serves its purpose. The variants on the concept of perfection would have been quite of a piece for two thousand years, had they not been confused with other, kindred concepts. The chief of these was the concept of that which is the best: in Latin, "excellentia" ("excellence"). In antiquity, "excellentia" and "perfectio" made a pair; thus, for example, dignitaries were called "perfectissime", just as they are now called "excellency." Nevertheless, these two expression of high regard differ fundamentally: "excellentia" is a distinction among many, and implies comparison; while "perfectio" involves no comparison, and if something is deemed perfect, then it is deemed so in itself, without comparison to other things. Gottfried Wilhelm Leibniz, who thought much about perfection and held the world to be the best of possible worlds, did not claim that it was perfect. Paradoxes The parallel existence of two concepts of perfection, one strict ("perfection", as such) and the other loose ("excellence"), has given rise, perhaps since antiquity but certainly since the Renaissance, to a singular paradox: that the greatest perfection is imperfection. This was formulated by Lucilio Vanini (1585–1619), who had a precursor in the 16th-century writer Joseph Juste Scaliger, and they in turn referred to the ancient philosopher Empedocles. Their argument, as given by the first two, was that if the world were perfect, it could not improve and so would lack "true perfection", which depends on progress. To Aristotle, "perfect" meant "complete" ("nothing to add or subtract"). To Empedocles, according to Vanini, perfection depends on incompleteness ("perfectio propter imperfectionem"), since the latter possesses a potential for development and for complementing with new characteristics ("perfectio complementii"). This view relates to the baroque esthetic of Vanini and Marin Mersenne: the perfection of an art work consists in its forcing the recipient to be active—to complement the art work by an effort of mind and imagination. The paradox of perfection—that imperfection is perfect—applies not only to human affairs, but to technology. Thus, irregularity in semiconductor crystals (an imperfection, in the form of contaminants) is requisite for the production of semiconductors. The solution to the apparent paradox lies in a distinction between two concepts of "perfection": that of regularity, and that of utility. Imperfection is perfect in technology, in the sense that irregularity is useful. Perfect numbers Perfect numbers have been distinguished ever since the ancient Greeks called them "teleioi." There was, however, no consensus among the Greeks