Things you may not know about Principia Mathematica

I recently got a chance to examine some original, first edition prints of Newton’s Philosophiæ Naturalis Principia Mathematica, also known as Principia (pronounced prin-kip-ee-a). The first edition was printed in 1687 and the version I saw was worth about $350,000, so I think it’s safe to say that it’s the most valuable thing I've ever held. Unfortunately, I only got to hold the first edition and couldn't actually open it because the spine was too fragile and the archivists were worried that opening it would damage it further. I did get to open and look through the second and third edition prints though, which were published in 1713 and 1726, respectively. All three of these editions are significant because Newton was still alive and made the corrections and alterations himself (although surprisingly few differences are found between the first and third editions). After donning white gloves and carefully leafing through them, I noticed some interesting things. The whole reason that Principia was even published was because Edmond Halley (whom Halley’s Comet is named after) convinced Newton to publish. Newton was a notoriously quiet and shy person, although it’s clear that Halley had a great deal of respect for him. In fact, the first edition of Principia even had an “Ode to Newton” written by Halley. It’s a poem that waxes lyrically about how Newton has elucidated the natural world by discovering the laws that govern it. Here’s an excerpt:

The inmost place of the heavens, now gained,
Break into view, nor longer hidden is
The force that turns the farthest orb. The sun
Exalted on his throne bids all things tend
Toward him by inclination and descent,
Nor suffer that the courses of the stars
Be straight, as through the boundless void they move,
But with himself as centre speeds them on
In motionless ellipses. Now we know
The sharply veering ways of comets, once
A source of dread, nor longer do we quail
Beneath appearances of bearded stars.

The “source of dread” Halley mentions is a reference to an old belief that seeing a comet was a sign of the impending death of the emperor. What surprised me about his ode was that it’s quite poetic for a scientist! Imagine buying a book today on quantum physics and seeing a poem in the beginning written by another scientist about how it has changed our understanding of the physical world. Actually, I think I would like that. But Halley didn't just use his ode to admire the science of Principia. He also used it as an opportunity to suck up to Newton. This is the last stanza:

Then ye who now on heavenly nectar fare,
Come celebrate with me in song the name
Of Newton, to the Muses dear; for he
Unlocked the hidden treasuries of Truth:
So richly through his mind had Phoebus cast
The radiance of his own divinity.
Nearer the gods no mortal may approach.

I think “ye who on heavenly nectar fare” is the best synonym for “astronomer” ever. But look at the original Latin text of that last stanza:

Notice how Newton’s name is in small caps. Where else have you seen that?

In most printings of the Bible, the word “Lord” is printed with small caps, which signifies respect because it indicates that the original text used the word yahweh. Some old books use small caps when referring to a shortened version of a long name, like “Dᴏɴ Qᴜɪxᴏᴛᴇ de La Mancha”, but “Issac Newton” isn’t a very long name, so I am left wondering if Halley did that to try to show a little extra respect to Newton.

Another thing I noticed that surprised me was the location of Newton’s Three Laws. You might think that these laws would be distributed throughout the book’s hundreds of pages with plenty of explanation and derivation working up to the revelation of each one. Instead, you find this:

Lex I, Lex II, Lex III, in two pages. And look how early it is! By page 12 of his book, Newton has defined Newtonian physics. I thought that was a pretty interesting organizational and stylistic choice. The rest of the book deals with implications of these laws, derivations and corollaries, and later his Law of Gravitation. The last section uses all of these tools to derive Kepler’s law’s of planetary motion to their more accurate form (Kepler had arrived at them empirically and his version was missing a few terms).

Principia has lovely diagrams, including this fold out of a comet:

All of the proofs in the book are geometric proofs, because calculus didn't exist yet (at least not formally):

However, I noticed a tantalizing glimpse of something that looked like a Riemann sum:

Bernhard Riemann wouldn't be born for another 100 years after this.

When Principia was published, it was not without its critics. Later editions included a section at the end that defended Newton against those who accused him of not properly explaining the tides or other physical phenomena. Obviously those objections did not stand the test of time.

Like all scientific books at the time, Principia was written in Latin. However, Newton’s notes are all written in English and can be seen online at the Cambridge Digital Library. It is amazing to me how modern his writing sounds even though it was written in the 17th century. If you’d like to see a scan of the Latin version of Principia, the Cambridge Digital Library has that available here. An free English translation is hosted here by Archive.org.