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Lessons from Reading Clavius Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Lessons from reading Clavius Anders O. F. Hendrickson Concordia College Moorhead, MN MathFest, Pittsburgh August 5, 2010 Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Outline 1 Christopher Clavius, S.J. 2 Calendar Reform 3 Lessons from Clavius 4 Conclusion Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Christopher Clavius, S.J. (1538–1612) Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Clavius’s life Born in Bamberg c. 1538 1555 received into the Society of Jesus by St. Ignatius Loyola 1556–1560 studied philosophy at Coimbra 1561–1566 studied theology at the Collegio Romano 1567–1612 professor of mathematics at Collegio Romano 1570 published Commentary on the Sphere of Sacrobosco 1574 published edition of Euclid’s Elements c. 1572–1582 on papal calendar commission c. 1595 retired from teaching, focused on research 1612 died in Rome Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Clavius as teacher As a teacher, Clavius Taught elementary (required) courses in astronomy Led a seminar for advanced students Fought for status of mathematics in the curriculum Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Calendar Reform: Solar How to keep the calendar in synch with solar year (365.24237 days, equinox to equinox): Julian calendar: 365.25 days Gregorian calendar: omit 3 leap days every 400 years; hence 365.2425 days Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Calendar Reform: Lunar Easter is the first Sunday after the first full moon on or after the vernal equinox. How to forecast phases of the moon to find Easter Old solution, Metonic cycle, no longer matched the actual moon’s phases New solution: a complicated scheme to adjust the Metonic cycle involving Golden numbers Epacts Tables, tables, and more tables Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion History of the reform 1570’s: Gregory XIII convenes a calendar commission, including Clavius 1582 papal bull Inter Gravissimas reforms the calendar IG 9 explains changes to the solar calendar IG 10 says the lunar calendar is being changed, but doesn’t explain the details; refers the reader to an explicatio 1582 Clavius publishes six “canons” (34 pp.) explaining the lunar calendar (up to A.D. 4999); refers to explicatio for details 1603 Clavius finally publishes that 600-page explicatio Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Our goal We know Clavius taught introductory courses. We know the six canons of 1582 were written for non-mathematicians (a pedagogical text). Our Goal In reading the six canons, look for glimpses into Clavius’s classroom. Caveat This is just a reading by an interested mathematician. All translations are my own. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Our goal We know Clavius taught introductory courses. We know the six canons of 1582 were written for non-mathematicians (a pedagogical text). Our Goal In reading the six canons, look for glimpses into Clavius’s classroom. Caveat This is just a reading by an interested mathematician. All translations are my own. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Our goal We know Clavius taught introductory courses. We know the six canons of 1582 were written for non-mathematicians (a pedagogical text). Our Goal In reading the six canons, look for glimpses into Clavius’s classroom. Caveat This is just a reading by an interested mathematician. All translations are my own. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Our goal We know Clavius taught introductory courses. We know the six canons of 1582 were written for non-mathematicians (a pedagogical text). Our Goal In reading the six canons, look for glimpses into Clavius’s classroom. Caveat This is just a reading by an interested mathematician. All translations are my own. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Use of examples We shall make this clear by means of examples. To the year 1582 after the correction corresponds the capital letter D in the table of the equation, and the Golden number is then 6. Again, in the year 1583 (already corrected) the Golden number is 7, and to it in the table of the equation corresponds the same capital letter D. Next, to the year 4218 in the table of the equation corresponds the letter l, and the Golden number is 1. Moreover to the year 1710 corresponds the capital letter C in the table of the equation, and the Golden number is again 1. Again, to the year 1912 corresponds the capital letter B in the table of the equation, and the Golden number is 13. Wherefore . The capital letter C corresponds also to the year 1715 in the table of the equation, and the Golden number is 6. Finally, to the year 1916 corresponds the capital letter B . (pp. 22–23) Observation 1 Clavius illustrates every step of his algorithm with many examples. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Perpetual table of the cycle of Epacts P l C c P F f s M i A a m D d ∗ 11 22 3 14 xxv/25 6 17 28 9 20 1 12 23 4 q G g t N k B b n E e r H h u 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 Table of the equation of the perpetual cycle of Epacts Year of the Lord Year of the Lord Year of the Lord N 1 A 2200 q 3600 Leap year P 320 Leap year u 2300 p 3700 P 500 Leap year A 2400 Leap year n 3800 a 800 Leap year u 2500 n 3900 b 1100 Leap year t 2600 n 4000 Leap year c 1400 Leap year t 2700 m 4100 10 days subtracted t 2800 Leap year l 4200 D 1582 s 2900 l 4300 D 1600 Leap year s 3000 l 4400 Leap year C 1700 r 3100 k 4500 C 1800 r 3200 Leap year k 4600 B 1900 r 3300 i 4700 B 2000 Leap year q 3400 i 4800 Leap year B 2100 p 3500 i 4900 We shall make this clear by means of examples. To the year 1582 after the correction corresponds the capital letter D in the table of the equation, and the Golden number is then 6. If therefore in the perpetual table of the cycle of Epacts you assign the Golden number 1 to the cell of the lower-case letter a, which is the third [to the left] from the cell of the capital letter D, and the Golden number 2 to the following cell to the right, and so on, the Golden number 6 of the proposed year 1582 will fall in the cell of Epact 26, which will show the New Moons in the Calendar from the Ides of October of that year. (pp. 22–23) Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Perpetual table of the cycle of Epacts P l C c P F f s M i A a m D d ∗ 11 22 3 14 xxv/25 6 17 28 9 20 1 12 23 4 q G g t N k B b n E e r H h u 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 Table of the equation of the perpetual cycle of Epacts Year of the Lord Year of the Lord Year of the Lord N 1 A 2200 q 3600 Leap year P 320 Leap year u 2300 p 3700 P 500 Leap year A 2400 Leap year n 3800 a 800 Leap year u 2500 n 3900 b 1100 Leap year t 2600 n 4000 Leap year c 1400 Leap year t 2700 m 4100 10 days subtracted t 2800 Leap year l 4200 D 1582 s 2900 l 4300 D 1600 Leap year s 3000 l 4400 Leap year C 1700 r 3100 k 4500 C 1800 r 3200 Leap year k 4600 B 1900 r 3300 i 4700 B 2000 Leap year q 3400 i 4800 Leap year B 2100 p 3500 i 4900 We shall make this clear by means of examples. To the year 1582 after the correction corresponds the capital letter D in the table of the equation, and the Golden number is then 6. If therefore in the perpetual table of the cycle of Epacts you assign the Golden number 1 to the cell of the lower-case letter a, which is the third [to the left] from the cell of the capital letter D, and the Golden number 2 to the following cell to the right, and so on, the Golden number 6 of the proposed year 1582 will fall in the cell of Epact 26, which will show the New Moons in the Calendar from the Ides of October of that year. (pp. 22–23) Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion Perpetual table of the cycle of Epacts P l C c P F f s M i A a m D d ∗ 11 22 3 14 xxv/25 6 17 28 9 20 1 12 23 4 q G g t N k B b n E e r H h u 15 26 7 18 29 10 21 2 13 24 5 16 27 8 19 Table of the equation of the perpetual cycle of Epacts Year of the Lord Year of the Lord Year of the Lord N 1 A 2200 q 3600 Leap year P 320 Leap year u 2300 p 3700 P 500 Leap year A 2400 Leap year n 3800 a 800 Leap year u 2500 n 3900 b 1100 Leap year t 2600 n 4000 Leap year c 1400 Leap year t 2700 m 4100 10 days subtracted t 2800 Leap year l 4200 D 1582 s 2900 l 4300 D 1600 Leap year s 3000 l 4400 Leap year C 1700 r 3100 k 4500 C 1800 r 3200 Leap year k 4600 B 1900 r 3300 i 4700 B 2000 Leap year q 3400 i 4800 Leap year B 2100 p 3500 i 4900 We shall make this clear by means of examples.
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