tmblg:
(via thisrecording)
Newton’s Naturalis Principia Mathematica
One semester in college, I worked as a personal assistant to the director of our library. One of the rare joys of the position was being responsible for the ball park appraisal of volumes being added to, or put on display from, our rare books collection (for the purposes of insurance.)
Highlights include handling first editions of Gustave Doré’s illustrated Divine Comedy and Paradise Lost, as well as the first volume of the second edition of the Principia, published in 1713, which added Newton’s own corrections as well as his infamous concluding General Scholium, easily worth $50,000 in solid condition.
The first lemma of Book I (On the Motion of Bodies), in which Newton lays the groundwork for the mathematical concept of a “limit” (and gives rise to the possibility of his Calculus) remains one of the singly most formative fragments of thought in my academic life. It reads:
Quantities, and also ratios of quantities, which in “any finite time” constantly tend to equality, and which before the end of that time approach so close to one another that their difference is less than any given quantity, become ultimately equal.
And an excerpt from one of my (many) papers in which it was prominently featured:
Lemma 1 recasts the description of motion as a strictly geometric problem. We can now describe what is happening in an indefinitely small division of time without ever having to actually find such an instant or attempt to go there. The limit effectively achieves Galileo’s ‘one fell swoop’ and strips away the complication of flowing time. We are now able to freeze-frame our previously cinematographical perception of phenomena and extract the underlying ratiometric intelligibility as pure planar geometry. The force displacing a body along a curvilinear path has entered into proportion with a given line in a planar figure.
Consequently, by means of Lemma 1, acting as a revolution of perspective, we find ourselves removed from inherently empirical motion (as an a posteriori synthesis of both space and time) and again in the synthetic a priori comforts of pure mathematics.