Writings of Anatolius of Alexandria. Paschal Canon of
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Text edited by Rev. Alexander Roberts and James Donaldson and
first published by T&T Clark in Edinburgh in 1867. Additional
introductionary material and notes provided for the American
edition by A. Cleveland Coxe, 1886.
The Paschal Canon of Anatolius of Alexandria. [1128]
Translator's Biographical Notice
[a.d. 230-270-280.] From Jerome [1129] we learn that Anatolius
flourished in the reign of Probus and Carus, that he was a native of
Alexandria, and that he became bishop of Laodicea. Eusebius gives a
somewhat lengthened account of him, [1130] and speaks of him in terms
of the strongest laudation, as one surpassing all the men of his time
in learning and science. He tells us that he attained the highest
eminence in arithmetic, geometry, and astronomy, besides being a great
proficient also in dialectics, physics, and rhetoric. His reputation
was so great among the Alexandrians that they are said to have
requested him to open a school for teaching the Aristotelian
philosophy in their city. [1131] He did great service to his
fellow-citizens in Alexandria on their being besieged by the Romans in
a.d. 262, and was the means of saving the lives of numbers of them.
After this he is said to have passed into Syria, where Theotecnus, the
bishop of Caesareia, ordained him, destining him to be his own
successor in the bishopric. After this, however, having occasion to
travel to Antioch to attend the synod convened to deal with the case
of Paul of Samosata, as he passed through the city of Laodicea, he was
detained by the people and made bishop of the place, in succession to
Eusebius. This must have been about the year 270 a.d. How long he held
that dignity, however, we do not know. Eusebius tells us that he did
not write many books, but yet enough to show us at once his eloquence
and his erudition. Among these was a treatise on the Chronology of
Easter; of which a considerable extract is preserved in Eusebius. The
book itself exists now only in a Latin version, which is generally
ascribed to Rufinus, and which was published by Aegidius Bucherius in
his Doctrina Temporum, which was issued at Antwerp in 1634. Another
work of his was the Institutes of Arithmetic, of which we have some
fragments in the theologou'mena tes arithmetikes, which was published
in Paris in 1543. Some small fragments of his mathematical works,
which have also come down to us, were published by Fabricius in his
Bibliotheca Graeca , iii. p. 462.
I.
As we are about to speak on the subject of the order of the times and
alternations of the world, we shall first dispose of the positions of
diverse calculators; who, by reckoning only by the course of the moon,
and leaving out of account the ascent and descent of the sun, with the
addition of certain problems, have constructed diverse periods, [1132]
self-contradictory, and such as are never found in the reckoning of a
true computation; since it is certain that no mode of computation is
to be approved, in which these two measures are not found together.
For even in the ancient exemplars, that is, in the books of the
Hebrews and Greeks, we find not only the course of the moon, but also
that of the sun, and, indeed, not simply its course in the general,
[1133] but even the separate and minutest moments of its hours all
calculated, as we shall show at the proper time, when the matter in
hand demands it. Of these Hippolytus made up a period of sixteen years
with certain unknown courses of the moon. Others have reckoned by a
period of twenty-five years, others by thirty, and some by eighty-four
years, without, however, teaching thereby an exact method of
calculating Easter. But our predecessors, men most learned in the
books of the Hebrews and Greeks,--I mean Isidore and Jerome and
Clement,--although they have noted similar beginnings for the months
just as they differ also in language, have, nevertheless, come
harmoniously to one and the same most exact reckoning of Easter, day
and month and season meeting in accord with the highest honour for the
Lord's resurrection. [1134] But Origen also, the most erudite of all,
and the acutest in making calculations,--a man, too, to whom the
epithet chalkente's [1135] is given,--has published in a very elegant
manner a little hook on Easter. And in this book, while declaring,
with respect to the day of Easter, that attention must be given not
only to the course of the moon and the transit of the equinox, but
also to the passage (transcensum) of the sun, which removes every foul
ambush and offence of all darkness, and brings on the advent of light
and the power and inspiration of the elements of the whole world, he
speaks thus: In the (matter of the) day of Easter, he remarks, I do
not say that it is to be observed that the Lord's day should be found,
and the seven [1136] days of the moon which are to elapse, but that
the sun should pass that division, to wit, between light and darkness,
constituted in an equality by the dispensation of the Lord at the
beginning of the world; and that, from one hour to two hours, from two
to three, from three to four, from four to five, from five to six
hours, while the light is increasing in the ascent of the sun, the
darkness should decrease. [1137] ... and the addition of the twentieth
number being completed, twelve parts should be supplied in one and the
same day. But if I should have attempted to add any little drop of
mine [1138] after the exuberant streams of the eloquence and science
of some, what else should there be to believe but that it should be
ascribed by all to ostentation, and, to speak more truly, to mad ness,
did not the assistance of your promised prayers animate us for a
little? For we believe that nothing is impossible to your power of
prayer, and to your faith. Strengthened, therefore, by this
confidence, we shall set bashfulness aside, and shall enter this most
deep and unforeseen sea of the obscurest calculation, in which
swelling questions and problems surge around us on all sides.
II.
There is, then, in the first year, the new moon of the first month,
which is the beginning of every cycle of nineteen years, on the six
and twentieth day of the month called by the Egyptians Phamenoth.
[1139] But, according to the months of the Macedonians, it is on the
two-and-twentieth day of Dystrus. And, as the Romans would say, it is
on the eleventh day before the Kalends of April. Now the sun is found
on the said six-and-twentieth day of Phamenoth, not only as having
mounted to the first segment, but as already passing the fourth day in
it. And this segment they are accustomed to call the first
dodecatemorion (twelfth part), and the equinox, and the beginning of
months, and the head of the cycle, and the starting-point [1140] of
the course of the planets. And the segment before this they call the
last of the months, and the twelfth segment, and the last
dodecatemorion, and the end of the circuit [1141] of the planets.
And for this reason, also, we maintain that those who place the first
month in it, and who determine the fourteenth day of the Paschal
season by it, make no trivial or common blunder.
III.
Nor is this an opinion confined to ourselves alone. For it was also
known to the Jews of old and before Christ, and it was most carefully
observed by them. [1142] And this may be learned from what Philo,
and Josephus, and Musaeus have written; and not only from these, but
indeed from others still more ancient, namely, the two Agathobuli,
[1143] who were surnamed the Masters, and the eminent Aristobulus,
[1144] who was one of the Seventy who translated the sacred and holy
Scriptures of the Hebrews for Ptolemy Philadelphus and his father, and
dedicated his exegetical books on the law of Moses to the same kings.
These writers, in solving some questions which are raised with respect
to Exodus, say that all alike ought to sacrifice the Passover [1145]
after the vernal equinox in the middle of the first month. And that is
found to be when the sun passes through the first segment of the
solar, or, as some among them have named it, the zodiacal circle.
IV.
But this Aristobulus also adds, that for the feast of the Passover it
was necessary not only that the sun should pass the equinoctial
segment, but the moon also. For as there are two equinoctial segments,
the vernal and the autumnal, and these diametrically opposite to each
other, and since the day of the Passover is fixed for the fourteenth
day of the month, in the evening, the moon will have the position
diametrically opposite the sun; as is to be seen in full moons. And
the sun will thus be in the segment of the vernal equinox, and the
moon necessarily will be at the autumnal equinox.
V.
I am aware that very many other matters were discussed by them, some
of them with considerable probability, and others of them as matters
of the clearest demonstration, [1146] by which they endeavour to
prove that the festival of the Passover and unleavened bread ought by
all means to be kept after the equinox. But I shall pass on without
demanding such copious demonstrations (on subjects [1147] ) from
which the veil of the Mosaic law has been removed; for now it remains
for us with unveiled face to behold ever as in a glass Christ Himself
and the doctrines and sufferings of Christ. But that the first month
among the Hebrews is about the equinox, is clearly shown also by what
is taught in the book of Enoch. [1148]
VI.
And, therefore, in this concurrence of the sun and moon, the Paschal
festival is not to be celebrated, because as long as they are found in
this course the power of darkness is not overcome; and as long as
equality between light and darkness endures, and is not diminished by
the light, it is shown that the Paschal festival is not to be
celebrated. Accordingly, it is enjoined that that festival be kept
after the equinox, because the moon of the fourteenth, [1149] if
before the equinox or at the equinox, does not fill the whole night.
But after the equinox, the moon of the fourteenth, with one day being
added because of the passing of the equinox, although it does not
extend to the true light, that is, the rising of the sun and the
beginning of day, will nevertheless leave no darkness behind it. And,
in accordance with this, Moses is charged by the Lord to keep seven
days of unleavened bread for the celebration of the Passover, that in
them no power of darkness should be found to surpass the light. And
although the outset of four nights begins to be dark, that is, the
17th and 18th and 19th and 20th, yet the moon of the 20th, which rises
before that, does not permit the darkness to extend on even to
midnight.
VII.
To us, however, with whom it is impossible for all these things to
come aptly at one and the same time, namely, the moon's fourteenth,
and the Lord's day, and the passing of the equinox, and whom the
obligation of the Lord's resurrection binds to keep the Paschal
festival on the Lord's day, it is granted that we may extend the
beginning of our celebration even to the moon's twentieth. For
although the moon of the 20th does not fill the whole night, yet,
rising as it does in the second watch, it illumines the greater part
of the night. Certainly if the rising of the moon should be delayed on
to the end of two watches, that is to say, to midnight, the light
would not then exceed the darkness, but the darkness the light. But it
is clear that in the Paschal feast it is not possible that any part of
the darkness should surpass the light; for the festival of the Lord's
resurrection is one of light, and there is no fellowship between light
and darkness. And if the moon should rise in the third watch, it is
clear that the 22d or 23d of the moon would then be reached, in which
it is not possible that there can be a true celebration of Easter. For
those who determine that the festival may be kept at this age of the
moon, are not only unable to make that good by the authority of
Scripture, but turn also into the crime of sacrilege and contumacy,
and incur the peril of their souls; inasmuch as they affirm that the
true light may be celebrated along with something of that power of
darkness which dominates all.
VIII.
Accordingly, it is not the case, as certain calculators of Gaul
allege, that this assertion is opposed by that passage in Exodus,
[1150] where we read: "In the first month, on the fourteenth day of
the first month, at even, ye shall eat unleavened bread until the
one-and-twentieth day of the month at even. Seven days shall there be
no leaven found in your houses." From this they maintain that it is
quite permissible to celebrate the Passover on the twenty-first day of
the moon; understanding that if the twenty-second day were added,
there would be found eight days of unleavened bread. A thing which
cannot be found with any probability, indeed, in the Old Testament, as
the Lord, through Moses, gives this charge: "Seven days ye shall eat
unleavened bread." [1151] Unless perchance the fourteenth day is not
reckoned by them among the days of unleavened bread with the
celebration of the feast; which, however, is contrary to the Word of
the Gospel which says: "Moreover, on the first day of unleavened
bread, the disciples came to Jesus." [1152] And there is no doubt as
to its being the fourteenth day on which the disciples asked the Lord,
in accordance with the custom established for them of old, "Where wilt
Thou that we prepare for Thee to eat the Passover? "But they who are
deceived with this error maintain this addition, because they do not
know that the 13th and 14th, the 14th and 15th, the 15th and 16th, the
16th and 17th, the 17th and 18th, the 18th and 19th, the 19th and
20th, the 20th and 21st days of the moon are each found, as may be
most surely proved, within a single day. For every day in the
reckoning of the moon does not end in the evening as the same day in
respect of number, as it is at its beginning in the morning. For the
day which in the morning, that is up to the sixth hour and half, is
numbered the 13th day of the month, is found at even to be the 14th.
Wherefore, also, the Passover is enjoined to be extended on to the
21st day at even; which day, without doubt, in the morning, that is,
up to that term of hours which we have mentioned, was reckoned the
20th. Calculate, then, from the end of the 13th [1153] day of the
moon, which marks the beginning of the 14th, on to the end of the
20th, at which the 21st day also begins, and you will have only seven
days of unleavened bread, in which, by the guidance of the Lord, it
has been determined before that the most true feast of the Passover
ought to be celebrated.
IX.
But what wonder is it that they should have erred in the matter of the
21st day of the moon who have added three days before the equinox, in
which they hold that the Passover may be celebrated? An assertion
which certainly must be considered altogether absurd, since, by the
best-known historiographers of the Jews, and by the Seventy Elders, it
has been clearly determined that the Paschal festival cannot be
celebrated at the equinox.
X.
But nothing was difficult to them with whom it was lawful to celebrate
the Passover on any day when tile fourteenth of the moon happened
after the equinox. Following their example up to the present time all
the bishops of Asia--as themselves also receiving the rule from an
unimpeachable authority, to wit, the evangelist John, who leant on the
Lord's breast, and drank in instructions spiritual without doubt--were
in the way of celebrating the Paschal feast, without question, every
year, whenever the fourteenth day of the moon had come, and the lamb
was sacrificed by the Jews after the equinox was past; not
acquiescing, so far as regards this matter, with the authority of
some, namely, the successors of Peter and Paul, who have taught all
the churches in which they sowed the spiritual seeds of the Gospel,
that the solemn festival of the resurrection of the Lord can be
celebrated only on the Lord's day. Whence, also, a certain contention
broke out between the successors of these, namely, Victor, at that
time bishop of the city of Rome, and Polycrates, who then appeared to
hold the primacy among the bishops of Asia. And this contention was
adjusted most rightfully by Irenaeus, [1154] at that time president
of a part of Gaul, so that both parties kept by their own order, and
did not decline from the original custom of antiquity. The one party,
indeed, kept the Paschal day on the fourteenth day of the first month,
according to the Gospel, as they thought, adding nothing of an
extraneous kind, but keeping through all things the rule of faith. And
the other party, passing the day of the Lord's Passion as one replete
with sadness and grief, hold that it should not be lawful to celebrate
the Lord's mystery of the Passover at any other time but on the Lord's
day, on which the resurrection of the Lord from death took place, and
on which rose also for us the cause of everlasting joy. For it is one
thing to act in accordance with the precept given by the apostle, yea,
by the Lord Himself, and be sad with the sad, and suffer with him
thatsuffers by the cross, His own word being: "My soul is exceeding
sorrowful, even unto death; " [1155] and it is another thing to
rejoice with the victor as he triumphs over an ancient enemy, and
exults with the highest triumph over a conquered adversary, as He
Himself also says: "Rejoice with Me; for I have found the sheep which
I had lost." [1156]
XI.
Moreover, the allegation which they sometimes make against us, that if
we pass the moon's fourteenth we cannot celebrate the beginning of the
Paschal feast in light, [1157] neither moves nor disturbs us. For,
although they lay it down as a thing unlawful, that the beginning of
the Paschal festival should be extended so far as to the moon's
twentieth; yet they cannot deny that it ought to be extended to the
sixteenth and seventeenth, which coincide with the day on which the
Lord rose from the dead. But we decide that it is better that it
should be extended even on to the twentieth day, on account of the
Lord's day, than that we should anticipate the Lord's day on account
of the fourteenth day; for on the Lord's day was it that light was
shown to us in the beginning, and now also in the end, the comforts of
all present and the tokens of all future blessings. For the Lord
ascribes no less praise to the twentieth day than to the fourteenth.
For in the book of Leviticus [1158] the injunction is expressed
thus: "In the first month, on the fourteenth day of this month, at
even, is the Lord's Passover. And on the fifteenth day of this month
is the feast of unleavened bread unto the Lord. Seven days ye shall
eat unleavened bread. The first day shall be to you one most
diligently attended [1159] and holy. Ye shall do no servile work
thereon. And the seventh day shall be to you more diligently
attended [1160] and holier; ye shall do no servile work thereon."
And hence we maintain that those have contracted no guilt [1161]
before the tribunal of Christ, who have held that the beginning of the
Paschal festival ought to be extended to this day. And this, too, the
most especially, as we are pressed by three difficulties, namely, that
we should keep the solemn festival of the Passover on the Lord's day,
and after the equinox, and yet not beyond the limit of the moon's
twentieth day.
XII.
But this again is held by other wise and most acute men to be an
impossibility, because within that narrow and most contracted limit of
a cycle of nineteen years, a thoroughly genuine Paschal time, that is
to say, one held on the Lord's day and yet after the equinox, cannot
occur. But, in order that we may set in a clearer light the difficulty
which causes their in credulity, we shall set down, along with the
courses of the moon, that cycle of years which we have mentioned; the
days being computed before in which the year rolls on in its
alternating courses, by Kalends and Ides and Nones, and by the sun's
ascent and descent.
XIII.
The moon's age set forth in the Julian Calendar.
January, on the Kalends, one day, the moon's first (day); on the
Nones, the 5th day, the moon's 5th; on the Ides, the 13th day, the
moon's 13th. On the day before the Kalends of February, the 31st day,
the moon's 1st; on the Kalends of February, the 32d day, the moon's
2d; on the Nones, the 36th day, the moon's 6th; on the Ides, the 44th
day, the moon's 14th. On the day before the Kalends of March, the 59th
day, the moon's 29th; on the Kalends of March, the 60th day, the
moon's 1st; on the Nones, the 66th day, the moon's 7th; on the Ides,
the 74th day, the moon's 15th. On the day before the Kalends of April,
the 90th day, the moon's 2d; on the Kalends of April, the 91st day,
the moon's 3d; on the Nones, the 95th day, the moon's 7th; on the
Ides, the 103d day, the moon's 15th. On the day before the Kalends of
May, the 120th day, the moon's 3d; on the Kalends of May, the 121st
day, the moon's 4th; on the Nones, the 127th day, the moon's 10th; on
the Ides, the 135th day, the moon's 18th. On the day before the
Kalends of June, the 151st day, the moon's 3d; on the Kalends of June,
the 152d day, the moon's 5th; on the Nones, the 153d day, the moon's
9th; on the Ides, the 164th day, the moon's 17th. On the day before
the Kalends of July, the 181st day, the moon's 5th; on the Kalends of
July, the 182d day, the moon's 6th; on the Nones, the 188th day, the
moon's 12th; on the Ides, the 196th day, the moon's 20th. On the day
before the Kalends of August, the 212th day, the moon's 5th; on the
Kalends of August, the 213th day, the moon's 7th; on the Nones, the
217th day, the moon's 12th; on the ides, the 225th day, the moon's
19th. On the day before the Kalends of September, the 243d day, the
moon's 7th; on the Kalends of September, the 244th day, the moon's
8th; on the Nones, the 248th day, the moon's 12th; on the Ides, the
256th day, the moon's 20th. On the day before the Kalends of October,
the 273d day, the moon's 8th; on the Kalends of October, the 247th
day, the moon's 9th; on the Nones, the 280th day, the moon's 15th; on
the Ides, the 288th day, the moon's 23d. On the day before the Kalends
of November, the 304th day, the moon's 9th; on the Kalends of
November, the 305th day, the moon's 10th; on the Nones, the 309th day,
the moon's 14th; on the Ides, the 317th day, the moon's 22d. On the
day before the Kalends of December, the 334th day, the moon's 10th; on
the Kalends of December, the 335th day, the moon's 11th; on the Nones,
the 339th day, the moon's 15th; on the Ides, the 347th day, the moon's
23d. On the day before the Kalends of January, the 365th day, the
moon's 11th; on the Kalends of January, the 366th day, the moon's
12th.
XIV.
The Paschal or Easter Table of Anatolius.
Equinox
Moon
Easter
Moon
1. Sabbath
XXVI
XVth before the Kalends of May, i.e., 17th April
XVIII
2. Lord's Day
VII
Kalends of April, i.e., 1st April
XIV
3. IId Day (ferial)
XVIII
XIth before the Kalends of May, i.e., 21st April
XVI
4. IIId Day
XXIX
Ides of April, i.e., 13th April
XIX
IVth Day
X
IVth before the Kalends of April, i.e., 29th April
XIV
Vth Day
XXI
XIVth before the Kalends of May, i.e., 27th March
XVI
7. Sabbath2
II
VIth before the Kalends of April, i.e., 27th March
XVII
8. Lord's Day
XIII
Kalends of April, i.e., 1st of April
XX
9. IId Day
XXIV
XVIIIth before the Kalends of May, i.e., 14th March
XV
10. IIId Day
V
VIIIth before the Ides of April, i.e., 6th April
XV
11. IVth Day
XVI
IVth before the Kalends of April, i.e., 29th March
XX
12. Vth Day
XXVII
IIId before the Ides of April, i.e., 11th April
XV
13. VIth Day
VIII
IIId before the Nones of April, i.e., 3rd April
XVII
14. Sabbath
XX
IXth before the Kalends of May, i.e., 23rd April
XX
15. Lord's Day
I
VIth before the Ides of April, i.e., 8th April
XV
16. IId Day
XII
IId before the Kalends of April, i.e., 31st March
XVIII
17. IVth Day2
XXIII
XIVth before the Kalends of May, i.e., 18th April
XIX
18. Vth Day
IV
IId before the Nones of April, i.e., 4th April
XIV
19. VIth Day
XV
VIth before the Kalends of April, i.e.
Now, then, after the reckoning of the days and the exposition of the
course of the moon, whereon the whole revolves on to its end, the
cycle of the years may be set forth from the commencement). [1162]
This makes the Passover (Easter season) circulate between the 6th day
before the Kalends of April and the 9th before the Kalends of May,
according to the following table:
XV.
This cycle of nineteen years is not approved of by certain African
investigators who have drawn up larger cycles, because it seems to be
somewhat opposed to their surmises and opinions. For these make up the
best proved accounts according to their calculation, and determine a
certain beginning or certain end for the Easter season, so as that the
Paschal festival shall not be celebrated before the eleventh day
before the Kalends of April, i.e., 24th March, nor after the moon's
twenty-first, and the eleventh day before the Kalends of May, i.e.,
21st April. But we hold that these are limits not only not to be
followed, but to be detested and overturned. For even in the ancient
law it is laid down that this is to be seen to, viz., that the
Passover be not celebrated before the transit of the vernal equinox,
at which the last of the autumnal term is overtaken, [1163] on the
fourteenth day of the first month, which is one calculated not by the
beginnings of the day, but by those of the moon. [1164] And as this
has been sanctioned by the charge of the Lord, and is in all things
accordant with the Catholic faith, it cannot be doubtful to any wise
man that to anticipate it must be a thing unlawful and perilous. And,
accordingly, this only is it sufficient for all the saints and
Catholics to observe, namely, that giving no heed to the diverse
opinions of very many, they should keep the solemn festival of the
Lord's resurrection within the limits which we have set forth.
XVI.
Furthermore, as to the proposal subjoined to your epistle, that I
should attempt to introduce into this little book some notice of the
ascent and descent of the sun, which is made out in the distribution
of days and nights. The matter proceeds thus: In fifteen days and half
an hour, the sun ascending by so many minutes, that is, by four in one
day, from the eighth day before the Kalends of January, i.e., 25th
December, to the eighth before the Kalends of April, i.e., 25th March,
an hour is taken up; [1165] at which date there are twelve hours and
a twelfth. On this day, towards evening, if it happen also to be the
moon's fourteenth, the lamb was sacrificed among the Jews. But if the
number went beyond that, so that it was the moon's fifteenth or
sixteenth on the evening of the same day, on the fourteenth day of the
second moon, in the same month, the Passover was celebrated; and the
people ate unleavened bread for seven days, up to the twenty-first day
at evening. Hence, if it happens in like manner to us, that the
seventh day before the Kalends of April, 26th March, proves to be both
the Lord's day and the moon's fourteenth, Easter is to be celebrated
on the fourteenth. But if it proves to be the moon's fifteenth or
sixteenth, or any day up to the twentieth, then our regard for the
Lord's resurrection, which took place on the Lord's day, will lead us
to celebrate it on the same principle; yet this should be done so as
that the beginning of Easter may not pass beyond the close of their
festival, that is to say, the moon's twentieth. And therefore we have
said that those parties have committed no trivial offence who have
ventured either on anticipating or on going beyond this number, which
is given us in the divine Scriptures themselves. And from the eighth
day before the Kalends of April, 25th March, to the eighth before the
Kalends of July, 24th June, in fifteen days an hour is taken up: the
sun ascending every day by two minutes and a half, and the sixth part
of a minute. And from the eighth day before the Kalends of July, 24th
June, to the eighth before the Kalends of October, 24th September, in
like manner, in fifteen days and four hours, an hour is taken up: the
sun descending every day by the same number of minutes. And the space
remaining on to the eighth day before the Kalends of January, 25th
December, is determined in a similar number of hours and minutes. So
that thus on the eighth day before the Kalends of January, for the
hour there is the hour and half. For up to that day and night are
distributed. And the twelve hours which were established at the vernal
equinox in the beginning by the Lord's dispensation, being distributed
over the night on the eighth before the Kalends of July, the sun
ascending through those eighteen several degrees which we have noted,
shall be found conjoined with the longer space in the twelfth. And,
again, the twelve hours which should be fulfilled at the autumnal
equinox in the sun's descent, should be found disjoined on the sixth
before the Kalends of January as six hours divided into twelve, the
night holding eighteen divided into twelve. And on the eighth before
the Kalends of July, in like manner, it held six divided into twelve.
XVII.
Be not ignorant of this, however, that those four determining
periods, [1166] which we have mentioned, although they are
approximated to the Kalends of the following months, yet hold each the
middle of a season, viz., of spring and summer, and autumn and winter.
And the beginnings of the seasons are not to be fixed at that point at
which the Kalends of the month begin. But each season is to be begun
in such way that the equinox divides the season of spring from its
first day; and the season of summer is divided by the eighth day
before the Kalends of July, and that of autumn by the eighth before
the Kalends of October, and that of winter by the eighth before the
Kalends of January in like manner. [1167]
Fragments of the Books on Arithmetic. [1168]
What is mathematics?
Aristotle thinks that all philosophy consisted of theory and practice,
[1169] and divides the practical into ethical and political, and the
theoretic again into the theological, the physical, and the
mathematical. And thus very clearly and skilfully he shows that
mathematics is (a branch of) philosophy.
The Chaldaeans were the originators of astronomy, and the Egyptians of
geometry and arithmetic....
And whence did mathematics derive its name? Those of the Peripatetic
school affirmed that in rhetoric and poetry, and in the popular music,
any one may be an adept though he has gone through no process of
study; but that in those pursuits properly called studies, [1170] none
can have any real knowledge unless he has first become a student of
them. Hence they supposed that the theory of these things was called
Mathematics, from ma'thema, study, science. And the followers of
Pythagoras are said to have given this more distinctive name of
mathematics to geometry, and arithmetic alone. For of old these had
each its own separate name; and they had up till then no name common
to both. And he (Archytas) gave them this name, because he found
science [1171] in them, and that in a manner suitable to man's study.
[1172] For they (the Pythagoreans) perceived that these studies dealt
with things eternal and immutable and perfect, [1173] in which things
alone they considered that science consisted. But the more recent
philosophers have given a more extensive application to this name, so
that, in their opinion, the mathematician deals not only with
substances [1174] incorporeal, and falling simply within the province
of the understanding, [1175] but also with that which touches upon
corporeal and sensible matter. For he ought to be cognisant of [1176]
the course of the stars, and their velocity, and their magnitudes, and
forms, and distances. And, besides, he ought to investigate their
dispositions to vision, examining into the causes, why they are not
seen as of the same form and of the same size from every distance,
retaining, indeed, as we know them to do, their dispositions relative
to each other, [1177] but producing, at the same time, deceptive
appearances, both in respect of order and position. And these are so,
either as determined by the state of the heavens and the air, or as
seen in reflecting and all polished surfaces and in transparent
bodies, and in all similar kinds. In addition to this, they thought
that the man ought to be versed in mechanics and geometry and
dialectics. And still further, that he should engage himself with the
causes of the harmonious combination of sounds, and with the
composition of music; which things are bodies, [1178] or at least
are to be ultimately referred to sensible matter.
What is mathematics?
Mathematics is a theoretic science [1179] of things apprehensible by
perception and sensation for communication to others. [1180] And
before this a certain person indulging in a joke, while hitting his
mark, said that mathematics is that science to which Homer's
description of Discord may be applied.--
"Small at her birth, but rising every hour,
While scarce the skies her horrid (mighty) head can bound,
She stalks on earth and shakes the world around." [1181]
For it begins with a point and a line, [1182] and forthwith it takes
heaven itself and all things within its compass.
How many divisions are there of mathematics?
Of the more notable and the earliest mathematics there are two
principal divisions, viz., arithmetic and geometry. And of the
mathematics which deals with things sensible there are six divisions,
viz, computation (practical arithmetic), geodesy, optics, theoretical
music, mechanics, and astronomy. But that neither the so-called
tactics nor architecture, [1183] nor the popular music, nor physics,
nor the art which is called equivocally the mechanical, constitutes,
as some think, a branch of mathematics, we shall prove, as the
discourse proceeds, clearly and systematically.
As to the circle having eight solids and six superficies and four
angles.... What branches of arithmetic have closest affinity with each
other? Computation and theoretical music have a closer affinity than
others with arithmetic; for this department, being one also of
quantity and ratio, approaches it in number and proportion. [1184]
Optics and geodesy, again, are more in affinity with geometry. And
mechanics and astrology are in general affinity with both.
As to mathematics having its principles [1185] in hypothesis and
about hypothesis. Now, the term hypothesis is used in three ways, or
indeed in many ways. For according to one usage of the term we have
the dramatic revolution; [1186] and in this sense there are said to
be hypotheses in the dramas of Euripides. According to a second
meaning, we have the investigation of matters in the special in
rhetoric; and in this sense the Sophists say that a hypothesis must be
proposed. And, according to a third signification, the beginning of a
proof is called a hypothesis, as being the begging of certain matters
with a view to the establishment of another in question. Thus it is
said that Democritus [1187] used a hypothesis, namely, that of atoms
and a vacuum; and Asclepiades [1188] that of atoms [1189] and
pores. Now, when applied to mathematics, the term hypothesis is to be
taken in the third sense.
That Pythagoras was not the only one who duly honoured arithmetic, but
that his best known disciples did so too, being wont to say that "all
things fit number." [1190]
That arithmetic has as its immediate end chiefly the theory of
science, [1191] than which there is no end either greater or nobler.
And its second end is to bring together in one all that is found in
determinate substance. [1192]
Who among the mathematicians has made any discovery?
Eudemus [1193] relates in his Astrologies that Oenopides [1194]
found out the circle of the zodiac and the cycle [1195] of the great
year. And Thales [1196] discovered the eclipse of the sun and its
period in the tropics in its constant inequality. And Anaximander
[1197] discovered that the earth is poised in space, [1198] and
moves round the axis of the universe. And Anaximenes [1199]
discovered that the moon has her light from the sun, and found out
also the way in which she suffers eclipse. And the rest of the
mathematicians have also made additions to these discoveries. We may
instance the facts--that the fixed stars move round the axis passing
through the poles, while the planets remove from each other [1200]
round the perpendicular axis of the zodiac; and that the axis of the
fixed stars and the planets is the side of a pentedecagon with
four-and-twenty parts.
Footnotes
[1129] Euseb., Hist. Eccles., vi. 11. [Narcissus must have been born
about A.D. 121. Might have known Polycarp.]
[1130] Ibid., vi. 46. [Narcissus lived till A.D. 237, and died a
martyr, aged 116.]
[1131] He was a pupil of Pantaenus, continued under Clement, and
defended Origen against the severity of Demetrius. Two dates which are
conjectural are adjusted to these facts. I find it difficult to
reconcile them with those implied by Eusebius.]
[1132] Circulos [Note the reference to Hippolytus.]
[1133] Gressus. Vol. v. p. 3: also Bunsen, i. pp. 13, 281.]
[1134] [It seems probable that the hegemony which Alexandria had
established in all matters of learning led to that full recognition of
it, by the Council of Nicaea, which made its bishop the dictator to
the whole Church in the annual calculation of Easter. Vol. ii. 343.]
[1135] i.e., "smith" or "brasier," probably from his assiduity.
[1136] Lunae vii. Perhaps, as Bucher conjectures, Lunae xiv., fourteen
days, &c.
[1137] The text is doubtful and corrupt here.
[1138] Aliquid stillicidii.
[1139] [The Church's Easter-calculations created modern astronomy,
which passed to the Arabians from the Church. (See Whewell's Inductive
Sciences.) They preserved it, but did not improve it, in Spain.
Christianity re-adopted it, and the presbyter Copernicus new-created
it. The court of Rome (not the Church Catholic) persecuted Galileo;
but it did so under the lead of professional "Science,'" which had
darkened the human mind, from the days of Pythagoras, respecting his
more enlightened system.]
[1140] The word is aphesis, which Valesius makes equivalent to
apheteria, the rope or post from which the chariots started in the
race, and so = starting-point.--Tr.
[1141] 0 periodou.
[1142] pro`s auton--others read pro', before them.
[1143] Anatolius writes that there were two Agathobuli with the
surname Masters; but I fear that he is wrong in his opinion that they
were more ancient than Philo and Josephus. For Agathobulus, the
philosopher, flourished in the times of Adrian, as Eusebius writes in
his Chronicon, and after him Georgius Syncellus.--Vales.
[1144] Aristoboulou tou pa'nu--Rufinus erroneously renders it
Aristobulum ex Paneade, Aristobulus of Paneas. Scaliger also, in his
Animadversiones Eusebianae, p. 130, strangely thinks that the text
should be corrected from the version of Rufinus. And Bede, in his De
Ratione Computi, also follows the faulty rendering of Rufinus, and
writes Aristobulus et Paniada, as though the latter word were the
proper name of a Jewish writer, finding probably in the Codex of
Rufinus, which he possessed, the reading Aristobulus et Paneada, which
indeed is found in a very ancient Paris manuscript, and also in the
Codex Corbeiensis. But that that Aristobulus was not one of the
seventy translators, as Anatolius writes, is proved by Scaliger in the
work cited above. This Aristobulus was also surnamed dioa'skalos, or
Master, as we see from the Maccabees, ii. 1. For I do not agree with
Scaliger in distinguishing this Aristobulus, of whom mention is made
in the Maccabees, from the Peripatetic philosopher who dedicated his
Commentaries on the Law of Moses to Ptolemy Philometor--Vales. [See
vol. ii. p. 487, and Elucidation II. p. 520, same volume, this
series.]
[1145] ta' diareteria tho'ein.
[1146] kuriaka`s apodei'xeis--Christophorsonus renders it ratas;
Rufinus gives validissimas assertiones. The Greeks use ku'rios in this
sense, kupi'ai di'kai, doxai, &c., decisive, valid, judgments,
opinions, &c.
[1147] The text gives apaiton perie'j3retai, &c.; various codices
read ap' au'tosn, &c. Valesius now proposes ulas apaiton o
perie'j3retai, I shall pass on without...for the veil is removed from
me.
[1148] An apocryphal book of some antiquity, which professes to
proceed from the patriarch of that name, but of whose existence prior
to the Christian era there is no real evidence. The first author who
clearly refers to it by name is Tertullian. [Vol. iii. p. 62, and iv.
380.]
[1149] xiv. luna. The Romans used the phrase luna prima, secunda,
&c., as meaning, the first, second day, &c., after new moon.--Tr.
[1150] Exod. xii. 18, 19.
[1151] 0 Exod. xii. 15; Levit. xxiii. 6.
[1152] Matt. xxvi. 17; Mark xiv. 12; Luke xxii. 7.
[1153] But the text gives 12th.
[1154] [Vol. iii. p. 630. The convenire ad of Irenaeus is thus
shown to be geographical, not ecclesiastical. Vol. i. pp. 415, 569.]
[1155] Matt. xxvi. 38.
[1156] Luke xv. 6.
[1157] Lucidum.
[1158] Levit. xxiii. 5-7.
[1159] Celeberrimus, honoured, solemn.
[1160] Solemn.
[1161] 0 [The sanctification of the Lord's Day is thus shown to be a
Christian principle. The feast of Easter was the Great Lord's Day, but
the rule was common to the weekly Easter.]
[1162] Annorum circuli principium inchoandum est.
[1163] In quo autumnalis novissima pars vincitur.
[1164] Lunae orsibus.
[1165] Diminuitur. [This year (1886) we have the lowest possible
Easter.]
[1166] Temporum confinia.
[1167] [Compare what is said of Hippolytus, vol. v. p. 3, this
series. See the valuable work of Professor Seabury on the Calendar,
ed. 1872.]
[1168] Fabricius, Biblioth. Graeca, ed. Harles, vol. iii. p. 462.
Hamburg, 1793.
[1169] theori'as kai` pra'xeos.
[1170] mathe'mata.
[1171] to` epe'mata.
[1172] ma'thesin.
[1173] ei`likrine, absolute.
[1174] ulen.
[1175] noete'n.
[1176] theoretiko's.
[1177] 0 tou`s pro`s allela lo'gous.
[1178] so'mata, substances.
[1179] episte'me theoretike'.
[1180] pro`s te`n ton hupopipto'nton do'sin.
[1181] Iliad, iv. 442-443 (Pope).
[1182] semei'ou kai` grammes.
[1183] to` harchitektoniko'n.
[1184] analogias.
[1185] archa's, beginnings.
[1186] peripeteia, reversal of circumstances on which the plot of a
tragedy hinges.
[1187] 0 A native of Abdera, in Thrace, born about 460 B.C., and,
along with Leucippus, the founder of the philosophical theory of
atoms, according to which the creation of all things was explained as
being due to the fortuitous combination of an infinite number of atoms
floating in infinite space.
[1188] A famous physician, a native of Bithynia, but long resident
in great repute at Rome in the middle of the first century B.C. He
adopted the Epicurean doctrine of atoms and pores, and tried to form a
new theory of disease, on the principle that it might be in all cases
reduced to obstruction of the pores and irregular distribution of the
atoms.
[1189] onkois.
[1190] [Wisd xi. 20; Ecclus. xxxviii. 29 and xlii. 7.]
[1191] te`n epistemonike` theori'an.
[1192] sulle'bden katalabein po'sa te orisme'nej3 ousiaj3
sumbe'beken.
[1193] A native of Rhodes, a disciple of Aristotle, and editor of
his works.
[1194] A native of Chios, mentioned by Plato in connection with
Anaxagoras, and therefore supposed by some to have been a contemporary
of the latter sage.
[1195] peristasin, revolution.
[1196] Of Miletus, one of the sages, and founder of the Ionic
school.
[1197] 0 Of Miletus, born 610 B.C., the immediate successor of Thales
in the Ionic school of philosophy.
[1198] mete'oros.
[1199] Of Miletus, the third in the series of Ionic philosophers.
[1200] apechousin alle'lon.
[1201] De Decret. Nic. Syn., 25, Works, vol. i. part i. p. 230.
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