Part 2

How to resolve a triangle of which the base is known to be 10 yards, and two of its angles. Well, we have said above that the sum of the three angles is always the same, equal to 180 degrees, having on one side 84, and on the other 95, that makes together 84 by 95, equal to 179 degrees. The difference between this number and 180 is 1 degree, therefore the angle ABC measures one degree.

We know that an angle of one degree corresponds to a distance of 57 yards. Multiply the base of our triangle by 57 yards and you obtain a distance of the church from the points A and B, 10 by 57, equal to 570 yards. Nothing is more simple than this.

The smaller the measured angle the further off the object will be. As seen in our figure, the upright lines, _m o_, _m’ o’_, _m, o,_, do not vary, but according to their distances from point C, they form various angles, _ac_, _a’c’_, _a,c,_, becoming smaller and smaller.

A graphometer is not always to be had. When approximate distances only are required, the following contrivance may be used. Trace on a cardboard of large size a semi-circumference which one divides first into 180 equal parts, then each of these is divided again in 2, 3, 4 divisions, etc., according to the size given to the circumference, which constitutes a large protractor.

To measure an angle place the cardboard upright in an horizontal position, supporting it by the center of the semi-circumference by means of a screw fixed on a stick. Then proceed as stated above.

From a pin stuck in the center mark the spot where the visual ray passes, go to A and to B, and you get approximately the desired result.

Practical Tracing of a Meridian Line.

The meridian line of a place is an imaginary line passing through this place and the center of the sun, when the latter is at the highest point of the arc of the circle, which it daily describes. At that very moment it is noonday exactly at the place in question.

As the position of the earth changes from day to day, the sun does not every day touch the meridian line at noon; sometimes it is in advance, sometimes behind.

Various instruments have been invented to indicate in a practical manner the meridian of a place. We owe the following construction to Mr. E. Brunner of the longitudinal office.

On a window-sill in a southerly position, fix in a solid, permanent manner, a small cupful of quicksilver; cover it with a lid made of varnished metal, and pierced in its center by a small round hole about a quarter of an inch in diameter. This lid must fit well, but not too tightly, so as to permit its being lowered in close proximity to the surface of the quicksilver.

When the window is open the solitary ray reflected on the mercury will be projected on the ceiling of the room. At the exact noonday the center of the mirror and the center of the reflected image are in the meridian plane. It remains only to be traced.

At the moment of its passage one marks in B, for example, a spot corresponding to the center of the reflected image; one knocks a small nail there, and with a string connect this point with another outside the window, so that the string passes through the center of the diaphragm, M. The line, B M, is the meridian plane. From A, suspend a lead-line which meets the string, B M.

All you have to do now is to join on the ceiling the points, A B, and continue them to D. A black thread may be stretched to serve as the line, and this is the meridian required.

To get the mean time one has only to note the exact passage, and deduct the corrections given in various astronomical papers.

To Measure the Height of a Mountain.

One can, without instruments, take the height of a building or a mountain, provided you are able to measure their base. A yardstick and two ordinary sticks are enough. Suppose the height of the tower, E F, is to be taken.

Some distance off plant a stick, a yard high, A B; one yard from this we plant another and longer one, C D. Measure exactly the distance, B F, and applying the eye at A, we aim at the summit of the tower, E; mark on the stick, C D, the point where the visual ray meets the stick, _i.e._, point G.

Then, by measuring the distance, D G, and subtracting one yard you get G I, and may be expressed in the following statement:

A H : A I :: E H : G I

In the given example let us suppose that A H = 150 yards, A I will, of course, be equal to one yard; G I =, say four fifths of a yard; the problem will be: 150 yards : 1 yard :: _x_ : four fifths of a yard. Work the sum out, and the value of _x_ is 120 yards.

Having taken our lease, A H, at one yard from the ground, we must add one yard to 120, making 121 yards, which is the height of the tower wanted.

To Take Up Four Knives with One.

Here is one more trick of equilibrium, which appears to be interesting enough to find a place among these experiments.

We need not give any long explanations, for our figure fully illustrates the way in which it has to be executed.

First place a knife straight before you, then two others which you place, blade upon blade, over the first. Finally, the two last ones are arranged transversely, their blades passing over those of the two knives put down in the second instance, and below the blade of the first knife.

By taking hold of the handle of the first knife, you can lift them up all at once without breaking the equilibrium.

The Tack in the Ceiling.

To nail a tack in the ceiling without hammer, using a ladder or chair to reach it, seems as impossible as pulling the moon down from the sky. Yet, with a little cleverness, it is quite an easy thing to do.

Place a tack, head downwards, on a half dollar, then place a small piece of tissue paper over it, so that the point of the tack passes through.

Then turn the sides of the paper down round the coin. Throw the whole, point upwards, violently against the ceiling, trying to keep this projectile of a new description from turning over on its course.

With a little practice the knack is soon acquired. The tack enters the ceiling, the violence of the shock tears the paper, which, carried away by the coin, falls to the ground.

Suppose you have a light object to suspend on the ceiling; you may do it in this manner without much trouble. Simply tie a thread to the tack, the object being attached to the other end.

If the projectile is well thrown the tack will go right in, and stick very firmly.

The Jumping Pea.

Take an unbroken straw, four or five inches long, not closed by knots, but forming a tube, and about one twentieth of an inch in diameter.

Divide one of its extremities to a length of about half an inch in four, five or six parts, which separate slightly, so as to form a truncated cone.

After having thus prepared the straw, take a dry pea, with a larger diameter than that of the tube, and place it in the cone. Hold the tube upwards, and blow into it at the opposite end.

The pea will be forced upward by the air column which you blow into the tube. It will remain suspended in the air as long as the interior pressure continues, then fall back into the arms of the cone.

To vary that experiment pass a long pin through the pea, the point of which is turned into the tube. When well thrown up, the pea can be maintained at a distance of two or three inches from the mouth of the straw. According to the stronger or weaker blast of breath, the pea will go up or down.

To Acquire a True Eye.

Here is a peculiar and clever recreation, easily performed, though at first sight it may appear difficult.

Put a tumbler upside down. By means of bread crumbs, fix a match vertically on the top. On the edge of the table place another match, partly raised on a piece of cork or wood.

Stoop down and aim at the vertical match on the glass, so that the one on the table is in the exact line of fire.

When you think it is aimed straight, give it a fillip on the lower end, it will shoot up and touch the one placed on the glass if the aim be good.

If you succeed, you may congratulate yourself on having good eyes--a very desirable gift if you should have to handle a gun, as a soldier or a sportsman.

The Air-Tight Stopper.

How many times has it happened to you, when wanting to cork a bottle, that the intended cork was too large to enter the neck?

What have you done? Cut the cork all round, and obtained, but imperfectly, the desired end.

Next time when the same occasion arises, turn the difficulty in this way: Instead of attacking the sides, cut four notches, bevel-shaped, into the cork as shown in the figure.

Treated in this manner your cork will fit, and close the bottle hermetically.

The Fusee Rocket.

For this you only want a simple match box. Take out a match, and holding it on to the box as shown in figure, _i.e._, hold the box a little slanting, between the thumb and forefinger, and place the match head downwards on the side of the emery paper, where the match ignites when rubbed against.

With medium force press on the match and with the other hand give it a fillip in the direction indicated by the arrow.

The little missile will fly into the air all ablaze, and fall down at a distance of four, five, or even six yards.

With a little practice you will succeed each time. It looks like a small rocket, especially when done in complete darkness.

Be careful to make the experiment only where there is no danger of setting anything on fire.

A Novel Table Mat.

To construct this original table mat 6 objects, always at hand when table is laid for a meal, are required; 3 knives and three tumblers of equal size and arrange the tumblers upside down, in the form of a triangle, and on each of them rest the handle of a knife. Cross the blades so that the first laid passes over the second and the second over the third, this latter passing over the first X.

The blades sustain themselves and you may place on them a dish or any other heavy object, without being afraid of a collapse.

The arrangement is sufficiently shown in the design with out requiring more detailed explanation.

Geometrical Paper Band.

Take a band of paper, say a postal wrapper; you observe that it has two lines and two surfaces (interior surface and exterior surface.) The problem is to arrange it so that it presents only one line and one surface. It may seem improbable, yet it is possible as you will see. Cut the band and gum together again the two pieces thus separated, after having turned over one of them as shown in figure as above. Arranged in this manner the paper has but one line and one surface, for it has the aspect of a screw without end.

Photographic Camera.

Here is a simple way to construct a camera for a pocket photographic apparatus.

Cut out of strong cardboard a piece of about 2 to 2-1/4 inches square. In the middle cut out a circle a little smaller than the lens with which you cover it, so that this lens holds on the edge of the hole.

Cut out also two triangles of cardboard, having one side equal to the square, and a length in proportion to the focus of the lens; say for a simple lens of 3 inch focus, and one inch diameter, a length of one and a half inches.

Paste the two triangles on the square at A and B, their base C must hold a rectangular mirror of the same dimensions as the side C of the square and the side of the triangles. On side D fix a roughened glass pane, or instead, a thin transparent sheet of paper; tissue paper for example.

Cut a black piece of cardboard as indicated in Fig. 3 C; the dotted lines indicate the sides to be turned down. This shade is fixed on to the camera.

Pass through the holes, S S, an iron rod or a long needle, which must pass likewise through the upper angle of the triangles, forming the sides, (Fig. 1). When your lens has been fixed on the round hole of the square your camera is complete.

The shade produces complete obscurity so that the operator can see in the middle of the camera the object or person he wishes to photograph.

In order to fix it on the photographic apparatus, one may fasten a wire, in the form of an elongated U, just below the mirror at E.

The Phantom Needle.

You know that when you sit at a window with a looking-glass in your hand, you can catch a beam of sunlight on the glass and throw it into the eyes of a person on the other side of the street.

What have you done in this case? You answer at once that you have bent the sunlight out of its course and turned it in another direction. If the glass were not there it would fall in a straight line on the window seat. This bending out of the straight line is called reflection.

Now for an experiment; cut a small round piece of cork, not quite half an inch thick. Run a needle into its center and place it in a tumbler two-thirds full of water, needle downwards.

Looking down on the cork you cannot see the needle. Now alter your position, and stoop down so that your eye is on level with the table on which the glass stands. Then you will perceive the needle to be on the top of the cork.

This apparent topsy-turveydom is called total reflection. The needle is reflected on the top of the water, and as the ray from your eye meets the top of the water, you see the needle, as it were, on the top of the cork.

Amphitrite.

At fairs, and in halls of mysteries a variety of optical illusions are presented. Under the name of Amphitrite, the spectacle is sometimes of a woman who seems to rise from the deep, moves about in the empty space, apparently without being sustained by anything or anybody.

She seems completely isolated in mid-air. She turns about, sometimes in a circle, moving now the legs, then the arms. Then after several graceful evolutions in all directions, she stands straight and descends rapidly, seemingly precipitated into a decorated scenery representing the ocean.

The illusion is produced in this way: Behind a well-stretched muslin curtain, M M, is painted D D, with the sky and clouds, below a canvas representing the sea. In front, in the direction of G G, is a mirror, without quicksilver back, inclined at an angle of forty-five degrees.

Below the mirror is a round table moving on a pivot, and on this the actress, who takes the part of the Amphitrite, lays down.

In executing various movements, the table in turning, reflects in the glass the image of the person on whom a vivid light is thrown. The spectators placed at S see the image on the canvas at the back, D D. When the time comes for making the lady disappear altogether, the table, which glides on rails, is drawn off the stage, and Amphitrite seems to plunge into the waters. It is by this process that the specters and ghosts at the theaters are produced.

You can perform this illusion, based on the reflection of the light at home, in reducing its construction to the simple proportions of a small theater of marionettes.

Optical Illusions.

Illusions of the eye are numberless, and afford a wide field for experiment. For example, if you ask any one wearing a silk high hat, to what height he thinks his hat would reach if placed on the ground against the wall or door. Nine times out of ten the mark of the height guessed will be half as much again, at least a third over the real height of the hat.

Again, represents two triangles. Ask which is the one whose center is the better indicated. Every one will say, “triangle A.” Well, every one will be wrong, it is B. Take a pair of compasses and you will easily prove it.

The same occurs with the above figure. The two parallelograms, A B, are absolutely equal, and yet A appears to be larger than B. The two lines, A and B are both of equal length; yet B seems a third longer than A.

The sides, AB, CD, BD of the middle figure, BE, AM, EM, etc., are equal, yet it seems to the eye that the surface, A B E M, is longer than the square A B C D.

There is another deception the eye is liable to. On a sheet of paper trace several circles, having the same center. Place the sheet on your thumb and turn it horizontally, it will then seem to you as if the rounds turned, though you watch with the utmost attention, the illusion will be complete.

In order to terminate this series, which can be varied infinitely, we will, in our turn, ask you this question: Which is the tallest man of the three personages appearing in the adjoining figure? Is it the first, the last, or the middle one?

Try to find out without any instrument of course, simply by the aid of your eyes which you suppose exact and true. It will appear to you at first sight that the artist has made a mistake, and has made a bad drawing. The last _seems_ the tallest, whereas the first seems shortened.

However, measure with a pair of compasses, and the illusion will at once disappear. The draughtsman was not mistaken; the first _is_ the tallest, and the two others go diminishing in height.

* * * * *

This terminates our experiments on optical illusions and you will now enter upon another field of knowledge altogether.

The Insensible Coin.

Cut a piece of cardboard about six inches long, and by sticking the extremities together with a pin, or with gum, form a circle or ring. Balance it carefully on the neck of a wine bottle or decanter, and on the top of the ring place a dime, exactly over the neck of the bottle. Now the trick to be performed is to take off the ring so that, without touching it, the coin falls into the bottle. On the inner side of the ring give a sharp knock with the finger, or, better still, with the thumb and forefinger, as in shooting a marble, as shown in figure. The ring will come off, and the coin which on account of its inertia, does not participate in the movement, will infallibly fall into the bottle. It is absolutely necessary to strike the interior of the circle, because in striking it from the outside one would not get any result at all, on account of the elasticity of the cardboard.

The Asses’ Bridge.

Every schoolboy knows which is the famous geometrical theorem, commonly called the Asses’ Bridge, and which is propounded as follows:

The square constructed on the hypotenuse of a right angled triangle is equivalent to the sum of the squares constructed on the two other sides.

If we had only to propound this terrible theorem, it would be an easy matter, but the question is to prove it by A and B, and by means of the triangles, similar angles, equivalents, etc. Well, instead of all this, we give here a very simple way to prove the truth; if not quite pedagogic, it is none the less real.

Trace on a piece of cardboard or thick paper a square, and divide into 49 parts. This done, cut it out in following the big lines. Take out on the center one division, which add to the small square, and then construct the figure 2.

The right-angled triangle A C D will be found by the sides of the three squares, and the sum of the two small squares constructed on the two sides of the triangle will be equivalent to the great square constructed on the hypotenuse. Effectively:

Square No. 1 has 9 divisions. Square No. 2 has 16 divisions. -- Together 25 ==

And the square No. 3 has also 25 divisions. Therefore the theorem is proved.

Another Way to Prove the Preceding Theorem.

In a square A B D C, trace four similar and equal triangles; cut them out and dispose them as shown in Fig. 1. You will have in the middle an empty space forming a great square, which just has one of the sides of the hypotenuse of the right-angled triangle A E B.

Trace the outlines of this square and remount the triangles one against the other, H C E, against A E B, and C D G, against B F G, you will get the Fig. below.

The successively covered and uncovered parts of the two squares have not changed in extent. But this time the uncovered part is formed of the two squares 2 and 3 which correspond to those constructed on the two other sides of the triangle, A E B.

This very simple demonstration has the advantage of being applicable to any rectangle.

Indented Angles.

Given two sheets of paper of the same size and form of a rectangle, fold them both in four equal parts, one lengthwise and the other sideways, as shown below.

When so folded take a fourth part off each. Part A in the figures. The question is now to cover quite exactly one of the remaining surfaces with the other, in cutting the latter in two perfectly equal parts.

To resolve this question take the surface (parts A having been detached), with which it is intended to cover the other, fold it again into equal parts, but this time in the opposite way to the one in which it was folded first, as indicated in Fig. 4: cut it out then in following the dotted line, F L, formed by the marks of the fold; this done one will obtain two parts absolutely equal, F L.

In order to cover the other surface, Fig. 3, all that is now necessary is to lower the angles, viz.: angle A’, must be in front of angle A, angle B’ in front of angle B, and angle C’ in front of angle C. When the angles are lowered in this way, the two surfaces will be quite similar, and can be covered one by the other.

This experiment can be made either with one or the other sheet, in lowering or raising the angles.

In the example shown here it is the fourth Figure which is destined to cover the other one.

When the operation is terminated as indicated above, part M, of Fig. 4, will be at M’ of Fig. 3, and part O at O’ of the covert figure.

A Cheap Shooting Gallery.

With a whalebone stay busk, make a bow and draw a target on a card. For the arrows, divide lengthwise a steel nib, choosing long shaped ones in the form of a lance and fix each part at the end of a match. You now have complete a saloon shooting gallery, inoffensive and sufficiently recreative at least for your smaller friends.

The Coin in Equilibrium.

Here is a curious demonstration of the balancing of bodies having their center of gravity displaced by a counterpoise.

We propose to keep a coin horizontally in equilibrium on the rim of a tumbler, and it must rest on the glass only by its extreme edge, as shown by the figure which gives the complete demonstration.

Take a silver dollar and place it between the prongs of two forks covering each other, then place the edge of the coin upon the glass and draw the handles of the forks together, or distend them till the whole are balanced. The center of gravity will then be at the point of contact, and you may give a slight swing without the risk of breaking the equilibrium thus obtained.

The Submerged Coin.

In order to make the previous experiment more significant, you may present it also in the following manner: In a soup plate place a coin; beside the latter an inverted glass, then pour water into the plate just to cover the coin. You then inform the spectators that you will withdraw the coin from the plate without wetting your fingers. You will meet with a great deal of disbelief from many of your friends looking on. Leave them in doubt as to the success of your operation.