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A Fahrenheit-Celsius Activity

by Vincent Mallette
Copyright © 1999 Inwit Publishing, Inc.

In South Bend, Indiana I saw a temperature sign on a bank cycling back and forth between the following two displays:



What I took to be a hyphen in front of the eleven simply appeared and disappeared, appeared and disappeared. It took me a couple of minutes to figure out what was going on. Can you?

Answer: The temperature display of this particular bank’s sign gave the temperature in both Fahrenheit and Celsius, alternately. A more normal display would have been something like 68 and 20 where the 68 and the 20 flash alternately. The 68 of course is degrees Fahrenheit and the 20 is degrees C (68°F = 20°C).1 But wait...what does the sign show when the temperature sinks to 11° F? It happens that 11° F = -11.666 °C. Presumably the bank’s circuitry is not "finessed" to round off the display in some rational way, so it sees -11.666... °C as -11° C, just knocking off the decimals. This also proves that the Fahrenheit temperature is primary to the bank’s machine. For had it seen minus 11 Celsius internally, it would have translated that as 12.2 Fahrenheit, and that would have rounded off by their primitive method to 12! So in that case I would have seen -11 and 12 cycling back and forth.

More observations. Can you program a calculator or computer to find the exact number at which Fahrenheit and Celsius have the same absolute value, that is, without regard to a positive or negative sign? Here’s the guts of a program for a Casio:

(F-32) ´ 5 ¸ 9 ® C : Abs C ® F : Dsz A : Go to 1

("A" is just a dummy variable to return the iteration operator to the beginning.) Anyway, after 40 some-odd passes this program yields ½ C ½ = ½ F ½@ 11.42857143

( ½ ½ means absolute value of whatever is between the bars, that is, without regard to sign. So you see that if your checking account was overdrawn to minus $16.23, then ½ $-16.23½ = $16.23, and you’d have money again, a neat little trick that the bank might not go along with.)

But there’s more...what happens if the temperature falls to -40°F? Actually it has never been -40°F in South Bend, Indiana, and I doubt if the bank sign could handle that low a temperature in any case.2 but we’ll pretend. I’ll leave this to you to prove: it turns out that -40°F = -40°C, exactly. So our imaginary bank sign would show no change whatsoever at minus 40; it would just sit there displaying -40 over and over. Somehow I think the citizens of South Bend wouldn’t even notice, at forty below!

While we’re on the subject, though, have you ever wondered where these fool temperature scales get their numbers, their fixed points? Celsius is supposed to be so scientific, with the freezing and boiling points of water tying down the zero and 100.3 Actually it’s no more scientific 4 than Fahrenheit, and its degrees are nearly twice as large as Fahrenheit’s, giving it a poor resolution unless you go to decimals. But where did Fahrenheit get his zero and 100; what is their significance? Well, the zero came from mixing snow and salt to get the lowest possible temperature. Fahrenheit wanted to avoid negative temperatures as much as possible in daily life, and he pretty well succeeded — Fahrenheit temperatures below 0 are uncommon year round unless you live at the poles. The 100, though, is more mysterious. The standard story is that Fahrenheit first chose 96 as the human body temperature (measured in the armpit) because 96 has a lot of divisors, ten to be exact. Then the 100 just fell wherever it fell and has no significance. However, the story that the 100 represents is a little more colorful. As cows were pretty common all over Europe in the 18th century, the rectal temperature of cows was chosen as a repeatable standard — more so than human body temperature measured under the arm.

Now that we’ve brought up the subject, let’s talk a bit about human body temperature. Many clinical thermometers bought in the drugstore have a red line or an arrow at 98.6. There’s good reason, however, to believe that most well-fed, healthy folks have a modal temperature between 98 and 98.6. According to this interpretation, the hallowed 98.6 would be the high end of the range of normal, and 98.4 would be closer to a "normal" temperature. Supposedly the sacred number came about this way: nearly a hundred years ago there was a project to measure the body temperature of thousands of healthy people. The average came out to be 36.88 degrees Celsius (the scale was then called "centigrade"). The physician in charge sensibly rounded this to 37 ° C — and this is exactly 98.6 F. After that, every book on earth it seems trumpeted 37 or 98.6. However, go back about 100 years, before this survey, and it’s a different story: I have here in my hand Alfred Gage’s little book, Elements of Physics, published in Boston in 1889, and on page 152 it says, "Blood heat...98°." Likewise the 1899 edition of Edwin Edser’s classic, Heat, reads "...the blood of a healthy man... 98° F." And so on. Did you know that prior to 1866 you had to hold the thermometer in your mouth for 20 minutes to get a reading? In that year Sir Thomas Allbutt, a physician, got tired of hearing his patients go "Mmm...mmm...mmm" for a third of an hour and he invented the fast-reading medical thermometer.5

Before we leave this subject I should mention that there is another temperature scale, still used in certain benighted pockets of Europe and Eurasia. This is the Réaumur scale. Réaumur was a French jack-of-all-trades who took a vacation from his work on iron metallurgy, insects, and gastric juices to saddle us with yet another temperature scale, in which, for no reason known to me, the boiling point of water is designated as 80. The freezing point of water on his scale was sensibly zero, so to convert to Fahrenheit you multiply the Réaumur by 2.25 and add 32.6

Afterword: Neither Fahrenheit nor Celsius "invented" the thermometer. The liquid-in-glass expansion thermometer "was probably invented by Ferdinand II, Grand Duke of Tuscany, around 1654".7 "Fahrenheit" as a term in English for Herr Fahrenheit’s thermometer scale was first seen in 1753, some 17 years after the German physicist died. Celsius’ scale was called "centigrade" for nearly 150 years, starting in English-speaking countries about 1801; in 1948 the Ninth General Conference on Weights and Measures decided to attach Celsius’ name to the centigrade scale just as Fahrenheit’s was attached to his; Celsius had then been dead for 204 years. It took decades for me to stop saying centigrade and start saying Celsius, though I was only eight years old when the change was officially made. For technical reasons I can’t go into, should the "centigrade" scale be revived it would differ from the Celsius by 0.01°, because of the way it was originally defined.8

What, then, is temperature? "What the thermometer measures, roughly speaking, is an average mechanical energy of the particles constituting the system under study....higher temperature is a macroscopic manifestation of more energetic molecular and submolecular motion".9

1 Apparently the bank thought there would be no confusion about which was which, because the display didn’t even put up a C and F!

2 All the mercury thermometers in town would already have been frozen; mercury solidifies at -37.9 ° F. You can tie your shoe with a string of mercury at minus 40. (Of course, I realize that these advertising display signs use electrical thermometers, but I doubt if the standard model goes to -40.)

3 In one of the great scientific insanities of all time, Celsius first ascribed 100 to the freezing point of water, and zero to its boiling point! [The Timetables of Science — by Alexander Hellemans and Bryan Bunch (NY: Simon and Schuster, 1988), p. 184]. Rationality prevailed, and the two numbers were switched shortly after Celsius died.

4 "...there is no fundamental principle of physics involved in the fixing of an ordinary thermometric scale. The choice of a scale is purely arbitrary...." [Heat for Advanced Students — by Edwin Edser, revised by N. M. Bligh (London: Macmillan and Co., Limited, 1936), p. 12]. The only really scientific temperature scale is the absolute scale of Lord Kelvin, which starts at absolute zero. You can use any size degrees as long as you count up from absolute zero; when Celsius-size degrees are used it’s called the Kelvin scale; when Fahrenheit-sized degrees are used it’s called the Rankine scale. It’s not just because of starting at absolute zero that these scales are considered scientific; an additional feature is that the ratio of any two temperatures K or R is also "the ratio of the heat absorbed to the heat rejected by an ideal heat engine working between those two temperatures." You can’t say this of Celsius, for all its zero and 100 "scientific-ness."

5 The Timetables of Science — by Alexander Hellemans and Bryan Bunch (New York: Simon and Schuster, 1988), p. 339.

6 You need to know this if you read a lot of Russian novels; Dostoyevsky and Tolstoy were always having their characters opine about "degrees frost" in Réaumur. "Degrees frost" is degrees below the freezing point of water. Hence "10 degrees frost Réaumur" is -10 °R, or (-10 ´ 2.25) + 32 = 9.5 ° F — bone chilling even with plenty of vodka inside you. By the way, the phrase "degrees frost" always means Réaumur in a Russian novel, whether it says so or not; in Britain, however, degrees frost just means degrees below zero Fahrenheit. So 10 degrees frost in London would be 22 ° F. James Joyce mentions Réaumur in his Ulysses; he speaks of thousands of degrees below zero Réaumur, showing his knowledge of an obscure thermometric scale but his ignorance of physics.

7 Robert Weinstock in Encyclopedia of Physics, Second Edition — ed. by R. G. Lerner and G. L. Trigg (New York: VCH Publishers, Inc., 1991), p. 1246

8 Merriam-Webster Third New International Dictionary, Unabridged; articles "Celsius" and "centigrade." Also see American Institute of Physics Handbook, Third Edition — ed. by Dwight E. Gray (New York: McGraw-Hill Book Company, 1972), page 4-2 and Table 4a-1

9 Robert Weinstock in Encyclopedia of Physics, Second Edition — ed. by R. G. Lerner and G. L. Trigg (New York: VCH Publishers, Inc., 1991), p. 1248

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