The torquetum (or turketum): Was it an observing instrument?

Introduction

The torquetum is referred to at the turn of the 14th century in the writings of astronomers who made observations to test the Toledan Tables and who eventually contributed to the compilation of the Parisian Alfonsine Tables. However, the extant sources are not sufficient to determine if this instrument played a significant role in contemporaneous astronomical observations. In fact our sources do not include records of comprehensive or large-scale observations made with the torquetum. This holds true also for the following centuries, as far as the 1560s. The scarceness of data strongly impedes the assessment of the significance of the torquetum in comparison to other instruments used in ancient observational astronomy. This has also led to conspicuous differences of opinion among modern researchers on the uses of the torquetum. Some find this instrument hardly useful for positional astronomy and describe it either as a mechanical analogue computer for transforming between celestial coordinates, ¹ or as an instrument more suitable for demonstration purposes than for serious astronomical observation, ² or, to cite Anthony J. Turner, ‘an example of conspicuous intellectual consumption [rather] than a much used instrument’. ³ So far, however, I have not found any analyses of observational procedures linked with the torquetum, ⁴ nor any, even if very elementary, theory of the instrument, that is, the quantitative discussion of how errors inherent in the process of measurement and in the instrument itself could influence the results of observation. The aim of the present study is to supplement our knowledge about the torquetum with the above mentioned elements, which in turn should allow us to take a more precise stance in the critical discussion and answer the question asked in the title. Let us first consider the construction of the torquetum.

The construction of the torquetum: Written sources and artefacts

The construction of the torquetum can be analysed on the basis of some manuscripts and prints from the first half of the 16th century (some of which contain illustrations), and a few extant artefacts the earliest of which comes from the 15th century. ⁵ The instrument was first described at the end of the 13th century by two Latin authors: Franco de Polonia and Bernard de Verdun. ⁶ According to these sources the torquetum could be used to determine altitude–azimuth, as well as equatorial or ecliptic coordinates. For the purpose, the instrument should be fully deployed. Indeed such position of the instrument is visible on the drawing extant in the Paris copy of Bernard de Verdun’s manuscript (Figure 1). Based on this drawing, we can distinguish three major elements of the construction. First, a horizontal plate called tabula orizontis (A) used to place the torquetum in the plane of the horizon and in the plane of the meridian. Secondly, moving upwards, another plate (tabula aequinoctialis) (B) representing the plane of the equator and therefore tilted to the local horizon at the co-latitude (90° minus the geographical latitude). Affixed to this plate is a round disc (basilica) (C) which – while sliding on this plate – rotates around the axis of the celestial equator which goes through its centre (P1). The turn of basilica represents the daily motion of the celestial sphere. A circular disc of the ecliptic (tabula orbis signorum) (D) is fixed to basilica and tilted to it at the angle which equals the obliquity of the ecliptic. The ecliptic circle turns along with the turn of basilica. An axis running through the centre of the ecliptic circle marks the axis of the ecliptic (P2). The upper part of the torquetum rotates around this axis. The upper part consists of an alidade (turnus) (E) rotating in the plane of ecliptic, and therefore indicating longitudes. Perpendicular to the alidade and in its centre, parallel to the axis of the ecliptic, there is a small pillar with yet another round disc called crista or the comb of a chicken (F). This vertical circle has another alidade (alidada circuli magni) (G), which, while rotating, can show latitudes with respect to the plane of the ecliptic. Both alidades have pinholes. Additionally, the second alidade has a loosely hanging vertical half circle (semis) with an attached plumb-line (perpendiculum) (Figure 2). It can show latitudes with respect to the horizon. The torquetum – set like this – represents the essential circles of spherical astronomy.

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Figure 1.

The drawing of the torquetum in the treatise by Bernard de Verdun Sic facies instrumentum ad modum trunqueti. . ., BnF Lat. 7333, f. 49v. Courtesy of Bibliothèque nationale de France–Gallica.

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Figure 2.

Semis with a plumb-line. Bernard de Verdun, Sic facies instrumentum ad modum trunqueti. . ., BnF Lat. 7333, f. 50r. Courtesy of Bibliothèque nationale de France–Gallica.

The two oldest extant torqueta come from the 15th century. The first one was made in Nuremberg and purchased there in 1444 by Nicholas of Cusa; it is still preserved in the St. Nikolaus-Hospital in Bernkastel-Kues on the Mosel River (Figure 3). ⁷ The second torquetum was constructed in 1487 by Hans Dorn for Martin Bylica of Olkusz. It is presently held by the Jagiellonian University Museum Collegium Maius in Cracow (Figure 4). Neither the torquetum of Nicholas of Cusa, nor the torquetum of Bylica are preserved as a whole, and in both cases certain elements have been reconstructed. In the case of Bylica’s torquetum we have the originals of all major plates. At the end of the 19th century Ludwik Antoni Birkenmajer completed detailed and accurate measurements of the extant plates of Bylica’s torquetum, carefully assessing the scales. ⁸ In 2019 I had an opportunity to disassemble and assemble again Bylica’s torquetum as well as to test its usefulness for ‘observing purposes.’ I shall refer to Birkenmajer’s description of the torquetum as well as my own experience with it to offer a detailed account of this instrument.

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Figure 3.

Nicholas of Cusa’s torquetum, 1444. The diameter of the instrumental ecliptic: 36 cm. Courtesy of St. Nikolaus-Hospital/Cusanusstift, Bernkastel-Kues. Photo Erich Gutberlet.

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Figure 4.

Martin Bylica’s torquetum, 1487. Height: 71 cm, base: 41 × 56.5 cm. Courtesy of the Jagiellonian University Museum Collegium Maius, Cracow. Photo Grzegorz Zygier.

Let us begin with the lower plate, that is tabula orizontis (see Figure 4). It is a rectangle measuring 56.5 × 41 cm. The plate has a sundial engraved upon it. This dial is hardly related to the functions of the torquetum but it contains the information that it had been set for the latitude of 50°, which corresponds to Cracow (though the torquetum was probably made in Hungary). On both sides of the dial, there are two identical linear scales which allow one to set the equatorial plate at a geographical latitude of 20°–60° (with up to 1° accuracy). This can be done with two moveable supports. There is no meridian line on the horizontal plate that could be adjusted to the local meridian. Instead there is a compass. A line is engraved below the compass needle. This line is tilted from the plate’s axis of symmetry by 9.5° eastwards. This was a way of accounting for the magnetic declination while setting the instrument in the plane of the meridian indicated by a compass.

There are two plates in the plane of the ecliptic: tabula aequinoctialis and basilica. There are two circular scales engraved on tabula aequinoctialis. The inner scale shows the division of the equatorial circle into 360° with arcs marked every 5°. The outer scale represents the division into hours, with two sets of 12 equal hours, based on the usual assumption that an hour corresponds to 15°. Inside the scales there is a basilica. It has a diameter of 37 cm which is also an inner diameter of the scale tabula aequinoctialis. There is a zodiacal circle on its edge, and each of the zodiacal signs is divided into 30° (Figure 5). However, in this case the arcs corresponding to relevant signs are not equal. This is confirmed by the results of the measurements of the length of the chords made by Birkenmajer. These results can be divided into three groups (Table 1).

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Figure 5.

A fragment of the basilica of Martin Bylica’s torquetum. Courtesy of the Jagiellonian University Museum Collegium Maius, Cracow. Photo Grzegorz Zygier.

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Table 1.

The length of the chords of the zodiacal signs in Martin Bylica’s torquetum.

Zodiacal sign	Mean value of the chord (mm)	Central angle α (°)
Aries, Virgo, Libra, Pisces	89.18	27;51.2
Taurus, Leo, Scorpius, Aquarius	95.73	29;56.8
Gemini, Cancer, Sagittarius, Capricornus	102.75	32;12.0

Based on the average length of the chords, Birkenmajer calculated corresponding central angles. Next he noticed that the values were related as follows:

for ε = 23;30°, and λ = 30°, 60° and 90°. This means that the scale allows the user to read the right ascension for a given longitude. ⁹

The circular disc of the ecliptic (tabula orbis signorum) is tilted against the plane of the equator and has a diameter of 38 cm. There are two concentric scales on its edge (Figure 6). The outer scale is divided into 12 zodiacal signs at each 30°. The inner scale represents 12 months: February has 28 days, whereas the remaining months 30 or 31 days. The arcs corresponding to individual months vary not only due to different numbers of days. Birkenmajer measured central angles corresponding to all months and obtained the results shown in Table 2. ¹⁰

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Figure 6.

A fragment of the tabula orbis signorum of Martin Bylica’s torquetum. Courtesy of the Jagiellonian University Museum Collegium Maius, Cracow. Photo Grzegorz Zygier.

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Table 2.

Central angles corresponding to twelve months on the tabula orbis signorum of Martin Bylica’s torquetum.

Month	Number of days	Central angle (°)	Daily mean motion of the Sun (′)
January	31	31;28.4	60.916
February	28	28;08.6	60.307
March	31	30;21.9	58.771
April	30	29;05.7	58.190
May	31	29;43.7	57.539
June	30	28;34.6	57.153
July	31	29;38.7	57.377
August	31	29;59.8	58.061
September	30	29;27.7	58.923
October	31	31;07.2	60.232
November	30	30;36.0	61.200
December	31	31;47.6	61.535

If we divide the values of angles by the number of days in a given month, we arrive at the daily mean motion of the Sun in a given month, consistent with an eccentric model of solar motion. As one could expect in this case, the variable velocity of the Sun has its minimum when it is at apogee (near summer solstice) and reaches a maximum when at perigee (near winter solstice). Furthermore, one can calculate the approximate values of the eccentricity and the solar apogee on this basis. Birkenmajer obtained for the maximum solar equation the value of 2°9′, and for the solar apogee the value Cancer 3.8° or 93.8°. The latter value results from the position of the vernal equinox on the outer scale in relation to the annual inner scale: Aries 0° matches 12.2 March. Consequently, this scale would allow one to fix the alidade (turnus) rotating in the plane of the ecliptic in a position corresponding to the computed longitude of the Sun.

The outer scale is divided into 12 zodiacal signs at each 30° and allows one to determine (with the help of the alidade) the longitude of heavenly bodies in relation to the actual ecliptic.

There is a round disc on the ecliptic alidade called crista which rotates with it around the axis of the ecliptic. This disc has a dimeter of 34 cm (see Figure 4). Additionally a moveable semicircle is affixed to the alidade attached to cristia. The full circle had four graduation from 0° to 90° and was used to determine the latitude. The semicircle has two graduations from 0° to 90°, made in a similar way, which allow one to determine altitude with a plumb line.

How to make observations with the torquetum: Written sources

At this point let us set aside the material culture and turn to written sources. One can distinguish five early textual traditions of describing the torquetum and its use. ¹¹ I will discuss them briefly, recommending Richard Kremer’s work to those interested in details. The best known source is the treatise about the torquetum ascribed to Franco de Polonia. Some of the manuscripts of this treatise feature the date 1284. The authorship of second treatise on the torquetum is ascribed to Bernard de Verdun, and the text is dated to the 1280s. The third description of the instrument can be found in an anonymous text which has been transcribed from a single manuscript and published by Emmanuel Poulle. ¹² The fourth text was written by Johannes Regiomontanus in 1469 and printed in 1544. ¹³ The fifth text has been recently identified by Kremer. ¹⁴ It is a single manuscript composed in the 1480s by a little-known Viennese astronomer, Johann de Epperies, and dedicated to King Matthias Corvinus of Hungary. To this corpus of texts one can add a slightly later description of the torquetum by Peter Apian, an experienced astronomer, printed several times in the years 1533–40. ¹⁵

All these sources seem to a large extent similar. Occasionally, as in the case of Epperies’s treatise, they contain more or less accurate paraphrases from earlier texts. The distinguishing feature of Regiomontanus’s text is the classification of astronomical instruments in the preface, wherein he differentiates between portable and fixed instruments, with the torquetum listed in both categories, and gives some practical recommendations. Apian offers a very detailed description of the construction of the instrument and the best iconographic material, though the drawings of the torquetum and some of its elements can be found in mediaeval manuscripts, too. However the aim of the present study is not a comparative analysis of the treatises on the torquetum. Instead I shall strive to extract from these sources the kind of data that should be indicative of the quality of the observations made with the torquetum. ¹⁶

Franco and Bernard set the pattern of dividing the description of the torquetum into two parts: first, the description of the construction of the instrument and, secondly, a kind of a manual explaining how to solve certain astronomical problems with this instrument. (This organisation informs also subsequent texts, with the exception of the anonymous treatise published by Poulle, which offers the description of the torquetum only). Some interesting comments can be found in both parts.

Let us begin with the question of the stability of a fully deployed torquetum. The text ascribed to Franco de Polonia puts much emphasis on the proper foundation for the base of the instrument, that is for the tabula orizontis. The foundation should be made of stone or wood, well-levelled and marked with the line of the local meridian. ¹⁷ To this base, one should affix the horizontal table in such a way that it cannot move. I have not come across such advice in the text by Bernard de Verdun, nor in the anonymous text published by Poulle. However, a similar remark can be found in Regiomontanus’s account. There is yet another bit of practical advice in Franco’s text. While describing the connection of the basilica with the equatorial plate by means of a shaft going through the centre of both plates, Franco stresses that the shaft should be stuck fast so the basilica can rotate only stubbornly. Indeed these are the two key points ensuring the stability of the instrument which – even though its construction mirrored the great wheels of the heavenly sphere – was devoid of spherical symmetry.

In the case of a torquetum that was to be used in various latitudes, there is yet another problem associated with the stability of the equatorial plate, as regards the horizontal plate. Franco mentions a rod affixed to the equatorial plate, but he says nothing about the manner of anchoring the rod in the horizontal plate. Despite the fact that in Cracow we have not just one but two rods, the solution seems very unstable and poses a threat of a sudden closing of an instrument. A more stable solution, based on two metal wedges, is proposed in Nicholas of Cusa’s torquetum. A similar solution can be found in Apian’s Introductio geographica (and later in the Astronomicum Caesareum) (Figure 7).

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Figure 7.

The construction that ensures a stable positioning of the equatorial plate. Apian, Introductio geographica, 1533. Courtesy of the Ossolineum.

In Franco’s text (and later in Epperies’s text), the appropriate placing of the torquetum in position for observations depends on fixing together the moveable plates of the instrument. Consequently, whenever levelling of the torquetum is required, the text instructs as follows: Place the equatorial plate on the horizontal plate. Fasten the zodiacal alidade on the equatorial plate and turn it so that it points to the east-west direction. Set the alidade of the crista along the ecliptic line of the cristia. If the plumb-line divides the semicircle into halves, both the equatorial plate and the horizontal plate are set parallel to the horizon. The torquetum featured on the title page of Apian’s treatise on the comet of 1532 is probably set in this position (Figure 8). In this reduced form, the instrument could be used to measure the altitude of the comet and its tail – this is the simplest of the observations that can be made with the torquetum. It must be noticed, however, that the altitude of the heavenly body could be also measured with a fully deployed torquetum. For this purpose, one has to read the same angle delineated by the plumb line on the semicircle. Both Franco and Bernard describe such use of the torquetum to measure the altitude.

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Figure 8.

The torquetum adjusted to work in the horizontal coordinate system. Apian, Ein kurtzer bericht der Observation vnnd vrtels des Jüngst erschinnen Cometen. . ., 1532. Courtesy of Österreichischen Nationalbibliothek in Vienna.

A stable torquetum, which has been aligned to the plane of the local meridian and set for the geographical latitude, can be used in a more sophisticated way to measure the ecliptic longitudes and latitudes of heavenly bodies. This calls for setting the instrumental ecliptic to the actual position of the ecliptic. For this purpose a reference object is necessary. During the day the obvious choice is the Sun. Both Franco and Bernard mention the Sun in this context. However, their instructions are very general and boil down to the recommendation that the ecliptic alidade should point to the Sun. Franco refers to the ray of Sun passing through both pinholes. Bernard mentions that the alidade should not cast any shadow.

In fact the necessary procedure is complex because it calls for simultaneous turning of two parts of the torquetum, that is the joint equatorial and ecliptic plates and the alidade, around two different axis, which necessitates a sequence of iterations. Regiomontanus described in detail how to tackle this problem.

Having deployed it [the torquetum], turn the zodiac in either direction and likewise turn separately crista, till the ray of sunshine goes straight through both holes of the bigger rings. ¹⁸ At this point the head of the alidade on the zodiac will show the relevant position of the Sun. The head of the alidade refers to this end of the alidade which is directed towards the ecliptic. Even though this [procedure] seems easy and quick to perform, it leads to some uncertainty due to two different movements of the combined elements of the instrument i.e. the crista and the zodiac. You can prevent this from happening by the following useful method. Turn the zodiac along with the crista, till the shadow of the plate of the latitude is a straight line, which happens when the surface of the plate of the latitude, assuming it is set accurately, points to the center of the solar disc. At this point you shall see the ray of sunshine going through the front ring, close to the rear ring. If the hole in the second ring is accurately filled, the head of the alidade indicates the position of the Sun on the zodiac. If this ray goes above the hole in the second ring, the head of the alidade needs to be moved towards the part of the zodiac which is more tilted, and as before, while turning the zodiac, one has to see if the ray going through the front ring enters through the hole of the second ring. If after the first attempt, the ray going through the front ring goes below the hole in the rear ring, one has to move the head of the alidade towards the upper part of the zodiac, while monitoring if the ray of sunshine shines precisely through both holes in the rings. ¹⁹

Similar inconveniences will appear if the instrumental ecliptic is set to the Moon, or a planet or a star at night, assuming that we know their ecliptic coordinates. In this case one has to additionally set the alidade to account for the latitude of the reference object. Significantly, Epperies is the only author who mentioned the two types of pinholes of the alidade of the crista: one with a bigger diameter for observing stars and planets, and one with a smaller diameter for ‘letting in the rays of sunshine’. However, he does not offer any details of the construction that would help us to understand how these two types of pinholes could operate on a single alidade nor the actual diameters of the sighting holes. Whatever is the case, this remark stems probably from the experience gained while observing, as openings that are too small may make a precise measurement impossible. ²⁰

A different problem arises in connection with observations of the Moon due to a relatively big angular dimeter of its disc. Franco gives the simple advice of directing the alidade of the crista in such a way that one can see the Moon through the pinholes. This advice is obviously too general. Apian recommends setting the alidade with a pinhole towards the centre of the Moon, and if the centre cannot be seen, to try to imagine it. ²¹ Bernard does not mention pinholes, and his instruction is also very general. The Moon’s longitude is set by directing towards the Moon the alidade of the ecliptic (turnus). Subsequently we direct towards the Moon the alidade of the crista, thereby obtaining the latitude of the Moon. Regiomontanus recommends that one first bisect the lunar disc while it is viewed along the surface of the crista, and then direct towards the centre of the lunar disc the alidade of the crista. This is undoubtedly the most useful advice.

However, let us return to the procedure of setting the instrumental ecliptic to the Sun. To simplify it, one can set the alidade to the computed longitude of the Sun and turn the main body of the instrument in such a way that the Sun would appear in the line of the alidade. It seems that Franco allows for such a solution and recommends using first the double scales on tabula orbis signorum. In this way the longitude of the Sun for each day can be read. In Bernard’s text, this problem is discussed vaguely.

Considering the two methods of aligning the ecliptic of the instrument with the sky, the first one seems purely observational, whereas the second one clearly dependent on the quality of the solar theory and astronomical tables. No matter which one is used, it can be the first step to observe at night. Franco, Bernard and Regiomontanus describe how the Moon can be used for this purpose as a connecting link between the sky by day and by night. ²² The Moon can be seen on the sky together with the Sun. Therefore aligning the instrumental ecliptic to the Sun allows one to measure the ecliptic longitude of the Moon. In this case, after sunset, one can use the Moon for proper alignment of the instrumental ecliptic. This should allow the measurement of the coordinates of a star or a planet. This methods requires appropriate corrections for the Moon’s motion and for its parallax. Bernard does not mention this at all. Franco takes into account the motion of the Moon only, pointing to the necessity of correcting the position of the Moon by 0.5° for each hour (read on the scale of the instrument when the torquetum is aligned to the Moon). Regiomontanus devotes more attention to this method than Franco and Bernard, but he thinks it is ‘less accurate’ due to the ununiform motion of the Moon (‘which in one equal hour sometimes exceeds 0.5°, and sometimes not’) and its parallax. ²³

Regiomontanus believes that the more reliable method is the use of ‘a possibly accurate clock that shows also the minutes of equal hours’. The clock is to determine the difference in time between aligning the instrumental ecliptic to the Sun during the day and starting observations after sunset. After turning the basilica by an angle that corresponds to the time elapsed (one degree is four minutes), the ecliptic of the instrument should reach the required position. ²⁴ It must be stressed that the employment of this method is possible due to the specific construction of the torquetum where the basilica turns inside the scale of the tabula aequinoctialis.

Epperies and Apian do not address the question of the differing operations with the ecliptic for daytime and nighttime observations. They assume that in order to set the ecliptic of the torquetum at night one should rely on the ecliptic coordinates of a star or a planet, measured or calculated beforehand. They recommend also using the clock. In the case of Epperies, the clock may be used to find the ascendant, which in turn allows one to set the right position of the ecliptic. Apian recommends using the time measured by the clock to align the position of the Sun on the scale of basilica with the scale of tabula aequinoctialis, which is similar to Regiomontanus’s method.

Setting aside the text by Regimontanus, it seems that none of the above treatises on the torquetum is a comprehensive source of practical suggestions about minimising the error of aligning the ecliptic. These suggestions could have referred to direct handling of the instrument. This includes the necessity of the simultaneous motion around two axes to keep the ecliptic aligned, the problem of sighting the Sun or the Moon through the pinholes of the alidade and, finally, to accounting for various astronomical phenomena such as the ununiform motion of the Moon and its parallax. The treatises written at the turn of the 16th century (Regiomontanus, Epperies and Apian) recommend the use of the mechanical clock in this challenging procedure. As mentioned before, the use of the clock was possible due to the construction of the torquetum but such a modification was still laden with great uncertainty at that time, and it had been impossible to use it earlier.

Having aligned the ecliptic of the instrument, one can start measuring the coordinates of planets, stars and other heavenly bodies. A theory of the instrument allows us to understand how an inaccurate position of the ecliptic of the torquetum impacts the results of measurements.

The armillary astrolabe versus the torquetum and a theory of the instrument

At this point it is worthwhile to recall another instrument that, like the torquetum, represents the essential circles of spherical astronomy. I refer here to the armillary astrolabe, which served for direct observations of the ecliptic coordinates of heavenly bodies from Antiquity to the end of the 16th century. The best known description of the armillary astrolabe can be found in Ptolemy’s Almagest (Figure 9). It is usually assumed that the torquetum was invented to do away with the complexities of the armillary astrolabe, with its several concentric rings. This is to a certain extant confirmed by Richard of Wallingford who in 1326 introduced the instrument he invented and called ‘rectangulus,’ a device even simpler than the torquetum, as it was composed of rods rather than plates or rings:

We designed the rectangulus to obviate the tedious and difficult work of making an armillary [sphere] . . . We conceived it as a means of determining the paths and places of the planets and fixed stars . . . together with all the other problems which may be solved by the armillary, the astrolabe, or the turketum. ²⁵

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Figure 9.

The system of rings of the armillary astrolabe: ring 1 with sighting pinnules yy slips within ring 2, the inner latitude ring; ring 3 is the ecliptic ring; ring 4 – the sostitial colure ring; 5 – the external latitude ring; 6 – the meridian ring; xx and zz are, respectively, the celestial and ecliptic poles.

Even if on the whole the torquetum was easier to construct that the armillary astrolabe, the problem of the accuracy of the observations made with the torquetum has not been addressed so far. In fact, the problems stemmed precisely from the construction of the torquetum. As we have already seen, the necessary requirement of making observations of the ecliptic longitude and latitude was aligning the instrumental ecliptic with the ecliptic plane. The very construction of the torquetum made it difficult. To the contrary, it was relatively easy in the case of the armillary astrolabe, irrespective if one chose the Sun, the Moon or a star.

The manner of joining the rings of the armillary astrolabe makes it easy to adjust their position with one fluent movement. If we observe the Sun, the shape of the rings allows us to see when the Sun finds itself in the plane of the ring. To cite Ptolemy:

Then we rotated the ring through the poles until the intersection [of ring 5 and ring 3] marking the Sun’s position was exactly facing the Sun, and thus both the ecliptic ring and [the ring] which goes through the poles of the ecliptic cast its shadow exactly on itself. ²⁶

In the case of observations conducted during the night, locating the heavenly body along the plane of the measuring ring can be done with adequate accuracy, even without the necessity of using pinholes. ²⁷

Taking into account their aim and construction, the torquetum and the armillary astrolabe are very similar when it comes to the analysis of the impact of errors inherent in the process of measurement, and in the instruments themselves, on the results we can obtain with these devices. Their complex construction combines in an identical way three systems of coordinates: local (azimuth and altitude), equatorial (hour angle and declination) and ecliptical (longitude and latitude). The quantitative theory of the instrument for the armillary astrolabe was published by Jerzy Dobrzycki. ²⁸ This theory could be to a large extant applied also to the torquetum.

In general, one has to analyse first the way in which instrumental errors, such as the uncertainty of geographical latitude and inclination of the ecliptic, and errors resulting from the position of the instrument in the local system of coordinates (azimuth and latitude) influence the position of the instrumental pole on the celestial sphere. The spherical quadrangle shown in Figure 10 defines the limits of the displacement of the instrumental pole (in equatorial coordinates) as the summation of these errors. In an ideal situation the zenith Z_d of the instrument overlaps with the zenith point Z, and the instrumental pole P_i with the north celestial pole P. In reality, the instrumental meridian does not match the local meridian. The position P_i will be determined by the joint impact of errors of the instrument settings: INC – inclination ZZ_i and DEV – deviation Z_iZ_d, represented by spherical coordinates ψ and dZ, as well as AZI and LAT, that is, the errors in azimuth and geographical latitude, respectively. These errors result both from the setting of the torquetum in the observational position, and from the lack of knowledge of the accurate values of local geographical latitude φ and the obliquity of the ecliptic ε. Ultimately the position of the instrumental pole P_i can be defined as the arc PP_i = dP at the angle χ to the point P_i.

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Figure 10.

Instrumental errors (the joint impact of the errors of the settings of the torquetum in an observational position and inaccurate values φ and ε) cause that the local meridian NPZS does not match the instrumental meridian P_iZ_d.

The next step calls for investigating the extent to which the results of the observations reflect the propagation of the longitude error resulting from the imprecise setting of the instrumental ecliptic of the actual instrument with displaced pole. Additionally, in the case of the torquetum, as in the case of the armillary astrolabe, we have to take into account two methods of setting the instrumental ecliptic: first, a purely observational method of determining the Sun’s position on the ecliptic; and second, setting the instrument to a theoretical longitude. In both cases solving relevant systems of equations allows us to show the longitude error as a function of the longitude depending on the assumed instrumental errors (dP – the displacement of the instrumental pole from its true position, and dε – the error of the obliquity of the instrumental ecliptic) and the hour angle of the Sun τ (the angle is measured westward from the local meridian).

In his calculations Jerzy Dobrzycki assumed dP = 0.25° and dε = 0.25°. The value of dP is naturally chosen arbitrarily, but it seems relevant particularly in the case of the torquetum, due to the absence of the symmetry of the instrument as regards the zenith–nadir line. It is worth emphasising that a likely source of an error was the uncertainty of geographical latitude. For example, Regiomontanus claims that the torquetum which he describes was made for the latitude of Esztergom which he assumed to be 47°30′. ²⁹ Thus the error in the tilt of the plate representing the celestial equator amounts to 0.28°, since the modern latitude of Esztergom is 47°47′. Significantly, in the torquetum of Martin Bylica of Olkusz the latitude could be set with the maximum accuracy of 1°.

Setting aside other factors, dε is determined by the error of the obliquity resulting from the assumed current value for the inclination of the equator with respect to the ecliptic and by the possible error in the construction of the instrument. A peculiar bit of information can be found in an anonymous text printed by Poulle. The author claims that the instrumental ecliptic of the torquetum should be tilted to the instrumental equator at the 2ε angle, because the centres of these two plates are apart. ³⁰ At this point he offers value 2ε = 47°7′, which is twice 23°33.5′, the obliquity used in the Toledan Tables. The same value of the angle of the instrumental ecliptic with the instrumental equator, and with the same arguments, can be found in Epperies. ³¹ Obviously the instrument constructed in such a way would be useless.

Franco and Bernard refrain from specifying any values of the angle between these two plates. Regiomontanus asserts it equals almost 23°30′, ³² which matches the value ε = 23°28′ featuring in the Epitome (I, 17). Apian indicates the value of 23°30′. ³³ Both numbers are less than 0.1° different from the value derived from the modern theory. Furthermore, if anybody had wished to build the torquetum in 1284 relying on the obliquity from the Toledan Tables, the error would have been be also less than 0.1° since according to modern theory, for year 1284 the obliquity was about 23°32′.

Figure 11 shows the distribution of the longitude error Δλ as a function of the longitude λ for the first, purely observational method of aligning the instrumental ecliptic to the Sun. ³⁴ As mentioned before, the assumed instrumental errors dP and dε amount to 0.25°. Considering the symmetry as regards the meridian, the diagram shows Δλ for 0° ⩽ τ ⩽ 120°. The same diagram will be obtained for τ = 0° + 180° and so forth. In general, this leads us to a conclusion that in the case of the first method of aligning instrumental ecliptic with the ecliptic plane, accurate results cannot be expected even with relatively small errors of the displacement of the instrumental pole by 0.25° and the same value of the obliquity error. This refers in particular to times near the solstices. Only observations made near sunrise or sunset (τ ≈ 90° or 270°) around the equinoxes (λ ≈ 0° or 180°) can be expected to have relatively small errors. Interestingly, in Regiomontanus’s text we shall find remarks matching the conclusions reached on the basis of Figure 11. Setting the instrumental ecliptic at the correct position with the first method requires the observer to begin at sunset. ³⁵ Regiomontanus is aware of this though he writes about it in a different place: ‘One has to remember that when the Sun approximates the tropics, observations become unreliable’. ³⁶

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Figure 11.

The distribution of the longitude error Δλ as a function of the longitude λ for the first purely observational method of aligning the instrumental ecliptic to the Sun.

A different picture emerges when the instrument is used in conjunction with a calculated position of the Sun. In this case the results of observations are affected by the error of the solar longitude given by the theory, practically irrespective of observing conditions (unless observations are made near sunrise or sunset ³⁷ ). However, the above can be true only if it is possible to align precisely the position of the Sun marked on the instrumental ecliptic and the Sun on the sky. In the case of the torquetum this procedure was more difficult to complete than in the case of the armillary astrolabe, which I have already mentioned before.

Finally we must note that the asymmetry in the construction of the torquetum may cause uncontrolled changes in the position of the instrumental pole. This in turn may lead into a situation when even in the case of a series of observations made over the same night, with an unchanged setting of the torquetum, there will be no single value of the displacement of the instrumental pole.

The torquetum and historical observations

The treatise on the torquetum ascribed to Franco de Polonia ends with a declaration that thanks to this instrument it will be possible to compare the actual positions of the planets and stars with the Toledan Tables. Unfortunately, so far I have not come across any set of regular measurements of stellar and planetary longitudes and latitudes made with this instrument in the 14th or 15th century. The observations of heavenly bodies associated with the torquetum are very few and appear to be rather random. Interestingly, some of the oldest observations pertain not to stars or planets but to comets. I refer here to the observations of the comet made in January and February 1299 by Peter of Limoges, ³⁸ and similar observations of the comet (1/P Halley) made in September and October 1301 and reported in an anonymous manuscript which is also ascribed to Peter. ³⁹ In both cases longitude and latitude were measured to a degree. ⁴⁰ Irrespective of the above, the observations of comets made many centuries ago are obviously not the best source to verify the accuracy of observations made with the torquetum.

John of Murs praised the torquetum (turqueto) and astrolabe (astrolabio) as especially reliable observational instruments yielding ‘truth’. ⁴¹ However John of Murs does not mention the name of the torquetum while listing instruments which he used in his astronomical observations, the earliest of which came from 13 March 1318, when he was still a student in the faculty of arts in Paris. ⁴² John of Murs would observe the Sun’s meridian altitude with an instrument called kardaga, set in the meridian line, ⁴³ or with an adequate instrument (instrumentum ydoneum) or grand instrument (instrumentum magnum). To observe a solar eclipse he would use good astrolabes (astrolabia bona).

The torquetum is also associated with the observations of the comet of 1456 (1/P Halley). An Italian astronomer, Paolo Toscanelli, left a manuscript with his observations of comets between 1433 and 1472. Jane L. Jarvis noticed that in the case of the comet of 1456, Toscanelli changed his observational technique, and used an instrument which gave him readings directly in longitude and latitude (earlier he would determine the position of the comet by alignment with nearby stars). Jarvis put forward a hypothesis that this could be the armillary astrolabe or the torquetum. ⁴⁴ Subsequent authors eagerly followed her suggestion and arbitrarily opted for the torquetum ⁴⁵ even though there is no evidence to support this claim. Significantly, longitudes and latitudes noted by Toscanelli had an accuracy of 5′ and 10′.

Regiomontanus tried to revive the idea of the torquetum as an observing instrument. However, we do not know about any observations by Regiomontanus which could be undeniably interpreted as made with the torquetum. In section 15 of his treatise on the torquetum Regiomontanus writes about the observations of the angular distances of the planets from the Sun at their last sightings near the Sun. ⁴⁶ He offers even values – with accuracy to a degree (and with an exception of Jupiter where half of a degree is given) – but it is not clear if these results have been obtained with the torquetum. For the purpose of measuring angular distances of heavenly bodies Regimontanus would use rather the armillary astrolabe. He tested also the Jacob’s staff. ⁴⁷

Subsequent sources with references to the use of the torquetum pertain again to the observations of comets. The first comet is the one of 1531, and therefore again 1/P Halley. The comet was observed between 15 and 26 August by Johannes Schöner of Nuremberg, the publisher of the works by Regiomontanus. Schöner owned the torquetum at least from 1524. ⁴⁸ He presented his thoughts about the comet along with the observations in a short work printed the same year. ⁴⁹ Initially Schöner determined the position of the comet with respect to nearby stars without an instrument (star occultations, position as regards the line connecting two stars). On 17, 18, 23 and 25 August Schöner used the torquetum. He recorded the longitude and latitude of the comet with one-degree accuracy, and only twice read the longitude with the accuracy of 0.5°. However, he did not offer any details of the observations, but in all cases he recorded the azimuth of the comet, also in degrees. ⁵⁰

The observations made by Schöner and his instrument, that is, the torquetum, are mentioned by Apian in his elaborate report on his own observations of comet of 1531, appended to his Practica for the year 1532. ⁵¹ Apian’s work gained historical significance because he was first to offer an impressive evidence for the antisolar nature of cometary tails. The next 2 years were dominated by the comets of 1532 and 1533. The first of them was discussed by Apian in his second work as of 1532 featuring an illustration shown in Figure 8. Ultimately all Apian’s observations of the comets of 1531, 1532, 1533 together with the observations of some definitely less spectacular comets of 1538 and 1539 were collected in his Astronomicum Caesareum (1540). ⁵²

Apian’s reports on the observations of the comets precede the treatise on the torquetum. The torquetum is also mentioned at the beginning of the section introducing the observations of the first of the five comets, that is, the comet of 1531. However, at this point Apian explains why he used in his observations the meteoroscope (meteoroscopium), and not the torquetum, even though the observations with the latter instrument ‘would be very easy’, but he would not have been able to determine the length of the tail of the comet. ⁵³ (Apian’s meteoroscope was described also in Astronomicum Caesareum, it was a quadrant divided by the grid derived from a particular stereographic projection). ⁵⁴ The reports on the observations of the comets of 1538 and 1539 are very short in comparison with the reports on the first three comets. There is limited data about the ecliptic coordinates of each of the comets, measured with 1° accuracy as in the case of Schöner. ⁵⁵ Perhaps Apian used the torquetum to this end but he does not inform us about it.

The last attempt to make practical use of the torquetum was made by the Landgraff Wilhelm IV of Hesse in his observatory in Kassel. In the years 1560–1563 he used the torquetum to measure the positions of 58 stars. Wilhelm IV wrote the stellar longitudes and latitudes which he measured in his copy of the Astronomicum Caesareum. This list served as the basis for the execution of the globe of the Kassel planetary clock. Jan Hendrik Leopold published a transcription of this list with, however, an altered order of the stars. ⁵⁶ A detailed account of the circumstances of creating this list and the observational method which was used for this purpose, along with a partial attempt at assessing the quality of the results of the observations, have been offered by Karsten Gaulke and Michael Beck. ⁵⁷ The work by Gaulke and Beck is based on rich source material. I shall refer to it briefly to discuss the manner of making observations before I start analysing the list of 58 stars.

The procedure used by Wilhelm IV was as follows. First he measured the midday solar altitude (taking into account the solar parallax to which he ascribed the value of 2′). This allowed him to determine the declination of the Sun and, by solving a relevant spherical triangle, to obtain the solar longitude. ⁵⁸ The second observation was made before sunset. The increase of the solar longitude was determined from the time elapsed from noon, measured by a mechanical clock. Once the ecliptic of the torquetum was set, the longitude of the Moon or Venus was measured. Then one of these heavenly bodies was used at night to set the torquetum to measure ecliptic coordinates of stars.

Thus Wilhelm IV followed the suggestion of Regiomontanus and Apian to use the clock while observing with the torquetum. He also used Venus as an intermediary body for daily and nightly observations, a method earlier used by another astronomer from Nuremberg, Bernard Walther, observing with the armillary astrolabe. ⁵⁹ It is worth recalling that the observations made by Walther were published by Schöner together with Regiomontanus’s treatise on the torquetum in the Scripta (1544).

The list of the stars of Wilhelm IV allows us to assess the precision and accuracy of the observations made with his torquetum (see Appendix for details). For the 56 stars the mean error in longitude amounts to 6′ ± 0.8′, and the standard deviation of the error is 7′. The mean error in latitude amounts to −8′ ± 1.1′, and the standard deviation is 7′.

Wilhelm IV gave us an interesting possibility of comparing the accuracy and precision of the observations made with the torquetum in his successive observational programmes. Hence in the years 1566–67 he compiled a list of 58 stars (the list is not identical with the previous list though a significant number of stars features in both of them), completing measurements with the azimuthal quadrant and using clocks. ⁶⁰ In this case he would determine the declination and the right ascension of a star. It turned out that the standard deviation of the error in right ascension amounted to 2.2′, whereas the standard deviation of the error in declination equalled 4.4′, ⁶¹ and therefore these measurements proved more precise than those made with the torquetum. Interestingly, the mean error in right ascension equalled −22′, and, consequently, it was bigger than the mean error in longitude (6′) in the observations made with the torquetum. The observations from the years 1566–67 were used by Wilhelm IV to calculate the ecliptic coordinates of stars. A large mean error in right ascension led to bigger errors in longitude and latitude than in the case of the coordinates of stars measured with the torquetum, ⁶² and yet the quadrant ensured more coherent results.

The analysis of the list of stars from the years 1560–63 invites a discussion of yet another interesting problem. In the section on the theory of the instrument, I have noted that considering the basic technical features of the torquetum one can apply to it the theory of the armillary astrolabe. However, we can also try to compare these instruments with regard to observations. There are two extant sets of measurements of ecliptic coordinates of stars which reveal the ultimate limitations of the armillary astrolabe: one compiled by Bernard Walther of Nurmberg at the turn of the 16th century and one by Jarosław Włodarczyk compiled in the 1980s. ⁶³ In the case of the observations made by Walther the standard deviation of the error in longitude amounts to 8′–9′, and the standard deviations of the error in latitude amounts to 10′–11′. ⁶⁴ In the case of the observations made with a modern wooden replica of the armillary astrolabe, the standard deviation of the error in longitude amounts to 7′–12′. ⁶⁵ The values obtained by Wilhelm IV with the torquetum are very similar to those above.

However it would be far too early to treat this similarity as a proof of the usefulness of the torquetum for positional measurements. There are reasons to believe that the torquetum used by Wilhelm IV exemplified a return to the use of rings as elements of the construction, and therefore ran counter to the rationale which underlined the 13th–century efforts to construct the torquetum as an alternative to existing devices. The torquetum from Kassel can be seen on the portrait of Landgraff Wilhelm IV and his wife, Sabina von Württemberg, painted by Kaspar van der Borcht in 1577 (Figure 12). ⁶⁶ The construction remains more asymmetrical than in the case of the armillary astrolabe, and yet we can see the instrument built of rings and not of plates. However, it must be emphasised that we cannot be absolutely certain that the torquetum depicted in the painting was used to compile the list of stars in the years 1560–63. Based on the inventory made in 1573, Wilhelm IV owned two copies of the torquetum. ⁶⁷ We do not know what the other torquetum looked like.

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Figure 12.

The torquetum in Kassel. Courtesy of Museumslandschaft Hessen Kassel, Gemäldegalerie Alte Meister. Photo Ute Brunzel.

On the other hand, the completed analysis of observational errors strongly suggests that the torquetum used in the observations had a similar construction to that of an armillary astrolabe, and therefore looked like the instrument shown in the painting. Perhaps the other torquetum had a more traditional form? Perhaps it was used to complete the observations of the comet of 1558, an enterprise reported in accordance with the earlier observational tradition of Peter of Limoges or Schöner as follows:

. . . a comet appeared, which on the 20th August was observed by the Most Illustrious Prince and Lord WILHELM, LANDGRAVE OF HESSE, with a torquetum, around the ninth hour, at 21 Degrees into Virgo, with a latitude 31 Degrees from the Ecliptic. . . ⁶⁸

Apart from the list of the stars observed by the torquetum and the picture of it, there are also some specific comments on Wilhelm IV’s instrument. These comments were made by Tycho Brahe who not only saw the instrument but also observed with it. This is how Tycho recalled his visit to Kassel in his letter to Christoph Rothmann as of 21 February 1589 ⁶⁹ :

It is a different story with the torquetum, an instrument used for observing by His Lordship [Wilhelm IV]. Even if it were of the right size, it still has a complex construction consisting of too many different parts. A mere attempt of aligning it involves multiple things which are hard to control simultaneously and during observations. In consequence if one takes care of a single thing, other things get easily neglected which may lead to a major error. The instrument is too clumsy due to its weight because its settings do not ensue the rotational balance around the axes. Out of all astronomical instruments devised by the ancients, this one is most difficult to operate and least reliable even though its construction appears clever enough . . . Furthermore while His Lordship’s torquetum, which was used to complete these observations, was made of strong brass, it was still too tiny, as I have myself discovered 15 years ago when I stayed a couple of days with His Lordship in Kassel. He would have two such torqueta then but none of them was of sufficient size, and both had been already thought by His Lordship as outdated. There was no way they could show reliable positions of stars even when His Lordship would assist. And thus it would easily happen that while observing with this torquetum which is not big enough I would make an error of several degrees . . . ⁷⁰

The portrait of Wilhelm IV and his wife allows us to understand what Tycho meant when he referred to the ‘too tiny’ torquetum. We know that the quadrant, shown next to the torquetum in the painting, has a radius of two feet, that is, approximately 50 cm. ⁷¹ Therefore we can assume that the rings of the torquetum had a similar diameter. The torquetum of Wilhelm IV was ‘too tiny’, and yet it is the biggest of the torqueti whose dimensions we know or can estimate.

Notes on the dimensions of the torquetum

There is a great number of manuscript copies of the early torquetum treatises. ⁷² However the sources which would confirm the existence of the instruments built on the basis of such descriptions are few. One can risk the hypothesis that the elaborate design of the torquetum led to the situation that by the middle of the 15th century the overall number of the instruments, manufactured and in use, was significantly smaller that the number of descriptions of the torquetum circulating in Europe. The latter conclusion has been prompted by the inspection of a few drawings which feature along the descriptions of the torquetum. These drawings have some obvious errors such as, for example, placing the ecliptic circles on the same axis as the equatorial plate. Such error can be also seen in the drawing appended to the treatise preserved in Bodleian Library in Oxford and ascribed to Franco de Polonia. ⁷³ As a result the ecliptic axis around which the crista should revolve is not perpendicular to the ecliptic plate. A similar error can be seen in Figure 1. ⁷⁴

Furthermore, the list of the extant torqueti is very short. Even if we take into account these instruments that were once catalogued and then lost. Additionally, all of them come from the 15th and 16th century, and therefore the first 150 years of the written tradition of the torquentum does not have a single material representation.

Such artefacts are important from the point of view of our investigation as they help to reach conclusions about the dimensions of the constructed instruments. This information is missing from the written sources, and at the same time it is vital for the assessment of the usefulness of the torquetum in various observations. There are also two representations of the torquetum in paintings which make it possible to assess the size of the instruments. The first painting is The Ambassadors (1533) by Hans Holbein the Younger, ⁷⁵ now in the National Gallery in London, whereas the second is the previously mentioned double portrait of Landgraff Wilhelm IV and his wife (1577) by Kaspar van der Borcht. We have also woodcuts from Apian’s treaty on the comet of 1532 (Figure 8), where the torquetum (if it is indeed a reduced torquetum) is shown next to the figure of a man. However, we cannot be sure if the right proportions were kept in this case.

The list below, compiled chronologically, features the dimensions of the ecliptic disc in the above mentioned copies of the torquetum. In cases where data are not available, an estimated value is given, based on the information about other parts of the instrument.

Out of the 10 instruments on the list, the torquetum of Wilhelm IV is obviously the biggest one and we know that it was used for determining ecliptic coordinates of stars. Most of the determined longitudes and latitudes had an accuracy of 5′ which may suggest the degrees on the scale were divided every 10′. This however does not have to be the true, as 17 out of 116 endings are not multiples of 5′. Out of these 17, five are a multiple of 6′ (0.1°). In the remaining 12, 8′ is overrepresented and appears as many as six times, the remaining endings are 3′, 22′, 33′, 38′, 43′ and 52′. Jarosław Włodarczyk observed with the armillary astrolabe, where the longitude ring had a diameter similar to that of the torquetum of Wilhelm IV, that is, approximately 60 cm, and therefore 1° was 5 mm wide. Even though the longitude scale was graduated into 360°, without subdivisions, most of the readings were noted to 0.1°. ⁸⁵

The two oldest instruments happen to be also the biggest instruments of the remaining nine copies. Their longitude scales are graduated into 360°, without subdivisions. In those instruments 1° calibration is spaced approximately 3 mm apart. In the case of these two instruments, it would be certainly possible to obtain the accuracy of 0.5°. In the case of the rest of the torqueti 1° of the longitude is approximately 1.5 mm wide.

What is also worth mentioning is the historical tendency. Setting aside the torquetum of Wilhelm IV, the instruments made in the second half of the 16th century (including the torquetum on Holbein’s painting), due to their size, appear to be instruments used for educational or representational purposes rather than devices meant to be employed in advanced astronomical observation.

Conclusion

Almost half a century ago John D. North contended:

One frequently encounters references to the instrument [torquetum] in fourteenth- and late thirteenth-century writings, and there can be little doubt that the instrument was taken seriously and used for observation from the time of its introduction into Europe. ⁸⁶

However, it seems that a substantial number of mediaeval manuscripts is not a sufficient reason to confirm the arrival of a new instrument that significantly contributed to astronomy of the time. The instructions on how to use the torquetum contain very few recommendations stemming from the actual use of the instrument, and, at the same time, they recycle the information from the two earliest treatises on the torquetum, which are rather enigmatic in this respect. The only text which stands out in this context is the one by Regiomontanus. However, it was written in 1469 and printed as late as 1544. The text might have led to the construction of at least two instruments, those of Schöner and of Wilhelm IV, which were used in astronomical observations. Notwithstanding the above, in the second half of the 16th century when observational astronomy was in full swing, the torquetum became a representational instrument, perhaps used for educational purposes. This is evidenced by the miniaturisation of the latest surviving copies.

The evidence against the widespread use of the torquetum in advanced positional astronomy can be found in scarce extant observations and their specificity. Apart from the list of 58 stars from the years 1560–63, compiled in Kassel, in the years 1299–1558 one can point to merely several very short observational series or individual observations made with the torquetum. Moreover, all of them pertain to the comets, and the registered celestial coordinates had the accuracy of 1°, sporadically of 0.5°.

The theory of the instrument shows that the torquetum could be used for determining the ecliptic coordinates exclusively by way of differential observations, that is, when for the purpose of setting the instrumental ecliptic a reference luminary of known longitude was used. At the same time, from the point of view of advanced positional astronomy, the torquetum was doomed to failure from the very beginning due to the lack of rotational symmetry and the continuous alterations of the inclination of its plates in relation to the horizon. Each new motion of one of the plates could result in the alteration of the instruments’ position for observation. A distinct evidence of the weakness of the construction of the torquetum is the copy owned by Wilhelm IV, depicted on the painting. In this version, the torquetum adheres to the well-established tradition of positional astronomy, and plates are replaced with rings.

Consequently the torquetum could be and was used for observations, but it could not compete with classical instruments described in Ptolemy’s Almagest, including the armillary astrolabe, the very instrument that the torquetum was supposed to replace. Could it be used to test astronomical tables? In some cases yes, but at least parallel verifying potential was offered by more simple devices, including naked eye observations. In the case of the torquetum, its complex construction did not contribute to the accuracy of prospective results.