Fraudo the Frog?

This was the very first Ockham's talk that I ever did, back in 1985, as near as I can recall -- I wrote it on my trusty Commodore PET, with 32k of RAM. I found something odd, rang Robyn Williams, expecting that he would do an interview with me, and instead he asked me to write a script. "If it goes 13 minutes and 20 seconds, we'll use it in Ockham's," he said. "If it is longer or shorter, we'll use it in the Science Show."

It seems that with intro and outo and other bits, the ideal time for a 15-minute talk is 13 minutes and twenty seconds: for a fast talker like myself, that is about 2300 words: enough to develop a reasonable thesis, and so I have been drawn, over the years, to return to the format, over and over again. But this one has always been a favourite: it appeared in the second collection of Ockham's talks that was printed, in 1987.

All tribes have their myths. The young are brought up on these myths, and they are expected to live by these myths. The strongest of all the myths of the Science Tribe is the one about the Scientific Method. The elders say that a scientist starts out with a particular idea or rule about how things work, variously called a Theory, a Law, a Principle, an Hypothesis, or even a Conjecture.

By careful consideration, the scientist is then able to make certain predictions about what will happen if the idea is right, given some new assumption. Then all that is needed is a couple of quick experiments to see how the scientist's predictions stand up. Before long the rule, along with its assumptions, can be judged, and given either the All-Clear, or the Order of the Boot.

Not that the All-Clear necessarily means that the rule is correct, they explain. All we can really say is that such-and-such seems a bit more likely, or seems to be an acceptable approximation. Newton's Laws are still Laws, Einstein notwithstanding, because Newton lets us send rockets to the Moon, Mars, Halley's Comet, and beyond, without needing to say, "left hand down a bit to allow for relatively".

In the real world, the myths don't work. Laws don't spring, fully-formed, from the sweaty brows of scientists. Laws start out with somebody doing a bit of data-snooping.

Data-snooping involves making lists of measurements, and poring over them to see if there is any mathematical relationship or pattern or trend that might give us a hint about the rule that lies beneath the measurements. We assume that there is some rule, but in the absence of any real knowledge, we must try all the tricks.

Measured values, their squares and cubes, square roots and cube roots, their products, progressions, logarithms, sines, tangents, and other exotic mathematical functions are all thrown in, even fractions involving combinations of functions. Heavy stuff, but well worth it, if only we can make a breakthrough.

If there is a pattern, the next step is to explore the relationship further. Are there any missing values in the range of values studied? Can we extrapolate beyond the range? If we can, we must predict some values, and then go looking for them.

Johann Balmer did this when he found a relationship linking four of the hydrogen lines in the visible spectrum. It seemed that there should be another line, right on the edge of the ultraviolet, a line of which he had no prior knowledge. When he checked, the predicted line was there, just where he said it would be, and Balmer's rather odd little equation was confirmed. Score one point to data snooping.

Sometimes, though, the confirmation can be misleading, as happened in the case of Bode's Law. Now before we start, Bode's Law isn't a Law, and it wasn't even Bode's: other people had said it before him. Those problems apart, it isn't a bad sort of Bode's Law: in fact, it is the best Bode's Law that we'll ever have.

Start with the numbers zero, one, two, four, eight, and so on, triple each number, add four, and then divide by ten. This recipe gave Bode a series of numbers that closely matched the orbits of the known planets, measured in astronomic units. One astronomic unit, of course, is the distance from the Earth to the Sun.

Bode's values had no logical cause, there was one value with no matching planet, and there were slight discrepancies, but this was basic data-snooping, and the fit was quite impressive. Mercury is at 0.39, while Bode predicted 0.4, Venus is at 0.72, Bode predicted 0.7, and Earth is exactly at Bode's 1.0 position.

Mars is at 1.52, while Bode's Law says 1.6, there was nothing found at Bode's next point, 2.8, but Jupiter is spot on at 5.2, and Saturn is at 9.54, not far off Bode's value of 10 astronomic units. It seemed a terrible pity about that gap at 2.8. Maybe that was why most people ignored Bode's Law at first.

Then Herschel found Uranus 19.2 A.U. from the sun, close to Bode's next value, 19.6. So people looked at the gap between Mars and Jupiter again, and found the asteroid Ceres, at 2.77 A.U. Later, other asteroids were found, leading people to believe for a while that the asteroids were the remains of a broken-up planet.

Bode's Law, as I said before, is not a Law of Science. Even though it's an elegant pattern, and even though it predicted unknown events, both inside and outside the range of observations, Bode's Law was of no great use, and the pattern could not be tied in with any theory of Why Things Are.

Maybe we could ignore Neptune which fails to fit the pattern, squeeze Pluto into the next value, and look for Planet X at the value after that, but we don't. There seems to be no future in doing it, and it's hard to fiddle with figures which are accurately and publicly known.

That's the problem with data-snooping, though: there will always be a temptation to bend the facts. The data are often derived from experiments under the investigator's control, and if it helps to make the data fit better, well why not? We can even give it a more likeable name: let's call it fudging.

Now fudging can vary from unconsciously biased observation through massaging the data to outright fraud. Sometimes the fudging is legitimate, as when R.A. Millikan, of Oil-Drop Experiment fame, practised and practised until he got his technique right, and his results consistent. Millikan ignored his early results, and said so. You couldn't really say he cheated.

Mendel rejected one set of results, repeated the experiment, accepted the second set of figures, and said so. Mendel also fiddled mildly with a few other things and said nothing. However charitably you look at it, Mendel came perilously close to cheating.

Dulong and Petit massaged their data, and faked some more. They said nothing about this: there can be no doubt at all that they cheated. This is a serious charge, so I shall devote the rest of this talk to proving my case.

Dulong and Petit came on the chemical scene in 1819 with a law that linked atomic weights and specific heats. All elements, they said in effect, had the same heat capacity, about 25 joules per mole per Kelvin. This was a useful approximation at the time, but it hasn't been needed much since about 1830. Maybe that's why their skulduggery has gone unremarked till now.

Their Law was useful in the 1820s because proper chemical theories needed the accurate atomic weights, or at least accurate comparative atomic weights. Structures, formulae, valency, the Periodic Table were all impossible until these basic values were worked out.

In the early days, the atomic weights of metals were determined by a variety of methods, including electrolysis of soluble salts, and the reduction of metal oxides to metals. The problem of multiple valencies meant that there were conflicting values available for a number of metals. Nobody knew which values to take: was copper twice as heavy as oxygen, or was it four times as heavy? There seemed to be no solution.

Up bobbed Dulong and Petit with a law that made the measurement of atomic weight a simple matter. The Gallic duo noted that specific heats were easy to determine, and advised that the product of atomic weight and specific heat was a constant for all elements. They even presented a table of values to show how this worked out.

Now their claim was better made for metallic elements: the only non-metal in their table was sulfur. On their data, it sort of fits, but with modern data, it sticks out like a sore thumb on the butcher's scales.

Their data are not immediately examinable: for one thing, the atomic weights are all given in terms of oxygen, which is taken to have a value of one. So when I started to examine Dulong and Petit's figures, I suspected nothing more than a bit of minor fudging, and I certainly found that quickly enough.

Putting it simply, where the product of atomic weight and specific heat is too low on modern data, Dulong and Petit's specific heat value is too high. Where the product is higher than the average, their specific heat value is too low.

This could be excused by saying that the measured values for specific heat in 1819 weren't fixed: even at the end of the nineteenth century, published values varied quite a lot. There are just two give-aways in the table, two uncontestable whistle-blowings that tell us what was really going on.

The first of these is the information on cobalt. This, we are assured, has a relative weight of 2.46, and a specific heat of 0.1498. Translating the relative weight, we get an atomic weight of about 39.36. That wasn't right, I thought, reaching for the CRC Handbook. Sure enough, it gives an atomic weight for cobalt of 58.93, almost 50% higher.

When I found this, I looked again at the specific heats quoted by Dulong and Petit and blow me down! the modern figure was only two-thirds the one they used. How fortunate they were, two cancelling errors like that! Curious, too, seeing that cobalt has valencies of 2 and 3, so that the error in the relative weight is almost, well, totally predictable. What a shame that the same can't be said for the error in the specific heat.

In 1981. I had the pleasure of unravelling a fraudulent set of evaluations of a computer-based education system, just at the time when I was first reading Eugene Kamin's marvellous expos‚ of Cyril Burt's work.

I came to the part where Kamin suggests that "a benign Providence appears to have smiled on Professor Burt's labours". I fell in love with the expression, and I had used several variations on it. Now seemed to be the time to revive that phrase. I started to look more carefully at the rest of the data.

I am not, I must confess, familiar with the chemistry of tellurium, and I wasn't really expecting another find. But there it was: the relative weight of tellurium, 4.03, gives us an atomic weight of 64.5, about half the accepted modern value of 127.6. And would you believe it? The specific heat had gone from today's .048 to .0912, almost double. Funny, that, especially as tellurium has valencies of 2 and 4.

I still can't explain why they had problems with cobalt: measuring its specific heat should be quite straightforward. But I think I can shed a bit of light on the tellurium problem. You see, tellurium is unpleasant stuff, and it gets absorbed through the skin, ever so easily. So do its compounds. And once you've absorbed the tellurium, you exhibit something that the CRC Handbook calls "tellurium breath". You smell of old garlic, and it lasts for months. So I wouldn't really blame any experimenter who chose to avoid tellurium in the lab, and no bad jokes about French food either, thanks.

So there we have it. Just to nail it home, I calculated the correlation coefficient of the specific heat fudge factors against the shift needed to get a perfect fit, and got a correlation coefficient of -.71, which is significant at the 1% level, and consistent with the other evidence.

I conclude that the two scientists massaged their data. They made two errors in atomic weights, each consistent with a wrong guess at the valency of a metal, and in each case, the specific heat was grossly wrong, and just happened to cancel out the errors. This was even less likely than the convenient little errors elsewhere. And if they guessed wrongly on cobalt and tellurium, how many of the other figures were just luckier guesses?

Overall, I think that their practices stink worse than tellurium breath, but does it matter? Their "Law" helped the early chemists to bypass with confidence a sticky problem, so even if they did cheat, it served a useful purpose. I can't help wondering, though: how many of our cherished and established facts of science were born of similar fraud?

Additional data


Dulong and Petit's data

D&P at. wt.

Modern equivalent data


Sp heat

A. W.



Sp. heat

A. W.










































































































The things to look out for here, the tell-tale signs of cheating are in bold:

* the 'fudging' of the specific heats of lead and sulfur, to make them fit better, and

* the serious errors in the figures for tellurium and cobalt, matched by errors the other way in the specific heats.

This is one of a set of talks which were originally heard on ABC Radio National in Australia. All of the talks are copyright © Peter Macinnis, 2001, but permission will be readily granted on request for educational and most non-profit purposes. Contact Peter Macinnis specifying the talk(s) you want, and the purpose to which they will be put. For the rest of the talks, go to Six Months of Sundays.