EXTRACTED (by J R Stockton for www.merlyn.demon.co.uk) FROM "A Budget of Paradoxes", Volume I (of II), by Augustus De Morgan, found at URL http://www.gutenberg.org/files/23100/23100-h/23100-h.htm
To give a better chance of the explanation being at once produced, next time the real full moon and Easter Day shall fall together, I insert here a summary which was printed in the Irish Prayer-book of the Ecclesiastical Society. If the amusement given by paradoxers should prevent a useless discussion some years hence, I and the paradoxers shall have done a little good between us—at any rate, I have done my best to keep the heavy weight afloat by tying bladders to it. I think the next occurrence will be in 1875.
In the years 1818 and 1845, Easter Day, as given by the rules in 24 Geo. II cap. 23. (known as the act for the change of style) contradicted the precept given in the preliminary explanations. The precept is as follows:
"Easter Day, on which the rest" of the moveable feasts "depend, is always the First Sunday after the Full Moon, which happens upon or next after the Twenty-first Day of March; and if the Full Moon happens upon a Sunday, Easter Day is the Sunday after."
But in 1818 and 1845, the full moon fell on a Sunday, and yet the rules gave that same Sunday for Easter Day. Much discussion was produced by this circumstance in 1818: but a repetition of it in 1845 was nearly altogether prevented by a timely reference to the intention of those who conducted the Gregorian reformation of the Calendar. Nevertheless, seeing that the apparent error of the Calendar is due to the precept in the Act of Parliament, which is both erroneous and insufficient, and that the difficulty will recur so often as Easter Day falls on the day of full moon, it may be advisable to select from the two articles cited in the note such of their conclusions and rules, without proof or controversy, as will enable the reader to understand the main points of the Easter question, and, should he desire it, to calculate for himself the Easter of the old or new style, for any given year.
1. In the very earliest age of Christianity, a controversy arose as to the mode of keeping Easter, some desiring to perpetuate the Passover, others to keep the festival of the Resurrection. The first afterwards obtained the name of Quartadecimans, from their Easter being always kept on the fourteenth day of the moon (Exod. xii. 18, Levit. xxiii. 5.). But though it is unquestionable that a Judaizing party existed, it is also likely that many dissented on chronological grounds. It is clear that no perfect anniversary can take place, except when the fourteenth of the moon, and with it the passover, falls on a Friday. Suppose, for instance, it falls on a Tuesday: one of three things must be done. Either (which seems never to have been proposed) the crucifixion and resurrection must be celebrated on Tuesday and Sunday, with a wrong interval; or the former on Tuesday, the latter on Thursday, abandoning the first day of the week; or the former on Friday, and the latter on Sunday, abandoning the paschal commemoration of the crucifixion.
The last mode has been, as every one knows, finally adopted. The disputes of the first three centuries did not turn on any calendar questions. The Easter question was merely the symbol of the struggle between what we may call the Jewish and Gentile sects of Christians: and it nearly divided the Christian world, the Easterns, for the most part, being Quartadecimans. It is very important to note that there is no recorded dispute about a method of predicting the new moon, that is, no general dispute leading to formation of sects: there may have been difficulties, and discussions about them. The Metonic cycle, presently mentioned, must have been used by many, perhaps most, churches.
2. The question came before the Nicene Council (A.D. 325) not as an astronomical, but as a doctrinal, question: it was, in fact, this, Shall the passover be treated as a part of Christianity? The Council resolved this question in the negative, and the only information on its premises and conclusion, or either, which comes from itself, is contained in the following sentence of the synodical epistle, which epistle is preserved by Socrates and Theodoret. "We also send you the good news concerning the unanimous consent of all in reference to the celebration of the most solemn feast of Easter, for this difference also has been made up by the assistance of your prayers: so that all the brethren in the East, who formerly celebrated this festival at the same time as the Jews, will in future conform to the Romans and to us, and to all who have of old observed our manner of celebrating Easter." This is all that can be found on the subject: none of the stories about the Council ordaining the astronomical mode of finding Easter, and introducing the Metonic cycle into ecclesiastical reckoning, have any contemporary evidence: the canons which purport to be those of the Nicene Council do not contain a word about Easter; and this is evidence, whether we suppose those canons to be genuine or spurious.
3. The astronomical dispute about a lunar cycle for the prediction of Easter either commenced, or became prominent, by the extinction of greater ones, soon after the time of the Nicene Council. Pope Innocent I met with difficulty in 414. S. Leo, in 454, ordained that Easter of 455 should be April 24; which is right. It is useless to record details of these disputes in a summary: the result was, that in the year 463, Pope Hilarius employed Victorinus of Aquitaine to correct the Calendar, and Victorinus formed a rule which lasted until the sixteenth century. He combined the Metonic cycle and the solar cycle presently described. But this cycle bears the name of Dionysius Exiguus, a Scythian settled at Rome, about A.D. 530, who adapted it to his new yearly reckoning, when he abandoned the era of Diocletian as a commencement, and constructed that which is now in common use.
4. With Dionysius, if not before, terminated all difference as to the mode of keeping Easter which is of historical note: the increasing defects of the Easter Cycle produced in time the remonstrance of persons versed in astronomy, among whom may be mentioned Roger Bacon, Sacrobosco, Cardinal Cusa, Regiomontanus, etc. From the middle of the sixth to that of the sixteenth century, one rule was observed.
5. The mode of applying astronomy to chronology has always involved these two principles. First, the actual position of the heavenly body is not the object of consideration, but what astronomers call its mean place, which may be described thus. Let a fictitious sun or moon move in the heavens, in such manner as to revolve among the fixed stars at an average rate, avoiding the alternate accelerations and retardations which take place in every planetary motion. Thus the fictitious (say mean) sun and moon are always very near to the real sun and moon. The ordinary clocks show time by the mean, not the real, sun: and it was always laid down that Easter depends on the opposition (or full moon) of the mean sun and moon, not of the real ones. Thus we see that, were the Calendar ever so correct as to the mean moon, it would be occasionally false as to the true one: if, for instance, the opposition of the mean sun and moon took place at one second before midnight, and that of the real bodies only two seconds afterwards, the calendar day of full moon would be one day before that of the common almanacs. Here is a way in which the discussions of 1818 and 1845 might have arisen: the British legislature has defined the moon as the regulator of the paschal calendar. But this was only a part of the mistake.
6. Secondly, in the absence of perfectly accurate knowledge of the solar and lunar motion (and for convenience, even if such knowledge existed), cycles are, and always have been taken, which serve to represent those motions nearly. The famous Metonic cycle, which is introduced into ecclesiastical chronology under the name of the cycle of the golden numbers, is a period of 19 Julian years. This period, in the old Calendar, was taken to contain exactly 235 lunations, or intervals between new moons, of the mean moon. Now the state of the case is:
19 average Julian years make 6939 days 18 hours.
235 average lunations make 6939 days 16 hours 31 minutes.
So that successive cycles of golden numbers, supposing the first to start right, amount to making the new moons fall too late, gradually, so that the mean moon of this cycle gains 1 hour 29 minutes in 19 years upon the mean moon of the heavens, or about a day in 300 years. When the Calendar was reformed, the calendar new moons were four days in advance of the mean moon of the heavens: so that, for instance, calendar full moon on the 18th usually meant real full moon on the 14th.
7. If the difference above had not existed, the moon of the heavens (the mean moon at least), would have returned permanently to the same days of the month in 19 years; with an occasional slip arising from the unequal distribution of the leap years, of which a period contains sometimes five and sometimes four. As a general rule, the days of new and full moon in any one year would have been also the days of new and full moon of a year having 19 more units in its date. Again, if there had been no leap years, the days of the month would have returned to the same days of the week every seven years. The introduction of occasional 29ths of February disturbs this, and makes the permanent return of month days to week days occur only after 28 years. If all had been true, the lapse of 28 times 19, or 532 years, would have restored the year in every point: that is, A.D. 1, for instance, and A.D. 533, would have had the same almanac in every matter relating to week days, month days, sun, and moon (mean sun and moon at least). And on the supposition of its truth, the old system of Dionysius was framed. Its errors, are, first, that the moments of mean new moon advance too much by 1 h. 29 m. in 19 average Julian years; secondly, that the average Julian year of 365¼ days is too long by 11 m. 10 s.
8. The Council of Trent, moved by the representations made on the state of the Calendar, referred the consideration of it to the Pope. In 1577, Gregory XIII submitted to the Roman Catholic Princes and Universities a plan presented to him by the representatives of Aloysius Lilius, then deceased. This plan being approved of, the Pope nominated a commission to consider its details, the working member of which was the Jesuit Clavius. A short work was prepared by Clavius, descriptive of the new Calendar: this was published in 1582, with the Pope's bull (dated February 24, 1581) prefixed. A larger work was prepared by Clavius, containing fuller explanation, and entitled Romani Calendarii a Gregorio XIII. Pontifice Maximo restituti Explicatio. This was published at Rome in 1603, and again in the collection of the works of Clavius in 1612.
9. The following extracts from Clavius settle the question of the meaning of the term moon, as used in the Calendar:
"Who, except a few who think they are very sharp-sighted in this matter, is so blind as not to see that the 14th of the moon and the full moon are not the same things in the Church of God?... Although the Church, in finding the new moon, and from it the 14th day, uses neither the true nor the mean motion of the moon, but measures only according to the order of a cycle, it is nevertheless undeniable that the mean full moons found from astronomical tables are of the greatest use in determining the cycle which is to be preferred ... the new moons of which cycle, in order to the due celebration of Easter, should be so arranged that the 14th days of those moons, reckoning from the day of new moon inclusive, should not fall two or more days before the mean full moon, but only one day, or else on the very day itself, or not long after. And even thus far the Church need not take very great pains ... for it is sufficient that all should reckon by the 14th day of the moon in the cycle, even though sometimes it should be more than one day before or after the mean full moon.... We have taken pains that in our cycle the new moons should follow the real new moons, so that the 14th of the moon should fall either the day before the mean full moon, or on that day, or not long after; and this was done on purpose, for if the new moon of the cycle fell on the same day as the mean new moon of the astronomers, it might chance that we should celebrate Easter on the same day as the Jews or the Quartadeciman heretics, which would be absurd, or else before them, which would be still more absurd."
From this it appears that Clavius continued the Calendar of his predecessors in the choice of the fourteenth day of the moon. Our legislature lays down the day of the full moon: and this mistake appears to be rather English than Protestant; for it occurs in missals published in the reign of Queen Mary. The calendar lunation being 29½ days, the middle day is the fifteenth day, and this is and was reckoned as the day of the full moon. There is every right to presume that the original passover was a feast of the real full moon: but it is most probable that the moons were then reckoned, not from the astronomical conjunction with the sun, which nobody sees except at an eclipse, but from the day of first visibility of the new moon. In fine climates this would be the day or two days after conjunction; and the fourteenth day from that of first visibility inclusive, would very often be the day of full moon. The following is then the proper correction of the precept in the Act of Parliament:
Easter Day, on which the rest depend, is always the First Sunday after the fourteenth day of the calendar moon which happens upon or next after the Twenty-first day of March, according to the rules laid down for the construction of the Calendar; and if the fourteenth day happens upon a Sunday, Easter Day is the Sunday after.
10. Further, it appears that Clavius valued the celebration of the festival after the Jews, etc., more than astronomical correctness. He gives comparison tables which would startle a believer in the astronomical intention of his Calendar: they are to show that a calendar in which the moon is always made a day older than by him, represents the heavens better than he has done, or meant to do. But it must be observed that this diminution of the real moon's age has a tendency to make the English explanation often practically accordant with the Calendar. For the fourteenth day of Clavius is generally the fifteenth day of the mean moon of the heavens, and therefore most often that of the real moon. But for this, 1818 and 1845 would not have been the only instances of our day in which the English precept would have contradicted the Calendar.
11. In the construction of the Calendar, Clavius adopted the ancient cycle of 532 years, but, we may say, without ever allowing it to run out. At certain periods, a shift is made from one part of the cycle into another. This is done whenever what should be Julian leap year is made a common year, as in 1700, 1800, 1900, 2100, etc. It is also done at certain times to correct the error of 1 h. 19 m., before referred to, in each cycle of golden numbers: Clavius, to meet his view of the amount of that error, put forward the moon's age a day 8 times in 2,500 years. As we cannot enter at full length into the explanation, we must content ourselves with giving a set of rules, independent of tables, by which the reader may find Easter for himself in any year, either by the old Calendar or the new. Any one who has much occasion to find Easters and movable feasts should procure Francœur's tables.
12. Rule for determining Easter Day of the Gregorian Calendar in any year of the new style. To the several parts of the rule are annexed, by way of example, the results for the year 1849.
I. Add 1 to the given year. (1850).
II. Take the quotient of the given year divided by 4, neglecting the remainder. (462).
III. Take 16 from the centurial figures of the given year, if it can be done, and take the remainder. (2).
IV. Take the quotient of III. divided by 4, neglecting the remainder. (0).
V. From the sum of I, II, and IV., subtract III. (2310).
VI. Find the remainder of V. divided by 7. (0).
VII. Subtract VI. from 7; this is the number of the dominical letter
|1||2||3||4||5||6||7||(7; dominical letter G).|
VIII. Divide I. by 19, the remainder (or 19, if no remainder) is the golden number. (7).
IX. From the centurial figures of the year subtract 17, divide by 25, and keep the quotient. (0).
X. Subtract IX. and 15 from the centurial figures, divide by 3, and keep the quotient. (1).
XI. To VIII. add ten times the next less number, divide by 30, and keep the remainder. (7).
XII. To XI. add X. and IV., and take away III., throwing out thirties, if any. If this give 24, change it into 25. If 25, change it into 26, whenever the golden number is greater than 11. If 0, change it into 30. Thus we have the epact, or age of the Calendar moon at the beginning of the year. (6).
When the Epact is 23, or less.
XIII. Subtract XII., the epact, from 45. (39).
XIV. Subtract the epact from 27, divide by 7, and keep the remainder, or 7, if there be no remainder. (7)
When the Epact is greater than 23.
XIII. Subtract XII., the epact, from 75.
XIV. Subtract the epact from 57, divide by 7, and keep the remainder, or 7, if there be no remainder.
XV. To XIII. add VII., the dominical number, (and 7 besides, if XIV. be greater than VII.,) and subtract XIV., the result is the day of March, or if more than 31, subtract 31, and the result is the day of April, on which Easter Sunday falls. (39; Easter Day is April 8).
In the following examples, the several results leading to the final conclusion are tabulated.
|XII.||16||4||23||20||13||0 say 30|
13. Rule for determining Easter Day of the Antegregorian Calendar in any year of the old style. To the several parts of the rule are annexed, by way of example, the results for the year 1287. The steps are numbered to correspond with the steps of the Gregorian rule, so that it can be seen what augmentations the latter requires.
I. Set down the given year. (1287).
II. Take the quotient of the given year divided by 4, neglecting the remainder (321).
V. Take 4 more than the sum of I. and II. (1612).
VI. Find the remainder of V. divided by 7. (2).
VII. Subtract VI. from 7; this is the number of the dominical letter
|1||2||3||4||5||6||7||(5; dominical letter E).|
VIII. Divide one more than the given year by 19, the remainder (or 19 if no remainder) is the golden number. (15).
XII. Divide 3 less than 11 times VIII. by 30; the remainder (or 30 if there be no remainder) is the epact. (12).
When the Epact is 23, or less.
XIII. Subtract XII., the epact, from 45. (33).
XIV. Subtract the epact from 27, divide by 7, and keep the remainder, or 7, if there be no remainder, (1).
When the Epact is greater than 23.
XIII. Subtract XII., the epact, from 75.
XIV. Subtract the epact from 57, divide by 7, and keep the remainder, or 7, if there be no remainder.
XV. To XIII. add VII., the dominical number, (and 7 besides if XIV. be greater than VII.,) and subtract XIV., the result is the day of March, or if more than 31, subtract 31, and the result is the day of April, on which Easter Sunday (old style) falls. (37; Easter Day is April 6).
These rules completely represent the old and new Calendars, so far as Easter is concerned. For further explanation we must refer to the articles cited at the commencement.
The annexed is the table of new and full moons of the Gregorian Calendar, cleared of the errors made for the purpose of preventing Easter from coinciding with the Jewish Passover.
The second table (page 370) contains epacts, or ages of the moon at the beginning of the year: thus in 1913, the epact is 22, in 1868 it is 6. This table goes from 1850 to 1999: should the New Zealander not have arrived by that time, and should the churches of England and Rome then survive, the epact table may be continued from their liturgy-books. The way of using the table is as follows: Take the epact of the required year, and find it in the first or last column of the first table, in line with it are seen the calendar days of new and full moon. Thus, when the epact is 17, the new and full moons of March fall on the 13th and 28th. The result is, for the most part, correct: but in a minority of cases there is an error of a day. When this happens, the error is almost always a fraction of a day much less than twelve hours. Thus, when the table gives full moon on the 27th, and the real truth is the 28th, we may be sure it is early on the 28th.
For example, the year 1867. The epact is 25, and we find in the table:
When the truth is the day after + is written after the date; when the day before, -. Thus, the new moon of March is on the 6th; the full moon of April is on the 18th. 
I now introduce a small paradox of my own; and as I am not able to prove it, I am compelled to declare that any one who shall dissent must be either very foolish or very dishonest, and will make me quite uncomfortable about the state of his soul. This being settled once for all, I proceed to say that the necessity of arriving at the truth about the assertions that the Nicene Council laid down astronomical tests led me to look at Fathers, Church histories, etc. to an extent which I never dreamed of before. One conclusion which I arrived at was, that the Nicene Fathers had a knack of sticking to the question which many later councils could not acquire. In our own day, it is not permitted to Convocation seriously to discuss any one of the points which are bearing so hard upon their resources of defence—the cursing clauses of the Athanasian Creed, for example. And it may be collected that the prohibition arises partly from fear that there is no saying where a beginning, if allowed, would end. There seems to be a suspicion that debate, once let loose, would play up old Trent with the liturgy, and bring the whole book to book. But if any one will examine the real Nicene Creed, without the augmentation, he will admire the way in which the framers stuck to the point, and settled what they had to decide, according to their view of it. With such a presumption of good sense in their favor, it becomes easier to believe in any claim which may be made on their behalf to tact or sagacity in settling any other matter. And I strongly suspect such a claim may be made for them on the Easter question.
I collect from many little indications, both before and after the Council, that the division of the Christian world into Judaical and Gentile, though not giving rise to a sectarian distinction expressed by names, was of far greater force and meaning than historians prominently admit. I took note of many indications of this, but not notes, as it was not to my purpose. If it were so, we must admire the discretion of the Council. The Easter question was the fighting ground of the struggle: the Eastern or Judaical Christians, with some varieties of usage and meaning, would have the Passover itself to be the great feast, but taken in a Christian sense; the Western or Gentile Christians, would have the commemoration of the Resurrection, connected with the Passover only by chronology. To shift the Passover in time, under its name, Pascha, without allusion to any of the force of the change, was gently cutting away the ground from under the feet of the Conservatives. And it was done in a very quiet way: no allusion to the precise character of the change; no hint that the question was about two different festivals: "all the brethren in the East, who formerly celebrated this festival at the same time as the Jews, will in future conform to the Romans and to us." The Judaizers meant to be keeping the Passover as a Christian feast: they are gently assumed to be keeping, not the Passover, but a Christian feast; and a doctrinal decision is quietly, but efficiently, announced under the form of a chronological ordinance. Had the Council issued theses of doctrine, and excommunicated all dissentients, the rupture of the East and West would have taken place earlier by centuries than it did. The only place in which I ever saw any part of my paradox advanced, was in an article in the Examiner newspaper, towards the end of 1866, after the above was written.
A story about Christopher Clavius, the workman of the new Calendar. I chanced to pick up "Albertus Pighius Campensis de æquinoctiorum solsticiorumque inventione... Ejusdem de ratione Paschalis celebrationis, De que Restitutione ecclesiastici Kalendarii," Paris, 1520, folio. On the title-page were decayed words followed by "..hristophor.. C..ii, 1556 (or 8)," the last blank not entirely erased by time, but showing the lower halves of an l and of an a, and rather too much room for a v. It looked very like E Libris Christophori Clavii 1556. By the courtesy of some members of the Jesuit body in London, I procured a tracing of the signature of Clavius from Rome, and the shapes of the letters, and the modes of junction and disjunction, put the matter beyond question. Even the extra space was explained; he wrote himself Clauius. Now in 1556, Clavius was nineteen years old: it thus appears probable that the framer of the Gregorian Calendar was selected, not merely as a learned astronomer, but as one who had attended to the calendar, and to works on its reformation, from early youth. When on the subject I found reason to think that Clavius had really read this work, and taken from it a phrase or two and a notion or two. Observe the advantage of writing the baptismal name at full length.
 See note 348, page 160.
 Sir Nicholas Harris Nicolas (1799-1848) was a reformer in various lines,—the Record Commission, the Society of Antiquaries, and the British Museum,—and his work was not without good results.
 See note 98, page 69.
 In the Companion to the Almanac for 1845 is a paper by Prof. De Morgan, "On the Ecclesiastical Calendar," the statements of which, so far as concerns the Gregorian Calendar, are taken direct from the work of Clavius, the principal agent in the arrangement of the reformed reckoning. This was followed, in the Companion to the Almanac for 1846, by a second paper, by the same author, headed "On the Earliest Printed Almanacs," much of which is written in direct supplement to the former article.—S. E. De Morgan.
 It may be necessary to remind some English readers that in Latin and its derived European languages, what we call Easter is called the passover (pascha). The Quartadecimans had the name on their side: a possession which often is, in this world, nine points of the law.—A. De M.
 Socrates Scholasticus was born at Constantinople c. 379, and died after 439. His Historia Ecclesiastica (in Greek) covers the period from Constantine the Great to about 439, and includes the Council of Nicæa. The work was printed in Paris 1544.
 Theodoretus or Theodoritus was born at Antioch and died about 457. He was one of the greatest divines of the fifth century, a man of learning, piety, and judicial mind, and a champion of freedom of opinion in all religious matters.
 He died in 417. He was a man of great energy and of high attainments.
 He died in 461, having reigned as pope for twenty-one years. It was he who induced Attila to spare Rome in 452.
 He succeeded Leo as pope in 461, and reigned for seven years.
 Victorinus or Victorius Marianus seems to have been born at Limoges. He was a mathematician and astronomer, and the cycle mentioned by De Morgan is one of 532 years, a combination of the Metonic cycle of 19 years with the solar cycle of 28 years. His canon was published at Antwerp in 1633 or 1634, De doctrina temporum sive commentarius in Victorii Aquitani et aliorum canones paschales.
 He went to Rome about 497, and died there in 540. He wrote his Liber de paschate in 525, and it was in this work that the Christian era was first used for calendar purposes.
 See note 259, page 126.
 Johannes de Sacrobosco (Holy wood), or John of Holywood. The name was often written, without regard to its etymology, Sacrobusto. He was educated at Oxford and taught in Paris until his death (1256). He did much to make the Hindu-Arabic numerals known to European scholars.
 See note 36, page 44.
 See note 45, page 48.
 The Julian year is a year of the Julian Calendar, in which there is leap year every fourth year. Its average length is therefore 365 days and a quarter.—A. De M.
 Ugo Buoncompagno (1502-1585) was elected pope in 1572.
 He was a Calabrian, and as early as 1552 was professor of medicine at Perugia. In 1576 his manuscript on the reform of the calendar was presented to the Roman Curia by his brother, Antonius. The manuscript was not printed and it has not been preserved.
 The title of this work, which is the authority on all points of the new Calendar, is Kalendarium Gregorianum Perpetuum. Cum Privilegio Summi Pontificis Et Aliorum Principum. Romæ, Ex Officina Dominici Basæ. MDLXXXII. Cum Licentia Superiorum (quarto, pp. 60).—A. De M.
 Manuels-Roret. Théorie du Calendrier et collection de tous les Calendriers des Années passées et futures.... Par L. B. Francœur,... Paris, à la librairie encyclopédique de Roret, rue Hautefeuille, 10 bis. 1842. (12mo.) In this valuable manual, the 35 possible almanacs are given at length, with such preliminary tables as will enable any one to find, by mere inspection, which almanac he is to choose for any year, whether of old or new style. [1866. I may now refer to my own Book of Almanacs, for the same purpose].—A. De M.
Louis Benjamin Francœur (1773-1849), after holding positions in the Ecole polytechnique (1804) and the Lycée Charlemagne (1805), became professor of higher algebra in the University of Paris (1809). His Cours complet des mathématiques pures was well received, and he also wrote on mechanics, astronomy, and geodesy.
 Albertus Pighius, or Albert Pigghe, was born at Kempen c. 1490 and died at Utrecht in 1542. He was a mathematician and a firm defender of the faith, asserting the supremacy of the Pope and attacking both Luther and Calvin. He spent some time in Rome. His greatest work was his Hierarchiæ ecclesiasticæ assertio (1538).