Anders O. F. Hendricksonfaculty.cord.edu/ahendric/presentations/PittsburghClavius.pdf · Gregorian...

Christopher Clavius, S.J. Calendar Reform Lessons from Clavius Conclusion

Lessons from reading Clavius

Anders O. F. Hendrickson

Concordia CollegeMoorhead, MN

MathFest, PittsburghAugust 5, 2010

Outline

1 Christopher Clavius, S.J.

2 Calendar Reform

3 Lessons from Clavius

4 Conclusion

Christopher Clavius, S.J. (1538–1612)

Clavius’s life

Born in Bamberg c. 15381555 received into the Society of Jesus by St. IgnatiusLoyola1556–1560 studied philosophy at Coimbra1561–1566 studied theology at the Collegio Romano1567–1612 professor of mathematics at Collegio Romano1570 published Commentary on the Sphere of Sacrobosco1574 published edition of Euclid’s Elementsc. 1572–1582 on papal calendar commissionc. 1595 retired from teaching, focused on research1612 died in Rome

Clavius as teacher

As a teacher, ClaviusTaught elementary (required) courses in astronomyLed a seminar for advanced studentsFought for status of mathematics in the curriculum

Calendar Reform: Solar

How to keep the calendar in synch with solar year(365.24237 days, equinox to equinox):

Julian calendar: 365.25 daysGregorian calendar: omit 3 leap days every 400 years;hence 365.2425 days

Calendar Reform: Lunar

Easter is the first Sunday after the first full moon on or after thevernal equinox.

How to forecast phases of the moon to find EasterOld solution, Metonic cycle, no longer matched the actualmoon’s phasesNew solution: a complicated scheme to adjust the Metoniccycle involving

Golden numbersEpactsTables, tables, and more tables

History of the reform

1570’s: Gregory XIII convenes a calendar commission,including Clavius1582 papal bull Inter Gravissimas reforms the calendar

IG 9 explains changes to the solar calendarIG 10 says the lunar calendar is being changed, but doesn’texplain 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 details1603 Clavius finally publishes that 600-page explicatio

Our goal

We know Clavius taught introductory courses.We know the six canons of 1582 were written fornon-mathematicians (a pedagogical text).

Our GoalIn reading the six canons, look for glimpses into Clavius’sclassroom.

CaveatThis is just a reading by an interested mathematician.

All translations are my own.

Our goal

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 tableof the equation, and the Golden number is then 6. . . .

Again, in the year 1583 (already corrected) the Golden number is 7, and to it inthe 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, andthe Golden number is 1. . . .

Moreover to the year 1710 corresponds the capital letter C in the table of theequation, and the Golden number is again 1. . . .

Again, to the year 1912 corresponds the capital letter B in the table of theequation, and the Golden number is 13. Wherefore . . . .

The capital letter C corresponds also to the year 1715 in the table of theequation, and the Golden number is 6. . . .

Finally, to the year 1916 corresponds the capital letter B . . . (pp. 22–23)

Observation 1Clavius illustrates every step of his algorithm with manyexamples.

Perpetual table of the cycle of EpactsPP ll CC c P F f s M i A aa m DD d∗∗ 1111 22 3 14 xxvxxv/2525 6 17 28 9 20 1 12 23 4

q G g t NN k BB b n E e r H h u15 2626 77 18 29 10 21 2 13 24 5 16 27 8 1919

Table of the equation of the perpetual cycle of EpactsYear of the Lord Year of the Lord Year of the Lord

N 1 A 2200 q 3600 Leap yearP 320 Leap year u 2300 p 3700P 500 Leap year A 2400 Leap year n 3800a 800 Leap year u 2500 n 3900b 1100 Leap year t 2600 n 4000 Leap yearc 1400 Leap year t 2700 m 4100

10 days subtracted t 2800 Leap year l 4200DD 15821582 s 2900 l 4300D 1600 Leap year s 3000 l 4400 Leap yearC 1700 r 3100 k 4500C 1800 r 3200 Leap year k 4600B 1900 r 3300 i 4700B 2000 Leap year q 3400 i 4800 Leap yearB 2100 p 3500 i 4900

To the year 15821582 after the correction corresponds the capital letter D in the table of the equation, and theGolden number is then 66. If therefore in the perpetual table of the cycle of Epacts you assign the Goldennumber 1 to the cell of the lower-case letter a, which is the third [to the left] from the cell of the capital letterD, and the Golden number 2 to the following cell to the right, and so on, the Golden number 6 of theproposed year 1582 will fall in the cell of Epact 26, which will show the New Moons in the Calendar from theIdes of October of that year.

(pp. 22–23)

Again, in the year 15831583 (already corrected) the Golden number is 77, and to it in the table of the equationcorresponds the same capital letter D. For since this year is not found in the table, the next smaller one is tobe sought, namely 1582, to which the capital letter D corresponds. Assigning therefore the Golden number1 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 thecapital letter D, and the Golden number 2 to the following cell to the right, and so on, the Golden number 7of the proposed year will fall in the cell of the Epact 7, which will show the New Moons that year.

(pp. 22–23)

Next, to the year 4218 in the table of the equation corresponds the letter ll, and the Golden number is 11.Therefore if in the table of Epacts you assign that year’s Golden number 1 to the cell of the letter u, which isthe third to the left from the cell of the letter l, you will find the Epact 19 of that year.

(pp. 22–23)

Moreover to the year 1710 corresponds the capital letter CC in the table of the equation, and the Goldennumber is again 11. Wherefore if you assign that year’s Golden number 1 to the first cell of the capital letter Pin 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)

Again, to the year 1912 corresponds the capital letter BB in the table of the equation, and the Golden numberis 1313. Wherefore if you assign the Golden number 1 to the cell of the capital letter N in the perpetual table ofEpacts, 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 thesecond cell. Therefore the Epact will then be 11.

(pp. 22–23)

The capital letter C corresponds also to the year 1715 in the table of the equation, and the Golden numberis 6. Assigning therefore the Golden number 1 to the cell of the capital letter P in the table of Epacts, whichis 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 twoEpacts, xxv and 25, expressed in different scripts. But because the Golden number, 6, is less than 12, theformer Epact, xxv, is to be taken for the year 1715.

(pp. 22–23)

Finally, to the year 1916 corresponds the capital letter B in the table of the equation, and the Golden numberis 17. Wherefore if the Golden number 1 were given to the cell of the letter N in the table of Epacts, which isthe third from the cell of the capital letter B, and the Golden number 2 to the following cell, etc., returning tothe beginning of the table, the Golden number 17 of the proposed year will fall upon the same cell of theletter F, below which the two Epacts xxv and 25 of different scripts are set. And because the Golden number17 is greater than 11, the latter Epact, 25, is to be taken for the year 1916.

(pp. 22–23)

Use of examples

Observation 2The examples are carefully selected to illustrate difficulties,proceeding from the simplest to most complicated, showingonly one or two “twists” at a time.

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 583years are to be sought in the table. Since they are not contained in it, again the next smaller year in thetable, 500, is to be taken, and its Golden number 6, which having been added to the previously foundGolden number 12, the number 18 will be made. After this the 83 years which remain are to be taken in thetable, but since they are not found, the next smaller year in the table, 80, is to be taken, and its Goldennumber 4. Once this is added to the golden number 18 previously formed, the number 22 will be made, fromwhich if 19 are subtracted, 3 remain. Afterwards the 3 remaining years are to be taken in the table, and theGolden 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 Goldennumber for the year 1583 will be 7. (pp. 17–18)

Observation 3Clavius walks the reader through a complete example at leastonce, in complete precision and utter clarity.

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, 10001000, is to be taken, and its Golden number 12. Then the remaining 583years are to be sought in the table. Since they are not contained in it, again the next smaller year in thetable, 500500, is to be taken, and its Golden number 6, which having been added to the previously foundGolden number 12, the number 18 will be made. After this the 83 years which remain are to be taken in thetable, but since they are not found, the next smaller year in the table, 8080, is to be taken, and its Goldennumber 4. Once this is added to the golden number 18 previously formed, the number 22 will be made, fromwhich if 19 are subtracted, 3 remain. Afterwards the 33 remaining years are to be taken in the table, and theGolden 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 Goldennumber 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, correspondingto the year 10001000, and to it I add the Golden number 6 which corresponds to the year 500500, and I sum up thenumber 18. Then I add the Golden number 14 corresponding to the year 9090 to the Golden number 18 thusobtained, and I produce the number 32, from which 19 having been subtracted, the number 13 remains, towhich I join the Golden number 5 corresponding to the year 55 and I fashion the number 18. To this finally if Iwill add 1, I will have 19 for the Golden number of the year 1595. (pp. 17–18)

Observation 4As he repeats similar examples, Clavius speeds up his delivery.

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.

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.

Vocabulary

Observation 5Clavius employs a wealth of synonyms to express addition andsubtraction, mitigating the mind-numbing effect of so manycalculations.

Familiarity

Let it be required to find (inveniendus sit) the number of the Solarcycle for the year 1000. (p. 25)

Thus youyou see from the year 1000 all the way to the year 10000 theGolden number 12 is always to be added to the preceding Goldennumber, and 19 is to be subtracted when it can be subtracted, . . .(p. 19)

II take first the number of the Solar cycle 0 from the line of the year7000, and II add it to the number of the Solar cycle 14 found from theline of the year 70, and II produce the number 14. (p. 26)

Observation 6Clavius uses the second person, and even the first personsingular, to create an informal, conversational tone.

Familiarity

Multiple perspectives

Golden Numbers: Approach #1Table of the cycle of the Golden number, taking its beginning

from the year of correction 15826 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 year1582, the second one, which is 7, to the following year 1583,and so on forever, until it comes to the year whose Goldennumber you seek, returning to the beginning of the tablehoweversooften you shall have run through it. For the cell, inwhich the proposed year falls, will give the desired Goldennumber.

Golden Numbers: Approach #2BUT since it is very laborious and vexing to count off so many years inthe aforesaid table, and to repeat it so often, until one comes to theyear whose Golden number is sought, especially indeed if theproposed year is far off from the year 1582, we have constructed thisother following table. . .

Years Golden Years Golden Years Golden Years Goldenof the Number of the Number of the Number of the NumberLord Add 1 Lord Add 1 Lord Add 1 Lord Add 1

1 1 300 15 50000 11 7000000 122 22 400 1 60000 17 8000000 123 3 500 6 70000 4 9000000 44 4 600 11 80000 10 10000000 155 5 700 16 90000 16 20000000 116 6 800 2 100000 3 30000000 77 7 900900 77 200000 6 40000000 38 8 10001000 1212 300000 9 50000000 189 9 2000 5 400000 12 60000000 141010 1010 3000 17 500000 15 70000000 1020 1 4000 10 600000 18 80000000 630 11 5000 3 700000 2 90000000 240 2 6000 15 800000 5 100000000 1750 12 7000 8 900000 8 200000000 1560 3 8000 1 1000000 11 300000000 1370 13 9000 13 2000000 3 400000000 1180 4 10000 6 3000000 14 500000000 990 14 20000 12 4000000 6 600000000 7

100 5 30000 18 5000000 17 700000000 5200 10 40000 5 6000000 9 800000000 3

1 1 300 15 50000 11 7000000 122 22 400 1 60000 17 8000000 123 3 500 6 70000 4 9000000 44 4 600 11 80000 10 10000000 155 5 700 16 90000 16 20000000 116 6 800 2 100000 3 30000000 77 7 900900 77 200000 6 40000000 38 8 10001000 1212 300000 9 50000000 189 9 2000 5 400000 12 60000000 141010 1010 3000 17 500000 15 70000000 1020 1 4000 10 600000 18 80000000 630 11 5000 3 700000 2 90000000 240 2 6000 15 800000 5 100000000 1750 12 7000 8 900000 8 200000000 1560 3 8000 1 1000000 11 300000000 1370 13 9000 13 2000000 3 400000000 1180 4 10000 6 3000000 14 500000000 990 14 20000 12 4000000 6 600000000 7

100 5 30000 18 5000000 17 700000000 5200 10 40000 5 6000000 9 800000000 3

1 1 300 15 50000 11 7000000 122 22 400 1 60000 17 8000000 123 3 500 6 70000 4 9000000 44 4 600 11 80000 10 10000000 155 5 700 16 90000 16 20000000 116 6 800 2 100000 3 30000000 77 7 900900 77 200000 6 40000000 38 8 10001000 1212 300000 9 50000000 189 9 2000 5 400000 12 60000000 141010 1010 3000 17 500000 15 70000000 1020 1 4000 10 600000 18 80000000 630 11 5000 3 700000 2 90000000 240 2 6000 15 800000 5 100000000 1750 12 7000 8 900000 8 200000000 1560 3 8000 1 1000000 11 300000000 1370 13 9000 13 2000000 3 400000000 1180 4 10000 6 3000000 14 500000000 990 14 20000 12 4000000 6 600000000 7

100 5 30000 18 5000000 17 700000000 5200 10 40000 5 6000000 9 800000000 3

Golden Numbers: Approach #3By far the easiest way to find the Golden number of whatsoeveryear is through Arithmetic precepts, in the following way. To theproposed year of the Lord let 1 be added, and let the sum bedivided 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

Observation 7Clavius has immense patience with his readers.

Observation 8Clavius presents multiple ways of doing the same task as anaid to understanding.

Observation 7Clavius has immense patience with his readers.

Observation 8Clavius presents multiple ways of doing the same task as anaid to understanding.

The absurd

The tables for computing Golden numbers and Dominicalletters run through the year A.D. 800,000,000. (p. 18)

Observation 9Clavius may be using “absurd” extreme cases as a way to keephis readers’ interest.

Conclusion

Observations1 Many, many examples2 Examples are carefully selected to illustrate difficulties one

at a time.3 Complete precision and utter clarity4 Gradually speeding up delivery5 A wealth of synonyms and broad vocabulary6 First and second person pronouns; informal,

conversational tone.7 Patience!8 Multiple ways of doing the same task as an aid to

understanding9 “Absurd” extreme cases

Conclusion

These techniques are not new to us, nor probably toClavius.Nevertheless it’s fascinating to see one’s own pedagogy inthe pages of a 16th century Latin document.

Thank you.

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