1. It is most certain that there are infinite numbers, since at
whatever number you think an end must be made I say not only that it
can be increased by the addition of one, but, however great it is,
and however large a multitude it contains, by the very method and
science of numbers it can not only be doubled but even multiplied.
2. Each number is limited by its own proper qualities, so that no
one of them can be equal to any other. Therefore in relation to one
another they are unequal and diverse, and the separate numbers are
each finite, and all are infinite.
ON GEOMETRY
INTRODUCTION
In spite of the high development of geometry among the Greeks it
never took root as a pure science in the western Roman world,[249]
and neither the various practical applications of its principles
nor its use as a disciplinary educational subject sufficed to
fasten thoughtful attention upon it; in consequence, it lost almost
its entire content. As it appears in the four writers who treat
of it in later Roman and early medieval times, Martianus Capella,
Boethius,[250] Cassiodorus, and Isidore, it furnishes a striking
commentary upon the intellectual conservatism that could retain
without a suspicion of criticism a subject that was no longer
anything but empty form.
[249] Cantor, _Vorlesungen über Geschichte der Mathematik_, vol.
i, p. 521.
[250] The authenticity of the work on geometry that has been
handed down under Boethius’ name is questioned. (See Cantor,
_ibid._, pp. 536 _et seq._) It contains the complete proof
of only three of Euclid’s propositions. It also contains
calculations of areas of geometrical figures. See edition of
Friedlein (Leipzig, 1867).
The substance of Isidore’s _De Geometria_ comes with little change
from Cassiodorus. It is noteworthy that these two writers have
nothing that does not go with the subject according to the modern
conception of it, and do not follow the example of their predecessor
Martianus Capella,[251] in whose account of the seven liberal arts
the void caused by the loss of the proper content of geometry is
filled with geography.
[251] _Cf._ Martianus Capella’s definition: “Geometria vocor
quod permeatam crebro admensamque tellurem eiusque figuram,
magnitudinem, locum, partes et stadia possim cum suis rationibus
explicare neque ulla sit in totius terrae diversitate partitio
quam non memoris cursu descriptionis absolvam.” Eyssenhardt, 198,
30.
TRANSLATION[252]
[252] The whole of Isidore’s _De Geometria_ is here given,
with the exception of a few passages that are untranslatable.
It is given as a whole to enforce attention to the loss of the
traditional content, partial or complete, which was so striking
a feature of all the members of the quadrivium in early medieval
times.
Book III, Chapter 10. On the inventors of geometry and its name.
1. The science of geometry is said to have been discovered first by
the Egyptians, because when the Nile overflowed and all their lands
were overspread with mud, its origin in the division of the land by
lines and measurements gave the name to the art. And later, being
carried further by the keenness of the philosophers, it measured the
spaces of the sea, the heavens, and the air.
2. For, having their attention aroused, students began to search into
the spaces of the heavens, after measuring the earth; how far the
moon was from the earth, the sun itself from the moon, and how great
a measure extended to the summit of the sky; and thus they laid off
in numbers of stades with probable reason the very distances of the
sky and the circuit of the earth.
3. But since this science arose from the measuring of the earth, it
took its name also from its beginning. For _geometria_ is so named
from the earth and measuring. For the earth is called γῆ in Greek,
and measuring, μέτρον. The art[253] of this science embraces lines,
intervals, magnitudes, and figures, and in figures, dimensions and
numbers.
[253] _Hujus ars disciplinae_. _Ars_ may be equal to ‘hand-book’
here.