In his main book "Discorsie Dimostrazioni Matematiche, Intorno a Due Nuove Scienze" published in 1638 by Elsevier in Leiden, Galileo Galilei, "the Father" of modern science, put the material science and strength of materials on the first place. He introduced the notions of stress and strength that have been fundamental since then. Moreover, in unison with Plato's theory of forms he found out the perfect shape of a force-bent beam we call today equistrong. This discovery laid the foundation for search of other perfect elastic bodies as a continuation of Galilei's work. There are no theorems of existence for equistrong bodies so that the quest for them is like a gold-digging. In what follows, the shapes of the following heavy, equistrong beams were found out: a) beam of constant thickness and of variable width, simply supported at both ends, b) beam clamped at one end and loaded at the other end while having either constant thickness and variable width, or constant width and variable thickness, and c) equistrong shape of the profile of aircraft wings accounting for gravity and lift loads. The shape of equistrong rod at buckling under a compressive force is found in the Euler's problem. Equistrong structures possess minimum weight for given safety factor or maximum safety factor for given weight.