"Are they the same?"

"No," the boy said, "I got the other wrong somewhere."

"S'posin' you had him right," the puzzle-maker said, "it took you hour.

Ordinary figures you did him in thirty-two seconds."

"I see," said Eric, "it's another case of wonderful but not wonderful enough, isn't it?"

"Exactly. Here," the other continued, reaching down a ma.n.u.script portfolio, "is every kind of numbers ever made. You find that the Hindu--or wrongly called Arabic--numerals are the only ones wonderful enough for modern uses."

Thoroughly interested, the boy sat down with this big ma.n.u.script book.

Weird schemes of numeration rioted over the pages, from the Zuni finger and the Chinese knuckle systems to the latest groups of symbols, used in modern higher mathematics, of which the boy had not even heard. It was noon before he realized with a start that the morning was gone.

"Oh, Dan!" he said reproachfully, "we haven't done anything to-day."

"Never mind," said the old man, "we get a start after a while."

That afternoon, when the boy settled down to do some work on his own account, he felt a much greater friendliness to the mere look of figures. They seemed like old friends. Before, a figure had only been something in a "sum," but now he felt that each one had a long history of its own. Little did he realize that the biggest step of his mathematics was accomplished. Never again would he be able to look at a page of figures with revulsion. They had come to life for him.

The next morning, Eric found the old puzzle-maker busy with a chess-board.

"Aren't we going to do any work to-day, either?" he asked, disappointedly.

"Soon as I finish," the old man answered. "Get pencil and paper. As I move knight from square to square, you draw."

Shrugging his shoulders slightly, but not so noticeably that the puzzle-maker could see, Eric obeyed. It seemed very silly to him. But as the knight went from square to square in the peculiar move which belongs to that piece in chess, the boy was amazed to find a wonderful and fascinating geometrical design growing under his hand.

"Another way, too," said the old man thoughtfully, the instant the figure was finished, not giving the boy a chance to make any comment.

And, without further preface he started again. This time an even stranger but equally perfect design was formed.

"But that's great!" said Eric, "how do you know it's going to come out like that! I wonder if I could do it?"

"Try him," the puzzle-maker answered, getting up from the board. For half an hour Eric moved the knight about, but never got as perfect an example as the old man.

"Are there only those two ways?" said the boy at last.

"Over thirty-one million ways of moving the knight so that he occupies each square once," was the reply. "Every one makes a different design."

"I'll try some this evening," said the boy. "But it's funny, too. Why does it always make a regular design?"

"You want to know? Very well." And the puzzle-maker quietly explained some of the most famous mathematical problems of all time, working them out with the chessmen and the board.

"You know what they call him, magic?" queried the old man.

"Magic! No!" exclaimed Eric p.r.i.c.king up his ears at the word. "Tell me about it, Dan."

"Numbers all friends, live together, work together," the puzzle-maker answered. "I show you." And, taking pencil and paper, he dotted down in forms of squares and cubes rows and rows of figures. "Add him up," he said, "up and down, cross-wise, any way. He all make same number."

"They do, sure enough," said Eric, after testing half a dozen magic squares, "but how do you do it? Do you have to remember all those figures and just where they go?"

"Don't remember any of him," the other answered. "He has to go so."

"But I can't make them come that way," exclaimed the boy, after trying for a few minutes. "What's the trick?"

"All friends," repeated the old man, and in his curiously jolting speech he told Eric the startling links that are found in the powers of numbers. As soon as he had the principle clearly in mind, the boy found that there was no great difficulty in making up the most astonishing magic squares.

As the winter drew on, and calls for help on the stormy waters increased, the opportunities for sessions with the shrewd old mathematician grew fewer, but Eric stuck fast to his promise to spend all the time he could afford with his instructor. He was keenly disappointed that the puzzle-maker showed such an absolute disregard of the actual things the boy wanted to prepare for in his examinations. But Eric had been rigidly trained by his father in the sportsmanlike att.i.tude of never complaining about any arrangement he had made himself, and he paid for his coaching out of his small earnings without a word. In order to make up for what he inwardly felt was lost time, he worked by himself at his books in such few minutes as he was able to s.n.a.t.c.h from his life-saving duties. And, although he was tired almost to exhaustion, many and many a day, he found that even in that work he was getting along quite well.

Eric could never get his eccentric teacher to look at the books required in his preparatory work. What was more, he had a feeling that he couldn't really be getting much good from his hours spent with Dan, because he enjoyed them so much. Early schooldays had made him a.s.sociate progress with discomfort.

For example, one day Dan showed him tricks with cards--and then explained the mathematics of it, making the most puzzling mysteries seem only unusual applications of very simple principles. Another day, the puzzle-maker told him of curious problems of chance, by dice, by lotteries, and so forth, and almost before Eric realized what the old man was driving at, the essential ideas of insurance and actuary work were firmly fixed in his mind.

It was not until a couple of weeks before the expected close of navigation that the puzzle-maker said,

"Let me see book!"

Astonished at the now unexpected request, Eric handed him the much bethumbed volume over which he had struggled so hard. The old man skimmed through its pages, nodding his head from time to time and mumbling in a satisfied way. Then, like a man driving in a nail, he pounded Eric with question after question. He seemed to be asking them from the book, but Eric knew that none of the problems had their origin in it, for they dealt with the work he had been doing in the little cottage by the sea. Yet to almost every one the boy returned a correct answer, or at least, one which was correct in its approach. For two long hours the puzzle-maker questioned him, without ever a minute's let up.

At the end of it, Eric was as limp as a rag. At last the old man laid down the book.

"When your examination is?" he asked.

"Next June," the boy replied.

"You can pa.s.s him now."

Eric stared at the old man with wild surprise in his gaze and with a down-dropped jaw.

"But I haven't even started on the second half of the book," he said.

"And I've got to do it all!"

"You pa.s.s him now," was the quiet answer. "The second part--you have done him, too. Learn rules, if you like. No matter. You know him. See!"

He showed the very last set of examples in the book and Eric recognized problems of the kind he had been doing, all unwitting to himself.

"Mathematics not to learn," he said, "he is to think. You now can think.

To know a rule, to do sum--bah! he is nothing! To know why a rule and because a sum--he is much. You do him."

In the few remaining visits that Eric paid the puzzle-maker, he found the old man's words to be quite true. Having learned the inside of mathematics, its actual workings seemed reasonable. The clew gave Eric the sense of exploring a new world of thought instead of being lost in a tangled wilderness.

Meantime, he had become absolutely expert in every detail of the station. His particular delight was the capsize drill. The keeper had got the crew trained down to complete the whole performance within fifty seconds from the time he gave the order. The boat had to be capsized, every man underneath the boat. Then they had to clamber on the upturned boat, right it again, and be seated on the thwarts with oars ready to pull before the fiftieth second was past. It was quick work, and although only a drill, was as exciting as the lad could wish. Two or three times, one of the men, who wasn't quite as quick as the rest, got "waterlogged" and the crew had to help him up. When that occurred, there was an awful howl.

Once, only once, Eric delayed the drill about two seconds and it was weeks before he overcame his sense of shame at the occurrence. But, before the winter finally closed down, Eric was as able a coast-guardsman as any on the Great Lakes. It was well that he was, for a day was coming which would test his fort.i.tude to the full.

Navigation had been lessening rapidly, and the boy was beginning to think about Thanksgiving Day. They were just sitting down to supper, when one of the men came in with haste.

"Heard anything of a wreck round Au Sable way?" he asked breathlessly.

"No," said the keeper, "what did you hear?"