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CLBMENT MACKROWJ\4.I.N.A.

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76

THH

NAVAL ARCHITECT'S AND SHiPBUiLDERS

POCKET-BOOK

OF

jpormulae, fiulcs?, anti Cables;

AXD

MARINE ENGINEER'S AND SURVEYOR'S

HANDY BOOK OF REFERENCE

BY

CLEMENT MACKROW â€¢

NAVAL DRAtGHTSMAN

MEMBER OF THE IKSTITCTIOX OF NATAL ARCHITECTS

r^trli etfitton, acbiiSrt

(sr^Ij

LONDON

CROSBY LOCKWOOD AXD CO.

7 STATIONERS'- HALL COXJET, LrTDGATE HILL

1884

L0ND05 : POINTED BT

HPOTTiSWOOOE AND CO.. SKW-BTRKKT SQUAUE

AKU TAULIAJIEST STUEliT

PREFACE

The object of this work is to supply tiie great ^raiit

which has long been experienced by nearly all who are

connected professionally with shipbuilding, of a Pocket-

Book which should contain all the ordinary Formulae,

Kales, and Tables required when working out necessary

calculations, which up to the present time, as far as the

Author is aware, have never been collected and put into

so convenient a form, but have remained scattered

through a number of large works, entailing, even in

referring to the most commonly used Formulae, much

waste of time and trouble. An effort has here been

made to gather all this valuable material, and to con-

dense it into as compact a form as possible, so that the

Kaval Architect or the Shipbuilder may always have

ready to his hand reliaV)le data from which he can solve

the numerous problems which daily come before him.

How far this object has been attained may best be judged

by those who have felt the need of such a work.

Several elementary subjects have been treated more

fully than may seem consistent with the character of the

book. This, however, has been done foi* the benefit of

those who have received a practical rather than a theo-

retical training, and to whom such a book as this would

be but of small service were they not first enabled to

gather a few elementary principles, by which means they

may learn to use and understand these Formulae.

836451

^^' PREFACE.

In justice to those authors whose works have beer

consulted, it must be added that most of the Rules and

Formulae here given are not original, although perhaps

appearing in'^a new shape with a view to ma kin f^ them

simpler.

There are many into whose hands this work will fall

who are well able to criticise it, both as to the usefulness

and the accuracy of the matter it contains. From such

critics the Author invites any corrections or fresh mate-

rial which may be useful for future editions.

CLEMENT MACKROW.

London : July 187'.).

NOTE TO THE THIRD EDITION.

The rapid sale of the first and second editions of this

work has shown that the efforts made to supply a much

felt want have in some measure succeeded, and the

present opportunity has been taken of thoroughly revising

it, so as to make it more worthy of the confidence it has

received. Many strangers to the Author have taken a

generous interest in the book by making suggestions, kc,

which have, where possible, been carried out ; and it is

hoped that the same kindly interest in it will continue to

be shown.

CLEMENT MACKROW.

London : A2)Hl 1884.

CONTENTS.

PAGE

Algebraical Signs axd Symbols 1-3

Decimal Fractions . 4-6

Practical Geometry 7-19

Trigonometry 20-28

Tables of Circular Measure of Angles. . . . 29-31

Mensuration of Superficies 32-41

Mensuration of Solids 41-iG

Mensuration of the Surfaces of Solids .... 47

Circumferences and Areas of Circles .... 48-66

Areas of Segments of Circles 67-09

Centres and Moments of Figures 70-81

ToN'NAGE Rules and Tables 82-102

Board of Trade Regulations for Ships .... 103-114

English Weights and Measures 115-122

Metrical Weights and Measures 122-136

Decimal Equivalents of English Weights and Mea-

sures 1.57-140

Foreign ^Ioney, Weights, and Measures . . . 141-142

Mechanical Principles 143-145

Centre of Gravity of Bodies 146

Laws of Motion 147-151

Displacement of Ships 151-158

Centre of Gravity of Ships 159-161

Stability of Ships 1C2-177

Waves 178-181

VI CONTENTS.

PAUE

KoLLiNO 182-186

Propulsiox of Vessels . , , 186-209

Distances down Courses of Rivers, etc. . . , 210-219

Steering 220-221

Squares, Cubes, Roots, and Reciprocals of Numbers . 222-206

Evolution . . 267-268

Weight and Strength of Materials .... 269-305

Values of Whitworth's Gauges, etc. . . j . . 305-310

Useful Numbers often used in Calculations . 311-319

Riveting as employed in H.M.S. 'Hercules' . , . 321-324

Sines, Tangents, Secants, etc 325-332

Masting and Rigging Ships 333-360

French and English Vocabulary . . . â€ž . 361-376

Hyperbolic Logarithms 377-381

Wages and Percentage Tables, etc 382-389

Logarithms of Numbers 390-426

Strength of Materials, etc 427-451

Hydraulic and Miscellaneous Formul.e . . . . 452-456

Conic Sections, Catenary, etc 457-461

Mechanical Powers, Work, etc â€¢ 462-467

Board of Trade Regulations for Marine Boilers, etc. 467-478

Particulars, Weights, etc., of Marine Engines . . 479-483

Seasoning ^vnd Preserving Timber 484-485

Timber Measures 486-488

Bricklaying, Plastering, Painting, etc. . . . 488-490

Varnishes, Lacquers, Dipping Acids, Ce:ments, etc. . . 490-495

Miscellaneous Recipes, Tables, etc. . . â€¢ . 495-512

INDEX . 513

MACKEOW^

POCKET BOOK

OF

FOEilULJ:, EULES, AND TABLES

FOE

MYAL ARCHITECTS J^D SHIP-BUILDEES.

â€¢o*-

SIGNS AND SYMBOLS.

The following are some of the signs and symbols commonly

used in algebraical expressions : â€”

= This is the sign of equality. It denotes that the quantities

so connected are equal to one another ; thus, 3 feet = 1 yard.

+ This is the sign of addition, and signifies plus or more ;

thus, 4 + 3 = 7.

â€” This is the sign of subtraction, and signifies minus or less ;

thus, 4-3 = 1.

X This is the sign of multiplication, and signifies multiplied

by or into ; thus, 4 x 3= 12.

-f- This is the sign of division, and signifies di\ided by ; thus,

4-r2 = 2.

{} [] These signs are called brackets, and denote that the

quantities between them are to be treated as one quantity; thus,

5{3(4+2)-6(3-2)}=5(18-6) = 60.

This sign is called the bar or Adnculum, and is sometimes

iised instead of the brackets ; thus, 3(4 + 2) â€” 6(3 â€” 2) x 5 = 60.

Letters are often used to shorten or simplify a formula.

Thus, supposing we wish to express length x breadth x depth, we

might put the initial letters only, thus, ^ x J x t?, or, as is usual

when algebraical symbols are employed, leave out the sign x

between the factors and write the formula l.b.d.

When it is wished to express division in a simple form ths

â€¢F *f" 2/

divisor is written xmder the dividend; thus, (a; + y) -rz = -â€¢

z

Z SIGNS AND SYMBOLS.

! , t ' , ', , These a7e signs of proportion ; the sign : = is

to, the sign : : = as ; thus, 1:3 : ! 3 : 9, 1 is to 3 as 3 is to 9.

< This sign denotes less than ; thus 2 < 4 signifies 2 is less

than 4.

> This sign denotes more than : thus 4 > 2 signifies 4 is more

than 2.

*.' This sign signifies because.

.*. This sign signifies therefore. Sec.: '.' 9 is the square of

3 /. 3 is the root of 9.

'-^ This sign denotes difference, and is placed between two

quantities when it is not known which is the greater ; thus

{x '^ y) signifies the difference between x and y.

, , These signs are uÂ«ed to express certain angles in

degrees, minutes, and seconds : thus 25 degrees 4 minutes 21

seconds would be expressed 25Â° 4' 21".

JVote. â€” The two latter signs are often used to express feet and

inches ; thus 2 feet 6 inches may be written 2' 6".

n/ This sign is called tlie radical siffn, and placed before a

quantity indicates that some root of it is to be taken, and a

small figure placed over the sign, called the exponent of the root,

shows what root is to be extracted.

Thus 2^a or Va means the square root of a.

^a â€ž cube â€ž

4/Â« ,, fourth â€ž

-^^ This denotes that the square root of a has to be taken

and divided by h.

This denotes that 1) has to be divided by the square

root of a.

y

This denotes that the square root of a+ J has to be

a-\- d

divided by the square root of a-^d. It may also be written

thus, / , or -.

S/ a^rd ^/a + d

QC This is anotlier sign of proportion. Ex.: a(X:h; that is,

a varies as or is proportional to h.

oo This sign expresses infinity; that is, it denotes a quantity

greater than an}^ finite quantity.

This sign denotes a quantity infinitely small, nought.

Z This sign denotes an angle. Ex. : l ah would be written*

the angle ab.

SIGNS AND SYMBOLS. 3

L This sign denotes a right angle.

_L This sign denotes a perpendicular; 2iS, at L cd, i.e. aZ> is

perpendicular to cd.

A This sign denotes a triangle; thus, Aal?c, i.e. the triangle

izbc.

II This sign denotes parallel to. Ux.: ab \i cd would be

â€¢written, ab is parallel to cd.

f or F These express a function ; as, a =/> ; that is, ^ is a

function of x or equals x.

f This is the sign of integration ; that is, it indicates that the

^expression before which it is placed is to be integrated. AVhen

the expression has to be integrated twice or three times the sign

is repeated (thus,//, /(/'); but if more than three times an index

is placed above it (thus,/'").

Dord These are the signs of differentiation ; an index placed

above the sign (thus, d-) indicates the result of the repetition

of the process denoted hy that sign.

2 This sign (the Greek letter sigma) is used to denote that

the algebraical sum of a quantity is to be taken. It is com-

monly used to indicate the sitm of finite differences, in nearly the

same manner as the symbol/.

[jl This sign is sometimes^ used instead of tt, being a modifi-

cation of the letter C, for circtimference.

n This sign is sometimes used instead of e, being a modifi-

cation of the letter B, for base.

g This sign is used to denote the force of gravit}'^ at any

given latitude.

TT The Greek letter pi is invariabl}'' used to denote 3-14159;

that is, the ratio borne by the diameter of a circle to its circum-

ference.

As the letters of the Greek alphabet are of constant recur-

rence in mathematical formulae it has been deemed advisable to

append the following table : â€”

A a

Alpha.

I I

Iota.

P p

Eho.

B 3

Beta.

K K

Kappa.

2 o-s

Sioma,

r 7

Gamma.

A K

Lambda.

T T

Tau.

A S

Delta.

M IX

Mu.

T V

Upsilon.

E 6

Epsilon.

N V

Nu.

* <p

Phi.

z C

Zeta.

H 1

Xi.

X X

Chi.

H 7,

Eta.

O

Omicron.

Y y\>

Psi.

e d

Theta.

n IT

Pi.

D. o)

Omega.

E 2

SUBTRACTION AND MULTIPLICATION OF DECIMALS.

DECIMAL FRACTIONS.

Decimal Fractions are those which have 10, 100, 1000, &c.,

for a denominator, and are expressed by writing the numerator

only and placing a point before it on the left hand.

J^tL. J.. TOâ€” i-' 100 ' "â€¢ 1000 OtKl.

Ex.2, xo^"'^* iM~'^^* 1000 ~'^^^-

Ex.d. 113-3 = 113^ = iig^ = ie^.

ExA. 113-03 =^'^j'^ = ^-^ = iiÂ§g2.

Addition of Decimals.

Rule. â€” Arrange the numbers so that all the decimal points

come directly under one another; add them together as in

wliole numbers, and point off as many figures for decimals as

are equal to the greatest number of decimals in any of the given

numbers.

Ex. : Add together 3-79, -117, 87-225, 478-91.

3-79

â€¢117

87-225

478-91

570-042'. Ans.

Subtraction of Decimals.

Rule. â€” Place the numbers under one another, as in addition ;

subtract as in whole numbers, keeping the decimal point in the

remainder directly under those above it.

Ex. : From 97*378

- take 46-4972

50^8808. Ans.

Multiplication of Decimals.

Rule. â€”Multiply the factors together, as in whole numbers ;

then point off from the product as many decimal places as there

are in both factors, supplying any deficiency by annexing ciphers

to the left hand.

Ex. 1. Mult. 4-735 Ex. 2. Mult. -04735

by_^374 by -0374

18940 18940

33145 33145

14205 14205

1-770890. Atis. -001770890. Ans.

DIVISION AND EEDUCTION OF DECIMALS. S

Division of Decimals.

EULE. â€” Remove the decimal point in the dividend as many-

places to the right as there are decimal places in the divisor ;

supply any deficiency by annexing ciphers. Then make the

divisor a whole number, and proceed as in the division of

simple numbers, and the quotient will contain as many decimal

places as are used in the dividend.

Ex. 1. Divide 74-23973 by 6-12. Ex. 2. Divide -7423973 by 612.

612) 7423-973 (12-130. Ans. 612) -742.8973 (-0012130. Ans.

612 612

1303 1303

1224 1224

799 799

612 612

1877 l877

1836 1836

413 413

To Reduce ajn'y Fraction to a Decimal.

Rule. â€” Annex ciphers to the numerator till it be equal to or

greater than the denominator ; divide by the denominator, as

in division of decimals, and the quotient will be the decimal

required.

Ex. 1. Reduce ^fs to a decimal.

256) 7-00000000 (-02734375. Ans.

512

1880

1792

880

Ex. 2. Reduce ^ to a decin

768

12) 7-00000000

1120

1024

â€¢58333333. A7is.

960

768

1920

1792

1280

1280

1

To Reduce Numbers of Different Denominations

into Decimals.

Rule 1. â€” Reduce the given weight or measure, &c., into the

lowest denomination given, for a dividend ; then reduce the

b EEDUCTION OF DECIMALS.

integer into the same denomination for a divisor ; tlie resulting-

fraction, changed to a decimal, will be the decimal required.

Rule 2. â€” Divide the least denomination by such a number as

will reduce it to the next greater; to the decimal so obtained

prefix the given number of the same denomination ; then divide

by such a number as will reduce it to the next greater; thus

proceed till it be reduced to the decimal of the required integer.

Ex. 1 to Rule 1. â€” Reduce 2 cwt. 3 qrs. 21 lbs. to the decimal

of a ton.

2 cwt. 3 qrs. 21 lbs. .329 lbs. -, ,^o - *

- ^ = â– ;; â€” = -l-168/o ton:

1 ton 2240 lbs.

or, by Rule 2â€” | 7) 21-0 lbs.

2^ i 4 ) 3-0

4 ) 3-7o qr s.

2 0) 2-9375 cw ts.

Ans. '146875 ton.

Kc. 2 to Rule 1. â€” Reduce 2 ft. 9 in. to the decimal of a yard.

^i!:AiE: = 33jn. = . 916666 yard;

1 yard 36 in. ^

or, by Rule 2â€” 12 ) 9 in .

3 ) 2-75 fee t

Ans. â€¢91666 yard.

To Find the Value of any Decimal.

Rule. â€” Multiply the given decimal by the number of parts

contained in the next lesser denomination, and point off from

tlie product as many figures as the decimal consists of. Mul-

tiply the remaining decimal by the number of parts in the next

lesser denomination, and point off as many decimals in the

product as before. Proceed thus till you have brought out the

least known parts of the integer.

Ex. 1. What is the value of Ex. 2. What is the value of

â€¢146875 of a ton ? -91666 of a yard ?

â€¢146875 -91666

20 3

cwts. 2-937500 feet 274998

4 J^

qrs. 3-750000 in. 899976

28

lbs. 21-0000 000

A7i8. = 2 cwts. 3 qrs. 21 lbs.

Ans. = 2 ft. 9 in.

GEOMETRY.

PRACTICAL GEOMETRY.

1. From any given point in a straight line

to erect a lyerpendicular . (Fig-. 1.)

On each side of the point a in the line from

which the perpendicular is to be erected set off

equal distances Kb, A.c ; and from h and e as

centres, with any radius greater than Kb or Ar,

describe arcs cutting each other at d.,d' \ a line

drawn through dd' will pass through the point

A, and Kd will be perpendicular to he.

2. To erect a lyerpendicular at or near the

end of a litw. (Fig. 2.)

With any convenient radius, and at any

distance from the given line AB, describe an

arc, as BAG, cutting the given point in a ;

through the centre of the circle N draw the

line BXC : a line drawn from the point A,

cutting the intersection at c, will be the

required perpendicular.

3

Fig. 1.

d-'k

To divide a line into any number of eq^ual paH&. (Fig. 3.)

h ^'-

/

L.'

3

Fig. 3.

L^-'

-&

a.-

'B

Frc4. 4.

Let AB be the given straight

line to be divided into a number of

equal parts ; through the points

A and B draw two parallel lines AC

and DB, forming any convenient

angle with AB ; upon AC and DB set

off the number of equal parts re-

quired, as A-1, 1-2, ko,., B-1, 1-2, &c ; "

join A and d, 1 and 3, 2 and 2, 3 and 1, c and B, cutting AB

in a, b, and c, which will thus be divided into four equal parts.

4. To find the length of any given arc of a circle. (Fig. 4.)

With the radius Kd, equal to one-

fourth of the length of the chord of the

arc AB, and from A as a centre, cut the

arc in c ; also from B as a centre with

the same radius cut the chord in b ;

draw the line cb, and twice the length

of the line cb is the length of the arc nearly

5. To draw from or to the cir-

cumference of a circle lines tend-

ing towards the coitre, when the

centre is inaccessible. (Fig. 5.)

Divide the given portion of

the circumference into the

desired number of parts ; then

with any radius less than the dis-

tance of two parts, describe arcs cutting each other as Al, cl, k.c.;

Fm. -

-8

GEOMETRY.

draw the lines b1, c2, &c., which will lead to the centre, as required.

To draw the end lines XT', Tr from B and E, with the same radii

as before describe the arcs r', r, and with the radius Bl, from A

as centre, cut the former arcs at r', r, lines then drawn from Ar'

and F;* will tend towards the centre, as required.

Jo. To desonhe an arc of a circle of large radius, (i'ig. 6.)

Fig. 6. Let A, B, c be the three points through

B which the arc is to be drawn ; join ba

and BC ; then construct a flat trian-

gular mould, having two of its edges

perfectly straight and making with

each other an angle equal to ABC.

Each of the edges should be a little

longer than the chord AC. In the points A, c fix pins ; and fix a

pencil to the mould at B, and move the mould so as to keep its

edges touching the pins at A and c, when the pencil will describe

the required axe.

7. Another method. (Fig. 7.)

^ Fig- 7. Draw the chord ADC, and

E iw >_ B a TTf r draw ebf parallel to it: bisect

the chord in D and draw db per-

pendicular to AC ; join ab and

BC ; draw AE perpendicular to

â– '^ â€¢ 2 D 2 1 c XB and CF perpendicular to BC ;

also draw An and cn periDcndicular to AC ; divide AC and ef

into the same number of equal parts, and An. cn into half that

number of equal parts ; join 1 and 1, 2 and 2, also B and .?, s,

and B, and t, t ; through the points where they intersect

describe a curve, which will be the arc required.

8. To describe an ellipse, the transverse and conjugate diameters

being given. (Fig. 8.)

- Let AB be the transverse and CD the

conjugate diameters, bisecting each other

at right angles in the centre E ; from C as

a centre, with EA as radius, describe arcs

cutting AB in F and f', which will be the

foci of the ellipse ; between E and f

set off any number of points, as 1, 2 (it

is advisable that these points should

be closer as they approach f).

From F and f', with radius Bl, describe the arcs g, g', g", g'".

From F and f', with radius Al, describe the arcs H, h', h", h'",

intersecting the arcs G, G'. g", g'" in the points I, i, i, I, which will

be four points in the curve.

Then strike arcs from F, f' first with A2, then with b2 ;

these radii intersecting will give four more points. Proceed

in this way with all the points between E and F; the curve of

the ellipse must then be traced through these points by hand.

GEOMETRY.

Fig. 10.

9. Another metlwd. (Fig. 9.)

At 0, the intersection of the two dia-

meters, as a centre, with a radius equal to

the difference of the semi-diameters,

describe the arc ab, and from Z> as a

centre with half the chord tea describe

the arc cd ; from o as centre with the

distance od cut the diameters in dr, dt ;

draw the lines rs, rs, ts, ts ; then from r

and t describe the arcs sds, scs ; also from

d and d describe the smaller arcs SAS, SBS, which will complete

the ellipse required.

10. To draw a tangent and a perpendicular to an elUjyse at

any point. (Fig. 10.)

Let G be the point ; from F, f', the

two foci of the ellipse, draw straight

lines through G and produce them ;

hisect the angle made by the produced

parts, by GH, then GH is perpendicular

to the curve ; at G bisect the angle

formed by FG and f'g produced, by IJ,

then IJ will be the tangent to the curve

at G, and it will also be perpendicular

to GH.

\\. To describe an elliptic arc, the span and height being given.

(Fig. 11.)

Bisect with a line at right angles the

chord or span AB ; erect the perpendicular

AQ, and draw the line QD equal and parallel

to AC ; bisect AC in c, and AQ in n ; make

CL equal to CD, and draw the line lcQ ; draw

also the line nSD, and bisect SD with a line

KG at right angles to it, and meeting the

line LD in G ; draw the line gkq, and make

Gp equal to CK, and draw the line 6/^2 ;

then from G as centre with the radius

GD describe the arc sd2, and from K and p as centres with the

radius ak describe the arcs as and 2b, which complete the arc,

as required.

12. Another method. (Fig. 12.)

Bisect the chord ab, and fix at right

angles to it a straight guide, as be ; prepare

of any material a rod or staff equal to half

the length of the chord, as def ; at a

distance from the end of the staff, equal

to the height of the arc, fix a pin e, and at

the extremity a tracer/; move the staff,

keeping its end to the guide and the fixed

10

GEOMETRY.

pin to the chord, and the tracer will describe a half of the-

arc required.

13. To obtain hj measwrement the length of any direct liney

tlwvgh intercepted by some viateHal object. (Fig. 13.)

Fig. 13. Suppose the distance between

A and B is required, but the

straight line is intercepted by

the object G. On the point d with

any convenient radius describe

the arc cc', and make the arc

twice the radius do in length ;

through c' draw the line dc'e, and

â€¢on e describe another dire ff equal in length to the radius dc ;

draw the line efr equal to efd\ from r describe the arc g'g^

equal in length to twice the radius rg ; continue the line through

rg to B : then A and b will make a right line, and de or er will

equal the distance between dr^ by which the distance between

AB is obtained, as required.

Fig. 14.

14. To ascertain the distance geometrically of an inaccessible-

object on a level plane. (Fig. 14.)

Let it be required to find the distance

between A and b, a being inaccessible.

Produce AB to any point d, and bisect

BD in c ; through d draw DÂ«, making

any angle with da, and take DC and db

respectively and set them off on T>a as

T)b and d^' ; join Be, cb, and Ab ; through

E, the intersection of B^ and cb, draw

.DEF meeting aZ> in F ; join BF and pro-

duce it till it meets Da in a: then ab will be equal to AB, the

distance required.

Fig. 15.

15. Another method. (Fig. 15.)

Produce ab to any point d ; draw the line

jyd at any angle to tlie line ab ; bisect the

line Dd in c, through which di-aw the line B&^

and make cb equal to bc ; join AC and db,

and produce them till they meet at a : then

ba will equal ba, the distance required.

GEOilETEY.

11

distance hetn-een two objects, both b&ing

16. To measure the

inaccessible. (Fig. 16.)

Let it be required to find the distance

between the points a and b, both being in-

accessible. From any point c draw any line

cc, and bisect it in D ; produce a.o and bc, and

prolong them to E and F ; take the point E in

the prolongation of A^. and draw the line ed^,

making T>e equal to de.

In like manner take the point f in the

prolongation of bc, and make D/ equal to df ;

produce ad and ec till they meet in Â«, and also

produce bd and /c till they meet in h : then

the distance between the points a and b equals

the distance between the inaccessible points

A and B.

17. To inscribe any regular polygon in a

given circle. (Fig. 17.)

Divide any diameter ab of the circle abd

into ai> many equal parts as the polygon is

required to have sides ; from A and B as

centres, with a radius equal to the diameter,

describe arcs cutting each other in c ; draw

the line CD through the second point of divi-

sion on the diameter ab, and a line drawn

from D to A is equal to one side of the poly-

gon required.

18. To cut a beam of the strongest section from

any round piece of timber. (Fig. 18.)

W

Umi ARCHITECT^ ^ SHifBUlLDEJS

- POCKET-BOOift I

CLBMENT MACKROWJ\4.I.N.A.

UC-NRLF

B 3 137 DbT

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CROSBY IDCRWOOD & :CÂ§H

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76

THH

NAVAL ARCHITECT'S AND SHiPBUiLDERS

POCKET-BOOK

OF

jpormulae, fiulcs?, anti Cables;

AXD

MARINE ENGINEER'S AND SURVEYOR'S

HANDY BOOK OF REFERENCE

BY

CLEMENT MACKROW â€¢

NAVAL DRAtGHTSMAN

MEMBER OF THE IKSTITCTIOX OF NATAL ARCHITECTS

r^trli etfitton, acbiiSrt

(sr^Ij

LONDON

CROSBY LOCKWOOD AXD CO.

7 STATIONERS'- HALL COXJET, LrTDGATE HILL

1884

L0ND05 : POINTED BT

HPOTTiSWOOOE AND CO.. SKW-BTRKKT SQUAUE

AKU TAULIAJIEST STUEliT

PREFACE

The object of this work is to supply tiie great ^raiit

which has long been experienced by nearly all who are

connected professionally with shipbuilding, of a Pocket-

Book which should contain all the ordinary Formulae,

Kales, and Tables required when working out necessary

calculations, which up to the present time, as far as the

Author is aware, have never been collected and put into

so convenient a form, but have remained scattered

through a number of large works, entailing, even in

referring to the most commonly used Formulae, much

waste of time and trouble. An effort has here been

made to gather all this valuable material, and to con-

dense it into as compact a form as possible, so that the

Kaval Architect or the Shipbuilder may always have

ready to his hand reliaV)le data from which he can solve

the numerous problems which daily come before him.

How far this object has been attained may best be judged

by those who have felt the need of such a work.

Several elementary subjects have been treated more

fully than may seem consistent with the character of the

book. This, however, has been done foi* the benefit of

those who have received a practical rather than a theo-

retical training, and to whom such a book as this would

be but of small service were they not first enabled to

gather a few elementary principles, by which means they

may learn to use and understand these Formulae.

836451

^^' PREFACE.

In justice to those authors whose works have beer

consulted, it must be added that most of the Rules and

Formulae here given are not original, although perhaps

appearing in'^a new shape with a view to ma kin f^ them

simpler.

There are many into whose hands this work will fall

who are well able to criticise it, both as to the usefulness

and the accuracy of the matter it contains. From such

critics the Author invites any corrections or fresh mate-

rial which may be useful for future editions.

CLEMENT MACKROW.

London : July 187'.).

NOTE TO THE THIRD EDITION.

The rapid sale of the first and second editions of this

work has shown that the efforts made to supply a much

felt want have in some measure succeeded, and the

present opportunity has been taken of thoroughly revising

it, so as to make it more worthy of the confidence it has

received. Many strangers to the Author have taken a

generous interest in the book by making suggestions, kc,

which have, where possible, been carried out ; and it is

hoped that the same kindly interest in it will continue to

be shown.

CLEMENT MACKROW.

London : A2)Hl 1884.

CONTENTS.

PAGE

Algebraical Signs axd Symbols 1-3

Decimal Fractions . 4-6

Practical Geometry 7-19

Trigonometry 20-28

Tables of Circular Measure of Angles. . . . 29-31

Mensuration of Superficies 32-41

Mensuration of Solids 41-iG

Mensuration of the Surfaces of Solids .... 47

Circumferences and Areas of Circles .... 48-66

Areas of Segments of Circles 67-09

Centres and Moments of Figures 70-81

ToN'NAGE Rules and Tables 82-102

Board of Trade Regulations for Ships .... 103-114

English Weights and Measures 115-122

Metrical Weights and Measures 122-136

Decimal Equivalents of English Weights and Mea-

sures 1.57-140

Foreign ^Ioney, Weights, and Measures . . . 141-142

Mechanical Principles 143-145

Centre of Gravity of Bodies 146

Laws of Motion 147-151

Displacement of Ships 151-158

Centre of Gravity of Ships 159-161

Stability of Ships 1C2-177

Waves 178-181

VI CONTENTS.

PAUE

KoLLiNO 182-186

Propulsiox of Vessels . , , 186-209

Distances down Courses of Rivers, etc. . . , 210-219

Steering 220-221

Squares, Cubes, Roots, and Reciprocals of Numbers . 222-206

Evolution . . 267-268

Weight and Strength of Materials .... 269-305

Values of Whitworth's Gauges, etc. . . j . . 305-310

Useful Numbers often used in Calculations . 311-319

Riveting as employed in H.M.S. 'Hercules' . , . 321-324

Sines, Tangents, Secants, etc 325-332

Masting and Rigging Ships 333-360

French and English Vocabulary . . . â€ž . 361-376

Hyperbolic Logarithms 377-381

Wages and Percentage Tables, etc 382-389

Logarithms of Numbers 390-426

Strength of Materials, etc 427-451

Hydraulic and Miscellaneous Formul.e . . . . 452-456

Conic Sections, Catenary, etc 457-461

Mechanical Powers, Work, etc â€¢ 462-467

Board of Trade Regulations for Marine Boilers, etc. 467-478

Particulars, Weights, etc., of Marine Engines . . 479-483

Seasoning ^vnd Preserving Timber 484-485

Timber Measures 486-488

Bricklaying, Plastering, Painting, etc. . . . 488-490

Varnishes, Lacquers, Dipping Acids, Ce:ments, etc. . . 490-495

Miscellaneous Recipes, Tables, etc. . . â€¢ . 495-512

INDEX . 513

MACKEOW^

POCKET BOOK

OF

FOEilULJ:, EULES, AND TABLES

FOE

MYAL ARCHITECTS J^D SHIP-BUILDEES.

â€¢o*-

SIGNS AND SYMBOLS.

The following are some of the signs and symbols commonly

used in algebraical expressions : â€”

= This is the sign of equality. It denotes that the quantities

so connected are equal to one another ; thus, 3 feet = 1 yard.

+ This is the sign of addition, and signifies plus or more ;

thus, 4 + 3 = 7.

â€” This is the sign of subtraction, and signifies minus or less ;

thus, 4-3 = 1.

X This is the sign of multiplication, and signifies multiplied

by or into ; thus, 4 x 3= 12.

-f- This is the sign of division, and signifies di\ided by ; thus,

4-r2 = 2.

{} [] These signs are called brackets, and denote that the

quantities between them are to be treated as one quantity; thus,

5{3(4+2)-6(3-2)}=5(18-6) = 60.

This sign is called the bar or Adnculum, and is sometimes

iised instead of the brackets ; thus, 3(4 + 2) â€” 6(3 â€” 2) x 5 = 60.

Letters are often used to shorten or simplify a formula.

Thus, supposing we wish to express length x breadth x depth, we

might put the initial letters only, thus, ^ x J x t?, or, as is usual

when algebraical symbols are employed, leave out the sign x

between the factors and write the formula l.b.d.

When it is wished to express division in a simple form ths

â€¢F *f" 2/

divisor is written xmder the dividend; thus, (a; + y) -rz = -â€¢

z

Z SIGNS AND SYMBOLS.

! , t ' , ', , These a7e signs of proportion ; the sign : = is

to, the sign : : = as ; thus, 1:3 : ! 3 : 9, 1 is to 3 as 3 is to 9.

< This sign denotes less than ; thus 2 < 4 signifies 2 is less

than 4.

> This sign denotes more than : thus 4 > 2 signifies 4 is more

than 2.

*.' This sign signifies because.

.*. This sign signifies therefore. Sec.: '.' 9 is the square of

3 /. 3 is the root of 9.

'-^ This sign denotes difference, and is placed between two

quantities when it is not known which is the greater ; thus

{x '^ y) signifies the difference between x and y.

, , These signs are uÂ«ed to express certain angles in

degrees, minutes, and seconds : thus 25 degrees 4 minutes 21

seconds would be expressed 25Â° 4' 21".

JVote. â€” The two latter signs are often used to express feet and

inches ; thus 2 feet 6 inches may be written 2' 6".

n/ This sign is called tlie radical siffn, and placed before a

quantity indicates that some root of it is to be taken, and a

small figure placed over the sign, called the exponent of the root,

shows what root is to be extracted.

Thus 2^a or Va means the square root of a.

^a â€ž cube â€ž

4/Â« ,, fourth â€ž

-^^ This denotes that the square root of a has to be taken

and divided by h.

This denotes that 1) has to be divided by the square

root of a.

y

This denotes that the square root of a+ J has to be

a-\- d

divided by the square root of a-^d. It may also be written

thus, / , or -.

S/ a^rd ^/a + d

QC This is anotlier sign of proportion. Ex.: a(X:h; that is,

a varies as or is proportional to h.

oo This sign expresses infinity; that is, it denotes a quantity

greater than an}^ finite quantity.

This sign denotes a quantity infinitely small, nought.

Z This sign denotes an angle. Ex. : l ah would be written*

the angle ab.

SIGNS AND SYMBOLS. 3

L This sign denotes a right angle.

_L This sign denotes a perpendicular; 2iS, at L cd, i.e. aZ> is

perpendicular to cd.

A This sign denotes a triangle; thus, Aal?c, i.e. the triangle

izbc.

II This sign denotes parallel to. Ux.: ab \i cd would be

â€¢written, ab is parallel to cd.

f or F These express a function ; as, a =/> ; that is, ^ is a

function of x or equals x.

f This is the sign of integration ; that is, it indicates that the

^expression before which it is placed is to be integrated. AVhen

the expression has to be integrated twice or three times the sign

is repeated (thus,//, /(/'); but if more than three times an index

is placed above it (thus,/'").

Dord These are the signs of differentiation ; an index placed

above the sign (thus, d-) indicates the result of the repetition

of the process denoted hy that sign.

2 This sign (the Greek letter sigma) is used to denote that

the algebraical sum of a quantity is to be taken. It is com-

monly used to indicate the sitm of finite differences, in nearly the

same manner as the symbol/.

[jl This sign is sometimes^ used instead of tt, being a modifi-

cation of the letter C, for circtimference.

n This sign is sometimes used instead of e, being a modifi-

cation of the letter B, for base.

g This sign is used to denote the force of gravit}'^ at any

given latitude.

TT The Greek letter pi is invariabl}'' used to denote 3-14159;

that is, the ratio borne by the diameter of a circle to its circum-

ference.

As the letters of the Greek alphabet are of constant recur-

rence in mathematical formulae it has been deemed advisable to

append the following table : â€”

A a

Alpha.

I I

Iota.

P p

Eho.

B 3

Beta.

K K

Kappa.

2 o-s

Sioma,

r 7

Gamma.

A K

Lambda.

T T

Tau.

A S

Delta.

M IX

Mu.

T V

Upsilon.

E 6

Epsilon.

N V

Nu.

* <p

Phi.

z C

Zeta.

H 1

Xi.

X X

Chi.

H 7,

Eta.

O

Omicron.

Y y\>

Psi.

e d

Theta.

n IT

Pi.

D. o)

Omega.

E 2

SUBTRACTION AND MULTIPLICATION OF DECIMALS.

DECIMAL FRACTIONS.

Decimal Fractions are those which have 10, 100, 1000, &c.,

for a denominator, and are expressed by writing the numerator

only and placing a point before it on the left hand.

J^tL. J.. TOâ€” i-' 100 ' "â€¢ 1000 OtKl.

Ex.2, xo^"'^* iM~'^^* 1000 ~'^^^-

Ex.d. 113-3 = 113^ = iig^ = ie^.

ExA. 113-03 =^'^j'^ = ^-^ = iiÂ§g2.

Addition of Decimals.

Rule. â€” Arrange the numbers so that all the decimal points

come directly under one another; add them together as in

wliole numbers, and point off as many figures for decimals as

are equal to the greatest number of decimals in any of the given

numbers.

Ex. : Add together 3-79, -117, 87-225, 478-91.

3-79

â€¢117

87-225

478-91

570-042'. Ans.

Subtraction of Decimals.

Rule. â€” Place the numbers under one another, as in addition ;

subtract as in whole numbers, keeping the decimal point in the

remainder directly under those above it.

Ex. : From 97*378

- take 46-4972

50^8808. Ans.

Multiplication of Decimals.

Rule. â€”Multiply the factors together, as in whole numbers ;

then point off from the product as many decimal places as there

are in both factors, supplying any deficiency by annexing ciphers

to the left hand.

Ex. 1. Mult. 4-735 Ex. 2. Mult. -04735

by_^374 by -0374

18940 18940

33145 33145

14205 14205

1-770890. Atis. -001770890. Ans.

DIVISION AND EEDUCTION OF DECIMALS. S

Division of Decimals.

EULE. â€” Remove the decimal point in the dividend as many-

places to the right as there are decimal places in the divisor ;

supply any deficiency by annexing ciphers. Then make the

divisor a whole number, and proceed as in the division of

simple numbers, and the quotient will contain as many decimal

places as are used in the dividend.

Ex. 1. Divide 74-23973 by 6-12. Ex. 2. Divide -7423973 by 612.

612) 7423-973 (12-130. Ans. 612) -742.8973 (-0012130. Ans.

612 612

1303 1303

1224 1224

799 799

612 612

1877 l877

1836 1836

413 413

To Reduce ajn'y Fraction to a Decimal.

Rule. â€” Annex ciphers to the numerator till it be equal to or

greater than the denominator ; divide by the denominator, as

in division of decimals, and the quotient will be the decimal

required.

Ex. 1. Reduce ^fs to a decimal.

256) 7-00000000 (-02734375. Ans.

512

1880

1792

880

Ex. 2. Reduce ^ to a decin

768

12) 7-00000000

1120

1024

â€¢58333333. A7is.

960

768

1920

1792

1280

1280

1

To Reduce Numbers of Different Denominations

into Decimals.

Rule 1. â€” Reduce the given weight or measure, &c., into the

lowest denomination given, for a dividend ; then reduce the

b EEDUCTION OF DECIMALS.

integer into the same denomination for a divisor ; tlie resulting-

fraction, changed to a decimal, will be the decimal required.

Rule 2. â€” Divide the least denomination by such a number as

will reduce it to the next greater; to the decimal so obtained

prefix the given number of the same denomination ; then divide

by such a number as will reduce it to the next greater; thus

proceed till it be reduced to the decimal of the required integer.

Ex. 1 to Rule 1. â€” Reduce 2 cwt. 3 qrs. 21 lbs. to the decimal

of a ton.

2 cwt. 3 qrs. 21 lbs. .329 lbs. -, ,^o - *

- ^ = â– ;; â€” = -l-168/o ton:

1 ton 2240 lbs.

or, by Rule 2â€” | 7) 21-0 lbs.

2^ i 4 ) 3-0

4 ) 3-7o qr s.

2 0) 2-9375 cw ts.

Ans. '146875 ton.

Kc. 2 to Rule 1. â€” Reduce 2 ft. 9 in. to the decimal of a yard.

^i!:AiE: = 33jn. = . 916666 yard;

1 yard 36 in. ^

or, by Rule 2â€” 12 ) 9 in .

3 ) 2-75 fee t

Ans. â€¢91666 yard.

To Find the Value of any Decimal.

Rule. â€” Multiply the given decimal by the number of parts

contained in the next lesser denomination, and point off from

tlie product as many figures as the decimal consists of. Mul-

tiply the remaining decimal by the number of parts in the next

lesser denomination, and point off as many decimals in the

product as before. Proceed thus till you have brought out the

least known parts of the integer.

Ex. 1. What is the value of Ex. 2. What is the value of

â€¢146875 of a ton ? -91666 of a yard ?

â€¢146875 -91666

20 3

cwts. 2-937500 feet 274998

4 J^

qrs. 3-750000 in. 899976

28

lbs. 21-0000 000

A7i8. = 2 cwts. 3 qrs. 21 lbs.

Ans. = 2 ft. 9 in.

GEOMETRY.

PRACTICAL GEOMETRY.

1. From any given point in a straight line

to erect a lyerpendicular . (Fig-. 1.)

On each side of the point a in the line from

which the perpendicular is to be erected set off

equal distances Kb, A.c ; and from h and e as

centres, with any radius greater than Kb or Ar,

describe arcs cutting each other at d.,d' \ a line

drawn through dd' will pass through the point

A, and Kd will be perpendicular to he.

2. To erect a lyerpendicular at or near the

end of a litw. (Fig. 2.)

With any convenient radius, and at any

distance from the given line AB, describe an

arc, as BAG, cutting the given point in a ;

through the centre of the circle N draw the

line BXC : a line drawn from the point A,

cutting the intersection at c, will be the

required perpendicular.

3

Fig. 1.

d-'k

To divide a line into any number of eq^ual paH&. (Fig. 3.)

h ^'-

/

L.'

3

Fig. 3.

L^-'

-&

a.-

'B

Frc4. 4.

Let AB be the given straight

line to be divided into a number of

equal parts ; through the points

A and B draw two parallel lines AC

and DB, forming any convenient

angle with AB ; upon AC and DB set

off the number of equal parts re-

quired, as A-1, 1-2, ko,., B-1, 1-2, &c ; "

join A and d, 1 and 3, 2 and 2, 3 and 1, c and B, cutting AB

in a, b, and c, which will thus be divided into four equal parts.

4. To find the length of any given arc of a circle. (Fig. 4.)

With the radius Kd, equal to one-

fourth of the length of the chord of the

arc AB, and from A as a centre, cut the

arc in c ; also from B as a centre with

the same radius cut the chord in b ;

draw the line cb, and twice the length

of the line cb is the length of the arc nearly

5. To draw from or to the cir-

cumference of a circle lines tend-

ing towards the coitre, when the

centre is inaccessible. (Fig. 5.)

Divide the given portion of

the circumference into the

desired number of parts ; then

with any radius less than the dis-

tance of two parts, describe arcs cutting each other as Al, cl, k.c.;

Fm. -

-8

GEOMETRY.

draw the lines b1, c2, &c., which will lead to the centre, as required.

To draw the end lines XT', Tr from B and E, with the same radii

as before describe the arcs r', r, and with the radius Bl, from A

as centre, cut the former arcs at r', r, lines then drawn from Ar'

and F;* will tend towards the centre, as required.

Jo. To desonhe an arc of a circle of large radius, (i'ig. 6.)

Fig. 6. Let A, B, c be the three points through

B which the arc is to be drawn ; join ba

and BC ; then construct a flat trian-

gular mould, having two of its edges

perfectly straight and making with

each other an angle equal to ABC.

Each of the edges should be a little

longer than the chord AC. In the points A, c fix pins ; and fix a

pencil to the mould at B, and move the mould so as to keep its

edges touching the pins at A and c, when the pencil will describe

the required axe.

7. Another method. (Fig. 7.)

^ Fig- 7. Draw the chord ADC, and

E iw >_ B a TTf r draw ebf parallel to it: bisect

the chord in D and draw db per-

pendicular to AC ; join ab and

BC ; draw AE perpendicular to

â– '^ â€¢ 2 D 2 1 c XB and CF perpendicular to BC ;

also draw An and cn periDcndicular to AC ; divide AC and ef

into the same number of equal parts, and An. cn into half that

number of equal parts ; join 1 and 1, 2 and 2, also B and .?, s,

and B, and t, t ; through the points where they intersect

describe a curve, which will be the arc required.

8. To describe an ellipse, the transverse and conjugate diameters

being given. (Fig. 8.)

- Let AB be the transverse and CD the

conjugate diameters, bisecting each other

at right angles in the centre E ; from C as

a centre, with EA as radius, describe arcs

cutting AB in F and f', which will be the

foci of the ellipse ; between E and f

set off any number of points, as 1, 2 (it

is advisable that these points should

be closer as they approach f).

From F and f', with radius Bl, describe the arcs g, g', g", g'".

From F and f', with radius Al, describe the arcs H, h', h", h'",

intersecting the arcs G, G'. g", g'" in the points I, i, i, I, which will

be four points in the curve.

Then strike arcs from F, f' first with A2, then with b2 ;

these radii intersecting will give four more points. Proceed

in this way with all the points between E and F; the curve of

the ellipse must then be traced through these points by hand.

GEOMETRY.

Fig. 10.

9. Another metlwd. (Fig. 9.)

At 0, the intersection of the two dia-

meters, as a centre, with a radius equal to

the difference of the semi-diameters,

describe the arc ab, and from Z> as a

centre with half the chord tea describe

the arc cd ; from o as centre with the

distance od cut the diameters in dr, dt ;

draw the lines rs, rs, ts, ts ; then from r

and t describe the arcs sds, scs ; also from

d and d describe the smaller arcs SAS, SBS, which will complete

the ellipse required.

10. To draw a tangent and a perpendicular to an elUjyse at

any point. (Fig. 10.)

Let G be the point ; from F, f', the

two foci of the ellipse, draw straight

lines through G and produce them ;

hisect the angle made by the produced

parts, by GH, then GH is perpendicular

to the curve ; at G bisect the angle

formed by FG and f'g produced, by IJ,

then IJ will be the tangent to the curve

at G, and it will also be perpendicular

to GH.

\\. To describe an elliptic arc, the span and height being given.

(Fig. 11.)

Bisect with a line at right angles the

chord or span AB ; erect the perpendicular

AQ, and draw the line QD equal and parallel

to AC ; bisect AC in c, and AQ in n ; make

CL equal to CD, and draw the line lcQ ; draw

also the line nSD, and bisect SD with a line

KG at right angles to it, and meeting the

line LD in G ; draw the line gkq, and make

Gp equal to CK, and draw the line 6/^2 ;

then from G as centre with the radius

GD describe the arc sd2, and from K and p as centres with the

radius ak describe the arcs as and 2b, which complete the arc,

as required.

12. Another method. (Fig. 12.)

Bisect the chord ab, and fix at right

angles to it a straight guide, as be ; prepare

of any material a rod or staff equal to half

the length of the chord, as def ; at a

distance from the end of the staff, equal

to the height of the arc, fix a pin e, and at

the extremity a tracer/; move the staff,

keeping its end to the guide and the fixed

10

GEOMETRY.

pin to the chord, and the tracer will describe a half of the-

arc required.

13. To obtain hj measwrement the length of any direct liney

tlwvgh intercepted by some viateHal object. (Fig. 13.)

Fig. 13. Suppose the distance between

A and B is required, but the

straight line is intercepted by

the object G. On the point d with

any convenient radius describe

the arc cc', and make the arc

twice the radius do in length ;

through c' draw the line dc'e, and

â€¢on e describe another dire ff equal in length to the radius dc ;

draw the line efr equal to efd\ from r describe the arc g'g^

equal in length to twice the radius rg ; continue the line through

rg to B : then A and b will make a right line, and de or er will

equal the distance between dr^ by which the distance between

AB is obtained, as required.

Fig. 14.

14. To ascertain the distance geometrically of an inaccessible-

object on a level plane. (Fig. 14.)

Let it be required to find the distance

between A and b, a being inaccessible.

Produce AB to any point d, and bisect

BD in c ; through d draw DÂ«, making

any angle with da, and take DC and db

respectively and set them off on T>a as

T)b and d^' ; join Be, cb, and Ab ; through

E, the intersection of B^ and cb, draw

.DEF meeting aZ> in F ; join BF and pro-

duce it till it meets Da in a: then ab will be equal to AB, the

distance required.

Fig. 15.

15. Another method. (Fig. 15.)

Produce ab to any point d ; draw the line

jyd at any angle to tlie line ab ; bisect the

line Dd in c, through which di-aw the line B&^

and make cb equal to bc ; join AC and db,

and produce them till they meet at a : then

ba will equal ba, the distance required.

GEOilETEY.

11

distance hetn-een two objects, both b&ing

16. To measure the

inaccessible. (Fig. 16.)

Let it be required to find the distance

between the points a and b, both being in-

accessible. From any point c draw any line

cc, and bisect it in D ; produce a.o and bc, and

prolong them to E and F ; take the point E in

the prolongation of A^. and draw the line ed^,

making T>e equal to de.

In like manner take the point f in the

prolongation of bc, and make D/ equal to df ;

produce ad and ec till they meet in Â«, and also

produce bd and /c till they meet in h : then

the distance between the points a and b equals

the distance between the inaccessible points

A and B.

17. To inscribe any regular polygon in a

given circle. (Fig. 17.)

Divide any diameter ab of the circle abd

into ai> many equal parts as the polygon is

required to have sides ; from A and B as

centres, with a radius equal to the diameter,

describe arcs cutting each other in c ; draw

the line CD through the second point of divi-

sion on the diameter ab, and a line drawn

from D to A is equal to one side of the poly-

gon required.

18. To cut a beam of the strongest section from

any round piece of timber. (Fig. 18.)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63