much as reason. When a proposition is
stated, we have to imagine the demonstration; that is, we have to find
upon what proposition already known the new one depends, and from all
the consequences of this known principle select just the one required.
According to this method the most exact reasoner, if not naturally
inventive, must be at fault. And the result is that the teacher,
instead of making us discover demonstrations, dictates them to us;
instead of teaching us to reason, he reasons for us, and exercises only
our memory.

Make the diagrams accurate; combine them, place them one upon another,
examine their relations, and you will discover the whole of elementary
geometry by proceeding from one observation to another, without using
either definitions or problems, or any form of demonstration than
simple superposition. For my part, I do not even pretend to teach
Emile geometry; he shall teach it to me. I will look for relations,
and he shall discover them. I will look for them in a way that will
lead him to discover them. In drawing a circle, for instance, I will
not use a compass, but a point at the end of a cord which turns on a
pivot. Afterward, when I want to compare the radii of a semi-circle,
Emile will laugh at me and tell me that the same cord, held with the
same tension, cannot describe unequal distances.

When I want to measure an angle of sixty degrees, I will describe from
the apex of the angle not an arc only, but an entire circle; for with
children nothing must be taken for granted. I find that the portion
intercepted by the two sides of the angle is one-sixth of the whole
circumference. Afterward, from the same centre, I describe another and
a larger circle, and find that this second arc is one-sixth of the new
circumference. Describing a third concentric circle, I test it in the
same way, and continue the process with other concentric circles, until
Emile, vexed at my stupidity, informs me that every arc, great or
small, intercepted by the sides of this angle, will be one-sixth of the
circumference to which it belongs. You see we are almost ready to use
the instruments intelligently.

In order to prove the angles of a triangle equal to two right angles, a
circle is usually drawn. I, on the contrary, will call Emile's
attention to this in the circle, and then ask him, "Now, if the circle
were taken away, and the straight lines were left, would the size of
the angles be changed?"

It is