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UCSB LIBRARY
THE
MODERN MECHANIC:
SCIENTIFIC GUIDE AND CALCULATOR
COMPRISING
RULES AND TABLES IN THE VARIOUS
DEPARTMENTS OF MECHANICAL
SKILL AND LABOR
BY WILLIAM GRIER,
CIVIL ENGINEER.
How have we obtained this great superiority over th?se poor savages? Becans*
Science has been at work, for many centuries, to diminish th amount of our
mental labor, by teaching us the easiest mode of calculation.
RESULTS or MACHINERY
BOSTON:
HIGGINS, BRADLEY AND DAYTON,
20 WASHINGTON STREET.
CONTENTS.
Air pump 222
Air vessel 228
Animal strength 273
Arithmetic 17
Artificers' work 108
Barometer 218
Beam, working to form 157
Bramah's press 180
Catenary 99
Collision 118
Conic sections 92
Contraction, marks of 39
Cotton spinning 290
Cube root, extraction of 35
Cycloid 98
Drawing instruments 80
Drawing, mechanical 86
Eccentric 259
Ellipse 9497
Floating bodies 1 82
Ply wheel 260
Forces, central 146
Forces, parallelogram of. 119
Forcing pump 227
Fractions, decimal 22
Fractions, vulgar 17
Friction 275
Geometry 57
Governor .' 1-18
( J ravity, centre of 134
Gravity, specific 182
Gyration, centre of. 142
Heat 238
Heights, measurement of 219
Horses power 2o t
Hyd/odjh arnica 195
Hydrostatics , 1 75
Hyperbola 9(597
Inclined plane, the 132
Joists 163
Journals 162
Lever, the 121
Lifting pump 226
Machines 278
Materials, strength of 148
Materials, weight of 1 88
Measures and weights 51
Mechanics 115
Mensuration 100
Mill-weight's table 212
Motion, accelerated 117
Motion, uniform 116
Numbers, compound 26
Oscillation, centre of 137
Parabola 94 97
Parallel motion 261
Pendulum 137
Percussion, centre of 141
Pipes, contents of. 224
Pneumatics 216
Position 48
Powers, mechanical 121
Powers and roots 33
Progressions 44
Proportion, compound 43
Proportion, simple 41
Pulley, the 130
Pumps 22 1
Railways 266
Rotation 142
Screw, the 133
Sector, the 83
Shafts 159
sliding rule 36
S juare ro'>t. extr.u-tt'jii of. ... 33
Steam . . 243
CONTENTS.
1 Pg*
Steam engine 248
Steam vessels 269
Suction pump 223
Syphon, the 220
Thermometer 238
Timber, measurement of 107
Water, motion of 195
Water, pressure of 1 75
Water wheels v 204
Wedge, the 133
Weights and measures 51
Weight of materials 188
Wheels 165
Wheel and axle 124
Wind 229
Windmill, horizontal 230
Windmill, vertical . . '. 233
TAB L ES.
Alcohol, vapour of 247
Capacities for heat 241
Circular segments 1 03
Cohesion 150
Crushing 151
Drawing paper 56
Elasticity and strength of tim-
ber 149
Gauge points 264
Gravities, specific 183
Heat, effects of 240
Iron plate 189
Iron rod 191
Iron, Swedisjj, weight of. .... 188
Iron, wrougk-t, weight of. .... 1 88
Lateral strength 151
Level, difference of. 209
Machines, power of 289
Metals, weight of. 189 190
Mechanical effect 288
Millwright's 212
Pipe, cast iron 191
Pipes, content of. 226
Pitch of wheels 168
Platonic bodies 1 05
Polygons 101
Proportions 47
Shaft journals 1 62
Specific heat 242
Steam, elasticity of. 245 246
Steam vessels 270
Traction, force of 269
Water, discharge of. . 199200
202203
Weight of materials 193
Weights and measures 51
Wheels 173
I Wheels, teeth of 175
Wind, force of 230
1 Windmill sails 234
INTRODUCTION.
IT is our intention, in these introductory panes, to make a tew
observations on the nature of scientific knowledge, which may be
useful to the younj reader in enabling him to understand 'more
clearly the subjects contained in the volume, and in guarding him
against the adoption of false theory, or the wasting of his time
in inquiries which can terminate in no useful result. Such intro-
ductory observations are rendered the more necessary, as a correct
knowledge of the subjects to which they relate, is the only sure
foundation on which there can be raised a solid superstructure
of science.
It is a general opinion that scientific knowledge is entirely dif-
ferent from all other kinds of knowledge ; or that it requires for
its cultivation a constitution of mind only to be met with here
and there in the great family of mankind ; and what is said of the
poet is also thought of the philosopher th-.it lie is burn, not made.
All men are certainly not equally endowed with capacities for the
acquisition of scientific knowledge, but there are few men indeed
who are totally unprivileged. The man who would relinquish
scientific pursuits merely because he had no hope of reaching
the eminence of a Newton, a Watt, or a Davy, is no better than
him, who, in despair of ever obtaining a share of wealth equal
to that of the rich inheritor of the land, would cease to make any
honest exertion to raise himself from a state of the most squalid
wretchedness. We would not be understood by this to bring the
acquisition of knowledge into invidious comparison with the
acquisition of wealth the one is in every case a godlike employ-
ment, but the other is often the concomitant of vice.
The young mechanic should be made well aware that the
knowledge of the man of science differs from the knowledge of
1* 5
6 INTRODUCTION.
ordinary men, not so much in kind as in degree ; and the know-
ledge which guides the little boy in the erection of his summer-
house, constitutes a part of that knowledge which guides the best
architect in the erection of the most splendid edifice. The boy
raises his paper kite in the air, with no other end in view save
his own amusement he has learned to do so by seeing other
boys do the same, and by trials he linds that the kite will fly
better in a moderate wind than in a perfect culm, and that the
weight at the tail may be too heavy or too light, and he regulates
his actions accordingly : so f;\r he is a little philosopher. /Y man
raises a kite knowing all that the boy know, but he raises it with
a view of determining the state of the atmosphere so far as
electricity is concerned, for which purpose, instead of employing
the hempen cord, which was sulFicient for the purpose of the boy,
ne employs a metallic wire, which he knows by experience will
conduct the electricity from the clouds to the earth, and thus
effects his design. In this respect the knowledge of the man is
more extensive than that of the boy, but, this additional knowledge
has been obtained exactly in the same way as the knowledge of
the boy, that is to say, by experience. Kven the Indian, unlearned
as he seems to be, is in some respects a philosopher. He sees
daily that the paddle of his canoe is to appearance broken when
he puts it into the water; but it is only to appearance, for by re-
peated trials, he finds that the paddle is as whole when in the
water as when out oi it. He kno\vs also, by repeated trials,
that the fish, while it shoots along through the clear flood, does
not appear to be where it really is ; for though the most unerring of
marksmen, yet if he throws his dart directly at the point where
the fish appears, he will certainly miss it. In vain will he try to
strike the fish on the same principles as he strikes the bird flying
in the air; but he finds, that when he directs his dart to a line which
is nearer to him than that in which the fish seems to move, he will
strike the fish. The Indian remembers the circumstance of his
paddle, and other circumstances of a like kind, and concludes that,
when bodies are viewed through water, they do not seem to be in
the place in which they really are. When he knows and acts upon
this principle, he is a man of science so far as this is concerned
The man of science, indeed, as we commonly understand that
appellation, knows much more than this : he knows that many
other substances have a like effect in changing the apparent
INTRODUCTION. 7
position of objects wnen seen through them nat one r* x>u> a
greater and another a less change, and by rer rated trmls 1e as ,er-
rains the actual amount of their changes by measurement, and
can subject them to the most rigid calculation ; all of which
knowledge is obtained in the same way as that of the Indian, but
is more extensive.
An examination of facts is the foundation of all true science ;
but science does not consist in a mere examination of facts.
They must be compared with each other, and the general circum-
stance of their agreement carefully marked. When we have
compared several facts together, and find that there is one general
circumstance in which they agree, this one circumstance becomes,
as it were, a chain by which they are all linked together. This
general circumstance of agreement, when expressed in language,
is what is called a law. For instance, it is a law that all bodies,
when left to fall freely, will tend to the earth ; and this law has
been framed by us, because in all cases which we have examined
this has been the case ; and the term gravity, by which this law
is designated, is nothing else than a name invented to express a (
circumstance in which we have found innumerable facts to agree.
It was known for a very long time truflwater would not rise in a
sucking pump to a height of more than thirty-two feet, and this
was said to take place because nature abhorred a vacuum. The
reason given was afterwards found to be false, yet the knowledge
of the fact was exceedingly useful in the construction of pumps
for lifting water. About the middle of the seventeenth century,
Toricelli, the pupil of Galileo, made experiments on the subject,
and found that fluids would rise in tubes or in sucking pumps
higher in proportion as they were lighter; and collecting all the
facts together, he concluded that the fluids were forced up by the
pressure of the atmosphere, and thus laid down one of the most
hnportant laws of physical science. A collection of such laws
which refers to some particular class of objects, when properly
arranged, becomes what is called a theory. Thus we see that a
theory, properly so called, is founded on an examination of par-
ticular facts, and of course cannot refer to any other but those
facts which have been examined ; or, if it is attempted so to do,
it is no longer a theory, but an hypothesis or supposition.
Hypotheses although they ought not to be relied upon, are never*
theless useful, as in our endeavours to discover whether they t*
8 INTRODUCTION.
true or false, we may at last ascertain the class of facts to which
they belong, and thus arrive at the true theory.
In the examination of facts, it is to be observed, that we must
depend on the information derived through the medium of the five
senses, that is, the senses of seeing hearing touching tasting
and smelling; for it is only by bodies affecting these organ*
fhat the properties of matter become known to us ; and all that
the mind does is to compare and classify the information thus
derived.
It is a common error to suppose that many of our greatest
inventions and discoveries were made by accident. Many
wonderful anecdotes are told in support of this assertion ; but
the very circumstance of their exciting our wonder is sufficient
to show that they are out of the common course of our experience,
and that, therefore, before they are received, they ought to undergo
a careful examination. A multitude of facts might be adduced
to prove that knowledge is more regularly progressive than is
commonly imagined. Far be it from us to detract from the
merits of those great men who have, from time to time, benefited
mankind by their important discoveries ; but from a survey of
the history of science, wfr*are led to the conviction, that where-
ever a new path has been struck out in the great field of truth,
that path has been previously prepared by former inquirers. Had
Kepler not discovered the three fundamental laws of the planetary
motions, it is highly probable that the Principia of Newton never
would have issued from the pen of that illustrious man ; and
had it not been for the brilliant discoveries of Dr. Black on the
subject of heat, it is probable that Watt never would have made
his improvements on the steam engine, that invaluable distributer
of power. It is not unlikely, however, from the state of know-
ledge in the days of Newton, that, independent of the exertions
of his mighty mind, the knowledge contained in the Principia
would soon after have been given to the world by some one or
more individuals and the like may be said of the inventions of
James Watt.
The great lesson which we would wish the young mechanic to
.earn from these observations is that great discoveries are nevei
made without preparation that previous knowledge is necessary
to turn what are called accidental occurrences to good account.
And when he is told that the law of gravitation was suggested
INTRODUCTION. 9
to Newton by the falling of an apple from a tree in his garden
or that the invention of the cotton jenny was suggested to liar
greave hy the circumstance of a common spinning-wheel conti-
nuing in its ordinary motion while in a state of falling to the
ground let him be well assured, that, had the minds of Newton
and Hargreave not been previously stored with knowledge, these
discoveries never would have been made by them. Apples and
spinning wheels bad fallen a thousand and a thousand times, but
the knowledge necessary to turn these circumstances -to good
account was first concentrated in the minds of these two illus-
trious benefactors of mankind.
In Smith's 'Wen kh uf Nations it is related that the ingenious
apparatus for opening and shutting the valves of the steam engine
was introduced by the accident of an idle boy having fastened a
brick as a counterweight to the bandies which opened and shut
tin. valves, and thus allowed him time to leave the machine and
go to play. This simple trick of an idle boy, it is said, gave rise
to the apparatus which superseded the constant attendance of a
person while the engine was at work. This, however romantic,
is not the fact the invention originated in necessity, no doubt,
but it was begun and perfected by a thorough mechanic, Mr. H
Brighton, about the year 1717.
While we are on this subject we cannot pass over another very
common prejudice, which we conceive has a very hurtful tendency
on the progress of the young mechanic. We allude to the pride
that some men take in boasting that all their knowledge is ori-
ginal ; or that they are self-taught. This is, in other words,
stating, that no assistance has been taken either from teachers or
books ; and goes only to prove, that the knowledge of the indivi-
dual so circumstanced must be very limited indeed. The unas-
sisted exertions of one man must be very feeble, when compared
with the collected exertions of the many who have gone before
him in the career of discovery. That man must know little of
geometry who has not availed himself of the use of Euclid's
Elements, or some work of a similar nature; and the Elements
of Euclid would have been meager and confined, had he not
availed himself of the discoveries of his contemporaries and pre-
decessors. A like remark may be made on the cultivation of
every department of knowledge ; and to those whom we are new
10 INTRODUCTION.
addressing we say learn from others all that you possibly can,
and when you have done so, try to correct and improve what you
have obtained. We know of no dishonourable means of acquiring
knowledge, and therefore wherever we meet it we are disposed
to respect it, even though it should not contain one particle of
originality, if such be possible ; for it is not easy to conceive
how any man should be in possession of useful knowledge, and
not make some, new application of it ; and a new application of
an old principle is certainly one constituent of originality. "With
a knowledge of what others have done, that workman will bo
less likely to waste his time in enterprises which may ruin him
by their failure, or in speculations which are unsupported by the
principles of science.
In the museum of the mechanics' class of the university founded
hv the venerable Anderson of Glasgow, there is preserved the
model of a machine to procure a perpetual motion. For the con-
trivance and execution of this beautiful specimen of workmanship,
we are, we believe, indebted to an ingenious clock-maker of Dun-
dee, who has proven himself a master in the use of his tools.
But had he been acquainted with the first principles of mechanics,
or with the nature and failure of the various attempts which had
been made before his time for the same purpose, he would have
seen the utter folly of his enterprise, and would have spent the
seven years which he occupied in the construction of this truly
beautiful model in some more useful employment. These seven
years might have been devoted to the construction of timepieces
which would have been of infinite service to the commerce and
navigation of his country in guiding the lonely mariner when
far away on the billow in determining the exact distance and
direction of the part for which he is bound whereas, the model
of his perpetual motion is preserved in the museum as a lasting
monument of this clock-maker's ignorance, perseverance, and
handicraft.
It is another common error to suppose that genius alone can
make a man a great mechanic, a great chemist, or a great any
thing. Some one makes the remark, that every man is more
than half humanity; and we do believe that the differences of
the degrees of knowledge of different men arise more from their
difference of application than from original differences of capa*
INTRODUCTION. 11
city. Let, therefore, the young workman earnestly try lo learn
and we do assure him that he will make advances which will he
proportional to his application.
This book has been written with the view of assisting the
young workman in obtaining a knowledge of the calculations
connected with machinery. The first part is devoted to such
parts of arithmetic as workmen generally require, and in which
they are most commonly deficient. Nor is this deficiency to be
wondered at, since the school books in our language contain,
generally speaking, no explanation of the nature of the rules
which they give, and are, moreover, embarrassed with so many
divisions and subdivisions, that the mind of the scholar is per-
fectly perplexed, nor can it lay hold of the great leading principles
which pervade the whole system. As this is the great instrument
used throughout the book, we have endeavoured to make its use
and management easily understood. The examples which we
have given are indeed few and simple ; but, if carefully consi-
dered, they will be found sufficient to establish the principle.
The mere habit of calculation cannot be said to constitute a
knowledge of arithmetic ; it is easily obtained, but is of no avail
without the principles. This is well illustrated by an occurrence
of but recent date. To construct a set of mathematical tables
requires, not only a knowledge of principles, but also immense
calculation. M. De Pronney was desired by the government of
France to construct a very large set of such tables ; a task which
would require the labour of a mathematician for many years.
But Pronney fell upon an expedient which was every way worthy
of a man of science. A change in the fashions of the Parisians
had thrown about five hundered wig-makers idle, and Pronney
contrived at once to give employment to these barbers, and at
the same time to serve the purposes of science. He digested the
principles of the calculation of these tables into short and simple
rules, and printed forms of them, which he gave into the hands
of these workmen, who, in a few months, produced a set of tables,
the most correct and extensive that ever has been made. The
peruke-makers may, so far as the construction of the tables was
concerned, be regarded as mere machines, under the guidance of
M. de Pronney. The same principle has been of late years car-
ried to a far greater extent by our countryman, Professor Babbage,
who has invented a machine by which logarithms and astrono-
12 INTRODUCTION.
mica) tables may be calculated and printed with the ttos lunerring
certainty, thus obviating the necessity of employing either calcu
lators or compositors. Let not these statements induce you,
however, to neglect the practice of calculation ; on the contrary,
improve yourself in it wherever you can, but be also careful to
learn the principle.
In that part devoted to geometry, we have given such informa-
tion without demonstration as was necessary to the right under-
standing of the rest of the book ; and the like may be said of the
conic sections, mensuration, and useful curves. Thus far the
book may be said to be a compend of certain branches of the
mathematics. It is hoped that the reader, to whom such studies
are new, will not be contented lo stop here ; but will be induced
to investigate these subjects in theory ; and for such as may be
desirous of entering on such a course of study, where there is
nothing to be met with but unsophisticated truths connected
together by a chain of the most beautiful relations, we intend to
offer a few words of well-meant advice as to the order and means
of prosecuting such studies.*
In the first place, let the Elements of Euclid be studied so far
as the end of the first book, in the course of which it should be
borne in mind, that there is nothing really difficult to be met with.
The greatest difficulty is, we believe, this, that, to a proposition
which is so simple as to be almost self-evident, there is often
* In a very creditable work, recently published, " Stuart's History oi
the Steam Engine, " it is stated that mathematics is not necessary to
make a great mechanic, and Watt is cited as an instance. The instance
chosen is most unfortunate for the author's assertion. Watt was de-
scended from a family of mathematicians, and inherited in the highest
degree the genius of his ancestors. One instance will sufficiently prove
this. With a desire to determine what relation the boiling point bore to
the pressure of the atmosphere on the surface of the water, he made
several experiments with apothecaries' phials, and having found the rela-
tion between the pressure and temperature of ebullition, under different
circumstances, he laid the temperatures down as abscissae, and the pres-
sures as orrtinates, and thus found a curve whose equation gave that weF.
known formula, the equation of the boiling point. No man but a mathe*
matician of high attainments would have thought of such a method of
proceeding. To this we may add, that mechanics is a branch of mathe
matics ; tor, as Sir Isaac Newton has defined it, " mechanics is the geo-
metry of motion."
INTRODUCTION. 13
attached a long demonstration, which is apt to lead the reader to
suppose that there is really something mysterious in it, which he
does not understand. This proceeds from the fact, that itoften
requires a greater deal of circumlocution to show the connection
of simple propositions with first principles, compared with propo-
sitions which are more complex ; but we have no hesitation in
saying, that if the steps of the propositions are carefully consi-
dered, one by one, they will be easily understood, and will lead
at last to perfect conviction ; for, as Lord Brougham has well
observed, " Mathematical language is not only the simplest and
most easily understood of any, but the shortest also ;" and Euclid
has transmitted to posterity a specimen of the purest mathemati-
cal language. Of Euclid's Elements, there are various editions.
Those of Simpson and Playfair are generally used in this country,
and are deservedly popular. That of Dr. Thomson is a very
valuable work, and very correct. But we beg to recommend to
the workman the edition of Mr. Robert Wallace, of Glasgow,
both for its execution and cheapness. The demonstrations are
clear and short ; many new propositions are added, and the con-
nection of theory with practice is never omitted where it can be
introduced.
When the first book of Euclid has been read, the study of al-
gebra should be commenced, on which subject there are few good
treatises to be found. That which we think best is the treatise
of Euler, a book which has come from the hand of a master, and
is therefore characterized by great simplicity. Another good
book is the treatise of Saunderson. Let either of these works,
or others if they cannot be had, be read carefully so far as to
equations of the second degree. If any one part of this depart-
ment can be said to be difficult, it is that of powers and roots,
which is a subject of the greatest importance; and should, on
that account, receive the most careful attention; and, if the trea-
tise of Euler be used, we have no hesitation in saying, that little
difficulty will be experienced. Jt may be necessary to observe,
that attention should be paid all along to the intimate connection
of arithmetic and algebra, which will tend to the better under-
standing of them both. Having advanced thus far, Euclid must
again be returned to ; and, after revising the first book, re.d on
to the sixth inclusive. Occasional revision of the algebra is
recommended, and an advancement as far as equations of the
2
14 INTRODUCTION.
third degree ; after which Euclid may be read to the termination.
The study of trigonometry may then be introduced; on which
subpect we have various works of various merits. The treatise
prefixed to Brown's Logarithmic Tables may be employed ; and
when it is understood, and the management of the logarithmic
tables acquired, the works of Gregory, Lardner, or Thomson may
be consulted ; the last is the most simple. After the study of
trigonometry, Simpson's conic sections may be read with advan-
tage.
Perhaps it may be a kind of relief at this stage, to see some-
thing of the application of mathematics to mechanics, and, for
this purpose, the work of Keil on Physics, or the article Mecha-
nics, Hutton's Mathematics, Tegg's edition. The neat little
treatise of Mr. Hay of Edinburgh will answer the same purpose
exceedingly well. But for the purpose of obtaining a good know-
ledge of theoretical mechanics, a more extensive knowledge of
mathematics than we have hitherto supposed becomes absolutely
necessary. A knowledge of the method of fluxions and fluents,
or the differential and integral calculus, which bear a strong ana-
logy to each other, and which have been employed for similar
purposes. The simplest work on fluxions, and we believe the
best, is the treatise of Simpson ; and this may be followed by a
perusal of Thomson's differential and integral calculus. With
this preparation the student may now go on to read the first vo-
lume of Gregory's Mechanics, a book in which, we believe, he
will find ample satisfaction. The second volume of this excellent
work is almost entirely popular, and can cause no difficulty what-
ever. Another work, well worthy of a perusal, is that of Sir
John Leslie : we allude to his Natural Philosophy ; a book which,
though neither strictly mathematical, nor strictly popular, yet
contains much valuable information communicated in both ways.
Indeed all the works of this great man, although much has been
said against them as to the floridness of their style, will, never-
theless, be found to amply repay the trouble of a perusal.
THE
MODERN MECHANIC.
ARITHMETIC.
VULGAR FRACTIONS.
1. IN many cases of division after the quotient is ob-
tained? there is a remainder, which is placed at the end of
the quotient, above a small line with the divisor under it :
thus 88 divided by 12 gives the quotient 7 and remainder 4,
which is written 12) 88 (7 T 4 2- Now, this T \' s ca M e( l a frac-
tion ; and it is written in this way to show that 4 ought to
be divided by 12 ; and in all cases where we meet with num-
bers written in this form, we conclude that the number above
the line is to be divided by that under the line. This
should be well borne in mind, as it is of the greatest use in
obtaining a clear notion of fractions.
2. A fraction is said to express any number of the equal
parts into which one whole is divided. It consists of two num-
bers one placed above and the other below a small line.
The upper number is called the Numerator, because it
numerates how many parts the fraction expresses ; and the
under number is called the Denominator, because it ex-
presses or denominates of what kind these parts are ; or,
in other words, the denominator shows into how many parts
one inch, foot, yard, mile one whole any thing is sup-
posed to be divided ; and the numerator shows how many
of these parts are taken : as ^ f a foot. The denominator
shows that the foot is here divided into 12 equal parts
(inches ;) and the numerator 4, shows that four of these
parts are taken (4 inches.)
2* 17
18 AKITHMETIC
3. If the numerator had been equal to the denominator,
as ^4, then the value of the fraction would have been one
whole (foot;) and the numerator, being divided by the
denominator, gives 1 as a quotient. In the fraction }| of a
foot, the numerator is greater than the denominator, and
the value of the fraction is greater than one : for the foot
being divided into twelve equal parts, (inches,) and fourteen
such parts (inches) being expressed by this fraction, its
value is more than one foot ; and the numerator being
divided by the denominator, gives \^ . Again, /^ of a foot
is just six inches, or one-half foot ; and had the foot been
divided into two equal parts, one of these parts would have
been equal to T 6 5 , or 5 is equal to T *v. From this we may
conclude, that when the numerator is equal to, less, or
greater than the denominator, the value of the fraction is
equal to, less, or greater than one whole. It is, then, not the
numbers which express the numerator and denominator of
a fraction, but the relation they bear to each other, that
determines the real value of a fraction. , f , f , -,"2, are ah
equal, although expressed by different numbers, the deno-
minators of all the fractions being respectively doubles of
their numerators.
4. From what has been said, it will easily be seen* that,
if we multiply or divide both terms of any fraction by the
same number, a new fraction will be found, equal to the
first ; thus, ; multiply both terms by 2, we get T 8 ff , or
divide them by 2, f , and these again by 2, . All who
know any thing of a common foot-rule will understand this,
at sight.
5. The first use which we shall make of the principle last
stated, is to bring two or more fractions to the same deno-
minator, and that without altering their real values. For
example, take f and | of a foot. Multiply both terms of the
first fraction f by the denominator of the second, 4 : we get
j%. Next multiply both terms of the second fraction by the
denominator of the first fraction, that is, | by 3 : the result
is -j\. Now it will^be seen (from No. 4) that these two
fractions, T 8 ^ and 7 9 ^, are equal to the two f and |, with
this additional advantage, however, that they have the same
denominator, 12 : the great use of which will be seen here
after. A like process is employed in the case of three or
more fractions : thus, , |, , multiply the terms of the
first fraction by 4 and 5, the denominators of the second
VULGAK FHACTIONS. 18
and third, we get |- ; next multiply the second | by 3 and
5, the denominators of the first and third, we next get - ;
lastly) multiply the third by the denominators of the first and
second, 3 and 4, we get |. It will be useful to look over
what we have done. In obtaining the numerators of the
new fractions, we have multiplied each numerator in the
former fractions by all the denominators except its own ;
and so also for the denominators. But 3 multiplied by 4,
and 4 multiplied by 3, are the same thing, viz. 12 : so, like-
wise, 3 multiplied by 4 multiplied by 5 is 60, and will be 60 in
whatever order we take them 3 by 4 by 5, or 4 by 3 by 5, or
5 by 3 by 4 ; when, therefore, we have obtained one deno-
minator, it is sufficient. Hence the usual rule to reduce
fractions to a common denominator : Multiply each nume-
rator by all the denominators except its own for new nume-
rators, and all th denominators together for the common
denominator.
6. We are now prepared to add two or more fractions
together. It is very easy to see how we may add f and |
of an inch, and that their sum is ; but it is not quite so
evident how we are to add f and | of a foot. If we had
them, however, of one denomination, the difficulty would
vanish. By No. 5, bring them to a common denominator
they stand thus : T \ and -^, or 8 and 9 inches ; add the
numerators, and under their sum place the denominator, }|- ;
divide the numerator by the denominator, (No 1,) the quo-
tient is 1 T V, or one foot five inches. The reason of bring-
ing them to a common denominator is, that we cannot add
unlike quantities together : and we do not add the denomi-
nators, their ojply use being to show of what kind the quan-
tities are. The rule, then, is bring the fractions to a
common denominator, add the numerators together, and
under their sum place the common denominator.
7. In subtraction we bring the fractions to a common
dei ominator, and taking the lesser from the greater of the
two numerators, place under their difference the common
denominator. The reirson of this may be easily inferred
from (No. 6) | subtracted from 5, when brought to a com-
mon denominator, T 6 g from T 8 j the 1 difference is T 2 ? , equal to
i, by No. 4.
8. To take one number as often as there are units in
another, is to multiply the one number by the other. To
multiply 4 by 2, is to take the number four two times, an
20 ARITHMETIC.
there are two units in 2 ; and to multiply 4 by |, is to take
four one-half times, or the half of four, as there is only half
a unit in the fraction 5. This may be thought so simple,
that it need not be stated ; but, let it be observed, that it
explains a fact in the multiplication of fractions, which
many excellent practical arithmeticians do not understand ;
viz. how that, when we multiply by a fraction, the product
is less than the number multiplied. If the fraction 5 is to
be multiplied by ?, (let the fractions both refer to an inch,)
this is taking 5 (inch) ? times, or taking the one-fourth part
of one-half inch, which is one-eighth. The product 5 is
obtained by this simple process : multiply the numerators
together for a new numerator, and the denominators to-
gether for a new denominator ; the new fraction will be the
product. That this is true in general may be shown by
taking other fractions, thus: of f, *he product by the
rule is ^, which may be simplified by dividing the nume-
rator and denominator by the same number, on the principle
of No. 4 ; if 4 be the divisor, the result is -, which is the
same as %. Now, that % is the real product of | by f , may
be shown thus : divide a line AB
into six equal parts ; take two of c
these parts, and join them by A r
CD. Divide CD into four parts,
and it will be seen that the two parts of this line CD are just
equal to one division on the line AB ; or f of CD is equal
to of AB ; so that,,! f f ls ! ^' ne ru l e > then, is general.
9. Division is the reverse of multiplication; hence, to
divide in fractions, invert the divisor, and proceed as in
multiplication. Thus, to divide by |, insert the divisor
4, it becomes ^, which, multiplied by 5 gives 5 multiplied by
}, equal to f ; and by dividing, to make the fraction less,
we obtain f, which, by No. 1, is just 2 or twice. This is
he quotient ; and it is easily seen, if these fractions relate
to a foot, that there are 2 quarters or twice of a foot, in
one-half foot, or 5.
10. We have now endeavoured* to explain the nature of
the fundamental rules of vulgar fractions, as simply as pos-
sible ; but instances often occur, where it is necessary to
prepare for these operations ; first, where whole numbers
are concerned ; and secondly, where the fractions are large,
and, consequently, not so easily managed.
11. As to the first, where whole numbers are concerned.
VULGAR FRACTIONS. 21
t is to be observed, that when unit, or 1, is used, either to
multiply or divide a number, it does not change the value
of that number. Thus, 6 multiplied by 1 is 6, and 6 divided
by 1 is 6. According to the principle shown in No. 1, we
may write the number 6 in this way, f , without altering
its real value with this advantage, that we have it now in
the form of a fraction. We shall illustrate this by a few
examples, and show that numbers, whether whole or frac-
tional, are in this department of arithmetic managed by the
same rules.
Add 8 to |, here we write them and I, which, brought
to a common denominator, are, 3 T 2 , I their sum is 3 ? s ; then
by No. 1, divide the numerator by the denominator, we get
8|, the number we set out from. 7 j, which is read seven
and a third, may on the same principle be put in the form
of a common fraction : for it is 7 wholes added to 5 part of
a whole, and may be thus written, | and 5, equal to a - and
whose sum is \ 2 ; divide the 22 by the 3, the result is 7j,
the first number. This very simple principle is often used,
and is embraced in the following rule multiply the whole
number by the denominator of the fraction, add the nume-
rator, and under the sum place the denominator.
12. When the fractions are very large, it becomes neces-
sary to bring them to a simple form, not only that we may
more easily see their value, but that they may be more
readily operated upon. Thus, j\ is not so simple nor so
easily managed as T T j, and the one fraction is just equal in
value to the other; for, by No. 4, the numerator and deno-
minator of 7 6 2 being both divided by 6, gives T 'j. Also,
.ji^Pg., when 100 is used as a divisor, gives ? V- Whenever
we can find a number which will divide both terms of the
fraction without remainders, we ought to employ it, and thus
make the fraction simpler in form, though of exactly the
same value. The divisor thus used to simplify fractions, is
usually called the common measure, and may frequently :
found at sight, although sometimes there is no such number
at all. Thus, in ~ , it is seen at once that 2 is the common
measure ; but in the fraction | no such common measure
can be found : consequently, the fraction cannot be made
more simple. Sometimes, also, two or more numbers will
divide the fraction ; thus, f may be divided by 4 or by 2
the greatest is preferred, because it brings the fraction to
the lowest terms at once. When this cannot be obtained at
22 ARITHMETIC.
sight, the following rule may be employed : Divide the
greater term by the less ; if these leave any remainder,
divide the last term by it ; and thus go on dividing the last
divisor by the last remainder, and that divisor which leaves
no remainder is the greatest common measure. This rule
may be applied to the following example :
1 470 By the above rule.
2205 1470 ) 2205 ( 1
1470
~735) 1470 (2
1470
735 is the common measure ; therefore,
735 ) ( f , the simple form of the fraction
DECIMAL FRACTIONS.
13. LET us examine the number 3333, (three thousand,
three hundred, thirty and three.) The same figure is used,
but for every place it is removed towards the left, its value
is increased ten times ; and consequently, if we begin at
the left hand side, and go on towards the right, we see that
every figure has a value ten times less than the same figure
placed one place nearer the left, each number expressing
tenth parts of the number next it to the left. Hundreds are
just tenth parts of thousands ; tens are tenth parts of hun-
dreds ; and units are tenth parts of tens, &c. Now. the
same 3333, with a point placed before any of its figures,
would still have the same property of each figure towards
the right, having a tenth part of the value it would have
had in the next place towards the left : that is to say, the point
has no effect in altering the relative value of the figures ; but
it has this effect, that the figure which stands at its right
hand would signify units : thus, 33-33, where we have
the same figures as before, with a point placed betwixt the
middle two, and from what has been said, we conclude that
DECIMAL FRACTIONS. 23
the 3 to the left of the point is units. From this it follows
that the next 3 on the right of the point is tenth parts of
unity, and the 3 following that again tenth parts of a tenth
part of unity, or hundredth parts. Had it been written
thus : 3.333, the last three to the right of the point would
have been a tenth less again, &c ; so that all the figures
that follow the point to the right are less than units, conse-
quently, they are fractional ; and from their decreasing by
tenths each place, they are called Decimal fractions from
the Latin word decem, ten. Thus, then, T \ may be written '3.
14. It is to be observed here, that the use of the cipher
(0) is in decimals quite similar to what it is in whole
r. ambers, where its only use is to remove some figure from
the units' place, and therefore alter its value tenfold. Thus,
in the number 40, the cipher of itself signifies nothing, but
serves to remove the 4 to the tens' place. Had it been 04
here the cipher is of no use, because there is no figure to
remove beyond it from the units' place. The same is true of
any number of units. Now, we have seen that '3 is just T 3 ^
and, from what has been said, it will follow, that .03 is three
hundredth parts, or T 7 , as the cipher in '03 removes the '3
a place farther from the units' place towards the right, and
(No. 13) makes it ten times less in value than it would have
been had it been one place nearer the left ; or, it is now
tenth parts of a tenth part. For the same reason '003 is
the same as -j-^^.
15. The number 33 is read thirty and three, and '33
is read three tenths and three hundredths, or sometimes
thirty-three hundreds. Now, T 3 , added to jfa give (No.
6) fVisV which, simplified, is T 3 ^, (No. 4.) If we wished to
write jjjVo in the other form, it is done simply thus : point
in tenth's place, in hundredth's place, and 3 in thousandth's
place ; that is, -003. Take, now, - t \ and T |^ ; adding, then,
by No. 6, we get-j^, ^, simplified ~ s , which, written with the
point, is simply *46. We may now see, that any number
placed after the decimal point is a fraction ; which may be
expressed by a numerator which is that number, and a de-
nominator consisting of 1, with as many ciphers annexed
as there are figures in the numerator : thus, '3034 is the
same thing as T VVoV
16. These simple statements being understood, all that
follows will be easy. The principle being kept in mind,
that the numbers to the one side of the point have the same
24 ARITHMETIC.
relation to one another as those on the other, e\ery figure
on the one side of the point as well as on the other, being
ten times greater than it would have been in the next place
to the right, and ten times less than in that to the left.
17. To add decimal fractions, we proceed just as.in whole
numbers, placing units under units, and consequently points
under points, and carrying to each new column to the left,
by 1 for every ten in the column already added. As ^ may
be written T S 7 or *5 ; 7 may, therefore, be written 7'5 ; 4^
may be written 4-5. Now, add 7*5 and 4-5 by
the rule we have given, and we will obtain a' 7"5
result which must be correct, as may be 4-5
proved by principles laid down in the former 12'0
chapter. Here we have kept the points under
each other, and put a point in the answer just under the
others, and the sum is 12, with no decimal fraction. Take 7 J
and bring it to the form of a common vulgar fraction, by the
principle, No. 1 1 , and it will be y ; do so likewise with 4 and
we get | ; they have a common denominator, and add them
by No. 6, we have 2 ^ 4 , now, this fraction, by No. 4, is equal
to l ^, or 12, the same as before. Take now 135*7, and 1-23,
and -764, and 9-102, and 8-003, and -035; to find their
sum. Here we place, as before, all the points under each
other, and proceed as in addition of whole numbers, carry-
ing by tens and pointing the sum in the line under the other
points :
135-7
1-23
764
9-102
8-003
035
154-834
18. Subtraction is managed in like manner as in common
numbers, the same attention being paid to the points.
Thus, subtract 33-785 from 1967 : 32 ;
they are placed thus, and subtracted as 1967*320
in whole numbers, the point in the 33-785
answer being placed in a line with the 1933-535
others. It is to be observed, that there
are more decimal places in the under number than 'n th"
DECIMAL FRACTIONS. 25
upper, and the deficiency may be supplied by adding ciphers
to the upper line, which, as there is no significant figure
beyond, does not alter the value of the number.
19. Multiplication of decimal fractions is performed as in
whole numbers, paying no attention to*the points until the
product is obtained, when we point off as many places from
the right hand side of the product, as there are decimal
places in both the multiplicand and the number which mul-
tiplies, or multiplier. Thus, multiply
36-42 by 4-7. Here -174 are pointed 36-42
off as decimals, as there are two deci- 4*7
mal places in the multiplicand and one 25494
in the multiplier in all three. That 14568
this rule is correct, may be inferred 17 1-174
from the results of a former example in
No. 8. Here we multiplied 4 by , and found the pro-
duct to be 2 : now, is equal to T * T , which may be written
5 ; then let us multiply 4 by -5, as directed
above, and we will find th same result, 2 ;
where, by principle of No. 14, the cipher
being pointed off, there remains 2 a whole 2-0
number.
20. Division may be properly defined, the finding of one
number (the quotient), such, that when multiplied by
another (the divisor), will give a product equal to a third
(the dividend). The dividend may thus be viewed as the
product of the quotient and divisor ; hence, the quotient and
divisor should, together, contain as many decimal places as
the dividend. This being observed, the rule will be easily
followed : Divide as in whole numbers, and when the quo-
tient is obtained, point off from the right as many places
for decimals as those of the divisor want of those in the
dividend. Divide 22-578 by 48-6,
the quotient 4-6f|.| is obtained by 48-6)22'578(4-6f f.f
common division, and pointed thus,
because the divisor wants only one decimal place to have
as many as the dividend. In many cases, when the quo-
tient is obtained, there will not be as many figures as make
up the number of decimal places required ; here we must
place one or more ciphers betwixt the point and the quo-
tient figures, so as to make up the number required. Thus,
divide 1-0384 by 236, the quotient is 44 only two places,
whereas there should be four decimals in the quotient ;
8
26 AUITHMETIC.
because there are four in the dividend and none in tha
divisor. We, therefore, place the quotient thus, -0044 ,
and to prove that this is the true quotient, we have only to
multiply it by the divisor, and the product being the same
as the dividend, the operation must be correct.
21. From the great facility with which decimal fractions
may be managed, it is very desirable that we could bring
vulgar fractions to the same form, in order that they might
more easily be wrought with. Now, this may be done on
the principles already laid down : take the fraction --, and,
on the principle of No. 4, multiply both terms by 1000, it
then becomes J , which is equal to I ; divide (No. 4) both
numerator and denominator by 8 ; then 8) --$ ( T Wo which
last fraction is expressed in the decimal notation thus, (on
the principle of No. 15,) '125, which, from the way it has
been derived, must be equal to G. This may, however, be
found more immediately thus : add as many ciphers to the
numerator as you find necessary, and divide by the denomi-
nator thus, 8)1000(-125. If we have only to add one
cipher before we get a quotient figure, we put a point in
the quotient ; but if more, then we put as many ciphers in
the quotient after the point. Thus, -j ; 25)100('04, and
TJ is just T ^, or -04.
22. In many cases the quotient would go on without end;
but it is to be observed, that it is not necessary to continue
any operation in decimals, at least in mechanical calcula-
tions, beyond three or four places, as ten thousandth parts
are seldom necessary to be considered in practice. For
similar reasons, it is unnecessary to give rules for repeating
and circulating decimals : i. e. decimal numbers, when the
same figures recur in some order thus, '3333, or, 142142,
&c., carry them to four places, and it is all that is neces-
sary.
Other applications of these principles will be found in the
next chapter, on compound numbers.
COMPOUND NUMBERS.
23. IN mechanical calculations, we are often concerned
with weights and measures, and it is necessary to know how
to operate with the numbers which express these. The rule
COMPOUND NUMBKRS. 27
given in books of arithmetic are generally very long, and,
therefore, not very easily understood ; yet the steps of the
operation are simple. We shall therefore show the mode of
procedure, in some very easy examples, and the reader will
find no difficulty in applying the principles he may thus im-
bibe to cases more complex.-
24. If we have to add 9 yards 2 feet 6 J' ds - feet "*
inches, to 2 yards 1 foot 3 inches, 8 yards
feet 1 1 inches, long measure. Then we ^
must in this, as in all other cases of com-
pound addition, arrange them in order, -20 1 8
the greater towards the left hand, and the lesser towards
the right ; and there must be a column for every denomina-
tion of weight or measure, in which column the respective
quantities must stand, so that feet will stand under feet, inch-
es under inches, pounds under pounds, and ounces under
ounces, &c. Add now the column toward the right, which
in this example amounts to 20 inches, or 1 foot 8 inches, we
therefore put down the 8 inches under the column of inches,
and add the 1 foot to the column of feet, which comes to 4
feet ; that is, 1 yard and 1 foot. The 1 foot is put down un-
der the column of feet, and the 1 yard is added, or carried, as
it is usually called, to the column of yards, whose sum is 20.
If we have to add 2 tons tons cwt quar lbs> oz<
2 cwt. 1 quar. 17 Ibs. 10 3 2 1 17 10
oz. avoirdupois, to 12 tons 12 10 2 2
10 cwt. 2 Ibs. 2 oz., 2 cwt. 02 1 18 3
1 quar. 18 Ibs. 3 oz., and Q Q Q 911
9 Ibs. 1 1 oz. ; then, from -^ ^ 3 ^ ^
what was remarked above,
they will be put down as in the margin. Then the sum of
the right hand column is 26 oz., which is 1 Ib. 10 oz., we
put down the 10 in the column of oz., and carry the 1 Ib. to
the column of Ibs. which is next ; and this when added
comes to 47 Ibs., that is, 1 quar. and 19 Ibs. ; the 19 is put
in the column of Ibs. and the 1 is carried to that of quars.,
which comes to 3, which not amounting to 1 cwt. we put
down the 3 in the column of quars. and carry nothing to
the column of cwts., which, when added, amounts to 14, this
we put down, and, as it does not amount to 20 cwt. or 1 ton,
we carry nothing to the column of tons ; and when this co-
lumn is added, its sum is 14.
25. In Subtraction the same principle of arrangement is to
28 ARITHMETIC.
be observed, and the lesser quantity is to be put under the
greater. If we have to subtract 1 ton 13 cwt. 2 quars. 17 Ibs
12 oz., from 9 tons 8 cwt. tons cwt quars lbs oz
1 quar. 4 lbs 7 oz. avoirdu- 98147
pois, they are arranged as j 13 2 17 12
in the margin- We begin ~ j^ % 14 TT
to subtract at the lowest
denomination, viz. oz. 12 oz. from 7 oz. we cannot, bu
we add a Ib. or 16 oz. to the 7, which is supposed to be
borrowed from the column of lbs. which stands next it,
towards the left ; now 16 added to 7 makes 23, and 12 from
23 leaves 11, which is put down .in the column of oz. Now
we must pay back to the column of lbs. the pound or 16oz.
which we borrowed, therefore, it is 18 from 4. Here we
have to borrow from the column of quars., and 1 quar.
being 28 lbs. we borrow 28, then 28 and 4 are 32, there-
fore 18 from 32 leaves T4, which is put down, and the 1
quar. paid back to the column of quars.; 3 from 1, we
cannot, and must borrow 1 cwt. or 4 quars., therefore 3
from 5 and 2 remains, which is put down. Add 1 to 13 for the
1 cwt. that was borrowed, then 14 from 8, we cannot, but
borrow 20 from the next column, then 14 from 28 and 14
remains. Pay back to the column of tons the 1 ton, or 20
cwt. which we borrowed, then 2 from 9 and 7 remains, which
is put down.
The same principle holds in other examples, the only va
riation being that the numbers to be borrowed from the next
higher column, will depend upon the relative values of these
columns, which may be known by examining a table of the
particular weight or measure to which the example may refer.
26. In Multiplication, which is only a short way of perform-
ing addition in particular cases ; the principles are nearly
similar : thus, to multiply 3 tons 2 cwt. 2 quars. 6 lbs. 10
oz. by 3; they are arranged tons cwt quars lbs> oz
as in margin. 1 hen the first 3 2 2 6 10
product is 30 oz. or 1 Ib. 3
which is carried to the co- Q ^ r^r rr*
lumn of lbs., and 14 oz.,
which is put down in the column of oz. The product of
the lbs. is 18, and the one Ib. carried is 19, which no
amounting to 28 lbs. or 1 quar., nothing is to be carried to
the column of quars. The product of the quars. is 6,
which is 1 cwt. to be carried and 2 quars. to be put down.
COMPOUND NUMBERS. 29
The product of cwts. is 6, and the one carried from the
former column makes 7, nothing being carried ; the co-
lumn of tons is 9. By examining the following examples,
and referring to the tables of weights and measures, the
general application may be easily inferred. See Appendix
to Arithmetic.
Degrees, min. seconds. yds. feet. inch. 8th parts,
23 14 17 17 2 9 6
6 ' 5^
139 ^5 42 89 _2 _0 _6
Carry by "60 60 ~3 12 ~8
27. It may not be out of place here to notice, Duodeci-
mal, or what is commonly called Cross Multiplication ; which
is very useful to artificers in general, in measuring timber, &c.
The foot is divided into 12 inches, each inch into 12 parts,
and each part again into 12 seconds ; these last, however,
are so small, that they are generally neglected in calculation.
If we wish to find the surface of a plank, whose breadth
is 1 foot 7 inches, and length 8 feet 5 8 5
inches, we place the one under the other, \ 7
feet under feet, inches under inches, &c., Q ^
as in the margin. Multiply the inches . , ~ , .
and feet in the upper line, by the feet
in the under line, placing the product ^
of the inches, under the inches, and that of the feet, under
the feet. Then multiply the inches and feet, of the upper
line, by the inches in the under line, placing the product
one place further towards the right, and carry by twelves
where necessary ; as in this example, 7 times 5 is 35, that
is, two twelves and 11 over; the 11 is put down, and the
2 added to the product of the next column, 7 times 8 is
56, and the 2 carried makes 58, that is four twelves and
10 over ; the 10 is put down, and the 4 carried to the next
column. These are now added, observing again to carry
by twelves.
feet. inch. feet. inch, parts.
47 35 4 6
8 4 12 3 4
36 8
164
38
424
6
8
10
1
6
11
9
6
434
3
11
3*
30 ARITHMETIC.
The feet in the example are square feet, but the inches
are not square, as might be thought at first sight, but 12th
parts of a square foot ; and also the numbers standing in
the third place, are 12th parts of these 12 parts of a foot,
and so on.
28. Before we consider the Division of compound num-
bers, it will be necessary to attend a little to the nature of
reduction. This is usually thought by beginners to be very-
perplexing, but a little attention to the principle, will ob-
viate all this apparent difficulty.
In every lineal foot there are 12 inches, and therefore
there will be 12 times as many inches, in any number of
feet, as there are feet; thus, in 8 feet there are 8 times 12,
that is, 96 inches. In every Ib. avoirdupois there are 16
ounces, therefore in 18 Ibs. there are 18 times 16, that is,
288 ounces. So that we multiply the higher denomina-
tion, by that number of the lower which makes one of the
higher, and the product is the number of the lower contained
in the number of the higher, which we multiply. In the pre-
vious examples, feet and pounds are the higher denomina-
tions, and inches and ounces are the lower. From these
remarks it will be easy to see, how we proceed in rinding
the number of parts of an inch contained in 3 yards 2
feet 7 inches, and J- parts, long measure. Bring the yards
to feet, 3 multiplied by 3 are 9, to which we add the 2
feet, which make 11. This brought to inches, is 11 mul-
tiplied by 12, or 132, to which we add the 7 inches, making
139. This brought to parts gives 139, multiplied by 8,
that is, 1112, to which we add the 5 eighth parts, making
1117 the answer.
The examples subjoined are managed in a like manner;
the multipliers varying with the kind of weight or mta-
eure.
cwt. quar. Ibs. acres. roods. poles.
27 1 22 22 3 24
4 mult. 4 mult.
108 quars. 88 roods
1 add 3 add
109 quars. 91 roods
28 mult. 40 mult.
3052 Ibs. 3640 poles
22 add 24 add
3074 Ibs. 3664 poles.
COMPOUND NUMBERS. 31
The work is reversed, when we wish to ascertain how
many of a higher denomination are contained in any num-
ber of a lower. Thus, in 1440 inches, long measure, there
will be one foot for every 12 inches, we therefore divide
1440 by 12, and the quotient will be the number of feet,
that is, 120 feet. Then there is no remainder, but if there
had, it would have been of the same kind with the dividend,
that is, inches. In the same way lind how many tons, cwts.
quars. and Ibs., are contained in 12345678 oz.
oz. in 1 lb. 16 ) 12345678 ounces.
Ibs. in 1 quar. 28 ~)771604 Ibs. 14 oz.
quars. in cwt. 4 )27557 quars. 8 Ibs.
cwt. in 1 ton 20 )6889 cwt. 1 quar.
344 tons 9 cwt.
The answer therefore is 344 tons 9 cwt. 1 quar. 8 lb. 14
oz. which may be proved by reducing the work to ouncef
by the method given above.
29. It is frequently of great use, to express compound
numbers fractionally ; thus, so many feet and inches as the
fraction of a yard. What fraction of a yard is 2 feet 8
inches ? Now, from what has been said on vulgar fractions,
it will be easily seen that one yard is here the unit, or de-
nominator of the fraction, which must of course be brought
to inches. Now there are 36 inches in one yard, which
must be the denominator of the fraction, and the numera-
tor will be the quantity taken ; that is, 2 feet 8 inches re-
duced to inches, or 32 inches. The fraction therefore is
ff, or simplilied , which, turned into a decimal, is 0*8888,
one yard being 1. So likewise, what fraction of a cwt. is
2 qrs. 14 Ibs. 3 oz.? This last reduced to ounces is 1123,
which is the numerator of the fraction, and the denomina-
tor is 1 cwt. reduced to oz., or 1792 oz. ; the fraction is
therefore jif ?, which is expressed decimally 0-6264. We
think that these examples will be sufficient to show the
mode of procedure, and it remains for us to consider the
reverse of this; to estimate the value of such fractions in
terms of the weight or measure to which they refer.
3C. It will be easily seen, that one-half of a foot is twelve
times greater than one-half of an inch, or that any given
part of a foot, is a twelve times greater part of an inch ; thus,
5 of a foot is y of an inch ; so that to bring any fraction of
32 ARITHMETIC.
a foot to the fraction of an inch, we have only to multiply the
numerator by 12. So likewise $ of a pound avoirdupois, is
*/, of an ounce, and ^ of a yard is -^ of a foot, or 3 -j of an
.'nch; and if we divide the numerator by the denominator,
we get in the last example ^ of a yard, equivalent to 7y
inches.
What is the value of 5 of 1 cwt. ? By applying the fore-
going principle it will be found that 5 of 1 cwt. is A of a
quar., or a 28 times greater part of 1 lb., that is l ^- 2 ; that is
37 5 Ibs. also | of 1 lb. is 16 times 5 of an ounce, or y,
equal to 5| ounces.
31. It will generally be found best to express these deci-
mally, thus, the last example will be i of a cwt. or 0.333
of a cwt., or 1.333 of a quar., or 37.666 of a pound. Thus
it appears that any fraction of a cwt. is 4 times greater than
a like fraction of a quarter, and any fraction of a quarter
is 28 times greater than a similar fraction of a pound ;
hence, to reduce a fraction of a higher to its value in a
lower denomination, we multiply the numerator of the frac-
tion, by that number which expresses how many of the lower
are contained in one of the higher, while the denominator
remains unaltered. On the other hand, to bring a fraction
from a lower to a higher denomination, the numerator re-
mains the same ; but we multiply the denominator by that
number which expresses how many of the lower is contained
in one of the higher. Thus i of an inch is J ff of a foot, or
-j-^g- of a yard ; or expressed in decimals 0.3333 of an inch,
or 0.0277 of a foot, or 0.00924 of a yard.
32. On a like principle the value of a decimal expressing
weight or measure, may be determined, simply by multiply-
ing the decimal by that number of the next lower denomi
nation, which is contained in one of the higher, and cutting
off the proper number of decimals in the product, thus :
37689 of a cwt.
4
1.50756 quarters.
28
14.21168 pounds.
3.38688
Here it will be observed, that the integers or whole num
POWERS AND ROOTS. 33
bers cut off are not multiplied, and the value of .37689 of
a cwt. is 1 quar. 14 Ibs. 3.386 oz.
We will conclude this chapter on compound numbers,
with some remarks on Division. The same arrangement ia
to be observed here as in addition ; the greater quantity be-
ing towards the left of the lesser.
Let it be required to divide 13 yards 2 feet 8 inches by
4. We say 4 in 13, 3 times and 1 over, that is one yard,
which must be reduced to feet, the next lower denomination;
that is 3 feet, and the 2 feet are five feet now 4 in 5, 1 and
1 over, which last being a foot, must be reduced to inches ;
it is therefore 12 inches, and the 8 make 20 ; then 4 in 20,
times ; the answer therefore is 3 yards, 1 foot, 5 inches.
yards. feet. inch. yards. feet. inch.
3) 16 2 9(5 1 11
15
~T
3 mult.
3
2 add
T
3
2
12 mult
24
9 add
3) 33
33
POWERS AND ROOTS.
32. THE square of any number is the product of that
number multiplied by itself: thus, the square of 2 is 4, the
square of 4 is 16, the square of 5 is 25, <fcc. The cube of
any number is the product of that number multiplied twice
by itself : thus, the cube of 2 is 8, the cube of 3 is 27, the
cube of 4 is 64, &c. On the other hand, when we talk 9 s
34 ARITHMETIC.
the square and cube roots of any numbers, we mean such
numbers that, when squared or cubed, will produce these
numbers : thus, 2 is the square root of 4, 3 is the square root
of 9, and 4 is the square root of 16, &c. In like manner, 3
is the cube root of 27, 4 the cube root of 64, 5 the cube root
of 125, &c. The cube and cube root are said to be of higher
order than the square and square root ; and there are higher
orders than these, with which we shall not concern ourselves,
as they will not occur in our calculations. The method of
raising any number to the square and cube powers, will be
sufficiently obvious from what has been said above ; but the
method of extracting the square and cube roots is not by any
means so easy. We shall give the rules for the extraction
of these roots ; and as they are long, we would recommend
the beginner to compare carefully each step in the example,
with that part of the rule to which it refers ; and by doing
so attentively, he will find that the greater part of the diffi-
culty will vanish.
33. The rule for extracting the square root is this :
First Commencing at the unit figure, point off periods
of two figures each, till all the figures in the given number
are exhausted. The second point will be above hundreds
in whole numbers, and hundredths in decimals.
Second If the first period towards the left be a complete
square, then put its square root at the end of the given num-
ber, by way of quotient ; and if the first period is rtot a com-
plete square, take the square root of the next less square.
Third Square this root now found, and subtract the
square from the first period ; to the remainder annex the
next period for a dividend, and for part of a divisor double
the root already obtained.
Fourth Try how often this part of the divisor now found
is contained in the dividend, omitting the last figure, and
annex the quotient thus found, not only to the root last
found, but also to the divisor, last used.
Fifth Then multiply and subtract, as in division ; to the
remainder bring down the next period, and, adding to the
divisor the figure of the root last found, proceed as before
Sixth Continue this process till all the figures in the
given number have been used ; and if any thing remain,
proceed in the same manner to find decimals adding two
ciphers to find each figure.
;.s AND ROOTS.
The square root of 365 is required.
305(19-1049
1
29
9
205
2(U
400
38204
4
190000
152816
382089
9
3718400
3438801
382098 I 279599
'fiie square root of 2 to six places of decimals is required.
. 2 ( 1-414213
1
24
4
100
96
281
1
400
281
2824
4
11900
11296
28282
2
60400
56564
282841
1
383600
282841
2828423 j 100759
34. The easiest rule for the extraction of the cube root
is tliis :
By trials, take the nearest cube to the given number,
whether it be greater or less, and call it the assumed cube :
thus, if 29 was the given cube whose root was to be ex-
tracted, then, 3 times 3 times 3, or 27, is the nearest less
cube, and 4 times 4 times 4, or 64, is the nearest greatest
cube ; 27 is the nearer of the two, therefore, 27 is the as-
sumed cube.
Add double the given cube to the assumed cube, and
multiply this sum by the root of the assumed cube, and this
product divided by the given cube, added to twice the
39 ARITHMETIC,
assumed rube, Trill give a quotient which will be th r<e*
quired root, nearly,
By using, in like manner, the cube of the last nnswer, as
an assumed root, and proceeding in the same manner, we
will get a second answer nearer the truth than the first, aiuS
*o on.
Find the cube root of 21 035-8.
If 20 is assumed, its eube is 8000 ; if 30, its cube is 27000,
the one a great deal too small and the other too great : let us
therefore try some number between them, as 27 ; the cube
of this is 19683, which we shall call the assumed cube ; then,
twice the assumed cube is 39366 twice the given cube i
12071-6.
Therefore, the sum of the given cube and twice the as-
sumed cube is 60401-8, and the sum of the assumed cube
and twice the given cube is 61754-6.
Wherefore, by the rule,
61754-6
27
4322822
1235092
60401-8) 1667374-2(27-6047
This quotient is the root nearly ; and by using 27*6047 in
the same way that we used 27, we will get an answer still
nearer the true root. For a Table of Powers and Root*,
see Grier's Mech. Diet.
THE SLIDING RULE.
35. We are indebted for the invention of this useful in
strument to Edmond Gunter. It is a kind of logarithmic
table, whose great use is to obtain the solution of arithme-
tical questions by inspection, in the multiplication, division,
and extraction of the roots of numbers. It consists of two
equal pieces of boxwood, each 12 inches long, joined toge-
ther by a brass folding joint. In one of those pieces there
is a brass slider. On the face of this instrument, there are
engraven four lines, marked by the letters A, B, O, and D :
at the beginning of each line, the lines A and I) being
THE SLIDING RULE. 37
marked on the wood part of the rule, and B and C on the
brass slider.
36. Before the use of the sliding rule can be explained,
it is necessary that a correct idea, should be formed of the
method of estimating the values of the several divisions on
these lines. Let it be observed, then, that whatever value
is given to the first 1 from the left, the numbers following,
viz. 2, 3, 4, 5, Ac., will represent twice, thrice, four times,
<fcc., that value. If 1 is reckoned 1 or unity, then 2, 3, 4, &c.,
will just signify two, three, four, &c. ; but if 1 is reckoned
iO, then 2, 3, 4, <fcc., will represent 20, 30, 40, &c. If the
first 1 is reckoned 100. then 2, 3, 4, &c., will represent 200,
300, 400, &c. The value of the 1 in the middle of the line
Is- always ten times that of the first 1 ; the value of the
second 2 is ten times that of the first. 2 : so that if the value
of the first 1 be 10, that of the second 1 will be 100; the
first 2 will he 20, and the second 2 will be 200, <fec. The
value of these divisions being understood, we may now at-
tend to the minute divisions between these. Now, on the
lines A, B, and C, there are 50 small divisions betwixt 1 and
2, 2 and 3, 3 and 4, &c. ; and it follows, from the nature of
the larger divisions, that if the first 1 be reckoned 1, or
unity, each of these small divisions between 1 and 2, 2 and 3,
&c., will be -j'g, or '02 ; and supposing still the first 1 to be
unity, then the small divisions from the second 1 to 2, 2 to
3, <kc., will each be ten times greater than a T ' 5 , or P 02, that
is, each of them will be -^, or j, or '2. In the same way,
if the first 1 represents 100, the -first 2 will be 200 ; if the
second 1 be 1000, the second 2 will be 2000, <fec. ; and on
the same principle as above the small divisions or 50th parts
will represent each ^ of 100, or 2, in the *:rst half, or from
the first 1 to 2, 2 to 3, &c., and 3 ' ff of 1000, or 20, in the second
half; or from the second 1 to the second 2,2 to 3, &c.
37. These divisions being understood, we may proceed to
show the method of using this rule in the solution of arith-
metical questions.
38. To find the product of two numbers :
Move the slider, so that 1 on B stands against one of the
factors on A ; then the product will be found on the line A,
against the other factor on the line B.
Thus, to find the product of 3 by 8 :
Set 1 on B to 3 on A ; then against 8 on B will be found
the product 24 on A.
4
38 ARITHMETIC.
For trie product of 34 by 16 :
Set 1 on B against 16 on A, then look on B for 34, and
igainst it on the line A will be found the product 544.
39. To find the quotient of two numbers :
This may be done in two ways, either set 1 on the slider
B against the divisor on A, then against the dividend on A
the quotient will be found on B. Or, set the divisor on B
against 1 on A, then the quotient will be found on A against
the dividend on B ; therefore, in general, it is to be remem-
bered, that the quotient, must always be found on the same
line on which 1 was taken, and the divisor and dividend on
the other line.
Thus, to find the quotient of 96 divided by 6 :
Move the slider till 1 on B stands against 6 on A ; then
the quotient 16 will be found on B against the dividend
96 on A.
In like manner, to find the quotient of 108 divided by 12,
we may take the latter form of the rule, thus :
Set 12 on B against 1 on A ; then on the line A will be
found the quotient 9 agains! 96 on B.
40. To solve questions in the rule of three or simple pro-
portion
Set the first term on the slider B to the second on A ;
th$i on the line A wll be found the fourth term, standing
against thj third term on B.
If 4 Ibs of brass cost 36 pence, what will 12 Ibs. cost ?
Move the slider so, that 4 on B will stand against 12 on
A ; then against 36 on B will be found the fourth term 108
on A.
41. To extract the square root:
Move the slider so, that the middle division on C, which
is marked 1, stands against 10 on the line D, then against the
given number on C the square root will be found on D.
It is to be observed before applying this rule, that if the
given number consists of an even number of places of figures,
as two, four, six, &c., it is to be found on the left hand part
of the line C ; but if it consists of any odd number of places,
as three, five, seven, &c., it is to be found on the right hand
side of C, 1 being the middle point of the line.
To find the square root of 81 :
Here the number of places are even, being two ; therefore,
the number 81 is sought for on the left hand side of the
line C
.MAKKS 01' CO NTH ACTION. 39
Set 1 on C against 10 on D ; then against 81 on C will
be found 9, the square root on D.
For the square root of 144 :
Set 1 on C to 10 on D ; then against 144 on C will be
found the square root 12 on D.
42. To rind the area of a board or plank :
The rule is, to multiply the length by the breadth, the
product will be the area ; therefore, by the sliding rule,
Set 12 on P against the breadth in inches on A ; then on
the line A will be found the surface in square feet, against
the length in fe (; t on the line B.
To find the area of a phnk 18 inches broad and 10 feet
3 inches lonjj :
Move the slider so that 12 on B stands against 18 on A ;
then will 10| on B stand against 15* on A, which 15| is
square feet.
This may be proved by cross multiplication.
10 3
_1 G
10 3
5 1 6
15 4 6
43. For the solid content of timber.
The rule is to multiply length, breadth, and thickness all
together.
Set the length in feet on C to 12 on D ; then on C will
be found the content in feet against the square root of the
product of the depth and breadth in inches on D.
What is the content of a square log of timber, the length
of which is ten feet, and the side of its square base is 15
'nches.
Set 10 on C against 12 on D ; then will 15 on D stand
against the content 15| on C.
44. Other particulars on the measurement of timber wil
be given hereafter, when we come to Mensuration.
MARKS OF CONTRACTION.
45. WE earnestly request that particular attention be paid
to this chapter, not because it is difficult, but because it is of
the greatest importance to the clear understanding of what
4C ARITHMETIC.
follows in this book, and contributes greatly towards its
shortness and simplicity.
46. When we mean to say that one thing is equal to an-
other, we use this mark = thus, 3 added to 5 = 8, is read
thus, 3 added to 5 is equal to 8.
47. But the words, added to, may also be represented by
the mark + thus, 3 + 5 = 8, is read, 3 added to, or plus, 5 is
equal to 8.
48. So likewise the difference of two numbers may be
represented by the mark , which is a short way of express-
ing the word subtract, thus, 5 3=2, is read from 5 sub-
tract 3 the difference is equal to 2 ; and thus, 3 + 6 2=7
is a short way of writing, to 3 add 6 and subtract 2, the result
is equal to 7.
49. After the same manner the mark x is used instead
of the words multiply by, thus, 3x2 = 6, is read 3 multi-
plied by 2 is equal to 6.
50. To show that the operation of division is to be per-
formed this mark is sometimes used, viz. -=-, which is a short
way of writing the words, divided by, thus, 15-r-3 = 5, is read
15 divided by 3 is equal to 5 : but we will in general place
the divisor below a line with the dividend above it, on the
principle stated in vulgar fractions, thus, y =5 the same
as 15-7-3=5.
51. The square of any number or quantity is marked by a
small a placed at its upper right hand corner, thus, 3 8 =9 is
read, the square of 3 is 9. The cube is marked by a 3 placed
in the same way, as 3 3 =27, that is, the cube of 3 is 27.
The square root is noted in a similar manner by the frac-
tion ^ placed in the same way, as 9^ = 3, and so likewise the
cube root, as 27* = 3 ; but the square root is often denoted
byv/placed before the number or quantity, thus, v /9 = 9*=3,
and the cube root, in like manner, by <$f, thus, v ! /27 =27^=3.
52. Parenthes'es ( ) are used to show that all the numbers
within them are to be operated upon as if they were only one ;
thus, 3 + 2x5, means that 3 is to be added to the product of
2 and 5, that is, the amount of this is 13 ; but (3 + 2) x5,
means that 3 and 2, that is, 5, is to be multiplied by 5, and
the result will be 25 ; a very different thing from what it
was before, which arises entirely from the use of paren-
theses. In like manner 3 + 2 3 =7, but (3+2) 2 =25 ; here
as in every other case, the whole of the numbers wittnn
the parentheses are taken as one whole, and as such **
. 41
affccted by whatever is without the paretttUese*. The
same thing is often marked by drawing a line over all the
numbers or quantities to be taken as one whole ; thus, instead
of (3-f2)x5, we may write 3-f2x5; also (6x4) 3x2
is the same as 6x4 3 X 2, both being equal to 42.
f>3. The rule for tin; measurement of the surface of timber,
given in our remarks on the sliding ruic, may be expressed
thus, length xbreadtk s area; and the rule for simple pro-
portion, to be given in the next chapter, may also be writtea
thus :
Second term X third term,
=iourth term.
first tenu,
54. It is obvious that this is merely a kind of short hand
which might he carried still farther ; for instance, in the
last example we rnijrla make F stand for the first term,
S for the second, T for the third, and for the last, aad
the rule woul'i then bo
55. We again insist that the young reader will read this
chapter carei'uiiy over.
PROPORTION.
56. When four numbers following eaoh other are such
that the first is as many times greater or less than the
second, as the third is greater or less than the fourth, they
are said to be in proportion; thus, 2, 4, 3, 6, usually written
thus, 2 : 4 : : 3 : 6 ; the taark : being put for the words, is
to, and : : for, as, so that this would he read, "2 is to 4 as 8
is to 6. Here the first is half the second, and the third is
half the fowtVi, and they are therefore in proportion; bu<
they may be arranged otherwise ant! yet he in proportion,
thus, 4 : 2 : : 6 : *. where the first is twice as large as the
second, and the third is twice as lane as tfce fourth. In ali
cnses the two middle terms are called the means, and the
two oster terms are called fhe extremes. The product of
"the two means is equal to that of the two extremes, thus i
the last example, 2x6 = 1-2, and 4x3 = 12. Now, if w
wanted ttae last terra, to wit, 3, it couid easily he &ead by
42
means of this property of numbers in proportion. If we h;*4
only three terms given, ns 4 : 2 : 6, to find the fourth in
proportion, which is the last extreme, and 4 is the nrst
extreme. Now, we must lind $uh a number, that, when
multiplied by 4, the product will be equal to the product of
the means ; 2x6=12, to find such a number we have only,
by the definition of division, to divide the product of the
two means, viz. 12 by the first extreme 4, and the quotient
3 will be the answer. So universally 6:9:: 12: where
the last term will be found, as before, by multiply ing- the
two means 12x9=108, and dividing the product 108 by
the first extreme 6, the quotient will be the last extreme 18,
hence 6:9:: 12,: 18. The rute may be expressed simply
thus: let F stand for the first term, S the second, T the
third, and the last, then we have = , and this
r
rule holds true whether the numbers be whole or fractional ;
and here it may be observed, that it will in most, if not i
all cases, be best to turn all vulgar fractions, when they
occur, into deciaials ; thus, 2 : 3 : : 6-1 : or f : V : : V :
2* = -
31 = y = 3-666 ^ 2-5 : 3-666 : : S'25 :
>i 5 O.^ "1
"1 2 43
)| = U = 3-666 I
% = 6-25 J
Here the mode of determining- the fourth term is the
same in all; the two means being x, and their product --,
by the first term. This is usually called the rule of three,
and is of the utmost utility in practical arithmetic. We
shall now show how it is to be applied.
If we pay 40 pence for 2 feet of wood, how much will we
pay for 6 feet at the same rate ? Here it is clear we will
pay in proportion to the quantity of wood ; for as many
times as we have 2 feet, \ve will pay so many times 40 pence ;
that is, the price will be in proportion to the quantity of
wood. So that we may say,, as the one quantity of wood is
to another quantity, so will be the price of the first quantity
to the price of the second. Hence the terms in the question;
will stand arranged thus : 2 : 6 :: 40 : 120, which term 120
is the price oi' 6 feet, and is found by the rule given above t
57. In every question in simple proportion, there wiB
always be thsee terras, one of *vbtth io tlve same kind
COMI'Ut Mi) I'KOI'OUTION. 4J
with the answer sought, whether it be money, measure,
time, force, or any thing, which term in the question we
put in the third place ; :i< in the h^t question the answer
w:is to be money, and therefore the money in the question,
40 pence, was placed as the third term. When this is done,
we next consider whether the answer will be greater or less
than the third term, and place the greater or less of the
other two terms next it in the second place, and the other
one first, as the answer may require ; after which, employ
the rule given above to find the answer.
58. As, for ex'ample, 40 men will'do a piece of work in
15 days, in how many days will 20 men do the same?
Here the answer must be days ; consequently, 15 goes in
the third term, and 20 men will take more time than 40 to
do it, therefore we must put the greatest in the second place,
and the least in the first ; and it therefore stands thus :
20 : 40 : : 15 : the answer 30, which is found by the rule.
40X15 ^30.
COMPOUND PROPORTION.
51 COMPOUND PROPORTION depends entirely on the sam
principles as simple proportion. For instance, if 2 feet of
fir cost 40 pence, what will G feet of mahogany cost, 3 feet
of mahogany being equal in value to 9 of fir. Here we
may find the price of the 6 feet of mahogany as if they
were fir, and it comes out, by the last article, 120 pence,
but 3 is to 9 as the price of fir is to that of mahogany ;
therefore we put the 120, the price of 6 feet of fir, in the
third term, and state the proportion, 3:9:: 120 : 360, the
price of 6 feet of mahogany. The same would have been
more easily found by stating it thus :
6 : 54 : : 40 : 360. Ans.
where the proportion/ are stated under each other, and
multiplied together, which produces 3x2 = 6 and 6x9
= 54, two terms of a new proportion, in the simple rule,
where 4( is the third term ; and this is only the particular
example of a general rule, where we may have as many
44 ARITHMETIC.
proportions as we please reduced to the form of a simple
question in the rule of three. As, therefore, that quantity
which is of the same kind with the required answer is put
in the third term, the rest will be found to go in pairs ;
two expressing relation of price, two relation of quality,
two relation of time, which must be put in proper order in
the first and second terms, as directed for simple propor-
tion. When this is done, all the first terms of these several
proportions are to be multiplied together for a new first
erm, all the second terms together for a new second term,
which being placed with the third, in the form of simple
proportion, and operated upon as there directed, will give
the answer.
Forty boys are set to dig a trench in summer ; 14 spade-
fuls can be dug in summer for 12 in winter; 6 men can do
as much as 13 boys ; and 16 men can do it in 104 days in
winter : how long will the boys take ? Here the answer
is to be, how many days? We have in the question 104
days ; the third term, relative of difficulty, 14 spadefuls
and 12 spadefuls ; of strength, 6 men to 13 boys ; relation
of numbers, 16 to 40 ; which will be stated thus :
Relation of number, 40 : 16") makes the timeless.
Relation of difficulty, 14 : 12 y :: 104 makes the time less.
Relation of strength, 6:13J makes the time greater
Product, 3360 : 2496 : : 104 : 77 T V F days, Ans.
ARITHMETICAL AND GEOMETRICAL PROPORTIONS
AND PROGRESSIONS.
60. THE subject of this chapter is often referred to in
elementary books on mechanical science ; and for this rea-
son, we shall draw the attention of the reader, for a little
while, to the subject.
61. When we inquire as to the difference of two num-
bers, we inquire for their arithmetical ratio ; but when we
inquire as to the quotient of two numbers, we inquire for
their geometrical ratio. Thus, 12 3 = 9 and 12 -j- 3 =
4 ; here 9 is the arithmetical ratio of 12 and 3, and 4 is the
geometrical ratio of the same numbers. From this it will
be seen, that ratio and relation are terms which have the
same signification.
PROPORTIONS AND PROGRESSIONS. 45
62. When four numbers follow each other, and are such
that the difference of the first two is the same as, or equal
lo, the difference of the last two, these numbers are said to
be in arithmetical proportion; thus the numbers 12, 7, 9, 4,
form an arithmetical proportion, because the difference of
12 and 7 is the same as the difference of 9 and 4, both being
5. The numbers in an arithmetical proportion may be
varied in their position, but still the result will be an arith-
metical proportion ; for instance, 12, 7, 9, 4, may h.e written
12, 9, 7, 4, or 9, 12, 4, 7 : but the most remarkable pro-
perty of arithmetical proportion is this, that the sum of the
first and last terms is always equal to the sum of the second
and third ; thus, 12 + 4 = lt> and 9 + 7 = 16; and from
this it evidently follows, that to find the fourth term, we
add the second and third terms together," and from their
sum subtract the first; the remainder is the fourth term.
63. An arithmetical progression is a series of numbers
such, that, in taking any three numbers in succession, the
difference of the first and second is the same as the differ-
ence of the second and third ; thus, 1, 2, 3, 4, 5, 6, 7, 8, or
14, 12, 10, 8, 6, 4, 2, where the difference of the succeeding
numbers in the first is 1, and in the second 2. As the num-
bers in the first increase from the beginning, it is called an
increasing arithmetical series, or progression, and as they
decrease, from the beginning, in the second example, it is
called a decreasing arithmetical progression, or series.
64. Let us place any one of these progressions above
itself, in this manner :
2
14
4
12
6
8
8
10
6
12
4
14
2
16 16 16 16 16 16 16
writing the same progression as increasing and decreasing 1
the respective terms of the one being directly under the re
spective terms of the other in columns, as above, the lowest
line of the three being the sums of the several columns,
which are all seen to be 16. Now, it will be obvious, that
the first column consists of the first and last terms of the
series, 2, 4, 6, <fec., with their sum, which is 16; the second
column consists of the first but one and the last but one of
the terms of the same series, together with their sum, which
is likewise 16. The third column consists of the first but
two and the last but two terms, with their sum, which agaic
16 AK
is 16. We may therefore infer, that, in an arithmetical
progression, the sum of any two terms, equally distant from
the first and last, is equal to the sum of any other two terms
which are equalry distant from the first and last, or equal to
the sum of the first and last. It will also be seen, that the
under line, or sum of the two series, is therefore equal to
twice the sum of one of the progressions. Now, there are
seven sixteens, or 112, which is twice the sum of the pro-
gression^therefore 2)112(56 is the sum of the progression.
65. It is also apparent, that if any term be wanting, that
term may be found by adding the common difference, or
arithmetical ratio, of the progression, to the term going be-
fore the term sought, or subtracting it from the term which
follows, if the series is increasing, but the reverse if decreas-
ing. Thus, 2, 4, 8, the term awanting between the 4 and 8,
may be supplied, either by adding the common difference,
2, to the 4, or subtracting it from the 8, and we thus get 6.
The same may be found by taking the sum of 'the terms on
each side of the term sought, and dividing by 2 ; thus, 4 -(- 8
= 12, then 2)12(6, the same as before ; so, likewise, 3, 5,
7, 9, 13. To fill up the term awanting between 9 and 13,
we have 9 + 13 = 22, therefore 2)22(11, which is the
number sought, and it is called the arithmetical mean.
66. The quotient of two numbers is their geometrical
ratio, and thus a fraction, as j\, expresses the ratio of 6 to
12, and therefore 1 : 2 : : 6 : 12 is the same thing as 5 = T ^.
We thus get another view of the rule of three, and it is use-
ful to view any subject of this kind in different ways, as by
so doing we acquire a more accurate and extensive know-
ledge of its nature and application. The limits of this book
will not permit us to dwell on this subject, as we have dis-
cussed the subject of proportion in a former chapter.
67. In a series of progression of numbers, as 2, 4, 8, 16,
32, 64, where the quotient of any term, and that which fol-
ows it, is equal to the quotient of any other term, and that
which follows it, such progression is said to be geometrical.
68. Let us take the geometrical progression, 2, 6, 18, 54.,
162, and write it as we did the arithmetical, both as in :a-
f reasing and decreasing series, thus :
2 6 18 54 162
162 54 18 6 2
324 324 324 324 324
P.iOI'OIM ION.-, AND
47
Here we observe that the product of the terms of each
column is the same, whatever column we take; and we
arrive at a knowledge of the fact, that the product of the
first and last terms is the same as the product of any other
two terms, one of which is as many places distant from the
first as the other is dista"* from he last term.
69. If one term in t 1 **. above se.'ics were wanting, for in-
stance the second, that is. t, take the terms on each side of
it, and find their product, 2 x IB = 36, now the square root
of this, or 6, will be the number sought, which is called
the geometrical mean. In like manner we might Knd the
.etrical mean between 18 and 162 ; thus, 18 x 102 =
29 IB, th.e square root of which is 2916.| = 54, the number
sought. The geometrical mean is sometimes called the
mean proportional.
70. The sum of any geometrical series may be found
thus :
i The greater extreme X ratio) less extreme,
-= the sum of series,
ratio 1
thus the sum of the last series is
(162x3) 2 480 2 484
3-1 = ' -T- = 1T = 242 ' the sum '
71. Terms relating to proportion often occur in books
read by mechanics, of which it would be useful to know
the signification; and, to prevent their being misapplied,
we give the following illustration. If there be four num-
bers in proportion, as 4 : 16 :
Directly, 4
Alternately, 4
Inversely 16
C Compounded,.. .4 -f- 16
{That is 20
5 Divided, 4
{That is 12
("Con verted,.. ..4
{That is, 4
Also, 4
That is 4
$ Mixed, 4 4- 16
{That is, 20
To these may be added, duplicate rulio, or ratio of th
16:
3 : 12, then,
16
3 12
3
16
12
4
12
3
16
16
3 4- 12
12
16
15
12
-16
16
3 12
12
16
9
12
164-4
3
12 4- 3
20
3
15
16 4
3
12 3
12
3
9
4 16
34-
12
3 12
12
15
9
48 ARITHMETIC.
squares ; triplicate ratio, or ratio of the cubes ; sub-dupli-
cate ratio, or ratio of the square roots ; and sub-triplicate
ratio, or ratio of the cube roots.
POSITION.
72. Posrnox is a rule in which, from the assumption of
one or more fatae answers to a problem, the true one is
obtained.
7H. It admits of two varieties, single position and double
position.
74. In single position the answer is obtained by one as-
sumption ; in double position it is obtained by two.
75. Single position maybe applied in resolving problems,
in which the required number is any how increased or dimi-
nished in any given ratio ; such as when it is increased or
diminished by any part of itself, or when it is multiplied
or divided by any number.
76. Double position is used, when the result obtained by
increasing or diminishing the required number in a given
ratio, is increased or diminished by some number which ig
no known part of the required number ; or when any root
or power of the required number, is either directly or in-
directly contained in the result given in the question.
SINGLE POSITION.
77. Rule. Assume any number, and perform on it the
operations mentioned in the question as being performed on
the required number. Then, as the result thus obtained is
to the assumed number, so is the result given in the ques-
tion to the number required.
Exam. Required a number to which if one half, one-
third, one-fourth, and one-fifth of itself be added, the sum
may be 1644.
Suppose the number to be 60 : then, if to 60 one-
half, one-third, one-fourth, and one-fifth of itself be
added, the sum is 137. Hence, according to the rule, as
137 : 1644 : : 60 : 720, the number required. The truth
of the result is proved by adding to 720 one-half, one-third,
<fec., of itself, and the sum is found to be 1644. The num-
ber 60 was here assumed, not as being near the truth, but
POSITION. 49
being a multiple of 2, 3, 4, and 5; and in this way the
operation was kept free from fractions. By the assumption
of any other number, however, the answer would have
been found correctly, but not so easily. The reason of the
operation is so obvious as not to require illustration.
DOUBLE POSITION.
78. Rule. Assume two different numbers, and perform
on them separately the operations indicated in the question.
Then, as the difference of the results thus obtained is to
the difference of the assumed numbers, so is the difference
between the true result and either of the others to the cor-
rection to be applied to the assumed number which gave
this result. Add the correction to this number, if the cor-
lesponding result was too small ; otherwise, subtract it.
79. A more general rule is this. Having assumed two
different numbers, perform on them separately the opera-
tions indicated in the question, and find the errors of thj
results. Then, as the difference of the errors, if both results
be too great or both too little, or as the sum of the errors,
if one result be too great and the other too small, is to the
difference of the assumed numbers, so is either error to the
corrections to be applied to the number that produced that
error.
80. When any root or power of the required number is
in any way contained in the result given in the question,
the preceding rules will only give an approximation to the
required number. In this case the assumed numbers should
be taken as near the true answer as possible. Then, to ap-
proximate the required number still more nearly, assume
for a second operation the number found by the first, and
that one of the two first assumptions which was nearer the
true answer, or any other number that may appear to be
nearer it still. In this way, by repeating the operation as
often as may be necessary, the true result may be approxi-
mated to any assigned degree of accuracy.
81. It may be further observed also, that the method of
double position, besides its use in common arithmetic, is of
rruch utility in algebra, affording in many cases a very con-
venient mode of approximating the roots of equations, and
rinding the value of unknown quantities in very compli-
rated expressions, without the usual reductions.
82 Exam. 1. Required a number, from which, if 2 be
5
$0 AKITHMKTir
subtracted, one-third of the remainder will be 5 less than
half the required number.
Here, suppose the required number to be 8, from which
take 2, and one-third of the remainder is two. This being
taken from one-half of 8, the remainder is 2, the first result.
Suppose, again, the number to be 32, and from it take 2 :
one-third of the remainder is 10, which being taken from
the half of 32, the remainder is 6, the second result. Then,
the difference of the results being 4, the difference of the
assumed numbers 24, and the difference between 5, the true
result,, and 6, the result nearest it, being 1 ; as 4 : 24 : : 1 : 6,
the correction to be subtracted from 32, since the result 6
was Joo great. Hence, the required number is 26.
83. Exam. 2. If one person's age be now only four
times as great as another person's, though 7 years ago it
was six times as great ; what is the age of each ?
Here, suppose the age of the younger to be 12 years ; then
will the age of the older be 48. Take 7 from each of these,
and there will remain 5 and 41, their ages 7 years ago.
Now, 6 times 5 is 30, which, taken from 4i, leaves an error
of 1 1 years. By supposing the age of the younger to be 1 5,
and proceeding in a similar manner, the error is found to
be 5 years. Hence, as 6, the difference of the errors, (both
results being too small,) is to 3, the difference of the as-
sumed numbers, so is 5, the less error, to 2|, the correction ;
which, being added to 15, the sum, 172, is the age of the
younger, and consequently that of the older must be 70.
Both the rules above given for double position depend
on the principle, that the differences between the true and
the assumed numbers, are proportional to the differences
between the result given in the question and the results
arising from the assumed numbers. This principle is quite
correct in relation to all questions which in algebra would
be resolved by simple equations, but not in relation to any
others ; and hence, when applied to others, it gives only
approximations to the true results. The subject is of too
little importance to claim further illustration in this place.
84. Exam. 3. Required a number to which, if twice
its square be added, the sum will be 100.
It is easy to see that this number must be between 6 and
7. These numbers being assumed, therefore, the sum of 6
and twice its square is 78, and the sum of 7 and twice its
square is 105. Then, as 10578 : 76 : 105100 -18
U'KICIirS AM) MKAS1IKKS. 51
which, beintr taken from 7. the remainder* G - 82, is the
required number, n"-.r!v. ' 'J'o this let twice i:.< square he
added, and tlie result b*90'8448. Then. :is 10599-8448 :
7 6-82 : : 105 100 : '1740 ; which, bein<r taken from 7,
the remainder is (V8254, the required number still mor
nearly ; and if the operation were repeated with this and
the former approximate answer, the required number would
be found true for seven or ei^ht figures.
APPENDIX TO A R IT H M E T I C,
CtlNTAINMNR
TABLES OF WEIGHTS A.\U MEASURES.
ENGLISH.
AVOIRDUPOIS WEIGHT.
Drachms.
10 = 1 Ounce.
256 = 16 = 1 Pound.
7168 = 44H = 28 = 1 Quarter.
286782 = 1792 = 1 12 = 4 = 1 Cwt. .
573440 = 35840 = 2240 = 80 = 20 = 1 Ton.
Tons are marked t. ; hundred weights, r.wt. ; quartets,
yr. , pounds, Ib. ; ounces, oz. ; and drachm?, dr.
TROY WEIGHT.
Grains.
24 = I Pennyweight.
480 = 20 = 1 Ounce.
5760 = 240 = 12 = 1 Pound.
Pounds are marked, lb.; ounces, oz. ; pennyweights,
dwt. ; and grains, gr.
LONG MEASURE.
Barley corns.
'3 = 1 Inch.
36 = 12 = 1 Foot.
108 = 36 = 3 = 1 Yard.
594 =^ 198 = 16-5= 5-5= 1 Pole.
23760 = 7920 = 660 =220 = 40 = 1 Furlong
190080 = 63360 =5280 =1760 =320 = 8 = 1 Mile,
SQUARE MEASURE.
Inches.
144 = 1 Foot.
1296 = 9 = 1 Yard.
39204 = 272| = 30j = 1 P. le.
1568160 = 10890 = 1210 = 40 = 1 Rood.
6272640 = 43560 = 4840 = 160 = 4 = 1 Acre.
SOLID MEASURE.
Inches.
1728 = 1 Foot.
46656 = 27 = 1 Yard.
WINE MEASURE.
Pints.
2 =s 1 Quart.
8 = 4 = 1 Gallon.
336 = 168 = 42 = 1 Tierce.
504 = 257 = 63 = 1-5= 1 Hogshead.
672 = 336 = 84 = 2 = 1-5= 1 Puncheon.
1008 = 504 = 126 = 3 =2 = 1-5= 1 Pipe.
2016 = 1008 = 252 = 6 =4 =3 =2 = 1 Tun
ALE AND BEER MEASURE.
Pints.
2 = 1 Quart.
8 = 4 = 1 Gallon.
72 = 36 = 2=1 Firkin.
144 = 72 = 18 = 2 = 1 Kilderkin.
288 = 144 = 36 = 4=2 = 1 Barrel.
432 = 216 = 54 = 6 = 3 = 1-5= 1 Hogshead.
576 = 288 = 72 = 8=4=2 = 1-5= 1 Puncheon
864=432 = 108 = 12=6=3 =2 = 1-5=1 Bu *
DRY MEASURE.
Pints.
8 = 1 Gallon.'
16 = 2 = 1 Peck.
64 = 8 = 4=1 Bushel.
256 = 32 = 16 = 4 = 1 Coom.
512 = 64 = 32 = 8 = 2 = 1 Quarter.
2560 = 320 = 160 = 40 = 10 = 5=1 Wey.
5120 = 640 = 320 = 80 = 20 = 10 2 = 1 Last
WKIGHTS AND MEASURES. 53
TIME.
GO seconds = 1 minute, 00 minutes = 1 hour,
24 hours = 1 clay, 365] days = 1 year, nearly.
THE CIRCLE.
The circle is divided into 360 equal parts, called degrees.
Seconds.
60 = 1 Minute.
360 = 60 = 1 Degree.
32400 = 5400 = 90 = 1 Quadrant.
129600 = 21600 = 300 = 4 = 1 Circumference.
Degrees, minutes, and seconds, are marked , ', "; as,
4 5' 6" 4 degrees, 5 minutes, 6 seconds.
REMARKS OX ENGLISH WEIGHTS AND MEASURES.
Troy weight is used frequently by chemists, and also in
weighing gold, silver, and jewels ; but all metals, except
gold and silver, are weighed by avoirdupois weight.
175 troy pounds are equal to 144 avoirdupois pounds.
175 troy ounces = 192 avoirdupois ounces.
14 oz., 11 dwt., 15.J grs. troy = 1 Ib. avoirdupois.
18 dwt., 5.j gr. troy = 1 oz. avoirdupois.
3 miles long measure = 1 league.
69-pV English miles = 60 geographical miles.
1089 Scottish acres = 1369 English acres.
A chaldron of coals in London = 36 bushels, and weighs
3136 Ibs. avoirdupois, or nearly 1 ton, 8 cwt.
The ale gallon contains 282 cubic inches, and the wine
gallon contains 231 cubic inches the wine gallon being to
the ale gallon nearly as 1 Ib. avoirdupois to 1 Ib. troy.
By an Act of Parliament passed in 1824, and carried
into execution in 1826, imperial weights and measures were
introduced by this.
The pound troy contains 5760 grains.
The pound avoirdupois contains 7000 grains.
The imperial gallon contains 277'274 cubic inches.
The bushel (dry measure) contains 2218-192 cubic
inches.
To find the value of the old in terms of the new, or the
reverse, the following table of multipliers is given.
5*
54 AKITHMETIC.
Dry. Wine. Ale.
To convert the old into new x 0-96943 0-83311 1-01704
To convert new into old x 1-03153 1-20032 0-98324
Example.fi. What is the value in imperial measure, of
32 wine gallons old measure ?
83311 x 32 = 26-65952 imperial gallons.
In like manner 4 bushels imperial measure = 1*03153 X
4 = 4-12612 old or Winchester bushels.
FRENCH WEIGHTS AND MEASURES.
Old System.
English Troy Grains.
The Paris Pound = 7561
Ounce = 472-5625
Gros = 59-0703
Grain = -8204
Eng. Inches.
The Paris Royal Foot of 12 Inches = 12-7977
The Inch = 1-0659
The Line, or one-twelfth of an inch = -0074
Eng. Cubical Feet.
The Paris Cubic Foot = 1-211273
The Cubic Inch = -000700
MEASURE OF CAPACITY.
The Paris pint contains 58-145 English cubical inches,
and the English wine pint contains 28-875 cubical inches ;
or the Paris pint contains 2-0171082 English pints ; there-
fore to reduce the Paris pint to the English, multiply by
2-0171082.
New System.
MEASURES OF LENGTH.
English Inches.
Millimetre = -03937
Centimetre = -39370
Decimetre = 3-93702
Metre = 39-37022
Decametre = 393-70226
VVKIGHTS AND
55
Hecatometre = 3937-02260
Chiliometre = 3'J. HO "22601
Myriometre = 3P37iW'26014
M. P. Y. Ft. In
A "Decametre is = 00 10 2 9-7
A. Hecatometre = )09 1 '1
A Chiliometre = 04 213 1 10-2
A Myriometre = 6 1 156 -0
Eight Chiliometres are nearly 5 English miles.
MEASURES OF CAPACITY.
English Cubic Inches.
Millilitre = -06102
Centilitre = -61024
Decilitre = 6-10244
Litre = 61-02442
Decalitre = 610-24429
Hecatolitre = 6102-44238
Chiliolitre = 61024-42878
Myriolitre = 610244-28778
A Litre is nearly 2g wine pints.
14 Decilitres are nearly 3 wine pints.
A Chiliolitre is a tun, 12-75 wine gallons.
WEIGHTS.
English Grains.
= -0154
= -1544
= 1-5444
= 15-4440
= 154-4402
= 1544-4023
= 15441-0234
== 154440-2344
A Decagramme is 6 dwts. 10-44 gr. troy ; or 5-65 dr
Avoirdupois.
A Hecatogramme is 3 oz. 8-5 dr. avoirdupois.
A Chiliogramme is 2 Ibs. 3 oz. 5 dr. avoirdupois.
A Myriogramme is 22 1'15 oz. avoirdupois.
100 Myriogrammes are 1 ton, wanting 32-8 Ibs.
Milligramme-
Centigramme
Decigramme
Gramme
Decagramme
Hecatogramme
Chiliogramme (Kilogram)-
Mvriofframme
56 ARITHMETIC.
AGRARIAN MEASURES
Are, . square Decametre = 3-95 Perches.
Hecatare = 2 Acres, 1 Rood, 30'1
Perches.
FIR WOOD.
Decistre, l-10th Stere = 3-5315 cub. ft. Erg
Stere, 1 Cubic Metre = 35-3150 cub. ft.
DIVISION OF THE CIRCLE.
100 seconds = 1 minute.
100 minutes = 1 degree.
100 degrees = 1 quadrant.
4 quadrants = 1 circle.
THE ENGLISH DIVISION.
60 seconds = 1 minute.
60 minutes = 1 degree.
360 degrees = 1 circle.
DIMENSIONS OF DRAWING PAPER IN FEET AND INCHES.
Ft. In. Ft. In.
Demy 1 7 X 1 3
Medium 1 10 X 16
Royal 20x17
Super royal 23X17
Imperial 25X19*
Elephant 2 3| X 1 10^
Columbier 2 9| X 1 11
Atlas 29x22
Double elephant 34x22
Wove antiquarian 44x27
GEOMETRY.
DEFINITIONS.
1. A POINT is (hat which has position,
but no magnkude, nor dimensions ; neither
length, breadth, nor thickness.
2. A Line is length, without breadth or
thickness.
3. A Surface or Superficies, is an exten-
sion or a figure of two dimensions, length
and breadth ; 'out without thickness.
4. A Body or Solid, is a figure of three
dimensions, namely, length, breadth, and
depth or thickness.
5. Lines are either Right, or Curved, or mixed of these
two. ^
6. A Right Line, or Straight Line, lies all in the same
direction, between its extremities ; and is the shortest dis-
tance between two points.
When aline is mentioned simply, it means a Right Lins
7. A Curve continually changes its di-
rection between its extreme points.
8. Lines are either Parallel, Oblique,
Perpendicular, or Tangential.
9. Parallel lines are always at the same
perpendicular distance 4 and they never
meet, though ever so far produced.
10. Gbliqr.e lines change their distance,
and would meet, if produced on the side
of the least distance.
11. One line is Perpendicular to an-
other, w'lien it inclines not more on the
one side than the other, or when the angles
on both sides of it are equal.
12. A line or circle is Tangential, or
;is a t:m<retit to a circle or other curve,
when it touches it without cutting, whea
oth xre
f
58
ARITHMETIC.
13. An Angle is the inclination or open-
ing of two lines, having different direc-
tions, and meeting in a point.
14. Angles are Right or Oblique, Acute or Obtuse.
15. A Right Angle is that which is made
by one line perpendicular to another. Or
when the angles on each side are equal to
one another, they are right angles.
16. An Oblique Angle is that which is
made by two oblique lines ; and is either
less or greater than a right angle.
17. An%A.cute Angle is less than a right
angle.
18. An Obtuse Angle is greater than a
right angle.
19. Superficies are either Plane or Curved.
20. A Plane Superficies, or a Plane, is that with which
a right line may, every way, coincide. Or, if the line
touch the plane in two points, it will touch it in every point.
But, if not, it is curved.
21. Plane Figures are bollhded either by right lines or
curves.
22. Plane figures that are bounded by right lines have
names according to the number of their sides, or of their
angles ; for they have as many sides as angles ; the lea?t
number being three.
23. A figure of three sides and angles is called a Tri-
angle. And it receives particular denominations from the
relations of its sides and angles.
24. An Equilateral Triangle is that
whose three sides are all equal.
25. An Isosceles Triangle is that which
has two sides equal.
26. A Scalene Triangle is that whose
three sides are all unequal.
27. A Right-angled Triangle is that
which has one right angle.
DKFINITIOJfS.
28. Other triangles are Oblique-angled,
and are eilher obtuse or acute.
29. An Obtuse-Angled Triangle has one
obtuse aniilf.
30. An Acute-angled Triangle has all
its three angles acute.
31. A figure of four sides and angles is called a Quad-
rangle, or a Quadrilateral.
32. A Parallelogram is a quadrilateral which has both
its pairs of opposite sides parallel. And it takes the fol-
lowing particular names, viz. Rectangle, Square, Rhombus,
Rhomboid.
33. A Rectangle is a parallelogram,
having right angles.
34. A Square is an equilateral rec-
tangle ; having its length and breadth
equal.
35. A Rhomboid is an oblique-angled
parallelogram.
36 A Rhombus is an equilateral rhom-
boid ; having 4 all its sides equal, but its
angles oblique.
37. A Trapezium is a quadrilateral
which hath not its opposite sides parallel.
38. A Trapezoid has only one pair of
opposite sides parallel.
39. A Diagonal is a line joining any
two opposite angles of a quadrilateral.
40. Plane figures that have more than four sides are, in
general, called Polygons ; and they receive other particular
names, according to the number of their sides or angles.
Thus,
41. A Pentagon is a polygon of five sides ; a Hexagon
of six sides; a Heptagon, seven; an Octagon, eight; a
Nonagon, nine ; a Decagon, ten ; an Undecagon, eleven ;
and a Dodecagon, twelve sides.
42. A Regular Polygon has all its sides and all its
angles equal. If they are not both equal, the polygon is
43. An Eauikteral Triangle is also a regular figure of
60
GEOMETRY
Ihree sides, and the Square is one of four : the former being
also called a Trigon, and the latter a Tetragon.
44. Any figure is equilateral, when all its sides are
equal : and it is equiangular when all its angles are equal.
When both these are equal, it is a regular figure.
45. A Circle is a plane figure bounded
by a curve line, called the Circumference,
which is everywhere equidistant from a
certain point within, called its Centre.
The circumference itself is often called a Circle, and
also the Periphery.
46. The Radius of a circle is aline drawn
from the centre to the circumference.
47. The Diameter of a circle is a line
drawn through the centre, and terminating
at the circumference on both sides.
48. An Arc of a circle is any part of the
circumference.
49. A Chord is a right line joining the
extremities of an sfrc.
50. A Segment is any part of a circle
hounded by an arc and its chord.
51. A Semicircle is half the circle, or a
segment cut off by a diameter.
The half circumference is sometimes
called the Semicircle.
52. A Sector is any part of a circle
which is bounded by an arc, and two radii
drawn to its extremities.
53. A Quadrant, or Quarter of a circle,
is a sector having a quarter of the circum-
ference for its arc, and its two radii are
perpendicular to each other. A quarter
of the circumference is sometimes called
a Quadrant.
A
as
DEFINITIONS.
54 The Height or Altitude of a figure
is a perpendicular let fall from an angle,
or its vertex, to the opposite side, called
the base.
55. In a right-angled triangle, the side
opposite the right angle is called the Hy-
potenuse ; and the other two sides are
called the Legs, and sometimes the Base
and Perpendicular.
56. When an angle is denoted by three
letters, of which one stands at the angular
point, and the other two- on the two sides,
that which stands at the angular point is
read in the middle. Thus, DAE.
57. The circumference of every circle is supposed to be
divided into 360 equal parts, called degrees ; and each de-
gree into 60 minutes, each minute into 60 seconds, and so
on. Hence a semicircle contains 180 degrees, and a quad-
rant 90 degrees.
58. The measure of an angle, is an arc
of any circle contained between the two v v x ^~* s V r
lines which form that* angle, the angular ;' N v / \
point being the centre ; and it is estimated \ /
by the number of degrees contained in *' -''
that arc.
59. Lines, or chords, are said to be
Equidistant from the centre of a circle,
when perpendiculars drawn to them from
the centre are equal.
60. And the right line on which the
Greater Perpendicular falls, is said to be
farther from the centre.
61. An Angle in a segment is that
which is contained by two lines, drawn
from any point in, the arc of the segment,
to the two extremities of that arc.
6'2. An Angle on a segment, or an arc, is that which is
contained by two lines, drawn from any point in the oppo
sue or supplemental part of the circumference, to the extre
unties of the arc, and containing the arc between them.
6
f
32
GEOMETRY.
63. An Angle at the circumference, is
that whose angular point or summit is
anywhere in the circumference. And an
angle at the centre, is that whose angular
point is at the centre.
64. A right-lined figure is Inscrihed in a
circle, or the circle Circumscribes it, when
all the angular points of the figure are in
the circumference of the circle.
65. A right-lined figure Circumscribes a
circle, or the circle is Inscribed in it, when
all the sides of the figure touch the circum-
ference of the circle.
66. One right-lined figure is Inscribed in
another, or the latter Circumscribes the
former, when all the angular points of the
former are placed in the sides of the latter.
67. A Secant is a line that cuts a circle,
lying partly within and partly without it.
68. Two triangles, or other right-lined figures, are said
to be mutually equilateral, when all the sides of the one are
equal to the corresponding sides of the other, each to each :
and they are said to be mutually equiangular, when the
angles of the one are respectively equal to those of the
other.
69. Identical figures, are such as are both mutually equi-
lateral and equiangular ; or that have all the sides and all
the angles of the one, respectively equal to all the sides and
all the angles of the other, each to each ; so that, if the one
figure were applied to, or laid upon the other, all the sides
of the one would exactly fall upon and cover all the sides
of the other ; the two becoming as it were but one and the
same figure.
70. Similar figures, are those that hnve all the angles of
the one equal to all the angles of the other, each to each,
and the sides about the equal angles proportional.
71. The Perimeter of a figure, is the sum of all its sides
taken together.
72 A Proposition, is something which is either proposed
THEOKEMS.
63
to be done, or to be demonstrated, and is either a problem
or a theorem.
73. A Problem, is something proposed to be done.
71. ATheorem, is something proposed to be demonstrated.
75. A Lemma, is something which is premised, or de-
monstrated, in order to render what follows more easy.
76. A Corollary, is a consequent truth, gained imme-
diately from some preceding truth, or demonstration.
77. A Scholium, is a remark or observation made upon
something going before it.
THEOREMS.
1. In the two triangles ABC, DEF, if
the side AC be equal to the side DF, and
the side BC equal to the side EF, and the
angle C equal to the angle F ; then will
the two triangles be identical, or equal in
all respects.
2. Let the two triangles ABC, DEF, have the angle A
equal to the angle D, the angle B equal to the angle E, and
the side AB equal to the side DE ; then these two triangle*
will be identical.
3. If the triangle ABC have the side
AC equal to the side BC ; then will the
angle B be equal to the angle A.
The line which bisects the vertical angle
of an isosceles triangle, bisects the base,
and is also perpendicular to it.
Every equilateral triangle, is also equiangular, or has all
its angles equal.
4. If the triangle ABC, have the angle
A equal to the angle B, it will also have
the side AC equal to the side BC.
Every equiangular triangle is also equi-
lateral. *
5. Let the two triangles ABC, ABD,
have their three sides respectively equal,
viz. the side AB equal to AB, AC to AD,
and BC to BD ; then shall the two triangles
be identical, or have their angles equal,
GEOMETRY.
viz. those angles that are opposite to the equal sides ; viz
the angle BAG to the angle BAD, the angle ABC to the
angle ABD, and the angle C to the angle D.
6. Let the line AB meet the line CD ;
then will the two angles ABC, ABD, taken
together, be equal to two right angles.
7. Let the two lines AB, CD, intersect
in the point E ; then will the angle A EC
be equal to the angle BED, and the angle
AED equal to the angle CEB.
8. Let ABC be a triangle, having the
side AB produced to D ; then will the out-
ward angle CBD be greater than either
of the inward opposite angles A or C.
9. Let ABC be a triangle, having the
side AB greater than the side AC ; then
will the angle ACB, opposite the greater
side AB, be greater than the angle B, op-
posite the less side AC.
10. Let ABC be a triangle ; then will
the sum of any two of its sides be greater
than the third side ; as, for instance, AC
-f CB greater than AB.
1 1 . Let ABC be a triangle ; then will the
difference of any two sides, as AB AC,
be less than the third side BC.
12. Let the line EF cut the two parallel
lines AB, CD ; then will the angle AEF
be equal to the alternate angle EFD.
13. Let the line EF, cutting the two
lines AB, CD, make the alternate angles
AEF, DFE, equal to each other ; then
wii! AB be parallel to CD.
65
14. Let the line EF cut the two paral-
lel lines AB, CD ; then will the outward
angle EGB be equal to the inward opposite
an trie GHD, on the same side of the line
KF; and the two inward angles BGH,
GHD, taken together, will be equal to two
right angles.
15. Let the lines AB, CD, be each of
them parallel to the line EF ; then shall
the lines AB, CD, be parallel to each
other.
16. Let the side AB, of the triangle
A.BC, be produced to D ; then will the out-
ward angle CBD be equal to the sum of
ihe two inward opposite angles A and C.
17. Let ABC be any plane triangle;
then the sumof the three angles A-fB + C
is equal to two right angles.
If two angles in one triangle, be equal
to two angles in another triangle, the third
angles will also be equal, and the two triangles equiangii
lar.
If one angle in one triangle, be equal to one angle in
another, the sums of the remaining angles will also be
equal.
If one angle of a triangle be right, the sum of the other
two will also be equal to a right angle, and each of them
singly will be acute, or less than a right angle.
The two least angles of every triangle are acute, or eacQ
less than a right angle.
18. Let ABC D be a quadrangle ; then
the sum of the four inward angles, A-f-B
-fC-f D is equal to four right angles.
19. Let ABCDE be any figure; then
the sum of all its inward angles, A + B-f-
C + D + E, is equal to twice as many
right angles, wanting four, as the figure
has sides.
6*
66
GEOMETKY.
20. Let A., B, C, &c., be the outward
angles of any polygon, made by. producing
all the sides; then will the sum A + B +
C-j-D + E, of all those outward angles, be
equal to four right angles.
21. I%AB, AC, AD,' <fec., be lines drawn
from the given point A, to the indefinite
ine BE, of which AB is perpendicular;
then shall the perpendicular AB be less
than AC, and AC less than AD, &c.
22. Let ABCD be a parallelogram, of
which the diagonal is BD ; then will its
opposite sides and angles be equal to each
other, and the diagonal BD will divide it
into two equal parts, or triangles.
If one angle of a parallelogram be a right angle, all the
other three will also be right angles, and the parallelogram
a rectangle.
The sum of any two adjacent angles of a parallelogram
is equal to two right angles.
2,3. Let ABCD be a quadrangle, having the opposite sides
equal, namely, the side AB equal to DC, and AD equal to
BC ; then shall these equal sides be also parallel, and the
figure a parallelogram.
24. Let AB, DC, be two equal and parallel lines; then
will the lines AD, BC, which join their extremes, be also
equal and parallel.
25. Let ABCD, ABEF, be two paral-
lelograms, and ABC, ABF, two triangles,
standing on the same base AB, and between
ihe same parallels AB, DE ; then will the
parallelogram ABCD be equal to the paral-
lelogram ABEF, and the triangle ABC
equal to the triangle ABF.
Parallelograms, or triangles, having the same base and
altitude, are equal. For the altitude is the same as the
perpendicular or distance between the two parallels, which
is everywhere equal, by the definition of parallels.
Parallelograms, or triangles, having equal bases and :il-
titudes, are equal. For if the one figure be applied wilh
its base on the other, the bases will coincide or be the same,
because they are equal: and so the two figures, having the
same base and altitude, are equal.
n f H G
DO
THKOUKMS.
26. Let ABCD be a parallelogram, and
ABE a triangle, on the same base AH, and
between the same parallels AB, I)l<- ; then
will the parallelogram ABCI) be double
the triangle ABE, or the triangle half the
parallelogram.
A triangle is equal to half a parallelogram of the same
base and altitude, because the altitude is the perpendicular
distance between the parallels, which is everywhere equal,
by the definition of parallels.
If the base of a parallelogram be half that of a triangle,
of the same altitude, or the base of a triangle be double that
of the parallelogram, the two figures will be equal to each
other.
27. Let BD, FH, be two rectangles,
having the sides AB, BC, equal to the
sides EF, FG, each to each; then will the
rectangle BD be equal to the rectangle
FH.
28. Let AC be a parallelogram, BD a
diagonal, EIF parallel to AB or DC, and
OIH parallel to AD or BC, making AI,
1C, complements to the parallelograms
EG, HF, which are about the diagonal DB:
then will the complement AI be equal to the complement
1C.
29. Let AD be the one line, and AB
the other, divided into the parts AE, EF,
FB ; then will the rectangle contained by
AD and AB, be equal to the sum of the
rectangles of AD and AE, and AD and
EF, and AD and FB : thus expressed, AD . AB=AD . AE
+AD . EF+AD . FB.*
If a right line be divided into any two parts, the square
of the whole line, is equal to both the rectangles of the
whole line and each of the parts.
* Instead of the mark X a point is often used ; thus, length X
breadth = area, is the same as length . breadth = area. Instead of Jhe
parenthesis, a stroke is often used; thus, (first -j- last) -7- 2= arith-
metical mean, is the same thing as first -f- last -f- 2 = arithmetical
mean. For the square root this mark ^/ is sometimes used, and for the
cube root /, &c.
68
GEOMETRY.
30. Let the line AB be the sum of any
two lines AC, CB ; then will the square A
of AB be equal to the squares of AC, CB,
together with twice the rectangle of AC
. CB. That is, AB a =AC a +CB 3 -f 2AO B
.CB.
If a line be divided into two equal parts ; the square of
the whole line will be equal to four times the square of half
the line.
31. Let AC, BC, be any two lines, and
AB their difference ; then will the square
of AB be less than the squares of AC, BC,
by twice the rectangle of AC and BC.
Or, AB 3 = AC 3 -f BC 2 2AC . BC.
32. Let AB, AC, be any two unequal
lines; then will the difference of the
squares of AB, AC, be equal to a rect-
angle under their sum and difference.
That is, AB 3 - AC 3 = AB -f- AC .
AB AC.
33. Let ABC be a right-angled tri-
angle, having the right angle at C ; then
will the square of the hypotenuse AB, be
equal to the sum pf the squares of the
other two sides AC, CB. Or AB 3 =AC 3
-fBC 3 .
34. Let ABC be any triangle, having CD
perpendicular to AB ; then will the differ-
ence of the squares of AC, BC, be equal
to the difference of the squares of AD,
BD ; that is, AC 2 BC 3 =AD 2 BD 3 . A u B D A B
35. Let ABC be a triangle, obtuse-angled at B, and CD
perpendicular to AB.; ihen will the square of AC be greater
than the squares of AB, BC, by twice the rectangle of AB,
BD. That is, AC 3 =AB 2 +BC 2 -f 2AB . BD.
36. Let ABC be a triangle, having the angle A acute, and
CD perpendicular to AB ; then will the square of BC, ba
less than the squares of AB, AC, by-twice the rectangle of
AB, AD. That is, BC 2 =AB 3 +AC 3 2AD AB.
THEOREMS.
69
37. Let ABC be a triangle, and CD the
line drawn from the vertex to the middle
of the base AB, bisecting it into the two
equal parts AD, DB : then will the sum
of the squares of AC, CB, be equal to twice
the sum of the square of CD, AD ; or AC a
-|-(;B 8 =2CD 2 +2AD a .
38. Let ABC be an isosceles triangle,
and CD a line drawn from the vertex to
any point D in the base : then will the
square of AC, be equal to the square of
CD, together with the rectangle of AD and
DB. That is AC S =CD 2 +AD . DB.
39. Let ABCD be a parallelogram,
whose diagonals intersect each other in E :
then will AE be equal to EC, and BE to
ED ; and the sum of the squares of AC,
BD, will be equal to the sum of the squares
of AB, BC, CD, DA. That is,
AE=EC, and BE=ED,
and AC 2 -f BD*=AB 2 +BC 2 +CD a +DA 2
40. Let AB be any chord in a circle,
and CD a line drawn from the centre C to
the chord. Then, if the chord be bisected
in the point D, CD will be perpendicular
to AB.
41. Let ABC be a circle, and D a point
within it ; then if any three lines, DA, DB,
DC, drawn from the point D to the cir-
cumference, be equal to each other, the
point D will be the centre.
42. Let two circles touch one another internally in the
point ; then will the point and the centres of those circles
be all in the same right line.
43. Let two circles touch one another externally at the
point ; then will the point of contact and the centres of the
two circles be all in the same right line.
44. Let AB, CD, be any two chords at
equal distances from the centre G ; then
will these two chords AB, CD, be equal
to each other.
70
GEOMETRY.
45. Let the line ADB be perpendicular
to the radius CB of a circle ; then shall
AB touch the circle in the point D only.
46. Let AB be a tangent to a circle,
and CD a chord drawn from the point of
contact C ; then is the angle BCD mea-
sured by half the arc CFD, and the angle
ACD measured by half the arc CGD.
47. Let BAG be an angle at the circum-
ference ; it has for its measure, half the
arc BC which subtends it.
48. Let C and D be two angles in the
same segment ACDB, or, which is the
same thing, standing on the supplemental
arc AEB ; then will the angle C be equal
to the angle D.
49. Let C be an angle at the centre C,
and D an angle at the circumference, both
standing on the same arc or same chord
AB ; then will the angle C be double of
the angle D, or the angle D equal to half
the angle C.
50. If ABC or ADC be a semicircle,
then any angle D in that semicircle, is a
right angle.
51. If AB be a tangent, and AC a chord,
and D any angle in the alternate segment
ADC ; then will the angle D be equal to
the angle BAG made by the tangent and
chord of the arc AEG.
52. Let ABCD be any quadrilateral in-
scribed in a circle ; then shall the sum of
the two opposite angles A and C, or B
and D, be equal to two right angles.
THEOREMS.
71
53. If the side AB, of the quadrilateral
ABCD, inscribed in a circle, be produced
to E : the outward angle DBE will be
equal to the inward opposite angle C.
54. Let the two chords AB, CD be
parallel ; then will the arcs AC, BD, be
equal; or AC=BD.
55. Let the tangent ABC be parallel to
the chord DF ; then are the arcs BD, BF,
equal; that is, BD=BF.
56. Let the two chords AB, CD, inter-
sect at the point E ; then the angle AEC,
or DEB, is measured by half the sum of
the two arcs AC, DB.
57. Let the angle E be formed by two
secants EAB and ECD ; this angle is
measured by half the difference of the two
arcs AC, DB, intercepted by the two se-
cants.
58. Let EB, ED, be two tangents to a
circle at the points A, C ; then the angle
E is measured by half the difference of the
two arcs CFA, CGA.
59. Let the two lines AB, CD, meet
each other in E ; then the rectangle of AE,
EB, will be equal to the rectangle of CE,
ED. Or, AE . EB = CE . ED.
When one of the lines in the se-
cond case, as DE, by revolving about
the point E, comes into the position
of the tangent EC or ED, the two
points C and D running into one ;
72
GEOMETRY.
then the rectangle of CE, ED, becomes the square of CF
because CE and DE are then equal. Consequently, th*
rectangle of the parts of the secant, AE . EB, is equal to
the square of the tangent, CE 3 .
60. Let CD be the perpendicular, and
CE the diameter of the circle about the
triangle ABC ; then the rectangle CA,
CB is = the rectangle CD . CE.
61. Let CD bisect the angle C of the
triangle ABC ; then the square CD 2 -f the
rectangle AD . DB is = the rectangle
AC . CB.
62. Let ABCD be any quadrilateral in-
scribed in a circle, and AC, BD, its two
diagonals ; then the rectangle AC . BD is
= the rectangle AB . DC + the rectangle
AD . BC.
63. Let the two triangles ADC, DEF,
have the same altitude, or be between the
same parallels AE, CF ; then is the sur-
face of the triangle ADC, to the surface
of the triangle DEF, as the base AD is
to the base DE. Or, AD : DE : : the tri-
angle ADC : the triangle DEF.
64. Let ABC, BEF, be two triangles
having the equal bases AB, BE, and whose
altitudes are the perpendiculars CG, FH ;
then will the triang-ie ABC : the triangle
BEF : : CG : FH.
Triangles and parallelograms, when their bases are equal,
are to each other as their altitudes ; and by the foregoing
one, when their altitudes are equal, they are to each other
as their bases ; therefore, universally, when neither are
equal, they are to each other in the compound ratio, or as
the rectangle or product of their bases and altitudes.
65. Let the four lines A, B, C, D, be
proportionals, or A : B : : C : D ; then
will the rectangle of A and D be equal to
the rectangle of B and C ; or the rectangle
i . D - B . C.
THEOUEMS.
73
66. Let DE he parallel to the side BC
of the trianolc ABC ; then will AD : DB
: : AE : EC.
AB : AC : : AD : AE,
AB : AC : : BD : CE.
67. Let the angle ACB, of the triangle
\.BC, be bisected by the line CD, making
.he angle r equal to the angle s : then will
the segment AD be to the segment DB,
as the side AC is to the side CB. Or, AD
: DB : : AC : CB.
68. In the two triangles ABC, DEF, if
AB : DE : : AC : DF : : BC : EF ; the
two triangles will have their correspond-
ing angles equal.
69. Let ABC, DEF, be two triangles,
having the. angle A = the angle D, and
the sides AB, AC, proportional to the sides
DE, DF ; then will the triangle ABC be
equiangular with the triangle DEF.
70. Let ABC be a right-angled tri-
angle, and CD a perpendicular from
the right angle C to the hypotenuse
AB ; then will
CD be a mean proportional between AD and DB ;
AC a mean proportional between AB and AD ;
BC a mean proportional between AB and BD.
71. All similar figures are to each other, as the squares
of their like sides.
72. Similar figures inscribed in circles, have their like
sides, and also their whole perimeters, in the same ratio
as the diameters of the circles in which they arc inscribed.
73. Similar figures inscribed in circles are to each other
as the squares of the diameters of those circles.
74. The circumferences of all circles are to each other
as their diameters.
75. The areas or spaces of circles, are to each other as
the squares of (heir diameters; or of their radii.
7(1. The area of any circle, is equal to the rectangle of
Half its circumference and half its diameter.
1 To bisect a line AB ; that is, to divide it into two
equal parts.
c
From the two centres A and B, with
any equal radii, describe arcs of circles,
intersecting each other in C and D ;
and draw the line CD, which will bi-
sect the given line AB in the point E.
2. To bisect an angle BAG.
From the centre A, with any radius,
describe an arc cutting off the equal
lines AD, AE ; and from the two centres
D, E, with the same radius, describe
arcs intersecting in F ; then draw AF,
which will bisect the angle A as re-
quired.
3. At a given point C, in a line AB, to erect a per
pendicular.
From the given point C, with any ra- ^
dius, cut off any equal parts CD, CE, of
the given line ; and, from the two centres
D and E, with any one radius, describe
arcs intersecting in F ; then join CF,
tvhich will be perpendicular as required.
OTHERWISE.
When the given point C is near the end of the line.
From any point D, assumed above
the line, as a centre, through the given
point C describe a circle, cutting the
given line at E ; and through E and
the centre D, draw the diameter EDF;
then join CF, which will be the per-
oendicular required.
A E
PROBLEMS.
75
4. From the giver, point A; to let fall a perpendicular on
a given line BC.
From the given point A as a centre,
w'uh any convenient radius, describe an
arc, cutting the given line at the two
points D and E ; and from the two cen-
tres D, E, with any radius, describe two
arcs, intersecting at F ; then draw AGF,
which will be perpendicular to BC as re-
quired.
OTHERWISE.
When the given point is nearly opposite the end of the
line.
From any point D, in the given line .
BC, as a centre, describe the arc of a cir-
cle through the given point A, cutting
BC in E : and from the centre E, with
the radius EA, describe another arc, cut-
ting the former in F; then draw AGF,
which will be perpendicular to BC as re-
quired.
5. At a given point A, in a line AB, to make an angle
equal to a given angle C.
E
From the centres A and C, with any
one radius, describe the arcs DE, FG.
Then, with radius DE, and centre F, de-
scribe an arc, cutting FG in G. Through
G draw the line AG, and it will form the
angle required.
A F B
6. Through a given point A, to draw a line parallel to a
s:iven line BC.
From the given point A draw a line AD
to any point in the given line BC. Then
draw the line EAF, making the angle at A
equal to the angle at D (by prob. 5) ; so
shall EF be parallel to BC as required.
E A
76
GEOMETRY.
D
7. To divide a line AB into any proposed number of
equal parts.
Draw any other line AC, forming any
angle with the given line AB ; on which
set off as many of any equal parts AD,
DE, EG, FC, as the line AB is to be
divided into. JoinBC; parallel to which r
draw the other lines FG, EH, DI ; then
.these will divide AB in the manner required. For those
parallel lines divide both the sides AB, AC, proportionally.
8. To find a third proportional to two given lines AB, AC
Place the two given lines AB, AC, A B
forming any angle at A ; and in AB take
also AD equal to AC. Join BC, and
draw DE parallel to it ; so will AE be
the third proportional sought.
9. To find a fourth proportional to three lines, AB, AC,
AD.
Place two of the given lines AB, A B
AC, making any angle at A; also
place AD on AB. Join BC ; and pa-
rallel to it draw DE ; so shall AE be
the fourth proportional as required.
D B
10. To find a mean proportional between two lines, AB
BC.
Place AB, BC, joined in one straight'
line AC : on which, as a diameter, de-
scribe the semicircle ADC ; to meet which
erect the perpendicular BD ; and it will
be the mean proportional sought, between
AB and BC.
B C
O B C
11. To find the, centre, of a circle.
Draw any chord AB ; and bisect it
perpendicularly with the line CD, which
will be a diameter. Therefore CD bi-
sected in O, will give the centre, as re-
quirec
PROBLEMS.
77
12. To describe the circumference of a circle thiough
three given points, A, B, C.
From the middle point B draw chords
BA, BC, to the two other points, and bi-
sect these chords perpendicularly by lines
meeting in O, which will be the centre.
Then from the centre O, at the distance
of any one of the points, as OA, describe
a circle, and it will pass through the two
other points B, C, as required.
13. To draw a tangent to a circle, through a given
point A.
When the given point A is in the cir-
cumference of the circle : join A and the
centre ; perpendicular to which draw
BAG, and it will be the tangent.
14. On a given line B to describe a segment of a circle,
to contain a given angle C.
At the ends of the given line make
angles DAB, DBA, each equal to the
given angle C. Then draw AE, BE per-
pendicular to AD, BD ; and with the
centre E, and radius EA or EB, describe
a circle ; so shall AFB be the segment
required, as any angle F made in it will
be equal to the given angle C.
15. To cut off a segment from a circle, that shall contain
a given angle C.
Draw any tangent AB to the given cir-
cle ; and a chord AD to make the angle
DAB equal to the given angle C ; then
DEA will be the segment required, any
angle E made in it being equal to the
given angle O
7*
76
GEOMETRY.
16. To make a triangle with three given lines, AB, AC
BC.
With the centre A, and distance AC,
describe an arc. With the centre B, and
distance BC, describe another arc, cutting
the former in C. Draw AB, BC, and
ABC will be the triangle required.
17. To inscribe a circle in a given triangle ABC.
Bisect any two angles A and B, with
the two lines AD, BD. From the inter-
section D, which will be the centre of the
circle, draw the perpendiculars DB, DF,
DG, and they will be the radii of the cir-
cle required.
18. To describe a circle about a given
triangle ABC.
Bisect any two sides with two per-
pendiculars DE, DF, and their inter-
section D will be the centre.
19. To inscribe an equilateral triangle in a given circle
Through the centre C draw any di-
ameter AB. From the point B as a
centre, with the radius BC of the
given circle, describe an arc DCE.
Join AD, AE, DE, and ADE is the
equilateral triangle sought.
20. To inscribe a square in a given circle.
Draw two diameters AC, BD, cross-
ing at right angles in the centre E.
Then join the four extremities A, B,
C, D, with right lines, and these will
form the inscribed square ABCD.
79
21. To describe a square about a given circle.
Draw two diameters AC, BD, cross-
ing at right angles in the centre E.
Then through their four extremities
draw FG, III, parallel to AC, and FI,
GH, parallel to BD, and they will
form the square FGHI. I H
22. To inscribe a circle in a regular polygon.
A
Bisect any two sides of the poly-
gon by the perpendiculars GO, FO,
and their intersection O will be the
centre of the inscribed circle, and OG
or OF will be the radius.
23. To describe a circle about a regular polygon.
Bisect any two of the angles, C
and D, with the lines CO, DO ; then
their intersection O will be the centre
of the circumscribing circle ; and OC,
or OD, will be the radius.
24. To make a square equal to the sum of two or mart
given squares.
Let AB and AC be the sides of two
given squares. Draw two indefinite
lines AP, AQ, at right angles to each
other ; in which place the sides AB,
AC, of the given squares ; join BC ;
then a square described on BC will be
equal to the sum of the two squares
described on AB and AC.
P B
25. To make a square equal to the difference of two givtn
squares.
Let A T ^ and AC, taken in the same
straight L;ie, be equal to the sides of
the two given squares. From the
centre A, with the distance AB, de-
Bcribe a circle; ami make CD perpen-
dicular to AB, meeting the circumference in D; so shall a
c B
sc
GEOMETRY.
square described on -CD be equal to AD 2 AC 2 , or AB 9 -
AC 2 , as required.
26. To make a triangle equal to a given pentagon
ABCDE.
D
Draw DA and DB, and also EF,
CG, parallel to them, meeting AB pro-
duced at F and G : then draw DF and
DG ; so shall the triangle DFG be
equal to the given pentagon ABCDE.
F A B G
27. To make a square equal to a given rectangle ABCD
Produce one side AB, till BE be
equal to the other side BC. On AE
as a diameter describe a circle, meet-
ing BC produced at F ; then will BF
be the side of a square BFGH, equal
to the given rectangle BD, as required.
r>
A H
APPENDIX TO GEOMETRY.
INSTRUMENTS.
28. To facilitate the construction of geometrical figures
we add a short description of a few useful instruments
which do not belong to the common pocket-case.
29. Let there be a flat ruler, AB, from R
one to two feet in length, for which the
comnjon Gunter's scale may be substi-
tuted ; and, secondly, a triangular piece
of wood, , 6, c, flat, and about the same
thickness as the ruler : the sides ab and
V of which are equal to one another, and form a right
angle at b. For the convenience of sliding, there is usually
a hole in the middle of the triangle, as may be seen in the
figure.
30. By me#ns of these simple instruments many very
useful geometrical problems may be performed. Thus, to
draw a line through a given point parallel to a given line.
INSTRUMENTS. 51
Lay the triangle on the* paper so that one of its sides will
coincide with the given line to which the parallel is to be
drawn ; then, keeping the triangle steady, lay the ruler on
the paper, with its edge applied to either of the other sides
of the triangle ; then; keeping the ruler firm, move the tri-
angle along its edge, up or down, to the given point ; the
side of the triangle which was placed on the given line will
always keep parallel to itself, and hence a parallel may be
drawn through the given point.
31. To erect a perpendicular on a given line, and from
any given point in that line, we have only to apply the
ruler to the given line, and place the triangle so, that its
right angle shall touch the given point in the line, and one
of the sides about the right angle, placed to the edge of the
ruler the other side will give the perpendicular required.
32. If the given point be either above or below the line,
the process is equally easy. Place one of the sides of the
triangle about the right angle on the given line, and the
ruler on the side opposite the right angle, then slide the tri-
angle on the edge of the ruler till the given point from
which the perpendicular is to be drawn is on the other side,
then this side will give the perpendicular.
33. Other problems may be performed with these instru-
ments, the method of doing which it will be easy for the
reader to contrive for himself.
34. When arcs of circles of great diameter are to be
drawn, the use of a compass may be substituted by .a very
simple contrivance. Draw the chord of the arc to be de-
scribed, and place a pin at each ex-
tremity, A and B, then place two
rulers jointed at C, and forming an
angle, ACB = the supplement of half
the given number of degrees ; that is
to say, the number of degrees which the arc whose chord
given is to contain, is to be halved, and this half being sub-
tracted from 180 degrees, will give the degrees whicli form
the angle at which the rulers are placed, that is; the angle
ACB. This being done, the edges of the rulers are moved
along against the pins, and a pencil at C will describe the
arc required.
35. Large circles may be described by a contrivance
equally simple. On an axle, a foot or a foot and a half
82 GEOMETRY.
long, there are placed two wheels, M and
F, of which one is fixed to the axle, name-
ly, M, and the other is capable of being
shifted to different parts of the axle, and,
by means of a thumb-screw, made capable
of being fixed at any point on the axle.
These wheels are of different diameters, say of 3 and 6
inches, the fixed wheel F being the largest. This instru-
ment being moved on the paper, the circles M and F will
roll, and describe circles of different radi . the axle will al-
ways point to the centre of these circle*, and there will be
this proportion :
As the diameter of the larg-e wheel
Is to the difference of the diameters of the two wheels,
So is the radius of the circle to be described by the
large wheel
To the distance of the two wheels on the axle.
36. If the diameters of the wheels are as above stated,
and it is required to describe a circle of 3 feet radius, then
from the above proportion we have 6:6 3 : : 3 feet or
36 inches : 18 inches = the distance of the two wheels, to
describe a circle 6 feel in diameter.
37. It may be observed, that it will be best to make the
difference of the wheels greater if large circles are to be de-
scribed, as then a shorter instrument will serve the purpose.
38. We will conclude this appendix, by making a few
remarks on the Diagonal Scale and Sector, the great use
of the lattei jf which, especially, is seldom explained to
the young mechanic.
39. The diagonal scale to be found on the plain scale in
common pocket-cases of instruments, is a contrivance for
measuring very small divisions of lines ; as, for instance,
hundredth parts of an inch.
40. Suppose the accompanying cut to E A
represent an enlarged view of two di-
visions of the diagonal scale, and the
bottom and top lines to be divided into
two parts, each representing the tenth
part of an inch. Now, the perpendicular
lines BC, AD, are each divided into ten
equal parts, which are joined by the
crossing lines, 1, 2, 3, 4, &c., and the diagonals BF, DE,
INSTRUMENTS. 83
nre drawn as in the figure. Now, as the division FC is
the tenth part of an inch, and as the line FB continually
approaches nearer and nearer to BC, till it meets it in B,
it will follow, that the part of the line 1 cut off by this
diagonal will be a tenth part of FC, because Bl is only one-
tenth part of BC ; so, likewise, 2 will represent two-tenth
parts, 3, three-tenth parts, and so on to 9, which is nine-
tenth parts, and 10, ten-tenth parts, or the whole tenth of
an inch ; so that, by means of this diagonal, we arrive at
divisions equal to tenth parts of tenth parts of an inch, or
hundredths of an inch. With this consideration, an exami-
nation of the scale itself will easily show the whole matter.
It may be observed, that if half an inch and the quarter of
an inch be divided, in the same manner, into tenths and
tenths of tenths, we may get thus two-hundredth and four-
hundredth parts of an inch.
THE SECTOR.
41. THIS very useful instrument consists of two equal
rulers each six inches long, joined together by a brass
folding joint. These rulers are generally made of boxwood
or ivory; and on the face of the instrument, several lines
or scales are engraven. Some of these linjes or scales pro-
ceed from the centre of the joint, and are called sectorial
lines, to distinguish them from others which are drawn
parallel to the edge of the instrument, similar to those on
the common Gunter's scale.
42. The sectorial lines are drawn twice on the same face
of the instrument; that is to say, each line is drawn on both
legs. Those on each face are,
A scale of equal parts, marked L,
A line of chords, marked C,
A line of secants, marked S,
A line of polygons, marked P, or Pol.
These sectnrial lines are marked on one face of the instru-
ment; and on the other there are the following:
A line of sines, marked S,
A line of tangents, marked T,
A line of tangents to a less radius, marked L
84 GEOMETRY.
Tins last line is intended to supply the defect of the former
and extends from about 45 to 75 degrees.
43. The lines of chords, sines, tangents, and secants, but
not the line of polygons, are numbered from the centre,
and are so disposed as to form equal angles at the centre ;
and it follows from this, that at whatever distance the sec-
tor is opened, the angles which the lines form, will always
be respectively equal. The distance, therefore, between 10
and 10, on the two lines marked L, will be equal to the
distance of 60 and 60 on the two lines of chords, and also
to 90 and 90 on the two lines of sines, &c., at any particu-
lar opening of the sector.
44. Any extent measured with a pair of compasses, from
the centre of the joint to any division on the sectorial lines,
is called a lateral distance; and any extent taken from a
point in a line on the one leg, to the like point on the
similar line on the other leg, is called a transverse or
parallel distance.
With these remarks, we shall now proceed to explain the
use of the sector, in so far as it is likely to be serviceable to
mechanics.
USE OF THE LINE OF LINES.
45. This line, as was before observed, is marked L, and
its uses are,
To Divide a line into any number of equal parts : Take
the length of th^line by the compasses, and placing one of
the points on that number in the line of lines which denotes
the number of parts into which the given line is to be
divided, open the sector till the other point of the com-
passes touches the same division on the line of lines marked
on the other leg ; then, the sector being kept at the same
width, the distance from 1 on the line L on the one leg,
to 1 on the line L on the other, will give the length of one
of the equal divisions of the given line to oe divided.
Thus, to divide a given line into seven equal parts : take
the length of the given line with the compasses, and setting
one point on 7, on the line L of one of the legs, move the
other leg out until the other point of the compasses touch
7 on the line I, of that leg ; this may be called the trans-
verse distance of 7 on the line of lines. Now, keeping the
sector at the same opening, the transverse distance of 1
will be the length of one of the 7 equal divisions of the
INSTRUMENTS. f>5
giver line ; the transverse distance of 2 will be two of these
divisions, &c.
46. It will sometimes happen, that the line to be divided
will be too long for the largest opening of the sector ; and
in this case we take the half, or third, or fourth of the line,
as the case may be ; then the transverse distance of 1 to 1,
will be a half, a third, or a fourth of the required equal part.
47. To divide a given line into any number of parts that
shall have a certain relation or proportion to each other:
Take the length of the whole line to be divided, and placing
one point of the compasses at that division on the line
of lines on one leg of the instrument which expresses
the; smn of all the parts into which the given line is to be
divided, and open the sector till the other point of the
compasses is on the corresponding division on the line of
lines of the other leg. This is evidently making the sum
of the parts into which the given line is to be divided a
transverse distance; and when this is done, the proportional
parts will be found by taking, with the same opening of
the sector, the transverse distances of the parts required.-
To divide a given line into three parts, in the proportion
of 2, 3, 4: The sum of these is 9; make the given line a
transverse distance between 9 and 9 on the two lines of
lines; then the transverse distances of the several numbers
2, 3, 4, will give the proportional parts required.
48. To find a fourth proportional to three given lines:
Take the lateral distance of the second, and make it the
transverse distance of the first, then will the transverse dis-
tance of the third be the lateral distance of the fourth; then,
let there be given 6:3:: 8, make the lateral distance of
3 the transverse distance of 6 ; then will the transverse
distance of 8 be the lateral distance of 4, the fourth propoi-
tional required.
49. This sector will be found highly serviceable in draw-
ing plans. For instance, if it is wished to reduce the draw-
ing of a steam engine from a scale of 1^ inches to the foot,
to another of f to the foot. Now, in lg inches there are */
parts; so that the drawing will be reduced in the propor-
tion of 12 10 5. Take the lateral distance of 5, and keep
the compasses at this opening ; then open the sector till the
points of the compasses mark the transverse distance of 12 ;
keep now the sector at this opening, and any measure taken
on the drawing, to be copied and laid off on the sector as a
8
86 MECHANICAL DRAWING
lateral distance, the transverse distance taken from that
point will give the corresponding measure to be laid down
in the new drawing.
50. If the length of the side of a triangle, of which we
have the drawing, is to he reckoned 45 ; what are the
lengths of the other two sides ? Take the length of the
side given, by the compasses, and open the sector till the
measure be the transverse distance of 45 to 45 ; then the
lengths of the other sides being applied transversely, will
give their numerical lengths.
USE OF THE LINE OF CHORDS.
51. By means of the sector, we may dispense with the
protractor. Thus, to lay down an angle of any number of
degrees: take the radius of the circle on the compasses,
and open the sector till this becomes the transverse distance
of 60 on the line of chords ; then take the transverse dis-
tance of the required number of degrees, keeping the sec-
tor at the same opening; and this transverse distance being
marked off on an arc of the circle whose radius was taken,
will be the required number of degrees.
We will not enter farther on the use of the sectorial lines,
as what we have said will, we hope, be found sufficient for
the purposes of the practical mechanic.
MECHANICAL DRAWING AND PERSPECTIVE.
52. A FLAT rectangular board is first to be provided, of
any 'convenient size, as from 18 to 30 inches long, and from
16 to 24 inches broad. It may be made of fir, plane tree,
or mahogany ; its face must be planed smooth and flat, and
the sides and ends as nearly as possible at right angles to
each other the bottom of the board and the left side should
be made perfectly so ; and this corner shouL! be marked,
so that the stock of the square may be always applied to
the bottom and left hand side of the board. To prevent
the board from casting, it is usual to pannel it on the back
or on the sides.
53. A T square must also be provided, such that by
AND PERSPECTIVE. 87
means of a thumb-screw fixed in the stock, it may be made
to answer either the purposes of a common square, or bevel,
the one-half of the stock being movable about the screw,
and the other fixed at right angles on the blade. The blade
ought to be somewhat flexible, and equal in length to the
length of the board.
54. Besides these, there will be required a case of mathe-
matical instruments ; in the selection of which it should be
observed, that the bow compass is more frequently defect-
ive than any of the other instruments. After using any of
the ink feet, they should be dried ; and if they do not draw
properly, they ought to be sharpened and brought to an
equal length in the blade, by grinding on a hone.
55. The colours most useful are, Indian ink, gamboge,
Prussian blue, vermilion, and lake. With these, all colours
necessary for drawing machinery or buildings may be
made ; so that, instead of purchasing a box of colours, we
would advise that those for whom this book is intended
should procure these cakes separately : the* gamboge may
be bought from an apothecary a pennyworth will serve
a lifetime. In choosing the rest, they should be rubbed
against the teeth, and those which feel smoothest are of the
best quality.
56. Hair pencils will also be necessary, made of camel's
hair, and of various sizes. They ought to taper gradually
to a point when wet in the mouth, and should, after being
pressed against the finger, spring back.
57. Black-lead pencils will also be necessary. They
ought not to be very soft, nor so hard that their traces can-
not be easily erased by the Indian rubber In choosing
paper, that which will best suit this kind of drawing is
thick, and has a hardish feel, not very smooth on the sur-
face, yet free from knots.
58. The paper on which the drawing is to be made, must
be chosen of a good quality and convenient size. It is then
to be wet with a sponge and clean water, on the opposite
side from that on which the drawing is to lie made. When
the paper absorbs the water, which may be seen by the
wetted side becoming dim, as its surface is viewed slantwise
against the light, it is to be laid on the drawing board
v/ith the wetted side next the board. About half an inch
must be turned up ou a straight edge all round the paper
88 MECHANICAL DRAWING
and then fastened on the board. This is done because th
paper when wet is enlarged, and the edges being fixed or.
the board, act as stretchers when the paper contracts by
drying. To prevent the paper from contracting before the
paste has been sufficiently fastened by drying, the paper is
usually wet on the upper surface, to within half an inch of
the paste mark. When the paper is thoroughly dried, it
will be found to lie firmly and equally on the board, and is
then fit for use.
59. If the drawing is to be made from a copy, we ought
first to consider what scale it is to be drawn to. If it is to
be equal in size to, or larger than the copy ; and a scale
should be made accordingly, by which the dimensions of
the several parts of the drawing are to be regulated. The
diagonal scale, a simple and beautiful contrivance, will be
here found of great use for the more minute divisions ; and
whenever the drawing is to be made to a scale of 1. inch,
inch, | inch to the foot, a scale should be drawn of 20 or
30 equal parts ; the last of which should be subdirided into
12, and a diagonal scale formed on the same principles as
the common one, but with eight parallels and 12 diagonals,
to express inches and eighths of an inch. For making such
scales to any proportion, the line L on the sector will be
found very convenient.
60. Great care should be taken in the penciling, that an
accurate outline be drawn, for on this much of the value
of the picture will depend. The pencil marks should be
distinct, yet not heavy, and the use of the rubber should be
avoided as much as possible, as its frequent application
ruffles the surface of the paper. The methods already
given for constructing geometrical figures will be here
found applicable, and the use of the T square, parallel
ruler, &c., will suggest themselves whenever they require
to be employed.
61. The drawing thus made of any machine or building
is called a plan. Plans are of three kinds a ground plan,
or bird's-eye view, an elevation or front view, and a per-
spective plan.
62. When a view is taken of the teeth of a wheel, with
the circumference towards the eye, the teeth appear to be
nearer as they are removed from the middle point of the
circumference opposite the eye, and it may not be out of
AND PERSPECTIVE. , 89
place here to give the method of representing them on
paper: IfABbe the circumference of
a wheel as viewed by the eye, and it is ^T
required to represent the teeth as they
appear on it. Only half of the circum-
ference can be seen in this way at one
time, consequently we can only represent the half of the
teeth. On AB describe a semicircle, which divide into
half as many equal parts as the wheel has teeth ; then from
each of these points of division draw perpendiculars to the
wheel AB, then will these perpendiculars mark the relative
places of the teeth.
63. When the outline is completed in pencil, it is next
to be carefully gone over with Indian ink, which is to be
rubbed down with a little water, on a plate of glass 01
earthenware so as to be sufficiently fluid to flow easily out
of the pen, and at the same, time have a sufficient body of
colour. While drawing the ink lines, the measurement
should all be repeated, so as to correct any error that may
have slipped during the penciling. The screw in the
drawing pen will regulate the breadth of the strokes ; which
should not be alike heavy ; those strokes being the heaviest
which bound the dark part of the shades. Should any line
chance to be wrong drawn with the ink, it may be taken
out by means of a sponge and water, which could not be
done if common writing ink were employed.
65. In preparing for colouring it is to be observed, that
a hair pencil is to be fixed at each end of a small piece of
wood, made in the form of a common pencil, one of which
is to be used with colour, and the other with water only.
If the colour is to be laid on, so as to represent a flat sur-
face, it ought to be spread on equally, and there is here no
jise for the water brush ; but if it is to represent a curved
surface, then the colour is to be laid on the part intended
.o be shaded, and softened towards the light by washing
with the water brush. In all cases it should be borne in
*nind, that the colour ought to be laid on very thin, other-
wise it will be more difficult to manage, and will never
make so fine a drawing.
66. In colours even of the best quality, we sometimes
meet with gritty particles, which it is desirable to avoid
Instead of rubbing the colour on a plate with a little water
8*
90 MECHANICAL DUAWING
as is usual, it will be better to wet the colour, and rub it on
the point of the forefinger, letting the dissolved part drop
off the finger on to the plate.
67. In using the Indian ink, it will be found advanta-
geous to mix it with a little blue and a small quantity of lake,
which renders it much more easily wrought with, and this
is the more desirable as it is the most frequently used of all
the other colours in Mechanical Drawing, the shades being
all made with this colour.
The depth and extent of the shades will depend on
various circumstances on the figure of the .object to be
shaded, the position of the eye of the observer, and the
direction in which the light comes, <fce. The position of
the eye will vary the proportionate size of any object in a
picture when drawn in perspective. Thus, if a perspective
view of a steam engine is given, the eye being supposed to
be placed opposite the end nearest the nozzles, an inch of
the nozzle rod will appear much larger than an inch of the
pump rod which feeds the cistern ; but if the eye is sup-
posed to be placed opposite the other end of the engine,
the reverse will be the case. But in drawing elevations
and ground plans of machinery, every part of the machine
is drawn to the proper scale an inch or foot in one part
of the machine, being just the same size as an inch or foot
in any other part of the machine. So that by measuring
the dimensions of any part of the drawing, and then apply-
ing the compass to the scale, we determine the real size of
the part so measured. Whereas, if the view were given in
perspective, we would be obliged to make allowance for the
effect of distance, &c.
68. The light is always supposed to fall on the picture
at an angle of forty-five degrees, from which it follows, that
the shade of any object, which is intended to- rise from the
plane of the picture, or appear prominent, will just be equal
in length to the prominence of the object.
69. The shades, therefore, should be as exactly measured
as any other part of the drawing, and care should be taken
that they all fall in the proper direction, as the light is sup-
posed to come from one point only.
70. It is frequently of great use for the mechanic to take
a hasty copy of a drawing, and many methods have been
given for this purpose by machines, tracing, &c. Wa
give the following as easy, accurate, and convenient.
AND PEUSPECTIVK. 91
Mix equal parts of turpentine and drying oil, and with a
rag lay it on a sheet of good silk paper, allowing the paper
to lie by for two or three days to dry, and when it is so it will
be fit for use. To use it, lay it on the drawing to be copied,
and the prepared paper being nearly transparent, the lines
of the drawing will be seen through it, and may be easily
traced with a black-lead pencil. The lines on the oiled
paper will be quite distinct when it is laid on white paper.
Thus, if the mechanic has little time to spare, he may take
a copy and lay it past to be recopied at his leisure.
Care and perseverance are the chief requisites for attain-
ing perfection in this species of drawing. Every mechanic
should know something of it, so that he may the better un-
derstand how to execute plans that may be submitted to
him. or make intelligible to others any invention he him-
self may make.
CONIC SECTIONS.
DEFINITIONS.
A CONE is a solid figure having a circle for its base and
terminated in a vertex ; it may be conceived to be formed
by the revolution of a triangle about one of its sides.
Conic Sections are the figures made by a plane cutting a
cone. According to the different positions of the cutting
plane there arise five different figures or sections, namely,
a triangle, a circle, an ellipse, an hyperbola, and a parabola :
the last three of which only are peculiarly called Conic
Sections. If the cutting plane pass through the vertex of
the cone, and any part of the base, the section will be a
triangle ; as VAB, fig. 1. If the plane cut the cone parallel
*o the base, or make no angle with it, the section will be a
circle ; as fig. 2. The section DAB is an ellipse when the
cone is cut obliquely through both sides, or when the plane
is inclined to the base iu a less angle than the side of the
cone is, fig. 3. The section is a parabola, when the cone
is cut by a plane parallel to the side, or when the cutting
plane and the side of the cone make equal angles with the
base, fig. 4. The section is an hyperbola, when the cutting
plane makes a greater angle with the base than the side of
the cone makes, fig. 5. And if all the sides of the cone be
continued through the vertex, forming an opposite equal
cone, and the plane be also continued to cut the opposite
cone, this latter section will be the opposite hyperbola to
ihe former.
92
DEFINITIONS. 93
The Vertices of any section, are the points where the
cutting plane meets the opposite sides of the cone, or the
ides of the vertical triangular section.
Hence the ellipse and the opposite hyperbolas, have each
two vertices ; but the parabola only one ; unless we consider
the other as at an infinite distance.
The Axis, or Transverse Diameter, of a conic section, is
the line or distance between the vertices.
Hence the axis of a parabola is infinite in length.
The centre is the middle of the axis.
Hence the centre of a parabola is infinitely distant from
the vertex. And of an ellipse, the axis and centre lie
within the curve ; but of an hyperbola, the axis and centre
lie without it.
A Diameter is any right line drawn through the centre,
and terminated on each side by the curve ; and the extremi-
ties of the diameter, or its intersections with the curve, are
its vertices.
Hence all the diameters of a parabola are parallel to the
axis, and infinite in length. Hence also every diameter of
the ellipse and hyperbola has two vertices ; but of the para-
bola, only one ; unless we consider the other as at an infi-
nite distance.
The Conjugate to any diameter, is the line drawn through
the centre, and parallel to the tangent of the curve at the
vertex of the diameter.
Hence the conjugate of the axis is perpendicular to it.
An Ordinate to any diameter, is a line parallel to its con-
jugate, or to the tangent at its vertex, and terminated by
the diameter and curve.
Hence the ordinates of the axis are perpendicular to it.
An Absciss is a part of any diameter contained between
its vertex and an ordinate to it.
Hence, in the ellipse and hyperbola, every ordinate has
two determinate abscisses ; but in the parabola only one ;
the other vertex of the diameter being infinitely distant.
The Parameter of any diameter is a third proportional
to that diameter and its conjugate, in the ellipse and hyper-
bola, and to one absciss and its ordinate in the parabola.
The Focus is the point in the axis where the ordinate is
equal to half the parameter.
The ellipse and hyperbola have each two foci ; but
parabola only one.
94 CONIC SECTIONS.
PROBLEMS FOR THE CONIC SECTIONS.
THE PARABOLA.
1. Given two abscisses A and B, together with the ordi-
nate of A, to find the ordinate of B.
*/ absciss B x ordinate A _
: T = ordinate B.
\/ absciss A
Ex. An absciss is 9, and its ordinate is 16, it is required
to find the ordinate of another absciss 36.
^ 36 X 16 6 X 16
= = 32, the required ordinate.
V' "
2. Given the ordinate and absciss, required the para-
meter.
ordinate 3
- : = parameter.
absciss
Ex. The ordinate being 12 and absciss 6, then,
12 a 144
= = 24 = the parameter required.
3. To find the length of the curve of a parabola, cut off
by a double ordinate to the axis.
v/ (ordin. 8 -f abs. 2 ) x 2 = the length of the curve.
Ex. The length of the double ordinate being 12 and the
absciss 2, then,
\/ (6 a + f 2 2 ) x 2 = 12-858 = the length of the curve.
NOTE. This rule is sufficiently correct for practice, but
will not apply when the absciss is greater than the half
ordinate.
THE ELLIPSE.
1. To find an ordinate, we have the proportion :
As the transverse axis is to the conjugate, so is the square
root of the product of the two abscisses, to the ordinate.
Ex. The transverse axis being 60, the conjugate 45, the
one absciss 12, and the other 48, then,
60 : 45 : : </ (48 X 12) : 18 = the ordinate required.
PROBLEMS. 95
2. To finci the absciss.
v/ (the half conju. a ordin. s ) x trans, axis
conjugate axis.
distance between the ordinate and centre of the axis, which
being added to the half axis, will give the greater absciss
or being subtracted, will give the shorter absciss.
Ex. One axis being 20 and the other 15, what are the
abscisses to the ordinate whose length is 6.
= 6 = the distance from the centre,
15
wherefore 10 -f 6 = 16 * the longer absciss, and 10
6 = 4 = the shorter.
3. To find the conjugate axis.
As x/(one absciss x other absciss) is to their ordiaate,
so is the transverse axis to the conjugate.
Ex. The transverse axis being 200, the ordinate 60
one absciss is 40 and the other 160, then,
^/(leO X 40) : 60 : : 200 : 150 = the conjugate axis.
4. To find the transverse axis.
Take the square root of the difference of the squares of
the ordinate and half conjugate, and add to this the half
conjugate if the lesser absciss is used, but subtract the .half
conjugate if the greater absciss is used. In either case call
the result of this part of the operation M, then,
conjugate x absciss x M
-ji = transverse axis,
ordinate *
Ex. If the ordinate be 20, the lesser absciss 14, and the
conjugate 50, then by the above,
s/(25 3 20 2 ) -f 25 = 40 = M.
50 X 14 X 40
--^ = 70 = the transverse axis.
5. To find the circumference of an ellipse.
sum of the sq. of the two axes\
2 ) X 3-1416 = circumfer
Ex. The one axis being 24 and the other 18, then.
24 s -t- 18 3 \
3. J X 3-1416 = 66-643 = circumference.
86 CONIC SECTIONS.
THE HYPERBOLA.
1. To find the ordinate.
As the transverse axis is to the conjugate ; so is the square
root of the product of the two abscisses, to the ordinate.
Ex. The transverse axis being 24, the conjugate 21, and
the absciss 8 ; then,
24 : 21 : : v/(32 X 8) : 14 = the ordinate.
2. To find the abscisses.
v/ford. 9 + half conjn.*)X trans, axis
= distance between
conjugate
the ordin. and centre. Then this distance, added to the
half transverse, gives the greater absciss ; or, subtracted
from it, the less.
Ex. The transverse axis being 40, the conjugate 32,
and the ordinate 12; then,
x /(12 a + 16')X40 . .
~ i = 25 = distance from the middle of
3Z
the transverse. Wherefore, 25 -f 20 = 45 = the greatef
absciss ; and 25 20 = 5 = the lesser.
3. To find the conjugate.
ordinate X transverse axis
=-: " r = conjugate.
v/(productof the abscisses)
Ex. The transverse axis being 144, the lesser absciss
48, and its ordinate 84 ; then,
84 x 144
v/(192 X 48) = = con J u g ate required.
4. To find the transverse.
Take the half conjugate, and, according as the lesser or
greater absciss is used, add it to, or subtract it from, the
square root of the sum of the squares of the half conjugal
and of the ordinate, and call this result m ; then,
abscissa x conjugate x m
P *~f = the transverse axis.
ordinate 2
Ex. The conjugate being 18, the lesser absciss 10, and
'ts ordinate 12; then,
9 + */(9* + 12 2 ) = 9 -f 15 = 24 = m;
10 X 18 x 24
= 30 = the transverse axis.
Lm
PROBLEMS.
\
Descriptions of Conic Sections on a Plane.
1. Parabola. Let AB be a v w .
right line and C a point with-
out it, and DKF a ruler in the
same plane with the line and
point, so that one side, as DK,
be applied to AB, and KF
coincide with the point C ;
on F, fix one end of the
thread FNC, and the other at
the point C ; and Jet part of
the thread, as FN, be brought to the side KF by a pin N ,
then let the square DKF, be removed from B to A, applying
its side DK dose to BA; and in the mean time the thread
will be always applied to the side KF; and by the motion
of the pin N there will be described a curve called a semi-
parabola. Then bringing the square to its first position
moving from B to H the other semi-parabola will be
described.
2. Ellipse. If two points, as
A and B, be taken in any plane,
and in them is fixed a thread
longer than the distance, between
them, and this be extended by
means of a pin C ; and the pin
be moved round from any point till it return back again
to the same place, the thread being extended all the while,
the figure described is an ellipse.
3. Hyperbola. If to the point A, one end of the ruler
AB be placed, so that about that point as a centre it may
freely move ; and if to the other end B is fixed the ex-
tremity of the thread
BDC shorter than the
ruler A13, and the other
end of the thread fixed
in the point C, coincid-
ing with the side of the
ruler AB :n the same
place with the given
point A ; let part of the
thread BD be brought
to the side of the ruler
98
CONIC SECTIONS.
A.B by the pin D ; then let the ruler be moved about the
point A from C to T, the thread extended, and the re-
maining part coinciding with the side of the ruler; by the
motion of the pin D a semi-hyperbola will be described
The ellipse returns into itself: but the parabola and hyper-
bola are unlimited.
USEFUL CURVES.
*
THE Cycloid is a very useful curve ; and may be defined,
the curve formed by a nail in the rim of a wheel, while il
moves along a level road. The cycloid may be described
on paper, thus : If the circumference of a circle be rolled
on a right line, beginning at any point A, and continued
till the same point A arrive at the line again, making just
one revolution, and thereby measuring out a straight line
ABA equal to the circumference of the circle, while the
point A in the circumference traces out a curve line
ACAGA: then this curve is called a cycloid; and some
of its properties are contained in the following iemma:
If the generating or revolving circle be placed in the
middle of the cycloid, its diameter coinciding with the axis
AB, and from any point there be drawn the tangent CF,
the ordinate CDE perpendicular to the axis, and the chord
of the circle AD ; then the chief properties are these :
The right line CD = the circular arc AD ;
The cycloidal arc AC = double the chord AD ;
The semi-cycloid ACA= double the diameter AB, and
Tne tangent CF is parallel to the chord AD.
If the ball of a pendulum be made to move in a cycloid,
its vibrations will be isochronous, or, they will all be per-
formed in the same time. The cycloid is also the line of
swiftest descent, or, a body will fall through the arc of this
curve, from one given point to another, in less time than
through any other path. See Centre of Oscillation
CONIC SECTIONS. 99
The Catenary is that curve which is formed by a chain
or chord of uniform texture, when it is hung upon tw<?
points, and left to hang freely, without any restraint. It
matters not whether these points of suspension be in the
same horizontal line or not, or whether the chain be slack
or tight; still the fcurve will be a catenary. A knowledge
of this curve is very useful in the construction of suspension
bridges See the chapter on Strength of Material*.
MENSURATION.
DEFINITIONS.
To the definitions in geometry the lollowmg are aduud,
in order to make the subject of mensuration understood.
1. A. prism is a solid, of which the sides are parallelo-
grams, and the ends equal, similar, and parallel plane
figures. The figure of the ends gives the name to the
prism ; if the ends are triangular, the prism is triangular,
&c. If the sides and ends of a prism be all equal squares,
the prism is called a cube ; and if the base or ends be a
parallelogram, the prism is called a parallelopipedon. The
cylinder is a round prism, having circular ends.
2. The pyramid has any plane figure for its base, and
its sides triangles, of which all the vertices meet in a point
at the top, called the vertex of the pyramid.
3. A sphere or globe is a solid bounded by one continued
surface, every point of which surface is equally distant from
a point within the sphere, called the centre. The diameter
or axis of a sphere, is any line which passes through its
centre, and is terminated at both ends by the circumference
4. A prismoid has its two ends as any unlike parallel
plane figures of the same number of sides ; the upright
sides being trapezoids.
5. A spheroid is a solid resembling the figure of a sphere,
but not exactly round one of its diameters being longer
than the other ; and, likewise, a conoid is like a cone, but
has not its sides straight lines but curved.
6. A spindle is a solid formed by the revolution of some
curve round its base.
7. The axis of a solid is a straight line drawn through
the solid, from the middle of one end to the middle of the
opposite end.
8. The height of a solid is a line drawn from the vertex,
perpendicular to the base, or the plane on which the base rests.
9. The segment of a solid is a part cut off by a plane,
parallel to the base ; and the frustum is the part remaining
after the segment is cut off.
100
MENSURATION.
101
SURFACES.
1. For the area of a square, rhombus, or rhomboid.
Base X height = area.
Ex. The base of a rhombus is 16, the height 9 ; there-
fore, 16 X 9 = 144 = area.
2. For the area of a triangle.
5 (base X height) = area.
Ex. The base of a triangle is 2j, and height 7<3 ; there-
fore, d (2-25 X 7-5) = 8-437, the area.
3. For the area of a frapczoid.
5 (sum of the two parallel sides) x height = area.
Ex. In a trapezoid one of the parallel sides is 16s, tb<?
other is 14], and the height or perpendicular distance be-
tween them is 7 ; therefore,
I (16-125 + 14-25) x 7 = 106-3125, the area.
4. For any right-lined figure of four or more unequal
sides.
Divide it into triangles, by lines drawn from various
angles ; find the area of each ; then, the sum of these areas
will be the area of the whole figure.
5. For a regular polygon.
Inscribe a circle ; then, 5 (radius of insc. circle x length
of one side x number of sides) = area.
Ex. In a polygon of 8 sides, the length of a side is 16,
and radius of inscribed circle 19-2 ; then (3x16x8) =
1230, the area.
The following table will greatly facilitate the solution of
questions connected with polygons.
iid
Nime of
An?. F at
Ang. C of
Ana.
A.
B.
c.
3
Trigon
120
60
0-4330127
2-
1-73
579
4
Tetragon
90
90
1-0000000
1-41
1-412
705
6
Pentagon
72
108
1-7204774
1-238 1-174
852
6
Hexagon
60
120
25980762
1-156
= ftJiu>
= Leo#*
of side.
7
Heptagon
51 :
128*
3-6339124
1-11
867
1-16
8
Octagon
45'
135
4-8284271
1-08
765
1-307
9
Nonagon
40
140
6-1818242
1-062-681
1-47
10
Decagon
36
144
7-6942088
1-05
616
1-625
11
Urulecagon
32 T 8 T
147J\
9-3656405
1-04
561
1-777
12
Dodecagon
30
150
11-1961524
1-037
515625
1-94
1
102 MENSURATION.
The first column of this table gives the number of sides
of the polygon ; the second, the name ; the uses of the
third and fourth will be explained in the note at the bottom
of the page,* and the uses of the rest will appear by the fol-
lowing rules and examples. The answers found are only
approximate, but come sufficiently near the truth for all
practical purposes.
Side of polygon 3 x No. column AREA = area.
Ex. In *a figure of 10 equal sides, the length of one
side being 8, we have 8' 2 = 8 x 8 = 64 ; hence 64 X
7-6942088 = 492-4293632 = the area.
Take the length of a perpendicular, drawn from the
centre to one of the sides of a polygon, and multiply this
by the numbers in column A, the product will be the radius
of the circle that contains the polygon.
Ex. If the length of a perpendicular drawn from the
centre to one of the sides of an octagon be 12, then 12 x
1*08 = 12-96 = radius of circumscribing circle.
The radius of a circle multiplied by the number in
column B, will give the length of the side of the correspond-
ing polygon which that circle will contain. Suppose, for
an octagon, the radius of a circle to be 12-96, then 12-96
X '765 = 9-9144 = the length of one side of the inscribed
polygon of 8 sides.
The length of the side of a polygon multiplied by the
corresponding number in the column C, will give the radius
of circumscribing circle. Thus the length of one side of a
decagon being 10; then 10 x 1'625 = 16-25 = radius
of circumscribing circle.
6. For the circle.
1st, diameter X 3-1416 = circumference;
* The third and fourth columns of the table of polygons will greatly
facilitate the construction of these figures by the aid of the sector. Thus,
if it be required to describe a polygon of eight sides, then look in column
Angle F, opposite Octagon, and you find 45. With the chord of 60 on
the sector as radius describe a circle, then taking the length 45 on the
same line of the sector, mark this distance off on the circumference,
which being repeated round the circle, will give the points of junction
of the sides cf the octagon. The fourth column of the table gives the
angle in degrees, which any < wo adjoining sides of the respective figurei
make with each other.
MENSURATION.
103
, circumference
2</ ' -TI416- = diameler;
3d, circumference x radius = area.
Ex. In a circle whose diameter is 14 inches, we have
1st, 14 x 3-1416 = 43-9824, the circumference;
43-9824
*d,
3-1416
= 14, the diameter ;
14
8d, diameter 2 = radius, so ~ = 7 = radius. Then
(43-9824) X 7 = 153-9384, the area.
7. For the length of the arc of a circle.
Radius x -079577 X number of degrees = length of are.
Ex. If the radius be 12, ind number of degrees 22, then
12 X -079577 X 22 = 21-008328, the length.
8. For the area of a circular sector.
Radius X 5 length of arc.
Ex. The radius being 12, and length of arc 2 1-008328 ;
then, 12 X 10-504164 = 126-049968, the area.
9. For the area of a circular segment.
TABLE OF THE AREAS OF CIRCULAR SEGMENTS.
H.
AIM.
H.
Arta.
H. A
H
A-e.
01
001329
14
06683:5
27 -171089
40
293369
02
003748
15
073874
28 -100019
41
303187
03
006865
16
081112
29
189047
42
313041
04
010537
17
088535
30
198168
43
322928
05
014681
18
096134
31
207376
44
332843
06
019239
19
103900
32
216666
45
3-1 27*2
07
024168
20
111823
33
226033
46
352742
08
029435
21
119897
34
235 i73
47
362717
09
035011
2*
128113
35
244980
48
372704
10
040875
23
136465
36
251550
49
382099
11
047005
24
.144944
37
264)78
50
392699
12
053385
25
153546
38
273861
001
000042
13
059999
26
162263
39
2S3592
002
000119
This may be done easily by the help of the preceding
table ; to use which, divide the height of the segment by
the. diameter of the circle, and look for the quotient in the
104 MENSURATION.
column H, opposite to which will be found a number in
column AREA, which multiplied by the square of the dia-
meter will give the area of the segment. Should the height
of the segment be greater than the diameter, find by the
foregoing rule the area of the remaining segment, and by
subtracting this from the area of the whole circle, the area
of the greater segment will be found.
18
EA Let the height be 18 and diameter 48, then = -37;
which, in the column marked H in col. AREA, corresponds
to -264178 ; hence 48 2 x '264178 = 608-6661 = the area.
10. For the area of a cycloid.
Area of generating circle x 3 = area of cycloid.
Ex. The diameter of generating circle being 10, then
(10 x 3-1416) x V X 3 = 235-619, the area of cycloid.
11. For the area of a parabola.
(Base x height) X | = the area.
Ex. The base being 20, and height 6 ; then,
20 x 6 X | = 80, the area.
12. 'For the area of an ellipse.
(Long axis x short axis) x -7854 = area.
Ex. The -greater axis being 300, and lesser 200 ; then,
300 x 200 x '7854 = 47124, the area.
SOLIDS.
1. For the surface and content of a prism or cylinder.
1st. Area of two ends + length X perimeter = surface.
2d. Area of base x height = content.
The circumference of a cylinder is 6, and its length 9
inches ; what is the surface and content ?
The area of each end is 2-85 ; therefore 2 X 2'85 ==
5-7 = the area of the two ends, and then 5'7 -f- (6 X 9)
= 59-7 = the area of the whole cylinder. Also, 2-85 X
9 = 25-65 = content.
2. For a cone or pyramid.
1st. k (slant height x perimeter of base) + area of base
= surface.
MENSURATION.
105
2</. 3 (area of base x perpend, height) = content.
Ex. Shun height is 10, perimeter of base 16; then,
(10 X )6 = 80 -f 10 = 96, surface of a four-sided pyra-
mid, whose side at the base is 4.
The area of the base of a cone being 147 - 68, and per-
pendicular height 14,
Then | (U\ 147*68) = 689-17, content.
3. For a cube or parallelopiped.
Isf, The sum of tli areas of all the sides = surface.
2d. Length X breadth x depth = content.
Ex In a parallelopiped the length 30, breadth 6, and
depth 4.
30 x 6 x 4 = 720, content, and 648 = the surface.
It is worthy of remembrance that one cubic foot contains
1728 cubic inches. 22,000 cylindric, 3300 spherical inches,
and 66 conical. The cone, sphere, and cylinder, are as 1,
2, and 3.
4. For regular or platonic bodies, or bodies of equal sides
1st. Linear edge 2 x tabular number of figures for sur-
face = surface.
2(1. Linear edge 3 X tabular number of figures for soli-
dity = content.
No. of Sides.
Name.
Multiplier for
Surface.
" Multiplier for
Solidity.
4
6
8
12
20
Tetrahedron,
Hexahedron,
Octahedron,
Dodecahedron,
Icosahedron,
1-7320508
6-0000000
3-4641016
20-6457288
8-6602540
0-1178513
1-00000
0-4714045
7-6631189
2-181695
Ex. In an Octahedron the length of the ridge of a side
is 5, therefore 5* x 3-4641016 = 86-6025 = surface, and
5 3 x -4714045 = 58-9255, the solidity.
5. For the surface of a sphere and segment.
/Diameter 2 X 3-1416 = surface of the whole sphere.
Ex. If the diameter be 36, then 36" x 3- 14 16=4071 '504
square inches = surface.
Height of segment X diameter of sphere x 3-1416 =*
surface of segment.
106 MENSURATION.
Ex. The diameter of the sphere being 12, and the
height of segment 6, then
6 Xl2 X 3-1416 = 226-1952 = surface of spheric segment.
6. For the. content of a sphere and spheric segment.
Diameter 3 X 0-5236 = content.
Ex. If the diameter of a sphere be 2 inches, then 2 8
X 0-5236 = 4-1888 = the content.
(radius of segment's base a x 3 -f- height of segment 8 ) x
height x -5236 = content of segment.
Ex. If the height of a spheric segment be 2, and radius
of base 6, then
(6 a X 3 + 2 a ) X 2 X -5236 = 117-2864 = content
7. For the solidity of a steroid.
Revolving axis 3 X fixed axis x -5236 = content.
NOTE. If the spheroid revolVe round the greater axis,
it is said to be oblate ; if round the lesser, oblong.
Ex. The two axes of a spheroid are 24 and 18; there-
fore,
24 S X 18 X -5236 = 5428-56 = content if it be oblate.
18 3 X 24 X -5236 = 4071-5 = content if it be oblong.
8. For the solidity of a parabolic conoid.
Area of base X half the height = content.
Ex. The height being 18, and the diameter of base 24,
then the area of the base therefore is 452-39 ; hence
452-39 X 9 = 4071-51 the content.
9. For the frustum of a cone or pyramid.
(perim. of one end -}- perim. of the other end) x slant height
~2~
= surface.
Ex. In the frustum of a triangular pyramid the peri-
meter of one end is 25, that of the other 36, and the slant
height is 10 ; therefore,
(25 + 36) x 10
= 305 = the surface.
!c
^(area of one end + ar. of other) + area of one end + ar. of other
3T~
X height = content.
Ex. A log of wood is 20 feet long ; its ends are squares,
MENSURATION. 107
f which the sides are respectively 12 and 16 inches ; there-
fore,
- < _ con(en ,
TIMBER MEASURE.
EXAMPLES of timber measuring have already been given in
the department allotted to arithmetic, but it is necessary to
be here somewhat more particular. The surface of a plank
is found :
1st. By multiplying the length by the breadth. When
the board tapers gradually, add the breadth at both ends
together, and take the half of this sum for the mean
breadth.
2d. By the sliding rule. Set the length in inches on
B to 12 on A, and against the length in feet on B will be
the area in square feet and decimals on A.
Ex. A board is 12 feet 6 inches long and 1 foot 3
inches broad ; hence,
12 : 6
1 : 3
T2 : 6
3:1:6
15 : 7 : 6
1st. For the content of squared timber, length x mean
breadth X mean thickness = content.
2rf. By the sliding rule. Find the mean proportional
between the breadth and thickness, then set the length on
C to 12 on D, and against the mean proportional on D the
solid content on C. If the mean proportional be in feet,
reduce to inches.
Ex. A log is 24 feet long, the mean depth and breadth
being each 13 inches.
1 : 1
1 : 1
1:2:1
24
28 : 2 :
10S MENSURATION.
For round timber. 1st. Take one-fourth of the mean
girth and square it, this multiplied by the length will give
die content.
2d. By the sliding rule. Set the length in feet on C
to 12 on D, then against the quarter girth in inches on D,
will be the content on C.
This gives no allowance for bark, but there is usually a
deduction made of about an inch to the foot of quarter girth.
The rule given above gives the customary, but not the true
ontent ; the following gives the true content.
One-fifth of the girth squared and multiplied by twice the
length = content.
Ex. The mean girth of a tree being 5 feet 8 inches, and
its length 18 feet, the two rules will apply as below:
4) 5 : 8 (1 : 5 5) 5 : 8 (1 : 1 : 7
1 ; 5 1 ; 1 ; 7
2:0:1 T~: 3 : 4 : 6
18 36
36 : 1 : 6 46 : 1 : 6
Trees very seldom have an equal girth throughout, one
end being generally much smaller than the other : the girth
taken above is the mean girth ; that is to say, the girths of
both ends added together, and their sum halved for the
mean girth. It is to be observed, however, that, if the
difference of the girths is great, it will be best to find the
content of the tree as if it $vere a conic frustum. The
method of using the sliding rule in the measurement of tim-
ber has been given before.
ARTIFICERS' WORK.
ARTIFICERS compute the contents of their works by
several different measures ; as, glazing and masonry by the
foot; painting, plastering, paving, &c., by the yard of 9
square feet; flooring, partitioning, roofing, tiling, &c., by
the square of 100 square feet; and brickwork, either by
a yard of 9 square feet, or by the perch, or square rod or
pole, containing 2721 square feet, or 30.1 square yards,
being the square of the rod or pole of 16s feet of 5^ yards
long. As this number 272.j is troublesome to divide by,
the 1 is often omitted in practice, and the content in feet
divided only by the 272. But when the exact divisor
ARTIFICERS' WORK. 109
272] is to be used, it will be easier to multiply the feet by
4, and then divide successively by 9, 11, and 11. Also to
divide square yards by 30], first multiply them by 4, and
then divide twice by 11.
BRICKLAYERS' WORK. Brickwork is estimated at the
rate of a brick and a half thick. So that, if a wall be more
or less than this standard thickness, it must be reduced to
it, as follows : Multiply the superficial content of the wall
by the number of half bricks in the thickness, and divide
the product by 3. The dimensions of a building are usu-
ally taken by measuring half round on the outside, and half
round on the inside ; the sum of these two gives the com-
pass of the wall, to be multiplied by the height, for the
content of the materials. Chimneys are by some measured
as if they were solid, deducting only the vacuity from the
hearth to the mantel, on account of the trouble of them.
And by others they are girt or measured round for their
breadth, and the height of the story is their height, taking
the depth of the jambs for their thickness. And in this
case, no deduction is made for the vacuity from the floor
to the mantel-tree, because of the gathering of the breast
and wings, to make room for the hearth in the next story.
To measure the chimney shafts, which appear above the
building, gird them about with a line for the breadth, to
multiply by their height. And account their thickness
half a brick more than it really is, in consideration of the
plastering and scaffolding. All windows, doors, &c., are
to be deducted out of the contents of the walls in which
they are placed. But this deduction is made only with
regard to materials ; for the whole measure is taken for
workmanship, and that all outside measure too, namely,
measuring quite round theoutside of the building, being
in consideration of the trouble of the returns or angles.
There are also some other allowances, such as double
measure for feathered ffahle ends, &c.
Ex. The end wall of a house is 28 feet 10 inches long",
"ai\'\ 55 tVet 8 inches high, to the eaves: 20 feet high is 2'
bucks thick, other 20 feet high is 2 bricks thick, and the
remaining 15 feet 8 inches is H brick thick; above which
is, a triangular gable, 1 brick thick, and which rises 42
courses of bricks, of which every 4 courses make a foot.
What is the whole content in standard measure ?
Ans. 253-626 yards.
10
110 MENSURATION
MASONS' WORK. To masonry belong all sorts of stone-
work ; and the measure made use of is a foot, either super-
ficial or solid. AValls, columns, blocks of stone or marble,
&c., are measured by the cubic foot; and pavements,
slabs, chimney-pieces, &c., by the superficial or square
foot. Cubic or solid measure is used for the materials,
and square measure for the workmanship. In the solid
measure, the true length, breadth and thickness, are taken,
and multiplied continually together. In the superficial,
there must be taken the length and breadth of every part
of the projection, which is seen without the general upright
face of the building.
Ex. In a chimney-piece, suppose the
Length of the mantel and slab, each 4 feet 6 inches ;
Breadth of both together, 3 2
Length of each jamb, 4 4
Breadth of both together, 1 9
Required the superficial content. Ans. 21 feet, 10 inch.
CARPENTERS' AND JOINERS' WORK. To this branch
belongs all the wood-work of a house, such as flooring,
partitioning, roofing, &c. Large and plain articles are
usually measured by the square foot or yard, &c., but
enriched mouldings, and some other articles, are often
estimated by running or lineal measures, and some things
are rated by the piece.
In measuring of joists, it is to be observed, that only one
of their dimensions is the same with that of the floor ; for
the other exceeds the length of the room by the thickness
of the wall and | of the same, because each end is let into
the wall about f of its thickness.
No deductions are made for hearths, on account of the
additional trouble and waste of Viaterials.
Partitions are measured from wall to wall for one dimen-
sion, and from floor to floor, as far as they extend, for the other.
No deduction is made for door-ways, on account of the
trouble of framing them.
In measuring of joiners' work, the string is made to ply
close to every part of the work over which it passes.
The measure for centering for cellars is found by making
a string pass over the surface of the arch for the breadth,
and taking the length of the cellar for the length ; but in
groin centering, it is usual to allow double measure, on
account of their extraordinary trouble.
ARTIFICERS' WORK. Ill
In roofing, the length of the house in the inside, to-
gether with ~ of the thickness of one gable, is to be con-
sidered as the length ; and the breadth is equal to double
the length of a string which is stretched from the ridge
down the rnfic-r, and along the eaves-board, till it meets with
the top of the wall.
For staircases, take the breadth of all the steps, by making
a line ply close over them, from the top to the bottom, and
multiply the length of this line by the length of a step, for
the whole area. By the length of a step is meant the
length of the front and the returns at the two ends ; and
by the breadth, is to be understood the girth of its two outer
surfaces,, or the tread and riser.
For the balustrade, take the whole length of the upper
part of the hand-rail, and girt oVer its end till it meet the
top of the newel post, for the length ; and twice the length
of the baluster upon the landing, with the girth of the hand-
rail, for the breadth.
For wainscoting, take the compass of the room for the
length ; and the height from the floor to the ceiling, making
the string ply close into alb the mouldings for the breadth. T
Out of this must be made deductions for windows, doors,
and chimneys, &c., but workmanship is counted for the
whole, on account of the extraordinary trouble.
For doors, it is usual to allow for their thickness, by add-
ing it unto both the dimensions of length and breadth, and
then to multiply them together for the area. If the door
be paneled on both sides, take double its measure for the
workmanship ; but. if the one side only be paneled, take
the area and its half for the workmanship. For the sur-
rounding architrave, sfird it about the outermost parts for
its length ; and measure over it, as far as it can be seen
when the door is open, for the breadth.
Window-shutters, bases, &c., are measured in the same
manner.
In the measuring of roofing for workmanship alone,
Holes for chimney-shafts and skylights are generally de-
dusted. But in measuring for work and materials, they
commonly measure in all skylights, luthern-lights, and
holes for the chimney-shafts, on account of their trouble
and waste of materials.
Ex. To how much, at Qs. per square yard, amounts the
wainscoting of a room ; the height, Baking in the cornice
MENSURATION.
and mouldings, being 12 feet 6 inches, and the whole com-
pass 83 feet 8 inches ; also three window- shutters are each
7 feet 8 inches by 3 feet 6 inches, and the door 7 feet by 3
feet 6 inches ; the door and shutters, being worked on both
sides, are reckoned work and half work ?
Ans. 36, 12s. 2%d.
SLATERS' AND TILERS' WORK. In these articles, the
content of a roof is found by multiplying the length of the
ridge by the girth over from eaves to eaves ; making allow-
ance in this girth for the double row of slates at the bottom,
or for how much one row of slates or tiles is laid over an-
other. When the roof is of a true pitch, that is, forming
a right angle at top, then the breadth of the building with
its half added, is the girth over both sides. In angles
formed in a roof, running from the ridge to the eaves, when
(he angle bends inwards, it is called a valley; but when
outwards, it is called a hip. Deductions are made for
chimney-shafts or window-holes.
Ex. To how much amounts the tiling of a house, at
25s. 6d. per square ; the length being 43 feet 10 inches,
and the breadth on the flat 27 feet 5 inches, also the eaves
projecting 16 inches on each side, and the roof of a true
pitch? 24, 9s. 5|rf.
PLASTERERS' WORK. Plasterers' work is of two kinds,
namely, ceiling which is plastering upon laths and ren-
''ering, which is plastering upon walls ; which are measured
separately.
The contents are estimated either by the foot or yard, or
square of 100 feet. Enriched mouldings, <fcc., are rated by
running or lineal measure.
Deductions are to be made for chimneys, doors, win-
dows, &c. But the windows are seldom deducted, as the
plastered returns at the top and sides are allowed to com
pensate for the window opening.
Ex. Required the quantity of plastering in a- room, the
length being 14 feet 5 inches, breadth 13 feet 2 inches, and
height 9 feet 3 inches to the under side of the cornice,
which girts 85 inches, and projects 5 inches from the wall
on the upper part next the ceiling deducting only for a
door 7 feet by 4.
Ans. 53 yards 5 feet 3 inches of rendering,
18 5 6 of ceiling,
39- O of cornice.
ARTIFICERS' WORK. 113
PAINTERS' WORK. Painters' work is computed in square
yards. Every part is measured where the colour lies ; and
the measuring line is forced into all the mouldings and
corners.
Windows are done at so. much apiece. And it is usual
to illow double measuYe for carved mouldings, &c.
Ex. What costs the painting of a room at 6d. per yard ;
its length being 24 feet 6 inches, its breadth 16 feet 3
inches, and height 12 feet 9 inches ; also the door is 7 feet
by 3 feet 6 inches, and the window-shutters to two windows
each 7 feet 9 inches by 3 feet 6 inches, but the breaks of
the windows themselves are 8 feet 6 inches high, and 1 fo.ot
3 inches deep deducting the fire-place of 5 feet by 5 feet
6 inches ? Ans. dB3, 3s. 10|rf.
GLAZIERS' WORK. Glaziers take their dimensions either
in feet, inches, and parts ; or feet, tenths, and hundredths.
And they compute their work in square feet.
In taking the length and breadth of a window, the cross
bars between the squares are included. Also, windows of
round or oval forms are measured as square, measuring
them to their greatest length and breadth, on account of the
waste in cutting the glass.
Ex. Required the expense of glazing the windows of a
house at 134. a foot; there being three stories, and three
windows in each story.
The height of the lower tier is 7 feet 9 inches,
'.. of the middle 6 6
of the upper 5 3|
and of an oval window over the door 1 10 k
the common breadth of all the windows being 3 feet 9
inches. Ans. 12, Is. 8-irf.
PAVERS' WORK. Pavers' work is done by the square
yard. And the content is found by multiplying the length
by the breadth.
Ex. What will be the expense of paving a rectangular
courtyard, "whose length is 63 feet, and breadth 45 feet ;
in which there is laid a footpath of 5 feet 3 inches broad,
running the whole length, with broad stones, at 3s. a yard
tbj rest being paved with pebbles, at Is. 67. per ynrd ?
Ans. 40, 5.10</.
TLTTMBERS' WORK. Plumbers' work is rated at so much
a pound, or else by the hundred weight, of 112 pounds.
Sheet lead used in roofing, sputtering, &c., is from 7 to 13
10*
114 MENSURATION.
lb. to the square foot. And a pipe of an inch bore is com-
monly 13 to 14 lb. to the yard in length.
Ex. What cost the covering and guttering a roof with
lead, at 19s. the cwt. ; the length of the roof being 43 feet,
and breadth or girth over it 32 feet the guttering 60 feet
long, and 2 feet wide, the former 9 lb., and the latter 8 lb
to the square foot ? Ans 113, 3s. Sid
MECHANICS.
DEFINITIONS.
1. A BODY is any quantity of matter collected together.
2. Whatever communicates, or has a tendency to com-
miniciite motion to a body, is called a force.
3. That department of knowledge which comprehends a
statement of the effects of forces on bodies, is called Mecha
nics. If a body be put in motion by the action of one or
more forces, the consideration of the circumstances of this
body belongs to that branch of Mechanics called Dynamics;
but if two or more forces act on a body in such a way that
they destroy each other's effects, and the body remains at
rest, or in equilibrium, the consideration of the circum-
stances of a body, in this case, belongs to that department
of Mechanics called Statics.
4. The density of matter, is the quantity of matter con-
tained in any body compared with its bulk. Thus lead is
denser than cork.
5. The weight of a body, is its quantity of matter, with-
out regard to its bulk.
6. When we speak of some given space, which a moving
body passes over in a given time, we speak of the velocity
of the body. If a body moves over one foot of space in one
second of time, it is said to have a velocity of one foot in
the second ; and its velocity would be increased to the
double, if it passed over two feet in one second of time.
7. If, while the body is in motion, the velocity continues
the same, the body is said to have a uniform motion ; but
if, while the body moves onward, the velocity continually
increases, it is said to have an accelerated motion ; and, on
the/other hand, if during the progress of the body in motion,
the velocity continually decreases, the body is said to have*
a retarded motion. '
8. The quantity of matter in a moving body, multiplied
by the velocity with which it moves, is called the quantity
of motion, or momentum of the body.
115
116
MECHANICS.
9. Gravity is that force by which all bodies endeavour to
descend towards the centre of the earth.
AXIOMS, OR PLAIN TRUTHS.
IF a body be at rest, it will remain at rest ; and if in mo-
tion, it will continue that motion, uniformly in a straight
line, if it be not disturbed by the action of some external
cause.
The change of motion takes place in the direction in
which the moving force acts, and is proportional to it.
The action and reaction of bodies upon one another, are
equal.
LAWS OF MOTION.
Uniform motion is caused by the action of some force.
by one impulse, on the body : and if
b signify the quantity of matter to be moved,
f the force which caused the body's motion,
v the velocity with which the body moves,
m the momentum of the body in motion,
s the space passed over by the moving body,
t the time of describing that space ;
and if b = 3, m = 6, v = 2,/ = -6, s = 4, t = 2 : then
the figures in the examples will show the application of the
theorems.
b:
m
s :
1
v :
t :
m
v
m :
t X
m
T :
s
THEOREMS.
fxt
3:
6 :
6 :
4 :
2:
2:
6
EXAMPLES.
6 6x2 6X2
V
b X v
b x v
tx
V '
s
bx
s
s
2
6
6
2
6
"3
4
' 2 '
: 3 x2
: 3 X2
X2- 2
4 4
3X4
t
b X
s
3
2
X4
t
m t
x/
X
2
6 2X6
s
7
* X b
b
f
b
s X b
b
3
46
rt* * O
-^ O
4x3
3
4x3
v
m
f
2
' 6
6
MOTION. 117
OF ACCELERATED MOTION.
If the moving force continues to act all the while that the
body is in motion, then that motion will be uniformly ac-
celerated : such is the case with bodies falling to the earth,
as the force of gravity acts constantly. Now, it has been
found by experiment, that a body falling through free space,
in the latitude of London, will, by the force of gravity, fall
through 16*095 feet in the rirst second of time ; and as forces
are measured by the effects ihey produce, this 16'095 may
be taken as the measure of the force of oravity ; and as this
quantity does not differ materially from 16 feet", we shall
neglect the fraction '095 in our calculation of the circum-
stances of falling bodies.
The subjects of consideration here are, the time that the
falling body is in motion, the spar*: it falls through in that
tinie. and the velocity which it has acquired in falling
through that space, or that velocity with which it would
continue to move, supposing gravity to cease its action, and
the motion of the body becoming uniform.
The time is always supposed to be taken in seconds, and
the space in feet.
The velocity acquired = 32 x time of falling:,
or = x /(64 x space fallen through)
_,, . rrii- the velocitv acquired
The time of falling = ~
040
I/the space fallen through \
%' 16 '
the velocitv acquired *
Tne space fallen through = ^
or = time, 2 x 16.
Ex. If a body falls through 100 feet, then
-v/(64 x 100) = 80 = the velocity acquired
80
r = 2-^v = 2-5 = the time of falling.
If /the space described be 64 feet, then
Iff* 4 \
I- - = 2 = the time of falling,
32 x 2 = 64 = the velocity acquired.
If the space descended be 400, then
118 MECHANICS.
v/(400 x 64) = 160 = the velocity acquired,
L -^ - = 5 = the time of falling.
o
If the times be as 1, 2, 3, 4, 5, &c.
The velocities will be as 1, 2, 3, 4, 5, &c.
And the spaces as 1, 4, 9, 16, 25, &c.
The space for each time as 1, 3, 5, 7, 9, &c.
COLLISION OF BODIES.
IF two bodies, A and B, in motion, weigh respectively 5
and 3 Ibs., and their velocities respect-
ively 3 and 2 before they strike, - - - - - .
then will 3 X 5 be the momentum of
A, and 2x3 that of B, before the stroke; also, 5 -f 3 =
8 is the sum of their weights ; then, 1st. If the bodies move
the same way, the quotient arising from the division of the
sum of the momentums of the two bodies, by the sum of
their weights, will give the common velocity of the two
bodies after the stroke. 2d. If the bodies move contrary
ways, then the quotient arising from the division of the
difference of their momentums, by the sum of their
weights, will give the common velocity after the stroke.
3d. If one of the bodies be at rest, then the quotient of the
momentum of the other body, divided by the sum of the
weights of the two bodies, will give the common velocity
after the stroke. Hence, assuming the numbers given above,
1 e ( f*
we have, in the first case, = 2| ; in the second
8
- - = 1| ; and in the third - = 1, as the common
8 8
velocity after the stroke.
When the bodies are perfectly elastic, the theorems be-
come more complicated.
If the weight of the one body be A, and the velocity V ;
the weight of the other body B, and its velocity v : then,
1st. If the bodies move in the same direction before the
stroke,
(2Bxt>) (A-BxV) , . . .
- - = the velocity of A after the stroke.
A-f-r>
_ the velocity of B after the stroke.
A+L*
PARALLELOGRAM OF FORCES. 19
2</. If B move in the contrary direction to A before the
stroke,
(A-B)xV-2xBxV
i = velocity of A after the stroke.
(A B) xy+2+AxV
- - = velocity of B after the stroke.
3d. If the body B had been at rest before it was struck
by A, then
A I>
X V = the velocity of A after the stroke.
X V = the velocity of B after the stroke.
A + B
2 A
A x B
Ex. If the weight of an elastic body A be 6 Ibs., and
its velocity 4, and the weight of another body B be 4 Ibs.,
and its velocity 2; then we have these results: in the first
case,
(2x4x2) + (6 4x4) , . , .
= -8 = velocity of A alter the stroke.
6 + 4
(2x6x4) + (6 4X2)
i = 5-2 velocity of B after the stroke.
The sum of these two velocities, viz. 5-2 and -8 = 6, which
was the sum of the velocities 2 and 4 before the stroke ;
and this is a general law. The reader may exercise him-
self with the rules for the other cases.
It is to be observed, that when non-elastic bodies, that is,
bodies which have no spring, strike, they will both move
in the direction of the motion of that body which has the
greater momentum ; but if they are elastic, they will recoil
after the stroke, and move contrary ways.
THE COMPOSITION AND RESOLUTION OF FORCES.
j IF a body be acted upon
By two forces, one of which
would cause it to move from
A to B in any given time,
and the other would cause
it to move from A to C m
12! MECHANICS
the same time ; then if these forces act upon the body at
one instant, it will move in neither .of the lines AB, AC,
but in the line AD, which is the diagonal of the parallelo-
gram of which the two lines AB and AC are containing
sides ; and by the action of the two forces, the body will be
found at D, at the end of the time that it would have, been
found at B or C, by the action of either of the forces singly
This important fact in mechanical science, is usually called
the parallelogram of forces. From this statement it will
be seen, that if we have the quantity and direction of any
two forces urging a body at the same instant, we can find
the resulting motion, both in quantity and direction.
It will not be difficult to understand, that if the two forces
which act upon a body, act not at an angle, but in the same
straight line, and in contrary directions, the resulting
motion will be in that straight line, and in the direction of
the greater force ; but if the forces be equal, the body will
remain at rest. If, while a body A is urged by a force in
the direction AB, which would carry it to A, it be acted
on by another force in the direction AC which would carry
it to C, and a third force in the direction DA, which would
carry it over a space as great as that from D to A, these
being the sides and diagonals of a parallelogram, the body
A will remain at rest. Also, if a body A has a tendency to
move in the direction AB, but is counteracted by a force
DA, and if we wish to keep the body A from moving,
altogether, we must apply another force AC, forming the
other side of the parallelogram of which AB is one side and
AD the diagonal.
If there be three forces acting on a body at the same time,
make the sides of a parallelogram represent any two of
them ; then the diagonal of this parallelogram, together
with the third force as the two sides of another parallelo-
gram, will give a diagonal which will be the result of the
three forces acting* at once on the body.
If the two forces which urge the body, both produce a
uniform motion, the resulting motion will be in a straight
line; tout if one of them act by impulse, which would pro-
duce a uniform motion, and the other act constantly so as
to produce an accelerated motion, the resulting motion will
be in a curve. Thus, if the ball of a cannon were sent in
a horizontal direction, it would never deviate from this
straight line unless acted on by some external force. The
THE LEVER. 12}
force of gravity acts on the body constantly, so as to draw
it to the earth, by a uniformly accelerated motion ; and the
result is, that the ball will move in a curve, and this curve
may bo easily shown to be that of the parabola. The re-
sistance of the air being taken into account together with
these circumstances, constitute the basis of the science of
gunnery.
We shall give a simple example, to show the application
of the former part of this subject. One force will cause
the body A to move 20 miles in a day, and another, acting
at right angles, will cause it to move 18 miles a day; draw
these lines 20 and 18 from the line of lines on the sector,
as the sides AB, AC, of a parallelogram, and complete it:
draw the diagonal, then measure it, and it will be found
to be 26'9, the resulting motion ; and the angle being
measured, will give the direction. There are other methods
of doing this by calculation, but this is simple, and is suffi-
cient to show the principle.
MECHANICAL POWERS.
1. A MACHINE is any instrument employed to regulate
motion, so as to save either time or force. No instrument
can be employed by man so as to save both time and force ;
for it is a maxim in mechanics, that whatever we gain in
the on" of these two, must be at the expense of the other.
2. The simple machines, or those of which all others are
constructed, are usually reckoned six : the lever, the wheel
and axle, the pulley, the inclined plane, the wedge, and
the screw. To these the funicular machine is sometimes
added.
3. The weight signifies the body to be moved, or th
resistance to be overcome ; and the power is the force em
ployed to overcome that resistance, or move that body
They are frequently represented by the first letters of theii
.names, W and P.
THE LEVER.
4. A LEVER is an inflexible bar, either straight or bent,
supposed capable of turning round a fixed point, called the
fulcrum
1]
122 MECHANICS.
According to tho relative positions of tho weight, power
and fulcrum, on the lever, it is said to be of three kinds,
viz. when the fulcrum is somewhere betwixt the weight and
power, it is of the first kind ; when the weight is between
the power and the fulcrum, it is of the second kind ; and
when the power is between the weight and the fulcrum, it
is of the third kind : thus,
5. 1st. -
6. 3d.
7. 3d.
8. In the first and second kinds there is an advantage of
power, but a proportionate loss of velocity ; and in the third
kind, there is an advantage in velocity, but a loss of power.
9. When the weight X its distance from the fulcrum =
the power x its distance from the fulcrum, then the lever
will be at rest, or in equilibrio ; but if one of these pro-
ducts be greater than the other, the lever will turn round
the fulcrum in the direction of that side whose product is
the greater.
10. In all the three kinds of levers, any of these quanti-
ties, the weight or its distance from the fulcrum, or, the
power or its distance from the fulcrum, may be found from
the rest, such, that when applied to the lever, it will remain
at rest, or the weight and power will balance each other.
weight X its dist. from fulc.
11. -re 2 7 TT-J = power.
dist. of power from fulc.
power x its dist. from fulc.
dist. of weight from fulc.
, weight x dist. weight from fulc.
13. =dist. power from ful.
power
powerxdist.power from fulc. ..
14. =dist. weight from fulc.
weight.
15. In the first kind of lever, the pressure upon the ful-
crum = the sum of weight and power; in the second and
third = their difference.
16. If there be several weights on both sides of the ful-
crum, they may be reckoned powers on the one side of the
fulcrum, and weights on the other. Then, if the sum of
the product of all the weights x their distances from tho
THE LEVER. 123
fulcrum he = to the sum of the products of all the power*
X their distances from the fulcrum, the lever will be at rest,
if not, it will turn round the fulcrum in the direction of that
side whose products are gre;>:
17. In these calculations, the weight of the lever is not
taken into account ; but if it is, it is just reckoned like any
other weijrht or power acting at the centre of gravity.
18. When two, three, or more levers act upon each other
in succession, then the entire mechanical advantage which
they give, is found by taking the product of their separate
advantages.
19. It is to be observed, in general, before applying these
observations to practice, that if the lever be bent, the dis-
tances from the fulcrum must be taken, as perpendiculars
drawn from the lines of direction of the weight and power
to the fulcrum.
Ex. In a lever of the first kind, the weight is 16, its
distance from the fulcrum 12, and the power is 8 ; there-
fore, by No. 13 of this chapter,
= 24, the distance of power from the fulcrum.
8
In a lever of the second kind, a power of 3 acts at a distance
of 12 ; what weight can be balanced at a distance of 4
from the fulcrum ? Here, by No. 12,
3 x 12 o w '
= 9, weight.
In a lever of the third kind, the weight is 60, and its dis-
tance 12, and the power acts at a distance of 9 from the
fulcrum.; therefore, by No. 11,
60 X 12 .
= 80, the power required.
if
If there be a lever of the first kind, having three weights,
7, 8, and 9, at the respective distances of 6, 15, and 29,
from the fulcrum on one side, and a power of 17 at the dis-
tance of 9 on the other side of the fulcrum ; then a power is
,to be applied at the distance of 12 from the fulcrum, on the
last mentioned side : what must that power be to keep the
l^ver in balance ?
Here (6 x 7) + (15 x 8) + (29 x 9) = 423 = the
effect of the three weights on the one side of the ful-
crum ; and 17 X 9 = 153 = the effect of the power on the
other side. Now, it is clear that the effect of the weight is
124
MECHANICS.
far grei ter than the effect of the power ; and the difference
423 1 53 = 270 requires to be balanced by a power ap-
plied at the distance of 12, which will evidently be found
by dividing 270 by 12, which gives 22'5, the weight re-
quired.
20. The Roman steel-yard is a lever of the first kind, so
contrived that only one movable weight is employed.
The common weighing balance is also a lever of the first
kind. The requisites of a good balance are : that the points
of suspension of the scales and the centre of motion, or ful-
crum of the beam, be all in one straight line that the arms
of the beam be equal to each other in every respect that
they be as long as possible that the centre of gravity of
the beam be a very little below the centre of motion that
the beam be balanced when the scales are empty, &c.
But we may ascertain the true weight of any body even by
a false balance, thus : weigh the body first in one scale,
then in the other, and multiply their weights together ;
then the square root of this product will be the true weight.
THE WHEEL AND AXLE.
21. THE wheel and axle is a kind of lever, so contrived
as to have a continued motion about its fulcrum, or centre
of motion, where the power acts at the circumference of the
wheel, whose radius may be reckoned one arm of the lever,
the length of the other arm being the radius of the axle,
on which the weight acts. If the power acts at the end
of a handspike fixed in the rim of the wheel, then this in-
creases the leverage of the power, by the length of the
handspike.
The wheel and axle consists of a wheel
having a cylindric axis passing through its
centre. The power is applied to the cir-
cumference of the wheel, and the weight
to the circumference of the axle.
In the wheel and axle, an equilibrium
takes place when the power multiplied by
the radius of the wheel, is equal to the
weight multiplied by the radius of the
axle ; or, P : W : : CA. : CB.
For the wheel and axle being nothing else but a lever
so contrived as to have a continued motion about its ful-
THE WHEEL AND AXLE. 125
cnim C, the arms of which may be represented by AC and
BC, therefore, by the property of the lever, P : W : : CA
:CB.
If the power does not act at right angles to CB, but
obliquely, draw CD perpendicular to the direction of the
power, then, by the property of the lever, P : W : : CA :
CD.
22. It will be easily seen, that if two wheels fastened
together and turning round the same centre, be so adjusted,
that while they turn round they will coil on their cir-
cumferences strings, to which weights are suspended ;
one of those wheels being larger than the other, the larger
wheel will coil up a greater length of the string than the
smaller one will do in the same time, and this will depend
either on the radii or circumferences of the two wheels.
The velocity of the weight will be in proportion to the
length of string coiled in a given time ; therefore, the ve-
locity of the weight will be greater as the wheel is larger.
Now, as in the lever we saw that a small weight required a
great velocity to balance a large weight with a small velo-
city, we may infer, that the rules given for levers will also
apply to the wheel and axle ; since the velocity of any body
on a lever depends upon its distance from the fulcrum.
Ex. A weight of 13 Ibs. is to be raised at a velocity
of 14 feet per second, by a power whose velocity is 20 feet
per second ; how great must that power be ?
13x14
-- = 9-1, the power required.
If the velocity of the weight, be to that of the power, as
14 to 20, and the radius of the axle on which the weight is
coiled be 7 ; then,
20 x 7
= 10, radius of wheel on which the power acts.
If a weight of 36 Ibs, is to be raised by an axle 3 inches
diameter ; what must be the power applied at the end of a
handspike 4 inches long, fixed in the rim of the wheel con-
nected with the axle, the wheel being 6 inches diameter ?
/Here the handspike will increase the distance of the
power from the fulcrum, and will add to the diameter of th
wheel twice its own length ; therefore, 8 -f- 6 = 14 ;
hence, ] 4 : 3 : : 36 : 7'77, the power required to keep the
weight in equilibrio.
11*
126 MECHANICS.
23. Wheels acting on each other by teeth or bands, may
be easily calculated in the same way. Thus, if a wheel
which has 30 teeth, drives another of 10 teeth, it is evident,
that as the larger wheel has three times as many teeth as
the smaller, the smaller wheel will be turned round three
times for once that the larger one is turned round ; so that
the velocities of the wheels will be inversely as their num-
ber of teeth. In like manner, it is clear, that if the larger
wheel drives the smaller not by teeth but by a band, their
revolutions will be inversely as their circumferences.
Ex. The number of teeth in one wheel are 160, and
in another driven by it are 20, and the larger wheel makes
12 revolutions in a minute ; how many does the smaller one
make ?
20 : 160 : : 12 : 96 = the number of turns which the
smaller wheel makes in a minute.
24. The larger wheel is usually called the wheel, driver,
or leader, and the smaller one is called the pinion, driven
wheel, or follower.
25. Let us now see what would be the action of two
wheels and a pinion. If the first wheel contains 80 teeth,
the pinion 12 teeth, and second wheel 36 teeth. Place the
first wheel and the pinion on the same axis, so that they
move together, one revolution of the one being in the same
time as a revolution of the other, and the pinion drives
the second wheel. If the first wheel makes 16 revolutions
in a minute, the pinion will do the same, and the pinion
drives the second wheel ; therefore, 36 : 12 : : 16 : 5 3
= the velocity of the second wheel. Place these so, that
the teeth of the first wheel act in the teeth of the pinion,
and these again act in the teeth of the second wheel. If
the first wheel make, as before, 16 turns in a minute, then
the pinion will make 12 : 80 : : 16 : 106 T 8 ^ = in a minute ;
consequently, the revolutions of the second wheel will be
36 : 12 : : 106 T R ? : 35*55 = turns of the second wheel in
a minute.
26. When there are a number of
wheels A, B, C, D, E, acting on
the respective pinions o, b, c, r/, e,
as then the effect of the whole may
be found thus : if the letters which
epresent the wheels and pinions be
understood to signify the number of teeth of each,
THE WHEEL A>'P AXLE. 127
power xAxBxCxDxE
= weight.
x6xcx</xe
If the velocity oi' the first wheel be used instead of the
power applied, then this rule will give the resulting velo-
city instead of the weight.
Ex. If the numbers of the teeth of the wheels are 9, 6
9, 10, 12, and those of the pinions 6, 6, 6, 6 ; then if the-
power applied be 14 Ibs., we have
14 x 9 X 6 X 9 X 10 X 12
= 105 Ibs., the weight.
6x0x6x6x6
And, by the remark under the rule, if the first make 14
revolutions in the minute, the speed of the last will be 105
in the same time.
The same rule will apply to the case where the wheels
act on each other by ropes or straps, if the circumferences
of the wheels and pinions are used for the number of teeth.
27. It often happens, in the construction of machinery,
that two shafts must be connected by means of toothed
wheels, in such a way, that the one shaft's velocity shall
bear a certain proportion to that of the other shaft ; and we
must determine the numbers of leeth in each of the con-
necting wheels and pinions.
Take the respective numbers of teeth in the pinions at
pleasure, and multiply all these together, and their product
again by the number of turns that the one shaft is to make
for one turn of the other shaft. Take, HOAV, this product,
and find all the numbers which will divide it without a re-
mainder, or divide its divisors without a remainder always
excepting the number 1. Arrange all these in one line, and
separate them into parcels or bands, each containing as many
numbers, or factors (as they are called) as you please ; but
observing, that there must be as many bands as there are
wheels required ; then the product of the numbers in each
band will five the number of teeth in the respective wheels.
Thus, if one shaft is to turn 720 times for another shaft's
once, and there be interposed 4 pinions, one of which is
fixed to the end of the one shaft, each pinion having- 6 teeth
or leaves : then, 6x6x6xGx 720 ; all the divisors or
f/tors of which arc 3. 2. 3, 2, 3. 2, 3, 2, 2, 2, 3, 5. 2, 2, 3
divided into 4 bands at pleasure, give the number of
teeth in the wheels. Thus,
128 MECHANICS.
f2x3x5 =30, f3x3x5 = 45,
p-tv, J 2x2x2x3 = 24, n j 3X2X2X2X2 = 48,
Elther ^2x2x3x3 = 36, M 3x3x2 =18,
1^2x2x3x3 = 36, ^3x2x2x2 =24.
The application of what we laid down may he thus illus-
trated. In finding the number of teeth in the wheels of an
orrery, we extract from M arm's Mechanical Philosophy.
" There is considerable difficulty in proportioning the num-
ber of teeth in wheels for clocks, orreries, &c. the problem
indeed is indeterminate ; we shall, however, give an ex-
ample, that will point out a method by which any ingenious
mechanic may complete a piece of machinery, such as an
orrery, so as to show, at all times, in what part of its orbit
any planet is. The following example is for Mercury;
this planet goes round the sun in 87d. 23h.; now, as the
hour hand of a clock goes round twice in 24 hours,
it will make 175 }-j revolutions in 87d. 23h. For the
fraction ||-, take any multiple of the denominator plus
or minus unity, and make it the third term of the propor-
472
tion ; thus say, as 12 : 11 : : 515 : 472 nearly ; for -
o 1 o
is one unit less in each than a multiple of }*- by 43 =
516
r 472 90597
hence the revolutions become 175 = . Now the
515 olo
only difficulty remaining, is to find proper factors or divi-
sors that will divide the numerator and denominator
without a remainder, in order to determine the number of
teeth and leaves in the wheels and pinions. For the
numerator, the best method I have found is to make trial
of the numbers 2 X 5 or 10, as often as we can, and if we
do not succeed, to try successively the prime numbers 3,
7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, &c. I find by
trial the numerator will break into the factors 101 X 39 X
23 =90597, I conclude then that these numbers 101, 39,
23, may be the number of teeth in three wheels. I can
easily break the denominator into the numbers 103 and 5;
but as 103 is too large for the teeth in a pinion, and being
a prime number, another number must be sought for that
will answer the purpose better. Again say, as 12 : 11 :
1 f*fy Q
1825 : 1673. the revolutions now become 175 ^ or
1S2)
THE WHEEL AND AXLE. 129
321048
. Hence I find by trial that the numerator (321048)
1825
can be broken into the factors 91 X 72 X 49 = 321048,
which may be three wheels having that number of teeth in
each. Again, the denominator of the fraction, or 1825, is
capable of being broken into the factors 73 X 5 X 5 -=
1825. Now the product of the number of teeth in all the
wheels, divided by the product of the number of teeth in
all the pinions, will give the revolutions. For example,
32104 -r- 1825 = 175 revolutions, llh. Om. Is. 58 thirds,
which does not exceed the 87d. 23h. (or 175] | revolutions)
by two seconds. The numbers last found for the wheels
and pinions, may be transformed by multiplication into
98X91X72 144X98X91
more convenient numbers, as - =
73X10X5 73X10X10
= 175r. llh. Om. Is. 58th. either of which will be a train
of wheel-work proper for such a motion, and this train may
be conveniently attached to the pinion of the hour-wheel
of a clock. The reason for finding a new fraction, will
appear evident ; for if we take the original number
175|i- = 2 }| 1 , we shall find it impossible to break the
numerator into factors without leaving a fraction, which is
inconsistent with wheel-work, as nothing but whole num-
bers will answer the purpose. It is obvious that the higher
we take a multiple of ji the nearer we approach to the true
time of revolution, provided we can break the numerator
and denominator into proper numbers for the teeth and
leaves of the wheels and pinions. It is necessary to ob-
serve, that there must be either three wheels and three
pinions, or, if the numbers when broken be too large, if we
can break them into five wheels and five pinions, it will be
the same thing ; because as the hands of a clock go round
with the sun, that motion would make two wheels and two
pinions (attached Co the pinion on the hour wheel) go round
the contrary way to what they ought; but three or five will
answer the intended purpose."
28. As the subject of wheel-work is of the greatest im-
portance to mechanics, we shall resume il in a more
advanced part of this work, where it maybe more properly
introduced.
130
MECHANICS.
THE PULLEY.
29. IF a rope or string pass round the groove or rim of a
wheel, movable round an axle, with a power ?i the one end
of the string or rope, and a weight at the other, such a
machine is called a Pulley. The axis of the pulley may be
either fixed or movable. If the axis of the pulley be
fixed, it only serves to change the direction of the power's
action ; but if it be movable, the power acts with an ad-
vantage of two to one.
The accompanying engraving exhibits various forms of
the pulley. AB is a beam from which they are suspended.
No. 1, is the fixed pulley in which there is no other ad-
vantage gained than that the power. P and weight W move
in a contrary direction. No. 2, is a movable pulley, in
which the power P by moving upwards raises the pulley,
to the block of which the weight W is attached ; but the
one end of the string being attached to the beam AB, the
power must move twice as fast as the weight, and there
will be a gain of power proportional. No. 3, is a combi-
nation of two movable pulleys, in which the gain of power
will be four ; and No. 4 is a combination of two fixed and
two movable pulleys, in which the gain of power will be
the same as in No. 3.
30. If in a system of pulleys, where each pulley is em-
braced by a cord, attached at one end to a fixed point, and
at the other to the centre of the movable pulley next above
it, and the weight is hung to the lowest pulley ; then the
effect of the whole will be = the number 2 multiplied by
itself, as .many times as there are movable pulleys in the
system : thus, if there be 4 movable pulleys, then 2x2
THE PULLEY. 131
< 2 X2 =10: wherefore, if the weight be one lb., it will
oe sustfined by a power of one oz. avoirdupois.
31. When there are any number of movable pulleys on
one block, and as many on a fixed block, the pulleys are
called Sheeves, and the system is called a Muffle ; and the
weight is to the power inversely as one is to twice the
number of movable pulleys in the system, or
the weight to be raised
twice the number of mov. pulleys ~
Ex. In a muffle where each block has 4 sheeves, one
block being h'xed and the other movable, a weight of 112
Ibs. is to be raised ; how great must be the power ?
112
-p = 14 Ibs., the power required.
If a power of 236 Ibs. is to be applied to a tackle con
nected with two blocks of pulleys, otic fixed, consisting of
6, and another movable, of 5 pulleys ; what weight can be
raised ? (Here the rule above must be reversed.)
Therefore 236 X 10 = 2360 Ihs., the weight.
REMARK. In all the above cases of the pulley, the strings,
cords, or ropes, are supposed to act parallel to each other ;
when this is not the case, the relation of power and weight
may be found by applying the principle of the parallelo-
gram of forces ; thus, draw ab in tbe direction of the
power's action and of that length, taken from
a scare of equal parts, which expresses the
quantity of that power ; next, draw bd a per-
pendicular to the horizon, and from draw
ad parallel to be, the direction of the string,
which is fastened at c: then the power is to
the weight, as ba is to bd; and the strain on the hook at c,
is as ad to db, these lines being all measured on the same
scale of equal parts.
It may be further observed, that the pulley is a species
of lever of the second kind ; where the point at which the
string is fastened may be called the fulcrum ; the axis of
the pulley the place of the weight, and the place of the
power the other end of the lever ; or, the diameter of the
puiley may be reckoned the length of the lever f the weight
being in the middle.
132 MECHANICS
THE INCLINED PLANE.
32. WHEN a power acts on a body,
on an inclined plane, so as to keep that C
body at rest ; then the weight, the
power, and the pressure on the plane,
will be as the length, the height, and
the base of the plane, when the power acts parallel to tht
plane ; that is,
The weight f "] AC,
The power -< will be as >BC,
The pressure on the plane (_ J AB.
These properties give rise to the following rules :
weight X height of plane
power = ,
length 01 plane
power x length of plane
weight = : , jr-i
height of plane
weight X base of plane
pressure on the plane =
length of plane
33. The force with which a body endeavours to descend
down an inclined plane, is as the height of the plane.
When the power does not act parallel
to the plane, then from the angle C of
the plane, draw a line perpendicular
to the direction of the power's action ;
then the weight, the power, and the B *
pressure on the plane, will be as AC, CB, AB.
When the line of direction of the power is parallel to the
plane, the power is least.
34. If two bodies, on two Inclined planes, sustain each
other, by means of a string over a pulley, their weights
will be inversely as* the lengths of the planes.
35. In the exercises on inclined planes, it is often neces-
sary to find the length of the base, and height, or length of
the plane. Any two of these being given, the third may
be found and this is done on the principle stated in Geo-
metr) r , that the hypotenuse 3 of a right-angled triangle (the
length of the plane) is equal to the base a + height 3 .
Ex. The height of an inclined plane is 20 feet, and its
length TOO ; what is the pressure on the plane of a weight
of 1000 Ibs.? Here we must first ascertain the base,
(100 a 20 2 ) k = 97-98 = the base of the plane ; and from
THE INCLINED PLANE. 133
what has been said above, 100 : 1000 :: 97'98 : 979-8 the
pressure upon the plane; also 100 : 20 : : 1000 200, the
power necessary to keep the body from rolling down the
plane.
If a wagon of 3 cwt. on an inclined railway of 10 feet to
the 100, be sustained by another on an opposite railway of
10 feet to 90 of an incline ; what is the weight of the second
wagon ? Here 100 : 90 : : 3 : 2-7 cwt. = the weight oi
the second wagon.
36. The space which a body describes upon an inclined
plane, when descending on the plane by the force of
gravity, is to the space which it would fall freely in the
same time, as the height is to the length of the plane ; and
the spaces being the same, the times will be inversely in
this proportion.
Ex. If a body roll down an inclined plane 320 feet long,
and 26 feet in height; what space will it pass down the
plane in one second, by the force of gravity alone ?
320 : 26 : : 16 : 1-3 foot = the answer.
This subject, as connected with railways, will be resumed
when we come to treat of friction and railways.
THE WEDGE.
37. THE wedge is a triangular prism, formed either of
wood or metal, whose great use is to split or raise timber,
stones, &c.
The circumstances in which it is applied are such that
it is not easy to devise a general rule to determine the
amount of its action. The wedge has a great advantage
over all the other mechanical powers, in consequence of the
way in which the power is applied to it, namely, by per
cussion, or a stroke, so that by the blow of a hammer,
almost any constant pressure may be overcome.
THE SCREW.
38. THE screw is a kind of continued inclined plane,
being an inclined plane rolled about a cylinder the
height of the plane being the distance between the centres
of two threads, and its length the circumference ; hence,
12
134 MECHANICAL CENTRES
the rule to find the power of a screw pressing either up-
wards or downwards, is as the distance between two threads
of the screw is to the circumference where the power i?
applied : thus., if the distance of the centres of two threads
of the screw be | of an inch, and the radius of the hand-
spike attached to the screw be 24 inches ; the circumference
j>( the screw will be 150J- inches, nearly: therefore,
| : 150| : : 1 : 603} ; and if the power applied be 150
Ibs., the force of the screw will therefore be 603 X 150
== 90480 Ibs.
39. REMARKS ON THE MECHANICAL POWERS. The me-
chanical powers may be variously modified and applied,
but still they form the elements of all machinery. In our
calculations of their effects, we have not made allowance
for friction, or the resistance arising from one body rubbing
against another a subject which will be discussed hereafter.
The justice of the remark made before, will now be seen
to hold generally, that of the two velocity and power >
whatever we gain in the one, we lose in the other ; or, as
power and weight are opposed to each other, there will
always be an equilibrium between them, when the power
X its velocity = the weight x its velocity, that is, when
the momentum of the one is equal to the momentum of the
other.
All the advantage that we can obtain from the mechanical
powers, or their combinations, is to raise great weights, or
overcome great resistances, and this must be done at the
expense of time; or, to generate rapid velocities, as in
turning-lathes, or cotton-spinning machinery, and this is
done at the expense of power.
MECHANICAL CENTRES.
1. THESE are the centres of gravity, oscillation, percus-
sion, and gyration.
THE CENTRE OF GRAVITY.
2. THERE is a certain point in every body, or system of
bodies connected together ; .which point, if suspended, the
THE CENTRE OF GRAVITY. 135
body or system of bodies will remain at rest when acted
upon by the force of gravity alone ; this point is called
the Centre of Gravity. If a body or system of bodies be
suspended by any other point than the centre of gravity,
such body or system of bodies will move round that point,
until the centre of gravity be in a vertical line with the
point of suspension. If a body be sustained from falling
by two forces, the lines of direction in which these two
forces act, will meet in the centre of gravity of the body,
or, in the vertical line which passes through it.
3. It is often useful in calculation to consider the whole
weight of a body as placed in its centre of gravity, but it
is to be remembered, that gravity and weight do not
signify the same thing gravity is the force by means of
which bodies, if left to themselves, fall to the earth in
directions perpendicular to the earth's surface ; weight, on
the other hand, is the resistance or force which must be
exerted, to prevent a given body from obeying the law of
gravity.
4. To find the centre of gravity of any plane figure, me-
chanically : Suspend the figure by any point near its edge,
and mark the direction of a plumb-line hung from that
point, then suspend it from some other point, and mark
the direction of the plumb-line in like manner. The
centre of gravity of the figure will be in that point where
the marks of the plumb-line cross each other. For instance,
if we wish to find the centre of gravity of the arch of a
bridge, we draw the plan upon paper to a certain scale,
cut out the figure, and proceed with it as above directed ;
and by means of the plumb-line from the points of sus-
pension, its centre of gravity will be found ; whence, by
measuring the relative position of this centre in the plan
by the scale, we may determine by comparison its position
in the structure itself.
5. We can find the centre of gravity of many figures by
calculation.
6. The centre of gravity of a line, parallelogram, prism,
cylinder, circle, circumference of a circle, sphere, and
regular polygon, is the geometrical centre of these figures
respectively.
ff. To find the centre of gravity of a triangle draw a line
from any angle to the middle of the opposite side, then f
13t MECHANICAL CENTRES.
of this line from the angle will be the position of the centre
of gravity.
8. For a trapezium, draw the two diagonals, and find
the centres of gravity of each of the four triangles thus
formed, then join each opposite pair of these centres of
gravity, and the two joining lines will cut each other in
the centre of gravity of the figure.
9. For the cone and pyramid, the centre of gravity is in
the axis, at the distance of ? of the axis from the vertex
10. For the arc of a circle,
radius of circle X chord of arc - ,
. = distance of the
length ot arc
centre of gravity from the centre of the circle.
11. For the sector of a circle,
2 X chord of arc x radius of circle
: ? = distance of
3 x length of arc
the centre of gravity from the centre of the circle.
12. For a parabolic space, the distance of the centre of
gravity from the vertex is of the axis.
13. For a paraboloid, the centre of gravity is f of the
axis from the vertex.
14. For two bodies, if at each end of a bar a weight be
hung, the common centre of gravity will be in that point
which divides the bar, in the same ratio that the weights
of the bodies bear to each other, and this point will be
nearest the heavier body.
Examples. If the line drawn from the middle of the base
of a triangle to the opposite angle be 15, then we have
15
X 2 = 10 = the distance of the centre of gravity from
o
the vertical angle.
If the height of a cone be 24 inches, then we have
24
X 3 = 18 = the distance of the centre of gravity from
the vertex.
If the length of the arc of a circle be 157-07, and the
chord 153-07, and radius 200 ; then,
200 X 153-07
1 ^7-07 == "***" = distance of the centre of
gravity from the centre of the circle.
If there be the sector of a circle of which the chord
OSCJU.ATiON AND I'M ISC 1VSJON. 137
radius, and length of :irc, arc the same as in the last ex
anipli-, we have
2 X 153-07 X 200 lonQ
3 X l-)?-07 ~ = = distance of the
centre of gravity from the centre of the circle.
In a parabolic space, if the axis be 25 inches long, then
25
X 3 = 15 = the distance of the centre of gravity from
o
the centre.
30
In a parabdoid, if the axis be 30, then we have X 2
3
= 20 = the distance of the centre of gravity from the vertex.
A bar of wood, 24 feet long, has a weight suspended at
each end, that at one end being 16 Ibs., and the other 4 :
then, we have 20 : 24 : : 16 : 19-2
and 20 : 24 : : 4 : 4-8
the distances of the weights from the common centre of
gravity, the greater weight being least distant. Hence we
see, that 19-2 -f 4-8 = 24, the whole length of the bar;
and also 4x19-2 = 16x4-8 = 76-8 ; so that the prin-
ciple of virtual velocities, stated before, holds good here also ;
and here it may be observed, that it is of the greatest im-
portance to trace any leading principle of this kind through
its various applications, as it serves to link together and
harmonize the whole, and enables us to apply and remember
it with greater facility.
It is often necessary to determine the centre of gravity
experimentally, as in many cases it cannot be conveniently
done by calculation. To maintain the firmness of any
body resting on a base, it is necessary that the perpendicu-
lar drawn from the centre of gravity of the body, to the
base on which it rests, be within that base ; and the body
will be the more difficult to overset, the nearer that per-
pendicular is to the centre of the base, and the more ex-
tensive the base is, compared to the height of the centre of
gravity.
T* HE CENTRE OF OSCILLATION. THE PENDULUM, AND
CENTRE OF PERCUSSION.
1. THE centre of oscillation in a vibrating body, is that
point in the axis of vibration, in which, if the whole matter
12*
138 MECHANICAL CENTRES.
contained in the body were collected, and acted upon by
the same force, it would, if attached to the same axis of
motion, perform its vibrations in the same time. The
centre of oscillation is always situated in the straight line
which passes through the centre of gravity, and is perpen-
dicular to the axis of motion. It will be seen by these
remarks, that the subject of pendulums must be considered
here.
2. In theory, a simple pendulum is a single weight, con-
sidered as a point, hanging at the lower extremity of an
inflexible right line, having no weight, and suspended from
a fixed point or centre, about which it vibrates, or oscil-
lates ; a compound pendulum, on the other hand, consists
of several weights, so connected with the centre of suspen-
sion, or motion, as to retain always the same distance from
it, and from each other.
3. If the pendulum be inverted, so that the centre of
oscillation shall become the centre of suspension, then the
former centre of suspension will become the centre of os-
cillation, and the pendulum will vibrate in the same time :
this is called the reciprocity of the pendulum ; and it is a
fact of the greatest utility, in experimenting on the lengths
of pendulums.
4. Of the simple pendulum we may observe, that its
length, when vibrating seconds, must in the first place be
determined by experiment, as it vibrates by the action of
gravity, which force differs at different distances from the
pole of the earth. By the latest experiments, the length of
the seconds' pendulum in the latitude of London, has been
found to be 39-1393 inches, or 3-2616 feet; the length at
the equator is nearly 39-027, and at the pole 39-197 inches.
The length for the latitude of London may be taken for all
places in Britain, without any material error.
5. The times of vibration of two pendulums, are directly
proportional to the square roots of the lengths of these pen-
dulums.
6. Thus : what will be the time of one vibration of a
pendulum of 12 inches long at London?
x/39-1393 : x/ 12 :: I : 0-5537 = time of one vibration.
If the pendulum be 36 inches long,
v/39-1393 : v/ 36 :: 1 : 0-9599 = time of one vibration.
7. The lengths of the pendulums are to each other in-
OSCILLATION, AND PERCUSSION. 139
trersely as the squares of the numbers of vibrations made in
a given time.
What is the length of a pendulum vibrating half-seconds
or making 30 vibrations in a minute ?
(60) a : (30) a :< 39-1393 : 9-7848 = length in inches.
The length of a pendulum to make any given number of
vibrations in a minute, may be easily found by the following
short rule :
140850
i- -i " -- = length.
number of vibrations 3
Thus a pendulum to make 50 vibrations in a minute, will
be
140850 140850 _
SO* 2500
8. All the rules for simple pendulums may be expressed
as follows :
The time of one vibration in seconds of any pendulum is
_ 1 _
number of vibrations in one second
I
or
|/ the length of the pendulum\
\> 39-1393
Exam. If the number of vibrations of a pendulum be
6256, then
- = 1-598 = the time of one vibration.
"o25o
Or, if the length of the pendulum be 100 inches, then
The length of a pendulum in inches is
= 39-1393 x time of one vibration 8 ;
39-1393
or - --
number of vibrations'
Exam. If the time of one vibration be 1'598 ; find the
length. 39-1393 X l'598 a = 100, length of pend.
Or. if the number of vibrations in a second be as above,
6856, then we have
39-1393
. 6256 a = 100, length of pendulum.
14C MECHANICAL CENTRES.
The number of vibrations in a second may be found thus :
| 39-1393
. . r f j- : = number ol vibrations ;
\length of pendulum
or, the number of vibrations in a second is
time of one vibration
Tf the time of one vibration be, as above, 1*598 ; then
- = '6256, number of vibrations ;
i *oy o '
or, if the length of 100, we have"
~) =
When a clock goes too fast or too slow, so that it shalj
lose or gain in twenty-four hours, it is desirable to regulate
the length of the pendulum so that it shall go right. The
pendulum bob is made capable of being moved up or down
on the rod by means of the screw. If the clock goes too
fast, the bob must be lowered, and if too slow, it must be
raised ; and we have this rule : number of threads in an
inch of the screw X the time in minutes that the clock
loses or gains in 24 hours ; this product divided by 37 will
give the number of threads that the bob must be screwed
up or down, so that the clock shall go right.
Ex. If the rod have a screw 70 threads in the inch, and
the pendulum is too long, so that the clock is 12 minutes
slow in 24 hours ; then we have
2 x 70 x 12
= 45 Jf = threads we must raise the bob,
o7
so that the clock shall go right.
9. It is often desirable that a pendulum should vibrato-
seconds, and yet be much shorter than 39-1393 inches;
Avhich may be done by placing one bob on the rod above
the centre of suspension, and another below it : then, having
the distances of the weights from the centre of suspension,
we may find the ratio which the weights should bear to
each other by this rule. Call D the distance of the lower,
and d the distance of the upper weight, from the centre of
suspension ; then,
39-1393 x D D s
OSCILLATION ANU PERCUSSION. 141
a number which, when multiplied by the lower weight,
will give the hi-licr. 1) anil d are. taken in inches.
Ex. In a pendulum having two bobs, the one 12 inches
oelow the centre of suspension, and the other 9-6 inches
above the same centre, the lower weight being 8 ounces ;
what is the upper weight?
39-1393 X 12 12 3 = .
39-1393 x 9-6 + 9-tt*
then, 0-696 X 8 = 5-568 ounces = the weight of the
upper bob.
10. If a common walking-stick be held in ^ie hand, and
struck against a stone, at different points of its length, it
will be found that the hand receives a shock when it is
struck at any part of the stick, but at one particular point,
at which, if the stick be struck, the hand will receive no
shock. This point is called the centre of Percussion, and is
usually defined thus : The centre of percussion is that point
in a body revolving about an axis, at which, if it struck an
immovable obstacle, all the motion of the body would be
destroyed, so that it would incline neither way after the
stroke.
11. The distance of the centre of percussion from the
axis of motion, is the same as the distance of the centre of
oscillation from the centre of suspension ; and the same
rules serve for both centres. See Oscillation.
12. The distance of either of these centres from the axis
of motion, is found thus :
13. If the axis of motion be in the vertex of the figure,
and the motion be flatwise ; then,
14. In a right line, it is = * of its length ;
In an isosceles triangle = | of its height ;
In a circle = f of its radius ;
In a parabola = i of its height.
15. But if the bodies move side wise, we have it
In a circle = | of the diameter ;
In a rectangle suspended by one angle = f of the
diagonal.
16. In a parabola suspended by its vertex,
= 4 axis + i parameter;*
hot if suspended by the middle of its base,
= ^ axis -f 5 parameter.
142 MECHANICAL CENTRES.
3 X arc X radius
17. In the sector of a circle =
18. In a cone = f axis -f
4 x chord
(radius of base)'
5 x axis
10 T u 2 x , T ,
19. In a sphere = r r + radius -}- </, where
5 (a -f radius)
e? is the length of the thread by which it is suspended.
20. We have given these rules for the sake of reference
but we shall illustrate by examples the most useful.
Examples. What must be the length of a rod without a
weight, so mat when hung by one end it shall vibrate
seconds ?
To vibrate seconds, the centre of oscillation must be
39-1393 inches from that of suspension; hence, as this
must be f of the rod, 2:3:: 39-1393 : 58-7089 inches =
the length of the rod.
What is the centre of percussion of a rod 46 inches long?
^ X 46 = 30| inches from the axis of motion.
In an isosceles triangle, suspended by one angle, and
oscillating flatwise, the height is 24 feet ; what is the dis-
tance of the centre of percussion from the axis of motion ?
| X 24 = 18 feet.
In a sphere the diameter is 14, and the string by which
the sphere is suspended is 20 inches ; therefore,
2 v 7 fl 98
7 + 20= - + 27 = 27-725 ;
5 (20 + 7) ' 135
so that the centre of oscillation or percussion is farther from
the axis of motion than the centre of the sphere, by 7'725
inches.
THE CENTRE OF GYRATION AND ROTATION.
21. IT will be seen, that the last two centres refer to
bodies in motion round a fixed axis, and belonging to the
same class : there is yet another centre to be considered,
of the utmost impof tance to the practical mechanic. We
saw, in determining the centre of oscillation, that we were
finding a point in which, if all the matter of the body were
collected, the motion would be the same as that of the body
which motion was caused by the action of gravity ; but
GYRATION AND ROTATION. 143
when the body is put in motion by some other force than
gravity, the point in question becomes the centre of Gyra-
tion. The centre of ny ration may therefore be defined,
that point in a body or system of bodies revolving round
an axis, in which point, if all the matter in the body or
system of bodies were collected, the same number of revo-
lutions in a given time would be generated by the applica-
tion of a given force, as would be generated by the same
force applied to the body or system of bodies itself.
22. The position of the centre of gyration is a me?wi pro-
portional between the centres of oscillation and gravity.
23. The centre of gyration of the following bodies may
be found by these rules :
24. For a straight line or cylinder, whose axis of motion
is in one end, = length x 0*5775.
25. For a cylinder or plane of a circle, revolving about
the axis, or the circumference about the diameter, = radius
X 0-7071.
26. For the plane of a circle about its diameter = 5
'radius.
27. For the surface of a sphere about its diameter =a
radius X -8165.
28. For a solid sphere or globe, about its diameter =
radius X -6324.
29. For the circumference of a circle upon a perpendicu-
lar axis passing through the centre = radius.
Ex. What is the distance of the centre of gyration
from the centre of motion, of a rod 58*7089 inches long?
Here 58-7089 X -5775 = 33-9044.
In a wheel of uniform thickness, revolving about its axis,
the diameter is 36 inches ; hence 18 X '7071 = 12 = dis-
tance of the centre of gyration from the axis.
In a solid globe revolving about its diameter, which is
2 feet, the distance of the centre of gyration is = 12 X
6324 = 7-5888 inches.
30. Effects are proportional to their causes ; the motion
'generated in any body is proportional to the force which
.produces that motion ; hence we see, that all constant forces
may be compared to the force of gravity. And it is often
u/eful to know the time in which a revolving body of a
certain weight, acted upon by a known constant force, will
acquire a given velocity. The principles we have laid
.44 MECHANICAL CENTRES.
in discussing the inclined plane, will here be found
serviceable.
As the weight of the body moved,
fs to the weight or force causing it to move,
So is the length of an inclined plane, such that the
given force would just support the body upon it,
To the height of the plane.
l^ow, if in a wheel 6 feet diameter, whose weight, 400 Ibs.,
if turned by a force of 56 Ibs., acting at the distance of 18
inches from its centre of motion, its centre of gyration being
b feet from the same centre ; what will be the time required
to give by this force a velocity of 20 feet per second at the
centre of gyration. Here, by the lever,
18 X 56
-gg =lttlbs.-
the force exerted at the centre of gyration. We now wish
to know the length of time in which a body would acquire
a velocity of 20 feet per second, on an inclined plane,
whose length is to its height as 400 is to 16^f ; wherefore,
by the laws of falling bodies, we have
16-8
the time required to fall perpendicularly ; therefore, by the
.nclined plane, we have, 20 : 400 : : -525 : 10-5 = the
riiie required.
31. All the circumstances comprehended under this kind
< 'fotatory motion, may be expressed by the following rules :
*<et W express the weight of a wheel,
F, the force acting upon the wheel,
D, the distance of the force from the axis of motion,
G, the distance of the centre of gyration from the
axis of motion,
t. the time the force acts,
v, the velocity acquired by the revolving body in that
time.
Gj<^W^X v _ G X W X v _ D
D~X~~rX 32 " ~ F X t X 32" ~
F X D X t X 32 _ F X D X t X 32 _
W X v G X v
G X W X v F X D X t X 32 _
F X D X 32 ^ * G X W
GYRATION AND ROTATION. 145
It is to be observed, before applying these rules, that the
number of turns of a revolving body in a minute are often
given, and it is required to find the velocity of feet per
second. A wheel of 8 feet diameter, for instance, makes 12
revolutions in a minute ; how many iVet does a nail in its
circumference pass over in a second? Here, 8 x 3-1416
= 25-1328 feet the nail passes through in one revolution,
but 25-1328 X 12 = 301-5936 = the feet it passes through
in a minute; hence, 60)301-5936(5-0265, the velocity in
ft. per second. The whole may be expressed shortly thus :
8 X 3-1416 X 12
= o-026a.
60
Ex. What must be the weight of a fly-wheel that makes
12 revolutions in a minute, whose diameter is 8 feet, urged
by a force of 84 Ibs. at its rim, acting for 6 seconds, the
distance of the centre of gyration being 3 feet 6 inches ?
84 X 4 x 6 x 32
*5 X 5-0265
In a wheel which is 2 tons weight, and 12 feet diameter,
the centre of gyration is 6 feet from the centre of rotation,
the velocity with which this wheel moves is 10 feet per
second ; what force must be applied for 8 seconds, at the
distance of 3 feet from the centre, to generate that velocity ?
What is the distance of the centre of gyration from the
centre of motion of a fly-wheel, the force which moves the
wheel being 2 cwt., acting at the distance of 7 feet from
the centre of motion, and for 10 seconds, the weight of the
wheel being 2 tons, and its velocity 8 feet per second ?
Here 2 tons = 50 cwt. .
2X7X10X 12 Ml . t ,. 4 ,
- = Hi feet, distance of centre of gyration.
oU X O
What is the velocity acquired by a fly-wheel acted upon
by a force of 84 Ibs., at the distance of 4 feet from the axis,
the time in which the force has been acting is 7 seconds, the
weight of the wheel 1A tons, and the distance of the centre
of gyration 5 feet from the centre of motion? Here lj
ton = 30 cwt. = 3360 Ibs. ; therefore,
84 X 4 X 7 X 32
K oon = **'"* ^ eet P er secon d u 16 velocity
5 X oooU
acquired by the wheel.
13
146 CENTRAL FORCES.
CENTRAL FORCES.
1. INTIMATELY connected with the foregoing subject is
that of central forces, the nature of which maybe illus-
trated by a very simple instance. When a boy causes a stone
in a sling to revolve round his hand, the stone is kept from
flying off by the strength of the string, which continually
draws the stone, as it were, to the hand or centre of motion ;
but if the string is let go, or breaks, then the stone will fly
off in a straight line, by means of its centrifugal force ;
the strength of the string, while it restrains this tendency,
is called the centripetal force : when both forces are spoken
of they are jointly called central forces.
2. When a body revolves round a fixed centre, the cen-
tripetal force may sometimes be the cohesion of the par-
ticles of which the body is composed, or sometimes it may
be the power of some attracting body such as gravity in
the case of the planets.
3. In talking of the angular velocity of a revolving body,
we mean not the space which is passed over in a given time,
out the number of degrees, minutes, &c., that the body de-
scribes in a certain time, whether the circle be large or
small. Thus, a body moving in a circle of 10 feet diameter,
may have an angular velocity of 15 in a second, so may
also another body moving in a circle of three feet diameter ;
they will complete their respective circles in the same time,
but the actual spaces they pass through are very different ;
that is, their angular velocities are the same, but their actual
velocities are not.
4. The central forces are as the radii of the circles
directly, and the squares of the times inversely, afso the
squares of the times are as the cubes of the distances.
When a body revolves in a circle by means of central forces,
its actual velocity is the same as it would acquire by falling
through half the radius by the constant action of the centri-
petal force. From these facts the following rules for cen-
tral forces are derived.
_ veloc. of rev. body 9 X weight of body
5. - .. -.-. r , ~ r-^ - = centrif. force.
radius of circle of revolution X 32
velocity of revol. body 8 x weight of body
6. J - r j |L 1 = radius of
centrifugal force x 32
the circle of revolution.
CENTRAL, FORCES. 147
centril. force x 32 X rad. circle
7. -- r- i~ " = weight of the re-
veloc. ot revolving body'
volving body.
l/rad. circle x 32 x centrifugal force\
8.. I -/* velocity.
\ v weight
9. There will be no difficulty in applying what has been
said to practice.
There are two fly-wheels of the same weight, one of
which is 10 feet diameter, and makes 6 revolutions in a
minute ; what must the diameter of the other be to revolve
3 times in a minute ? Here 6 a : 3 a : : 10 : 2-5 = the
diameter of the second.
What is the centrifugal force of the rim of a fly-wheel,
its diameter being 12 feet, and the weight of the rim 1 ton,
making 65 turns in a minute ?
8 X 3-1416 X 65 = 4Q . 84 =
the velocity in feet per second ; hence,
40-84 8 X 1
32X6
the tendency to burst.
Let us employ the centre of gyration. If the fly above
mentioned is in two halves, which are joined together by
bolts capable of supporting 4 tons in all their positions, the
whole weight of the wheel is 1 5 tons, the circle of gyration
is 5-5 feet from the axis of motion ; what must be its velocity
so that its two halves may fly asunder? The force tending
to separate the two halves will be 5 of the whole force ;
wherefore, by the rule,
X 4 * 5 5 ' 8 X 2 = 3-636 = the velocity,
11 X 3-1416 = 34-5576 = circumference of circle of gy-
ration, wherefore, 34-5576 : 30-636 : : 60 : 53-191 revolt!
tions in a minute.
10. The steam engine governor, or conical pendulum ;
action the principle of central forces. It is so constructed,
that when the balls diverge, or fly outwards, the ring on
the upright shaft is raised, and that in proportion to the in-
crease of the velocity, squared ; or, the square roots of the
148 CENTRAL FORCES.
d. stances of the ring from the top, corresponding to two
velocities, will be as these velocities.
Ex. If a governor makes 6 revolutions in a second,
when the ring is 16 inches from the top ; what will be the
distance of the ring when the speed is increased to 10 revo-
lutions in the same time ? The balls will diverge more,
consequently the ring will rise and the distance from tlie
top become less ; therefore, we have
10 : 6 :: v' 16 or 4 : 2-4;
which, squared, gives 5*76 inches, the second distance of
the ring from the top. See Steam Engine.
11. We shall elsewhere introduce other particulars on
rotation and central forces.
STRENGTH OF MATERIALS, MACHINES,
MODELS, &c.
MATERIALS are exposed to four different kinds of strain :
1st. They may be torn asunder, as in the case of ropes
ana stretchers. The strength of a body to resist this kind
of strain is called its Resistance to Tension, or Absolute
strength.
2d. They may be crushed or compressed in the direction
of their length, as in the case of columns, truss beams, &c.
3d. They may be broken across, as in the case of joists,
rafters, &c. The strength of a body to resist this kind of
strain is called its Lateral strength.
4th. They may be twisted or wrenched, as in the case
of axles, screws, &c.
Extensive and accurate experiments are necessary to
determine the several measures of these strengths in the
different materials; ami when this is done, the subsequent
calculations become comparatively easy. We shall, there-
fore, in the first place, lay down the results of the experi-
ments of practical men.
STRENGTH OF MATERIALS.
149
A.
TABLE OF THF FLEXIBILITY AND STRENGTH OF TIMBER.
Name of the Wood.
u
E
s
c
Teak,
818
9657802
2462
15555
Poon,
596
6759200
2221
14787
English oak,
598
3494730
1181
9836
Do.
435
5806200
16.72
10853
Canada oak,
588
8595864
1766
11428
Dantzic oak,
724
4765750
1457
7386
Adriatic oak,
610
3885700
1583
8808
Ash,
395
6580750
2026
17337
Beech,
615
5417266
1556
9912
Elm,
509
2799347
1013
5767
Pitch pine,
588
4900466
1632
10415
Red pine,
605
7359700
1341
10000
New English fir,
757
5967400
1102
9947
Riga fir,
588
5314570
1108
10707
Do.
3962800
1051
Mar forest fir,
588
2581400
1144
9539
Do.
403
3478328
12.62
10691
Larch,
411
2465433
653
Do.
518
3591133
832
Do.
518
4210830
1127
7655
Do.
518
4210830
1149
7352
Norway spar,
648
5832000
1474
12180
NOTE. The extensive use of the above table will be
shown hereafter.
U. The ultimate strength. E. Lateral strength. S
Transverse strength. C. Cohesion.
B.
T^ble showing the weight that will pull asunder a prism
one inch square.
Ibs. Ibs.
Cast gold, 22000 ' Bismuth, 29000
Cast silver, 41000 I Good brass, 51000
13*
150
STRENGTH OP MATERIALS.
Ibs.
Anglesea copper, 34000
Swedish copper, 37000
f^nst ifrvn .. . ^OOOO
It*.
Bar iron, ordinary," 68000
Do. Swedish, 84000
Tt-ir tr>f>l anft . ..190000
COMPOSITIONS
Gold 5, copper 1,
Silver 5, copper !,
Swed. copper 6, tin
Block tin 3, lead 1,
Tin 4, lead 1, zinc
OF
-50000
...48500
1,. 64000'
...10200
1,. 13000
...45000
Do. razor temper,150000
Cast tin, Eng. block, 5200
Dn n-rsjin .. fi'lOO
7.\nf. :... 2flOO
c.
The same from Rennie :
Weight that would tear
it asunder in Ibs.
Length in feet that would
break with its own weight
Cast steel, 134256
Swedish iron,-"." 72064
English iron, 55872
Cast iron, 19096
Cast copper, 19072
Yellow brass, 17958
Cast tin, 4736
Castlead,"-- 1824
39455
19740
16938
6110
5092
5180
1496
306
Good hemp rope,
Do. one inch diam.
6400 18790
5026 18790
D.
The cohesive force of a square inch of iron ; from dif-
ferent experimentists.
Ibs.
.
Ibs.
...fil fiOfl
Dn
Dn
...fifVTTP
.. . R4QfiO
Tin
... f^^ffR
, . ,(\i nni
T) n
.. . ^04.79
Fin
Dn.
. 5SOOO
..1fi9S5
STRENGTH OP MATERIALS. 151
E.
Table of the lateral strength of the following materials
one foot long, and one inch square.
Weight that will
break them.
Weight which they CM
tear with safety.
O-jt .
. 20Q -
1 1ft
A rri*>rir;in whitfimnp..
. 9OR _
C.'.t -
F.
The force necessary to crush one cubic inch.
Aberdeen granite, blue,
Very hard freestone,
Black Limerick limestone,
Compact limestone,
Craigleith stone,
Dundee sandstone,
Yorkshire paving stone,
Redbrick, 1817
Pale red brick, 1265
Chalk, 1127
Cubes of one-fourth of an inch.
Iron cast vertically, 11140
horizontally, 10110
Cast copper, 7318
Cast tin, 9.66
Cast lead, 483
Having made these statements, we shall proceed to show
how, by the assistance of theoretical results, they may be
applied to the wants of the practical engineer.
The absolute strength of ropes or bars, pulled length
wise, is in proportion to the squares of their diameters.
All cylindrical or prismatic rods are equally strong in every
part, if they are equally thick, but if not, they will break
where the thickness is least.
'The lateral strength of any beam or bar of wood, stone,
met^il. &c., is in proportion to its breadth X its depth 8 .
In square beams the lateral strengths are in proportion to
the cubes of the sides, and in general of like-sided beams
as the cubes of the similar sides of the section.
152 STRENGTH OP MATERIALS.
The lateral strength of any beam or bar, one end being
fixed in the wall and the other projecting, is inversely as
the distance of the weight from the section acted upon ;
and the strain upon any section is directly as the distance
of the weight from that section.
If a projecting beam be fixed in a wall at one end, and
a weight be hung at the other, then the strain at the end in
the wall, is the same as the strain upon a beam of twice
the length, supported at both ends and with twice the
weight acting on its middle. The strength of a projecting
beam is only half of what it would be, if supported at both
mds.
If a beam be supported at both ends, and a weight act
upon it, the strain is greatest when the weight is in the
middle ; and the strain, when the weight is not in the
middle, will be to the strain when it is in the middle, as
the product of the weight's distances from both ends, is to
the square of half the length of the beam. Take any two
points in a beam supported at both ends ; call one of these
points a and the other b ; then a weight hung at a will
produce a strain at 6, the same as it would do at a if hung
at b.
In a beam supported at the ends p c
A. and B ; the strain at C, with the
whole weight placed there, is to the strain at C with the
whole weight placed equally between C and P, as AC is
to AP x 5 PC ; and the strain at C by a weight placed
equally along AP, is to the strain at C by the same weight
placed on C, as AP is to AC.
If beams bear weights in proportion to their lengths,
either equally distributed over the beams or placed in similar
points, the strains upon the beams will be as their lengths 2 .
If a beam rest upon two supports, and at the same time
be firmly fixed in a wall at each end, it will bear twice as
much weight as if it had lain loosely upon the supports ;
and the strain will be everywhere equal between the
supports.
In any beam standing obliquely, or in a sloping direction,
As strength or strain will be equal to that of a beam of the
same breadth, thickness, and material, but only of the
length of the horizontal distance between the points of
support.
Similar plates of the same thickness, either supported at
STHENGTII OF MATERIALS. 153
the ends or all round, will carry the same weight either
uniformly distributed or laid on similar points, whatever be
their extent.
The strength of a hollow cylinder, is to that of a solid
cylinder of the same length and the same quantity of mat-
ter, as the greater diameter of the hollow cylinder is to the
diameter of the solid cylinder ; and the strength of hollow
cylinders of the same length, weight, and material, are as
their greater diameters.
The lateral strength of beams, posts, or pillars, are dimi-
nished the more they are compressed lengthwise.
The strength of a column to resist being crushed is
directly as the square of the diameter, provided it is not so
long as to have a chance of bending. This is true in metals
or stone, but in timber the proportion is rather greater than
the square.
The strength of homogeneous cylinders to resist being
twisted round their axes, is as the cubes of their diameters ;
and this holds true of hollow cylinders, if their quantities
of matter be the same.
PROBLEMS.
To find the strength of direct cohesion :
Area of transverse section in inches x measure of cohe-
sion = strength in Ibs. to resist being pulled asunder.
Ex. In a square bar of beech, 3 inches in the side, we
have 3 X 3 X 9912 = 89208 Ibs.
NOTE. The measure of cohesion for timber is taken
from col. C, table A, and for other materials, from tables B
or C.
In a beam of English oak, having four equal sides, each
side being four inches, we have
4 x 4 x 9836 = 157376 Ibs., the strength.
In a rod of cast steel, 2 inches broad and Ik inch thick,
we have 2 X U X 134256 = 402768 Ibs., the strength.
What is the greatest weight which an iron wire - v of an
inch thick will bear ?
The area of the cross section of such wire will be '007854,
hence we have -007854 x 84000 = 659-736 Ibs.
/To find the ultimate transverse strength of any beam :
When the beam is fixed at one end and loaded at the
other then the dimensions being in inches,
154 STRENGTH OF MATERIALS.
breadth x depth 2 x transverse strength
= the ultimate
length of beam
transverse strength.
NOTE. In column S, Table A, will be found the trans-
verse strength of timber, and in table E, that of iron, <fec. ,
and let it be observed, that when the beam is loaded uni-
formly, the result of the last rule must be doubled.
What weight will break a beam of Riga fir, fixed at one
end and loaded at the other, the breadth being 3, depth 4,
and length 60 inches ?
886 Ibs.
What weight uniformly distributed over a beam of
English oak would break it, the breadth being 6, depth 9,
and its length 12 feet?
x a = ime Ibs.
144
If the number be taken from table F, we must use the
length in feet.
When the beam is supported at both ends, and loaded in
the centre,
tabular value of S, tab. A X depth 3 X breadth X 4 _
length
the weight in pounds.
NOTE. When the beam is fixed at one end and loaded
in the middle, the result obtained by the rule must be in-
creased by its half. When the beam is loaded uniformly
throughout, the result must be doubled. When the beam
is fixed at both ends and loaded uniformly, the result must
be multiplied by three.
Ex. What weight will it require to break a beam of
English oak, supported at both ends and loaded in the
middle, the breadth being 6, and depth 8 inches, and length
12 feet?
1672 X 8 a X 6 X 4
144
By using table E :
depth 3 x breadth X tabular number
length in feet
PROHLEMS. 155
Ex. What weight will a cast iron bar bear, 10 feet
ong, 10 inches deep, and 2 inches thick, laid on its edge *
10 8 x 2 x 1090
10
The same on its broad side :
2- x 10 x 1090
= 21800 Ibs.
= 4360 Ibs.
10
To find the breadth to bear a given weight.
length x weight
rr- 5 - rs = breadth,
number m table L x depth 3
What must be the breadth of an oak beam, 20 feet long
and 14 inches deep, to sustain a weight of 10000 Ibs. ?
20 X 10000
- = 4-8o inches = the breadth.
14 2 x 209
To find the length :
depth 2 X breadth x tabular number
- = length,
weight
In a beam 1 ft. deep and 4 in. broad, the weight being
5000 Ibs. ; then we have, if the beam be made of Memel fir,
12 3 X 4 X 130
= 14 - 97 feet, length required.
5000
To find the depth :
|/ length X weight \ .
\ \tabular number x breadth/
We wish to support a weight of 2000 Ibs. by a beam of
American pine ; what is its depth, its length being 20 feet
and breadth 4 inches ?
2000X20\
-7 ) = >/ (145) = 12 inches, nearly.
t>y x 4
To find the deflection of a beam fixed at one end, and
loaded at the other :
length of beam in inches 3 X 32 x weight
f tab. numb. E (in table A) x breadth x depth 3
flection in inches.
J(OTE. If the beam be loaded uniformly, use 12 instead
of 32 in the rule.
If a weight of 300 be hung at the end of an ash bar fixed
156 STRENGTH OF MATERIALS.
m a wall at one end, and five feet long, it being 4 inchei
square : what is its deflection ?
60 3 X 32 X 300
= 1-23 inches = the deflection.
6580750 X 4 x 4 3
If the beam be supported at both ends and loaded in the
middle :
length (in inches) 3 X weight .
tab. numb*. (E, table A) X breadth x depth 3
NOTE. When the beam is firmly fixed at both ends, the
deflection will be f of that given by the rule.
Ex. If a beam of pitch pine, 8 inches broad, 3 inches
thick, and thirty feet long, is supported at both ends and
loaded in the centre with a weight of 100 Ibs. ; what is its
deflection ?
360 3 x 100
= 4-407 inches, deflection.
4900466 X 8 x 3 3
If the beam had been firmly fixed at both ends, the de-
flection would have been
4-408 X | = 2-938 inches.
If the beam had been supported at both ends, and loaded
uniformly throughout, the deflection would have been
4-408 X !- = 2-754.
To find the ultimate deflection of a beam <f timber
before it breaks :
length (in inches} 8
-p--^ 3 p = ultimate deflection,
tab. numb. U (table A) X depth
What is the ultimate deflection of a beam of ash, 1 foot
broad, 8 inches deep, and 40 feet long ?
AQ(\2
- = 72-72lnches, the ultimate deflection.
To find the weight under which a column placed verti-
cally will begin to bend, when it supports that weight :
tab. numb. E (table A) x least thickness 3 x greatest x -2056
length (in inches) 2
= weight in pounds. It will be found by the application
of this rule, that it will require 40289-22 Ibs. to bend a
beam of English oak 20 ft. long, 6 in. thick, and 9 in. broad.
BEAMS.
WE take the liberty here of introducing a short extract
from Messrs. Hann and Dodds' Mechanics, on the subject
BEAMS.
157
of beams. " In the construction of beams, it is necessary
that their form should be such that they will be equally
strong throughout. If a beam be fixed at one end, and
loaded at the other, and the breadth uniform throughout its
length, then, that the beam may be equally strong through-
out, its form must be that of a parabola. This form is
generally used in the beams of steam engines."
Dr. Young and Mr. Tredgold have considered that it will
answer better, in practice, to have some straight-lined
figure to include the parabolic form ; and the form which
they propose is to draw a tangent to the point A of the
parabola ACB.
To draw a parabola.
Let CB represent the
length of the beam, and
AB the semi-ordinate, or
half the base ; then, by
the property of the para-
bola, the squares of all
ordinates to the same
diameter are to one an-
other as their respective abscisses. Now, if we take CB
= 4 feet, and AB = 1 foot, we may proceed to apply this
property to determine the length of the semi-ordinates
corresponding to every foot in the length of the beam, aa
follow :
CB
that is, 48
AB a
12 a
CF
35
EF 8 ;
108 = EF 2 ;
the square root of which is 10'4 nearly = EF.
CG: GH a ;
24 : 72 = GH 3 ;
And CB : AB a
48 : 12 3
the square root of wh ch is 8-5 nearly = GH.
CB:AB 9 : CI : IK 8 ;
48 : 12 9 : 12 : 36 = IK 3 ;
the square root of which is 6 inches = IK.
Now, if we take CL = 6 inches,
then CB : BB 9 :: CL : LM 9 ;
48 : 12 9 :: 6 : 18 = LM 3 ;
the square root of which is 4-24, which is very near 4|
inches = LM. Now, if any flexible rod be bent so as just
to touch the tops A, E, H, K, M, of the ordinates, and the
vertex O, then the form of this rod is a parabola.
To draw a tangent to any point A of a parabola:
From the vertex G of the parabola draw CD perpendicu
14
158 STRENGTH OF MATERIALS.
ar to CB, and make it equal to AB ; then join AD, and
the right line AD will be a tangent to the parabola at the
point A ; that is, it touches the parabola at that point. ' In
the same manner, we may draw a tangent to the parabola
at any other point, by erecting a perpendicular at the vertex
equal to half the semi-ordinate at that point.
When a beam is regularly diminished towards the points
that are least strained, so that all the sections are similar
figures, whether it be supported at each end and loaded in
the middle, or supported in the middle and loaded at each
end, the outline should be a cubic parabola.
When a beam is supported at both ends, and is of the
same breadth throughout, then, if the load be uniformly
distributed throughout the length of the beam, the line
bounding the compressed side should be a semi-ellipse.
The same form should be made use of for the rails of a
wagon-way, where they have to resist the pressure of a
load rolling over them.
MODELS. The relation of models to machines, as to
strength, deserves the particular attention of the mechanic.
A model may be perfectly proportioned in all its parts as a
model, yet the machine, if constructed in the same propor-
tion, will not be sufficiently strong in every part ; hence,
particular attention should be paid to the kind of strain the
different parts are exposed to ; and from the statements
which follow, the proper dimensions of the structure may
be determined.
If the strain to draw asunder in the model be 1, and if
the structure is 8 times larger than the model, then the
stress in the structure will be 8 3 = 512. If the structure is
6 times as large as the model, then the stress on the struc-
ture will be 6 3 = 216, and so on ; therefore, the structure
will be much less firm than the model ; and this the more,
as the structure is cube times greater than the model. If
we wish to determine the greatest size we can make a ma-
chine of which we have a model, we have,
The greatest weight which the beam of the model can
bear, divided by the weight which it actually sustains = a
quotient which, when multiplied by the size of the beam in
the model, will give the greatest possible size of the same
beam in the structure.
Ex. If a beam in the model be 7 inches long, and bear
a weight of 4 Ibs., but is capable of bearing a weight of 26
SHAFTS 159
bs. ; what is the greatest length which we can make tha
corresponding beam in the structure ? Here
therefore, 6'5 X 7 = 45'5 inches.
The strength to resist crushing, increases from a model
to a structure in proportion to their size, but, as above, the
strain increases as the cubes ; wherefore, in this case also,
the model will be stronger than the machine, and the
greatest size of the structure will be found by employing
the square root of the quotient in the last rule, instead of
the quotient itself; thus,
If the greatest weight which the column in a model can
bear is 3 cwt., and if it actually bears 28 Ibs., then, if the
column be. 18 inches high, we have
wherefore, 3'464 x 18 =? 62'352 inches, the length of the
column in the structure.
SHAFTS.
THE strength of shafts deserves particular attention ;
wherefore, instead of incorporating it with the general sub-
ject, strength of materials, we have allotted to it a separate
chapter under that head.
When the weight is in the middle of the shaft, the rule is
[/ weight in Ibs. X length in feet\
3 H / = diameter in inches.
\ v 500
This is to be understood as the journal of the shaft, the
body being usually square.
What is the diameter of a shaft 12 feet long, bearing a
weight of 6 cwts., the weight acting at the middle ?
672 x 12\ _
2-525 inches.
500
If the weight be equally diffused, we have, the weight in
Ibs. x length ; extract the cube root and divide by 10; the
quotient is the diameter.
f Thus, take the last example, then 672 x 12 = 8064;
' the cube root of which is 20*05, which divided by 10 gives
2'005, the diameter of the shaft.
16C STRENGTH OP MATERIALS.
If a cylindrical shaft have no other weight to sustain be-
sides its own, the rule is, v/(-007 X length 3 ) = diameter:
thus, if a shaft having only the stress of its own weight be
10 feet long;
v/('007xl0 3 ) = 2-645 the diameter of the shaft in inches.
For a hollow shaft supporting so many times its own
weight, we have
I/-012 x length 3 x No. times its own weight\
\ v 1 + inner diameter 2 /
outer diameter in inches.
For wrought iron shafts find the diameter by the forego
ing rules, which apply to cast iron, then multiply by -935,
and for oak shafts the multiplier is 1-83, and for fir 1-716.
Ex. What is the diameter of a cast iron shaft 12 feet
long, and the stress it bears being twice its own weight?
Here we have,
v' (-012 X 12 3 x 2) = 6-44 inches.
For wrought iron, using the multiplier,
6-44 x '935 = 6-0215,
and for oak, using the multiplier,
6-44 x 1-83 = 11-3852,
and for fir, we have
6-44 X 1-716 = 11-05104.
A rule often used in practice, though by no means a cor-
rect one, for determining the diameter of shafts is this.
The cube root of the weight which the shaft bears taken in
cwts. is nearly the diameter of the shaft in inches. It will
be found safe in practice, to add one-third more to this
result.
If a cast metal shaft has to bear a weight of 1| ton, that
is, 30 cwts., then we have,
# 30 = 3-107 inches by this rule ;
and supposing it 12 feet long, we will apply the other rule,
we have,
3360 X 12\
wo) = 4 ' 319 -
We have now considered the strength of shafts, so far as
regards their power to resist lateral pressure by weight act-
ing on them ; we have now to consider their power to
resist torsion or twisting.
V/ J llll
ifc
SHAFTS. 161
For cylindrical shafts, we have,
240 X No. of horses' power \
''No. of revolutions in a minute/
ihe diameter of the shaft in inches.
This rule is for cast iron ; and it may be used for
wrought iron*by multiplying the result by -963, or for oak
oy 2-238, or for fir by 2-06.
If the shaft belong to a 7 horse power engine, and the
etrap turns 11^ times in a minute,
3 ( -= } = 5-267 inches diameter for cast iron.
\ \ 11-5 /
For fir, 5-267 X 2*06 = 10-85.
For oak, 5-267 X 2*38 = 12-535.
And for wrought iron, 5-267 X -963 = 5-0719.
NOTE. This rule comes from the best authority, anc
gives perfectly safe results, though some employ 340, in
stead of 240, as a multiplier, which gives a greater diameter
to the shaft. We may compare the two :
== 5-916,
whereas the other was 5-267 something more than half an
inch of difference.
It is to be remembered, that these rules relate to the
iihafts of first movers, or the shafts immediately connected
with the moving power. But these shafts may communi-
cate motion to other shafts, called second movers, and these
again to others, called third movers, and so on. The dia-
meters of the second movers may be found from those of
Ihe first, by multiplying by -8, and those of the third
movers, by multiplying by -793, thus, if the diameter of
the first mover be 5-267, then that of the second will be
5-267 X -8 = 4-2136, and that of the third mover will be
5-267 X -793 = 4-1767.
One material may resist, much better than another, one
kind of strain ; but expose both to a different kind of strain,
and that which was weakest before may now be the strong-
"est. This may be illustrated in the case of cast and wrought
icon. The cast iron is stronger than the wrought iron when
exposed to twisting or torsional strain, but the malleable
iron is the stronger of the two when they are exposed to
14*
162
STRENGTH OF MATERIALS.
lateral pressure. We shall subjoin a few results of experi-
ments on the weight which was necessary to twist bars j
close to the bearings.
Oast metal, ............. 9
Do. vertical cast, ..... 10
Cast steel, .............. 17
Shear steel...... ....... 17
Blister steel, ........... 16
oz.
17
10
9
1
11
English iron wrought,. 10
Swedish iron wjought, 9
Hard gun rnetal, 5
Brass bent, 4
Copper cast, 4 5
lb. 01
2
8
11
It would appear that the strength of bodies to resist torsion
is nearly as the cubes of their diameters.
REMARKS. The rules and statements we have now given
will often find their application in the practice of the engi-
neer. On the proper proportioning of the magnitude of
materials to the stress they have to bear, depends much of
the beauty of any mechanical structure ; and, what is of fai
greater moment, its absolute security. We will, in the Ap
pendix to this book, give some examples of the application
of these principles to practice.
TABLE OF THE DIAMETERS OF SHAFT JOURNALS.
10
20
30
40
50
60
70
80
90
100
5
5-9
4-7
4-1
3-7
3-5
3-3
3-1
3-0
2-9
2-7
6
6-3
5-0
4-4
4
3-7
3-5
3-4
3-2
3
2-9
7
6-6
5-2
4-6
4-2
3-9
3-6
3-5
3-4
3-3
3-1
8
6-9
5-5
4-8
4-4
4-1
3-9
3-7
3-5
3-4 3-3
9
7-2
5-7
5
4-5
4-2
4
3-7
3-6
3-5
3-4
10
7-4; 5-9
5-2
4;7
4-4
4-1
3-9
3-7
3-6
3-5
15
8-5
7-0
6-0
5-5
5-1
4-6
4-5
4-3
4-2
4-0
20
9-3
7-4
6-6
5-9
5-6
5-2
5-0
4-6
4-5
4.4
30
10-7
8-4
7-4
6-9
6-5
5-9
5-7
5-5
5-2
5-0
40
11-7 9-5
8-3
7-4
6-9
6-6
6-2
5-9
5-7
5-6
50
12-610-0
9-0
8-0
7-4
7-2
6-8
6-5
6-2
5-9
60
13-610-8
9-3
8-6
7-7
7-4
7-2
6-8
6-7
6-4
In the preceding table of the diameters of the shafts of first
movers, the number of horses' power of the engine is given
in the left-hand column, and the number of revolutions the
shaft makes in a minute is given in the top column. Then,
to use the table, we have only to look for the power of the
engine in the side column, and the number of turns the shaft
JOISTS AND ROOFS. 163
makes in a minute in the line which runs across the top,
and where these columns meet will be found the diameter
of the shaft in inches. The table is constructed for cast
iron, and first movers ; the rules for finding the second and
third have been given above, as also for finding equally
strong shafts of other materials.
This table answers for first movers only. It may, how-
ever, be made to give results for second and third movers,
by using the multipliers for that purpose, formerly given.
What is the diameter of the journal of the shaft of the
first mover in a 30 horse power engine, the shaft making
40 revolutions in a minute? Here, by looking in the table,
in the side column of horses' power, w<i find 30, and in the
top column of revolutions, we find 40, and whert* these
columns meet, we find 6 - 9 = the diameter of the first
mover, in inches ; wherefore, the second mover of this
power and velocity will be = 6 - 9 X '8 = 5-52 inches ; and,
in like manner, the third mover will be = 6 - 9 x "64 =
4'416 inches = the diameter of the third mover to the same
power and speed.
JOISTS AND ROOFS.
JOISTS should increase in strength in proportion to the
squares of their lengths ; for instance, a joist 10 feet long
should be four times as strong as another joist 8 feet long,
similarly situated; because 8* : 16*: : 1 : 4. From what
has been previously stated, it will easily appear, that the
stress on a beam or joist supported at both ends, increases
towards the middle, where it is greatest ; it therefore fol-
lows, that a beam should be strengthened in proportion to
the increasing strain ; and, as it would not be easy to add
to the thickness of a beam towards the middle, which would
destroy the levelness of the floor, a good substitute may be
to fasten pieces to the sides of the joist, and thus increase
its breadth ; thus causing the beam to taper, in breadth,
from the centre to the ends. In this way joists may be
made much stronger than they usually are of the same
'length, and out of the same quantity of timl er. It may
^Iso be observed, that joists are twice as strong when firmly
fixed in the wall, as when loose ; but it is to be remarked,
th%t they have, when fixed, a far greater tendency to shake
the wall. It is also to be remarked, that a joist is four times
stronger when supported in the middle.
164 JOISTS AND ROOFS.
If the letter L represent the length of some known joist,
whose strength has been tried, and D its depth, and T its
thickness ; and if another joist is required of equal strength
with the former, when similarly situated ; whose length is
represented by /, its depth by c/, and its thickness by t; we
have the following rules :
x l , I/D ' x T x 1
D a x /' X T .. /</* x / x I
D'xT
If a joist 30 feet long, 1 foot deep, and 3 inches thick, be
sufficient in one case, what must the depth of a beam be,
similarly placed, whose length is 15 feet, its depth and
thickness bearing the same proportion to each other, as in
the former beam ? Here, by the first theorem, we have,
= ' 6298 fect = 7 ' 55 inohes
the depth ; and therefore 12 depth : 3 thickness : : 7*55 :
1-88 the breadth.
If the given beam be, as in the last example, 12 inches
deep, 3 thick, and 30 feet long, and the required beam, of
the same strength, is 8 inches deep, and 6 inches thick, then
by the 4th we have,
If a joist, whose length is 30 feet, depth 12 inches, and
thickness 8, is given, to find the depth of another of equal
strength, only 6 inches thick, and 28-28 feet long? Here,
by the 2d, we have,
f!2 a x 3 X 2S-28 3
\l -- 6 x SO -- = incnes > l " e depth.
To find the thickness from the same circumstances, we
nave by the 3J,
12 8 x 2S-28 3 x 3
go v. or>a - == ^ inches, the thickness.
O X oU
The same remarks hold true to a certain extent in roof-
ing. A high roof is both heavier and more expensive than
d low roof, as they will always be as the squares of the
lengths of the couple-legs, so far as the scantling is con-
cerned ; but the slates and other materials increase in weight
WHEELS. 165
ind expense as the length of the couple -legs simply. High
roofs have, however, the advantage of being less severe
upon the walls than low ones, that is to say, so far as a
tendency to push out the walls is concerned. To obtain
the length of the rafter from that of the span, a common
rule is to multiply the span by -66, which gives the length
of the rafter; thus, 14 feet of span gives 14 x '66 = 9-24
feet, the length of the rafter.
NOTE. The numbers in the tables of the strength of
materials are such as will break the bodies in a very short
time ; the prudent artist, therefore, will do well to trust no
more than about one-third of these weights ; also great
allowance must be made for knotty timber, and such as is
sawn in any part across or obliquely to the fibres.
WHEELS.
IN page 136 we promised again to resume the subject of
wheel-work ; and we now proceed to consider, in the first
place, the formation of the teeth of wheels.
A Cog-wheel is the general name for any wheel which
has a number of teeth or cogs placed round its circumference.
A Pinion is a small wheel which has, in general, not
more than 12 teeth ; though, when two toothed wheels act
upon one another, the smallest is generally called the pinion ;
so is also the trundle, lantern, or wallower.
When the teeth of a wheel are made of the same material
and formed of the same piece as the body of the wheel, they
are called teeth ; when they are made of wood or some other
material, and fixed to the circumference of the wheel, they
are called cogs ; in a pinion they are called leaves ; in a
trunflle, staves.
The wheel which acts is called a leader, or driver ; and
the wheel which is acted upon by the former is called a
foBower, or the driven.
When a wheel and pinion are to be so formed that the
.pinion shall revolve four times for the wheel's once, then
they must be represented by two circles, whose diameters
a/e to one another, as 4 to 1. When these two circles are
so placed that they touch each other at the circumferences,
then the line drawn joining their centres, is called the lina
166 WHEELS.
of centres, and the radii of the two circles the proportions*
radii.
These circles are called, by mill-wrights in general,
pitch-lines.
The distances from the centres of two circles to the ex-
tremities of their respective teeth, are called the real radii,
and the distances between the centres of two contiguous teeth
measured upon the pitch-line, is called the pitch of the wheel.
Two wheels acting upon one another in the same plane,
are called spur geer. When they act at an angle, they are
called bevel geer.
Teeth of wheels and leaves of pinions require great care
and judgment in their formation, so that they neither clog
the machinery with unnecessary friction, nor act so irre-
gularly as to produce any inequalities in the motion, and a
consequent wearing away of one part before another. Much
has been written on this subject by mathematicians, who
seem to agree that the epicycloid is the best of all curves
for the teeth of wheels. The epicycloid is a curve differing
from the cycloid formerly described, in this, that the gene-
rating circle instead of moving along a straight edge, moves
on the circumference of another circle.
The teeth of one wheel should press in a direction per-
pendicular to the radius of the wheel which it drives. As
many teeth as possible should be in contact at the same
time, in order to distribute the strain amongst them ; by
this means the chance of breaking the teeth will be dimi-
nished. During the action of one tooth upon another, the
direction of the pressure should remain the same, so that
the effect may be uniform. The surfaces of the teeth in
working should not rub one against another, and should
suffer no jolt either at the commencement or the termination
of their mutual contact. The form of the epicycloid satis-
fies all these conditions ; but it is intricate, and the involute
of the circle is here substituted, as satisfying equally these
conditions, and as being much more easily described.
Take the circumference ABC of the
wheel on which it is proposed to raise
the teeth, and let a be a point from
which one surface of one tooth is to
spring, then fasten a string at B, such
that when stretched and lying on the
circumference shall reach to a/ fix a
WHEELS. lb'7
pencil at a, and keeping the string equally tense, move
the pencil outwards, and it will describe the involute of the
circle which will form the curve for one side of the tooth.
Fasten the string at B so that its end, to which the pencil
is fixed, be at the point from which the other face of the
tooth is to spring and proceed as above ; then the curve
of the other side of the tooth will be formed ; and the figure
of one of the teeth being determined, the rest may be traced
from it.
The teeth of the pinion are formed in like manner.
The observation of practical men has furnished us with
a method of forming teeth of wheels, which is found to an
swer fully as well in practice as any of the geometrical
curves of the mathematician.
We have the pattern here of
the segment of a wheel with
cogs fixed on in their rough
state, and it is required to bring
them to their proper figure :
they are consequently understood to be much larger than
they are intended to be when dressed. The arc 6, 6, is the
circumference of the wheel on which the bottoms of the
teeth and cogs rest. Draw an arc a, a, on the face of the
teeth for the pitch line of their point of action ; draw also
rf, rf, for their extremities or tops. When this is done, the
pitch circle is correctly divided into as many equal parts as
there are to be teeth. The compasses are then to be opened
to an extent of one and a quarter of those divisions, and with
this radius arcs are described on each side of every division
on the pitch line a, a, from that line to the line d, d. One
point of the compasses being set on c, the curve f, g, on
one side of one tooth, and o, n, on the other sides of the
other are described. Then the point of the compasses being
set on the adjacent division k, the curve /, m, will be de-
scribed : this completes the curved portion of the tooth e.
The remaining portion of the tooth within the circle a, a,
is bounded by two straight lines drawn from g and m to-
wards the centre. The same being done to the teeth all
round, the mark is finished, and the cogs only require to be
dressed down to the lines thus drawn.
/ It will be easy to determine the diameter of any wheel
having the pitch and number of teeth in that wheel given
Thus, a wheel of 54 teeth having a pitch of 3 inches, we
163
WHEELS.
have 54
quently,
X 3 = 162 inches, the circumference, conse-
162
= 51-5 inches diameter, nearly.
3-1416
or about 4 feet 3 inches.
In the following table we have given the radii of wheels
of various numbers of teeth, the pitch being one inch. To
find the radius for any other pitch, we have only to multiply
the radius in the table by the pitch in inches, the product
is the answer. Thus for 30 teeth at a pitch of 3| inches,
we have 4-783 x 3-5 = 16-74 inches, the radius.
1 2
3
4
5
6
7
8
9
10
1-668
1-774 1-932
2-089
2-247
2-405
2-563
2-721
2-879
3-038
20
3-196
3355 3-513
3-672
3-830
3-989
4-148
4-307
4-465
4-624
30
4-783
4-942 5-101
5-260
5-419
5-578
5-737
5-896
6-055
6-214
40
6-373
6-532 6-643
6-850
7-009
7-168
7-327
7-486
7-695
7-804
50
7-963
8-122 8-231
8-440
8-599
8-753
8-962
9-076
9-235
9-399
60
9-553
9-7121 9-872
10-031
10-190
10-349
10-508
10-662
10-826
10-935
70
11-144
11-303 11-463
11622
11-731
11-940
12-099
12-758
12-417
12-676
80
12-735
12-895 13-054
13-213
13370
13531
13690
13-849
14-008
14-168
90
14-327
14-4H6 14-645
14-804
14-963
15-122
15281
15-441
15-600
15-759
100
15-918
16-072 16-236
16-305
16-554
16713
16-873
17-032
17-191
17-350
110
17-504
17-6U8 17-987
17-827
18-146
18-305
18-464
18-623
18-782
18-941
120
19-101
19-260 19-419
19-578
19-737
19-896
20-055
20-2 f4
20-374
20-533
130
20-692
20-851 21-010
21-169
21-328
21-48S
21-647
21-806
21-460
22-124
140
22-283
22-44222-602
22-761
22920
23074
23-238
23397
23-556
23-716
150
23875
24-03424-193
24-352
24-511
24620
24-830
24-989
25-148
25-307
160
25-466
25-625 25-784
25-944
26-103
26-262
26-421
26-580
26-739
26-894
170
27-058
27-217 27-376
27-535
27-694
27-853
27-931
28-172
28-331
28-490
180
28-699
28-80828-967
29-126
29-286
29-445
29-604
29-763
29-922
30-086
190
30-241
30-400 30-559
30-718
30-S77
31-036
31-196
31-355
31-514
31-673
200
31-832
31-992 32-150
32-310
32-469
32-628
32-787
32-846
33-105
33-264
210
33-424
3358333-742
33-901
34-060
34-219
34278
34-537
34-697
34-856
220
35-015
35-174 35-333
35-492
35-652
35-811
35-970
36-129
36-288
36-447
230
36-607
36-76636-925
37-084
37-243
37-402
37-561
37-720
37-880
38-039
240
38-198
38-357 38-516
38-725
38-835
38-994
39-153
39312
39-471
39-631
250
39-790
39-949 ! 40-108
40-262
40-426
40-585
40-744
40-904
41-063
41-222
260
41-381
41-54U41-699
4i-858
42-019
42-177
42-336
42-495
42-654
42-813
270
42-973
43-13243-291
43-450
43-609
43-768
43-927
44-087
44231
44-405
280
44-564
44-723 44-882
45-042
45201
45360
45519
45-678
45837
45-996
290
46-156146 315,46-474
46-633
16-792
46-751
47-111
47-270
47-429
47-583
1 1
This will be found a very useful table in abridging calcu-
lation, for instance, if we wish to find the radius of a wheel
having 132 teeth, we look for 130 at the left-hand side
column, and 2 at the top, and where these columns meet,
we find the number 21-010, which, if the pitch of the
wheel be 2 inches, we multiply by 2|.
WHKKl.s. 169
21-010 x 2-5 = 52-525 inches, radius of required wheel.
An easy practical rule Cor the same purpose is the fol-
lowing :
Take the pitch by a pair of compasses, and lay it off on
a straight line, seven times, divide this line into eleven
equal parts ; each will be equal to four ot' the radius, which
is supposed to consist of as many parts as the wheel has
teeth.
Let the pitch be two inches, and the number of teeth 60
t-ien the diagram will show how to lay it down.
1 2 :* 4567
i~~2T~3~~4" ~5~6 7. 8 9 10 11
C D
4, 8, 12, 16, &c.
The upper line is the pitch laid off seven times, and
forming AB, which is divided into 11 equal parts, one
of which, CD, being repeated for every four teeth in
the wheel, that is, in this case, fifteen times, will give the
radius.
The same may be done by calculation, going by the
principles of the rule, thus,
2x7 = 14, then = 1-272, which divided by 4
gives 0-318 = the value of of the radius ;
4 60
wherefore, -318 x 60 = 19-08.
By the table we have,
9-552 X 2 = 19-104,
the difference in the two results being
19-104 19-08 = -024, or twenty-four thousandth
parts of an inch.
Reversing the operation, let it be required to find the
pitch, the radius of the wheel being 19-104, and number
of teeth 60.
We have --* -318, then -318 x 4 = 1-272,
60
and 1-272 X 11 = 13-992. Now this is the whole line AB,
13*992
and therefore, = 1-998, which is so verv nearly
/ 7
two inches, the difference being 2 1-998 = -002 oi an
inch, we ought in practice to take two as the pitch.
15
170 WHEELS.
A little reflection on the part of the reader will show
*? 11 6Qfi
that since = -636, and = 1-571, and = *159
117 4
Tve have,
(1) pitch x '159 x number of teeth = radius.
/n\ radius
' number of teeth x '159 =
radius
(3) - ; - TTT: = number of teeth.
v ' pitch X '159
Thus,
(1) 2 X '159 X 60 = 19-08 = radius,
19
19
( 3 ) ^ - ;W = 60 = number of teeth.
v ' 2 x -159
NOTE. The number -16 may be employed instead of
159, being easily remembered. These rules are approxi-
mate, and the error diminishes as the number of teeth in-
creases. The true pitch is a straight line, but these rules
give it an arc of the circle, which passes through the centre
of the teeth, whereas it should be the chord of the arc.
An eminent writer on clock-work gives the following
rules regarding wheels and pinions :
(A) As the number of teeth in the wheel + 2-25,
Is to the diameter of the wheel,
So is the number of teeth in the pinion + 1-5,
To the diameter of the pinion.
A wheel being 12 inches diameter, having 120 teeth,
drives a pinion of-20 leaves ; wherefore,
120 + 2-25 = 122-25 and 20 + 1-5 = 21-5,
Then 122-25 : 12 : : 21-5 : 2^104 = the diameter of the
pinion.
(B) As the number of teeth in the wheel + 2-25,
Is to the wheel's diameter,
So is 3 (teeth in wheel -f leaves in pinion)
To the distance of their centres.
A wheel's diameter being 3'2 inches, number of teeth 96,
the leaves in the pinion being 8, then,
104
96 + 2-25, = 98-25 and (96 -f 8) = - - = 52.
171
Hence, 98-25 : 3'2 : : 52 : 1-6936 = the distance which
the centres ought to have.
The strength of wheels is a subject which has occupied
the attention of the most eminent practical engineers, but
the rules they have given us are entirely empirical, that is
to say, the result of experiment.
The strength of the teeth will vary with the velocity of
the wheel, the strength in horses' power at a velocity ol'
2-27 feet per second, will be
breadth of the tooth x its thickness"
- = power,
length of tooth
Required the strength in horses' power of a tooth 4 inches
broad, 1-3 inches thick, and 1-6 inches long, at a velocity
of 2-27 feet per second, here we have
4 X l"3 a
= 4-225, the horses' power at a velocity of 2 27.
1'6
The power at any other velocity may be found by pro-
portion, thus the same at 6 feet per second.
2-27 : 6 :: 4-225 : 11-1 = horses' power a* i, velocity
of 6 feet per second.
The thickness of a tooth x 2-1 = the pitch
The thickness of a tooth x 1'2 = length.
Ex. The thickness of a tooth being l in- hos, then we
have
1-5 X 2-1 = 3-15 = the pitch.
1-5 X 1-2 = 1-8 = the length
The breadth in practice is usually 2'5 times the pitch.
The arms of wheels generally taper from the axle to the
rim, because they sustain the greatest stress towards the
axle. It is obvious, that the more numerous the arms of a
wheel are, they each suffer a proportionately less strain, as
the resistance will be diffused over a greater number.
The power acting at the rim X length of :rm 3
number of arms X 2656 X 0-1
and cube of depth.
Ex. If the force acting at the extremity of the arm of a
wheel be 16 cwt.; the radius of the wheel being 5 feet, and
the number of arms 6, then we have 16 X 112 = 1792 Ibs.
.== the force ; wherefore,
1792 X 5 3 224000
r^T656^roT == -159F6 * 14 ' breadth and CUbe f
depth.
172 WHEELS.
Now, let us suppose that the breadth is two inches, we
Amst divide this 140 by it, whence,
140
- = 70, the cube of the depth,
9
ai.d the cube root of 70 will be found = 4*121, which is
the depth of each arm.
When the depth at the axis is intended to be double of
the depth at the rim, the number 1640 is to be used in the
rule instead of 2't56.
The tables vhich follow will be found in the highest
degree useful to <he practical mechanic.
VHEKLS.
173
ABLE OF PITCHES OF WHEELS IN ACTUAL USE IN MILL
WORK.
v . v. s. 5 a
3 ffl Ol O> (D 2 >
? "i - ? -i 9 9
X
B
*< I ! ! I 5 CD a tp g>
ca : : : : : P -7- --7
fr
Honet' power.
-,- l-i-
Breadth of teeth in incho.
_____,; _ ,i , : ,i ^:
<iOCOOOOi *>. W W 00 63 SO
COCO
Revolves per
minute.
ODQOO'OlO9taW^.COfcS*fcO r. - r.
H- H-" H- 1 >
OO>-^-O500O<J<lO 6S WOO
-i i^COiO _^ ^. tC JC
_ tOQDO ODOO-^Oi
Cl W 62
OOtS
' Teeth.
CO
o cotocn oopcpfcOH-apo "*?
to co H- ^i co
Revolves per
tOf- CCK/i 00(0
Breadth proportional lo 10
horses' power,aiid prticot
velocity.
" o en TO c) totoaoc
QOW-^ltntt en en
tionalto 10liors'[iower, I
at 3 f. p. second, that is, i
reduciDealltheezamplei '
to the same deaom.
15*
174 WHKELS.
EXPLANATION OF REFERENCES, &c., IN THE FOREGO-
ING TABLE.
1 The only defect in this geering, which has been 1 6 years at work,
is the want of breadth in the spur-wheel and pinion : they ought to have
been 6 inches or more, as they will not last half so long as the bevel-
wheels and pinions connected with them.
3 Has been 1 6 years at work. The teeth are much worn.
3 Has been 16 years at work. This geering is found rather too nar-
row for the strain, as it is wearing much faster than the rest of the wheels
in the same mill.
4 and 5 This wheel has wooden teeth, and has been working for three
years.
6 This is a better pitch for the power than the following.
7 This pitch has been found to be too fine.
In the foregoing table the wheels are all reduced to what
may be called one denomination. 1st. By proportioning
all their breadths to what they should be, to have the same
strength, if the resistance were equal to the work of a steam
engine of ten horses' power. 2d. By supposing their pitch-
lines all brought to the same velocity of three feet per se-
cond, and proportioning their breadth accordingly. This
particular velocity of three feet per second has been chosen,
because it is the velocity very common for overshot wheels.
Such cases as appear to have worn too rapidly, are marked,
which may tend to discover the limit in point of breadth.
TABLE OF PITCHES.
THE succeeding table of pitches of wheels was drawn up
in the following manner : The thickness of the teeth in
each of the lines is varied one-tenth of an inch. The
breadth of the teeth is always four times as much as their
thickness. The strength of the teeth is ascertained by
multiplying the square of their thickness into their breadth,
taken in inches and tenths, &c. The pitch is found by
multiplying the thickness of the teeth by 2-1. The num-
ber that represents the strength of the teeth, will also repre-
sent the number of horses' power, at a velocity x of about
four feet, per second. Thus, in the table where the pitch
is 3'15 inches, the thickness of the teeth 1-5 inches, and
the breadth 6 inches, the strength is valued at 13| horses
power, with a velocity of four feet per second at the pitch
line.
WHEELS.
175
A Table of Pitches of Wheels, ivith the breadth and thick
ness of the teeth, and the. corresponding number of
horses' power, moving at the pitch-line at the "ate of
three, four, six, and eight feet, per second.
Pitch in
inches.
Thick-
ness of
inches.
Briathh
of tef>th
ill inches.
Strengih of
ll'rlhjiT lll>.
of horses'
]* nvcr. at 4
cund.
Horses'
power at 3
feet per
second.
Horses'
po*er at 6
feet per
second.
Horses'
power at 3
feet per
second.
3-99
1-9
7-6
27-43
20-57
41-14
54-85
3-78
1-8
7-2
23-32
17-49
34-98
46-64
3-57
1-7
6-8
19-65
14-73
29-46
39-28
3-36
1-6
6-4
16-38
12-28
24-56
32-74
3-15
1-5
6-
13-5
10-12
20-24
26-98
2-94
1-4
5-6
10-97
8-22
16-44
21-92
2-73
1-3
5-2
8-78
6-58
13-16
17-34
2-52
1-2
4-8
6-91
5-18
10-36
13-81
2-31
1-1
4.4
5-32
399
7-98
10-64
2-1
1-0
4-
4-0
3-0
6-0
8-0
1-89
9
3-6
2-91
2-18
4-36
5-81
1-68
8
3-2
2-04
1-53
3-06
3-08
1-47
7
2-8
1-37
1-027
2-04
2-72
1-26
6
2-4
86
64
1-38
1-84
1-05
5
2-
5
375
75
1-
HYDROSTATICS.
HYDROSTATICS comprehends all the circumstances of the
pressure of non-elastic fluids, as water, mercury, <fcc., and
of the weight and pressure of solids in them, when these
fluids are at rest. Hydrodynamics, on the other hand, refers
to the like circumstances of fluids in motion.
The particles of fluids are small and easily moved among
themselves.
Motion or pressure in a fluid is not in one straight line
in the direction of the moving force, but is propagated in
every direction, upwards, downwards, sidewise, and oblique.
From this property it is, that water will always tend to
come to a level, for if two cisterns be filled with water, the
one 10 feet deep, and the other 6, there will be more pres-
sure on the bottom of the 10 feet, than the 6 feet cistern ;
and, if the bottoms of both cisterns be on a level, and a
pipe be made to communicate between them, then the water
in the deep cistern will exert a greater pressure than that
in the other, and will cause the other to rise till their pres-
sures become equal, that is, when their surfaces come to a
level ; and this will hold true, however different the sur-
faces of the two cisterns may be in area. Hence, if water
be communicated through pipes between any number of
places, it will rise to the same level in all the places, whe-
ther the pipes be straight or bent, wide or narrow ; and any
fluid surface will rest only when that surface is level.
If a vessel contain water, the pressure on any point in
the sides or bottom, is proportional to the perpendicular
height of the fluid, above that point, in the side or bottom.
The pressure of a fluid upon a horizontal base, is equal
to the weight of a column of the fluid, of the area of the
base multiplied by the perpendicular height of the fluid,
whatever be the shape of the containing vessel : so that by
a long and very small pipe, the strongest casks or vessels
176
HYDROSTATICS. 177
may be burst asunder by the pressure of a very small quan-
tity of water.
Ex. Into a square box a tube is fixed, so that it shall
stand perpendicularly ; the area of the bottom of the box
is 9 square feet, and the height of the top of the tube above
the bottom of the box is 5 feet, and therefore the pressure
on the bottom is 5 X 9 = 45 cubic feet of water. Now
the weight of one cubic foot of water is found to be very
nearly 1000 ounces avoir., therefore, 45 X 1000 = 45,000
ounces, = 1 ton, 5 cwt. qrs. 12 Ibs. 8 oz.
The content in imperial gallons of any rectangular cis-
tern may be found thus,
cistern's content in cubic feet X 6-232,
or cistern's content in inches x '003607,
cistern's content in cubic inches
277-274
content in imperial gallons.
From these rules, which are approximate, it is easy to
see that of the three, the length, breadth, and depth of a
cistern, any two being given the third may be found, so
that the yessel shall contain any given number of gallons,
thus,
number of gallons
-= the third
any two dimensions in feet X 6-232
dimension in feet.
For the content in gallons of a cylindric vessel,
diameter 8 X length X 4-895,
if the dimensions are in feet, but if the diameter be in
inches, use '034 instead of 4-895, and should both dimen-
sions be in inches, use -002832, or divide by 352-0362.
Also when the length and diameter are in feet,
number of gallons
length x 4-895
number of gallons
= diameter
= length.
diameter 2 x 4-895
For a sphere we have diameter 3 X 3-263 = content in
gallons, the diameter being in feet, but when the diameter
s in inches, use the number '001888. These rules may be
illustrated by the following examples.
The length of a cistern being 8 feet, its breadth 4-5, and
depth 3, then will its content be 8 X 4-5 X 3 = 108 cubic
178 HYDROSTATICS.
feet, hence 108 X 6-232 = 673-06 gallons may be con-
tained in it.
It is required that a cistern should contain 1000 gallons,
but must not exceed 10 feet in length and 5 in breadth,
wherefore,
'1000 1000
10 x 5 x 6-232 ~ 3TF6 =
A cylinder is 6-5 feet long and 3 inches diameter, there-
fore 6-5 X 3 2 X -034 = 1-989 gallons that it will contain.
A pipe is to be made 20 inches in length, what must be
its diameter so that it shall contain 5 gallons ?
x 354
20
= 9*4 inches.
The quantity of pressure upon any plane surface on which
a fluid rests, is equal to the pressure upon the same plane
placed horizontally at the depth of its centre of gravity.
If any plane surface, either vertical or inclined, be placed
in a fluid, the centre of pressure of the fluid on the plane
is at the centre of percussion, the surface of the fluid being
supposed the centre of motion. Thus it will be found that
in a cistern whose sides are vertical, the centre of pressure
on the sides is two-thirds from the top, which is also the
centre of percussion.
To ascertain the whole pressure on a flood-gate, or other
surface exposed to the pressure of water, a very near ap-
proach to the truth may be made by these rules the breadth
and depth being taken in feet.
31-25 X breadth X depth 3 = pressure in Ibs.
2727 X breadth X depth 3 = pressure in cwts.
If the gate be wider at the top than bottom,
/breadth at top breadth at bottom\
31-25 X (- -) + breadth
\ 9 f
at bottom X depth 3 = pressure in Ibs. ; and -2727, used in
stead of 31-25, will give the pressure in cwts., nearly
Exam. What is the pressure upon a rectangular flood-
gate, whose breadth is 25 feet, and depth below the surface
of the water 12 feet?
31-25 X 25 X 12 a == 112500 Ibs. pressure.
If the breadth* at top be 28 feet, that at bottom 22, and
the height 12, as before, then,
HYDROSTATICS. 179
oa 22
31-25 X - 4- 22 X 12 a = 108000 Ibs. pressure.
3
The weight of a cubic foot of river water is about ^ of
a cwt. The pressure at the depth of 30 feet is about 13
Ibs. to the square inch. And at the depth of 36 feet the
pressure is about 1 ton to the square foot. The weight of
an imperial <rallon of water is about 10 Ibs.
Ex. What is the pressure at the depth of 120 feet on a
square inch ?
30 : 120 : : 13 : 52 = the pressure, and at the same
depth, 36 : 120 : : 1 : 3i tons on the square foot.
It is not difficult to see that the strength of the vessels
or pipes which contain or convey water must be regulated
according to the pressure.
The thickness of pipes to convey water must vary in pro-
portion to the height of the head of water X diameter of
pipe -T- the cohesion of one square inch of the material of
which the pipe is composed.
By experiment it has been found that a cast iron pipe 15
inches diameter and | of an inch thick of metal, will be
sufficiently strong for a head 600 feet high. A pipe of oak
15 inches diameter and 2 inches thick, is sufficient for a
head of 180 feet. When the material is the same, the
thickness of the material varies with the height of head X
diameter of pipe
Ex. What must be the thickness of a cast iron pipe 10
inches diameter for a head of 360 feet ?
360 X 10 X !
600~X~15~~ = T * a " * thlckness -
If the same pipe is to be made of oalc, then
300 x 10 X 2
- = 2f thickness in inches.
When conduit pipes are horizontal and made of lead,
their thicknesses should be 2|, 3, 4, 5, 6, 7, 8 lines, when
ihe diameters are 1, 1|. 2, 3, 4, 6, 7 inches and when
the pipes are made of iron, their thickness should be 1, 2,
3f 4, 5, 6, 7, 8 lines, when their diameters are 1,2, 4, 6,
B, 10, 12.
The plumber should be aware that the tenacity of lead is
increased four times, by adding 1 part of zinc to 8 of lead.
When the vessel which contains the water has, besides the
180 HYDROSTATICS.
pressure arising from the weight of the water, to resist an
additional pressure exerted by some force on the water, as
in Bramah's press, where the pressure exerted by means of
a force pump on the water in a small tube, which commu-
nicates with a large cylinder, is, by the principles stated
before in this chapter, multiplied on the piston of the
cylinder as often as the area of the tube is contained in the
area of the piston of the cylinder. If the area of the tube
( be one inch, the area of the piston 92 inches, and if the
pressure on the water in the tube be 16 Ibs., then the pres-
sure on the piston will be 16 X 92 = 1472 Ibs.
The annexed figure and description taken from the
Popular Encyclopedia, 'will give a clearer idea of the
operation of this press. " Here AB is the bottom of a
hollow cylinder, into which a piston
P is accurately fitted. Into the bot- f p
torn of this cylinder there is intro-
duced a pipe C leading from the
forcing pump D ; water is supplied
to this pump by a cistern below, from
which the pipe E is led, being fur- D
nished with a valve opening upwards
where it is joined to the pump barrel. E
Where the pipe C enters into the pump barrel there is alsi
a valve opening outwards into the pipe ; consequently,
when the piston D rises, this valve shuts, and the valve at
ihe cistern pipe opens, and the fluid rises into the pump
barrel. The top of the piston rod, P, is fixed in the bottom
of the board on which the goods are laid, and when the
piston rises the goods are pressed against the top of the
framing of the machine. When the piston begins to de-
scend, the cistern valve shuts, and the water is forced
through the pipe C into the large cylinder AB ; and by the
law of fluids before alluded to, whatever pressure be exerted
by the piston D on the surface of the water in the pump,
will be repeated on the piston of the large cylinder AB as
many times as the area of the small piston I) is contained
in the area of the large piston AB ; that is, if the area of
the pump-piston were one square inch, and that of the
cylinder 100 inches, and if the piston were forced down
with a pressure of 10 Ibs., then the whole pressure on the
bottom of the piston AB will be 10 times 100, that is, 1000
Ibs. When the page which is now before the reader was taken
HYDROSTATICS. 181
wet off the types, it was all deeply indented in consequence
of the pressure of the printing press ; but after being dried, it
was subjected to the action of Bramah's press, by which
process, as will be seeYi, these indentations have been nearly
obliterated. In the press by which this has been accom-
plished, the pump has a bore of three-fourths of an inch in
diameter, and the cylinder one of eight inches, their areas
are therefore to one another, as 9- 16th to 64, (the squares
of the diameters,) that is, as 1 is to 113; hence if the
pressure upon the pump-cylinder be 56 Ibs., (which can be
easily effected by boys,) the pressure upon the piston of
the large cylinder will be 56 X 113, that is, 6'328 Ibs.
This astonishing power has also been employed in the
construction of cranes."
To ascertain the thickness of metal necessary for the
cylinder of such presses, this rule will serve :
pressure per square inch X radius of cylinder
cohesion of the metal per square in. pressure
thickness of metal necessary for the cylinder to sustain the
pressure. The pressure being in Ibs.
NOTE. The cohesive force of a square inch of cast iron
is 18,000 Ibs.
What is the thickness of metal in a cast iron press whose
cylinder is 12 inches diameter, the pressure being 1-5 tons
on the circular inch ?
A circular inch is to a square inch as 0'7854 to 1, there-
fore 1'5 tons per circular inch = 1'9 per square inch =
4256 Ibs.
Here we have
18000 4256
What is the thickness of metal in a press of yellow brass,
whose cylinder is 10 inches in diameter, and which is in
tended for a pressure of 2 tons to the square inch ?
The cohesive force of yellow brass being 17958, we have
by the same rule,
2 tons = 4480 Ibs.
- = T66 inches, the thickness of the
17958 4480
metal.
182 HYDROSTATICS.
When the diameter remains the same, the thickness ap-
pears to increase with the increase of pressure=
FLOATING BODIES.
WHEN any body is immersed in water, it will, if it be of
the same density of the water, remain suspended in any
place ; but if it be more dense than the water it will sink,
and if less dense it will float.
Bodies immersed and suspended in a fluid lose the weight
of an equal bulk of the fluid, and the fluid acquires the
weight that the body loses : also, bodies floating on a fluid
lose weight in proportion to the quantity of fluid they dis-
place.
When a body floats upon the water, it will sink in the
water till the water which is displaced be equal in weight
to the weight of the body.
When, a body floats on a fluid, it will only be at rest
when the centre of gravity of the body and the cen're of
gravity of the displaced fluid are in the same vertical line ;
and the lower the centre of gravity is, the more stable will
the body be.
The buoyancy of casks, or the load which they will carry
without sinking, may be estimated at about 10 Ibs. to the
ale gallon, or 282 cubic inches of the content of the cask.
SPECIFIC GRAVITY.
SPECIFIC gravity is the relative weight of any body of a
certain bulk, compared with the weight of some body taken
as a standard of the same bulk. The standard of compari-
son is water ; one cubic foot of which is found to weigh
1000 ounces avoir, at a temperature of 00 of Fahrenheit,
so that the weight expressed in ounces of a cubic foot of any,
body, will be its specific gravity, that of water being 1000.
To determine the specific gravity.
If a body be a solid heavier thaft water Weigh it first
carefully in air, and note this weight; then immerse it in
water, and in this state note its weight. Then divide the
body's weight in air by the difference of the weights in air
and water, the quotient is the specific gravity.
SPECIFIC GRAVITY.
183
If a body be a solid lighter than u'atcr Tie a piece of
metal to it, so that the compound may sink in water then
to the weight of the solid itself in air, add the weight of the
metal in water, and from this sum subtract the weight of the
compound in water, which difference makes a divisor to a
dividend, which is the weight of the solid in air, then the
quotient will he the specific gravity.
If (lie body be a Jlvid Take a solid, whose specific
gravity is known, and that will sink in the fluid ; then take
the difference of the weights of the solid in and out of the
fluid, and multiply this difference by the specific gravity
of the solid ; then, this product divided by the weight of
the body in air, will give the specific gravity of the fluid.
On these principles there has been constructed tables of
specific gravities, one of which we insert. The column,
specific gravity, may be taken to represent the weight cf
a cubic foot.
TABLE OF SPECIFIC GRAVITIES.
METALS.
Specific Gravity,
Arsenic, 5763
Cast antimony, 6702
Cast zinc, 7190
Cast iron, 7207
Cast tin, 7291
Bar iron 7788
Cast nickel, 7807
Cast cobalt, 7811
Hard steel, 7816
Soft steel, 7833
Cast brass, 8395
Cast copper 8788
Specific Gravity
Cast bismuth, 9822
Cast silver, 10474
Hammered silver, ... 10510
Cast lead, 11352
Mercury, 13568
Jewellers' gold, 15709
Gold coin, 17647
Cast gold, pure, 19258
Pure gold, hammered, 19361
Platinum, pure, 19500
Platinum, hammered, 20336
Platinum wire, 21041
STONES, EARTHS, ETC.
Brick 2000
Sulphur, 2033
Stone, paving, 2416
Stone, common, 2520
Pebble, 2664
Slate 2672
Marble, 2742
Chalk 2784
Granite, red, 2654 Basalt, 2864
Glfess, green, 2642 Hone, white razor, ... 2876
Glass, white, 2892 Limestone, 3179
Glass, bottle, 2733
184
HYDROSTATICS.
RESINS, ETC.
Specific Gravity.
Specific GrTi*y
Wax, 897
Tallow, :.... 945
Bone of an ox, 1659
Ivory, 1822
LIQUIDS.
Air at the earth's sur-
face,
If
Oil of turpentine, 870
Olive oil, 915
Distilled water, 1000
Sea water, 1028
Nitric acid, 1218
Vitriol.., . 1841
WOODS.
Cork, 246
Poplar, 383
Larch, 544
Elm and new English fir,556
Mahogany, Honduras,- -560
Willow, 585
Cedar, 596
Pitch pine, 560
Pear tree, 661
Walnut, 671
Fir, forest, 694
Elder, 695
Beech, 696
Cherry tree, 715
Teak, 745
Maple and Riga fir, 750
Ash and Dantzic oak, "760
Yew, Dutch, 788
Apple tree, 793
Alder, 800
Yew, Spanish, 807
Mahogany, Spanish, 852
Oak, American, 872
Boxwood, French, 912
Logwood, 913
Oak, English, 970
Do. sixty years cut,- -1170
Ebony, 1331
Lignum vitae, 1333
Specific gravity of gases, that of atmospheric air being
= 1.
Hydrogen, 0-0694
Carbon, 0-4166
Steam of water, 0-481
Ammonia, 0-5902
Carburetted hydrog., 0-9722
Azote, 0-9723
Oxygen, 1-1111
Muriatic acid, 1-2840
Carbonic acid, 1'5277
Alcohol vapour, 1-6133
Chlorine, 2-500
Nitrous acid, 2-638
Sulphuric acid, 2-777
Nitric acid, 3-75
Oil of turpentine, 5-013
NOTE. The specific gravity of atmospheric air at a tem-
perature of 60 Fah. and barometric column 30 inches is
1-22 according to M. Arago, and in round numbers we may
regard water as 825 times heavier than air.
SPECIFIC GRAVITY. 185
The preceding table will be found of the utmost use in
determining the weight and magnitude of bodies.
To find the magnitude of a body from its weight :
weight of body in ounces
. r^ = content in cubic feet.
its specific grav. in table
How many cubic feet are in one ton of mahogany ?
Here 20 x 112 x 16 = 35840 ounces in a ton ; therefore.
35840
= 64 cubic feet.
oou
Had the timber been fir, then
-= 64-46 cubic feet.
ODD
Or English oak :
35840
= 36-94 cubic feet.
7/ U
To find the weight of a body from its bulk :
cubic feet x specific gravity = weight in ounces.
What is the weight of a log of larch, 14 feet long, 2
broad, and 1$ thick?
Here 2-5 X 1'25 X 14 = 43-75; then,
43-75 x 544 = 23800 ounces = 13 cwt. 1 qr. 3 Ibs. 8 oz.
What is the weight of a cast iron ball, 2 inches diameter?
Here the content of the globe will be 2 s x -5236 = 4-1888
cubic inches = -00242 feet, and then -00242 X 7271 =
17-29 ounces = 1-08 Ibs.
A bullet of lead of the same magnitude would be -00242
X 11344 = 27-44 ounces = 1-71 Ibs.
If we wish to determine the quantity of two ingredients
in a compound which they form,
Let H be the weight of the heavy body.
A, its specific gravity.
L, the weight of the lighter body.
/, its specific gravity.
C, the weight of the compound.
c, its specific gravity.
Then,
(c /) x h
W - f. -- x C = H.
(h /) x c
also,
-
16*
186 HYDI50.STAT10S. -
Ex. A mixture of gold and silver weighed 170 Ibs. and
its specific gravity was 15030 ; hence
h (by the table) == 19326. / == 10744
c = 15630 C = 170 Ibs. wherefore, by the rule,
(19326 15630) x 10744 _ 39709824
(1932ft 10744) X 15630 * 134136660 X
= -296 X 170 = 50-32 Ibs. of gold ;
consequently there will be 170 50-32 = 119-68 Ibs. of
silver.
The weight of bodies their magnitudes and also their
quantities in a compound, may thus be found by means of
a table of specific gravities ; and for the more expeditious
calculation in practice we add the following memoranda:
430-25 cubic inches of cast iron weigh one cwt., as also
397-60 of bar iron, 368-88 of cast brass, 352-41 of cast
copper, and 372-8 of cast lead.
14-835 cubic feet of common paving stone weigh one ton,
as also 14-222 of common stone, 13-505 of granite, 13-070
of marble, 64-46 of elm, 64 of Honduras mahogany, 51-65
of fir, 51-494 of beech, 42-066 of Spanish mahogany, and
36-205 of English oak.
For wrought iron square bars, allow 100 inches in length
of an inch square to a quarter of a cwt.
A similar cast iron bar would require 9 feet in length for
a quarter of a cwt. One foot in length of an inch square
bar weighs 3-|- Ibs. also the breadth and thickness being
taken in, th of an inch, and the length in feet.
length x breadth X thickness X 7
= = the weigh!
in avoirdupois pounds.
Ex. An iron bar 10 feet long, 3 inches broad, and 2j
thick. Here 3 inches = 24, and 2 5 = 20-8ths ; therefore,
10 X 24 x 20 x 7 .
= 233 Ibs.
144
For the weight of a cast iron pipe :
The length being taken in feet, the diameter and thick
ness of metal in inches, then we have
0-0876 X length x thickness x (inner diameter 4
thickness) = the weight in cwts.
For a leaden pipe the rule is,
0-t382 X length X thickness x (inner diameter -f
thickness) = the weight in cwts.
SPECIFIC GRAVITY. 187
NOTE. The weight of a cast iron pipe is to a leaden
pipe of the same dimensions nearly as 7 is to 11.
Ex. If the inner diameter or bore of a cast iron pipe
be 3 inches, and its thickness $ of an inch ; what is the
weight of 14 feet of it?
0876 X 14 x $ x (3 + ) = -99645 cwt. = 3 qrs.
27 Ibs. 9 oz.
A leaden pipe is 12 feet long, the bore is 4 inches, and
thickness of metal | of an inch ; therefore,
1382 X 12 X * X (4 + 4-) = 1-762 cwt. = 1 cwt. 3 qrs. 1 Ib.
For the weight of the rim of a fly-wheel. Let D be the
diameter of the fly, exclusive of the rim, taken in inches ;
then take the difference of this and the diameter of the fly,
including the rim, and call this difference d, T being the
thickness of the rim of the fly, from side to side, then W3
have
0073 x T x d x (D + cT) = the weight of the rim in cwts,
Ex. If the interior diameter of the fly be 100 inches
= D, half the difference of the exterior and interior dia-
meter 5 = c/, hence if the rim is 10 inches broad, as the
exterior diameter will then be 110, and let the thickness
of the rim be 4 inches = T, then,
0073 < 4 X 5 X (100 + 5) = 15-33 cwts.
188
HYDROSTATICS.
TABLE A.
Of the weight of 1 lineal foot of Swedish iron, of all breadths and
thicknesses, from 1 tenth of an inch to 1 inch, in pounds and deci'
mal parts.
1
2
3
4
5
6
7
8
270
541
9
1-0
lOths of
inches.
034
068
101
135
169
203
237
304
338
1
135
203
270
338
406
473
608
676
2
304
406
507
609
710
811
913
1-014
3
541
676
811
947
1-082
1-217
1-352
4
845
1-014
1-183
1-352
1-521
1-690
5
1-217
1-420
1-623
1-826
2-029
6
1-657
1-893
2-130
2-367
7
2-164
2-434
2-657
8
2-739
3-043
9
3-381
1-0
TABLE B.
Of the weight of 1 lineal foot of Swedish iron, of all breadths and
thicknesses, from 1 inch to 6 inches, in pounds and decimal parts.
1 u 14 1 11
2
1
3
3*
4
5
6
in.
3-38 4-23 5-07 5-91
6-76
8-45
10-14
11-83
13-52
16-91 20-29
1
5-29 6-34, 7-40
8-45
10-56
12-68
14-79 16-91
21-13 25-36
1*
7-60 8-87
10-14J12-67
15-21
17-75 20-29
25-36
30-43
35Td
ii
i*
(10-35
11-83
13-52
14-78
17-75
20-71
23-67
29-58
i
16-91
20-29
23-67
27-05
33-81
40-51
2
21-13
25-36
29-58
35-50
33-81
42-26
50-72
2*
30-43
40-57
50-72
60-86
3
41-42
47-34
59-1 6 [71-00
3*
54-10
67-62
81-14
4
5
T
84-52
101-44
121-72
SPECIFIC GRAVITY.
189
TABLE C.
Of the weight of 1 superficial foot of Swedish iron plate from IQbth
part of an inch thick to one inch.
Thickness in
parts of an inch.
Weight in Ibs.
Thickness in
parts of a a inch.
Weight in Ibs.
01
406
1
4-057
02
811
2
8-114
03
1-217
3
12-172
04
1-623
4
16-232
05
2-029
5
20-286
06
2-434
6
24-344
07 ..
2-840
7
28-401
08
3-246
8
32-458
09
3-651
9
36-516
10
4-057
1-
40-573
TABLE D.
Of Multipliers for the other Metals, whereby their weights may bf
found from the above Tables.
Metals.
Multi-
pliers.
Metals.
Multi-
pliers.
Platinum, laminated
2-846
2-503
2-486
2-47
1-457
1-350
1-344
1-136
1-132
Copper, cast
1-128
1-096
1-080
1-003
1-
980
925
960
937
Brass wire
Pure gold, hammered
Steel
Iron, Swedish
Pure silver, hammered
- cast i
Pewter
, hammered . .
Tin, cast
190
HYDROSTATICS.
TABLE E.
Table of the weight of one square foot of different rt.etals in various
thicknesses, in pounds and decimal parts.
Thickness
in 16tlis of
ninch.
Mai. Iron.
Swed.
Mai. Iron,
English.
Cast Iron.
Copper.
Bns,
Lead.
1
2-535
2-486
2-345
2-860
2-738
3-693
2
5-070
4-972
4-690
5-720
5-476
7-386
3
7-605
7-458
7-035
8-580
8-214
11-079
4
10-140
9-944
9-380
11-440
10-952
14-772
5
12-675
12-130
11-725
14-300
13-690
18-465
6
15-216
14-916
14-670
17-160
16-428
22-158
7
17-851
17-402
16-415
20-020
19-166
25-851
8
20-280
19-888
18-760
22-880
2t;904
29-544
9
22-815
22-774
21-105
25-740
24-642
33-237
10
25-350
24-260
23-450
28-600
27-380
36-930
11
27-885
26-746
25-795
31-460
30-118
40-623
12
30-410
29-232
28-140
34-320
32-856
44-316
13
32-945
31-718
30-485
37-180
35-594
48-009
14
35-480
34-204
32-880
40-040
38-332
51-702
15
38-015
36-690
35-225
42-900
41-170
55-405
16
40-550
39-176
37-570
45-760
43-908
59-098
TABLE F.
Tabh of the weight of 1 foot in length ofmal/eable Iron rod, from
one-fourth to 6 inches diameter.
Diam.
Weight.
Diam.
Weight.
Diam.
Weight.
Diam.
Weisht.
Inch.
)bs.
Inch.
Ibs.
Inch.
Ibs.
Inch.
Ibs.
*
163
U
8-01
31
27-65
41
59-06
1
368
U
9-2
3|
29-82
41
62-21
5
654
2
10-47
ai
32-07
5
65-45
1
1-02
2
11-82
31
34-4
51
68-76
n
J
1-47
2|
13-25
31
36-81
51
72-16
i
2
2|
14-76
3i
39-31
51
75-63
1
2-61
2.1
16-36
4
41-89
5^
79-19
i*
3-31
2|
18-03
4*
44-54
5S
82-83
H
4-09
21
19-79
4*
47-28
51
86-56
if
4-94
21
21-63
4|
50-11
51
90-36
ii
5-89
3
23-56
4
53-01
6
94-25
u
6-91
3
25-56
4j
56
SPECIFIC GRAVITF.
.91
TABLE G.
Jbble of int. weight of cast iron Pipes, 1 foot long, und of different
thicknesses.
Diam. of
bore.
i
Inch.
i
Inch.
Inch.
Inch.
1
Inch.
I
Inch.
1
Inch. '
Inch.
Ib
Ib.
Ib.
Ib.
Ib.
Ib.
Ib.
1
3-06
5-06
7-36
9-97
12-89
16-11
19-63
U
3-68
5-98
8*- 59
11-51
14-73
18-25
22-09 '
4-29
6-9
9-82
13-04
16-56
20-4
24-54
H
4-91
7-83
11-05
14-57
18-41
22-55
27-
2
5-53
8-75
12-27
16-11
20-25
24-7
29-45
21
6-14
9-66
13-5
17-64
22-09
26-84
31-85
Si
6-74
10-58
14-72
19-17
23-92
28-93
34-36
2.?
7-36
11-5
15-95
20-7
25-71
31-14
36-81
3
7-98
12-43
17-18
22-19
27-62
33-29
39-28
31
8-59
13-34
18-35
23-78
29-45
35-44
41-72
3d
9-2
14-21
19-64
25-31
31-3
37-58
44-18
31
9-76
15-19
20-86
26-85
33-13
39-73
46-63
4
10-44
16-11
22-1
28-38
34-98
41-88
49-1
41
11-1
17-08
23-37
29-97
36-87
44-08
51-6
4
11-66
17-94
24-54
3 1 -4 4
38-65
46-17
54-
4!
12-27
18-87
25-77
32-98
40-5
48-32
56-45
5
12-80
19-78
26-99
34-51
42-33
50-46
59-
51
13-5
20-71
28-23
36-05
44-18
52-62
61-36
5i
14-11
21-63
29-45
37-58
46-02
54-76
63-81
C 3
5?
14-73
22-5,5
30-68
39-12
47-86 56-91
66-27
6
15-34
23-47
31-91
40-65
49-7
59-06
68-73
61
15-95
24-39
33-13
42-18
51-54
61-21
72-
61
16-57
25-31
34-36
43-72
53-39
63-36
73-41
6f
17-18
26-23
:<r>-r>!)
45-26 55-23
65-28
76-1
7
17-79
27-15
36-82
46-79
56-84
67-65
78-53 ;
71
18-41
28-08
38-05
48-1
58-91
69-79
81- !
71
19-03
29-
39-05
49-86 60-74
71-95
83-45
71
19-64
29-69
40-5
51-38 G2-r>9
74-09
86- :
8
20-02
30-83
41-71
52-92 61-42
76-23
88-35 ;
. 'Q >
. S 4
20-86
31-74
42-95
54-45
'66-26
78-38 90-81
8.|
21-69
32-9
44-4
56-21
68-33
80-76 93-49
* 8|
22-09
33-59
45-4
57-52
69-95
82-68 95-72
ft
22-71
34-52
46-64
59-07
71-8
84-84
98-18'
192
HYDROSTATICS.
Diini.
of bore.
iL.
Inch.
Inch.
Inch.
Inch.
f
Inch.
Inch.
Inch.
lb.
lb.
lb.
lh.
11).
lb.
lb.
8i
23-31
35-43
47-86
60-59
73-63
86-97
100-63
9*
23-93
36-36
49-09
62-13
75-47
89-13
103-1
9|
24-55
37-28
50-32
(j:Mi6
77-32
91-28
105-54
10
25-16
38-2
51-54
65-2
79-16
93-42
108
10i
25-77
39-11
52-77
66-73
80-99
95-57
110-44
10*
26-38
40-04
54
68-26
82-84
97-71
113
10|
27
40-96
55-22
69-8
84-67
99-86
115-35
11
27-62
41-88
56-46
71-33
b6-,";2
102-01
117-81
Hi
38-22
42-8
57-67
72-86
88-35
104-15
120-26
"i
28-84
43-71
58-9
74-39
90-19
106-3
122-71
11*
29-45
44-64
60-13
75-93
92-04
108-45
125-18
12
30-06
45-55
61-35
77-46
93-6
110-6
127-6
Diam.
of bore.
Inch.
Inch.
Inch.
1
Inch.
H
Inch.
U
Inch.
li
Inch.
1*
Inch.
2
Inch.
Inch.
lb.
lb.
lb.
lb.
lb.
lb.
lb. "
lb.
lb.
12*
63-5
97-3
114
132
149
167
205
243
285
13
66
101
118
137
154
173-5
212
252
294
18*
68-4
104-8
122
141-5
160
179
219
260
304
14
75
108-2
126
146
165
185
227
269
314
'*!
73-4
112-3
130
151
170
192
234
277
324
15
75-8
115-7
135
156
176
198
242
286
334
15*
78-1
119
139
.161
181
204
250
295
344
16
80-7
123
143
166
187
211
257
303
355
16*
83-1
126-5
147
170-1
192
217
264
312
363
17
85-5
130
152
178-5
198
223
271
322
376
17*
87-8
133-5
157
180-5
203
229
278
330
383
18
90-5
137
161
185
209
235
285
338 '
393
18*
93
140-5
166
190
217
241
293
347
402
19
95-5
144-8
169
195
222
247
300
354
412
19*
97-8
148-5
174
200
227
253
307
363
422
20
100
152
178
205
233
259
315
372
432
| 20*
102-5
156 ': 183
210
238
26
323
381
442
The following TABLE of the weight of different sub-
stances used in building and engineering requires no ex
planation.
SPECIFIC GRAVITY.
193
NamMofBodim.
Weight of a
ubic foot in
ounces.
VfeifM of a
cubic foot in
p'.Ulllt.
Wright of a
cubic inch in
ounces.
Weight of a
cubic inch in
(ouil'l-.
Nuabcr oT I
cubic inche* uu
, a pound. 71
Copper, cast....
Copper, sheet
8788
8915
8396
7271
7631
11344
7833
7816
7190
7292
9880
8784
1520
1250
2000
2416
2672
2742
3160
2880
945
240
544
556
660
696
745
760
852
970
870
915
932
927
1000
1028
1015
1026
J13568
549-25
557-18
52 -1-7 5
151-43
47(>-i)3
709-00
489-56
488-50
449-37
455-75
619-50
649-00
95-00
78-12
125-00
151-00
167-00
171-37
197-50
180-00
59-00
15-00
34-00
34-75
41-25
43-50
46-56
47-50
53-25
60-62
54-37
57-18
58-25
57-93
62-50
64-25
63-43
64-12
848-00
5-086
5-159
4-852
4-203
4-410
6-456
1-527
4-517
4-156
4-215
5-710
5-0775
8787
7225
1-156
1-396
1-544
1-585
1-826
i-r.fU
5462
138
315
321
382
403
431
440
493
561
503
529
539
536
578
594
587
593
7-851
3178
3225
3037
263
276
4103
2833
2827
26
2636
3585
3177
055
0452
0723
0873
0967
0991
1143
1042
0342
0087
0197
0201
024
0252
027
0275
0308
0351
0315
0331
0337
03352
03617
0372
0367
037
4908
3-146
3-103
3-293
3-802
3-623
2-437
3-530
3-537
3-845
3-790
2-789
3-147
18-190
22-120
13-824
11-443
10-347
10-083
8-750
9-600
29-258)
115-200!
50-823
49-726
41-890
39-724
37-113
36-370
32-449
28-505
31-771
30-220
29-665
29-288
27-648
26-894,
27-24$
26-949
2-037
Steel, soft
Coal
Stone, paving
\Tirh1p. ..
rjiocc .
Pnrk .
F,l m
Pine, pitch
Teak
AVi . .
Mahogany
Oak .
Oil of turpentin
Linseed oil ...
Spirits, proof--
- Water, distilled
Tir . .
17
194 HYDROSTATICS.
The foregoing tables and rules will be found of the ut-
most service, in the ready calculation of the weight of
materials commonly used in engineering.
What is the weight of a bar of Swedish iron 16. feet long,
3 inches broad, and !! inch thick?
By table B, 3-38 is the weight of a piece of Swedish
iron, of one foot long and one inch square, wherefore,
3-38 x 16 X 3 = 162-24; and then for the fraction -1,
in table A, we have for the weight of 1 foot by ! of an
inch square = -034; hence, -034 x 3 x 16 = 16-32;
wherefore the sum of the two = 162-24 + 16-32 = 178-56
Ibs., the weight/
If we wish the weight of an equal bar of cast iron, we
must employ the multipliers in table D ; hence,
178-56 X '925 = 165-168.
If we wished it for lead, the multiplier from the same
table being 1-457, we have,
178-56 x 1-457 = 260-1619 Ibs., &c., &c,
Then if lead were 1 penny per pound, the price of such
a bar would be
The following practical rules are often useful and may
be easily remembered.
For round bars of iron,
diameter (m) 3 X length in ft. X 2-6 = weight of
wrought iron in Ibs.
diameter (m) a X length in ft. x 2-48 = weight of
cast iron bars in Ibs.
A cylindrical bar is 2 inches diameter and 29 inches long,
therefore, 2 s x 2-5 x 2-6 = 26 Ibs. if it be wrought iron,
but if cast, 2 s x 2-5 x 2-48 = 24-8 Ibs.
Multiply the sum of the exterior and interior diameters
of a cast iron ring by the breadth and thickness of the rim,
and also by 0074, he results will be the weight in cvvts.
HYDRODYNAMICS.
As hydrostatics embraces the consideration of fluids at
rest, so hydrodynamics or hydraulics comprehends the cir-
cumstances of fluids in motion. Of this science, little,
comparatively speaking, is yet known ; but as it is of the
utmost importance to man, we will endeavour to lay before
our readers a statement of the more important results of
recent inquiry into it.
If a fluid move through a pipe, canal, or river, of various
breadths, always filling it, the velocity of the fluid at dif-
ferent parts will be inversely as the transverse sections of
these parts.
Thus let there be a canal, AB, of various breadths at
different places, then will the velocity in the portion ab be
to that of the velocity in erf, as the area of the cross section
at cd is to that at ab, and the velocity at ef will be to that
at cd as the area at cd is to the area at ef, being always in
inverse proportion.
Suppose the velocity at ab 10 feet per second, and the
area there 100 feet, then if the area at cd be 25 feet, we
have 25 : 100 : : 10 : 40 feet, the velocity of the water
at erf; and if the area at efbe 50 feet, then 50 : 25 : : 40
: 20 feet, the velocity at ef, the canal being kept con-
tinually full.
The quantity of water that flows through a pipe, or in a
canal or river, at any part, is in proportion to the area
multiplied by the velocity at that part.
The calculation of the motion of rivers is often of the
hig/iest utility to the engineer. This is sometimes done by
Jhe employment of very intricate formulas, but such
methods, if easier could be found, would evidently be in-
195
196 HYDROl/rNAMICS.
consistent with the nature of this work. The method which
we shall give is simple, and will be found to answer all the
purposes of the practical man.
In a river, the greatest velocity is at the surface and in
jthe middle of the stream ; from which it diminishes toward
the bottom and sides, where it is least.
The velocity at the middle of the stream may be ascer
tained, by observing how many inches a body floating with
the current passes over in a second of time. Gooseberries
will fit this purpose exceedingly well; if they are not at
hand, a cork may be employed.
Take the number of inches that the floating body passes
over in one second, and extract its square root ; double this
square root, subtract it from the velocity at top, and add
one, the result will be the velocity of the stream at the
bottom.
And these velocities being ascertained, the mean velocity,
or that with which if the stream moved in every part,
it would produce the same discharge, may be found =
the velocity at top v/velocity at top + '5.
Exam. If the velocity at the top and in the middle of
the stream, be 36 inches per second, then, 36 (2 X \/36)
-f 1 =36 12 + 1 = 25 = the least velocity, or the
velocity at bottom. And the mean velocity will be =
36 v/36 -f -5 = 36 6 + -5 = 30-5.
When the water in a river receives a permanent increase
from the junction of some other river, the velocity of the
water is increased. This increase in velocity causes an in-
crease of the action of the water on the sides and bottom,
from which circumstance the width of the river will always
be increased, and sometimes, though not so frequently, the
depth also. By the reason of this increased action of the
water on the bottom, the velocity is diminished until the
tenacity of the soil or the hardness of the rock afford a
sufficient resistance to the force of the water. The bed of
the river then changes only by very slow degrees, but the
bed of no river is stationary.
It is of the greatest use to know the amount of the action
of any stream on its bed, and for this purpose a knowledge
of the nature of the bed and of the velocity at bottom, are
absolutely necess-ary.
Every kind of soil has a certain velocity which will insure
the stability of the bed. A less velocity would allow the
HYDRODYNAMICS. 197
waters to deposit more of the matter which is carried with
the current, and a greater velocity would tear up the chan-
nel. From extensive experiments it has been found, that
a velocity of 3 inches per second at the bottom, will just
begin to work upon the fine clay used for pottery, and,
however firm and compact it may be, it will tear it up.
A velocity of 6 inches will lift fine sand 8 inches, will lift
coarse sand (the size of linseed) 12 inches, will sweep
along gravel 24, will roll along pebbles an inch diameter
and 3 feet at bottom, will sweep along shivery stones the
size of an egg.
When water issues through a hole in the bottom or side
of a vessel, its velocity is the same as that acquired by a
body falling through free space from a height equal to that
of the surface of the water above the hole.
The most correct rule for ascertaining the velocity of
water running through pipes and canals is this :
57 X height of head X diam. of pipe\ 1
Vlength of pipe X 57 X diam. of pipe/
the velocity in inches with which the water will issue from
the orifice. All the measures are understood to be taken in
inches.
Exam. If there be a reservoir of water whose depth is
6 feet, having a tube 1 foot long and 2 inches bore, open
so as to let the water escape at a distance of 18 inches from
the bottom, then we have, 6 X 12 = 72 = whole depth
of water on the reservoir, and 72 18 = 54, the height
of the head of the fluid above the orifice, wherefore by the
rule,
v' (4-5) X 231 = 2-121 X 23s = 49-49 inches per second,
the velocity of the water. And, by multiplying this result
by the area of the orifice, we get the quantity discharged in
one second hence, if the pipe be circular, we have,
2-5 2-5 x 3-1416 .
= 1-25 = radius, and = naif
^
tircu inference = 1-9635 = area of orifice, hence, 49-49 X
1-9635 = 97-173 cubic inches.
The* quantity of water that flows out of a vertical rectan-
gular aperture, that reaches as high as the surface, is | of
17*
198 HYDRODYNAMICS.
the quantity that would flow out of the same aperture,
placed horizontally at the depth of the tiase.
When water issues out of a circular aperture in a thin
plate placed on the bottom or side of a reservoir, the stream
is contracted into a smaller diameter, to a certain distance
from the orifice. The vein is smaller at the distance of half
the diameter of the orifice where the area of the section .of
the vein is {-{? of that of the orifice, and at the above point
the stream has the velocity given by theory, so that to ob-
tain the quantity of water discharged, we multiply the
velocity by the area of the orifice, and { of this will be
the true result. -When the water issues through a short
tube, the vein of the stream will be less contracted than in
the former case, in the proportion of 16 to 13. But when
the water issues through an aperture which is the frustum
of a cone, whose greater base is the aperture, the height of
the conic frustum = one half the diameter of the aperture
and the area of the small end to that of the large end, as
10 : 16 ; then, in this case, there will be no contraction of
the vein ; and from this it may be inferred, that, when a
supply of water is required, the greatest possible from a
given orifice, this form should be employed.
To determine the quantity of water discharged by a
small vertical or horizontal orifice, the time of discharge,
and the height of the fluid in the vessel, when any two
of these quantities are known.
Let A represent the area of the small orifice, W the
quantity of water discharged ; T the time of discharge, H
the height of fluid in the vessel, and # = 16-087 feet, the
space described by gravity in a second.
Then we have,
W = 2 x A x * N/# X H
4.
2
X
t
X
W
x H
w s
H =
4 x g X t 2 X A 3
By means of these formulas we may determine the quan-
tity of water W which is discharged in the same time T,
from any other vessel in which A' is the area of the orifice,
HYDRODYNAMICS.
199
and H the altitude of the fluid ; for since t and g are con
etant, we shall have
W : W'= A,/ H : A' -,/ H'.
Table shou'iag the quantify of JTater discharged in one
Minute by Orifices differing in form and position.
aslant
H( a iL-!it of the
Fluiil above
the centre of
the orifice.
Form and position of the Orifice.
Diameter of
the orifice.
No. of cubic
inches dis-
charged in a
minute.
Ft. in. lin.
Lines.
11 8 10
Circular and Horizontal,
6
2311
Circular and Horizontal,
12
9281
Circular and Horizontal,
24
37203
Rectangular and Hori-
-
zontal,
12 by 3
2933
Horizontal and Square,
12 side
11817
Horizontal and Square,
24 side
47361
900
Vertical and Circular,
G
2018
Vertical, and Circular,
12
8135
400
Vertical and Circular,
6
1353
Vertical and Circular,
12
5436 ,
507
Vertical and Circular,
12
628
From these results we may conclude,
1. That the quantities of water discharged in equal times
by the same orifice from the same head of water, are very
nearly as the areas of the orifices ; and,
2. That the quantities of water discharged in equal times
by the same orifices under different heads of water, are
nearly as the square roots of the corresponding heights of
the water in the reservoir above the centres of the orifices.
3. The quantities of water discharged during the same
time by different apertures under different heights of water
in the reservoir, are to one another in the compound ratio
of the areas of the apertures, and of the square roots of
'the heights in the reservoirs.
This general rule may be considered as sufficiently cor-
rect for ordinary purposes ; but, in order to obtain a great
Jflegree of accuracy, Bossut recommends an attention to
he three following rules.
I. Friction is the cause, that, of several similar orifices
200
HYDRODYNAMICS.
the smaLest discharges less water in proportion than those
which are greater, under the same altitudes of water in the
reservoir.
2. Of several orifices of equal surface, that which has the
smallest perimeter ought, on account of the friction, to give
more water than the rest, under the same altitude of water
in the reservoir.
3. That, in consequence of a slight augmentation which
the contraction of the fluid vein undergoes, in proportion
as the height of fluid in the reservoir increases, the expense
ought to be a little diminished.
Table of Comparison of the Theoretic with the Real dis
charges from an orifice one inch in diameter.
Constant
height of the
water in the
reservoir
above the
centre of the
orifice.
Theoretical dis-
charges through
acircular orifice
one inch in dia-
meter.
Real discharges
in the same time
through the
same orifice.
Ratio of the theoretical
to the real discharges.
Paris feet.
Cubic inches.
Cubic inches.
1
4381
2722
1 to 0-62133
2
6196
3846
1 to 0-62073
3
7589
4710
1 to 0-62064
4
8763
5436
1 to 0-62034
5
9797
6075
1 to 0-62010
6
10732
6654
1 to 0-62000
7
11592
7183
1 to 0-61965-
8
12392
7672
1 to 0-61911
9
13144
8135
1 to 0-61892
10
13855
8574
1 to 0-61883
11
14530
8990
1 to 0-61873
12
15180
9384
1 to 0-61819
13
15797
9764
1 to 0-61810
14
16393
10130
1 to 0-61795
15
16968
10472
1 to 0-61716
1
2
3
4
It appears from this table, that the real as well as the
theoretical discharges are nearly proportional to the square
roots of the heights of the fluid in the reservoir. Thus
for the heights 1 and 4, whose square roots are as 1 to 2
feet, the real discharges are 2722 and 5436, which are to
one another as 1 to 1-997, very nearly as 1 to 2.
HYDRODYNAMICS. SOi
Let it be required to determine the quantity of water dis-
charged from an orifice of 3 inches in diameter, under an
altitude of 30 feet. Then, since the real quantities dis-
charged are in the compound ratio of the orifices, and the
square roots of the altitudes of the water, and since the
theoretical discharge by an orifice 1 inch in diameter, under
an altitude of 15 feet is 16968 cubical inches in a minute,
we have 1 ^/ 15 : 9 v/ 30 = 16968: 215961, the theoreti-
cal discharge. But the theoretical is to the real discharge
as 1 to -62, the above value being diminished in that ratio,
gives 133309 cubic inches for the real quantity, of water
discharged by the orifice.
The following formulae have been given by M. Prony
as deduced from the preceding experiments of Bossut,
Q = 0-61938 AT ^/ 2 g H,
A being the area of the orifice in square feet, H the altitude
of the fluid in feet, T the time, g the force of gravity at
the end of a second, and Q the quantity of water in cubic
feet. As ^/ 2 g is a constant quantity, and is equal to
7*77125, we have
Q = 4-818 AT */ H for orifices of any form.
If the orifices are circular, and if d represents their dia-
meter, then
Q = 3-7842 d* T </ H.
From the second of these equations we obtain
A Q
Till T </ H
T. 5
4-818 A v/ H
H Q
(4-818 AT)*
These formulae will be found to give very accurate r*
suits ; but if we wish to obtain a still higher degree of ac-
curacy, we must not use the mean co-efficient 0*6194, but
the one in the table which comes nearest to the circum-
stances of the case. Thus if the head of water happens tc
be small, such as 1 foot, then we must take the co-efficient
fl-62133, and if it happens to be great, we must use the least
'co-efficient 0-61716.
202
HYDRODYNAMICS.
Table containing the quantity of Water discharged over a weir.
Depth of the up-
per edije of the
wasteboard below
the surf-tee in Eng-
lish inches.
Cubic feet of water
discharged in a minute
b_y every inch of the
wasteboard.accorJii.g
to Du Buat's formula.
Cubic fe t of water dis-
board ace >rding to expe-
riments ir adu in Scotland.
1
0,403
0,428
2
1,WO
1,211
3
2,095
2,226
4
3,225
3,427
5
4,507
4,789
6
5,925
6,295
7
7,466
7,933
8
9,122
9,692
9
10,884
11,564
10
12,748
13,535
11
14,707
15,632
12
16,758
17,805
13
18,895
20,076
14
21,117
22,437
15
23,419
24,883
16
25,800
27,413
17
28,258
30,024
18
30,786
32,710
Talk containing the quantities of Water discharged by Cylindrical
Tubes one inch in diameter and of different lengths, whether the
Tubes were inserted in the bottom or in the sides of the vessel.
Constant altitude of the fluid above the superior base of the tube
11 feet 8 inches and 10 lines.
Lengths of the Tubes expressed in
lines.
charged in a minute.
The tube filled with the \ jj
issuing fluid f
J iy
The tube not filled Avith -
the issuing fluid 3
12274
12188
12168
9282
IIYDKODVNAMICS.
203
Table, of comparison of lite Theoretical u-itft the Rial Dixtliurgefi from
an additional Tube of a cylindrical form, one inch in diameter ana
tico inches long.
foment alli-u.lt
TJieorrtical dis-
Ri-il ilivhargei in the
it the Wi-n in
chir^.5 'hrnn^li a
* ilnc: tinir l.y > cy-
Ratio of the throrcticil
itHivr !(]< (t, n.
circular orifice one
nu h ID diameter.
in iliiiin-'ir .iii.l livo
to the real discharge*
of the i<n&ce.
iuchlong.
t tris ft-ll.
Cubic- inrhn.
Cubic iii.-lii-s.
1
4381
3539
1 to 0-81781
2
0190
5002
1 to 0-80729
3
7589
6126
1 to 0-80724
4
8763
7070
1 to 0-80681
5
9797
7900
1 to 0-80638
6
10732
8654
1 to 0-80638
7
11592
9340
1 to 0-80573
&
12392
9975
1 to 0-80496
9
13144
10579
1 to 0-80485
10
13855
11151
1 to 0-80483
11
14530
11693
1 to 0-80477
12
15180
12205
1 to 0-80403
13
15797
12699
1 to 0-80390
14
16393
13177
1 to 0-80382
15
16968
13620
1 to 0-80270
1
2
3
4
Hence it follows, that the velocity in English inches will
be V = 22-47 v/ H for additional tubes.
M. Prony has given the following formulae, as deduced
from the preceding table.
\ 4-9438 'I
4-9438 T v/ H
H- Q
~ (4-9438 d* T) a
The resistance that a body sustains in moving through a
fluid is in proportion to the square of the velocity.
The resistance that any plane surface encounters in rnov-
irfg through a fluid with any velocity, is equal to the weight
of a column whose height is the space a body would have
'J04 HYDRODYNAMICS.
to fall through in free space to acquire that velocity, and
whose base is the surface of the plane.
Ex. If a plane 16 inches square, move through water at
the rate of 13 feet per second ; then,
13 2
64~ =
the space a body would require to fall through free space to
acquire a velocity of 13 per second, wherefore, as 2'6 feet
=31-2 inches, we have 16 x 31-2 = 499-2 cubic inches=
the column of matter whose height and base are required ;
therefore, since 1728 cubic inches = 1 cubic foot of water
weighs 1000 ounces, we have 1728 : 499-2 : : 1000 : 288
ounces = 18 Ibs. which is the amount of resistance met
with by the plane at the above velocity.
As action and reaction are equal and contrary, it is the
same thing whether the plane moves against the fluid, or
the fluid against the plane.
WATER WHEELS.
MOTION is generally obtained from water, either by ex-
posing obstacles to the action of its current, or by arresting
its progress during part of its descent, by movable buckets.
Water-wheels have three denominations depending on
their particular construction, undershot, breast, and over-
shot. If the water is to act on the wheel by its weight, it
is delivered from the spout as high on the wheel as possible,
that it may continue the longer to press the buckets down ;
but when it acts on the wheel by the velocity of the stream,
it is made to act on the float-boards at as low a point as
possible, that it may have acquired previously the greatest
velocity. In the first case, the wheel is said to be overshot,
in the second, undershot. The overshot wheel is the most
advantageous, as from the same quantity of water it gives a
greater power, but it is not always that we can employ an
overshot wheel from the smallness of the fall. When this
is the case, we must deliver the water farther down than
the top of the wheel, and, in this case, it becomes a breast-
wheel, and partakes in some degrees of the properties of
the overshot. When we cannot employ a breast wheel,
we must have recourse to the undershot, which is the least
WATER WHEELS.
205
powerful of all. The force of a stream of water against
the floats of an undershot wheel is equal to a column of
water, whose base is the section of the stream in that place,
and height the perpendicular height of the water to the
surface. Where the quantity of water is given, its force
against the floats of the wheel is directly proportional to
the velocity of the stream, or the square root of the heigh!
of the surface. These remarks hold true only when the
water is allowed to escape from the float boards, after it
has struck them. For if the floats be too near each other,
thcnihe water from one float will be sent back and obstruct
the progress of the next float.
Engraved representations of the three forms of the water
wheel are given in plate 1st. Fig. 1 is a representation
of the undershot ; fig. 2 of the breast ; and fig. 3 of the
overshot water wheel. The floats of the undershot as
likewise of the breast wheel are flat, those of the latter
being fitted so nearly to the water way that little of the
fluid is allowed to escape between their edges and the stone
or brick work, as may be seen in the figure. The over-
shot wheel is furnished with buckets instead of floats, so
constructed that they shall retain as much as possible of
the water from the time they receive it until they arrive al
the lowest point, where each bucket should be emptied,
since if any water be carried by the bucket in its ascent it
will be just so much unnecessary weight that the wheel has
to lift. The following geometrical construction will show
the method of forming the buckets so that there shall be the
greatest possible advantage derived from the overshot wheel.
This bucket is formed of three planes ;
AB is in the direction of the radius of
the wheel, and is called the start, or
shoulder ; BC is called the arm, and CH
the u'ritf. These buckets are so con-
structed, that when AB makes an angle
of 35 with the vertical diameter of the
.wheel, the line AD is horizontal ; and the
area of the figure ADCB is equal to that
tofFCBA; so that as much water is re-
tained in the bucket in this position as would fill FCBA ;
the whole of the water is not discharged until CD becomes
horizontal, which takes place when the line AB is very
near the lowest point.
18
206 HYDRODYNAMICS.
To find the velocity of the water acting upon the wheel,
^(height of the fall x 64-38) the velocity in feet per
second.
Ex. If the height of the fall be 14 feet, then we have
*/(14x64-38) = .v/ 901-32=30-02 feet per second, nearly.
To find the area of the section of the stream,
The number of feet flowing in 1 second
velocity in feet per second
the section of the stream in square feet.
Ex. If there be 40 feet flowing in a second, and the
velocity of the stream is 5 feet per second, then,
i=8 =
5
the area of the section of the stream in square feet.
To calculate the power of the fall :
Area of section of stream where it acts upon the wheel X
height of fall x 62| = the number of Ibs. avoir, the wheel
can sustain, acting perpendicularly at its circumference, so
as to be in equilibrium. If this number of Ibs. which keeps
the wheel at rest be diminished, the wheel will move.
If the wheel move as fast as the stream, it is clear that
the water would have no effect in moving it, if the wheel
were to move faster than the stream, the water would -be a
positive hindrance to its motion ; and it can only be ad-
vantageous when the velocity of the stream is greater than
that of the wheel. There is a certain relation between the
velocity of the wheel and that of the stream, at which the
effect will be the greatest possible or a maximum.
The effect of an undershot wheel is a maximum when
the velocity of the wheel is 5 of the velocity of the stream.
Ex. If the area of the cross section of a stream be 6
feet, and its velocity 4 feet per second, and a fall of 16 feet
can be procured, then we have 4x6=24, the number of
cubic feet flowing per second :
v/ (16x64-38)=32, the velocity of the water at the end
of the fall : .
= I, the section of the stream at the end of the fall in
o2
square feet :
| X 16 X 62| = 750 lbs.= the weight which the wheel
will sustain in equilibrium
WATER WHEELS. 207
Now, the effective velocity of the stream is the difference
between the velocities of the stream and wheel, and the
wheel's velocity being 3 of that of the stream, the difference
or effective velocity will be ; now, the power of the
stream is as the square of the effective velocity, and the
square of f is -J. We must multiply the power of the fall
as above calculated by this , and also by 5, in order that
the wheel may move with the proper velocity ; hence, 750
X xl = lll' Ibs. raised through 10| feet per second, the
velocity of the wheel, which is 5 of 32 the velocity of the
stream. An undershot water wheel is capable only of rais-
ing 2^- of the weight of the water to the height of the fall.
From numerous experiments on water wheels, it has been
found, that in practice the water not being allowed to es-
cape from the floats immediately after it has impinged upon
them, the maximum effect is, when the velocity varies be-
tween j and I, that of the water being nearly ^. There
is another deviation from theoretical result, in consequence
of the water not being allowed to escape immediately from
the float-boards, as the water is heaped up to about 2| times
its natural height, and thus acts partly by its weight, and
partly by its force in consequence of which it happens,
that a well-constructed undershot water wheel, instead of
raising *2 4 T of the weight of the water expended on the
height of the fall, will raise 3.
The effective head being the same, the effect of the wheel
will depend on the quantity of water expended ; and the
quantity of water being the same, the 'effect of the wheel
depends on the height bf the head of the fall.
The section of the stream being the same, the effect will
be nearly as the cube of the velocity.
Overshot water wheel. If th* water in the buckets of an
overshot wheel be supposed to be equally diffused over half
the circumference of the wheel, then the whole weight of
the water in the buckets is to its power to turn the wheel
as 1 1 to 7.
, An overshot water wheel will raise nearly as much water
to the height of the fall, as is expended in driving the
tfheel : if the height of the fall be reckoned from the bucket
that receives the water to the bucket that discharges if.
According to the last experiments, the velocity of an over-
shot wheel should be between 2 and 4 feet per second 5or
208 HYDRODYNAMICS.
all diameters of wheels. A breast wheel partakes of the
properties of the two foregoing, as part of its action de
pends on the velocity, and part on the weight of the water
which moves it.
Circumstances will regulate which of these three species
of water wheels is to be employed. For a large supply of
water with a small fall, the undershot wheel is the most ap-
propriate. For a small supply of water with a large fall, the
overshot ought to be employed. Where both the quantity
of water and height of fall are moderate, the hreast wheel
must be used.
Before erecting a water wheel, all the circumstances must
be taken into account, and our calculations made accordingly.
We must measure the height of head velocity, and area of
stream, &c., to do which a slight knowledge of levelling will
be required. What follows will make this subject suffi-
ciently plain.
Levelling. A pole about 10 feet long must be procured,
:md also a staff about five feet long, on the top of which is
fixed a spirit level with small sight holes at the ends, so
that when the spirit level is perfectly horizontal, the eye
may view any object before it through the sights in a per-
fectly horizontal line. If you have to measure the perpen-
dicular distance between the bottom and top of a hill, for
instance ; place the level staff on the side of the hill in such
a way that when the level is truly set, the top of the hill
may be seen through the sights. Keep the level in this
position and look the contrary way, then cause some person
to place the 10 feet staff before the sight further down the
hill, and looking through the sights to the staff, cause the
person to move his finger up or down the staff until the
finger be seen through the sights, and mark the position of
the finger on the staff. Keep your ten feet staff in the
same place, and carry your level staff down the hill to a
convenient distance, then fix it in the same way as before;
and looking through the sights at the ten feet staff, cause
the person to bring his finger towards the bottom of the
staff, and move his finger up or down the staff in the same
way until it be seen through the sights, and mark the place
of -the finger. Then the distance between the two linger
marks added to the height of the level staff, will be the
perpendicular distance between the place where the level
staff now stands and the top of the hill. The process is
WATER WHEELS.
209
perfectly simple, and it will not be difficult to repeat it
oftener if the height of the hill requires it.
This process will give what is called the apparent level,
which however is not the true level. Two stations are on
the same true level when they are equally distant from the
centre of the earth. The apparent level gives the objects
in the same straight line, but the true level gives the line
which joins them as part of a circle whose centre is the
centre of the earth. In small distances there is no sensible
difference between the true and apparent level of any two
objects. When the distance is one mile, the true level will
be about 8 inches different from the apparent level. This
will serve well enough to remember, but more correctly
speaking it is 7*962 inches for one mile, and for all other
distances the difference of the two levels will be as the
square of the distance. Thus at the distance of two miles
u will be,
I 3 : 2 s : : 8 : 32 inches, or 2 feet 8 inches nearly.
These circumstances must be strictly observed in the
formation of canals, railways, &c., &c.
The following table will save the trouble of calculation.
The distances are measured on the earth's surface.
Distance
yards.
Allowance
in
inches.
Distance
measured in
miles.
Allowance iu
fert and
inches.
100
0-026
k
200
0-103
5
2
300
0-231
1
4
400
0-411
1
8
500
0-643
2
2 8
600
0-925
3
6
700
1-260
4
10 7
800
1-645
5 16 7
900
2-081
6
23 11
1000
2-570
7
32 6
1100
3-110
8
42 6
1200
3-701
9
53 9
1300
4^344
10
66 4
1400
5-038
11
80 3
1500
5-784
12
95 7
1600
6-580
13
112 2
1700
7-425
14
130 1
IS*
21C HYDRODYNAMICS.
Construction of a water wheel. To find the centre of
gyration of a water wheel, take the radius of the whee]
and the weight of its arms, rim, shrouding, and float
boards. Then call the weight of the rim R, which must
be multiplied by the square of the radius, and the pro-
duct be doubled and then carried out. Next the weight
of the arms called A must be multiplied by the square
of the radius, and be doubled and carried out as before.
Then the weight of the water in action called W must
be multiplied by the square of the radius and carried
out. If these products be added together into one sum
they will form a dividend. For a divisor, double the sum
of the weights of the rim and the arms, and add the weight
of the water to them. Divide the dividend by the divisor.
and the square root of the quotient will be the radius of
gyration.
Ex. In a wheel 24 feet diameter The weight of the
arms is 2 tons, the shrouding and rims 4 tons, and the
water in action 2 tons ; hence, by the above,
R = 4 tons x 12 a X 2 = 1152
A = 2 tons x 12 a x 2 = 576
W = 2 tons x 12 3 = 288
Their sum 2016 dividend, and
2 X (4 -f 2 + 2) = 16, the divisor.
The answer, J^^h = /126 = 11-225.
Tables for the more ready performance of calculations
for water wheels are usually given in books of Mechanics ;
the construction and use of which we shall now proceed to
explain.
1. Find, by measuring and levelling, the height of the
fall of water which is reckoned from its upper surface to
the middle of the depth of the stream, where it acts upon
the float-boards.
2. Find the velocity acquired by the water in falling
through that height, which is done thus : multiply the
height of the fall by 64-38, extract the square root of the
product which would be the velocity of the stream if there
were no friction, but to allow for friction take away of
this result for tke true velocity.
WATKK WHEELS. 211
3. Find the velocity that ought to be gi\en to the float-
Doards, by taking 4 f lne velocity of the water, which
product will be the number of feet the float-boards have to
pass through in one second of time to produce the maxi-
mum effect.
circumference of wheel
velocity of the float-boards
the number of seconds that the wheel takes to make one
turn.
4. Divide 60 by the last number. The quotient is the
number of revolutions the wheel makes in one minute.
5. Divide 90 by the last quotient, the new quotient is the
number of turns of the millstone for one of the wheel : 90
being the number of turns that a millstone of five feet dia-
meter ought to make in a minute.
6. As the number of turns of the wheel in a minute
Is to the number of turns of the millstone in a minute,
So is the number of staves in the trundle
To the number of teeth in the spur-wheel, avoiding
fractions.
7. The number of turns of the wheel in a minute x
the number of turns of the millstone for one turn of
the wheel = the number of turns of the millstone per
minute.
Or, by a different method, multiply the number of teeth
in the spur-wheel by the number of turns of the water-
wheel per minute, and divide this product by the number
of staves in the trundle, the quotient is the number of turns
of the millstone per minute.
In this way has the following table been constructed for
a water-wheel of 1 5 feet diameter, the millstone being 5
feet diameter and making 90 tuns in one minute.
213
HYDRODYNAMICS
A MILLWRIGHT S TABLE,
In which the Velocity of the Wheel is Three-Seve.it/is of the Velocity
of the Wafer, allowance being made fur Ihe Effects <-f Friction en
the Velocity of the Stream for a Wheel of Fifteen Feet diameter.
r
Height of
the fall of
water.
Velocity of
the watei
per second.
Velocity of
wheel per
second,
hein* It-Ttlii
of that of
the water.
Revolutions
nf Ihe
"nmutT
Numlier of
revolutions i f
for one of
UM
Uhr.l.
Ti-eth in Ihe
wheel, and
':! in Ihe
trundle.
R.-V) utions
of the mill-
minute bv
(he*catavgi
,-ind teeth.
3
- =
A * .
S'~5
J= Ji
1^1
1
2.
8
IfJ
u. g.-
8
f 3 "3
Ifi
res
c g.-
*|
3 c =
> s.?
Oj O U
PS 2 u
1
7-62
3-27
4-16
21-63
130 6
90-07
2
10-77
4-62
5-88
15-31
92 6
90-16
3
13-20
5-66
7-20
12-50
100 8
90-00
4
15-24
6-53
8-32
10-81
97 9
89-67
5
17-04
7-30
9-28
9-70
97 10
90-02
6
18-67
8-00
10-19
8-83
97 11
89-86
7
20-15
8-64
10-99
. 8-19
90 11
89-92
8
21-56
9-24
11-76
7-65
84 11
89-80
9
22-86
9-80
12-47
7-22
72 10
89-68
10
24-10
10-33
13-15
6-84
82 12
89-86
11
25-27
10-83
13-79
6-53
85 13
90-16
12
26-40
11-31
14-40
6-25
75 12
90-00
13
27-47
11-77
14-99
6-00
72 12
89-94
14
28-51
12-22
15-56
5-78
75 13
89-77
15
29-52
12-65
16-13
5-58
67 12
90-06
16
30-48
13-06
16-63
5-41
65 12
90-06
17
31-42
13-46
17-14
5-25
63 12
89-99
18
32-33
13-86
17-65
5-10
61 12
89-72
19
33-22
14-24
18-13
4-96
60 12
90-65
20
34-17
14-64
18-64
4-83
58 12
90-09
It is desirable that the millwright should possess short
easy rules, which would answer the purposes of practice
rather than the conditions of mere theory. The following
will be found useful, as they give the power with allowance
for friction and waste of water.
WATER WHEELS. 213
For an undershot :
Height of fall X quantity of water flowing per minute
DOOQ
the number of horses' power which the effect is equal to.
For an overshot :
Power of an undershot X 2| = horses' power.
For a breast-wheel :
Find the power of an undershot from the top of the fall
to where the water enters the bucket ; then for an overshot
for the rest of the fall the sum of these two is the power
of the breast wheel.
NOTE. The quantity of water flowing per minute, ana
the height of the fall are both taken in feet.
Ex. What power can be obtained from an undershot
wheel the fall being 25 feet, the section of the stream
being 9 feet, and the velocity of the water 18 feet per
minute?
9 x 18 X 25 4050
5000 5000
one horse power being unit.
And an overshot in the same situation would be '81 X
2-5 == 2-025 horses' power.
And if, in a breast wheel, the water enters the bucket 10
. r eet from the top of the fall, then we have,
10 X 8 X 9 ol _ 720 o , _ 1800-0 _
~~5000~ ~~5000 X : 5000 =
for an overshot, and for the undershot we found it before
81 ; hence, '36 + '81 = 1-17 horses' power for the breast
wheel.
BARKER'S MILL.
IN plate 1st, fig. 4, we have given a view of Barker's
mill, where CD is a vertical axis, moving on a pivot at D,
and carrying the upper millstone m, after passing through
an opening in the fixed millstone C. Upon this axis is
fixed a vertical tube TT communicating with a horizontal
rube AB, at the extremities of which A, B, are two aper-
tures in opposite directions. When water from the mill-
course MN is introduced into the tube TT, it flows out of
the apertures A, B, and by the reaction or counterpressure
of the issuing water, the arm AB, and consequently the
214 HYDRODYNAMICS.
whole machine, is put in motion. The bridge-tree ab is
elevated or depressed by turning the nut c at the end of
the lever cb. In order to understand how this motion is
produced, let us suppose both the apertures shut, and the
tube TT filled with water up to T. The apertures A, B,
which are shut up, will be pressed outwards by a force
equal to the weight of a column of water whose height is
TT, and whose area is the area of the apertures. Every
part of the tube AB sustains a similar pressure ; but as
these pressures are balanced by equal and opposite pres-
sures, the arm AB is at rest. By opening the aperture at
A, however, the pressure at that place is removed, and
consequently the arm is carried round by a pressure equal
to that of a column TT, acting upon an area equal to that
of the aperture A. The same thing happens on the arm
TB ; and these two pressures drive the arm AB round in
the same direction. This machine may evidently be ap-
plied to drive any kind of machinery, by fixing a wheel
upon the vertical axis CD.
This ingenious machine has not been much employed,
even in those situations for which it is best adapted ; partly,
we suspect, from the millwright's riot having in his posses-
sion sufficiently simple rules for its construction ; as the
theory of Barker's mill, simple as its construction and action
may appear, is not by any means well developed. In the
mean time the following directions may be found useful to
the mechanic.
1. Make each arm of the horizontal tube, from the centre
of motion to the centre of the aperture of any convenient
length, not less than ^ of the perpendicular height of the
water's surface above these centres.
2. Multiply the length of the arm in feet by -61365, and
the square root of this product will be the proper time for a
revolution, in seconds ; then adapt the other parts of the
machinery to this velocity ; or,
If the time of a revolution be given, multiply the square
of this time by 1-6296 for the proportional length of the
arm in feet.
Multiply together the breadth, depth, and velocity per
second of the race, and divide the last product, 14-27 X
the square root of the height ; the result is the area of
either aperture ; or, multiply the continual product of the
breadth, depth, and velocity of the race, by the square root
BARKER'S MILL. 215
of the height, and by the decimal -07, the last product
divided by the height will give the area of the aperture.
Multiply the area of either aperture by the height of the
head of water, and this product by 55*795 (or, in round
numbers, 56) for the moving 1 force, estimated at the centres
of the apertures in Ibs. avoirdupois.
Ex. If the fall be 18 feet from the head to the centre
of the apertures, then the arm must not be less than 2 feet,
as^of 18 = 2, and,/ (2 x -61305) = v/( 1-22730) = 1-107
the time of a revolution in seconds; also, the breadth
of the race being 17 inches, and depth 9, and the velocity
of the water 6 feet per second, here we have,
17 in. = 1-41 feet, and 9 in. = -75 feet, then
1-41 x '75 x 6 = 6-34 = the area of section of the
race x velocity of water ; hence,
6-39 x N/ 18 X -07 = 1-896 = the area of the aperture
in inches ; and,
1-876 x 18 X 56 = 1909 Ibs. the moving force.
The following dimensions have been employed in prac-
tice with success. The length of arm from the centre pivot
to the centre of the discharging hole, 46 inches ; inside
diameter of the arm, 13 inches ; diameter of the supplying
pipe, 2 inches ; height of the working head of water 21
feet above the level of the discharge. When a machine
of these dimensions, and in such circumstances, was not
loaded and had one orifice open, it made 115 turns in a
minute.
PNEUMATICS.
PNEUMATICS comprehends the knowledge of the proper-
ties of common air and elastic fluids in general.
Air is capable of being compressed to almost any degree,
that is, may be forced into a space infinitely smaller than
the space which it commonly occupies, and this is effected
by additional pressure. When this additional pressure is
taken away, the air will regain, by its elasticity, its former
magnitude. Were it not for this circumstance, the subject
of this chapter might have been introduced when we dis-
cussed the equilibrium and motion of water and fluids,
which are non-elastic or incompressible, as their fundamen-
tal laws are the same. It has, indeed, been found oy recent
experimenters, that water, mercury, &c., are compressible,
but to a very limited degree ; so that although the distinction
of elastic and non-elastic fluids is not absolutely correct, it
is yet sufficiently so to retain Pneumatics, in elementary
arrangement, as a distinct branch of science.
The air or atmosphere is a fluid body which surrounds
(he earth, and gravitates on all parts of its surface.
The mechanical properties of air are the same as other
elastic fluids, and being the most common, inquiries in
pneumatics are generally confined to this fluid.
The air has weight. A cubic foot of it weighs 1 P 2857
ounces at the surface of the earth, or, as some state it, 1'222.
The air being an elastic fluid, it is compressible and ex-
pansible, and its degrees of compression and expansion are
proportional to the forces or weights which compress it.
All the air near the earth's surface is in a state of com-
pression, in consequence of the weight of the atmosphere
which is above it.
As the less weight that presses the air compresses it the
less, or causes it to be less dense, and as the higher we
rise in the atmosphere there will be the less weight, so the
higher we go in the atmosphere the air will be the less
dense.
2Jfi
OF THE ATMOSPHERE. 217
The spring or elasticity of the air is equal to the weight
of the atmosphere above it, and they will produce the same
effects since they always sustain and balance each other.
It' the density of the air be increased by compression, its
spring or elasticity is also increased, and in the same pro-
portion.
By the pressure and gravity of the atmosphere on the
surface of fluids such as water, they are made to rise in
pipes or vessels, where the spring or pressure Within is
taken off or diminished. This fact, a knowledge of which
is applied to a multitude of useful purposes, will uot be
difficult of explanation. L*it a tube 3 feet long bo filled
with water, the tube being open at one end and close at the
other; one unacquainted witn the subject might naturally
expect that if this tube were held perpendicularly with the
open end downmost, the water would flow out of *he tube
by reason of its weight. But if we consider all the c.ireum-
stances, we will see that this can only happen on certain
conditions. The water has a tendency to fall to thr earth
in consequence of its weight, but then the air of the aL:no-
sphere, which we have stated before as also possessed of
weight, presses upon the surface of the water at the r.pen
end of the tube ; and as the pressure of fluids of t-Il kinds
is exerted in every direction, it follows, that the \\r will
have a tendency to force the water up the tube. Now the
pressure of the atmosphere at the surface of the earth is
about 15 Ibs. for every square inch, which is therefore the
force by which the water will be pressed up the tube by
the action of the air. A column of water 3 feet high does
not exert such a pressure on the base ; wherefore , as the
pressure upwards is greater than the pressure downwards,
the water will remain suspended in the tube.
Let us now take a tube 36 feet long, similar to the
former, filled with water and inverted in the same way as
before, it will now be found that a part of the water will
tlow out of the tube, the reason of which will be easily seen.
, It was stated under Hydrostatics, that the pressure of a
column of water 30 feet high was equal to 13 Ibs. on the
/quare inch. So that we see, that the pressure of the air
will keep 30 feet of the water in the tube, but it will keep
more, for the pressure of the air is 15, and that of 30 feet
of water is only 13 ; and as the pressure of the water will
be as its depth, we say, 13 : 15 : : 30 : 34, which, there-
19
PNEUMATICS.
fore, is the greatest height at which the water will be sup-
ported by the pressure of the atmosphere.
For the purpose of arriving at this conclusion of the
effect of the pressure of the atmosphere, we might have
employed a much shorter tube if we had used a heavier
fluid than water, for instance, mercury. Now the cubic
foot of mercury weighs 13600 ounces, and a cubic inch will
be found, 13600
= 7-866 ounces,
or nearly 8 ounces, that is about half a pound avoirdupois ;
therefore 30 inches will weigh 15 Ibs., hence, the atmo-
sphere will balance by its pressure 30 inches of mercury.
Thus we have arrived at the principle of the barometer, or
weather glass, as it is commonly called. The pressure of
the air at the surface of the earth is not always constant,
but varies within certain limits. The mean pressure is
about 14 Ibs. to the square inch, and the corresponding
height of the mercury in the barometer will therefore be
15 : 14 : : 30 : 28 inches.
It will appear evident, from what has been said before,
that as the higher we. ascend in the atmosphere there will
be less pressure, and therefore the mercury in the barometer
will fall, and this fact has been used as a means of measur-
ing heights by the barometer. If the air were of the same
uniform density up to the top of the atmosphere as it is at
the earth's surface, we might very easily determine its
height, for the specific gravity of air being to that of water
as 1-222 to 1000, nearly, we have this proportion, 1'222 :
1000 : : 33-25, (the mean height of a water barometer in
feet,) : 27200 feet, which is very nearly 5| miles ; but by
a process which proceeds on correct principles, .the height
of the atmosphere has been estimated at about 50 miles.
The law of the diminution of density at different heights in
the atmosphere is this, that if the heights increase in arith-
metical progression, the densities will decrease in geometri-
cal progression; for instance, if the density at the surface
of the earth be called 1, and if at the height of 7 miss it
be called 4 times rarer than at
14 16,
21 it will be 64 times rarer,
28 256,
35 1024,
PRESSURE OK THE ATMOSPHERE. 219
and in this way it might be shown, that at the height of one-
half the diameter of the earth, one cubic inch of atmospheric
air of the density which we breathe, would expand so much
as to fill the bounds of the solar system.
Many eminent men have investigated this subject, and
derived theorems of great use for determining altitudes by
the barometer. Some of these are exceedingly complex
and unfitted for a work of this nature : that of Sir J. Leslie
is the most simple, and gives results sufficiently near the
truth for all ordinary purposes.
As the sum of the heights of the mercury at the bottom
and top of the mountain is to the difference of the heights,
so is 52000 to the altitude of the mountain in feet.
At the bottom of a hill the barometer stood at 29'8, and
at the top 27*2, wherefore,
29-8 + 27-2 = 57 = the sum,
and 29-8 27-2 = 2-6 = the difference ;
hence, 57 : 2*6 : : 52000 : 2372 feet, the height of the
mountain nearly.
When air becomes denser, its elastic force is increased,
and that in proportion. Thus, when air is compressed into
half its bulk, its elastic force will be double of what it was
before.
It will, therefore, be easy to calculate the elastic force
of air compressed any number of times ; thus, if, by any
means, we condense the air in a vessel into of the space
which it occupied when not confined, it will press on the
inside of the vessel with a force of 15 X 3 = 45 Ibs. on
every square inch. It must be remembered, however, that
the atmosphere presses with a force of 15 Ibs. on each square
inch of the outside of the vessel, which therefore counter-
acts so much of the force of the condensed air within the
real pressure, therefore, is 45 15 = 30 Ibs. It is clear,
then, that whatever be the degree of condensation of the
enclosed air, we must always deduct the pressure of the at-
mosphere to ascertain its true effect. The young mechanic
will easily understand what is meant by the phrase a
pressure of 2, 3, 4, or any number of atmospheres, one
Atmosphere being understood as exerting a pressure of 15
Ibs. on the square inch, two atmospheres 30, and three 45,
&c. When the air is by any means entirely taken out of
sny vessel, there is said to be a vacuum in that vessel.
What is the whole amount of pressure on the inside sur-
220 PNEUMATICS.
(ace of a sphere, which contains air condensed to | of its
natural bulk, and is 6 inches in diameter within. Here,
by mensuration, we have, 6- X 3-1416 = 113-0976 = the
surface of the inside of the sphere and 15 x 4 15 =
45 = the pressure on a square inch, therefore, 113-0976 X
45 = 5089-3920 Ibs. on the inner surface of the globe.
Here the globe is supposed to be in a vacuum.
In a cylinder 6 feet long, and closed at the bottom, a pis-
ton is thrust down to the distance of one foot from the bot-
tom, the cylinder being 24 inches in diameter, then, by the
rules in mensuration, the area of the piston will be found to
be 452-4 inches, the diameter of the piston being 24 inches,
and the cylinder being 6 feet long, and the piston being
pressed down to 1 foot from the bottom, the air will be com-
pressed into -g- of its former bulk, and its elastic force will
be 6 times greater than it was before. At first it was 15 Ibs.
to the square inch, but now it will be 15 X 6 = 90 on the
square inch, and one atmosphere being deducted for the
contrary pressure of the atmosphere above the piston, the
pressure is 90 15 = 75 Ibs. to the square inch, where-
fore, 452-4 x 75 = 33930 Ibs., the force by which the
piston will be pressed upwards.
THE SYPHON.
A SYPHON, or, as it is frequently written, siphon, is any
bent tube.
If a syphon be filled with water and inverted, so that the
bend shall be uppermost, then if the legs be of equal
length, and it be held so that the two lower ends of the
syphon are on a level, then we will find that if the perpen-
dicular height of the bend of the tube above the level of the
two ends be not more than 32 or 33 feet, the water will re-
main suspended in the tube. It will not be difficult to see
how this happens, for the atmosphere pressing on the water
at the orifice of the tube at each extremity, presses the
water up the tube with a force capable of raising it 33 feet ;
but in the case supposed, the orifices and the legs are equal,
and do not exceed the limit of 32 or 33 feet, therefore,
since the pressure on one orifice is the same as the pressure
on the other, there will be an equilibrium and the water
in the one leg has no more power to move than that in the
other.
If we now suppose the syphon to be inclined a little, so
PUMPS. 221
that the twc orifices shall not be on a level, or what is the
same thing, if we suppose the length of the one leg to be
greater than that of the other, we will find that the equi-
librium will be no longer maintained ; and the water will
flow out of the orifice which is lowest. For although the
air presses equally on both orifices with a force of 15 Ibs.
to the square inch, yet the contrary pressures downwards
by the weight of the water are not equal, therefore motion
will ensue where the power of the water is greatest. If
the shorter leg be immersed in a vessel of water, and the
syphon be set a running, the water will flow out of the
lower end of the syphon, until the other end be no longer
supplied. Instead of filling the syphon with water, as has
been supposed above, a common practice is to apply the
mouth to the lower orifice, and by sucking, exhaust the air
in the tube, which diminishes the pressure at the other
orifice, and consequently the action of the atmosphere will
force the water in the vessel up the tube of the syphon and
fill it, and it will continue to act in the same way as before.
PUMPS.
A PUMP is a machine used for exhausting vessels con-
taining air, or for raising water, sometimes by means of
the pressure of the atmosphere, sometimes by the condensa-
tion of air, and sometimes by a combination of both.
It may be necessary here to explain what is meant by the
term valve, that our remarks on the action of the pump may
be rendered more intelligible.
A valve is usually defined to be a close lid affixed to a
tube or opening in a vessel, by means of a hinge or some
sort of movable joint, and which can be opened only in one
direction. There are various kinds of valves. The clack
valve consists merely of a circular piece of leather covering
tlje hole or bore of the pipe which it is intended to stop,
.and moving on a hinge, sometimes a part of itself, and
sometimes made of metal. The butterfly valve, which is
superior to the clack valve, consists of two pieces of leather
each formed into the shape of a half circle ; they are at-
Aached by hinges on their diameters, or straight parts, to a
bar that crosses the centre of the orifice to be closed. The
button or conical valve consists of a plate of brass ground
in such a way as exactly to fit the conical cavity in which it
lies. Sometimes valves are made in the form of pyramids
19*
222
PNEUMATICS
IB
consisting of four triangular flaps which form the "sides of
the pyramid, and move upon hinges which are placed round
the edge of the orifice to be closed. The tops of these
flaps must all meet accurately in the middle of the orifice
and are supported by four bars which meet in the centre.
The action of the air pump may be thus explained. Le'
R be the section of a glass
bell, called a receiver, closed
at the top T, but open at the
bottom, and having its lower
edge ground smooth, so as
to rest in close contact with
a smooth brass plate, of
which SS is a section. In
the middle is an opening A,
which communicates by a-
tube AB with a hollow cy-
linder or barrel, in which a
solid piston P is moved.
The piston rod C moves in
an air-tight collar D, and at
the bottom of the cylinder a valve V is placed, opening
freely outward, but immediately closed by any pressure
from without. There is thus a free communication between
the receiver R, the tube AB, and the exhausting barrel BV.
When the piston CP is pressed down, and has passed the
opening at B, the air in the barrel BV will be enclosed, and
will be compressed by the piston. As it will thus be made
to occupy a smaller space than before, its density, and con-
sequently its elasticity, will be increased. It will therefore
press downwards upon the valve V with a greater force
than that by which the valve is pressed upwards by the
external air. This superior elastic force will open the
valve, through which, as the piston descends, the air in tke
barrel will be driven into the atmosphere. If the piston be
pushed quite to the bottom, the whole air in the barrel will
be thus expelled. The moment the piston begins to ascend,
the pressure of the air from without closes the valve V com-
pletely, and, as the piston ascends, a vacuum is left beneath
it; but, when it rises beyond the opening B, the air in the
receiver R and the tube AB expands, by its elasticity, so
as to fill the barrel BV. A second depression of the piston
will expel the air contained in the barrel, and the process
PUMPS. 223
may be continued at pleasure. The communication be-
tween the barrels and the receiver may be closed by a stop-
cock at G. In consequence of the elasticity of the air it
expands and fills the barrel, diffusing itself equally through-
out the cavity in which it is contained. The degree of
rarefaction produced by the machine may, hojivever, be
easily calculated. Suppose that the barrel contains one-
third as much as the receiver and tube together, and, there-
fore, that it contains one-fourth of the whole air within the
valve V. Upon one depression of the piston, this fourth
part will be expelled, and three-fourths of the original quan-
tity will remain. One-fourth of this remaining quantity
will in like manner be expelled by the second depression
of the piston, which is equal to three-sixteenths of the ori-
ginal quantity. By calculating in this way, it will be found
that after thirty depressionsAf the piston, only one 3096th
part of the original quantity will be left in the receiver.
Rarefaction may thus be carried so far that the elasticity of
the air pressed down by the piston shall not be sufficient to
force open the valve.
We now proceed to the consideration of the common
suction pump. This pump consists of a hollow cylinder A,
of wood 'or metal, which contains a piston B,
stuffed so as to move up or down in the cy-
linder easily, and yet be air tight : to this
piston there is attached a rod which will reach
at least to the top of the cylinder when the
piston is at the bottom. In the piston there
is a valve which opens upwards, and at the
bottom of the cylinder there is another valve
C also rising upwards, and which covers the
orifice of a tube fixed to the bottom of the cy-
linder, and reaching to the well from whence
the water is to be drawn. This tube is commonly called
the suction tube, and the cylinder, the body of the pump.
From what has been said of the pressure of the atmo
sphere, it will not be difficult to understand how this ma-
diine operates. For when the piston is at the bottom of the
cylinder, there can be no air, or at least very little, between
it and the valve C, for as the piston was pushed clown, the
valve in it would allow the air to escape instead of being
condensed, and when it is drawn up, the pressme of the air
would shut 'he valve, and there would be a vacuum produced
ffl
224 PNEUMATICS.
in the body of the cylinder when the piston arrived at the
top. But the air in the cylinder being very much rarifisd,
the pressure of the valve C on the water at the bottom will
be greatly less than that of the external atmosphere on the
surface of the water in the well ; therefore, the water will
be pressed up the pump to a height not exceeding 32 or
33 feet. As the valves shut downwards, the water is pre-
vented from returning, and the same operation being
repeated, the water may be raised to any height, not
exceeding the above limit, in any quantity.
The quantity of water discharged in a given time, is de-
termined by considering that at each stroke of the piston a
quantity is discharged equal to a cylinder whose base is the
area of a cross section of the body of the pump, and height
the play of the piston. Thus, if the diameter of the cylin-
der of the pump be 4 inches^nd the play of the piston 3
feet, then, by mensuration, we have to find the content of a
cylinder 4 inches diameter, and 3 feet high now, 4 inches
is the f of a foot, or -333, hence, -333 a x '7854 = -110999
X '7854 = -08796 = the area of the cross section of the
cylinder in square feet; hence, -08796 X 3 = -2639 = the
content of the cylinder in cubic feet = the quantity in
cubic feet of water discharged by one stroke of the piston.
Now, a cubic foot of water weighs about 63-5 Ibs. avoirdu-
pois, wherefore, -2639 x 63-5 = 16-756 Ibs. avoirdupois,
and an imperial gallon is equal to 10 Ibs. of water ; whence,
dividing the above number 16-756 by 10, we get the num-
ber of ale gallons = 1-6756. The piston, throughout its
ascent, has to overcome a resistance equal to the weight of
a column of water, having the same base as the area of the
piston, and a height equal to the height of the water in the
body of the pump above the water in the well.
In our calculations of the effects of the pump, it will be
necessary to determine the contents of pipes, for which
purpose the following simple rules will serve.
Diameter of pipe in inches 8 = number of avoirdupois
pounds contained in 3 feet length of the pipe.
If the last figure of this be pointed off as a decimal, the
result will be the number of ale gallons, and if there be
only one figure this is to be considered as so many tenths
of an ale gallon : ale gallons x 282 = the. number of cubic
inches.
Thus, in a pipe 5 inches diameter, we have,
PUMPS. 225
5 9 = 25 = number of avoirdupois pounds contained in 3
feet of the pipe 2-5 = the number of ale gallons and 2'5 x
282 = 705 cubic inches.
These rules find the content for three feet in length of
the pipe, the content for any other length may be found by
proportion; thus, for a pipe 6 inches in diameter, and IV
feet long ; we have, 6 a = 30 = pounds of water avoir, con-
tained in the pipe to the length of 3 feet ; therefore,
3 : 12 : : 36 : 144 = the number of pounds in 12 feel
length, and,
14-4 = ale gallons, and 14'4 X 282 = 4060-8 = the
cubic inches in 12 feet length.
The resistance which is opposed to a pump rod in raising
water, is equal to the weight of a column of water whose
base is the area of the piston, and height the height of the
surface of the water in the body of the pump above the
surface of the water in the well, together with the friction
and the piston and piston rod.
Suppose the body of the pump to be 6 inches in diifmeter,
and the height to which the water is raised be 30 feet, and
also the weight of the piston and rod is 10 Ibs., and the
friction is - of the whole weight of the water.
Now, fi 8 = 36 = the Ibs. avoirdupois of 3 feet of the
column of water, but the column is 30 feet, therefore, 3 :
30 : : 36 : 360 Ibs., the weight of the whole column. To
this we must add the effect of friction, which we have sup-
posed to be y of the weight of the water ; hence,
^fid
' = 72 Ibs., and this must be added to the weight of
D
the column of water, which gives 360 + 72 = 432 Ibs. the
whole amount of resistance arising from the weight of the
water and friction ; to this must be added the weight of the
piston and pump rod, therefore, 432 -\- 10 = 442 = the
whole resistance opposed to the rising of the piston, any
thing greater than this will raise it.
In the construction of pumps it is usual to employ a lever
to work the piston, which gives an advantage in power ;
*and if in the case estimated above, we employ a lever who? e
'arms are in the proportion of 10 to 1, the pump might te
wrought with a force of 44*2 Ibs., or we may say 45 Ibs.
For the convenience of workmen we insert the following
table. It has been calculated on the supposition that the
handle of the pump is a lever which srives an advantage on
22C
PNEUMATICS.
the piston rod of 5 to 1, and that a man can, with a pump
30 feet long, and a 4 inch bore, discharge 27'5 wine gallons
(oil measure) in a minute. And if it be required to find
the diameter of a pump that a man could work with the
same ease as the above pump at any required height above
the surface of the well, this table will give the diameter of
l-ore, and the quantity of water discharged in a minute.
Height of (lie pump
above the surfc.ee of
the mill in feet.
Diameter of the bore
where the piston
works iu iuchei.
Water discharged per
gallons and pints.
10
6-93
81 6
15
5-66
54 4
20
4-90
40 7
25
4-38
32 6
30
4-
27 2
35
3-70
23 3
40
3-46
20 3
45
3-27
18 1
50
3-10
16 3
55
2-95
14 7
60
2-84
13 5
65
2-72
12 4
70
2-62
11 5
75
2-53
10 7
80
2-45
10 2
85
2-38
9 5
90
2-31
9 1
95
2-25
8 5
100
2-19
8 1
We stated before that water could not be raised to a
greater height than 32 feet by means of the kind of pump
we have described, and it may seem strange that this table
extends to 100 ; but these are pumps acting on a different
principle, by means of which water may be raised to any
height, and whose action will be considered before we
leave this subject.
The lifting pump. This pump, like the suction pump, has
. two valves and a piston, both opening upwards ; but the
valve in the cylinder, instead of being placed at the bottom
of the cylinder, is placed in the body of it, and at the height
PtJMPS.
227
where the water is intended to be delivered. The bottom
of the pump is thrust into the well a considerable way,, and
if the piston be supposed to be at the bottom, it is plain,
that as its valve opens upwards, there will be no obstruction
to the water rising in the cylinder to the height which it is
in the well ; for, by the principles of Hydrostatics, water
will always endeavour to come to a level. Now when the
piston is drawn up, the valve in it will shut, and the water
in the cylinder will be lifted up ; the valve in the barrel
will be opened, and the water will pass through it, and can-
not return, as the valve opens upwards ; another stroke
of the piston repeats, the same process, and in this way the
water is raised from the well : but the height to which it
may be raised, is not in this, as in the suction pump, limited
to 32 or 33 feet. To ascertain the force necessary to work
this pump, we are to consider that the piston lifts a column
of water whose base is the area of the. piston, and height
the distance between the level of the water in the well and
the spout, at which the water is delivered. Thus, -if the
diameter of the pump's bore be 4 inches, and the height
of the spout above the level of the well = 40 feet, then we
have 4 2 = 16 Ibs. in three feet of the barrel; wherefore,
3:40::16:213j Ibs. the weight of the water, and the
friction and weight of the piston and rod must be added to
this, to find the whole force necessary. If the friction be
reckoned, as it usually is, ][, then we have,
wherefore, 213 -|- 42 = 255; as we have neglected frac-
tions we may reckon it 256, and if the weight of the piston
and rod be 20 Ibs. the whole will be 256 -f 20 = 276 Ibs..
the whole force necessary to balance the piston ; any thing
greater than this will raise it.
The forcing pump remains to be consi-
dered. The piston of this pump has no
valve, but there is a valve at the bottom of
,the cylinder, the same as seen at A. In the
side of the cylinder, and immediately above
tl/e valve B, there is another valve A open-
ing outwards into a tube which is bent up-
wards to the height H at which the water
is to be delivered. When the piston is
raised, the valve in the bottom of the pump
228 PNEUMATICS
opens, ana a vacuum being produced, the water is pressed
up into the pump on the principle of the sucking pump
But when the piston is pressed down, the valve A at the
bottom shuts, and the valve B at the side which leads into
the ejection pipe opens, and the water is forced up the tube.
When the piston is raised again the valve B shuts, and the
valve A opens. The same process is repeated, and the
water is thrown out at every descent of the piston, the dis
charge therefore is not constant.
It is frequently required that the dis-
charge from the pump should be continu-
ous, and this is effected by fixing to the
top of the eduction pipe an air vessel.
This air vessel consists of a box AB, in
the bottom of which there is a valve C
opening upwards into the box. This valve
covers the top of the eduction pipe D. A
tube, E, is fastened into the top of the box,
which reaches nearly to the bottom of the box, it rises out
of the box, and is furnished with a stop cock. If the stop
cock be shut, and the water be sent by the action of the
pump into the air vessel, it cannot return because of the
shutting of the valve at the bottom of the box ; and be-
cause of the space occupied by the water, the air in the box
is condensed, and will consequently exert a pressure on the
water in the air vessel. If the water fill three-fourths of
the box, then the air will be compressed so as to exert four
times its original force ; and the stop cock being opened,
the water will be forced up the tube, with a force which
will send it one less than as many times 32 feet as the air
is compressed, that is, in the case supposed 3 X 32 = 96
feet. On this principle it is that jets of fountains act.
The air vessel may therefore be considered as a magazine
of power, and so long as there is as much water forced into
the fir vessel by pumping, as there is forced out by the
pressure of the air, there will be a constant jet of water.
The force necessary to raise the piston in this pump, is
found exactly in the same way as for the suction purnp.
And the force necessary to depress the piston, is found by
taking the weight of a column of water, whose height is
equal to the height of the spout where the water is de-
livered above the level of the piston, before it begins to
descend Thus, if the piston when raised is 26 feet above
WINDMILLS. 229
the level of the well, and the spout is 63 feet above the same
level, therefore, the height of the column* is 63 26 = 37
feet; and supposing the diameter of the ejection pipe to
be 5 inches, we have for 3 feet of the pipe 5 a = 25 Ibs..
wherefore for 37 feet we have,
3 : 37 :: 25 : 308 1 Ibs.
The weight of the piston and its rods oppose the raising of
the piston, but assist in depressing it.
The power applied to the piston rod of a suction pump
should be an intermitting power, otherwise there will be a
waste of .power ; but in a forcing pump the power must be
continued throughout equable. A single stroke steam en-
gine will be best to raise water by the sucking, and a
double stroke by a forcing pump. The piston rod of a
forcing pump should be loaded with a weight sufficient to
balance a column of water, whose base is the section of the
piston, and whose height is the excess of the height of the
spout from the level of the water in the cistern above 68
feet. This will cause a regular application of power when
this pump is wrought with a steam engine.
WIND AND WINDMILLS.
WE have seen the effect of the pressure of air, arising
from its weight and elasticity when at rest ; it now remains
for us to consider its effects when put in motion, as in the
case of wind.
Were it not for the irregularity in direction and force of
the wind, it would be the most convenient of all the first
movers of machinery, but even as it is, its efficacy may be
taken advantage of, and deserves our consideration.
The force with which wind strikes against a surface, is
as the square of the velocity of the wind. This simple
theorem is so nearly true that it may be employed without
fear of error.
The force in avoirdupois pounds with which the wind
strikes on any surface on which it acts perpendicularly may
be found by using the rule,
surface struck x velocity of wind 3 x '002288 ;
\vhere the surface and velocity of wind are taken in feet,
amd the time 1 second. If the wind moves at the rate of
30 feet per second, and the surface exposed to it. action be
14 feet square, then, 14 x 30 3 x '002288 = 28-8288.
From this statement it might appear at first sight, that iu
20
830
PNEUMATICS.
the case of mills winch act by the impulse of wind on re- .
volving surfaces called sails it might appear, we say, that
the greater quantity of sail exposed to the action of the
wind, the greater would be the effect of the machine. But
this has been found not to hold : it would appear that the
wind requires space to escape. The sails of the windmill
may be supposed to intercept a cylinder of wind ; and it
would seem, that when the whole cylinder is intercepted,
the effect of the machine is diminished ; and it is concluded
from experiments, that the sails should not intercept above
seven-eighths of the cylinder of the wind.
We here subjoin a tabular view of the effects of wind at
different velocities.
Table showing the pressure of the Wind for the following Velocities.
Velocity of the Wind.
Force upon 1 square foot in
pounds avoir.
Miles in 1 hour.
Feet in 1 second.
1
1-47
005
2
2-93
020
3
4-40
044
4
5-87
079
5
7-33
123
10
14-67
492
15
22-00
1-107
20
29-34
1-968
25
36-67
3-075
30
44-01
4-429
35
51-34
6-027
40
58-68
7-873
45
66-01
9-963
50
73-35
12-300
60
88-02
17-715
80
117-36
31-490
100
146-70
49-200
Windmills are constructed either so that the sails shall
move in a horizontal plane, or in a plane nearly vertical ;
their former are called horizontal, and the latter vertical
windmills. In plate 2,, fig. 1 and 2, we have given a plan
and section of a horizontal windmill, on an improved con-
WINDMILLS. 231
struction. HH are the side walls of an octagonal building
which contains the machinery. These walls are surmounted
by a strong timber framing GG, of the same form as the
building, and connected at top by cross-framing to support
the roof, and also the upper pivot of the main vertical
shaft AA, which has three sets of arms, BB, CC, DD,
framed upon it at that part which rises above the height
of the walls. The arms are strengthened and supported
by diagonal braces, and their extremities are bolted to
octagonal wood frames, round which the vanes or floats EE
are fixed, as seen in outline in fig. 2, so as to form a large
wheel, resembling a water wheel, which is less than the
size of the house by about 18 inches all round. This space
is occupied by a number of vertical boards or blinds FF,
turning on pivots at top and bottom, and placed obliquely,
so as to overlap each other, and completely shut out the
wind, and stop the mill, by forming a close case surround-
ing the wheel ; but they can be moved altogether upon
their pivots to allow the wind to blow in the direction of a
tangent upon the vanes on one side of the wheel, at the
time the other side is completely shaded or defended by
the boarding. The position of the blinds is clearly shown
at FF, fig. 2. At the lower end of the vertical shaft AA,
a large spur-wheel aa is fixed, which gives motion to a
pinion c, upon a small vertical axis rf, whose upper pivot
turns in a bearing bolted to a girder of the floor n. Above
the pinion c, a spur-wheel e is placed, to give motion to
two small pinions f, on the upper ends of the spindles g,
of the millstone n. Another pinion is situated at the
opposite side of the great spur-wheel aa, to give motion to
a third pair of millstones, which are used when the wind
is very strong ; and then the wheel turns so quick as not
to need the extra wheel e to give the requisite velocity to
the stones. The weight of the main vertical shaft is borne
by a strong timber 6, having a brass box placed on it to
receive the lower pivot of the shaft. It is supported at its
'ends by cross-beams mortised into the upright posts 66, as
sjnown in the plan, fig. 2. A floor or roof 1 1 is thrown
across the top of the brick building to protect the machinery
from the weather, and to prevent the rain blowing down
the opening through which the shaft descends, a broad cir-
cular hoop K is fixed to the floor, and is surrounded by
another hoop or case L, which is fixed to the arms DD of
23 PNEUMATICS.
the wheel. This last is of such a size, as exactly to go
over the hoop K, without touching it when the wheel turns
round. By this means, .the rain is completely excluded
from the upper room M, which serves as a granary, being
fitted up with the bins mm, to contain the different sorts
of grain which is raised up by the sack-tackle. A wheel i
is fixed on the main shaft, having cogs projecting from both
sides. Those at the under side work into a pinion on tho
end of the roller K, which is for the purpose of drawing
up sacks. Another pinion is situated above the wheel i,
which has a roller projecting out over the flap-doors seen
at p, in fig. 2, to land the sacks upon. The two pinions
mm, fig. 2, are turned by the great wheel act, and are for
giving motion to the dressing and bolting machines, which
are placed upon the floor N, but are not shown in the
drawing, being exactly similar to the dressing machines
used in all flour-mills. The cogs upon the great wheel a
are not so broad as the rim itself, leaving a plain rim about
three inches broad. This is encompassed by a broad iron
hoop, which is made fast at one end to the upright post b ;
the other being jointed to a strong lever n, to the extreme
end of which a purchase o is attached, and the fall is made
fast to iron pins on the top of a frame fixed to the ground.
This apparatus answers the purpose of the brake or gripe
used in common windmills to stop their motion. By pull-
ing the fall of the purchase 0, it causes the iron strap to em-
brace the great wheel, and produces a resistance sufficient
to stop the wheel. The mill can be regulated in its motion,
or stopped entirely, by opening or shutting the blinds F,
which surround the fan-wheel. They are all moved at
once by a circular ring of wood situated just beneath the
lower ends of the blinds upon the floor 1 1, being connected
with each blind by a short iron link. The ring is moved
round by a rack and spindle which descend into the mill
room below, for the convenience of the miller. The mode
of bringing the sails back against the wind, which Mr.
Beatson invented, is, perhaps, the simplest and best for
that end. He makes each sail AI, fig. 3, to consist of six
or eight flaps or vanes, AP b 1, b I c 2, <fcc., moving upon
hinges represented by the dark lines, AP b 1, c 2, &c., so
that the lower side b 1 of the first flap wraps over the hinge
or higher side of the second flap, and so on. When the
wind, therefore, acts upon the sail AI, each flap will press
WINDMILLS. 233
upon the hinge of the one immediately below it, and the
whole surface of the sail will he exposed to its action.
But when the sail AI returns against the wind, the flaps
will revolve round upon their hinges, and present only
their edjjes to the wind, as is represented at EG, so that
the resistance occasioned by the return of the sail must be
greatly diminished, and the motion will be continued by
the great superiority of force exerted upon the sails in the
position AI. In computing the force of the wind upon the
sail AI, and the resistance opposed to it by the edges of the
flaps in EG, Mr. Beatson finds, that when the pressure
upon the former is 1872 pounds, die resistance opposed by
the latter is only about 3G pounds, or j 2 part of the whole
force ; but he neglects the action of the wind upon the
arms, CA, &c., and the frames which carry the sails, be-
cause they expose the same surface in the position AI, as
in the position EG. This omission, however, has a ten-
dency to mislead us in the present case, as we shall now
see ; for we ought to compare the whole force exerted
upon the arms, as well as the sail, with the whole resistance
which these arms and the edges of the flaps oppose to the
motion of the windmill. By inspecting the figure it will
appear, that if the force upon the edges of the flaps, which
Mr. Beatson supposed to be 12 in number, amounts to 36
pounds, the force spent upon the bars CD, DG, GF, FE,
&c., cannot be less than 60 pounds. Now, since these bara
are acted upon with an equal force, when the sails have the
position AI, 1872 -f 60 = 1932 will be the force exerted
upon the sail AI, and its appendages, while the opposite
force upon the bars and edges of the flaps when returning
against the wind will be 36 -f- 60 = 96 pounds, which is
nearly ^ of 1932, instead of -j- 1 ^ as computed by Mr. Beat
son. Hence we may see the advantages which will pro
bably arise from usinff a screen for the returning sail instead
of movable flaps, as it will preserve not only the sails, but
the arms and the frame which supports it, from the action
of the wind.
f Figures 4 and 5, plate 2d, represent the most improved
form of the vertical windmill ; aaua, are the vanes or sails
of the mill, which communicate motion to the wind-shaft b
and the crown wheel c; f/, the centre wheel which conveys
this motion along the shaft e to the spur-wheel f; g, a
wheel, or trundle, on the end of the spindle of the upper
20*
234 fx\ fcuiviA TIL a.
or turning millstone ; i, the case in which the millstones
are placed ; k, the bridge-tree which supports the spindle
of the turning-stone ; /, another wheel, or trundle, on the
end of the shaft m, which conveys the motion lower down
the building to another spur-wheel n; this spur-wheel puts
other two millstones in motion at pleasure, in the same
manner as the former ; o, the brake, or rubber, for stopping
the mill, it operates by friction ; p, the governor for regu-
lating the motion, by opening or shutting the wind-boards
on the vanes ; q, the director which carries round the roof
with the wind, by keeping the vanes always at right angles
to it. On the spindle of this director is placed an endless
screw, .working into a wheel which turns a shaft having a
pinion fixed at the other end of it. This pinion works into
another wheel connected with the rack pinion, which puts
the whole roof in motion.
The wind does not act perpendicularly on the sails of a
wind-mill, but at a certain angle, and the sail varies in the
degree of its inclination at different distances from the
centre of motion, in resemblance to the wing of a bird ;
this is called the weathering of the sail. The angles of
weathering have been found by Smeaton as follows. The
radius being divided into 6 equal parts, and the first part
from the centre being called 1, the last 6.
Distance from Angle with Angle with the
the centre. the axis. plane of motion.
1 72 18
2 71 19
3 72 18
4 74 16
5 77 12
6 83 7
Smeaton gives the following maxims for the construction
rf windmills.
1. The velocity of the windmill sails, whether unloaded
or loaded, so as to produce a maximum, is nearly as the
velocity of the wind, their shape and motion being the same.
2. The load at the maximum is nearly but somewhat less
than, as the square of the velocity of the wind, the shape
and position of the sails being the same. 3. The effects of
the same sails at a maximum are nearly but somewhat less
than, as the cubes of the velocity of the wind. 4. The load
WINDMILLS.
233
of the same sails at the maximum is nearly as the squares,
antl their effects as the cubes of their number of turns in a
given time. 5. When the sails are loaded so as to produce
a maximum at a given velocity, and the velocity of the
wind increases, the load continuing the same, then, when
the increase of the velocity of the wind is small, the effect
will be nearly as the squares of the velocities ; but when
the velocity of the wind is double, the effects will be nearly
as 10 to 27rl ; and when the velocities compared are more
than double of that where the given load produces a maxi-
mum, the effect increases only as the increase of the velo-
city of the wind. 6. If sails are of a similar figure and
position, the number of turns in a given time will be in-
versely as the radius of length of the sail. 7. The load at
a maximum that sails of a similar figure and position will
overcome, at a given distance from the centre of motion,
will be as the cube of the radius. 8. The effect of sails of
similar figure and position are as the square of the radius.
Rules for modelling, the sails of Windmills.
The accompanying cut is the
front view of one sail of a wind-
mill. The length of the arm
AA, called by workmen the
whip, is measured from the
centre of the great shaft B, to
the outermost bar 19. The breadth of the face of the whip
A next the centre, is -j 1 ^ of the length of the whip, and its
thickness at the same end is | of the breadth ; and the back
side is made parallel to the face for half the length of the
whip : the small end of the whip is square, and "at its end
is 1-1 6th of the length of the whip, or half the breadth at
the great end.
From the centre of the shaft B, to the nearest bar 1 of
the lattice is l-7th of the whip, the remaining space of
6-7 ths of the whip is equally divided into 19 spaces ; l-9th
of one of these spaces gives the size of the mortice, which
must be made square.
To prepare the whip for mortising, strike a gauge score at
about three-quarters of an inch from the face on each side,
and the gauge score on the leading side, 4, 5, will give the
face of all the bars on each side ; but on the other side the
faces of all the bars will fall deeper than the gauge score,
236 PNEUMATICS.
according to a certain rule, which is this : Extend the
compasses to any distance at pleasure, so that 6 times tlfat
extent may be greater than the breadth of the whip at the
seventh bar. Set off these six spaces upon a straight line
for a base, at the end of which raise a perpendicular ; set
off the same six spaces on the perpendicular, and divide the
two spaces on the perpendicular which are farthest from
the base, each into 6 equal parts, so that these two spaces
will contain 13 points. Join each of these 13 points with
the end of the base farthest from the perpendicular.
To apply this scale to any given case, set off the breadth
of the whip at the last bar (that is the bar at the extremity
of the sail) from the centre of the scale, along the base to-
wards the perpendicular, and at this point raise a perpen-
dicular to cut the oblique line nearest the base ; also set off
the breadth at the seventh bar in the same manner, and at
this point raise a perpendicular to cut off the thirteenth
oblique line. Now, from the point where the first of these
two perpendiculars cuts the first oblique line from the base,
to the intersection of the second perpendicular with the
thirteenth oblique line, there is drawn a line joining the
two points of intersection ; and perpendiculars being drawn
from the points where this joining line cuts the oblique
lines to the base, will be the several distances of the face
of each bar from the gauge line. These distances give a
difference, set off for each bar to the seventh, which must
be set off for all the rest to the first. The length of the
longest bar is -| of the whip.
We now proceed to show the method of weathering the
sails. Draw AB = the length of the vane, BC its breadth,
and BCD the angle of the weather at the extremity of the
vane, equal to 20 degrees. With the length of the vane
AB, and breadth BC, construct the isosceles triangle ABC ;
and from the point B, draw BD perpendicular *,o CB, then
BD is the proper depth of the vane.
Divide the line AB into any number of parts, say four
at these divisions draw the lines 1 E, 2 F, 3 G, &c
WINDMILLS. 237
parallel to the line BC. Also, from the points of division,
1, 2, 3, &c., draw the lines 1 i, 2 k, 3 1, &c. perpendicular
to 1 E, 2 F, 3 G, &c., all of them equal in length to BD.
Join Ei, Fk, Gl, &c., then the angles 1 Ei, 2 Fk, 3 Gl,
&c., are the angles of the weather for these divisions of the
vane ; and if the triangles be conceived to stand perpen-
dicular to the paper, the angles i, k, 1, and D, denoting the
vertical angles, the hypotenuses of these triangles will
give a perfect idea of the weathering of the vane as it recedes
from the centre.
HEAT, STEAM, &c,
IT would be out of place in a work of this nature to
enter into a minute detail respecting the nature of heat ; in
this section, therefore, we shall confine ourselves to a de-
scription of the more important of its mechanical properties.
Heat expands bodies, that is, increases their dimensions.
Different bodies expand differently by the application of
the same quantity of heat. With the same degree of heat,
solids expand less than liquids, and liquids less than gases.
On the principle that bodies expand by heat, is con-
structed the Thermometer. The action of this instrument
is very simple. It consists of a small glass tube with a
hollow bulb at one end, and at the other end it is closed.
The bulb is filled with mercury, as likewise a part of the
tube, the other portion of the tube being entirely deprived
of air. When heat is applied to the bulb of the thermome-
ter, the mercury expands and rises in the tube, and accord-
ing to the degree of heat applied to it, so will the mercury
rise. To the tube there is attached a divided scale, to de-
note the degrees of heat by the rising of the mercury, which
scale is thus formed. The bulb of the thermometer is put
into 'melting ice, and the height of the mercury is marked
om the scale ; this is called the freezing point, and num-
bered 32. The bulb is then put into boiling water, and the
height of the mercufy in the tube is marked upon the scale
and numbered 212 this is called the boiling point. The
space betwixt these two points on the scale is divided into
180 equal parts, called degrees, and the scale is then ex-
tended both above and below these points. This is the
scale commonly used in this country, and is known by the
name of its inventor, Fahrenheit. But the French and many
philosophers in Britain use a thermometer having a scale
of much more simple construction, called, from the nature
of its divisions, the Centigrade scale. The freezing point,
which in Fahrenheit is marked 32, is in the Centigrade
238
HEAT. 239
marked or zero ; and the boiling point, in Fahrenheit
marked 21 2, is in the Centigrade marked 100. In Reaumur's
thermometer the freezing point is murked 0, and the boiling
point 80.
Let F represent Fahrenheit, R Reaumur, and C Centi-
grade, then we have the following rules for converting the
degrees of any one of these thermometers into the corres-
ponding temperature, as marked in the others :
(1.) F = C x 1-8 -f 32.
q ft
(2.) F = + 32.
(5.) R _
(6.) R = C x 0-8.
Thus 185 Fahrenheit's will be found to correspond to
85 of the Centigrade, and 68 of Reaumur's thermometer.
(1.) 85 x 1-8 + 32 = 185.
(2.) !L^ + 32 = 185 .
(3.) "^ _ 85.
(40 - 85.
(5.) * X (185 -32) = 68.
(6.) 85 X 0-8 = 68.
, 'There are many other particulars regarding the thermo-
meter which it would be inconsistent with the design of
these pages to consider : what we have said will be sufficient
forxhe understanding of what is hereafter to follow on the
suoject of steam, &c.
Before we introduced the subject of the thermometer, we
stated the fact of the expansion of bodies by heat. Bars
of the following substances, whose length at 'a temperature
240 HEAT.
of 32 was 1, were heated to 212 Fahrenheit, and expanded
so as to become,
Cast iron, 1-00110940
Steel, 1-00118990
Copper, 1-00191880
Brass, 1-00188971
This is the expansion in length ; the expansion in length,
breadth, and thickness, will be found by multiplying the
above numbers by 3.
The effects of different degrees of heat on different bodies,
according to Fahrenheit's scale, are shown below.
Cast iron thoroughly melted, 20577
Cast iron begins to melt, 17977
Greatest heat of a common smith's forge, 17327
Flint glass furnace, strongest heat, 1 5897
Welding heat of iron, (greatest) 13427
Swedish copper melts, 4587
Brass melts, 3807
Iron red hot in the twilight, 884
Heat of a common fire, 790
Iron bright red in the dark, 752
Zinc melts, 700
Mercury boils, 672
Lead melts, 594
The surface of polished steel becomes uniformly
deep blue, 580
The surface of polished steel becomes a pale
straw colour, 460
Tin melts, 442
A mixture of 3 tin and 2 lead melts, 332
Hgat passes through different bodies with very different
degrees of velocity, and according to the rapidity or slow-
ness with which heat passes through any body, it is said to
be a good or a bad conductor of heat. The conducting
power of copper being 1, that of brass will be 1, iron, 1-1,
tin, 1-7, lead, 2-5. The densest bodies are generally the
'best conductors of heat ; but this is not ufijy.ersal, as platina,
one of the densest of all metals, is one of the worst con-
ductors. Earthy substances are much inferior to metals in
their conducting power, and the worst conductors of all are
the coverings of animals.
When heated bodies are exposed to the air they lose por-
tions of their heat by projection in right lines into space
from all parts of their surface. This is called the radiatior
I
HEAT. 241
of heat. Bodies which radiate heat best have the power of
absorbing it in the same proportion, and the least power
of reflecting it ; hence, in leading steam through a room,
it would be absurd to use black pipes, because, in that case,
much of the heat would escape by radiation before the
steam- would be carried to the place where it was to be used.
If the steam is used to heat the apartment, black pipes are
the best. Hence the cylinder of a steam engine ought to
be polished, but the condenser should not. Vessels in-
tended to receive heat should be black.
The comparative quantities of heat existing in different
bodies may be ascertained by marking the time which equal
quantities of them require to cool a certain number of de-
grees, reckoning their capacities for heat to be as these
times estimated by the volume ; or, if divided by the spe-
cific gravity of the substance, by the weight.
It is necessary here to distinguish carefully between what
is called the specific heat of a body, and its capacity for heat,
these two terms being often confounded. If we take two
bodies at the same temperature, and expose them to the
action of a greater heat, it will be found that one body
will have absorbed a greater quantity of heat than the
other, by the time that they have acquired an equal tern
perature ; and the amount of this additional heat, referred
to some standard, is denominated the specific heat of the
body. Thus if it be found that it requires 1 degree of heat
to raise water from one temperature, T, to another tempera-
ture, f^and if to produce the same change of temperature
in steam-it requires 0-847 degrees, then is 0-847 the specific
heat of steam, water, as the standard, being 1-000. The
capacity of one body for heat compared to another is not
the relative quantities of heat required to raise them a
certain number of degrees, but the absolute quantities con-
tained in them at the same temperature.
CAPACITIES OF BODIES FOR HEAT.
GASES.
Atmospheric air, 1 -7900
Aqueous vapour 1-550C
Carbonic acid gas, .'.1-0454
LIQUIDS.
Alcohol, 1-0860
W/iter, 1-0000
21
842
HEAT.
Solution of muriate of soda, 1 in 10 of water, -9360
Sulphjric acid, diluted with 10 parts water, -9250
Solution of muriate of soda in 6'4 of water, -9050
Olive oil, t -7100
Nitric acid, specific gravity 1-29895, .-6613
Sulphuric acid, with an equal weight of water, -6050
Nitrous acid, specific gravity 1 -354, -5760
Linseed oil, -5280
Oil of turpentine, '4720
Quicksilver, specific gravity 13-30, -0330
SOLIDS.
Ice, , -9000
White wax, -4500
Quicklime, with water, in the proportion of 16 to 9, -4391
Quicklime, -3000
Quicklime saturated with water, and dried, -2800
Pit coal -2800
Pit coal, -2777
Rust of iron, -2500
Flint glass, specific gravity 287, -1900
Iron, -1300
Hardened steel, -1230
Soft bar iron, specific gravity 7'724, -1190
Brass, specific gravity 8-356, -1160
Copper, specific gravity 8-785, '1140
Sheet iron, -1099
Zinc, specific gravity 8-154, -1020
White lead, -0670
Lead, -0352
Specific heats. Specific heats.
Specific heat of water equal 1. Specific heat of water equal 1.
Bismuth, 0-0288 Tellurium, 0-09 12
Lead, -0-0293 Copper, 0-0949
Gold, 0-0298 Nickel, 0-1035
Platinum,.- 0-0314 | Iron 0-1100
Tin 0-0514 Cobalt, 0-1498
Silver, 0-0557 Sulphur, 0-1880
Zinc, 0-0927
HEAT. 243
Large quantities of heat must enter into bodies, and be
concealed, to enable them to pass from the solid to the fluid
state, or from the fluid state to that of vapour. Thus the
quantity of heat necessary to convert any given weight of
ice into water* would raise the same weight of water 14(
degrees of Fahrenheit. This quantity of heat is not sensi
ble, but is, as it were, kept hid or laltn! : nor can it be de
tected by the touch, or by application <if the thermometer.
Every addition of heat applied to water in a fluid state,
raises the temperature until it arrives at the boiling point ;
but however violently the fluid may boil, it does not become
hotter, nor does the steam that arises from it indicate a
greater degree of heat than the water: hence, <i large pro-
portion of the heat must enter into the steam and become
latent. The quantity of heat that becomes latent in steam,
was found by Dr. Black to be 810 decrees of Fahrenheit.
Under the common pressure of the atmosphere at the
surface of the earth, (15 Ibs. on the square inch,) water
cannot be raised above a temperature of 212 Fahr. : but
when exposed to greater pressure, by being confined in a
vessel, the water may be raised to a much higher degree of
heat, and if, in this state of confinement, the heat applied
be insufficient to cause the water to boil : if the vessel should
be open, steam will rush out, and the water which remains
will fall in temperature to 212. On the contrary, water
boils at very low temperatures when the pressure is dimi-
nished ; as in an exhausted receiver, nr at the tops of
mountains.
When the temperature of steam is reduced, it assumes
again the fluid form, and the quantity of latent heat set free
by steam in passing to the state of water, has been found,
by Mr. Watt, to.be 945 degrees. He also found that a
cubic inch of water may be converted into a cubic foot of
steam ; and that when this steam is condensed, by injecting
cold water, tlie latent heat which the steam gives out in
passing to the fluid state, would be sufficient to heal 6 cubic
inches of water to the temperature of 212, or the boiling
point. It is generally considered that steam raised from
boiling water occupies 18 hundred times as much space as
the water did from which it was raised, and instead of
making the latent heat of steam 810, as Dr. Black found it,
mjore correct experiments show it to be 1000, at the ccm
nion pressures of the atmosphere ; but the latent heat of
244 HEAT.
steam is inversely proportional to the degree of pressure
under which it is produced ; that is, the latent heat is
greatest where the pressure is least, and least where the
pressure is greatest.
It has lately been discovered that the sensible heat and
latent heat of steam at any one temperature added together,
give a sum which is constant ; that is to say, which is the
sum of the sensible and latent heat of any other tempera-
ture, or under any other pressure. Now, the sensible heat
of steam at the ordinary pressure of the atmosphere is
212 32 = 180 ; and the latent heat has been found to be
1000, their sum is 1180, which is the constant sum of the
latent and sensib'e heats of steam under any other pressure.
Thus, at the temperature of 248, where the elastic force of
the steam is equal to two atmospheres, or a pressure of 30
Ibs. on the square inch, the sensible heat will be 248 32
= 216, wherefore the latent heat is 1180 216 = 964,
and so of the other temperatures.
It has also been found that while the elasticity of steam
increases in geometrical progression, with a ratio of 2, the
latent heat diminishes with a ratio of T0306, differing hot
very materially from a unit.
Many experiments have been made to ascertain the
elastic force of steam of various temperatures. The most
valuable of them are those recently made by the French
academicians, the results of which are given below in a
tabular form ; and the practical man will duly estimate the
value of this gift of science.
The following simple rule is easily remembered and ap-
plied, and comes near enough to the truth for all practical
uses.
/temperature + lOOV
) = the force of the
v 177
steam in inches of mercury. Thus if the temperature be
307, then,
307 + 100 _.
~T77~
then 2-3 X 2-3 x 2-3 x 2-3 x 2-3 X 2-3 = 148-0359, thin
divrJed by 30, g\ves the atmospheres,
148-0359
= 4'93 atmospheres.
HEAT
245
TABLE OF THE ELASTICITY OF STEAM,
BY M. ARAGO AND OTHERS.
Elasticity of
Irani, the
piw. at the
a!ni'i-|.!irrc
being 1.
temp, in de*. of
Fahrenheit.
Elasticity of
team, the
in--, "f 'In;
atnirwphere
being 1.
Corresponding
temp, in deg, of
Fahrenheit.
1
212
13
380-66
u
231
14
386-94
2
250-5
15
392-86
2
263-8
16
398-48
3
275-2
17
403-83
3d
285
18
408-92
4
293-7
19
413-78
4d
300-3
20
418-46
5
307-5
21
422-96
5
314-24
22
427-25
6
320-30
23
431-42
6d
326-26
24
435-56
7
331-7
25
439-34
71
336-86
30
457-16
8
341-78
35
472-73
9
350-78
40
486-59
10
358-78
45
499-24
11
366-85
50
510-6
12
374
in constructing this table the results were derived from
experiments up to 24 atmospheres, after which the formula
which follows was employed.
E = (l +T + 0-7153) 5
Where E represents the elasticity, and T the temperature,
by the centigrade thermometer, regarding 100 as unity,
and T the excess of tepiperature above 100 It maybe
observed that this formula is more accurate in veiy high
temperatures than for low.
21*
246
HEAT.
ELASTIC FORCE OF STEAM, BY DR. URE.
_ F.lasHr II _
Elastic
Elastic
Elastic
Temp.
force, j lem P-
Temp.
force
Temp.
force.
24
0-170
155
8-500
242 3
53-600
281-8
104-400
32
0-200
160
9-600 245
56-340
283-8
107-700
40
0-250
165
10-800 245-8
57-100
285-2
112-200
50
0-360
170
12-050 218-5
60-400
287-2
114-800
55
0-416!
175
13-550 250
61-900
289
118-200
60
0-516
180
15-160
251-6
63-500
290
120-150
65
0-630JIIS5
16-900
254-5
66-700
292-3
123-100
70
0-726
190
19*900
255
67-250
294
126-700
75
0-860'
195
21-100
257-5
69-800
295
129-000
80
1-010!
200
23-600
260
72-300
295-6
130-400
85
1-170!
205
25-900
260-4
72-800
297-1
133-900
90
1-360'
210
28-830
262-8
75-900
298-8
137-400
95
1-640
212
30-000
264-9
77-900
300
139-700
100
1-860!
216-6
33-400
265
78-040
300-6
140-900
105
2-100!
220
35-540
267
81-900
302
144-300
110 12-456
221-6
36-700
269
84-900
303-8
147-700
115
2-820
225
39-110
270
86-300
305
150-560
120
3-300!
226-3
40-100
271-2
88-000
306-8
155-400
125 (3-830
230
43-100
273-7
91-200
308
157-700
130 4-366:
230-5 43-500
275
93-480
310
161-300
135 5-070
234-5
46-800
275-7
94-600
311-4
164-800
140 i5-770;
235
47-220
277-9
97-800
312
167-000
145
6-600
238-5 50-300
279-5
101-600
312
165-5
150
7-530
240 51-700J280
101-900
Before we describe the application of steam in the
steam engine, we shall briefly allude to some other useful
purposes to which it has been subjected. It has been as-
certained that one cubic foot of boiler will heat about 2000
feet of space, in a cotton mill, to an average heat of about
70 or 80 Fahr. It has also been proved that one square
foot of surface of steam pipe is adequate to the warming of
200 cubic feet of space. This quantity is adapted to a well
finished, ordinary brick or stone building. Cast iron pipes
are preferable to all others for the diffusion of heat, the pipes
being distributed within a few inches of the floor. Steam
is also used extensively for drying muslin and calicoes.
Large cylinders are filled with it, which, diffusing in the
apartment a temperature of 100 or 130, rapidly dry the
suspended cloth. Experience has shown that bright dyec 1
yarns, like scarlet, dried in a common stove heat of 128,
have their colour darkened, and acquire a harsh feel ;
while similar hanks, laid on a steam pipe heated up to 165
HEAT.
247
retain the shade and lustre they possessed in the moist state.
Besides, the people who work in steam drying rooms are
healthy, while those who were formerly employed in he
stove heated apartments, became, in a short time, sickly
and emaciated. The heating, by steam, of lar<_ r e quantities
of water or otheY liquids, either for baths or manufactures,
may be effected in two ways : The steam pipe may be
plunged, with an open end, into the water cistern ; or the
steam may be diffused around the liquid in the interval be-
tween the wooden vessel and the interior metallic case.
Elastic force of vapour of alcohol of a specific gravity of
0-813, water being 1.
Alcohol of S. G. 0-813.
Temp.
Force of vap.
Temp.
Force of yap.
3-3
0-40
180-0
34-73
40-0
0-56
18-2-3
36-40
45-0
0-70
185-3
39-90
50-0
0-86
190-0
43-20
55-0
I'OO
193-3
46-60
60-0
1-23
196-3
50-10
65-0
1-49
200-
53-00
70-0
1-7(1
20(5-0
60-10
75-0
2-10
210-0
65-00
80-0
2-45
214-0
69-36
85-0
2-93
21i-0
72-20
90-0
3-40
220-0
78-50
95-0
3-90
225-0
87-50
100-0
4-50
230-0
94-10
105-0
5-20
232-0
97-10
110-0
6-00
236-0
103-60
115-0
7-10
233-0
106-90
1-20-0
8-10
210-0
111-24
1-25-0
9-25
214-
118-20
130-0
10-60
247-0
122-10
135-0
12-15
248-0
126-10
140-0
- 13-90
249-7
131-40
145-0
15-95
250-0
132-30
150-0
18-00
252-0 138-60
155-0
20-30
254-3 143-70
160-0
22-60
258-6 151-60
165-0
25-40
260-0 155-20
170-0
28-30
262-0 161-40
173-0
30-00
264-0
166-10
178-3
33-50
248 STEAM ENGINE.
THE STEAM ENGINE.
IT is not consistent with the plan of this book, that we
should enter into minute details as to all the modifications
and departments of the steam engine? a subject which
would of itself occupy a large volume. We shall, however,
attempt to explain the leading principles on which this in-
valuable machine operates, so that the mode of calculating
its effects may be the more clearly comprehended.
The engine of Newcomen consists of a hollow cylinder
furnished with a solid piston. This piston is attached
to a rod, the top of which is connected with a large beam,
resting upon a fulcrum in the centre. To the other end
of this large beam, called the working beam, the pump rod
is attached. When steam is admitted into the bottom of
the cylinder, it will, by the superiority of its elastic force
above the pressure of the atmosphere, assisted by the coun-
teraction of the weight of the pump rod, cause the piston
to rise to the top of the cylinder. But when the piston
arrives at this point, cold water is injected into the cylinder,
by which the steam is condensed, and a vacuum formed,
then the pressure of the air on the top of the piston will
cause it to descend to the bottom of the cylinder. The
steam is again injected and again condensed, and thus the
operation of the machine is continued. This is called the
atmospheric engine. It is liable to this objection, that
there is a great waste of steam, and consequently of fuel
incurred in consequence of the steam being condensed in
the cylinder, since the cylinder must be heated to a certain
temperature, before the steam which it contains can exert
a sufficient elastic force, and the admission of cold water
cooling it down below this temperature, a considerable
quantity of steam is employed in again raising its heat to
the proper point.
In order to obviate this defect, the* illustrious WATT
made such arrangements as enabled him to condense the
steam in a separate vessel, and thus to maintain a uniform
temperature in the cylinder. By this great improvement
the effect of the same quantity of steam was increased in
about the proportion of 12 to 7. Such was the principle
of Watt's single-acting engine ; but he afterwards so ar-
ranged the structure of the machine as to admit the steam
STKAM ENGINE. 249
alternately above and below the piston, and still to con-
dense it in a separate vessel, as will be understood from the
description of the engraving, plate III, which will be given
a little farther on. This form of the steam engine is called
the double-acting line-pressure engine.
The steam engine was further improved by Mr. Watt,
by his shutting off the steam when the piston had passed
through a portion of its stroke, by which means the acce-
lerated motion of the piston is counteracted, from the elastic
force of the steam diminishing: during its expansion. This
is the principle of what is called the expansive engine.
In the kigk-presfttre steam engine, the steam, of high
temperature, is admitted into the cylinder alternately above
and below the piston; but instead of being condensed, H is
allowed to escape into the atmosphere. In this engine,
which is the most simple in its construction, the steam acts
by its elastic force alone.
The construction of the low-pressure double-acting steam
engine, will be understood in its more minute details, from
the following description.
Plate III is a side elevation of a low-pressure portable
double-acting steam enginfe, in which the boiler and the
other principal parts are drawn in section.
After the flame frflm the furnace A passes under the
whole bottom surface of the boiler, it enters the flue C,
from which it escapes into a flue running up one side of the
boiler ; from this side flue it passes into the end flue D,
which carries it into a flue running along the other side of
the boiler ; and from this last the smoke is conducted into
the chimney E. The bridge B helps to spread the flame
over the bottom of the boiler. When the furnace is cleaned,
the plate between the end of the furnace bars and the bridge
can be drawn forward by means of two handles, (one of
which only is shown,) in order that the .cinders may be
pushed over the end of the furnace bars into the ashpit.
If one of the gauge cocks, FF, is opened, it will emit
steam ; and the other cock if opened will blow out water
. if the boiler be just as full of water as it ought to be. As
these cocks stop up sometimes, a wire may be passed down
through them, if the part above the key is not bent over.
^The writer should always stand somewhere between the
'dotted lines passing below the ends of the gauge cocks.
, <" is a small valve opening inwards, placed in the man-hole
250 STEAM ENGINE.
i
door, to keep the sides of the boiler from being pressed
together by the force of the atmosphere, if the steam should
happen to be suddenly condensed by the water that feeds
the boiler. HH is the feed pipe, and the small valve sus-
pended from the point O, of the lever K, regulates the
quantity of water passing into the boiler; the lever which
works the feed valve is connected by means of a rod to the
float I, which rises or falls along with the water in the
boiler, and this opens or shuts the valve, according as the
water stands low or high in the boiler. The pipe L con-
ducts the water into the feed pipe from a cistern fixed
above the boiler house, which is kept full by means of the
hot water pump, which takes in water from the hot well.
The cistern on the top of the boiler house should be large
enough to fill the boiler, as also the large cistern on which
the engine stands, if they should happen to be empty at any
time. The pipe M carries away any overplus water from
the feed pipe. NN is the pipe which conveys the steam
from the boiler into the nozles, and the safety valve is
placed above the bend in it. Q is a section of the cylinder,
showing also the outside of the metallic piston. The oblong
opening, near the top of the condenser R, admits a jet of
cold water to condense the steam after it has acted in the
cylinder. The injection cock is boiled to the outside of
the oblong opening, and the water which is forced through
it into the condenser by the pressure of the atmosphere is
taken from the large cistern on which the engine stands ;
this cistern is always kept nearly full by the cold water
pump. The hot and cold water pumps are both wrought
off the same spindle P, fixed in the working beam, a pump
being attached to each end of the spindle. The foot valve
S is placed between the condenser R, and the air pump T.
The bucket shown in the air pump is not sectioned. The
valve in the air pump bucket, and the discharging valve,
which opens into the hot well on the top of the air pump,
have each a shallow flat-bottomed recess turned on the top,
so as to fit nicely the flat-bottomed disks X and W ; the
one disk is keyed on the air pump rod, and the other is
fixed by means of studs and nuts to khe hot well ; as it
gives more water way, it is an improvement to have the
recess in the valve, rather than in the disk. If each valve
had not a recess turned in it to contain a quantity of water,
which, as it is forced out by the disk, reduces the momen-
STEAM ENGINE. 251
turn of the valve by degrees ; the stroke of the valve on its
disk or guard would be very great, and the parts would
soon work out of order. The pipe which carries away the
water that is pumped out of the condenser by the air pump,
is shown near the top of the hot well, on the side farthest
from the cylinder.
The way in which the tire is regulated, is as follows :
When the steam gets too strong, the water in the boiler
rises in the feed pipe, and carries up the float W ; and as
the float is connected by a chain and a pulley with the
damper V, the damper descends into the flu*e, and reduces
the draught in the furnace, and the force of the steam.
Again, if the steam gets too low, the float falls and raises
the damper, to increase the draught. The two pulleys
which form the connexion between the damper and the
float are both fixed on one shaft ; on atcount of the one
being placed exactly behind the other, one of them only
can be seen.
As the balls YY are carried round along with the rod Z,
when the engine is going too quick, the balls by their cen-
trifugal force fly out, and the rods and levers in connexion
with them shut more or less a valve at A', in the steam
pipe ; if the engine goes too slow, the balls fall down, and
open this valve to give the engine more steam to bring up
its motion. The rod B', and the lever C', form part of the
connexion with the valve in the steam pipe and the go-
vernor.
It is clear, that the power of the steam engine will de-
pend upon the energy of the steam, 1st. Steam of two
atmospheres will, other things being equal, produce double
the effect of steam of one atmosphere. 2d. the force of the
steam remaining the same, the power of the engine will
depend on the extent of surface acted upon, that is, on the
area of the piston. 3rf. these two circumstances remaining
the same, the power of the engine will depend on the
velocity with which the piston moves.
For the sake of illustration, let us suppose that steam is
admitted into the cylinder, so as to press down the piston
with the force of one hundred pounds, and that the length
of the stroke is five feet ; and suppose that the end of the
piston red is attached to a beam whose fulcrum is in the
Jcentre, and that to the other end of the beam there is
attached a weight of any thing less than one hundred
252 STEAM ENGINE.
pounds, there being no friction. By the descent of the
piston, the weight at the end of the beam will be raised 5
feet; therefore it follows, that 100 pounds raised 5 feet
during one descent of the piston, will express the mechanic-
al effect of the engine. The reader will easily perceive
that the weight at the end of the beam must be somewhat
less than 100 pounds, for as it acts contrary to the power
of the piston, if they were equal the machine would be at
rest. If we suppose the area of the piston double of what
t was before,, other things being the same, the engine
would raise 200 pounds through the same space of 5 feet
in the same time : and the same effect would evidently
ensue if we supposed the area of the piston to remain as it
was at first, but the force of the steam to be doubled. If
the area of the piston and force of steam be the same as at
first, but the length of stroke doubled, then the mechanical
effect of the engine will be 100 Ibs. raised 10 feet high
during one descent of the piston ; and if the descents be
performed in the same time, this engine will be double the
power of the first.
Let us proceed now to actual cases. In the common
low-pressure steam engine of Watt, steam is admitted into
the cylinder whose elastic force is somewhere about that
of the. atmosphere, which we have all along supposed to be
15 Ibs. to the square ir.ch ; but friction and imperfect
vacuums tend to diminish this pressure, and the effective
pressure may be reckoned only four-fifths of this. If the
pressure of the steam is diminished by its one-fifth part,
which is 3 Ibs. to the square inch, then will the effective
pressure be 12 Ibs. to the square inch. The working
pressure is generally reckoned at 10 Ibs. to the circular
inch, and Smeaton only makes it 7 Ibs. The effective
pressure we have taken is between these extremes, being
equivalent to 9-42 Ibs. to the circular inch.
Mr. Tredgold gives the following table, which will show
how the power of the steam, as it issues from the boiler, is
distributed. In an engine which has no condenser :
The pressure on the boiler being lO'OOO
1. The force necessary for producing
motion of the steam in the cylinder- *0069
2. By cooling in the cylinder and pipes -0160
3. Friction of piston and waste '2000
STKA.M iJNGJ.Nj;. 253
4. The force required to expel the steam
into the atmosphere '0069
5. The force expended in opening the
valves, and friction of the parts of an
engine -0622
6. By the steam being cut off before the
end of the stroke -1000
Amount of deductions 3920
Effective pressure- 6080
In one which has a condenser :
The pressure on the boiler being 1000
1. By the force required to produce motion
of the steam into the cylinder 007
2. By the cooling in the cylinder and pipes 016
3. By the friction of the piston and loss 125
4. By the force required to expel the steam ;
through the passages 007
5. By the force required to open and close
the valves, raise the injection water, and
overcome the friction of the axes 063
6. By the steam being cut off before the end
of the stroke 100
7. By the power required to work the air-
pump 050
368
632
If we now suppose a cylinder whose diameter is 24 inches,
the area of this cylinder, and consequently the area of the
piston in square inches, will be,
24 a x '7854 = 452-39.
Let us also make the supposition that steam is admitted
into the cylinder of such power as exerts an effective pres-
sure on the piston of 12 Ibs. to the square inch ; therefore,
452-39 X 12 = 5428-68 Ibs., the whole force with which
the piston is pressed. If we now suppose that the length
' of the stroke is five feet, and the engine makes 44 single
or 22 double strokes in a minute, then the piston will move
through a space of 22 x 5 x 2 = 220 feet in a minute :
<end from what has been said before, it will not be difficult
to see, that the power of the engine will be equivalent to a
-weight of 5428 Ibs. raised through 220 feet in a minute.
22
254 STEAM ENGINE.
This is the most certain measure of the power of a steam
.engine. It is usual, however, to estimate the effect as equi
valent to the power of so many horses. This method,
however simple and natural it may appear, is yet, from
differences of opinion as to the power of a horse, not very
accurate ; and its employment in calculation can only be
accounted for on the ground, that when steam engines were
first employed to drive machinery, they were substituted
instead of horses ; and it became thus necessary to estimate
what size of a steam engine would give a power equal to
so many horses.
There are various opinions as to the power of a horse.
According to Smeaton, a horse will raise 22,916 Ibs. one
foot high in a minute. Desaguliers makes the number
27,500 ; and Watt makes it larger still, that is, 33,OOC
There is reason to believe that even this number is too
small, and that we may add at least 11,000 to it, which gives
44,000 Ibs. raised one foot high per minute.
Now, in the case above, we found that the engine of 24
inch cylinder, would raise 5428 Ibs. through the space of
220 feet in one minute ; and it is easily seen that it could
raise 220 x 5428 Ibs. through one foot in the same time,
therefore, 220 X 5428 = 1194160 Ibs. raised through one
foot in one minute, is the effective power of the engine ;
and from these considerations it will be easy to find the
power according to the different estimates of a horse's
power. For,
1194160
22Q16 = 52 horses power,
according to Smeaton.
1194160
TTSOO" = 43 horses P wer '
according to Desaguliers.
1194160
33000" = 3
according to Watt.
H94160
44000 = 27 horses' power,
according to the usual estimate.
The reader will have no difficulty in forming a general
rule for estimating the power of a steam engine. (The
STEAM ENGINE. 255
effective pressure on each square inch x the area of piston
in square inches X length of stroke in feet X number of
strokes per minute) -5- 44000 = the number of horses'
ppwer of the engine.
What is the power of a low-pressure engine, whose
cylinder is 30 inches diameter, length of stroke 6 feet,
making 16 double strokes in the minute ?
NOTE. An easy rule to find the area of the piston in
s>quare inches, is this,
The diameter x circumference
- = area.
4
Here we have,
30 X (30 x 3-1416) 2827-44
= 706-86,
4 4
equa the area of the piston in square inches ; and 12 the
effective pressure, 6 the length of stroke, 16 the number of
double strokes in a minute ?
706 86 Xl2x6xl6x2_ 1628605*44 _
44000~~ 44000
horses' power.
If the cylinder of a higrh-pressure steam engine has a
piston of 5 inches diameter, with a twelve inch stroke,
making 32 double strokes in a minute ; steam being ad-
mitted of an elastic force equivalent to 7 atmospheres on the
inside of the cylinder. Its effective pressure will be 7 X
15 = 105 Ibs. to the square inch without friction; but al-
lowing one-fifth for friction, the effective pressure will be
105 21 = 84 Ihs. to the square inch.
5 X (3-1416 X 5)
here - - - - = 19-63 the area of the piston :
19-63 X 84 x 1 X 32 x 2 _ 105530-88
44000 " 44000
horses', power.
A convenient rule for finding the power of a high-pres-
sure engine, is let P be the force of the steam in the
boiler, A the area of the piston, and V the velocity of the
piston in feet per minute, then,
0-9 P _ 6 x A X V
44000
^ The pressure of the steam in a boiler is 30 Ibs. per
square inch, the diameter of cylinder 12 inches, length of
25tl STEAM ENOINE.
stroke 3 feet, and the engine making 30 double strokes put
minute. Here the area of piston will be 113-097, tho
velocity of piston = 3 x 30 X 2 = 180 feet per minute
and since 0-9 x 30 6 = 21, then,
0-9 x 30 6 X 113-097 X 180 427506-66
44000
9*7 horses' power.
We might simplify this rule still farther on the consi
deration, that the divisor 44000 may be viewed as the de
nominator of a fraction whose numerator is one, and by
converting this into a decimal, and multiplying by it, we
might avoid the necessity of division.
Since - = -0000227, hence we may devise the rule.
44000
Effective pressure of steam X area of piston in square
inches x length of stroke in feet x number of strokes per
minute x 227 ; and from the product cutting off seven
places as decimals ; the horses' power of the engine.
This is for a single stroke engine for a double stroke
engine the multiplier is 227 X 2 = 454.
If the cylinder be 42 inches diameter, and the piston
moves 210 feet per minute, then the engine being low
pressure, we have,
area of cylinder equal 1385-44 ; hence 227 X 1385-44 X
210 X 12 = 792527097 :
and the seven figures cut off as decimals, leave 79 horses'
power.
These are at best but approximations, and for safety it
might be advisable that a lower- number than 12 should be
employed, as the effective pressure of the steam ; the num
her 10 may be used as being easily managed, and coming
near the truth ; and thus the above rule may be simplified
by neglecting the pressure of the steam, and cutting off six
places for decimals instead of seven, as there is reason to
believe that the above results will answer only ponies in-
stead of strong horses.
The stroke of an engine is commonly reckoned equal to
one complete revolution of the crank shaft, and therefore
double the length of the cylinder, and it has been stated
by Mr. Thomas Tredgold, that to ascertain the velocity of
the piston when the engine performs at its maximum, we
may employ the rule,
STEAM ENGINE. 257
120 X v/ length of stroke = velocity.
If an engine has a two feet stroke, then,
120 X N/ 2 = 120 x 1-4142 = 109-704,
or we may say 170, as the velocity of the piston per minute
in feet ; wherefore as the engine has a single stroke of 2
feet we have,
170
= 42s strokes in the minute.
4
If an engine have a four feet stroke, then we have,
120 X v/ 4 = 120 X 2 = 240 =
the velocity of the piston per minute ; and,
240
= 30, equal the number of strokes per minute,
o
The safety valves of most of the steam engines in this
part of the country, are generally loaded with a weight of
from 3 to 4 Ibs. to the square inch of their area; let us
take 3 5 Ibs. in the present instance. The temperature of
steam necessary to balance this pressure, is, according to the
best experiments, 223 degrees of Fahrenheit's thermometer.
But besides this sensible heat, there is a quantity of latent
heat not indicated by the thermometer, and which can only
be detected when the steam passes, by condensation, into
the fluid state ; as the latent heat is then given out. Now,
if the latent heat of the steam at the above temperature, be
found on the principle stated in our remarks on heat, that the
sensible and latent heats of steam at all temperatures, when
added together, make a constant quantity ; we will find that
the latent heat of steam at this temperature is 989. The real
quantity of heat then in the steam is 223 -f 989= 1212
degrees. We will not be far from the truth in supposing,
that one cubic foot of this steam will, when condensed into
water, measure one cubic inch ; and the steam is supposed
to be condensed by the injection of cold water. Now it is
evident, that the temperature of the water formed by the
condensation of the steam, will be somewhere between the
temperature of cold waierandthe boiling point. Say that
the temperature of the injected water is 50 degrees, and that
the temperature of the water arising from the condensation
of the steam is 100. We must deduct the 100 degrees from
Lj:e heat of the uncondensed steam, that is, 1212 100 =
H112, which is left to be communicated to the injection
22*
258 STEAM ENGINE.
water , and since each cubic inch of the cold water requires
50 of heat to raise it to the temperature of the water found
after the condensation of the sieam, therefore,
1 1 12
= 22 T 3 , cubic inches
50
of water necessary to condense one cubic foot of steam to
the temperature of 100, the injected water being 50.
From these considerations may be Serived a rule for de-
termining the quantity of water necessary to condense any
quantity of steam, at any given temperature.
Total heat of the steam temperature of warm water
temp, of warm water temp, of cold water
quantity of steam in cubic feet = the quantity of cold water
in cubic inches necessary to produce the effect.
Let us illustrate this by an example. What quantity of
cold water will it require of the temperature of 60, to con-
dense 8 cubic feet of steam, of the temperature of 223, to
water at 90 ? The whole heat is as before, 989 -f 223 =
1212, wherefore by the rule,
1212 90
X 8 = 299-2 cubic inches =
90 60
299-2
= *17 of a cubic foot of water.
1728
From this it will be easy to determine how much water
must be discharged by the pump which feeds the condenser,
in order that a proper vacuum may be formed.
From practice it would appear that about 26 cubic inches
of cold water for condensing should be used for each cubic
foot of the capacity of the cylinder.
We may infer from observation, that the engines com-
monly in use require betwixt 3| and 4 gallons of cold water
per minute for each horse's power. If the water is return-
ed as it is in some engines, then a greater quantity will be
necessary. Now, in the usual construction of engines, the
pump rod which supplies the condenser with cold water,
is fixed halfway between the end of the beam and the cen-
tre ; hence, the length of its stroke is one-half that of the
piston in the large cylinder: therefore, if there be a 40
horse power engine, the length of whose stroke is 6 feet,
the length of the stroke of the pump will be 3 feet.
Now an imperial wine gallon occupies a space of 277'274
STKAM KNGI.M:- 259
cubic inches, and 71 Dillons will occupy a space of 277'274
X 7'5 = 2079-555 cubic inches; and as the engine is 40
horses' power, there must be discharged in one minute,
2079-555 x 40 = 83182-2 cubic inches,
and if the engine makes 30 strokes per minute, then
83182-2
= 277-274 cubic inches
30
discharged at one stroke : but the stroke is 3 feet long, and
it remains only to find what must be the diameter of a
pump's bore, whose length is 36 inches, so that its capacity
shall be 2772 ; hence we find that,
2772
-^- = 7/ inches,
nearly equal to the area of the pump's bore ; now the area
of circles are to each other as the squares of their diame-
ters, and the area of a circle whose diameter is 9, is 63-6:
therefore,
63-6 : 77 : : 0- : JM,
the square root of which will be the diameter of the pump,
and will be found = 9-9 inches.
With respect to the fly wheel,
Horses' power of engine x 2000
Velocity of circumfer. wheel in feet per second-
the weight of the fly wheel in cwts.
If the diameter of the fly of a 30 horse power engine be
20 feet, and make 18 revolutions per minute, then,
20 X 3-1416 = 62-832 =
circumference in feet, and 62-832 X 18 = 1128-97 feet,
the space which the circumference moves through in one
minute ; hence,
1128-97
=18-81 feet per second;
30 x 2000 60000
18-81- = 353* == 169 CWtS '
r= 8 tons 9 cwts. the weight of the fly.
In the working of the valve of a steam engine, an eccentric
wheel is often employed, and it becomes necessary to cal-
culate the degree of eccentricity necessary to give a certain
length of stroke. The eccentric wheel's radius mav bo
260 STEAM ENGINE.
easily found; thus, suppose the length of stroke required is
20 inches, and the diameter of the shaft on which the wheel
is screwed is 5 inches, and the thickness of metal required
to key on the wheel 21 inches. Take the half of the re-
quired stroke, that is, 10 inches, as the distance of the cen-
tre of the shaft from the centre of required wjieel, and add-
ing to this the half thickness of the shaft = 2,1 inches, as
likewise the thickness of metal necessary for keying = 21,
then 10 + 21 + 21 = 15 inches, ihe radius of the wheel.
Now let E be = the radius of the eccentric wheel L = the
length of the eccentric rod, and / -= the length of the bar
between the other end of this rod and the slide ; and let e
= the length of slide; then,
E = L _*_? r == /xE
/ e
I x E L x e
P /
L E
Suppose the length of the stroke of the slide e 6 inches,
the length of the slide rod / = 5 inches, and the radius of
the eccentric = 24 inches = E, then the length of the rod
5 X 24
L = = 20 inches.
6 ,
The other rules are wrought on the same principle
We have before spoken of the governor while treating of
central forces and rotation. It remains for us here only
to observe, that the governor performs in one minute half
as many revolutions as a pendulum, whose length is the
perpendicular distance between the plane in which the balls
move and the centre of suspension. Thus, if the distance
between the point of suspension and the plane in which the
balls move be 28 inches :
[/ 39-1386 \
.]( ^ ) = I 1 182 vibrations in a second from the
nature of the pendulum ; hence,
1-182
- = 0-591, the revolutions of the governor in a se-
cond, or 0-591 x 60 = 35-46 in one minute.
The piston rod of a steam engine may be made to move
up and down in a right line in various ways. The rod may
be made to terminate in a rack, the teeth of which act in
S1EAM ENGINE 2611
the teeth of an arched head of the .oAg lever, called the
working beam : but the most efficacious of all contrivances
of this kind, is that of Watt, commonly called the parallel
motion This contrivance is founded on geometrical prin-
ciples, which it would be inconsistent with the plan of this
work to consider ; we shall therefore simply describe the
contrivance of this illustrious mechanic.
The working beam has an alternating circular motion
ound its centre A, and it is clear that the points B ami G
ill have a circular motion round the common centre A.
et the point B be exactly in the middle, between the
centre and end of the beam. Let there be a bar or rod
CD, of the same length as AB, capable of moving round
the centre C, by means of a pivot. The other end of this
rod is" attached by means of a pivot, to the rod DB. Now,
by the alternate rising and falling of the beam, the points
B and D will move in circular arches, but the middle point
P of the connecting rod BD, will move upwards and
djwnwards in a vertical straight line, or at least so very
neatly so, as the difference cannot be perceived. Now, to
this point P, there is attached the end of the pump rod,
which will, of course, follow the direction of the impelling
point, and move in a straight line. For the purpose of
communicating a similar motion to the other piston rod,
conceive another rod CP' introduced, of the same length as
BD, and its extremities moving likewise on pivots. The
oiston rod of the cylinder is attached to the point P', and
.this point moves quite in the same way as the point P.
The only difference in the motion of these two points will
jje, that the point P' will move twice as fast as the point P,
or will, in the same time, move twice as far.
262 STEAM ENGINE.
The length of the links are made = 4 to 5, the length of
the stroke being 1, according to circumstances, the longef
link being preferred when practicable. From the length
of the links must be determined the position of the radius
bar, for the vertical distance between the centres of motion
of the working beam and the radius bar must be equal to
the length of a link.
When the parallel bar is not more than one-half of the
working beam's radius, then,
Let B = radius of the beam,
P = length of parallel bar,
S = length of stroke,
R = length of radius bar; we have
B 2Px(iS)" R
B ^B' [iS] 2 ) X2P "*
Suppose the length of the beam from the centre = 12
feet, the length of stroke 6, and of parallel bar 5 feet,
that is, B = 12, S = 6, and P = 5, then,
B 2 P X (S) a = 12 10 X (^6) 3 = 18
= the dividend ; then, B v/(B 3 [|S] a ) x 2 P = 12
v/(12 a [16] 2 ) x 10 = 12 11-62 X 10 = 0-38
X 10 = 3-8 the divisor, wherefore,
18
- + 5 = 9-74 = the length in feet of the radius bar.
3'8
When the parallel bar is more than half the length of the
radius of the beam, the rule is,
2P-BX(|S)
B (B a [|S] 2 ) x 2P
by which rule it will be found that when the length of
stroke and radius bar are each 6, and the radius of beam 10
feet, the length of radius bar will be 2-75 feet.
Many rules have been given for the quantity of fuel
necessary for the production of steam, but they cannot be
depended on, so many circumstances must be taken undei
consideration the quality of material used for fuel and
the mode of constructing the fireplace.
It has been found that 3 cwt. of Newcastle coals are
equivalent to 4 cwt. of Glasgow co?ls, or 9 cwt. of wood,
or 7 cwt. of culm. A chaldron of coals in London contains
36 bushels, and weighs 3136 Ibs., or nearly 1 ton. 8 cwt.
.
STEAM ENGINE. 263
It would appear, that in the common low-pressure steam
engines, the eonsumpt of coal per hour for 1 horse power,
is about 16 Ibs., of wood 56 Ibs.,- and of culm 35 Ibs.
These statements are given somewhat large, and by proper
regulation much less fuel might serve.
In the boiler there are certain proportions generally ob-
served. The width, depth, and length, are as the numbers
1, 1-1, 2-5. So that if the width be 5 feet, then the depth
will be 1-1 x 5 = 5 feet 6 inches ; and the length 5 x 2*5 =
12 ft. 6 in. ; and the whole content of the boiler will be,
5 x 5-5 X 12-5 = 343-75 cubic feet.
Now Boulton and Watt allow 25 cubic feet of space in the
boiler for each horse power ; and according to this estimate,
343'75
= 13 and a fraction, the number of horses' power
of this engine for which this boiler would be fitted. Some,
instead of computing the size of boiler in this way, allow
5 square feet of surface of water for each horse's power;
but in all cases, it is common to make the boiler of a size
fitted for an engine of at least 2 horses' power more than
that to which it is applied.
There are two ways of loading the safety valve of a
boiler ; the one by placing a weight on the top of it, and
the other by causing the weight to act on the valve by a
lever.
When the weight is placed upon the valve ; area of
valve X pressure per square inch = whole weight, and also
whole weight
= pressure per square inch.
area of valve
Thus, if a weight of 50 Ibs. be placed upon a valve whose
area is 10 inches, then the pressure per square inch is
= 5 Ibs. pressure per square inch.
When the weight acts by a lever, it is placed at one end
the fulcrum being at the other, and the valve connected
with the lever somewhere between them ; this, then, is a
simple case of the lever. Hence, if the length of the lever
be 24 inches, the diameter of the valve 3 inches, (its area
will be 7,) the distance between the fulcrum and the valve
3 inches, then to give 60 Ibs. pressure per square inch on
Ahe valve 60 x 7 = 420 Ibs. the whole pressure on the
valve, and
264
STEAM ENGINE,
420 x 3
60 Ibs. will be the weight hung at the
24 3
end of the lever to give the required pressure.
To find the action of the weight of the lever divide its
whole length by the distance of the valve from the fulcrum,
and multiply the quotient by half the weight of the lever.
The following rules for calculations connected with the
steam engine are extracted from a useful little compendium
lately published by Mr. Templeton, of Liverpool. These
rules we have inserted here, not so much for their superior
accuracy, as from a desire to present our readers with
methods by which they may approximate to the true results
by means of the sliding rule. It is to be observed that the
term gauge point is used to denote the number to be taken
on the line stated in the rule.
Length of
strtfke in
ft. and in.
Gauge point.
Length of
stroke in
ft. and in.
Gauge point.
2
295
6
392
2 6
318
7
41
2 9
322
8
414
3
33
MARINE ENGINES.
3 6
335
3
3
4
343
3 6
31
4 6
355
4
317
5
385
4 6
326
RULE. Set the gauge point upon C to 1 upon D, and
against the number of horses' power upon C, is the diame-
ter in inches upon D ; or, against the diameter in inches
upon D, is the number of horses' power upon C.
Ex. 1. What diameter must a cylinder be with a 4 feet
stroke, to be equal to 20 horses' power?
Set 343 upon C to 1 upon D ; and against 20 upon C is
24'2 inches diameter upon D..
Ex. 2. What number of horses' power will an engine
be equal to, when the cylinder's diameter is 19 inches and
stroke 3 feet ?
19 3 X "7854 x 7-25 x 192 _ 394672-7328 _
33000 33000
11*96 or 12 horses' power nearly.
i.
STEAM ENGINE. 265
The proportion of parts of a high-pressure steam en-
gine. The U:ngih of the stroke should, if possible, be
twice its diameter. 'The velocity in feet per minute should
be 103 times the square root of the length of the stroke in
feet. Anil, as 4S-M) is to the velocity thus found, so is the
area of the cylinder to the area of the steam passages.
The proportions of the parts of an atmospheric en-
gine. The length of the cylinder should be twice the dia-
meter. The velocity in feet per minute should be ninety-
eight times, the square root of the length of 'the stroke in
feet. The area of the steam passages will be as 4800 is to
the velocity in feet per minute, so is the area of the cylin-
der to the area of the steam passage. If the area of the
" cylinder in feet be multiplied by half the velocity in feet,
and that product by 1"23 added to 1'4 divided by the dia-
meter in feet, the result divided by 1480 will give the cubic
feet of water required for steam per minute. If the num-
ber of times the quantity of water required for injection
must be greater than that required for steam, in general it
will be about twelve times the quantity, but it had better be
a little in defect than excess. The aperture for the injec-
tion must be such that the above quantity of water will be
injected during the time of the stroke. In order that the
injection be sufficiently powerful at first, the head should
be about three times the height of the cylinder ; and making
the jet apertures square, the area should be the 850th part
of the area of the cylinder. The conducting pipe should
be about four times the diameter of the jet.
The piuportions of the parts of a single-acting low-
pressure engine. The length of the cylinder should be
twice its diameter. The velocity of the piston in feet per
minute should be ninety-eight times the square root of the
length of the stroke. The area of the steam passages should
be equal* to the area of the cylinder, multiplied by the velo-
city of the piston in feet per minute, and divided by 4800.
The air pump should he one-eighth of the capacity of the
cylinder, or half the diameter and half the length of the
strike of the cylinder, and the condenser should be of
the same capacity. The quantity of steam will be found
bv multiplying the area of the cylinder in fet by half the
velocity in feet; with an addition of one-tenth for cooling
f ami waste, and this divided by the volume of the steam
corresponding to its force in the boiler, gives the quantity
23
266 RAILWAYS.
of water required for steam per minute, from whence the
proportions of the boiler may be determined.* At the com-
mon pressure of two pounds per circular inch on the valve,
the divisor will be 1497. The quantity of injection water
should be twenty-four times that required for steam, and
the diameter of the injection pipe one-thirty-sixth of the
diameter of the cylinder. The valves in the air pump bucket
should be as large as they can be made, and the discharge
and foot valves not less than the same area.
Summary of proportions of a double engine, working
at full pressure. The length of a cylinder should be twice
its diameter ; for a cylinder having this proportion exposes
less surface to condensation than any other enclosing the
same quantity of steam. The area of the steam passages *
should be about one-fifth of the diameter of the cylinder;
or their area should be equal to the area of the cylinder,
multiplied by the velocity of the piston in feet per minute,
and divided by 4800. The diameter of the air pump should
be about two-thirds of the diameter of the cylinder, and
half the length of stroke ; and the larger the passages
through the air bucket and the discharging flap are, the
better. The quantity of water for injection should be about
283 times that required for steam, or about 26 cubic inches
to each cubic foot of the contents of the stroke of the pis-
tog. Watt considered a wine pint, or 28g cubic inches,
quite sufficient. There should be 62 times as much water
in the boiler as is introduced at one feed.
These proportions are taken from Tredgold's valuable
treatise on the steam engine.
RAILWAYS, STEAMBOATS, &c.
IT has been deduced from very extensive experiments or
the Liverpool and Manchester railways, that the 'effective
power of a locomotive engine is about '3 of the pressure of
the steam on the piston, on the calculated power of the
engine being 1. In one case, for instance, a cylinder 21
inches diameter was used, the elasticity of steam in the
boiler was 30 Ibs. to the square inch, above the pressure
* To 459 add the temperature in degrees, and multiply the sum by
76-5. Divide the product by the force of the steam in inches of mer-
cury, and the result will be the space in feet the steam of a cub'c foot
of water will occupy.
RAILWAYS. . -867
ol the atmosphere. The length of the rail, which was in-
clined, \vas :H65 feet, and the height 24 feet. The time
of drawing 6 loaded wagons, each weighing 9010 Ibs. up the
rail, was 570 seconds, during which time the engine made
444 single strokes, each 5 feet long. Now,
21 s x '7854 = 346-36 = the area of the piston in square
inches, wherefore, 346'36 x 30 = 10390 Ihs. = the pres-
sure of steam upon the piston, whose stroke was 5 feet,
and number of strokes in the given time 444; hence 444 x
3 == 2220 feet = the space through which the power 10390
has traversed; therefore, 10390 x 2220 = 23065800 Ibs.
= the impelling power of the engine. Now, it was found
that the actual weight including resistance moved, wa*
'7124415 Ihs.; then,
7124415
which will give the effect about 30-9 per
230uobOO
cent., but the foregoing number may be taken as a safe me-
dium, that is, 30 per cent or -3.
The amount of retardation, arising from steam carriages
moving on railways, has been estimated thus ;
Loaded carriages weighing altogether 8522 Ibs. the fric-
tion amounted to 50 Ibs., or the ^-$ part of the weight. In
empty carriages weighing 2586 Ibs., the friction amounted
to 10 Ibs., or the ^TS P art f tne weight; and the friction
may be regarded as a constant retarding force. Wrought
iron rails seem from a multitude of experiments to be much
better than those of cast-iron, as they are more durable and
cause less friction.
The Rocket was tried, weighing 4 tons and 5 cwt., to it
there was attached a tender with water and coals, weighing
3 tons, 2 cwt. quar. 2 Ibs. ; and two carriages loaded
with stones, weighing 9 tons, 10 cwt. 3 qr. 26 Ibs., making
in all .17 tons. At full speed she moved at the rate of 30
miles, in 2 hours, 6 minutes, 9 seconds, or 14| per hour
at the end of stage, about 6 miles ; and the greatest velocity
was 29 1 miles per hour. The quantity of water used 92'6
cubic feet, and it required 11/j Ibs. of coke for each cubic
foot of steam.
In the Rocket the boiler is cylindrical, with flat ends 6
feet long, and 3 feet 4 inches in diameter. To one end of
the boiler there is attached a square box as a furnace, 3 feet
Jong by 2 feet broad, and about 3 feet deep at the bottom
of this box five bars are placed, and the box is entirely
surrounded with a casting, except at the bottom and the
"8 RAILWAYS.
side next the boiler. Betwixt the casting and the box
there is left a space of about 3 inches, which is kept con-
stantly filled with water. The upper half of the boiler is
used .as a reservoir for steam; the under half being kept
filled with water, and through this part copper tubes reach
from one end to the other of the boiler, being open to the
fire box at one end, to the chimney at the other ; these
tubes are 25 in number, each being 3 inches in diameter.
The cylinders were each 8 inches in diameter, and one was
at each side of the boiler; the piston had a stroke of 16^
inches. The diameter of the large wheels was 4 feet 85
inches. The area of the surface of water, exposed to the
radiant heat of the fire, was 20 square feet, being that sur
rounding the fire box or furnace ; and the surface exposed
to the heated air or flame from the furnace, or what may
be called communicative heat, is 117'8 square feet.
The average velocity of the Rocket may be stated at 14
miles per hour, and during one hour she evaporates 18'24
cubic feet of steam, with a consumpt of about 17'7 Ibs. of
coke for each cubic foot of water.
An empirical rule has been given for the ascertaining of
the quantity of fuel necessary for steam carriages, which
may be useful.
The weight of the load X 51 '55 -f weight of carriages
898
the quantity of coals required to carry one mile, but a near
approximation to the truth may be to allow 2 Ibs. for every
ton for one mile.
Iron railroads are of two descriptions. The flat rail, or
tram road, consists of cast iron plates about 3 feet long,
4 inches broad, and 5 an inch or 1 inch thick, with a
flaunch, or turned up edge, on the inside, to guide the
wheels of the carriage. The plates rest at each nd on
Ptone sleepers of 3 or 4 cwt. sunk into the earth, and they
are joined to each other so as to form a continuous horizon-
tal pathway. They are, of course, double; and the distance
between the opposite rails is from 3 to 4| feet, according
to the breadth of the carriage or wagon to be employed
The edge rail, which is found to be superior to the tram
rail, is made either of wrought or cast iron ; if the latter b* 1
used, the rails are about 3 feet long, 3 or 4 inches broad,
and from 1 to 2 inches thick, being joined at the ends by
cast metal sockets attached to the sleepers. The upper
edge of the rail is generally made with a convex surface
RAILWAYS.
269
to which the wheel of the carriage is attached by a groove
made somewhat wider. When wrought iron is used, which
is in many respects preferable, the bars are made of a
smaller size, of a wedge shape, and from 12 to 18 feet
long; but they are supported by sleepers, at the distance
of every 3 feet. In the Liverpool railroad the bars are 15
feet long, and weigh 35 Ibs. per lineal yard. The wagons
in common use run upon 4 wheels of from 2 to 3 feet in
diameter. Railroads are either made double, 1 for going
and 1 for returning; or they are made with sidings^ where
the carriages may pass each other. See M'Cullocli's Diet.
Table showing the effects nf a force of traction of 100 pounds, ai
different velocities, on canals, railroads, and turnpike roads.*
Velocity of motion.
Load moved by a power of 100 ll.
Miln per
hour.
Feet per
second.
On a Canal.
On a level Railway.
On a level
Turnpike Road.
Total mass
moved.
UKful ef-
fect.
Total man
moveJ .
L'Kfal ef-
fect.
Total man ! Dseful ef-
moved. I feet.
2
3
3-66 55,500 39,400
4-40 38,542 27,361
Ibt.
14,400
14,400
10,800
10,800
Us. i Hi.
1,800.1,350
1,800 1,350
3d
5-13 28,316 20,100
14,400
10,800
1,800 1,350
4
5-86 21,680 15,390
14,400 10,800
1,800 1,350
5
7-33 13,875 9,850
14,400 10,800
1,800 1,350
6
8-80 9,635
6,840
14,400 10,800
1,800 1,350
7
10-26 7,080
5,026
14,400 10,800
1,800 1,350
8
11-73
5,420
3,848
14,400 10,800
1,800 1,350
9
13-20
4,282 3,040
14,400 10,800
1,800 1,350
10
14-66! 3,468
2.462
14,400 10,800 1,800 1,350
13-5
19-9
1,900
1,359
14,400
10,800 1,800,1,350
The subject of steam vessels has been investigated by
different engineers, on mathematical principles, but the
calculations which their rules direct are by far too intricate
for a work of this nature. We will, however, insert a state-
ment of the proportions, &c.. of several steamboats already
made, which will doubtless be acceptable to the practical
man, and those who wish to investigate the theory will find
' ample material in the work of Tredgold.
* The force of traction on a canal varies a< the square of the velocity
but the mechanical power necessary to move the boat is usually reckoned
Ao increase as the cube of the velocity. On a railroad or turnpike, the
force of traction is constant; but the mechanical power necessary to
move the carriage, increases as the velocity.
23*
870
STEAMBOATS.
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ANIMAL STRENGTH. 273
The rule for determining the tonnage is according to law.
but by no means according to correct principles. It is as
follows :
Take the length = L from the back of the main stern
?ist to the fore part of the main stem< beneath the bow-
sprit, and subtract from it the length of the engine room
= E, and from the remainder subtract three-fifths of B =
the breadth of the vessel taken from outside to outside of
the planks at the widest part of the vessel, whether it be
above or below the wales, and divide this last remainder by
188 ; the quotient multiplied by the square of B will give
the register tonnage, or,
Wherefore the length being 162 feet, the length of engine
-oom 47, and the breadth of the vessel 32, then,
tonnage.
ANIMAL STRENGTH.
%
THERE is a certain load which an animal can just bear
but cannot move with it, and there is a certain velocity with
which an animal can move but cannot carry any load. In
these two circumstances it is clear, that the exertion of the
animal can be of no avail as a mover of machinery. These
are, as it were, the extremes of the animal's exertion,
where its effect is nothing ; but between these two extremes,
there must be weights and velocities with which the animal
can move, and be more or less efficient.
If one man travel at the rate of three miles an hour, and
jarry a load of 56 Ibs., and another move at the rate of 4
miles an hour and carry a load of 42 Ibs., the speed of
the first is 3, and the load 56, the useful effect may there-
fore be estimated as the momentum = 168. The other
Carries only 42 Ibs., but travels at the rate of 4 miles an
hyur ; therefore, in the same way, his useful effect will be
nyur ; tnei
r x 42 =
X 42 = 168, the same as before: hence the effect of
274 ANIMAL STRENGTH.
these twc men are the same. It will not be difficult to
show, that in the same time they perform the same quantity
of work. For the first will in six hours carry 56 Ibs. 3x6
= 18 miles, as he travels at the rate of 3 miles an hour;
and if he be supposed to carry a different load, hut of the
3ame weight every rnile, he will in the six hours have car-
ried altogether 18 X 56 = 1008 Ibs. ; but the other carries
in the same way, 4 times 42 Ibs. every hour, that is 168
Ibs. in one hour therefore in 6 hours he will have carried
168 x 6 = 1008 Ibs., the same as the other.
It will now be seen what is meant by the phrase useful
effect, and from what has been observed above, we will be
led to conclude, that when the load is the greatest which
the animal can possibly bear ; the useful effect is nothing,
because the animal cannot move ; and when the animal
moves with its greatest possible speed, the useful effect will
also be nothing, for then the animal can carry no load ;
and it becomes a very useful problem to determine where
between these two limits, the load and speed are so related
that the useful effect of the animal will be the greatest. By
investigation it has been found that the maximum effect of
an animal will be when it moves with 5 of its greatest
speed, and carries ^ths of the greatest load it can bear.
Thus, if the greatest speed at which a man could travel
or run, without a load, be 6 miles per hour ; and if the
greatest load which he can "bear, without moving, be 2
cwt., then this reduced to Ibs. is 280 Ibs.. hence,
- = 124-4 Ibs. = the load, and = 2 miles, the
y 3
speed with which a man will produce the greatest useful
effect.
Sir John Leslie gives a formula for a horse's power, in
traction, in which he denotes the velocity in miles per
hour, I (12 - V) 2 by which it will be found that if a
horse begins this pull with a force = 144 Ibs., he would
draw 100 at the rate of 2 miles, 64 at 4, and 36 at 6 ; the
greatest effect being at 4 miles per hour.
The French employ a measure of animal action which
they denominate a Dynamical unit, which is a cubic metre
of water raised to the height of a metre.
There are so many causes operating to produce variations
in animated beings even of the same kind, that it is difficult,
if not impossible, 10 form a correct estimate of the amount
FRICTION. 275
of any one particular class, or the comparative strength of
different classes, hence we find great differences in the
results of different experimenters.
Gregory has estimated the average force of a man at rest
to he 70 Ibs., and his utmost walking velocity, when un-
loaded, to be 6 feet per second ; and that a man will pro-
duce the greatest mechanical effect in drawing, when the
weight was 31^ '.os., with a velocity of 2 feet per second.
But this is not the most advantageous way of applying the
strength of men, although it has been found to be the bes\,
war of employing the strength of horses. Robertson Bu-
chanan states, that the mechanical effects of men in work-
ing a pump, in turning a winch, in ringing a bell, and
rowing a boat, are as the numbers 100, 107, 227, and 248.
According to Hatchette, of a man working at the cord of a
pulley to raise the ram of a pile engine = 50 dynamical
units. A man drawing water from a well by means of a
cord 71 ; a man working at a capstan 116. The dynami-
cal unit being, as stated before, equivalent in English mea-
sure to 2208 Ibs., or 4 hogsheads of w.iter raised to the
height of 3-281 feet in a minute; these things being con-
sidered, the above results will give the average strength of
men per day.
We meet with similar difficulties in estimating the
strength of horses. According to Desagulicrs and Smea-
ton, 1 horse equal to 5 men. According to Bossut, 1 horse
equal to 7 men. Schulze makes it 14 men; and Bossut
states, that I ass is equivalent to 2 men. It is also stated
by Amontons, that 2 horses yoked in a plough exert a
power of 150 Ibs. See the section on the Steam Engine.
FRICTION.
WE have considered the effects of the first movers of
machinery, and we must now direct our attention to the
subject of Friction, which, as we have frequently noticed,
tends to diminish these effects. On this subject it is not
.OIK- intention to dwell long, as all the researches that have
been hitherto made in this branch of mechanical science,
a/e not of such a nature as to furnish means of deducing
satisfactory laws. The resistance arising from one surface
276 FRICTION.
rubbing against another is denominated friction ; and it is
the only force in nature which is perfectly inert its ten-
dency always being to destroy motion. Friction may thus
be viewed as an obstruction to the power of man in the
construction of machinery ; but, like all the other forces
in nature, it may, when properly understood, be turned to
his advantage, for friction*is the chief cause of the stability
of buildings or machinery, and without it animals could not
exert their strength.
The friction of planed woods and polished metals, with-
out grease, 4m one another, is about one-fourth of the pres-
sure.
The friction does not increase on the increase of the rub-
bing surfaces.
The friction of metals is nearly constant.
The friction of woods seems to increase after they are
some time in action.
The friction of a cylinder rolling down a plane, is in-
versely as the diameter of the cylinder.
The friction of wheels is as the diameter of the axle di-
rectly, and as the diameter of the wheel inversely. The
following hints may be of use for the purpose of diminish-
ing friction.
The gudgeons of pivots and wheels should be made of
polished iron ; and the bushes or collars in which they
move should be made of polished brass. In small and
delicate machines, the pivots or knife edges should rest on
garnet. Oily substances diminish friction swine's grease
and tallow should be used for wood, but oil for metal.
Black lead powder has been used with effect for wooden
gudgeons. The ropes of pulleys should be rubbed with
tallow.
As to the friction of the mechanic powers. The simple
lever has no such resistance, unless the place of the ful-
crum be moved during the operation. In the wheel and
axle the friction on the axis is nearly as the weight, the
diameter of the axis, and the angular velocity it is, how-
ever, very small. When the sheaves rub against the block?
the friction of the pulley is very great. In most, if not in
all screws, the friction of the screw is equal to the pres-
sure the square threaded screw is the best.
In the inclined plane, the friction of a rolling body is
far less than that of a sliding one.
FRICTION. 277
To estimate the amount of the friction of a carriage upon
a railway, we have,
P X T
P = friction,
in which rule P signifies the power that will keep the
wagon on the plane, independent of friction, T the time of
descent without friction, both of which are to he deter-
mined by the laws of the inclined plane given before : and
t is the time of actual descent of the wagon or carriage.
There is a loaded carriage on a railroad 120 feet in
length, having an inclination of one foot to the hundred.
The carriage, together with its load, weighs 4500 Ibs.
Now, the height of the plane may be found by the princi-
ples of geometry, from the proportion of similar triangles.
IOC : 120 : : 1 : i'2 == the height of the plane ; and by
the laws of falling bodies, and the properties of the inclined
plane,
J4? 1 X 120 = -2731 X 120 = 32-772 = the time
^( 16
in seconds in which the carriage would descend the plane
without friction and by the properties of the inclined
plane, 100 : 1 : : 4500 : 45 = the force that sustains the
carriage, without friction, from rolling down the plane :
wherefore, by the rule,
45 ; 20-421 = the friction in pounds",
60
which retards the carriage in rolling down the railway.
OF MACHINES IN GENERAL,
THEIR REGULATION AND COMPARATIVE EFFECTS.
A. MACHINE, howsoever complicated it may be, is nothing
else than an organ or instrument placed between the work-
men, or source of force or power, whatever it may be, and
the work to be done. Machines are used chiefly for three
reasons. 1. To accommodate the direction of the moving
force to that of the resistance which is to be overcome. 2,
To render a power, which has a fixed and certain velocity,
effective in performing work with a different velocity. 3,
To make a moving power, with a certain intensity, capable
of balancing or overcoming a resistance of a greater in-
tensity.
These objects may be accomplished in different ways,
either by using machines which have a motion round some
fixed point, as the three first mechanic powers ; or by those
which furnish, to the resistance to be moved, a solid path
along which it may be impelled, as is the case in the last
three mechanic powers : hence some authors have reduced
the simple machines to two the lever and inclined plane.
Simplicity in the construction of machines cannot be too
warmly recommended to the young engineer; for com-
plexity increases the friction and expense, and endangers
the chance of success from the derangement of the parts.
In consequence of friction, it is well known, that no
machine can overcome a resistance without an expense of
the power of the first mover, and as the more complicated
the machine is, the greater will the friction be ; so also will
the machine be less powerful. If two machines be con-
structed, the one simple and the other complex, and be
such, that the velocity of the impelled point is to the
velocity of the working point in the same proportion in
both ; then will the simple machine be the most powerful.
The methods of communicating motion from one point
to another are infinitely diversified ; and we, in the last
278
MACHINERY. 219
caapter, gave an account of the best of these which have
hitherto been invented. We confine ourselves in the
mean time to a few general remarks on the construction of
machinery.
When heavy stampers are to be raised in order to drop
on matter to be pounded, the wipers by which they are
raised should be of such a form, that the stampers may be
r.ised by a uniform pressure, or with a motion as nearly as
p.ssible uniform. If this is not the case, and the wiper is
merely a pin sticking out of the axis, the stamper will be
"arced into motion at once, which will occasion violent jolts
n the machine, together with great strains on its moving
parts and points of support. But if gradually lifted, no
inequality will be felt at the impelled point of the machine.
The judicious engineer will therefore avoid, as much as
possible, all sudden changes of motion, especially in any
ponderous part of a machine.
When several stampers, pistons, or other reciprocal
i/iovers are to be raised and depressed, common sense
teaches us to distribute their times of action in a uniform
n:anner, so that the machine may be always equally loaded
with work. When this is done, and the observations in
the foregoing paragraph attended to, the machine may be
made to move almost as smoothly as if there were no re-
ciprocations in it. Nothing shows the ingenuity or skill
of the contrives more than the simple yet effectual con-
trivances for obviating those difficulties which are unavoid-
able, from the .nature of the work to be done by the
machine, or of the power applied. There is also much
ingenuity required in the management of the moving
power, when it is such as does not answer the kind of mo-
tion required ; for instance, ,n employing a power which
necessarily reciprocates to produce a motion which shall be
uniform, as in the employment of a steam engine to drive
a cotton mill. The necessky of reciprocation of the first
mover causes a waste of much power. The impelling
power is wasted first in imparting, and then in destroying
a vast quantity of motion in the working beam. The en-
gineer will see the necessity of erecting the mover in a
f separate building from the machinery moved, which pre-
vents the great shaking and speedy destruction of the
/buildings.
The gudgeons of a water wheel should ne\ er rest on tho
28C MACHINERY
building, but should be placed on a separate erection ; and
if this is not practicable, blocks of oak should be placed
below them, which tend to soften all tremors, like* the
springs of a carriage.
It will often conduce to the equality of motion of ma-
chinery, to make the resistance unequal, to accommodate
the inequalities of the moving power. There are some
beautiful specimens of this kind in the mechanism of the
human body.
It is always desirable, that the motion of a machine should
be regular, when this can be effected ; and we now pro-*
seed to state the various methods that have heretofore been
employed for producing regularity in the motion of the
machine, both as regards the reception and distribution of
power.
Even supposing that the first mover is perfectly constant
and equable in its action, the machine may not be regular
in its movement, from the irregularity of the resistance to
be overcome. But still, if both the power and the resist-
ance were perfectly regular, the machine would not be
perfectly uniform in its motion ; for there are particular
positions in which the moving parts of a machine are more
efficacious than in others, as in the crank for instance :
hence the energy of the first mover will be unequally trans-
mitted, and irregularity in the motion of the machine will
consequently follow. The motion of some machines bears
a constant tendency to accelerate, others to retard ; and
others alternately to accelerate and retard ; and perhaps
in no case whatever can the motion of a machine be said to
be perfectly uniform. But of this we will speak more at
large when we come to treat of the maximum effect of
machines.
We intend to confine our attention chiefly to the regula-
tors of machinery employed in the steam engine, making
occasional remarks on others as we go along.
For the purpose of regulating the moving power, the
conical pendulum or governor is commonly employed.
The nature of this beautiful contrivance has been described
under central forces, and alluded to in our remarks on the
steam engine. The ring on the shaft acts upon a lever of
the first kind, whose other end opens or shuts a valve, which
is fixed in the pipe that admits the steam from the boiler to
the cylinder ; and according to the degree of opening or
MACHINERY. 281
shutting of tliie valve, and consequently the divergence or
convergence of the halls, or the velocity of the shaft, will
be the quantity of steam admitted to the cylinder. The
governor is frequently applied to the water wheel, and acts
in a similar way by a board or valve in the shuttle, which
delivers the water to the wheel. So likewise in the wind-
mill, it is employed to furl or unfurl more or less sail.
Sometimes the governor is found inadequate to the regu-
lation of the machine, and another contrivance of great
power and simplicity is introduced. The machine is made
to work a pump, which tends continually to fill a cistern
with water. From this cistern there proceeds an eduction
pipe, leading to the reservoir, from which the water is
drawn by the pump. Tliis simple contrivance is so regu-
lated, that when the machine goes with its proper velocity,
the pump throws just as much water into the cistern as the
ejection pipe draws from it; consequently, the water in the
cistern remains at the same level. But if the machine goes
too fast, then the pump will throw in more water than is
let out by the ejection pipe, wherefore the level of the
water will rise in the cistern. If the machine goes too
slow, the level of the water will in like manner fall. Now,
on the surface of the water in the cistern, there is a float
which rises or falls with the surface of the water ; and is
thus made to answer the same purpose as the ring of the
governor. It may be observed, that the delicacy of this
kind of regulator will depend, in a great measure, upon
the smallness of the surface of the water which supports the
float ; for then a small difference between the supply and
discharge, will cause a greater difference in the elevation
or depression of the surface, than if the surface were large.
To procure a constant supply of steam in the steam en-
gfne, it is necessary that the water in the boiler be always
at the same level. To effect this purpose, there is a lever
fixed on a support, on the top of the boiler, to one end of
which lever there is attached a slender rod, which descends
into the boiler, and is terminated by a float, which rests
on the surface of the water in the boiler. To the other
end of the lever, there is attached another rod, to the end
of which is affixed a valve, opening and shutting the orifice
i of a pipe which leads into the boiler. The top of the pipe,
/where the valve is placed, opens into a cistern of water,
is supplied by a pump driven by the engine itself.
24*
282 MACIIINEUY.
When the water in the boiler falls below its common level
in consequence of the formation of steam, the float falls
with it, and consequently depresses that side of the lever
to which the float rod is attached ; the other arm rises and
opens the valve at the top of the pipe, which leads from
the cistern into the boiler, and thus admits water until the
float rises to the proper height, and then the valve is closed.
In this beautiful contrivance, the water is not supplied to
the boiler in jolts, but the float and valve continuing in a
state of constant and quick vibration, the supply is rendered
quite constant.
There is a very ingenious contrivance called the Tacho-
meter, from its use as a measure of small variations in
velocity, which is often employed in the steam engine and
other machinery. The simplicity of this contrivance will
render its action easily understood. If a cup with any
fluid, as. mercury, be placed on a spindle, so that the brim
of the cup shall revolve horizontally round its centre, then
the mercury in the cup will assume a concave form, that is,
the mercury will rise on the sides of the cup, and be de-
pressed in the middle ; and the more rapid the motion of
the cup is, the more will the surface of the mercury differ
from a plane. Now, if the mouth of this cup be closed, and
a tube inserted in it, terminated in the cup by a ball-shaped
end, and half filled with some coloured fluid, as spirits of
wine and dragon's blood ; then it is clear, that the more
the surface of the mercury is depressed, the more the fluid
in the tube will fall, and vice versa : consequently, the
rapidity or slowness of the motion of the cup, will be indi-
cated by the height of the coloured fluid in the tube ; and
thus it becomes a measure of small variations in velocity.
In the steam engine, we also find an apparatus for regu-
lating the strength of the fire of the boiler, which apparatus
is called the self-acting damper. There is a tube inserted
into the boiler, reaching nearly to the bottom, which tube is
open at both ends. Now, from the principles of Pneuma-
tics, it is plain, that the greater the pressure of the steam in
the boiler is, the water will be pressed to the greater height
in this tube. The water in the tube supports a weight, to
which there is attached a chain going over two wheels ; and
to the other end of the chain is attached a plate r , which
slides ovei ihe mouth of the flue which leads into the fire.
These thii.gs are so formed, that the rising of the weight
MACHINERY. 283
in the tube will cause more or less of the flue to he covered
by the plate ; and thus increase or diminish the quantity
of air which feeds the fire. Now, if there is too much steam
produced, there will be a greater pressure on the surface
of the water in the boiler, and it will be forced up the tube
the weight in the tube will be raised, and consequently the
plate at the other end of the chain will fall, and cover more
of the mouth of the flue, and thus diminish the quantity
of air which feeds the fire ; and there will consequently be
generated in the boiler a less quantity of steam.
We come now to speak of the nature and use of the fly
wheel. A fly in mechanics may be defined to be a heavy
wheel or cylinder, which moves rapidly upon its axis, and
is applied to a machine for the purpose of regulating its
motion.
We have already stated that there are many circum-
stances which tend to render the motion of a machine ir-
regular variation in the energy of the first mover, whether
it be water, wind, steam, or animal strength variation in
the resistance or work to be done and variations in the
efficacy of the machine itself, arising from the nature of its
construction, whereby it is of necessity more effective in
one position than in another. We have already seen how
many of these irregularities are compensated, and we are
now come to speak of the fly, which is the simplest and
most effective of them all. The principle on which the fly
acts is that of inertia, one of the most important of the first
principles of mechanical science. At any one given time,
a body must be in one or other of these two states rest or
motion. And let any body be in one or other of these two
state*, it has no power within itself to change it, if it be
at rest, it has no power to put itself in motion and if in
motion, it has no power in itself either to increase, diminish,
or destroy that motion. From a knowledge of this fact,
and from what was stated before on the momentum, or
moving force of a body, that it is the quantity of matter
multiplied by the velocity of the moving body the nature
of the operation of the fly will be easily understood.
As the fly wheel, to do its office effectually, must have
a considerable velocity, it is clear that its rim, which has a
considerable weight, must also have a considerable momen-
turn, and consequently a considerable power to overcome
. snv tendency either to increase or retard its motion.
284 MACHINERY.
To apply these observations to actual cases, let us sup-
pose th'at a single horse drives a gin. .When the gin has been
set in motion, the animal cannot exert a uniform strength-
there will be occasional increases and relaxations in the
velocity of the gin ; but suppose a fly wheel to be added,
then, whenever the animal slackened its exertions, the
machine would have a tendency to move slower, but the
momentum which the fly had acquired, would overcome this
tendency to retardation, and -continue the motion of the
machine at the same rate as before, until the animal had
recovered iiself so as to exert the same strength as before.
So, likewise, if the animal exerted an extraordinary pull,
the inertia of the wheel would oppose a resistance which
would check the tendency to increase in the velocity of the
gin. In this way the fly wheel regulates the motion of the
gin, whether the animal takes occasional rests, or makes
occasional extraordinary exertions. It is evident that the
fly would operate in the same way, if the first mover were
steam, water, or wind, and that the other regulators which
we have described, are merely assistants to the fly wheel.
Variations in the resistance, or work to be performed,
are also compensated by the fly wheel. For instance, in a
small thrashing mill without a fly. When the machine is
not regularly fed with the corn, there will be an occasional
resistance, which will have a sensible effect on the whole
train of the machinery, even the water wheel itself; which
irregularity, may, however, be avoided by the introduction
of a fly, as its inertia will procure equality of motion : but
it may be observed, that when the machine is large, there
will be less necessity for a fly, as the inertia of the machine
itself will then effect the same purposes.
It was before stated, that even supposing the first mover
and resistance to be perfectly uniform, the machine itself is
liable to variations in energy at different positions. It was
seen, for instance, that a crank is more effective in one
position than another ; but the momentum communicated
to the fly, when the crank is in the most effective position,
will carry the crank past its least effective position. There
are many cases, however, where there are irregularities of
motion proceeding from the nature of the machinery, which
could be compensated better than with a fly. Thus, if a
bucket is to be drawn from the bottom of a coal pit, which
is 60 fathoms in depth: the weight of bucket beiug 14
MACHINERY.
285
cwt., and the chain by which it is coiled up round the cylin
tier weighing 8 Ibs. to every fathom, it is plain, that when
the bucket is at the bottom, not only the weight of the
bucket, but also the weight of the chain, will require to be
overcome in the raising of the bucket. Now the weight of
the chain is 60 + 8 = 480 Ibs., and the amount of the
weight of the bucket is 14 cwt. or 1568 Ibs. ; hence 1 568 +
480 = 2048 Ibs. ; but the weight of the chain will always
be getting less as it is coiled round the cylinder, until the
bucket comes to the cylinder, when the chain will be all
coiled, and there will remain only the weight of the bucket.
Now, the use of a fly may be advantageously dispensed
with, if the barrel on which the chain is coiled is formed
like a cone ; the diameter of the barrel thus increasing with
the uniform diminution of the weight.
The effect of the fly wheel in accumulating force, has led
some to suppose that there is, positively, a creation of force
in the fly ; but this is a mistake, for it is only, as it were, a
magazine of power, where there is no force but what has
been delivered to it. The great use of the fly wheel is thus
to deliver out at proper intervals, that force which has been
previously communicated to it ; and although there is ab-
solutely a small loss of power by the use of the fly, yet this
is more than compensated by its utility as a regulator.
The motion of machines may, as stated before, be re-
duced to three kinds. That which is gradually accelerated,
which generally takes place at the commencement of a
machine's action: that which is entirely uniform: that
which is alternately accelerated and retarded. The nearer
that the motion of a machine approaches to uniformity, the
greater will he the quantity of work done.
In order that the few remarks, which we intend to make
on the effect of machines, may be clearly understood, we
desire the reader to attend to the following definitions.
The impelled point of any machine, is that point at which
the force which moves the machine, may be considered as
applied as the piston of a steam engine, or the float board
of a water wheel.
The working point, on the contrary, is that point where
the resistance may be supposed to act.
The velocity of the moving power is the same as the ve-
locity of the impelled point, the velocity of the resistance
is the same as the velocity of the working point.
886 MACHINERY.
The performance or effect of a machine is measured by
the resistance or work performed, (calculated by weight,)
multiplied by its velocity, which is, in other words, the mo-
mentum of the working point. The momentum of impulse,
on the other hand, is measured by the energy of the first
mover, (also estimated by weight,) multiplied by the ve-
locity of the impelled point.
These definitions being understood, we proceed to a
simple statement of principles.
When any power is made to act in opposition to a resist-
ance, by means either of a simple or compound machine ;
which machine will be in a state of rest, when the velocity
of the power is to that of the resistance as the weight of
the resistance is to that of the power. In this state of things
the machine can do no work, because it has no motion ; but
if the power is increased, so as to overcome the resistance,
the machine will have an accelerated motion so long as
the power exceeds the resistance. If the power should
diminish, the machine would accelerate less and less, until
its motion became uniform. The same effect would ne-
cessarily follow, if the resistance increased, a circumstance
which may arise from various causes. From the resistance
of the air, which increases with an increase of velocity ;
and also from friction, which often increases with the in-
crease of velocity. Hence we find, that machines have
commonly a tendency to become uniform in their motion.
We have seen before, while treating of the water wheel,
that the velocity of the floats of the undershot wheel, must
be less than the velocity of the stream. For, when the
float board is at rest, the water will impinge on it with the
greatest possible effect ; but so soon as the float begins to
move, then it leaves the water, as it were, and does not re-
ceive the whole impetus of the stream ; and if the velocity
of the float were equal to that of the stream, it is clear that
the water would have no effect upon it at all ; and, as was
stated before, there is a certain relation between the velocity
of the wheel and that of the stream, at which the effect will
be a maximum. This is not confined to the water wheel,
but is common to all machines, as we have seen illustrated
in the steam engine.
We have seen before, that the maximum effect of an
animal was, when its velocity was one-third of its creates
possible speed, and the load which it bore, or the resistance
MACHINERY. '287
which it overcame, was equal to four-ninths of its greatest
possible load.
The following tables (A and B) constructed from the re-
sults of Dr. Kobison, will be useful to the mechanic.
Table A contains the least proportions between the velo-
cities of the impelled and working points of a machine ; or
between the levers by which the power and resistance act.
The use of this table is very simple, for suppose we
wished to raise 3 cubic feet of water per second, by means
of a water wheel, whose radius was 8 feet, (= the length
of the lever by which the power acts,) and the power which
moves the wheel being 6 cubic feet of water per second.
Employ this rule :
Power,
: x 10 = a number,
Resistance,
which look for in column M, and against it in column N,
will be found a number which, when multiplied by the
length of lever at which the power acts, will give the lengtK
of lever at which the resistance should act.
Wherefore, in the above case,
/
X 10 = 20, the number corresponding to which is
0-732051, hence 0-732051 X 8 = 5-856408 = the radius
of the axle at which the resistance or work to be done acts.
This table will be found very useful in the construction
of machines ; but they are frequently already constructed,
and it becomes then necessary for us to regulate the power
and resistance in order to produce a maximum effect, with-
out making -\ny alteration in the machine. For this pur-
pose we employ table B, in order to show the use of which
*re give the following rule and example :
Length of lever of resistance,
T-J = a number, which, when
Length of lever of power,
found in column O, will stand against a number in column
P: such, when multiplied by the energy of power, will give
the proper energy of resistance. Thus, if a man exerts a
constant force of 56 Ibs. on the handle of a capstan, whose
leverage is 4 feet, and the barrel is one foot in radius, then
we have.
=- a number, which will be found in column O, cor
4 4
288
MACHINERY.
Desponding to which will be found, in column P, the num-
ber 1*8885 ; wherefore, by the rule,
1-8885 x 56 = 105-756 = the resistance which the man
in these circumstances, can overcome with the greatest ad-
vantage, or with the maximum mechanical effect.
TABLE A.
TABLE B.
M
N
M
N
1
0-048809
20
0-732051
2
0-095445
21
0-760682
3
0-140175 -
22
0-788854
4
0-183216
23
0-816590
5
0-224745
24
0-843900
6
0-264911
25
0-870800
7
0-303841
26
0-897300
8
0-341641
27
0-923500
9
0-378405
28
0-949400
10
0-414211
29
. 0-974800
11
0-449138
30
1-000000
12
0-483240
40
1-236200
13
0-516575
50
1-449500
14
0-549193
60
1-645600
15
0-581139
70
1-828400
16
0-612451
80
2-200000
17
0-643168
90
2-162300
18
0-673320
100
2-.3 16600
19
0-702938
o
P
P
*
1-8885
7
0-03731
1
3
1-3928
8
0-03125
3
0-8986
9
0-02669
1
0-4142
10
0-02317
2
0-1830
11
0-02037
3
0-1111
12
0-01809
4
0-0772
13
0-01622
5
0-0587
14
0-01466
6
0-0457
15
0-01333
MACHINERY.
28S
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290 COTTON SPINNING.
It is not by any means an easy matter to estimate the
relative quantities of work done by different machines.
Their effects are generally stated as equivalent to so many
horses' power, and the following data are commonly given:
One horse's power, at a maximum, is equivalent to the
raising of 1000 Ibs. 13 feet high in one minute. In cotton
factories, 100 spindles, with preparation, are allowed to each
horse power for spinning cotton yarn twist, or five time?
that number of spindles, with preparation, for mule yarn,
No. 48 ; and if it be No. 110, ten times that number of
spindles, with preparation and the power-loom factories
12 beams with subservient machinery.
Thus a steam engine on Watt's principle, having a cylin-
der of 30 inches diameter, and a stroke of 6 feet, making
21 double strokes per minute, will give, by the usual cal-
culation,
7854 X 30 g X 10 X 6 X 21 X 2 _
44000
40 horses' power. Hence such an engine will drive 4000
spindles cotton yarn twist, or 20,000 spindles mule twist,
No. 48, or 40,000 mule twist spindles, No. 110, or 480
power-looms in each of which cases subservient or pre
paratory machinery is included.
RULES FOR COTTON SPINNERS.
IN the following calculations the reader is supposed to be
acquainted with the construction of the various machines
employed in the cotton manufacture, so that the rules are
only intended to assist the memory of the practical man in
cases of particular difficulty. The effects of shafts, belts,
drums, pulleys, pinions, and wheels, in varying velocity,
depend upon the principles established when treating of the
mechanical powers, and the calculations connected with
them may be easily performed by the rules given in that
section.
To find the draught on the spreading machine, count the
number of teeth of the wheel on the end of the feeding roller
shaft, calling it the first leader, and also the number of teeth
on the pinion which it drives, calling it the first follower,
and in like manner reckon all the leaders and followers on
COTTON SPINNING. 291
to the last follower i. e. the wheel on the calender roller
shaft, omitting all intermediate wheels, then,
product of leaders x diam. calender roller
product of followers x feeding roller
If the teeth of the leaders be 160, 22, and 20, and those
of the followers 90, 22 and 40 ; the diameter of calender
roller 5, and feeding roller 2 inches ; then,
160 X22 X20X5
- = 2-26 = the draught.
90 x 22 x 40 x 2
The reader will have no difficulty in applying the prin-
ciple of this rule to the calculation of the draught of other
machines in cotton manufacture.
To find the number of twists per inch given to the rove
by the fly frame :
Turns of front roller per minute x its circumference =
length of rove produced in one minute, dividing the turns
of the spindle per minute by that product, gives the number
of twists on the rove per inch.
Let the revolutions of the front roller per minute be 100,
and the circumference 4 inches, then 100 x 4 = 400 inches
= 33 feet 4 inches of rove produced in a minute, where-
fore, if the spindle revolve 600 times in a minute, then,
600
= 1'5 twists per inch.
The proper diameter of the taking-out pulley, or men-
doza pulley of the stretching frame that shall regulate the
motion of the carriage to the delivery of the rove, may be
found by taking the product of the diameter of the front
roller X the number of teeth in the mendoza wheel, and
dividing by the number of teeth in the front roller pinion,
and subtracting from the quotient the diameter of the men-
doza bond. Thus if the diameter of the front roller be 1|
inches, the diameter of the mendoza bond 5 inch, the teeth
in front roller pinion 20, and in mendoza wheel 110, then,
110 x U 137-5
i i = _ I = 6-8 -5 = 6-3 inches
<0 *U
= the diameter of mendoza pulley.
/The revolutions of the spindle of the throstle may be
found thus :
292 COTTOTS SPINNING.
turns of cylinder per minute x its diameter
diameter of wharve
A cylinder of 7*5 inches diameter makes 450 revolutiona
per minute, and the diameter of the wharve is 1 inch,
nence,
450 X 7'5
- - - = 3375 = turns of the spindle per minute.
To find the draught of the roller of the jenny, take the
product of the teeth of the front roller pinion X the grist
pinion X diameter of back roller for a divisor, and take
the product of the diameter of front roller X the number of
teeth of the crown wheel x those of the back roller wheel
for a dividend, then the dividend divided by the divisor will
give the draught. Thus if the teeth of the crown wheel be
72, back roller wheel 56, front roller pinion 18, and grist
pinion 24, the diameter of front roller 1 inch, and of back
roller |, then,
72 x 56 x 1
18X 24X|
In order to determine the size of yarn from hank rove,
we must first find the quantity of rove given out by the
roller during one stretch, which is = the whole length of
stretch the inches gained, and calling this the divisor,
the dividend will be found by taking the product of the
number of hank rove X the length of the stretch X the
draught, the quotient will be the size of yarn produced.
Thus, if the draught be as found above 10*666, the stretch
56, the gaining of carriage 5 inches, and the rove 5 hank,
then,
10-66 X 5 x 56 r
- - - - - = 58-52 = size of yarn.
56 5
To find the effect of a change of the grist pinion on the
jenny.
Take the product of the pinion producing a known size
of yarn, and call it the dividend, if this be divided by any
other number of yarn, the quotient will be the correspond-
ing grist pinion ; or if another grist pinion be used as a
divisor, the quotient will be the corresponding size of yarn
produced. Thus if No. 70 yarn be produced by a pinion
of 24 teeth, then,
COTTOP; SPINNING. 293
24 x 7-
- = 28 = the number of teeth in a grist pinion
that shall produce yarn No. 60 ; and also
24 x 70
= 42 = the number of yarn that shall be pro-
duced by a grist pinion of 42 teeth.
Take the product of the diameter of the front roller X
the teeth of the mendoza wheel, and divide by the teeth of
the pinion on the front roller that drives the mendoza
wheel. From the quotient thus found, subtract the diame-
ter of the mendoza band, and the remainder is the diameter
of a pulley that will move the carriage out with the same
speed as the yarn passes through the front rollers. When
this is found, the diameter of such a pinion as will give a
certain gain on the stretch may be found by multiplying "
the last result by the full length of the stretch, and divide :
the product bv the difference of the length of the stretch
and the gaining required.* Thus, if the length of stretch
be 56 inches, the gain upon stretch 5 inches, the diameter
of the front roller 1 inch, and of the mendoza band f- of an
inch, the number of teeth on the mendoza wheel 112, and
on the front roller pinion 18, then,
^- = 6-22 -625 = 5-595 = the
diameter of mendoza pulley, to move the carriage uniformly
with the delivery of the front roller, and
56X5-595 313-32
- = = 6-14 = the diameter of men
56 5 51
doza pulley to move the carriage with a gain of five inche*
on the stretch.
The number of twists given to cotton yarn varies with
the quality of the fibre of the wool, the fineness of the yarn,
and whether it be intended for warp or weft. But omitting
the variation necessary for difference in the length of fibre,
which is comparatively trifling, the number of twists in the
inch will vary with the square root of the No. of the yarn,
or a go<^l practical rule is this,
x/No. of yarn x 3-75 for the twists per inch cf warp
yarn, and
v/No. of yarn X 3-25 for wefts.
25*
294 COTTON SPINNING
Th^s for No. 36 warps, we have,
v/36 X 3-75 = 6 x 3-75 = 22'5 twists per inch
And for No. 64 wefts,
v/64 X 3-25 = 8 X 3'25 = 26 twists per inch.
When cotton yarn is put up in hanks or spindles, it is
coiled upon' a reel, one revolution of which takes up 54
inches of thread, and this length of yarn is denominated a
thread.
54 in. = lj yards = 1 thread or round of the reel.
120 = 80 = 1 skein or ley.
840 = 560 = 7 == 1 hank or No.
15120 = 10080 = 126 = 18 = 1 spindle.
Cotton yarn is sold by weight, and its fineness is esti-
mated by the No. of hanks in a pound. Thus, No. 20
yarn contains 20 hanks, or 20 X 840 yards = 16800 yards
in one pound ; No. 64 contains 64 hanks or 64 x 840 =
53760 yards of thread in a pound ; consequently the
diameter of the thread of No. 64 must be much less than
the diameter of the thread of No.^0.
When the yarn is in cops the fineness is determined by
reeling a few hanks, and by finding their weight, the No.
of the yarn may be found by proportion ; thus if a spindle
be reeled, and its weight found to be 4 ounces 8 drachms,
then by proportion, since there are 18 hanks in a spindle,
and 16 ounces in a pound, and 16 drachms in an ounce,
we have,
4 : 16 :: 18 : 64 = the number of the yarn; or,
288
^-r ? .. . = No. of yarn ;
weight of a spindle in oz.
and,
288
-=^ j SB weight of a spindle in ounce*.
SQUARE AND CUBE ROOTS.
295
' Mix
Square root
Cube net
V
Square root.
Cube root.
No.
Squire root.
Cot* root.
1
1-
1-
49
7-
3-659
97
9-8488
4.594
2
1-4142
.1-259
50
7-0710
3-684
9S
9-8994
4-610
3
1-7320
1-442
7-1414
3-708
99
9-9498
4-626
4
2-
1-587
52
7-2111
3-732
100
10-
4-641
5
2-2360
1-709 || 53
7-2801
3-756
101
10-0498
4-657
6
2-4494
1-817
54
7-3484
3-779
102
10-0995
4-672
7
2-6457
1-912
55
7-4161
3-802
103
10-1488
4-687
8
2-8284
2-
56
7-4833
3-825
104
10-1980
4-702
9
3-
2-080
67
7-5498
3-848
105
10-2469
4-717
10
3-1023
2-154
58
7-6157
3-870
106
10-2956
4-732
11
3-3 HiG
2-223
59
7-0811
3-892
107
10-3440
4-747
12
3-4 (141
2-289
60
7-7459
8-914
108
10-3923
4-762
13
3-C055
2-351
61
7-8102
3-936
109
10-4403
4-776
14
3-7416
2-410
62
7-8740
3-957
110
10-4880
4-791
15
3-8729
2-466
63
7-9372 3-979
111
10-5356
4-805
16
4-
2-519
64
8-
4-
112
10-5830
4-820
17
4-1231
2-571
65
8-0622
4-020
113
10-6301
4-834
18
4-2426
2-620
66
8-1240
4-041
114
10-6770
4-848
19
4-3588
2-668
67
8-1853
4-061
115
10-7238
4-862
20
4-4721
2-714
68
8-2462
4-081
116
10-7703
4-876
21
4-5825
2-758
69
8-3066
4-101
117
10-8166
4-890
22
4-6904
2-802
70
8-3666
4-121
118
10-8627
4-904
23
4-7958
2-843
71
8-4261
4-140
119
10-9087
4-918
24
4-8989
2-884
72
8-4852
4-160
120
10-9544
4-932
M
5-
2-924
73
8-5440
4-179
121
11-
4-946
26
5-0990
2-962
74
8-6023
4-198
122
11-0453
4-959
27
5-1961
3-
75
8-6602
4-217
123
11-0905
4-973
28
5-2915
3-036
76
8-7177
4-235
124
11-1355
4-986
29
5-3851
3-072
77
8-7749
4-254
125
11-1803
5-
30
5-4772
3-107
78
8-8317
4-272
126
11-2249
5-013
31
5-5677
3-141
79
8-8881
4-290
127
11-2694
5-026
32
5-6568
3-174
80
8-9442
4-308
128
11-3137
5-029
33
0-7445
3-207
81
9-
4-326
129
11-3578
5-052
S4
5-8309
3-239
82
9-0553
4-344
130
11-4017
5-065
3.->
5-9160
3-271
83
9-1104
4-362
131
11-4455
5-078
96
6-
3-301
84
9-1651
4-379
132
11-4891
5-091
37
6-0827
3-332
85
9-2195
4-396
133
11-5325
5-104
38
6-1644
3-361
86
9-2736
4-414
134
11-5758
5-117
39 G-2449
3-391
9-3-273
4-431
135
11-6189
5-129
40 6-3245
8419
9-3808 4-447
136
11-6019
5-142
41 6-4031
3-448
89 9-4339 4-464
137
11-7046
5-155
42 6-4807
3-476 | 90 9-4868 4-481
138i 11-7473:5-167
43 6-5574
44 (;<;:<:.:
3-503 : 91 1 9-5393 ; 4-497
3530 92 9-5916,4-514
139 11--,
140 ll-83vil
5-180
5-192
45
6-7082
3-556
93
9-043I) 4-530
141 11-8743
5-204
46
6-7823
3583
94 9-U953 4-546
142
119163
5-217
47 6-8556
3608
95
9-7467 4-562
143
11-9582
5-229
48 6-9282
3-634 96
9-7979
4-578
144
12-
5-241
25*
296
SQUARE AJCD CUBE ROOTS.
No.
Square root.
Cube root.
HO.
Square root.
Cube root.
No.
Square root.
Cab* root
145
12-0415
5-253
193
13-8924
5-778
241
15-5241
6-223
146
12-0830
5-265
194
13-9283
5-788
242
15-5563
6-231
147
12-1243
5-277
195
13-9642
5-798
243
15-5884
6-240
148
12-1655
5-289
196
14-
5-808,
244
15-6204
6-248
149
12-2065
5-301
197
14-0356
5-818
245
15-6524
6-257
150
12-2474
5-313
198
14-0712
5-828
246
15-6843
6-265
151
12-2882
5-325
199
14-1067
5-838
247
15-7162
6-274
152
12-3288
5-336
200
14-1421
5-848
248
15-7480
6-282
153
12-3693
5-348
201
14-1774
5-857
249
15-7797
6-291
154
12-4096
5-360
202
14-2126
5-867
250
15-8113
6-299
155
12-4498
5-371
203
14-2478
5-877
251
15-8429
6-307
156
12-4899
5-383
204
14-2828
5-886
252
15-8745
6-316
157
12-5299
5-394
205
14-3178
5-896
253
15-9059
6-324
158
12-5698
5-406
206
14-3527
5-905
254
15-9373
f 333
159
12-6095
5-417
207
14-3874
5-915
255
15-9687
&-341
160
12-6491
5-428
208
14-4222
5-924
256
16-
6-349
161
12-6885
5-440
209
14-4568
5-934
257
16-0312
6-357
162
12-7279
5-451
210
14-4913
5-943
258
16-0623
6-366
163
12-7671
5-462
211
14-5258
5-953
259
16-0934
6-374
164
12-8062
5-473
212
14-5602
5-962
260
16-1245
6382
165
12-8452
5-484
213
14-5945
5-972
261
16-1554
6-390
166
12-8840
5-495
214
14-6287
5-981
262
16-1864
6-398
167
12-9228
5-506
215
14-6628
5-990
263
16-2172
6-406
168
12-9614
5-517
216
14-6969
6-
264
16-2480
6-415
169
13-
5-528
217
14-7309
6-009
265
16-2788
6-423
170
13-0384
5-539
218
14-7648
6-018
266
16-3095
6-431
171
13-0766
5-550
219
14-7986
6-027
267
16-3401
6-439
172
13-1148
5-561
220
14-8323
6-036
268
16-3707
6-447
173
13-1529
5-572
221
14-8660
6-045
269
16-4012
6-455
174
13-1909
5-582
222
14-8996
6-055
270
16-4316
6-463
175
13-2287
5-593
223
14-9331
6-064
271
16-4620
6-471
176
13-2664
5-604
224
14-9666
6-073
272
16-4924
6-479
177
13-3041
5-614
225
15-
6-082
273
16-5227
6-487
178
13-3416
5-625
226
15-0332
6-091
274
16-5529
6-495
179
13-3790
5-635
227
15-0665
6-100
275
16-5831
6-502
180
13-4164
5-646
228
15-0996
6-109
276
16-6132
6-510
181
13-4536
5-656
229
15-1327
6-118
277
16-6433
6-518
182
13-4907
5-667
230
15-1657
6-126
278
16-6733
6-526
183
13-5277
5-677
231
15-1986
6-135
279
16-7032
6-534
184
13-5646
5-687
232
15-2315
6-144
280
16-7332
6-542
185
13-6014
5-698
233
15-2643
6-153
281
16-7630
6-549
186
13-6381
5-708
234
15-2970
6-162
282
16-7928
6-557
187
13-6747
5-718
235
15-3297
6-171
283
16-8226
6-565
188
13-7113
5-728
236
15-3622
6-179
284
16-8522
6-573
189
13-7477
5-738
237
15-3948
6-188
285
16-8819
6-580
190
13-7840
5-748
238
15-4272
6-197
286
16-9115
6-588
191
13-8202
5-758
239
15-4596
6-205
287
16-9410
6-596
192
13-8564
5-768
240
15-4919
6-214
288
16-9705
6-603
SQUARE AND CUBE ROOTS.
297
;:
8quurool.
'ubr root.
No.
Square root.
?ute root.
No.
Square root.
Cub* roc 1
289
17-
6-611
337
18-3575
6-958
385
19-6214
7-274
290
17-0293
6-<>19
338
18-3847
6-965
386
19-6468
7-281
291
17-0587
6-626
339
18-4119
6-972
387
19-6723
7-287
292
17-0880
6-634
340
18-4390
6-979
388
19-6977
7-293
293
17-1172
6-641
341
18-4661
6-986
389
19-7230
7-299
294
17-1464
6-649
342
18-4932
6-993
390
19-7484
7-306
295
17-1755
6-656
343
18-5202
7-
391
19-7737
7-312
296
17-2046
6-664
344
18-5472
7-006
392
19-7989
7318
297
17-2336
6-671
345
18-5741
7-013
393
19-8242
7-324
298
17-2626
6-679
346
18-6010
7-020
394
19-8494
7-331
299
17-2916
6-686
347
18-6279
7-027
395
19-8746
7-337
300
17-3205
6-694
348
18-6547
7-033
396
19-8997
7-343
301
17-3493
6-701
349
18-6815
7-040 :
397
19-9248
7-349
302
17-3781
6-709
350
18-7082
7-047
398
19-9499
7-355
303
17-4068
6-716
351
18-7349
7-054
399
19-9749
7-361
304
17-4355
6-723
352
18-7616
7-060
400
20-
7-368
305
17-4642
6-731
353
18-7882
7-067
401
20-0249
7-374
306
17-4928
6-738
354
18-8148
7-074
402
20-0499
7-380
307
17-5214
6-745
355
18-8414
7-080
403
20-0748
7-386
308
17-5499
6-753
356
18-8679
7-087
404
20-0997
7-39S
309
17-5783
6-760
357
18-8944
7-093
405
20-1246
7-398
310
17-6068
6-767
358
18-9208
7-100
406
20-1494
7-404
311
17-6351
6-775
359
18-9472
7-107
407
20-1742
7-410
312
17-6635
6-782
360
18-9736
7-113
408
20-1990
7-416
313
17-6918
6-789
361
19-
7-120
409
20-2237
7-422
314
17-7200
6-796
362
19-0262
7-126
410
20-2484
7-428
315
17-7482
6-804
363
19-0525
7-133
411
20-2731
7-434
316
17-7763
6-811
364
19-0787
7-140
412
20-2977
7-441
317
17-8044
6-818
365
19-1049
7-146
413
20-3224
7-447
318
17-8325
6-825
366
19-1311
7-153
414
20-3469
7-453
319
17-8605
6-832
367
19-1572
7-159
415
20-3715
7-459
320
17-8885
6-839
368
19-1833
7-166
416
20-3960
7-465
321
17-9164
6-847
369
19-2093
7-172
417
20-4205
7-470
322
17-9443
6-854
370
19-2353
7-179
418
20-4450
7-476
323
17-9722
6-861'
371
19-2613
7-185
419
20-4694
7-482
324
18-
6-868
372
19-2873
7-191
420
20-4939
7-488
325
18-0277
6-875
373
19-3132
7-198
421
20-5182
7-494
326
18-0554
6-882
374
19-3390
7-204
422
20-5426
7-5dO
327
18-0831
6-889
375
19-3649
7-211
423
20-5669
7-506
328
18-1107
6-896
376
19-3907
7-217
424
20-5912
7-512
329
18-1383
6-903
377
19-4164
7-224
425
20-6155
7-518
330
18-1659
6-910
378
19-4422
7-230
426
20-6397
7-524
331
18-1934
6-917
379
19-4679
7-236
427
20-6639
7-530
332
18-2208
6-924
380
19-4935
7-243
428
20-6881
7-536
333
18-2482
6-931
381
19-5192
7-249
429
20-7123
7-541
334
18-2756 6-938
382
19-5448
7-255
; 430
20-7364
7-547
335
18-3030 6-945
383
19-5703
7-262
431
20-7605
7-553
836 18-3303 6*853
384
19-5969
7-268 1) 438
20-7848
7-559
89b
SQUARE AND CUBE ROOTS.
Mo.
Squire root.
Cuberdbt.
No.
Squire root.
Cube root.
No.
Square root.
Cube root.
433
20-8086
7-565
481
21-9317
7-835
529
23-
8-087
434
20-8326
7-571
482
21-9544
7-840
530
23-0217
8-092
435
20-8566
7-576
483
21-9772
7-846
531
23-0434
8-097
436
20-8806
7-582
484
22-
7-851
532
23-0651
8-102
437
20-9045
7-588
485
22-0227
7-856
533
23-0867
8-107
438
20-9284
7-594
486
22-0454
7-862
534
23-1084
8-112
439
20-9^23
7-600
487
22-0680
7-867
535
23-1300
8-118
440
20-9761
7-605
488
22-0907
7-872
536
23-1516
8-123
441
21-
7-611
489
22-1133
7-878
537
2:M732
8-128
442
21-0237
7-617
490
22-1359
7-883
538
23-1948
8-133
443
21-0475
7-623
491
22-1585
7-889
539
23-2103
8-138
444
21-0713
7-628
492
22-1810
7-894
540
23-2379
8-143
445
21-0950
7-634
493
22-2036
7-899
541
23-2594
8-148
446
21-1187
7-640
494,
22-2261
7-905
542
23-2808
8-153
447
21-1423
7-646
495
22-2485
7-910
543
23-3023
8-158
448
21-1660
7-651
496
22-2710
7-915
544
23-3238
8-163
449
21-1896
7-657
497
22-2934
7-921
545
23-3452
8-168
450
21-2132
7-663
498
22-3159
7-926
546
23-3666
8-173
451
21-2367
7668
499
22-3383
7-931
547
23-3880
8-178
452
21-2602
7-674
500
22-3606
7-937
548
23-4093
8-183
453
21-2837
7-680
501
22-3830
7-942
549
23-4307
8-188
454
21-3072
7-685
502
22-4053
7-947
550
23-4520
8-193
455
21-3307
7-691
503
22-4276
7-952
551
23-4733
8-198
456
21-3541
7-697
504
22-4499
7-958
552
23-4946
8-203
457
21-3775
7-702
505
22-4722
7-963
5f)3
23-5159
8-208
458
21-4009
7-708
506
22-4944
7-968
554
23-5372
8-213
459
21-4242
7-713
507
22-5166
7-973
555
23-5584
8-217
460
21-4476
7-719
508
22-5388
7-979
556
23-5796
8-222
461
21-4709
7-725
509
22-5610
7-984
557
23-6008
8-227
462
21-4941
7-730
510
22-5831
7-989
558
23-6220
8-232
463
21-5174
7-736
511
22-6053
7-994
559
23-6431
8-237
464
21-5406
7-741
512
22-6274
8-
560
23-6643
8-242
465
21-5638
7-747
513
22-6405
8-005
561
23-6854
9-247
466
21-5870
7-752
514
22-6715
8-010
562
23-7065
8-252
467
21-6101
7-758
515
22-6936
8-015
563
23-7276
8-257
468
21-6333
7-763
516
22-7156
8-020
564
23-7486
8-262
469
21-6564
7-769
517
22-7376
8-025
565
23-7697
8-267
470
21-6794
7-774
518
22-7596
8-031
566
23-7907
8-271
471
21-7025
7-780
519
22-7815
8-036
567
23-8117
8-276
472
21-7255
7-785*
520
22-8035
8-041
568
23-8327
8-281
473
21-7485
7-791
521
22-8254
8-046
569
23-8537
8-286
474
21-7715
7-796
522
22-8473
8-051
570
23-8746
8-291
475
21-7944
7-802
523
22-8691
8-056
571
23-8956
8-296
476
21-8174
7-807
524
22-8910
8-062
572
23-9165
8-301
477
21-8403
7-813
525
22-9128
8-067
573
23-9374
8-305
478
21-8632
7-818
526
22-9346
8-072
574
23-9582
8-310
)479
21-8860
7-824
527
22-9564
8-077
575
23-9791
8-315
480
21-9089
7-829
5^8
22-9782
8-082
576
24-
8-320
SQUARE AND CUBE ROOTS.
299
No.
Square Tool.
i
Cube root.
fc
Square root.
Cube root.
No.
S pre root.
Cub. mot.
577
24 0208
8-325
625
25-
8-549
673
25-9422
8-763
578
24-0416
8-329
G2C, ^5-0199
8-554
674
25-9615
8-767
579
24-0624
8-334
<;-,'7,
26-0399
07 5
25-9807
8-772
580
24-0831
8-339
628
25-^)599 8-563
676
26-
8-776
581
24-1039
8-344
629
25-0798
8-5C8 <;-,?
26-0192
8-780
582
24-1246
8-349
630
25-0998
8-6T2
678
26-0384
8-785
583
24-1453
8-353
631
25-1197
8-577
2()-(i:>"/(i
8-789
584
24-1660
8-358
632
25-13!*6
8-581
680
26-0768
8-793
585
24-1867
8-363
633
25-1594
8-586
t;si
26-0959
8-797
586
24-2074
8-368
634
25-1793
8-590
26-1151
8-802
587
24-2280
8-372
635
25-1992
8-595
683
26-1342
8-806
588
24-2487
8-377
636
25-2190
8-599 J684
26-1533
8-810
589
24-2693
8-382
637
25-2388
8-604
685
26-1725
8-815
590
24-2899
8-387
638
25-2586
8-608
IkSli
26-1916
8-819
591
24-3104
8-391
639
25-2784
8-613
687
26-2106
8-823
592
24-3310
8-396
640
25-2982
8-617
688
26-2297
8-828
593
24-3515
8-401
641
25-3179
8-622
689
26-2488
8-832
594
24-3721
8-406
642
25-3377
8-626
690
26-2678
8-836
595
24-3926
8-410
643
25-3574
8-631
61)1
26-2868
8-840
596
24-4131
8-415
644
25-3771
8-635 i)!>2
26-3058
8-845
597
24-1335
8-420
645
25-39B8
8-640 r,!::
26-3248
8-849
698
24-4540
8-424
646
25-4165
8-644 694
26-3438
8-8M
599
24-4744
8-429
647
25-4361
8-649 695
26-3628
8-857
600
24-4948
8-434
648
25-4558
8-653 69<>
26-3818
8-862
601
24-5153
8-439
649
25-4754
8-657
897
26-4007
8-866
602
24-5356
8-443
650
25-4950
8-662
098
26-4196
8-870
603
24-5560
8-448
651
25-5147
8-666
6<J9
26-4386
8-874
604
24-5764
8-453
652
25-5342
8-671
700
26-4575
8-879
605
24-5967
8-457
653
25-5538
8-675
701
26-4764
8-883
606
24-6170
8-462
654
25-5734
8-680 702
26-4952
8-887
607
24-6373
8-466
956
25-5929
B-684
703
26-5141
8-891
608
24-6576
8-471
656
25-6124
8-688
704
26-5329
8-895
609
24-6779
8-476
657
25-6320
8-693
705
26-5518
8-900
610
24-6981
8-480
658
25-6515
8-697
706
26-5706
8-904
611
24-7184
8-485
659
25-6709
8-702
707
26-5894
8-908
612
24-7386
8-490
660
25-6904
8-706
708
26-6082
8-912
613
24-7588
8-494
661
25-7099
8-710
709
26-6270
8-916
614
24-7790
8-499
662
25-7293
8-715
710
26-6458
8-921
615
24-7991
8-504
663
25-7487
8-719
711
26-6645
8-925
61G
24-8193
8-508
664
25-7681
8-724
71-2
26-6833
8-929
617
24-8394
8-513
665
25-7875
8-728
713
26-7020
8-933
618
24-8596
8-517
666
25-8069 8-732 714
26-7207
8-937
619
24-8797
8-522
667
25-8263 8-737
715
26-7394
8-942
620
24-8997 8-527
668
25-8456 8-741
716
26-7581
8-945
621
24-9198 8-531
669
25-8650 8-745
717
26-7768
8-950
622
24-9399 8-536
670
25-8843 8-750
718
26-7955
8-954
623
24-959^
671
25-9036 8-754 719
26-8 14.1
8-958
624
24-9799 ; 8-545
672
25-9229 8-759 720
20-3328
8-962 .
300
SQUARE AND CUBE ROOTS.
1
1 No.
Square root.
Cube root.
No.
Square root.
Cube root.
No.
Square root.
Cube root.
721
26-8514
8-966
769
27-7308
9-161
817
28-5832
9-348
722
26-8700
8-971
770
27-7488
9-165
818
28-6006
9-352
723
26-8886
8-975
771
27-7668
9-169
819
28-6181
9-356
724
26-9072
8-979
772
27-7848
9-173
820
28-6356
9-359
72ft
26-9258
8-983
773
27-8028
9-177
821
28-6530
9-363
726
26-9443
8-987
774
27-8208
9-181
822
28-6705
9-367
727
26-9629
8-991
775
27-8388
9-185
823
28-6879
9-371
728
26-9814
8-995
776
27-8567
9-189
824
28-7054
9-375
729
27-
9-
777
27-8747
9-193
825
28-7228
9-378
730
27-0185
9-004
778
27-8926
9-197
826
28-7402
9-382
731
27-0370
9-008
779
27-9105
9-201
827
28-7576
9-386
732
27-0554
9-012
780
27-9284
9-205
828
28-7749
9-390
733
27-0739
9-016
781
27-9463
9-209
829
28-7923
9-394
734
27-0924
9-020
782
27-9642
9-213
830
28-8097
9-397
73ft
27-1108
9-024
783
27-9821
9-216
831
28-8270
9-401
736
27-1293
9-028
784
28-
9-220
832
28-8444
9-405
737
27-1477
9-032
785
28-0178
9-224
833
28-8617
9-409
738
27-1661
9-036
786
28-0356
9-228
834
28-8790
9-412
739
27-1845
9-040
787
28-0535
9-232
835
28-8963
9-416
740
27-2029
9-045
788
28-0713
9-236
836
28-9136
9-420
741
27-2213
9-049
789
28-0891
9-240
837
28-9309
9-424
742
27-2396
9-053
790
28-1069
9-244
838
28-9482
9-427
743
27-2580
9-057
791
28-1247
9-248
839
28-9654
9-431
744
27-2763
9-061
792
28-1424
9-252
840
28-9827
9-435
745
27-2946
9-065
793
28-1602
9-256
841
29.
9-439
746
27-3130
9-069
794
28-1780
9-259
842
29-0172
9-442
747
27-3313
9-073
795
28-1957
9-263
843
29-0344
9-446
748
27-3495
9-077
796
28-2134
9-267
844
29-0516
9-450
749
27-3678
9-081
797
28-2311
9-271
845
29-0688
9-454
750
27-3861
9-085
798
28-2488
9-275
846
29-0860
9-457
751
27-4043
9-089
799
28-2665
9-279
847
29-1032
9-461
752
27-4226
9-093
800
28-2842
9-283
848
29-1204
9-465
753
27-4408
9-097
801
28-3019
9-287
849
29-1376
9-468
754
27-4590
9-101
802
28-3196
9-290
850
29-1547
9-472
755
27-4772
9-105
803
28-3372
9-294
851
29-1719
9-476
756
27-4954
9-109
804
28-3548
9-298
852
29-1890
9-480
757
27-5136
9-113
805
28-3725
9-302
853
29-2061
9-483
758
27-5317
9-117
806
28-3901
9-306
854
29-2232
9-487
759
27-5499
9-121
807
28-4077
9-310
855
29-2403
9-491
760
27-5680
9-125
808
28-4253
9-314
856
29-2574
9-494
761
27-5862
9-129
809
28-4429
9-317
857
29-2745
9-498
762
27-6043
9-133
810
28-4604
9-321
858
29-2916
9-502
763
27-6224
9-137
811
28-4780
9-325
859
29-3087
9-505
764
27-6405
9-141
812
28-4956
9-329
860
29-3257
9-509
765
27-6586
9-145
813
28-5131
9-333
861
29-3428
9-513
766
27-6767
9-149
814
28-5306
9-337
862
29-3598
9-517
767
27-6947
9-153
815
28-5482
9-340
863
29-3768
9-520 j
768
27-7128
9-157
816
28-5653
9-344
864
29-3938
9-524
SQUARE AND CUBE ROOTS.
301
No.
Square root.
"ube root.
No.
Squirt root.
Cube root.
No.
S<jirp mot.
Cube root.
865
29-4108
9-528
910
30-1662
955
30-9030
9-847
866
29-4278
9-531
911
30-1827
9-694
9S6
30-9192
9-851
867
29-4448
9-535
912
30-1993
9-697
957
30-9354
9-854
868
29-4618
9-539
913
30-2158
9-701
958
30-9515
9-857
869
29-4788
9-542
914
30-2324
9-704
959
30-9677
9-861
870
29-4957
9-546
915
30-2489
9-708
960
30-9838
9-864
871
29-5127
9-550
916
30-2654
9-711
961
31-
9-868
872
29-5296
9-553
917
30-2820
9-715
962
31-0161
9-871
873
29-5465
9-557
918
30-2985
9-718
963
31-0322
9-875
874
29-5634
9-561
919
30-3150
9-722
964
31-0483
9-87S
875
29-5803
9-564
920
30-3315
9-725
965
31-0644
9-881
876
29-5972
9-568
921
30-3479
9-729
966
31-0805
9-885
877
29-6141
9-571
922
30-3644
9-732
967
31-0966
9-888
878
29-6310
9-575
923
30-3809
9-736
968
31-1126
9-892
879
29-6479
9-579
924
30-3973
9-739
969
31-1287
9-895
880
29-6647
9-582
925
30-4138
9-743
970
31-1448
9-898
881
29-6816
9-586
926
30-4302
9-746
971
31-1608
9-902
882
29-6984
9-590
927
30-4466
9-750
972
31-1769
9-905
883
29-7153
9-593
028
30-4630
9-753
973
31-1929
9-909
884
29-7321
9-597
929
30-4795
9-757
974
31-2089
9-912
885
29-7489
9-600
930
30-4959
9-761
975
31-2249
9-915
886
29-7657
9-604
931
30-5122
9-764
976
31-2409
9-919
887
29-7825
9-608
932
30-5286
9-767
977
31-2569
9-922
888
29-7993
9-611
933
30-5450
9-771
978
31-2729
9-926
889
29-8161
9-615
<J34
30-5614
9-774
979
31-2889
9-929
890
29-8328
9-619
935
30-5777
9-778
980
31-3049
9-932
891
29-8496
9-622
936
30-5941
9-782
981
31-3209
9-936
892
29-8663
9-626
937
30-6104
9-785
982
31-3368
9-939
893
29-8831
J9-629
938
30-6267
9-788
983
31-3528
9-943
894
29-8998
9-633
939
30-6431
9-792
984
31-3687
9-946
895
29-9165
9-636
940
30-6594
9-795
985
31-3847
9-949
896
29-9332
9-640
941
30-6757
9-799
986
31-4006 9-953
897
29-9499
9-644
942
30-6920
9-802
987
31-4165 ; 9-956
898
29-9666
9-647
943
30-7083
9-806
988
31-4324 9-959
899
29-9833
9-651
944
30-7245
9-809
989
31-4483
9-963
900
30-
9-654
945
30-7408
9-813
990
31-4642 9-966
901
30-0166
9-658
946
30-7571
9-816
991
31-4801 9-969
902
30-0333
9-662
947
30-7733
9-820
992
31-4960 9-973
903
30-0499
9-665
948
30-789fi
9-823
993
31-5119 9-976
904
30-0665
9-669
949
30-8058
9-827
994
31-5277 9-979
906
30-0832
9-672
950
30-8220
9-830
995 31-5436 9-983
906
30-0998
9-676
951
30-8382
9-833
996 31-5594 9-986
907
30-1164
9-679
952
30-8544
9-837
997
31-5753 9-989
908
30-1330
9-683
953
30-8706
9-840
998
31-5911
9-993
| 909 30-1496
9-686 || 954
30-8868
9-844
999
31-6069
9-996
302 RECIPES.
USEFUL RECIPES FOR WORKMEN.
For Lead. Melt one part of block tin, and when in a
state of fusion add two parts of lead. If a small quantity
of this, when melted, is poured out upon the table, there
will, if it be good, arise little bright stars upon it. Resin
should be used with this solder.
For Tin, Take four parts of pewter, one of tin, and one
of bismuth ; melt them together, and run them into thin
slips. Resin is also used with this solder.
For Iron. Good tough brass, with a little borax.
CEMENTS.
A very strong glue is made by adding some powdered
chalk to common glue when melted; and a glue which will
resist the action of water may be formed by boiling one
pound of common glue in two quarts (English measure) of
skimmed milk.
Turkey Cement. Dissolve five or six bits of mastich, as
large as peas, in as much spirit of wine as will dissolve it.
In another vessel dissolve as much isinglass, (which has
been previously soaked in water till it is softened and
swelled,) in one glass of strong whisky; add two small bits
of gum galbanum, or ammoniacum, which must be rubbed
or ground till dissolved, then mix the whole by the assist-
ance of heat. It must be kept in a stopped phial, which
should be set in hot water when the cement is to be used-
For turners, an excellent cement is made by melting in
a pan over the fire one pound of resin, and when melted
add a quarter of a pound of pitcli : while these are boiling
add brick dust, until, by dropping a little upon a cold stone,
you think it hard enough. In winter it is sometimes found
necessary to add a little tallow.
In joining the flanches of iron cylinders or pipes, to
withstand the action of boiling water and steam, great in-
convenience is often felt by the workmen for want of a
durable cement. The following will be found to answer:
Boiled linseed oil, litharge, and white lead, mixed up to a
proper consistence, and applied to each side of a piece of
flannel, linen, or even pasteboard, and then placed between
the pieces before they are brought home, as it is c illed, or
joined.
RECIPES. 303
For Steam Engines an excellent cement is as follows :
Take of sal ammoniac two ounces, sublimed sulphur one
ounce, and cast iron filings or fine turnings one pound ; mix
them in a mortar, and keep the powder dry. When it is
to be used mix it with twenty times its quantity of clean
iron turnings, or filings, and grind the whole in a mortar,'
then wet it with water, until it becomes of a convenient
consistence, when it is to be applied to the joint ; after a
time it becomes as hard and strong as any other part of the
metal.
LACQUERS AND VARNISHES.
Old Varnish is made by pouring, by little and little, half
a pound of drying oil on a pound of melted copal, constantly
stirring with a piece of wood. When the copal is melted,
take the mixture off the fire and add a pound of Venice
turpentine ; then pass the whole through a linen cloth.
When the varnish gets thick by keeping, add a little Venice
turpentine ; and if it be wished of a dark colour, amber
should be used instead of copal.
Black varnish for iron is made of twelve -parts of amber,
twelve of turpentine, two of resin, two of asphaltum, and
six of drying oil.
For cabinet work and musical instruments a varnish may
be made thus ; Take four ounces of gum sandarack, two
ounces of lack, the same of gum mastich, and an ounce of
gum elemi ; dissolve them in a quart of the best whisky ;
the whole being kept warm when they are dissolved, add
half a gill of turpentine.
Lacquer is a varnish to be laid on metal, for the purpose
of improving its appearance or preserving its polish. The
lacquer is laid on the surface of the metal with a brush : the
metal must be warm, otherwise the lacquer will not spread.
For brass a good lacquer may be made thus : Take one
ounce of turmeric root ground, and half a drachm of the best
dragon's blood ; put them in a pint of spirits of wine,
(English measure,) and place them in a moderate heat,
shaking them for several days. It must then be strained
through a linen cloth, and being put back into the bottle,
three ounces of good seed-lack, powdered, must be added.
The mixture must again be subjected to a moderate heat,
and shaken frequently for several days, when it is again
strained, and corked tightly in a bottle for use.
304 RECIPES.
STAINING WOOD AND IVORY.
Fellow. Diluted nitric acid will often produce a fine
yellow on wood ; but sometimes it produces a brown, and
if used strong it will seem nearly black.
Red. A good red may be made by an infusion of Brazil
wood in stale urine, in the proportion of a pound to a gal-
lon This stain is to be laid on the wood boiling hot ; and
before it dries it should be laid over with alum water. For
the same purpose a solution of dragon's blood in spirits of
wine may also be used.
Mahogany colour may be produced by a mixture of mad-
der, Brazil wood, and logwood, dissolved in water and put
on hot. The proportions must be varied by the artist ac-
cording to the tint required.
Slack. Brush the wood several times over with a hot
decoction of logwood, and then with iron lacquer, or, if this
cannot be had, a strong solution of nut galls.
Ivory may be stained blue thus : Soak the ivory in a
solution of verdigris in nitric acid, which will make it
green, then dip it into a solution of pearlash boiling hot,
and it will turn blue.
To stain ivory black the same process as for wood may
be employed.
Purple may be produced by soaking the ivory in a solu-
tion of sal ammoniac into four times its weight of nitrous
acid.
To make Edge-Tools from Cast Steel and Iron. This
method consists in fixing a clean piece of wrought iron,
brought to a welding heat, in the centre of a mould, and
then pouring in melted steel, so as entirely to envelope the
iron ; and then forging the mass into the shape required.
To colour Steel Slue. The Steel must be finely polish-
ed on its surface, and then exposed to a uniform degree
of heat. Accordingly, there are three ways of colouring :
first, by a flame producing no soot, as spirit of wine ; se-
condly, by a hot plate of iron ; and thirdly, by wood ashes.
As a very regular degree of heat is necessary, wood ashes
for fine work bears the preference. The work must be
covered over with them, and carefully watched ; when the
colour is sufficiently heightened, the work is perfect. This
colour is occasionally taken ofF with a very dilute marina
acid
RECIPES 305
To distinguish Steel from Iron. The principal cha-
as Aers by which steel may be distinguished from iron are
GS follow :
1. After being polished, steel appears of a whiter, light
JSty hue, without the blue cast exhibited by iron. It also
takes a higher polish.
2. The hardest steel, when not annealed, appears granu-
lated, but dull, and without shining fibres.
3. When steeped in acids, the harder the steel is of a
darker hue is its surface.
4. Steel is not so much inclined to rust as iron.
5. In general, steel has a greater specific gravity.
6. By being hardened and wrought, it may be rendered
much more elastic than iron.
7. It is not attracted so strongly by tke magnet as soft
iron. It likewise acquires magnetic properties more slowly,
but retains them longer, for which reason steel is used in
making needles for compasses, and artificial magnets.
8. Steel is ignited sooner, and fuses with less degree of
heat than malleable iron, which can scarcely be made to
fuse without the addition of powdered charcoal ; Jby which
it is converted into steel, and afterwards into crude iron.
9. Polished steel is sooner tinged by heat, and that with
higher colours, than iron.
10. In a calcining heat, it suffers less loss by burning
than soft iron does in the same heat and the same time. In
calcination a light blue flame hovers over the steel, either
with or without a sulphureous odour.
11. The scales of steel are harder and sharper than those
of iron ; and consequently more fit for polishing with.
12. In a white heat, when exposed to the blast of the
bellows among the coals, it begins to sweat, wet, or melt,
partly with light-coloured and bright, and partly with red
sparkles, but less crackling than those of iron. In a melt-
ing heat, too, it consumes faster.
13. In the vitriolic nitrous, and other acids, steel is vio-
lently attacked, but is longer in dissolving than iron. After
maceration, according as it is softer or harder, it appears of
8 lighter grey or darker colour ; while iron, on the other
band, is white.
26*
UCS8 LIBRARY
A 000 606 496 8