328 Proceedings of the Royal Society 
to the nature of space that follow therefrom, are hereby assumed 
that — (1) The sum of the angles of any triangle can never exceed 
two right angles ; (2) If the sum of the angles of any one triangle 
is two right angles, then the sum of the angles in every triangle is 
two right angles. But independently of the theory of parallels, this 
is in substance as far as we can go. If we assume that the sum of 
the angles of any triangle is less than two right angles, then we arrive 
at the conclusion that this sum depends on the area of the triangle, 
the defect from two right angles being less the less the area, and the 
same for all triangles of the same area, consequently therefore propor- 
tional to the area of the triangle. The effect of this assumption on 
the theory of parallels is very remarkable. Defining parallels as 
straight lines in the same plane that do not intersect (this is not the 
definition adopted in recent books, such as that of Frischauf, above 
named, but that is a mere question of words), we find that there are 
an infinite number of straight lines passing through the same point 
all parallel to a given straight line ; that through one point on one 
of a pair of parallels only one straight can be drawn that makes the 
alternate angles equal ; that parallel straight lines are not equi- 
distant ; that the locus of the points equidistant from a given 
straight line is not a straight but a curved line ; that equal parallelo- 
grams on the same base cannot be between the same parallels, and 
so on. All this, and much more, is shewn by Kelland to form part 
of a system of geometry as logical as Euclid’s. 
As far as can be gathered from the memoir, and the form of the 
demonstrations, all but the fundamental propositions (the mere idea 
in fact) is Kelland’s own work. It is characteristic of the man that 
he was in the habit of treating this subject in his class lectures. 
Clearness in dealing with the fundamental principles of mathe- 
matical science was one of the virtues of Kelland’s thinking and 
teaching. His text-book on Algebra is distinguished over other 
text-books in present use by its attempts to give a rational account 
of the first principles of the subject. The same readiness to grasp 
a new elementary idea, and trace its consequences, is exemplified by 
the fact that he took up Quaternions with his class in the Univer- 
sity, and so late as 1873 published, in conjunction with Professor 
Tait, an excellent elementary treatise on this branch of mathe- 
matics. (See the Preface to that work.) 
