Part 95
Let the number thought on be 7 Then the double of this is 14 And 4 added to it makes 18 This multiplied by 5 is 90 And 12 added to it is 102 And this multiplied by 10 is 1020 From which deducting 302 There remains 718,--
which, by striking off the last two figures, gives 7,--the number thought on.
_To tell the Number a Person has fixed upon, without asking him any Questions._
The person having chosen any number in his mind, from 1 to 15, bid him add one to it, and triple the amount. Then,
If it be an even number, let him take the half of it, and triple that half; but if it be an odd number, he must add 1 to it, and then halve it, and triple that half.
In like manner let him take the half of this number, if it be even, or the half of the next greater, if it be odd; and triple that half.
Again, bid him take the half of this last number, if even, or of the next greater, if odd; and the half of that half in the same way; and by observing at what steps he is obliged to add 1 in the halving, the following table will shew the number thought on:
1--0--0 -- 4-- 8 2--0--0 --13-- 5 3--0--0 -- 3--11 1--2--0 -- 2--10 1--3--0 -- 8-- 0 1--2--3 -- 6--14 2--3--0 -- 1-- 9 0--0--0 --15-- 7
Thus, if he be obliged to add 1 only at the first step, or halving, either 4 or 8 was the number thought on; if there were a necessity to add 1 both at the first and second steps, either 2 or 10 was the number thought on, &c.
And which of the two numbers is the true one may always be known from the last step of the operation; for if 1 must be added before the last half can be taken, the number is in the second column, or otherwise in the first, as will appear from the following examples:
Suppose the number chosen to be 9 To which, if we add 1 The sum is 10 Then the triple of that number is 30 1. The half of which is 15 The triple of 15 is 45 2. And the half of that is 23 The triple of 23 is 69 3. The half of that is 35 And the half of that is 18
From which it appears, that it was necessary to add 1 both at the second and third steps, or halvings; and therefore, by the table, the number thought on is either 1 or 9. And as the last number was obliged to be augmented by 1 before the half could be taken, it follows also, by the above rule, that the number must be in the second column; and consequently it is 9.
Again, suppose the number thought on to be 6 To which, if we add 1 The sum is 7 Then the triple of that number is 21 1. The half of which is 11 The triple of 11 is 33 2. And the half of that is 17 The triple of 17 is 51 3. The half of that is 26 And the half of that half is 13
From which it appears, that it was necessary to add 1 at all the steps, or halvings, 1, 2, 3, therefore, by the table, the number thought on is either 6 or 14.
And as the last number required no augmentation before its half could be taken, it follows also, by the above rule, that the number must be in the first column; and consequently it is 6.
_A curious Recreation, usually called--The Blind Abbess and her Nuns._
A blind abbess visiting her nuns, who were twenty-four in number, and equally distributed in eight cells, built at the four corners of a square, and in the middle of each side, finds an equal number in every row, containing three cells. At a second visit, she finds the same number of persons in each row as before, though the company was increased by the accession of four men. And coming a third time, she still finds the same number of persons in each row, though the four men were then gone, and had each of them carried away a nun.
_Fig. 1._ +-----+ |3 3 3| |3 3| |3 3 3| +-----+
_Fig. 2._ +-----+ |2 5 2| |5 5| |2 5 2| +-----+
_Fig. 3._ +-----+ |4 1 4| |1 1| |4 1 4| +-----+
Let the nuns be first placed as in fig. 1, three in each cell; then when the four men have got into the cells, there must be a man placed in each corner, and two nuns removed thence to each of the middle cells, as in fig. 2, in which case there will evidently be still nine in each row; and when the four men are gone, with the four nuns with them, each corner cell must contain four nuns, and every other cell one, as in fig. 3; it being evident, that in this case also, there will still be nine in a row, as before.
_Any Number being named, to add a Figure to it, which shall make it divisible by 9._
Add the figures together in your mind which compose the number named; and the figure which must be added to this sum, in order to make it divisible by 9, is the one required.
Suppose, for example, the number named was 8654; you find that the sum of its figures is 23; and that 4 being added to this sum will make it 27; which is a number exactly divisible by 9.
You therefore desire the person who named the number 8654, to add 4 to it; and the result, which is 8658, will be divisible by 9, as was required.
This recreation may be diversified, by your specifying, before the sum is named, the particular place where the figure shall be inserted, to make the number divisible by 9; for it is exactly the same thing, whether the figure be put at the end of the number, or between any two of its digits.
_A Person having made choice of several Numbers, to tell him what Number will exactly divide the Sum of those which he has chosen._
Provide a small bag, divided into two parts; into one of which put several tickets, numbered 6, 9, 15, 36, 63, 120, 213, 309, or any others you please, that are divisible by 3, and in the other part put as many different tickets marked with the number 3 only.
Draw a handful of tickets from the first part, and, after shewing them to the company, put them into the bag again; and having opened it a second time, desire any one to take out as many tickets as he thinks proper.
When he has done this, open privately the other part of the bag, and tell him to take out of it one ticket only.
You may then pronounce, that this ticket shall contain the number by which the amount of the other numbers is divisible; for, as each of these numbers is some multiple of 3, their sum must evidently be divisible by that number.
This recreation may also be diversified, by marking the tickets in one part of the bag with any numbers which are divisible by 9, and those in the other part of the bag with the number 9 only; the properties of both 9 and 3 being the same; or if the numbers in one part of the bag be divisible by 9, the other part of the bag may contain tickets marked both with 9 and 3, as every number divisible by 9 is also divisible by 3.
_To find the Difference between any two Numbers, the greater of which is unknown._
Take as many 9's as there are figures in the less number, and subtract the one from the other.
Let another person add that difference to the larger number; and then, if he take away the first figure of the amount, and add it to the remaining figures, the sum will be the difference of the two numbers, as was required.
Suppose, for example, that Matthew, who is 22 years of age, tells Henry, who is older, that he can discover the difference of their ages.
He privately deducts 22, his own age, from 99, and the difference, which is 77, he tells Henry to add to his age, and to take away the first figure from the amount.
Then if this figure, so taken away, be added to the remaining ones, the sum will be the difference of their ages; as, for instance:
The difference between Matthew's age and 99, is 77 To which Henry adding his age 35 ---- The sum will be 112 And 1, taken from 112, gives 12 Which being increased by 1 -- Gives the difference of the two ages 13 And, this added to Matthew's age 22 -- Gives the age of Henry, which is 35
_A Person striking a Figure out of the Sum of two given Numbers, to tell him what that Figure was._
Such numbers must be offered as are divisible by 9; such, for instance, as 36, 63, 81, 117, 126, 162, 207, 216, 252, 261, 306, 315, 360, and 432.
Then let a person choose any two of these numbers, and after adding them together in his mind, strike out any one of the figures he pleases, from the sum.
After he has done this, desire him to tell you the sum of the remaining figures; and that number which you are obliged to add to this amount, in order to make it 9, or 18, is the one he struck out.
For example, suppose he chose the numbers 126 and 252, the sum of which is 378.
Then, if he strike out 7 from this amount, the remaining figures, 3 and 8, will make 11; to which 7 must be added to make 18.
If he strike out the 3, the sum of the remaining figures, 7 and 8, will be 15; to which 3 must be added, to make 18; and so in like manner, for the 8.
_By knowing the last Figure of the Product of two Numbers, to tell the other Figures._
If the number 73 be multiplied by each of the numbers in the following arithmetical progression, 3, 6, 9, 12, 15, 18, 21. 24, 27, the products will terminate with the nine digits, in this order, 9, 8, 7, 6, 5, 4, 3, 2, 1; the numbers themselves being as follows, 219, 438, 657, 876, 1095, 1314, 1533, 1752, and 1971.
Let therefore a little bag be provided, consisting of two partitions, into one of which put several tickets, marked with the number 73; and into the other part, as many tickets numbered 3, 6, 9, 12, 15, 18, 21, 24, and 27.
Then open that part of the bag which contains the number 73, and desire a person to take out one ticket only; after which, dexterously change the opening, and desire another person to take a ticket from the other part.
Let them now multiply their two numbers together, and tell you the last figure of the product, and you will readily determine, from the foregoing series, what the remaining figures must be.
Suppose, for example, the numbers taken out of the bag were 73, and 12; then, as the product of these two numbers, which is 876, has 6 for its last figure, you will readily know that it is the fourth in the series, and that the remaining figures are 87.
_A curious Recreation with a Hundred Numbers, usually called the Magical Century._
If the number 11 be multiplied by any one of the nine digits, the two figures of the product will always be alike, as appears from the following example:--
11 11 11 11 11 11 11 11 11 1 2 3 4 5 6 7 8 9 -- -- -- -- -- -- -- -- -- 11 22 33 44 55 66 77 88 99
Now, if another person and yourself have fifty counters apiece, and agree never to stake more than ten at a time, you may tell him, that if he will permit you to stake first, you will always undertake to make the even century before him.
In order to this you must first stake one, and remembering the order of the above series, constantly add to what he stakes as many as will make one more than the numbers 11, 22, 33, &c. of which it is composed, till you come to 89; after which, the other party cannot possibly make the even century himself, or prevent you from making it.
If the person who is your opponent have no knowledge of numbers, you may stake any other number first, under 10, provided you afterwards take care to secure one of the last terms, 56, 67, 78, &c.: or you may even let him stake first, provided you take care afterwards to secure one of these numbers.
This recreation may be performed with other numbers; but, in order to succeed, you must divide the number to be attained, by a number which is an unit greater than what you can stake each time; and the remainder will then be the number you first stake. Suppose, for example, the number to be attained is 52, and that you are never to add more than six; then dividing 52 by 7, the remainder, which is 3, will be the number you must stake first; and whatever the other stakes, you must add as much to it as will make it equal to 7, the number by which you divided; and so on.
_A Person in Company having privately put a Ring on one of his fingers, to Name the Person, the Hand, the Finger, and even the Joint on which it is placed._
Desire a third person to double the number of the order in which the wearer of the ring stands, and add 5 to that number, then multiply that sum by 5, and to the product add 10. Let him then add 1 to the last number, if the ring be on the right hand, and 2 if on the left, and multiply the whole by 10: to this product he must add the number of the finger, beginning with the thumb, and multiply the whole again by 10. Desire him then to add the number of the joint; and lastly, to increase the whole by 35.
This being done, he is to declare the amount of the whole, from which you are to subtract 3535; and the remainder will consist of four figures, the first of which will give the place in which the person stands, the second the hand, 1 denoting the right, and 2 the left hand, the third number the finger, and the fourth the joint.
EXAMPLE.
Suppose the person stands the second in order, and has put the ring on the second joint of the little finger of the left hand:
Double the order is 4 Add 5 -- 9 Multiply by 5 -- 45 Add 10 -- 55 Number for left hand 2 -- 57 Multiply by 10 ---- 570 Number of finger 5 ---- 575 Multiply by 10 ---- 5750 Number of joint 2 ---- 5752 Add 35 ---- 5787 Subtract 3535 ---- 2252
Hence it will appear that the first 2 denotes the second person in order, the second 2 the left hand, 5 the little finger, and 2 the second joint.
_To make a Deaf Man hear the Sound of a Musical Instrument._
It must be a stringed instrument, with a neck of some length, as a lute, a guitar, or the like; and before you begin to play, you must by signs direct the deaf man to take hold with his teeth of the end of the neck of the instrument; for then, if one strikes the strings with the bow one after another, the sound will enter the deaf man's mouth, and be conveyed to the organ of hearing through a hole in the palate, and thus the deaf man will hear with a great deal of pleasure the sound of the instrument, as has been several times experienced; nay, those who are not deaf may make the experiment upon themselves, by stopping their ears so as not to hear the instrument, and then holding the end of the instrument in their teeth, while another touches the strings.
_When two Vessels or Chests are like one another, and of equal Weight, being filled with different Metals, to distinguish the one from the other._
This is easily resolved, if we consider that two pieces of different metals, of equal weight in air, do not weigh equally in water, because that of the greatest specific gravity takes up a lesser space in water; it being a certain truth, that any metal weighs less in water than in air, by reason of the water, the room of which it fills; for example, if the water weighs a pound, the metal will weigh in that water a pound less than in the air: this gravitation diminishes more or less, according as the specific gravity of the metal is greater than that of the water.
We will suppose, then, two chests perfectly like one another, of equal weight in the air, one of which is full of gold, and the other of silver; we weigh them in water, and that which then weighs down the other must needs be the gold chest, the specific gravity of gold being greater than that of silver, which makes the gold lose less of its gravitation in water than silver. We know by experience, that gold loses in water about an eighteenth part only, whereas silver loses near a tenth part; so that if each of the two chests weighs in the air, for example, 180 pounds, the chest that is full of gold will lose in the water ten pounds of its weight; and the chest that is full of silver will lose eighteen: that is, the chest full of gold will weigh 170 pounds, and that of silver only 162.
Or, if you will, considering that gold is of a greater specific gravity than silver, the chest full of gold, though similar and of equal weight with the other, must needs contain a less bulk, and consequently it contains the gold.
_To find the Burden of a Ship at Sea, or in a River._
It is a certain truth, that a ship will carry a weight equal to that of a quantity of water of the same bulk with itself; subtracting from it the weight of the iron about the ship, for the wood is of much the same weight with water; and so, if it were not for the iron, a ship might sail full of water.
The consequence of this is, that, however a ship be loaded, it will not totally sink, as long as the weight of its cargo is less than that of an equal bulk of water: now, to know this bulk or extent, you must measure the capacity or solidity of the ship, which we here suppose to be 1000 cubical feet, and multiply that by 73 pounds, the weight of a cubical foot of sea-water; then you have in the product 73,000 pounds for the weight of a bulk of water equal to that of the ship; so that in this example, we may call the burden of the ship 73,000 pounds, or 36-1/2 tons, reckoning a ton 2,000 pounds, that being the weight of a ton of sea-water; if the cargo of this ship exceeds 36-1/2 tons, she will sink; and if her loading is just 73,000 pounds, she will swim very deep in the water upon the very point of sinking; so that she cannot sail safe and easy, unless her loading be considerably short of 73,000 pounds weight; if the loading come near to 73,000 pounds, as being, for example, just 36 tons, she will swim at sea, but will sink when she comes into the mouth of a fresh water river; for this water being lighter than sea-water will be surmounted by the weight of the vessel, especially if that weight is greater than the weight of an equal bulk of the same water.
_To Measure the Depth of the Sea._
Tie a great weight to a very long cord, or rope, and let it fall into the sea till you find it can descend no further, which will happen when the weight touches the bottom of the sea: if the quantity or bulk of water, the room of which is taken up by the weight, and the rope, weighs less than the weight and rope themselves; for if they weigh more, the weight would cease to descend, though it did not touch the bottom of the sea.
Thus one may be deceived in measuring the length of a rope let down into the water, in order to determine the depth of the sea; and therefore, to prevent mistakes, you had best tie to the end of the same rope another weight heavier than the former, and if this weight does not sink the rope deeper than the other did, you may rest assured that the length of the rope is the true depth of the sea; if it does sink the rope deeper, you must tie a third weight, yet heavier, and so on, till you find two weights of unequal gravitation, that run just the same length of the rope, upon which you may conclude, that the length of the wet rope is certainly the same with the depth of the sea.
_Method of Melting Steel, and causing it to Liquefy._
Heat a piece of steel in the fire, almost to a state of fusion, then holding it with a pair of pincers or tongs, take in the other hand a stick of brimstone, and touch the piece of steel with it: immediately after the contact, you will see the steel melt and drop like a liquid.
_How to dispose two little Figures, so that one shall light a Candle, and the other put it out._
Take two little figures of wood or clay, or any other materials you please, only taking care that there is a little hole at the mouth of each: put in the mouth of one a few grains of bruised gunpowder, and a little bit of phosphorus in the mouth of the other, taking care that these preparations are made beforehand.
Then take a lighted wax candle, and present it to the mouth of the figure with the gunpowder, which, taking fire, will put the candle out; then present your candle, having the snuff still hot, to the other figure; it will immediately light again by means of the phosphorus.
You may propose the same effects to be produced by two figures drawn on a wall with a pencil or coal, by applying with a little starch, or water, a few grains of bruised gunpowder to the mouth of one, and a bit of phosphorus to the mouth of the other.
_The Camera Obscura, or Dark Chamber._
We shall here give a short description of this optical invention; for though it is very common, it is also very pleasing: but every one knows not how to construct it.
Make a circular hole in the shutter of a window, from whence there is a prospect of the fields, or any other object not too near: and in this hole place a convex glass, either double or single, whose focus is at the distance of five or six feet: the distance should not be less than three feet; if it be, the images will be too small, and there will not be sufficient room for the spectators to stand conveniently; on the other hand, the focus should never be more than fifteen or twenty feet, for then the images would be obscure, and the colouring faint; the best distance is from six to twelve feet:--take care that no light enters the room but by this glass: at a distance from it, equal to that of its focus, place a pasteboard, covered with the whitest paper; this paper should have a black border, to prevent any of the side rays from disturbing the picture; let it be two feet and a half long, and eighteen or twenty inches high; bend the length of it inwards to the form of part of a circle, whose diameter is equal to double the focal distance of the glass: then fix it on a frame of the same figure, and put it on a moveable foot, that it may be easily fixed at that exact distance from the glass where the objects paint themselves to the greatest perfection: when it is thus placed, all the objects that are in the front of the window will be painted on the paper in an inverted position; this inverted position of the images may be deemed an imperfection, but it is easily remedied; for if you stand above the board on which they are received, and look down on it, they will appear in their natural position; or if you stand before it, and, placing a common mirror against your breast in an oblique direction, look down in it, you will there see the images erect, and they will receive an additional lustre from the reflection of the glass: or place two lenses in a tube that draws out: or, lastly, if you place a large concave mirror at a proper distance before the picture, it will appear before the mirror in the air, and in an erect position, with the greatest regularity, and in the most natural colours.