set one's cap at

To hear her rant, one would have supposed, who had not seen him, that her lank-haired, grimly partner, was the prettiest youth in the county of Dublin, and that all the comely lasses in Chapelizod and the country round were sighing and setting caps at him […]

How he has escaped marriage until now I cannot tell: the number of caps set in his direction would furnish a warehouse.