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1 Yeti: I'm starting the ritual to restart the universe. "Ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn..."
2 Isaac Newton: I should probably forget alchemy and return to studying light and motion. I suppose turning lead into gold is impossible.
3 Ishmael: Oh, no, you just need to remove 3 protons from the nucleus.
3 Isaac Newton: Really?? Hmmm... What's a proton?
3 Yeti: Iä! Iä! Shub-Niggurath!
4 [sound]: FWACKOOOM!
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It's interesting that for hundreds, if not thousands, of years, many people were convinced that you could transform lead into gold if only one could figure out how to do it. The search for the secret was part of a pseudo-mystical study known as alchemy.
During the Scientific Revolution in the 17th and 18th centuries, systematic studies of the properties of matter and how it changed from one form to another eventually culminated in the Chemical Revolution. The French chemist Antoine Lavoisier laid the groundwork for modern chemistry by establishing the law of conservation of mass in a reaction and providing the first correct explanation for the process of combustion. This paved the way for the establishment of the chemical elements and the unchangeability of the atoms that comprised them.
By the 19th century, the dream of turning lead into gold was dead. Lead was lead, gold was gold, and no force known to man could change one into the other. This was no longer merely an expression of a lack of knowledge, but rather an expression of the certainty that chemistry was now understood, and that elements could not be transmuted.
For roughly a hundred years, this was the state of human knowledge about transmuting base metals into gold.
Then in 1896, Henri Becquerel made a very strange discovery. Minerals containing a rare element known as uranium had a strange effect on plates prepared for the recently developed art of photography. They caused foggy images to appear on the plates. The very next year, J. J. Thomson made another strange discovery in his investigation of the mysterious "cathode rays" that apparently emerged from the negative electrode of a vacuum tube (itself recently invented thanks to the emerging science of electricity).
[Aside: Notice how these discoveries all cascade off other discoveries and technologies recently invented. Much of science is this way. There are simply things we cannot discover until we have a certain level of technology. Technology enables the expansion of scientific investigation into fields previously unfathomable or unknown. Nobody could possibly have predicted that photography and electricity technology would lead to the things about to be described in the next paragraphs. This is why science needs access to new technology, even if nobody can think of any possible reason for it. The discoveries still waiting to be made could be as profound and world-changing as, well, you're about to see...]
Together, Becquerel and Thomson's findings led, very rapidly, to the astonishing discovery that atoms were not indivisible - they were made of smaller particles which carried electric charge. Within the mind-bogglingly short space of 15 years from Becquerel's initial discovery, Ernest Rutherford had proposed his model of the atom - a model still taught as the first simple, yet more-or-less correct, model of atomic structure to secondary school students today. With this model came the idea that elements were determined by the number of protons in their atomic nucleus.
A proton is a particle carrying a positive electric charge. The number of protons in an atom determines what element it is. Gold atoms have 79 protons; lead atoms have 82.
But that's not all. Becquerel's discovery of 1896 was the discovery of radioactivity. Radioactivity is the natural emission of subatomic particles from an atomic nucleus. When they eject particles, the number of protons in the nucleus changes. You heard right. In other words, the atom changes from one element to another one. Not only is it possible to transmute one element into another, nature does it all the time, without any help.
It is, theoretically, possible to turn lead into gold. All you have to do is remove three protons from the nucleus of a lead atom, and it will be a gold atom. This is not necessarily easy, but it is possible, and we now know, in principle, how it could be done. As it happens, lead is not the easiest starting point, since each proton you remove takes effort, so as far as I know nobody has bothered to actually do this yet. But people have done experiments that have turned atoms of mercury (80 protons) into gold, by removing one proton.
Not only have we proved that turning base metals (such as mercury) into gold is possible, we have actually done it.
This is, admittedly, a fairly trivial use of atomic and nuclear physics. These discoveries, completely unheralded and unthought of not much more than 100 years ago, have led to much of the technology that surrounds us today: nuclear power generation, radiation therapy for medical treatments, and (the big one) basically anything and everything to do with electronics and computers.
And now I can make use of an electronic computer, to write these words, and send them all around the world in a fraction of a second, to thousands of people, who can read about the strangely fulfilled promise of alchemy. Isn't that more amazing than simply turning lead into gold?
 While strictly true, the atom of gold you will create just by removing protons will be radioactively unstable. To create a stable atom of gold, you also have to remove a few neutrons.
Modern scientist: Yes, we can. But it would cost thousands of times more than the value of the gold, and the gold would be poisonous in a way that if you had it anywhere near you it would kill you.
Alchemist: So... where does gold come from in the first place?
Modern scientist: Exploding stars.
Alchemist: You are kidding me.
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