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Learning from creationists; radiocarbon dating

Radiocarbon dating only takes us back some 50,000 years. This makes it a much smaller threat to Young Earth creationists than, say, lead-uranium dating, which takes us back billions of years. So why do creationists single it out for attack? Because there are indeed problems with the most simple-minded application of the method, and it does not matter to the creationist that these problems have long since been solved. Creationists think, and argue, more like lawyers than like scientists. In the courtroom, changing your story under cross-examination will destroy your credibility, and yet this is what scientists do all the time. Scientists accept that even the most well-established findings are subject to revision and refinement; lawyers, like theologians, seek certainty whether the data justify it or not.

SymTalk

Leonard Sym’s presentation to Glasgow Skeptics in the Pub, 21 March 2016

This post is derived from a talk given by my friend Leonard Sym to Glasgow Skeptics in the Pub, and like Leonard I will follow Rapoport’s rules for debate, which specify that one should first summarise one’s opponents’ position in terms they would accept, next, list points of agreement, then point out what one has learnt from one’sopponents, and only at that stage embark on criticism.

I start with a simplified summary of the principles behind radiocarbon dating, without which the discussion would be meaningless. Most radiometric methods depend on measuring the amount of a parent radioactive isotope present in the sample, and the amount of the daughter into which it decays. Add up the amount of parent still present and the amount of daughter, and that gives you the amount of parent present initially.[1] If you know the rate constant for the decay, you now have enough information to work out how long has passed since the parent material was put in place. You can determine the rate constant by taking a known amount of parent, and counting the number of decays per second, as measured with a Geiger counter or a more reliable and up-to-date instrument such as fluorescence counter.

In the case of radiocarbon dating, the parent is carbon-14 and the daughter is nitrogen-14, which is lost from the sample.[2] So we can’t find the initial amount of parent in the way described above, because we don’t know the amount of daughter. This seems like a dead end, until we remember where carbon-14 comes from. Carbon-14 is formed in the upper atmosphere by the effects of cosmic ray bombardment on nitrogen, is rapidly converted to carbon-14 dioxide, and then mingles with the rest of the CO2 in the atmosphere (see Figure). If we assume a steady rate of bombardment, that means we will have a steady rate of production of carbon-14, and a steady state abundance of carbon-14 in the atmosphere, where the amount decaying each year is equal to the amount being formed.

Now consider what happens during the life of an organism, and after its death. As long as it is alive and metabolising, it will exchange carbon with its environment, taking it in directly as carbon dioxide by photosynthesis (for a plant) or indirectly as food (for an animal). At this stage, the proportion of carbon present as carbon-14 will be directly dependent on that in the atmosphere. But as soon as it stops metabolising, it stops exchanging, and the proportion present starts decaying according to the radioactive decay law, with a halflife of 5730 years. So it looks as if we can just use the proportion present in the atmosphere right now as a measure of the initial proportion, and compare it with the proportion remaining.

radiocarbon_sub1

Production and decay of carbon-14**

So far, so good. Now let me list the creationists’ objections:

1) As in all radiometric dating, the decay rate is assumed to be constant. What if this isn’t true?

2) The production rate is assumed constant. But this is unrealistic, since the intensity of cosmic ray bombardment is known to change over time

3) For 150 years, and especially in the last 50 years, we have been adding carbon dioxide from fossil fuels to the atmosphere, diluting the radiocarbon since all the radiocarbon in the fossil fuels will have long since decayed

4) Considerable amounts of carbon-14 were added to the atmosphere by nuclear testing in the 1950s, further undermining the assumptions

5) What if carbon-14 is less readily taken up than carbon-12 by plants? Won’t this undermine the reasoning?

6) We can check the method by applying it to materials whose age we know, but this will only serve where we have a good historical record, and this record only goes back, at best, some 5000 years

7) The Genesis flood, which in Young Earth accounts is responsible for the formation of our fossil fuel deposits, would have further upset the clock by burying huge amounts of carbon-12. Moreover, it could have been associated with an increase in the rate of carbon-14 production, making pre-flood specimens look much older than they really are.

LibbyBookWith the exception of the first and last, all these objections have some degree of plausibility, but unfortunately for the creationists they have all long since been answered, many of the answers being set out by Willard F. Libby, inventor of the method, in his 1955 book on the subject.

1) Radiometric decay constants are just not the kind of thing that could change, unless everything else changes at the same time. We have known since the work of George Gamow in 1928 that radioactive decay is what is known as a quantum mechanical tunnelling effect, and that its rates depends on such things as the strength of nuclear and electrical forces, the mass of fundamental particles, and Planck’s Constant h, which gives the scale for all quantum mechanical phenomena. If any of these had been different, we would not have had the same kind of physics and chemistry that we have today. But we know from their structure that ancient rocks were formed under the same rules as we have today, because they contain the same kinds of elements combined to make the same kinds of minerals. The creationists have published theoretical curves for changes in decay constants, but these have no basis in science, and are generated merely to make observations fit the biblical timeline.

2) From the outset, radiocarbon dating has relied on calibration, using objects for which dates were known from historical records, then tree ring counting extending back 10,000 years. This method works because all but the outermost layers of a tree are metabolically inert, and out of circulation. The most recent calibration comes from organic debris in varves (annual layers of sediment) deposited in a lake (Lake Suigetsu, in Japan) that happens to be free of turbulent inflows. This has made possible the establishment of a calibration curve going back 52,800  years.

Radiocarbon_bomb_spike.svg

Near doubling of atmospheric C14 in the Southern hemisphere, as the result of nuclear testing

3) and (4) There have indeed been major disruptions since 1950, but no one uses radiocarbon dating for such recent material. The situation in 1950 is regarded as a baseline, material from that year is the standard for comparison, and 1950 is the “present” in conventional dating of “years before present”

5) It will surprise many people to learn that plants really do take up carbon-14 less readily than carbon-12. One of the lies you were probably told at school is that all isotopes of the same element have precisely the same chemical properties. This is not true, and generally speaking, heavier isotopes are slightly more sluggish in their chemical reactions. this gives rise to the process known as isotopic fractionation.

NorthRonadsaySheep

These North Ronaldsay sheep, which feed on seaweed, will show different isotopic fraction from sheep fed on grass

This effect has been measured for photosynthesis. In addition to very small amounts of radiocarbon, atmospheric carbon dioxide contains roughly 1% of the stable isotope carbon-13, the remainder being carbon-12. Carbon dioxide in plants is, as expected, slightly depleted in carbon-13 relative to carbon-12, and the effect is far from trivial; around 27 thousands of carbon-13 abundance for most kinds of plant. We expect the effect to be twice as large for carbon-14, which, using the known 5730 year halflife of carbon-14, corresponds to 435 years; not trivial when dating historical artefacts. However, exactly the same effect will apply to the material used to set up the calibration, and the errors will systematically cancel out. Ideally, the fractional abundance of carbon-13 should be measured, as well as that of carbon-12, to calibrate out any minor fractionation effects, and this is less arduous than it sounds because nowadays carbon-14 abundance is measured by direct counting in a mass spectrometer rather than, as in the original studies, indirectly inferred from sample radioactivity.

For plants, it is straightforward to match like with like. Not so for material derived from animals, where the total amount of isotopic fractionation will depend on their diets, and also on what they have been eating.

6) This objection would have had some force in 1946, when the method was newly developed. However, as already explained, we now have direct calibration back to 52,800 years before present, beyond which the amount of remaining carbon-14 is so small that using current techniques the method becomes useless.

7) This is pure special pleading. If carbon-12 had been buried in the flood, the appropriate amount of carbon-14 would have been buried with it. And the ideas of a changed rate of production or decay have been dealt with under (1) and (2) above.

There are other “objections” based on the obvious fact that organisms like cave water snails, alive right now but deriving their carbon from limestone, will have radiocarbon apparent ages measured in thousands of years. I have discussed this before. And marine specimens will always contain less carbon-14 than terrestrial specimens of the same age, because of the time it takes for mixing between the atmosphere and surface waters, and again between surface waters and the depths.

So how should we respond to the self-styled “creation scientist”? The first, and most difficult, thing is to realise that he is been perfectly sincere. He is certain that his reading of the Bible is true; but the facts of geology are also true; and it is therefore his mission to create an account that reconciles the two. If this means the mountains must have skipped like rams, then that is how they must have skipped. He will feel no more absurd at this point, than the cosmologist feels in invoking a time when our Universe was smaller than a tennis ball and its temperature was trillions of degrees. Counter-arguments will be dismissed as so many minor anomalies that will no doubt be explained away in due course. If the creationist repeats long-refuted claims, that is because he believes that there are refutations of the refutation, even if he cannot immediately call them to mind, or does not have time to explain them properly. He will remember the weaknesses of his opponents’ arguments, and attack them, while suppressing the recollection of their strengths, and in the process he will create, and then triumphantly destroy, a series of straw men. You and I of course would never do such things, but your friends might when it comes to defending emotionally precious but logically fragile beliefs; consider, for example, what passes for political discussion in your favourite pub or chatroom.

And what does this mean for debating with creationists? Simply this: don’t do it. Such a debate, unlike a discussion between people willing to learn from each other, is a zero-sum game. He will project simplicity, sincerity, and certainty, and when you come to reply, you will sound as if you are making excuses. He will present anomalies (did I mention those 2000-year-old water snails?), and when you explain the special circumstances, you will be the one who seems guilty of special pleading. His followers will end up confirmed in their convictions, as will yours, and those in the middle will come away confirmed in their own initial conviction that there are two sides to the story, both worth hearing. Which there aren’t.

But does that mean that we can learn nothing from the creationists? Not at all. In terms of Rappaport’s rules of debate, the scientific community had already come up with arguments (1) through (6), and taken the necessary countermeasures, and so cannot be said to have learnt from the creationists. But both Leonard and I have learnt a great deal from examining the creationist claims. Be smart, and learn from everyone.*

1] It is of course necessary to eliminate errors caused by the movement of material, or the presence of daughter in the initial material. There are standard techniques for doing this, for instance by measuring non-radiogenic isotopes of the daughter material, and, these days, by microsampling of single crystalline grains

2] Even if it is not lost as N2 gas, it will be undetectable against the background of organic nitrogen compounds already present

*Ben Zomah, Mishnah Pirkei Avot 4a

** extra credit for spotting (a) the misleading labelling in the diagram (h/t John Gribbin), (b) the reference in the text to Psalm 114

There are other creationist objections to radiocarbon dating, based on sample contamination or simple misinterpretation of data, but these have been discussed elsewhere  and need not detain us.

Lecture scene from Glasgow Skeptics in the Pub Facebook page. Atmospheric carbon-14 diagram public domain, by Hokanomono via Wikipedia. North Ronaldsay sheep by Liz Burke, CC BY-SA 2.0, https://commons.wikimedia.org/w/index.php?curid=3499298. Radiocarbon cycle schematic from ANU Radiocarbon Dating Lab materials

How to lie about radiometric dating, evolution, and even nuclear physics

Have you heard the one about the live snail with a carbon-14 age of 3000 years? Or the lava erupted in 1800 in Hawaii with a potassium-argon age in the millions? It’s all true, true I tell you. But does this signify a major problem with radiometric dating?

Spoiler: no.

I don’t know who first dug up these examples, but they were popularised by the creationist comic-book writer Jack Chick, in a publication called “Big Daddy”. The first page, available here if you’re lucky (the links to Chick Publications only seem to work at random), shows a well-primed creationist student arguing with a singularly ill-informed biology professor. The professor has been leading such a sheltered life that he’s never met these creationist arguments before. And he doesn’t understand anything about evolution or dating of rocks or embryology or indeed anything else. Surprise! the student wins! A skilled cartoonist, Jack Chick manages to squeeze the largest number of fallacies into the smallest number of words. There is a crib sheet at the end of this post, listing all the fallacies I spotted myself; I just reached double figures but there may be more.

Evoa

dating

gillarch

vestigial

Of course, it doesn’t help that the Professor doesn’t know anything about whale ancestors:

WhaleAncestors

Or that the student is allowed to make the most absurd statements unchallenged, on the basis of a video by Kent Hovind:

evoc

But there’s more! At the end of page 1, which is also the end of your free sample I’m afraid, the student converts the Professor by pointing out that no one has ever actually seen gluons:

Gluons

Gluons2

But fear not; an answer is at hand, in the very next frame:

Gluons3
So Jesus must be the force that holds the atomic nucleus together. Convinced by this reasoning, the Professor accepts Jesus, announces that as a result he can no longer teach evolution, and is sacked.

Jack Chick, by the way, has just published another comic book at the age of 92. In it, a bright young man from a good Christian (i.e. creationist) home is seduced by Satan into believing in evolution, and when we last see him is heading straight for damnation. In the words of one of Satan’s many horned helpers, “Joe trusted evolution, not God, and became a jobless party animal.” And a criminal and a drug addict, and covered himself with tattoos, and died and went to hell. Tragic, and so easily avoidable.

I never managed to get to Page 13 of Big Daddy, which is what we really need; link (if it works for you) here. It didn’t work for me, but you’ll find a description of the contents by someone called Honus at talkorigins, and I’ve seen some of the relevant cartoons reproduced elsewhere. So you can either take Honus’s and my word for it, or go online to Chick Publications and buy 25 copies (minimum purchase) of the tract, which I am not about to do.

The really remarkable thing about the tract is that it actually gives the primary literature references to the results that is discussing. And the briefest perusal of this primary literature will show why the papers that Chick refers to, far from undermining radiometric dating, actually reinforce it.

That snail was not 3000 years old, but that really was its apparent radiocarbon age, because it was exchanging calcium carbonate in its shell with mineral calcium carbonate. And that makes all the difference, so you need to take such features of the environment into account.

Many readers will be familiar with the principle of carbon-14 dating. Carbon-14 decays with a half-life of 5730 years. Nonetheless, the fraction of carbon-14 in the atmosphere stays roughly constant (or did before we started adding to it by nuclear weapons testing, and diluting it with carbon dioxide from fossil fuels). That is because the upper atmosphere is bombarded with cosmic rays, which cause nuclear reactions that convert nitrogen-14 (stable) to carbon-14. Mixing distributes this radiocarbon through the atmosphere, where it is taken up by plants and, in due course, animals. As long as you are alive, you are part of the circulating pool of carbon, but as soon as you die, the carbon-14 in your body starts decaying. Of course, cosmic ray intensity is not really constant over a long period, but we can calibrate carbon-14 dates by comparison with carbon in tree rings (dendrochronology). The tree ring correction is small for most purposes, but matters for things like precise dating of Egyptian dynasties.

The point, of course, is that the carbon in the lettuce being fed to the snails is part of the general pool, but the carbon in calcium carbonate minerals is radiochemically dead, having been out of circulation for a long time. What the paper really showed was that the snail exchanges carbonate in its shell with carbonate from dissolved minerals, giving a spurious depletion of radiocarbon in the snail. You will find the story in Science, 1963, p. 637 (paywall, sorry, but summary here).

What about these rocks in Hawaii? Here again the paper is behind a pay wall, but if you follow this link it will take you to the title and abstract, which is all you need. In fact, the title alone is all you need: “Radiogenic helium and argon in ultramafic inclusions from Hawaii”. Inclusions. And in case that’s not clear enough, the abstract tells you that the work is all about the dating of xenoliths. Xeno- foreign, as in xenophobia; lith rock, as in monolith. Look at the paper in more detail, if you can get access to it, and you will find that the excess argon is only found in bubbles of fluid within the rock, that bits of rock that aren’t bubbly don’t show any, and that there is circumstantial evidence that the argon comes from deep within the Earth’s mantle, not radioactive decay in the lava itself.

Anomalies happen all the time in geology. They are, in the original sense of the expression, exceptions that prove the rule; if there were no rule, we would not consider them exceptional. Uranium-lead and potassium-argon dates of rocks usually agree, but not if the rock has been so strongly heated that argon gas can escape. Whole rock dates can be misleading, as in the example of the Hawaiian volcano, if the rock has been contaminated from some source, in this case fluid from the mantle. So far from undermining the method, these anomalies add further information about the sample. In much the same way, radiocarbon dates will be anomalous if some of the carbon comes from inorganic sources, as in the case discussed above, and the anomaly might even be used to tell us something about the specimen’s history and diet.

Now here’s the bit that I really don’t understand. What is going on in Jack Chick’s mind, when he gives us this stuff? I assume that he is an honest person of goodwill, who is doing his best. He really believes that because I and most readers here accept the fact of evolution, we are going to be punished in hell for ever. Being a kindly man, he really doesn’t want that to happen, so he is doing his very best to convince us of the error of our ways.

So why does he do it by pointing us towards papers that say the very opposite of what he says they say? I can only speculate that this is the result of what psychologists call confirmation bias, which leads to interpreting new information, however perversely, in a way that supports what you already think. And when we come to creationism, the motivation for bias is extreme. Remember that we are talking about people who really believe (a) that if you don’t accept salvation through Jesus you are going to go to hell, and (b) that the doctrine of salvation through Jesus only makes sense if the biblical Fall is a historical fact. The papers I’ve mentioned above show that under certain rather special circumstances, radiometric dating will give you the wrong answer unless you take those circumstances into account. Young Earth creationists, knowing that their entire worldview depends on refuting radiometric dating, pounce on these examples as evidence that the method is unreliable. Which of course it is, if you don’t do it right. So what?

All of which gives me uncomfortable pause for two reasons. If creationists are so blinded by confirmation bias, what hope is there of reasoning with them? And if I see my intellectual opponents displaying confirmation bias, completely oblivious to what they are doing, what makes me think that I am any different?

h/t Sensuous Curmudgeon for tip-off about Jack Chick’s latest. Whale ancestors illustrated (Ambulocetus and Pakicetus) copyright JGM Thewissen; may be reproduced for non-commercial educational purposes.

IncredibleCrib sheet: Definition as obfuscation. Misdefinition of science to exclude all indirect inference (although even Young Earth creationists accept the fact of an Ice Age on geological evidence). Macroevolution, if the word means anything, means major change, and this takes more time than we have been watching. So of course we’ve never seen it. Similar fossils do indeed imply similar ages, but the order of these ages has been known for nearly 200 years on the basis of stratigraphy, and absolute ages established for over 100 years now by radiometric dating. Polystrate fossils were explained in 1868; the explanation is much the same today. New Scientist really did point out in 1997 that it is silly to carry on using Haeckel’s highly questionable drawings, as some still do, when we now have a much more detailed information. But, as explained in Alice Roberts’s Incredible Unlikeliness of Being and many other places, the gill folds on the human embryo really are homologous to the folds on that of a fish. They just develop rather differently, explaining such oddities as the tortuous path of our vagus nerves. As for the whale’s pelvis having “nothing to do with walking on land”, by 1999 we already had extensive series of fossils linking whales to their terrestrial ancestors; there is an excellent review here by one of the scientists involved in Evolution Education and Outreach (free download), and whales evolution also features in an excellent video here . The development of secondary functions (exaptation) is commonplace. Thus mammals’ ear bones are vestigial relics of reptiles’ rear jawbones. Creationists often argue, as here, that natural selection can only remove, and not add. This riddle was solved 120 years ago, with the discovery of mutation. Mutations supply novelty; selection winnows it. Creationists agree in explaining away pre-modern human fossils, such as Lucy and numerous others already known by 1999, as being either apes, or humans. Unfortunately, they can never agree on which is which. And, something that I think believers in particular should find offensive, the theological absurdities of the final frame.

Glasgow’s Intelligent Design Director has ”open mind” on age of Earth

Al Furqan Mosque, Glasgow, in whose Community Hall Dr Noble spoke last Friday (November 28) on ‘Intelligent Design: Myth or Reality?’

See how many errors of fact and logic you can find in what Dr Noble, Director of Glasgow’s own Centre for Intelligent Design, said when my friend and Scottish Secular Society colleague, Garry Otton, asked him about the age of the Earth. This was on the occasion of his visiting a mosque as part of the activities of Scotland’s Interfaith Council, which receives £9,000,000 a year of taxpayer money. I offering him  space to reply, but he tells me that “I really don’t think this kind of speculative trivia deserves a considered response.”

Dr Noble said that the scientific consensus is about 3 billion years old, but there is a lot of uncertainty about all scientific things and some think the Earth’s only thousands of years old. He thinks the Earth might be old, but human beings might be “much younger than most scientists would accept”. A geologist has shown him a piece of rock, dated as 300 million years old, but containing a seam of coal carbon-14 dated at 40,000 years. And carbon dating only goes back to around 50,000 years, so [he said with heavy emphasis] “the error is not in the date of the coal.” All methods depend on judgements about initial conditions that we have no way of knowing. The scientific consensus can be very arrogant, so he doesn’t have a serious problem with an old Earth, but is not completely convinced.

Here is my list of errors; let me know if you spot others:

First, and least important, that number, 3 billion years. It should of course be around 4. 5 billion years.

From the fact that carbon-14 is useless beyond 50,000 years, Dr Noble infers that the error is “not in the date of the coal.” The exact opposite is true. The 50,000 limit is the limit of the method, showing that it cannot be used for more ancient deposits.

Actually, it has long been known that ancient coal cannot be dated by carbon-14, because it gives spurious and erratic young(ish) ages, between 20,000 and 50,000 years. This has thwarted attempts to use it as a standard background (see e.g. this 1939 paper). One percent contamination by contemporary material of the area sampled will reduce the apparent age to less than the 40,000 years that Dr Noble quotes, and other potential problems include the presence of modern bacteria (well established by 1931) and carbon-14 generation in the coal from the nitrogen-14 present, as an indirect effect of the radioactive decay of heavy elements in the coal.

I don’t know where Dr Noble got his numbers from, but they are identical with those quoted in a 2002 Talkorigins article, which discusses and dismisses the alleged anomaly. A fuller and more recent critical analysis of claims of detecting carbon-14 in ancient materials  can be found at http://www.asa3.org/ASA/education/origins/carbon-kb.htm#22 (ASA in this address stands for American Scientific Affiliation, an organisation of Christians who reject attempts such as Dr Noble’s to discard the plain science of evolution and an ancient Earth; the author, Kirk Bertsche, holds a Ph.D. for his work on radiocarbon methods, and an MA in Exegetical Theology from Western Seminary, Portland, Oregon).

As for our judgments about initial conditions, we can check these in at least four separate ways. Firstly, careful mineralogy, picking out well isolated samples or even well-defined individual crystallites. Secondly, available since the 1940s, what are known as isochron methods, which use non-radiogenic isotopes as internal markers. Thirdly, SHRIMP (sensitive high resolution ion microprobe) methods, available since 1980, which allow assessment even of selected regions within crystallites, giving valuable information about thermal history and element mobility; I have a friend who does this for a living. Finally, the good agreement of disparate methods (this was mentioned to Dr Noble) provides cross-validation, and the rare occasions when this does not occur (for example in rocks with anomalously young potassium-argon ages) are themselves highly informative (in this example, regarding subsequent heating episodes). Radiometric Dating, A Christian Perspective, by Roger Wiens,  principal investigator of the Mars Curiosity Rover’s chemical laser analysis team, gives an excellent perspective on this and other alleged problems.

It is just not true that there is a lot of uncertainty about all scientific things. There is no real uncertainty about the age of the Earth, any more than there is real uncertainty about the existence of the atoms of which it is composed. Dr Noble himself would be very upset if I were to allege that there is a lot of uncertainty about the abstruse area of inorganic chemistry in which he obtained his own Ph.D. over 40 years ago.[1] And yes, we should keep an open mind, but not so open that our brains fall out.[2]

So how did Dr Noble, who evidently still thinks of himself as a scientist, come to make such a concatenation of elementary errors? I would suggest confirmation bias, selecting evidence that supports a view already adopted for very different reasons. And in this case, that view is identified by his wish to take seriously the possibility that human beings might be “much younger than most scientists would accept”. Dr Noble belongs to a church that believes in the “entire trustworthiness” of Scripture, and perhaps he looks kindly on the idea that human beings were specially created in the past few thousand years. But if this is the case, the biological science that shows we are sister species to chimpanzees, and the Earth science that dates undeniably human skulls back to more than 100,000 years ago, must be denounced as unreliable. The rest follows.

Added note: Some commentators have inferred that Dr Noble is a six-day creationist. I, however, interpret his careful distinction between the age of the Earth and the age of humanity a showing a more broad-minded perspective than that, one which also grants credibility to day-age and gap versions of creationism.  Here day-age accepts the order of events in Genesis, but allows that each “day” may refer to an era. “Gap” allows geological tme in between Genesis 1:1 and Genesi 1:2. Both were originally attempts by conservative theologians to accommodate biblical literalism to early 19th century geology, and both, crucially, accept a historical Eden a few thousand years ago, and separate creation of kinds. I deliberately chose impeccably Christian sources for the geological background, to highlight the possibility that Dr Noble’s hermeneutics may be as selective as his geology.

1] The Preparation and Properties of Tungsten Hexafluoride Derivatives, Glasgow, 1970.

2] For the history of this phrase see here.

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