From Chesterton's Blue Cross, describing a detective's tactic when said detective finds himself without a clue as to the whereabouts of the criminal he's after:
In such cases he reckoned on the unforeseen. In such cases, when he could not follow the train of the reasonable, he coldly and carefully followed the train of the unreasonable. Instead of going to the right places--banks, police stations, rendezvous-- he systematically went to the wrong places; knocked at every empty house, turned down every cul de sac, went up every lane blocked with rubbish, went round every crescent that led him uselessly out of the way. He defended this crazy course quite logically. He said that if one had a clue this was the worst way; but if one had no clue at all it was the best, because there was just the chance that any oddity that caught the eye of the pursuer might be the same that had caught the eye of the pursued. Somewhere a man must begin, and it had better be just where another man might stop. Something about that flight of steps up to the shop, something about the quietude and quaintness of the restaurant, roused all the detective's rare romantic fancy and made him resolve to strike at random. He went up the steps, and sitting down at a table by the window, asked for a cup of black coffee.
The person described is a French police chief who is on a hunt after a notorious criminal. I get the gist of the passage on the whole, but the bolded phrase puzzles me.