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Colleftion

A Memorial to the Founder
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THE

COMPLETE MEASURER:

f / / <^/-^ />/7

WHOLE ART OF MEASURINd^

IN TWO PARTS.

The Firft PART teaching

Decimal Arithmetick, with the ExtraSfion of
the Sc^ARE and Cube Roots :

And alfo the Mtdtiplication of Feet and Inches^ commonly
called Cross Multiplication.

The Second PART teaching to
Meafure all Sorts of Superficies and Solids,

by Decimals ; by Crcfs Multiplication, and by Scale and
' Coinpajj'es : Alio the Works of feveral Artificers,
relating to -5 /^//^//To-: and the Meafuring of ^o^^/v/and
Timber, Shewing the common Errors.

And fome Practical Questions,
The Sixtee7ith EDITION revifed and correded.

To which is added,

^« Appendix, i. OyGauging. 2. OyLand-Meafurino-,

Very ufcful for all Tradefmen j efpecially Carpenters, Bricklayers,
Plafterefs, Painters, Joiners, Gialicrs, Mafons, (^c.

By William II a w n e y, Philomath,
Recommended by the Rev. Dr, John Harris, F, R, S,

LONDON:

Printed for J. F. and C. Rivington, T. Loncmaw,
B. Law, S. Bladon, G. G. J. and J. Robinson,
j. Johnson, W. Goldsmith, R. Baldwin, J.
and J. Taylor, E. Newbery, Scatcherd and
Whitaker, and G, and T. Wilkie, 1709.


T Have perufed this BOOK, and
recommend it to the Publick
as a very ufeful One.

J. Harris, D.D.


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7 S;;/ '«> *--'\-"^ y. 'VV\ Nj^ v^^ > V w -x t; ^^ -^av


■j -^


THE


PRE F A C E.


HAVING perufed feveral Books concern-
ing the Menluration of Sup-rficies and
Solids, and the Woiks of Artificers re-
lating to Buiidin;^ ; but not finding any one Book
fo perfect, as to give any tolerable Satisfad^ion to
a Learner; and i having pradiiifed and taughc
Meafuring for feveral Years, and thereby gained
Experience and Knowledge in that Art, having
learned fome Things from one Author, and fome
Things from another, I began to think of diged-
ing my Thoughts into fome fuch Method as might
give a Learner full Satisfaction, without being at
the Charge of buying fo many Books ; and being
importuned thereunto by fome Friends, I fell ta
work, and at lalt brought them to that Perfedlion
you here find in the following Work.

T. As to the Decimal Arithmctkk^ I have bfefi
as concife as the Matter would well bear, to make
it plaiti.

A 2 2. As


iv P R E F A C E.

2. As to the Multiplying of Feet and Inches,
commonly calied Crofs Multiplication^ my Me-
thod differs from that which is tifually taught in
other Authors, as being (i think) much ihorter
and plainer.

3. In meafuring of Superficies and Solids, I have
given the Demonftration of the Rules, which I
tnought might be very acceptable to the Inge-
nious ; for, inded, I always look upon the Writ-
ing of a Rule without a Demonfbration (in any
Part of the Mathematicks) to be but lame and
tlefective ; for want of knowing the Reafon of
the Rule, a Learner may commit great Errors j
befides, when a Learner knows the Reafon of the
Rules, he may retain them better in his Memory,
The Rule for meafuring a Prifmoid andCylindrcid,
I had out of Mr. Everard^s Art of Gauging \ but
ihe Reafon he does not (hew, neither have 1 found
it in any other Author; but that the Method is
true, i have endeavoured to make plain.

The Demonftration of the Rules for finding the
Area of an Ellipfis and Parabola; alfo the De-
monftration of the Rules for finding the folid Con-
tent of the Fruftum of a Cone and Pyramid, the
Solidity of a Globe of a Spheroid, a Parabolic Co-
noid, and of a Parabolic Spindle, and their Fruf-
tums, I had from the ingenious Mr. lVard\ Toung
Mathematician' s Guide \ where the curious and in-
genious Reader may fee many other Demonftra-
tions algebraically performed. I have alfo demon-
ftrated the Rule for finding the Solidity of a Globe,
out of Pardie's Elements of Geometry (Book the
5th, Art. the 33d) publifhed mEngliJh with many

Additions,


PREFACE. V

Additions, by the Reverend Dr. Harris^ F. R. S.
and the fame is alfo done out of Sturmius's Ma^
thefts Enudeata ; fo that the ingenious Reader may
ufe which of thofe Ways he likes belt.

The Scale fuppofed to be ufed in all the Opera-
tions, is the Line of Numbers, commonly called
Guhters Lijte^ wh-ich is upon the ordinary Two-
Feet or Eighteen-Inch Rules, commonly ufed by
the Carpenters, iVlafons, <^c. becaufe I thought it
needlefs, as well as impertinent, to write the Ufe
of Sliding-Rules, or any other particular Scales,.
they being fuliiciently treated of by feveral Au-
thors ; viz. by the above-named Mr. Everard^ in
his Art of Gauging above-mentioned, where yon
have the Ufe of a Sliding-Rule in Arichmetick,
Geometry, in Meafuring of Superficies and Solids,
Gauging, isc, Likewife Mr. Hunt has written
largely of the Ufes of hi^ Sliding Rule, in Arith-
metick. Geometry, T; igonometry,- Gauging,,
Dialling, ^c. There are feveral others who have
explained the Ufe of their own Rules; fo that
the more carious Reader may find full Satisfac-
tion in thofe Authors-
One Thing I have omitted in the Boole, which-
I think may not be very improperly Inferted in this
Place; that is, how to find a Number upon the
Line. If the Number you would find confilis
only of U/iits, then the Figures upon the Line re-
prefent the Number fought: Thu.<^, if the Number-
be 1,2, 3, ^c. then I, 2, 3, ^c. Upon the I.rcprefents the Number fought. But if the Num-
ber confiits of tv/o Figures, that is, of UnJts and
Tens, then the Figure upon the Rule flands for

A 3, tdia-


VI


PREFACE.


the Ten?, and the large Divlfions {land for the
Units; thus, if 34 were to be found upon the
Line, the Figure 3 upon the Line is 30, and 4 of
the large Divifions (counted forwards) is the Point
reprefenting 34.; and if 340 were to be found, it
will be at the Came Point upon the Line; and if
304. were to be found, then the 3 upon the Line is
300, and four of the fmaller Divifions (counted
forward) is the Point reprefenting 304. If the
Number confifts of four Places, or Thoufands,
then the Figure upon the Line ftands for Thou-
fands, and the laiger Divifions are Hundreds, the
IcfTer Divifions are Tens, and the tenth Parts of
thofe lefl'er Divifions are Units. Thus, if 2735
were to be found, then the 2 to 2000 ; and the
7 larger Divifions (counted forward) is 700 more ;
und 3 of the lefTer Divifions is 30 more; and half
of one of the lefier Divifions is 5 more, which is
the Point reprefenting 2735. You muftremember,
that between each Figure upon the Line there are
10 Parts, which I call the larger Divifions ; and
each of thofe larger Divifions are fubdivided (or
fuppofed fo to be) into 10 other Parts, which I call
the fmailer Divifions ; and each of thofe Parts fup-
pofed fo to be fubdivided again into 10 other Parts,
i3c. You muil alfo remtmber, that if i in the
Middle <){ the Line Hands only for i, then i at
the upper End will be lo, and i at the lower
End v-iil only be -j^-^; but if i at the lower End
figniiies I, then i in the Middle ftands for 10,
and 1 at the upper End is ico, ^c.

There is one Thing more which I would have
nw Reader to undeiiland ; and that is, hovv^ to find


PREFACE. vli

all fiich proportional Numbers made ufe of in the
Proportions about a Circle, and of a Cylinder, and
in other Places; which Thing may be of good Ufe
to know how to corrc6t a Number, which may
happen to be falfe printed, or to enlarge any Num-
ber to more decimal Places, for more Exadlnefs ;
for though I have mentioned what fuch Numbers
are, yet I have not fhewn how to find them, which
a Learner may be a little at a Nonplus to do;
tkough they are tafily found by the Rules there
laid down. I fhall therefore give two or three
Examples, in this Place, of finding fuch Num-
bers, which may enable my Reader to find out
the reft.

And, firft, let it be required to find the Area of
a Circle, whofe Diameter is an Unit.

By the Proportion o^ Van Culen^ If the Diame-
ter be 1, the Circumference will be 3.14159265,
^c. of which 3.1416 is fufficicnt in moft Cafes.
Then the Rule teaches to multiply half the Cir-
cumference by half the Diameter, and the Pro-
du<51: is the Area : That is, multiply 1.5708 by .5
{yi%. half 3.1416 by half i) and the Product is
.^854, which is the Area of the Circle, whofe
Diameter is i.

Again ; if the Area be required when the Cir-
cumference is I, fiift find what the Diameter will
be, thus : 3. 141 6 : to i : : fo is i to .31H309,
which is the Diameter when the C i re u inference
is I. Then multiply half .318309 by half i ; that
is.i59i54by .5, and the Produ6lis .079577, which
is the Area of a Circle whofe Circumference is i.

If


vm F R E F A C E.

If the Area be given, to find the Side of the
Square equal, you need but extra<5t the Square
Root of the Area given, and it is done : So the
Square Root of .7854 is .8862 which is the
Side of a Square tquai when the Diameter is i.
And if you extract che Square Root of .079577,
it will be .2821, which is the Side of the Square
equal to the Circle whofe Circumference is i.

If the Side of a Square within a Circle be re-
quired, if you fquare the Semidiameter, and
double that Squaie, and out of that Sum extract
the Square Root, that fhall be the Side of the
Square which may be inlcribed in that Circle ; fo
if the Diameter of the Circle be i, then the half
is .5; whi:-h fquared, is ,25; and this, doubled,
is .5, whofe Square Root is .7071, the Side of the
Square infcribed.

Again; If the Diameterof a Globe be i. to find'
the Solidity. In Sc6\. XI. Chap. II. it is demon-
flrated, that the Globe is | of a Cylinder of the
fame Diameter and Altitude : Thus, if the Cylin-
der's Diameter be i, and its Altitude or Length be
alfo I, find the Solidity thereof, and talce | of it,>
and that will be the Solidity of the Globe re-
quired. Now if the Diameter be i, the Area of
the Circle^ or Bafe of the Cylinder, is .7854. (as
is above fliewn) which multiplied by i, the Alti-
tude of the Cylinder, and the Produ6l is alfo. 7854,
the Solidity of the Cylinder ; | whereof is .5236,
which is the Solidity of the Globe, whole Diame-
ter is In

Froai


PR E F A C E. IX

From what has been faid, the Reader may
eafily perceive how all other proportional Num-
bers are found, and may examine them at his
Pleafure.

I fhall not enlarge any farther upon the Matter,
but leave the Book to fpeak for it/elf; and if it
prove beneficial to tlie ingenious Pradlitioncrs,
I have my Defire. Sa, wifhing my ingenious
Reader good Succefs in his Endeavours, not
doubting but he will jreap Profit hereby ; which
that he may, is the hearty Defire of his Well-
wiflier,


W, HAWNEY,


CON^


C O N T E N T S,


PARTI.

Chap. Page

1. J.T/'H AT a Decimal FraSlion is i

i I . ' Redu6lio7i of Decimals 3

]ir. Addition of Decimals 9

IV. Suhtradion of Decimals 10

N . Muliiplicai ion of Decimals II

Ibid. Contrasted Multipltcation 14.

VI. Divijion of Decimals 1 9
\\y\dL, ContraHed Di'vifon 26

VII. ExtraJIion of the Square Roct 33
Vllf. Extra^ion of fh Cube R^ot 43
IX. Crofs Multiplication 58

P A R T II. C H A P. I.

Sea. page

1 . Of a Square 7 I

2. Of a Parallelogram 73

3. Of a Rhombus ' 74

4. 0/"<3 R homicides 75

5. O/'^rt Triangle 76

6. 0/ « Trapezium 82

7. Of irregular Figures 84

8. Of regular Polygons 86

9. Cy^ Circle 90
10. Of a Semicircle I 10
J I . O/' « ^ladrant I i I
Ibid. To find the Length of the Arch Line I i 2
Ibid. ^Ji* halving the Chord and 'verfed Sine, tofind\

the Diameter J -*

1 2 . Of the SeSlor of a Circle 1 1 6

1 3 . Of a Segjnent of a Circle I 1 8

14. Of co?npcund Figures ■ 122

15. Of an Ellipfsy or OvaJ 1 24

16. Of a Parabola 128

CHAP,


CONTENTS.

CHAP. II.

Of Solid Meajure.

1. Of a Cube 134

2. Of a ParalJelopipedon 137
Z^Ofa Prifm 140

4. Of a Pyramid 144

5. 0/^ Cylinder 1 54

6. O/^ Co«^ 156

7. Of the Frufum of a Pyramid 1 63

8 . 0/ //6^ Frufum of a Cone 1 7 z

9. Of a Pr if mo id 1 76
10. Oy'i? Cylindroid iBo
I I. (i>/'^ /2 Sphere i or Globe 182

12. Of a Spheroid 19^

13. O/'^ FarahoLic Conoid 19^

14. 0/« Parabolic Spindle 20 1

CHAP. III.

- 57;^ Meafur'ing of J Forks relating to Building,

Sea. Page

1. Of Carpenters JfWk 206

2. Of Bricklayers Work 21 I

3. OfPlafterers Work 223

4. Of Joiners Work 225

5. Of Painters Work 228

6. Of Glafiers Work 229

7. OfMafons Work ^32

C H A P. 17.

Seft- Page

1. Of Board-Meafure 235

2» Of fquared Timber 237

3. Of uneq^ualfi^uared Timber 245

4. 0/


CONTENTS.

Se£l. • Page

4. Of round Timher 'With equal Bafes 249

5. Of round Timber ivitb unequal Bafes 258

6. Of the Fi've regular Bo aits'' , 263

7. 'Of irregular Solids 211

CHAP. V.

Pra^ical ^efions 273

APPENDIX.

Se£l. Page

1. Of Gauging 309

2. Of Land-Meafuring 335


ERRATA.


Page 2.3, line i, ior ffthiVtzA fourth.^ p, 78, in the fig. for 78,
r. 7.8. p. 85, in the fig. for 66, r. 6.6. p. S7, for 12.2 in the fig.
r. 12.64. p. 122, in the fig. for iSc,, r. ir.9, and for 35, r. 3,5.
p. 2035 in the fig. for a^ r. d.


THE


THE


Complete Measurer


PART I.


CHAP. I.

No tat ion of Decimals.


A Decimal Fradlion is an artificial Way
of fetting down and exprcfling Natural,
or Vulgar fradlions, as whole Numbers 2
And whereas the Denominators of Vul-
gar Fractions are divers, the Denomina-
tors of Decimal Fradions are always certain : For a
Decimal Fradion hath always for its Denominator aa
Unit, with a Cypher or Cyphers annexed to it, and
mull therefore be either lo, loo, looo, loooo, ^r.
and confequently in writing down a Decimal Fr,ad,i-
on, there is no Neceflity for writing down the Deno-
minator ; as by bare Infpedion, it is certainly known,
confiding of an Unit with as miny Cyphers annexed
to it as there are Places (or Figures) in the Nume-
rator.

B Example,


2 Notation (?/ Decimals. Part I.

Example. This Decimal Fraftion -fV^ may be writ-
ten thus .25, its Denominator being known to be an
Unit with two Cyphers; becaufe there are two Fi-
gures in the N umeracor. In like Manner, -J5V0 may
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