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TRANSACTION'S OR THE TEXAS ACADEMY OR SCIENCE. 
must determine, but which so obtained, is found such, without 
noteworthy error, that the geometry assumed by all for practical 
measurements more than suffices, even if in itself it be not rigor- 
ously true. This means either that this system is found in Nature 
by chance, or else that in it all distances accessible to us are still 
infinitesimal. 
In general must every proposition that the imaginary geometry, 
if it is applied to lines of great extent, gives about the elements of 
a magnitude, necessarily lead to the rules of the ordinary geom- 
etry, because in this case only the first powers are retained of the 
numbers which represent the lines, and consequently everywhere 
only their ratios enter into the equations. Such for example are, 
that the distances between two perpendiculars to a straight are 
everywhere equal; that the perpendicular with its extremity de- 
scribes a straight line; that the circle with increasing diameter 
passes over into a straight line. 
Of all known theorems of this sort one must give first rank to 
that which carries with it the dependence of the ratio of lines upon 
the angles; at least here the simplicity in the idea corresponds to 
our first impressions; but also that is all that can be said in its de- 
fence; every other judgment is either false or superficial. 
Again it is not possible to find objectionable, that with the im- 
mediate dependence of lines upon angles a magnitude enters which 
is just as arbitrary as the choice of the unit. To this we may 
answer, that nothing prevents taking in the equations not the 
ratios of the lines to one of those therein considered, but the ratios 
to a magnitude determined in any way in Nature. That I have 
shown in the Imaginary Geometry by giving equations where all 
lines appear in relation to a single one to be determined from ob- 
servations, if these should suffice. 
I consider it not necessary to analyze in detail other assump- 
tions, too artificial or too arbitrary. 
Only one of them yet merits some attention; the passing over of 
the circle into a straight line. However, the fault is here visible 
b eforehand in the violation of continuity, when a curve which does 
not cease to be closed however great it may be transforms itself 
directly into the most infinite straight, losing in this way an essen- 
tial property. In this regard the imaginary geometry fills in the 
interval much better. In it if we increase a circle all of whose 
diameters come together at a point, we finally attain to a line such 
that its normals approach each other indefinitely, even though they 
can no longer cut one another. 
