From centripetal forces to conic orbits: a path through the early sections of Newton’s Principia

In this study, we test the security of a crucial plank in the Principia’s mathematical foundation, namely Newton’s path leading to his solution of the famous Inverse Kepler Problem: a body attracted toward an immovable center by a centripetal force inversely proportional to the square of the distance from the center must move on a conic having a focus in that center. This path begins with his definitions of centripetal and motive force, moves through the second law of motion, then traverses Propositions I, II, and VI, before coming to an end with Propositions XI, XII, XIII and this trio’s first corollary. To test the security of this path, we answer the following questions. How far is Newton’s path from being truly rigorous? What would it take to clarify his ambiguous definitions and laws, supply missing details, and close logical gaps? In short, what would it take to make Newton’s route to the Inverse Kepler Problem completely convincing? The answer is very surprising: it takes far less than one might have expected, given that Newton carved this path in 1687.     

Plotinus’s conception of unity and multiplicity as the root to the medieval distinction between lux and lumen

Plotinus resolved the paradox of the immanent transcendence, characterizing the relation between the One and the universe, through his theory of the two energeiai. According to this doctrine, all existents have an internal activity and an external activity: the internal activity comprises the true essence and substance of each being; the external activity is emitted outwards as its image. The source of the emission is thus present in the lower layer of being by virtue of its manifold images. The prominence given to light in elucidating this solution led to a distinction between two types of lights: an original light, corresponding to the internal energeia of every existent, and a secondary light, which is the outflow and image of the first light, existing outside of the luminous body.This paper demonstrates the striking similarity between these two Plotinian lights and the concepts of lux and lumen developed by two thirteenth-century philosophers: Robert Grosseteste and Albertus Magnus. Moreover, the paper contends that the purpose of these two medieval concepts of light was identical to what Plotinus had in mind when he first made the distinction: to account for the relation between the one and the many.     

The censoring of Galileo’s Sunspot Letters and the first phase of his trial

Galileo’s Sunspot Letters, published in 1613, underwent extensive censorship before publication. It seems likely that the Roman Inquisition had charge of the pre-publication review of Galileo’s work, rather than the usual organ, the Master of the Sacred Palace. A study of that process demonstrates that the issue to which the censors objected was Galileo’s use of the bible, not his allegiance to Copernicus. In the course of the first phase of Galileo’s trial, orchestrated by one of the most powerful Cardinal Inquisitors, two propositions allegedly drawn from the book were judged either “formally heretical” or “at least erroneous in the faith.” These judgments might have come not from the published book but from the Inquisition’s censorship of its drafts. They supported Galileo’s silencing in 1616.