Mathematics
Geometry, Mechanics, and the Language of Nature
There is no certainty where one can neither apply any of the mathematical sciences nor any of those which are based upon the mathematical sciences.
— Leonardo da Vinci
Overview
Mathematics was, for Leonardo, the language in which nature writes its laws. He was not a theoretical mathematician — he had no formal training in algebra and struggled with Latin — but his geometric intuition was extraordinary. He filled pages with attempts to square the circle, devised ingenious proofs about surfaces and solids, and declared mechanics "the paradise of the mathematical sciences."
His mathematical work is intertwined with everything else: the proportions of the human body, the perspective of painters, the behavior of water, the flight of birds. For Leonardo, mathematics wasn't a separate discipline — it was the skeleton inside every other subject.
Mechanics: The Paradise
Where mathematics bears fruit
Mechanics is the paradise of the mathematical sciences because by means of it one comes to the fruits of mathematics.
MS. E 8 v.
"The paradise of the mathematical sciences" — because mechanics is where abstract math becomes real. Levers, pulleys, forces, flight. Leonardo was an applied mathematician before the term existed. -D
On the Infinite
A question that has no end
What is that thing which does not give itself, and which if it were to give itself would not exist?
It is the infinite, which if it could give itself would be bounded and finite, because that which can give itself has a boundary with the thing which surrounds it in its extremities, and that which cannot give itself is that which has no boundaries.
C.A. 131 r. b
On Surface and Boundary
Where bodies end and air begins
Surface is the touching-part of the extremities of bodies, that is it is made by the extremities of the body of the air, together with the extremities of the bodies which are clothed by this air, and it is that which completes and forms with this air the boundary of the bodies surrounded by the air, and completes this air with the bodies clothed by it, and it does not participate either in the body which surrounds it or in that which is surrounded by it.
C.A. 182 r. a
A thing which moves acquires as much space as it loses.
MS. E 7 v.
Leonardo was self-taught in mathematics, and it shows in both the best and worst senses. He sometimes reinvented things that had been known for centuries. But he also made connections that no trained mathematician would have made, precisely because he wasn't constrained by formal training. His friendship with Luca Pacioli — whose De Divina Proportione Leonardo illustrated — shows how seriously he took the subject. -D