“Oh, the cab,” groaned Loewe—­“that clue failed.  The police must have mistaken the number.  They telephoned immediately to all the police stations, and a watch was set, with the result that number 72,863 was stopped as it was going home for the night.  But it then turned out that the cab had not been off the rank since eleven o’clock, and the driver had been in the shelter all the time with several other men.  But there is a clue; I have it here.”

Mr. Loewe’s face brightened for once as he reached out for the bandbox.

“The houses in Howard Street,” he explained, as he untied the fastening, “have small balconies to the first-floor windows at the back.  Now, the thief entered by one of these windows, having climbed up a rain-water pipe to the balcony.  It was a gusty night, as you will remember, and this morning, as I was leaving the house, the butler next door called to me and gave me this; he had found it lying in the balcony of his house.”

He opened the bandbox with a flourish, and brought forth a rather shabby billycock hat.

“I understand,” said he, “that by examining a hat it is possible to deduce from it, not only the bodily characteristics of the wearer, but also his mental and moral qualities, his state of health, his pecuniary position, his past history, and even his domestic relations and the peculiarities of his place of abode.  Am I right in this supposition?”

The ghost of a smile flitted across Thorndyke’s face as he laid the hat upon the remains of the newspaper.  “We must not expect too much,” he observed.  “Hats, as you know, have a way of changing owners.  Your own hat, for instance” (a very spruce, hard felt), “is a new one, I think.”

“Got it last week,” said Mr. Loewe.

“Exactly.  It is an expensive hat, by Lincoln and Bennett, and I see you have judiciously written your name in indelible marking-ink on the lining.  Now, a new hat suggests a discarded predecessor.  What do you do with your old hats?”

“My man has them, but they don’t fit him.  I suppose he sells them or gives them away.”

“Very well.  Now, a good hat like yours has a long life, and remains serviceable long after it has become shabby; and the probability is that many of your hats pass from owner to owner; from you to the shabby-genteel, and from them to the shabby ungenteel.  And it is a fair assumption that there are, at this moment, an appreciable number of tramps and casuals wearing hats by Lincoln and Bennett, marked in indelible ink with the name S. Loewe; and anyone who should examine those hats, as you suggest, might draw some very misleading deductions as to the personal habits of S. Loewe.”

Mr. Marchmont chuckled audibly, and then, remembering the gravity of the occasion, suddenly became portentously solemn.

“So you think that the hat is of no use, after all?” said Mr. Loewe, in a tone of deep disappointment.

“I won’t say that,” replied Thorndyke.  “We may learn something from it.  Leave it with me, at any rate; but you must let the police know that I have it.  They will want to see it, of course.”