time. You’ve said yourself that we are better engineers than half the young snots that come up from Earth.”
“Granted. But it’s not enough. Oh, you can learn most things on your own but I want you to have a formal, disciplined, really sound grounding in mathematics.”
“Huh? Why, we cut our teeth on differential equations!”
Pollux added, “We know Hudson’s Manual by heart. We can do a triple integration in our heads faster than Hazel can. If there’s one thing we do know, it’s mathematics.”
Roger Stone shook his head sadly. “You can count on your fingers but you can’t reason. You probably think that the interval from zero to one is the same as the interval from ninety-nine to one hundred.”
“Isn’t it?”
“Is it? If so, can you prove it?” Their father reached up to the spindles on the wall, took down a book spool, and inserted it into his study projector. He spun the selector, stopped with a page displayed on the wall screen. It was a condensed chart of the fields of mathematics invented thus far by the human mind. “Let’s see you find your way around that page.”
The twins blinked at it. In the upper left-hand corner of the chart they spotted the names of subjects they had studied; the rest of the array was unknown territory; in most cases they did not even recognize the names of the subjects. In the ordinary engineering forms of the calculus they actually were adept; they had not been boasting. They knew enough of vector analysis to find their way around unassisted in electrical engineering and electronics; they knew classical geometry and trigonometry well enough for the astrogating of a space ship, and they had had enough of non-Euclidean geometry, tensor calculus, statistical mechanics, and quantum theory to get along with an atomic power plant.
But it had never occurred to them that they had not yet really penetrated the enormous and magnificent field of mathematics.
“Dad,” asked Pollux in a small voice, “what’s a ‘hyper-ideal’?”
“Time you found out.”
Castor looked quickly at his father. “How many of these things have you studied, Dad?”
“Not enough. Not nearly enough. But my sons should know more than I do.”
It was agreed that the twins would study mathematics intensively the entire time the family was in space, and not simply under the casual supervision of their father and grandmother but formally and systematically through I.C.S. correspondence courses ordered up from Earth. They would take with them spools enough to keep them busy for at least a year and mail their completed lessons from any port they might touch. Mr. Stone was satisfied, being sure in his heart that any person skilled with mathematical tools could learn anything else he needed to know, with or without a master.
“Now, boys, about this matter of cargo—”
The twins waited; he went on: “I’ll lift the stuff for you—”
“Gee, Dad, that’s swell!”
“—at cost.”
The twins suddenly looked cautious. “How do you figure ‘cost’?” Castor asked.
“You figure it and I’ll check your figures. Don’t try to flummox me or I’ll stick on a penalty. If you’re going to be businessmen, don’t confuse the vocation with larceny.”
“Right, sir. Uh…we still can’t order until we know where we are going.”
“True. Well, how would Mars suit you, as the first stop?”
“Mars?” Both boys got far-away looks in their eyes; their lips moved soundlessly.
“Well? Quit figuring your profits; you aren’t there yet.”
“Mars? Mars is fine, Dad!”
“Very well. One more thing: fail to keep up your studies and I won’t let you sell a tin whistle.”
“Oh, we’ll study!” The twins got out while they were ahead. Roger Stone looked at the closed door with a fond smile on his face, an expression he rarely let them see. Good boys! Thank heaven he hadn’t been saddled with a couple of obedient, well-behaved little nincompoops!
When the twins reached
“Granted. But it’s not enough. Oh, you can learn most things on your own but I want you to have a formal, disciplined, really sound grounding in mathematics.”
“Huh? Why, we cut our teeth on differential equations!”
Pollux added, “We know Hudson’s Manual by heart. We can do a triple integration in our heads faster than Hazel can. If there’s one thing we do know, it’s mathematics.”
Roger Stone shook his head sadly. “You can count on your fingers but you can’t reason. You probably think that the interval from zero to one is the same as the interval from ninety-nine to one hundred.”
“Isn’t it?”
“Is it? If so, can you prove it?” Their father reached up to the spindles on the wall, took down a book spool, and inserted it into his study projector. He spun the selector, stopped with a page displayed on the wall screen. It was a condensed chart of the fields of mathematics invented thus far by the human mind. “Let’s see you find your way around that page.”
The twins blinked at it. In the upper left-hand corner of the chart they spotted the names of subjects they had studied; the rest of the array was unknown territory; in most cases they did not even recognize the names of the subjects. In the ordinary engineering forms of the calculus they actually were adept; they had not been boasting. They knew enough of vector analysis to find their way around unassisted in electrical engineering and electronics; they knew classical geometry and trigonometry well enough for the astrogating of a space ship, and they had had enough of non-Euclidean geometry, tensor calculus, statistical mechanics, and quantum theory to get along with an atomic power plant.
But it had never occurred to them that they had not yet really penetrated the enormous and magnificent field of mathematics.
“Dad,” asked Pollux in a small voice, “what’s a ‘hyper-ideal’?”
“Time you found out.”
Castor looked quickly at his father. “How many of these things have you studied, Dad?”
“Not enough. Not nearly enough. But my sons should know more than I do.”
It was agreed that the twins would study mathematics intensively the entire time the family was in space, and not simply under the casual supervision of their father and grandmother but formally and systematically through I.C.S. correspondence courses ordered up from Earth. They would take with them spools enough to keep them busy for at least a year and mail their completed lessons from any port they might touch. Mr. Stone was satisfied, being sure in his heart that any person skilled with mathematical tools could learn anything else he needed to know, with or without a master.
“Now, boys, about this matter of cargo—”
The twins waited; he went on: “I’ll lift the stuff for you—”
“Gee, Dad, that’s swell!”
“—at cost.”
The twins suddenly looked cautious. “How do you figure ‘cost’?” Castor asked.
“You figure it and I’ll check your figures. Don’t try to flummox me or I’ll stick on a penalty. If you’re going to be businessmen, don’t confuse the vocation with larceny.”
“Right, sir. Uh…we still can’t order until we know where we are going.”
“True. Well, how would Mars suit you, as the first stop?”
“Mars?” Both boys got far-away looks in their eyes; their lips moved soundlessly.
“Well? Quit figuring your profits; you aren’t there yet.”
“Mars? Mars is fine, Dad!”
“Very well. One more thing: fail to keep up your studies and I won’t let you sell a tin whistle.”
“Oh, we’ll study!” The twins got out while they were ahead. Roger Stone looked at the closed door with a fond smile on his face, an expression he rarely let them see. Good boys! Thank heaven he hadn’t been saddled with a couple of obedient, well-behaved little nincompoops!
When the twins reached
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