Lunar or Later

Moon: From 4.5 billion years ago to the present: Owners’ Workshop Manual, David M. Harland (Haynes 2016)

It was a clever idea: to put out a guide to the Moon in the same format as one of Haynes’ famous car-maintenance manuals. And the execution matched the idea. This is a detailed and interesting history of selenological speculation and lunar exploration, all the way from the ancient Greeks to the Apollo missions and beyond.

Except that there hasn’t been much beyond the Apollo missions. As the book’s final page notes:

On 31 December 1999 National Public Radio in the United States asked Sir Arthur C. Clarke, renowned for forecasting many of the developments of the 20th century, whether anything had happened in the preceding 100 years that he never could have anticipated. “Yes, absolutely,” he replied without a moment’s hesitation. “The one thing that I never would have expected is that after centuries of wonder and imagination and aspiration, we would have gone to the Moon… and then stopped.” (“Postscript”, pg. 172)

And we’ve been stopped for some time. Neil Armstrong died in 2012, forty-three years after that “small step for a man” and “giant leap for mankind” in 1969. But David M. Harland ends on an optimistic note: he thinks that “The Moon is humanity’s future.” It will be our gateway to the rest of the solar system and perhaps even the stars.

But it will be more than just a gateway. There is still a lot we don’t understand about our nearest celestial neighbour and big surprises may still be in store. One thing we do now understand is that the scarred and pitted lunar surface got that way from the outside, not the inside. That is, the moon was bombarded with meteors, not convulsed by volcanoes. But that understanding, so obvious in hindsight, took a long time to reach and it was actually geologists, not astronomers, who promoted and proved it (ch. 5, “The origin of lunar craters”). It was the last big question to be settled before the age of lunar exploration began.

Previously scientists had looked at the Moon with their feet firmly on the ground; at the end of the 1950s, they began to send probes and robotic explorers. Harland takes a detailed look at what these machines looked like, how they worked and where they landed or flew. Then came the giant leap: the Apollo missions. They were an astonishing achievement: a 21st-century feat carried out with technology from the 1960s, as Harland puts it. Yet in one way they depended on technology much earlier than the 1960s: pen and paper. The missions relied on the equations set out in Newton’s Principia Mathematica (1687). Newton had wanted to explain, inter multa alia, why the Moon moved as it did.

By doing that, he also explained where a spacecraft would need to be aimed if it wanted to leave the Earth and go into orbit around the Moon. His was a great intellectual achievement just as the Apollo missions were a great technological achievement, but he famously said that he was “standing on the shoulders of giants”. Harland begins the book with those giants: the earlier scientists and mathematicians who looked up in wonder at the Moon and tried to understand its mysteries. Apollonius, Hipparchus and Ptolemy were giants in the classical world; Galileo, Brahe and Kepler were giants in the Renaissance. Then came Newton and the men behind the Apollo missions.

Are there more giants to come? The Moon may be colonized by private enterprise, not by a government, so the next big names in lunar history may be those of businessmen, not scientists, engineers and astronauts. But China, India and Japan have all begun sending probes to the Moon, so their citizens may follow. Unless some huge disaster gets in the way, it’s surely only a matter of time before more human beings step onto the lunar surface. Even with today’s technology it will be a great achievement and more reason to marvel at the Apollo missions. And the Apollo photographs still look good today.

There are lots of those photographs here, with detailed discussion of the men and machines that allowed them to be taken. The Moon is a fascinating place and this is an excellent guide to what we’ve learned and why we need to learn more.

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De Minimis Curat Rex?

Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, Amir Alexander (Oneworld 2014)

Infinitesimal is an entertaining read on a fascinating topic: the pioneers of a new form of mathematics and those who opposed them. Amir Alexander claims that “the ultimate victory of the infinitely small helped open the way to a new and dynamic science, to religious toleration, and to political freedoms unknown in human history” (Introduction, pg. 14).

It’s an extraordinary claim and I don’t think he manages to provide extraordinary proof for it. In fact, he probably gets cause-and-effect reversed. Is it likelier that new mathematics opened minds, dynamized science and transformed politics or that open minds created new forms of mathematics, science and politics? I’d suggest that support for the new mathematics was a symptom, not a cause, of a new psychology. But Alexander makes a good case for his thesis and there is no doubt that the world was changed by the willingness of mathematicians to use infinitesimals. Calculus was one result, after all. The book begins in Italy and ends in England, because the pioneers lost in Italy:

For nearly two centuries, Italy had been home to perhaps the liveliest mathematical community in Europe. … But when the Jesuits triumphed over the advocates of the infinitely small, this brilliant tradition died a quick death. With Angeli silenced, and Viviani and Ricci keeping their mathematical views to themselves, there was no mathematician left in Italy to carry on the torch. The Jesuits, now in charge, insisted on adhering close to the methods of antiquity, so that the leadership in mathematical innovation now shifted decisively, moving beyond the Alps, to Germany, England, France and Switzerland. (ch. 5, “The Battle of the Mathematicians”, pg. 178)

Why were the Jesuits involved in an esoteric mathematical dispute? You might say that de minimis curat Loyola – Ignatius Loyola (1491-1556), founder of the Jesuits, cared about anything, no matter how small, that might undermine the authority of the Church. In the view of his successors, the doctrine of indivisibles did precisely that: “in its simplest form, the doctrine states that every line is composed of a string of points, or ‘indivisibles’, which are the line’s building blocks, and which cannot themselves be divided” (Introduction, pg. 9).

Indivisibles must be infinitesimally small, or they wouldn’t be indivisible, but then how does an infinitesimal point differ from nothing at all? And if it isn’t nothing, why can’t it be divided? These paradoxes were familiar to the ancient Greeks, which is why they rejected infinitesimals and laid the foundations of mathematics on what seemed to them to be solider ground. In the fourth century before Christ, Euclid used axioms and rigorous logic to create a mathematical temple for the ages. He proved things about infinity, like the inexhaustibility of the primes, but he didn’t use infinitesimals. When Archimedes broke with Greek tradition and used infinitesimals to make new discoveries, “he went back and proved every one of them by conventional geometrical means, avoiding any use of the infinitely small” (Introduction, pg. 11).

So even Archimedes regarded them as dubious. Aristotle rejected them altogether and Aristotle became the most important pre-Christian influence on Thomas Aquinas and Catholic philosophy. Accordingly, when mathematicians began to look at infinitesimals again, the strictest Catholics opposed the new development. Revolutionaries like Galileo were opposed by reactionaries like Urban VIII.

But the story is complicated: Urban had been friendly to Galileo until “the publication of Galileo’s Dialogue on the Copernican system and some unfavourable political developments” (pg. 301). So I don’t think the mathematics was driving events in the way that Alexander suggests. Copernicus didn’t use them and the implications of his heliocentrism were much more obvious to many more people than the implications of infinitesimals could ever have been. That’s why Copernicus was frightened of publishing his ideas and why Galileo faced the Inquisition for his astronomy, not his mathematics.

But Amir’s thesis makes an even more interesting story: the tiniest possible things had the largest possible consequences, creating a new world of science, politics and art. In Italy, two of the chief antagonists were Galileo and Urban; in England, two were the mathematician John Wallis (1616-1703) and the philosopher Thomas Hobbes (1588-1679). Alexander discusses Wallis and Hobbes in Part II of the book, “Leviathan and the Infinitesimal”. Hobbes thought that de minimis curat rex – “the king cares about tiny things”. Unless authority was absolute and the foundations of knowledge certain, life would be “nasty, brutish and short”.

However, there was a big problem with his reasoning: he thought he’d achieved certainty when he hadn’t. Hobbes repeatedly claimed to have solved the ancient problem of the “quadrature of the circle” – that is, creating a square equal in size to a given circle using only a compass and an unmarked ruler. Wallis demolished his claims, made Hobbes look foolish, and strengthened the case for religious toleration and political freedom. But I don’t think this new liberalism depended on new mathematics. Instead, both were products of a new psychology. Genetics will shed more light on the Jesuits and their opponents than polemics and geometry textbooks from the period. Alexander’s theory is fun but flawed.

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