1

QUEST. 56. If a cube of wood, floating in common water, have three inches of it dry above the water, and 40 inches dry when in sea water; it is proposed to determine the magnitude of the cube, and what sort of wood it is made of?

Ans. the wood is oak, and each side 40 inches.

QUEST. 57. An irregular piece of lead ore weighs, in air 12 ounces, but in water only 7; and another fragment weighs in air 144 ounces, but in water only 9; required their comparative densities, or specific gravities?

Ans. as 145 to 132.

QUEST. 58. An irregular fragment of glass, in the scale, weighs 171 grains, and another of magnet 102 grains; but in water the first fetches up no more than 120 grains, and the other 79: what then will their specific gravities turn out to be? Ans. glass to magnet as 3933 to 5202,

or nearly as 10 to 13.

QUEST. 59. Hiero, king of Sicily, ordered his jeweller to make him a crown, containing 63 ounces of gold. The workmen thought that substituting part silver was only a proper perquisite; which taking air, Archimedes was appointed to examine it; who on putting it into a vessel of water, found it raised the fluid 8-2245 cubic inches: and having discovered that the inch of gold more critically weighed 10.36 ounces, and that of silver but 5-85 ounces, he found by calculation what part of the king's gold had been changed. And you are desired to repeat the process.

Ans. 28.8 ounces.

QUEST. 60. Supposing the cubic inch of common glass weigh 1-4921 ounces troy, the same of sea-water 59542, and of brandy 5368; then a seaman having a gallon of this liquor in a glass bottle, which weighs 3-841b out of water, and, to conceal it from the officers of the customs, throws it overboard. It is proposed to determine, if it will sink, how much force will just buoy it up? Ans. 14-1496 ounces.

QUEST. 61. Another person has half an anker of brandy of the same specific gravity as in the last question; the wood of the cask suppose measures of a cubic foot; it is proposed to assign what quantity of lead is just requisite to keep the cask and liquor under water? Ans. 89-743 ounces.

QUEST. 62. Suppose, by measurement, it be found that a man-of-war, with its ordinance, rigging, and appointments, sinks

sinks so deep as to displace 50000 cubic feet of fresh water;
what is the whole weight of the vessel ?

Ans. 1395 ton's.

QUEST. 63. It is required to determine what would be the height of the atmosphere, if it were every where of the same density as at the surface of the earth, when the quicksilver in the barometer stands at 30 inches; and also, what would be the height of a water barometer at the same time? Ans. height of the air 291660 feet, or 5.5240 miles, height of water 35 feet.

१

i

QUEST. 64. With what velocity would each of those three fluids, viz. quicksilver, water, and air, issue through a small orifice in the bottom of vessels, of the respective heights of 30 inches, 35 feet, and 5.5240 miles, estimating the pressure by the whole altitudes, and the air rushing into a vacuum?

Ans. the veloc. of quicksilver 12.681 feet.

the veloc. of water
the veloc. of air.

47-447
1369-8

QUEST. 65. A very large vessel of 10 feet high (no matter what shape) being kept constantly full of water, by a large supplying cock at the top; if 9 small circular holes, each of an inch diameter, be opened in its perpendicular side at every foot of the depth: It is required to determine the several distances to which they will spout on the horizontal plane of the base and the quantity of water discharged by all of them in 10 minutes;

✔36

6.00000

and the quantity discharged in 10 min. 123-8849 gallons.

Note. In this solution, the velocity of the water is supposed to be equal to that which is acquired by a heavy body in falling through the whole height of the water above the orifice, and that it is the same in every part of the holes.

QUEST.

1

QUEST. 66. If the inner axis of a hollow globe of copper, exhausted of air, be 100 feet; what thickness must it be of, that it may just float in the air ?

Ans. 02688 of an inch thick.

QUEST. 67. If a spherical balloon of copper, of 금이 of an inch thick, have its cavity of 100 feet diameter, and be filled with inflammable air, of of the gravity of common air, what weight will just balance it, and prevent it from rising up into the atmosphere? Ans. 21273lb.

QUEST. 68. If a glass tube, 36 inches long, close at top be sunk perpendicularly into water, till its lower or open end be 30 inches below the surface of the water; how high will the water rise within the tube, the quicksilver in the common barometer at the same time standing at 291⁄2 inches?

Ans. 2-26545 inches.

QUEST. 69. If a diving bell, of the form of a parabolic conoid, be let down into the sea to the several depths of 5, 10, 15, and 20 fathoms; it is required to assign the respective heights to which the water will rise within it: its axis and the diameter of its base being each 8 feet, and the quicksilver in the barometer standing at 30.9 inches?

Ans. at 5 fathoms deep the water rises 2.03546 feet,

at 10

at 15

at 20

3-06393

3-70267

4-14653

VOL. II.

34

ON

ON THE NATURE AND SOLUTION OF EQUATIONS
IN GENERAL.

1. In order to investigate the general properties of the higher equations, let there be assumed between an unknown quantity x, and given quantities a, b, c, d, an equation constituted of the continued product of uniform factors: thus

(x-a) × (x-b) x (x−c) x(x-d) = 0. This, by performing the multiplications, and arranging the final product according to the powers or dimensions of x, be

Now it is obvious that the assemblage of terms which compose the first side of this equation may become equal to nothing in four different ways; namely, by supposing either x = a, or x = b, or x = c, or x = d; for in either case one or other of the factors x-a, x-b, x-c, x-d, will be equal to nothing, and nothing multiplied by any quantity whatever will give nothing for the product. If any other value e be put for x, then none of the factors e-a, e-b, e-c, e-d, being equal to nothing, their continued product cannot be equal to nothing. There are therefore, in the proposed equation, four roots or values of; and that which characterizes these roots is, that on substituting each of them successively instead of x, the aggregate of the terms of the equation vanishes by the opposition of the signs + and-.

The preceding equation is only of the fourth power or degree; but it is manifest that the above remark applies to equations of higher or lower dimensions: viz. that in general an equation of any degree whatever has as many roots as there are units in the exponent of the highest power of the unknown quantity, and that each root has the property of rendering, by its substitution in place of the unknown quantity, the aggregate of all the terms of the equation equal to nothing.

It must be observed that we cannot have all at once x = a, x = b, x = c, &c. for the roots of the equation; but that the particular equations x - a = 0, x-b = 0, x − c = 0, &c. obtain only in a disjunctive sense. They exist as factors in

the

1

:

the same equation, because algebra gives, by one and the same formula, not only the solution of the particular problem from which that formula may have originated, but also the solution of all problems which have similar conditions. The different roots of the equation satisfy the respective conditions: and those roots may differ from one another, by their quantity, and by their mode of existence.

It is true, we say frequently that the roots of an equation are x = a, x = b, x = c, &c. as though those values of x existed conjunctively; but this manner of speaking is an abbreviation, which it is necessary to understand in the sense explained above.

2. In the equation a all the roots are positive; but if the factors which constitute the equation had been x + a, x + b, x + c, x + d, the roots would have been negative or subtractive. Thus

x2+a) x3 +ab x2 +abc) x+abcd=0.... (Β)

+abd

has negative roots, those roots being x =

x= - c, x= - d: and here again we are to apply them

disjunctively.

3. Some equations have their roots in part positive, in part

negative. Such is the following:

x3-a) x2+ab) x+abc=0.

+c-bc

}

(C)

Here are the two positive roots, viz. x = a, x = b; and one negative root, viz. x = -c: the equation being constituted of the continued product of the three factors, x-a=0, x-b =0, x+c=0.

From an inspection of the equations A, B, c, it may be inferred, that a complete equation consists of a number of terms exceeding by unity the number of its roots.

4. The preceding equations have been considered as formed from equations of the first degree, and then each of them contains so many of those constituent equations as there are units in the exponent of its degree. But an equation which exceeds the second dimension, may be considered as composed of one or more equations of the second degree, or of the third, &c. combined, if it be necessary, with equations of the first degree, in such manner, that the product of all those constituent equations shall form the proposed equation. Indeed,