Everyone in grade school writes a book report on something they haven’t completely read (or read at all). Well, I’m long past, but here’s a book worth buying for the preface alone — Visual Differential Geometry and Forms by Tristan Needham.

His earlier book Visual Complex Analysis (another book I haven’t read completely) is a masterpiece (those parts I’ve managed to read) with all sorts of complex formulas and algebraicism explained visually. It’s best to be patient, as Needham doesn’t even get to complex differentiation until Chapter 4 (p. 188) The Amplitwist concept.

On page xviii of the preface, Needham describes the third type of calculus Newton invented, and the one he used in the great Principia Mathematica of 1687.

The first one of 1665 was basically the manipulation of power series (which I’ve never heard of. Have you?)

The second was one we all study in school, dx dy and all that.

Newton called the third method of calculus “the synthetic method of fluxions.”

Bishop Berkeley had a field day with them. “And what are these fluxions? The velocities of evanescent increments? And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them ghosts of departed quantities?”

Laugh not at the good Bishop; Berkeley California is named for him (as long as the woke don’t find out).

Needham gives a completely nontrivial example of the method. He shows geometrically how the derivative of tan(theta) is 1 + [ tan(theta)]^2,.

The diagram will not be reproduced here but suffice it to say you’ll have to remember a lot of your geometry, e.g. two lines separated by a very small angle are essentially parallel, the radius of a circle is perpendicular to the tangent, what a similar triangle is.

The best thing about it is you can actually visualize the limit process taking place as a triangle shrinks and shrinks and eventually becomes ‘almost’ similar to another. No deltas and epsilons for Newton (or Needham). There is some algebra involving the ratios of the sides of similar triangles, but it’s trivial.

Buy the book and enjoy. You’ll never think of differentiation the same way. Your life will be better, and you might even meet a better class of people.