Leonard Euler on the Center of Similitude

In the past Euler has shown considerable interest in specific cases of similarity in which one figure is constructed from another in some way (consider, e.g., E-129, Opera Omnia I.27 130-180 “Investigatio curvarum quae evolutae sui similes producant”, in which the objects of study are planar curves which are similar to their own evolutes). In the current work, he considers any two similar figures and seeks a point that “relates similarly to either figure” (“ad utramque figuram similiter referatur”, Section 1). Such a point he calls the center of similitude of the two figures. For the case of figures in the plane, Euler manages to make this notion precise and he gives a simple geometrical means of locating centers of similitude. For figures in space, Euler equivocates somewhat on what it means for two figures to be “related similarly” to a point, and this equivocation results in some flawed attempts to locate a center of similarity for similar figures in three dimensions. Nevertheless Euler’s efforts, insofar as they are successful, anticipate some nineteeethcentury work in geometric transformation theory; and insofar as they fail, they provide a useful illustration, both for undergraduates and for their instructors, of the need for the notion of a geometric transformation in order to clarify and organize certain geometric intuitions.