Clement Mackrow.

The naval architect's and shipbuilder's pocket-book of formulae, rules, and tables and marine engineer's and surveyor's handy book of reference online

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Online LibraryClement MackrowThe naval architect's and shipbuilder's pocket-book of formulae, rules, and tables and marine engineer's and surveyor's handy book of reference → online text (page 1 of 63)
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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.



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

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.


London : July 187'.).


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.

London : A2)Hl 1884.



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



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 . . . „ . 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 • 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. . . • . 495-512

INDEX . 513









The following are some of the signs and symbols commonly
used in algebraical expressions : —

= 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.

— 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) — 6(3 — 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

•F *f" 2/

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



! , 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 u«ed to express certain angles in
degrees, minutes, and seconds : thus 25 degrees 4 minutes 21
seconds would be expressed 25° 4' 21".

JVote. — 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 „ cube „

4/« ,, fourth „

-^^ 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.


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.


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

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

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 : —

A a




P p


B 3




2 o-s


r 7












E 6




* <p


z C


H 1




H 7,




Y y\>


e d


n IT


D. o)


E 2



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.. TO— i-' 100 ' "• 1000 OtKl.

Ex.2, xo^"'^* iM~'^^* 1000 ~'^^^-
Ex.d. 113-3 = 113^ = iig^ = ie^.

ExA. 113-03 =^'^j'^ = ^-^ = ii§g2.

Addition of Decimals.

Rule. — 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

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


570-042'. Ans.

Subtraction of Decimals.

Rule. — 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. —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 of Decimals.

EULE. — 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. — 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

Ex. 1. Reduce ^fs to a decimal.

256) 7-00000000 (-02734375. Ans.



Ex. 2. Reduce ^ to a decin


12) 7-00000000


•58333333. A7is.





To Reduce Numbers of Different Denominations

into Decimals.

Rule 1. — Reduce the given weight or measure, &c., into the
lowest denomination given, for a dividend ; then reduce the


integer into the same denomination for a divisor ; tlie resulting-
fraction, changed to a decimal, will be the decimal required.

Rule 2. — 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. — Reduce 2 cwt. 3 qrs. 21 lbs. to the decimal

of a ton.

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

- ^ = ■ ;; — = -l-168/o ton:

1 ton 2240 lbs.

or, by Rule 2— | 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. — Reduce 2 ft. 9 in. to the decimal of a yard.

^i!:AiE: = 33jn. = . 916666 yard;
1 yard 36 in. ^

or, by Rule 2— 12 ) 9 in .

3 ) 2-75 fee t
Ans. •91666 yard.

To Find the Value of any Decimal.

Rule. — 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
•146875 of a ton ? -91666 of a yard ?

•146875 -91666

20 3

cwts. 2-937500 feet 274998
4 J^

qrs. 3-750000 in. 899976


lbs. 21-0000 000
A7i8. = 2 cwts. 3 qrs. 21 lbs.

Ans. = 2 ft. 9 in.



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.


Fig. 1.

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

h ^'-




Fig. 3.





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. -



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

■'^ • 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.


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



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
•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 D«, 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.



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 «, 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.)

Online LibraryClement MackrowThe naval architect's and shipbuilder's pocket-book of formulae, rules, and tables and marine engineer's and surveyor's handy book of reference → online text (page 1 of 63)