over a period of only 25 days (Table 1). This feature of the sundial aroused the astonishment of Pastor Heinrich Hein, who wrote, For what reasons this clock was made, that during the winter could only show a few hours, I cannot yet guess.
70
If the sundial operated in the full range of hours, the hour line XI, in the days following 11 March, would have to have fallen onto an extension on the ceiling of the cloister. Unfortunately, there is no record of such an extension of the solar dial, and the repairs of the cloister roof carried out in 1911 destroyed any possibility of establishing this fact by visual inspection in situ.
It is possible that there were two projection systems. This alternative hypothesis for explaining the discrepancy in the slope of the hour lines is at least putatively corroborated by the historical reference of Hein as to the presence of two mirrors. Taking this into account and consulting Figure 11, it can be concluded that the first mirror could have served to indicate the hours before and soon after noon, i.e., the time lines XI the XII, the latter designation no longer in evidence, and line No. 2. The second group of lines, which could have been determined by means of another mirror, would then consist of the line designated by II, line No. 3, and, in common for both groups, line No. 2. According to this, somewhat tricky, hypothesis, line No. 2, identified as 13h07m59s, combines these two systems and would be identified by both mirrors. In attempting to justify this hypothesis, one should answer the questions of why the two mirrors were used and where the second mirror would have been located. The considerations in the next section, summarized in Table 3, show that the shadow of the castle’s tower does not interfere in making observations until 14h47m24s.
Covering of the mirror by the shadow cast by the castle tower in the days corresponding to the observation of extreme calendar lines.
Time is expressed as local solar time.
We are unfortunately unable to rule out this hypothesis because of the repainting of the hour lines at a time when no one remembered anything of the principles of operation of the solar dial: during works carried out at the castle between 1865 and 1866, partitions dividing the cloister were removed and the instrument renewed. Further renovations of the cloister in 1911 filled in missing spaces in the plaster and repainted the day lines and the clock. 71
If one looks at the time corresponding to the individual values of Local Hour Angle (LHA) collected in Table 2, one can see significant discrepancies between Roman markings of lines and calculated moments of time. And so, for the line marked as XI hour corresponds only to 10 o’clock, while the afternoon line marked as II hour corresponds to nearly 16 o’clock.
One would think that this discrepancy arises from the widespread range of hour lines which could be the result of a tilting of the mirror, which could cause the occurrence of the “keystone” phenomenon. 72 In order to check such a possibility, the coordinates (x, z) of the sunlight spot on the plane of the instrument, defined by equation (B1), were transformed by means of the matrix R = RZ(γ)·RY(β)·RZ(α), which is a product of the rotation matrices RZ and RY defining rotations in R3 around the axes 0x and 0y, respectively, 73 where Ω = (α, β, γ) are the Euler’s angles describing an arbitrary orientation of the mirror. 74 The resulting function was fitted, as before, to a set of points corresponding to individual hour lines. Unfortunately, no coherent description of all hour lines was obtained that would lead to receiving moments of time compliant with the Roman symbols of the outermost hour lines.
In the present state of research, it is at least difficult, if not impossible, to settle the question of the hour lines in an unambiguous manner. Nevertheless, in Table 2, a striking coincidence with markings of lines XI and II is seen of time instants, obtained by dividing by 30o the values of LHA corresponding to them.
The conditions of the illumination of mirror
The castle’s tower is the architectural element, whose shadow affects the operation of the solar dial (Figure 3). In the days of Copernicus, the tower had no cupola, and its height measured relative to the level of the courtyard was Hw = 29.06 m. Because the tower is circular, with radius Rw = 4.9 m, different parts of the tower are located at various distances from the wall of the cloister, so the instantaneous position, y(δ, t), and height, z(δ, t), of the shadow can be calculated only in an approximate way (see Appendix C in the online edition, equations (C1) and (C2)).
The distance, D, from the centre of the tower to the outer wall of the cloister on which the instrument resides is 40.42 m. On this basis, the position of the tower’s shadow in the plane of the wall of the cloister can be determined on the dates corresponding to the day lines shown in the solar dial (Figure 12). On the initial (δ = −15.49°) and final (δ = 9.07°) days of observations recorded on the solar dial, the position and the size of the shadow of the tower alter, as illustrated in Figures 13 and 14. It can be noticed that the average width of the shadow cast by the tower on the walls of the cloister is, in the early hours of the afternoon, practically constant at 9.44 ± 0.01 m.
The extent and width of the shadow of the tower against the wall of the cloister, from 12:00 to 17:00 of the apparent solar time on the first day of observation, δ = −15.49°. Black horizontal lines indicate the position of the solar dial, and the black circle represents the mirror.
The extent and width of the shadow of the tower against the wall of the cloister from 14:00 to 17:00 of the apparent solar time on the last day of observation,δ = 9.08°. Black horizontal lines indicate the position of the solar dial, and the black circle represents the mirror. Before 14:00, the shadow of the tower does not reach the wall of the gallery.
The angular aperture ω(R, r) as a function of the radius of the mirror R and distance r of the point in the solar dial from the mirror. The solid curves, from the bottom to the top, correspond to mirrors of radius R = 2, 3, and 4 cm, respectively. The horizontal dashed line corresponds with an angular aperture equal to that of the sun,
Analysis of the drawings shows that, on the first day of observation, the height of the tower shadow on the cloister wall reaches the height of the mirror in the early hours of the morning, whereas on the last day it is in the afternoon. The exact times t at which the shadow covers and uncovers the mirror may be obtained by solving equations (C1) and (C2) for the corresponding values of the declination of the sun, δ, and the assumed location of the mirror at the point y(δ, t), z(δ, t). These results, together with those for ensuing partial obscuration of the mirror by the shadow of the tower, are summarized in Table 3.
The accuracy of the solar dial
The search for an answer to this question can be reduced to two issues: what are the deviations of the line painted on the solar dial from the actual track of the spot of light, and what is the effective “thickness” of the line resulting from the impact of various factors?
As we have noted before, all day lines plotted on the solar dial are straight lines, not hyperbolas as follows from the theory. Deviations from the hyperbola show up especially at the bottom centre of the dial where their curvature should be the most obvious (compare Figures 6 and 8).
A constant value for the declination was assumed. It was calculated from the daily traces of the rays of the sun (see Table 1). The solar dial was illuminated from 10:00 until 15:00, and around the equinox, the rate of change of the declination of the sun does not exceed 1′ per hour. During this period, the daily change in the declination of the sun is almost linear (see Figure 6). This justifies the assumption about a negligible variation in value of the declination for a day line recorded during this period.
The next factor that influences the precision of determination of the day lines is atmospheric refraction. This makes astronomical objects appear to be situated at a greater than their actual height above the horizon. The effect of refraction may be calculated from the empirical formula given by Saemundsson for the change, Δh. 75 For the sun, whose heights over a horizon in the place of observation change in range h0 = 20o to 40o, its apparent height may be overestimated by Δh = 0.046o to 0.020o. These values correspond from 3.8 to 8.6 percent of the diameter of the sun’s disc, respectively. No correction for this effect was taken into account in the present analysis.
Some of the factors resulting from geometrical optics and photometry have a much greater effect on the precision of localization of the spot of the light projected onto the solar dial. These factors include the finite dimensions of the light spot and its increase with distance from the mirror, along with its increasing deformation, reduction of brightness, and lowering of contrast resulting from the non-uniform intensity of the light in the area of the spot. These factors will be addressed in sequence.
In the preceding discussion, we have implicitly assumed that the sun is a point source of light. In fact, the angular size, ω0, of the solar disc is about 0.54°. The finite size of the light spot projected by the mirror contributes to the deterioration of the accuracy of the determination of the position of reflected light on the solar dial.
As a result of the divergence of the beam of sunlight reflected from the flat mirror, the cross-sectional radius of the conical beam of sunlight increases linearly with its distance from the mirror (Figure 4). It leads to the increase of the light beam cross-sectional area, which is (approximately in this case) proportional to the square of the distance from the mirror what causes a corresponding diminution of the light intensity in the spot.
In addition, the edge of the mirror defines the effective aperture limiting the angular size of the beam reaching the solar dial. For an output angular aperture given by
A beam of light does not usually fall onto the solar dial perpendicularly but at an angle ξ ≥ 0. This produces a deformation of the image, which takes the shape of an ellipse, more or less elongated depending on its position in the dial. The shape and size of the ellipse are characterized by its semi-axes (see Appendix D in the online edition, equation (D3)). The shapes of the projected light spots calculated on this basis are shown in Figure 15. The inner circle defines the size of the mirror used. The smallest freckle is formed when the light beam is projected perpendicularly onto the dial in the horizontal plane (A = 33.557o – dashed line ellipse). This situation corresponds to a distance of the spot from the mirror of 4.25 m. On the other hand, the freckle reaches its greatest size on the left edge of the instrument (A = 340.950o – long dash line ellipse), in this case the distance of the spot from the mirror is greatest, reaching a value of 7.07 m.
The elliptical shape of the light spot for characteristic locations on the solar dial: dotted line – at the right end of the dial (A = 59.949°), dashed – the sun’s rays fall perpendicularly onto the dial (A = 33.557°), dot-dash line – the sun passes through the local meridian (A = 180°), and the largest long dash ellipse – at the left edge of the dial (A = 340.950°). For these calculations, it was assumed that the sun’s height, h, was 20° and the mirror was of radius 0.04 m (the solid circle in the middle).
The lines plotted on the solar dial have a width of about 1 cm, so they are much thinner than the size of the light spot. In constructing the dial, the centre of the spot had to be determined fairly accurately. This could be difficult, especially on the left edge of the dial where the eccentricity of the elliptic spot is greatest.
The effects discussed in this section, arising from the laws of geometrical optics, impose conflicting constraints on the size of the mirror. In the case of a small mirror, the resulting spot would be dim and the light contrast low. On the other hand, a large mirror would produce a brighter spot of greater size. These, however, are not the only factors limiting the accuracy of the observations.
A glance at the solar dial makes it clear that it allows for determination of the equinox only to an accuracy of 1 day. As stated earlier, the time as a variable was eliminated from the description of calendar lines. The location and shape of a line depend only on a single, fixed value of the sun’s declination or, as may be preferred, on the longitude of the sun assigned to each day line in the fitting procedure. Due to the effect of this overarching use of such a single value, the determination of the time of equinoxes, and thus the length of the tropical year, would seem to be rather imprecise. This raises the question as to whether one can make better use of the information contained in the solar dial. To carry out a more precise analysis of the accuracy of the instrument, it is necessary to reintroduce time as a variable associated with the moment of a defined equinox.
The apparent geocentric longitude of the sun as a function of the Julian Day was given by Jean Meeus. 76 It is corrected for nutation and light aberration and ensures an accuracy of Δλ ≈ 0.01°. A scrutiny of the course of the function of longitude of the sun through the year may be a helpful starting point in restoring the time variable. Starting from zero, the longitude rises steadily up to 360°, which it reaches at the subsequent vernal equinox, and at this point it drops instantly to zero again, whereupon the cycle is repeated. Each of the values of the longitude of the sun summarized in the eighth column of Table 1 is represented as a single point of the function of the apparent longitude of the sun that uniquely defines the moment of time related to a chosen vernal equinox.
The examination of the time accuracy of the solar dial presented here includes the majority of the observations recorded on it. The idea is to investigate pairs of calendar lines located symmetrically on either side of the equinox line (Table 4). Formally, such an evaluation of the accuracy of the instrument is relative and not linked to the date of a specific equinox. To make them definite, however, the values of λ may be associated with a specific date of an equinox. As an example, the year 1517 was selected. On this basis, the assumption was made that day line No. 9 corresponds to the Julian Day of the vernal equinox in 1517, 77 i.e., 2.27521175928 × 106 JD.
The longitudes of the sun, and the Julian Days corresponding to them, related to the vernal equinox of 1517 for selected pairs of calendar lines corresponding to observations made symmetrically before and after the equinox.
The numbering of lines in pairs is consistent with that used in Table 1.
The idea is to present individual observations as points on a plane with the coordinates: time in Julian Days and longitude of the sun in degrees of arc. However, as has already been pointed out, the ordinate of such a point is the only known coordinate, so that the location of it in the plane is somewhat arbitrary. To remove this ambiguity, two different methods were applied. The first method is to delineate an appropriate period on the time axis corresponding to the difference between the moments of observation and the vernal equinox. The average interval ΔT = 5.074d between successive observations recorded on the solar dial was chosen as the unit of this measure. The second method is to match the function of the apparent geocentric longitude of the sun to the ordinate λ resulting from observation. Both methods uniquely define the moment of time to be attributed to the specific observation of longitude. The points (Ti, λi) resulting from the first and the second approaches are presented in Figure 16: boxes corresponding to the first method, circles to the second. For the purpose of analysing the accuracy of the instrument, pairs of points were selected. The points forming a particular pair situated on two branches of the function of the apparent geocentric longitude of the sun are connected by a straight line defined by the following equation
where λ is the longitude of the sun and T is the time that separates given observation from the vernal equinox, while indices i and j denote the values of variables in the pair measured, respectively, before and after the vernal equinox.
The dash-dot line represents the function of the longitude of the sun, and its vertical part denotes the calculated instant of the vernal equinox of 1517. The circles and boxes define the observational points obtained by the first and second methods of the calculation of abscissa discussed in the text. The symbols in the upper part of the diagram denote the observations made before the vernal equinox, whereas symbols in the lower part correspond to observations made after the equinox, and they form the next cycle of the sun’s ecliptic length variability. The pairs of points are connected by straight lines defined by equation (3). On the scale of this diagram, the calculated and observationally determined timings of the equinox nearly overlap.
It is obvious from the linearity of equation (19) and the method of selection of points which they link that – in theory – all these lines should intersect at a single point corresponding to the moment of the equinox. However, solving the systems of equations specified by equation (19) for each pair of observations yields 10 different values, collated in Table 4. The size of the areas of intersections of lines connecting pairs of points defines the uncertainty of the observational determination of the moment of the equinox. The coordinates of intersections of the lines, obtained for both, mentioned earlier methods of introducing the time coordinate in Figure 16, the first by measuring the average time intervals between observations ΔT, and the second that follows from the observed longitude of the sun are presented in Table 5.
The coordinates (JD, λ) of the points of intersection of the lines connecting pairs of points that correspond to the calendar lines registered before and after the equinox.
The time measured in Julian Days is given with an accuracy of 100th of a second and the longitude of the sun in degrees of arc is presented with an accuracy of 100 of a second of arc.
Averaging of the data compiled in Table 5 gives two values for the instant of the March equinox of 1517: µ(T1) = 2.27521181532 × 106 JD and µ(T2) = 2.27521120002 × 106 JD, for the first method and the second method, respectively. The longitude of the sun may also be similarly estimated and the values obtained are µ(λ1) = 178.19586° and µ(λ2) = 172.69594°, respectively. It is obvious that the mean values of the longitude of the sun have no physical meaning in this case, but their deviation from 180° may be used as an indicator of the accuracy of the solar dial. The numerical values of the parameters estimated for two distributions of the intersection points resulting from different methods used for the reintroduction of the time coordinate are summarized in Table 6, and Figure 17 displays their graphical interpretation.
Summary results of the statistical analysis of the precision of calendar lines for both methods of the re-introduction of the time discussed herein.
LMT: Local Mean Time.
The enlarged view of the area of intersection of lines connecting pairs of points corresponding to the longitudes of the sun registered on the solar dial (Figure 16). The circles correspond to the points of intersection of specific lines. The central symbol, □, denotes the mean value of the coordinates of the intersection and the error bars indicate corresponding standard deviations. The covariance ellipse for one σ, with half-diameters: p1 = 0.3116 and p2 = 4.8797, is represented by a solid line, the 95 percent confidence ellipse by a dotted line. The vertical dash-dot line indicates the theoretically determined instant of the equinox of 1517 in Olsztyn’s Local Mean Time, while the dashed one corresponds to the same moment in UTC.
Because the scale of Figure 17 is much larger than the scale of Figure 16 and its time axis includes only 3 JD, the substantial disproportionalities between the standard deviations of the means and the sizes of the covariance ellipses obtained in both cases are clearly visible. The one σ covariance ellipses are drawn with solid lines and the 95 percent confidence ellipses, enlarged by a factor of 2.4477, are drawn with dotted lines. The circles in the areas of the larger ellipses represent the data generated by the second method. As is to be expected, every of these distributions of the points of intersection is correlated. It means that there is a linear dependence between coordinates λ and T that define every intersection point. Both bivariate distributions have very similar values of Pearson’s correlation coefficients, ρ, of −0.8253 and −0.8621, correspondingly. 78 The vertical dash-dot and dashed lines indicate the theoretically determined instant of the equinox of 1517 in Olsztyn’s Local Mean Time (LMT) and Coordinated Universal Time (UTC), respectively.
It is interesting to compare the mean value of the time of the equinox resulting from the studies of day lines recorded on the solar dial with the value for the year 1517 computed on the basis of the Meeus algorithm. 79 According to such calculations, the spring equinox in 1517 occurred on Sunday, 11 March, according to the Julian calendar, at 6h8m58.579s UTC. 80 Taking into account the offset ΔT = 1h21m53,832s from UTC, which results from the location of the instrument at geographical longitude λ = 20.4743°E, the equinox took place at 7h30m52.411s LMT. This value is extremely close to the result of 7h29m39.78s obtained with the first of the methods discussed. This convergence results from the fact that the first method directly propagates a posteriori information obtained from the calculation of the instant of the equinox, by assigning delays defined by multiples of the mean interval between observations recorded on the solar dial to the appropriate values of the observed longitude. According to the second method, on the other hand, the equinox took place the day before at 16h43m38.44s LMT so that it precedes the theoretically determined instant by 14h47m13.96s. As can be seen in Figure 17, all but one of the points of intersection are grouped to one side of the line indicating the moment of equinox. This demonstrates the occurrence of systematic errors, which result, probably, from the systematic delay of the observed longitude of the sun when compared with corresponding values calculated based on the algorithm (see Figure 16).
Concluding remarks
The aim of this study was to produce as accurate as possible a mathematical model of the solar dial considered to be the only surviving relic of Copernicus’ scientific activity. In the analysis of this observational instrument, the geometrical and environmental conditions that impact on its functioning were considered. Discussed in the previous sections, astronomical properties of the solar dial were supported by surviving to the present-day artefact. However, since there is no description of this instrument which was drawn up by its constructor, both the purpose for which it was constructed and the intention of its architect remain in the realm of speculation.
To the most easily ruled-out option is that this construction fulfilled the functions of a clock and calendar regulating the rhythm of life in the castle: Pastor Heinrich Hein had early on pointed out the absurdity of the construction of a sundial that shows the time only in a specific month of the year. 81 Seeking further for an explanation of the purpose of its construction, a more confident suggestion may be made regarding the interpretation of the calendar lines, the meaning of which is undeniable. Most probably, the solar dial served to determine the time of the vernal equinox and the length of the tropical year. If the instrument was indeed authored by Copernicus who resided at the castle in the years 1517–1519, its purpose would match Copernicus’ interests at that time which aimed at collecting observational data necessary for confirming the theory of the solar motion and for contributing to the discussion about the reform of the calendar.
The above analysis allows us to formulate some hypotheses concerning the construction and use of the solar dial. The instrument would be of little use as a sundial and calendar regulating the rhythm of life in the castle. Its area, despite its large size, covers only a fraction of a day and a small part of a year. Besides, as the days go by, the effective area of the instrument’s operation changes. 82 The unprecedented system of hour lines also does not indicate on the practical application of this device. A detailed enquiry allows to alienate the essential properties of the solar dial as an observational instrument and to contend that observations were recorded indelibly on the wall and averaged by interpolation. This means that observations collected during consecutive observational seasons were registered on the solar dial only in an integrated form.
The analysis of digitized calendar lines, preserved on fragments of the solar dial, showed that all of them, not only the equinox line, are exactly straight, though not parallel. This finding is at odds with the theory that is the foundation of the hyperbolic shape of the path of reflected light on the dial. The deviations of the theoretical curves from the equivalent lines on the solar dial are shown in Figure 7 and are also manifest in the quality of the fits presented in Figure 6. The actual azimuth (AT = 123°33′36″) of the wall chosen as the site for the solar dial contributed to the inaccurate delineation of calendar curves by making them similar to straight lines. For a wall perpendicular to the local meridian (AT = 90°), the construction of the instrument would not be subject to such a fault.
It seems reasonable, therefore, to ask the question as to how the lines on the wall were drawn, if the study of the structure of paints and plaster of the solar dial has not revealed unequivocally any marks mapping daily tracks of the light spot. One might suppose that only some points on the dial were determined on the basis of gnomonic projection, and the remaining segments of lines were determined by linear interpolation. This could be done, in practice, with a taut cord fixed at the observationally determined points.
In the still open dispute as to whether the lines of historical sundials were plotted based on observations or on calculations, the solar dial in Olsztyn seems, from these facts, to be an example of the first option, even though, as is almost certain, Copernicus had the knowledge required for developing a theoretical description of the hyperbolic profiles of the day lines.
The considerations relating to the seasonal and daily conditions of solar illumination of the solar dial lead to the conclusion that the location of the instrument has been chosen in an entirely ad hoc manner. The reason for this may have been the difficulty in finding a more suitable place in the castle (and the necessity of the haste in its implementation that resulted from the expectation of a period of measurement lengthy in comparison with the limited time during which Copernicus was fulfilling the function of administrator of the Warmian Chapter’s estates). It may be suspected that the obstacles arising from the architecture of the castle to prevent direct observations of the sun, together with low number of sunny days due to the poor weather conditions in Olsztyn, were factors that led Copernicus to construct a device enabling observations of the sun’s position which could be utilized for long periods of time, not necessarily at noon, and even in his absence.
With the considerations presented here, and in the lack of any mention of the solar dial in Copernicus’ manuscripts, it can be concluded that the instrument was an innovative experiment, which was eventually terminated without attaining, from a scientific point of view, any significant results. This lack of success may be attributed to an inappropriate mode of operation of the solar dial and to an accuracy of measurement insufficient to achieve the outcomes expected.
Supplemental Material
Supplemental material for The Solar Dial in the Olsztyn Castle: Its Construction and Relation to Copernicus
Supplemental material for The Solar Dial in the Olsztyn Castle: Its Construction and Relation to Copernicus by Jacek P. Szubiakowski and Jarosław Włodarczyk in Journal for the History of Astronomy
Footnotes
Acknowledgements
J.P.S. would like to express his gratitude to Janusz Cygański, the former director of the Museum of Warmia and Mazury, for the invitation to join the team researching the solar dial and for stimulating discussions and interesting comments. He also thanks Robert E. Dale for proofreading the draft of the article and many critical comments. J.P.S. and J.W. thankfully acknowledge the assistance of Paweł Sobotko in collecting the materials on the history of the instrument as well as the anonymous reviewers in achieving the final shape of the manuscript.
Notes on Contributors
Jacek P Szubiakowski is an assistant professor in the Department of Physics and Computer Methods at University of Warmia and Mazury in Olsztyn, Poland, whose current research focuses on issues of molecular and chemical physics and history of science. He is also director of the Olsztyn Planetarium and Astronomical Observatory.
Jarosław Włodarczyk is a professor at the Institute for the History of Science, Polish Academy of Sciences, Warsaw. His research focuses on the history of observational astronomy, the relation between observations and astronomical theories, and the cultural context(s) of astronomy. He is currently taking part in a collaborative project on the correspondence of Johannes Hevelius.
Notes
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