measuring land. The invention of it is ascribed by some to the Chaldeans
and Babylonians, by others to the Egyptians, who were obliged to determine
the boundaries of their fields after the inundation of the Nile, by
geometrical measurements. According to Cassiodorus, the Egyptians either
derived the art from the Babylonians, or invented it after it was known
to them. Thales, a Phœnician, who died 548 years B.C., and Pythagoras
of Samos, who flourished about 520 B.C., introduced it from Egypt into
Greece. In elementary geometry, Euclid of Alexandria, as everybody knows,
is particularly distinguished. Archimedes measured the sphere, and after
him other philosophers prosecuted the science with the utmost assiduity.
In Italy, where the sciences first revived after the dark ages, several
mathematicians were distinguished in the 16th century. The French, and
after them the Germans, followed; while in England, Hook, Newton, and
others, carried the science to the highest pitch of usefulness, and
through its aid made the most prodigious discoveries. It is not, however,
our province to enter into a long disquisition on the subject, but simply
to set before the young reader some of the more curious properties of
the science, that he may be excited to study it for himself; and we will
promise him that should he devote his mind to its study, he will be amply
repaid for any amount of labor he may bestow upon it.
GEOMETRICAL DEFINITIONS.
In geometry a _point_ is said to have neither breadth, length, nor
thickness. A _line_ is the distance between two points; parallel lines
always keep at the same distance from each other. A _right_ line is what
is commonly called a straight line. A _curve_ is a line which continually
changes its direction. An _angle_ is the inclination or opening of two
lines meeting in a point. A _figure_ is a bounded space, and is either
a superficies or a solid. A _triangle_ is a figure with three sides and
three angles. A _square_ has four equal sides, and four right angles.
A _circle_ is a plane figure bounded by a curved line running into
itself. Its diameter is a straight line drawn from one extremity of
its circumference to the other, and its center is equally distant from
every part of the circumference. A _solid_ is any body which has length,
breadth, and thickness; and a sphere is a solid, terminated by a convex
surface, every part of which is at an equal distance from a point within,
called its center.
THE FIVE GEOMETRICAL SOLIDS.
[Illustration: _Fig. 1._]
[Illustration: _Fig. 2._]
[Illustration: _Fig. 3._]
[Illustration: _Fig. 4._]
The following figures will show how the five geometrical solids may be
cut out of a piece of cardboard. Where the lines are drawn the board is
to be partly cut through with a penknife, so as to render the angles
of the models as sharp and as straight as possible. The edges which
require joining are to be fastened together with a slip of thin paper and
gum dissolved in just sufficient water to bring it to the consistence
of treacle. Fig. 1 will form a tetrahedron, a figure with four sides,
each shaped like an equilateral triangle. Fig. 2 forms a cube or
hexahedron. Fig. 3 an octohedron, with eight triangular sides. Fig. 4,
a dodecahedron, with twelve sides shaped like pentagons, with five equal
sides. Fig. 5, an isocahedron, with twenty sides, formed of equilateral
triangles.
[Illustration: _Fig. 5._]
HOW TO MAKE FIVE SQUARES INTO A LARGE ONE WITHOUT ANY WASTE OF STUFF.
Suppose you have five squares of cloth, or anything else, as in Fig. 7;
find the center of one side of four of these squares, and cut them from
that point to the opposite corner, then place the perfect square in the
centre, and the other pieces round, as seen in Fig. 8.
[Illustration: _Fig. 7._]
[Illustration: _Fig. 8._]
DECEPTIVE VISION.
[Illustration]
The following sleight shows how easily the eye may be deceived. Take a
piece of pasteboard, an inch and a half in width, and five inches in
length, and divide it by inked lines into thirty squares, then cut it from
corner to corner, so as to form two triangles. After this cut off the top
of these triangles at C and D,[13] and arrange the pieces in this manner:--
[Illustration]
On counting the squares in the first figure, there appear to be thirty,
but the other arrangement of the same card seems to contain thirty-two. It
does so, however, only in appearance, but it is only a very correct eye
that can detect the imperfection.
THE CARPENTER PUZZLED.
[Illustration]
A carpenter having a piece of mahogany of a triangular form, (see Fig.)
wished to know how he could make it up to the best advantage. His first
idea was to make an oblong square table of it, but he found that if he
did so the waste of the wood would be very great. After consideration he
discovered that the most economical method of using the wood would be to
form it into an oval. To make this oval contain as much wood as possible,
he proceeded in the following manner: Let B G D be the triangular piece of
wood; take G H one half of the base, and divide the triangle by drawing a
line from H to B. Take G H in the compasses, and set it off on one of the
sides from G to E, draw the line E F, and the point I will be the center
of the oval; draw K L parallel to E F, and at the same distance from the
center as the base G. The points A and C are found by dividing the line
from E to K and drawing A C, or by drawing the dotted lines D A and G
C through the center at I. These points being found, the oval must be
completed by the eye of the draughtsman.
THE BRICKLAYER PUZZLED.
[Illustration: _Fig. 1._]
A bricklayer had to construct a wall, whose length in the direction A B C
was twenty-four feet. The one half of this wall, namely from B to C, had
to be built over a piece of rising ground, so that the base of this part
of the wall would necessarily be more than twelve feet. In making out his
account he charged more for this half of the wall than for that which was
built on level ground from A to B. A geometrician assured him that the
square contents of both portions of the wall were exactly alike; which may
be proved in the following manner:--
[Illustration: _Fig. 2._]
[Illustration: _Fig. 3._]
[Illustration: _Fig. 4._]
Cut two pieces of cardboard, in the form shown in Figs. 2 and 3, to
represent the two parts of the wall; lay the piece representing the
straight wall on the curved piece, and it will be found that the angles
which project at A and B will exactly fill up the spaces at E and F.
The piece of board representing the straight wall may thus be found to
be exactly sufficient to form a piece equal to that representing the
curved wall. You may then lay the curved piece upon the straight one, and
reversing the experiment prove that the curved piece is capable of forming
a rectangular piece equal to the other.
TRIANGULAR PROBLEM.
Take four square pieces of pasteboard of the same dimensions, and divide
them diagonally, that is, by drawing a line from two opposite angles,
as in the figures, into eight triangles. Paint seven of these triangles
with the prismatic colors, red, orange, yellow, green, blue, indigo,
violet, and let the eighth be white. To find how many chequers or regular
four-sided figures, different either in form or color, may be made out of
these eight triangles.
[Illustration]
First, by combining two of these triangles there may be formed, either the
triangular square A, or the inclined square B, called a Rhomb. Secondly,
by combining four of the triangles the large square C may be formed, or
the long square D, called a parallelogram. Now the first two squares,
consisting of two parts out of eight, may each of them by the eighth
rank of the triangle be taken twenty-eight different ways, which makes
fifty-six. And the last two squares, consisting of four parts, may each be
taken by the same rank of the triangle seventy times, which makes 140.
TO FORM A SQUARE.
Take a piece of card of the shape and size or proportions of the
subjoined, and cut it into three parts, and with these three form a
perfect square.
[Illustration]
To do this, cut it in the direction of the dotted lines, and it will then
be easy to lay down the pieces to form a perfect square.
SQUARING THE CIRCLE.
"Squaring the circle," as it is called, is the puzzle of puzzles, and
there are many persons who fancy this can be accomplished, as there are
also many who believe that they can discover "perpetual motion."
The meaning of this phrase _squaring_ is scientifically expressed by the
term finding the quadrature of the circle; that is, the act of producing a
square equal to a given circle; and many persons but slightly acquainted
with mathematics have puzzled their brains to effect this object. The
Cardinal de Cusa rolled a cylinder over a plane, till the point which
was first in contact with the plane touched it again; and then, by a
train of reasoning very unmathematical, he endeavored to determine the
length of the line thus described. Oliver de Serras worked a circle,
and also a triangle equal to an equilateral triangle, inscribed within
the circle, and imagined that the former was exactly equal to two of
the latter, forgetting that the double of this triangle is equal to the
hexagon inscribed within the circle, and therefore smaller than the circle
itself. A Frenchman challenged the world, and deposited 10,000 livres as
a stake, that he could accomplish the feat. He reduced the problem to
the mechanical process of dividing a circle into four quarters, and then
turning these with their angles outwards, so as to form a square, which
he asserted to be equal to the circle; this however was soon proved to be
ridiculous.
Some persons have taken a piece of pasteboard, and cutting it out into a
circular form, and by cutting that circular disc into pieces of a square
form and definite dimensions, and fitting the same turned pieces one
into the other, have come _near_ to a notion of the superficial area of
a circle. But this kind of demonstration is purely mechanical, and is
neither geometrical nor scientific, and, is in fact, no demonstration
according to mathematics. For if we take the pieces of card, however
exactly they may appear to be formed, and examine them with a microscope,
we shall soon find that none of them are geometrically true, nor of the
same length or breadth, and therefore the conclusion arrived at is a false
one.
[Illustration]
The early mathematicians, in their attempts to solve this problem,
generally proceeded on the following plan. If we draw a square exterior
to a circle, that is, touching the square in four points, each side
of the square being equal to the diameter of the circle, we can soon
convince ourselves that the boundary of the square will be greater than
the circumference of the circle, and the area of the former greater than
that of the latter. But if the square be drawn within the circle, so
that only the four corners touch it, then it is equally evident that the
circle is larger, both in boundary and area, than the square. By this
proceeding, we arrive at the conclusion that a circle is _smaller_ than a
square _external_ to it, and _larger_ than one _internal_ to it. Let us
next suppose that we draw a regular pentagon, that is, a figure of five
equal sides, exterior to the circle, and touching it on five points; then
it is evident that as the circle is wholly contained within the pentagon,
it must be smaller than that which contains it. But if the pentagon be
described within the circle, touching it at the five angular points, then
of course the circle is larger than the pentagon which it contains.
Now, in geometry, my young readers must bear in mind, the exact periphery
or circumference, and the exact area of any figure bounded by straight
lines, may be determined with rigorous accuracy; and if we draw two
polygons--say of one hundred sides, one within and one without the
circle--we can ascertain the exact area of those polygons, and affirm
that the area of a circle is greater than a certain amount, and less than
another certain amount. These two amounts, if the number of the sides of
the polygon be so large as we here suppose, may be so very nearly alike,
that either one will give the area of the circle with great closeness.
[Illustration]
By some such means as these Archimedes found that if the diameter of a
circle be called 7, then the circumference will be nearly 22; and that
if the square of the diameter be 14, then the area of the circle will be
equal to about 11; but this computation was slightly in error, and gave to
the area of the circle too great a measure by about one three-thousandth
part of the whole. At a later period, however, a European mathematician,
named Metius, discovered a method which makes an extraordinary approach
to accuracy, and is at the same time easily remembered. He found that if
the diameter be considered equal to 113, then the circumference would
equal 355; or if we multiply the square of the radius by 355, and divide
it by 113, the area will be given. Now this method is so very nearly
correct, that the area of a circle one foot in diameter is given within
the fifty-thousandth part of a square inch.
Other mathematicians have carried the approximation still further. Ludolph
Van Ceulen worked it out to 36 places of figures, showing that if the
diameter be 1, the circumference will be
3.14,159,265,358,979,323,846,264,338,327,950,288.
or that if the last figure be 8, the result will be a little below the
truth, and if 9, a little above it.
Since this, Mr. Sharp, an English mathematician, carried the approximation
to 72 places of figures; Mr. John Machin to 100 figures, and eclipsed all
others. M. de Lagny worked it out to 128 places of figures, and of the
degree of _nearness_ to which this computation brings the proportion,
Montucla says, "If we suppose a circle, the _diameter_ of which is a
thousand million times greater than the distance between the sun and
the earth, the error in the proportion of the circumference would be a
thousand million times less than the thickness of a hair."
But after all, none of these computations are quite correct; they all
deviate from the truth, and bring us to the conclusion that there are
no numbers or collection of numbers which will give the exact ratio of
the circumference, or of the area of a circle to its diameter. We offer
this explanation on the subject to our young friends that they may not
be puzzled by the question; and that should they be asked to square the
circle, or hear any one assert that he can do so, they may be able to show
that they are "awake" to the question, and know how to explain it.
[Illustration]
FOOTNOTES:
[Footnote 13: _I. e._, at the fourth square from the right angle.]
PRACTICAL PARADOXES AND PUZZLES.
[Illustration]
A puzzle is not solved, impatient sirs,
By peeping at its answer, in a trice--
When Gordius, the plow-boy, king of Phrygia,
Tied up his implements of husbandry
In the far-famed knot, rash Alexander
Did not undo, by cutting it in twain.
Paradoxes and Puzzles, although by many persons looked upon as mere
trifles, have, in numerous instances, cost their inventors considerable
time, and exhibit a great degree of ingenuity. We can readily imagine
that some of the complicated puzzles in the ensuing pages may have been
originally constructed by captives, to pass away the hours of a long and
dreary imprisonment; thus does the misery of a few frequently conduce
to the amusement of many. We look upon a Paradox as a sort of superior
riddle, and a tolerable Puzzle, in our opinion, takes precedence of a
first-rate rebus. There is often considerable thought, calculation,
patience, and management, required to solve some of these strange enigmas;
and we have, ere now, followed the mazes of a Puzzle so ardently, as to
be entirely absorbed in devising means to extricate ourself from its
bewildering difficulties; and felt almost as much pleasure in eventually
achieving victory over it, as we have in conquering an adversary at some
superior game of skill. It is "in good sooth, a right dainty and pleasant
pastime," to watch the stray wanderings of another person attempting to
elucidate a Paradox, or perform a Puzzle, with which one is previously
acquainted. It is laughable to see him elated with hope at the apparent
speedy end of his troubles, when you know that, at that moment, he is
actually farther from his object than he was when he began; and it is no
less amusing to watch his increasing despair, as he conceives himself
to be getting more and more involved, when you are well aware that he
is within a single turn of a happy termination of his toils; but what a
mirthful moment is that, when there being only two ways to turn, the one
right and the other wrong, as is usually the case, he takes the latter,
and becomes more than ever
"Pozed, puzzled, and perplexed."
Puzzles are by no means of modern origin; the Sphynx puzzled the brains
of some of the heroes of antiquity, and even Alexander the Great, as it
is written, made several essays to untie the knot with which Gordius, the
Phrygian king, who had been raised from the plow to the throne, tied up
his implements of husbandry in the temple, in so intricate a manner, that
universal monarchy was promised to the man who could undo it: after having
been repeatedly baffled, he, at length, drew his sword, considering that
he was entitled to the fulfillment of the promise, by cutting the Gordian
knot.
1. THE CHINESE CROSS.
[Illustration]
Have six pieces of wood, bone, or metal, made of the same length as No.
6, in the above figures, and each piece of the same size as No. 7. It is
required to construct a cross, with six arms, from these pieces, and in
such a manner that it shall not be displaced when thrown upon the floor.
The shaded parts of each figure represent the parts that are cut _out_ of
the wood, and each piece marked _a_ is supposed to be facing the reader,
while the pieces marked _b_ are the _right_ side of each piece turned over
_towards_ the left, so as to face the reader. No. 7 represents the end of
each piece of wood, &c., and is given to show the dimensions.
2. THE PARALLELOGRAM.
[Illustration]
A parallelogram, as in the illustration, fig. 1, may be cut into two
pieces, so that by shifting the position of the pieces, two other figures
may be formed, as shown by figs. 2 and 3.
3. THE DIVIDED GARDEN.
[Illustration]
A person let his house to several inmates, who occupied different floors,
and having a garden attached to the house, he was desirous of dividing it
among them. There were ten trees in the garden, and he was desirous of
dividing it so that each of the five inmates should have an equal share of
garden and two trees. How did he do it?
4. THE ENDLESS STRING.
[Illustration]
Now, sir, your coat is off!
And see--
Your right-hand pocketed!
So let it be:
While o'er your arm
An endless string--
Some three yards round--
Hangs like a sling.
Take the string off--
But, just for fun,
It must be done
Keeping your right-hand in its place,
And not a smile must stir your face.
Until you find this puzzle out,
No coat shall wrap your back about.
5. CHINESE MAZE. THE WILLOW-PATTERN PLATE.
[Illustration]
Ye fair ones who, in continent or isle,
Long for delights which love alone can bring;
Whilst ruby lips display affection's smile,
Haste through the maze, and reach the "wedding ring"
The sweet Koong-see, whose spirit hovers near,
Shall watch thee wand'ring through the doubtful way;
And when thou showest aught of hope or fear,
Shall whisper to thee, as thy footsteps stray!
6. THE VERTICAL LINE PUZZLE.
Draw six vertical lines, as below, and, by adding five other lines to
them, let the whole form nine.
[Illustration:
| | | | | |
| | | | | |
| | | | | |
]
7. THE THREE RABBITS.
Draw three rabbits, so that each shall appear to have two ears, while, in
fact, they have only three ears between them.
8. THE ACCOMMODATING SQUARE.
Make eight squares of card, then divide four of them from corner to
corner, so that you will now have twelve pieces. Form a square with them.
9. THE CIRCLE PUZZLE.
Draw a circle upon a piece of paper, and thrust a pin through it without
crossing the circle, or thrusting it downward through the center.
10. THE CARDBOARD PUZZLE.
Take a piece of cardboard or leather, of the shape and measurement
indicated by the diagram, cut it in such a manner that you yourself may
pass through it, still keeping it in one piece.
[Illustration:
+-----------+
| 3 Inches. |
| |
|6 |
| |
|I |
|n |
|c |
|h |
|e |
|s |
|. |
| |
+-----------+
]
11. THE BUTTON PUZZLE.
[Illustration]
In the center of a piece of leather make two parallel cuts with a
penknife, and just below a small hole of the same width; then pass a piece
of string under the slit and through the hole, as in the figure, and tie
two buttons much larger than the hole to the ends of the string. The
puzzle is, to get the string out again without taking off the buttons.
12. THE QUARTO PUZZLE.
[Illustration:
|\ /|
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \/ |
| |
| |
| |
+--------------+
]
Divide this figure into four equal parts, each of the same figure.
13. THE PUZZLE OF FOURTEEN.
[Illustration]
Cut out fourteen pieces of paper, card, or wood, of the same size and
shape as those shown in the diagram, and then form an oblong with them.
14. THE SQUARE AND CIRCLE PUZZLE.
[Illustration:
+---------------+
|o o o|
| |
| |
|o oo o|
| oo |
| |
| |
|o o o|
+---------------+
]
Get a piece of cardboard, the size and shape of the diagram, and punch in
it twelve circles or holes in the position shown. The puzzle is, to cut
the cardboard into four pieces of equal size, each piece to be of the same
shape, and to contain three circles, without cutting into any of them.
15. THE SCALE AND RING PUZZLE.
[Illustration]
Provide a thin piece of wood of about two inches and a half square; make a
round hole at each corner, sufficiently large to admit three or four times
the thickness of the cord you will afterwards use, and in the middle of
the board make four smaller round holes in the form of a square, and about
half an inch between each. Then take four pieces of thin silken cord, each
about six inches long, pass one through each of the four corner holes,
tying a knot underneath at the end, or affixing a little ball or bead to
prevent its drawing through; take another cord, which, when doubled, will
be about seven inches long, and pass the two ends through the middle
holes _a a_, from the front to the back of the board, (one cord through
each hole,) and again from back to front through the other holes _b b_;
tie the six ends together in a knot, so as to form a small scale, and
proportioning the length of the cords, so that when you hold the scale
suspended, the middle cord, besides passing through the four center holes,
will admit of being drawn up into a loop of about half an inch from the
surface of the scale; provide a ring of metal or bone, of about three
quarters of an inch in diameter, and place it on the scale, bringing the
loop through its middle; then, drawing the loop a little through the scale
toward you, pass it, double as it is, through the hole at the corner A,
over the knot underneath, and draw it back; then pass it in the same way
through the hole at corner B, over the knot, and draw it back; then,
drawing up the loop a little more, pass it over the knot at top, and
afterward through the holes C and D in succession, like the others, and
the ring will be fixed.
16. THE HEART PUZZLE.
[Illustration]
Cut a piece of thin wood the shape indicated by the diagram, and having
perforated it as above, draw a piece of string, with a smaller heart
attached at the end, through No. 1, pass it behind, and bring it through
2 before, and through 3, and so on to 6, when a loop must be made so as to
enclose that part of the string which runs from 2 to 3. The puzzle is to
remove the string from the large heart altogether, without unfastening the
loop.
Care should be taken to avoid twisting or entangling the string. The
length of the string should be proportioned to the size of the heart; if
you make the heart two inches and a half high, the string when doubled
should be about nine inches long.
17. THE CROSS PUZZLE.
[Illustration]
Cut three pieces of paper to the shape of No. 1, one to the shape of No.
2, and one to that of No. 3. Let them be of proportional sizes. Then place
the pieces together so as to form a cross.
18. THE YANKEE SQUARE.
[Illustration]
Cut as many pieces of each figure in cardboard as they have numbers marked
on each; then form the pride of the American army.
19. THE CARD PUZZLE.
[Illustration]