### Studying Newton via Densmore

We are lucky when a piece of scholarship is so lovingly done, it becomes the key that unlocks the door to a classic text. Such is the case with Dana Densmore’s and William H. Donahue’s Newton’s Principia: The Central Argument.

I am making another stab at understanding Newton’s Principia Mathematica. Thus I add to the amount of time spent over the past 14 years trying to understand what Newton was up to.

Understanding Newton’s Principia presents two problems.

One, he bases his exposition on the geometry of Euclid’s Elements and Apollonius’ Conics. If you want to understand Newton in his terms, you must know Euclid and Apollonius. Elements and Conics are ancient Greek mathematics texts, which are not our modern day mathematics with its algebraic and numerical manipulations making things much easier to calculate. The Greeks understood concepts such as amount, magnitude, and number in geometrical terms that are somewhat alien to our modern way of thinking about mathematics. That is not to say you cannot learn basic geometry from Euclid, for the modern course of high school geometry is Euclid reworked to take advantage of modern notation, algebra, and number systems. Modern notation was not fully available to Newton even though he invented the calculus. Even so, he eschewed using the more sophisticated notation of his time for the proofs of his propositions. He was possibly or partially motivated to see if his peers in the scientific world could understand his system even when he presented it in terms of mathematics they were most familiar with.

Second, Newton’s demonstrations are extremely terse. They were written for the major scientific figures of his day. He assumed they would easily fill in the gaps. Such is not the case for the modern student. One can wander through the wilderness for years trying to understand his opening lemmas on the calculus he uses in the rest of the book—or at least not fully understand what he meant and demonstrated.

The student with a mathematics degree might not have a better shot than the student without one in terms of studying Newton. The mathematics student might automatically respond to Newton by trying to translate his treatise into modern mathematics and physics notation, which is, in a way, to miss Newton. Those who take the classical route from Euclid through Apollonius and Galileo might have the better shot since they will be steeped in the original notation and concepts. Of course, knowing modern mathematics and ancient mathematics is the best blend.

Help is out there though in overcoming these two stumbling blocks. Dana Densmore comes to the rescue with her Newton’s Principia: The Central Argument. Densmore provides an excellent translation of the Principia along with detailed commentary on Newton’s demonstrations. She fills in the gaps with references to the appropriate propositions from Euclid and Apollonius. Her goal is to present Principia’s Book III, On the System of the World. To do this she carefully explains all the propositions from Book I that are needed to understand Book III. Donahue adds diagrams to those used in the original Principia and the new diagrams make the exposition easier to understand. In fact, Newton recommended that if one wanted to understand his work, you should study Book I, and then tackle his system of the world in Book III.

With Densmore to the rescue, all that is left is a lot of hard work and the excitement of discovering Newton.

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