Wednesday, August 22, 2012

One Beer's law too many

Some people may think that Beer’s law has to do with underage drinking, and that August Beer is what comes before OktoberFest. Beer’s law is, however, one of the coolest laws of photometry, and August Beer is the guy who it is named after. (For a complete discussion of how it got that name, skip to the end of this blog post.

This blog post is a re-enactment of a seminal experiment that a preeminent researcher reported on back in 1995. This phenomenal scientist has had such a profound influence on the worlds of printing and colorimetry, that I am tirelessly committed to the promulgation of his work. I am speaking, of course, about myself.  
Experimental setup
The picture below details the equipment to be used in this experiment. At left is a constant current power supply, which provides power for the blue Luxeon LED. This LED shines into the optical assembly, which is supported by one of the biggest books I have on color science. At the far right is the sensor for an expensive light sensor, with the control unit show on the expensive black carpet. The observant reader will no doubt be impressed by the huge expense that I must have gone through to dig this pile of junk out of my basement.
Expensive equipment used for this experiment
The lights were turned down and the system calibrated so that the light meter read 100.0 banana units when there was nothing between the light source and the detector, as shown below.
Expensive optical stuff, bored, with nothing to read
Now the party begins. I cracked open a cold one and set it in the beam. Note that the reading has dropped to 90.0 banana units, indicating that 10.0 banana units of light got caught by the amber fluid and never quite made it home. I can definitely identify with these photons.
Same set up, but with one sample cell
As they say, you can’t milk a camel while standing on one leg, so let’s order another one. But before it gets set down on the bar, let’s take a guess at what the light meter will read. Hmmm…. The first sample dropped it by 10.0, so it would make sense that the second one would so the same. My guess at the results: 80.0.
Same set up, only this time with two samples
For those of you who agreed with my guess, it was commendable, but wrong. There was indeed a pattern established, but not the one you were thinking of. Why did it go down to 81.0, instead of 80.0? For every 100 photons that entered the first sample, 10 of them were absorbed, and 90 were transmitted on to the second sample. Upon reaching the second sample, the same probabilities apply. Of the 90 photons that made it to the second sample, 90% of those made it out, so that there were 81.
You now know Beer’s law.
But just to make sure the concepts are all down, let’s take this one step further. How about three samples? 90% X 90% X 90% = 72.9%, as verified by the highly sensitive experimental set up below.
Results for three samples
One last thing… Can you guess what kind of beer was used?
Miller Lite – the official beer of color scientists everywhere
Disclaimers – Do not try this experiment at home. I am a trained professional. The mixing of beer and scientific equipment is not recommended. No beer was wasted in the photoshoot for this blog. I cannot say the same for the scientist who performed the experiment.
Who invented Beer’s law, anyway?
Some folks may have just assumed that Beer’s law was named after William Gosset, who was a pioneer in statistics, and worked for Guinness. That would be a good guess, since he was a smart guy. It would have been just like him to have a really cool law of physics named after him, since he invented the t test, which was named after Student, which was actually his pen name. But that’s another interesting story.
The guess is unfortunately wrong, since Beer’s law was named after August Beer. This is yet another in my series of mathematical misnomers.
This law of physics was first discovered by the father of photometry Pierre Bouguer in 1729. August beer didn’t discover this law until over a century later in 1852. Beer worked with  Johann Heinrich Lambert on a book (“Introduction to the Higher Optical”) that was published in 1860. So naturally, the law has become known as “Beer’s law”, “Beer-Lambert law”, “Beer Lambert-Bouguer law”, “Lambert-Bouger law”, “Lambert’s law”, and “Bob”.
Why is it known in the printing industry as “Beer’s law”? There are two key influences that led to this egregious misnomer. The first was a landmark 1967 book by J.A.C. Yule, “Principles of Color Reproduction”. Any book with the word “reproduction” in the title is apt to move quickly. I just checked with Amazon. They only have two copies left.
The second thing that probably had an even greater effect was the frequent use of the eponym by the eminent applied mathematician, color scientist, mathematics historian, and all around nice looking guy, John “the Math Guy” Seymour. He has made no bones about why he decide on this name among all the potential candidates. I quote here from his paper delivered at the 2007 Technical Association of the Graphic Arts:
Since there seems to be little agreement about who is responsible for which law, I have chosen to refer to the statement that optical densities of filters add as Beer’s law. My decision is not based on historical evidence, but on the gedanken I introduced in a paper given at IS&T (Seymour, 1995). In this, I demonstrated the law by using a varying number of mugs filled with beer. My hope is that my further corruption of already corrupt historical fact will help remember the law!
Brilliant words by a brilliant man, indeed.

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