16 
TRANSACTIONS OF THE TEXAS ACADEMY OF SCIENCE. 
I have made a first trial of it in algebra; I am now to undertake 
it for geometry. 
Pure analysis, without any intermixture of synthesis, ought to 
be introduced in geometry only after haying established equations 
for every sort of dependence, and given expressions for all geo- 
metric magnitudes. 
We can not comprehend a geometric magnitude without meas- 
uring it, an operation which we can not effect, rigorously speaking, 
either for curves or for curved surfaces. In fact, however little 
may be the parts of a curve, they do not cease to be curves, and 
consequently they can not be compared with a straight; just as 
parts of a curved surface are not comparable with portions of a 
plane. 
, From another side, lines straight or curved, planes and curved 
surfaces do not exist in nature; we encounter only bodies, so that 
all the rest, created by our imagination, exist only in theory. 
Lagrange admits in principle the proposition of Archimedes, 
that on a curve one can always take two points so near that the arc 
between these points may be considered as being greater than its 
chord, and smaller than the broken line formed by the two tan- 
gents touching at its extremities. (Theorie des fonctions analy- 
tiques, par Lagrange.) 
Such a proposition is, in fact, necessary, but it destroys by itself 
the primitive idea of measuring curves with straights. The same 
thing happens when one proposes to measure curved surfaces with 
planes. 
Thus the evaluation of the length of a curve represents not at 
all the rectification of the curvature; but it has for aim the finding 
of a limit to which the magnitude that would be obtained by a real 
measuring would approach the more as this measure became more 
exact. Now measuring is made more exactly as the chain em- 
ployed has smaller links; it is done altogether exactly when in 
place of a chain a perfectly flexible thread is used. This is why 
in geometry one is obliged to prove that as the subdivisions are in- 
creased the sum of tangents decreases and that of chords increases, 
until these two sums differ indefinitely little from the common 
limit toward which they tend, and which is considered as the 
length of the curve. 
It is then clear that exhaustion by such rules accords so much 
the more with direct measuring as this is the more exact. From 
this is also seen on what is founded the proposition of Archimedes. 
As we have reasoned on lines, we should reason on curved sur- 
faces, not pretending at all that very small parts of tliese latter 
can be planified. 
