On the Determination of the Solar Rotation Elements i, {\Omega} and Period using Sunspot Observations by Ru{\dj}er Bo\v{s}kovi\'c in 1777

In September 1777, Ru{\dj}er Bo\v{s}kovi\'c observed sunspots for six days. Based on these measurements, he used his own methods to calculate the elements of the solar rotation, the longitude of the node, the inclination of the solar equator and the period. He published a description of the methods, the method of observation and detailed instructions for calculations in the second chapter of the fifth part of the Opera in 1785. In this paper, Bo\v{s}kovi\'c original calculations and repeated calculations by his procedure are published. By analysing the input quantities, procedures, and results, the input quantities of the error, and the calculation results are discussed. The reproduction of Bo\v{s}kovi\'c calculations is successfully reproduced and we obtained very similar results. The conclusion proposes a relationship of Bo\v{s}kovi\'c research with modern astronomy.


Introduction
Ruđer Josip Bošković (1711 -1787) was a Croatian Jesuit priest with a broad interest in various scientific fields (Dadić, 1998;James, 2004;MacDonnell, 2014). He made important contributions to mathematics, astronomy, physics, geodesy, cartography, archeology, civil engineering, and philosophy. Moreover, he was a successful poet and diplomat, and invented/improved some optical and astronomical instruments.
Astronomy was, however, one of his main research fields (Kopal, 1961;Špoljarić and Kren, 2016;Špoljarić and Solarić, 2016). R. Bošković investigated, both theoretically and observationally, many astronomical phenomena: solar rotation and sunspots, transit of the planet Mercury before the Sun, the aberration of stars, eclipses, comet and planetary orbits, among others. He was also a founder of the Brera Observatory in Italy, which confirms a dominant interest of R. Bošković in astronomy.
In his first scientific paper (Boscovich, 1736), Bošković introduced and described two methods for the determination of the solar rotation elements: the solar rotation axis position in space and the rotational period. The direction of the solar axis in space is determined by the angle of inclination (i) between the ecliptic plane and the equatorial plane of the Sun and by the ecliptic longitude (Ω) of the ascending node of the solar equator, which is the angle, in the ecliptic, between the equinox direction and the direction where the solar equator intersects the ecliptic from the South, i.e. in the sense of rotation (Stix, 2002). In his last extensive astronomical work (Boscovich, 1785), Bošković described his third method for the determination of the solar rotation elements, and applied the methods to his own sunspot observations performed in Sens, near Paris in 1777.
Solar rotation elements belong to the fundamental astronomical properties of the Sun (Stix, 2002). A precise knowledge of the solar rotation axis position in space is important for data reduction of solar observations and transformation of the measured coordinates of objects on the Sun into heliographic ones (Wöhl, 1978;Stark and Wöhl, 1981;Balthasar et al., 1986;Balthasar et al., 1987;Reinsch, 1999;Stix, 2002). Solar rotation can be expressed in terms of the rotational period (in days) or in terms of the angular velocity (in degrees per day). The investigation of solar rotation is very important within solar physics. The Sun rotates differentially, which means that the angular velocity is a function of the heliographic latitude, the depth and the time (Howard, 1984;Schröter, 1985;Beck, 1999), which is possible since the Sun is composed of plasma and does not rotate as a rigid body. Moreover, research of the solar differential rotation is important, as it is closely related to the solar magnetohydrodynamical dynamo, which, according to present concepts, plays a significant role in generating and maintaining solar magnetic activity (Stix, 2002;Charbonneau, 2020).
The three main experimental methods to measure solar rotation are the tracer method, the Doppler method and the helioseismological method (Beck, 1999). R. Bošković (Boscovich, 1785) used the tracer method, which is also the oldest method for the solar rotation determination. The method consists of following the positions of any recognizable objects on the Sun, e.g. sunspots, in time. Then one possibility is to determine the solar angular velocity of rotation by dividing the difference in position with the elapsed time, and this is the procedure used by R. Bošković.
There is a large quantity of observational evidence which shows that the solar rotation changes in time (Brajša et al., 1997;Brajša et al., 2006;Wöhl et al., 2010) and that this variation is related to the solar activity cycle (Jurdana-Šepić et al., 2011;Ruždjak et al., 2017). For this reason, it is important to discover, collect and reduce as many historical sunspot observations as possible Nogales et al., 2020).
In this present work, we describe the three methods of R. Bošković for the determination of the solar rotation elements. We repeat the original calculation of R. Bošković applied to his own sunspot observations with the aim to fully understand the method and to reproduce his published results.

Methods of Ruđer Bošković
for the determination of the inclination i, the longitude of the node Ω, and the period of the solar rotation T using sunspot observations Ruđer Bošković developed three methods for the determination of the solar rotation elements: the inclination i, the longitude of the node Ω, and the period of solar rotation: the graphical method, and two numerical methods using the planar and the spherical trigonometry (Boscovich, 1785, Préface, №4-9, pages 77-79). The original formulas and descriptions of figures for his methods are presented in Opuscule II (Boscovich, 1785, №10-19, pages 79-85). At the end of Tomus V, the table of figures 1. to 9. and the extract (Boscovich, 1785, Extrait, starting page 444, §.II. Du second Opuscule, №25-№33, pages 456-461) are presented.
The methods for determining the elements of solar rotation, based on the observation of sunspots, Ruđer Bošković published in 1736 in his first dissertation De maculis solaribus (About sunspots) in Latin (Boscovich, 1736). The formulas for the methods of Ruđer Bošković are presented in the Préface (Boscovich, 1785, Préface, pages 81-84). The example is performed with logarithmic tables presented in twelve Roman numbered tables Tab. I.

The method for Ω
The method for the determination of the longitude of the node Ω, the intersection of the ecliptic and the solar equator, uses two positions of the same sunspot on the equal latitude. The observation of the same sunspot on the same latitude practically is not reliable, but it can be mathematically simulated using three sunspot positions, the one on the left, and the two on the right of the maximal sunspot latitude (Boscovich, 1785, §.IV., №45). The numerical solution of the method is the ratio of the differences of ecliptic latitudes and longitudes.

The method for i
The planar trigonometric method for the determination of the inclination of the solar equator regarding the ecliptic uses two sunspot positons with the longitude of the node that is already known. The method uses one planar triangle for inclination determination: the planar graphical construction and the planar trigonometry solution (Boscovich, 1785, §.V., №53).

The method for the period
The method for the solar rotation period determination uses two sunspot positions and already known elements: the longitude of the node and the inclination of the solar equator. The method uses spherical trigonometry for determination of the angle between two declination arcs from two sunspot positons to the pole in the equatorial coordinate system. The period of rotation is calculated from the angular velocity (Boscovich, 1785, §.VI.).
production of his results in this present work are: Table  22 to Table 33. In 3.4. Present work results, only input and output calculation data are displayed instead of larger tables. Tables in the present work are formatted as much as possible like the original tables using fonts and formatting rules (upright, italic, bold). In the beginning (Boscovich, 1785, №3) Bošković described what we need for the reproduction of his results: the sunspot observations, the astronomic almanac (URL 1), Opuscule II (Boscovich, 1785) and logarithmic tables. The formulas are trigonometric. The original application of the formulas uses logarithmic rules, where, for example, multiplication transforms into the addition of logarithm factors: log(a•b)=log(a)+log(b), and then the result of multiplication is (a•b)=10 [(log(a) + log(b)] . Today, we do not use logarithmic tables for this type of calculation.
In September 1777, the observations were made by Bošković himself in Sens. He described the methods, formulas, and the example of determination of solar rotation elements: the solar equator inclination i, the longitude of the node Ω, and the period of solar rotation T (Boscovich, 1785, §.I.- §.XIV., pages 75-169). In Opuscule II, he numbered the paragraphs continuously with Arabic numbers №1 to №165, and chapters with Roman numbers §.I. to §.XIV. Opuscule II has the appendix Appendice where paragraphs were numbered with Arabic numbers №1 to №20, pages 170-178. There, all the ob-servations he made in September 1777 are given for four sunspots (Boscovich, 1785, Appendice, pages 170-178).
In Opuscule II (Boscovich, 1785, №4), his observation procedure was described. Bošković made observations himself using a telescope equipped with a micrometre for the determination of the differences of declination and rectascension of an observed sunspot and a precise pendulum for measuring time (Préface, №4).
Bošković's description of the observed sunspot one: The sunspot was clearly recognizable, regular, of medium size, so that the thread of the telescope could pass through its center 1 . We can conclude that the sunspot could be type A, B, C, D or G according to The Zurich Classification System of Sunspot Groups (URL 4).
For six days he observed one medium-sized sunspot in early afternoon about 3 p.m. He observed the sunspot in a series of five measurements. He observed passing times of the solar disk edges of the sunspot, and the vertical distance of the sunspot from the northern edge of the solar disk. He determined the horizontal distance of the sunspot from the solar disk center using passing times of the edges and the sunspot through the vertical line.
In the field of view of the fixed telescope (see Figure  1), he observed the passing times of the solar disk t 1 , the sunspot t t , and the solar disk t 2 through the vertical line. At the same time, he measured the vertical position of the sunspot from the northern edge of the solar disk A: he put the fixed horizontal line of a micrometre at the sunspot, and then he measured the position of the northern edge of the solar disk with the mobile line of the micrometre. He converted the vertical distance measured using the micrometre into angular seconds using the constant of the micrometre C. He determined constant C relatively to the apparent diameter of the solar disk described in section 3.2 Astronomic almanac data. He measured the vertical distance of the sunspot from the solar disk edge in angular seconds using the micrometre. The values: A is the distance of the sunspot from the northern edge of the solar disk (bord boreal); observed the times of passage of the vertical line (thread, reticule): the time t 1 of the 1 st edge (1 bord), the western (right) edge of the solar disk; the time t t of the sunspot (tache), the time t 2 of the 2 nd edge (2 bord), the eastern (left) edge of the solar disk; and B is the time difference B=t t -(t 1 +t 2 )/2 (see Figure 1). The difference A is measured with a micrometre, and B he determined as the time difference of the sunspot moment t t and the solar disk centre (S) moment t S =(t 1 +t 2 )/2, B=t t -(t 1 +t 2 )/2=t t -t S . In this paper, the observations are presented in Table 1 (Bos- Table 1: The records of the observed sunspot of six day observations: the 1 st line: September 12 th , 1777; the 2 nd line: north edge (bord boreal) with its arithmetic mean the far right (milieu); the 3 rd line through the 5 th lines: the observed times of passing the vertical line: the 1 st edge (1 bord) -t 1 , the sunspot (tache) -t t , and the 2 nd edge (2 bord) -t 2 ; and the 6 th line: the difference (Différence) with its arithmetic mean the far right (milieu); September 12 th and 15 th , 1777 have the wrong values in boxes (Boscovich, Table 2) are: the northern border arithmetic mean A (bord boreal, milieu); the 1 st edge in the 1 st series, the observation beginning time (1 bord); the 2 nd edge in the last, 5 th series, the observation ending time (2 bord); and the difference arithmetic mean B (Différence, milieu), and the constant of the micrometre C. The constant of the micrometre C is determined empirically by observations and data in astronomic almanac (URL 1, page 108) described in the section 3.2.

Astronomic almanac data
The solar rotation element reproduction includes: the solar equator inclination i, the longitude of the node Ω, and the period of the solar rotation. The astronomic almanac Connoissance des temps (URL 1) contains all the needed additional data besides his observations for the reproduction of the results in the present work. The astronomic almanac data that Bošković used are: 1. the positions of the Sun and the correction for the mean solar time (see Table 3); 2. the longitude of the Sens from Paris (see Table 4); 3. the apparent diameter of the Sun (see Table 5 and Table 6); and 4. the inclination of the ecliptic (see Table 7).
Daily input data for Tab. I. are boldface in Table 3, later derived in Table 20: the solar longitude (Longitude du Soleil); the solar declination (Déclinaison du Soleil); and Correction for the mean solar time (Temps moyen au Midi vrai) (Connoissance des temps, URL 1, pages 102-103).
Bošković took the longitude of Sens from Paris (0* h 3 m 48 s or.) from the Table of meridian differences in hours and degrees from l'Observatoíre Royal de Paris (see Table 4). In the table, the latitudes and the meridian differences have one of two prefixes: * (determined by Academia) or † (determined by other astronomers), and suffix or. (east of the Paris meridian) or oc. (west of the Paris meridian).
Bošković determined the constant of the micrometre C empirically. On September 11 th , 1777 with the fixed line of the micrometre, he observed the solar disk edge and with the mobile line of the micrometre, he measured the vertical size of the solar disk. The apparent diameter of the solar disk was measured by a large number of observations. These measurements differed very little from Table 2: Derived observation data: the constant of the micrometre is C; and for each date: the observation beginning time: the 1 st edge in the 1 st series (1 bord); the observation ending time: the 2 nd edge in the last, the 5 st series (2 bord); the northern border arithmetic mean A (bord boreal, milieu); and the difference arithmetic mean B (Différence, milieu) (Boscovich, 1785, pages 87-89).     For the same day, the date September 11 th , 1777, we discovered that he made the linear interpolation of the apparent diameter of the Sun. Diameters of the Sun from the Table 5 are: for September 7 th , 1777 diameter is D ʘ7 =31'52.2'', and for September 13 th , 1777 diameter is D ʘ13 =31'55.3''. The linear interpolation reproduces the apparent solar diameter D ʘ =31'54.26'' that he rounded in whole seconds D ʘ =31'55''=31•60''+55''=1915'' (URL 1, page 108). Finally, Bošković determined the constant of the micrometre C=1915''/1237 units and logC=0.189799 (see Table 8, the second column, the first row).

Observation data for the calculations
Another diameter of the Sun in the astronomic almanac is D ʘ =31'57.5'' (URL 1, page 260, DIMENSIONS
In the next sections, there are Bošković's results in Table 8 to

Bošković's results
Bošković presented his work in 12 tables assigned with Roman numbers. In the tables, the input and the output data are presented in boldface. In subsections 3.3. Bošković's results and 3.4. Present work results, of the present work, we determined: the inclination i, the longitude of the node Ω, and the period of solar rotation. Some tables in the original have no units in the table headers that we added here.
The first independent part in Tab. I. (see Table 8) is the determination of the time T.M. of the observed sunspot. The second step is the determination of the centre of the solar disk, the solar longitude (lon.ʘ) and the solar declination (dec.ʘ). The centre of the solar disk is the origin for the determination of the sunspot position in the ecliptic coordinate system: the longitude lon.t and the latitude lat.B.t.
Tab. I. (Boscovich, 1785, page 166) presents a calculation of the position of the centre of the solar disk, the longitude (lon.ʘ) and declination (dec.ʘ), and then the position of the sunspot on the solar disk lon.t and lat.B.t 2 , and T.M., the last piece of data at the end of the Tab. I. The Tab. I. is the example for September 12 th , 1777 (see Table  8). The calculation is repeated for the other 5 days of observation September 13 th , 15 th , 16 th , 17 th , and 19 th , 1777.
The input data for Tab. I. (see Table 8) are presented by boldface: 1. the derived observation data in Table 2 for each day of the observation: the beginning and the 2 For the ecliptic latitude of the sunspot Bošković used two notations: lat.t in the Tab. I. and lat.B.t in the Tab. II. (Boscovich, 1785, pages 167 and 168 respectively). ending time of daily observations, A vertical distance of the sunspot from northern edge of the solar disk determined by the position of the telescope micrometre, B the longitude -the time difference of the sunspot from the centre of the solar disk, and the constant of the micrometre C; and 2. the astronomic almanac data: The longitude of Sens from Paris 0* h 3 m 48 s or. in Table 4; the position of the centre of the solar disk, the longitude (lon.ʘ) and the declination (dec.ʘ), and the correction of time (Temps moyen au Midi vrai) in the Table 3, the apparent solar diameter 1915'' determined using a linear interpolation of the values in Table 5; and the inclination of the ecliptic ε=23°28' that Bošković rounded to whole angular minutes from Table 7.
The time T.M. is the arithmetic mean of the time t 1 (1 bord) in the first series and the time t 2 (2 bord) in the fifth (the last) series in a day (see Table 1) corrected to the Paris meridian using the time difference of Sens from the astronomic almanac in Table 4 (URL 1, pages 263 and 268, the column Différ. Des Méridiens, en Temps.) and then he corrected that true solar time of Paris meridian to the mean solar time using the correction for each day of the observation in Table 3 (URL 1, page 103, Temps moyen au Midi vrai). The abbreviation j means franc. jour -day. The time difference of Sens is 3 minutes 48 seconds eastern from Paris. The results of Tab. I. (see Table 8) for each day of observations are derived in Tab. II. (see Table 9): the moment of observation T.M. and the sunspot positon: the longitude lon.t and the latitude lat.B.t (Boscovich, 1785, page 167).

Tab. III. and Tab. IV.
The longitude of the node Ω Bošković denoted with N. From Tab. II. (see Table 9) he took three positions of   Table 10) he determined D twice in combination with position 1 before the sunspot culmination regarding the ecliptic and then with position 2 for the pair of sunspot positions 3 and 5 after the culmination, the example for the first pair is given in Tab III (see Table 10) (Boscovich,

1785, Tab. III. Bin 3 & 5, page 167).
He made the procedure for another three pairs of the positions of the sunspot and presented them in the Tab. IV. (see Table 11): the eight D values, the sum, and the arithmetic mean long.D in Tab. IV. (see Table 11). He discussed the results and decided to remove the 4 th and the 6 th value, and take into account six other values and he determined another arithmetic mean long.D (Boscovich, 1785, №114, pages 136-137). The arithmetic mean long.D increased for 3 s gives the longitude of the node Ω presented in units: sign of Zodiac (1 s =30°), degrees and minutes N=D+3 s =11 s 10°21'+3 s =14 s 10°21'-12 s = 2 s 10°21' that we converted in angular degrees and minutes N=(2•30+10)°21'=70°21'. The final longitude of the node Ω=N=2 s 10°21'=70°21' Bošković used for determination of the inclination i, and the period of the solar rotation.

Tab. V. and Tab. VI.
Bošković determined the inclination of the solar equator i using the positions of five pairs of one sunspot. The input data are the sunspot positions from Tab. II. (see Table 9): the longitudes B, B' (lon.t), the latitudes BC and B'C' (lat.B.t), and the longitude of the node N determined in the Tab. IV. (see Table 11); the output is the inclination of the solar equator i. The example for the first pair B for the 3 rd day and B' for the 6 th day he presented in the Tab. V., (see Table 12). He performed the procedure for the five pairs of the positions presented in the Tab. VI. (see Table 13) where he presented the sum 38°40' and the arithmetic mean of the inclination of solar equator i=7°44' (Boscovich, 1785, page 168).
The determination of the periods of solar rotation uses all the values determined before: D in Tab. IV. (see Table  11), B in Tab. II. (see Table 9), and i in Tab. VI. (see Table 13). Tab. VII. (see Table 14) and Tab. VIII. (see Table 15) determine auxiliary values CP'D for each day and then in the Tab. IX. (see Table 16) and in the Tab. X. (see Table 17) the sidereal period of solar rotation T' and then the synodic ones T'' in Tab. XI. (see Table 18). The auxiliary value CP'D determined in the Tab. VII. (see Table 14) for six days of one sunspot is in Tab. VIII. (see Table 15). Tab. IX. (see Table 16) determines T' from six pairs of T.M. and the values CP'D from Tab. VIII. (see Table 15). The arithmetic mean of T' is the sidereal period of solar rotation. Finally, Tab. XI. (see Table 18) determines the synodic solar period T''.
The calculation of the sidereal and the synodic periods of the solar rotation is performed in two steps: 1. The CP'D in Tab. VII. (see Table 14), and Tab. VIII. (see Table 15); and 2. The T' in the Tab. IX. (see Table 16) for six pairs of observations using the mean solar time  T.M. from the second column of Tab. II. (see Table 9). The arithmetic mean T'=26.77 days is given in Tab. X. (see Table 17), and T''=28.89 days in Tab. XI. (see Ta-       ble 18) using the arithmetic mean of T' from Tab. X. (see Table 17).

Tab. XII.
In Tab. XII. (see Table 19) the calculations of the longitude of the node N and then the inclination of solar equator i are presented, using positions of one sunspot in three different sunspot observations. The example for days 1, 3 and 6 is in Tab. XII. (see Table 19), (Boscovich, 1785, page 169). The input data are three positions of the same sunspot longitudes B, B', B'' (lon.t), and latitudes BC, B'C', B''C'' (lat.B.t) in Tab. II. (see Table  9). Two calculations with two combinations of three sunspot positions in the upper half of the Tab. XII. (see Table 19) are equal to the procedure of calculation with two sunspot positions in Tab. V. (see Table 12). In the furthest right column of the Tab. XII. (see Table 19), there are four angles SD'D, SD'D'', SD''D', and G'D'G used for further calculation of the longitude of the node N=2 s 14°03', and the inclination i=6°49'.

Present work results
In the present work, we determined the time T.M. and the positon of the sunspot for six days of observations in Tab. I. (see Table 22). The input data are the derived observation data from Table 2: the constant of the micrometre C=1915''/1237, the inclination of the ecliptic ε=23°28', and the difference from the Paris meridian ∆t Sens =3 m 48 s or., and the astronomic almanac data for the solar longitude (lon.ʘ), and the solar declination (dec.ʘ), the correction for the mean solar time (Temps moyen au Midi vrai) derived in the Table 20.
Determination of the position of the sunspot uses the apparent solar radius R ʘ for each day of observation. The B''= 1 11 9 N = 2 14 3 Bošković put 36, it should be 46, typographic mistake.   apparent solar diameter for every seventh day in September 1777 is given in the astronomic almanac Table 5 (URL 1, page 108). Linear interpolation of the apparent solar diameter D ʘ determines the daily apparent solar diameter and then divided by two gives us the apparent solar radius R ʘ in Table 21. For the days of observation, the apparent radius is given in bold italic. The observation input data: t 1 , t 2 , A, B, and the apparent solar radius R ʘ determined in Table 21 with position of the Sun (lon.ʘ, dec.ʘ), and the correction for the mean solar time in Table 20 are the input data for Tab. I. (see Table 22).
We reproduced the mean solar time T.M. differently than Bošković did. In the present work, calculation of the solar rotation period uses both results of T.M.: the one Bošković published in Tab. II. (see Table 9) and the corrected T.M. in Tab. II. (see Table 23). Both values of T.M. reproduce the solar rotation period twice.
The present work T.M., and the present work sunspot positions lon.t and lat.B.t in Tab. II. (see Table 23 Table 9). We assume that the results are different because Bošković used the wrong table in an astronomic almanac for the correction of the true solar time to the mean solar time.
For the reproduction of the solar rotation elements determination, we used Bošković's original values from Tab. II. (see Table 9). In that way, we used the same input data as Bošković did and we can compare the results.

The mean solar time T.M. and solar rotation periods
Bošković used the values for mean solar time of T.M. from Tab. II. (see Table 9), which are not equal to the values we determined in the present work. We deter-
Modern statistics can eliminate from the results those values that deviate more from the predetermined value. Bošković invented his own L1 fitting method that considers absolute values of differences from arithmetic   1, page 103).

Bošković's and the present work results
The results we reproduced using the original formulas are very similar to the values that Bošković published in 1785. We can conclude that we successfully reproduced Bošković's example (Boscovich, 1785, pages 166-169) in this present work and presented it in Table 38.

Discussion
The first results of the present work reproduced solar rotation elements using input data T.M. and sunspot positions lon.t. and lat.B.t that Bošković determined in Tab. II. (see Table 9) (Boscovich, 1785, Tab. II., page 167). The translation of the old-French text revealed a missing link between the observation data and Tab. I. (see Table 8) that we had in the beginning of the research.
Ruđer Bošković presented a detailed report of all the steps for obtaining the results: the solar equator inclination i, the longitude of the node Ω and period of solar rotation T using logarithmic tables, the astronomical almanac Connoisance des Temps (URL 1) and his Opuscule II (Boscovich, 1785, №3, pages 76-77). In September 1777, during a period of six days, in order for the observations of one sunspot to begin, the first step was the determination of the time, and the sunspot position, the longitude and the latitude of the sunspot (Boscovich, 1785, Tab. II., page 167). There are many issues resolved during the research: the issues of time, observation input data control, the longitude of the node, apparent diameter of the Sun, the inclination of the ecliptic ε, and positions and mean solar time of the sunspot, missing formulas in some steps.

Time
The time issue should take into account historical epoch, 18 th century, when Bošković made the observations and his example. At that time, he used a pendulum and a telescope with a micrometre for precise angle measurements. All the observations were made at about 3 p.m., that means after upper solar culmination, the true solar
On September 19 th , 1777 the difference between series 3 and 4 was negative ∆ 4 =-12''. That cannot be real. The first time t 1 is near 2 h 42'00'' and the last time t 2 is near 2 h 43'00'', and we could presume that the times in the third series could be one minute less (

The longitude of the node N=Ω
Bošković discussed the differences of the 8 values arithmetic mean of long.D in Tab. IV. (see Table 11). He identified the differences of the 4 th and the 6 th values that are too far from the others, more than 2°. The arithmetic mean of six other values D=11 s 10°21', and new differences were less than 2° (see Table 37). The final longitude of the node Ω=N=2 s 10°21'=70°21'=N. Furthermore, Bošković added long.D pair 3&4 with values 21°17' and 18°04'. The new total sum is of 131°45', divided by 10 and the arithmetic mean is D=11 s 13°09'. The longitude of the node using 10 values is N=2 s 13°09'=73°09'. He concluded that the result is very near to the longitude of the node through three points N=2 s 14°03' in Tab. XII. (see Table 19) (Boscovich, 1785, №115, page 137).     Table 9) and the reproduction in present work Tab. II. (see Table 23) have differences, presented in Table 41. The differences are not significant: -3 m <∆T.M.<3 m , -5'<∆lon.t<6' and 0'<∆lat.B.t<2'. The periods T' and T'' derived from T.M. in Table 36 are almost the same. For lon.t and lat.B.t, we have not derived results yet, but the differences are not substantial, so we suggest further research. In this present work, we determined the positions using the corrected input data discussed in 4.2. Observation input data control. For the reproduction of the results in this present work, we used Bošković's original results in Tab. II. (see Table 9). That way, we can compare the solar rotation elements in Bošković's example and in the present work.

Conclusions
The most time-consuming part of this research involves discovering "the calculation chains" for each computational process. In the beginning, many elements of the chains were missed. Later, the gaps were identified and filled, and now we have the whole chains for every part of the calculation. The most challenging part of the research was discovering the parts where the original formulas were missing. We reconstructed these formulas using Bošković's results integrated in spreadsheets for calculations. The results are presented here and later critically discussed.
Bošković determined the solar rotation elements using his own observations of one sunspot over a period of six days in September 1777. He determined the mean solar time T.M., and six positons of the sunspot, the longitude and the latitude, its ecliptic coordinates lon.t and lat.B.t in the Tab. II. (see Table 9).
We reproduced the solar equator inclination i, the longitude of the node Ω, and the period of the solar rotation T with Bošković's original formulas. In the present work, the results for the one sunspot observed over a period of six days are given. We successfully reproduced the whole original work (Boscovich, 1785, pages 166-169) resulting in very similar results in this present work.
Ruđer Bošković determined the mean solar time of T.M. and the geocentric positions of one sunspot, and then the ecliptic coordinates based on observations of the trajectory on the solar disk over 6 days in Tab. I. and Tab. II. Based on the mean solar time and ecliptic coordinates of the sunspot trajectory in six days of observation (see Table 9), Bošković determined the elements of solar rotation with his own methods: longitude of the node Ω, inclination of the solar equator towards the ecliptic i, and the period of solar rotation sidereal T' and synodic T''. Bošković determined the longitude of the node Ω on the basis of two methods: 1. the method using two positions of the same sunspot (2.1. The method for Ω) with 6, 8 and 10 pairs: Ω 6 =70°21' (6 pairs), Ω 8 =71°32' (8 pairs), Ω 10 =73°09' (10 pairs), and 2. the method based on three positions of one sunspot (2.4. The method for i and Ω) Ω 136 =74°03' (positions 1, 3 and 6). The inclination of the ecliptic i was determined by: 1. the method based on two positions of the same sunspot and the known longitude of the node (2.2. The method for i) based on five pairs i=7°44', and 2. the method based on three positions of one sunspot (2.4. The method for i and Ω) i=6°49'. The rotation period was determined by method 2.3. The method for the period from 6 pairs of spots sidereal T'=26.77 days and synodic T''=28.89 days (3.3. Bošković's results).
We reproduced the same example in the same way with Bošković's methods: mean solar time T.M. and the geocentric coordinates of the sunspot and then the ecliptic coordinates of the sunspot. We reproduced the elements of solar rotation with the original Bošković mean solar times and ecliptic coordinates so that we could compare the numerical values in Bošković's example, since our mean solar time and ecliptic sunspot positions are slightly different from Bošković's results (see Table  41). Using Bošković's methods with the same mean solar times and ecliptic coordinates of one and the same sunspot, we determined the longitude of the node Ω on the basis of two methods: 1. method using two positions of the same sunspot (2.1. The method for Ω) with 6, 8 and 10 pairs: Ω 6 =70°21' (6 pairs), Ω 8 =71°32' (8 pairs), Ω 10 =73°12' (10 pairs), and 2. the method for three positions of one sunspot (2.4. The method for i and Ω) Ω 136 =74°03' (positions 1, 3 and 6). The inclination of the ecliptic i was determined by: 1. the method based on two positions of the same sunspot and the known longitude of the node (2.2. The method for i) based on five pairs i=7°45', and 2. the method based on three positions of one sunspot (2.4. The method for i and Ω) i=6°48'. The rotation period was determined by method 2.3. The method for the period of 6 pairs of sunspots sidereal T'=26.77 days and synodic T''=28.89 days. Periods of solar rotation were also determined based on our mean solar times T.M. (see Table 34) and determined almost identical values of T'=26.76 days and synodic T''=28.87 days of the period of solar rotation (3.4. Present work results).