“Where was the fellow who tried to head us off?” asked Hal.

“When?”

“At any time.”

“We located him at various points along the river.  No doubt he was on a boat up to the very last when the two were very near together.”

“Where was the island operator when he sent his last message?  Did you get the one in which he confessed the affair was a hoax?”

“Yes.  But he did not send that message.  It was sent by the other fellow.”

“How do you know?”

“That was plain.  Did you not notice his peculiar manner of sending?  All three of us noticed that.”

“Did you pick up any more from them since then?”

“Not a dot.”

Hal then asked the obliging amateur to indicate as nearly as possible the location of the island from which the messages came.  The latter did as requested, and Hal marked the point on the chart of the St. Lawrence River carried by the Catwhisker.  This closed the wireless interview.  Hal promised to report back to the Rockport amateur any further developments of interest and tapped “goodnight” with his key.

“Well, your two main points have been proved, Mr. Perry,” Bud announced as all three boys removed the receivers from their ears.

“What are they?” asked the man thus addressed.

“Mathematics and geography.”

Mr. Perry smiled.

“Yes,” he said “I could hardly have hoped for so remarkable a demonstration of my theory.  You boys have solved the geography of this problem with the aid of some very clever mathematics.  But what branch of mathematics is it?”

“We didn’t do it ourselves,” Hal reminded.  “It was those three amateurs with their loop aerials.”

“Wasn’t it more mechanical than mathematical?” Cub inquired meditatively.  “Those radio compasses make me think of a surveyor’s instrument.”

“Oh, pshaw, my boy, don’t spoil everything,” pleaded the last speaker’s father.  “I’m afraid you’ve missed the big point.  Mathematics is the biggest factor in all mechanics.  Bud, I thought from the way you spoke that you grasped the situation completely.  Can’t you help Bob and Hal out?  By means of what branch of mathematics was that island of our Canadian Crusoe located?”

“Geometry,” replied Bud confidently.

Cub snapped his finger with an impatient jerk of his long right arm.

“Of course!” he exclaimed in disgust.  “Every branch of mathematics I ever heard of, except geometry, went buzzing through my head.  I was trying to recall something in algebra that would fit this case.”

“Oh, Cub,” laughed Hal; “algebra is all x’s and y’s and z’s over z’s and y’s and x’s,”

“I admit I’m a chump,” Cub grinned with a shrug of self-commiseration; “but say, let’s draw those geometrical lines on our chart and see if we get the same result those radio compass fellows got.”