Flipping back through the pages of this book, you can see how important geometric figures were in the development of calculus. The figures become more sophisticated as the truths they reveal become deeper; Figure 6.5 of Leibniz, for example, goes to the heart of the connections within calculus, but falls just shy of being an impenetrable maze of lines. Leibniz, as much as anyone in his day, desired to push calculus past the point where its truths are a consequence of diagrams. The notation he invented allowed this, and we use many of his symbols today.

Leibniz describes differentials

The notation of Leibniz underwent a maturing process similar to that of calculus generally. This brief treatment does not attempt to tell the whole story, focusing instead on the final payoff of his efforts.

Leibniz interpreted a curve, like the one in Figure 7.1, as the ratio (at each point) of the curve's vertical motion to its horizontal motion. Mark off equally-spaced divisions on the horizontal, associating each mark (such as A)with the point (B) on the curve directly above it; then mark the corresponding point (C) on the vertical axis. Where the curve has a small vertical rate of change, the points on the vertical axis crowd together.

Leibniz viewed the distances between the marks in Figure 7.1 as differences (for example, we may see AD as OD – OA) and he chose the notation d to represent such distances.