The Clockwork Universe (42 page)

41
One prominent historian calls it “incomprehensible” that Greek mathematicians never conceived of graphs. But neither did their intellectual descendants for well over a thousand years. Even an enormous hint went unnoticed. Monks in the Middle Ages invented musical notation, which meant they no longer had to commit countless chants to memory. “The musical staff was Europe's first graph,” noted the historian Alfred Crosby, but several more
centuries
would pass before scientists saw that they, too, could use graphs to depict changes in time.

42
The question of whether vacuums could exist spurred long, angry debates. The invention of the air pump did not settle the debate, in the view of Leibniz and some others, because even if a jar no longer contained
air
it might still contain some more ethereal fluid. Leibniz and Descartes both maintained that the very notion of a vacuum was nonsensical—how could there be a place containing nothing at all, when the meaning of the word
place
is “the location where something happens to be”? Newton and Pascal insisted just as vehemently that vacuums were real. Descartes contended, cattily, that the only vacuum was in Pascal's head.

43
If the first block were 1 inch thick, the next
½
inch, then
⅓
,
¼
,
1
/
5
, and so on, the tower
would
climb infinitely high (although it would rise excruciatingly slowly).

44
A sequence
may
attain its goal. The sequence 1, 1, 1, . . . has the number 1 as its limit. But a “typical” sequence draws ever nearer to its goal without actually touching it. The sequence .9, .99, .999, . . . never reaches its limit, which is the number 1.

45
Gilbert and Sullivan's Major-General knew it well, along with much else. “About binomial theorem I'm teeming with a lot o' news / With many cheerful facts about the square of the hypotenuse.”

46
Unbeknownst to Leibniz, the English mathematician and astronomer Thomas Harriot had been the first to discuss binary numbers, decades before. But Harriot never published any of his work, and his papers went unseen until the late 1700s. It turns out that Harriot had recorded a number of other firsts as well; Harriot turned a telescope to the sky a few weeks before Galileo did.

47
Sometimes the right notation can even hint at a deep, surprising insight. Simply using decimal notation, and then adding column by column, suggests that 1 + .1 + .01 + .001 + . . . = 1.11111 . . . , and not infinity.

48
Most people “know” not just that an apple fell but that it bonked Newton on the head.

49
Another exotic import, tea, had arrived at about the same time, although coffee caught on first. On September 25, 1660, Pepys wrote in his diary that “I did send for a cup of tee (a China drink) of which I never had drank before.”

50
The statement
if a planet travels in an ellipse
,
then it follows an inverse-square law
is different from the statement
if a planet follows an inverse-square law
,
then it travels in an ellipse.
It might have been that one was true but the other was not.
If someone owns a dog
,
then he owns a pet
is true;
if someone owns a pet
,
then he owns a dog
is not. In this case it was clear to Newton (though bewilderingly obscure to others) that if one statement was true, the other had to be true as well.

51
Sixteen hundred years before Newton, Plutarch wrote that Archimedes grew so absorbed in his thoughts that he “would often forget his food and neglect his person” and have to be “carried by absolute violence to bathe.”

52
In time, this bewilderment died away. Darwin noted impatiently that, although his critics demanded that he explain where intelligence and awareness come from, nobody demanded a similar account of gravity. “Why is thought being a secretion of brain more wonderful than gravity a property of matter?” he asked.

53
The debate over whether we should look at scientists' characters and motives, or if it is only their findings that matter, continues today. “Science doesn't work because we're all nice,” a NASA climatologist declared in November 2009, in the midst of a dispute over global warming. “Newton may have been an ass, but the theory of gravity still works.”