The _results_ of reasoning may be hit upon by accident. The stereoscope was actually a result of reasoning; it is conceivable, however that a man playing with pictures and mirrors might accidentally have hit upon it. Cats have been known to open doors by pulling latches, etc. But no cat, if the latch got out of order, could open the door again, unless some new accident of random fumbling taught her to associate some new total movement with the total phenomenon of the closed door. A reasoning man, however, would open the door by first analyzing the hindrance. He would ascertain what particular feature of the door was wrong. The lever, e.g., does not raise the latch sufficiently from its slot--case of insufficient elevation: raise door bodily on hinges! Or door sticks at bottom by friction against sill: raise it bodily up! How it is obvious that a child or an idiot might without this reasoning learn the _rule_ for opening that particular door. I remember a clock which the maid-servant had discovered would not go unless it were supported so as to tilt slightly forwards. She had stumbled on this method after many weeks of groping. The reason of the stoppage was the friction of the pendulum-bob against the back of the clock-case, a reason which an educated man would have analyzed out in five minutes. I have a student's lamp of which the flame vibrates most unpleasantly unless the chimney be raised about a sixteenth of an inch. I learned the remedy after much torment by accident, and now always keep the chimney up with a small wedge. But my procedure is a mere association of two totals, diseased object and remedy. One learned in pneumatics could have abstracted the _cause_ of the disease, and thence inferred the remedy immediately. By many measurements of triangles one might find their area always equal to their height multiplied by half their base, and one might formulate an empirical law to that effect. But a reasoner saves himself all this trouble by seeing that it is the essence (_pro hac vice_) of a triangle to be the half of a parallelogram whose area is the height into the entire base. To see this he must invent additional lines; and the geometer must often draw such to get at the essential property he may require in a figure. The essence consists in some _relation of the figure to the new lines_, a relation not obvious at all until they are put in. The geometer's genius lies in the imagining of the new lines, and his sagacity in the perceiving of the relation. =Thus, there are two great points in reasoning.= _First, an extracted character is taken as equivalent to the entire datum from which it comes; and_, _Second, the character thus taken suggests a certain consequence more obviously than it was suggested by the total datum as it originally came._ Take these points again, successively.