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. 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. 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. For since this year is not found in the table, the next smaller one is to be sought, namely 1582, to which the capital letter D corresponds. Assigning therefore the Golden number 1 to the cell of the lower-case letter a in the table of Epacts, 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 7 of the proposed year will fall in the cell of the Epact 7, which will show the New Moons 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. 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. For since this year is not found in the table, the next smaller one is to be sought, namely 1582, to which the capital letter D corresponds. Assigning therefore the Golden number 1 to the cell of the lower-case letter a in the table of Epacts, 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 7 of the proposed year will fall in the cell of the Epact 7, which will show the New Moons 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. 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. For since this year is not found in the table, the next smaller one is to be sought, namely 1582, to which the capital letter D corresponds. Assigning therefore the Golden number 1 to the cell of the lower-case letter a in the table of Epacts, 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 7 of the proposed year will fall in the cell of the Epact 7, which will show the New Moons 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. 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. For since this year is not found in the table, the next smaller one is to be sought, namely 1582, to which the capital letter D corresponds. Assigning therefore the Golden number 1 to the cell of the lower-case letter a in the table of Epacts, 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 7 of the proposed year will fall in the cell of the Epact 7, which will show the New Moons 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
We shall make this clear by means of examples. Next, to the year 4218 in the table of the equation corresponds the letter l, and the Golden number is 1. Therefore if in the table of Epacts you assign that year’s Golden number 1 to the cell of the letter u, which is the third to the left from the cell of the letter l, you will find the Epact 19 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
We shall make this clear by means of examples. Moreover to the year 1710 corresponds the capital letter C in the table of the equation, and the Golden number is again 1. Wherefore if you assign that year’s Golden number 1 to the first cell of the capital letter P in the table of Epacts, which is the third from the capital letter C, you will find ∗ for the Epact 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
We shall make this clear by means of examples. Again, to the year 1912 corresponds the capital letter B in the table of the equation, and the Golden number is 13. Wherefore if you assign the Golden number 1 to the cell of the capital letter N in the perpetual table of Epacts, which is the third from the capital letter B, and the Golden number 2 to the following cell to the right, and so on, coming back to the beginning of the table, the proposed year’s Golden number 13 will fall in the second cell. Therefore the Epact will then be 11. (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
We shall make this clear by means of examples. The capital letter C corresponds also to the year 1715 in the table of the equation, and the Golden number is 6. Assigning therefore the Golden number 1 to the cell of the capital letter P in the table of Epacts, which is the third from the cell of the capital letter C, and the Golden number 2 to the following cell to the right, etc., the Golden number 6 of the proposed year will fall in the cell of the letter F, below which are placed two Epacts, xxv and 25, expressed in different scripts. But because the Golden number, 6, is less than 12, the former Epact, xxv, is to be taken for the year 1715. (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
We shall make this clear by means of examples. Finally, to the year 1916 corresponds the capital letter B in the table of the equation, and the Golden number is 17. Wherefore if the Golden number 1 were given to the cell of the letter N in the table of Epacts, which is the third from the cell of the capital letter B, and the Golden number 2 to the following cell, etc., returning to the beginning of the table, the Golden number 17 of the proposed year will fall upon the same cell of the letter F, below which the two Epacts xxv and 25 of different scripts are set. And because the Golden number 17 is greater than 11, the latter Epact, 25, is to be taken for the year 1916. (pp. 22–23) Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Use of examples
Observation 2 The examples are carefully selected to illustrate difficulties, proceeding from the simplest to most complicated, showing only one or two “twists” at a time. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
The matter will be made more clear by examples. Again let it be proposed to find the Golden number for the year 1583. Because this year is not in the table, the next smaller year in the table, 1000, is to be taken, and its Golden number 12. Then the remaining 583 years are to be sought in the table. Since they are not contained in it, again the next smaller year in the table, 500, is to be taken, and its Golden number 6, which having been added to the previously found Golden number 12, the number 18 will be made. After this the 83 years which remain are to be taken in the table, but since they are not found, the next smaller year in the table, 80, is to be taken, and its Golden number 4. Once this is added to the golden number 18 previously formed, the number 22 will be made, from which if 19 are subtracted, 3 remain. Afterwards the 3 remaining years are to be taken in the table, and the Golden number 3 corresponding to them; once they are added to the Golden number 3 most recently found, the number 6 is formed, to which finally if 1 is added, as is prescribed in the head of the table, the Golden number for the year 1583 will be 7. (pp. 17–18)
Observation 3 Clavius walks the reader through a complete example at least once, in complete precision and utter clarity. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
The matter will be made more clear by examples. Again let it be proposed to find the Golden number for the year 1583. Because this year is not in the table, the next smaller year in the table, 1000, is to be taken, and its Golden number 12. Then the remaining 583 years are to be sought in the table. Since they are not contained in it, again the next smaller year in the table, 500, is to be taken, and its Golden number 6, which having been added to the previously found Golden number 12, the number 18 will be made. After this the 83 years which remain are to be taken in the table, but since they are not found, the next smaller year in the table, 80, is to be taken, and its Golden number 4. Once this is added to the golden number 18 previously formed, the number 22 will be made, from which if 19 are subtracted, 3 remain. Afterwards the 3 remaining years are to be taken in the table, and the Golden number 3 corresponding to them; once they are added to the Golden number 3 most recently found, the number 6 is formed, to which finally if 1 is added, as is prescribed in the head of the table, the Golden number for the year 1583 will be 7. Finally let the Golden number of the year 1595 be sought. I take first the Golden number 12, corresponding to the year 1000, and to it I add the Golden number 6 which corresponds to the year 500, and I sum up the number 18. Then I add the Golden number 14 corresponding to the year 90 to the Golden number 18 thus obtained, and I produce the number 32, from which 19 having been subtracted, the number 13 remains, to which I join the Golden number 5 corresponding to the year 5 and I fashion the number 18. To this finally if I will add 1, I will have 19 for the Golden number of the year 1595. (pp. 17–18)
Observation 4 As he repeats similar examples, Clavius speeds up his delivery. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
The matter will be made more clear by examples. Again let it be proposed to find the Golden number for the year 1583. Because this year is not in the table, the next smaller year in the table, 1000, is to be taken, and its Golden number 12. Then the remaining 583 years are to be sought in the table. Since they are not contained in it, again the next smaller year in the table, 500, is to be taken, and its Golden number 6, which having been added to the previously found Golden number 12, the number 18 will be made. After this the 83 years which remain are to be taken in the table, but since they are not found, the next smaller year in the table, 80, is to be taken, and its Golden number 4. Once this is added to the golden number 18 previously formed, the number 22 will be made, from which if 19 are subtracted, 3 remain. Afterwards the 3 remaining years are to be taken in the table, and the Golden number 3 corresponding to them; once they are added to the Golden number 3 most recently found, the number 6 is formed, to which finally if 1 is added, as is prescribed in the head of the table, the Golden number for the year 1583 will be 7. Finally let the Golden number of the year 1595 be sought. I take first the Golden number 12, corresponding to the year 1000, and to it I add the Golden number 6 which corresponds to the year 500, and I sum up the number 18. Then I add the Golden number 14 corresponding to the year 90 to the Golden number 18 thus obtained, and I produce the number 32, from which 19 having been subtracted, the number 13 remains, to which I join the Golden number 5 corresponding to the year 5 and I fashion the number 18. To this finally if I will add 1, I will have 19 for the Golden number of the year 1595. (pp. 17–18)
Observation 4 As he repeats similar examples, Clavius speeds up his delivery. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
The matter will be made more clear by examples. Again let it be proposed to find the Golden number for the year 1583. Because this year is not in the table, the next smaller year in the table, 1000, is to be taken, and its Golden number 12. Then the remaining 583 years are to be sought in the table. Since they are not contained in it, again the next smaller year in the table, 500, is to be taken, and its Golden number 6, which having been added to the previously found Golden number 12, the number 18 will be made. After this the 83 years which remain are to be taken in the table, but since they are not found, the next smaller year in the table, 80, is to be taken, and its Golden number 4. Once this is added to the golden number 18 previously formed, the number 22 will be made, from which if 19 are subtracted, 3 remain. Afterwards the 3 remaining years are to be taken in the table, and the Golden number 3 corresponding to them; once they are added to the Golden number 3 most recently found, the number 6 is formed, to which finally if 1 is added, as is prescribed in the head of the table, the Golden number for the year 1583 will be 7. Finally let the Golden number of the year 1595 be sought. I take first the Golden number 12, corresponding to the year 1000, and to it I add the Golden number 6 which corresponds to the year 500, and I sum up the number 18. Then I add the Golden number 14 corresponding to the year 90 to the Golden number 18 thus obtained, and I produce the number 32, from which 19 having been subtracted, the number 13 remains, to which I join the Golden number 5 corresponding to the year 5 and I fashion the number 18. To this finally if I will add 1, I will have 19 for the Golden number of the year 1595. (pp. 17–18)
Observation 4 As he repeats similar examples, Clavius speeds up his delivery. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
The matter will be made more clear by examples. Again let it be proposed to find the Golden number for the year 1583. Because this year is not in the table, the next smaller year in the table, 1000, is to be taken, and its Golden number 12. Then the remaining 583 years are to be sought in the table. Since they are not contained in it, again the next smaller year in the table, 500, is to be taken, and its Golden number 6, which having been added to the previously found Golden number 12, the number 18 will be made. After this the 83 years which remain are to be taken in the table, but since they are not found, the next smaller year in the table, 80, is to be taken, and its Golden number 4. Once this is added to the golden number 18 previously formed, the number 22 will be made, from which if 19 are subtracted, 3 remain. Afterwards the 3 remaining years are to be taken in the table, and the Golden number 3 corresponding to them; once they are added to the Golden number 3 most recently found, the number 6 is formed, to which finally if 1 is added, as is prescribed in the head of the table, the Golden number for the year 1583 will be 7. Finally let the Golden number of the year 1595 be sought. I take first the Golden number 12, corresponding to the year 1000, and to it I add the Golden number 6 which corresponds to the year 500, and I sum up the number 18. Then I add the Golden number 14 corresponding to the year 90 to the Golden number 18 thus obtained, and I produce the number 32, from which 19 having been subtracted, the number 13 remains, to which I join the Golden number 5 corresponding to the year 5 and I fashion the number 18. To this finally if I will add 1, I will have 19 for the Golden number of the year 1595. (pp. 17–18)
Observation 4 As he repeats similar examples, Clavius speeds up his delivery. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Vocabulary
Clavius’s Vocabulary for Addition
2 + 3 = 5 addo I add 3 to 2. summa The sum is 5. compono I compose 5. conficio I make 5. conflo I forge 5. adicio I throw 3 onto 2. appono I set 3 beside 2. adiungo I join 2 and 3. efficio I fashion 5 from 2 and 3. procreo 2 and 3 beget 5. fio 5 comes to be. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Vocabulary
Clavius’s Vocabulary for Subtraction
6 − 2 = 4 subtraho I pull away 2. rejicio I throw away 2. detractio I pull down 2. residui 4 remain. supersunt 4 are left over. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Vocabulary
Observation 5 Clavius employs a wealth of synonyms to express addition and subtraction, mitigating the mind-numbing effect of so many calculations. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Familiarity
Let it be required to find (inveniendus sit) the number of the Solar cycle for the year 1000. (p. 25)
Thus you see from the year 1000 all the way to the year 10000 the Golden number 12 is always to be added to the preceding Golden number, and 19 is to be subtracted when it can be subtracted, . . . (p. 19)
I take first the number of the Solar cycle 0 from the line of the year 7000, and I add it to the number of the Solar cycle 14 found from the line of the year 70, and I produce the number 14. (p. 26)
Observation 6 Clavius uses the second person, and even the first person singular, to create an informal, conversational tone. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Familiarity
Let it be required to find (inveniendus sit) the number of the Solar cycle for the year 1000. (p. 25)
Thus you see from the year 1000 all the way to the year 10000 the Golden number 12 is always to be added to the preceding Golden number, and 19 is to be subtracted when it can be subtracted, . . . (p. 19)
I take first the number of the Solar cycle 0 from the line of the year 7000, and I add it to the number of the Solar cycle 14 found from the line of the year 70, and I produce the number 14. (p. 26)
Observation 6 Clavius uses the second person, and even the first person singular, to create an informal, conversational tone. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Familiarity
Let it be required to find (inveniendus sit) the number of the Solar cycle for the year 1000. (p. 25)
Thus you see from the year 1000 all the way to the year 10000 the Golden number 12 is always to be added to the preceding Golden number, and 19 is to be subtracted when it can be subtracted, . . . (p. 19)
I take first the number of the Solar cycle 0 from the line of the year 7000, and I add it to the number of the Solar cycle 14 found from the line of the year 70, and I produce the number 14. (p. 26)
Observation 6 Clavius uses the second person, and even the first person singular, to create an informal, conversational tone. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Familiarity
Let it be required to find (inveniendus sit) the number of the Solar cycle for the year 1000. (p. 25)
Thus you see from the year 1000 all the way to the year 10000 the Golden number 12 is always to be added to the preceding Golden number, and 19 is to be subtracted when it can be subtracted, . . . (p. 19)
I take first the number of the Solar cycle 0 from the line of the year 7000, and I add it to the number of the Solar cycle 14 found from the line of the year 70, and I produce the number 14. (p. 26)
Observation 6 Clavius uses the second person, and even the first person singular, to create an informal, conversational tone. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Multiple perspectives
Golden Numbers: Approach #1 Table of the cycle of the Golden number, taking its beginning from the year of correction 1582 6 7 8 9 10 11 12 13 14 15 16 17 18 19 1 2 3 4 5
The first number of the table, which is 6, is given to the year 1582, the second one, which is 7, to the following year 1583, and so on forever, until it comes to the year whose Golden number you seek, returning to the beginning of the table howeversooften you shall have run through it. For the cell, in which the proposed year falls, will give the desired Golden number. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Multiple perspectives
Golden Numbers: Approach #2 BUT since it is very laborious and vexing to count off so many years in the aforesaid table, and to repeat it so often, until one comes to the year whose Golden number is sought, especially indeed if the proposed year is far off from the year 1582, we have constructed this other following table. . . Years Golden Years Golden Years Golden Years Golden of the Number of the Number of the Number of the Number Lord Add 1 Lord Add 1 Lord Add 1 Lord Add 1 1 1 300 15 50000 11 7000000 1 2 2 400 1 60000 17 8000000 12 3 3 500 6 70000 4 9000000 4 4 4 600 11 80000 10 10000000 15 5 5 700 16 90000 16 20000000 11 6 6 800 2 100000 3 30000000 7 7 7 900 7 200000 6 40000000 3 8 8 1000 12 300000 9 50000000 18 9 9 2000 5 400000 12 60000000 14 10 10 3000 17 500000 15 70000000 10 20 1 4000 10 600000 18 80000000 6 30 11 5000 3 700000 2 90000000 2 40 2 6000 15 800000 5 100000000 17 50 12 7000 8 900000 8 200000000 15 60 3 8000 1 1000000 11 300000000 13 70 13 9000 13 2000000 3 400000000 11 80 4 10000 6 3000000 14 500000000 9 90 14 20000 12 4000000 6 600000000 7 100 5 30000 18 5000000 17 700000000 5 200 10 40000 5 6000000 9 800000000 3 Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Multiple perspectives
Golden Numbers: Approach #2 BUT since it is very laborious and vexing to count off so many years in the aforesaid table, and to repeat it so often, until one comes to the year whose Golden number is sought, especially indeed if the proposed year is far off from the year 1582, we have constructed this other following table. . . Years Golden Years Golden Years Golden Years Golden of the Number of the Number of the Number of the Number Lord Add 1 Lord Add 1 Lord Add 1 Lord Add 1 1 1 300 15 50000 11 7000000 1 2 2 400 1 60000 17 8000000 12 3 3 500 6 70000 4 9000000 4 4 4 600 11 80000 10 10000000 15 5 5 700 16 90000 16 20000000 11 6 6 800 2 100000 3 30000000 7 7 7 900 7 200000 6 40000000 3 8 8 1000 12 300000 9 50000000 18 9 9 2000 5 400000 12 60000000 14 10 10 3000 17 500000 15 70000000 10 20 1 4000 10 600000 18 80000000 6 30 11 5000 3 700000 2 90000000 2 40 2 6000 15 800000 5 100000000 17 50 12 7000 8 900000 8 200000000 15 60 3 8000 1 1000000 11 300000000 13 70 13 9000 13 2000000 3 400000000 11 80 4 10000 6 3000000 14 500000000 9 90 14 20000 12 4000000 6 600000000 7 100 5 30000 18 5000000 17 700000000 5 200 10 40000 5 6000000 9 800000000 3 Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Multiple perspectives
Golden Numbers: Approach #2 BUT since it is very laborious and vexing to count off so many years in the aforesaid table, and to repeat it so often, until one comes to the year whose Golden number is sought, especially indeed if the proposed year is far off from the year 1582, we have constructed this other following table. . . Years Golden Years Golden Years Golden Years Golden of the Number of the Number of the Number of the Number Lord Add 1 Lord Add 1 Lord Add 1 Lord Add 1 1 1 300 15 50000 11 7000000 1 2 2 400 1 60000 17 8000000 12 3 3 500 6 70000 4 9000000 4 4 4 600 11 80000 10 10000000 15 5 5 700 16 90000 16 20000000 11 6 6 800 2 100000 3 30000000 7 7 7 900 7 200000 6 40000000 3 8 8 1000 12 300000 9 50000000 18 9 9 2000 5 400000 12 60000000 14 10 10 3000 17 500000 15 70000000 10 20 1 4000 10 600000 18 80000000 6 30 11 5000 3 700000 2 90000000 2 40 2 6000 15 800000 5 100000000 17 50 12 7000 8 900000 8 200000000 15 60 3 8000 1 1000000 11 300000000 13 70 13 9000 13 2000000 3 400000000 11 80 4 10000 6 3000000 14 500000000 9 90 14 20000 12 4000000 6 600000000 7 100 5 30000 18 5000000 17 700000000 5 200 10 40000 5 6000000 9 800000000 3 Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Multiple perspectives
Golden Numbers: Approach #3 By far the easiest way to find the Golden number of whatsoever year is through Arithmetic precepts, in the following way. To the proposed year of the Lord let 1 be added, and let the sum be divided by 19. For the number which remains from the division, will be the Golden number of the proposed year.
Golden number = (year + 1) mod 19 Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Multiple perspectives
Observation 7 Clavius has immense patience with his readers.
Observation 8 Clavius presents multiple ways of doing the same task as an aid to understanding. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Multiple perspectives
Observation 7 Clavius has immense patience with his readers.
Observation 8 Clavius presents multiple ways of doing the same task as an aid to understanding. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
The absurd
The tables for computing Golden numbers and Dominical letters run through the year A.D. 800,000,000. (p. 18)
Observation 9 Clavius may be using “absurd” extreme cases as a way to keep his readers’ interest. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Conclusion
Observations 1 Many, many examples 2 Examples are carefully selected to illustrate difficulties one at a time. 3 Complete precision and utter clarity 4 Gradually speeding up delivery 5 A wealth of synonyms and broad vocabulary 6 First and second person pronouns; informal, conversational tone. 7 Patience! 8 Multiple ways of doing the same task as an aid to understanding 9 “Absurd” extreme cases Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Conclusion
These techniques are not new to us, nor probably to Clavius. Nevertheless it’s fascinating to see one’s own pedagogy in the pages of a 16th century Latin document. Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion
Thank you.