**Reinher of Paderborn**was master and principal of the Paderborn cathedral school, and canon and dean of the Paderborn cathedral. This is supported by fifteen documents testified by him and by copies of his treatise

*Computus emendatus*Ⓣ.

In 1171 Reinher published his treatise

*Computus emendatus*Ⓣ, an improved calculation method for calculating the date of Easter. In the 12

^{th}century, it was obvious that the new moon in the sky appeared several days earlier than it was recorded in the calendar, represented by the church. The church followed the Easter calendar tables, developed for the years 532-626 by the abbot Dionysius Exiguus (died c. 540) on instruction by Pope John I (523-526). As Easter and the movable feasts stand in a special relation to the spring full moon, there existed a particular difficulty in calculating the date of Easter to match the solar year of about 365 days and the lunar year of about 354 days. There is no simple relation between the length of these two types of years hence they were to be calculated as exactly as possible.

Reinher was very interested in raising the prestige of the church in the field of calendrical calculations.

Because of his theological interest he looked at the evident error of the calendar caused by a misunderstood lunar cycle; the error in the solar cycle he had set aside. An innovation for this period was his introduction to the Jewish calendar knowledge, based on this knowledge he analyzed the mistakes of the previous calculation of the lunar year and he described in detail the steps for a correct approach. Reinher compared the lunar cycles of the Hebrews and of Dionysius Exiguus. Both assumed that 235 lunar cycles amounted to 19 years. Reinher used a lunar orbit of the Hebrews consisting of 29 days, 12 hours and 793 parts, each hour corresponds to 1080 parts. He calculated the lunar orbit of Dionysius by dividing every 19 years by 235; and he obtained 29 days, 12 hours and 174 parts, each hour corresponds to 235 parts. Consequently, in 315 years the two calculations differed by more than one day.Qui fidem catholicam impugnant, gaudent, quod errorem inveniunt in compoto, quem tenet ecclesia. Putant enim et affirmant etiam in aliis nos errare.

(The fighters against the Catholic faith are pleased that they find an error in Computus, which the church represents. They believe and assure that we are also wrong in other respects.)

In the western world Reinher was the first who has used the Indian number system instead of the Roman numeral system for mathematical and astronomical work - because of the advantage of writing and arithmetic.

The scientific achievements of Reinher were extremely advanced for his time. Other western sources reported the Indian numerals and the Algorismus, such as theIn designatione numerorum, figuris plerumque utimur aliis quam latinis, propter scribendi et computandi compendium.

(When naming the numbers we use mostly of the other characters than Latin because of the advantage of writing and arithmetic.)

*Salzburger Computus*(1143), the note-book of Hugo von Lerchenfeld or the annals of Regensburg (1174-1197), the

*Computus*of Master Chonrad (1200, in a revision of 1396), manuscripts (late 12

^{th}century) of the monastery Prüfening and the monastery Salem (liber algorizmi), the

*Massa compoti*of Alexandre de Villedieu (c. 1200) or the arithmetic book

*Liber abaci*of Leonardo Fibonacci (1202). But no one before Reinher used the new number system in a scientific treatise.

Reinher did not use the Indian numerals as new typefaces for the Roman numerals, such as Gerbert of Aurillac, later Pope Sylvester II, had done with the apices. He understood the new idea of the Indian number system, and earlier than many others of his time he recognized how the use of Indian numerals simplified the arithmetic. Reinher made his calculations with our familiar decimal system. But his mathematical vocabulary was that of Boethius (480-524) and not that of Plato Tiburtinus (c. 1134), as van Wijk [1, p. 73] (he used coacervare for 'to add') and Honselmann [2, p. 123] (he used coacervare and aufferre for 'to add' and 'to subtract') claimed. Since both were not yet able to scan a text, they were unaware that Reinher had used more than once addere and subtrahere, e.g. addiderimus (lib. 2.4) and subtraxerimus (lib. 2.15).

Reinher reported (lib. 1.19) about the special planetary alignment in the night of 13 September 1170, when Mars and Jupiter for a time appeared as only one star, a conjunction in the zodiac sign of Gemini, which is proved by the tables of Schoch (1873-1929) for the 13./14.9.1170. Incidentally, the same sample year (4930) also is used by Moses Maimonides (1138-1204) in his book of recognition (Mishneh Torah).

The calendar was adapted to the solar year and to the lunar year by the Gregorian calendar reform (1582). With regard to the lunar year the mistake was that 235 synodic months by 1h28m15s were shorter than 19 Julian years. This caused an error of 1 day in 310.028329 years. How exactly the 19-year lunar cycle was calculated by Reinher, shows a comparison with the values determined by him. Reinher calculated the 19-year lunar cycle - according to the Hebrew calendar - with 6939.68962 days. According to him, the mistake of Dionysius Exiguus is 1 day in 314.683706 years.

By comparing the tables of Reinher with his own *New Tables for the Reduction of Jewish Dates*, Van Wijk found a deviation of 6 hours, but he said approvingly [12]:-

The scriptUnder this restriction it must be said that the tables of Reinher are admirably designed and calculated with a lot of care.

*Computus emendatus*Ⓣ of Reinher of Paderborn is preserved in five manuscripts. They are named according to their storage locations short HANN (Niedersächsische Landesbibliothek Hannover, IV, 373, 3v-10v, 12

^{th}century), LEID (Leiden, B. der Rijksuniversiteit, BPL 191 E, 129r-140r, 12

^{th}century), PRAG (Praha, Státni knihovna, XIII.F.8 (2346), 105r-127, 13/14

^{th}century), VATI (Vatican City, B. Apostolica Vaticana, Vatlat. 3124, 33ra-43ra, 13/14

^{th}century), ADMO (Stiftsbibliothek Admont, Austria, Cod. 442 [HMML Pr. No. 9503] - Bl. 38ra-51va., 14

^{th}century). LEID, ADMO, PRAG have been known for some time, HANN and VATI were rediscovered by Herold [5]. The copies HANN and LEID begin with a praise for Reinher.

For the former esteem of the manuscripts of Reinher it is worth mentioning that the enthusiasts - of broader collections of manuscripts - apparently had an interest to have Reinher's text bound together with other valuable texts, such asIncipit praefatio magistri Reinheri decani Patherbornensis, perspicacissimi calculatoris, in compotum emendatum.

(Preface of Master Reinher, dean of Paderborn, the most ingenious mathematician, for improved Computus.)

*Massa compoti*of Alexandre de Villedieu and

*De anni ratione*of Johannes de Sacrobosco.

In the 15^{th} century the manuscript of Reinher is proven to have been known, although we do not know the handwriting used. At the Council of Basel 1431-1449, the treatise of Reinher was in many ways helpful. Cusanus and the Cistercian monk Hermann Zoestius of the convent of Marienfeld belonged to the calendar committee and pushed for a calendar reform. Both quoted in their tracts from the work of Reinher. Zoestius described Reinher as author, but Cusanus made Albertus Magnus the author.

In 1951 van Wijk [12, p. 5] rediscovered up the work of Reinher:-... quae, cum obvient praeceptis legalibus, intentioni patrum et scandalum in fide parant, quia dicit Albertus, quod inimici fidei et de hoc gloriantur, quoniam ex illo errore nosin aliis pariformiter errare subsannant.

(... since this is contrary to the law and to the intent of the fathers, it also served as a nuisance in the faith, because Albertus says that the enemies of faith also rejoice themselves about it, because they mock on the basis of this error that we would be in the same way being wrong in other things.)

Honselmann [9, p. 126] remarked that:-It not only possesses irresistible charm, which is for all time the nature of a well-planned and well written scientific treatise, rather it is the first work about calendar knowledge that uses modern figures and it is the first Western source, which gives information about the modern computus of the Jews.

Incidentally, the later mathematician Karl Theodor Wilhelm Weierstrass (1815-1897) was a student of the Akademisches Gymnasium (now Theodorianum), which in its tradition goes back to the cathedral school of 799.... the former Cathedral School in Paderborn may be regarded as one of the most advanced of its time with respect to the mathematical teaching because no other can make it probable that there has been already calculated with Arabic numerals previously.

Borst [2, p. 83] gave the *Computus emendatus* Ⓣ of Reinher the rating:-

... of an equally moving as forgotten brilliant achievement of medieval science.

**Article by:** Werner Herold, University of Paderborn.

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