A History of Mathematics- From Mesopotamia to Modernity

Geometries andSpace 197

3. Space and infinity

As lines (so loves) oblique may well
Themselves in every corner greet:
But ours so truly parallel,
Though infinite, can never meet. (Andrew Marvell,The Definition of Love, in 1972, p. 50)

The point has already been made that the axioms of Euclid’s geometry do not properly apply to
what Greeks thought of as ‘space’; and this distinction between the two objects of study seems
to have remained constant until about 1600. That lines can indeed be infinite is, on the other
hand, apparently assumed by Marvell in the quote above (about 1650), a neat poetic formulation
of postulate 5. Something had changed in the way space was thought about which was to pose
further problems. Geometry had not only to be ‘self-evident’ in terms of some idea of common
sense, it was necessary that it should describe the world.
It would appear that Thomas Bradwardine (fourteenth century), in a typically scholastic
approach, was among the first to consider the idea of infinite space—if not as where we live
then as a property of God:

God must imagine the site of the world before creating it; and since it is absurd to imagine a limited empty space, what

God imagines is the infinite space of geometry...‘Indeed, He coexists fully with infinite magnitude and imaginary

extension and with each part of it’. (Torelli 1978, p. 28, quoting Bradwardine,De causa Dei)

Of course, such speculations coexisted with the more extreme ones such as whether God could

create a triangle whose angles did not add to two right angles; but in terms of a unification of

geometric space with the actual universe, the idea gained ground through more radical early

modern thinkers such as Giordano Bruno (sixteenth century), Descartes, and finally Newton. Was

Descartes’s need for infinite space (‘the extension of the world is indefinite’) related to his revolution

in geometry? It would not appear so, since Descartes’s plane is still an abstract one, a copy of Euclid’s

with numbers introduced. [And if you look back to the extracts from theGeometryin Chapter 6,

you will see that he needs parallels to introduce the numbers.] Rather, it is a consequence of his

physicallaw, equivalent to Newton’s First Law, that

a freely moving body will always continue to move in a straight line—thereby perpetually performing the construction

demanded by Euclid’s second postulate—and this would be impossible if every distance in the world were less than or

equal to a given magnitude. (Torelli 1978, p. 24, referring to Descartes Princ. Phil)

We noted in Chapter 7 the importance of Greek geometry for Newton’s later work, specifically

thePrincipia. Indeed, Newton went further than Descartes by constructing a vast scheme of how

all matter in the universe behaved. Here the universe was explicitly identified with the space of

Euclidean geometry, in which straight lines have indefinite extension. (Infinite, if you are freer

in your language.) Again, it was impossible for the laws of physics to work without such an

identification, but it is important to stress that this was relatively new. If Newton had to some extent

borrowed the idea from Descartes he certainly made it much more explicit in the whole geometric

and deductive structure of thePrincipia. With Descartes and Newton, a great step forward is

achieved, in that geometrycanapply to physical space. As for the application of postulate 5,

it means that any inclined lines (in the same plane) will meet somewhere in the universe. The

drawback of this is that questions about geometry may become identical with questions about the