The elements of non-Euclidean plane geometry and trigonometry, by H. S. Carslaw.

6, 7]

SACCHERI AND LEGENDRE

15

thesis of the Acute Angle, there are an infinite number of rays
which have a common perpendicular with the line b. These
rays obviously cannot intersect the line b.
There is no last ray of this set, although the length of the
common perpendicular decreases without limit; but there is
a lower limit to the set.
Also, proceeding from the line AB, we have a set of rays which
intersect the line b. There is no last ray of this set; but there
is an upper limit to the set.
The upper limit of the one set and the lower limit of the other,
he showed to be one and the same ray.
Thus, there is one ray, the line a,, which divides the pencil
of rays into two parts, such that all -he rays on the one side
of the line a1, beginning with AB, intersect the line b; and all
the rays on the other side of the line a,, beginning with the
line AX, perpendicular to the line AB, do not intersect b.
The line a1 is the boundary between the two sets of rays, and
is asymptotic to b.
The result which Saccheri obtained is made rigorous by the
introduction of the Postulate of Dedekind. According to that
postulate a division of the two classes such as is described above
carries with it the existence of a ray separating the one set of
lines from the other.
This ray, which neither intersects b nor has with it a common
perpendicular, is the right-handed (or left-handed) parallel of
Bolyai and Lobatschewsky to the given line.
~ 7. The Work of Legendre (1752-1833).
The contribution of Legendre must also be noticed. Like
Saccheri, he attempted to establish the truth of Euclid's
Postulate by examining in turn the Hypothesis of the Obtuse
Angle, the Hypothesis of the Right Angle, and the Hypothesis
of the Acute Angle. In his work these hypotheses entered as
assumptions regarding the sum of the angles of a triangle.
If the sum of the angles of a triangle is equal to two right
angles, the Parallel Postulate follows; at any rate, if we
assume, as Euclid did, the Postulate of Archimedes.*
Legendre thus turned his attention to the other two cases.
He gave more than one rigorous proof that the sum of the
angles of a triangle could not be greater than two right angles.
*Cf. Heath's Euclid, vol. i. pp. 218-9.