Let ε < 0.

05.10.09

Parable of units

Filed under: Science humor — Travis @

Few questions have made physicists lose their sense of humor more often that that debated by a committee in Paris in 1932: electromagnetic units. One topic was “Are the flux density B and magnetizing force H quantities of the same kind? Is their ratio mu a pure numeric, or should it be treated as a dimensional quantity?” The committee was divided along national lines, with the British on one side and the French on the other.

Although we pay less regard to the authority of the past, the controversy is by no means dead, because one system of electromagnetism tends to be incorporated into the MKS system of units. This article appears, incongruously, in a 600-page Festschrift dedicated in 1968 to Georg Busch and published in a volume of Helvetica Physica Acta.

On a related note, one recommended British unit of thermal conductivity, useful for calculating the heat transmission of walls, is the BTU/hr/ft2/cm/oF.

A parable of units

–H. B. G. Casimir

Once upon a time there was in a faraway country a great, great kitchen in which many cooks plied their trade and in which there was a great profusion of pots and pans and kettles and cauldrons and bowls and basins of every size and kind and description. Some of these vessels were empty but others contained eggs or rice or apples or spices and many other delectable things. Now the cooks, if they were not busy broiling and baking and cooking and frying and preparing sundy soups and sauces, amused themselves with philosophical speculation and so it came to happen that the art of tagenometry (from the Greek, for frying-pan) was developed to great perfection. Sometimes it was even referred to as panmetry, the art of measuring everything, but the ignorant scullions, misinterpreting the word, promptly also spoke abot potmetry, much the same way in which the tranatlantic chefs have supplemented the hamburger with a cheeseburger.

To every vessel tagenometry assigned a volume V. This was measured in cubic inches and determined by measuring dimensions with great precision and by then applying the formulae of solid gemetry or in case of irregular shapes by numerical integration on a beanheaded abacus. But to every vessel there was also assigned an entirely different quantity, the volumetric displacement W. This was measured in gallons and determined by filling the vessel with water, pouring out the water, weighing said water in pounds avoirdupois, correcting for temperature and dividing by 10. The ratio of volumetric displacement to volume was referred to as the volumetric constant, e = W/V. In the course of time it became clear that this volumetric constant had the same value for every empty vessel; this became known as the volumetric constant of empty space, e0. But for other vessels the volumetric constant behaved in an erratic way. It changed after thermal treatment, or simply with time; it depended on the speed of measurement. Also the dynamic behavior of moving non-empty pans posed curious problems.

One day a wise man entered the kitchen and after having listened to the worried cooks he said: “I can solve your problems. There is really only one tagenometric quantity, let us call it the volume and measure it in cubic centimeters. Weighing water will give the same value for an empty vessel if you take the weight in grams. So your volumetric constant of empty space is just unity. But in a non-empty pan part of the volume is occupied by edibles like potatoes or pears or plums; let us call this valume P. Then, with the water-method you determine VP. In many cases P will be proportional to V, that is, P = kV. Then the water-weight volume, your volumetric displacement, is W = VkV = (1 – k)V, and hence e = 1 – k. What you should study is P and its independnece on the constitution and preparation of the victuals. And instead of studying the dynamics of a non-empty pan, you should study the motion of the things it contains.”

The cooks understood, yet they looked crestfallen. “But our beautiful units,” they said. “What about our goldplated pounds and ounces and drams? Look at that wonderful half-perch in yon corner, neatly subdividied into 99 inches. It would be ill-convenient to change all that.” The wise man smiled. “There is no real need to change, ” he said. “As long as you are sure to remember that e0 is just a way to change from one unit to another and that P and k are the only physically relevant quantities, you can work in any system of units you like.”

The years went by. The wise man had died, new generations of cooks worked in the kitchen and got restive over the principles of tagenometry. “How crazy,” they said. “Isn’t it obvious that V and W are quite different quantities, since they are determined in quite different ways? And why should the volumetric constant of empty space be unity? Is a pot of rice not just as good or better than an empty pot?” These protests prevailed. It was decided at an international congress that even if volume and volumetric displacement were identical in magnitude the one should be measured in Euclid — this being a cubic centimeter — the other in Archimedes. The volumetric displacement of empty space — although equal to unity — had the dimension Archimedes/Euclid. And after having created order in this way, the new generation has returned to inches and pounds, and brands as reactionary anyone who heeds the wise lessons of the wise man.

That is how today’s cooks spend their moments of leisure; let us hope that their cuisine will not suffer.

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