Geometry, it has been satisfactorily shown, had a purely empirical
origin. It appears that the earliest geometrical formulæ which have
been discovered belong to ancient Egypt, and that all these formulæ
served a useful purpose. The oldest of them are concerned with the
measurements of areas, a class of problem which the yearly sinking
of the Nile rendered of great importance. The formulæ obtained
by the ancient Egyptians were usually wrong, although they were
approximately correct; they evidently rested on no theoretical basis,
but were compendious statements of the results of somewhat rough
measurements, a point of view which is borne out by the fact that no
proof, nor even an attempt at a proof, is anywhere hinted at. So far
as the evidence goes, it seems to be established that geometry, as
consisting of logical deductions from stated premises, began with the
Greeks. A number of theorems of a fair degree of complexity had been
developed before they were reduced to a system; before, that is, the
assumptions on which they were based were made explicit. The task of
discovering the necessary and sufficient assumptions on which a system
of geometry rests is one of the greatest difficulty; the necessary
combination of subtlety and rigour is rare. The great systematisation
of Greek geometry was effected, of course, by Euclid, and although
his reduction of the system to its essential assumptions was not
final, his performance was such as to awaken the admiration of great
mathematicians in every succeeding century. But there is one point in
which this great reduction is notably imperfect--the so-called parallel
axiom. It says, essentially, that through a given point only one line
can be drawn parallel to a given straight line. It was felt, even
by the earliest commentators on Euclid, that this postulate did not
possess quite the same degree of self-evidence as was manifested by the
others. It was necessary, they felt, to give a proof of this postulate;
they attempted to improve on Euclid’s work in a number of minor ways,
but it was the parallel axiom which they were most concerned to revise;
the proof of this postulate should be contained, they thought, in the
other postulates. The attempts to supply this proof were all fruitless,
and the sixth century was reached with this nine-hundred-years-old
disfigurement still persisting. For some time after the sixth century
the world rested from Euclid’s parallel axiom; indeed, it rested from
geometry altogether, and the old empirical outlook of the Egyptians,
and even their formulæ, again became current. But the Greek culture
penetrated to the Arabs, and with the Greek culture came the riddle of
Euclid’s axiom. Again proofs were attempted; a famous attempt is that
of Nasir Eddin, who flourished in the thirteenth century. In 1663 John
Wallis made the important discovery that unless the parallel axiom be
assumed, similar figures of different sizes are not possible, that
is to say, that if we are to assume that _shape_ is independent
of _size_, then we must assume Euclid’s parallel axiom. Many of
these attempts brought out points of interest, but none of them were
successful. In the year 1733, however, the whole research took on a new
complexion with the publication of Girolamo Saccheri’s _Euclides ab
omni naevo vindicatus_. The importance of this work consists in the
fact that, although it was written to vindicate Euclid’s parallel axiom
once for all, it contains the first real outline of a non-Euclidean
geometry.
Saccheri was a Jesuit, and it was in 1690, while he was teaching
grammar in Milan, that he first studied the _Elements_ of Euclid.
He was a man of very great acumen, and when he, in turn, succumbed to
the spell of the parallel postulate, he brought to bear on it a more
subtle and rigorous logic than had yet been applied to it. Thirty-six
years before he published his treatise on Euclid he had published a
book on logic which gives him a high place as a logician. In it he
is particularly concerned with investigating the compatibility of
different assumptions or postulates. His method was to determine
whether a member of a group of postulates is independent of the others
by finding a particular case in which the postulate in question is not
true while all the others remain true. If such a case can be found, it
is obvious that the postulate in question cannot be deduced from the
others, else it would be true whenever they were true. This was the
method he applied to the parallel postulate of Euclid. He showed that
the parallel postulate is equivalent to saying that the three interior
angles of a triangle are equal to two right angles. He proceeds,
therefore, in accordance with his method, to develop the consequences
of supposing them less than, or greater than, two right angles. In
the latter case he succeeds in showing that we are led to impossible
conclusions, since he assumed, as everybody assumed for more than a
century after, that the straight line is of infinite length. But in the
former case, the hypothesis that the interior angles of a triangle are
together less than two right angles, Saccheri, although he struggled
very hard, did not succeed in falling into contradictions. He does not
seem to have had the boldness necessary completely to trust his own
logic, but the fact remains that, accepting the rest of Euclid’s axioms
and denying the parallel axiom, he developed a logically consistent
geometry.
There is reason to suppose that Saccheri’s work had some influence on
subsequent thought, although its full significance was certainly not
perceived. The parallel axiom continued to be investigated, and the
total effect of all these efforts was to induce a doubt concerning
the absolute _necessity_ of the Euclidean geometry. Such a doubt
was very daring; for two thousand years the postulates of Euclid had
been accepted as absolutely true; the fact of their existence had
profoundly influenced philosophy, and, indeed, theology. But the
doubt persisted and grew, until finally, early in the nineteenth
century, a perfectly logical and consistent non-Euclidean geometry,
one explicitly denying the parallel postulate, was published to the
world. As so often happens, the great step was taken by two men
independently of one another, Lobatschewski, a Russian, and Bolyai, a
Hungarian. It appeared, however, that both had been preceded by that
great mathematical genius, Gauss, although he had been too timid to
publish his conclusions. The new geometry developed the consequences of
that one of Saccheri’s alternatives which supposed the interior angles
of a triangle to be less than two right angles. The whole outlook on
geometry now assumed a new complexion. Riemann tried the effect of
denying the infinity of the straight line and of developing Saccheri’s
other alternative. He found he was led to no contradictions. But with
Riemann’s work we come to a yet further extension of geometry--the
extension to space of four, five, or any number of dimensions. And
these investigations, which seemed for some time to constitute the most
gratuitous, although the most profound and subtle, exercises of the
mind, have now received their complete justification by flowering into
the Generalised Principle of Relativity.