"The first geometric formula is this: only straight lines may be drawn from one given point to another.
"Starting from this axiom, which the hand alone can demonstrate, I have fixed two points upon the blackboard and have connected them by means of a vertical. My pupils try to do the same between the dots they have upon their paper, but with some the vertical descends to the right of the point and with others, to the left, to say nothing of those whose hand diverges in all directions. To arrest these various deviations which are often far more defects of the intelligence and of the vision, than of the hand, I have thought it wise to restrict the field of the plane, drawing two vertical lines to left and right of the points which the child is to join by means of a parallel line half way between the two enclosing lines. If these two lines are not enough, I place two rulers vertically upon the paper, which arrest the deviations of the hand absolutely. These material barriers are not, however, useful for very long. We first suppress the rulers and return to the two parallel lines, between which the idiot learns to draw the third line. We then take away one of the guiding lines, and leave, sometimes that on the right, sometimes that on the left, finally taking away this last line and at last, the dots, beginning by erasing the one at the top which indicates the starting point of the line and of the hand. The child thus learns to draw a vertical without material control, without points of comparison.
"The same method, the same difficulty, the same means of direction are used for the straight horizontal lines. If, by chance, these lines begin well, we must await until the child curves them, departing from the centre and proceeding to the extremity _as nature commands him_, and because of the reason which I have explained. If the two dots do not suffice to sustain the hand, we keep it from deviating by means of the parallel lines or of the rulers.
"Finally, have him trace a horizontal line, and by uniting with it a vertical ruler we form a right angles. The child will begin to understand, in this way, what the vertical and horizontal lines really are, and will see the relation of these two ideas as he traces a figure.