Vršac Circles are a set of five, practically, concentric circles located in a plain of western Serbia. There have not been any systematic archaeological research in the zone but some exploration work indicate a possible date for the earthwork to be around five thousand years ago. In this paper, we present a sequential analysis of the sizes (diameters) of the circles that may obey, with the help of a gnomonic approach (the "gnomonic factor"), an algorithm involving the number of the circle and the height of the gnomon. The first assumption is that the smallest radius would be obtained from the shadow of a gnomon on winter solstice at noon; the others by the addition to the first a multiple of 1.5 times the gnomon’s height.At the final stage of the analysis we discuss a possible calendrical meaning of the circles. Our results, we consider, call the archaeologist to look at this place for confirmation or rejection of these hypothesis. They may support subsequent development of ideas for the history of mathematics.