2. Arithmetic was not practised methodically by the setting of sums to be worked out by the pupils, but consisted chiefly in demonstrating the more obvious properties of numbers, such as odd, even, prime, perfect, etc., together with many fanciful absurdities.[1] Operations with figures were indicated verbally in a disconnected manner; multiplication tables to be learnt by heart had not been invented; the higher rules and decimal fractions were unknown.
3. Systematic astronomy at this period and for long after, as is well known, was conceived of on false principles which, whilst admitting of the correct solution of some problems, such as the prediction of eclipses, left the vastness of the universe and its physical constitution totally unapprehended. All the heavenly bodies were regarded as mathematically, if not teleogically, disposed about the earth, to which as a centre even the fixed stars, at varying and immeasurable distances as they are, were constrained fantastically by a revolving sphere of crystal.[2] The reasoning, however, by which
- [Footnote: to say that black was white"; Lactantius, Div. Inst., iii, 24; Epitome,
39. "The earth stands firm on water [going back to Thales] and does not turn"; Chrysostom, Genesis, Hom. xii, 3, 4 (in Migne, iv, 101); In Titum Hom. iii, 3 (in Migne, xi, 680); cf. Cosmas Ind., op. cit., x, for other theological authorities on cosmology.]), they regarded the objective universe as only one of them, and had no idea that myriads of systems similar to that in which they lived lay before their eyes.]
- ↑ Such as that five represents the world, being made up of three and two, which typify male and female respectively; or that seven equates Minerva, the virgin, neither contained or containing; and other Pythagorean notions; see M. Capella, vii, and the arithmetic of Boethius.
- ↑ Such is the well-known system elaborated by Hipparchus and Ptolemy, but the Pythagoreans put the sun at the centre, though without definite reasons and with imaginative details; see Diogenes Laert. and Delambre's Hist. Astron. Ant. Although Democritus, Epicurus, and others held that there were an infinite number of worlds ([Greek: kosmoi
