To make a showy display, to attract attention. The term originated in America and comes from mowing, a “swath” being the amount cut by one big sweep of the scythe. It was transferred to human showoffs by the mid-nineteenth century. “How he was a strutting up the sidewalk—didn’t he cut a swath!” wrote Ann S. Stephens in High Life in New York (1843). It is heard less often today, but has not quite died out.