The apparatus is represented in the accompanying figure.  An iron ruler, I, parallel with the axis of the X’s, is fixed upon a drawing-board, and is provided with a longitudinal groove in its upper surface.  In this groove move two rollers, which, in the center of the piece that connects them, carry two brass T-squares that are parallel with each other and at right angles with the first, or parallel with the axis of the Y’s.  Between these two rulers move two carriages, the first of which (nearest the axis of the X’s) carries a point, A, designed to follow the contour of the curve to be integrated, while the second, which is placed further away, is provided at the center with a drawing-pen, A’, whose point is guided by two equidistant wheels, R, R’, that roll over the paper in such a way as to have their plane parallel with a given straight line, and that have always a direction such that the tangent of the point’s angle with the axes of the X’s is constantly proportional to the ordinate of the primitive curve.

The carriages are rendered very movable by substituting rolling for a sliding friction of the axes.  To this effect, the extremities of the axes of the wheels that support and guide them are made thin, and roll over the plane surface of recesses formed for the purpose in the lateral steel surfaces of the carriages, while the circumference of the wheels rolls in grooves along the two T-squares.

These latter are, on the one hand, carried by rollers that run in the groove of the iron, I, and, on the other, by a single roller that runs over the paper.  At right angles with one of these bars is fixed a divided ruler, through one point of which continually passes a third ruler, whose extremity pivots upon the point, A, of the first carriage.

When the divided ruler is placed upon the axis of the X’s, and the point, A, of this carriage is following the contours of the figure to be integrated, the tangent of the angle made by the inclined ruler with the axis of the X’s will be proportional to the ordinate of the figure.  The wheels, R and R’, of the drawing-pen, A’, of the second carriage must move parallel with this ruler.  In order to obtain such parallelism, we employ a parallelogram formed as follows:  Two gear-wheels of the same diameter are fixed upon the ruler that ends at the point, A, of the first carriage, and their line of centers is parallel with the latter.  The second carriage likewise carries two drums equal in diameter to those of the toothed wheels.  These are fixed, and their line of centers must remain constantly parallel with the line of centers of the gear-wheels, and consequently with the straight line which passes through the point, A. This parallelism is obtained by means of a weak steel spring, or of a silken thread passing over the four wheels, the two first of which (the gear-wheels) hold it taut by means of a barrel and spring placed in the center of one of them.

The edge of the wheels, R, R’, of the second carriage prevents the latter from giving way to the traction of the threads, permitting it thus to move only in the direction of their plane.