But, even if it were the case that the natural laws had no longer any secret for us, we
could still only know the initial situation approximately. If that enabled us to predict
the succeeding situation with the same approximation, that is all we require, and we
should say that the phenomenon had been predicted, that it is governed by laws. But
it is not always so; it may happen that small differences in the initial conditions
produce very great ones in the final phenomena. A small error in the former will
produce an enormous error in the latter. Prediction becomes impossible, and we
have the fortuitous phenomenon.” (Poincaré 1914, pp. 67–68)
20.6.2 Extract from Lorentz
“In Fig. 43 [note: corresponds to Fig. 20.2 of the chapter], we see a copy of fifteen
months of the somewhat faded original output, divided for display purposes into
three five-month segments.
The chosen variable is an approximate measure of the latitude of the strongest
westerly winds; a high value indicates a low latitude. There is a succession of
“episodes,” in each of which the value rises abruptly, remains rather high for a
month or so, and then drops equally abruptly, but the episodes are not identical and
are not even equal in length, and the behavior is patently non-periodic. At one point
I decided to repeat some of the computations in order to examine what was
happening in greater detail. I stopped the computer, typed in a line of numbers
that it had printed out a while earlier, and set it running again. I went down the hall
for a cup of coffee and returned after about an hour, during which time the computer
had simulated about two months of weather. The numbers being printed were
nothing like the old ones. I immediately suspected a weak vacuum tube or some
other computer trouble, which was not uncommon, but before calling for service I
decided to see just where the mistake had occurred, knowing that this could speed up
the servicing process. Instead of a sudden break, I found that the new values at first
repeated the old ones, but soon afterward differed by one and then several units in
the last decimal place, and then began to differ in the next to the last place and then
in the place before that. In fact, the differences more or less steadily doubled in size
every four days or so, until all resemblance with the original output disappeared
somewhere in the second month. This was enough to tell me what had happened: the
numbers that I had typed in were not the exact original numbers, but were the
rounded-off values that had appeared in the original printout. The initial round-off
errors were the culprits; they were steadily amplifying until they dominated the
solution. In today’s terminology, there was chaos. It soon struck me that, if the real
atmosphere behaved like the simple model, long-range forecasting would be impossible. The temperatures, winds, and other quantities that enter our estimate of
today’s weather are certainly not measured accurately to three decimal places,
and, even if they could be, the interpolations between observing sites would not have
similar accuracy. I became rather excited, and lost little time in spreading the word
to some of my colleagues. In due time, I convinced myself that the amplification of
20 A Combination of Historical Physics Documents and Other Teaching Tools for the. . . 259
could still only know the initial situation approximately. If that enabled us to predict
the succeeding situation with the same approximation, that is all we require, and we
should say that the phenomenon had been predicted, that it is governed by laws. But
it is not always so; it may happen that small differences in the initial conditions
produce very great ones in the final phenomena. A small error in the former will
produce an enormous error in the latter. Prediction becomes impossible, and we
have the fortuitous phenomenon.” (Poincaré 1914, pp. 67–68)
20.6.2 Extract from Lorentz
“In Fig. 43 [note: corresponds to Fig. 20.2 of the chapter], we see a copy of fifteen
months of the somewhat faded original output, divided for display purposes into
three five-month segments.
The chosen variable is an approximate measure of the latitude of the strongest
westerly winds; a high value indicates a low latitude. There is a succession of
“episodes,” in each of which the value rises abruptly, remains rather high for a
month or so, and then drops equally abruptly, but the episodes are not identical and
are not even equal in length, and the behavior is patently non-periodic. At one point
I decided to repeat some of the computations in order to examine what was
happening in greater detail. I stopped the computer, typed in a line of numbers
that it had printed out a while earlier, and set it running again. I went down the hall
for a cup of coffee and returned after about an hour, during which time the computer
had simulated about two months of weather. The numbers being printed were
nothing like the old ones. I immediately suspected a weak vacuum tube or some
other computer trouble, which was not uncommon, but before calling for service I
decided to see just where the mistake had occurred, knowing that this could speed up
the servicing process. Instead of a sudden break, I found that the new values at first
repeated the old ones, but soon afterward differed by one and then several units in
the last decimal place, and then began to differ in the next to the last place and then
in the place before that. In fact, the differences more or less steadily doubled in size
every four days or so, until all resemblance with the original output disappeared
somewhere in the second month. This was enough to tell me what had happened: the
numbers that I had typed in were not the exact original numbers, but were the
rounded-off values that had appeared in the original printout. The initial round-off
errors were the culprits; they were steadily amplifying until they dominated the
solution. In today’s terminology, there was chaos. It soon struck me that, if the real
atmosphere behaved like the simple model, long-range forecasting would be impossible. The temperatures, winds, and other quantities that enter our estimate of
today’s weather are certainly not measured accurately to three decimal places,
and, even if they could be, the interpolations between observing sites would not have
similar accuracy. I became rather excited, and lost little time in spreading the word
to some of my colleagues. In due time, I convinced myself that the amplification of
20 A Combination of Historical Physics Documents and Other Teaching Tools for the. . . 259