Understanding Solar Activity after the Maunder Minimum: Sunspot Records by Rost and Alischer

The Maunder Minimum was a period with significantly reduced solar activity between 1645 and 1715, approximately. The transition between the low solar activity in the Maunder Minimum and the subsequent “normal” regime of solar activity was gradual. However, there are discrepancies in the solar activity level from sunspot number indices and solar activity proxies in that period. Among the contemporaneous observers, Johann L. Rost and Sebastian Alischer were two key sunspot observers to understand the solar activity in this transition just after the Maunder Minimum. We have revised all their sunspot records, counting the number of groups and individual sunspots to derive reliable data for the solar activity level for the period 1716–1726. We found significant misinterpretations of the sunspot group counting assigned to these astronomers in the existing group number databases. Our new group sunspot counting significantly reduces the number of groups for Rost and Alischer’s observations compared to entries in existing databases. Furthermore, our sunspot number estimates (obtained from the active day fraction methodology) of the maximum amplitude of Solar Cycles −3 and −4 are significantly lower than the amplitudes according to the official sunspot number, but they are compatible with sunspot number values obtained from solar activity proxies such as radioisotopes. Our result would imply that solar activity after the Maunder Minimum recovered more gradually and with a lower intensity than previously considered.


Introduction
The earliest sunspot observations were made with the naked eye mainly in the East Asian civilizations for millennia (Wittmann & Xu 1987;Yau & Stephenson 1988;Hayakawa et al. 2017).Some hundreds of these historical observations have been available since then (Wittmann & Xu 1987;Yau & Stephenson 1988;Vaquero et al. 2002).Systematic sunspot observations have been carried out only since the beginning of the 17th century with the use of telescopes as astronomical instruments (Vaquero & Vázquez 2009;Arlt & Vaquero 2020).Thus, the first known telescopic sunspot record was made by Harriot in 1610 December, and then, other records were made by Scheiner, Galileo, Malapert, Mögling, and others (Herr 1978;Neuhäuser & Neuhäuser 2016;Carrasco et al. 2019aCarrasco et al. , 2020Carrasco et al. , 2022b;;Vokhmyanin et al. 2020Vokhmyanin et al. , 2021;;Hayakawa et al. 2021a).
Many of these first sunspot observations were collected by Rudolf Wolf, director of the Zurich Observatory, in the mid-19th century (Clette et al. 2014).Wolf used these records to create the relative sunspot number (with data from 1700), which is defined as R z = k • (10 • g + s), where k is a calibration factor between the reference and a secondary observer, g is the number of sunspot groups, and s is the number of individual spots.This series was maintained by the Zurich Observatory until 1980 when the Royal Observatory of Belgium took this responsibility (Clette et al. 2007;Stenflo 2016).More recently, Hoyt & Schatten (1998) carried out another great compilation of sunspot groups to define the group sunspot number: R g = 12.08•g.Unlike the relative sunspot number, the group sunspot number includes values from 1610 taking the first sunspot records into account.New versions of the sunspot number and group number series have been recently published (Clette & Lefèvre 2016;Svalgaard & Schatten 2016;Usoskin et al. 2016;Chatzistergos et al. 2017;Bhattacharya et al. 2024).
Within the telescopic era, we witnessed a grand solar minimum, "the Maunder Minimum" (Eddy 1976;Usoskin et al. 2015), that is, an important period for the understanding of solar dynamo, long-term solar activity, and the solar influence on Earth (Charbonneau 2020;Chatzistergos et al. 2023;Karak 2023;Usoskin 2023;Chatzistergos 2024).This is the only grand minimum period recorded in the telescopic era (Usoskin 2023).It has some particular characteristics with respect to other periods of solar activity.The solar activity level in this period (1645-1715) was abnormally low (Carrasco et al. 2015(Carrasco et al. , 2021(Carrasco et al. , 2022a;;Usoskin et al. 2015;Vaquero et al. 2015b;Hayakawa et al. 2021b).Some of the other particularities of this period were, for example, a strong hemispheric asymmetry since sunspots were observed mainly in the southern hemisphere (Ribes & Nesme-Ribes 1993;Hayakawa et al. 2021c) and an apparent loss of coronal streamers (Hayakawa et al. 2021d).
A gradual transition of solar activity is seen between the low solar activity level in the Maunder Minimum and the "normal" solar cycles after it (Svalgaard & Schatten 2016;Hayakawa et al. 2021e, 2022.However, there is an evident discrepancy between the solar activity level obtained from indices based on direct solar observations, such as the SILSO10 sunspot number (source: WDC-SILSO, Royal Observatory of Belgium, Brussels) and the group number by Svalgaard & Schatten (2016), and those obtained from proxies such as 14 C cosmogenic radioisotope concentrations measured in ancient trees (Usoskin et al. 2021) and number of aurora nights (Rethly & Berkes 1963;Silverman 1992;Riley et al. 2015;Vaquero & Trigo 2015).Solar activity in Solar Cycles −3 and −4 was similar (even slightly higher) to that in the Dalton Minimum (Hayakawa et al. 2020;Carrasco 2021) according to sunspot number and group number (maximum amplitudes around 100 and 5 for annual sunspot number and group number, respectively).Instead, solar activity from radioisotopes and auroras shows an activity level significantly lower and, in addition, a more gradual transition with an increase in solar activity from Solar Cycle −4 to the mid-18th century.Furthermore, a possible gradual transition in solar activity at the onset of the Maunder Minimum was suggested by Vaquero et al. (2011) by reassessing sunspot records over the two cycles preceding the Maunder Minimum (Carrasco et al. 2019b(Carrasco et al. , 2022b)).
We highlight that the observational coverage of the sunspot observations throughout the 18th century is generally low (Vaquero et al. 2016;Clette et al. 2023).In particular, there are no sunspot records available in the databases in less than half of the days for the study period in this work (1716)(1717)(1718)(1719)(1720)(1721)(1722)(1723)(1724)(1725)(1726).Furthermore, it is significantly striking that the group number databases (Hoyt & Schatten 1998;Vaquero et al. 2016) assign daily numbers of groups higher than 30 to the German astronomer Johann Leonhard Rost (1688Rost ( -1727) ) in his sunspot observations made just after the Maunder Minimum.Some historical notes about him can be found in Gaab & Simons (2010).A priori, those values seem an evident misinterpretation of the sunspot records made by Rost. Hayakawa et al. (2021e) statistically compared their analyses on Johann Christoph Müller's sunspot drawings with Rost's records in the existing databases, suggesting a probable confusion of individual spot counts with group counts in the database.They called for philological analyses of Rost's original records.Thus, this work aims at revisiting Rost's original records and revising the real number of sunspot groups in Rost's records.Moreover, we have also revised Alischer's sunspot records, as they were included in the same documentary sources that include Rost's sunspot records.Thus, we have reexamined Rost and Alischer's sunspot reports in the period 1716-1726 to count the number of sunspot groups and individual sunspots.Given that a significant portion of the available observations during that period were conducted by Rost and Alischer, these key observers are essential for future recalibrations of the sunspot number series (Muñoz-Jaramillo & Vaquero 2019;Clette et al. 2023) enabling a more reliable characterization of the solar activity level after the Maunder Minimum.
The outline of this article is as follows.Section 2 includes a description of the documentary sources consulted in this work.The new sunspot counting is shown in Section 3, in addition to a comparison with the current counting included in the group number databases.A comparison between the sunspot observations made by Rost and Alischer is made in Section 4. Section 5 includes an estimation of the sunspot number for the period 1716-1726 considering our new sunspot counting and the active day fraction.A comparison between our estimations with different solar activity indices is analyzed in Section 6.Finally, the main conclusions of this work are discussed in Section 7.

Documentary Sources
The sunspot records made by Rost and Alischer published in the documentary sources consulted in this work were initially collected by Wolf in his database (Wolf 1860(Wolf , 1880)).Then, Hoyt & Schatten (1998) consulted those sources and included Rost's and Alischer's sunspot observations in their group sunspot number database.Later, Vaquero et al. (2016) incorporated the counting by Hoyt & Schatten (1998) for Rost and Alischer in their updated group number database.

A Description of the Documentary Sources
Most of the sunspot observations analyzed in this work are published in Sammlung von Natur-und Medicin-wie auch hierzu gehörigen Kunst-und Literatur-Geschichten (SNMKL hereafter).This documentary source includes a collection of observations of different natural phenomena, such as significant rain, cold and heat events that occurred in Europe, as well as publications related to medicine, art, and literature.Several scientists in central Europe, coordinated by Johhan Kanold (a doctor from Breslau, now Wrocław in Poland), were responsible for collecting and publishing the information in the aforementioned source.The publication of this documentary source was biannual.The first volume was published in 1717 (including records made in the second half of that year) and the last one in 1730, which included observations made in 1726 (Hirsch 1882).This documentary source can be consulted online, for example, on the website of the Bayerische Staatsbibliothek Digitale Bibliothek/Münchener Digitalisierungszentrum11 .We note that Hoyt & Schatten (1998) indicated in their metadata that "The supplement to this journal [SNMKL] contains sunspot observations, but we have been unable to locate a copy of it."We were able to locate that supplement and we ascertained that it does not include any additional information on sunspot records.Furthermore, we have consulted the other two sources with records by Rost collected by Wolf: Rost (1718), where there are some isolated observations made by Rost, and Rost (1727), where one sunspot observation made by C. A. Hausen is described.

Sunspot Records by Johann Leonard Rost
We have consulted Rost's sunspot observations in SNMKL and Rost (1718).Most of them are included in the first documentary source since it spans from 1718 December to 1726 March.Rost's records in this journal consist of annotations in German with no sunspot drawings.
The second source includes records of seven isolated observation days made by Rost between 1716 and 1718.These records consist of German notes describing the sunspot observations.Moreover, they are accompanied by three sunspot drawings for the observations corresponding to 1717 September 10-11 and November 8.As an example, Figure 1 shows Rost's notes of the sunspot observation on 1717 November 8 (Rost 1718, p. 380), and the drawing made by Rost for that date is shown in Figure 2 (left panel).The translation of the annotations made by Rost on that observation is: "1717 November 8th at 10.45 hr in the morning, with the most beautiful weather.When I was at home, I saw many spots on the Sun through a 5 foot [∼1.5 m] telescope.I drew and described them through an 11 foot [∼3.3 m] telescope such as: At about the east limb of the Sun, there was a nice spot E, which had three others below it, the middle one had a nucleus and a halo or a faint additional circle, with a dark spot above them.Below, there were five small spots G, which I did not see straight away because they seemed quite faint.Spots H were at the meridian, i.e., two spots were at the vertical AB line, of which the lower one was black, but the upper one was as a shadow with a small [spot] Q with a shadow on the right.The same was also observed under spots I and L. K and M were in a straight line, under which I observed two dark [spots].I observed [spot] N as a smoke, which was as two oblong nuclei together with small dark [spot] above them being in the bright and wide cloud.O and P consisted of coal-black, large/round nuclei, of which the one in P was larger than in O, and the cloud around it was very clear/ wide and bright to see.So, I saw 26 spots in the Sun in this observation."We note that the spots K mentioned in the text are actually those that seem to be defined by the letter V in the drawing, just below spot H.This fact may be due to a printing problem.Also, we have considered that one foot is equivalent to ∼0.3 m to calculate the length of the telescopes used by Rost (Noback & Noback 1851, p. 768).
Wolf (1860) noted that Professor Heiss found the collection of sunspot observations by Rost (and Alischer) published in  SNMKL, which was not known by Wolf until then.Therefore, as Hoyt & Schatten (1998) indicated, Heiss was the one who examined the journal, not Wolf. Wolf (1860) indicated that the observations made by Rost were carried out during the period 1718 December 22-1720 November 29.However, more sunspot records made by Rost than those indicated by Wolf are included in SNMKL.Most of these records not mentioned by Wolf are included in the databases by Hoyt & Schatten (1998) and Vaquero et al. (2016).Hayakawa et al. (2021c) explained that Rost used the instruments of the Eimmart Observatory.The telescope used by Rost to observe sunspots and eclipses had at least 2 feet (∼60 cm) in length with a good optic system.From the translation of the annotation shown in Figure 1, we can deduce that Rost used telescopes of 5 and 11 foot in length to observe sunspots.Unfortunately, the aperture of the telescope is unknown.More details of the observation method by Rost can be consulted in Hayakawa et al. (2021c).

Sunspot Records by Sebastian Alischer
Sebastian Alischer recorded his sunspot observations in a diary called Diaria macularum solarium according to the metadata by Hoyt & Schatten (1998).Alischer's notes on his observations were included in SNMKL.These notes are written in Latin for the period 1719-1721 and are accompanied by sunspot drawings for the years 1720 and 1721, while those for 1725-1726 are in German and no sunspot drawings are available.An example of the sunspot drawings made by Alischer can be seen in Figure 2 (right panel).
According to Wolf (1860), Alischer made sunspot observations between 1720 December 1 and 1721 June 30, and then between 1726 March 12 and May 13.Revisiting the source provided by Wolf (1860), Hoyt & Schatten (1998) extended the number of observations made by Alischer.Vaquero et al. (2016) incorporated into their updated group number database the counting by Hoyt & Schatten (1998).An important information provided by Wolf (1860) is that Alischer used a small telescope to carry out his observations.Unfortunately, we could not locate any information on the diameter or focal length of Alischer's telescope so far.
Alischer's records in 1727 in the group number databases (1727 January 7 and 24 and February 8 and 21) were recently analyzed by Hayakawa et al. (2022).Therefore, these records were not studied in this work.

Some Information on Other Sunspot Observers
We have found some observations made by other sunspot observers of that time described in the said documentary sources.There are general comments from three sunspot observers.Rost (1718, p. 386) indicates that Kirch observed several rotations of the same sunspots between 1684 April 26 and July 17.Wurzelbaur did not see any sunspots for the period 1710 October 29-1713 May 18 (Rost 1718, p. 384) and Liefmann observed spots 3 times within a 10-week period in 1725, for which he noted that sunspots crossed the solar disk in 10-11 days (Kanold 1726).

A New Revision of the Group Counts
We carried out a new sunspot group counting from the sunspot records included in the documentary sources previously described.The daily number of groups from Rost (red), Alischer (blue), Hausen (yellow), and Liefmann (orange) is shown in Figure 3 (top panel according to previous studies and bottom panel according to our recounts in the present work) along with all the sunspot observers included in Vaquero et al. (2016) database for the period 1716-1726.More details on the new sunspot group counting for each observer are provided below.The new group counting made in this work is available as supplementary material and also on the website of the Historical Archive of sunspot Observation12 (Table 1).
We note that since Rost's records were mostly descriptive, we defined a sunspot group when he described a cluster of sunspots.However, as Rost did not always clarify the distances of each sunspot cluster, it is frequently difficult to count sunspot groups, as we are not sure how Rost defined his sunspot clusters-for example, according to the different group types in the Waldmeier classification.Therefore, in many cases, our group counts show an upper limit rather than the actual value.An example of this fact is detailed in the following section.We added a comment in the column "Notes" of the file including the counting made in this work to indicate those dates when the number of groups is unclear.
Our recounting indicates that the highest number of groups for Solar Cycle −3 (the first cycle after the Maunder Minimum) is 12, in contrast to 36, which is included in the Vaquero et al. (2016) database for that cycle.Moreover, the shape of that cycle is now more evident with the rising phase until reaching the maximum number of groups in 1719-1720 (unlike the SILSO sunspot number series where it is set in 1717), the declining phase from the maximum to around 1724 and then a rising phase of a new solar cycle.We acknowledge that these are raw data.Also, we note that another perspective on the solar activity level and date of the maximum amplitude is obtained in Section 5 using the active day fraction method described by Carrasco et al. (2022a).
Rost and Alischer are two important observers, especially in 1719-1721 around the maximum of Solar Cycle −3, because many of their sunspot records were made in days when no more sunspot records were available.Removing Rost's and Alischer's records from Vaquero et al. (2016), the observational coverage would reduce from 45.9% to 31.8% for the period 1716-1726.In particular, the coverage would not change in the years 1716-1717 and 1723-1724, whereas it would decrease in 1718-1722 and 1725-1726.Table 2 lists these statistics.Also, the observational coverage for the period 1716-1726 slightly increases from 45.9% from Vaquero et al. (2016) to 47.1% including our new counting in Vaquero et al. (2016) database.Therefore, these astronomers are key to understanding solar activity just after the Maunder Minimum.

Rost
The group counting assigned to Rost in the existing databases is very striking since the daily number of groups is often above 10 reaching up to 36 groups on 1720 May 29 (Figure 3, top panel).Our analysis detected misinterpretations of the records because, generally, the number of groups assigned to Rost in the group number databases is actually the number of individual sunspots observed by Rost.As an example, Rost indicated on 1720 May 29: "K Then, I counted 36 spots this time" (Kanold 1720, p. 537).In this case, we count 10 groups instead of 36 groups assigned to Rost in the group number database on that date.Confusing individual spots with sunspot groups, as previously, leads to a significant overestimation of the solar activity level from Rost's sunspot observations.After our revision, the highest number of groups recorded by Rost is 12.In addition, the average of the number of groups assigned to Rost in Vaquero et al. (2016) is 6.9, whereas it is 3.5 from our counting.
The method used by Rost to note clusters of spots in his annotations was different from the modern group morphological classifications (McIntosh 1990).Sometimes, Rost counted as two different groups the same bipolar group.An example of this fact can be seen in the groups "M" and "N" recorded by Rost in the drawing in Figure 2 (left panel).Thus, our new group counting from Rost's observations can be considered as an upper limit of the number of groups observed by Rost.This  Note.The table indicates the year, month, and day of the observation; the number of groups and single sunspots recorded by each astronomer; the observer responsible for the observations; the documentary source where the record can be consulted; and some notes (if some important fact related to the observation is needed to be clarified).
(This table is available in its entirety in machine-readable form.) fact should be taken with caution for the reconstruction of solar activity as well as the difficulty to count groups from his notes since generally no distance between them is specified.The new group counting from Rost's records can be seen in red in Figure 3 (bottom panel).
On the other hand, the number of observations assigned to Rost in Vaquero et al. (2016) is 299 (Table 2).After our counting, the number of observations made by Rost is 295 (43 removed and 39 new records added).The records that were removed from our counting were because of misinterpretation of the observation dates included in the existing databases.We highlight that the observations assigned to Rost in 1720 December in the databases were actually made by Alischer.This fact was corrected in our new counting.Furthermore, we cannot obtain the number of groups of the observations corresponding to 1719 January 16, August 17, November 13, and December 27 and 1722 June 5, 16, 18, and 20 because the information provided in the notes is not enough to know it, although we note that Rost observed sunspots in those dates.

Alischer
Alischer sometimes recorded several observations for the same day.In these cases, we have selected for our counting: (1) the observation with the highest number of groups and (2) if the number of groups was the same, then we selected the observation with the highest number of individual spots.
We also found many misinterpretations of the sunspot records assigned to Alischer in the databases.As in the case of Rost, the number of individual sunspots recorded by Alischer was assigned to this observer in the group number databases as the number of groups recorded by him, instead of the real number of groups.An example can be seen in Figure 2 (right panel) where we count four groups and nine individual spots.However, nine groups, instead of nine individual spots, are assigned to Alischer in the group number database for that date.This produces an overestimation of the solar activity level from Alischer's data.For example, the highest number of sunspot groups recorded by Alischer is significantly reduced in our new counting (Figure 3, bottom panel).There are nine groups in Vaquero et al. (2016), whereas there are four groups in our counting.Moreover, the average of the number of groups assigned to Alischer in Vaquero et al. (2016) is 1.7, whereas it is 1.0 from our counting.
The number of observations assigned to Alischer for the study period 1716-1726 in Vaquero et al. (2016) is 520 (Table 2).In our new group counting, the number of observation days recorded by Alischer is 602 (18 removed and 100 new records added).The records that were removed from our counting were mainly because of misinterpretation of the observation dates included in the existing databases.In addition to those observations, we know that Alischer recorded sunspots on 1719 July 26-28, August 17 and September 28-30, and 1726 March 12, although it is not possible to determine the number of groups observed by Alischer on those dates from his records because there is no information on the position of the spots on the solar disk.

Other Sunspot Observers
We found that Wurzelbaur observed one group with two spots on 1719 April 11 and the same number of groups and spots on 1719 October 17.These observations are not included in the group number databases (Hoyt & Schatten 1998;Vaquero et al. 2016).Moreover, the existing group number databases wrongly assigned the observation date by Hausen on 1725 November 26, while the actual date was on 1725 November 16 (two groups and six individual spots).Therefore, that date should be corrected in future versions of the group number database according to our work.Also, Liefmann observed sunspots in 1725 (Kanold 1726), but he did not provide specific dates and number of groups.These were included in the group number databases with incorrect/ uncertain observation dates over 1725 June 1-11 and should be removed.Figure 3 and Table 3 include information on the aforementioned records.

A Comparison between Rost's and Alischer's Sunspot Records
We have carried out a comparison between the number of groups recorded by Rost and Alischer in common observation days.The number of days when both astronomers made sunspot observations is 130, all of them between 1719 (55 We have previously mentioned that Alischer used a small telescope to observe sunspots.Furthermore, Alischer noted on 1721 February 17 that Rost got more accurate observations because of longer and more excellent telescopes (Kanold 1722, p. 167).Therefore, a priori, we expect that the number of groups observed by Rost is, in general, higher than that observed by Alischer considering the same dates.This can indeed be seen in Figure 4.
The average of the daily number of groups recorded by Alischer in those common observation days is 1.3, whereas it is 4.0 according to Rost, that is, a factor around 3 higher.In addition, the number of days when the difference between the number of groups recorded by Rost and Alischer is greater than, for example, 3 and 5 is 53 and 21, respectively.Rost's records always show a greater or equal number of groups than Alischer's.In particular, the number of days when Rost and Alischer recorded the same number of groups was 26 (20% of the common observation days).Of these 26 days when the same number of groups was recorded, 11 days were spotless days.Thus, the difference in the counting from the records by both observers is significant.
We have also calculated the ratio between the Wolf number (WN) calculated from Rost's and Alischer's sunspot observations (using the formula: WN = 10•G + S, where G is the number of groups and S is the number of individual spots) and the number of groups recorded by these observers.The values obtained were 12.0 ± 0.8 in the case of Rost and 11.5 ± 0.9 in the case of Alischer.These values are similar to the ratio obtained by Hoyt et al. (1994), that is, ∼12.This means that the number of single sunspots per group recorded by Rost and Alischer was similar to that from the observers used by Hoyt et al. (1994) to equal the average of the group sunspot number and international sunspot number for the period 1874-1991.

An Estimation of the Sunspot Number for the Period 1716-1726
An active day is considered in this work when, at least, one observer recorded one sunspot on the solar disk in a given day.Then, the active day fraction is calculated as the percentage of active days with respect to the number of observation days for each year.We have estimated the annual sunspot number for the period 1716-1726 from the calculation of the active day fraction using the group number database by Vaquero et al. (2016) including our new counting from Rost's and Alischer's observations.We have followed the methodology proposed by Carrasco et al. (2022a) to estimate sunspot number values from the active day fraction.The best fit found by Carrasco et al. (2022a) for that estimation (only valid for active day fraction values <100%) is with the following third-degree polynomial: where SN is the sunspot number and ADF is the active day fraction.We have discarded the years 1722, 1723, and 1724 for the calculations because of the low number of observation days available for these years, which implies larger uncertainties.Moreover, we were able to incorporate a greater number of observation days than those in the group counting.That is because although it was not possible to obtain the exact number of groups from some Rost's and Alischer's records, the information they carried was sufficient to establish that sunspots were seen in those days (see Section 3 for more details).Figure 5 (top panel) compares the annual values of the Note.Note that the number of observation days only takes dates into account when it is possible to count the number of groups from the records of the astronomers.
sunspot number provided by SILSO to sunspot number estimates obtained from the original Vaquero et al. (2016) database as well as from the latter database including our new counting.The error bars were calculated by supposing that the observation days without records are all active days (upper limit) or spotless days (lower limit).Moreover, Table 4 includes the number of active and spotless days as well as the number of observation days per year according to this work and from the original database by Vaquero et al. (2016).
The sunspot number values obtained in this work are generally similar to those computed from the original Vaquero et al. (2016) database (Figure 5, top panel).In particular, some of our estimations are slightly lower than the ones using the Vaquero et al. (2016) database, that is, in 1719, 1720, and 1721.However, we acknowledge that some of the spotless days recorded only by Alischer could have been active days in case of, for example, Rost had observed on those same dates.As a result, some of our estimations would be underestimated.In fact, both Alischer and Rost recorded 0 groups for only 11 common days (as mentioned in Section 4), whereas Alischer recorded zero groups in 17 observation days when Rost observed sunspots.This means that roughly 60% of the spotless days recorded by Alischer, when Rost also observed on the same observation dates, were actually active days.
We have identified in Vaquero et al. (2016) all the spotless days in dates when only a sunspot observation by Alischer is available: 21 days in 1719, 60 days in 1720, and 83 days in 1721.Following the previous analysis, we have considered that 60% of those days were active to match Rost's observations.Thus, 13 of the 21 spotless days recorded only by Alischer in 1719 are considered active days, as well as 36 of 60 days in  1720 and 50 of 83 days in 1721.Then, we recalculated the active day fraction for the period 1716-1726 and the sunspot number taking this correction into account in our new counting and following the same methodology previously described.These new estimations of the sunspot number are represented by blue stars in Figure 5 (bottom panel).The sunspot number values using this correction are slightly higher in 1719, 1720, and 1721 than those without applying corrections.However, we acknowledge that these estimates are speculative.Estimations of the sunspot number obtained in our work are significantly lower than the SILSO sunspot number.In particular, the value of the maximum amplitude of Solar Cycle −3 according to our work (∼50) is roughly half of that of the SILSO sunspot number (∼100).This means that Solar Cycle −3 could have been significantly less active than, for example, the cycles of the Dalton Minimum.Our result would imply that the recovery of the low solar activity levels in the Maunder Minimum to "normal" solar activity levels recorded in the 18th century could have been more gradual with less intense activity than what is known so far.
We must also take into account that there are still some suspicious observations included in the group number databases assigned to some of the observers around the Maunder Minimum."Zero" sunspots are assigned to those observations but, generally, their objective was to make astrometric measurements, not sunspot observations (see examples and more details in, e.g., Clette et al. 2014).As an example of this fact for the period 1716-1726, continuous "zeros" are assigned in Vaquero et al. (2016) to Maraldi in 1716, which seems unreal even without looking at the original reports.Then, we applied the "strict model" defined by Vaquero et al. (2015a) in order to remove suspicious observations from our data set.That is, we have removed observations based on general statements and an inactive day is defined when at least two astronomers reported "zero" sunspots independently.We note that one day is also considered a spotless day if: (1) Rost reported no sunspots and (2) Alischer observed no sunspots when Rost did not observe on the same day.
Applying the same methodology to estimate the sunspot number, we obtain that the maximum amplitude of Solar Cycle −3 is in 1717 with a sunspot number value around 90.We highlight that uncertainty is very high in this case since the sunspot number values for the bottom and upper limits are around 15 and 95, respectively.This result would indicate that the solar activity level of Solar Cycle −3 could be similar to or slightly higher than that of the cycles in the Dalton Minimum.

Solar Activity in Solar Cycles −3 and −4
To contextualize how the transition of solar activity was between the end of the Maunder Minimum (Solar Cycle −4) to that after the Maunder Minimum (Solar Cycle −3), we have composed a series including the sunspot number values estimated by Carrasco et al. (2022a) for Solar Cycle −4 (1700-1715) and those estimated in this work for Solar Cycle −3 (1716Cycle −3 ( -1726)).Both works estimate the sunspot number from the active day fraction using the database by Vaquero et al. (2016).The upper and lower limit for the whole period 1700-1726 is calculated by supposing that the days without records in each year were all active or spotless days, respectively (as in Figure 5).Figure 6 depicts this series as well as the version of the series applying a correction to the spotless days recorded by Alischer in 1719-1721.Furthermore, this figure includes the yearly SILSO sunspot number (Clette et al. 2023), the annual sunspot number by Usoskin et al. (2021) obtained from 14 C measurements, and the 11 yr average of the number of Hungarian aurora nights by Rethly & Berkes (1963).Note that Hayakawa et al. (2021f) showed that the number of auroras included in the catalog by Rethly & Berkes (1963) would be an upper limit of the aurora number sighted in Hungary.This is because Rethly & Berkes (1963) explicitly confirmed probable contaminations of different phenomena in their auroral catalog.Thus, we should be slightly cautious about the homogeneity of these Hungarian auroral data sets, as suggested in Stangl & Foelsche (2021).
According to our estimations, Solar Cycle −4 had a maximum sunspot number amplitude between 10 and 15, whereas Solar Cycle −3 presents a maximum amplitude of around 40 with an upper limit of 55.Thus, at least, Solar Cycle −4 should be considered as part of the "Extended Maunder Minimum", as Vaquero & Trigo (2015) defined.Our  Usoskin et al. (2021) in gray downward triangles (1700-1726) and the sunspot number estimations obtained in this work for the period 1700-1726 in green circles and the same but including the correction factor of 60% for the number of active days from Alischer for the period 1719-1721 in blue stars.The 11 yr average of the number of auroral nights (upper limits) recorded by Rethly & Berkes (1963) is depicted in the line with orange squares.Shaded surfaces for our counting provide the uncertainty range by assuming that all days without observations are either spotless or all active, whereas that for Usoskin et al. (2021) represents the limits considering one standard deviation.
estimations are significantly lower than the solar activity level from the yearly SILSO sunspot number where both cycles present maximum amplitudes around 100, that is, 20% greater than the amplitudes of the solar cycles that occurred during the Dalton Minimum and slightly lower than, for example, the maximum amplitude of the previous Solar Cycle 24, which reached a peak of 113 in 2014.We note that if Solar Cycle −4 had a maximum amplitude of around 100, it should not be considered as a cycle of the Maunder Minimum since that solar activity level is too high to be part of a grand minimum period.
Likewise, our estimations are not in agreement with the group number series by Svalgaard & Schatten (2016) who obtained that the maximum amplitude of Solar Cycle −4 is similar to that of the solar cycles in the Dalton Minimum and Solar Cycle −3 had a maximum amplitude significantly higher (about double) than those cycles in the Dalton Minimum.
Our results are compatible with the annual sunspot number values obtained by Usoskin et al. (2021) from 14 C measurements and the auroral data where it can be seen a more gradual recovery of solar activity after the Maunder Minimum with sunspot number values significantly lower than those provided by SILSO.Thus, our estimations using direct solar observations are closer to the sunspot values obtained from solar activity proxies than those calculated from sunspot observations.We highlight that Carrasco et al. (2022a) used all the records included in Vaquero et al. (2016), including corrections by Hayakawa et al. (2021c), to carry out their sunspot number estimates.As previously, we have discarded the suspicious observations for the period 1700-1715 using the "strict model" proposed by Vaquero et al. (2015a).In this case, the sunspot number estimated for the maximum amplitude of Solar Cycle −4 is around 20 in 1705 and 1706 with values of lower and upper limits around 10 and 50, respectively.Although this estimate is larger than that obtained previously without discarding records and the sunspot number by Usoskin et al. (2021), it is also significantly lower than the peak for Solar Cycle −4 according to SILSO.In particular, the maximum amplitude for this cycle would be half or lower than those from SILSO for the Dalton Minimum, approximately.

Conclusions
J.H. Rost and S. Alischer made sunspot observations for the period 1716-1726, just after the Maunder Minimum.These observations were collected by Wolf and also included in the Hoyt & Schatten (1998) database and then in the most up-todate group number database by Vaquero et al. (2016).We note that the database by Vaquero et al. (2016) is updated with some frequency incorporating the analysis of group number data that are analyzed in the most recent published studies.The present work will be included in future versions of that group number database.
Rost and Alischer recorded most of the sunspot observations available in the databases around the maximum of Solar Cycle −3, the first cycle considered after the Maunder Minimum.For this reason, they are two key observers in understanding how the transition from low to normal solar activity was at the end of the Maunder Minimum.
We have found significant misinterpretations, which are affecting the observation dates and sunspot counting assigned to Rost and Alischer in the existing databases.Then, we revised all their sunspot records and presented a new counting of group numbers and single sunspots.This information will be important for future reconstructions of the sunspot number series because, in addition to an update of the number of sunspot groups recorded by both observers, we provide the number of individual sunspots, which is more infrequent information in the historical sunspot observations.Moreover, we have found observations made by these astronomers not included in the existing group number databases.
Sometimes, it is difficult to extract the number of groups from the notes recorded by the observers.Comparing some sunspot drawings made by Rost with his annotations on the number of groups and sunspots for the dates when the drawings were made, we realize that Rost sometimes counted a bipolar group as two different groups.Therefore, the group counting provided in this work would represent an upper limit of the number of groups observed by Rost.This should be taken into consideration for future recalibrations of the sunspot number series.
Our group counting for Rost and Alischer is significantly lower than that of Vaquero et al. (2016).The maximum number of groups in our counting is a third of that from Vaquero et al. (2016) in the case of Rost (36 versus 12) and roughly half in the case of Alischer (9 versus 4).Moreover, the average group number is also significantly lower in our counting for both Rost (6.9 versus 3.5) and Alischer (1.7 versus 1.0).Furthermore, comparing the sunspot observations made by Rost and Alischer, it can be seen that Rost systematically recorded a greater number of groups than Alischer because Rost used better and bigger telescopes, as Alischer noted.
Following the methodology by Carrasco et al. (2022a), we have estimated annual sunspot number values for the study period (1716-1726) from the active day fraction calculated using the database by Vaquero et al. (2016) but incorporating our new counting.We find a peak in Solar Cycle −3 around 50.Moreover, we have composed a sunspot number series for the period 1700-1726 including data from Carrasco et al. (2022a).The maximum amplitude estimated in the sunspot number for Solar Cycle −4 is around 10 with an upper limit of 15.Moreover, we applied the same methodology but discarded suspicious data from the group number databases using the "strict model" proposed by Vaquero et al. (2015a).In this case, we obtained that the peak in terms of sunspot number of Solar Cycle −4 could be around 20 and that of Solar Cycle −3 around 90.However, there is a large uncertainty in these estimations of the sunspot number due to low observational coverage.The lower and upper limits for Solar Cycles −3 and −4 are 15-95 and 10-50, respectively.
Our estimations are significantly lower than the maximum amplitudes provided by the SILSO sunspot number.Note that maximum amplitudes for Solar Cycles −3 and −4 according to SILSO are around 100.Furthermore, we highlight that the solar activity level of Solar Cycles −3 and −4 according to the SILSO sunspot number is not compatible with a grand minimum period.In addition, our estimations are not in agreement with the group number series by Svalgaard & Schatten (2016) who obtained a maximum amplitude for Solar Cycle −4 similar to that occurred in cycles of the Dalton Minimum and for Solar Cycle −3 significantly higher.
We note there is a discrepancy between the solar activity level over Solar Cycles −3 and −4 obtained from the solar activity levels calculated from sunspot observations and solar activity proxies, the latter showing a more gradual solar activity recovery with lower activity levels than the former.Therefore, our results are compatible with the upper limit of Hungarian auroral data (Rethly & Berkes 1963;Hayakawa et al. 2021f) and the sunspot number values by Usoskin et al. (2021) estimated from 14 C measurements.The main source of the discussed discrepancy between solar activity indices and proxies can be the misinterpretation of the sunspot records, which are included in the databases, and the low temporal coverage in some earlier periods of the telescopic era.
Our sunspot number series would imply that the recovery of the low solar activity levels in the Maunder Minimum to "normal" solar activity levels recorded in the 18th century could have been more gradual and with lower intensity than what is known so far.This result can have important implications for understanding the concept of "intermittency" in solar dynamo models (Charbonneau 2020).This term refers to the existence of periods with strongly suppressed solar activity randomly included within the period of "normal" solar activity.Therefore, it is fundamental to characterize solar activity as reliably as possible before and after a grand solar minimum, such as the Maunder Minimum, to understand the behavior of the solar dynamo at that time of "intermittency" and their transition to "normal" solar activity.
We emphasize the difficulty of obtaining a reliable solar activity level if our calculations are based on databases including some erroneous or uncertain records and a low observational coverage.Therefore, a database including reliable sunspot records is fundamental to characterize solar activity and to better understand its behavior.This work is evidence of the need to continue improving the existing sunspot databases.Furthermore, we need more efforts to find sunspot records not included in the databases, in particular, those made in the 18th century when the observational coverage was extremely low in some periods.
Two fundamental facts characterize solar activity after the Maunder Minimum.On the one hand, hemispheric symmetry was recovered after a long period of active regions in the southern hemisphere of the Sun (Vaquero et al. 2015b).On the other hand, as shown in this study, the recovery of the level of solar activity was very progressive after the end of the Maunder Minimum.Physicists who study the functioning of the Sun using solar dynamo models have these two important characteristics of solar activity to reproduce and explain with their models.

Figure 1 .
Figure 1.Textual sunspot records made by Rost on 1717 November 8 [Source: Rost 1718, p. 380].See Section 2.2 for the translation of this text and Figure 2 for a drawing of the same observation.

Figure 3 .
Figure 3. (Top panel) Daily number of sunspot groups recorded by all the observers included in the sunspot group number database by Vaquero et al. (2016; V16) for the period 1716-1726.The sunspot observations made by Rost, Alischer, Hausen, and Liefmann (only in top panel, see Section 3.3) are depicted in red, blue, yellow, and orange colors, respectively.(Bottom panel) The same but applying our group counting to the sunspot records made by the previous observers.

Figure 4 .
Figure 4. Daily number of sunspot groups recorded by Rost vs. Alischer in common observations days.Different colors represent the frequency that occurred in each combination according to the color bar on the right side of the figure.The diagonal line represents the slope line equal to one.

Figure 5 .
Figure 5.Comparison between the annual sunspot number provided by SILSO (black triangles) and estimations of the sunspot number obtained in this work (green circles) and: (top panel) from Vaquero et al. (2016; orange diamonds) and (bottom panel) from our counting but considering 60% of the spotless days as active days when only a sunspot observation by Alischer is available for the period 1719-1721 (blue stars).Error bars are calculated by supposing that the observation days without records are all active days (upper limit) or spotless days (lower limit).

Figure 6 .
Figure 6.Comparison between annual values from SILSO sunspot number in black upward triangles, from sunspot number byUsoskin et al. (2021) in gray downward triangles (1700-1726) and the sunspot number estimations obtained in this work for the period 1700-1726 in green circles and the same but including the correction factor of 60% for the number of active days from Alischer for the period 1719-1721 in blue stars.The 11 yr average of the number of auroral nights (upper limits) recorded byRethly & Berkes (1963) is depicted in the line with orange squares.Shaded surfaces for our counting provide the uncertainty range by assuming that all days without observations are either spotless or all active, whereas that forUsoskin et al. (2021) represents the limits considering one standard deviation.

Table 1
Some Examples Lines of the Sunspot Counting Made in This Work (Sammlung, 1720, vol.2, p. 56)Clouds did not let him observe properly

Table 2
Numbers of Annual Observation Days by Rost and Alischer According to the Vaquero et al. (2016) Database (V16) Listed in the Second and Fourth Columns, Respectively, and the Same but According to Our Work in the Third and Fifth ColumnsNote.In addition, the number of records removed from V16 because of misinterpretation (r) and the number of new records not included in V16 (n) are indicated in the third and fifth columns in brackets.Yearly observational coverage of V16 taking into account all the records is shown in the sixth column and the same but removing Rost and Alischer's observations from V16 in the seventh column.

Table 3
Summary of the Information on Sunspot Observers Analyzed in this Work for the Period 1716-1726 Compared with Vaquero et al. (2016; V16)

Table 4
Number of Active Days (AD), Spotless Days (SD), and Observation Days (NOBS) per Year for the Period 1716-1726 from Vaquero et al. (2016; V16) and Those Including the New Counting Made in This Work