GEORGE BRUCE HALSTED NEW ELEMENTS OF GEOMETRY.. 
15 
same representation which in our mind is connected immediately 
with the primary ideas. 
The principal advantage of analysis consists in this, that from 
the equations one goes always straight to the end proposed. 
Synthesis is in general subject to no universal rule, but one 
must necessarily make the beginning with it, in order finally, after 
one has found the equations, therewith likewise to reach the limit 
beyond which now all goes over into the science of numbers. 
For example, one demonstrates in geometry that two straights 
perpendicular to a third never meet; that the equality of triangles 
follows from that of certain of their parts. In vain would one seek 
to treat analytically propositions of this species, even as all the 
theory of parallels. One would never succeed, just as one would 
not be able to do without synthesis for measuring plane rectilineal 
figures, or solids terminated by plane surfaces. 
It is well understood that in synthesis we should use the helps 
offered by analysis; but it is incontestible that in the beginnings 
of geometry or mechanics, analysis can not serve as sole method. 
One may compress the circle of synthesis; but it is impossible to 
suppress it completely. 
It is not even necessary too much to hasten the substitution of 
analysis for synthesis, and the introduction of functions each time 
some dependence can be perceived between quantities, without yet 
knowing in what this dependence consists and how it expresses 
^ itself. 
With this restriction, analysis is the sole method which, intro- 
duced in the science to begin with notions where reasoning alone 
is thenceforth sufficient, extends the bounds of our knowledge in- 
definitely in all directions. 
Notions acquired in nature by the aid of our senses are without 
any doubt the primitive data. 
The mind can and should as much as possible reduce their num- 
ber, to make them afterwards the solid bases of the science. 
No one, however, uses synthesis from this point of view, observ- 
ing the rules we have just indicated. 
It has generally been chosen to introduce prematurely analysis, 
and to develop the notions formed naturally in our mind, and to 
which remains only to give a name, without entering profoundly 
enough into the explanation, and without considering the exacti- 
tude of the definitions. 
If facility and simplicity did not require to choose a method of 
instruction like that I recommend, its superiority in rigor would 
still make us sometimes employ it to g'reat advantage. 
