Aristotle’s Politics is another valuable book. This work begins with a history of human communities, starting with the family, then the most simple groups, then larger groups created for the sake of life (security and other like goods), and “finally,” as he says, “for the sake of a good life.” A good life is only possible in a community wherein most members are virtuous. Again there is a certain circularity of the argument, but perhaps this is unavoidable. In any event, to live in a town or city where most people are good, and the bad are kept under reasonable control, seems to be desirable.
Towns or cities are one thing; a nation is another. Aristotle understood this, too, and he concludes the Politics with a discussion of the best kind of government. He declares that a nation living under a constitution that is accepted by all or most citizens is the best. Being the eminently practical man that he was, he admitted that such a government is never likely to be perfect and, what is more, may find it difficult to continue to exist. He recognized that, first, the rich are always desirous of governing for their own benefit, and second that the people can always turn against them and become a mob, taking things into its own hands. As always, the mean between extremes is to be sought.
The Ethics and the Politics contain inconsistencies and seeming contradictions, some apparently missing connections, and other such defects. Thus they are not always easy reading. Choosing among translations doesn’t solve the problem. Even a good translation will not supply what is most needed, and that is a teacher to stand by our side and guide us as we read. A living teacher is best. Failing that, such a book as Aristotle for Everybody, by Mortimer J. Adler (my own teacher), is almost a necessity.
EUCLID
fl. c. 300 BCE
The Elements
Little is known about Euclid. What is known is mysterious. He probably lived and produced his major work around 300 BCE, in Alexandria, the city Alexander the Great had recently founded and that was considered the intellectual and commercial center of the Hellenic world. Euclid probably did tell Ptolemy I, who ruled Alexandria in his day, that “there is no royal road to geometry”—this in response to a question by the king as to whether there was not an easier way to learn the subject that might suit a busy man and monarch. It may also be true that Euclid once gave a coin to a student who, after his first lesson, inquired about the practical value of learning mathematics, saying contemptuously, “He must needs make gain in what he learns.”
If Euclid’s life is misty and unclear, his book, The Elements, known traditionally as The Thirteen Books of Euclid’s Elements, is clarity itself. Building on the work of all his forebearers, Euclid summed up and collated what was known in his day about geometry and also, as admiring Proclus said, “brought to incontrovertible demonstration the things which were only loosely proved by his predecessors.” As a result the Elements is both beautiful and right, infused with a brilliance that remains as wonderful today as it was when the book was first published and immediately became a classic.
“Euclid alone,” the poet Edna St. Vincent Millay wrote, “has looked on Beauty bare.” The statement makes a point about beauty—that it can exist shorn of its incorporation in things—that may be questioned. But the suggestion that mathematical truth is, somehow, the essence of the beautiful is surely interesting. Many, mathematicians and nonmathematicians alike, have felt the same way.
There is indeed something gratifying about a mathematical proof. If we assume this (set of mathematical truths), and if we follow certain rules of operation (i.e., definitions and axioms) that we have ourselves laid down in advance, then that which follows will have a certainty which is no less than that of the circling of the heavenly spheres. No less, I say; but this certainty is even greater. The planets in their revolutions are subject to the inevitable disturbances that affect all physical things. But mathematical entities are not physical. They are eternal, and eternally unchanged. The point which, according to the first Definition of Euclid’s First Book, “is that which has no part,” will never change and cannot change. Incommensurable magnitudes will remain incommensurable for ever, whatever we do, and the icosahedron (twenty-sided regular solid) inscribed in its sphere will always shimmer in our imaginations, or in that of God Himself, who cannot change it either.
The inscription over the door of Plato’s Academy was a warning: “Let no one enter here who has not studied geometry.” This was not because geometry was useful, although it was and is: ancient geometers used it for surveying, and modern engineers use it to build computers. Plato meant that geometry—in fact all mathematics—is intimately involved in making a good mind.
Much of what I have said here is no longer true, or no longer wholly true, of modern mathematics, which has its deep intellectual difficulties, as do all other sciences. But those difficulties do not apply to Euclid’s Elements: It is perfect, right, and true forevermore.
The Elements is a difficult, but not impossible, read. You may not want to read all thirteen books, but you should glance at them all, read the definitions at the beginning of each, sample their various propositions. They deal with different mathematical entities. Wander among the pages as if wandering among the galleries of a strange museum full of perfect works of art. Why did this artist create that work? You do not really know, at least the first time you look at it, but you are certain that he had a good reason. And you are aware that the whole museum hangs together, forms a unity unlike that of anything but mental places. Euclid’s book is a mental place, and one can do worse than to retreat to it. You will meet many other minds there and you will find them at peace.
ARCHIMEDES
287?–212 BCE
Scientific Writings
When Archimedes was born, in about 287 BCE in Syracuse, Sicily, mankind knew very little about the natural world it inhabited—at least understood very little scientifically about it. The distinction is important. A great deal was known, of course, about the lives of animals and plants, about the weather, about the stars, as well as about the behavior of human beings in all kinds of circumstances: hunger, fear, desire. The fact that societies had survived up to that point attests to their knowledge—rather, their lore—about the world around them. And the works of writers such as Homer, Aeschylus, and Euripides, to say nothing of Plato and Aristotle, attest to the knowledge that the Greeks (as well as other peoples) possessed of the human heart.
But for the most part, this was not scientific knowledge. The basic principles according to which things worked and acted on other things were still, for the most part, unknown. Good guesses had been made, as they had to be to get anything more than the simplest things done. But there is a difference between a guess, no matter how inspired, and a scientific principle or law, which is based on knowledge of why things work the way they do and that allows the scientist to predict with accuracy future occurrences of the same sort. The patient construction of the structure of natural laws as we know them today has been the major work of Western man for the past twenty centuries. For all intents and purposes, this work was begun by Archimedes. In many and important ways he was the first true scientist, and as a result of his work mankind knew much more, scientifically, about the natural world on the day of his death than it had on the day of his birth.