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Definition: Originally developed by the German physician, Samuel Hahnemann (1755-1843), homeopathy is based upon a principle of sympathetic magic: an illness can be cured by the use of minute quantities of some substance which produces symptoms similar to that of the illness itself. Thus, for example, to cure a stomach flu you have to take something which causes fever and something which causes nausea.
The best that can be said for homeopathic medicine is that it is unlikely to produce any harm itself, beyond the fact that it might cause people not to seek real medicine. The reason for this is the same reason why it has no medical value: the substances used, even if they had any medicinal value, are diluted too far to have any effectiveness. As a matter of fact, Hahnemann argued that the more diluted a substance becomes, the stronger it becomes.
A typical homeopathic dosage for some substance might be 30X. What does this mean? The basic method of homeopathy is dilution: you start with one part of some herb or substance and dilute it with ten parts water. Then you dilute that with ten parts water, resulting in a ratio of 1:100 of the original substance to water. Then you dilute that again, resulting in a ratio of 1:1000.
It does not take long to achieve extreme dilutions with this method, but a "dilution limit" is reached when only a single molecule of the original "medicine" remains in the prescribed dosage - beyond this point, there is simply nothing left to dilute. To achieve a dosage of 30X, you have to go through the above process 30 times, which means that the final ratio is 1 part "medicine" to 1,000,000,000,000,000,000,000,000,000,000 parts water.
To get just one molecule of the original substance, a person would have to drink 7,874 gallons of the diluted solution - far, far beyond the basic dilution limit. Another common dosage is 30C, which means means that the original medicine is diluted in one hundred parts water, not ten parts water, and this is repeated thirty times.
The resulting ratio will be one molecule of "medicine" for every 10^90 molecules of water - that is a 1 followed by 90 zeroes. But the number of atoms in the entire universe is about 10^80, or a 1 followed by 80 zeroes. This means that such "remedies" being suggested go beyond the dilution limit of the entire universe.
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