"Is not life itself a paradox?"
C.L. DODGSON, _Pillow Problems_.
"It is a wonderful age!" said Mr. Allgood, and everybody at the table
turned towards him and assumed an attitude of expectancy.
This was an ordinary Christmas dinner of the Allgood family, with a
sprinkling of local friends. Nobody would have supposed that the above
remark would lead, as it did, to a succession of curious puzzles and
paradoxes, to which every member of the party contributed something of
interest. The little symposium was quite unpremeditated, so we must not
be too critical respecting a few of the posers that were forthcoming.
The varied character of the contributions is just what we would expect
on such an occasion, for it was a gathering not of expert mathematicians
and logicians, but of quite ordinary folk.
"It is a wonderful age!" repeated Mr. Allgood. "A man has just designed
a square house in such a cunning manner that all the windows on the four
sides have a south aspect."
"That would appeal to me," said Mrs. Allgood, "for I cannot endure a
room with a north aspect."
"I cannot conceive how it is done," Uncle John confessed. "I suppose he
puts bay windows on the east and west sides; but how on earth can be
contrive to look south from the north side? Does he use mirrors, or
something of that kind?"
"No," replied Mr. Allgood, "nothing of the sort. All the windows are
flush with the walls, and yet you get a southerly prospect from every
one of them. You see, there is no real difficulty in designing the house
if you select the proper spot for its erection. Now, this house is
designed for a gentleman who proposes to build it exactly at the North
Pole. If you think a moment you will realize that when you stand at the
North Pole it is impossible, no matter which way you may turn, to look
elsewhere than due south! There are no such directions as north, east,
or west when you are exactly at the North Pole. Everything is due
south!"
"I am afraid, mother," said her son George, after the laughter had
subsided, "that, however much you might like the aspect, the situation
would be a little too bracing for you."
"Ah, well!" she replied. "Your Uncle John fell also into the trap. I am
no good at catches and puzzles. I suppose I haven't the right sort of
brain. Perhaps some one will explain this to me. Only last week I
remarked to my hairdresser that it had been said that there are more
persons in the world than any one of them has hairs on his head. He
replied, 'Then it follows, madam, that two persons, at least, must have
exactly the same number of 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. Filkins," 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.'"
"Then how do you pronounce the second day of the week?"
"Well, that I should pronounce 'Tuesday,' not 'Toosday.'"
"Would you really? I should pronounce it 'Monday.'"
"If you go on like this, Willie," said Uncle John, with mock severity,
"you will soon be without a friend in the world."
"Can any of you write down quickly in figures 'twelve thousand twelve
hundred and twelve pounds'?" asked Mr. Allgood.
His eldest daughter, Miss Mildred, was the only person who happened to
have a pencil at hand.
"It can't be done," she declared, after making an attempt on the white
table-cloth; but Mr. Allgood showed her that it should be written,
"£13,212."
"Now it is my turn," said Mildred. "I have been waiting to ask you all a
question. In the Massacre of the Innocents under Herod, a number of poor
little children were buried in the sand with only their feet sticking
out. How might you distinguish the boys from the girls?"
"I suppose," said Mrs. Allgood, "it is a conundrum--something to do with
their poor little 'souls.'"
But after everybody had given it up, Mildred reminded the company that
only boys were put to death.
"Once upon a time," began George, "Achilles had a race with a
tortoise--"
"Stop, George!" interposed Mr. Allgood. "We won't have that one. I knew
two men in my youth who were once the best of friends, but they
quarrelled over that infernal thing of Zeno's, and they never spoke to
one another again for the rest of their lives. I draw the line at that,
and the other stupid thing by Zeno about the flying arrow. I don't
believe anybody understands them, because I could never do so myself."
"Oh, very well, then, father. Here is another. The Post-Office people
were about to erect a line of telegraph-posts over a high hill from
Turmitville to Wurzleton; but as it was found that a railway company was
making a deep level cutting in the same direction, they arranged to put
up the posts beside the line. Now, the posts were to be a hundred yards
apart, the length of the road over the hill being five miles, and the
length of the level cutting only four and a half miles. How many posts
did they save by erecting them on the level?"
"That is a very simple matter of calculation," said Mr. Filkins. "Find
how many times one hundred yards will go in five miles, and how many
times in four and a half miles. Then deduct one from the other, and you
have the number of posts saved by the shorter route."
"Quite right," confirmed Mr. Allgood. "Nothing could be easier."
"That is just what the Post-Office people said," replied George, "but it
is quite wrong. If you look at this sketch that I have just made, you
will see that there is no difference whatever. If the posts are a
hundred yards apart, just the same number will be required on the level
as over the surface of the hill."
[Illustration]
"Surely you must be wrong, George," said Mrs. Allgood, "for if the posts
are a hundred yards apart and it is half a mile farther over the hill,
you have to put up posts on that extra half-mile."
"Look at the diagram, mother. You will see that the distance from post
to post is not the distance from base to base measured along the ground.
I am just the same distance from you if I stand on this spot on the
carpet or stand immediately above it on the chair."
But Mrs. Allgood was not convinced.
Mr. Smoothly, the curate, at the end of the table, said at this point
that he had a little question to ask.
"Suppose the earth were a perfect sphere with a smooth surface, and a
girdle of steel were placed round the Equator so that it touched at
every point."
"'I'll put a girdle round about the earth in forty minutes,'" muttered
George, quoting the words of Puck in _A Midsummer Night's Dream_.
"Now, if six yards were added to the length of the girdle, what would
then be the distance between the girdle and the earth, supposing that
distance to be equal all round?"
"In such a great length," said Mr. Allgood, "I do not suppose the
distance would be worth mentioning."
"What do you say, George?" asked Mr. Smoothly.
"Well, without calculating I should imagine it would be a very minute
fraction of an inch."
Reginald and Mr. Filkins were of the same opinion.
"I think it will surprise you all," said the curate, "to learn that
those extra six yards would make the distance from the earth all round
the girdle very nearly a yard!"
"Very nearly a yard!" everybody exclaimed, with astonishment; but Mr.
Smoothly was quite correct. The increase is independent of the original
length of the girdle, which may be round the earth or round an orange;
in any case the additional six yards will give a distance of nearly a
yard all round. This is apt to surprise the non-mathematical mind.
"Did you hear the story of the extraordinary precocity of Mrs. Perkins's
baby that died last week?" asked Mrs. Allgood. "It was only three months
old, and lying at the point of death, when the grief-stricken mother
asked the doctor if nothing could save it. 'Absolutely nothing!' said
the doctor. Then the infant looked up pitifully into its mother's face
and said--absolutely nothing!"
"Impossible!" insisted Mildred. "And only three months old!"
"There have been extraordinary cases of infantile precocity," said Mr.
Filkins, "the truth of which has often been carefully attested. But are
you sure this really happened, Mrs. Allgood?"
"Positive," replied the lady. "But do you really think it astonishing
that a child of three months should say absolutely nothing? What would
you expect it to say?"
"Speaking of death," said Mr. Smoothly, solemnly, "I knew two men,
father and son, who died in the same battle during the South African
War. They were both named Andrew Johnson and buried side by side, but
there was some difficulty in distinguishing them on the headstones. What
would you have done?"
"Quite simple," said Mr. Allgood. "They should have described one as
'Andrew Johnson, Senior,' and the other as 'Andrew Johnson, Junior.'"
"But I forgot to tell you that the father died first."
"What difference can that make?"
"Well, you see, they wanted to be absolutely exact, and that was the
difficulty."
"But I don't see any difficulty," said Mr. Allgood, nor could anybody
else.
"Well," explained Mr. Smoothly, "it is like this. If the father died
first, the son was then no longer 'Junior.' Is that so?"
"To be strictly exact, yes."
"That is just what they wanted--to be strictly exact. Now, if he was no
longer 'Junior,' then he did not die 'Junior." Consequently it must be
incorrect so to describe him on the headstone. Do you see the point?"
"Here is a rather curious thing," said Mr. Filkins, "that I have just
remembered. A man wrote to me the other day that he had recently
discovered two old coins while digging in his garden. One was dated '51
B.C.,' and the other one marked 'George I.' How do I know that he was
not writing the truth?"
"Perhaps you know the man to be addicted to lying," said Reginald.
"But that would be no proof that he was not telling the truth in this
instance."
"Perhaps," suggested Mildred, "you know that there were no coins made at
those dates.
"On the contrary, they were made at both periods."
"Were they silver or copper coins?" asked Willie.
"My friend did not state, and I really cannot see, Willie, that it makes
any difference."
"I see it!" shouted Reginald. "The letters 'B.C.' would never be used on
a coin made before the birth of Christ. They never anticipated the event
in that way. The letters were only adopted later to denote dates
previous to those which we call 'A.D.' That is very good; but I cannot
see why the other statement could not be correct."
"Reginald is quite right," said Mr. Filkins, "about the first coin. The
second one could not exist, because the first George would never be
described in his lifetime as 'George I.'"
"Why not?" asked Mrs. Allgood. "He _was_ George I."
"Yes; but they would not know it until there was a George II."
"Then there was no George II. until George III. came to the throne?"
"That does not follow. The second George becomes 'George II.' on account
of there having been a 'George I.'"
"Then the first George was 'George I.' on account of there having been
no king of that name before him."
"Don't you see, mother," said George Allgood, "we did not call Queen
Victoria 'Victoria I.;' but if there is ever a 'Victoria II.,' then she
will be known that way."
"But there _have_ been several Georges, and therefore he was 'George I.'
There _haven't_ been several Victorias, so the two cases are not
similar."
They gave up the attempt to convince Mrs. Allgood, but the reader will,
of course, see the point clearly.
"Here is a question," said Mildred Allgood, "that I should like some of
you to settle for me. I am accustomed to buy from our greengrocer
bundles of asparagus, each 12 inches in circumference. I always put a
tape measure round them to make sure I am getting the full quantity. The
other day the man had no large bundles in stock, but handed me instead
two small ones, each 6 inches in circumference. 'That is the same
thing,' I said, 'and, of course, the price will be the same;' but he
insisted that the two bundles together contained more than the large
one, and charged me a few pence extra. Now, what I want to know is,
which of us was correct? Would the two small bundles contain the same
quantity as the large one? Or would they contain more?"
"That is the ancient puzzle," said Reginald, laughing, "of the sack of
corn that Sempronius borrowed from Caius, which your greengrocer,
perhaps, had been reading about somewhere. He caught you beautifully."
"Then they were equal?"
"On the contrary, you were both wrong, and you were badly cheated. You
only got half the quantity that would have been contained in a large
bundle, and therefore ought to have been charged half the original
price, instead of more."
Yes, it was a bad swindle, undoubtedly. A circle with a circumference
half that of another must have its area a quarter that of the other.
Therefore the two small bundles contained together only half as much
asparagus as a large one.
"Mr. Filkins, can you answer this?" asked Willie. "There is a man in the
next village who eats two eggs for breakfast every morning."
"Nothing very extraordinary in that," George broke in. "If you told us
that the two eggs ate the man it would be interesting."
"Don't interrupt the boy, George," said his mother.
"Well," Willie continued, "this man neither buys, borrows, barters,
begs, steals, nor finds the eggs. He doesn't keep hens, and the eggs are
not given to him. How does he get the eggs?"
"Does he take them in exchange for something else?" asked Mildred.
"That would be bartering them," Willie replied.
"Perhaps some friend sends them to him," suggested Mrs. Allgood.
"I said that they were not given to him."
"I know," said George, with confidence. "A strange hen comes into his
place and lays them."
"But that would be finding them, wouldn't it?"
"Does he hire them?" asked Reginald.
"If so, he could not return them after they were eaten, so that would be
stealing them."
"Perhaps it is a pun on the word 'lay,'" Mr. Filkins said. "Does he lay
them on the table?"
"He would have to get them first, wouldn't he? The question was, How
does he get them?"
"Give it up!" said everybody. Then little Willie crept round to the
protection of his mother, for George was apt to be rough on such
occasions.
"The man keeps ducks!" he cried, "and his servant collects the eggs
every morning."
"But you said he doesn't keep birds!" George protested.
"I didn't, did I, Mr. Filkins? I said he doesn't keep hens."
"But he finds them," said Reginald.
"No; I said his servant finds them."
"Well, then," Mildred interposed, "his servant gives them to him."
"You cannot give a man his own property, can you?"
All agreed that Willie's answer was quite satisfactory. Then Uncle John
produced a little fallacy that "brought the proceedings to a close," as
the newspapers say.