hairs on their heads.' If this is a fact, I confess I cannot see it."
"How do the bald-headed affect the question?" asked Uncle John.
"If there are such persons in existence," replied Mrs. Allgood, " who haven't a solitary hair on their heads discoverable under a magnifying-glass, we will leave them out of the question. Still, I don't see how you are to prove that at least two persons have exactly the same number to a hair."
"I think I can make it clear," said Mr. Filkins, who had dropped in for the evening.
"Assume the population of the world to be only one million. Any number will do as well as another. Then your statement was to the effect that no person has more than nine hundred and ninety-nine thousand nine hundred and ninety-nine hairs on his head. Is that so? "
"Let me think," said Mrs. Allgood. " Yes — yes—that is correct."
" Very well, then. As there are only nine hundred and ninety-nine thousand nine hundred and ninety-nine different ways of bearing hair, it is clear that the millionth person must repeat one of those ways. Do you see? "
" Yes; I see that—at least I think I see it."
" Therefore two persons at least must have the same number of hairs on their heads; and as the number of people on the earth so greatly exceeds the number of hairs on any one person's head, there must, of course, be an immense number of these repetitions."
" But, Mr. Filkms," said little Willie Allgood,
" why could not the millionth man have, say, ten thousand hairs and a half?"
" That is mere hair-splitting, Willie, and does not come into the question."
"Here is a curious paradox," said George.
"If a thousand soldiers are drawn up in battle array on a plane"—they understood him to mean "plain"—"only one man will stand upright."
Nobody could see why. But George explained that, according to Euclid, a plane can touch a sphere only at one point, and that person only who stands at that point, with respect to the centre of the earth, will stand upright.
"In the same way," he remarked, "if a billiard-table were quite level—that is, a perfect plane—the balls ought to roll to the centre."
Though he tried to explain this by placing a visiting-card on an orange and expounding the law of gravitation, Mrs. Allgood declined to accept the statement. She could not see that the top of a true billiard-table must, theoretically, be spherical, just like a portion of the orange-peel that George cut out. Of course, the table is so small in proportion to the surface of the earth that the curvature is not appreciable, but it is nevertheless true in theory. A surface that we call level is not the same as our idea of a true geometrical plane.
"Uncle John," broke in Willie Allgood, "there is a certain island situated between England and France, and yet that island is farther from France than England is. What is the island?"
"That seems absurd, my boy; because if I place this tumbler, to represent the island, between these two plates, it seems impossible that the tumbler can be farther from either of the plates than they are from each other."
"But isn't Guernsey between England and France?" asked Willie.
"Yes, certainly."
"Well, then, I think you will find, uncle, that Guernsey is about twenty-six miles from France, and England is only twenty-one miles from France, between Calais and Dover."
"My mathematical master," said George, "has been trying to induce me to accept the axiom that 'if equals be multiplied by equals the products are equal.'"
" It is self-evident," pointed out Mr. Filkins. "For example, if 3 feet equal 1 yard, then twice 3 feet will equal 2 yards. Do you see?"
"But, Mr. Filkins," asked George, "is this tumbler half full of water equal to a similar glass half empty?"
"Certainly, George."
"Then it follows from the axiom that a glass full must equal a glass empty. Is that correct?"
"No, clearly not. I never thought of it in that light."
"Perhaps," suggested Mr. Allgood, "the rule does not apply to liquids."
"Just what I was thinking, Allgood. It would seem that we must make an exception in the case of liquids."
"But it would be awkward," said George, with a smile, "if we also had to except the case of solids. For instance, let us take the solid earth. One mile square equals one square mile. Therefore two miles square must equal two square miles. Is this so?"
"Well, let me see! No, of course not," Mr. Filkins replied, "because two miles square is four square miles."
"Then," said George, "if the axiom is not true in these cases, when is it true?"
Mr. Filkins promised to look into the matter, and perhaps the reader will also like to give it consideration at leisure.
"Look here, George," said his cousin Reginald Woolley: "by what fractional part does four-fourths exceed three-fourths?"
"By one-fourth!" shouted everybody at once.
"Try another one," George suggested.
"With pleasure, when you have answered that one correctly," was Reginald's reply.
"Do you mean to say that it isn't one-fourth?"
"Certainly I do."
Several members of the company failed to see that the correct answer is "one-third," although Reginald tried to explain that three of anything, if increased by one-third, becomes four.
"Uncle John, how do you pronounce 't-o-o'?" asked Willie.
"'Too,' my boy."
"And how do you pronounce 't-w-o'?"
"That is also 'too.'"
