Egberts — Modern Mathematics. 
155 
However, the want of all the modem methods never prevented 
jome of the older school of mathematicians from arriving at the 
, 9lutions of problems to which they had devoted themselves. In fact, 
ihese powerful intellects were able, no doubt, to really use the modern 
' lachinery intuitively. We see how great mathematicians frequently 
rrived at results, of whose correctness they were assured, but by 
leans of which they found it difficult to give an explanation ; as, for 
I'lstance, Laplace, whose second proof of Legendre's problem, in the 
mrth chapter of the third book of the Mecanique Celeste was shown 
) be altogether unsatisfactory by Liouville, in his Journal de MatM- 
itttiques^ t. ii., 1837. 
The modern ideas seem to place the true position of mathematics 
L a clearer light, both with respect to philosophy, on the one hand, 
lid physical science, that is experience, on the other. Mathematics 
rone seem to give us a firm standpoint of absolute truth, completely 
i ee from the mist of our subjectivity ; for although we arrive at the 
L'imary mathematical truths by means of experience, yet we have a 
fiUviction that they are necessary and do not depend upon that source, 
iill, however, disputes this; and, among other remarks on the 
'Object, observes that there exist no real things exactly conformable 
' the definitions of geometry; also, that we cannot conceive a line 
ithout breadth, or form a mental picture of such a line. This may, 
course, be conceded ; but it may seem a preferable view to regard 
iometrical conceptions rather as qualities than as actual embodiments 
I matter ; for instance, we may regard a surface as the limiting form 
a portion of matter. In any case, our inability to realize in nature 
3 so-called imaginary objects arises from the fact that we are finite 
lings, and cannot secure perfection in the infinitely small. 
Since the modern mathematical ideas have come into operation, 
ometry has broken away from, and become almost independent of, 
I perience — the original foundation on which it stood. In the Eacli- 
' m geometry we start from lengths and angles. The circle is the most 
: [portant curved line which comes under our notice, and these con- 
otions, at least from Eiemann's point of view, depend upon 
perience. But the modern geometry can be built up without any 
.■erence to lengths and angles, or of such a curve as a circle. We 
\i then assume certain properties of space, and hence deduce a con- 
tent geometry of lengths and angles. These assumptions may be 
Lde so as to give us either the Euclidian or the elliptic and other 
jids of space. In these kinds of space lengths and angles are the 
: ms assumed by certain invariants in the more general geometry. 
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