[1189] In quo autumnalis novissima pars vincitur.
[1190] Lunæ orsibus.
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; [1191] 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.
[1191] Diminuitur. [This year (1886) we have the lowest possible Easter.]
XVII.
Be not ignorant of this, however, that those four determining periods, [1192] 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.
[1192] Temporum confinia.
[1193] [Compare what is said of Hippolytus, vol. v. p. 3, this series. See the valuable work of Professor Seabury on the Calendar, ed. 1872.]
Fragments of the Books on Arithmetic. [1194]
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What is mathematics?
Aristotle thinks that all philosophy consisted of theory and practice, [1195] 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 Chaldæans 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, [1196] 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 mathema, 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 [1197] in them, and that in a manner suitable to man's study. [1198] For they (the Pythagoreans) perceived that these studies dealt with things eternal and immutable and perfect, [1199] 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 [1200] incorporeal, and falling simply within the province of the understanding, [1201] but also with that which touches upon corporeal and sensible matter. For he ought to be cognisant of [1202] 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, [1203] 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, [1204] or at least are to be ultimately referred to sensible matter.
What is mathematics?
Mathematics is a theoretic science [1205] of things apprehensible by perception and sensation for communication to others. [1206] 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." [1207]
For it begins with a point and a line, [1208] 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, [1209] 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. [1210] 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 [1211] 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; [1212] 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 [1213] used a hypothesis, namely, that of atoms and a vacuum; and Asclepiades [1214] that of atoms [1215] 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." [1216]
That arithmetic has as its immediate end chiefly the theory of science, [1217] 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. [1218]
Who among the mathematicians has made any discovery?
Eudemus [1219] relates in his Astrologies that OEnopides [1220] found out the circle of the zodiac and the cycle [1221] of the great year. And Thales [1222] discovered the eclipse of the sun and its period in the tropics in its constant inequality. And Anaximander [1223] discovered that the earth is poised in space, [1224] and moves round the axis of the universe. And Anaximenes [1225] 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 [1226] 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.
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[1194] Fabricius, Biblioth. Græca, ed. Harles, vol. iii. p. 462. Hamburg, 1793.
[1195] theorias kai praxeos.
[1196] mathemata.
[1197] to epistemonikon.
[1198] mathesin.
[1199] eilikrine, absolute.
[1200] hulen.
[1201] noeten.
[1202] theoretikos.
[1203] tous pros allela logous.
[1204] somata, substances.
[1205] episteme theoretike.
[1206] pros ten ton hupopiptonton dosin.
[1207] Iliad, iv. 442-443 (Pope).
[1208] semeiou kai grammes.
[1209] to architektonikon.
[1210] analogias.
[1211] archas, beginnings.
[1212] peripeteia, reversal of circumstances on which the plot of a tragedy hinges.
[1213] 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.
[1214] 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.
[1215] onkois.
[1216] [Wisd. xi. 20; Ecclus. xxxviii. 29 and xlii. 7.]
[1217] ten epistemoniken theorian.
[1218] sullebden katalabein posa te horismene ousia sumbebeken.
[1219] A native of Rhodes, a disciple of Aristotle, and editor of his works.
[1220] 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.
[1221] peristasin, revolution.
[1222] Of Miletus, one of the sages, and founder of the Ionic school.
[1223] Of Miletus, born 610 b.c., the immediate successor of Thales in the Ionic school of philosophy.
[1224] meteoros.
[1225] Of Miletus, the third in the series of Ionic philosophers.
[1226] apechousin allelon.
alexander_capp epistles anf06 alexander_capp-epistles From the Epistles of Alexander /ccel/schaff/anf06.vi.iv.html