14 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
termine one dimension for a curved line, and two for a curved 
surface? 
True, one need not require that length, breadth and height be 
perpendicular to one another; it suffices, if one has taken for them 
lines in different directions. Yet we meet also in this case pecu- 
liar difficulties. 
If we take as principle not to anticipate ideas to be developed 
later, the question is how then to express the condition that the 
three dimensions of a solid shall appertain to three straights in 
different planes? 
Further, one must not confuse the different direction of the two 
parts which go out from a break in the line, with the twofold ex- 
tension of a surface; finally one must set forth completely what is 
to be understood precisely by a direction and by an angle. 
In short, space, dimension, locus, solid, surface, line, point, direc- 
tion, angle, are words with which one begins geometry, with which 
one however never connects a clear idea. 
However, we may consider all these things from still another 
side. It is necessary to observe that here the obscurity in the idea 
is produced by the abstractness, which is superfluous in the appli- 
cation to actual measurement, and consequently is introduced use- 
lessly into the theory. 
Surfaces, lines, points, as geometry defines them, exist only in 
our imagination; when we treat the measurement of surfaces and 
lines, we apply thereto solids. 
For this reason we need only to speak of surfaces, lines, points, 
as we must think them in an actual measurement, and then we will 
hold ourselves only to just the ideas which are immediately con- 
nected in our understanding with the representation of bodies, to 
which our imagination is accustomed and which we can in nature 
immediately verify, without entering upon others, artificial and 
strange. 
But with these new ideas the science takes even at the beginning 
another direction, which it follows until it goes over into analysis, 
where then the procedure in exposition now again takes the ordi- 
nary aspect. I will endeavor to explain wherein consists this 
change. 
In mathematics one follows two methods, Analysis and Syn- 
thesis. 
Equations make the distinctive characteristic of. analysis. They 
serve as first basis for every judgment and in general lead to all 
conclusions. 
Synthesis, or the method of building up, requires exactly the 
