Generic trademark acquired substantial market dominance or mind share, such that the primary meaning of the genericized trademark becomes the product or service itself rather than an indication of source for the product or service. A trademark thus popularized has its legal protection at risk in some countries such as the United States and United Kingdom, as its intellectual property rights in the trademark may be lost and competitors enabled to use the genericized trademark to describe their similar products, unless the owner of an affected trademark works sufficiently to correct and prevent such broad use. Thermos, Kleenex, ChapStick, Aspirin, Dumpster, Band-Aid,
Hippocrates of Chios would be known too. Only much later was it proven (by Ferdinand von Lindemann, in 1882) that this approach had no chance of success, because the factor pi (π) is transcendental. The number π is the ratio of the circumference to the diameter of a circle, and also the ratio of the area to the square of the radius. In the century after Hippocrates at least four other mathematicians wrote their own "Elements", steadily improving terminology and logical structure. In this way Hippocrates' pioneering work laid the foundation for Euclid's "Elements" (c. 325 BC) that was to remain the standard