The metric properties discovered for a primitive figure remain applicable, without modifications other than changes of signs, to all correlative figures which can be considered to arise from the first. As stated by Lachlan (1893), the principle states that if, from the nature of a particular problem, a certain number of solutions are expected (and are, in fact, found in any one case), then there will be the same number of solutions in all cases, although some solutions may be imaginary.
For example, two circles intersect in two points, so it can be stated that every two circles intersect in two points, although the points may be imaginary or may coincide. The principle is extremely powerful (if somewhat difficult to state precisely), and allows immediate derivation of some geometric propositions from other propositions which may appear simpler and may be substantially easier to prove.
The continuity principle was first enunciated by Kepler and thereafter enunciated by Boscovich. However, it was not generally accepted until formulated by Poncelet in 1822. Formally, it amounts to the statement that if an analytic identity in any finite number of variables holds for all real values of the variables, then it also holds by analytic continuation for all complex values (Bell 1945). This principle is also called "Poncelet's continuity principle," or sometimes the "permanence of mathematical relations principle" (Bell 1945).
