| | DITCH. |
+ +----+----+---+---+-----+
| | | | | | |
| | | | | | |
| | | | | | |
| G | H | I | J | K | F |
+---+----+----+---+---+-----+
| 3 | 26 | 3 | 3 | 6 | 138 |
| 3 | 20 | 4 | 4 | 8 | 128 |
| 3 | 23 | 3 | 3 | 6 | 120 |
| 3 | 17 | 4 | 4 | 8 | 104 |
| 3 | 19 | 3 | 3 | 6 | 96 |
| 3 | 16 | 4 | 4 | 8 | 96 |
| 2 | 18 | 3 | 3 | 6 | 90 |
| 2 | 15 | 10 | 5 | 8 | 60 |
| 3 | 19 | 3 | 3 | 6 | 96 |
| 2 | 16 | 4 | 4 | 6 | 72 |
| 2 | 12 | 4 | 3 | 6 | 51 |
| 2 | 9 | 6 | 3 | 8 | 36 |
| 2 | 15 | 10 | 5 | 8 | 60 |
+---+----+----+---+---+-----+
SIMPLE METHODS OF TRACING FIELD WORKS, ON THE GROUND.
_1. Square redoubt._
Place pickets in a line (in length conformable to the side of the
intended work), at each end of which erect perpendiculars equal in
length to the side first marked out, and join the termination of
these lines; which will complete the perimeter of the redoubt.
_Note._—A perpendicular is raised on a given line, with a chain or
cord, by forming a right-angled triangle from the numbers 3, 4, and
5, or any multiples thereof, and extending the cord, &c., so that
the base may correspond with the base line of the pickets, and the
perpendicular be in the direction of the side required.—_Vide_
PRACTICAL GEOMETRY.
_2. Pentagonal redoubt._
With a chain, tape, or cord, construct, and lay down with pickets
five similar, and contiguous triangles, having their bases, which
form the sides of the pentagon, in the proportion of 47 to the other
two equal sides, the length of each of these being 40.
_3. Hexagonal redoubt._
From a central point with a chain, or line, construct, and lay down
with pickets six equilateral, and contiguous triangles, the bases of
which will form the required hexagon.
_4. Octagonal redoubt._
Construct a square (_vide_ No. 1), from the centre of each side
of which erect perpendiculars outwards, in length proportional to
the side as 13 to 60 (nearly 1 to 5); join the extremities, or
termination of the perpendiculars, to the angles of the square, which
will determine the sides of the octagon.
_Note 1._—The directions for the construction of the pentagonal,
and hexagonal redoubts are on a small scale; but the redoubts may
be increased by the equal extension of the interior sides of the
triangles, until the bases are sufficiently long for the periphery of
the work required.
_Note 2._—By means of the pocket sextant, prismatic compass, or
reconnoitring protractor, the pentagonal, hexagonal, and octagonal
redoubt may be thus traced on the ground. From a central point
place pickets at the requisite distance from each other, and in the
direction of lines drawn from the angle of the centre of the intended
work. (_Vide_ PRACTICAL GEOMETRY. _To find the angles at the centre,
and circumference of a polygon._) Extend these radii equally until
the relative distances between them are of the length required to
form the sides of the proposed equilateral redoubt.
_5. Front of fortification, for a Field fort._
Place pickets in a straight line, of the length required for the
front of the proposed field work; from the centre of which drop a
perpendicular inwards, making it for a square, pentagon, hexagon,
or octagon, respectively one-eighth, one-fifth, one-fourth, or
one-third of the exterior side. Direct the lines of defence from the
termination of the exterior side to the end of the perpendicular,
making the faces of the bastions two-sevenths of the exterior side,
and constructing the flanks perpendicular to, and joining the lines
of defence. Other fronts are traced by laying down the exterior
sides, at the angle of the circumference of the intended polygon
(_Vide_ PRACTICAL GEOMETRY) by means of the prismatic compass, &c.,
and then proceeding as directed for the former front.
PART X.
BRIDGES, AND PONTOONS.
BRIDGES.
_To find the number of planks required to form a float, to support a
given weight._
1st. Find the content of one plank (_vide Practical Geometry, Part
12_), and multiply it by the specific gravity of the wood; the
product will be the _weight of the timber_.
2nd. Multiply the same solid content by the specific gravity of
water: the product will be the _weight of an equal bulk of water_.
Then take the difference of these two products, or weight, and it
will be the weight one piece of timber will support without sinking.
_Hence by Proportion_, the number required to support the given
weight may be found.
_To find the number of casks required to form a raft to support a
given weight._
1st. Find the solid content of one cask in cubic inches (_vide
Practical Geometry_), and multiply it by the specific gravity of
water; the product will be the weight of a quantity of water of equal
bulk with the cask.
2nd. From this product, or weight, subtract the weight of the cask,
and the remainder will be the weight it will support without sinking.
_Then by Proportion_, the number required for the formation of the
raft may be found.
_To find the number of boats, or pontoons, required to support a
given weight._
The burthen a boat, or pontoon, will support without sinking beyond a
given depth (the form of the boat, or pontoon being known) must first
be found, thus—
1st. Find the solid content of the part to be sunk, in cubic feet
(_vide Practical Geometry, Part 12_), and multiply it by the specific
gravity of water (_vide Gravity, Part 12_).
2nd. Subtract this product from the weight of the boat, or pontoon,
and the remainder will be the burthen it will support without sinking
beyond the required depth.
_Then by Proportion_, the number required to support the given weight
may be computed.
_Note._—In the construction of bridges, should a rope require to
be extended across a rapid river, the coil should be placed in the
boat (instead of on shore) and be paid out as the boat advances.
PONTOONS.
Those called Blanshard’s (from their inventor, Colonel Blanshard,
Royal Engineers) are of two descriptions.
1.—LARGE PONTOONS.
_Displacement of water_, 97½ cubic feet, equals 6088 lb., or 54½ cwt.
_The buoyant power_ of a raft of two pontoons, its own weight
deducted, is 77 cwt., about one-half of which is a safe load. Each
raft, or one carriage load, forms 2 bays, or 20 ft. 8 in. of bridge:
its own weight will sink it about 7 or 8 inches. The crew of a raft
consists of 6 rowers, and 1 steersman.
_At open order_ the bridge will pass cavalry, field artillery, or
infantry, with closed files.
_At close order_ the bridge will pass any part of a heavy train.
_Dimensions, and weight of cylindrical pontoons, manufactured in the
arsenal at Woolwich._
+-------------------------+---------+-----------+---------+
| | Extreme | Diameter. | Weight. |
| | length. | | |
| +---------+-----------+---------+
| | ft. in. | ft. in. | cwt. |
| LARGE PONTOON, | | | |
| with hemispherical ends | 22 3 | 2 8 | 4 |
| SMALL PONTOON, | | | |
| with conical ends | 15 0 | 1 8 | 1¼ |
+-------------------------+---------+-----------+---------+
ONE CARRIAGE LOAD.
+---------------------+-------+--------------------------+------------+
| | | Dimensions. | |
| | +--------+--------+--------+ |
| |Number.|Length.*|Breadth.| Depth. | Weight. |
| +-------+--------+--------+--------+------------+
| | | ft. in.| ft. in.| ft. in.|cwt.qrs. lb.|
| Pontoons | 2 | 22 | 2 7 | 2 7 | 9 1 9 |
| Baulks | 12 | 14 2 | 5 | 3 | 5 1 1 |
| Chesses | 8 | 12 | 1 2 | 1½ | 1 3 19 |
| Half-chesses | 4 | 11 5 | | 1½ | 1 3 0 |
| Gunnels | 2 | 12 | 1 2 | 4 | 1 3 19 |
| Side-pieces | 5 | 10 4 | 5 | 2½ | 1 1 0 |
| Anchor | 1 | | | | 3 1 |
| Buoy | 1 | | | | 3 |
| Cable (20 fathoms) | 1 | 168 | 3 | 3 | 1 18 |
| Oars | 7 | 14 | | | 3 14 |
| Boat-hook | 1 | 14 | | | 7 |
| { Body | 3 | 8 9 | 4 | | 20 |
| Lashings { Gunnel | 4 | 15 | 1 | 1 |} 8 |
| { Carriage | 4 | 22 | 1 | 1 |} |
| Rack-sticks, and | | | | | |
| lashings | 8 | | | | 14 |
| Buoy-line | 1 | 60 | 1½ | 1½ | 5 |
| Breast-line | 1 | 60 | 1½ | 1½ | 5 |
| | +--------+--------+--------+------------+
| | | | | | 31 1 0 |
| Weight of Carriage | 1 | | | | 13 3 20 |
| | | | | +------------+
| Total | | | | | 45 0 20 |
+---------------------+-------+--------+--------+--------+------------+
* The length includes the hemispherical ends.
2.—SMALL PONTOONS.
Five pontoons, with their appurtenances, form the load of one
carriage. Length, 15 feet; diameter, 1 foot 7 inches; displacement,
27½ cubic feet; buoyancy, 1718 lb., or 15¼ cwt., from which deduct 2
cwt. for the weight of pontoon, and share of superstructure. A bridge
of this nature is so light that it may be made on shore, and carried
by hand entire. It will support as many men as can be placed on it,
and, by removing the chesses over the gunnels, it may be bent so as
to be passed without difficulty down a steep bank, or counterscarp.
ONE CARRIAGE LOAD.
+-----------------------+-------+--------------------------+-----------+
| | | Dimensions. | |
| | +--------+--------+--------+ |
| |Number.|Length.*|Breadth.| Depth. | Weight. |
| +-------+--------+--------+--------+-----------+
| | | ft. in.| ft. in.| ft. in.|cwt.qrs.lb.|
| Pontoons | 5 | 15 | 1 7 | 1 7 | 5 0 0 |
| Gunnels | 5 | 8 4 | 9 | 3 | 3 12 |
| Baulks | 30 | 6 6 | 3 | 1½ | 1 1 9 |
| Chesses | 20 | 8 | 1 4 | 1½ | 4 0 20 |
| Paddles, used as | | | | | |
| side-pieces | 5 | | | | 1 27 |
| Rack-sticks, and | | | | | |
| lashings | 10 | | | | 6 |
| Carriage lashings | 2 | | | | 5 |
| Anchor, and cable | 1 | | | | 2 0 |
| | +--------+--------+--------+-----------+
| | | | | | 12 1 23 |
| Weight of the Carriage| 1 | | | | 9 0 0 |
| | | | | +-----------+
| Total | | | | | 21 1 23 |
+-----------------------+-------+--------+--------+--------+-----------+
* The length includes the ends.
SCALING LADDERS.
Scaling ladders are made in portions, 12 feet, and 7½ feet long;
which are joined together by placing the end of one portion into
staples at the end of another, and securing them together with a
lashing of rope. Four men are sufficient to carry an 18 feet scaling
ladder.
PART XI.
FIREWORKS.
CANDLES, ROMAN.
When the case has been properly finished, and the end secured with
strong twine, ram in a little dry clay, then put in a small quantity
of corned powder, and over this a small piece of paper; after which
as much of the composition is to be put in as will, when rammed
down, fill the case about one-sixth of the length. Over this a piece
of paper is placed, covering about two-thirds of the diameter,
on which a little powder is laid, and then a ball, or star. Some
more composition is then poured in, and pressed lightly down till
the case is about one-third full, when it is to be rammed down by
gentle strokes; after which, put in another small piece of paper,
powder, and star, over which pour in some more composition, ram down
gently, and continue thus until the case is filled, when it will
contain about five stars; after which it is capped, and primed. Roman
candles should be placed in rows on a stand, some of them being
perpendicular, and others at angles, not, however, exceeding 45
degrees.
_Compositions._
1. Mealed powder, ½ lb. Sulphur, ½ lb. Saltpetre, 2½ lb. Glass dust,
½ lb.
2. Mealed powder, 1 lb. Charcoal, ½ lb.
3. Signal rocket composition.
CANDLES, BLUE.
Saltpetre, ¼ oz. Powder, ½ oz. Sulphur, 1 oz.
CRACKERS.
The case is made of cartridge paper, the dimensions required being
15 inches by 3½ inches. First fold down one edge, about ¾ of an inch
broad, then turn down the double edge about ¼ of an inch, and bend
back the single edge over the double fold, so as to form within a
channel, which is to be filled with mealed powder, not ground very
fine; the powder is then to be covered by the folds on each side, and
the whole is to be pressed by a flat ruler; and the part containing
the powder is to be folded into the remainder of the paper, every
fold being pressed down. The cracker is then doubled backwards, and
forwards in folds about 2¼ inches, which are pressed quite close, and
a piece of twine is passed twice round the middle across the folds,
and the joinings secured by causing the twine to take a turn round
the middle at each fold successively; one of the ends of the folds
may be doubled short under, which will produce an extra report; the
other must project a little beyond the rest for the purpose of being
primed.
EARTHQUAKE, ARTIFICIAL.
Mix together four pounds of iron sand, and four pounds of sulphur;
and after making it into a paste with water, bury it a little depth
in the ground. In ten or twelve hours, if the weather be warm,
the earth will swell up, and burst; flames will also issue out,
scattering around a yellow and black dust.
FIRES OF VARIOUS COLOURS.
_White fire_ is made by mixing powder with iron sand, or filings.
_Red fire_ is made by mixing powder with iron sand of the first order.
_Silver colour_ is made by introducing raspings of ivory.
_Pale white colour_ is made by mixing camphor with the composition.
_Lemon colour_ is produced from raspings of yellow amber.
_Russet colour_ is produced from crude antimony.
_Red flame_, inclining to brown, is produced by Greek pitch.
_Dusky flame_ is produced by black pitch.
_Bluish flame_ is produced by introducing sulphur in a moderate
quantity.
_Green flame_ is made from the introduction of sal ammoniac, and
verdigris.
_White flame_: saltpetre, sulphur, mealed powder, and camphor; the
saltpetre must be the chief part.
WILDFIRE, WHICH BURNS UNDER WATER.
Mix well together equal quantities of sulphur, naphtha, bitumen,
pitch, and gum.
GERBES.
Gerbes consist of strong cylindrical cases of thick paper, filled
with brilliant composition, and sometimes with balls, or stars.
Gerbes throw up into the air luminous and sparkling jets of fire;
and when arranged in a circular manner, as the radii of a circle,
they form what is called a fixed sun. The thickness of the cases for
brilliant fire must be a fourth part of the diameter, and for Chinese
fire a sixth part. The case is loaded on a nipple, having a point
equal in length to the same diameter, and in thickness to a fourth
part of it; but as it generally happens that the mouth of the jet
becomes larger than is necessary for the effect of the fire, the case
should be first charged by filling it to a height equal to a fourth
part of the diameter, with clay, which must be rammed down. When the
charge is completed with the composition, the case should be closed
with a tompeon of wood, and then choked. The train, or match, must
be of the same composition as that employed for loading, otherwise
the jet would be subject to burst. Charged cases may be pierced with
two holes near the neck, in order to have three jets; or, if a top
pierced with a number of holes be added to them, they will imitate
a bubbling fountain. Jets intended for representing sheets of fire
ought not to be choked. They must be placed in a horizontal position,
or inclining a little downwards.
_Composition for Gerbes, or jets of fire._
Key:
S = Saltpetre.
P = Pulverized powder.
Su = Sulphur.
C = Charcoal.
+-------------------------+------+------+----+----+--------------------+
| | | | | | |
| JETS. | S | P | Su | C | Iron sand, or |
| | | | | | filings. |
+-------------------------+------+------+----+----+--------------------+
| |lb.oz.|lb.oz.| oz.| oz.| oz.|
| ·4 of an inch or less | | | | | |
| interior diameter | | | | | |
| { Chinese fire | 1 | 1 | 8 | 2 | |
| { White fire | 1 | 8 | 3 | 2 | Sand, 1st order 8 |
| | | | | | |
| ·5 to 1 inch diameter | | | | | |
| { Brilliant fire | | 1 | | | Filings 5 |
| { White fire | 1 | | 8 | 2 | |
| { Chinese fire | 1 4 | | 5 | 5 | Sand, 3rd order 12 |
| | | | | | |
| { | 1 4 | | 7 | 5 | Mixed sand 12 |
| 1·1 to 1·5 diameter { | | 1 8 | | 5 | |
| { | 2 | 2 | | 8 | |
+-------------------------+------+------+----+----+--------------------+
The saltpetre, powder, and charcoal, are three times sifted through
a hair sieve; the iron sand is besprinkled with sulphur, after being
moistened with spirits of wine, that the sulphur may adhere to it,
and they are then mixed together. The sulphuretted sand is then
spread over the first mixture, and the whole is mixed with a ladle
only, for if a sieve were employed, it would separate the sand from
the other materials.
When sand larger than the 2nd order is used, the composition is
moistened with spirits of wine, so that it forms itself into balls,
and the jets are then loaded with them.
IRON FILINGS.
These filings must be quite clean, and free from rust. They must be
thoroughly mixed with the compositions of which they form one of the
ingredients; which, however, will not afterwards keep in good order
longer than a week, as the moisture contracted by saltpetre rusts the
filings, and destroys the effect they are intended to produce.
IRON SAND, OR POUNDED IRON.
Having broken a cast-iron plate, or iron pot, to pieces, on an anvil,
pulverize the fragments till the grains are not larger than radish
seed, then sift them through six graduated sieves to separate the
different sizes; and preserve these six different kinds in a very dry
place, in closely-corked bottles. The grains which pass through the
first or finest sieve, are called _Sand of the 1st order_, and those
that pass through the second sieve, _Sand of the 2nd order_, &c.
Compositions into which iron filings, or sand, are introduced, must
not be driven hard, for fear of accidents by explosion.
LEADERS, OR PIPES OF COMMUNICATION.
These are small tubes of paper, of lengths adapted to the distances
to which they are to extend. The paper is cut into slips two or three
inches broad, or sufficient to go four times round the formers, which
are about one-fourth of an inch diameter. Brass wire formers are the
best, and should be oiled to prevent the paper sticking. Quick match
is inserted in these tubes, but must be made to go in easily. The
quick match should project an inch beyond each end of the leader, and
should be inserted into the mouths of the cases of the fireworks with
a small quantity of mealed powder. The leaders must not be placed too
near, or cross each other so as to touch, as it may happen that the
fire from one may communicate to another, and destroy thereby the
intended arrangements.
MARROONS.
Marroons are boxes containing from 1 to 6 ounces of powder. They are
made either on a square, or round former, and the ends of the paper
are pasted down, and well welded round with kitted twine worked over
cross-ways. A hole is bored into the case, and a match inserted.
MEALING GUNPOWDER.
A small quantity of powder being placed on a table with a rim round
it, is rubbed down with a scored wooden mealer until all the grains
are broken, and it becomes sufficiently fine to pass through a lawn
sieve.
PASTE FOR REPRESENTING ANIMALS, ETC., IN FIRE.
Reduce sulphur to an impalpable powder, and having formed it into
a paste with starch, cover with it the figure intended to be
represented on fire, which must, however, have been previously coated
over with clay, to prevent it from being burnt. When covered with
the paste, besprinkle it, while still moist, with mealed powder; and
when perfectly dry, arrange some matches on the principal parts, so
that fire may be speedily communicated to every part of it. The same
paste may be employed to form festoons, garlands, &c., the flowers of
which might be imitated by fires of different colours, and could be
formed on the frieze of a piece of architecture, covered with clay to
preserve the stone, or wood from the effect of the fire. The Chinese
imitate grapes exceedingly well, by mixing powdered sulphur with the
pulp of the jejube, instead of flour-paste or starch.
PORTFIRES FOR ILLUMINATIONS.
The cases are made of three or four rounds of thin paper, the last
round being pasted; they are from two to five-eighths of an inch in
diameter, and from two to six inches long; they are pinched close
at one end, and left open at the other. In filling them, a small
quantity of the composition must be put in at a time, ramming it
lightly, so as not to break the case.
_Compositions._
1. Saltpetre, 1 lb.; sulphur, 8 oz.; mealed powder, 6 oz.
2. _Amber lights_: Mealed powder, 9 oz.; amber, 3 oz.
3. Saltpetre, 2 lb.; sulphur, 3 lb.; antimony, 1 lb.
4. Saltpetre, 3½ lb.; sulphur, 2½ lb.; mealed powder, 1 lb.;
antimony, ½ lb.; glass dust, 4 oz.; brass dust, 1 oz.
_Note._—_Compositions_ No. 3, and 4, driven 1¼ inch in a one ounce
case, will burn one minute.
RAIN, GOLD, AND SILVER.
Fill small paper cases, or goose quills, with the composition, and
place upon the mouth of each some moist powder, both to keep in the
composition, and to serve as a match. If the head of a rocket be
loaded with these cases or quills, a shower of fiery rain will be
produced at the extreme range of the rocket.
_Compositions._
_Gold rain._
1. Saltpetre, 1 lb.; mealed powder, 4 oz.; sulphur, 4 oz.; brass
dust, 1 oz.; sawdust, 2¼ oz.; glass dust, 6 drs.
2. Mealed powder, 12 oz; saltpetre, 2 oz.; charcoal, 4 oz.
3. Saltpetre, 8 oz.; sulphur, 2 oz.; glass dust, 1 oz.; antimony, ¾
oz.; brass dust, ¼ oz.; sawdust, 12 drs.
_Silver rain._
1. Saltpetre, 4 oz.; sulphur, mealed powder, and antimony, each 2
oz.; sal prunella, ½ oz.
2. Saltpetre, 8 oz.; sulphur, 2 oz.; charcoal, 4 oz.
3. Saltpetre, 1 lb.; antimony, 6 oz.; sulphur, 4 oz.
4. Saltpetre, 4 oz.; sulphur, 1 oz.; powder, 2 oz.; steel dust, ¾
oz.
ROCKETS, LINE.
Any rocket which is not very large, may be made to run along an
extended rope. For this purpose affix to the rocket an empty case,
and introduce therein the rope which is to carry it, placing the head
of the rocket towards that side to which it is intended to move. Two
rockets with an empty case may be similarly used, and may be made
to move in a retrograde direction by placing them with their heads
reversed, and a leader to communicate from the head of the rocket to
be first ignited to the tail of the second.
ROCKETS, SIGNAL.
_Composition._
Pulverized saltpetre, 4 lb.; sublimated sulphur, 1 lb.; dogwood
charcoal, 1 lb. 8 oz.
The charcoal is first pounded fine enough to pass through a wire
sieve (No. 3). The saltpetre and sulphur are each separately passed
through a fine hair sieve, then mixed well together with a copper
slice, and passed three times through the hair sieve. The charcoal is
then spread on a tray, and the saltpetre and sulphur sifted a fourth
time on it, and the whole being carefully mixed with a hard brush, is
afterwards passed four times through the wire sieve.
To each ladleful of composition, 25 blows are given for the pound,
and 21 for the half-pound rocket.
Twenty-eight ladlesful of composition (7½ oz.) are required to
complete the pound, and twenty-five (5 oz.) the half-pound rocket.
To prevent accidents in driving rockets, &c., the workman should keep
his body erect, the drift being well cleaned after each ladleful; and
while driving, it should be moved backward, and forward by a pair of
holders.
Rockets are driven 3½ calibres hollow, 1 calibre solid, and ½ calibre
with clay.
_Moulds._
Length. Diameter.
1 Pounder Exterior 14 inches 3·9
Interior 10·5 ” 1·7
½ Pounder Exterior 11·6 ” 2·3
Interior 8·5 ” 1·3
_Spindles._
Diameter.
Top. Middle. Bottom. Length.
1 Pounder ·2 ·35 ·5 6·4
½ Pounder ·2 ·3 ·4 5·2
_Drifts._
Length. Diameter. Hollowed. No.
{ 12·1 inches 1·1 6·5 1
1 Pounder { 10 ” 1·1 5·7 2
{ 7·3 ” 1·1 2·5 3
{ 6·8 ” 1·1 Solid 4
{ 9·3 ” ·9 5·4 1
½ Pounder { 7·1 ” ·9 2·3 2
{ 4 ” ·9 Solid 3
The rocket is primed with mealed powder, and spirits of wine.
When complete, the length of the pound rocket is 15½ inches; and the
half-pound 12 inches; the weight of the pound rocket and stick is 1¾
lb.; and the half-pound, 13 oz.
_Length of sticks for rockets._
1-Pounder rocket 8 feet.
Half-pounder 6 feet 4 inches.
_Star compositions._
No. 1.
Saltpetre, pulverized 8 lb.
Sulphur, sublimated 2 lb.
Antimony, pounded 2 lb.
Isinglass, dissolved 3½ oz.
Vinegar 1 quart.
Spirits of wine 1 pint.
2. _White stars._ Mealed powder, 4 oz.; saltpetre, 12 oz.; sulphur
vivum, 6 oz.; oil of spike, 2 oz.; camphor, 5 oz.
3. _Blue stars._ Mealed powder, 8 oz.; saltpetre, 4 oz.; sulphur, 2
oz.; spirits of wine, 2 oz.; oil of spike, 2 oz.
4. _Coloured, or variegated stars._ Mealed powder, 8 drs.; rockpetre,
4 oz.; sulphur vivum, 2 oz.; camphor, 2 oz.
5. _Brilliant stars._ Saltpetre, 3½ oz.; sulphur, 1½ oz.; mealed
powder, ¾ oz., worked up with spirits of wine.
6. _Common stars._ Saltpetre, 1 lb.; sulphur, 4 oz.; antimony, 4¾
oz.; isinglass, ½ oz.; camphor, ½ oz.; spirits of wine, ¾ oz.
7. _Tailed stars._ Mealed powder, 3 oz.; sulphur, 2 oz.; saltpetre, 1
oz.; charcoal, coarsely ground, ¾ oz.
8. _Drove stars._ No. 1. Saltpetre, 3 lb.; sulphur, 1 lb.; brass
dust, 12 oz.; antimony, 3 oz.
No. 2. Saltpetre, 1 lb.; antimony, 4 oz.; sulphur, 8 oz.
9. _Fixed pointed stars._ Saltpetre, 8½ oz.; sulphur, 2 oz.;
antimony, 1 oz. 10 drs.
10. _Stars of a fine colour._ Sulphur, 1 oz.; mealed powder, 1 oz.;
saltpetre, 1 oz.; camphor, 4 drs.; oil of turpentine, 4 drs.
The dry ingredients are well mixed, and sifted through a hair sieve:
the isinglass dissolved over a fire with vinegar, and the spirits
of wine afterwards added, and with which the dry composition is
thoroughly mixed. It is then formed on moulds, and a hole is left
in the middle to assist its ignition. Thirty-six stars are put in a
1-pounder, and twenty-two in a half-pounder rocket.
A great variety of figures may be represented in the air by attaching
to a large rocket several small rockets, or small cases filled with
the composition; or serpents may be attached to the rocket by means
of packthread.
SALTPETRE, PULVERIZED.
Sixteen pounds of refined lakepetre are put into a copper vessel, to
which four quarts of water are added. It is placed over a charcoal
fire to boil; as the water evaporates it is well stirred with
copper-shod spatulas or paddles, occasionally taking it off the fire,
until the evaporation ceases; and when brought to a fine powder it is
sifted through a hair sieve, and spread on paper to cool.
_To extract saltpetre from damaged gunpowder._
Dissolve the powder in warm water, filter the solution through fine
linen bags, and then evaporate the water by boiling it, until the
solution is of sufficient strength to crystallize.
SERPENTS, OR SQUIBS.
The case is made by rolling stout cartridge paper in slips of 6 or
8 inches in breadth three times round a former, and pasting down
the last fold. The case, having been choked at one end, is filled
about two-thirds with the composition, and a small piece of paper
is inserted, over which powder is placed, and this end is secured
with twine. At the other extremity, moist powder with touch-paper is
inserted. To introduce the composition into the case, a quill, cut
into the form of a spoon, may be employed, and the composition must
be rammed down by a few strokes of a mallet on an iron rod tipped
with brass.
_Composition._
1. lb. oz. | 2. lb. oz.
Mealed powder 1 8 | 1 0
Charcoal 4 | 1
Sulphur 1 | 0
Saltpetre 3 | 1¾
SHELLS, OR AERIAL GLOBES.
These globes are made of wood, and their thickness is equal to about
a twelfth part of their diameters. The usual charge is an ounce of
powder for a shell of 4 lb. weight, and 2 ounces for a shell of 8 lb.
They may be fired from any mortars that have not a chamber.
_To form the shell._
Two wooden hemispheres (with a fuze hole) are joined firmly together,
enclosing stars, squibs, rain, &c. A small quantity of powder is
inserted to explode the shell, by means of a fuze.
SHOWERS OF FIRE.
To form a shower of fire, mould small paper cases on an iron rod,
two-tenths of an inch in diameter, and 2½ inches in length. They must
not be choked, as it will be sufficient to twist the end of the case,
and having put the rod into it, beat it to make it assume its form.
When the cases are filled (which is done by immersing them in the
composition) fold down the other end, and then apply a match. They
must be fixed on a frame with leaders, to be fired simultaneously.
_Compositions._
_Chinese fire._ Mealed powder, 1 lb.; sulphur, 2 oz.; iron filings, 5
oz.
_Ancient fire._ Mealed powder, 1 lb.; charcoal, 2 oz.
_Brilliant fire._ Mealed powder, 1 lb.; iron filings, 4 oz.
The Chinese fire is the best of the above compositions.
SPECKIE.
Brilliant, and diversified displays of fireworks may be readily
exhibited by means of speckie, or lances. Illuminated designs of
figures, &c., are represented by affixing on a black board small
cases filled with various coloured compositions, to which leaders
must be attached. The cases are fastened on with glue, and red lead
mixed together.
_Compositions._
+--------------------+-------+-------+-------+-------+-------+
| | White.|Yellow.| Blue. |Yellow.|Yellow.|
+--------------------+-------+-------+-------+-------+-------+
| |lb. oz.|lb. oz.|lb. oz.|lb. oz.|lb. oz.|
| Saltpetre | 2 4 | 8 | 1 2 | 1 | 2 |
| Sulphur | 8 | 1 | 4 | 8 | ½ |
| Mealed powder | 4 | 12 | | 10 | 8 |
| Antimony, crude | 4 | | | | |
| Gum Succum | | 8 | | | |
| Antimony, prepared | | | 8 | | |
| Basket salt | | | | 6 | |
| Gamboge | | | | | 3 |
+--------------------+-------+-------+-------+-------+-------+
Stars, crosses, revolving suns, &c., being fixed on the wooden frame
with the speckie, will form a brilliant display.
SPUR FIRE.
_Compositions._
1. lb. oz. | 2. lb. oz.
Saltpetre 4 8 | 1
Sulphur 2 | 8
Lamp black 1 8 | 4 quarts.
The saltpetre and sulphur must be first sifted together, and then
put into a marble mortar with the lamp black. These ingredients must
be thoroughly mixed with a wooden pestle. The composition, if rubbed
too much, will be too fierce, and hardly show any stars; and on the
contrary, when not mixed enough, will be too weak, and throw out an
obscure smoke, and lumps of dross without any stars. This composition
is generally rammed in 1, or 2 ounce cases, about five, or six inches
long, but not driven very hard. Cases filled with spur fire may be
used in rooms without any danger of setting fire to the furniture,
&c., and some of them being placed round a transparent pyramid of
paper, and fired in a large room, make a very pretty appearance.
SUNS, OR WHEELS, FIXED, AND MOVEABLE.
None of the pyrotechnic inventions can be applied with so much
success in artificial fireworks, as suns, or wheels, of which there
are two kinds, fixed, and revolving.
FIXED SUNS.
Construct a circular piece of wood, into the circumference of which
screw 12 or 15 pieces in the form of radii, and to these attach jets
of fire, the mouth of each of which must be towards the circumference
of the frame; and leaders being affixed to all the jets, they will,
when ignited, produce the appearance of a radiated sun. The wheel is
fixed vertically. The jets may be arranged so as to cross each other
in an angular manner, in which case a _star_, or _cross of Malta_
will be formed. To produce a very brilliant effect, these suns may be
made with several rows of jets.
REVOLVING SUNS.
Provide a wooden wheel of the requisite size, and bring it into
perfect equilibrium around its centre, in order that the least
effort may make it turn round. Attach to the circumference of it
jets placed in the direction of the circumference; and affix leaders
of match to communicate the fire from jet to jet, according as may
be required. When fire is applied to one of the jets, the recoil
will immediately cause the wheel to revolve, unless it should be too
ponderous or large: therefore, when these suns are intended to be
of a considerable size, that is, when they consist of twenty jets,
fire must be communicated at the same time to the 1st, 6th, 11th, and
16th, from which it will proceed to the 2nd, 7th, 12th, and 17th, and
so on. Four jets will thus make the wheel revolve rapidly. If two
similar suns be placed one behind the other, and made to turn round
in a contrary direction, they will produce a very brilliant cross
fire.
For a sun 5 feet in diameter, the cases should be 8 oz., filled about
10 inches in length with composition.
_Compositions._
_Slow fire._
oz.
Saltpetre 4
Sulphur 2
Mealed powder 1½
_Illumination fire._
lb. oz.
Saltpetre 1 0
Sulphur 8
Mealed powder 6
_Dead fire._
oz.
Saltpetre 1¼
Sulphur ¼
Lapis Calemanaris ¼
Antimony ⅛
_Golden colour._
lb. oz.
Mealed powder 1 0
Charcoal, very good 2
_Brilliant fire._
lb. oz.
Mealed powder 6 0
Saltpetre 8
Sulphur 2
Iron Sand 12
_Red Chinese fire._
lb. oz.
Mealed powder 1 0
Saltpetre 1 0
Charcoal 4
Sulphur 4
Iron sand 2nd }
and 3rd order } 14
_White Chinese fire._
lb. oz.
Mealed powder 1 0
Saltpetre 1 0
Sulphur 8
Iron sand, 2nd & 3rd order 14
_Grey colour._
lb. oz.
Mealed powder 1 0
Saltpetre 4
Sulphur 2
Charcoal 1¼
Four ounce cases will be required for wheels of 14, or 16 inches;
if the wheels are larger, 8 oz., 1 lb., or even 2 lb. cases will be
required.
The Chinese compositions are intended for cases of nine-tenths of an
inch interior diameter, but they will be found to answer for cases as
low as four ounces.
TOUCH PAPER.
Dissolve saltpetre in spirits of wine, or water, more or less of the
saltpetre, according as the paper is to burn fast, or slow: then dip
into the solution blue paper, which, when well saturated, take out,
and dry for use. The touch-paper must be cut into slips, placed once
round the mouth of the firework, and the end of the paper outside the
case should be twisted to a point.
WHEELS, PIN, OR CATHERINE.
The pipe or case is made on a long wire former, about
three-sixteenths of an inch in diameter, into which the composition
is poured through a funnel, and shaken down. The case is then rolled
round a small circle of wood about one inch in diameter, and not more
than half an inch thick, with a hole through the centre of it for a
nail, or pin. One end of the case is to be pasted round the wood, and
each half turn of it secured with sealing-wax, or a strip of paper
pasted across the wheel. The end is then primed.
_Composition._
Mealed powder, 12 oz.; saltpetre, 3 oz.; sulphur, 1½ oz.
Two ounces of iron sand, or camphor, may be added, but it keeps
better without either.
PART XII.
MATHEMATICS.
_Mathematics_ is the science which treats of all kinds of quantity
whatever, that can be numbered, or measured.
_Arithmetic_ is that part which treats of numbering.
_Fractions_ treat of broken numbers, or parts of numbers.
_Algebra_ is the art of computing by symbols.
In this science, quantities of all kinds are represented by the
letters of the alphabet.
_Geometry_ is the science relating to measurement. By the assistance
of geometry, engineers, &c., conduct all their works, take the
distances of places, and the measure of inaccessible objects, &c.
_Characters, marks, or signs_,
which are used in arithmetic, and algebra, to denote several of the
operations, and propositions:
+ _signifies_ plus, or addition,
- minus, or subtraction,
× multiplication,
÷ division,
: :: : proportion,
= equality,
√ square root,
∛ cube root,
4^2 _denotes that_ 4 is to be squared.
4^3 _denotes that_ 4 is to be cubed.
ARITHMETIC.
REDUCTION.
_Reduction_ is the method of converting numbers from one name, or
denomination to another: or the method of finding the value of a
quantity in terms of some other higher, or lower quantity.
_To reduce from a higher to a lower denomination._
_Rule._—Multiply the given number by as many of the lower
denomination as make one of the greater;[40] adding to the product as
many of the lower denomination as are expressed in the given sum.
_Example._—In £6 15_s._ 5_d._, how many pence?
£. _s._ _d._
6 15 5
20
----
135
12
----
1625 _Answer._
----
_To convert from a lower to a higher denomination._
_Rule._—Divide the given number by as many of the lower denomination
as are required to make one of the greater.[41] Should there be any
remainder, it will be of the same denomination as the dividend.
_Example._—Convert 1625 pence into pounds, shillings, and pence.
12) 1625 pence
----
20) 135 5
--------
£6 15_s._ 5 _d._ _Answer._
THE RULE OF THREE, OR SIMPLE PROPORTION.
It is called the _Rule of Three_ because three numbers are given to
find a fourth. It is also called _Simple Proportion_, because the 1st
term bears the same proportion to the 2nd, as the 3rd does to the
4th. Of the three given numbers, two of them are always of the same
kind, or name, and are to be the 1st, and 2nd terms of the question;
the 3rd number is always of the same name, or kind as the 4th, or
answer sought; and in stating the question it is always to be made
the 3rd term. If the answer will be _greater_ than the 3rd term,
place the least of the other two given quantities for the 1st term;
but if the answer will be _less_ than the 3rd term, put the greater
of the two numbers, or quantities, for the 1st term.
_Rule._—State the question according to the above directions, and
multiply the 2nd and 3rd terms together, and divide this product by
the 1st, for the 4th term, or answer sought.
If the 1st and 2nd terms are not of the same denomination, they must
be reduced to it; and if the third term is a compound number, it must
be reduced to its lowest denomination before the multiplication, or
division of the term takes place.
_Note 1._—The operation may frequently be considerably abridged, by
dividing the 1st and 2nd, or the 1st and 3rd terms, by any number
which will exactly divide them, afterwards using the quotients,
instead of the numbers themselves.*
_Example._—If 2 tons of iron for ordnance cost £40, how many tons may
be bought for £360?
As £40 : £360 :: 2 tons : 18 tons.
(Thus 360 × 2) ÷ 40 = 18. The answer.
* Or thus, 9 × 2 = 18. The answer.
_Note 2._—_A concise method of ascertaining the annual amount of a
daily sum of money._
_Rule._—Bring the daily sum into pence, and then add together as many
pounds, half pounds, groats, and pence, as there are pence in the
daily sum, for the amount required. For leap year, add the rate for
one day.
_Example._—Required the annual amount of 2_s._ 6_d._ per diem.
2_s._ 6_d._ = 30_d._ 30 pounds.
15 = 30 half pounds.
10_s._ = 30 groats.
2_s._ 6_d._ = 30 pence.
-----------------------------
Annual amount (365 days) ... £45 12_s._ 6_d._
_Note 3._—To find the amount of any number of days’ pay, the daily
rate (under twenty shillings) being given.
The price of any article being given, the value of any number may be
ascertained in a similar manner.
_Rule 1._ When the rate (or price) is an even number, multiply the
given number by half of the rate, doubling the first figure to the
right hand for the shillings, the remainder of the product will be
pounds.
_Example._ Required the amount of 243 days’ pay, at 4_s._ per diem.
4/2 = 2 243
2
-------
£48 12_s._ _Ans._
_Rule 2._ When the price is an odd number, find for the greatest
number as before, to which add one-twentieth of the given number for
the odd shilling.
_Example._ What is the price of 566 pairs of shoes, at 7_s._ per pair.
566 2/0 ) 56/6
3 -------
-------- 28 6
169 16_s._ -------
28 6
--------
£198 2_s._ _Ans._
FRACTIONS.
_A fraction_ is a quantity which expresses a part, or parts of a
unit, or integer. It is denoted by two numbers placed with a line
between them.
_A Simple fraction_ consists of two numbers, called the numerator,
and denominator; thus,
3 numerator,
--
5 denominator.
_The Denominator_ is placed below the numerator, and expresses the
number of equal parts into which the integer is divided.
_The Numerator_ expresses the number of parts of the broken unit, or
integer; or shows how many of the parts of the unit are expressed by
the fraction.
_A Compound fraction_ is a fraction of a fraction, as ⅔ of ⅘.
_A Mixed number_ consists of a whole number with a fraction annexed
to it, as 4⅖.
_An Improper fraction_ has the numerator greater than the
denominator, as 6/5.
REDUCTION OF FRACTIONS.
is bringing them from one denomination to another.
_To reduce a fraction to its lowest terms._
_Rule._—Divide the numerator, and the denominator, by any number that
exactly divides them, and the quotients by any other number, till
they can be no longer divided by any whole number, when the fraction
will be in its lowest terms.
_Example._—Reduce 4032/6048 to its lowest terms.
Thus, (4)4032/6048 = (12)/1008/1512 = (6)/84/126 = (7)/14/21 = 2/3.
_Answer._
_To reduce an improper fraction to a whole, or mixed number._
_Rule._—Divide the numerator by the denominator, the quotient will
be the whole number; and the remainder (if any) the numerator of the
fraction, having the divisor for the denominator.
_Example._—Reduce 114/12 to a whole, or mixed number.
12 ) 114
--------
9-6/12 _Answer._
_To reduce a mixed number to an improper fraction._
_Rule._—Multiply the whole number by the denominator, and add the
numerator to the product, under which place the given denominator.
_Example._—Reduce 17⅝ to an improper fraction.
17⅝
8
---
141
---
8 _Answer._
_To reduce a compound fraction to a simple fraction._
_Rule._—Multiply all the numerators together for the numerator, and
all the denominators for the denominator.
_Example._—Reduce ⅜ of ⅙ of ½ of 9 to a simple fraction.
Numerators 3 × 1 × 1 × 9 27 9
----- ----- = -- = -- _Answer._
Denominators 8 × 6 × 2 × 1 96 32
_To reduce fractions of different denominators to equivalent
fractions, having a common denominator._
_Rule._—Multiply each numerator by all the denominators except
its own for the new numerators, and multiply all the denominators
together for a common denominator.[42]
_Example._—Reduce ⅜, ⅔, and ⅘ to fractions having a common
denominator.
3 × 3 × 5 = 45
2 × 8 × 4 = 80
4 × 8 × 3 = 96
8 × 3 × 5 = 120 _Answer_, 45/120, 80/120, and 96/120
ADDITION OF FRACTIONS.
_Rule._—Bring compound fractions to simple fractions; reduce all
the fractions to a common denominator, then add all the numerators
together, and place their sum over the common denominator. When mixed
numbers are given, find the sum of the fractions, to which add the
whole numbers.
_Example._—Add together ⅚, ¾, and 6½.
5 × 4 × 2 = 40 40/48 + 36/48 + 24/48 + 6 = 8-4/48
3 × 6 × 2 = 36 or, by cancelling, and dividing,[43]
1 × 6 × 4 = 24 10/12 + 9/12 + 6/12 + 6 = 8-1/12. _Answer._
6 × 4 × 2 = 48
SUBTRACTION OF FRACTIONS.
_Rule._—Prepare the quantities, as in addition of fractions. Place
the less quantity under the greater. Then, if possible, subtract the
lower numerator from the upper; under the remainder write the common
denominator, and, if there be whole numbers, find their difference as
in simple subtraction. But if the lower numerator exceed the upper,
subtract it from the common denominator, and to the remainder add the
upper numerator; write the common denominator under this sum, and
carry 1 to the whole number in the lower line.
_Example._— From 54-5/6 or 54-25/30
Take 25-5/15 or 25-10/30
-------
29-15/30 _Answer._
MULTIPLICATION OF FRACTIONS.
_Rule._—Reduce mixed numbers to equivalent fractions; then multiply
all the numerators together for a numerator, and all the denominators
together for a denominator, which will give the product required.
_Example._—Multiply ⅚, ⅜, and 2½ together.
⅚ × ⅜ × (2½ or) 5/2 = 75/96 _Answer._
DIVISION OF FRACTIONS.
_Rule._—Prepare the fractions, as for multiplication; then divide the
numerator by the numerator, and the denominator by the denominator,
if they will exactly divide; but if they will not do so, then invert
the terms of the divisor, and multiply the dividend by it, as in
multiplication.
_Example._—Divide 9/16 by 4½.
9/16 ÷ (4½ or) 9/2 = ⅛ _Answer._
RULE OF THREE IN FRACTIONS.
_Rule._—State the terms, as directed in “Simple proportion;” reduce
them (if necessary) to improper, or simple fractions, and the _two
first_ to the same denomination. Then multiply together the second
and third terms, and the first with its parts inverted, as in
division, for the answer.
_Example._—If 4⅕ cwt. of sugar cost £19⅞, how much may be bought for
£59⅝?
As 19⅞ : 59⅝ :: 4⅕
Or, 159/8 : 477/8 :: 21/5 : 12⅗ _Answer._
8/159 × 477/8 × 21/5 = 80136/6360 = 12⅗ cwt.
DECIMALS.
_A decimal fraction_ is that which has for its denominator an unit
(1), with as many ciphers annexed as the numerator has places; and it
is usually expressed by setting down the numerator only, with a point
before it, on the left hand. Thus, 5/10 is ·5; 25/100 is ·25; 25/1000
is ·025; ciphers being _prefixed_, to make up as many places as are
required by the ciphers in the denominator.
_A mixed number_ is made up of a whole number with some decimal
fraction, the one being separated from the other by a point, thus
3·25 is the same as 3-25/100 or 325/100.
_Ciphers on the right hand of decimals_ make no alteration in their
value; for ·5, ·50, ·500 are decimals having all the same value,
each being = 5/10. But when they are placed on the left hand, they
decrease the value in a tenfold proportion; thus, ·5 is 5/10; but ·05
is 5/100.
ADDITION OF DECIMALS.
_Rule._—Set the numbers under each other, according to the value of
their places, in which state the decimal separating points will all
stand exactly under each other. Then beginning at the right hand, add
up all the columns of numbers as in integers, and point off as many
places for decimals as are in the greatest number of decimal places
in any of the lines that are added; or place the point directly below
all the other points.
_Example._—Required the sum of 29·0146, 3146·5, 14·16, and 165.
29·0146
3146·5
14·16
165·
---------
_Answer_ 3354·6746
SUBTRACTION OF DECIMALS.
_Rule._—Place the numbers under each other according to the value of
their places. Then, beginning at the right hand, subtract as in whole
numbers, and point off the decimals, as in addition.
_Example._—Subtract 4·90142 from 214·81.
214·81
4·90142
---------
_Answer_ 209·90858
MULTIPLICATION OF DECIMALS.
_Rule._—Place the factors, and multiply them together, the same as if
they were whole numbers. Then point off in the product just as many
places of decimals as there are decimals in both the factors. But, if
there be not so many figures in the product, prefix ciphers to supply
the deficiency.[44]
_Example._—Multiply 32·108 by 2·5.
32·108
2·5
------
160540
64216
-------
80·2700 _Answer._
DIVISION OF DECIMALS.
_Rule._—Divide as in whole numbers, and point off in the quotient as
many places for decimals as the decimal places in the dividend exceed
those in the divisor. When the decimal places of the quotient are not
so many as the above rule requires, the deficiency is to be supplied
by prefixing ciphers. When there is a remainder after the division,
or when the decimal places in the divisor are more than those in
the dividend, then ciphers may be annexed to the dividend, and the
quotient carried on as far as required.
_Example._—Divide 234·7052 by 64·25.
64·25 ) 234·7052 ( 3·65 _Answer._
19275
------
41955
38550
------
34052
32125
-----
1927 _Remainder._
REDUCTION OF DECIMALS.
_To reduce a vulgar fraction to its equivalent decimal._
_Rule._—Divide the numerator by the denominator, as in Division of
Decimals, annexing ciphers to the numerator as far as necessary: and
the quotient will be the decimal required.
_Example._—Reduce 7/24 to a decimal.
24 = 4 × 6. Then 4)7·
-------
6)1·75
-------
·291666, &c.
_To find the value of a decimal, in terms of the inferior
denominations._
_Rule._—Multiply the decimal by the number of parts in the next lower
denomination, and cut off as many places to the right hand for a
remainder, as there are places in the given decimal. Multiply that
remainder by the parts in the next lower denomination, again cutting
off for another remainder as before. Proceed in the same manner
through all the parts of the integer; then the several denominations,
separated on the left hand, will make up the answer.
_Example._—What is the value of ·775 pounds sterling.
·775
20
------
Shillings 15·500
12
-----
Pence 6·000 _Answer_ 15_s._ 6_d._
_To convert integers, or decimals to equivalent decimals of higher
denominations._
_Rule._—Divide by the number of parts in the next higher
denomination, continuing the operation to as many higher
denominations as may be necessary.
_When there are several numbers, all to be converted to the decimal
of the highest_—
Set the given numbers directly under each other for dividends,
proceeding from the lowest to the highest; opposite to each dividend,
on the left hand, place such a number for a divisor as will bring it
to the next higher name. Begin at the uppermost, and perform all the
divisions, placing the quotient of each division, as decimal parts,
on the right hand of the dividend next below it; so shall the last
quotient be the decimal required.
_Example._—Convert 15_s._ 9¾_d._ to the decimal of a pound sterling.
4 | 3·
12 | 9·75
20 | 15·8125
£·790625 _Answer._
_Example._—Convert 1 dwt. to the decimal of a pound, Troy weight.
20 ) 1
-------
12 ) ·05 oz.
-----------
·004166 lb., &c., _Answer._
RULE OF THREE IN DECIMALS.
_Rule._—Prepare the terms, by reducing the fractions to decimals;
compound numbers to decimals of the higher denominations, or integers
of the lower; also the first, and second terms to the same name. Then
multiply, and divide, as in the Rule of Three, in whole numbers.
_Example._—If ⅜ of a yard of cloth cost £⅖, what will 5/16 of a yard
cost?
yd. yd. £. _s. d._
⅜ = ·375 As ·375 : ·3125 :: 4 : ·333 &c. or 6 8
4
------
⅖ = ·4 ·375)·12500 (·3333 &c.
1125 20
--------------
5/16 = ·3215 1250_s._ 6·666 &c.
1125 12
--------------
_Answer_, 6_s._ 8_d._ 125_d._ 7·999 &c. nearly 8_d._
DUODECIMALS.
By Duodecimals, artificers, &c., compute the content of their works.
_Rule._—Set down the two dimensions to be multiplied together one
under the other, so that feet may stand under feet, inches under
inches, &c.
Multiply each term in the multiplicand, beginning at the lowest, by
the feet in the multiplier, and set the result of each straight under
its corresponding term, observing to carry 1 for every 12, from the
inches to the feet. In like manner multiply all the multiplicand
by the inches, and parts of the multiplier, and set the result
of each term one place removed to the right hand of those in the
multiplicand: omitting, however, what is below parts of inches,
only carrying to these the proper number of units from the lowest
denominations. Or, instead of multiplying by the inches, take such
part of the multiplicand as those are of a foot.
Then add the two lines together for the content required.
_Example._—Multiply 14 feet 9 inches, by 4 feet 6 inches.
ft. in.
14 9
4 6
-------
59 0
7 4½
-------
66 4½ _Answer._
-------
TABLES OF WEIGHTS, AND MEASURES.
TROY WEIGHT.
24 grains 1 pennyweight.
480 20 1 ounce.
5760 240 12 1 pound.
AVOIRDUPOIS WEIGHT.
16 drams 1 ounce.
256 16 1 pound.
7168 448 28 1 quarter.
28672 1792 112 4 1 hundred weight.
573440 35840 2240 80 20 1 ton.
_Note._—1 lb. Avoirdupois weight equals 14 oz. 11 dwts. 15½ grs. Troy.
1 oz. ditto 18 dwts. 5½ do.
1 dr. ditto 27·34375 do.
APOTHECARIES’ WEIGHT.
20 grains 1 scruple.
60 3 1 dram.
480 24 8 1 ounce.
5760 288 96 12 1 pound.
WEIGHTS.
_To find the weight, for tonnage._
_Cattle_—
Divide the number by 3, for weight in tons.
_Sheep_ Average 60 lb. each.
Divide by 33, for weight in tons.
_Pigs_ Average 80 lb.
Divide by 15, for tons.
_Beer, or Ale_—
Barrel 3¼ cwt.
Hogshead 5¼ cwt.
_Oats_ Sack—24 stone.
Divide quarters by 5, for tons.
_Rum_—
Divide gallons by 224, for tons.
_Wine_ Cask—12 cwt.
_Rule for ascertaining the weight of Hay._
Measure the length and breadth of the stack; then take its height
from the ground to the eaves, and add to this last one-third of the
height from the eaves to the top: Multiply the length by the breadth,
and the product by the height, all expressed in feet; divide the
amount by 27, to find the cubic yards, which multiply by the number
of stones supposed to be in a cubic yard (viz., in a stack of new
hay, six stones; if the stack has stood a considerable time, eight
stones; and if old hay, nine stones), and you have the weight in
stones. For example, suppose a stack to be 60 feet in length, 30 in
breadth, 12 in height from the ground to the eaves, and 9 (the third
of which is three) from the eaves to the top; then 60 × 30 × 15 =
27000; 27000 ÷ 27 = 1000; and 1000 × 9 = 9000 stones of old hay.
LONG MEASURE.
12 inches 1 foot.
36 3 1 yard.
198 16½ 5½ 1 pole, perch, or rod.
7920 660 220 40 1 furlong.
63360 5280 1760 320 8 1 mile.
LAND MEASURE (_Length_).
7·92 inches 1 link.
100 links, or 22 yards 1 chain.
80 chains 1 mile.
69·121 miles 1 geographical degree.
LAND MEASURE (_Surface, or Superficial_).
62·7264 square inches 1 square link.
625 square links 1 square pole, or perch.
10000 square links 1 square chain.
2500 square links 1 square rood, or pole.
10 square chains 1 square acre.
100000 square links 1 square acre.
NAUTICAL MEASURE.
1 nautical mile 6082·66 feet.
3 miles 1 league.
20 leagues 1 degree.
360 degrees the earth’s circumference.
SQUARE MEASURE.
144 s. inches 1 s. foot
1296 9 1 s. yard.
39204 272¼ 30¼ 1 s. pole.
1568160 10890 1210 40 1 rood.
6272640 43560 4840 160 4 1 acre.
CUBIC MEASURE (_Measure of solidity_).
1728 cubic inches 1 cubic foot.
27 cubic feet 1 cubic yard.
_Note._—A cubic foot is equal to 2200 cylindrical inches, or 3300
spherical inches, or 6600 conical inches.
_Timber._
40 feet of round, and 50 feet of hewn timber make 1 _Ton_; 16 cubic
feet make 1 _Foot_ of wood; 8 feet of wood make 1 _Cord_.
_Water._
Maximum density 42 deg. Fahrenheit.
1 cubic foot of water 6¼ imperial gallons.
1 cylindric foot do. about 5 do.
1 cubic foot weighs 62·5 lb. avoirdupois.
1 cylindric do. do. 49·1
1 lineal do. (1 in. square) do. ·434
12·2 imperial gallons weigh 1 cwt.
224 do. do. 1 ton.
1·8 cubic feet do. 1 cwt.
35·84 do. do. 1 ton.
MEASURES OF CAPACITY.
69⅓ cubic in 2 pints 1 quart.
277¼ 8 4 1 gallon.
554½ 16 8 2 1 peck.
2218⅕ 64 32 8 4 1 bushel.
10¼ cubic ft. 512 256 64 32 8 1 quarter.
FRENCH MEASURES.
English
cubic inches.
Millilitre ·06103
Centilitre ·61028
Decilitre 6·10279
Litre, or cubic decimetre 61·02791
Decalitre 610·27900
Hectolitre 6102·79000
Kylolitre 61027·90000
Myrialitre 610279·00000
1 litre is nearly 2⅛ wine pints.
1 kilolitre 1 tun 12¾ wine gallons.
1 stere, or cubic metre 35·3171
English
feet.
Metre 3·281
” French feet, 3·07844
Millimetre. ·03937
Centimetre ·39371
Decimetre 3·93708
Metre 39·37079
Decametre 393·70790
Hectometre 3937·07900
Kilometre 39370·79000
Myriametre 393707·90000
8 kilometres are nearly 5 miles.
1 inch is ·0254 metre.
100 feet are nearly 30·5 metres.
INVOLUTION.
_Involution_ is the raising of powers from any given number, as a
root.
A _Power_ is a quantity produced by multiplying any given number,
called the _Root_, a certain number of times continually by itself.
Thus, 2 × 2 = 4, the 2nd power, or square of 2, expressed thus, 2^2.
_The index, or exponent of a power_ is the number denoting the
height, or degree of that power. Thus, 2 is the index of the 2nd
power.
Powers that are to be raised, are usually denoted by placing the
index above the root, or first power.
Thus 2^2 = 4, the 2nd power of 2.
_Example._—What is the 2nd power of 45?
45 × 45 = 2025 _Answer._
EVOLUTION.
_Evolution_ is the reverse of Involution, being the extracting, or
finding the roots of any given powers, or numbers.
_The Root_ of any number, or power, is such a number as being
multiplied into itself a certain number of times, will produce that
power.
Thus, 2 is the square root, or 2nd root of 4, because, 2^2 = 2 × 2 =
4; and 3 is the cube root, or third root of 27. But there are many
numbers of which a proposed root can never be exactly found; by means
of decimals, however, the root may be very nearly ascertained.
_Any power of a given number, or root_, may be found exactly by
multiplying the number continually into itself.
Those roots which only approximate are called _Surd-roots_; but those
which can be found, quite exactly, are called _Rational-roots_. Thus,
the square root of 3 is a surd root, but the square root of 4 is a
rational root, being equal to 2; also the cube root of 8 is rational,
being equal to 2, but the cube root of 9 is surd, or irrational.
Roots are sometimes denoted by writing the character √ before the
power with the index of the root against it. Thus, the 3rd, or cube
root of 20 is expressed by ∛20. When the power is expressed by
several numbers with the sign + or - between them, a line is drawn
from the top of the sign over all the parts of it; thus the cube (or
third) root of 45 - 12 is ∛(45 - 12) or thus ∛(45 - 12).
TO EXTRACT THE SQUARE ROOT.
_Rule._—Divide the given number into periods of two figures each,
by setting a point over the place of units, and another over the
place of hundreds, and so on over every second figure, both to
the left hand in integers, and right hand in decimals. Find the
greatest square in the first period on the left hand, and set its
root on the right hand of the given number, after the manner of
the quotient figure in division. Subtract the square thus found
from the said period, and to the remainder annex the two figures
of the next following period for a dividend. Double[45] the root
above-mentioned for a divisor, and find how often it is contained in
the said dividend, exclusive of its right-hand figure; and set that
quotient figure both in the quotient, and divisor. Multiply the whole
augmented divisor by this last quotient figure, and subtract the
product from the said dividend, bringing down to it the next period
of the given number, for a new dividend. Repeat the same process over
again—viz., find another new divisor, by doubling all the figures now
found in the root; from which, and the last dividend find the next
figure of the root as before; and so on through all the periods to
the last.
_To extract the square root of a fraction, or mixed number._
Reduce the fraction to a decimal, and extract its root.
Mixed numbers may be either reduced to improper fractions, and the
root extracted; or the fraction may be reduced to a decimal, then
joined to the integer, and the root of the whole extracted.
_Example._—To find the square root of 29506624.
29506624 ( 5432 The Root.
25
--------
104 | 450
4 | 416
-------------
1083 | 3466
3 | 3249
-------------
10862 | 21724
2 | 21724
TO EXTRACT THE CUBE ROOT.
_Rule 1._—By trials, or by the table of roots (_vide page_ 280), take
the nearest rational cube to the given number, whether it be greater,
or less, and call it the assumed cube.
2.—Then (_by the Rule of Three_),
As the sum of the given number, and double the assumed cube, is to
the sum of the assumed cube, and double the given number, so is the
root of the assumed cube, to the root required, nearly.
3.—Or as the first sum,
is to the difference of the given, and assumed cube,
so is the assumed root,
to the difference of the roots, nearly.
4.—Again, by using, in like manner, the cube of the root last found
as a new assumed cube, another root will be obtained still nearer.
Repeat this operation as often as necessary, using always the cube of
the last-found root, for the assumed root.
_Example._—To find the cube root of 21035·8.
By trials it will be found _first_, that the root lies between 20,
and 30; and, _secondly_, between 27, and 28. Taking, therefore, 27,
its cube is 19683, which will be the assumed cube. Then by No. 2 of
the Rule
19683 21035·8
2 2
----- --------
39366 42071·6
21035·8 19683·
------ --------
As 60401·8 : 61754·6 :: 27 : 27·6047 the Root, nearly.
Again for a second operation, the cube of this root is
21035·318645155832, and the process by No. 3 of the Rule will be
21035·318645, &c.
2
------------
42070·637290 21035·8
21035·8 21035·318645, &c.
------------ ------------
As 63106·43729 : diff. ·481355 :: 27·6047 :
: the diff. ·000210560
----------
consequently the root required is 27·604910560
TABLE OF SQUARES, CUBES, AND ROOTS.
+----+-------+---------+------------+----------+
| No.| Sqr. | Cube. | Sqr. root. |Cube root.|
+----+-------+---------+------------+----------+
| 1 | 1 | 1 | 1·0000000 | 1·000000 |
| 2 | 4 | 8 | 1·4142136 | 1·259921 |
| 3 | 9 | 27 | 1·7320508 | 1·442250 |
| 4 | 16 | 64 | 2·0000000 | 1·587401 |
| 5 | 25 | 125 | 2·2360680 | 1·709976 |
| 6 | 36 | 216 | 2·4494897 | 1·817121 |
| 7 | 49 | 343 | 2·6457513 | 1·912933 |
| 8 | 64 | 512 | 2·8284271 | 2·000000 |
| 9 | 81 | 729 | 3·0000000 | 2·080084 |
| 10 | 100 | 1000 | 3·1622777 | 2·154435 |
| 11 | 121 | 1331 | 3·3166248 | 2·223980 |
| 12 | 144 | 1728 | 3·4641016 | 2·289428 |
| 13 | 169 | 2197 | 3·6055513 | 2·351335 |
| 14 | 196 | 2744 | 3·7416574 | 2·410142 |
| 15 | 225 | 3375 | 3·8729833 | 2·466212 |
| 16 | 256 | 4096 | 4·0000000 | 2·519842 |
| 17 | 289 | 4913 | 4·1231056 | 2·571282 |
| 18 | 324 | 5832 | 4·2426407 | 2·620741 |
| 19 | 361 | 6859 | 4·3588989 | 2·668402 |
| 20 | 400 | 8000 | 4·4721360 | 2·714418 |
| 21 | 441 | 9261 | 4·5825757 | 2·758923 |
| 22 | 484 | 10648 | 4·6904158 | 2·802039 |
| 23 | 529 | 12167 | 4·7958315 | 2·843867 |
| 24 | 576 | 13824 | 4·8989795 | 2·884499 |
| 25 | 625 | 15625 | 5·0000000 | 2·924018 |
| 26 | 676 | 17576 | 5·0990195 | 2·962496 |
| 27 | 729 | 19683 | 5·1961524 | 3·000000 |
| 28 | 784 | 21952 | 5·2915026 | 3·036589 |
| 29 | 841 | 24389 | 5·3851648 | 3·072317 |
| 30 | 900 | 27000 | 5·4772256 | 3·107232 |
| 31 | 961 | 29791 | 5·5677644 | 3·141381 |
| 32 | 1024 | 32768 | 5·6568542 | 3·174802 |
| 33 | 1089 | 35937 | 5·7445626 | 3·207534 |
| 34 | 1156 | 39304 | 5·8309519 | 3·239612 |
| 35 | 1225 | 42875 | 5·9160798 | 3·271066 |
| 36 | 1296 | 46656 | 6·0000000 | 3·301927 |
| 37 | 1369 | 50653 | 6·0827625 | 3·332222 |
| 38 | 1444 | 54872 | 6·1644140 | 3·361975 |
| 39 | 1521 | 59319 | 6·2449980 | 3·391211 |
| 40 | 1600 | 64000 | 6·3245553 | 3·419952 |
| 41 | 1681 | 68921 | 6·4031242 | 3·448217 |
| 42 | 1764 | 74088 | 6·4807407 | 3·476027 |
| 43 | 1849 | 79507 | 6·5574385 | 3·503398 |
| 44 | 1936 | 85184 | 6·6332496 | 3·530348 |
| 45 | 2025 | 91125 | 6·7082039 | 3·556893 |
| 46 | 2116 | 97336 | 6·7823300 | 3·583048 |
| 47 | 2209 | 103823 | 6·8556546 | 3·608826 |
| 48 | 2304 | 110592 | 6·9282032 | 3·634241 |
| 49 | 2401 | 117649 | 7·0000000 | 3·659306 |
| 50 | 2500 | 125000 | 7·0170678 | 3·684031 |
| 51 | 2601 | 132651 | 7·1414284 | 3·708430 |
| 52 | 2704 | 140608 | 7·2111026 | 3·732511 |
| 53 | 2809 | 148877 | 7·2801099 | 3·756286 |
| 54 | 2916 | 157464 | 7·3484692 | 3·779763 |
| 55 | 3025 | 166375 | 7·4161985 | 3·802953 |
| 56 | 3136 | 175616 | 7·4893148 | 3·825862 |
| 57 | 3249 | 185193 | 7·5498344 | 3·848501 |
| 58 | 3364 | 195112 | 7·6157731 | 3·870877 |
| 59 | 3481 | 205379 | 7·6811457 | 3·892996 |
| 60 | 3600 | 216000 | 7·7459667 | 3·914867 |
| 61 | 3721 | 226981 | 7·8102497 | 3·936497 |
| 62 | 3844 | 238328 | 7·8740079 | 3·957892 |
| 63 | 3969 | 250047 | 7·9372539 | 3·979057 |
| 64 | 4096 | 262144 | 8·0000000 | 4·000000 |
| 65 | 4225 | 274625 | 8·0622577 | 4·020726 |
| 66 | 4356 | 287496 | 8·1240384 | 4·041240 |
| 67 | 4489 | 300763 | 8·1853528 | 4·061548 |
| 68 | 4624 | 314432 | 8·2462113 | 4·081656 |
| 69 | 4761 | 328509 | 8·3066239 | 4·101566 |
| 70 | 4900 | 343000 | 8·3666003 | 4·121285 |
| 71 | 5041 | 357911 | 8·4261498 | 4·140818 |
| 72 | 5184 | 373248 | 8·4852814 | 4·160168 |
| 73 | 5329 | 389017 | 8·5440037 | 4·179339 |
| 74 | 5476 | 405224 | 8·6023253 | 4·198336 |
| 75 | 5625 | 421875 | 8·6602540 | 4·217163 |
| 76 | 5776 | 438976 | 8·7177979 | 4·235824 |
| 77 | 5929 | 456533 | 8·7749644 | 4·254321 |
| 78 | 6084 | 474552 | 8·8317609 | 4·272659 |
| 79 | 6241 | 493039 | 8·8881944 | 4·290841 |
| 80 | 6400 | 512000 | 8·9442719 | 4·308870 |
| 81 | 6561 | 531441 | 9·0000000 | 4·326749 |
| 82 | 6724 | 551368 | 9·0553851 | 4·344481 |
| 83 | 6889 | 571787 | 9·1104336 | 4·362071 |
| 84 | 7056 | 592704 | 9·1651514 | 4·379519 |
| 85 | 7225 | 614125 | 9·2195445 | 4·396830 |
| 86 | 7396 | 636056 | 9·2736185 | 4·414005 |
| 87 | 7569 | 658503 | 9·3273791 | 4·431047 |
| 88 | 7744 | 681472 | 9·3808315 | 4·447960 |
| 89 | 7921 | 704969 | 9·4339811 | 4·464745 |
| 90 | 8100 | 729000 | 9·4868330 | 4·481405 |
| 91 | 8281 | 753571 | 9·5393920 | 4·497942 |
| 92 | 8464 | 778688 | 9·5916630 | 4·514357 |
| 93 | 8649 | 804357 | 9·6436508 | 4·530655 |
| 94 | 8836 | 830584 | 9·6953597 | 4·546836 |
| 95 | 9025 | 857375 | 9·7467943 | 4·562903 |
| 96 | 9216 | 884736 | 9·7979590 | 4·578857 |
| 97 | 9409 | 912673 | 9·8488578 | 4·594701 |
| 98 | 9604 | 941192 | 9·8994949 | 4·610436 |
| 99 | 9801 | 970299 | 9·9498744 | 4·626065 |
|100 | 10000 | 1000000 | 10·0000000 | 4·641589 |
+----+-------+---------+------------+----------+
PILING OF SHOT, AND SHELL.
Shot, and shells, are usually piled in horizontal courses, the base
being either an equilateral triangle, a square, or a rectangle. The
triangular, and square piles terminate each in a single ball, but the
rectangular pile finishes in a row of balls.
_To find the number of balls in a complete pile._
_Rule._—Add the three parallel edges together; then one-third of the
product of that sum, and of the number of balls in the triangular
face, will be the number sought.
_Note 1._—_The parallel edges_ in a _rectangular pile_ are the two
rows in length at the base, and the upper ridge. In the _square pile_
the same, except that the upper row is only a single ball. In the
_triangular pile_, one side of the base, the single ball at top, and
that at the back, are considered the parallel edges.
_Note 2._—_The number of balls in the triangular face_ is found by
multiplying half the number in the breadth at the base, by the number
in the breadth at the base _plus_ 1.
_Note 3._—In all piles the breadth of the bottom is equal to the
number of courses. In the oblong pile, the top row is one more than
the difference between the length, and breadth of the bottom.
_Example._—To find the shot in a triangular pile, the bottom row
consisting of 12 shot.
[Illustration: (triangular pyramid)]
Parallel edges. { 12
{ 1 12 ÷ 2 = 6
{ 1 12 + 1 = 13
-------
--- Triangular face 78
3 ) 14 4⅔
--- ----
4⅔ 312
--- 52
----
_Answer_ 364
_Example._—To find the shot in a square pile, the bottom row
consisting of 12 shot.
[Illustration: (square pyramid)]
12 12 ÷ 2 = 6
12 12 + 1 = 13
1 -----
---- 78
3 ) 25 8⅓
---- -----
8⅓ 624
---- 26
-----
_Answer_ 650
_Example._—To find the shot in an oblong pile, whose base consists of
18 shot in length, and 12 in breadth.
[Illustration: (rectangular pyramid)]
18 18 - 12 = 6
18 1
7 --
---- 7
3 ) 43
----
14⅓ 12 ÷ 2 = 6
---- 12 + 1 = 13
----
78
14⅓
----
312
78
26
-----
_Answer_ 1118
-----
_Triangular pile._
_Rule._—Multiply the base by the base _plus_ 1, this product by the
base _plus_ 2, and divide by 6.
_Square pile._
_Rule._—Multiply the bottom row by the bottom row _plus_ 1, and this
product by twice the bottom row _plus_ 1, and divide by 6.
_Rectangular, or oblong pile._
_Rule._—Multiply the breadth of the base by itself _plus_ 1; and this
product by three times the length of the base _plus_ 1, _minus_ the
breadth of the base, and divide by 6.
_In the following formulæ let the letter_ (L) _denote the number in
the bottom row, or the length; and_ (B) _the breadth of the lowest
course_.
_Triangular pile_ (L × (L + 1) × (L + 2)) / 6
_Square pile_ (L × (L + 1) × (2L + 1)) / 6
_Oblong pile_ (B × (B + 1) × (3L + 1 - B)) / 6
_The number of shot in any pile,_
(whose base does not exceed 21) may readily be ascertained by
referring to _the following Table, page 284_.
_For the square pile._—Look for the number of shot in the base, in
the first vertical column on the left hand, and also in the diagonal
column; and at their angle of meeting will be found the content
required.
Thus 20 base gives 2870.
_For the triangular pile._—Look for the number in the base row in the
diagonal column, and opposite to it will be found the content.
Thus 18 base gives 1140.
_For the oblong pile._—Look for the number in the length of the base
in the vertical column, and the breadth of the base in the diagonal
column, and at their angle of meeting will be found the content
required.
Thus 17 length, and 12 breadth, gives 1040.
_To find the number of balls in an incomplete pile._
Compute the number in the pile considered as complete; also the
number in the upper pile, or part wanting; and the difference between
the two piles thus found will be the number in the frustrum, or
incomplete pile.
_Table for computing the Content of any Pile, whose base row does not
exceed 21 balls._
+--+--+--------------------------------------------------------------
| 1| 2| 4
| +---+
| 2| 5| 3| 10
| +---+
| 3| 8| 14| 4| 20|
| +---+
| 4|11| 20| 30| 5| 35
| +---+
| 5|14| 26| 40| 55| 6| 56
| +---+
| 6|17| 32| 50| 70| 91| 7| 84
| +---+
| 7|20| 38| 60| 85|112|140| 8|120
| +---+
| 8|23| 44| 70|100|133|168|204| 9|165
| +---+
| 9|26| 50| 80|115|154|196|240|285| 10| 220
| +----+
|10|29| 56| 90|130|175|224|276|330|385| 11| 286
| +----+
|11|32| 62|100|145|196|252|312|375|440| 506| 12| 364
| +----+
|12|35| 68|110|160|217|280|348|420|495| 572| 650| 13| 455
| +----+
|13|38| 74|120|175|238|308|384|465|550| 638| 728| 819| 14| 560
| +----+
|14|41| 80|130|190|259|336|420|510|605| 704| 806| 910|1015| 15| 680
| +----+
|15|44| 86|140|205|280|364|456|555|660| 770| 884|1001|1120|1240| 16|
| +
|16|47| 92|150|220|301|392|492|600|715| 836| 962|1092|1225|1360|1496|
|
|17|50| 98|160|235|322|420|528|645|770| 902|1040|1183|1330|1480|1632|
|
|18|53|104|170|250|343|448|564|690|825| 968|1118|1274|1435|1600|1768|
|
|19|56|110|180|265|364|476|600|735|880|1034|1196|1365|1540|1720|1904|
|
|20|59|116|190|280|385|504|636|780|935|1100|1274|1456|1645|1840|2040|
|
|21|62|122|200|295|406|532|672|825|990|1166|1352|1547|1750|1960|2176|
++--+--+---+---+---+---+---+---+---+--+----+----+----+----+----+----+
15... | 816 ||
+----+ ||
16... | 17| 969 ||
| +----+ ||
17... |1785| 18|1140 ||
| +----+ ||
18... |1938|2109| 19|1330 ||
| +----+ ||
19... |2091|2280|2470| 20|1540 ||
| +----+ ||
20... |2244|2451|2660|2870| 21|1771 ||
| +----+ ||
21... |2397|2622|2850|3080|3311| 22|2024||
-------+----+----+----+----+----+----+----++
CORDAGE.
_Ropes_, _cables_, and all other descriptions of cordage are
distinguished by their circumference, thus a two-inch rope means a
rope two inches in circumference.
_To find the weight of a rope._
_First method._—Multiply the length in fathoms by the square of the
circumference, and divide the product by 480 for the weight in cwts.
_Example._—Required the weight of 110 fathoms of 3-inch rope.
3 × 3 × 110 = 990, which divided by 480, gives 2 cwt. 7 lb.
Weight required.
_Second method._—Divide the square of the circumference by 4, the
quotient will give the weight, in pounds, per fathom.
_Example._—What is the weight of a 3-inch rope per fathom?
3^2 ÷ 4 = 2¼ lb. Weight required.
_To find the strength of a rope, or the weight it will support._
_First method._—Square the circumference, and divide by 5, for the
number of tons which it will bear suspended from it.[46]
_Example._—What weight will 3-inch rope of the best description
support?
(3 × 3) / 5 = 9/5 = 1⅘ ton, or 4030 lb. Weight required.
_Second method._—Multiply the square of the circumference by 2, the
product will give the _practical weight in cwts_. that may be lifted
by it, or about half the breaking weight.
_Example._—What number of cwts. may be lifted by a 3-inch rope?
3^2 × 2 = 18 cwts. Weight required.
_The strain_, in pounds, _a rope will bear safely_ = girt^2 × 200}
” ” _a cable_ ” ” = girt^2 × 120}
nearly.
CHAINS.
_To find the weight of chains._
The square of the diameter of the link, measured in eighths of
inches, will give the weight of the chain, per fathom, in pounds.
_Example._—What is the weight per fathom of a ¾-inch chain?
¾-inch = 6/8; 6^2 = 36 lb. Weight per fathom.
_Or_, the weight per foot of the chain, multiplied by 24, will give
the weight per fathom of the chain, _nearly_. A chain cable with a
stay across the links will weigh about one-twelfth more than the
foregoing examples.
_To find the weight that may be safely lifted by a chain._
Divide the square of the diameter of the links, taken in eighths of
an inch by 8, and the quotient will give the number of tons that may
be lifted by the chain.
_Example._—What number of tons will a chain made of ¾-inch iron carry
with safety?
¾-inch = 6/8 6^2 = 36 36 / 8 = 4½ tons. Weight required.
_The safe strain_ is equal to about 8 tons, per square inch, of the
iron of which the chain is made.
_The stay across the link of a chain_ increases its strength about
one-sixth.
_When the chain is of great length_, a deduction, from the above
rules, must be allowed for the weight of it.
IRON RODS.
_To find the weight of round iron rods._
Divide the square of the diameter, in quarter inches, by 2, and the
quotient will give the weight in pounds, per yard.
_Example._—What is the weight of a yard of 1-inch round iron.
1 inch = 4 quarters 4^2 = 16 16 / 2 = 8 lb. Weight required.
_To find the weight of square rods._
The weight of round rods, of similar diameter, divided by ·7854 will
give the weight of the square rods.
_To find the weight that may be sustained, or lifted by round iron
rods._
Find the weight in pounds, per yard; two-thirds of which will give
the safe load, in tons.
A round iron rod of average quality of iron, one inch in diameter,
will be torn asunder by 16 tons; it will be perceptibly damaged by
half this strain, or 8 tons; its safe load will be one-third, or 5·33
tons.
TIMBER.
_To find the area, or superficial content of a plank._
Multiply the length by the mean breadth.[47]
_Example._—Required the content of a board whose length is 11 feet 2
inches, and breadth 1 foot 10 inches
ft. in. ft. in. ft. in.
11 2 × 1 10 = 20 5. Content required.
_To find the solid content of squared, or four-sided timber._
Multiply the mean breadth by the mean thickness, and the product by
the length, for the content, nearly.
_Note 1._—If the tree taper regularly from the one end to the other,
either take the mean breadth, and thickness in the middle, or take
the dimensions at the two ends, and half their sum will be the mean
dimensions; which, multiplied as by the above rule, will give the
content, nearly.
_Note 2._—If the piece do not taper regularly, take several different
dimensions, add them all together, and divide their sum by the number
of them, for the mean dimensions.
_Example._—Required the content of a piece of timber 16 feet long,
and side of square 14 inches.
ft. in. ft. in. ft. ft. in.
1 2 × 1 2 × 16 = 21 9. Content required.
_To find the solidity of round, or unsquared timber._
1. Multiply the square of the quarter girt (or the square of ¼ of the
mean circumference), by the length, for the content.
_Note._—When the tree is tapering, take the mean dimensions, either
by girting it in the middle for the mean girt, or at the two
ends, taking half the sum of the two; or by girting it in several
places, then adding all the girts together, and dividing the sum
by the number of them for the mean girt. But when the tree is very
irregular, divide it into several lengths, and find the content of
each part separately.
_Example._—Required the content of a tree, whose mean girt is 3·15
feet, and length 14½ feet.
3·15 / 4 = ·7875 ·7875 × ·7875 = ·62015625.
·62015 × 14·5 = 8·9922 feet of solid timber. The content required.
2. Find the mean area of a round tree, and multiply it by the length
for the content.
_To find the weight of a tree._
Find its content in feet, and multiply that by the specific gravity
of the wood.
(_Vide_ GRAVITY, _and Table of Specific Gravities_. _Page_ 318.)
_Example._—Required the weight of an elm-tree; whose mean girt is 5
feet, and length 60 feet.
5/4 = 1·25 1·25 × 1·25 = 1·5625.
1·5625 × 60 = 93·75. Content in feet.
TONNAGE.
Table of Tonnage, and Weight of _one_ of the following Carriages,
Carts, Waggons, Gyns, &c., used in land service.
+--------------------------------------+-------+---------+-------------+
| | Tonn- | Weight. | |
| | age. | | |
+--------------------------------------+-------+---------+-------------+
| |Tons. |Cwt. lb.| |
| | ft.| qrs. | |
| | | |} With |
|Carri- {24 pounder | 6 0|34 0 22|} bullock |
|ages. { 8 inch | 6 0|34 2 12|} pole and |
| T{ {For Iron {18 pounder | 4 39|27 2 9|} chain, |
| r{ {Ordnance.{12 ” 21 cwt. | 4 7|18 3 24|} weighing 2 |
| a{ | | |} qrs. 19 lb.|
| v{ {Howitzer {10 inch| 6 17|39 0 9| |
| ” { e{ { 8 ” | 5 37|33 2 0| Do. do. do.|
| ” { l{ {12 pounder | 5 33|22 0 10| |
| ” { l{ { 9 ” | 5 1|20 2 14| |
| ” { i{For Brass { 6 ” | 4 21|17 3 5| |
| ” { n{Ordnance. { {32 pr.| 5 29|23 3 13| |
| ” { g{ {Howitzer {24 ” | 5 6|21 0 17| |
| ” { { { {12 ” | 4 21|18 3 14| |
| ” { c{Ammunition waggon | 5 36|20 0 3| {For all |
| ” { o{ | | { natures. |
| ” { m{Forge | 5 38|19 1 0| |
| ” { p{Store-waggon (without spare | 5 11|18 1 10| |
| ” { l{ wheel) | | | |
| ” { e{Small arm ammunition waggon | 4 36|14 2 16| |
| ” { t{Rocket {12 pounder | 7 33|20 2 8| |
| ” { e{ { 6 ” | 5 17|20 1 20| |
| ” { .{Pontoon {Large | 3 30|42 2 13| |
| ” { { {Small | |22 2 5| |
| ” {Garrison, wood, common standing }| | | |
| ” { for 32 pounder of 25 cwt. }| 1 8| 8 0 7| |
|Capstan, crab | 0 31| 3 3 26| |
| {Forge, cavalry | 4 32|11 2 3| |
| {Hand | 1 10| 4 3 4| |
| {Hospital, conveyance | 3 16|10 2 20| |
|Carts {Sling | 3 38|16 1 17| |
| {Store | 3 16| 9 1 0| |
| {French | 1 32| 5 2 13| |
|Drugs {Large | 2 7|17 1 24| |
| {Small | 0 29| 5 0 4| |
|Gyns, {Large | 1 23| 9 2 22| |
|Triangle {Small | 1 2| 7 3 3| |
| {For 32 pounder | | | |
| { garrison carriage}| 0 26 | 6 0 12| Fir. |
| {Madras {For traversing }| | | |
|Platform. { { carriage with }| | | |
| { { tail-piece }| 1 23 |14 2 0| Teak. |
| {Mortar, Alderson’s pattern | 0 30 | 8 1 4| |
|Portable forge, and pack saddle, | | |
| in wooden case | 0 17 | 2 1 3| |
| {Flanders | 5 0 |16 1 25| |
|Waggons {Platform | 3 16 |21 3 18| |
| {Sling | 8 11 |31 3 26| |
|Waggons, hospital, Mr. Holmes’ {Large | 9 10 |21 0 0| |
| pattern {Small.| 6 30 |18 0 4| |
+--------------------------------------+-------+---------+-------------+
The calculation of tonnage for baggage, stores, &c., is by
measurement: _a Ton_, consisting of 40 cubic feet; but metals, and
very heavy articles are estimated by actual weight, without reference
to bulk.
_To ascertain the tonnage of sailing vessels, the hold being clear._
_Rule._—Divide the length of the upper deck between the afterpart of
the stem, and the forepart of the stern-post, into six equal parts.
_Depths._—At the foremost, the middle, and the aftermost of those
points of division, measure in feet, and decimal parts of a foot, the
depth from the under side of the upper deck to the ceiling at the
limber strake. In the case of a break in the upper deck the depths
are to be measured from a line stretched in a continuation of the
deck.
_Breadths._—Divide each of those three depths into five equal parts,
and measure the inside breadths at the following points—viz., at
one-fifth, and at four-fifths from the upper deck of the foremost,
and aftermost depths, and at two-fifths, and four-fifths from the
upper deck of the midship depth.
_Length._—At half the midship depth measure the length of the vessel
from the afterpart of the stem to the forepart of the stern-post;
then to twice the midship depth add the foremost, and the aftermost
depths for the sum of the depths; add together the upper, and lower
breadths at the foremost division, three times the upper breadth, and
the lower breadth at the midship division, and the upper, and twice
the lower breadth at the after division, for the sum of the breadths:
then multiply the sum of the depths by the sum of the breadths, and
this product by the length, and divide the final product by 3500,
which will give the number of tons for register.
If the vessel have a poop, or half deck, or a break in the upper
deck, measure the inside mean length, breadth, and height of such
part thereof as may be included within the bulkhead; multiply these
three measurements together, and dividing the product by 92·4, the
quotient will be the number of tons to be added to the result as
above found.
In order to ascertain the tonnage of open vessels, the depths are to
be measured from the upper edge of the upper strake.
_To ascertain the tonnage of steam vessels._
_Rule._—In addition to the foregoing rules, when applied for the
purpose of ascertaining the tonnage of any ship or vessel propelled
by steam, the tonnage due to the cubical content of the engine-room
must be deducted from the total tonnage of the vessel, as determined
by either of the rules aforesaid, and the remainder will be the true
register tonnage of the said ship or vessel.
_To determine the tonnage due to the cubical content of the
engine-room._
_Rule._—Measure the inside length of the engine-room in feet
and decimal parts of a foot, from the foremost to the aftermost
bulkhead, then multiply the said length by the depth of the ship or
vessel at the midship division as aforesaid, and the product by the
inside breadth of the same division at two-fifths of the depth from
the deck, taken aforesaid, and divide the last product by 92·4, and
the quotient will be the tonnage due to the cubical content of the
engine-room.
_To ascertain the tonnage of vessels when laden._
_Rule._—Measure, _first_, the length on the upper deck between
the afterpart of the stem, and the forepart of the stern-post;
_secondly_, the inside breadth on the under side of the upper deck,
at the middle point of the length; and, _thirdly_, the depth from the
under side of the upper deck down the pump-well to the sink; multiply
these three dimensions together, and divide the product by 130, and
the quotient will be the amount of the register tonnage of such ships.
MECHANICS.
_Mechanics_ is the science of forces, and the effects they produce
when applied to machines in the motion of bodies.
_Machine, or engine_, is any mechanical instrument contrived to move
bodies.
_Equilibrium_ is an equality of action, or force, between two or more
powers, or weights, acting against each other, by which they destroy
each other’s effects, and remain at rest.
_The centre of motion_ is the fixed point about which a body moves.
_The axis of motion_ is the fixed line about which it moves.
_The centre of gravity_ is a certain point on which a body (being
freely suspended) will rest, in any position.
_The whole momentum_ or quantity of force of a moving body, is the
result of the quantity of matter multiplied by the velocity with
which it is moved.
THE MECHANICAL POWERS.
_Power is compounded of the weight, or expansive force of a moving
body multiplied into its velocity._
The power of a body, which weighs 40 lb., and moves with the velocity
of 50 feet in a second, is the same as that of another body which
weighs 80 lb., and moves with the velocity of 25 feet in a second:
for the products of the respective weights, and velocities are the
same.
40 × 50 = 2000; and 80 × 25 = 2000.
_Power cannot be increased by mechanical means._
Power is applied to mechanical purposes—
1. By the lever; 4. By the inclined plane;
2. By the wheel, and axle; 5. By the wedge;
3. By the pulley; 6. By the screw;
which are the simple elements of all machines.
The whole theory of these elements consists simply in causing the
weight, which is to be raised, to pass through a greater or a less
space than the power which raises it; for, as power is compounded of
the weight, or mass of a moving body, multiplied into its velocity, a
weight passing through a certain space may be made to raise, through
a less space, a weight heavier than itself.
THE LEVER.
_The lever_ is the most simple of all machines, being only a straight
bar of iron, wood, &c., supported on, and moveable round a prop,
called the _fulcrum_.
_Case 1._—When the fulcrum of the lever is between the power, and the
weight.
_Rule._—Divide the weight to be raised by the power to be applied;
the quotient will give the difference of leverage necessary to
support the weight in equilibrio. Hence, a small addition either of
leverage, or weight, will cause the power to preponderate.
_Example 1._—A ball weighing 3 tons is to be raised by 4 men, who can
exert a force of 12 cwt.; required the proportionate length of lever?
3 tons = 60 cwt.; and 60/12 = 5
In this example, the proportionate lengths of the lever to maintain
the weight in equilibrio, are as 5 to 1. If, therefore, an additional
pound be added to the power, the power side of the lever will
preponderate, and the weight will be raised. But, although the ball
is raised by a force of only one-fifth of its weight, no power is
gained, for the weight passes through only one-fifth of the space.
The products, therefore, arising from the multiplication of the
respective weights, and velocities are the same.
_Example 2._—A weight of 1 ton is to be raised with a lever 8 feet in
length, by a man who can exert, for a short time, a force of rather
more than 4 cwt.; required at what part of the lever the fulcrum must
be placed?
(20 cwt.) / (4 cwt.) = 5; that is, the weight is to the power
as 5 to 1,
therefore, 8 / (5 + 1) = 1 foot and a third, from the weight.
_Example 3._—A weight of 40 lb. is placed 1 foot from the fulcrum of
a lever; required the power to raise the same, when the length of the
lever on the other side of the fulcrum is 5 feet?
(40 × 1) / 5 = 8 lb. _Ans._
_Case 2._—_When the fulcrum is at one extremity of the lever, and the
power at the other._
_Rule._—As the distance between the power, and the fulcrum is to the
distance between the weight, and the fulcrum, so is the effect to the
power.
_Example 1._—Required the power necessary to raise 120 lb., when the
weight is placed 6 feet from the power, and 2 feet from the fulcrum?
As 8 : 2 :: 120 : 30 lb. _Ans._
_Example 2._—A beam, 20 feet in length, and supported at both ends,
bears a weight of 2 tons at the distance of eight feet from one end;
required the weight on each support?
(40 cwt. × 8 feet) / (20 feet) = 16 cwt. on the support that is
furthest from the weight; and (40 × 12) / (20 feet) = 24 cwt.
on the support nearest to the weight.
_Case 3._—_When the weight to be raised is at one end of the lever,
the fulcrum at the other, and the power is applied between them._
_Rule._—As the distance between the power, and the fulcrum, is to the
length of the lever, so is the weight, to the power.
_Example._—The length of the lever being 8 feet, and the weight at
its extremity 60 lb., required the power to be applied 6 feet from
the fulcrum to raise it?
As 6 : 8 :: 60 : 80 lb. _Ans._
_Velocity is gained at the expense of power by the lever, and wheel,
and axle._
Note 1.—When two men are carrying a load on a pole between them,
the strongest man should have the weight placed nearer to him than
the other man.
Note 2.—_To carry guns, &c._—If the burden can be carried by four
men; after having made it fast to the middle of a large lever, fix
the extremities of this lever on two shorter levers, and place a
man at each of the points, C, D, E, F. _Vide plate, Mechanics, Fig.
1._ In _Fig. 2_, the weight is equally divided between eight men,
and in _Fig. 3_, between sixteen men.
THE WHEEL, AND AXLE.
The advantage gained is in proportion as the circumference of the
wheel exceeds that of the axle; therefore, the larger the wheel, and
the smaller the axle, the stronger is the power of this machine, but
then the weight will rise proportionally slower. A winch may be used
instead of a wheel, for in turning the winch the hand will describe
a circle, and there is no difference in the result, whether an
entire wheel be turned, or a single spoke which the winch as a lever
represents.
_Rule._—As the radius of the wheel is to the radius of the axle, so
is the effect, to the power.
_Example._—A weight of 50 lb. is exerted on the periphery of a wheel,
whose radius is 10 feet; required the weight raised at the extremity
of a cord wound round the axle, the radius being 20 inches.
(50 lb. × 10 feet × 12 inches) / (20 inches) = 300 lb. _Ans._
THE PULLEY.
_The pulley_ consists of a grooved wheel, called _a sheave_, moveable
on an axis, or gudgeon, and enclosed in a frame, or case, called a
_block_. By passing a cord over the pulley, a man will be enabled
_to draw up_ a weight equal to that which his own body supplies in
pulling downwards.
_By combining a number of pulleys_, as many assistants are obtained
as there are wheels: thus, two pulleys will have double the power
of one, because half the weight is sustained by the frame to which
one end of the cord is attached; but then it requires _double the
time_ to do the work. As the _friction of the pulley_ is very great,
particular attention must be paid that all the turns or kinks of a
rope be taken out, before it is made use of, and it should enter
easily into the grooves of the sheaves.
_Rule._—Divide the weight to be raised by twice the number of pulleys
in the lower block; the quotient will give the power necessary to
raise the weight.
_Example._—What power is required to raise 600 lb., when the lower
block contains six pulleys?
600 / (6 × 2) = 50 lb. _Ans._
TACKLES.
_Tackles_ are indispensable in the service of the artillery.
_The fall_ is the rope of which the tackle is composed; that end of
it which is fixed to the block is called _the standing part_, or
_end_; the other, which is pulled, or hauled on by the men, is called
_the running part_, or _end_; and the parts which pass from one block
to the other are called the _returns of the fall_.
_In all operations with tackles, the following directions should be
attended to_:—
1st. Make fastenings stronger than appears _actually_ necessary.
2nd. Examine the straps, and hooks of the blocks carefully.
3rd. Consider whether the cordage is new, half-worn, or almost worn
out.
4th. Attend to the seizings of the clinches, the sheet bends, the
_proper_ stoppering of the fall, the belaying the fall with two half
hitches.
5th. Be very distrustful of selvages applied on smooth worn rope.
6th. Do not allow ropes to be struck, or trampled on, when the weight
is suspended.
7th. The men should _stand as safe_ as the proper performance of the
various duties will permit.
8th. In pulling a rope, the men ought to place themselves in a right
line, and haul together. The most advantageous position for pulling
is down a slope, or in a descending position.
COMBINATION OF PULLEYS.
_A leading block_ is a fixed pulley, which alters the direction of
the power, but does not increase it: Power = Weight. On account of
friction the power must exceed the weight a little, in order to raise
it.
_Vide plate, Mechanics, Fig. 1._
_A whip_ is one moveable pulley, which increases the power without
altering the direction.
Power = ½ weight (or 2 to 1).—_Vide Fig. 2._
_A whip upon whip_ will afford the same purchase as a tackle having a
single and double block, and with much less friction.
_A gun tackle_ consists of two single blocks with fall fixed to the
one, then rove through the other, and then through the first. Power =
½ weight (or 2 to 1): or Power = ⅓ weight (or 3 to 1). _Vide Fig. 3,
and 4._
Two double blocks are generally used for very heavy guns.
_A luff tackle, or half watch tackle_, consists of one double and
one single block: the fall is fixed to the single, then rove through
first sheave of the double, then through sheave of single, and lastly
through second sheave of double block. Power = ⅓ weight (3 to 1): or
Power = ¼ weight (4 to 1). _Vide Fig. 5, and 6._
_A runner tackle_ is the same as a luff tackle, applied to the end
of a large rope, called a runner, which is rove through a single
block attached to a fixed point, or to a body that is to be moved, or
raised; the standing end of the runner being secured to another point.
Power is either 6 to 1, or 7 to 1, or 8 to 1.
_A gyn tackle_ consists of one triple and one double block: the fall
is fixed to the double, then rove through first sheave of triple,
then through first sheave of double, then through second sheave of
triple, then through second sheave of double, and lastly through
third sheave of triple block.
Power = ⅕ weight (5 to 1): or Power = ⅙ weight (6 to 1). _Vide Fig.
7._
If the moveable block of a tackle be strapped with a tail, it is
called a _tail_, or _jigger block_: and the tackle a _tail_, or
_jigger tackle_: a block with a hook strapped to it, and attached to
a selvage, answers the same purpose.
_Two double blocks_, with fall fixed to one of them, and then rove
through the sheaves of both blocks, will either give Power = ¼ weight
(4 to 1): or Power = ⅕ weight (5 to 1). _Fig. 8._
_Two triple blocks_, with fall fixed to one of them, then rove
through sheaves of both blocks, will either give power = ⅙ weight (6
to 1): or Power = ⅐ weight (7 to 1). _Fig. 9._
[Illustration: _System of Pullies._
Fig. 1 2 3 4 5 6 7 8 9]
[Illustration: _To carry Guns &c._
Fig. 1 2 3]
[Illustration: HEIGHTS AND DISTANCES.]
_In the system of pulleys_ (_vide plate, Mechanics_) the Power is
shown at the hooks of the moveable blocks, which are to be applied to
the bodies, or weights, requiring to be moved or raised. The strain
is also shown at the fixed blocks.
_In Fig. 3_, there are _three parts of the rope engaged_ in
supporting the weight—_viz._, _the parts marked 1, 1, 1_. Each of
them, hence, sustains _one-third_ of it, and the fall of the rope to
which the power is to be attached requires the Power = 1, if weight =
3. The same principle of calculation is applicable to all systems of
pulleys having one fixed block, any number of moveable wheels, and a
single rope over all the wheels. Hence, in such a system of pulleys,
gravity being applied, there will be an equilibrium, when the weight
is as many times the power as there are portions of the rope employed
in sustaining the weight. _For example_, in a system consisting of
six moveable sheaves, the same rope going over them all, there will
be 12 portions of the rope engaged; and to produce an equilibrium the
power must be equivalent to 1/12 the weight, no allowance being made
for friction.
From the foregoing observations, and by referring to the plate, it
will be seen that _each tackle has two applications_, differing in
power one from the other; _for example_, if the double block of a
luff tackle is fixed to a weight to be moved, and the single block
to a picket, or other fastening, _Fig. 6_, then, if _one man_ haul
on the fall, the power of _four men_ will be applied to the weight
(4 to 1), and the power of _three men_ to the picket; but if the
double block be fixed to the picket, _Fig. 5_, and the single, block
to the weight, then the force of only _three men_ will be applied
to the weight (3 to 1), and a power of _four men_ to the picket, or
fastening.
When the moveable block of one tackle is fixed to the fall of another
tackle, their respective powers are to be multiplied into each other
for the power of the combination: thus, if one luff tackle is fixed
to the fall of another luff tackle (the double blocks of both tackles
being moveable), the power will be 4 × 4 = 16 (16 to 1): in this, the
men haul through 16 feet to move the weight one foot; therefore if
the combination be increased until the men haul through 100 feet to
move the weight one foot, then the power would be 100 to 1.
The foregoing powers are, however, only true in theory, and are,
therefore, called _theoretical powers_: for owing to the great
friction of the pulleys, the stiffness of the ropes, &c., the actual
_practical powers_ are far less; so much so, that with a combination
giving a power of 48 to 1, a 24-pr. (2½ tons weight) suspended, can
scarcely overhaul the fall, the friction being so very great.
THE INCLINED PLANE.
_The inclined plane_ forms simply a gradual and sloping instead of a
sudden and perpendicular ascent, by which heavy bodies may be raised
to certain heights. The power necessary for raising a weight depends
on the difference between the length of the plane and the height to
be ascended. If the height be one-third of the length, then one pound
will lift three pounds. The force with which a rolling body descends
on an inclined plane is to the force of its absolute gravity, as the
height of the plane is to its length.
_Parbuckling a gun_ on skids unites the advantage of one moveable
pulley with that of the inclined plane.
_Rule._—As the length of the plane is to its height, so is the weight
to the power.
_Example._—Required the power necessary to raise 540 lb. up an
inclined plane, five feet long, and two feet high.
As 5 : 2 :: 540 : 216 lb. _Ans._
THE WEDGE.
_The wedge_ may be considered as two equally inclined planes joined
together at their bases. It has a great advantage over all the other
powers, arising from the force of percussion, or blow, with which
the back is struck; which is a force incomparably greater than any
dead weight, or pressure, such as is employed in other machines. The
largest masses of timber may by this means be riven, and vessels of
war, weighing many thousand tons, are lifted from their supports
by the power of a few men, exerted by blows of mallets on wedges
inserted for that purpose.
_The power of the wedge_ increases in proportion as its angle is
acute. In tools intended for cutting wood the angle is commonly about
30°; for iron from 50° to 60°; and for brass from 80° to 90°.
_Case 1._—_When two bodies are forced from one another, by means of a
wedge, in a direction parallel to its back._
_Rule._—As the length of the wedge is to half its back, or head, so
is the resistance, to the power.
_Example._—The breadth of the back, or head of the wedge, being three
inches, and the length of either of its inclined sides 10 inches,
required the power necessary to separate two substances, with a force
of 150 lb.
As 10 : 1½ :: 150 : 22½ lb. _Ans._
_Case 2._—_When only one of the bodies is moveable._
_Rule._—As the length of the wedge, is to its back, or head, so is
the resistance, to the power.
_Example._—The breadth, length, and force, the same as in the last
example.
As 10 : 3 :: 150 : 45 lb. _Ans._
THE SCREW.
_The screw_ is a spiral thread or groove cut round a cylinder, and
everywhere making the same angle with the length of it. The force
of a power applied to turn a screw round is to the force with which
it presses upward, or downward, setting aside the friction, as the
distance between two threads is to the circumference where the power
is applied; or the advantage gained is as much as the circumference
of a circle described by the handle of the winch exceeds the
interval, or distance, between the spirals of the screw. Hence the
force of any machine turned by a screw can readily be computed; for
instance, in a press driven by a screw, whose threads are each a
quarter of an inch asunder, and with a handle, to turn the screw,
four feet long; then, if the natural force of a man, by which he can
lift, pull, or draw, be 150 lb., and it is required to determine with
what force the screw will press when the man turns the handle with
his whole force; the diameter of the handle (_power_) being 4 feet,
or 48 inches, its circumference is 48 × 3·1416, or 150⅘ nearly; and
the distance of the threads being one-fourth of an inch, therefore
the power is to the pressure as 1 to (150⅘ × 4) = 603⅕, but the
power is equal to 150 lb., therefore as 1 : 603⅕ :: 150 : 90480,
and consequently the pressure is equal to a weight of 90480 lb.
independent of friction.
COMPOUND MACHINES.
Though each of the mechanical powers is capable of overcoming the
greatest possible resistance in theory, yet in practice, if used
singly for producing very great effects, they would frequently be so
unwieldy and unmanageable as to render it impossible to apply them.
For this reason it is generally found more advantageous to combine
them together, by which means the power is more easily applied, and
many other advantages are obtained. In all the mechanical powers, and
their combinations, and in all machines, simple as well as compound,
_what is gained in power is lost in time or velocity_; and _vice
versâ_, or in other words, the product of the power, and the space
through which it moves, is equal to the product of the weight, and
the space through which it moves in the same plane. Suppose that a
man, by means of a fixed pulley, raises a beam to the top of a house
in two minutes, it is clear that he will be able to raise six beams
in twelve minutes; but by means of a tackle with three lower pulleys,
he will raise the six beams at once with the same ease as he before
raised one, but then he will be six times as long about it, that is,
twelve minutes; thus the work is performed in the same time whether
the mechanical power is used, or not. But the convenience gained by
the power is very great; for if the six beams are joined in one, they
may be raised by the tackle, though it would be impossible to move
them by the unassisted strength of one man. No real gain of force is
obtained by mechanical contrivances; on the contrary, from friction
and other causes, force is always lost; but by machines a more
convenient direction can be given to the moving power, and so modify
its energy as to obtain effects which it could not otherwise produce.
FRICTION.
_Friction_ arises from the irregularities of the surfaces which move
upon one another. The surfaces of bodies of the same nature are
moved with more facility over each other than those of a dissimilar
nature. In proportion as the surfaces which are to be moved upon one
another are rough, a greater force is requisite to produce motion.
The same surfaces when under a greater pressure, are subject to still
further friction. A double pressure doubles the amount of friction,
a treble pressure trebles, and so on in nearly the same proportion.
When surfaces are moving along each other in the direction of their
grains, the friction is greater than when the direction of the grains
is at right angles. Friction is little influenced by the velocity
with which bodies move upon one another. Friction may be diminished
in various ways, as will appear by the result of the following
experiment with a block of square stone weighing 1080 lb.:—
1. In order to drag this stone along the floor of a quarry lb.
roughly chiselled, it required a force equal to 758
2. Over a floor of planks, ditto 652
3. Placed on a platform of wood, and dragged over a floor of
planks 606
4. After soaping the two surfaces of wood, which slide over
each other 182
5. Placed on rollers of three inches diameter, and moved along
the floor of the quarry 34
6. To drag it on these rollers over a wooden floor 28
7. Mounted on a wooden platform, and the same rollers placed
between the platform, and a plank floor 22
One of the most remarkable instances of the application of rollers
is the transport of the rock which now serves as the pedestal of the
equestrian statue of Peter the Great at St. Petersburg. This rock is
a single block of granite weighing 1217 tons. A railway was formed,
consisting of two lines of timber, furnished with hard metal grooves;
similar, and corresponding metal grooves were fixed to the under side
of the sledge, or frame, on which the stone was laid, and between
these grooves were placed spheres of hard brass, about six inches
in diameter. On these spheres the frame with its enormous load was
easily moved by sixty men working at capstans with triple purchase
blocks.
UNGUENTS.
Mr. G. Rennie found, from a mean of experiments, with different
unguents, on axles in motion, and under different pressures, that,
with the unguent tallow under a pressure of from 1 to 5 cwt., the
friction did not exceed 1/39th of the whole pressure; when soft
soap was applied, it became 1/34th; and with the softer unguents
applied, such as oil, hogs’ lard, &c., the ratio of the friction to
the pressure increased; but with the harder unguents, as soft soap,
tallow, and anti-attrition composition, the friction considerably
diminished; consequently, to render an unguent of proper efficiency,
the nature of the unguent must be measured by the pressure, or
weight, tending to force the surfaces together.
TRANSVERSE STRENGTH OF MATERIALS.
When a beam, of any material, is loaded, the surface in contact
with the load is _compressed_, and the opposite surface _extended_;
and there is a line between these, which is neither compressed, nor
extended, called _the neutral line_.
_If the depth of a beam be doubled_, the breadth, and length between
supports remaining the same, _its strength will be increased four
times_.
_If its breadth be doubled_, the other dimensions being as above,
_its strength will be doubled_.
By increasing the distance between the supports of any beam, its
strength is decreased in the same ratio; twice the distance between
the supports will weaken the beam one-half; half the distance between
the supports will enable it to bear twice the load.[48]
The same beam will bear twice the load, if, instead of being
concentrated in the middle, it be equally distributed over the whole
length of the beam.
If the load on a beam be placed near to one of the supports, instead
of in the middle, its effect will decrease in the ratio of its
proximity to the support.
Let S s represent the beam, W the load or weight in the middle, w the
weight near s; then the load which the beam will carry at the point
where w is placed will be found by the following proportion:—
As S w × w s : S W × W s :: W : w.