es a 
XXXVI. On the Theory of parallel Lines in Geometry. By 
James Ivory, M.A. F.R.S. 
Is laying down the elements of mathematical science, great dif- 
ficulties occur at the outset. In arithmetic we are immediately 
embarrassed with the doctrine of incommensurable quantities. 
In geometry, the manner of treating the subject of parallel lines 
is a blemish which the efforts of ancient and modern mathema- 
ticians have equally failed to remove. In the same science seme 
obscurity and even mistakes prevailed with respect to the equa- 
lity of solid figures, till accuracy and precision were introduced 
by the publication of Legendre’s Elements, one of the ablest and 
mosf original works that has appeared in modern times. The 
comparison of the pyramid with the prism must also have occa- 
sioned some perplexity to the first authors who wrote on geo- 
metry. In advancing further, new difficulties occur at every 
step; as when we would compare the lengths of curve and 
straight lines; or when we would determine the proportion be- 
tween curve and plane surfaces. 
In algebra some obscurity has arisen from what are called ne- 
gative quantities. But it would be very inaccurate to assimilate 
the seeming paradoxes and apparent contradictions that arise 
from the doctrine of negative quantities in algebra to the real 
difficulties that are met with in geometry. ‘The latter are un- 
avoidable, and inherent in the subject : the former originate from 
inaccurate phraseology, and the crude and unphilosophical man- 
ner of treating the elements of a branch of science comparatively 
new, and that, in no great space of time, has been almost im- 
measurably extended. There can be no better argument for 
the truth of what is here advanced, than to observe that the ab- 
surdities attending the use of the negative sign appear only in 
general discussions, and when the quantities affected with it are 
considered abstractly. It is only on such occasions that we hear 
of quantities less than nothing ; or that negative quantities are 
compared to debts, while positive quantities signify real posses- 
sions. These exceptionable modes of speaking are never intro- 
duced in the solution of particular problems. In such cases all 
the relations of the quantities considered are distinctly compre- 
hended, and the algebraist readily accomplishes his purpose by 
means of addition, and the no less clear operation of subtracting 
a less from a greater quantity. It is with these clear notions 
that the mind, in every investigation, sets out; and the diffi- 
culty consists in reconciling them with the generalizations not 
only permitted but required by the genius of algebra. 
Vol, 59. No. 287. March 1822. X Some 
