Pythagoras: His Life and Teaching, a Compendium of Classical Sources (75 page)

p. 347 note 323.
.

Also rendered the same by Friedlein,
Procli Diadochi In Primum Euclidis Elementorum Librum Commentarii
, p. 36. ‘Barocius' refers to Franciscus Barocius (the Latin name of Francesco Barozzi) who published a translation of Euclid's
Elements
in Venice in the year 1560.

p. 347 note 324.

Freidlein read this passage
. (Friedlein,
Procli Diadochi In Primum Euclidis Elementorum Librum Commentarii
, p. 36)

p. 347 note 326.
“Introduction to Arithmetic” The subsequent text, “so supply the Title, as a page. 30. 35. 44. 62. 76) cap. 4. refers to the Greek
which is the title of the book by Nicomachus by that name, known by its Latin name
Introductionis Arithmeticae.
In Chapter 3, Nicomachus quotes Plato's
Republic, Book VII, Chapter X
, wherein the four
mathemata
of the
quadrivium
(the “higher subjects”—Arithmetic, Geometry, Music and “Spheric,” i.e. Astronomy) are described and defended: “arithmetic for reckoning, distributions, contributions, exchanges and partnerships, geometry for sieges, the founding of cities and sanctuaries, and the partition of land, music for festivals, entertainment, and the worship of the gods, and the doctrine of the spheres, or astronomy, for farming, navigation and other undertaking, revealing beforehand the proper procedure and suitable season.”

Chapter 4 of Nicomachus' “Introduction to Arithmetic” clarifies the reasoning to select Arithmetic as the first of the four methods (Music, Mathematics, Geometry or Spheric) should be studied first. The Chapter is given in full below:

“Which then of these four methods must we first learn? Evidently, the one which naturally exists before them all, is superior and takes the place of origin and root and, as it were, of mother to the others. And this is arithmetic, not solely because we said that it existed before all the others in the mind of the creating God like some universal and exemplary plan, relying upon which as a design and archetypal example the creator of the universe sets in order his material creations and makes them attain to their proper ends; but also because it is naturally prior in birth inasmuch as it abolishes other sciences with itself, but is not abolished together with them. For example, ‘animal' is naturally antecedent to ‘man,’ for abolish ‘animal' and ‘man' is abolished; but if ‘man' be abolished, it no longer follows that ‘animal' is abolished at the same time. And again, ‘man' is antecedent to ‘schoolteacher,’ but if ‘schoolteacher' is nonexistent, it is still possible for ‘man' to be. Thus since it has the property of abolishing the other ideas with itself, it is likewise the older. Conversely, that is called younger and posterior which implies the other thing with itself, but is not implied by it, like ‘musician,’ for this always implies ‘man.’ Again, take ‘horse’; ‘animal' is always implied along with ‘horse,’ but not the reverse; for if ‘animal' exists,
it is not necessary that ‘horse' should exist, nor if ‘man' exists, must ‘musician' also be implied. So it is with the foregoing sciences; if geometry exists, arithmetic must also needs be implied, for it is with the help of this latter that we can speak of triangle, quadrilateral, octahedron, icosahedron, double, eightfold, or one and one-half times, or anything else of the sort which is used as a term by geometry, and such things cannot be conceived of without the numbers that are implied with each one. For how can ‘triple' exist, or be spoken of, unless the number 3 exists beforehand, or ‘eightfold' without 8? But on the contrary, 3, 4, and the rest might be without the figures existing to which they give names. Hence arithmetic abolishes geometry along with itself, but is not abolished by it, and while it is implied by geometry, it does not itself imply geometry.”

(Hoche,
,
Nicomachi Geraseni Pythagorei Introductionis Arithmeticae
, pp. 9-10, D'Ooge,
Introduction to Arithmetic, by Nicomachus of Gerasa
(pp. 181-190)

p. 347 note 342. Eustratius,
Ethica Nicomachea.1
& Servius,
Incipit Expositio Primae Eclogae
, VIII.

Cf. Virgil,
Ecologue
VIII:

“Threefold first I twine about thee these diverse triple-hued threads, and thrice round these altars I draw thine image: an odd number is god's delight.” (McKail,
The Eclogues and Georgics of Virgil
, p. 29)

p. 354 note 599. Meibomius seems to mistake the meaning of
, and therefore puts a point after
.