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.

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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.

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:

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.

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