In 1691, James Bernoulli proposed the following problem called elastica problem : "What shape of elastica, an ideal thin elastic rod in a plane, is allowed ?" Euler essentially solved the problem in 1744 by developing studies of variation problem and elliptic function theory. Their studies are regarded as prototypes of harmonic map theory, nonlinear differential theory, soliton theory, differential geometry, algebraic geometry, theory of moduil of elliptic curves and so on. In this article we mention their mathematical meaning with their historical background from viewpoint of pure and applied mathematics : Their studies started from concreteness to abstractand they applied constructed abstract theory to the concrete problem. We also introduce a current study of statistical mechanics of elasticas, which might be settled by knowledge of hyperelliptic function theory.