Our visitors arrived punctually, the third gentleman being, as we had supposed, Mr. Solomon Loewe.  Inspector Badger I had not seen before, and he now impressed me as showing a tendency to invert the significance of his own name by endeavouring to “draw” Thorndyke; in which, however, he was not brilliantly successful.

“I hope you are not going to disappoint Mr. Loewe, sir,” he commenced facetiously.  “You have had a good look at that hat—­we saw your marks on it—­and he expects that you will be able to point us out the man, name and address all complete.”  He grinned patronizingly at our unfortunate client, who was looking even more haggard and worn than he had been on the previous morning.

“Have you—­have you made any—­discovery?” Mr Loewe asked with pathetic eagerness.

“We examined the hat very carefully, and I think we have established a few facts of some interest.”

“Did your examination of the hat furnish any information as to the nature of the stolen property, sir?” inquired the humorous inspector.

Thorndyke turned to the officer with a face as expressionless as a wooden mask.

“We thought it possible,” said he, “that it might consist of works of Japanese art, such as netsukes, paintings, and such like.”

Mr. Loewe uttered an exclamation of delighted astonishment, and the facetiousness faded rather suddenly from the inspector’s countenance.

“I don’t know how you can have found out,” said he.  “We have only known it half an hour ourselves, and the wire came direct from Florence to Scotland Yard.”

“Perhaps you can describe the thief to us,” said Mr. Loewe, in the same eager tone.

“I dare say the inspector can do that,” replied Thorndyke.

“Yes, I think so,” replied the officer.  “He is a short strong man, with a dark complexion and hair turning grey.  He has a very round head, and he is probably a workman engaged at some whiting or cement works.  That is all we know; if you can tell us any more, sir, we shall be very glad to hear it.”

“I can only offer a few suggestions,” said Thorndyke, “but perhaps you may find them useful.  For instance, at 13, Birket Street, Limehouse, there is living a Japanese gentleman named Futashima, who works at Badcomb and Martin’s mother-of-pearl factory.  I think that if you were to call on him, and let him try on the hat that you have, it would probably fit him.”

The inspector scribbled ravenously in his notebook, and Mr. Marchmont—­an old admirer of Thorndyke’s—­leaned back in his chair, chuckling softly and rubbing his hands.

“Then,” continued my colleague, “there is in Rankin Street, Limehouse, a cab-yard, where another Japanese gentleman named Itu is employed.  You might find out where Itu was the night before last; and if you should chance to see a hansom cab there—­number 22,481—­have a good look at it.  In the frame of the number-plate you will find six small holes.  Those holes may have held brads, and the brads may have held a false number card.  At any rate, you might ascertain where that cab was at 11.30 the night before last.  That is all I have to suggest.”