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; [C 



THE 

IIECHANIOAL ENGINEER'S 
POCKET-BOOK 



A REFBRENCE-BOOK OF RULES, TABLES, DATA, 

AKD FORMULA, FOR THE USE OF 

ENGINEERS, MECHANICS, 

AND STUDENTS. 



BY 

WILLIAM KENT, A.M., M.E., 

CoMvUing Engineer, 
Member Amer, Soc*y Mechl. Sngrt. and Amer, Inst. Mining Slngra. 



J^IFTH EDITION, REVISED AND ENLARGED. 
EIGHTH THOUSAN^D. 



NEW YORK: 

JOHN WILEY & SONS. 

Lokdon: CJIAPMAN & HALL, Limited. 

1901. 






HARVARD COLLEGE LIBRARY 
Pr?-- 7:-- LSSRARY OF 

FRA\.. PEAB03Y MAGOUN 

IHt GIFT OF HIS SON 

MAY 8, 1929 



CopTniGHT, isoe^ 

BY 

WILLIAM KENT. 



Braunworth, Munn 6f Barbe , ^\ 
Printers and Binders . ^ 

Brooklyn, N. Y. -^ /^ 



PREPACK 



MoKA than twenty years ago the author began to follow 
the advice given by Nystrom : '* Every engineeer should 
make his own pocket-book, as be proceeds in study and 
practice, to suit his particular business." The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
niecbanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to this, a 
careful examination was made of the transactions of engi- 
neering societies, and of the most important recent works 
on mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor- 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it is believed, cause it to 
differ from other works of its class. In the first place it 
was considered that the field of mechanical engineering was 
so great, and the literature of the subject so vast, that as 
little space as possible should be given to subjects which 
especially belong to civil engineering. While the mechan- 
'Cal engineer must continually deal with problems which 
'>elong properly to civil engineering, this latter branch is 
so well covered by Trautwine*s ** Civil Engineer's Pocket- 
book" that any attempt to treat it exhaustively would not 
only fill no ** long-felt want," but would occupy space 
which should be given to mechanical engineering. 

Another idea prominently kept in view by the author has 
been that he would not assume the position of an ** au- 
:bontj" in giving rules and formuls for designing, but 
only that of compiler, giving not only the name of the 
originator of the rule, where it was known, but also the 
rolame and page from which it was taken, so that its 

m 



*^ PREFACE. 

derivation may be traced when desired. When different 
formulae for the same problem have been found they have 
been given in contrast, and in many cases examples have 
been calculated by each to show the difference between 
them. In some cases these differences are <|utte remark- 
able, as will be seen under Safety-valves ^nd Crank- pins. 
Occasionally the study of these differences has led to the 
author's devising a new formula, in which case the deriva* 
tion of the formula is given. 

Much attention has been paid to the abstracting of data 
of experiments from recent periodical literature, and numer* 
ous references to other data are given. In this respect 
the present work will be found to differ from other Pocket* 
books. 

The author desires to express his obligation to the many 
persons who have assisted him in the preparation of the 
work, to manufacturers who have furnished their cata« 
logues and given permission for the use of their tables, 
and to many engineers who have contributed original data 
and tables. The names of these persons are mentioned in 
their proper places in the text, and in all cases it has been 
endeavored to give credit to whom credit is due. The 
thanks of the author are also due to the following gentle- 
men who have given assistance in revising manuscript or 
proofs of the sections named : Prof. De Volson Wood, 
mechanics and turbines; Mr. Frank Richards, compressed 
air ; Mr. Alfred R. Wolff, windmills ; Mr. Alex. C, 
Humphreys, illuminating gas ; Mr. Albert E. Mitchell, 
locomotives ; Prof. James E. Denton, refrigerating-ina« 
chinery ; Messrs. Joseph Wetzler and Thomas W. Varlcy, 
electrical engineering ; and Mr. Walter S. Dix, for valuabU 
contributions on several subjects, and suggestions as to theii 
treatment. William Kent, 

Passaic, N. J., W>ri7, 1895. 

FIFTH EDITION, MARCH, 1900. 

Some typographical and other errors discovered in the fourt' 
edition have been corrected. New tables and some addition 
have been made under the head of Compressed Air. The nei 
(1899) code of the Boiler Test Committee of the America 
Society of Mechanical Engineers has been substituted for tli 
old (1885) code. W. K, I 



PREFACE TO FOURTH EDITION. 

In this edition many extensive alterations have been made. 
Much obsolete matter has been cut out and fresh matter substi- 
tuted. In the first 170 pages but few changes have been found 
necessary, but a few typographical and other minor errors have 
been corrected. The tables of sizes, weight, and strength of 
materials (pages 172 to 282) have been thoroughly revised, many 
entirely new tables, kindly furnished by manufacturers, having 
been substituted. Especial attention is called to the new matter 
on Cast-iron Columns (pages 250 to 253). In the remainder of 
the book changes of importance have been made in more than 100 
pagrs, and all typographical errors reported to date have been 
corrected. Manufacturers' tables have been revised by reference 
to their latest catalogues or from tables furnished by the manufac- 
turers especially for this work. Much new matter is inserted 
under the heads of Fans and Blowers, Flow of Air in Pipes, and 
Compressed Air. The chapter on Wire-rope Transmission (pages 
917 to 922) has been entirely rewritten. The chapter on Electrical 
Engineering has been improved by the omission of some matter 
that has become out of date and the insertion of some new matter. 

It has been found necessary to place much of the new matter of 
this edition in an Appendix, as space could not conveniently be 
made for it in the body of the book. It has not been found possi- 
ble to make in the body of the book many of the cross-references 
which should be made to the items in the Appendix. Users of the 
book may find it advisable to write in the margin such cross-refer- 
ences as they may desire. 

The Index has been thoroughly revised and greatly enlarged. 

The author is under continued obligation to many manufacturers 
who have furnished new tables and data, and to many individual 
engineers who have furnished new matter, pointed out errors in 
the earlier editions, and offered helpful suggestions. He will be 
glad to receive 5imilar aid, which will assist in the further 
Improvement of the book in future editions. 

William Kent. 

PA5SA1C, N. J., Seftemher^ 1898. 



CONTENTS. 



CFor Alphabetical Index see pace 1079.) 

MATHBMATIOS. 

Arithmetie. 

PAOB 

Arithmetical and Algebraical Signti. 1 

Greatest Common iNTisor. 9 

Least Common Multiple ft 

FractloiM 2 

Decimals 8 

Table. Decimal Equiyalents of Fractions of One Inch 8 

Table. Products of Fractions expressed in Dedmala 4 

Compound or Denominate Numbers 6 

Reduction Descending and Ascending 6 

BaCio and Proportion 5 

Involation, or Towers of Numbers 6 

Table. First Nine Powers of the First Nine Numbers 7 

Table. First Forty Powers of 8 7 

ETolution. Square Root • 7 

CubeBoot 8 

Alli«atlan 10 

Permutation 10 

Combination 10 

Arithmetical Progression 11 

Geometrical Progreesion 11 

Interest 18 

Disooont.. 18 

Compound Interest 14 

Compound Interest Table, 8, 4, fi, and 6 per cent 14 

Kqufttlon of Payments 14 

Partial Pajmeuts 16 

Annuities 16 

TaUes of Amount, Present Values, etc., of Annuities 16 

Weights and Measures. 

Long Measure 17 

OklLand Measure 17 

Nautical Measure 17 

Sqoare Measure 18 

Sdid or Cubic Measure 18 

Liqaid Measure 18 

The MiaersMnch 18 

Apothecaries* Fluid Measure 16 

Dry Measure 18 

SfatpplQg Measure 10 

AToirdnpois Weight. 10 

Troy W^bt. 10 

Apothecaries* Weight 10 

ToWeigli Correctly on an Incorrect Balance 10 

Circular Measure 80 

Measure of time 80 

V 



E 



u 



-s .^ 'i 



I I 



HARVARD COLLEGE LIBRARY 

fp::*: t-:-- lscpary of 

FRA-:.. FtAB02Y MAGOUN 

IHL GIFT OF HIS SON 

MAY 8, 1929 



CopTniGHT, isoe^ 

BY 

WILLIAM KENT. 



Braunworth, Munn ^ Barbc . '^\ 

Printers and Binders . ' ^^ 

Brooklyn, N. y. J ^ 



PREPACE. 



More than twenty years ago the author began to follow 
the advice given by Nystrom : " Every engineeer should 
make his own pocket-book, as be proceeds in study and 
practice, to suit his particular business.*' The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to this, a 
careful examination was made of the transactions of engi- 
neering societies, and of the most important recent works 
OQ mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor- 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it is believed, cause it to 
differ from other works of its class. In the first place it 
was considered that the field of mechanical engineering was 
so great, and the literature of the subject so vast, that as 
little space as possible should be given to subjects which 
especially belong to civil engineering. While the mechan- 
ical engineer must continually deal with problems which 
lelong properly to civil engineering, this latter branch is 
^o well covered by Trautwine*s ** Civil Engineer's Pocket- 
Ixxtk" that any attempt to treat it exhaustively would not 
onJy fill no "long-felt want," but would occupy space 
vhich should be given to mechanical engineering. 

Another idea prominently kept in view by the author has 
been that he would not assume the position of an '* au- 
iboritj'* in giving rules and formuls for designing, but 
only that of compiler, giving not only the name of the 
originator of the rule, where it was known, but also the 
rolume and page from which it was taken, so that its 

iii 



t- >• 1 ^^ '/ I I 



HARVARD COLLEGE LIBRARY 
Pn:" r;- LIBPA'RY OF 

FRAr. . FLAB03Y MAGOUN 

Iht GIFT OF HIS SON 

MAY 8, 1929 



COPTHIOHT, ISOB^ 
BY 

WILLIAM KENT. 



Braunwoith, Munn & Barbe '^\ 

Printers and Binders . ^ 



Brooklyn, N. Y. .^ 



•>) 



PREBACE. 



More than twenty years ago the author began to follow 
the advice gWen by Nystrom : "Every engineeer should 
make his own pocket-book, as be proceeds in study and 
practice, to suit his particular business." The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to this, a 
careful examination was made of the transactions of engi- 
neering societies, and of the most important recent works 
on mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor- 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it is believed, cause it to 
diScr from other works of its class. In the first place it 
was considered that the field of mechanical engineering was 
so great, and the literature of the subject so vast, that as 
little space as possible should be given to subjects which 
especially belong to civil engineering. While the mechan- 
xal engineer must continually deal with problems which 
belong properly to civil engineering, this latter branch is 
s^o well covered by Trautwine*s ** Civil Engineer's Pocket- 
book" that any attempt to treat it exhaustively would not 
"nly fill no " long-felt want," but would occupy space 
vhich should be given to mechanical engineering. 

Another idea prominently kept in view by the author has 
been that he would not assume the position of an " au- 
thority" in giving rules and formuls for designing, but 
only that of compiler, giving not only the name of the 
, originator of the rule, where it was known, but also the 
volume and page from which it was taken, so that its 

iii 



*^ PREFACE. 

derivation may be traced when desired. When different 
formulae for the same problem have been found they have 
been given in contrast, and in many cases examples have 
been calculated by each to show the difference between 
them. In some cases these differences are <|uite remark- 
able, as will be seen under Safety-valves ^nd Crank- pins. 
Occaaiooally the study of these differences has led to the 
author's devising a new formula, in which case the deriva- 
tion of the formula is given. 

Much attention has been paid to the abstracting of data 
of experiments from recent periodical literature, and numer* 
ous references to other data are given. In this respect 
the present work will be found to differ from other Pocket- 
books. 

The author desires to express his obligation to the many 
persons who have assisted him in the preparation of the 
work, to manufacturers who have furnished their cata* 
logues and given permission for the use of their tables, 
and to many engineers who have contributed original data 
and tables. The names of these persons are mentioned io 
their proper places in the text, and in all cases it has been 
endeavored to give credit to whom credit is due. The 
thanks of the author are also due to the following gentle- 
men who have given assistance in revising manuscript ot 
proofs of the sections named : Prof. De Volson Wood, 
mechanics and turbines ; Mr. Frank Richards, compressed 
air; Mr. Alfred R. Wolff, windmills; Mr. Alex. C 
Humphreys, illuminating gas ; Mr. Albert E. Mitchell 
locomotives ; Prof. James E. Denton, refrige rating-ma 
chinery; Messrs. Joseph Wetzler and Thomas W, Varlcv 
electrical engineering ; and Mr. Walter S. Dix, for valuabl 
contributions on several subjects, and suggestions as to tbei 
treatment. WiLUAM Kent. 

Passaic, N, J,, Aprils 1895. 

FIFTH EDITION, MARCH, 1900. 

Some typographical and other errors discovered in the fourt 
edition have been corrected. New tables and some addition 
have been made under the head of Compressed Air. The nei 
(1899) code of the Boiler Test Committee of the America 
Society of Mechanical Engineers has been substituted for th 
old (1885) code. W. K. 



PREFACE TO FOURTH EDITION. 

In this edition many extensive alterations have been made. 
Much obsolete matter has been cut out and fresh maiter substi- 
tuted. In the first 170 pages but few changes have been found 
necessary, but a few typographical and other minor errors have 
been corrected. The tables of sizes, weight, and strength of 
materials (pages 172 to 282) have been thoroughly revised, many 
entirely new tables, kindly furnished by manufacturers, having 
been substituted. Especial attention is called to the new matter 
on Cast-iron Columns (pages 250 to 253). In the remainder of 
the book changes of importance have been made in more than 100 
pagrs, and all typographical errors reported to date have been 
corrected. Manufacturers' tables have been revised by reference 
to their latest catalogues or from tables furnished by the manufac- 
turers especially for this work. Much new matter is inserted 
under the heads of Fans and Blowers, Flow of Air in Pipes, and 
Compressed Air. The chapter on Wire-rope Transmission (pages 
917 to 922) has been entirely rewritten. The chapter on Electrical 
Engineering has been improved by the omission of some maiter 
that has become out of date and the insertion of some new matter. 

It has been found necessary to place much of the new matter of 
this edition in an Appendix, as space could not conveniently be 
made for it in the body of the book. It has not been found possi- 
ble to make in the body of the book many of the cross-references 
which should be made to the items in the Appendix. Users of the 
book may find it advisable to write in the margin such cross-refer« 
ences as they may desire. 

The Index has been thoroughly revised and greatly enlarged. 

The author is under continued obligation to many manufacturers 
who have furnished new tables and data, and to many individual 
engineers who have furnished new matter, pointed out errors in 
the earlier editions, and offered helpful suggestions. He will be 
glad to receive .similar aid, which will assist in the further 
improvement of the book in future editions. 

Wjlliam Kknt. 

Passaic, S. J., Se^tembtr^ 1898. 



CONTENTS. 



(For Alphabetical Index see pace 1079.) 

MATHKMATIOS. 

Arltbmetlo. 

Arithmetical and Algebraical Bigna. ~"l 

Greatest Common Diviflor. S 

Least Oommon Multiple. 8 

Fractions 8 

Decfmala 8 

Table. Decimal Equivalents of Fractions of One Inch 8 

Table. Products of Fractions expressed in Decimals • 4 

Cbmpound or Denominate Numbers 6 

BeductioD Descending and Ascending 6 

Ratio and Proportion 6 

Involution, or rowers of Numbers 6 

Table. First Nine Powers of the First Nine Numbers 7 

Table. First Forty Powers of 8 7 

KvolutioD. Square Root 7 

CubeRoot 8 

AUigatloa 10 

Permutation 10 

Combination 10 

Arithmetical Progression 11 

Geometrical Progreasion • 11 

Interest 18 

DiKOOttnt. 18 

Compound Interest 14 

Compound Interest Table, 8, 4, 8, and 6 per cent 14 

Equatkm of Payments 14 

Partial Payments 15 

Annultlea 16 

Tables of Amount, Preaent Values, etc., of Annuities 10 

Weig^hta and Bleasures. 

LoagMeaaure 17 

Old Land Measure 17 

Nautical Measure 17 

Souare Measure 18 

Bokid or Cubic Measure 18 

Liquid Measure 18 

The Miners* Inch 18 

Apothecaries* Fluid Measure. 18 

Dry Measure 18 

Shipping Measure 10 

Avoirdupois Weight 10 

Troy Weigrht. 10 

Apotbec&Hes* Weight 10 

To Weigh Correctly on an Incorrect Balance 10 

Circular Measure SSO 

Measure of time 80 

V 



^^^ 3n./iJ? 




HARVARD 
COLLEGE 
LIBRARY 



^n^ 3^^,{)^ 




HARVARD 
COLLEGE 
LIBRARY 



THE 

MECHANICAL ENGINEER'S 
POCKET-BOOK 



, A REFEBENCM-BOOK OF RULES, TABLES, DATA, 
AND FORMULJB, FOR THE USE OF 
ENGINEERS, MECHANICS, 
i AND STUDENTS. 



BY 

WILLIAM KENT, A.M., M.E., 

Confuting Engineer, 
Member Amer, 8oc*y Mechl. Engrs, aiid Amer, Inst, Mining Engra. 



* MFTH EDITION, REVISED AND ENLARGED. 
I EIGHTH THOUSAND. 



NEW YORK: 

JOHN WILEY & SONS. 

London: CHAPMAN & HALL, Limited. 

1901. 



t >.-, -5^/ 7, c^/ 



u 



HARVARD COLLEGE LIBRARY 

pr-" , ,- L'BRARY OF 

FRA-:.. . I-EABODY MAGOUN 

THE GIFT OF HIS SON 

MAY 8. 1929 



ConrmaiiT, iflBB^ 

Br 

WILLIAM KENT. 



Braunworth, Munn ^ Barbe 
Printers and Binders 
Brooklyn, N. Y. \ 



.\^ 



' PREEACE. 

More than twenty years ago the author began to follow 
the advice given by Nystrom : '* Every engineeer should 
make his own pocket-book, as be proceeds in study and 
practice, to suit his particular business.'* The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to this, a 
careful examination was made of the transactions of engi- 
neering societies, and of the most important recent works 
on mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor- 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it is believed, cause it to 
differ from other works of its class. In the first place it 
was considered that the field of mechanical engineering was 
so great, and the literature of the subject so vast, that as 
little space as possible should be given to subjects which 
especially belong to civil engineering. While the mechan- 
<:al engineer must continually deal with problems which 
belong properly to civil engineering, this latter branch is 
ro well covered by Trautwine*s ** Civil Engineer's Pocket- 
>ook" that any attempt to treat it exhaustively would not 
>n]y fill no '* long-felt want," but would occupy space 
rbich should be given to mechanical engineering. 

Another idea prominently kept in view by the author has 
>^cn that he would not assume the position of an " au- 
bority" in giving rules and formuls for designing, but 
r/slj that of compiler, giving not only the name of the 
ri^inator of the rule, where it was known, but also the 
olume and page from which it was taken, so that its 

ill 



*^ PREFACE. 

derivation may be traced when desired. When different 
formuls for the same problem have been found they have 
been given in contrast, and in many cases examples have 
been calculated by each to show the difference between 
them. In some cases these differences are quite remark- 
able, as will be seen under Safety-valves ^nd Crank* pins. 
Occasionally the study of these differences has led to the 
Author's devising a new formula, In which case the deriva* 
tion of the formula is given. 

Much attention has been paid to the abstracting of data 
of experiments from recent periodical literature, and numer« 
ous references to other data are given. In this respect 
the present work will be found to differ from other Pocket- 
books. 

The author desires to express his obligation to the many 
persons who have assisted him in the preparation of the 
work, to manufacturers who have furnished their cata* 
logues and given permission for the use of their tables, 
and to many engineers who have contributed original data 
and tables. The names of these persons are mentioned in 
their proper places in the text, and in all cases it has been 
endeavored to give credit to whom credit is due. The 
thanks of the author are also due to the following gentle- 
men who have given assistance in revising manuscript or 
proofs of the sections named : Prof. De Volson Wood, 
mechanics and turbines ; Mr. Frank Richards, compressed 
air; Mr. Alfred R. Wolff, windmills; Mr. Alex. C. 
Humphreys, illuminating gas ; Mr. Albert E. Mitchell, 
locomotives ; Prof. James E. Denton, refrigerating-ma. 
chinery ; Messrs. Joseph Wctzler and Thomas W. Varlcy, 
electrical engineering ; and Mr. Walter S. Dix, for valuable 
contributions on several subjects, and suggestions as to thei] 
treatment. William Kent, i 

Passaic, N. J,, Aprils 1895. 

FIFTH EDITION, MARCH, 1900. \ 

Some typographical and other errors discovered in the fourd 
edition have been corrected. New tables and some additioii 
have been made under the head of Compressed Air. The nei 
(1899) code of the Boiler Test Committee of the America 
Society of Mechanical Engineers has been substituted for til 
old (1885) code. W. K 



PREFACE TO FOURTH EDITION. 

In ihis edition many extensive alterations have been made. 
Much obsolete matter has been cut out and fresh matter substi- 
tuted. In the first 170 pages but few changes have been found 
necessary, but a few typographical and other minor errors have 
been corrected. The tables of sizes, weight, and strength of 
materials (pages 172 to 282) have been thoroughly revised, many 
entirely new tables, kindly furnished by manufacturers, having 
been substituted. Especial attention is called to the new matter 
on Cast-iron Columns (pages 250 to 253). In the remainder of 
ifae book changes of importance have been made in more than 100 
pagrs, and all typographical errors reported to date have been 
corrected. Manufacturers' tables have been revised by reference 
to their latest catalogues or from tables furnished by the manufac- 
turers especially for this work. Much new matter is inserted 
under the heads of Fans and Blowers, Flow of Air in Pipes, and 
Compressed Air. The chapter on Wire-rope Transmission (pages 
917 to 922) has been entirely rewritten. The chapter on Electrical 
Engineering has been improved by the omission of some matter 
that has become out of date and the insertion of some new matter. 
It has been found necessary to place much of the new matter of 
this edition in an Appendix, as space could not conveniently be 
made for it in the body of the book. It has not been found possi- 
ble to make in the body of the book many of the cross-references 
which should be made to the items in the Appendix. Users of the 
book may find it advisable to write in the margin such cross-refer- 
ences as they may desire. 
The Index has been thoroughly revised and greatly enlarged. 
The author is under continued obligation to many manufacturers 
who have furnished new tables and data, and to many individual 
engineers who have furnished new matter, pointed out errors in 
the earlier editions, and offered helpful suggestions. He will be 
glad to receive similar aid, which will assist in the further 
improvement of the book in future editions. 

William Kknt. 

Passaxc, N. J., St^tembtr^ 1898. 



THE 

MECHANICAL ENGINEER'S 
POCKET-BOOK 



A nEFERENCE-BOOK OF RULES, TABLES, DATA, 

AND FORMULjE, FOR THE USE OF 

ENGINEERS, MECHANICS, 

AND STUDENTS. 



WILLIAM KENT, A.M., M.E., 

Conwiting Engineer, 
Member Amer. Soc'y Mechl, Enffrs. and Amer, Inti, Mining Engrs. 



MFTH EDITION, REVISED AND ENLARGED. 
EIGHTH THOUSAND. 



NEW YORK: 

JOHN WILEY & SONS. 

I^ondon: CJIAPMAN & HALL, Limited. 

1901. 



THE 

MECHANICAL ENGINEER'S 
POCKET-BOOK 



A REFERENCE-BOOK OF RULES, TABLES, DATA, 

AND FORMULA, FOR THE USE OF 

ENGINEERS, MECHANICS, 

AND STUDENTS, 



BT 



WILLIAM KENT, A.M., M.E., 

ConwiUing Engineer^ 
Member Amer, 8oc*y Mechl, Bngr9. and Amer. Inst, Mining JSngrt. 



JflFTH EDITION, REVISED AND ENLARGED. 
EIGHTH THOUSAND. 



NEW YORK: 

JOHN AVILEY & SONS. 

Ix)RDON: CPAPMAN & HALL. Limited. 

1901. 



£,.. 



'( ■=> -r I 1 



u- 



HARVARD COLLEGE LIBRARY 
Fn?"' 7-.- MBPARYOF 

FRA-: rtABODY MAGOUN 

^WL GIFT OF HIS SON 

MAY 8, 1929 



ConmiGHT, isgo; 

BY 

WILLIAM KENT. 



Braunworth, Muon ^ Bjrbc . ^ ^\ 

Printers and Binders . ' ^ 

Brooklyn, N. y. / "" 

'1 •>■ 



PREIACE. 



' More than twenty years ago the author began to follow 
the advice given by Nystrom : '* Every engineeer should 
make his own pocket-book, as be proceeds in study and 
practice, to suit his particular business." The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to this, a 
careful examination was made of the transactions of engi* 
neering societies, and of the most important recent works 
on mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor- 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it is believed, cause it to 
differ from other works of its class. In the first place it 
was considered that the field of mechanical engineering was 
so great, and the literature of the subject so vast, that as 
little space as possible should be given to subjects which 
especially belong to civil engineering. While the mechan- 
ical engineer must continually deal with problems which 
belong properly to civil engineering, this latter branch is 
so well covered by Trautwine's ** Civil Engineer's Pocket- 
book " that any attempt to treat it exhaustively would not 
only fill no "long-felt want," but would occupy space 
irhich should be given to mechanical engineering. 

Another idea prominently kept in view by the author has 
been that he would not assume the position of an ** au- 
thority'* in giving rules and formulae for designing, but 
only that of compiler, giving not only the name of the 
originator of the rule, where it was known, but also the 
Volume and page from which it was taken, so that its 

iii 



**^ PREFACE. 

»• 

derivation may be traced when desired. When different 
formula for the same problem have been found they have 
been given in contrast, and in many cases examples have 
been calculated by each to show the difference between 
them. In eome cases these differences are quite remark- 
able, as will be seen under Safety-valves ^nd Crank* pins. 
Occasionally the study of these differences has led to the 
author's devising a new formula, in which case the deriva- 
tion of the formula is given. 

Much attention has been paid to the abstracting of data 
of experiments from recent periodical literature, and numer« 
ous references to other data are given. In this respect 
the present work will be found to differ from other Pocket- 
books. 

The author desires to express bis obligation to the many 
persons who have assisted him in the preparation of the 
work, to manufacturers who have furnished their cata* 
logues and given permission for the use of their tables, 
and to many engineers who have contributed original data 
and tables. The names of these persons are mentioned in 
their proper places in the text, and in all cases it has been 
endeavored to give credit to whom credit is due. The 
thanks of the author are also due to the following gentle- 
men who have given assistance in revising manuscript or 
proofs of the sections named : Prof. De Volson Wood, 
mechanics and turbines ; Mr. Frank Richards, compressed 
air; Mr. Alfred R. Wolff, windmills; Mr. Alex. C. 
Humphreys, illuminating gas ; Mr. Albert E. Mitchell, 
locomotives ; Prof. James E. Denton, refrigerating-ma. 
chinery ; Messrs. Joseph Wetzler and Thomas W. Varlcy, 
electrical engineering ; and Mr. Walter S. Dix, for valuable 
contributions on several subjects, and suggestions as to tbeii 
treatment. William Kent. 

Passaic, N. J., Aprils 1895. 

FIFTH EDITION, MARCH, 1900. 

Some typographical and other errors discovered in the fourt] 
edition have been corrected. New tables and some additioni 
have been made under the head of Compressed Air. The ne« 
(1S99) code of the Boiler Test Committee of the American 
Society of Mechanical Engineers has been substituted for tK 
old (1885) code. W. K. 



PREFACE TO FOURTH EDITION. 

In ihis edilion many extensive alterations have been made. 
Much obsolete matter has been cut out and fresh matter substi- 
tuted. In the first 170 pages but few changes have been found 
necessary, but a few typographical and other minor errors have 
been corrected. The tables of sizes, weight, and strength of 
materials (pages 172 to 282) have been thoroughly revised, many 
entirely new tables, kindly furnished by manufacturers, having 
been substituted. Especial attention is called to the new matter 
on Cast-iron Columns (pages 250 to 253). In the remainder of 
the book changes of importance have been made in more than 100 
pagrs, and all typographical errors reported to date have been 
corrected. Manufacturers* tables have been revised by reference 
to their latest catalogues or from tables furnished by the manufac- 
turers especially for this work. Much new matter is inserted 
under the heads of Fans and Blowers, Flow of Air in Pipes, and 
Compressed Air. The chapter on Wire-rope Transmission (pages 
917 to 922) has been entirely rewritten. The chapter on Electrical 
Engineering has been improved by the omission of some matter 
that has become out of date and the insertion of some new matter. 

It has been found necessary to place much of the new matter of 
this edition in an Appendix, as space could not conveniently be 
made for it in the body of the book. It has not been found possi- 
ble to make in the body of the book many of the cross-references 
which should be made to the items in the Appendix. Users of the 
book may find it advisable to write in the margin such cross-refer- 
ences as they may desire. 

The Index has been thoroughly revised and greatly enlarged. 

The author is under continued obligation to many manufacturers 
who have furnished new tables and data, and to many individual 
engrineers who have furnished new matter, pointed out errors in 
the earlier editions, and offered helpful suggestions. He will be 
glad to receive similar aid, which will assist in the further 
improvement of the book in future editions. 

William Kbnt. 

Passaic, N. J., Sepiembtr^ 1898. 



CONTENTS. 



(For Alphabetical Index lee page 1079.) 

MATHBBIATIOS. 

Arithmetlo. 

PAGB 

Arithmetical and Algebraical Blgna 1 

Greatest Common DiTiaor. 8 

Least Common Multiple 8 

FractioDS 3 

Decimate 8 

Table. Decimal Equivalents of Fractions of One Inch 8 

Table. Products of Fractions expressed In Decimals 4 

Compound or Denominate Numbers 5 

Reduction Descending and Ascending 5 

Ratio and Proportion 5 

Involution, or Powers of Numbers 8 

Table. First Nine Powers of the First Nine Numbers 7 

Table. FIrat Forty Powers of 2 7 

Evohition. Square Booc 7 

CubeRoot 8 

Alligation 10 

Permutation 10 

Combination 10 

Arithmetical Progression 11 

Geometrical Progression 11 

Interest 18 

Discount 18 

Compound Interest 14 

Compound Interest Table, 8, 4, S, and 6 per cent 14 

Equation of Payments 14 

Partial Payments 15 

Annuities 16 

TkUea of Amount, Present Values, etc., of Annuities 16 

Weig^hts and Bleasares. 

toag Measure 17 

OldLand Measure 17 

Ksutical Meaaure 17 

Sqoare Measure 18 

Solid or Cubic Measure 18 

Liquid Measure 18 

The XineraMnch 18 

Apothecaries* Fluid Measure. 18 

DfT Measure 18 

Supping Measure 19 

Avoirdupois Weight. > 19 

Troy Weight. 19 

Apothecaries* Weij^t 19 

To Weigh Correctly on an Incorrect Balance 19 

Qrenlar Measure SO 

Measure of time 80 

V 



E 



L^ 



3 -• 



I / 



HARVARD COLLEGE LIBRARY 




Fr?" 7'.-- LICPA'RYOF 




FRAr . F-EAB03Y MAGOUN 




IHt GIFT OF HIS SON 




MAY 8, 1929 






3 ; 



BY 

WILLIAM KENT. 



Braunworth, Munn ^ Barbe . - A 
Printers and Binders • " ^ 

Brooklyn, N. y. i "^ 



PREIACE. 



More than twenty years ago the author began to follow 
the advice given by Nystrom : '* Every engineeer should 
make his own pocket-book, as be proceeds in study and 
practice, to suit his particular business." The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to this, a 
careful examination was made of the transactions of engi* 
neering societies, and of the most important recent works 
on mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor- 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it is believed, cause it to 
diflfer from other works of its class. In the first place it 
was considered that the field of mechanical engineering was 
so great, and the literature of the subject so vast, that as 
little space as possible should be given to subjects which 
especially belong to civil engineering. While the mechan- 
ical engineer must continually deal with problems which 
belong properly to civil engineering, this latter branch is 
so well covered by Trautwine's ** Civil Engineer's Pocket- 
book" that any attempt to treat it exhaustively would not 
only fill no "long-felt want," but would occupy space 
which should be given to mechanical engineering. 

Another idea prominently kept in view by the author has 
been that he would not assume the position of an ** au- 
thority " in giving rules and formulae for designing, but 
only that of compiler, giving not only the name of the 
originator of the rule, where it was known, but also the 
volume and page from which it was taken, so that its 

iii 



E 



I. 01 



HARVARD COLLEGE LIBRARY 
FR?" v:- L'BPARY OF 

FRAP .. . FtABODY MAGOUN 

IHt GIFT OF HIS SON 

MAY 8, 1929 



CopTniGiiT, isgo; 

Br 

WILLIAM KENT. 



Braunworth, Muon & Barbc . \ A 
Printers and Binders ■ ^^ 

Brooklyn, N. y. | 



PREIACE. 



[ More than twenty years ago the author began to follow 
the advice given by Nystrom : " Every engineeer should 
make his own pocket-book, as be proceeds in study and 
practice, to suit his particular business." The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to this, a 
careful examination was made of the transactions of engi- 
neering societies, and of the most important recent works 
on mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor- 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it is believed, cause it to 
differ from other works of its class. In the first place it 
was considered that the field of mechanical engineering was 
so great, and the literature of the subject so vast, that as 
little space as possible should be given to subjects which 
especially belong to civil engineering. While the mechan- 
ical engineer must continually deal with problems which 
belong properly to civil engineering, this latter branch is 
so well covered by Trautwine*s ** Civil Engineer's Pocket- 
book " that any attempt to treat it exhaustively would not 
>nly fill no ** long-felt want," but would occupy space 
irhich should be given to mechanical engineering. 

Another idea prominently kept in view by the author has 
been that he would not assume the position of an " au- 
thority" in giving rules and formulae for designing, but 
7nlj that of compiler, giving not only the name of the 
>ri£rinator of the rule, where it was known, but also the 
rolume and page from which it was taken, so that its 

m 



**^ PBEFACE. 

derivation may be traced when desired. When different 
formula for the same problem have been found they have 
been given in contrast, and in many cases examples have 
been calculated by each to show the difference between 
them. In some cases these differences are quite remark- 
able, as will be seen under Safety-valves ^nd Crank- pins. 
Occasionally the study of these differences has led to the 
author's devising a new formula, in which case the deriva- 
tion of the formula is given. 

Much attention has been paid to the abstracting of data 
of experiments from recent periodical literature, and numer« 
ous references to other data are given. In this respect 
the present work will be found to differ from other Pocket- 
books. 

The author desires to express his obligation to the many 
persons who have assisted him in the preparation of the 
work, to manufacturers who have furnished their cata* 
logues and given permission for the use of their tables, 
and to many engineers who have contributed original data 
and tables. The names of these persons are mentioned in 
their proper places in the text, and in all cases it has been 
endeavored to give credit to whom credit is due. The 
thanks of the author are also due to the following gentle- 
men who have given assistance in revising manuscript or 
proofs of the sections named : Prof. De Volson Wood, 
mechanics and turbines ; Mr. Frank Richards, compressed 
air ; Mr. Alfred R. Wolff, windmills ; Mr. Alex. C. 
Humphreys, illuminating gas ; Mr. Albert E. Mitchell, 
locomotives ; Prof. James E. Denton, refrigerating-ma« 
chinery ; Messrs. Joseph Wetzlcr and Thomas W. Varley, 
electrical engineering ; and Mr. Walter S. Dix, for valuable 
contributions on several subjects, and suggestions as to their 
treatment. William Kent. 

Passaic, N. J., Apriit 1895. 

FIFTH EDITION, MARCH, 1900. 

Some typographical and other errors discovered In the fourth 
edition have been corrected. New tables and some additions 
have been made under the head of Compressed Air. The new 
(1S99) code of the Boiler Test Committee of the American 
Society of Mechanical Engineers has been substituted for the 
old (1885) code. W. K. 



PREFACE TO FOURTH EDITION. 

In this edition many extensive alterations have been made. 
Much obsolete matter has been cut out and fresh matter substi- 
tuted. In the first 170 pages but few changes have been found 
necessary, but a few typographical and other minor errors have 
been corrected. The tables of sizes, weight, and strength of 
materials (pages 172 to 282) have been thoroughly revised, many 
entirely new tables, kindly furnished by manufacturers, having 
been substituted. Especial attention is called to the new matter 
on Cast-iron Columns (pages 250 to 253). In the remainder of 
the book changes of importance have been made in more than 100 
pagrs, and all typographical errors reported to date have been 
corrected. Manufacturers' tables have been revised by reference 
to their latest catalogues or from tables furnished by the manufac- 
turers especially for this work. Much new matter is inserted 
under the heads of Fans and Blowers, Flow of Air in Pipes, and 
Compressed Air. The chapter on Wire-rope Transmission (pages 
917 to 922) has been entirely rewritten. The chapter on Electrical 
Engineering has been improved by the omission of some matter 
that has become out of date and the insertion of some new matter. 

It has been found necessary to place much of the new matter of 
this edition in an Appendix, as space could not conveniently be 
made for it in the body of the book. It has not been found possi- 
ble to make in the body of the book many of the cross-references 
which should be made to the items in the Appendix. Users of the 
book may find it advisable to write in the margin such cross-refer* 
ences as they may desire. 

The Index has been thoroughly revised and greatly enlarged. 

The author is under continued obligation to many manufacturers 
who have furnished new tables and data, and to many individual 
engineers who have furnished new matter, pointed out errors in 
the earlier editions, and offered helpful suggestions. He will be 
glad to receive 5imilar aid, which will assist in the further 
improvement of the book in future editions. 

William Kknt. 

Passajc, N. J., September^ 1898. 



CONTENTS. 

(For Alphabetical Index lee page 1079.) 

MATHBBIATICS. 

Arithmetlo. 

PAOB 

AritlimeticBl and Algebraical Bigiw. 1 

Greatest Common DiTisor. 8 

Least Conmion Multiple. 8 

Fractions 8 

Decimals 8 

Table. Decimal Equivalents of Fractions of One Inch 8 

Table. Plroducts of Fractions expressed In Decimals 4 

Oompouod or Denominate Numbers 5 

Reduction Descending and Ascending 5 

Batio and Proportion 6 

Invoiation, or Poirers of Nonbers 8 

Table. First Nine Powers of the First Nine Numbers 7 

Table. First Forty Powers of 2 7 

ETolntion. Square Boot 7 

CubeBoot... 8 

AlUgatioa 10 

Permutation 10 

Combination 10 

Arithmetical Progression 11 

Geometrical Progression 11 

Interest 18 

DisKonnt. 18 

Compound Interest 14 

Compound Interest Table, 8, 4, S, and per cent 14 

Equation of Payments 14 

Partial PaymeuU IS 

Annuities 16 

Tsbles of Amount, Present Values, etc., of Annuities 16 

Weights and Bfeasares. 

I^ng Measure 17 

Old Land Measure 17 

Nautical Measure 17 

Square Measure 18 

Solid or Cubic Measure 18 

Liquid Measure 18 

The Miners* Inch 18 

Apothecaries* Fluid Measure. 16 

Dry Measure 18 

Shipping Measure 10 

Avoirdupois Weight 10 

Troy Weight. 10 

Apothecaries* Weisrht 10 

To Weigh Correctly on an Incorrect Balance .•• 10 

Circular Measure SO 

Measure of time 80 

V 



E,: 



-\-'>'-ilol 



u 



HAf?VARD COLLEGE LIBRARY 
Fr?-' T ;•■ L'GPARY OF 

FRA'... PEABODY MAGOUN 

lHi£ GIFT OF HIS SON 

MAY 8, 1929 



COPTRIOBT, 1891^ 
BY 

WILLIAM KENT, 



Braunworth, Munn fif Barbc .- > \ 
Printers and Binders •; ' ^ 

Brooklyn, N. y. ^ ^^ 



PREiACE. 

More than twenty years ago the author began to follow 
the advice given by Nystrom : " Every engineeer should 
make bis own pocket-book, as be proceeds in study and 
practice, to suit his particular business/' The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to this, a 
careful examination was made of the transactions of engi- 
neering societies, and of the most important recent works 
>n mechanical engineering, in order to fill gaps that might 
>e left in the original collection, and insure that no impor- 
ant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
ion of the Pocket-book that will, it is believed, cause it to 
iffer from other works of its class. In the first place it 
ras considered that the field of mechanical engineering was 
o gTCAt, and the literature of the subject so vast, that as 
ttle space as possible should be given to subjects which 
specially belong to civil engineering. While the mechan- 
al engineer must continually deal with problems which 
t\otig properly to civil engineering, this latter branch is 
> well covered by Trautwine's *' Civil Engineer's Pocket- 
>ok " that any attempt to treat it exhaustively would not 
ily fill no "long-felt want," but would occupy space 
dich should be given to mechanical engineering. 
Another idea prominently kept in view by the author has 
ten that he would not assume the position of an ** au- 
ority " in giving rules and formule for designing, but 
ily that of compiler, giving not only the name of the 
iginator of the rule, where it was known, but also the 
lume and page from which it was taken, so that its 

ill 



E ,, 



'( -5 '-i I 1 



L^ 



HAf?VARD COLLEGE LIBRARY 
FP?" V-- IJBPARY OF 

FRA". PlABODY MAGOUN 

THE GIFT OF HIS SON 

MAY 8. 1929 



coFTBioHT, Ian; 

BY 

WILLIAM KENT. 



Braunworth, Munn & Barbc .- ^ A 
Printers and Bindca-s .; ' ^^ 

Brooklyn, N. V. J O" 



c 

'' PREPACK 

I 

Moke than twenty years ago the author began to follow 
I the advice given by Nystrom : " Every engineeer should 

make his own pocket-book, as be proceeds in study and 
practice, to suit his particular business." The manuscript 
pocket-book thus begun, however, soon gave place to more 
-modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
'book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
arranging it in form for publication. In addition to this, a 
careful examination was made of the transactions of engi- 
neering societies, and of the most important recent works 
on mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor* 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it is believed, cause it to 
lififer from other works of its class. In the first place it 
ras considered that the field of mechanical engineering was 
)o great, and the literature of the subject so vast, that as 
fttle space as possible should be given to subjects which 
specially belong to civil engineering. While the mechan- 
rai engineer must continually deal with problems which 
elong properly to civil engineering, this latter branch is 
:> well covered by Trautwine's ** Civil Engineer's Pocket- 
>oic " that any attempt to treat it exhaustively would not 
ily fill no 'Mong-felt want,'* but would occupy space 
hicli should be given to mechanical engineering. 
Another idea prominently kept in view by the author has 
ren that he would not assume the position of an ** au- 
ority" in giving rules and formule for designing, but 
fjy that of compiler, giving not only the name of the 
i^inator of the rule, where it was known, but also the 
lume and page from which It was taken, so that its 

ill 



*^ PREFACE. 

t. 

derivation may be traced when desired. When different 
formula for the same problem have been found they have 
been given in contrast, and in many cases examples have 
been calculated by each to show the difference between 
them. In some cases these differences are <iuite remark- 
able, as will be seen under Safety-valves ^nd Crank* pins. 
Occasionally the study of these differentes has led to the 
author's devising a new formula, in which ease the deriva- 
tion of the formula is given. 

Much attention has been paid to the absiractlng of data 
of experiments from recent periodical literature, and numer« 
ous references to other data are given. In this respect 
the present work will be found to differ from other Pocket* 
books. 

The author desires to express his obligation to the many 
persons who have assisted him in the preparation of the 
work, to manufacturers who have furnished their cata- 
logues and given permission for the use of their tables, 
and to many engineers who have contributed original data 
and tables. The names of these persons are mentioned in 
their proper places in the text, and in all cases it has been 
endeavored to give credit to whom credit is due. The 
thanks of the author are also due to the following gentle- 
men who have given assistance in revising manuscript or 
proofs of the sections named : Prof. De Volson Wood, 
mechanics and turbines ; Mr. Frank Richards, compressed 
air; Mr. Alfred R. Wolff, windmills; Mr. Alex. C. 
Humphreys, illuminating gas ; Mr. Albert E. Mitchell, 
locomotives ; Prof. James £. Denton, re frige rating-ma« 
chinery ; Messrs. Joseph Wctzler and Thomas W. Varlcy, 
electrical engineering ; and Mr. Walter S. Dix, for valuable 
contributions on several subjects, and suggestions as to their 
treatment. WiLLlAM Kent. 

Passaic, N. J., A^ri/^ 1895. 

FIFTH EDITION, MARCH, 1900. 

Some typographical and other errors discovered in the fourtK 
edition have been corrected. New tables and some additions 
have been made under the head of Compressed Air. The new 
(1S99) code of the Boiler Test Committee of the American 
Society of Mechanical Engineers has been substituted for the 
old (1885) code. W. K. 



PREFACE TO FOURTH EDITION. 

In this edition many extensive alterations liave been made. 
Much obsolete matter has been cut out and fresh matter substi- 
tuted. In the first 170 pages but few changes have been found 
necessary, but a few typographical and other minor errors have 
been corrected. The tables of sizes, weight, and strength of 
materials (pages 172 to 282) have been thoroughly revised, many 
entirely new tables, kindly furnished by manufacturers, having 
been substituted. Especial attention is called to the new matter 
on Cast-iron Columns (pages 250 to 253). In the remainder of 
the book changes of importance have been made in more than 100 
pagrs, and all typographical errors reported to date have been 
corrected. Manufacturers' tables have been revised by reference 
to their latest catalogues or from tables furnished by the manufac- 
turers especially for this work. Much new matter is inserted 
under the heads of Fans and Blowers, Flow of Air in Pipes, and 
Compressed Air. The chapter on Wire-rope Transmission (pages 
917 to 922) has been entirely rewritten. The chapter on Electrical 
Engineering has been improved by the omission of some matter 
that has become out of date and the insertion of some new matter. 

It has been found necessary to place much of the new matter of 
this edition in an Appendix, as space could not conveniently be 
made for it in the body of the book. It has not been found possi- 
ble to make in the body of the book many of the cross-references 
which should be made to the items in the Appendix. Users of the 
book may find it advisable to write in the margin such cross-refer- 
ences as they may desire. 

The Index has been thoroughly revised and greatly enlarged. 

The author is under continued obligation to many manufacturers 
who have furnished new tables and data, and to many individual 
engineers who have furnished new matter, pointed out errors in 
the earlier editions, and offered helpful suggestions. He will be 
glad to receive similar aid, which will assist in the further 
improvement of the book in future editions. 

William Kent. 

Passaic, N. J., September^ 1898. 



CONTENTS. 



CFor Alphabetical Index see page 1070.) 

MATHEMATICS. 

Arlthmetle, 

PAOB 

Arithmetice] and Algebraical Sigmi. 1 

Greatest Common InTiaor. 2 

Least Common Multiple 8 

Ftactions 8 

Dpcimala 8 

Table. Decimal Squlvalents of Fractions of One Inch 8 

Table. Producta of Fractions ezpremed In Decimals 4 

Compound or Denominate Numbers 6 

Reduction Descending and Ascending 5 

Ratio and Proportion 6 

Invohitkm, or rowers of Numbers 6 

Table. First Nine Powers of the First Nine Numbers 7 

Tkble. First Forty Powers of 2 7 

Evolution. Square Boot 7 

CabeBoot 8 

Alligation 10 

Permutatloo 10 

Combination 10 

Arithmetical Progression U 

Geometrical Progression 11 

Interest 18 

Discount 18 

Compound Interest 14 

Compound Interest Table, 8, 4, 6, and 6 per cent 14 

Equation of Payments 14 

Partial Payments 15 

Annuities 16 

Tkblea of Amount, Present Valuea, etc., of Annuities 16 

Weights and Measures. 

Long Measure 17 

Old Land Measure 17 

Nautical Measure 17 

Square Measure 18 

Boiid or Cubic Measure 18 

Liquid Measure 18 

The Miners* Inch 18 

Apothecaries* Fluid Measure. 18 

Dry Measure. 18 

Shipping Measure 19 

Aroirdupois Weight. 10 

Troy Weight. 10 

Apothecaries* Weight 19 

To Weigh Correctly on an Incorrect Balance 19 

Circular Measure SO 

Measure of time 20 

V 



THE 

lECHAKICAL ENGINEER'S 
POCKET-BOOK 



A REFERENCE-BOOK OF RULES, TABLES, DATA, 

AND FORMULAE, FOR THE USE OF 

ENGINEERS, MECHANICS, 

AND STUDENTS. 



BT 

WILLIAM KENT, A.M., M.E., 

ConnUting Engineer, 
Member Amer, 8oc*y MechL Engrt. and Amer. Inst. Mining JBngrs. 



MFTH EDITION, REVISED AND ENLARGED. 
EIGHTH THOUSAND. 



NEW YORK: 

JOHN WILEY & SONS. 

Lokdon: chapman & HALL, Limited. 

1901. 



THE 

lECHANICAL ENGINEER'S 
POCKET-BOOK 



1 

J 



A REFERENCE-BOOK OF RULES, TABLES, DATA, 

AND FORMULJE, FOR THE USE OF 

ENGINEERS, MECHANICS, 

AND STUDENTS, 



BT 

WILLIAM KENT, A.M., M.E., 

Conntlting EngiJieer^ 
Member Amer. Soc*y MeckL JBngra, and Amer, Inti, Mining Bngrs. 



MFTH EDITION, REVISED AND ENLARGED. 
EIGHTH THOUSAND. 



NEW YORK; 

JOHN WILEY & SONS. 

Ia)Ndon: chapman & HALL. Limited. 

1901. 



t- >• I ^-'lol 



HARVARD COLLEGE LIBRARY 

pr-" T .' LIBRARY OF 

FRa: F£AE03Y MAGOUN 

IHt GIFT OF HIS SON 

MAY 8, 1929 



COPTRIOHT, IflQB^ 
BY 

WILLIAM KENT. 



Braunworth, Munn & Barbe 
Printers and Binders 
Brooklyn, N. Y. ' 



.■'C\ 



PREIACE. 

MoRB than twenty years ago the author began to follow 
the advice given by Nystrom : " Every engineeer should 
make his own pocket-book , as be proceeds in study and 
practice, to suit his particular business." The manuscript 
pocket-book thus begun, however, soon gave place to more 
modern means for disposing of the accumulation of engi- 
neering facts and figures, viz., the index rerum, the scrap- 
book, the collection of indexed envelopes, portfolios and 
boxes, the card catalogue, etc. Four years ago, at the re- 
quest of the publishers, the labor was begun of selecting 
from this accumulated mass such matter as pertained to 
mechanical engineering, and of condensing, digesting, and 
' arranging it in form for publication. In addition to this, a 
I careful examination was made of the transactions of engi* 
neering societies, and of the most important recent works 
on mechanical engineering, in order to fill gaps that might 
be left in the original collection, and insure that no impor- 
tant facts had been overlooked. 

Some ideas have been kept in mind during the prepara- 
tion of the Pocket-book that will, it Is believed, cause it to 
differ from other works of its class. In the first place it 
was considered that the field of mechanical engineering was 
so great, and the literature of the subject so vast, that as 
little space as possible should be given to subjects which 
especially belong to civil engineering. While the mechan- 
ical engineer must continually deal with problems which 
belong properly to civil engineering, this latter branch is 
so well covered by Trautwine*s " Civil Engineer's Pocket- 
book " that any attempt to treat it exhaustively would not 
only fill no ** long-felt want," but would occupy space 
which should be given to mechanical engineering. 

Another idea prominently kept in view by the author has 
been that he would not assume the position of an " au- 
thority" in giving rules and formule for designing, but 
only that of compiler, giving not only the name of the 
originator of the rule, where it was known, but also the 
rolume and page from which it was taken, so that its 

lii 



*^ PREFACE. 

derivation may be traced when desired. When different 
formula for the same problem have been found they have 
been given in contrast, and in many cases examples have < 
been calculated by each to show the difference between 
them. In some cases these differences are <iutte remark- 
able, as will be seen under Safety-valves ^nd Crank-pins, 
Occasionally the study of these differences has led to the 
author's devising a new formula, in which ease the deriva« 
tioa of the formula is given. 

Much attention has been paid to the abstracting of data 
of experiments from recent periodical literature, and numer* 
ous references to other data are given. In this respect 
the present work will be found to differ from other Pocket* 
books. 

The author desires to express his obligation to the many 
persons who have assisted him in the preparation of the 
work, to manufacturers who have furnished their cata« 
logues and given permission for the use of their tables, 
and to many engineers who have contributed original data 
and tables. The names of these persons are mentioned in 
their proper places in the text, and in all cases it has been 
endeavored to give credit to whom credit is due. The 
thanks of the author are also due to the following gentle- 
men who have given assistance in revising manuscript or 
proofs of the sections named : Prof. De Volson Wood, 
mechanics and turbines ; Mr. Frank Richards, compressed 
air; Mr. Alfred R. Wolff, windmills; Mr. Alex. C. 
Humphreys, illuminating gas ; Mr. Albert E. Mitchell, 
locomotives ; Prof. James E. Denton, refrigerating-ma« 
ciiinery ; Messrs. Joseph Wetzler and Thomas W. Varlcy, 
electrical engineering ; and Mr. Walter S. Dix, for valuable 
contributions on several subjects, and suggestions as to their 
treatment. WiLUAM Kent. 

Passaic, N. J,, Aprils 1895. 

FIFTH EDITION, MARCH, 1900. 

Some typographical and other errors discovered in the fourth 
edition have been corrected. New tables and some additions 
have been made under the head of Compressed Air. The nevr 
(1S99) code of the Boiler Test Committee of the American 
Society of Mechanical Engineers has been substituted for tho 
old (1885) code. W. K. 



PREFACE TO FOURTH EDITION. 

In this edition many extensive alterations have been made. 
Much obsolete matter has been cut out and fresh matter substi- 
tuted. In the first 170 pages but few changes have been found 
necessary, but a few typographical and other minor errors have 
been corrected. The tables of sizes, weight, and strength of 
materials (pages 172 to 282) have been thoroughly revised, many 
entirely new tables, kindly furnished by manufacturers, having 
been substituted. Especial attention is called to the new matter 
on Cast-iron Columns (pages 250 to 253). In the remainder of 
the book changes of importance have been made in more than 100 
pagrs, and all typographical errors reported to date have been 
corrected. Manufacturers' tables have been revised by reference 
to their latest catalogues or from tables furnished by the manufac- 
turers especially for this work. Much new matter is inserted 
under the heads of Fans and Blowers, Flow of Air in Pipes, and 
Compressed Air. The chapter on Wire-rope Transmission (pages 
917 to 922) has been entirely rewritten. The chapter on Electrical 
Engineering has been improved by the omission of some matter 
that has become out of date and the insertion of some new matter. 

It has been found necessary to place much of the new matter of 
this edition in an Appendix, as space could not conveniently be 
made for it in the body of the book. It has not been found possi- 
ble to make in the body of the book many of the cross-references 
which should be made to the items in the Appendix. Users of the 
book may find it advisable to write in the margin such cross-refer- 
ences as they may desire. 

The Index has been thoroughly revised and greatly enlarged. 

The author is under continued obligation to many manufacturers 
who have furnished new tables and data, and to many individual 
engineers who have furnished new matter, pointed out errors in 
the earlier editions, and offered helpful suggestions. He will be 
glad to receive similar aid, which will assist in the further 
improvement of the book in future editions. 

William Kent. 

Passaic. N. J., Stpttmber^ 1898. 



CONTENTS. 



(For Alphabetical Index see page 1079.) 

MATHEMATICS. 

Arithmetle. 

PAOB 

ArithmetiealandAlKebralcalSigiiB. 1 

Gnsalest Common DiTiaor. 9 

Least Oommon Multiple 8 

Fracrions 2 

D(«tmala 8 

Table. Dedmal Equivalents of Fractions of One Inch S 

Table. Products of Fractions expressed In Decimals 4 

Oomponnd or Denominate Numbers 6 

Reduction Descending and Ascending ft 

Ratk> and Proportion 6 

Involution, or Powers of Nmnbers 

Tftble. First Nine Powers of the First Nine Numbers 7 

Table. First Forty Powers of 2 7 

Evolution. Square Root 7 

CubeBoot 8 

Alligation 10 

Permutation 10 

Combination 10 

Arithmetical Progression 11 

Geometrical Progression 11 

Interest 18 

Discount. 18 

Compound Interest 14 

Compound Interest Table, 8, 4, 6, and 6 per cent 14 

Equation of Payments 14 

Partial Paymento 15 

Annuitiea 16 

Tables of Amount, Present Values, etc., of Annuities 16 

Weights and HeasuTes. 

Long Measure 17 

OldLand Measure 17 

Nautical Measure 17 

Square Measure 18 

Solid or Cubic Measure 18 

Uqnid Measure 18 

Tbe Miners* Inch 18 

Apothecaries* Fluid Measure. 18 

Dry MesMire 18 

Shipping Measure 19 

AvoirdupoU Weight. 19 

Troy Weight, 19 

Apothecariea' Weight 19 

To Weigh Correctly on an Incorrect Balance 19 

CIrcniar Measure 80 

Measure of time 20 

V 



VI ... CONTENTS. 

Board and Timber Measure iC 

Table. ConteDU in BVet of Joists, Scantlings, and Timber. 2( 

French or Metric Mensures 21 

British and French Equivalents. 2] 

Metric Conversion Tables , 2£ 

Compound Units. 

of Pressure and Weight 2^ 

of Water, Weight, and Bulk 21 

of Work, Power, and Duty Zi 

of Velocity 21 

of Pressure per unit area 21 

Wire and Sheet Metal Gaums 26 

Twist-drill and Steel-wire Gauges 2fi 

Music-wire Gauge 2S 

Circular-mil Wire Gauge 8G 

NewU.S. Standard Wire and Sheet Gauge, 1808 .. 80 

Decimal Gauge 88 

Alfl^ebra. 

Addition, Multiplication, etc 89 

Powers of Numbers 89 

Parentheses, Division 84 

Simple Equations and Problems 84 

Equations containing two or more Unknown Quantities 8.1 

Elimination . .. 8S 

Quadratic Equations 85 

heory of Exponents 36 

Binomial Theorem 80 

Geometrical Problems of Construction 87 

of Straight Lines...-. 87 

of Angles 88 

ofOircles 89 

ofTrlangles 41 

of Squares and Polygons 42 

oftheElllpee 46 

of the Parabola 48 

of the Hyperbola 49 

oftbeOycloid 49 

of the Tractriz or Schiele Anti-friction Curve SO 

oftheSpiral 00 

of the Catenary 51 

of the Involute. 52 

Geometrical Propositions 58 

Mensuration, Plane SurCaces* 

Quadrilateral, Parallelogram, eto 54 

Trapezium and Trapezoid. , 54 

Triangles 64 

Polygons. Table of Polygons 55 

Irregular Figures .... SB 

Properties of the Circle. S7 

Values of ir and its Multiples, etc 57 

B«latlons of arc, chord, etc fl8 

Belations of circle to inscribed square, etc 59 

Sectors and Segments. 50 

Circular Bing 50 

The Ellipse 59 

TheHeliz. 60 

The Spiral 00 

Mens oration , Solid Bodies. 

Prism...... 60 

Pyramid 80 

wedge 61 

The Prismoidal Formula 6i 

Rectangular Prlsmoid 61 

Cylinder 61 

Cone 61 



PIGS 

Sphere 61 

Spherical Triangle 61 

Spherical PoiygOD ...••.. 61 

^herical Zone 68 

Spherical Segment 69 

Spheroid or Ellipsoid 68 

Polyedron 62 

Cytindrical Bins 62 

Solkla of BevoluUoD 68 

SiriJidJea 63 

Frusutun of a Spheroid. 6S 

Parabolic Goooid 64 

Tolume of a Cask 64 

Irrefnilar Solids 64 

Plane Trigonometry. 

Solution of Plane Trlangtee 66 

Sine, TftsseDt, Secant, etc 65 

SilpM of the Trtffononvttric Functions 66 

TVigonometricalFormaln 66 

SohiHon of Plane Right-angled Triangles 66 

Solution of Oblique-angled Trianglee 66 

Analytieal Geometry. 

Ordlnates and Ahsctssaa 68 

EquatSona of a Straight Line, Intersections, etc 69 

Equations of the Circle 70 

Equations of the Ellipse TO 

Equations of the Parabola : 70 

Equationa of the Hyperbola. 70 

Logaritlunic Curves. 71 

DIlTerentlal Caleulns. 

Definttiona 7« 

I>Uferential8 of Algebraic Fonctlons .... 78 

Fonnulse for Differentiating 78 

Partial Differentials 78 

Integrals 78 

Formulsfor Integration 74 

Integration between Limits 74 

Quadrature of a Plane Surf ace. 74 

Ouadratore of Surfaces of Revolution 75 

Cubature of Volumes of Revolution 76 

Second, Third, etc.. Differentials 75 

ltaclanriB*s and Taylor's Theorems 76 

lf^Wi«^ aim! Minima 76 

Differential of an EziMnentlal Function 77 

Ixsgarlthms 77 

Differential Forms which have Known Integrals 78 

Exponential Functions 78 

Circular Functions. 78 

TbeQyelold 79 

Inteipml Calculus 79 

Uaihen&atleal Tables. 

BecipfYKsals of If umbers 1 to 8000 80 

Squares, Cubes, Square Roots, and Cube Roohi from 0.1 to 1600 86 

Squares and Cnbes of Decimals 101 

Fifth Rooto and Fifth Powers 102 

Circnmferenoes and Areas of Circles, Diameters 1 to 1000.... 103 

Ctrcnmferenoea and Areas of Circles, Advancing by Eighths from A to 

100 108 

Deefmala of a Foot Equivalent to Inches and Fractions of an Inch 118 

Ctrcnmferences of Circles in Feet and Inches, from 1 loch to 88 feet 11 

iocbee la diameter. 118 

Lengths of Circular Arcs, Degrees Given 114 

Lengths of Circular Arcs. Height of Arc Given 115 

Areas of the Segments of a Circle. lio 



Viii OONTENTa 

PAOK 

Spheres 116 

Oontents of Pipes and Cylinders, Cubic Feet and Gallons i:ao 

Cylindrical Vessels, Tanks, Cisterns, etc 181 

Oallons in a Number of Cubic Feet 1«3 

Cubic Feet in a Number of Gallons 123 

Square Feet iit Plates-S to 88 feet long and 1 inch wide 128 

Capacities of Rectangular Tanks in Gallons 125 

Numberof Barrels in CyUndrical Cisterns and Tanks 126 

Lof(arithms 1«7 

Table of LoRarltbms 129 

Hyperbolic J^iOgaritfams 156 

Natural Trigonometrical Functions 158 

Logarithmic Trigonometrical Functions ^ 168 

MATBKIAU. 

Chemical Elements 16& 

Specific Gravity and Weight of Materials 168 

Metals, ProperUes of 164 

The Hydrometer 165 

Aluminum 166 

Antimony 166 

Bismuth 166 

Cadmium 167 

Copper 167 

Gold 167 

Iridium 167 

Iron 167 

Lead 167 

Magnesium 168 

Manganese 168 

Mercury 168 

Nickel 168 

Platinum 168 

Silver 168 

Tin 168 

Zinc 168 

BUsoellaneoas Materials. 

Order of Malleabllitj, etc., of Metals 169 

FormulQ and Table for Calculating Weight of Bods, Plates, etc 169 

Measures and Weighu of Various Materials 169 

Commercial Sixes of Iron Bars 170 

Weights of Iron Bars. 171 

of Flat Rolled Iron ITS 

of Iron and Steel Sheets. 174 

of Plate Iron 175 

of Steel Blooms 17B 

of Structural Shapes 177 

Sizes and Weights of Carnegie Deck Beams 177 

^ '* Steel Channels 178 

" " ZBars 178 

" ** Penooyd Steel Angles 179 

u u u Tees lao 

** ** ** Channels 1«^ 

•♦ •• Roofing Materials 181 

" " Terra-cotta. 181 

" •• Tiles 181 

•• " Tin Plates 18i 

" •• Slates 18S 

** ** PineSblngles 189 

•• *' Skv.light Glass 184 

Weights of Various Rooi-coverlngs 1 8i 

** Cast-iron Pipes or Columns IKS 

l«ft. lengths 188 

-fltUngs. iset 

er and Gas-pipe 188 

" and thickness of Cast-iron Pipes ]8i 

Safe Pressures on Cast Iron Pipe 180 



*• PIpe-l 
'* Watei 



COKTElTTfl. IX 

PAGK 

Sbeetriron Hydraulic Pipe 101 

SUuidard Pipe Flanges 192 

Pipe Flanges and Cast-iron Pipe 198 

Standard Sises of Wrought-iron Pipe IM 

Wrottght-iron Welded Tubes 196 

RiTeted Iron Pipes 197 

Weight of Iron for Riveted Pipe 197 

Spiral BtTeted Pipe 198 

Seamless Brass Tubing 198, 199 

Coiled Pipes 199 

Braas^ Copper, and Zinc Tubing 200 

Lead and Tin-lined Lead Pipe SOI 

Weisrht <tf Copper and Brass Wire and Plates SOS 

^ Round Bolt Copper 208 

" Sheet and Bar Brass 208 

Composition of Rolled Brass 903 

Sizesof Shot 204 

Screw-thread, U. B. Standard 904 

Umit-gauges for Screw-threads 206 

Siaeoflron for Standard Bolts 206 

Sixes of Screw-threads for Bolts and Taps 207 

Set Screws and Tap Screws 208 

Standard Machine Screws 209 

Siaea and Weights of Nuts 209 

Weight of Bolts with Heads 210 

TnuSc Bolts 810 

W^eigfata of Nuts and Bolt-heads ^ 211 

Rivets 211 

Shoes of Tnmbuckles 211 

Washers 212 

Track Spikes 212 

RaawaySpikes 212 

BoatSplkes 212 

Wrought Spikes 213 

Wire Spikes 218 

CntNalfito 213 

Wire Nails ,. 214, 216 

Iroo Wire, Size, Strength, etc 216 

Galvanized Iron Telegraph Wire 217 

Tests of Telegraph Wire 217 

Copper Wire Table, B . W. Gauge 218 

^'^ '• " Edison or Circular Mil Gauge... 219 

•• " *• B.&S.Gauge 230 

losolated Wire 221 

Copper Telegraph Wire 221 

EJectricCables 221,222 

Galvanised Steel-wire Strand 223 

Steel-wire Cables for Vessels 223 

BpeeUcacions for Galvanized Iron Wire 224 

Strength of Piano Wire 884 

Ploiqcfa-steel Wire 224 

Wires of differen t metals 225 

Spedflcations for Copper Wire 226 

Cable-traction Ropes 226 

Wire Ropes 226, 227 

Floagrbsteel Ropes 227, 228 

Galvanized Iron Wire Rope 228 

Steel Hawsers 223,229 

Flat Wire Ropes 2-4;9 

Galvanized Steel Cables 230 

Streng^th of Chains and Ropes .. . 230 

Notes OB use of Wire Rope 281 

Locked Wire Rope 231 

Crane Chains 232 

Weights of Logs, Lumber, etc 232 

Siaea of Fh« Brick 288 

Fire Claj, Analysia ■ ... 284 

k Bricks 285 

286 



3t C0KTRKT8. 

Strength of Material!. 

^ PAOK 

Htreiid and Strain » ». ,.,., ^m 

EImUc Ltiiiit »,,., ,. j96 

Yield Point , ».... «? 

Modulus of Elasticity ,887 

Reiiillence ...*... »..».. 888 

Elastic Limit and Ultimate Stress ».», S38 

Repeated Stresses , ,». $18 

Hft«peat4*d Sli<K!ks ». 240 

Stresses due to Sudden Shocks , 841 

IncreasIuK Tensile Strength of Bars by Twisting » iM t 

Tensile Strength .» ,.»» »....» d44 

Measurement of Elongation » » S4^ 

Shapes of Test Specimens »...» w....k...» 843 

Ck>inpres8ive Strength ,» »,» , » 844 

Columns, Pillars, or Struts 816 

Hodgkinson^s formula , , 846 

Gordon's Formula »,.,» »*.•.....,» 947 

Moment of Inertia i.»..i...» »..».»»..... 847 

JUdl us of Gyration » ♦ , ,.... 847 

Elements of Usual Sections , ».,.. 848 

Strength of Cast-iron Columns 860 
ransverse Strength of Cast Iron Water-pipe Sal 

Safe Load on Cast-iron Columns i 8&8 

fiirengthof Brackets on Cast-iron Columns 858 

£cceiitric IxMidint; of Columns v.tt *... SM 

Wrought-iron Columns SJ55 

Built Columns 896 

Pbceniz Columns 867 

Working Form ulffi for Struts »... Aw 

Merriman's Formula for Columns > SAO 

Working Strains in Bridge Members %.. ^i 

Working Stresses for Steel ^ 166 

Resistance of Hollow Ci'linders io Collapse 864 

OoUapsine Pressure of Tubes or Flues $6& 

Formula for Corrugated Furnaces 866 

Transverse Strength t 866 

Formulae for Flexure of Beams ...i........ 96? 

Safe Loads on Steel Beams ».....».. ... 6B0 

Elastic Resilience •..*».....►» 8^0 

Beams of Uniform Strength » » •• ^St 1 

Properties of Rolled Structural Shapes. v % 878 

*' " Steel 1 BeaniH • .. . 876 

Spacing^of Steel I Beams «. .. ATI 

Properties of St H(^l Channels », iit 

•• "TSliapes % TTS 

" *' Angles S^78<k 

" ** Z bars 860 

Bise of Beams for Floors » »..» 886 

Flooring Material.. 661 

TieRod.sfor Brick Arches ».. 66t 

Torsional Strength » *%,.».. 9Sk 

Elastic Resistance to Torsion 881 

Combined Stresses 6tt 

Stress due to Temperature ..*•.,. ».. 9i6 

Strength of Flat Plates 886 

Strength of Unstayed Flat Surfaces ».. 8R4 

Unbraced Heads of Boilers «......«.. 266 

Thickness of Flat Cast-iron Plates *.,. 866 

Strength (if Stayed Surfaces 36i 

Spherical Shells and Domed Heads » 866 

Stresses Ih Steel Plating under Water Pressure »....%...... 86^ 

Thick Hollow Cylinders under Tension 86( 

Thin Cylinders under Tension 866 

Hollow Copier Balls 968 

Holding Power of Nails, Spikes, Bolts, and Scree's 969 

Cut verwM Wire Nails 9B6 

Strength of Wrought-iron Bolts 996 



OOKTBITTS. xi 

PAGE 

IniHal Strain on BolU S9S 

SUnd Pipes and their Deflign 998 

Ittveced£eel Water-pipes 8» 

If ^ yi»y^^f\M T» m% Xubes •..••••••■•■•t* •«••••■«• •>.•••••••••■•• 8M 

DrkaidT's Tests oCMAterisls SM 

Castlroa S9G 

IronCastings • 297 

Iron Bars, Porsings, etc S97 

Steel Balis JUuTTires 806 

Steel Axles, Shafts, Sprinc Steel 899 

Riveted Jointe 999 

Welds ^.... aOO 

Copper, Brass, Brooxe, etc 900 

Wire, Wira-rope •• 801 

Ropes, Hemp, sad Cotton 901 

Betting, GanTas 908 

Stones, Briclc, Cement 908 

Tensile Strength of Wire 908 

Watertown listing-machine Tests 909 

Rireted Joints 90S 

Wroocht'lron Bsrs, CompresBion Tests .•.......,* 904 

Steell^e-bsn T. 90i 

Wroofffat-iron Colomns • •••..• ... 905 

Cold Drawn Steel 908 

Amerioan Wood» 909 

Sbeartns Strength of Iron and Steel 908 

Holding Power of Boilei^tubes 807 

Chains, Weight, Proof Ttot, etc 807 

Wrought-iron Chain Cables. 806 

8lz«i«thofGHnB.... 906 

Copper at HtRfa Temperatnres 908 

Screngtb of limber 909 

Expansion of Timber • , 811 

Sbeartns Strength of Woods 818 

StreagtB <tf Bnck, Stone, etc .....••.••..«... 918 

^^ - Flsgging 819 

« ** Ume and Cement Mortsr 818 

XodQH of Etasticity of Ysrious Materials 814 

FSctoiB of asfetf 814 

Prvpertlcs of Cork 818 

Vukanixed India-rubber • 816 

Xf lolith orWoodstone 816 

AlomlaaaEi, Properties and Uses .•* • 917 

AUoTs. 

AHofv of Copper and Tin, Bronae 918 

Copper and Zinc, Brass , 881 

Variation in Stnngth of Broaae 921 

Copper*Un-sinc Alloys... .».....• ...888 

Uqnatlon or Separation of Metals 8x8 

AflfoyaaoedinBrsasFoondries .... 986 

Copper-tttokel AiJora 886 

Copper-«ino4ron AJioyB • 8UI 

TbblaBronae aW 

FboqiAior Bronae.. 



Akimiaam Brass • 929 

CsuOoa as to Strength of Alloys 989 

AJumiaiim hardened 8S0 

ABoys of Aluminum. Silicon, and Iron • 380 

Tttncsten-alumiaum Allogfs. 881 

Alumlaum-tin Alloys., 881 

Mangaoeae Alloys 981 

Msuiisnenn Bronae. • 981 

German SilTer ,888 

ABoysof Bismnth , 889 

rmoble Alloys 888 

~ - rHetalAllc^ys. 818 



Ill GOKTEKTS. 

PAOfl 

Alloys contatniDg Antlmonj 890 

White-metal AIlojs , 884 

Type-metal Uj 

Babbitt metals. 881 

Solders SS 

Ropes and Chains. 

Strength of Hemp, Iron, and Steel Ropes 33fi 

FlatBopes * , 83( 

WorkioK Load of Ropes and Chains 8:)fi 

Strength of Ropes and Chain Cables 34C 

Rope xor Hoisting or Transmission 3K 

Cordage, Technical terms of 84] 

Splicing of Ropes »4] 

Coal iioisting 84;! 

HanllaCordage. Weight, etc.. 844 

Knots, how to make 844 

Splicing Wh« Ropes 341 

Springs. 

Laminated Steel Springs 84'a 

Helical Steel Springs 34^ 

Carrying Capacity of Springs 84S 

Elliptical Springs 852 

Fhosphor-bronse Springs « 86:^ 

Springs to Resist Torsional Force 85^ 

Helical Springs for Cars, etc 859 

Riveted Joints. 

Fairbalm*s Experiments 854 

Loss of Strength by Punching 854 

Strength of Perforated Plates .....854 

Hand vs. Hydraulic Riveting 85S 

FormulsB for Pitch of Rivets 35*3 

Proportions of Joints 35C 

Efficiencies of Join ts 85fl 

Diameter of Rivetfl 860 

Strength of Riveted Joints •. 361 

Riveting Pressures 869 

Shearing Resistance of Rivet Iron 863 

Iron and Steel. 

Classification of Iron and Steel 884 

Grading of Pig Iron 365 

Influence of Silicon Sulphur, Phos. and Mn on Cast Iron 86fi 

Tests of Cast Iron 809 

Chemistiy of Foundiy Iron 870 

Analyses of Castings 87^3 

Strength of Cast Iron 874 

Speclflcations for Cast Iron 874 

Mixture of Cast Iron with Steel 87n 

Bessemerised Cast Iron 873 

Bad Cast Iron 875 

Malleable Cast Iron STB 

Wrought Iron 877 

CSieinistryof Wrought Iron 877 

Influence of Rolling on Wrought Iron 877 

Specifications for wrought Iron 87^ 

Stay-bolt Iron 879 

FormulsB for Unit Strains In Structures 379 

Permissible Stresses in Structures 881 

Proportioning Materials in Memphis Bridge 88:1 

Tenacity of Iron at High Temperatures SSI 

BJfect of Cold on Strength of Iron S8t 

Expansion of Iron by Heat 885 

Durability of Cast Iron 885 

Corrosion of Iron and Steel 886 

_ Preservative Coailugs; Paints, etc 887 



coNTBKra xiii 

PAOS 

Pou-oxidizinK Prooen of Annealing: 887 

Ibuigaiieae FlaUng of Iron 889 

8to«I. 

BBlatton between Chem. and Phys. PropertiM • - 889 

Variation In Strength 891 

Open-hearth 809 



Banleninff Soft Steel 808 

Effect of Cold BoUing 888 

Oomparlaott of FnlMaed and 8m«n Pleoea 8B8 

Treataoent of Structural Steel 8M 

Inflnenoe of Annealing upon Magnetic Oapadtj 8B8 

BpecUleatlona for Steel 807 

Boiler. Ship and Tank Flatea 800 

Steel for Springa, Azlea. etc 400 

XajOartwn be Burned out of Sterir 408 

Becaleaceficcof Steel '. 408 

Effectof NIckingaBar 408 

Bleetite ConductiTlt/ 408 

Spedfio QruriXj 408 



408 

Begregation In ] 

Earliest Usee for^tructuroa 406 

Steel CaatingB 406 

ManganeaeSted 407 

Hk:kel Steel 407 

Aluminnm Steel 400 

Chrome Steel 409 

Tnngaten Steel 400 

OompreaBed Steel 410 

CrudUe Steel 410 

Effect of Heat on Grain 41S 

» •• Hammering, eto 412 

Heating and Forging 419 

Ttanpering Steel 4ia 

MXOHANICS. 

Force, Unit of Foroe 411^ 

Inertia 415 

Mewton^aLawBof Motion 416 

Beaofaitlon of Forcea 415 

Parallelogram of Forces 410 

Moment of a Force. 410 

Statical Moment, StabiUty 417 

StabOItT of a Dam 417 

PandlelForcee 417 

Oooples 418 

EqniHhrlmnof Forces 418 

Oentre of OravitT 418 

rinerOa., 



fc of Inertia 419 

Oentre of Oyiution • 490 

Badlna of Gyration 490 

Centre of OacOlation 491 

Centre of Percuaalon 499 

The Pendulum 498 

Cboieal Pendulum 498 

Centrifugal Force 498 

Aooelmlion.; 498 

FaUlngBodiea 494 

Value of p. 494 

Angular veloci^ • 496 

Height doe to Velodiy 496 

Paraflelogram of Velocitiea 498 

MaflB 497 

Force of Acceleration 497 

Motion on InclhiedPliinea. 498 



:X1V CONTEKT& 

VteVIva 498 

Work, Foot-pound 4S8 

Power, Horse-power 429 

Energy 429 

Work of Acceleration 430 

Force of a Blow , 480 

Impact of Bodies 481 

Bnengy of Becoil of Guns 4S1 

Oonsenratlon of Energy 488 

Perpetual Motion 4^18 

Sfllciencyof aMachlne 488 

Aslmal-power, Man-power 488 

WorkofaHorse 484 

Man-wheel 484 

Horse-sin 484 

Bastotance of Vehldee 485 

Elements of Maohlnes. 

The Lever 486 

The Bent Lerer 486 

The Moving Strut 486 

The Toffgle-joiot 488 

The Inclinea Plane 487 

The Wedge 487 

TheScrew 437 

The Cam 488 

ThePulley 488 

Differontial PuUej iSS> 

DUTerential WlndJasB 489 

DifferenUal Screw 48C' 

WheelandAxle 489 

Toothed-wheel Gearing 489 

IMlesB Screw 4M 

Stresses In Fran&ed Struotures. 

Cranes and Derricks 440 

Shear Poles and Guys 448 

King Post Truss or Bridge. 448 

Queen Post Truss 448 

Burr Truss 44S 

Pratt or Whipple Truss 44S 

HoweTmss 445 

Warren Girder 445 

Roof Truss ; 4^ 

HBAT. 

Thermometeiv and Pyrometers 448 

Centigrade and Fahrenheit degrees compared 449 

Copper-ball Pyrometer 461 

Thermo*eleotnc FVrometer 451 

Tdniperatures in Furnaces 451 

Wlborgh Air pyrometer 458 

Seegers Fire-Clay Pyrometer 458 

Mesur6and Nouel's Pyrometer 468 

Uehling and BteinbarCs Pyrometer 468 

Air-thermometer , 454 

High Temperatures Judged by Color 454 

BotliDg-polnts of Substances 465 

Melting-points 456 

0nftof Heat 465 

Mechanical Equivalent of Heat 486 

Heat of Combustion 466 

Specific Heat 457 

Latent Heat of Fusion 459»461 

Expansion by Heat 468 

Absolute Temperature 461 

Absolute Zero 461 



OOSTTEiriBi XV 

PAGS 

Latent HeAt 461 

Latent Heat of Evaporation 4«d 

Total Heat of Evaporation 408 

Evaporation and Drying.. 408 

Evaporation from Beaervoirs 408 

Evaporation by the Kultlple System 468 

BeElstance to B<riling 468 

Manuf actui« of Salt 404 

Solubility of Salt and Sulphate of lime 404 

Salt CootenU of Brines 404 

Ooncentratlon of Sugar Solutions 465 

Evaporating by Kxhaiist Steam 466 

Drying in vacuum 466 

Radiation of Heat ; 467 

Conduction and Convection of Heat ..468 

Rate of Eztamal Oondnotion 460 

Steam-pipe Coverings 470 

Transmission through Plates 471 

** in Condenser Tubes 478 

" " Cast-iron Plates. 474 

" from Air or Gases to Water 4T4 

•• from Steam or Hot Water to Air 47B 

** through Walls of Buildings 478 

Thennodynamics 478 

PHTSICAI* PBOFERTI£S OF OASES. 

Ezpaaalon of Oases 4T9 

Boyle and ICarriotte's Law 479 

Law of Charles, Avogadro*s Law 479 

Saturation Point of vapors 480 

Law of Gaseous Pressure 480 

Flow of Gaaes 480 

Absorption byLtqDidB 480 

AIB. 

Properties of Air 481 

Air-manometer 481 

Pressure at Different Altitudes 481 

Barometric Pressures. 488 

Levelling br the Barometer and by Boiling Water 482 

To find Difference in Altitude 488 

Moisture in Atmosphere 488 

Weight of Air and Mixtures of Air and Vapor 484 

Specific Heat of Ahr 484 

Flow of Air, 

Flow of Air through Orifloes 484 

Flow of Air in Pipes 486 

Effect of Bends in Pipe 488 

Flow of Compressed Air 488 

Tables of Flow of Air 489 

Anemometer Measurements 491 

Equalization of Pipes 491 

Loss of Fraasure in Pipes 498 

Wind. 

Force of the Wfaid 498 

Wind Pressure hi Storms 496 

Windmills 495 

Capacity of Wtodmills 497 

Economy of Windmills 498 

ElectricFOwer from Windmills 499 

Compressed Air. 

499 

499 

600 



Heating of Air hw Compression 

Loss of Energy m Compressed Air. 
Volomea aparresBures » 



X?i CONTElirTS, 

PiOl 

Lofls due to Ezoess of Pressure 601 

Horse^wer Required for Corapressloo 501 

Table for Adiabaiic Compression ^ fiOsS 

Mean Effective Pressures • ...608 

Mean and Terminal Pressures 608 

Air-compressors 608 

Practical Results 606 

Efficiency of Oompressed-air Engines 606 

Requirements of Rock-drills 1 606 

Popp Oompressed-air System 607 

Small Compressed-air Motors 607 

Efficiency of Air-heatioK Stoves 607 

Efficiency of Compressed-air Transmission 60^ 

Shops Operated by Compressed Air 609 

Pneumatic Postal Transmission 609 

Mekarski Compressed-air Tramways 610 

Compressed Air Working Pumps in Mines 611 

Fans and Blowers. 

Centrifugal ^ans Sll 

Best Proportions of Fans 513 

Pressure due to Velocity 618 

Experiments with Blowers 614 

Quantity of Air Delivered 514 

Kfflciency of Fans and Positive Blowers 516 

Capacity of Fans and Blowers 517 

Table of Centrifugal Fans 518 

Engines, Fans, and Steam-coils for the Blower System of Heating. 519 

Sturtevant Steel Pressure-blower 519 

Diameter of Blast-pipes •••,. 519 

Efficiency of Fans •....•.•••••.•• •• 6S0 

Centrifugal Ventilators for Mines S3I 

Experiments on Mine Ventilators 6S8 

DiskFans , 694 

Air Removed by Exhaust Wheel 625 

Efficiency of Disk Fans 585 

Positive Rotary Blowers 520 

Blowing Engines 528 

Steam-let Blowers 607 

Steam-Jet for Ventilation 627 

BEATING AND VENTILATION. 

Ventilation BBS 

Quantity of Air Discharged through a Ventilating Duct. 590 

Artificial Cooling of Air 581 

Mine-ventilation 681 

Friction of Air in Underground Passages 581 

Equivalent Orifices 588 

Relative Efficiency of Fans and Heated Chimneys 683 

Heating and Ventilating of Large Buildings 684 

Rules for Computing Radiating Surfaces 688 

Overhead Steam-pipes 687 

Indirect Heating-surface 587 

Boiler Heating-surface Required 588 

Proportion of Grate-surface to Radiator-surface 538 

Steam-consumption in Car-heating 588 

Diameters of Steam Supply Mains 588 

Registers and Cold-air Ducts 530 

Physical Properties of Steam and Condensed Water 540 

Size of Steam-pipes for Heating 510 

Heating a Greenhouse by Steam 541 

Heating a Greenhouse by Hot Water 543 

Hot-water Heating 642 

Law of Velocltv of Flow 54« 

Proportions of Radiating Surfaces to Cubic Capacities 543 

Diameter of Main and Branch Pipes 543 

Rules for Hot-water Heating 544 

Arrangements of Mains 544 



OOHTBNTa Xvii 

FAOK 

fitowerSystetn of Hefttlng and Ventilating... <...*... 645 

KzperimenU with Aadiatore • 645 

Heating a Building to 70» P 645 

Beating by Electricity 646 

WATBR. 

Expansion of Water 647 

Wdglitof Water at different temperatures 547 

Preesure of Water due to its Weignt 640 

Head (X^rresponding to Pressures 640 

BoiiWpotot*;./.;iy.!i!iiii'.!!;i;!;ii!!i;;;".!;!'/;^!;;;'.!!;/r.*'.!!!'/.;!" eeo 

Freabtg-point 550 

Sea-water 640,560 

loe and Snow 590 

Specific Heat of Water 550 

Compressibility of Water 651 

Impurities of water........ 551 

Causes of Incmstation ^ 661 

Means for FreTeoting Incrustation ,, GS2 

Analyses of Boiler-scato 563 

Hardoess of Water 658 

Purifying Feed-water 554 

Softening Hard Water 665 

Hydraulics. Flow of Water. 

Fomuks for Discharge through Orifices 655 

Flow of Water from Orifloes 656 

Flow in Open and Closed Channels 657 

General Forroulae for Flow 557 

Table Fall of Feet per mile, etc .. 668 

Valnesof Vr for Circular Pipes 560 

Xntter*s Formula 650 

Motosworth^s Formula 56S 

Basin *8 Formula 668 

IV Arpy*s Formula 568 

Older FormuliB 664 

Velocity of Water in Open Channels 664 

Mean, Surface and Bottom Velocities 664 

Safe Bottom and Mean Velocities 665 

Reristance of Soil to Erosion 665 

Abrading and Transporting Power of Water 566 

Grade of Sewers 666 

BelatiQus of Diameter of Pipe to Quantity discharged 660 

Flow of Water in a90-iochPipe 566 

Velocitlesin Smooth Cast-iron Water-pipes 567 

Table of Flow of Water in Circular Pipes 668-578 

Loasof Head 573 

Flow of Water in Riveted Pipes 574 

Frictioiial Heads at given rates of discharge 677 

Effect of Bend andC^irves 578 

HydraoUc Orade-Une 678 

Flow of Water in House-service Pipes 678 

Air-bound Pipes 570 

VerticaJJeta 570 

Water Delivered through Meters 670 

Fire Streams 570 

Friction lioeses in Hose 580 

Head and Pressure Losses by Friction 580 

Loss of Pressure in smooth i^inch Hose 580 

Bated capacity of Steam Fire-engines 680 

Pressures required to throw water through Nozzles 681 

The Siphon 681 

Measurement of Flowing ISsiCer 682 

Pfesometer 682 

PItot Tnbe Gauge 588 

TbeVentori Meter 688 

Measurement of Discharge by means of Nozzles 684 



XVlll COKTBHIS. 

flow through Bedaiigular Oriflo«i 864 

MeuureoMiit of AQ Open Stream 064 

Minora* Inch Measuromente • 06B 

Flow of Water over Wein 086 

Francises Fonnula for Weirs 586 

Weir Table 687 

Baain^sExperimenta............. ..• 667 

Water»powefw 

^werofalUlorWater 586 

Horse-power of a Running Stream >■ K8B 

Currant Motors 660 

Horse-power or Water Flowing in a Tube 560 

Maximum BfflclencT of a Long Conduit 668 

Mlllpower f. 580 

Value of Wato^power 580 

The Power of Ooean Wavee 590 

UUliatlon of Tidal Power 600 

• Turbine Wheels. 

Pmportlonsof Turbines • 601 

Teste of Turbines 600 

Dimensions of Turbines 607 

The Felton Watei^wheel.. -••»•.. 507 

Pumps. 

Theoretieal eapadty of a pump 601 

Depth of Suction 604 

▲mount 01 Water raised by a Single-acting Lift-pump. 606 

Proportioning the Steam-cylinder of a Direct^ictlng Pump 600 

Speedof Water through Pipes and Pump -passages 600 

Sues of Direct-acting rumps 608 

TheDeanePump 606 

RAoienoy of Small Pumps • •».• 606 

The Worfchington Duplex Pump 604 

Speed of Piston .......% 665 

Speed of Water through ValTes- • 605 

Boiler-feed Pumps... ».•.... • 605 

Pump Valves 666 

Centrifugal Pamps 606 

Lawreooe Centrirugai Pumps 607 

Bflloienoy of Centrifugal and Reciprocating Pumps 605 

Vanes of Centrifugal Pumps 605 

The Centrifugal Pump used as a Suction Dredge 600 

Duty Trials of Pumping Sngines 600 

Leakage Tests of Pumps ..•••« 611 

Vaouum Pnmps • 616 

ThePulfloraeter... 61i 

TheJetPump ...614 

TlHsInlcwtor.... •• 614 

Airlift Pump 6l4 

The Hydrauiie Ram 614 

Quantity of Water Delivered by the Hydraulic Ram 515 

Hydraulio Pressure Transmission. 

Energy of Water under Pressure •.. 615 

Efllcienoy of Apparatus 615 

Hydraulic Presses 617 

tdrauHc Power in London 617 
drauUc Riveting Machines * 618 
draulio Forging 618 
» Aiken IntensHler 610 

HydrauHo Bagiae tV% 



ruBi*. 



liwory of Combustion 

Total Heat of Combustion.. 



COKTENTS. XIX 

• 

TAQM 

AttUyMBofGwMorOoailRistlOB «•• • «» 

TempenLCore of the Fire • *•»•••• •.•» ttO 

Classiiication of Solid Fuel 633 

Classification of Goals..... 634 

Analyses of Coals 6M 

^'««teni Lifnites 6S1 

Analyses oi Foreign Coals 6Si 

Nixon^s Navigation Coal 688 

SampIinfijDoal for Analyses ,...»•...• 632 

Itektive Value of Fine Sixes t»t 

Fmsed Fuel 683 

BelaUve Value of Steam Coals 688 

Approximate Heating Value of Coals ».» • 684 

iCiod of Furnace Adapted for Different Coals 885 

Downward-draught Fiimaces.. **..•.. 635 

CAiorimetric Tests of American Coals 636 

EvapoFatlTe Power of Bituminous Coals. 686 

Weathering of Coal ,....» 637 

Coke 687 

Szperlmentsin Coking »..•, «•..».. 687 

Coal Wasblns'. TT. : 688 

RecoTery of By-products In Coke manufacture 688 

Making Hard Coke 638 

Cieoeration of Steam from the Waste Heat and Gases from Coke-ovens. 638 

Products of the DistiUatfon of Coal • 689 

Wood as Fuel . 680 

Heating Value of Wood 689 

Compoiitioii of Wood 640 

CbarT!Tial * 640 

Yield of CfaKTCoal (krom a Cord of Wood 641 

Cbosmnptfon of Charcoal In Blast Furnaces. 641 

Abaorpnon of Water and of Oases by Charcoal 641 

Cmnposftion of Charcoals 64« 

Misceilajieous Solid Fuels 642 

Dost-f uel— Dust EzpIosiottS 643 

Peat or Turf 648 

Sawdust as Fuel *.%... 648 

Horse-manure as Fnd , » 618 

Wet Tan-bark as Fuel 648 

Straw as Fuel • ....w... 648 

Bsgawwe aa Fuel In Sugar Manufacture 648 

Petroleum* 

PHMlueUof DistillaUon 646 

Lima Petrotoora «.... 046 

Value of Fetroteum as Fuel.. ,«..« 645 

Ofl«;.OoalasFtael 046 

Fuel Gas* 

C^rtxmOaa 646 

Anthracite Gas 6«7 

BttumfDons Gas • .. 647 

Water Oas 648 

Pitidaoer-gas from One Ton of Coal 6¥9 

5atural Oas in Ohio and Indiana 649 

0*mbustion of Producer-gas 660 

Use of aceam in Producers 680 

Gas Fuel for Small Furnaces.... 651 

lUaminatlng^ Gas* 

Coal-cas 661 

Water-gas 694 

Analywes of Wster-pn and Coal ^as 668 

(^loriflc ESquirafents of Constituents 654 

Efficiency of a Water-gas Plant 654 

<^)ace Required fbr a Water-cas Plant 656 

fee^^atoe of Mtimiwitlne-gas 666 



211 OONTEKTS. 

PlOB 

Alloys contalDiDg Antimony ....^ 880 

White-metal Alloys 886 



KtEbii 



bitt metals. 886 

Solders 888 

Ropes and Chains. 

Strength of Hemp, Iron, and Steel Ropes «... 338 

FlatRopes , 889 

WorkiDff Load of Ropes and CSialns 839 

Strength of Ropes and Chain Gables 340 

Rope lor Hoisting or Transmission 840 

Cordage, TechnloU terms of 341 

Splicing of Ropes 341 

Coal Hoistlog 843 

ManllaCordage, Weight, etc 344 

Knots, how to make 844 

Splicing Wire Ropes 846 

Springs. 

Laminated Steel Springs 847 

Helical Steel Springs 847 

Carrying Capacity of Springs 849 

EUipUcai Springs 858 

Phosphor-bronze Springs « 9B2 

Springs to Resist Torsional Force 888 

Helical Springs for Cars, etc 858 

Riveted Joints. 

Falrbalm*s Experiments 854 

Loss of Strength by Punching 854 

Strength of Perforated Plates 854 

Hand TS. Hydraulic Riveting 855 

FormulsB for Pitch of Rivets 857 

Proportions of Joints 858 

Efficiencies of Joints 859 

Diameter of RIvetH . . 380 

Strength of Riveted Joints •. 861 

Riveting Pressures 362 

Shearing Resistance of Rivet Iron 868 

Iron and Steel. 

Classification of Iron and Steel 864 

Grading of Pig Iron 865 

Influence of Silicon Sulphur, Pbos. and Mn on Cast Iron 865 

Tests of Cast Iron 860 

Chemistry of Foundry Iron 870 

Analyses of Castings 87S 

Strength of Cast Iron 874 

Specltlcations for Cast Iron 874 

Mixture of Cast Iron with Steel 875 

Bessemerixed Cast Iron 875 

Bad Cast Iron 875 

Malleable Cast Iron 875 

Wrought Iron 877 

Chemistry of Wrought Iron 877 

Influenceof Rolling on Wrought Iron 877 

Specifications for Wrought Iron 878 

Stay-bolt Iron 879 

FormuliB for Unit Strains In Structures 879 

Permissible Stresses In Structures 881 

Proportioning Materials in Memphis Bridge 888 

Tenacity of Iron at High Temperatures 881 

Effect of Cold on Strength of Iron 881 

Expansion of Iron by Heat 885 

Durability of Cast Iron 885 

Corrosion of Iron and Steel 886 

, Preservative Coatings; Paints, etc 887 



COKTENTa Xlll 

PAOB 

Kou-ozidizinff Prooeas of AnDealiner 887 

SUngaoeae Putting of Iron 889 

Steel. 

fielAlion between CShem. and Phyi. FroportiM •. 889 

Variation in Strength 891 

Opeo*beart]i • 809 

** WT...... *'*^ 

ingSofI 

Effect of Cold RoUIni 



HaideDing Soft Steel 888 

Effect of Cold RoUIng 898 

Compartaon of FuU-elaed and Small Pfeoea , 



nent of Structural Steel 804 

Influence of Annealing upon Magnetks Capacity 808 

SpecifleayonsforStefa 807 

Boiler, Ship and Tank Plates 880 

Steel for Springs, Axles, etc 400 

May Carbon be Burned out of Stesl7 40S 

Becaleeoenceof Steel '. 408 

Bffectof Nlddngafiar 408 

Electric ConductiTity 408 

Specific QzaFitr 408 

Occaalonal Failures 408 

Segregation in In«>ts 404 

Earliest Uses forStmctures .. 406 

Steel Castings 406 

Manganese Steel 407 

Kickel Steel 407 

AlumiBum Steel 400 

Chrome Steel 400 

Tungsten Steel 400 

Compressed Steel 410 

Cmdble Steel 410 

Effect of Heat on Grain 418 

'" '* Hammering,eto 412 

Heating and Forging 418 

Tempering Steel 418 

MXGHANICS. 

Force. Unit of Foroe 411^ 

Inertia 415 

Newt<m*s Laws of Motion 416 

Besolutlon of Forces 415 

Parallelogram of Forces 418 

Moment of a Force 418 

Statical Moment, StabOity 417 

Stability of a Dam 417 

ParaUelForoes 417 

Couples 418 

Equilibrium of Forces 418 

Centre of OraTity 418 

Moment of Inerda 410 

Centre of Qyration 420 

Badlos of Oyratfon 480 

Centre of escalation 481 

Centre of Percussion.. 4S8 

The Pendulum 488 

Conical Pendulum 488 

Centrifugal Foroe 488 

Acoeleratlon.'. 488 

Falling Bodies 484 

Value ofo. 484 

Angular Velodty 485 

Height due to Velocity 485 

Parallelogram of Velocities 486 

Mass 487 

Foroe of Acceleration 427 

Motion on Inclined Planes. 488 

Momeotmu , 488 



211 CONTEKTS. 

PlOB 

Alloys containing Antimony , 880 

White-metal Alloyi 886 

TJrpe-metal 888 

Babbitt metals. 888 

Solders 888 

Bopes and Chains. 

Strength of Hemp, Iron, and Steel Ropes 838 

FlatRopes • , 880 

WorkiDff Load of Ropes and Chains 8:)9 

Strength of Ropes and Chain Gables 340 

Rope lor Hoisting or Transmission 340 

Cordage, Technical terms of 841 

Splicing of Ropes 341 

Coal Hoisting 843 

Manila Cordage. Weight, etc....... 344 

Knots, bow to make 814 

Splicing Wire Ropes 346 

Springs. 

Laminated Steel Springs 847 

Helical Steel Springs 847 

Carrying Capacity of Springs 849 

Elliptical Springs .. 8a8 

Phosphor-bronze Springs « 853 

Springs to Resist lx>rsiona1 Force SSS 

Helical Springs for Cars, etc 858 

Riveted Joints. 

Falrbalm*s Experiments 854 

Loss of Strength by Punching 854 

Strength of Perforated Plates 854 

Hand vs. Hydraulic Riveting 855 

FormulsB for Pitch of Rivets 857 

Proportions of Joints 358 

Efficiencies of Joints 850 

Diameterof Rivets 360 

Strength of Riveted Joints •. 361 

Riveting Pressures 868 

Shearing Resistance of Rivet Iron 868 

Iron and Steel. 

Classlflcation of Iron and Steel 864 

Grading of Pig Iron 865 

Influence of Silicon Sulphur, Phos. and Hn on Cast Iron 865 

Tests of Cast Iron 869 

Chemistry of Foundiy Iron 870 

Analyses of Castings 87S 

Strength of Cast Iron 874 

Bpeclflcations for Cast Iron 874 

Mixture of Cast Iron with Steel 87S 

Bessemerixed Cast Iron 875 

Bad Cast Iron 875 

Malleable Cast Iron 878 

Wrought Iron 877 

Chemistry of Wrought Iron 877 

Influenceof Rolling on Wrought Iron 877 

Speciflcat ions for Wrought Iron 878 

Stay-bolt Iron 879 

FormuUe for Unit Strains in Structures STB 

Permissible Stresses in Structures 881 

Proportioning Materials in Memphis Bridge 888 

Tenacity of Iron at High Temperatures 89 

Effect of Cold on Strength of Iron 888 

Expansion of Iron by Heat 885 

Burabillty of Cast Iron 885 

Corrosion of Iron and Steel 386 

, Preservative Coatings; Paints, etc 887 



COHMKTa Xlii 

PAOB 

ITou-oxidisiiiff Prooeas of ADneallnc 867 

Kanganeae Pliating of Iron 889 

Steel. 

BbIaUou between CShem. and PhyB. FropertiM - 889 

Variation in Strength 891 

Open-bearth 8» 

ingSoH 
rOoldl 



Hardening Soft Steel 888 

ESectdfOoldBoUing 808 

Comparison of FuU^ued and Small Pieoea . 



Treatment of Stmctural Steel 804 

Inflnenoe of Annealing upon Magnetic Oapadtj 886 

Specifksatfons f or StefQ 807 

Boiler. Ship and Tank Plates 899 

Steel for torings. Axles, etc 400 

May Oartwn be Burned out of Stesir 408 

Recalflsoenceof Steel '. 408 

Bffectof Nlckingafiar 408 

Electrio ConducdTity 408 

SpecsUlo OzafitT 408 

Oocasioiial FalTures 408 

Segregation in Ingots 404 

Barliest Uses forStructures 406 

Steel Castings 406 

Manganese Steel 407 

Nickel Steel 407 

Aluminum Steel 409 

Chrome Steel 409 

Tungsten Steel 409 

Compressed Steel 410 

Crucible Steel 410 

Effect of Heat OD Grain 418 

** ** Hammering, etc 418 

Heating and Fbrging 418 

Tempering Steel 418 

MBCHANICS. 

Force. Unit of Foroe 4tS 

Inertia 415 

Mewt<m*s Laws of Motion 415 

Resolution of Forces 415 

Parallelogram of Forces 416 

Moment of a Force 416 

Statical Moment, StablUty 417 

StabUity of a Dam 417 

PanillelForoes 417 

Couples 418 

Equilibrium of Forces 418 

Centre of Gravity 418 

Moment of InerUa 419 

Centre of Gyration 420 

Badios of Gyration 490 

Centreof escalation 481 

Centre of Percussion... 4S8 

The Pendulum 488 

Conical Pendulum 488 

Centrifugal Foroe 488 

AooelerMion.*. 488 

Falling Bodies 484 

Value ofo , 494 

Angular Velocity 485 

Height due to Velocity 485 

Paral]ek>gram of Velocities 486 

Mass 487 

Force of Acceleration 487 

Motion on Inclined Planes. 488 

Momwitnm , 488 



."XIY COKTEKTS. 

Vie Viva 4S 

Work, Foot-pound 49B 

Power, Horse-power 499 

Energy 429 

Work of Acceleration 480 

Force of a Blow 4ao 

Impact of Bodies 4S1 

Energy of Recoil of Guns 431 

Oonsenratlon of Energy 498 

Perpetual Motion 4»i 

SIBclencyof a Machine 48S 

Animal-power, Man-power 4Si 

Workof aHorse 484 

Man-wheel 484 

Horse-gin 484 

Resistance of Vehicles 48S 

Elements of Machines. 

The Lever 485 

The Bent Lever 436 

The Moving Strut 486 

The Toarie-Jolnt 486 

The Inclinea Plane 487 

The Wedge 487 

TheScrew 487 

The Cam 488 

ThePulley 488 

Differential PuUey 481) 

Differential Windlass 489 

Differential Screw 48(' 

WheelandAxle 489 

Toothed-wheel Gearing 489 

ftidless Screw 440 

Stresses In Framed Struotures. 

Cranes and Derricks 440 

Shear Poles and Guys 449 

King Post Truss or Bridge 449 

Queen Post Truss 449 

Surr Truss 448 

Pratt or Whipple Truss 448 

HoweTruss 446 

Warren Girder 445 

Roof Truss ; 446 

HEAT. 

Thermometers and Pyrometers 448 

Centigrade and Fahrenheit degrees compared 449 

Copper-ball Pyrometer 461 

Thermo-eleotno Pvrometer 451 

Temperatures in Furnaces 461 

Wlborgh Air Pvrometer 458 

Beegers Fire-clay Pyrometer 468 

Mesur^and Nouel's Pvrometer 468 

Uebling and Steinbart^s Pyrometer 458 

Air-thermometer 454 

High Temperatures judged by Color 464 

BolllDg-points of Substances 456 

MeUing-points 4U 

TTnitofHeat 406 

Mechanical Equivalent of Heat 496 

Heat of Combustion 466 

Speclflc Heat 497 

Latent Heat of Fusion 499,461 

Expansion by Heat 460 

Abnolute Temperature 461 

Absolute Zero 461 



OOKTEinSi XY 

PAOB 

Latent Heat 461 

Latent Heat of Evaporation 40^ 

Total Heat of Evanoration 4d8 

Evaporation and Drying: 4M 

Evaporation from Reservoirs 468 

Evaporation by the Multiple System 468 

Besistanoe to Boiling 468 

Manufacture of Salt 464 

SolubUity of Salt and Sulphate of lime 464 

Salt Contents of Brines 464 

Conoentratlon of Sugar Solutions 466 

Evaporating by Exhaust Steam 466 

Drying in vacuum 466 

Radiation of Heat .' 467 

Ck>nduction and Convection of Heat 468 

Rate of External ConduoUon 46Q 

Steam-pipe Coverings 4T0 

Transmission through Plates 471 

** in Condenser Tubes 478 

•* ** Cast-Iron Plates 474 

" from Air or Qaaes to Water 4T4 

•* from Steam or Hot Water to Air 476 

" through Walls of Buildings 478 

niermodynamics 478 

PHTSICAI^ PBOPERTIES OF OASES. 

Expansion of Gases 479 

Boyle and Marriotte's Lavr 479 

Law of Charles, Avogadro's Law 479 

Saturation Point of vapors 480 

Law of Osseous Pressure 480 

Flow of Oases 480 

Absorption byLiqalds 480 

AIB. 

Properties of Air 481 

Air-manometer 481 

Pressure at Different Altitudes 481 

Barometric Pressures 482 

Levelling by the Barometer and by Boiling Water 482 

To find Difierenoe In A I tltude 483 

Moisture In Atmosphere 488 

Weight of Air and Mixtures of Air and Vapor 484 

Specillc Heat of Air 484 

Flow of Air* 

Flow of Air through Orifices 484 

Flow of Air in Pipes 485 

Effect of Bends in Pipe 488 

Flow of Compressed Air 488 

Tables of Flow of Air 489 

Anemometer Measurements 491 

Equalization of Pipes 491 

Loss of Pressure in Pipes 498 

Wind. 

Force of the Wind 498 

Wind Pressure in Storms 495 

Windmills 495 

Capacity of Wind mills 497 

Economy of Windmills 496 

ElectrlcPower from Windmills 499 

Compressed Air. 

Heating of Air bj Compression 499 

Loss of Energy m Compressed Air 499 

Volumes and rressures , , 609 



lioos due to 11x0688 of Pressure ooi 

HorseHpower Required for Compression 601 

Table for Adiabatic CompressiOD i 5M 

Mean EffectWe Pressures 601 

Mean and Terminal Pressures 608 

Alr-oompressors 608 

Practical Results 6QS 

Efficiency of Compressed-air Engines 606 

Requirements of Rock-drills .,.', 606 

Popp Compressed-air System 607 

Small Compressed-air Motors 607 

Efficiency of Air-heating Stoves 607 

Efficiency of Compreesed-air Transmission 60f< 

Shops Operated by Compressed Air 600 

Pneumatic Postal Transmission 609 

Mekarski Compressed-air Tramways 610 

Compressed Air Working Pumps in Mines 611 

Fans and Blowers. 

Oentrifugal ffans 611 

Best Proportions of Fans 619 

Pressure due to Velocity &1S 

Experiments with Blowers 614 

Quantity of Air Delivered 614 

£fflciency of Fans and Positive Blowers 516 

Capacity of Fans and Blowers 517 

Table of Centrifugal Fans 618 

Engines, Fans, and Steam-coils for the Blower System of Heating. 610 

Sturtevant Steel Pressure-blower 610 

Diameter of Blast-pipes 619 

Efficiency of Fans 690 

Centrifugal Ventilators for Mines 621 

Experiments on Mine Ventilators S8i 

DiskFans B«4 

Air BemoTed by Exhaust Wheel 686 

Efficiency of Disk Fans 685 

Positive Rotary Blowers 886 

Blowing Engines 686 

Steam-jet Blowers 687 

Steam-Jet for Ventilation 687 

BKATING AND TJ&NTIUiTION. 

Ventilation 888 

Quantity of Air Discharged through a Ventilating Duct. 580 

Artificial Cooling of Air 681 

Mine-ventilation 681 

Friction of Air in Underground Passages 6S1 

Equivalent Orifices 688 

Relative Efficiency of Fans and Heated Chimneys 688 

Heating and Ventilating of Large Buildings 584 

Rules for Computing Radiating Surfaces 686 

Overhead Steam-pipes 687 

Indirect Heating-surface 5S7 

Boiler Heating-surface Required 688 

Proportion of Grate-surface to Radiator-surface 588 

Steam-consumption in Car-heating 688 

Diameters of Steam Supply Mains 589 

Registers and Cold-air Ducts 639 

Physical Prop«>rties of Steam and Condensed Water 540 

Size of Steam-pipes for Heating 540 

Heating a Greenhouse by Steam 541 

Heating a Greenhouse by Hot Water 54S 

Hot-water Heating 548 

Law of Velocitv of Flow 648 

Proportions of Radiating Surfaces to Cubic Capacities 548 

Diameter of Mai n and Branch Pipes 548 

Rules for Hot-water Heating 544 

Arrangements of Mains 644 



OONTEKTa Xvii 

Blower System of Reattng and VeDtilatlng.. M5 

Ezperimentfl with Radiators « 545 

Heating a Buildinfr to TO* F 545 

HeaUng by Electricity 540 

WATKR. 

Expansion of Water 647 

Weight of Water at different temperatures 547 

Pressure of Water due to its Weight 549 

Head Corresponding to Pressures 549 

Buoyancy 550 

Boiling-point , 560 

Freeauig-point 650 

Sea^water 549,560 

loe and Snow 650 

Specille Heat of Water 550 

OompressibUity of Water 551 

Impurities of Water... 551 

Causes of Incrustation ^ 551 

Means for PreveDting Incrustation 7, 5Kt 

Analyses of Boiler-seale 559 

Hardnees of Water 558 

Purifying Feed-water 554 

Softening Hard Water 655 

Hydraulics. Flow of Water. 

Fomuto for Discharge through Orifioes ... 555 

Flow of Water from Orifices 555 

Flow In Open and Closed Channels 657 

G«ieral Fx>rmuln for Flow 557 

Table Fall of Feet per mile, etc 558 

Vahiesof Vr for Circular Pipes 659 

Kutter's FormuU 569 

Molesworth's Formula 663 

Bazin^s Formula .. 563 

D*Arcy*s Formula 568 

Older FormulflD 561 

Velodty of Water in Open Channels 564 

Mean, Surface and Bottom Velocities 664 

Safe Bottom and Mean Velocities 665 

Resistance of Soil to Erosion 565 

Abrading and Transporting Power of Water 665 

Grade of Sewers 660 

Relations of Diameter of Pipe to QuanUty discharged 660 

Flow of Water in aSO-inchPipe 666 

Velocities in Smooth Cast-iron Water-pipes 567 

Table of Flow of Water in Circular Pipes 66&-678 

Lossof Head 578 

Flow of Water in Riveted Pipes 574 

Frictional Heads at given rates of discharge 577 

Effect of Bend andCurres •. 678 

Hydraulic Orade-line 678 

Flow of Water in House-service Pipes 678 

Air-bound Pipes 679 

VerticalJets 679 

Water Delivered through Meters 679 

FireStreams 679 

Friction Losses In Hose 680 

Head and Pressure Losses by Friction 580 

Loss of Pressure in smooth 3^-inch Hose 580 

Rated capacity of Steam Fire-engines 580 

Pressures required to throw water through Nozzles 68t 

The Siphon 581 

Measurement of Flowing 'VaCiBr 682 

Piezometer .T 582 

Pitot Tnbe Gauge 588 

The Venturi Meter 688 

Measurement of Dischaige by means of Nozzles 684 



PAOC 

flow through Reotangular Oriflo«i » • »••»».• MM 

Meaauromeat of AQ Open Stream •..• 664 

Minora* Inch MeAsuremeats..... • fi6S 

Flow of Water oTor Wein •...•• 08S 

Francises Formula for Weirs 686 

Weir Table 687 

Baain^s Experimenta •%••....»... 6iSf 

Watei>powefw 

Power of a VMl of Water 586 

fiorae-power of a Runnlog Stream ., 58B 

Current Motors 680 

HorsO'powerof Water Flowing in a Tube... . 68D 

Maximum Efficiency of a Long Conduit 660 

MIH.power :. 680 

Value of Water-power 800 

The Power of Ocean Waves 690 

UUliatlon of Tidal Power 000 

• Turbine Wheela, 

Proportions of Turbines • 601 

Tests of Turbines « 600 

Dimensions of Turbines » * •. 607 

Tlie Pelton Water-wheel - 607 

PnmpB. 

Theoretical capacity of a pump • •....«..••••.• 601 

Depth of Suction 604 

▲mount 01 Water raised by a Single-acring Lift-pump. . OQS 

Proportioning the Steam cylinder of a Direct-acting Pump 600 

Speedof Water through Pipes and Pump -passages 608 

Sues of Direct-acting rumps 603 

TheDeanePump 000 

ftAoienoy of Small Pumps O06 

The Wonhington Duplex Pump 004 

Speed of Pisoon .. 006 

Speedof Water through ValTes.... • 005 

Boiler-feed Pumps.... ..*•. • • 006 

Pnmp Valves 006 

Centrifugal Pumps 006 

lAwreooe Centrifugal Pumps 007 

Effioleaoy of Centrif^al and Reciprocating Pumps 606 

Vanes of Centrifugal Pumps 600 

The Centrifugal Pump used as a Suction Dredge 600 

Duty Trials of Pumping Engines 600 

Leakage Tests of Pumps ,... Oil 

Vaouum Pnmps.... • 613 

ThePulsometer.... 610 

IlieJetPump • ••..... 614 

Thelnieotor ..»• 614 

AfrllftPump 614 

The Hydraulic Ram 614 

Quantity of Water Delivered by the Hydraulic Ram Wi 

Hydraulic Pressure Transmission* 

Bn^rgy of Water under Pressure ...... 616 

EAcieDcy of Apparatus 616 

Hydraulic Presses 617 

Hydraulic Power in London • 617 

Hjrdraulic Riyeting Machines • • 616 

mrdraulic Forging 616 

1%e Aiken IntensHier 619 

Hydraulic Engine 610 



FUBIto 



Theory of Combustion 

Total Heat of Combustion.. 



COKTEKTfl. XIX 

• 

9AQM 

AmaysBBof GwMofOooibasllOB •« ••.*• «M 

Temperatun of the Fire »• »»»•••» •» ttO 

Classiflcation of Solid Fuel 6S3 

Classification of Goals 634 

Analjs^ee of Coals ^ 

Wostern Lfenites 681 

Analyses of Foreign Coals 6S1 

Kixon^s Navigation Coal 632 

SampUneCoal for Analyses •.•»••• ..» ^ 

B«kti¥e value of Fine Sixes 6Stt 

Pressed Fuel 682 

Belative Value of Steam Coals *. 663 

Approximate Heating Value of Coals •... » 634 

£ind of Furnace Adapted for Different Coals 6S& 

DowDward-draugbtFurnaces«„ »..«.. 635 

Caiorimetric Tests of American Coals 636 

Evaporative Power of Bituminous Coals. • • 636 

Weathering of Coal • 697 

Coke 637 

£xperimenta In Coking «...*.. 637 

Coil WashinK 7. 1 m 

Recovery of By-products in Coke manufacture 638 

Making Hard Cok:e » 638 

Generation of Steam from the Waste Heat and Gases from Coke-ovens. 638 

Products of the Distmatfon of Goal 689 

Woodasf\iel . 630 

Heating Value of Wood 6S9 

CompDBftion of Wood 640 

Charcoal 6« 

Yield of Ch&rooal fiDm a Cord of Wood 641 

Consmnption of Charcoal fn Blast Furnaces. • 641 

Absorption of Water and of Oases bj Charcoal 641 

Composition of Charcoals 64)1 

Miftcellaneous Solid Fuels 642 

Dust-fuel— Dust Ezplosiom 642 

Py»torTurf • 64« 

Sawdust as Fuel ...». .* 648 

Hone-manure as Fuel 648 

Wet Tan-bark as Fuel 648 

Straw as Fuel 648 

Bsffasse as Fuel in JSugar Manufacture 648 

Petroleum* 

Pn»duct80f DiGUllatloQ 645 

Uma Petroleum... •. «...»» .•«••....••..»... 640 

Value of Petroleum as Fad ••••••%..«« ».. 64& 

Oatv^OoalAsFnel » 646 

Fuel Gas. 

OnrtKmOaB 646 

Anthracite Gas 617 

Bituminons Gas 647 

Water Gas 648 

Produeer^gas from One Ton of Coal 049 

Natural Gas in Ohio and Indiana 649 

Combustion of Producer^as..... 680 

Use of Steam In Producers » 6S0 

Gas Fuel for Small Furnaces 661 

lUuminatiag Gas» 

Ooal'gas • 661 

Water-gas 668 

Analyses of Water-gas and Coal f^ns 668 

Oaioriflc Equivalents of Constituents 664 

KflftcleDcy of a Water-gas Plant 664 

^pace Required for a Water-gas Plant 666 

fbel-^ahieot Uhiminiiring'gas 666 



211 OONTEKTS. 

PAOB 

Alloys contatniog Antimony ,., 386 

White-metal Alloyi SM 

Type-metAl 886 

Babbitt metahk 886 

Solders 888 

Bopes and Chains. 

Strensthof Hemp, Iron, and Steel Ropes 338 

FlatRopes , 839 

Workine Load of Ropes and Chains S39 

Streneth of Ropes and Chain Cables 840 

Rope for Hoisting or Transmission SiO 

Cordage, Technical terms of 841 

Spliclnff of Ropes 341 

Coal Hoisting 848 

Manila Cordage. Weight, etc.. S44 

Knots, bow to make 814 

Splicing Wire Ropes 346 

Springs. 

Laminated Steel Springs 847 

Helical Steel Springs 847 

Carrying Capacity of Springs 849 

Elliptical Springs ., 852 

Phoepbor>bronze Springs « 853 

Springs to Resist Torsional Force S5S 

Helical Springs for Cars, etc 858 

Riveted Joints. 

Falrbalm*s Experiments 854 

Loss of Strength by Punching 854 

Strength of Perforated Plates 8M 

Hand vs. Hydraulic Riveting 8S6 

Formuln for Pitch of Rivets 867 

Proportions of Joints 358 

Efficiencies of Joints 858 

Diameter of RivetR .360 

Strength of Riveted Joints *. 861 

Rlvetuig Pressures 862 

Shearing Resistance of Rivet Iron 863 

Iron and Steel. 

Classlfleation of Iron and Steel 884 

Grading of Pig Iron 865 

Influence of Silicon Sulphur, Phos. and Mn on Cast Iron 865 

Tests of Cast Iron 869 

Chemistry of Foundiy Iron 870 

Analyses of Castings 873 

Strength of Cast Iron 874 

Specifications for Cast Iron 874 

Mixture of Cast Iron with Steel 875 

Bessemerized Cast Iron 875 

Bad Cast Iron 875 

Malleable Cast Iron 875 

Wrought Iron 877 

Chemistry of Wrought Iron 877 

Influenceof Rolling on Wrought Iron 877 

Speciflcations for wrought Iron 878 

Stoy-boltlron 879 

Formulas for Unit Strains in Structures 879 

Permissible Stresses in Structures « 8Bi 

Proportioning Materials in Memphis Bridge 8B3 

Tenacity of Iron at High Temperatures 888 

Effect of Cold on Strength of Iron 888 

Expansion of Iron by Heat 885 

Durability of Cast Iron 885 

Corrosion of Iron and Steel S86 

, Preservative Coatings; Paints, etc 887 



CONTBKTa Xiii 

5ou-ozidisiiiK Process of Annealing 887 

Kaoganeae Plating of Iron 889 

Steel. 

Belatlon between C9iem. and Phys. Fropertiei - 889 

Variation In Strength 891 

OpeiFliearth 809 



Hardening Soft Steel 888 

Effect of Gold Boiling 888 

Compariaon of FuUrtiaed aiid Small Pieces 898 

Treatment of Structural Steel 894 



Influence of Anuealiog upon Magnetic OaBadty . , 

SpecUloatlona for Steel 897 

BoOer. Ship and Tank Plates 889 

Steel for Springs, Aztes. etc 400 

Kay GariKm be Burned out of Stesir 408 

Recaloecence of Steel '. 409 

Effeetof Niddngafiar 408 

Eleetrle Conductirity 408 

Bpedfie GzaTitr 408 

Oocaaioiial Failures 408 

Segregation in InwMs 404 

EsriicBtUses forStnictuies 406 

Steel Castings 406 

Manganese Sted 407 

Nickel Steel 407 

Alnralnnm Sted 409 

Chnnne Steel 409 

Toncrten Steel ......••.••••..•.. • 409 

Oompressed Steei'.'/.r.IlII.lIII..'.!...I..l..I.I...l ....!...!.!'.*.'..!!*.!!!! 410 

Onidbie Steel 410 

Bffect of Heat on Qraln 41S 

** '* Hammering, etc 412 

Heating and Forging 418 

Tteipering Steel 41» 

MXGHANICS. 

Foroe,Unitof Foroe 41l( 

Inertia ...„ 415 

Newton's Laws of Motion 415 

Resolution of Forces 415 

ParsIldogFam of Forces 418 

Moment of a Force. 418 

Statical Moment, StabiUty 417 

StabiiilTroC aDam 417 

PsraUd Forces 417 

Ooupi« 418 

Bquilibriam of Forces 418 

Centre of Orarity 418 

TInerUa.. 



b of Inertia 419 

Centre of Gyration • 480 

BadinsofCgrration 490 

Oentreof Oscillation 481 

Centre of Percussion... 488 

Tbe Bendolnm 428 

Conical Pendulum 488 

Centrifugal Foroe 488 

Acceleration.-. 488 

FaUing Bodies 494 

Value of o. 494 

Angular Velocity 485 

Height due to Velocity 485 

Parallelogram of Velocities 486 

MsmTTT 487 

Force of Acceleration. 487 

I on Inclined PlMies. 488 



J 



Ill OOKTEKTS. 



Alloys containing Antimony. , 

White-metal Alloys 

Type-metal 

Babbitt metals. 

Solders 



Ropes and Chains. 

Strength of Hemp, Iron, and Steel Ropes , 

FlatRopes * ., 

Working Load of Ropes and Chains 

Strength of Ropes and Chain Cables 

Rope for Hoisung or Transmission 

Cordage, Technical terms of 

Splicing of Ropes 

Coal Hoisting 

Manila Cordi^. Weight, etc.. 

Knots, bow to make 

Splicing Wire Ropes 



Springs. 

Laminated Steel Springs 

Helical Steel Springs 



Carrying Capacity of Springs. 
ElUptlcai r ^^ 



EUiptlcfU Springs 

Phosphor-bronze Springs 

Springs to Resist lx>r8ional Force. 
Helical Springs for Cars, etc 



Riveted Joints. 

Falrbalm*s Experiments 

Loss of Strength by Punching 

Strength of Perforated Plates 

Hand ts. Hydraulic Riveting 

FormulsB for Pitch of Rivets 

Proportions of Joints 

Efficiencies of Joints 

Diameter of Rivets . . 

Strength of Riveted Joints *. 

RlTetuig Pressures 

Shearing Resistance of Rivet Iron 



Iron and Steel. 



Classiflca tion of Iron and Steel 

Grading of Pig Iron 

Influence of Silicon Sulphur, Phos. and Mn on Cast Iron. . 

Tests of Cast Iron 

Chemistry of Foundiy Iron 

Analyses of Castings 

Strength of Cast Iron 

Specifications for Cast Iron 

Mixture of Cast Iron with Steel 

Bessemerized Cast Iron 

Bad Cast Iron 

Malleable Cast Iron 

WrotKht Iron 

Chemistry of Wrought Iron 

Influence of Rolling on Wroughtlron 

Speciflcations for wrought Iron 

Stay-bolt Iron 

FormulsB for Unit Strains in Structures 

Permissible Stresses in Structures 

Proportioning Materials in Memphis Bridge 

Tenacity of Iron at High Temperatures 

Effect of Cold on Strength of Iron 

Expansion of Iron by Heat 

Durability of Cast Iron 

Corrosion of Iron and Steel 

, Preservative Coatings; Paiots« etc 



COKTBKTa Xiii 

FAOB 

5ou-ozidiminf Process of Annealing 887 

Kaaganeee Plating of Iron 880 

Steel. 

BeUtion between Oiem. and Pbys. Fropeitiei •• 8811 

Variation in Strength 891 

Open-hearth 809 

BaseeniBr ••••••.•• ••••••••••>••••••••••• ••• 809 

Hardening Soft Steel 888 

UtetoCOoldRoUIng 808 

OompariBon of FuU-ebed and Smatt Fleoea 888 

Treatnunt of Structural Steel 804 

Infloenoe of Annealing upon Magnetlo Ctopadtj. , 



iforSteel 807 

Boiler, Ship and Tank Plates 800 

ated for Springs, Axlea. etc 4M 

XajOartionbeBuniedoutofStesir 408 

Recalesoenceof Steel *. 408 

Effectof NickingaBar 408 

Eleetrie Conductivltj 408 

Specillc QraritT 408 

Occasional Fkdmres 408 

Segregation in Ingpca 404 

~~ " > Usee f orStructurM ...406 



Steel Caatings 406 

KaaeaeSteel. 



407 

Nickel Steel 407 

Atuminum Steel 400 

Chrome Steel 400 

Tnngrten Steel 400 

OompreaBed Steel 410 

CmcSle Steel 410 

Effect of Heat on Grain 418 

** '• Hammering, etc 412 

Beating and Forgtatg 418 

Tempering Steel 4ia 

KBCHANIC8. 

roree,Unitof Faroe 41» 

Inertia 415 

Newton's Laws of Motion 416 

Beaolntion of Forces • 415 

Parallelogram of Forces 410 

Komentof aForoe. 410 

Statical Moment, Stability 417 

Stability oC aDam 417 

ParallelForcea 417 

Ooaples 418 

Eqoilibriimi of Forces 418 

Omtre of OraTity 418 

Moment of Inertia 419 

Gbntre of Oyratton • 480 

Badios of Gyration 480 

CentreoT OscOlation 481 

Oentre of Percussion 483 

The Fendolom 488 

Conical Pendulom 428 

Oentrifugnl Force 488 

Acceleration.-. 488 

FUOing Bodies 484 

Value of o. 494 

Angular Velocity 485 

Height doe to Velocity 485 

Parallelogram of Vek)citiea 480 

Mass 487 

Force of Acceleration 487 

Motion on Inclined Planes. 488 



J 



:X1Y C01<rTEKI& 

Vis Viva 4i8 

Work, Foot-pound 4S8 

Power, Horse-power 4S9 

Energy 4» 

Work of Acceleration 480 

Force of a Blow 490 

Impact of Bodies 491 

Energy of Beooil of Quns 4S1 

Ctonaerration of Eneiigy 4SS 

Perpetual If otion 4% 

BAelencyof a Machine 4Si 

▲nlmal-power, Man-power 43S 

Workof aHorse 434 

Man-wheel 484 

Horse-gin 484 

Resistance of Vehicles 435 

Slements of Machines. 

The Lever 485 

TheBentLever 436 

The Moving Strut 436 

The Toffgle-joiat 436 

The Inclinea Plane 487 

The Wedge 487 

TheScrew 487 

The Cam 488 

ThePulley 488 

DUrerentlal PuUey 431^ 

Differential Windlass 489 

Differential Screw 48(^ 

WheelandAxle 48» 

Toothed-wheel Gearing 488 

Endless Screw 440 

SftrMses In Framed Stmotures. 

Cranes and Derricks 440 

Shear Poles and Ouys 442 

Xing Post Truss or Bridge 44S 

Queen Post Truss 449 

Burr Truss 443 

Pratt or Whipple Truss 448 

HoweTruss 44S 

Warren Qlrder 445 

Boof Truss : 440 

HKAT. 

Thermometers and Pyrometers 448 

Centigrade and Fahrenheit degrees compared 449 

Copper^Hill Pvrometer 461 

Thermo-eleotno fvrometer 451 

Temperatures in Furnaces 461 

Wiboivh Air Pvrometer 458 

Beegers Fire-clay Pyrometer 45S 

Mesurd and NouePs Pvrometer 458 

Uehlfng and SteinbarOs Pyrometer 493 

Air-thermometer , 454 

High Temperatures judged by Color 454 

Boil ing-polnts of Substances 455 

Melting-points 4S5 

Unltof Heat 485 

Mechanical Equivalent of Heat 466 

Heat of Combustion 456 

Specific Heat 457 

Latent Heat of Fusion 450,461 

Expansion by Heat 460 

Absolute Temperature 461 

Absolute Zero •• 461 



00HTE2n& XV 

_ PAQB 

Latent Beat 461 

Latent Heat of Eraporatloii Mi 

Total Heat of ETaporation 468 

Zvaporation and Drying 468 

XTaporation from Reservoirs 468 

Evaporation by the Multiple System 468 

BesistAooe to Boiling 468 

Manufacture of Salt 464 

Solubility of Salt and Sulphate of lime .'. 464 

Salt Contenta of Brines 464 

Oonoentration of Sugar Solutions 466 

Evaoorating by Exhaust Steam 466 

Drying in vacuum 466 

SadlatioD of Heat .' 467 

Oondoctkm and Convection of Heat 468 

Rate of External Oonduotion 460 

Steam-pipe Coverings 470 

Transmission through Plates 471 

•• in Condenser Tubes 478 

•* •* Cast-iron Plates. 474 

" from Air or Gases to Water 474 

•* from Steam or Hot Water to Air 476 

•• through Walls of Buildings 478 

niermodjnamics 478 

PHT8ICAI« PKOPERTI£S OF GASBS. 

Expaufon of Oases 479 

Boyle and Maniotte^s Law 470 

Law of Charies, Avogadro's Law 479 

Saturation Point of vapors 460 

Law of Gaseous Pressure 480 

Flow of Gases 480 

Absoiptloii by Liquids 480 

AIR. 

Properties of Air 481 

Air-manometer 481 

Pressure at Different Altitudes 481 

Barometric Pressures 488 

Levelling by the Barometer and by Boiling Water 482 

To find Imferenoe in Altitude 488 

Moisture in Atmosphere 488 

Weight of Air and Mixtures of Air and Vapor 484 

^>ecillc Heat of AJr 484 

Flow of Alr» 

Flow of Air through Orifices 464 

Flow of Air in Pipes 485 

Effect of Bends in Pipe 488 

Flow of Compressed Air 488 

Tables of Flow of Air 489 

Anemometer Measurements 491 

EquaJizatioo of Pipes 49I 

Loss of Pressure in Pipes 498 

Wind. 

Foroe of the Wfaid 498 

Wind Pressure in Storms 496 

WindmOOtf 405 

Capacity of WlndmOIs 497 

Economy of Windmills 496 

Electric rower from Windmills 499 

Compressed Air. 

Heating of Air b^ Compression 409 

Loss of Energy in Compressed Air 499 

ToiuiQM and Pressures » .j «... 600 



xvi coiirrBNTs. 

PAOl 

L0B8 due to Ezoen of Pressure 60 

Horae^wer Required for Compression 50 

Table for Adiabatic Oompression » 50 

Mean Eftective Pressures 60 

Mean and Terminal Pressures 5(X 

Air-compressors 60i 

Practical Results 50: 

Efficiency of Compressed-air Engines. • 60< 

Requirements of Rock-drills 1 50< 

Popp Compressed-air System 60! 

Small Compressed-air Motors 50^ 

Efficiency of AiivheatinK Stoves 50^ 

Efficiency of Compressed-air Transmission 60^ 

Shops Operated by Compressed Air 601 

Pneumatic Postal Transmission 60i 

Mekarski Compressed-air Tramways 6U 

Compressed Air Working Pumps in Mines 611 

Fans and Blowers. 

Centrifugall^ans 611 

Best Proportions of Fans 61S 

Pressure due to Velocity 51S 

Experiments with Blowers 6H 

Quantity of Air Delivered bU 

Efficiency of Fans and Positive Blowers 516 

Cap^icy of Fans and Blowers 617 

Table 01 Centrifugal Fans 618 

Engines, Fans, and Steam-coils for the Blower System of Heating. 519 

Sturtevant Steel Pressure-blower 510 

Diameter of Blastrpipes 51S 

Efficiency of Fans SiX 

Centrifugal Ventilators for Mines 6S1 

Experiments on Mine Ventilators 622 

DiskFans 684 

Air Removed by Exhaust Wheel 685 

Efficiency of Disk Fans 529 

Positive Rotary Blowers 5M 

Blowing Engines 680 

Steamjet Blowers 523 

Steam-jet for Ventilation 6S? 

BEATING AND TEMTII«ATION. 

Ventilation 688 

Quantity of Air Discharged through a Ventilating Duct 580 

Artificial Cooling of Air 58t 

Mine-ventilation 681 

Friction of Air in Underground Passages 68 

Equivalent Orifices 58 

Relative Efficiency of Fans and Heated Chimneys 68 

Heating and Ventilating of Large Buildings 6» 

Rules for Computing Radiating Surfaces.... St 

Overhead Steam-pipes 51 

Indirect Heating-surface 51 

Boiler Heating-surface Required 61 

Proportion of Grate-surface to Radiator-surface 51 

Steam-consumption in Car-hsaiing 5( 

Diameters of Steam Supply Mains 51 

Registers and Cold-air Ducts 51 

Physical Properties of Steam and Condensed Water 64 

Size of Steam-pipes for Heating 61 

Heati ng a Greenhouse by Steam 54 

Heating a Greenhouse by Hot Water 51 

Hot-water Heating 54 

Law of Velocitv of Flow H 

Proportions of Radiating Surfaces to Cubic Capacities 51 

Diameter of Main and Branch Pipes 51 

Rules for Hot-water Heating 54 

Arrangements of Mains • 54< 



00KTSNT8. XVll 

^ PAOK 

BIcnrer System of Heating ana Ventilating « M5 

^qMriments with Iftadiators * 545 

HeatiDgaBuildinirto70*F 645 

Beating by ElectHci^ 640 

WATER. 

ExpansfoD of Water 547 

Weight of Water at different temperatureB 547 

PresBore of Water due to its Weight 549 

Head Oorresponding to Preasares 549 

Buojancy 590 

Bofmig-potet 660 

Freenbg-potnt 550 

Sea-water 540,560 

IceandSnow 550 

Specific Heat of Water 550 

CoRipreesIbilitT of Water 651 

Imparities of Water 551 

Causes of Incrustation. ^ 561 

Means for PreTenting Incrustation 7, 558 



AnalTses of Boiler-scale . . 
Hardness c 



lof Water 553 

Purifying Feed-water 554 

Bofteolng Hard Water 665 

Hydraulics. Flow of Water. 

FomubB for Discharge through Orifices 565 

Flow of Water from Orifices 555 

Flow in Open and Closed Channels 557 

General FbrmulsB for Flow 557 

Tsble Fall of Feet per mile, etc .. 568 

Taluesof Vr for Qrcular Pipes 550 

Kntter*s Formula 560 

Xolesworth's Formula 562 

Bszin*s Formula .. 568 

D'Ansy's Formula 668 

Older Formulas 564 

Velocity of Water in Open Channels 664 

Hean. Surface and Bottom Velocities 664 

Safe BoUom and Mean Velocities 665 

Resistance of Boil to Erosion 565 

AbradinfT and Transporting Power of Water 566 

Orsde orSewers 666 

Relati4?ns of Diameter of Pipe to Quantity discharged 566 

Flow of Water in a SO-inch i>ipe 666 

VelocitSes in Smooth Caat-iron Water-pipes 567 

Table of Flow of Water in Circular Pipes 666-578 

Lossof Head , 578 

Flow of Water in Riveted Pipes 574 

Frictiooal Heads at Kiven rates of discharge 577 

Effect of Bend and Curres 678 

Hydraulic Grade-line 578 

Flow of Water in House-serrioe Pipes 578 

Air-bound Pipes 579 

VerticalJeta 570 

Water DellTered through Meters 679 

FlreStreama 579 

Friction Losses in Hose 580 

Head and Pressure Losses by Friction 580 

Loss of Pressure in smooth ^inch Hose 580 

Rated capacity of Steam Fire-engines 580 

P r e ssures required to throw water through Nozzles 581 

TbeSlpbon 581 

Measurement of Flowfaig HMSer 58S 

Piezometer .T 582 

PItot Tube Gauge 5M 

TheVentttri Meter 6W 

Measorement of Discharge by means of Nozzles • 686 



'J 



XVlU C0KTBK18. 

PAQB 

flow through BeotangttlarOiifloei.... •..•.•••> 684 

Moasuremont of AD Open Btream fiBi 

Mlnera* iDoh MeMuremeots •.. 065 

Flow of Water OTor W«lra 085 

FrandB'B Fonnula for Wein 586 

WelrTkble 687 

Barings Experiments 5d7 

Water-powefv 

FowerofaFaUof Watdr 588 

Horse-power of a BunnlDg Btroam K8B 

Currsnt Motors 68B 

Horge-power of Wator Flowing In a Tube 688 

Maximum Efficiency of a Long Conduit 580 

Mill-power S89 

Value of Wate^power 590 

Ttie Power of Ocean Waves •.... 599 

UUltntlon of Tidal Power OOO 

• Turbltte Wheels, 

Proportions of Turbines » •«.. 601 

Tests of Turbines 606 

Dlaiensions of Turbines • %••«••• 607 

The Pelton Water-wheel - 607 

Pumps. 

Theoretical capacity of a pump • • • 601 

Depth of Suction 602 

Amount 01 Water raised by a Slngleaciing Lift-pump. 60el 

Proportioning the 8teamcylinder of a Direct-acting Pump 6(tt 

Bpeedof Water through Pipes and Pump -passages 603 

Sues of Direct-acting rumps 603 

The Deane Pump 603 

Bttoienoy of Small Pumps 60S 

The Worfchington Duplex Pump 604 

Speed ofPisMm 606 

Speed of Water through ValTes 606 

Bollerfeed Pumps » 605 

Pump Valves 608 

Centrifugal PHmpB 606 

Lawrence Centrifugal Pumps 607 

Bflddenor of Centrifugal and Reciprocating Pumps 608 

Vanes of Centrifugal Pumps 600 

The Centrifugal Pump used as a Suction Dredge 609 

Duty Trials of Pumping Engines 60fl 

Leakage Tests of Pumps ,*•. 611 

Vacuum Pnmpe.... 614 

ThePulsometer..*. 6U 

TlieJetPump »•...». 6U 

Tht) Injector »*•••• 6U 

Air-lift Pump 6i4 

The Hydraulic Ram • 614 

Quantity of Water DeUvered by the Hydraulic Ram...., 014 

Hydraulic Pressure Transmiission, 

Energy of Water under Pressure • 61< 

Efficiency of Apparatus ....» 6ll 

Hydraulic Presses 61' 

Hydraulic Power in London • 6V 

Hydraulic Riveting Machines * Cli 

I^drauiic Forslnff 4S1 

The Aiken Intensiner «1 

hydraulic Engine • 61 

FUSIi. 

tlieoryof Combustion • •! 

Total Heat of OombusUon • »«•..••••«•» . «l 



C0KTEKT8. XIX 

PAOI 

AmljwmclQBamfiiCoiBaibaMcm i tttt 

Tempermciire of the Fire • •... m 

(lasilication of Solid Fuel 038 

CtasificatioQ of Goals 634 

▲Mly:MS of Ck>al8 6ii4 

Western Lifrnites 681 

ABAljaesof Foreign Coate 6S1 

NixiMi^fl Navigation Coal • 682 

SamplineCoal for Analyses »..••• ».» 6S3 

Aektive value of line Siies 688 

Praased Fuel 692 

BalaUve Value of Steam Coals 688 

A|»proxiinate Heating Value of Coals ».., • ..• 6S4 

Kindof Furnace Adapted for DifFereotCoaJs. 685 

Down want-draught Furnaces »«..•.. 635 

C^lorimetric Tests of American Coals 636 

EvapormtJ^e Power of Bituminous Coals. • ».... 686 

Weatberine of Goal , 637 

Ooke 68? 

fizperlmentfi In OoUng • ••.•,•..••...».... • 687 

GoiiWashliiff. rr. ; 688 

Recovery of Br-products In Coke manufacture • 688 

Making Hard Coke 638 

G«oeFation of Steam from the Wasie Heat and Gases from Coke-ovens. 638 

Prodocfs of the Distillation of Coal 68i 

WoodasFdel 68D 

Hmting Yaine of Wood 689 

CompoeiUoB of Wood « 640 

Charc«nl 640 

Yield of dmrooftl flrrni a Oofd of Wood 641 

Ooosumntion of Charcoal In Blast Fumacea 641 

Absorption of Water and of Oases by Charcoal 641 

CnmpaaltioD of Charcoals 64l( 

MiAceJlaneous Solid Fuels 642 

Dust-foel— Dust ExplosiODS 642 

PfeatorTorf 648 

Sawdust as Fuel »....• • 648 

HorBe-manare as Foel 643 

WetTto-barkasFuel.... 648 

Straw as Fuel ....... .... • 648 

Bi«aase as PuM In Sugar Hanufacture 648 

Petroleum* 

ProdocUof DleUnatloa 646 

Lima Petroieora «»...* ..• 646 

Value of Petroleum as Fwl « 646 

0ilc!s.O0ttlttsFttel 646 

Fuel Gas. 

OrbooOas 646 

Anthracite Oas 647 

Bttumioons Gas • 647 

WsterCkuB 648 

PtTKiuoer-gas from One Ton of Coal 649 

5&tunU Oas In Ohio and Indiana 649 

Otmibuscion of Producer-gas 6M 

Ui« of Steam hi Producers 690 

(>as Fuel for Small Furnaces 661 

niuminatioir Gas« 

Qial.|:aai 681 

Wacer-««B 6B8 

Analri"BR of Water-gas and Coal gas 663 

r^iorifle Equivalents of Constituents 664 

enctency of A Water-gas Plant 654 

•^lace Bequf red for a Water-gas Plant 666 

^Ml-imlnBOtllittmiBattliSVu 666 



;X1Y COKTEKTS. 

Vis Viva 498 

Work, Foot-pound 49B 

Power, Horse-power 499 

Energy 4S9 

Work of Acceleration 4ao 

Force of a Blow 4ao 

Impact of Bodies 481 

Energy of Beooil of Guns 441 

Ck>n8ervation of Energy 498 

Perpetual If otion 4.« 

BAeiencyof aMacbine 488 

▲nlmal-powerf Man-power 438 

WorkofaHorse 484 

Man-wheel 484 

Horse-gin 434 

Resistance of Vehicles 485 

Slements of Maohlnes. 

The Lever 485 

The Bent Lever 486 

The Moving Strut 486 

The Toe»le-toInt 486 

The Incunea Plane 487 

The Wedge 487 

TheScrew 487 

The Cam 488 

ThePuUey 488 

Differantlal PuUev 480 

Differential Windlass 48» 

Differential Screw 48S* 

Wheel and Axle 489 

Toothed-wheel Gearing 488 

BndkMS Screw 44<1 

Stresses in Framed Straotures. 

Cranes and Derricks 440 

Shear Poles and GHiys 442 

King Post Truss or Bridge 448 

Queen Post Truss 448 

Burr Truss 443 

Pratt or Whipple Truss 443 

HoweTross 446 

Warren Girder 445 

Boof Truss ; 446 

HKAT. 

Thermometers and Pyrometers 448 

Centigrade and Fahrenheit degrees compared 449 

Coppeivball Pyrometer 451 

Thermo-eleotno Pvrometer 451 

Temperatures in Fumaoes 461 

Wiborgh Air Pvrometer 458 

Seegers Fire-clay Pyrometer 458 

Me8ur6 and KouePs Pvrometer 458 

UehliDg and Steinbarrs Pyrometer 45S 

Air-thermometer , 454 

High Temperatures judged by Color 454 

Boiling-points of Subetances 455 

MelUDg-points 455 

UnitofHeat 456 

Mechanical Equivalent of Heat... 456 

Heat of Combustion 456 

Speclflc Heat 457 

Latent Heat of Fusion 459,461 

Expansion by Heat 460 

Absolute Temperature 461 

Absolute Zero 461 



G0NTE2n& XT 

PAGK 

Latent Heat 481 

Latent Heat of Evaporation M^ 

Total Heat of ETaporation «» 

ETaporation and Drying 448 

ByaporatioD from Resenroirs 468 

ETaporation by the Multiple System 468 

Reetstanoe to Boiling 468 

Manufacture of Salt 464 

SolubUity of Salt and Sulphate of lime 464 

Salt Contents of Brines 464 

Concentration of Sugar Solutions.... 466 

Eraporating by Exhaust Steam 466 

Drying in vacuum 466 

Radiation of Heat 467 

Conduction and Convection of Heat 468 

Rate of External Condootion.. 460 

Steam-pipe Coverings 470 

Transmission through Plates 471 

*' in Condenser Tubes 478 

•* *' Cast-iron Plates. 474 

** from Air or Qasies to Water 474 

•• from Steam or Hot Water to Air 476 

^ through Walls of Buildings 478 

Thermodynamics 478 

PHTSIGAI* PKOPERTIKS OF GASES. 

Expansion of Oases 479 

Boyle and Ifarriotte's Law 470 

Law of Charles, Avogadro*s Law 470 

Saturation Point of vapors 460 

Law of Gaseous Pressure 480 

Flow of Gases 480 

Absorpticni I7 Liquids 480 

AIR. 

Propertiee of Air 481 

Air-manometer 481 

Pressure at Different Altitudes 481 

Barometric P r ess ures 488 

Levelling bv the Barometer and by Boiling Water 488 

To find Plfference in Altitude 488 

Moisture in Atmosphere 488 

Weight of Air and Mixtures of Air and Vapor 484 

Specific Heat of Air 484 

Flow of Alr» 

Flow of Air through Orifices 484 

Flow of Air In Pipes 486 

Effect of Bends in Pipe 488 

Flow of Compressed Air 488 

Tkbles of Flow of Air 489 

Anemometer Measurements 491 

Equalization of Pipes 491 

Loss of Pressure in Pipes 498 

Wind. 

Force of the Wind 498 

Wind Pressure in Storms 495 

Windmills 406 

Capacity of Windmills 407 

Economy of Windmills 408 

ElectrlcPower from Windmills 409 

Compressed Air. 

Heating of Air bv Compression 400 

Loss of Energy m Compressed Air 400 

TohuoM and rassores , ^. 600 



Xvi CONTBNTa 

PIO] 

L0B8 due to Ezcem of Pressure 501 

Horse-power Required for Compression 601 

Table for Ad iabatio Cioinpression i 60:1 

Mean Effective Pressures 608 

Mean and Terminal Pressures 503 

Air-compressors 508 

Practical Results 508 

Efficiency of Oompressed-air Engines !.. 506 

Requii'emente of Rock-drills 1 506 

Popp CJompressed-air System 50^ 

Small Oompressed-air Motors 50! 

Efficiency of Air-heatinK Stoves 601 

Efficiency of Compressed-air Transmission 50^^ 

Shops Operated by Compressed Air 501] 

Pneumatic Postal Transmission 60fl 

Mekarski Compressed-air Tramways 51C 

Compressed Air Working Pumps in Mines 51] 

Fans and Blowers. 

Oentrifugall^ans 511 

Best Proportions of Fans 619 

Pi-essure due to Velocity 618 

Experiments with Blowers 51 4 

Quantity of Air Delivered 614 

Efficiency of Fans and Positive Blowers 616 

Capacity of Fans and Blowers 617 

Table of Centrifugal Fans 618 

Engines, Fans, and Steam-coils for the Blower System of Heating. 51S 

Sturtevant Steel Pressure-blower 51fl 

Diameter of Blast-pipes • 619 

Efficiency of Fans 600 

Centrifugal Ventilators for Mines 621 

Eaqperiments on Mine Ventilators 629 

DiskFans 604 

Air Removed by Exhaust Wheel 68S 

Efficiency of Disk Fans 68S 

Positive Rotary Blowers 626 

Blowing Engines 626 

Steam-let Blowers SS7 

Steam-Jet for Ventilation 627 

BEATING AND T£NTII^TION. 

Ventilation 528 

Quantity of Air Discharged through a Ventilating Duct 590 

Artificial Cooling of Air 531 

Mine-ventilation 631 

Friction of Air in Underground Passages 631 

Equivalent Orifices 633 

Relative Efficiency of Fans and Heated Chimneys 683 

Heating and Ventilating of Large Buildings 684 

Rules for Computing Radiating Surfaces 586 

Overhead Steam-pipes 687 

Indirect Heating-surface 537 

Boiler Heating-surface Required 688 

Proportion of Grate-surface to Radiator-surface 588 

Steam-consumption in Car-heating 638 

Diameters of Steam Supply Mains 539 

Registers and Cold-air Ducts 539 

Physical Properties of Steam and Condensed Water 540 

Size of Steam-pipes for Heating 510 

Heating a Greenhouse by Steam Ml 

Heating a Greenhouse by Hot Water 542 

Hot- water Heating 542 

Law of Velocity of Flow 642 

Proportions of Radiating Surfaces to Cubic Capacities 543 

Diameter of Main and Branch Pipes 543 

Rules for Hot-water Heating 544 

Arrangements of Mains 544 



00KTSKT8. XVII 

PAOB 

filower Systein of Heating and Ventilating * 545 

Experiments with lUdiators « 545 

Heating a BuildinR to 70* F 545 

Heating by Electricity 646 

WATER. 

Ezpanskm of Water 547 

Weight of Water at different temperatures 547 

PresBore of Water due to its Wefgbt 540 

Head Oorresponding to Pressures 549 

Buoyancy 550 

BoflW-point , 550 

Freediig-point 550 

Sea-water 549,550 

Ice and Snow 500 

foeciflc Heat of Water 560 

CompresBlbility of Water 551 

Imparities of Water.... 551 

Causes of Incrustation ^ 551 

Means for FreTeotIng Incrustation T, SBH 

Analyses of Boiler-scale 563 

Hardness of Water 558 

Purifying Feed-water 554 

Softening Hard Water 655 

Hydranlles. Flow of Water. 

FomnlsB for Discharge through Orifices 555 

Flow of Water from Orifices 565 

Flow in ^len and Closed Channels 557 

General Formulae for Flow 557 

Tsble Fsll of Feet per mile, etc 668 

Valnesof Vrforarcular Pipes 559 

Katter*s Formula 650 

Xolesworth^s Formula 503 

Bsxin^s Formula .. 668 

D*Arcy*s Formula 568 

Older Formule 564 

Velocity of Water hi Open Channels 564 

Mean. Surface and Bottom Velocities 564 

Safe Bottom and Mean VeioclUes 665 

Resistance of Soil to Erosion 665 

Abrading and Transporting Power of Water 666 

Grade of Sewers 566 

BehUiQns of Diameter of Pipe to Quantity discharged 566 

Flow of Water In aSO-lnchripe 666 

Veioclclesin Smooth Csst-Iron Water-pipes 667 

Table of Flow of Water In Circular Pipes 668-673 

liossof Head 678 

Flow of Water in Rireted Pipes 574 

FricU<»al Heads at given rates of discharge 677 

Effect of Bend and Curres 578 

HydrmuUc Grade-Une 678 

Flow of Water in House-senrloe Pipes 678 

Air-bound Pipes 670 

VertScalJets 670 

Water Delivered through Meters 570 

FlreStreams 670 

Friction Losses In Hose 680 

Head and Pressure Losses by Friction 580 

Loss of Pressure In smooth S^-inch Hose 680 

Bated capacity of Steam Fire-engines 680 

Pressures required to throw water through Nozisles 681 

The Siphon 681 

Measurement of Flowing WaCer 563 

Piesometer : 582 

PItot Tube Gauge 588 

TbeVenturi Meter 688 

Messufement oC Dischaige by means of Nozzles 684 



XVIH C0KTEK18. 

PAGB 

flow through Beofeangular Oiifloei 084 

MMMuremont of AD Open Stream 664 

Hlneni* iBoh Measurements flSB 

Flow of Water oTer Weirs » «... 685 

Francises Foimula for Weirs 686 

Weir Table 687 

Basin's Experiments • 6d7 

Watei>-powofv 

PowerofaFanofWatef 888 

HoFse-power of a Bunnlog stream ■> S80 

Current Motors 680 

Horse-power of Water Flowing in a Tube... 680 

Maximum Efflcienoy of a Long Conduit 889 

Miil.power 680 

Value of Wat«^pow0r *.•• •• 680 

The Power of Oceui Waves • 690 

UUltntionofTi(Ua Power OOO 

• Turbltte WhaeUk 

Proportions of Turbines • ••.•••.. 60t 

1>Mt8 of Turbines • *..••..» 648 

Dtmensions of Turbines •••••*.... 607 

The Felton Water-wheel %••••.»,.••••••,.. 647 

Pnmps. 

Theoretical capadty of a pump • 601 

Depth of Suction • 00^ 

Amount ol Water raised by a Sltigle-actl ng Lift-pump. 604 

Proportioning the Steam cylinder of a Direct-acting Tump 004 

Speed of Water through Pipes and Pump -passages 604 

Suees of Direct-acting rumps OOS 

The Deane Pump 604 

Rttcienoy of Small Pumps » * ».. 408 

The Worfchington Duplex Pump 404 

Speed of Piston ..»«»«» 406 

Speed of Water through Valves ..• 406 

Boiler-feed Pumps.. k .»»» • • 006 

Pvimp Valves 406 

Centrifugal Pumps 404 

Lawrence Centrifugal Pumps 407 

Bflddenoy of Centrifugal and Reciprocating Pumps 406 

Vanes of Centrifugal Pumps 404 

The Centrifugal Pump used as a Suction Dredge 409 

Duty Trials of Pumping Engines 400 

Leakage Tests of Pumps 411 

Vacuum Pnmps 414 

ThePulsoraeter... 614 

nie Jet Pump «. 614 

The Injector ..»••..» 614 

AtrllftPump 414 

The Hydraulic Ram •»..•.•• 414 

Quantity of Water Delivered by the Hydraulic Ram 614 

Hydranlio Pressure Transmission. 

Energy of Water under Pressure » .». 614 

fiAciency of Apparatus ...*» 414 

Hydraulic Presses «.... 617 

HydrauHc Power in London • , 417 

Hydraulic Riveting Machines » 418 

wdraulio Forslng 618 

The Aiken Intensiner « 419 

QydrauUc Engine ....» 614 



FVSIi. 



Theory of Combustion 

Total Heat of Combustion*, 



C0?TTEKT8. XIX 

• 

PAOS 

AMlyaeB of Gases ^OombufltiaB ••••••«• »».»• t^ 

Temperature of tbe Fire * • • .% m 

ClassificaUoDofSoUdFuel 628 

ClassUlcation of Ooala 634 

Analyses of Coals » 9114 

Wf«teni Llenites 681 

Analyses of Foreign Coals.... , ,•.•.....••..... 6S] 

KixuQ^s Navigation Coal « • 682 

SampUnirCoal for Analyses •••.«»•.••• ».* 683 

fiektiTe value of Fine Sizes * 688 

Pressed Fuel , 68'3 

ReJatiTe Value of Steam Coals 688 

Approximate Heating Value of Coals ....«» 684 

Kind of Furnace Adapted for Different Coals 685 

Downwani-draugbtFumacestk »..».. 635 

Calorimetric Tests of American Coals • 636 

EraporatiTe Power of Bituminous Coals • 636 

Weathering of Coal .,.•• »..•.. 637 

Coke 687 

Experiments In Coking ,..•.•...•...« •••.•.. 687 

Coal WashlnK. 1 688 

Recoveryof By-products In Coke manufacture 688 

Making Hard Ooke 688 

Generation of Steam from the Waste Heat and Gases from Coke-ovens. 638 

Phxluetsof tbeDistfUationofCoal 68i 

Wood as Fuel , . 680 

Heating Value of Wood , 689 

Oompinftion of Wood * 640 

Charvoal ,... ...». 640 

Yield of Charcoal from a Cord of Wood 641 

Coosmnption of Charcoal In Blast Furnaces. 641 

Ataaorption of Water and of Oases by Charcoal 641 

Ct>mpoBitlon of Charcoals Mt 

MisceilaDeousSolid Fuels 642 

Dusc-f oel— Dust ExplosiOBS 642 

PmtorTurf 648 

Sawdust as Fuel » , 648 

Borae-raanure as Fttel 618 

WetlHan-barkasFuel.... w. 648 

StmwasFuel • • 648 

Bsgaase as Fuel In Sugar Manufacture ••• 648 

Petrolennu 

ProducUof Distillaaoo 646 

lima Petroleiini » •••...«•.•.•..«•.•....»..,»... 646 

Value of Petroleum as Fwl 646 

Oil M. Goal as Fuel 646 

Fael Gas. 

QirttooQas 646 

AnthraciteOas 617 

Bitumioons Gas 647 

WaterOas 648 

Prodaoer'^as from One Ton of Goal 049 

Xatural Oaa in Ohio and Indiana 649 

Cvmbtutioo of Producer^gas 650 

Use of Steam in Producers « 690 

Gas Fuel for Small Furnaces 661 

lUaminatinir Oas« 

Onal-gas .«.••.• 661 

Vtier-fSaM 652 

Analywes of Water-gas and Coal gas 668 

Otloriflc Equivalents of Constituents 664 

Efficient of a Water-gas Plant 654 

(teace Bequf red fOr a Water-ipas Plant. 656 

AetralDBOtlllUttiiflMi&K^^ 606 






.IIV COUfTEKTS. 

Vis Viva 49S 

Work, Foot-pound 438 

Power, Horse-power 429 

Energy 439 

Work of Acceleration 480 

Force of a Blow 430 

Impact of Bodies 4«1 

Energy of BeooU of Quns 441 

Oonserration of Energy 4SS 

Perpetual If otion 43S 

BAeiencyof aMachlne 439 

Antmal-poweTf Man-power «» 

WorieofaHone 434 

Ifan-wheel 484 

Horse-gin 484 

Resistance of Vehicles 485 

Slements of Maohtnes. 

TheLeTor 485 

TheBent Lever 436 

The MoTlng Strut 486 

The Tofl»le-JoInt 486 

The Inclinea Plane 487 

The Wedge 487 

TheSorew 487 

The Cam 488 

ThePuUey 488 

Differential PuUey 481* 

Differential Windlass 489 

Differential Screw 4» 

Wheel and Axle 489 

Toothed-wheel Gearing 489 

Endless Screw 440 

Strssses in Framed Stmctiires, 

Cranes and Derricks 440 

Shear Poles and Ouys 44S 

King Post Truss or Bridge 44S 

Queen Post Truss 44S 

Burr Truss 443 

Pratt or Whipple Truss 445 

HoweTruss 44: 

Warren Qtrder 44! 

Roof Truss ; 44< 

HEAT. 

Thermometen and Pyrometers 44( 

Centigrade and Fahrenheit de grees compared 441 

Copper-ball Pyrometer 45: 

Thermo-etoctno Pvrometer 45 

Temperatures in Furnaoes 45 

Wlbofgh Air Pvrometer 4h 

Seegers Fire-clay Pyrometer 45 

lfeBur6and KouePs Pvrometer 45 

Uehling and Steiabart^s Pyrometer 45 

Air-thermometer , 45 

High Temperatures Judged by Color 45 

BoIling-pomtB of Substances 45 

Mel ting-points 4*) 

UnitofHeat 45 

Mechanical Equivalent <tf Heat... 45 

Heat of Combustion 45 

Speolflo Heat 45 

Latent Heat of Fusion 459, 46 

B xpaneion by Heat 46 

Absolute Temperature 46 

Absolute Zero • 46 



GOSTTSETEB. XY 

PAGK 

Latent Heat 461 

Latent Heat of Braporatlon 4^ 

Total Heat of Eraporation 408 

Evaporation and Drying 468 

Braporation from Reservoirs 468 

Eraporation by the Multiple System 468 

BesMtanoe to BfldliDg 468 

Manuf actuie of Salt 464 

Solubility of Salt and Sulphate of lime 464 

Salt Ck>ntent8 of Brines 464 

ConoKitration of Sugar Solutions 466 

Evaporatlns by Exhaust Steam 466 

Diyuig in vacuum 466 

Radiation of Heat 467 

GonducUon and Convection of Heat .. 468 

Rate of External Conduction 460 

Steam-pipe Coverings 470 

Transmission through Plates 471 

•• in Condenser Tubes 478 

" " Cast-iron Plates 474 

•• from Air or Gases to Water iU 

" from Steam or Hot Water to Air 476 

•• through Walls of Buildings 478 

Thermodynamics 478 

PHTSICAI* PKOPERTIKS OF GASES. 

Expansion of Oases 479 

Boyle and ]farrlotte*8 Law 479 

Law of Charles, Avogadro's Law 479 

Saturation Point of vapors 480 

Law of Gaaeotis Pressnre 480 

Flow of Gases 480 

Absorption hyIAqn\6s 480 

AIR. 

Properties of Air 481 

Air-manometer , 481 

Preesure at Different Altitudes 481 

Barometric Pressures 488 

Levelling br the Barometer and by Boiling Water 482 

To find iniference In Altitude 483 

Hofstnre In Atmosphere 488 

Weight of Air and Mixtures of Air aod Vapor 484 

SpeSlle Heat of Air 484 

Flow of Alr» 

Flow of Air through Orifices 484 

Flow of Afr in Pipes 485 

Effect of Bends in Pipe 488 

Flow of Compressed Air 488 

Tables of Flow of Air 489 

Anemometer Measurements 491 

Equalization of Pipes 491 

Loss of Pressure in Pipes 493 

Wind. 

Force of the Whid 498 

Wind Pressare in Storms 495 

Windmflls 496 

Capacity of Windmills 497 

Economy of Windmifls 496 

Electric rower from Windmills 499 

Compressed Air. 

Heating of Air b7 Compression 499 

Loss of Energy hi Compressed Air , 499 

Volwoef aoaProBsures i , 609 



Xvi CONTENTS. 

PIOI 

Loss due to Excess of Pressure 60! 

Horse-power Required for Compression 501 

Table for Ad iabatic Compression i fios 

Mean Effective Pressures fiOS 

Mean and Terminal Pressures MK 

Air-compressors fiOf 

Practical Results 6W 

Efficiency of Compressed-air Engines AM 

Requirements of Boclc-driUs ...1 60( 

Popp Compressed-air System 6a; 

Small Compressed-air Motors 60; 

Efficiency of Air-heatinK Stoves 60: 

Efficiency of Compressed-air Transmission 60^ 

Shops Operated by Compressed Air 50S 

Pneumatic Postal Transmission 60h 

Mekarski Compressed-air Tramways 61( 

Compressed Air Working Pumps in Mines 611 

Fans and Blowers. 

Centrifugal Fftns 611 

Best Proportions of Fans 61S 

Pi-essure due to Velocity 619 

Experiments with Blowers 614 

Quantity of Air Delivered tu 

Efficiency of Fans and Positive Blowers 516 

Capacity of Fans and Blowers 617 

Table of Centrifugal Fans 518 

Engines, Fans, and Steam-colls for the Blower System of Heating. 519 

Sturtevant Steel Pressure-blower 6lfl 

Diameter of Blast^pipes 61fl 

Efficiency of Fans 62C 

Centrifugal Ventilators for Mines 6S] 

Experiments on Mine Ventilators 62S 

DiskFans , 824 

Air Removed by Exhaust Wheel 52£ 

Efficiency of Disk Fans 62S 

Positive Rotary Blowers 6SC 

Blowing Engines 52C 

Steam-jet Blowers 527 

Steam-Jet for Ventilation 6Zi 

HEATING AND VENTII.ATION. 

Ventilation 628 

Quantity of Air Discharged through a Ventilating Duct. 58G 

Artificial Cooling of Air 581 

Mine-ventilation 581 

Friction of Air in Undeiiground Passages 58] 

Equivalent Orifices 589 

Relative Efficiency of Fans and Heated Chimneys 58fl 

Heating and Ventilating of Large Buildings 534 

Rules for Computing Radiating Surfaces SK 

Overhead Steam-pipes 68? 

Indirect Heating-surface 53? 

Boiler Heating-surface Required 636 

Proportion of Grate-surface to Radiator-surface 538 

Steam-consumption In Car-heating NS8 

Diameters of Steam Supply Mains 53S 

Registers and Cold-air Ducts 53S 

Physical Properties of Steam and Condensed Water 54G 

Size of Steam-pipes for Heating 6IC 

Heating a Greennouse by Steam 541 

Heating a Greenhouse by Hot Water 549 

Hot- water Heating 648 

Law of Velocity of Flow 548 

Proportions of Radiating Surfaces to Cubic Capacities 548 

Diameter of Main and Branch Pipes 548 

Rules for Hot-water Heating 644 

Arrangements of Mains 644 



CONTENTS XVil 

PAOB 

Slower System of Heating and Ventilating •. 545 

Expeiimente with Radiators ••... 545 

Heating a Buildinir to TO" F 545 

Heating by Electiricity 54« 

WATER. 

Expaaiioa of Water 547 

Weight of Water at different temperatures 547 

Preeaare of Water due to its Weight 549 

Head OorreepoDdtng to PresBures 549 

Buoyancy 550 

Bollbg-point 650 

Freezfiig-point 550 

Bea^water 549,550 

loe and Snow 550 

Bpedflc Heat of Water 550 

CompreesibilitT of Water 551 

Impurities of Water 551 

Causes of Incrustation ^ 561 

Means for Preventing Incrustation 7. SSH 

Analyies of Boiler-scale 56S 

Hardness of Water 553 

Porifying Feed-water 554 

Softening Hard Water 655 

Hydraulics. Flow of Water. 

Fomnte for Discharge through Orifices ... 665 

Flow of Water from Orifices 556 

Flow in Open and Closed Channels 557 

General F^mulflB for Flow 557 

Tsble Fall of_Feet per mile, etc 558 

Taluesof Vr for Qrcular Pipes 569 

Kntter*s Formula 669 

Molesworth's Formula 6es 

Basin's Formula 568 

D'Arcy*s Formula 568 

Older Formuls 564 

Velocity of Water in Open Channels 664 

Mean Surface and Bottom Velocities 664 

Safe Bottom and Mean Velocities 565 

Besistance of Sou to Erosion 665 

Abradiog and Transporting Power of Water 565 

Grade of Sewers 666 

BelatlQns of Diameter of Pipe to Quantity discharged 566 

Flow of Water in a 80-inch Pipe 566 

Veioeiilesin Smooth Castriron Water-pipes 567 

Table of Flow of Water in arcular Pipes 668-578 

LoesofHead 578 

Flow of Water in Riveted Pipes 574 

FricUonal Heads at given rates of discharge 577 

Effect of Bend and Curves 578 

Hydraulic Grade-line 578 

Flow of Water in House-service Pipes 578 

Air-bound Pipes 579 

VerticalJeU 579 

Water Delivered through Meters 579 

Fire Streams 579 

Friction Losses in Hose 580 

Head and Pressure Losses bj Friction 580 

Loss of Pressure in smooth 2^-ineh Hose 580 

Bated capacity of Steam Fire-engines 580 

Pressures required to throw water through KosBles 581 

The Siphon 581 

Measurement of Flowfaig Wafer 582 

Piesometer T 582 

Pttot Tnbe Gauge 588 

The Venturi Meter 688 

Measurement of Discharge by means of Nozzles 684 



XVlll C0KTBK18. 

_ PlOE 

flow through ReoftangularOrlfloet.. ••.»».• OM 

Hea«arem«Bt of aa Open Stream 664 

MfneraUnoh Measuremeats •.. OSS 

now of Water oTer Wein 1160 

FrandB^s Formula for Weirs fi86 

WeirTkble 687 

Baiiii'8 Experiments » fid7 

Waf;ei>powerw 

^werofsFUlofWater 568 

Horfle-power of a Runnloi; Stream ., G8B 

Current Motors » 689 

Horae-powerof Water Flowing in a Tube... • 66B 

Uaxlmum Effieiency of a Long Conduit 680 

MUlpower 68B 

Value of Watca^power •.... 600 

The Power of Ooean Waves » 690 

UtUfation of Tidal Fower OOO 

• Ttirbltte Wheels. 

Proportions of Turbines »•...•.. 601 

Tests of Turbines ••% 606 

Dimensions of Turbines ••»• 607 

llie Pelton Water-wheel - 607 

Pumps. 

Theoretical capadty of a pump • 601 

Depth of Suction 604 

JUnount oi Water raised by a Single-acting Lift-pump 60a 

Proportioning the Steam cylinder of a Direct-acting Pump OOt) 

Speed of Water through Pipes and Pump -passsges 603 

Sues of Dlrect>acting Pumps 603 

Tlie Deane Pump •....• • OOS 

Kflloienoy of Small Pumps * • ... ••*.. 606 

The Worfchlngton Duplex Pump • 604 

Speed of Piston ..>...» • 605 

Speed of Water through ValTes... » 605 

Boilerfeed Pumps • 605 

PnmpValvee « 606 

Centrifugal Pumps 606 

Lawrenoe Centrifugal Pumps 607 

EMoienqy of Centrifugal and Reciprocating Pumps 606 

Vanes of Centrifugal Pumps 600 

Tlie Centrifugal Pump used as a Suction Dredge 60D 

Duty Trials or Pumping Engines 600 

Leakage TesU of Pumps , 611 

Vacuum Pnmps.... • 616 

ThePul8ometer..» k... 61i 

IlieJetPump • • »»..»». 514 

The Injector •••••..•»..•». 614 

Air-lift Pump 514 

The Hydraulie Ram »> • 514 

Quantity of Water Delivered by the Hydraulic Ram..... 515 

Hydraulic Pressure Tranamission. 

Energy of Water under Pressure • 515 

fifliciency of Apparatus • 515 

ttydraulfc Presses 617 

Hydraulic Power in London 617 

Hydraulic Riveting Machines 518 

Hydraulic Forging 515 

1%e Aiken Intensmer 510 

Hydraulic Bngitte 515 



FUBIto 



Theory of Combustion 

Total Heat of Combustion., 



C0KTENT8. XIX 

PAOI 

AMlrnBorGMesofOombostta «» 

T»inper«ture of the Fire •• AH 

Clasftiflcatioo of Solid Fuel 628 

ClassiflcaUon of Coals 634 

Anelj-sas of Coals 624 

Western Lignites 631 

ADslysesof Foreign Coals >... 68) 

NixoQ^s Navigation Coal 6S2 

SsmpIinKCoal for Analyses » 683 

ftftkUve value of Fine Sixes 68li 

Pressed Fuel 68a 

Belatiye Value of Steam Coals 688 

A|»prox1mate Heating Value of Coals » 684 

Kind of Furnace Adapted for Different Coals 685 

DowDwafd-draugbt Furnaces. > ..».»..•.. 635 

Ghlorimetric Tesis of American Coals » 636 

£rsporative Power of Bituminous Coals. • • «.. 686 

Weathering of Coal... ,....« %... 697 

Coke 687 

Kzperiments in Oolcins • •. ,•..„ ..• .^•..•.. 687 

Coal Washing 77. ; m 

B<ecc»Tei7 of By-products In Coke manufacture 688 

Making Hard Coke 638 

(jtfoeraiion of Steam from the Waste Heat and Gases from Coke-ovens. 638 

Products of the DistlUatiOD of Coal 68» 

Wood as Fuel . 680 

HoatiDK Value of Wood 680 

Compoiltlon of Wood » 640 

ChaiTosl 640 

Yield of Charcoal ftom E Cord of Wood 641 

Consumption of Charcoal In BlaKt Furnaces. 641 

Absorption of Water and of Oases by Charcoal 641 

Oompositfon of Charcoals 64)1 

Mjaceilaneous Solid Fuels 643 

Dustrfuel— Dust Ezploslom 643 

Beat or Turf • 648 

Sawdust as Fuel ..*..»• 648 

Hone-raanure as Ftid .......•» 643 

Wee Tan-bark as Fuel 648 

Straw as Fuel 648 

Bagasse as Fuel la Sugar ICanufacture 648 

Petroleam* 

P^isduetaof DlsUIlation 646 

Lima Petroleum. ••• • • »..,%... 646 

Valueof FetroleumasFMl «.••*—* 646 

Oaea Gold as Fuel 646 

Fael Gas. 

O&rtxmGas • 646 

AnthrveiteGas 617 

Bttumiooas Gas 647 

WaterOas 648 

Pit>daoerogss from One Ton of Coal 649 

Natural Gas in Ohio and Indiana 640 

Oimbttstion of Prodnoer^gas 6!i0 

Use of Steam fan Producers 6S0 

Gas Fuel for Small Furnaces 651 

lUamlnatlng €kM» 

Ooal-gas , 661 

Water-ffas 660 

Analjwes of Water-gas and Coal gas 6S3 

Oaloriftc Equiralents of Constituents , 6M 

KSciency of a Water-gas Plant 664 

Hpace Required for a Water-^as Plant 656 

iSiel'^aliifl ol iBinwIniUllig-gas OM 



XIV GOKTEKT& 

VtoVlva 49S 

Work, Foot-pound 4M 

Power, Horse-power 499 

Energy 4» 

Work of Acceleration 480 

Force of a Blow 490 

Impact of Bodies 431 

Enency of BecoU of Guns 481 

OonseryatioD of Enency 49B 

Perpetual Motion 4» 

SIBciencyof aHacbine 489 

Animal-power, ]IIan>power 438 

WorkofaHorse 4S4 

Man-wheel 404 

Horse^n 484 

Besistanoe of Vehicles 485 

Blements of Machines. 

The Lever 485 

TheBentLsTer 438 

The Moving Strut 486 

The Toggle-Joint ■ 486 

The Incunea Plane 487 

The Wedge 487 

TheScrew 437 

The Cam 438 

ThePulloy 438 

DUTerantial Pulley 4SD 

Differential Windlass 489 

DUTerentJAl Screw 481' 

WheelandAxle 48» 

Toothed-wheel Gearing 488 

KwUesi Screw 4«0 

Stresses in Framed Struotnres. 

Cranes and Derricks 440 

Shear Poles and Guys US 

King Poet Truss or Bridge. 4lt 

Queen Post Truss 449 

Burr Truss 448 

Pratt or Whipple Truss 44S 

HoweTmss 445 I 

Warren Girder 44S 

Boof Truss ¥i 

HEAT. 

Thermometere end Pyrometers 44 

Centigrade and Fahrenheit degrees compared 44t 

Copper-hall Pvrometer 40 

Thermo-eleotno Pvrometer fil 

Temperatures in Fumaoes 4SI 

Wlborgh Air promoter 4S 

Seegers Fire-clay Pyrometer 4SS 

Mesur^and Kouel's Pyrometer 40 

Uehling and Stelnbart^s Pyrometer 49 

Air-thermometer , 4M 

High Temperatures judged by Color 4M 

Boiling-points of Subetaoces 4S 

Melting-points tf 

Unit of Heat .. «l 

Mechanical Equivalent of Heat fli 

Heat of Combustion el 

Specific Heat tSf 

Latent Heat of Fusion 4S>,«1 

Expansion by Heat # 

Abiiolute Temperature IB 

Absolute Zero « 



OONTESn& XV 

PAQK 

Latent Heat 461 

lAtent Heat of Eyaporation 46^ 

Total Heat of Evaporation 408 

ETaporatlon and Drying ..• 4(12 

Evaporation from Beeervoirs 468 

KTaporation by the Multiple System 468 

Reaisuuice to Boiling 468 

Manufacture of Salt 464 

SolubOity of Salt and Sulphate of Xime 464 

Salt Contents of Brineg 464 

Conoentration of Sugar Solutions.... 465 

ETaporatins by Exhaust Steam 466 

Vrymg in vacuum 466 

Radiation of Heat 467 

Oondoction and ConyecUon of Heat ...468 

Rate of External Conduction 460 

Steam-pipe Coverings 470 

Transmission through Plates 471 

'* in Condenser Tubes 478 

*• ** Cast-iron Plates 474 

** from Air or Oases to Water 474 

•• from Steam or Hot Water to Air 476 

*« through Walls of Buildings 478 

Thennodjnamics 478 

PHTSICAL PROPERTIES OF GASES. 

Expansion of Gases 470 

Bojleand Marriotte^s Lav 470 

Law of Charles, Avogadro*s Law 470 

Batnration Point of vapors 480 

Ijaw of Gaseous Pressure 480 

Flow of Gases 480 

Absorptfon by Liquids 480 

AIR. 

Properties of Air 461 

Air-manometer 481 

Pressure at Different Altitudes 481 

Barometric Pressures 489 

Levelling by the Barometer and by Boiling Water 482 

To find Imferenoe in Altitude 483 

Moisture in Atmosphere 488 

Weight of Air and Mixtures of Air and Vapor 484 

Specific Heat of Air 484 

Flow of Air* 

Flow of Air through Orifices 484 

Flow of Air in Pipes 485 

Effect of Bends in Pipe 488 

Flow of Compressed Air 468 

Tables of Flow of Air 480 

Anemometer Measurements 401 

Eqrualization of Pipes 401 

liOflS of Pressure in Pipes 408 

Wind. 

Force of the Wind 408 

Wind Pressure In Storms 406 

Windmills 405 

Capacity of Windmills 407 

Economy of Windmills 408 

KlectricPower from Windmills 400 

Compressed Air. 

Beatincr of Air bj Compression 400 

IxMB of Energy bi Compressed Air 400 

Volumes anaPreBBures * • .^ 600 



Xvi CONTENTS. 

PlOl 

Loss due to Excess of Presgure fiO] 

Horae-power Required for Compression 601 

Table for Ad labatic Compression i BOi 

Mean Effective Pressures 6Q9 

Mean and Terminal Pressures 603 

Air-compressors COS 

Practical Results SOS 

Efficiency of Compressed-air Engines 50e 

Requirements of Rock-drills 1 606 

Popp Compressed-air System 607 

Smflkil Compressed-air motors 607 

Efficiency of Air-heatinK Stoves 607 

Efficiency of Compressed-air Transmission 60^ 

Shops Operated by Compressed Air 60fl 

Pneumatic Postal Transmission 60d 

Mekarski Compressed-air Tramways 61C 

Compressed Air Working Pumps in Mines 61] 

Fans and Blowers. 

Centrifugal Fans 611 

Best Proportions of Fans 619 

Pressure due to Velocity 618 

Experiments with Blowers 614 

Quantity of Air Delivered 61<l 

Efficiency of Fans and Positive Blowers 516 

Capacity of Fans and Blowers 617 

Table of Centrifugal Fans 618 

Engines, Fans, and Steam-coils for the Blower System of Heating. 619 

Sturtevant Steel Pressure-blower 51fl 

Diameter of Blastrpipes 619 

Efficiency of Fans 82G 

Centrifugal Ventilators for Mines 621 

Experiments on Mine Ventilators 62S 

DlskFans 624 

Air Removed bv Exhaust Wheel 6S8 

Efficiency of Disk Fans 63S 

Positive Rotary Blowers &M 

Blowing Engines ftSC 

Steam-jet Blowers 6*7 

Steam-Jet for Ventilation 627 

HEATING AND VENTII.ATION. 

Ventilation 828 

Quantity of Air Discharged through a Ventilating Duct. 530 

Artlfldal Cooling of Air 681 

Mine-ventilation 681 

Friction of Air in Underground Passages 531 

Equivalent Orifices 688 

Relative Efficiency of Fans and Heated Chimneys 688 

Heating and Ventilating of Large Buildings 534 

Rules for Computing Radiating Surfaces 580 

Overhead Steam-pipes 637 

Indirect Heating-surface 687 

Boiler Heating-surface Required 588 

Proportion of Grate-surface to Radiator-surface 688 

Steam-consumption In Car-heating 638 

Diameters of Steam Supply Mains 539 

Registers and Cold-air Ducts 539 

Physical Properties of Steam and Condensed Water 540 

Size of Steam-pipes for Heating 610 

Heating a Qreenhouse by Steam 041 

Heating a Greenhouse by Hot Water 548 

Hot-water Heating 548 

Law of Velocitv of Flow 548 

Proportions of Radiating Surfaces to Cubic Capacities 543 

Diameter of Mai n and Branch Pipes 543 

Rules for Hot-water Heating 544 

Arrangements of Mains • 544 



CONTENTS Xvii 

-* PAOB 

Bbwer System of Heating and Ventilating 645 

Experiments with Radiators « 645 

Heating a Buildine to 70* F 645 

Heating by Electricity 646 

WATER. 

Expansion of Water 647 

W^ght of Water at different temperatures 547 

Preasare of Water due to Its Weigbt 649 

Head Oorrasponding to Pressures 649 

BoiSig-point*.!./.!l!l!I!I!iili!!lii!".r.y.;!;i!!!*!*^ '.;!!*.!;! ■".!**.;'.!*.!;.'*! 66o 

Freesmg-point 650 

Sea-water 649.560 

Ice and Snow 550 

Specific Heat of Water 650 

Compressibility of Water 651 

Impurities of water........ 651 

Causes of Incrustation ^ 661 

Means for Preventing Incrustation 7, 65'j 



Analyses of Boiler-scale., 
Hardness < 



lof Water 653 

Purifying Feed-water 654 

Softening Hard Water 665 

Hydranllcs. Flow of Water. 

Fomuls for Discharge through Orifices 665 

Flow of Water from Orifices 655 

Flow In Open and Closed Channels 557 

General Formuln for Flow 557 

Table Fall of_Feet per mile, etc 668 

Valuesof fr for Circular Pipes 559 

Kntter*8 Formula 659 

Molesworth's Formula 663 

Basin's Formula 568 

IV Arctr's Formula 668 

Older Formule 664 

Velocity of Water in Open Channels 664 

Mean, Surface and Bottom Velocities 664 

Safe Bottom and Mean Velocities 665 

Resistance of Soil to Erosion 665 

Abrading and Transporting Power of Water 665 

Grade of Sewers 666 

Belati^ns of Diameter of Pipe to Quantity discharged 666 

Flow of Water in a SO-inch Fipe 666 

Veioclclesin Smooth Cast-Iron Water-pipes 567 

Table of Flow of Water in Circular Pipes 668-678 

Lossof Head •578 

Flow of Water in Riveted Pipes 574 

Frictional Heads at given rates of discharge 577 

Effect of Bend and Curves * 678 

Hydraulic Grade-line 678 

Flow of Water in House-service Pipes 678 

Air-bound Pipes 679 

VertfcalJets 679 

Water Delivered through Meters 679 

FIreBtreams 679 

Friction Losses in Hose 580 

Head and Pressure Losses bv Friction 580 

Loss of Pressure in smooth 2^-inch Hose 580 

Rated capacity of Steam Fire-engines 680 

Pressures required to throw water through Kozssles 681 

The Siphon 681 

Measurement of Flowhig W«eier 688 

Piezometer 682 

Pilot Tube Gauge 683 

The Ventttri Meter 688 

Measurement of Dischai^ by means of Nozzles 684 



XVIU COKTEKT& 

PAQB 

flow through Beot&ngular Orifloet 684 

HeasurenMBt of aa Open Stream 064 

Mlnera* iBoh Measurements •• 06B 

Plow of Water oTor Weirs •*., 088 

Francis's Fonnula for Welfi 686 

WeirTkble 587 

Basin's Experiments 6d7 

Wafeer^powerw 

^werofaFhUof Water » Bd8 

Horfle-power of a Running 8tt>Bam M 

Current Motors 68B 

Hors6>power of Water Flowing in a Tube 680 

Maximum Efficiency of a Long Conduit 889 

Mill-power 689 

Value of Water-power • S90 

The Power of Ooean Waves 698 

UUliation of Tidal Fower 000 

• Ttirblne Wheels. 

Proportions of Turbines ••» ••• 001 

Tests of Turbines 600 

Dimensions of Turbines » 697 

llie Pelton Water-wheel • -• 807 

Pumps. 

Theoretical capacity of a pump • • 601 

0cpth of Suction 00< 

JUnoant 01 Water raised by a Single-acting Lift-pump ••.. . 60a 

Proportioning the Steam cylinder of a Direct-acting Pump 600 

Speed of Water through Pipes and Pump-passages • 009 

STses of Direct-acting Pumps OOS 

The Deane Pump 000 

Rffioienoyof Small Pumps. *........>. ••• 000 

The Worthington Duplex Pump » 004 

Speed of Piston »»..> 006 

Speed of Water through ValTes ••• 005 

Bollerfeed Pumps ...».»•••. • 006 

Pump Valves 006 

Centrifugal Pumps 000 

Lawreooe Centrifugal Pumps 007 

Bfficienqy of CentrifUral and Reciprocating Pumps 606 

Vanes of^Centrlfugal nimps 600 

The Centrifugal Pump used as a Suction Dredge 000 

Duty Trials of Pumping Engines «.. 000 

Leakage Tests of Pumps ,.. Oil 

Vacuum Pnmps 010 

ThePul8ometer..» 010 

The Jet Pump »..«.. 614 

Thelo^tor 014 

Air-lift Pump 014 

The Hydraulic Ram , 014 

Quantity of Water Delivered by the Hydraulic Ram eW 

Hydraulic Pressure Transmission. 

Energy of Water under Pressure ....•• 010 

fiflicieiiey of Apparatus 010 

Hydraulfo Presses .. . 017 

Hydraulic Power in London • 617 

HydrauHc Riveting Machines * 018 

Hydraulic Forging 018 

1%e Aiken Intensmer ^10 

HydrauHc Engine 610 



FUKIf 



Theory of Combustion 

Total Heat of Oombustioo.. 



CONTENTS. XIX 

• 

PAOI 

AnaljiMorGaanofOoiiibastta » «tt 

Temperature of the Fire »•••» ••»• tttt 

CUssificatioD of SoUd Fuel 628 

Ciasfitilcation of Coals 634 

Analys^ee of Coals 624 

Western Llraitee 681 

Aaaljses of Foreig:n Coals «. »... 681 

NixoD^s NavifraUon Coal 683 

SampIfDfrCoai for Analyses ,..•» ....,.» ». 682 

Hektive value of Fine Sizes 68ii 

Pressed Fuel 6$^ 

Belative Value of Steam Coals 683 

Approximate Heating Value of Coals • 684 

Kind of Furnace Adapted for Different Coals 686 

Downward-draufl:htFurnaceB«> .»»..%.. 635 

Cslorimetric Tests of American Coals 636 

£TBporatlve Power of Bituminous Coals ••..• ••.... 686 

Weathering of Coal... , 687 

Coke 637 

Experiments in Ookins »•.... ,..•. 687 

Coal WasWnar !?. ; 688 

Recovenr of By-products In Coke manufacture 688 

Making Hard Coke 688 

Generation of Steam from the Waste Heat and Gases from Coke-ovens. 638 

Products of the DisUUaiion of Coal 680 

WoodasFiiel 680 

Hoating Value of Wood 689 

CompOBitSon of Wood 640 

Charcoal ,... 640 

Yield of Cfaarcoftl ftom B Cord of Wood 641 

ConKumndon of Clmrcoai in Blast Furnaces. 641 

Absorption of Water and of Oases by Charcoal.... , 641 

Com positton of Charcoals 64« 

Miscellaneous Solid Fuels 642 

Dust-fuel— Dust BxplosioDS 642 

PteatorTurf 648 

Sswdust as Fuel • » ....» 648 

Borse-manure as Fuel 648 

Wet Tan-bark as Fuel.... 648 

StmwasFuel • 648 

Bsgasse as Fuel in Sugar lumufacture. 648 

Petrolenm* 

Praductoof DtotiUatloa 646 

UmaPtotroleam • 646 

Vsiue of Petroleum aaFMl 646 

OBet. Goal as Fuel 646 

Fael Gas. 

OifbonGas 646 

AnthnBiteOas 647 

Bttuminoas Gas 647 

Water Oas 648 

Producer-gas from One Ton of Coal 649 

Katnral Oas in Ohio and Indiana 649 

Combustion of Producer-gas 6M 

Use of Sieam In Producers 6S0 

Gas Fuel for Small Furnaces 661 

XUnminatlng Gas. 

Ooal-Kas , 661 

Water-gas 66t 

Analynes of Water-gas and Coal km 868 

Oiloriflc Equtralents of Constituents 654 

Efficiency of a Water-ffas Plant 664 

Space Required for a Water-yas Plant 656 

rasl-Talna of IBimiinitting-gas OM 



•XIV COKTEKT& 

VteVlTa 498 

Work, Foot-pound 4S8 

Power, Horse-power 4S9 

Energy 4» 

Work of Acceleration 480 

Force of a Blow 430 

Impact of Bodies 481 

Energy of Recoil of Quns 481 

Oonservatton of EnenKj 488 

Perpetual Motion 488 

SIBciencyof aMachlne 488 

Antmal-power, Man-power 488 

Workof aHorae 484 

Man-wheel 484 

Horge-gin 484 

RaelBtanoe of Vehicles 485 

Blements of Maehlnes. 

The Lever 485 

The Bent Lerer 436 

The Moving Strut 486 

The Toggle-joint 486 

The Inclined Plane 487 

The Wedge 487 

TheScrew 487 

The Cam 488 

ThePulley 488 

IMfferential Pulie/ 4m 

Differential Windlass 480 

Differential Screw 48S< 

Wheel and Axle 489 

1Vx>thed-wheel Gearing 480 

IkidleH Screw 44fl 

Stresses in Framed Struotnres. 

Cranes and Derricks 440 

Shear Poles and Guys 443 

King Post Truss or Bridge. 448 

Queen Poet Truss 44t 

Burr Truss 443 

Pratt or Whipple Truss 443 

HoweTruss 445 

Warren Girder 445 

Roof Truss : 4^ 

HEAT. 

ThermometeiB and Pyrometers 448 

Centigrade and Fahrenheit degrees compared 440 

Copper-ball Pvrometer 451 

Thermo-eleotno PVrometer 451 

Temperatures in Furnaces 451 

Wiborgh Air PVrometer 458 

Seegers Fire-clay Pyrometer 458 

Mesur^and Kouel's Pyrometer 458 

Uehling and Steinbart^s Pyrometer 458 

Air-thermometer , 454 

High Temperatures judged by Color 454 

Boiling-points of Substances 455 

MelUng-points 455 

Unitof Heat 465 

Mechanical Equivalent of Heat 466 

Heat of Combustion 456 

Specific Heat 457 

Latent Heat of Fusion 450,461 

Expansion by Heat 460 

Absolute Temperature 46t 

Absolute Zero 461 



C0NTEKT8. XV 

PAGK 

Latent Heat 461 

Latent Heat of Evaporation 46S 

Total Heat of Evaporation 469 

Evaporation and Drying 468 

Evaporation from Reservoirs 468 

Evaporation by the Multiple System 468 

Resistance to fioiling 468 

Manufacture of Salt 464 

Solubility of Salt and Sulphate of lime 464 

Salt Contents of Brines 464 

Concentration of Sugar Solutions 466 

Evanoratins by Exhaust Steam 466 

Drymg In vacuum 466 

Radiation of Heat 467 

Conduction and Convection of Heat ..468 

Rate of External Conduction 460 

Steam-pipe Coverings 470 

Transniiasion through Plates 471 

** in Condenser Tubes 478 

" ** Cast-iron Plates 474 

•• from Air or Gases to Water iU 

•• from Steam or Hot Water to Air 476 

•• through Walls of Buildings 478 

Thermodynamics 478 

PHTSICAIi PROPERTIES OF GASES. 

Expansion of Gases 470 

Boyle and Marriotte's Lav 470 

liaw of Charles, Avogadro's Law 470 

Saturation Point of vapors 480 

Law of Gaseous Pressure 480 

Flow of Oases 480 

Absorption by Liquids 480 

AIR. 

Properties of Air 481 

Air-manometer 481 

Pressure at DiflTerent Altitudes 481 

Barometric Pressures 489 

Levelling by the Barometer and by Boiling Water 482 

To find Differ enoe in A I ti tude 483 

Moisture in Atmosphere 488 

Weight of Air and Mixtures of Air and Vapor 484 

Specillc Heat of Air 484 

Flow of Air* 

Flow of Air through Orlfloes 484 

Flow of Air in Pipes 485 

Effect of Bends In Pipe 488 

Flow of Compressed Air 488 

Tables of Flow of Air 480 

Anemometer Measurements 401 

Equalisation of Pipes 401 

Loss of Pressure in Pipes 408 

Wind. 

Force of the Whid 408 

Wind Pressure in Storms 406 

Windmills 406 

Capacity of Windmills 407 

Economy of Windmills 408 

ElectricPower from Windmills 400 

Compressed Air. 

Heatlngof Air by Compression 400 

Loss of Energy in Compressed Air 400 

Volumes and Pressures , , 600 



XVi CONTENTS. 

PIGS 

Loss due to Excess of Pressure 60i 

Horse-power Required for Compression.... 60] 

Table for Adiabatic Compression i 602 

Mean Effective Pressures 60i 

Mean and Terminal Pressures 608 

Air-compressors • 608 

Practical Results 606 

Eflflciency of Compressed-air Engines. 606 

Requirements of Rock-driUs ..1 806 

Popp Compressed-air System 607 

Small Compressed-air Motors 607 

Efllciency of Air-heatinK Stoves 607 

Efficiency of Compressed-air Transmission 60R 

Shops Operated by Compressed Air 609 

Pneumatic Postal Transmission 609 

Mekarski Compressed-air Tramways 610 

Compressed Air Working Pumps in Mines • 611 

Fans and Blowers. 

Centrifugal fians 611 

Best Proportions of Fans 619 

Pressure due to Velocity 618 

Experiments with Blowers 614 

Quantity of Air Delivered 614 

Efficiency of Fans and Positive Blowers 516 

Capacity of Fans and Blowers 617 

Taole of Centrifugal Fans 618 

Engines, Fans, and Steam-coils for the Blower System of Heating. 619 

Sturtevant Steel Pressure-blower 519 

Diameter of Blast-pipes • 619 

Efficiency of Fans 620 

Centrifugal Ventilators for Mines 6SI 

Experiments on Mine Ventilators 688 

DiskFans 6»4 

Air Removed by Exhaust Wheel 685 

Efficiency of Disk Fans 685 

Positive Rotary Blowers 686 

Blowing Engines 686 

Steam-jet Blowers 687 

Steam-jet for Ventilation 687 

HEATING AND VENTII^TION. 

Ventilation 688 

Quantity of Air Discharged through a Ventilating Duct 680 

Artificial Cooling of Air 581 

Mine-ventilation 581 

Friction of Air In Underground Passages 581 

Equivalent Orifices 588 

Relative Efficiency of Fans and Heated Chimneys 688 

Heating and Ventilating of Large Buildings 684 

Rules for Computing Radiating Surfaces 686 

Overhead Steam-pipes 687 

Indirect Heating-surface 687 

Boiler Heating-surface Required 688 

Proportion of Grate-surface to Radiator-surface 588 

Steam-consumption in Car-hsatiiig 6S8 

Diameters of Steam Supply Mains 589 

Registers and Cold-air Ducts 539 

Physical Properties of Steam and Condensed Water 540 

Size of Steam-pipes for Heating 640 

Heating a Oreennouse by Steam 541 

Heating a Greenhouse by Hot Water 549 

Hot-water Heating 548 

Law of Velocitv of Flow 548 

Proportions of Radiating Surfaces to Cubic Capacities 543 

Diameter of Mai n and Branch Pipes 543 

Rules for Hot-water Heating 544 

Arrangements of Mains 544 



/ 



OOJSTTENTS* • Xvii 

^ PAOK 

Blower System of Heating and VentQatlng........ 645 

Sxperiments with Hadiatora • « 645 

Heating a Buildinir to 70* F c 545 

Heating by Electricity 546 

WATER. 

Exnansioii of Water 847 

W^ht of Water at different temperatures 547 

Pressure of Water due to ito Weignt 549 

Head Oorreepottdlng to Pressures 540 

Boiinig-potot*.!./.!IV.!!!l!li'.!!!Ii!'.".!*//.!!!l!!!*!!"/.!!l!!!r'.i!'. '/.!*//.];; 550 

Freeslng-point 550 

Sea-water 540,550 

Ice and Snow 650 

Specific Heat of Water 550 

Compresslbiltty of Water 651 

Impurities of water.... 551 

Oauaes of Incrustation ^ 561 

Means for Preventing Incrustation 65'^ 

Analyses of Boiler-scale 553 

HardnesB of Water 553 

Purifying Feed-water 654 

Softening Hard Water 655 

Hydranlies. Flow of Water. 

Fomnto for Discharge through Orifices 55S 

Flow of Water from Orifices 556 

Flow in Open and Closed Channels 657 

General Fx>rmul8B for Flow 657 

Table Fsli of_Feet per mile, etc 668 

Valuesof Vr for arcular Pipes 660 

Kutter*s Formula 660 

Molesworth's Formula 562 

Bazin 'a Formula 568 

IVArcy's Formula 668 

Older Formule 664 

Velocity of Water in Open Channels 664 

Mean. Surface and Bottom Teloclties 664 

Safe Bottom and Mean Velocities 665 

Resistance of SoU to Eroeion 666 

Abrading and Transporting Power of Water 665 

Grade ofSewers 666 

BelatiQns of Diameter of Pipe to Quantity discharged 666 

Flow of Water in a 80-inch Pipe 566 

Velocidesin Smooth Cast-iron Water-pipes 567 

Table of Flow of Water in Circular Pipes 668-678 

Loeeof Head 578 

Flow of Water in Riveted Pipes 574 

Frictional Heads at given rates of discharge 577 

Effect of Bend and Curves 678 

Hydraulic Grade-line 578 

Flow of Water in House-service Pipes 678 

Air-bound Pipes 579 

VertlcalJeta 670 

Water Delivered through Meters 670 

Fire Streams 670 

Friction Losses in Hose 660 

Head and Pressure Losses by Friction 580 

Loss of Pressure in smooth 2^inch Hose . 580 

Rated capacity of Steam Fire-engines 580 

Pressures required to throw water through Nozzles 581 

The Siphon 581 

Measurement of Flowfaig Wafer 582 

Piesometer : 582 

Pilot Tube Gauge 688 

The Venturt Meter 688 

Measurement of Discharge by means of Nozzles 684 



XVIU C0KTEK18. 

PAOB 

flow through BeoftangularOrifloet...*. 664 

Hea8urem«Bt of aa Open Stream 064 

Mfnem' iBoh Measurements • 065 

Plow of Water orer Weirs 068 

Francis's Formula for Weirs 686 

WeirTbble 687 

Basin's Experiments 6d7 

Wafeer^powerw 

^werofsFhUofWater M 

Horse-power of a Running Stream 6w 

Current Motors 680 

Horse-power of Water Flowing in a Tube 68D 

Maximum Efllclency of a Long Conduit 880 

MUlpower :. 680 

Value of Water-power , 500 

The Power of Ocean Waves • 690 

UUlbcation of Tidal Power OOO 

• Ttirblne Whaeli. 

Proportions of Turbines » 601 

Tests of Turbines 606 

Dimensions of Turbines ••••»•• 607 

The Pelton Water-wheel •••-• 607 

Pumps. 

Theoretical capaolty of a pump , • 601 

Depth of Suction 6<M 

JUnoant 01 Water raised by a Single-acting Lift-pump. 60(1 

Proportioning the Steamcy Under of a Direct-acting pump 6w 

Speedof Water through Pipes and Pump -passages 80) 

SuBes of Direct-acting Pumps 60S 

The Deane Pump 006 

EAoiency of Small Pumps. *......*»»• .» » ... ••*... 606 

The Wonhington Duplex Pump 604 

Speed of Piston »> 605 

Speedof Water through Valres-... 605 

BoUer-feed Pumps ...*»..*••. 006 

Pump Valves « 000 

Oentrlfueal Pumps 606 

Lawrenoe Centrifugal Pumps 007 

BMciency of CeotrifUgal and Reciprocating Pumps > 606 

Vanes ofCentrifugal Pumps 600 

Tlie Gentrifugal Pump used as a Suction Dredge 600 

Duty Trials of Pumping Engines 600 

Leakage Tests of Pumps 611 

Vacuum Pnmps 616 

ThePulsometer..* •».. 610 

TheJetPump ..,..614 

The loieofior. ....••..».. 614 

Air-lift Pump 614 

The Hydraulic Ram 614 

Quantity of Water Delivered by the Hydraulic Ram 615 

Hydraulic Pressure Transmission. 

Energy of Water under Pressure 616 

Emciency of Apparatus » .*... 616 

Hydraulfo Presses .. . 617 

Hydraulic Power in London • • 617 

Hydraulic Riveting Machines 616 

Hydraulic Forging 616 

The Aiken IntensKler... 610 

Hydraulic Engine 610 



FUKIi. 



tlieory of Combustion 

Total Heat of Gombustloo. 



CONTENTS. XIX 

PAGB 

AottlyBeBofGaflMofOombiMtlOB • • •• •••».%» W« 

Tempentture of the fire »••••.»•»••.•» ttS 

Classification of Solid Fuel 623 

ClaffiificatioQ of Ooala 634 

Analyses of Coals eoi 

Western XJsnites 681 

Analyses of Foreign Coals.... «.» «... 681 

Nizon^s Kavigatlon Coal 682 

SampUnirCoal for Analyses »••••• %.* ». 68S 

ftdctive value of Fine Slses * CSS 

Prassed Fuel 68^ 

Htlative Value of Steam Coals •• 688 

Approximate Heating Value of Coals », • ..» 634 

Kind of Furnace Adapted for Dilferent Coals. 685 

Downward-draught Furnaces. »..».. 635 

Oalorimetric Tests of American Coals 636 

BTaporatiye Power of Bltumi nous Coals •• • ».... 686 

Weathering of Coal... ,...,.»..• 687 

Coke 687 

Experiments in Coking •»••.•. ,•••.... »....» •• 687 

Coal Washing. TT. 688 

Hecovenr of By-products in Coke manufacture 688 

Making Hard Coke 688 

(veneration of Steam from the Wadte Heat and Gases from Coke-oyens. 638 

Products of the DistiUaiion of Coal » 680 

Wood as Fuel 680 

Heating Value of Wood 689 

Composition of Wood 640 

Charcoal ..640 

Yield of Chsrooal fkt>m a Cord of Wood 641 

Ooosumptiott of Clmreoai In BlaKt Furnaces. 641 

Absorption of Water and of Oases by Charcoal 641 

Cr)mposltion of Charcoals 64)1 

Miscellaneous Solid Fuels 642 

Dust-fuel^Dust Exploeioitt 642 

Peat or Turf • 64« 

Sawdust as Fuel ,..« »..« 618 

Horse-manure as Fuel •• » 648 

Wet Tan-bark as Fuel , 648 

Straw as Fuel * 648 

Bsgaase as Fuel In Sugar Hanufacture • 648 

Petroleum* 

Producteof DistlUatloo , 646 

UmaPtetroleora..*. • ••»..• •• •*.•»... 646 

Value <^Petx«leum as Fuel 646 

Oil M. Goal as Fuel , ...646 

Fuel Gas. 

Oirtxmaas 646 

Anthracite Gas 617 

BttuminoosGas 647 

WaterOaa 648 

Produeer^gas from One Ton of Coal 649 

NatunU Gas in Ohio and Indiana 649 

Combustion of Produoer^gas • 650 

?8e of Steam in Producers 690 

Gas Fuel for Small Furnaces 661 

Uluminatiog Gas» 

Ooal-gss ^ 661 

Water-gas 659 

Analyses of Water-gas and Coal gas 658 

C^lnriflc Equiralents of Constituents 654 

CAciettey of a Water-gas Plant 654 

Space Required for a Water-gas Plant 656 

ml-Tahieotldttminattiag-gas 0B6 



XX CONTENTS. 

PACIS 

Flow of Gas In Piped •«, .•••». • «••••... 697 

Senrice for Lamps ». .« «. : • 6S8 

Tenoperatare and Pressure ..*...•.* 6W 

rrotal Heat 669 

Latent Heat of Steam 650 

Latent Heat of Volume 660 

Specific Heat of Saturated Steam 660 

Density andVolume 660 

Superheated Steam 661 

RegnaulVs Experiments 661 

Table of the Properties of Steam 663 

Flow of Steam. 

Napler*s Approximate Rule 669 

Flow of Steam in Pipes ,.. 669 

Loss of Pressure Due to Radiation 6ri 

Resistance to Flow by Bends 673 

Siaes of Steam-pipes for Stationary EoRlnes 673 

Sixes of Steam-pipes for Marine Engines 674 

Stean& Pipes. 

BurstlnflT-teets of Copper Steam-pipes ••••• 674 

Thickness of Copper Steam-pipes... 075 

Reinforcing Steam-pipes 676 

Wire-wound Steam-pipes 675 

BiTcted Steel Steam-pipes 676 

Valves in Steam-pipeA 675 

Failure of a Copper Steam-pipe 676 

The Steam Loop 67V 

Loss from an Uncovered Steam-plpe cn 

THIS STEAM BOILBB. 

The Hon»*power of a Steam-boiler. 077 

Measures for Comparing the Duty of Boilers 078 

Steam-boiler Proportions 078 

Heating-surface 678 

Horse-power, Builders* Rating 679 

Grate-surface •., 680 

Areasof Flues 680 

Air.passages Throusrh Grate-bars 6R1 

Performance of Boilers 681 

Conditions which Secure Economy 68*^ 

Efficiency of a Boiler 688 

Tests of Steam-boilers 685 

Boilers at the Centennial Exhibllion 685 

Tests of Tubulous Boilers 686 

High Rates of Evaporation 687 

Economy Effected by Heating the Atr , 687 

Results of Tests with Different Coals 688 

Maximum Boiler Efficiency with Cumberland Coal 689 

Boilers Using Waste Gases 689 

Boilers for Blast Furnaces 68S 

Rules for Conducting Boiler Tests 696 

Table of Factors of Evaporaiiou 695 

Streuipth of Steam-boilers. 

Rules for Construction 700 

Shell-plate Form uln 701 

Rules for Flat Plates 7D] 

Furnace FormuliB 704 

Material for Stays 709 

Loads allowed on Stays 70S 

Girders 703 

Rules for Constructir^n of Boilere in Merchant Vessels in U. 8 706 



CONTENTS. j xxi 

PA« 

U.S. Rule for AIIowabtoPreMares 706 

Safe- working Pressures 707 

Rules GoTemiof? Inspection of Boilers In Philadelphia 708 

Fhies and Tubes for Steam Boilers 709 

Flat-stayed Surf aoes 709 

Diameter of Staj-bolts 710 

Strengtli of Stays 710 

Stay-bolts In Curved Surfaces 710 

Boiler Attachments, Famaces^ eto. 

Fusible Plugs 710 

Steam Domes 711 

Height of Furnace 711 

MeSianicat Stokers 711 

The Hawlev Down draught Furnace 719 

Under-feed Stokers 719 

Smoke Prevention 719 

Oas-flred Steam-boilerB 714 

Forced Combustion 714 

Fuel Economizers 715 

Incrustation and Scale 718 

Boiler-scale Compounds. 717 

BemoTalof Hard Scale 718 

Corrosion in Marine Boilers 719 

UseofZIno 790 

Effect of Deposit on Flues 790 

Dangerous Boilers • 790 

Safety Valves. 

Bnles for Area of Safety-valves 791 

Spring-loaded Safety-valves 794 

The Ii^ector* 

Equation of the Inieetor 795 

Performanoe of Injectors 796 

BoOsr-feeding Pumps 796 

Feed-water Heaters* 

StrafauiOMised by Cold Feed-water 787 

Steam Separators* 

Eflldeiioy of Steam Separators 798 

Determination of Moisture In Steam. 

Oolt Oslorimeter 799 

Throttling Calorimeters 799 

SeparaUng Calorimeters 780 

Identification of Drv Steam 780 

Usual Amount of Moisture in Steam 781 

Chlmnejrs. 

Chimney Draught Theory 781 

Force or Intensitv of Draught 789 

Bate of Combustion Due to Height of Chimney 783 

High Chimneys not Necessary 784 

Heights of Chimneys Required for Different Fuels ..734 

Table of Sise of Chimneys 784 

Protection of Chimney from Lightning 786 

Some Tall Brick Chimneys 787 

Stability of Chimneys 788 

Weak Chimneys 789 

Steel Chimneys 740 

Sheet-iron Chfanneys 741 

THIS STEAM ENGINE. 

Expansion of Steam 749 

Mean and Terminal Absolute Pressures 748 



J 



XXll CONTENTS. 

PAOK 

Oalcnlatfon Of Mean Effective Pressure....* • • 744 

Work of Steam in a Single Cylinder 746 

Measures for Comparing the Duty of Engines , ••., 748 

Efficiency, Thermal Uulis per Minute 749 

Beal Ratio of Expansion , 760 

Effect of Compression 751 

Clearance in Low and High Speed Engines 751 

Cylinder- condensation 752 

water-consumption of Automatic Cut-off Engines 753 

Experiments on Cylinder-condensation 753 

Indicator Diagrams 754 

Indicated Horse-power 755 

Rules for Estimating Horse'power 756 

Horse-power Constant 756 

Errors of Indicators 756 

Table of Engine Constants 756 

To Draw Clearanoeon Indicator-diagram 759 

To Draw Hyperbolic Curve on Indicator-diagram 759 

Theoretical Water Consumption .. 760 

Leakage of Steam 701 

Compound Engines. 

AdTantoges of Compounding ,« ,. 709 

Woolf and Receiyer Types of Engines 702 

Combined Diagrams •.., 764 

Proportions of Cylinders InCompound Engines 706 

Beceiyer Space 706 

Formula for Calculating Work of Steam.... 767 

Calculation of Diameters of Cylinders 768 

Triple-expansion Engines 709 

Proportions of Cylinders 709 

Annular Ring Method 709 

Rule for Proportioning Cylinders 771 

Types of Three-stage Expansion Engines 771 

Sequence of Cranks 772 

Velocity of Steam Through Passages 772 

8 uadruple Expansion Engines 772 
iameters of Cylinders of Marine Engines 773 

Pi-ogress in Steam-engines 7/3 

A Double-tandem Triple-expansion Engine 773 

Principal Engines, World's Columbian Exhibition, 1898 774 

Steam Eng^lne Economy. 

Economic Performance of Steam Engines 775 

Feed-water Consumption of Differ**nt Types 775 

Sisesand Calculated Performances of \ertical High-speed Engines 777 

Most Economical Point of Cut-off 777 

Type of Engine Used when Exhaust-steam is used for Heating 780 

Comparison of Compound and Single-cylinder Engines ..... 780 

Two-cylinder and Three-cy Under Engines 781 

Effect of Water in Steam on Efficiency 781 

Relative Commercial Economy of Compound and Triple-expansion 

Engines 781 

Triple-expansion Pumplng-eogines 782 

Test of aTripIe-expansion Engine with and without Jackets 783 

Relative Economy of Engines under Variable Loads 783 

Efficiency of Non-condenaing Compound Engines 784 

Economy of Engines under Varying Loads 784 

Steam Consumption of Various Sizes 785 

Steam Consumption in Small Engines 780 

Steam Consuniption at Various Speeds 780 

Limitation of Engine Speed 787 

Influence of the Steam Jacket 767 

Counterbalancing Engines 788 

Preventing Vibrations of Engines... 789 

Foundations Embedded in Air 789 

Cost of Coal for Steam-power 7B9 



0#NTENT8. XXlll 

PAOB 

Storinr Steam Heat •••«. 789 

Coet of Steam-power • 790 

Botary 8team««ii|;lnea» 

Steam Turbines •••• 991 

The Tower Spherfoal Eogine •• Ttti 

Dimensions of Parts of Bngines. 

^liDder 798 

Clearance of Piston 798 

ThlckDees of Cylinder 798 

Cjlinder Heads 794 

CyUnder-head Bolts 795 

Tbe Piston 795 

Piston Packiog-rings 796 

fit of Piston-rod 796 

Diameter of Piston-rods 797 

Piston-rod Guides 798 

The Connectiog-rod 799 

Connecting-rod Bnds 800 

Tapered &>nnecting-rod8 801 

TheOrank-pin ail 

Crosshead-pin or Wrist-ptai 804 

The Crank-arm 809 

The Shaft, Twistins Resistance . 806 

Besistance to Bendmg 808 

EquiTalent Twisting Moment 808 

Fly.wheel Shafts 809 

Length of Shaft-bearings 810 

Crank<«hafts with Centre-crank and Double-crank Arms 818 

Crank-shaft with two Cranks Coupled at 90* 814 

ValTe-fltem or VolTe-rod 815 

Size of Slot-link 815 

The Eccentric 816 

Tbe Eooentric-rod ....•• 816 

Rerersing-gear 816 

Engine-frames or Bed-plates 817 

Flywheels* 

Weight of Flv-wheels 817 

Oentrifugal Force in Fly-wheels 680 

Anns of Fly-wheels and Pulleys 880 

Diameters for Various Speeds • •• • 881 

Strains in the Rims 888 

Thickness of Kims 888 

A Wooden Rim Flywheel 884 

Wire-wound Fly-wheels • 694 

The SUde-TalTe. 

Definitions. Lap, Lead, eta 884 

Sweet's ValTe^iagram • 8S6 

The 2Seuner Valve-diagnun 887 

Port Opening.. 898 

Lead 829 

Inside Lead 8:29 

Ratio of Lap and of Fort-openlDg to Valve-travel 849 

Crank Angles for Connecting-rods of Different Lengths • 880 

Relative Motions of Crosshead and Crank 831 

Periods of Admission or Cut-off for Various Laps and Travels. 881 

Diagram for Port-opening, CutK>ff, and Lap 888 

Piston-valves 834 

Setting the Valves of an Engine 884 

To put an Engine on ite Centre • 884 

Link-motion • 834 

Goremors. 

Fendnlam or Fly-baU Gtovemors 886 

To Change the Speed of an Engine 887 



XXIT CONTENTS. 

PAO« 

Fly-wheel or Shaft-go^ernon • «.. 888 

Calculation of Springs for Shaft-governors 888 

Condensersy Air-pumps, Clronlating-pumps, etc. 

Tlie Jet Condenser 889 

Ejector Condensers 840 

The Surface Condenser.... 840 

Condenser Tubes • 840 

Tube-plates 841 

Spacing of Tubes 841 

Quantity of Cooling Water 841 

Air-pump 841 

Area through Valve-seats 84S 

drculating-pump 843 

Feed-pumps for Marlne-engioes 848 

An EvaporatlTe Surface Condenser. 844 

Continuous Use of Condensing Water 844 

Increase of Power by Condensers 846 

Evaporators and Distillers 847 

GAS, PBTBOI«BUM, AND HOT-AIB ENGINES. 

Gas-engines 847 

Efllciency of the Gas-engine 848 

Tests of the Simplex Gas Engine 848 

A 8S0-H.P. Gas-engine. 848 

Test of an Otto Gas-engine 849 

Temperatures and Pressures Developed 840 

Test of the Clerk Gas-engine 849 

Combustion of the Gas in the Otto Engine 849 

Use of Carburetted Air in Gas-enghies 849 

The Otto Gasoline-engine 850 

The Priestman Petroleum-engine 8G0 

Test of a 5-H.P. Priestman Petroleumrengine 850 

Naptharengines 851 

Hot-air or Caloric-eiigine& 851 

Test of a Hot^Ur Engine 861 

I«OCOMOTlVES. 

Efftciency of Locomotives and Resistance of Trains 861 

Inertia and Resistance at Increasing Speeds 868 

Efficiency of the Mechanism of a Locomotive •,. 864 

Slse of Locomotive Cylinders 854 

Size of Locomotive Boilers 855 

Qualities Essential for a Free-steaming Locomotive 866 

Wootten^s Locomotive 865 

Grate-surface, Smoke-stacks, and E^haust-noszles for Locomotives. . .. 865 

Exhaust Nozsles .. 856 

Fire-brick Arches. 860 

Size, Weight, Tractive Power, eta 860 

Leading American Types 868 

Steam Distribution for High Speed • 868 

Speed of Railway Trains. 880 

Dimensions of Some American Looomotives. 869-868 

Indicated Water Consumption 86S 

Locomotive Testing Apparatus .. 868 

Waste of Fuel in Locomotives 868 

Advantages of Compounding. .. 868 

Counterbalancing Locomotives • 864 

Maximum Safe Load on Steel Rails 685 

Narrow-guage Railways. e. 865 

Petroleum* burning Locomotives. 866 

Fireless Locomotives.... 860 

SHAFTING, 

Diameters to Resist Torsional Strain 867 

Deflection of Shafting.... 868 

Horse-power Transmitted by Shafting: . 863 

Tftble for Laying Out Shafung. 871 



CONTENTS. ^XXY 

puixsni 

PAOB 

Proportions of Pbllayt ••••^. ••••••• • 878 

Convexitr of Pulleys. •• .,« 874 

Cooo or B^p Pulloys. ,« • 874 

Theonr of Belts and Banda 870 

CentrfftiSBl Tension. • •.••••• 870 

Belting rrsctioe,FormuliB for Belting...... 877 

Hone-power of A Belt one inch wide 878 

A.F. Ka£le*8 Formula 878 

'Width or Belt for OiTen Hone-power. 879 

Taylor's Rules for Belting 880 

Koteson Belting... 888 

Lacing of Belts. 888 

Setting a Belt on Quarter-twist 883 

To Find the Lengtn of Belt. 884 

To Find the Ani^e of the Arc of Contact. 884 

To Find the Length of Belt when Closely Boiled 684 

To Flzid the Approximate Weleht of Belts .884 

Relations of the Size and Speeds of Driving and Driven Pulleys 884 

EtIIs of Tight Belts. 880 

Sag of Belts 885 

Arrangements of Belts and Pttll^s 885 

Careof Belts 880 

Strength of Belting. 880 

Adhesion, Independent of Diameter. 886 

EndiessBelts. 880 

Belt Data. 886 

Belt Dressing. 887 

Oemeot for Cioth or Leather 887 

Rubber Belting. 887 

GEARING. 

Pitch, Pfteh-efrde, eto 887 

Diametral and Circular Pitch 888 

ChordalFitch R89 

Diameter of Pitch-line of Wheels from 10 to 100 Teeth. 889 

Proportions of Teeth. 889 

Proportion of Qear-wheels 801 

Width of Teeth 891 

Bales for Oaleulating the Speed of Gears and Pulleys 891 

HiUing ChittecB for Interchangeable Geare 892 

Forms of the Teeth. 

The Cjrcioldsl Tooth 893 

The Involute Tooth 894 

Approzimrtlon by Circular Arcs 896 

Stepped Gears 89? 

TwtSed Teeth 897 

Spiral Gears 897 

worm Gearing • 897 

Teeth of Bevel-wheels ... 898 

Annular and Differential Gearing 898 

Effldeopy of Gearing 899 

Strength of Gear Teeth* 

Varloas F6nnul» for Strength 900 

ComparisoDotFormulo.. 008 

Maximum Speed of Gearing ...••• 906 

A Heavy ICachine-cut Spur-gear 906 

Frictlonal Gearing 905 

FrietloQal Grooved Gearing 906 

HOISTING. 

Weight and Strength of Cordage 906 

Wofting Strength of Blocks 906 



XXTi CONTENTS. 

PAOB 

Sflloleiicy of Cfhain-blocks 907 

ProporUona of Hooks ..•« .• • OOT 

power of HoUtiii« Euglnes ,.,, , OOS 

Effect of Slack Rope on Strain in Hoisting • 908 

Limit of Depth for Hoisting , 908 

Large Hoisti n g Reco rd s 906 

Pneumatic Hoisting « 909 

Counterbalancing of Winding-engines 909 

Belt Conveyors , , «#.••• t 91t 

Bands for Canning Grain , ,....,,.,,,.,.,, 9U 

Cranes. 

Classification of Cranes ' ».,...,•• 911 

Position of the Inclined Brace In a Jib Crane , , 919 

ALarge TravelUng-crane ,,., 919 

A 150-ton Pillar Crane , ,.,.•. ••.,..., 919 

Compressed-air Travelling Cranes ..••,..,.. 919 

Wtre-rope Hanlagv, 

Self-acting Inclined Plane ,..., «,,••. .,.,....,.. 918 

Simple Engine Plane ,,.• ,.„, 918 

Tail-rope System ., 918 

Endless Rope System , 914 

Wire-rope Tramways ««t.*.«**f.t«.«. ••.«••» 914 

Suspension Cableways and Cable Hoists , 915 

Stress in Hoisting-ropes on Inclined Planes 915 

Tension Required to rreyene Wire Slipping on Drunif. . • 910 

Taper Ropes of Uniform Tensile Strength , ,.«•,... 910 

Effect of Various Sized Drums on the Life of Wire Ropes • 917 

WIRB-ROPE TRANSMISSION. 

Elastic Limit of Wire Ropes 917 

Bending Stresses of Wire Ropes 918 

Horse-power Transmitted 919 

Diameters of Minimum Sheayes... ■ '. 919 

Deflections of the Rope WO 

Long-diatanoe Transmission 9Gi 1 

ROPE DRIVING. 

FormulsB for Rope Driving 999 

Horse-power of Transmission at Various Speeds , . , 9d4 

Sag of the Rope Between Pulleys , , , . • 925 

Tension on the Slack Part of the Rope 029 

Miscellaneous Notes on Rope»drlving 990 

FRICTION AND I«UBRICATION. 

Coefficient of Friction 998 

Rolling Friction 998 

Friction of Solids 9>« 

Friction of Rest 92S 

Laws of Un lubricated Friotlon 098 

Friction of Slidlug Steel Tires 928 

Coefficient of Rolling Friction , 0^9 

Laws of Fluid Friction 989 

Angles of Repose , 9-^ 

Friction of Motion 9*^ 

Coefficient of Friotlon of Journal 980 

Experiments on Friction of a Journal 081 

Coefficients of Friction of Journal with Oil Bath 039 

Coefficients of Frioiion of Motion and of Rest 089 

Value of Anti-friction Metals , 089 

Castrlron for Bearings 988 

Friction of Metal Under Steam-pressure OSU 

Morin*8 Laws of Friction ••*..• • .... 083 



CONTENTS. XXTXl 

PAOB 

Laws of Friction of welMubrlcated Journals 964 

Allowable PrsMuree on Bearing^Buif ace. MS 

Oil-prefl8ure in a Bearing 987 

Friction of Car-Journal firaaiies Wf 

Experinoents on Overheatingr of Bearings 088 

Moment of FrioUon and Work of Friction 088 

FlTot Bearings 089 

The Schiele Cunre 088 

Friction of A Flat FlTOt-bearing. 080 

XercuiyOMth Pivot MO 

Ball Bearings. 940 

FricUon Boners. 940 

Bearings for Vei7 High BotatlVe Speed 041 

Friction of Steam-engines , 941 

Distributioo of the Friction of Engines. 941 

IfUbricatloii. 

Durability of Lubricants 942 

Qualiiloations of Lubricants 948 

Amount of Oil to run an Engine 948 

Examination of Oils. 944 

F«ina. R. R. Specifloations 944 

Solid Lubricants 945 

Graphite, Soapstone, Fibre^rapbite, MetaUue • 045 

THE rOUMDBY. 

Cupola Practice.. M6 

Charging a Cupola 948 

enlarges in StOTC Foundries 949 

Resalts of Increased Driving. , 949 

Pressure Blowers 960 

IxMSoflroninMalUng 960 

Une of Softeners • 950 

Shrinkage of Castings. 961 

Weight of CasUngs from Weight of Pattem 958 

MouUOngSand 9Gii 

Foundiy Ladles 988 

THE MACHINE SHOP. 

Speed of Cutthig Tools 968 

*riible of Cutting Speeds. 954 

Speed of Turret Lathes ...... 954 

r^rms of Cutting Tools 955 

Rule for Gearing Lathes 955 

Change-gears for Lathes 056 

Ketrle Screw-threads.. 056 

Setthag the Taper in a Lathe. 056 

Speed of Drilling Holes , 066 

Speed of Twis^driUs. 057 

lOlling Cutters 057 

Speed of Cutters 968 

Brsolts with Milling-machines 959 

Killing with or Aff&lnst Feed 900 

Milling-machine v«. Planer 060 

Power Required for Machine Tods. 060 

Heavy Work on a Planer 960 

Honei>ower to mn Lathes 061 

Power used by Machine Tools. 968 

Power Required to Drive Mschinery 964 

Powernsed tai Machlne^hops. 966 

Abrasive Prooeases. 

The Cold Saw 006 

Beeae'sFttshig^lsk 960 

Cutting Stone with Wire . 966 

The Sand-blast ... 966 

Emery-wheels • 067-969 

Orindstonea 068-970 



XXVIU CONTENTS, 

Tarlons Took and Trooesseo, 

Taps for Machine-Borews. , • ~.~m 

TapDriUa.. , 9n 

Taper Bolts, Pins, Reamers, eta.. <,»> 978 

Punches, Dies, Presses 078 

Clearance Between Punch and Die.... 078 

Size of Blanks for Drawing^press 078 

Pressure of Drop-press.. -, 073 

Flow of Metals .r. 973 

Forcing and Shrinking Fits 079 

Efficiency of Screws «< 074 

Poweirs Screw-thread 978 

Proportioning Parts of Machine. • 075 

Keys for Qearing, etc • 075 

Holding-power of Set-screwB • .- 077 

Holding-power of Keys 078 

DTNAMOMETEICS, 

Traction Dynamometers 078 

The Prony Brake 078 

The Alden Dynamometer 070 

Capacity of Friction-brakes 060 

Transmission Dynamometers 980 

ICB MAKIlfO OB BEFBIGEBATIlfO MACHINES. 

Operations of a Refrigerator-machine 061 

Pressures, etc.. of Available Liquids 069 

Ice-melting EflTect 08S 

Echer-machines 06S 

Air-machines 068 

Ammonia Compression-machineB 068 

Ammonia Absorption-machines. 0B4 

SulphttiMliozide Machines. 066 

Performance of Ammonia Compression-machines. 066 

Economy of Ammonia Compression-machine 067 

Machines UsingVapor of Water 068 

Efficiency of aitefrigerating-machine 088 

Test Trials of Refrifcerating-machines 008 

Temperature Range 901 

Metering the Ammonia 002 

Properties of Sulphur Dioxide and Ammonia Qas 008 

Properties of Brine used to absorb Ref ligeratlng EflTect. 094 

Chloride-of-calcium Solution 004 

Actual Performances of BefirlgerattDg Machines. 

Performance of a 75-ton Ref rigerating-machine 004, 006 

Cylinderheating 007 

Tests of Ammonia Absorption-machine i 907 

Ammonia Compression-machine, Results of Tests •... 000 

Means for Applying the Cold 000 

Artlfloial Io« -manufacture. 

Test of the New York Hygeia loe-making Plant. 1000 

MABINE ENGINBEBINO. 

Rules for Measuring Dimensions and Obtaining Tonnage of Vessels. . .. 1001 

The Displacement of a Vessel 1001 

Coefficient of Fineness lOfti 

Coefficient of Water-lines 1008 

Resistance of Ships. 1008 

Coefficient of Performance of Vessels. 1008 

Defects of the Common Formula for Resistance 1008 

Rankine^s Formula. • lOOS 

Dr. Kirk's Method 1004 

To And the I. H. P. from the Wetted Surface 1006 

E. R. Mumford^a Method 1O08 

Belative Horse-power required for 'litferent Speeds of Vessels 1000 



COiJTENTS. IXIX 

PAOB 

BnifltiMioojMr Hono*powor for dlffsrent Spoods.. ••••••.•••••••• ••••••• 1000 

Results of Trials of Steam-Tesaels of Yarious Sixes 1007 

Speed on Ckuials, 1008 

Results of ProgresslTe Speed-trials Id laical Vessels. 1006 

Zst t matod Displaoement, Horse-power, ete., of Steam-Tessels of Tarious 



Tlie Sorow-propollMw 

8Iie of Sereir. • 1010 

Propeller Ooefflctents 1011 

£fflcieoc7 of the Propeller 1013 

Pitch-ratio and Slip for Screws of Standard Fonii..., 1018 

Results of Recent Researches. 1018 

The Paddle-wheel* 

Fsddle-wheel with Radial Floats. 1018 

Feathering Paddle-wheels • 1018 

EffldemgrofFaddle-wheels 1014 

Jet-propvlsioii. 

BesctioaoCaJet 1016 

Beeent PnMstlee In Harlne Bni^lnea. 

Forced Draught 1015 

Boilers.. 1015 

Piston-Talvea. 1016 

Steam-pipes 1016 

AuxiliaiT Supply of Fresh-water Eraporators. 1016 

Weir*s Feed-water Heater. 1016 

Passenger Steamers fitted with Twin-screws. 1017 

OomparatlTte Results of Working of Marine-engine, 187S, 1881, and 1801.. 1017 

WeJid^tof Three*etageEzpaDfdonenglne8 • 1017 

Partlonlars of Three-stage EzpansioD^Dgines. 1018 

OONSTBUCTION OF BUIU>ING8. 

Walls of Warehouses, Stores, Factories, and Stables 1010 

Strength of Floors, Roofs, and Supports. 1019 

Columns and Poets.... 1010, 10s» 

Fireproof Buildings 10*20 

Iron and Steel Columns 1020 

lintels. Bearings, and Supports. 1020 

Strains on Oirders and RfTcts. 1030 

Maxlmnm l4>ad on Floors 1031 

Strenctli of Floors 1031 

Safe Mtribated Loads on Southern-pine Beams .. 1038 

BLBCTRICAI. ENOIMEBBING. 
Standards of M easarement* 

CL 0. 8. t^ystem of Physical Measurement 1034 

Practical units used in Electrical Calculations 1034 

Relations of Various Units 1085 

EqniTalent Electrical and Mechanical Units 1036 

Analogies between Flow of Water and Electricity 10S7 

Analogy between the Ampere and Miner's Inch 1087 

Blectri«ttl Beslstanee* 

Laws of Sleotrleal Resistance 1088 

Equivalent Conductors 1088 

BectricalConductfri^ of Different Metals and Alloys 1088 

BelatiTeConductlrity of Different Metals lOSO 

Conductors and Insulators ...••• 1080 

Resistance Varies with Temperature , 1080 

Annealing 1080 

Standard of ResfstSDoe of Copper Wire 1000 

Eleoirio Oorrents. 

Ohm^Law 1000 

DIfldsdCbentti lOU 



XXX COKTKKTS. 

9Aam 

Conductors In Series • • • • 1061 

Internal Resistance • 1081 

Joint Resistance of TwoBraii6be8 ..••• 1088 

KlrchholTs Laws 108S 

Power of the Circuit •••. 1088 

Heat Generated by a Current • 1088 

Heating of Conductors 1033 

HeaUng of Wires of Cablet 1063 

Oopper-wire Table 1084» 1085 

HeaUngof Colls 1086 

Fusion of Wires • 1087 

BI«etrf« Trftiismlssloiu 

Section of Wire required for a Given Current 1038 

Constant Pressure 1038 

Short-circuiting 1039 

Economy of Electric Transmission 1039 

Table of Electrical Horse-powers 1041 

Wiring FormulflB for Incandencent Lighting ]04*J 

Wire Table for 100 and 600 Volt Circwts 1043 

Cost of Copper for Long-distance Transmission 1044 

Weifcht of Copper for Long-distance Transmission 1044 

Efficiency of Long-distance Ti-ansmission : 1045 

Systems of Electrical Distribution 1046 

Ismclency of a Combined Engine and Dynamo 1047 

Electrical Efficiency of a Generator and Motor — 1047 

Efficiency of an Electrical Pumping Plant 1048 

£leotrlc Hallways. 

Test of a Street-railway Plant 1046 

Bleetrle Lighting. 

Life of Incandescent Lamps 1040 

Life and Efficiency Tests of Lamps 1049 

Street Lighting 1049 

Lighting-power of Arc-lamps 1060 

Candle-power of the Arc-light 1050 

Blectric Welding ' 1051 

JBlectrlo Heaters 1058 

Electric Aooumulators or Storage-batteries. 

Sixes and Weights of Storage-batteries .' 1064 

Use of Storage-batteries in Power and Light Stations 1055 

Working Current of a Storage-cell 1055 

Slectro-chemloal Bqulvalents 10.VO 

Kleotrolysls 1056 

Bleotro-m agnets. 

Units of Electro-magnetic Measurement 1067 

Lines of Loops of Force. 1068 

Strength of an Electro^magnet 1058 

Force in the Gap between Two Poles of a Magnet )0'>9 

The Magnetic Circuit.. 1009 

Determining the Polarity of Electro-magnets 1050 

I>yiiaiiio*Electrlc Machines. 

Kinds of Dynamo-electric Machines as regards Manner of Winding 1060 

Current Generated by a Dynamo-electric Machine 1060 

Torque of an Armature 1061 

Electro-motive Force of the Armature Circuit 1061 

Strength of the Magnetic Field 1062 

Application to Designing of Dynamos 1068 

Permeability 1064 

Permissible Amperage for Magnets with Cotton-covered Wire . . , 1065 

Form ulflB uf Efficiency of Dynamos IOCS 

The Electric Motor 1066 

Table of Standard Belted Motors and Generators 1067 



CONTENTS. ■ XXXI 

API'KNDIX. 

Str«iig^1i of Timber. 

PAOK 

Safe Load on Wbite-oak Beams lUGi: 

Mathematics. 

Formula for Interpolation 1070 

Maxima and Minima without the Calculus lOTO 

Riveted JoiiitH. 

Pressure Required to Drive Hot Rivets 1070 

Heating and Ventilation. 

(Capacities for Hot-biast or Plenum Heating witU Fans and Blowers. .. 1071 

Water-wheels. 

Wat«T-powpr Plants OperatiitR under High Pressure ...... 1071 

FormulsB for Power of Jet Water-wheels 1079 

Gaa Fuel. 

Composition, Energy, etc., of Various Oases 10?2 

Steann-bollers. 

Rules for Steam-boiler Construction lOTS 

The Steam-engine. 

Current Practice in Engine Proportions 1074 

Work of Steam-turbines 1075 

BelativeCostof Different Sizes of Engines 1075 

I^iocomo lives. 

Resistance of Trains 1075 

Performance of a High-speed Locomotive 107ft 

Ijocomotive Link Motion 1077 

Gearing. 

ElBcieDcy of Worm Gearing 10?fi 



NAMES AND ABBREVIATIONS OF PERIODICALS 
AND TEXT B00K8 FJiEQUENTLY REFERRED TO 
IN THIS WORK. 



Am. Mach. American Machinist. 

App. Cyl. Mech. Appleton's Cyclopiedia of Mecbanlcii, Vols. I and IL 

Bull. I. & 8. A. Bulletin of the American Iron and Steel Association 
(Philadelphia). 

Burr'A Elasticity and Resistance of Materials. 

Clarlc, K T. D. 1>. K. Clark's Rules, Tables, and Data for Mechanical En- 
jrineers. 

Clarlc, S. E. D. K. Clark's Treatise on the Steam-engine. 

Engg. Engineering (London). 

Bng. News. EnKineeiing News. 

Engr. The Engineer (London). 

Fairbaim's Useful Informaiion for Engineers. 

Flynn's Irrigaiion Canals and Flow of water. 

Jour. A. C. L W. Journal of American Charcoal Iron Workera' Association. 

Jour. F. I. Journal of the Franklin Institute. 

Kapp's Electric Transmission of Energy. 

Lanza's Applied Mechanics. 

Merriman^s Strength of Materials. 

Modern Mechanism. Supplementary Tolume of Appleton*s CyclopSBdia of 
Mechanics. 

Proc. Inst. C. E. Proceedings Institution of CItII Engineers (London). 

Proc. Inst. M. E. Proceedings Institution of Mechanical Engineers (Lon- 
don). 

Peabody*s Thermodynamics. 

Proceedings Engineers* Club of Philadelphia. 

Rankine. S. E. Rankine's The Steam Engine and other Prime Movers. 

Rankitie*s Machinery and Mill work. 

Rankine, R. T. D. Rankine's Rules, Tables, and Data. 

Reports of U. S. Test Board. 

Reports of U. S. Testing Machine at Watertown, Massachusetts. 

Rontgen's Thermodynamics. 

Seatoii's Manual of Marine Engineering. 

Hamilton Smith, Jr.*8 Hydraulics. 

The Stevens Indicator. 

Thompson's Dynamo-electric Machln*»ry. 

Thurston's Manual of the Steam Engine. 

Thurston's Materials or Engineering. 

Trans. A. I. E E. Transactltins American Institute of Electrical Engineers. 

Trans. A. I. M. K. TrnnsactionH American Institute of Mining Engineers. 

Trans. A. S. O. E. Transactions American Society of Civil Engineers. 

Trans. A. S. M. E. Transactions American Soc'ty of Mechanical ElnglneerB 

Trautwlne's Civil Engineer's Pocket ik>ok. 

The Locomotive (Hartford, Connecticut). 

Unwinds Elements of Machine Design. 

Weishach's Mechanics of EngineerU)^ 

Wood's Resistance or Materiaia 

Wood's Tbermodjuamios. 

zxxli 



MATHEMATICS. 



Arttbmettcal and Alfl^brmical Slirns and AbbreTlatlons« 



^ plus (addition). 
-r positive. 

- miDus (subtraction). 

- negative. 

± plus or minus. 
T minus or plus. 
= equals. 
X multiplied by. 
ah or a.b = a x b. 
't- divided by. 
/ divided by. 



— = a/b = a-t-b. 



,M.= L». 



^ =:: 



.002 = 



_8_ 
10' •"*** •" 1000* 

V square root. 

V cube root. 

V 4th root. 

: is to, s so !s. : to (proportion). 

2; 4s8:6,as2isto4aoi88to6. 

: ratio; divided by. 

2 : 4. ratio of 2 to 4 =s 2/4. 
.*. therefore. 
> greater than. 
< le£« than. 
□ square. 
O round. 

» degrees, arc or thermometer. 

' minutes or feet. 

'' seconds or inches. 
""" accents to distinguish letters, as 

a', a", a'". 
oi, 09' Os* Ok« Atf- i'«*d a mib 1, a sub b. 

PtC. 

() [ 1 } } vincula, denoting 

that (he numbers enclosed are 
to be take n toge ther ; as, 

(a + 5)c = 4 + 8x5 = 85. 
a*, a*, a squared, a cubed. 
u^ a raised to thejtth power. 

a-i = !,«-. = 1. 
a a* 

10» = 10 to the 0th power = 1,000,000,- 

Rin. a rr the sine of a. 

ijn.-i a s the arc whose sine is a. 

sin. a-» = -j-i— 

sin. a. 

log. = logarithm. 

*°^e ^«: Vp. log. = hyperboUc loga- 
rithm. 



Z angle. 

L right angle. 

± perpeudkular to. 

sin., sine. 

COS., cosine. 

tang., or tan., tangent. 

sec., secant. 

▼ersin., versed sine. 

cot., cotangent. 

cosec, cosecant. 

covers., co- versed sine. 

In Algebra, the first letters of the 
alphabet, a, 2>, c, d, etc., are gener- 
ally used to denote known quantities, 
and the last letters, w^ or, y^ 2, etc., 
tmknown quantities. 

AbbreviatiwiB and Symbols com- 

monly used, 
d, differential (in calculus). 
y, integral (in calculus). 

y *, integral between limits a and 5. 

d, delta, difference. 

2. Sigma, sign of summation. 

*, pi, ratio of circumference of circle 

to diameter = S. 14159. 
y, a<;celeration due to gravity = 82.16 

ft. per sec. 

Abbreviations frequently used t» 

this Book. 
L., 1., length in feet and inchee. 
B., b., breadth in feet and inches. 
D., d., depth or diameter. 
H., h., height, feet and inches. 
T., t., thickness or temperatura 
v., v., velocity. 
F., force, or factor of safety. 
f., coefficient of friction. 
E., coefficient of elasticity. 
R., r., radius. 
W.,w., weight 
P.,j>., pressure or load. 
H.P., horse-power. 
I.H.P., indicated horse-power. 
B.H.P., brake horse-power, 
h. p., high pressure, 
i. p., intermediate pressure. 
1. p., low pressure. 

A. W. Q., American Wire Qauge 
(Brown & Sharpe). 

B. W.G., Birmingham Wire Gauge, 
r. p. m., or revs, permln., revolutions 

per minute. 



KATHEHATIC& 



ABITHMETia 

The user of this book is supposed to have had a training in arithmetic as 
well as in elementaiy algebra. Only those rules are given here which are 
apt to be easily forgotten. 

ORBATRST CORiniON IHBASVRE, OR GRBATE8T 
GOniJIEON DIVISOR OF TWO NUMBBR8. 

Rule*- Divide the greater Dumber by the less ; then divide the divinor 
by tile leniainder, and so on, dividing always the last divisor by the last 
remainder, until there is no remainder, and the last divisor is the greatest 
common measure required. 

I<BA8T COMMON M17I<TIPI<B OF TWO OR MORB 
N17MBBR8. 

Rule.— Divide the given numbers by any number that will divide the 
greaiettt number of them without a remainder, and set the quotients wiih 
the undivided numbers in a line beneath. 

Divide the second line as before, and so on, until there are no two numbers 
that can be divided ; then the continued product of the divisors and last 
quotients will give the multiple required. 

FRACTIONS. 

To reduce a eominon fyactlon to Its longest terms.— Divide 
botii Ufrma by tlielr greiiiest common divisor: if = f 
To cltaii|i:e an Improper fraction to a mixed numlMr. — 

Divide the numerator by the denominator; the quotient is the whole number, 
and the remainder place<l over the denominator is the fraction: V = ^V 
To change a mixed number to an Improper fWtetlon.— 

' '"ply the whole number by the denominator of the fraction; to the prod- 



uct add the numerator* place the sum over the denominator: 1{ = V. 

To express a irnole number In the form of a fraction 
ivltli a fflTcn denominator. ~Mu I ti ply the whole number by ilie 
given df noininaior, and place the product over that denominator: 18 = V- 

To reduce a compound to a simple fWictlon» also to 
multiply fractions.— Multiply Uie numerators together for a new 
numerator and the denominators together for a new denominator: 

8-4 8 .. 2^4 8 
-of- = -, also -Xj-j,. 

To reduce a complex to a simple Aractlon.— The numerator 
and denominator must each flrnt be given the form of a simple fraction; 
then multiply the numerator of the upper fraction by the denominator of 
the lower for the new numerator, and the denominator of the upper by the 
numerator of the lower for the new denominator: 

To dlTlde ftmctlons.—Reduoe both to the form of simple fraotiong, 
invert the divisor, and proceed as in multiplication: 

8-^^* = 8-^8- = 8^4==12- 

Cancellation of Aractlons.— In compound or multiplied fractions, 
divide any numerator and any denominator by any number which will 
divide them i)olh without remainder, Htrikine; out the nunibei'S thus divided 
and setting down the quotients in tlieir stead. 

To reduce ft*actlons to a common denominator. ^-Reduce 
each fraction to tlie form of a simple fraction; then multiply each numera- 



DECIMALS. 3 

tor by all the denominators except Ite own for the new numerators, and all 
the denominators together for the common denominator: 

1 1 8^21 14 18 
2' 8' 7 42' 42* 42' 

To mdd fyaetloDS.— Reduce them to a common deooroinator. then 
add Uie iiuiiieraiors and place their sum over the common denominator: 

118^ 21-1-14-1-18 ^68^ 
2^8 7 42 42 "' 

To ■vbtimct fkmcUoiis«~Reduce them to a common denominator, 
piibtriict the uumeratoni and place the difference over the common denomi- 
nator: , 

1 _ 8 _ 7-6 _ 2. 

2 7 14 "14* 



DECIlHAIiS. 

To add decimals.— Set down the flgures so that the decimal points 
are one akK>ve tiie other, then proceed as in simple addition: 18.764- .012 = 

To unMraet deelmals.— Set down the fl^riires so that the decimal 
poinrH art- nnt* utMve the other, then proceed as in simple subtraction: 18.75 
- .012 = 18 7**, 

To maltlply decimals.— Multiply as in multiplication of whole 
numbers, iben point off as many decimal places as there are in multiplier 
and muliiplicAnd taken tosretlier: 1.5 X .02 = .060 = .08. 

To dlTlde deelmals.— Divide as in whole numbem, and point off in 
the quotient as many decimal places as those in the dividend exceed those 
in Llie divisor. Ciphers mnst be added to the dividend to make its decimal 
places at least equal those in the divisor, and as many more as it is desired 
to have in the quotient: 1.5 -«- .25 = 6. 0.1 ■+■ 0.3 = 0.10000 -t- 0.3 = 0.8833 -h 

Decimal EqnlTalents of Fractions of One Inch. 



1-64 
1-8-.' 
8-64 
1-16 


.015625 ! 
.0:J125 
.046875 
.0625 


17-64 
9-32 
19-64 
6-16 


.265625 
.28125 
.296875 
.8125 


38-64 
17-82 
85-64 
0-16 


.515625 
.58125 
.546875 
.5625 


'jn-64 
25-82 
51-64 
18-16 


.765625 
.78125 
.796875 
.8125 


5-64 
:j-82 
7-*# 

1-8 


.078196 
.00875 
.109875 
.125 


21. 64 

11-82 

28-64 

S-8 


.828125 
.84875 
.850875 
.87? 


37-64 

19-82 

8»-64 

6-8 


.578126 
.59875 
.609375 
.685 


58-64 
27-^ 
56-64 

7-8 


.828125 
.84375 
.«S9875 
.875 


9-64 
5~:« 
11-61 
S-i6 


.140625 
.15685 
.171875 
.187ir 


25-64 
18-82 
27-64 
7-16 


.800625 
.40625 
.421875 
.4875 


4!-64 
ai-32 
48-64 
11-16 


.640625 
.65625 
.671875 
.6875 


57-64 
2d-32 
59-64 
16-16 


.890625 
.90625 
.921875 
.9375 


18-61 
7-8^ 

1.V64 
1-4 


.208125 
.21875 
.284875 
.25 


29-64 

15-82 

81-64 

l-« 


.453125 
.46875 
.484875 
.80 


45-04 

28-82 

47-64 

8-4 


.706125 
.71875 
.734375 
.75 


61-64 
31-82 
68-64 

1 


.953125 
.96875 
.984375 
1. 



To conwert a common fVmetlon Into a decimal.— Divide the 
nuiiifrator by the denominAbor, adding to che niimfraior as mnnv ciphers 
prpflxed by a decimal point as are n^cHssary to give the number of decimal 
places desired in the result: U = 1 <l0u0-i-8 = 0.8333 -f. 

To eoBwert a decimal Into a common fkmetlon.— Set down 
the decimal as a numerator, and place as the denominator 1 with as many 
cioberv annexed as there are decimal places in the numerator; eras^U^e 



ABITHMETIO. 



ooh" 






ill i 



•^o. 



g S 



2 § 
8 S 



§ 



< 



S s 
S fi 



o o 



iiiilil 



S 51 



HiSiiii 



iliiiiiggii 



m^iiiiiii 



H: 



i i 1 1 iT 



ill 

3 8 & 



S S S S S £: 

I i § § I i 



I i 



= s § 

o o o 



5 S 






ii 






'H-'^nSr-. 



COMPOUND KUMBEBS. 5 

d(H.-iuial point in the numerator, and raduoe the fraction tbua fonned to lia 
lowest temM: 

•* = io6'i' -^^ = 10000 '8' "~^- 

To redvee a reennrlnc decfmal to a coBmiOM flraetloii.— 

Subtract Ujo decimal Hffurea tliat do not recur from the whole decimal in- 
cluding one set of recurrinic flfcures; set down the remainder as the numer- 
ator or the fraction, and as many nines as there are reeurrinc figures, fol- 
lowed by as many ciphers as there are non-recurring figures, m toe denom- 
inator. Thus: 

.79064054, the recurring figures being 054. 
Subtract 79 

^^ =z (reduced to ito lowest terms) |>g. 

conpoinvD on denooeinatv nuhbess. 

Redaction desc^ndlnff .--To reduce a compound num ber to a low«*r 
denoro inailon. M uUiply the n umber by as many units of the lower denoml- 
nation as makes one of the higher. 

8 yards to inches: 8 X 86 = 108 Inches. 

.01 square feet to square inches: .04 X 144 s 6.70 sq. in. 

If tho given number Is in more than one denomination proceed in steps 
from the highest denomination to the next lower, and so on to the lowest, 
adding In the units of each denomination as the oper.itloii proceeds. 

8 yds. 1ft. 7 in. to inches: 8x 3 = 9, + 1 s 10, ]0xl8=> 190, + 7 s 1:27 in. 

llodaetloii ascendlns*— To express a number of a lower denomi- 
nation in terms of a higher, divide the number by the numb' r of units of 
the lower denomination contained In one of the next higher; the quotient is 
in the higher denomination, and the remainder, if any.ln the lower. 

1-/7 inches to higher denomination. 

is;-*- 12 = 10 feet + 7 inches; 10 feet i- 8 = 8 yardn + 1 foot. 

Ans. 8 yds. 1 ft. 7 in. 

To express the result in decimals of the higher denomination, divide the 
riven number by tbe number of units of the given denomination contained 
in one of the required denomination, carrying the result to as many places 
of dednutls as may be desired. 

197 loches to yards: 197 -i- 86 = 8|t = 8.5877 + yards. 
RATIO AND PROPORTION. 

Ratio is the relatiou of one number to another, as obtained by dividing 
one by the other. 

BAtioof9to4,or9 : 4a9/4=s1/8. 
Ratio of 4 to S, or 4 : 2 = 3. 

Proportion is the equality of two ratios. Ratio of 2 to 4 equals ratio 
of .1 CO 0, 2/4 = 8/6: expressed thus, 9 : 4 : : 8 : 0; read, 2 is to 4 as 3 is to 6. 

Tbe first and fourth terms are called the extremes or outer terms, the 
Moond and third the meftns or inner terms. 

The product of the means equals the product of the extremes: 

9 : 4 : : 8 : 6; 2 x 6 = 12; 8 X 4 = 19. 

Hence, given the first three terms to find the fourth, multiply the second 
and third terms together and divide by the first. 

2 : 4 : : 8 : what number? Ans. ^ ^ - = 6. 



6 ABITHHETIC. 

Alff«bralc expreMlon of proportion.— a : b : : e : d; ^ s -i;a(l 

.- ... be , be , ad ad 

= be; from which a=^;d=— ;o = — -; c = -r-. 

]VIeaift proportional between two given numbera, Ist and 2d. Is Ruch 
a number chat the ratio which the first bears to it equals the ratio which it 
bears to the second. Thus, 2 : 4 : : 4 : 8; 4 is a mean proportional between 
fi and 8. To find the mean proportional between two numbers, extract the 
square root of their product. 

Mean proportional of 2 and 8 = 1^2 X 8 = 4. 

fllnffle Rule of Tlireo ; or, finding the fourth term of a proportion 
when three terms are given.— Rule, as above, when the terms are stated in 
their proper order, multiply the second by the third and divide by the first. 
The difficulty is to state the terms in their proper order. The term which is 
of the same kind as the required or fourth term is made the third; the first 
and second must be lilce each other in kind and denomination. To deter- 
mine which is to be made second and which first requires a little reasoning. 
If an inspection of the problem shows that the answer should be greater 
than the third term, then the greater of the other two given terms should 
be made the second term— otherwise the first. Thus, 8 men remove 54 cubic 
feet of rock in a day; how many men will remove in the same time 10 cubic 
▼ards ? The answer is to be men— make men third term; the answer is to 
be more than three men. therefore make the greater quantity. 10 cubic 
yards, the second term ; but as it is not the same denomination as the other 
term it must be reduced, = 270 cubic feet. The proportion is then stated: 

8 X STO 
64 : 270 : : 8 : X (the required number) ; x = — r-r— = 15 men. 

The problem is more complicated if we increase the number of given 
terms. Thus, in the above question, substitute for the words '' in the same 
time " the words " in 8 days." First solve it as above, as if the work were 
to be done in the same time; then make another proportion, stating it thu^: 
If 15 men do it in the same time, it will take fewer men to do it in 8 days; 
make 1 day the 2d term and 8 days tlie first, tenn. S : 1 : : 15 men : 6 men. 

Compound Proportion^ or Double Rule of Tliree*— By this 
rule are nolved questions like the one just given, in which two or more stac- 
ingsare requinxi by the single rule of three. In it as in the single rule, 
there is one third term, which is of the same kind and dt nomination as ihe 
fourth or required term, but there may be two or more first and second 
terms. Bet down the third term, take each pair of terms of the same kind 
separately, and arrange them as first and second by the same reasoning as 
is adopted in the single rule of three, making the greater of ihe pair the 
second if this pair considered alone should require the answer to be 
greater. 

Set down all the first terms one under the other, and likewise all the 
second terms. Multiply all the first terms together and all the second terms 
together. Multiply the product of all the second terms by the third term, and 
divide this product by the product of all the first terms. Example: If 8 men 
remove 4 cubic yards in one day, working 12 hours a day, how many men 
working 10 hours a day will remove 20 cubic yards in 8 days Y 

Yards 4: 201 

Days 8 : 1 : : 8 men. 

Ho urs 10 ; 12 | 

Products 120 : 240 : : 8 : men. Ans. 

To abbreviate by cancellation, any one of the first terms may cancel 
either the third or anv of the second terms; thus. 8 in first cancels 8 in third, 
making it 1, 10 cancels into 20 making the latter 2, which into 4 makes it 2, 
which into 12 makes it 0, and the figures remaining are only 1 : 6 : : 1 : 6. 

IIVVOIilTTION, OR POWBRS OF NITnBKBS. 

InTolntlon is the continued multiplication of a number by itself a 
given numl)er of times. Tiie nunil>er is called the root, or first power, and 
the products are called powers. Ttie second power is ci^Ued the square and 



POWERS OP NUMBERS. 



the third power the cube. The operation mav be indicated without being 
perf ormea by writincr a small figure called tne index or exponent to the 
ri^ht of and a little above the root; thus, 8* =: cube of 8, = 27. 

To multiply two or more powers of the same number, add their exponents; 
thiis, «« X 2* = a». or 4 X 8 = 88 = 2». 

To divide two powers of the same number, subtract their exponents; thus, 

a» -I- 2« = 2> = S; 2* -•- 2* = 2"~* = — = -. The exponent may thus be nega- 
tive. 2* ■•> 2> = 2* = 1, whence the asero power of any number = 1. The 
first power of a number is the number itself. The exponent may be frac- 
tional, as 2^, 2i, which means that the root is to be raised to a power whose 
exponent is the numerator of the fraction, and the root whose sign in the 
denominator is to be extracted (see Evolution). The exponent may be a 
decimal, as 2***, 2^**; read, two to the five-tenths power, two to the one and 
five-tenths power. These powers are solved by means of Logarithms (which 
see;. 

Flrat Nine Ponrers of tUe First Nine Numbers. 



Ist 


•2d 


8d 


4th 


5th 


6th 


7th 


8th 


9th 


Pow'r 


Pow'r 


Power. 


Power. 


Power. 


Power. 


Power. 


Power. 


Power. 


1 


1 


1 


1 


1 


1 


1 


1 


1 


2 


4 


8 


16 


82 


64 


128 


256 


512 


8 


9 


27 


81 


248 


729 


2187 


6561 


19688 


4 


16 


64 


256 


1024 


4096 


16884 


65586 


262144 


5 


25 


125 


6;» 


8125 


15625 


78125 


890625 


1958125 


6 


86 


216 


1296 


7776 


46656 


279986 


1679616 


10077696 


7 


49 


848 


2401 


16807 


117649 


828548 


5764801 


40858607 


8 


64 


512 


4096 


8er68 


20SS144 


2097152 


16777216 


134217728 


9 


81 


729 


6681 


69049 


581441 


4782969 


48046721 


887420489 









The 


First Forty Ponrers of 3. 




i 


i 

> 


1 


> 


i 

1 


> 


1 


> 


1 







1 


9 


512 


18 


262144 


27 


184217728 


86 


68719476736 


1 


2 


10 


1024 


19 


624288 


28 


268435466 


37 


137438968472 


2 


4 


11 


2048 


20 


1048576 


20 


636870912 


88 


274877906944 


S 


8 


12 


4096 


21 


2007152 


:» 


1073741824 


89 


54971»813888 


4 


16 


18 


8192 


22 


4194804 


81 


2147483048 


40 


1099511627776 


5 


82 


14 


16884 


23 


8888606 


32 


4294967296 






6 


64 


15 


32768 


24 


16777216 


83 


8589984592 






7 


128 


16 


655.36 


25 


38554482 


84 


171T»«9184 






8 


256 


17 


181072 


26 


67108864 


36 


84350788868 







BFOIiVTION. 

Birolntlon is the finding of the root (or extracting the root) of any 
number the power of which is given. 

The sign y indicates that the square root is to be extracted : V V V'* (tie 
cube root, 4th root, nth root. 

A fractional exponent with 1 for the numerator of the fraction is also 
used to indicate that the operation of extracting the root is to be performed; 

thus, 2*, 2*= V2,\% 

When the power of a number i« indicated, the fnvohiMon not heiii); p4*r- 
furmcd, the extraction of any root of that power may also be indicated by 



8 AKITHMETIO. * 

dividing the Index of the poorer by the index of the root, Indicating: the 
division bj a fraction. Tims, extract the square root of the 6th power of 2: 

|/i^=:2' = 2^ = 8* = 8. 

The 6th power of 9» aa hi the table above, is 64 ; 4/64 m a 

Difficult problems in evolution are performed by logarithms, but the 
square root and the cube root may be extracted directly according to the 
rules given below. Tlie 4th root is the square root of the square root. The 
6th root is the cube root of the square root, or the square root of the cube 
root ; the 8th root is the cube root of Ute cube root • etc. 

To Kxtraet tbo Sqnare Root.— Point off the given number into 
periods of two places each, beghming with units. If there are decimals, 
point these off likewise, bejginning m the decimal point, and supplying 
as many ciphers as may be needed. Find the greatest number whoee 
square is less than the first left-hand period, and place it as the first 
figure in the quotient. Subtract its square from the left-hand neriod. 
and to the remainder annex the two figures of the second period for 
a dividend. Double the first figure of the quotient for a partial divisor ; 
find how many times the latter is contained in the dividend exclusive 
of the right-hand figure, and set the figure representing that number of 
times as the second figure in the quotient, and annex It to the right of 
the partial divisor, forming the complete divisor. Multiply this divisor bv 
the second figure In the quotient and subtract the product from the divi- 
dend. To the remainder bring down the next period and proceed as before. 
In each case doubling the figures in the root already found to obtain ihe 
trial divisor. Should the product of the second figure in the root by the 
completed divisor be greater than the dividend, erase rhe second filfrore both 
from the quotient and from the divisor, and substitute the next smaller 

Sture, or one small enough to make the product of the second figure by the 
visor less than or equal to the dividend. 

8.14168866S6I1.77846 + 

87W4 
1189 
84712515 
12429 
864218692 
71064 



85444'160866 
J141776 



85448511908936 
11772425 

To extract ths square root of a fraction, extract the root of numeratob 
and denominator separately. a/-t ^ j^ or first convert the fraction into a 

decimal,i/^=: /iSTf = .6666 -f . 

To Rztraet tbe Cube ]Koot«~Foint off the number Into periods of 
8 figures each, beginning at the right hand, or unites jpUce. Point off deci- 
mals In periods of 3 figures from the decimal point. Find the greatest cube 
that does not exceed tne left-band period ; write its root as the first figure 
In the required root. Subtract the cube from the left-hand period, and to 
the remainder bring down the next period for a dividend. 

Sqaare the first figure of the root; multiply by 300, and divide the product 
Into the dividend for a trial divisor ; write the quotient after the first figure 
of the root as a trial second figure. 

Complete the divisor by adding to SCO times the square of the first figure, 
80 times the product of the first by the second ftgur& and the square of the 
second figure. Multiply this divisor by the second figure; subtract the 
product from the remainder. (Should the product be greater than the 
remainder, the last figure of the root and the complete divisor are too lar^e ; 



CUBE BOOT. 9 

snbfltttote for Uie last figure the next anoaller number, aad ootreet tbe trial 
divhK>r aocordiDfrly.) 

To the remainder bring down the next period, and proceed as before to 
find the third figure of the root — that Is, square the two figures of the root 
already found; multiply bv 800 for a trial divisor, etc. 

If at any time the trial dmsor is greater than the diyldend, bringdown an- 
other period of 3 figures, and place in the root and proceed. 

The cube root of a number wlU contain as many figures as there are 
periods of S in the number. 

Siiorter Hetbods of BxtracUng: the €nbe Boot*— 1. From 
Wentworth^s Algebra: 



800 X 1« = 800 
80x1x8= 80 



«• = 



1,881,885,068,826 118845 
798 



_64_ I 168885 
800x19* s: ^4Srno\ 
80 X IS X 8 = lOdOl 
8*=: 9 

44989 
1060 



182887 



800 X ]88« = 45387tX) 
80 X 19S X 4 = 14760l 



90406988 



78M 



4568478 M8218904 
HTTOJ 2S2800B0885 
80O X 1284> 
80 X 1984 X 5 « 
5«=: 



457011995 



After the first two figures of the root are found the next trial divisor is 
f^jnd by bringing down the sum of the 80 and 4 obtained in completing the 
preceding diTisor , then adding the three lines connected by the brace, and 
annexing two ciphers. This method shortens the work in long ezamplet), as 
is seen in the case of the last two trial divisors, saving the labor of squaring 
193 and 1984. A further shortening of the work is made by obtaining the 
last two figures of the root by division, the divisor employed being three 
times the square of the part of the root already found; thus, after finding 
the first three figures: 

8 X 198< s 46387190496063145. 1 + 
—181548 ' 

984416 



74813 



The error due to tlie remainder if not sufficient to change the fifth figure of 
the root. 
2, Br Prof. H. A. Wood (atewfU Indicator, Julv, 1890): 

I. ^Ting sraarated the number into periods of three figures each, count- 
ing from toe right, divide by the square of the nearett root of the first 
period, or first two periods ; the nearett root is the trial root. 

II. To the quotient obtained add twice the trial root, and divide by 8. 
This gives the root, or first approximation. 

m. By using the first approximate root as a new trial root, and proceed- 
ing as before, a nearer approximation is obtained, which process may be 
repeated until the root has oeen extracted, or the approximation carried at 
far as desired. 



10 AKITHMETIO. 

ExAHPLB.-— Required the cube root of 80. The nearett ciibe to 20 is S*. 
8< = 9)20.0 
2.2 
«_ 
8)8^ 
2.7 IstT. B. 
».7«=: 7.29) 20.000 
2.748 
5.4 

8)8.148 
2.714, let ap. cube root 
«.714« a 7.865796)8 0.0000000 
2.7I62S84 
6.4g8 
8)6 .1482584 
2.7144178 2d ap. cube root 

Rbhark.— In the example it will be obeerved that the second term, or 
flr«t two figures of the root, were obtained by uslngr for trial root the root of 
the first period. Using:, in like manner, these two terms for trial roor« we 
obtained four terms of the root ; and these four terms for trial root save 
seven flgrures of the root correct. In that example the last figure efhould be 
7. Should we take these eight figures for trial root we should obtain at least 
fifteen figures of the root correct. 

To Extract a mffber Root tban tUe Cnbe*— The fourth root is 
the square root of the square root ; the sixth root Is the cube root of the 
square root or the square root of the cube root. Other roots are most con- 
Teniently found by the use of logarithms. 

ALLIGATION 

shows the value of a mixture of different ingredients when the quantity 
and value of each Is known. 

Let the ingredients be a, &, c, d, eta, and their respective values per unit 
v't ^« V$ «t etc. 

X = the sum of the quantities = a + b-{-e-{-dt etc. 

P ss mean value or price per unit of A. 

AP = ato -{■ bx + cy -{■ dz^ etc. 

^ aw-^-bx + cy + dz 

A 

PERIHUTATION 

shows in how many positions anv number of things may be arranged In a 
row; thus, the letters a, &, c may be arranged Id six positions, viz. ahc, acb, 
cctb, c6a, hoc, bca. 

Rule.— Multiply together all the numbers used in counting the things; thus, 
permutations of 1, a, and 8 = 1x2x8 = 6. In how many positions can 9 
things in a row be placed ? 

1X2X8X4X6X6X7X8X9 = 862880. 

COMBINATION 

shows how many arrangements of a few things may be made out of a 
greater number. Rule : Set down that figure which indicates the greater 
number, and after it a series of flgui*e8 diminishing by 1, until as many are 
set down as the number of the few things to be taken in each combination. 
Then beginning under the last one set down said number of few things ; 
then going backward set down a serien diminishing by 1 until arriving under 
the first of the upper numbers. Multiply together all the upper numbers to 
form one product, and all the lower numbers to form another; divide the 
upper product by the lower cno. 



GEOMETRICAL PROGRESSIOK. 11 

Row many combinations of 9 things can be made, taking 8 in each com- 
btuationr 

9X8X7 ^ ??1 « 84 
1X2X8 6 

ARrraniETICAIi PBOGRBSSIONy 

in a series of numbers, is a progressive increase or decrease io each succes- 
sive number by the addition or subtraction of the same amount at each step, 
as 1. 2, 3, 4, 5, etc., or 16, 12, 9, 6, etc. The numbers are called terms, and the 
equal increase or decrease the difference. Examples in arithmetical pro> 
gression may be solved by the foUowiog formulie : 

I-et a = first term, I = last term, d s common difference, n = number Of 
Urms, M =K sum of the terms: 

I = a + (n-l>l, --id±|/2d. + (a-|d),* 

2a 8 . jn^Dd 
«--«, *»+— 2 

. = ^»[3a + (n-l>cfl, = S +-8d"' 

= « + a)^, =^n[2i~(n-l)cl]. 



:I-(»-l)d, 



= |.V(' + i'')'-««^ 



2f 



I — a __ g^g - cm) 

^ = ^TH' •* n{n - 1)' 
It ^a* 2(nf ~ 8) 

= 2« - « - a' "" n(H - 1)' 



l-a 



d-2a±|/(2a-d)« + 8d* 



+ 1. = ad" 



2« 



2Z + </ ±|/(2l + d)« - 8ds 



= Z+a "" 2d 

4SiB01!IlBTAICAli PROGRB8SION, 

in a series of numbere, is a propressive increase or decrease in each sue* 
ceiwive number by the same multiplier or divl8or at each step, as 1, 2, 4, 8, 
16. etc., or :M3, 81, 27, 9. etc. The common multiplier is called the ratio. 

Lpt a =s first term, I = last term, r = ratio or constant multiplier, n » 
number of terms, m = any term, as 1st, 2d, etc., a = sum of the terms: 

i-ar^ ^ r r* - 1 

log I = log a + (tt - 1) log r, |(£ - I)* - * - a(« - o)« - 1 = a 

m = a»** "" ^ log m = log a + (m - 1) log r. 

r-1 • r-r - n-i- n-l.- 

y/ - Va 



r*-.r«-l 



12 



Aun iGTia 



o« 


1 




ys 


•VJ 




r* 


a a 


= 0. 




logl-lopfl 


+ 1. 




logr 






loga = log I ^ (n - 1) log r. 



=: * -** . log r = ?2«.izJ2»?. 



n-1 



log ? - log o 



' log (» - o) - log (8-1) 



+ 1. 



_ log [g + (r - l)g] - log g 
" log r 

^ log t - log [ tr - (r - 1)«) 
log r 



+ 1. 



Population of tlie ITnlted Statos* 

(A problem id geometrical progression.^ 



Tear. 
1860 
1870 
1880 
1890 
1900 
1906 
1910 



Population. 
81,448,821 

89.818,449^ 
50,156.788 

«»,e9s;B0 

76,*496,»» 

Est. 83,577,000 

** 91,554,000 



Increase In 10 Annual Increase. 
Years, per cent. per cent. 



S6.68 


8.89 


25.96 


8.88 


84.86 


8.86 


81.884 


1.994 




Est 1.840 


20.0 


•* 1.840 



Estimated Population in Each Yearfrovi 1870 to 1909. 
(Based on the above rates of increase, in even thousands.) 



1870.... 


89,818 


1880.... 


60,166 


1890.... 


62.622 


1900.... 


78,29.5 


1871 .. 


40.748 


1881... 


61,281 


1891. . 


68,871 


1901.... 


77,609 


1872. . 


41,699 


1888 ... 


52.433 


1802.... 


65,145 


1902 .. 


79,1-.»U 


1878.... 


4i,673 


1883.... 


58,010 


1893 ... 


66,444 


1909.... 


80,5K5 


18T4... 


48,070 


1884.... 


54,813 


1894... 


67,770 


1904.... 


82,007 


1875.... 


44.G90 


1886.... 


56,048 


1895 ... 


69,122 


19C5.... 


88,677 


1876.... 


45,878 


1886.... 


57,301 


1896.... 


70,500 


1906.... 


86,115 


1877.... 


4(J,H00 


1887.... 


58,588 


1897. . . . 


71,906 


1007.... 


86,6Kl 


1878 .. 


47,888 


1883... 


59.903 


1898.... 


78,841 


1908.... 


88.276 


1879.... 


49,011 


1889.... 


61,247 


1899.... 


74,803 


1909.... 


80,900 



The above table has been calculated by logarithms as follows : 
log J- = log I — log a -•- (II - 1), log m = log a + (mi — 1) log r 

Pop. 1900. . . . 76,296,220 log = 7.88SM988 = log I 

** 1890 . . 62,682,260 log = 7.7907285 = log a 



n&= ll,n 



difl. = 
1 = 10; diff. + 10 = 
add log for 1890 

log for 1891 : 
add again 



.0857708 
.00867708 
7.7967286 



= log r, 
= log a 



7.80530568 No. = 68.871 
.00857708 



log for 1898 7.81888256 No. = 65,145 . 

Compouiui Interest is a form of geometrical progression ; 
ing 1 plus the percentage. 



the ratio be- 



* Corrected by addition of 1 ,260.078, estimated error of the census of 1870, 
Census Bulletin No. 16. Dec. 12, 1890. 



DISCOUNT. LO 

INTBBB8T AND BISCOtJNT. 

Interest biiDone J paid for the ase of money for a given time; the fac 
tors are : 

p. the Ruu) loaned, or the principal: 
f, the time in years; 
r, the rate of fntereflt; 

t, the amount of interest for the ^ven rate and time; 
a = p + 1 = the amount of the principal with interest 
at the end of the time. 
Formulee : 

i = mterest = principal X time X rate per cent = t = j^; 

a = amount = principal + interest = p-r ^; 

lOOt 
r=rate = — ; 



p = principal = ?^=a.^^; 



* = time = . 

pr 

If the rate is expressed decimally as a per cent,— thus, 6 per cent = .06,— 
thb formulae become 

/ = p,-f,„ = p(l+rO; r = ±; * = ^; P = s = ,-frt- 

Roles for flndlnjir Interest*— Multiply the piincipal by the rate 
per annum divided by 100, aud by the time in years and fractions of a year. 
U the tln» te Kiven in day. intereet = pri5?'g^>|g|X »o. of dvt _ 

In banks interest is sometimes calculated on the basis of 800 days to a 
year, or 12 montlis of 80 days each. 
Short rules for interest at 6 per cent, when 360 davs are taken as 1 year: 
Multiply the principal by number of days and divide by 0000. 
Multiply the pnnci[>al bV number of months and divide by 200. 
Tlie interest of 1 dollar for one month is ^ cent. 

Intere»t of lOO Dollars for DlflTerent Ttmes and Rates, 

Time, t% %% A% h% ^% S% lOi 

lyear $2.00 $3.00 $4.00 $5.00 $6.00 $8.00 $10 00 

1 month .16} .25 .83} .4]| .50 .603 Ml 

1 day = ,iv year .00551 .0063} .01 lU -OlSSf .0166| .0222} .0277} 

1 day = ^ year .005479 .008219 .010959 .018699 .016438 .08191^$ .0273973 

DIseonnt if> interest deducted for payment of money before it is due. 

True discount is the difference between the amount of a debt pay- 
able at a future date w^ithout interest and its present worth. The present 
worth is that sum which put at interest at the legal rate will amount to the 
debt when it is due. 

To And the present worth of an amount due st future date, divide the 
amount by the amount of $1 placed at interest for the given time. Thtt dis- 
count equals the amount minus the present worth. 

What discount ^ould be allowed on $106 paid six months before it is due, 
interest being per cent per annum ? 

= $100 present worth, discount = 3.00. 

1 + 1 X .06 X 5 
2 

Bank discount is the amount deducted by a bank as interest on 
money loaned uu promissory notes. It is interest calculated not on tlie act- 
ual sum loaned, nut on the gross amount of the note, from wliich tlie dis- 
Cfiunt IS deducted in advance. It is also calculated on the basis of 30() da.^ s 
in the year, and for 8 (In some banks 4) days more than the time specified in 
the note. These are called days of grace, and the note is not payable ull 
the last of these days. In some States days of grace have been abolished. 



14 



AKITHMBTIC. 



What discount will be deducted by a bank fn discounting a note for $103 
tmyable 6 months hence ? Six months = 182 days, add 3 days grace = 135 
- , 108 X 186 „ ,„ 

Compound Interest*— In compound interest the interest Is added to 
the principal at the end of each year, (or shorter period if agreed upon). 

Letp = the principal, r = the rate expressed decimally, n = no of years, 
ftud a the amount : 



a = amount = p <1 + »")»; r = rate 



■VI 



1, 



log g - log p 



p = principal = ^^-^^ ; «o. of years = n = '^^^ ^^ ^ ^^ 



Compound Interest Table. 

(Value of one dollar at compound interest, compounded yearly, at 
8, 4, 5, and 6 per cent, from 1 to fiO years.) 



i 


t% 


i% 


^% 


W 


i 


t% 


4^ 


^ 


9% 


►* 










>* 










1 


1.08 


1.04 


1.06 


1.06 


16 


1.6047 


18780 


2.1829 


2.5408 


3 


1.0609 


1.0816 


1.1025 


1.1286 


17 


1.6528 


1.9479 


2.2920 


2.6928 


8 


1.095»7 


1.1249 


1.1576 


1.1910 


18 


1.7084 


2.0258 


2.4066 


2.8543 


4 


1.1255 


1.1609 


1.2155 


1.2626 


19 


1.7585 


2.1068 


2 5269 


8.02.'i6 


6 


1.1598 


1.2166 


1.2768 


1.8882 


20 


1.8061 


2.1911 


2.6588 


8.2071 


6 


1.1941 


1.2663 


1.3401 


1.4186 


21 


1.8008 


2.2787 


2.7869 


8.3995 


7 


1.S299 


1.81.50 


1.4071 


1.5036 


22 


1.9161 


2.8890 


2.9252 


8.60:36 


8 


1.2688 


1.8686 


1.4774 


l..Mn8 


88 


1.9786 


2 4647 


8.0715 


8.8197 





1.8048 


1.42^» 


1.5618 


1.6895 


24 


2.0828 


2.5638 


812251 


40487 


10 


1.8439 


1.4802 


1.6280 


1.7908 


25 


2.0937 


2.6658 


8.3864 


4.2919 


11 


1..3842 


1.5394 


1.7108 


1.8083 


30 


2.4272 


3.2484 


4.3219 


5 7435 


12 


1.4268 


1.6010 


1.79.58 


2.0122 


86 


2.8188 


8.9460 


5.5166 


7.6«R1 


13 


1.4685 


1.6651 


18856 


2 1329 


40 


8.2620 


4.8009 


7 0100 


10.-J858 


14 


1.5126 


1.7817 


1.9799 


2.2609 


45 


8.7815 


5.8410 


8.9860 


13.7646 


15 


1.5580 


1.8009 


2.0789 


2.3965 


CO 


4.3838 


7.1064 


11.6792 


18.4190 



At compound interest at 8 per cent money will double itself in 23^ year!^ 
at 4 per cent In 17^^ years, at 5 per cent in 14.2 years, and at 6 per cent in 
11.9 years. 

E41I7ATION OF PAYHIEFITS. 

By equation of payments we find the equivalent or average time In which 
one payment should be made to cancel a number of obligations due at dif- 
ferent dates ; also the number of days upon which to calculnte interest or 
discount upon a gross sum which is composed of several smaller sums pay- 
ab1f> nt different dates. 

Rnle.— Multiply each Item by the time of its maturity in days from a 
fixed date, taken as a standard, and divide the sum of the products by the 
sum of the items: the result is the average time in days from the standard 
date. 

A owes B $100 due in 80 days, $-300 due in 60 days, and $300 due in 90 days. 
In how many days may the whole be paid In one sum of $600 ? 

100 X 30 + 200 K 60 + 800 X 90 = 42,000; 42,000 h- 600 = 70 days, an$. 

A owes B $100, $200, an<l $300, which amounts are overdue respectively 80. 
60, and 90 days. If he now pays the whole amount, $600, how many days* 
interest should he pay ou limi snm y Au», 70 days. 



ANNUITIES. 



15 



PARTI All PAY1HBNT8. 

To compute interest on notes and bonds when partial payments haTe been 

Unlteil Stmt«s Role.— Find the amount of the principal to the time 
of I be flrat payment, and, subtracting the payment from it. And the amount 
of the reraamder as a new principal to the time of the next payment. 

If the payment is less than the interest, And the amount of the principal 
to the time when the sum of the imyments equals or exceeds the interest 
due, and subtract the sum of the payments from this amount. 

Proceed in this manner till the time of settlement. 

Note*— The principles upon which the precediuf; rule is founded are: 

1st. That payments must be applied first to discban^ accrued interest, 
and then the remainder. If any, toward the discharge of the principal. 

2d That only unpaid principal can draw interest. 

Mercantile niettaod.— When partial payments are made on short 
notes or interest accounts, business men commonly employ the following 
method : 

Find the amount of the whole debt to the time of settlement ; also find 
the amount of each payment from the time it was made to the time of set- 
tlement. Subtract the amount of payments from the amount of the debt; 
the remainder will be the balance due. 

ANNVITIES. 

An Annultir Is a fixed sum of money paid yearly, or at other equal times 
agrved upon. The values of annuities are calculated by the principles of 
compound interest. 

1. Let t denote interest on $1 for a year, then at the end of a year the 
amount will be 1 + i. At the end of n years it will be (1 + ^>*^. 

2. The sum which hi n years will amount to 1 is or (1+0" * or the 
present value of 1 due In n years. 

8. The amount of an annuity of 1 in any number of years n is 

4. The present value of an annuity of 1 for any number of years n is 
1 J(14-t)-n 

5 ITie annuity which 1 will purchase for any number of years n is 
i 



(1+0* 



i-(l-|-t)-»' 
6. The annuity which would amount to 1 in n years is 



(1+i)»-l 



Amounts, Present Talnes, etc., at &% Interest. 



Years 


(l+i)" 


(2) 

(1 + /)-»• 


(«) 

(1 4. i)n ^ 1 


(4) 

i-ci+o-* 


(6) 
t 


(6) 

t 




i 


i 


i-a+o-" 


a+t)*»-i 


1 

2 .... 

8 

4 

5 

6 

7 

8 

9 

10 


1.05 

l.lOiSS 

1.15:625 

1.215606 

1.276182 

1.840096 
1.407100 
1.477456 
l.S5l8i8 
1.688896 


.952%] 
.907029 
.669888 
.8*^702 
.'re3636 

.746815 
.710681 
.678889 
.644609 
.618913 


1. 

2.05 

8.1525 

4.3I0I25 

5.525631 

6.801918 
6.142006 
9.549109 
11.026564 
12.677898 


.962881 
1.859410 

3.645951 
4.329477 

5.075692 
5.786373 
6.4fi:«i3 
7.107822 
7.721735 


1.05 
.587805 
.867209 
.282012 
.230975 

.197017 
.i;'2820 
.154722 
.140690 
.129505 


1. 

.487806 
.317209 
.282012 
.180975 

.147018 
.122820 
.104782 
.090690 
.079506 



ABITHMETIC. 



,■« 




119.18 
101.08 
87.02 
75.87 
66.79 








to 




S?!SS8 






SisS98 






JO — Q0»^ 


am 








«0«J0»t-i0 


^ 








8SSSS 


c;^eodadtD 


^ 


sosgse 


gS?^2S 


S28S? 


2«S^S 


SSt:£S 


?ggs§ 


^SSSS 


^5£S$ 


9StiSS 


ss;;«" 


3! 

00 


^a5S$5 


^^Z^B 


$S'^SS 


SSSS9& 


&8S$S 


§gi?s 


ssssgs 


ssss<^ 


9$So£2S 


OIO*^0»I> 


5 












00 












01 
01 












essss 


5^^5«;: 


5SSf:S 


8fes:!:sj 


e8j&9 


l§ISS 


grggs 


^ssss 


i;9;;ssi 


gjQOjrOJO 








S55S;s:5 








S2J8SS 




£8Sc:3S 




e*eo"*ioo 


l> QO 0> © ^ 


9* CO "^ o «d 


t?22^«5 


S5?$«S 



WEIGHTS AND MBASURES. 



17 



TABLES FOR CAI^CULATING SINKING-FIJNIIS AND 
PBBSBNT TAI^ITES. 

Engineers aad others connected with municipal work and industrial enter* 
priiies often find it necessary to calculate payments to sinking-funds which 
vill provMe a sum of money sufficient to pay off a bond issue or other debt 
at the end of a giren period, or to determine the present value of certain 
annual charges. The accompanying tables were computed by Mr. John W. 
Hill, of Cincinnati, Eng^g News, Jan. 85, 1894. 

Table I (opposite page) shows the annual sum at rarious rates of interest 
required to net $1000 in from 2 to 50 years, and Table n shows the present 
value at various rates of interest of an annual charge of $1000 for from 5 to 
60 years, at five-year intervals and for lOO years. 

Tmhle II«— CapttaltaEatlon of Annuity of 81000 for 
flrom 5 to 100 Years* 



Rate of Interest, per cent. 



2» 



5 4,645 
10 8.758. 
1512,881. 
so! 15.589. 
25 18,4-^. 
I 



f.59 

1.58 

.15 

L48 

10038,614.81 



80 ;»,980.i 

85;sH,145.J 

40*^5.103.1 

45S 

60 28,989.^ 



4,679.60 
8,580.18 
11,937.80 
14,877.27 
17,418.01 

19.600.21 
21,487.04 
88,114.86 
84,518.49 
85,789.58 
81,598.81 



SH 



4.514.98 4,451.68 
6,816.45 8,110.74 
11,517.83,11,118.06 
14,212.181 18,590.81 
16,481.88,15,621.98 

18,891.86 17,891.86 
80,000.48 18,664.87 
8! ,354.83' 19,792.65 
82,495.83 80,719.89 
23,456.21 121,482. 06 
27,655.86 24,504.96 



4« 



4,889.91 
7.912.67 
10,789.48 
13,007.88 
14,828.12 

16,888.77 
17,460.89 
18,401.49 
19,156.84 
19,761 98 
21,949.21 



4,889.45 
7,721.78 
10,879.53 
12,468.13 
14,093.86 

15,878.36 
16,374.36 
17,159.01 
17,778.99 
18,855.86 
19,847.90 



6» 



4,268.09 
7,587.64 
10,087.48 
11,950.86 
18,413.88 

14,533.63 
15,890.48 
16.044.98 
16,547.65 
16,931.97 
18,095.68 



4,818.40 
7,860.19 
9,718.80 
11,469.96 
18,788.88 

13,764.85 
14,486.66 
15,046.81 
15,466.85 
15,761.87 
16^618.64 



WEIGHTS AND IfEASXJBES. 
Iions Measure.— Measures of Ijengtli* 

12 inches = 1 foot. 

8 feet = 1 yard. 

6^ yards, or 16^ feet = 1 rod, pole, or perch. 

40 poles, or 280 yardfl = 1 furlong. 

8 furlongs, or 1760 yards, or 5880 feet = 1 mile. 
8 miles = league. 

Additional measures of length in occasional use : 1000 mils = 1 inch; 
4 inches = 1 hand ; 9 inches = Ispan ; 8^ feet = 1 military pace ; 2 yards = 
1 fathom. 

Old I«and Measnre.— 7.98 inches = 1 link; 100 links, or 66 feet, or 4 
poles = 1 chain; lO chains = 1 furlong; 8 furlongs = 1 mile; 10 square ciiains 
= 1 acre. 

Nautical Measure. 

^te'milM^ ^^**^ ^^'' \ = * °*""^*' °^*^**' ^^ ^°°'-* 
3 nautical miles = 1 league . 

^ "Sltmi SilS' ^^ ^^'^^ \ = ^ ^^^ ^'^ ^* equator). 
860 degrees = circumference of the earth at the equator. 

♦The British Admiralty takes the round figure of 6080 ft. which is the 
lensth of the '* measured mile'' used in trials of vessels. The value varies 
from 6080.26 to 6088.44 ft. according to different measures of the earth's di- 
ameter. There to a difference of opinion among writers as to the use of the 
word ** knot ^* to meao length or a distance-Hsome holding that It should be 



18 ARITHMETIC. 

Square Memaure,- me^mnrem ofSnrfece. 

m sauare inches, or 1H3.35 circular I . . . 

Inches f = ^ square foot. 
9 square feet = 1 square yard . 

80i square yaixls, or 272J square feet = 1 square rod, pol<», or perclu 
4U square poles = i rood. 

4 roods, or 10 sq. chains, or 160 sq. ) 

poles, or 484U sq. yards, or 48M0 V = 1 acre, 

sq. feet, ) 

«0 acres = i square mile. 

An acre equals a square whose side is 208.71 feet. 

Circular Incli; Circular Jllll.-A circular inch is the area of a 

circle 1 inch in diameter = 0.7854 square inch. 

1 square inch = 1.2782 circular inches. 

A circular mil is the area of a circle 1 mil, or .001 inch in diamet«r- 
1000* or 1.000,000 circular mils = 1 circular inch. 

1 square inch = 1,278,288 circular mils. 

The mil, and circular mil are used in electrical calculations involving 
the diameter and area of wires. 

Solid or Cubic Rleaau re.— measures of VoIudbc. 

1728 cubic inches = 1 cubic foot. 
27 cubic feet = 1 cubic yard. 
1 cord of wood = a pile, 4x4x8 feet - 128 cubic feet 
1 perch of masonry = 16f X H X 1 foot = 24{ cubic feet 

Ijtquid measure. 

4 grills = 1 pint 

2 pints = 1 quart. 

A #itiai>ta — \ <r.ii/^ti i U. 8. 231 cubic inches. 

^ ^^^^ = ^ ^^^^ 1 Eng. 277.274 cubic inches. 

8U frallons = 1 barrel. 

43 fcaU'^ns = 1 tierce. 

2 barreifi, or 68 eallous = 1 hogshead. 
84 fifallons, or 2 tierces = 1 puncheon. 

2 hogsheads, or 126 gallons = 1 pipe or butt 
2 pipes, or 8 puncheons = 1 tun. 
The U. 8. gallon contains 281 cubic inches; 7.4805 gallons = 1 cubic foor. 
A cylinder 7 In. diam. and 6 in. high contains 1 gallon, very nearlv. or 230.9 
cubic inches. The British Impeiial gallon contains 277.S74 cubic Inches 
£= 1.20082 U. 8. fsrallnn. 

Tbe miner^s I ncli.— (Western U. 8. for measuring flow of a stream 
of water). 

The term Miner's Inch is more or less indefinite, for the reason that Call- 
fornfa water companies do not all use the same head above the centre of 
the aperture, and the inch varies from 1.86 to 1.78 cubic feet per minute 
each; but the most common measurement is through an apeilure 2 inches 
high and whatever length is required, and through a plank U inches thick. 
The lower edge of the aperture should be 2 inches above the bottom of the 
measuriug-box, and the plank 5 inches high above the aperture, thus mak- 
ing a 6-inch head above tue centre of the stream. Each square inch of ihis 
opening represents a miner's inch, which is equal to a flow of H cubic fee'; 
per minute. 

Apotbecartes' Fluid measure. 
60 minims = 1 fluid drachm. 

8 drachms, or437i grains, or 1.732 cubic inches = 1 fluid ounce. 

Dry measure^ IT. S, 

2 pints =: 1 quart. 
8 quarts = 1 peck. 
4 pecks = 1 bushel. 

used soJy to denoie a rate of speed. The length between knots on tlie log 
line is tIv ^f A nautical mile or 50.7 ft. when a half-minute glass is used; so 
that a speed of 10 knots is equal to 10 nautical miles per hour. 



WBIGHtS AKD ll£ASU&£d. 19 

Hie standard U. 8. bushel is the WiDchester bushel, which Is in cylinder 
form, 18i inches diameter and 8 Inches deep, and contains 2150.4^ cubic 
inches. 

A struck bushel contains 2150.4*2 cubic inches = 1.S445 cu. ft.: 1 cubic foot 
= 0.803!yS struck bushel. A heaped bushel is a cylinder 18^ inches diam- 
eter and 8 inches deep, with a heaped cone not less than 6 inches high. 
It is eqtial to Ij struck bushels. 

The British Imperial bushel is based on the Imperial gallon, and contains 
8 such gallons, or 2318. I9< cubic inches = 1.2887 cubic feet. The English 
quarter = 8 Imperial bushels. 

Capacity of a cylinder in U. 8. gallons = square of diameter, in inches X 
height in inches X .0084. (Accurate wlthhi 1 part in 100,000.) 

Capacity of a cylinder in U. 8. bushels = square of diameter in inches X 
height in inches X .0008662. 

SUpplng JHeasure. 

BegiMter Ton.— For register tonnage or for measurement of the entir*) 
internal capacity of a yessel : 

100 cubic feet = 1 register ton. 

This mimber is arbitrarily assumed to facilitate computation. 
Shipping Ton,— For the measurement of cargo : 

(1 U. S. shipping ton. 
40 cubic feet = •< 81.16 Imp. nushels. 
( 82.148 U. 8. '* 
( 1 British Rhipping ton. 
42 cubic feet = •< 82.719 Imp. bushels. 
1 88.75 U. 8. 
Carpenier^s £tt2e.— Weight a yessel will carry = length of keel X breadth 
at main beam X depth of hold in feet -i-96 (the cubic feet allowed for a ton). 
The result will be the tonnage. For a double-decker instead of the depth 
of the hold take half the breadth of the beam. 

WLemmnrem of Wetflit.'-ATolrdiipolM. or Commercial 
Welffht. 

16 drachms, or 487.5 grains = 1 ounce, oz. 
16 ounces, or 7000 grains = 1 pound, lb. 
28 pounds = 1 quarter, qr. 

4 quarters = 1 hundredweight, cwt. = 112 lbs. 

20 hundred weight = 1 ton of 2240 pounds, or long ton. 

3000 pounds = 1 net, or short ton. 

2801.6 pounds = 1 metric ton. 

1 stone = 14 pounds ; 1 quintal = 100 pounds. 

Troy mrelffbt. 

34 grains = 1 pennyweight, dwt. 

20 pennyweights = 1 ounce, 07.. = 480 grains. 

12 ounces = 1 pound, lb. = 5700 grains. 

Tray weight is used for weighing gold and silver. The grain Is 'he same 
in Avoirdupois, Troy, and Apothecaries^ weights. A carat, used in weighing 
««^pyMMifl -s 8.168 grains = .205 gramme. 

Apotbecarlea' ITelebt. 

20 grains = 1 .scruple, 3 
8 scruples = 1 drachm. 3 = 60 grains. 
8 drachms = 1 ounce, 1 = 480 grains. 

12 ounces = 1 pound, lb. = 5760 grains. 

To detcnnliie urbetber a lialance lias unequal arms.— 

Affer weighing an article and obtaining equilibrium. trauHpose the article 
anfl the freights. If the l>a1ance is true, it will remain in equilibrium ; if 
untrue, the nan suspended from the longer arm will descend. 

To ireligk eonreetlT om an Incorrect balance.— First, by 
sabstituUun. Put the article to be weighed in one pan of the balance and 



20 ARITHMETIC. 

couQlerpoiae it bv any convenient heavy articles placed on the other pan. 
Remove the article to be welched and subBUtute for it standard weif^rhta 
until equipoise is a^ain established. The amount of these weights is the 
weight of the article. 

Second, by transposition. Determine the apparent weight of the article 
as usual, then its apparent weight after transposing the article and the 
weights. If the diffei-ence is small, add half the difference to the smaller 
of the apparent weights to obtain the true weight. If Uie difference is 2 
per cent the error of this method Is 1 iiart in 10.000. For larger differences, 
or to obtain a perfectly accurate result, multiply the two apparent weights 
together and extract the square root of tlie product. 

Olrciilar IHemsiire* 

60 seconds, " = 1 minute, '. 
60 minutes, ' = 1 degree, ". 
00 degrees = 1 quadrant. 
860 '' = circumference. 

Time, 

60 seconds = 1 minute. 
60 minutes = 1 hour. 
S4 hours = 1 day. 
7 da3rs = 1 week. 
885 daySf 5 hours, 48 minutes, 48 seconds = 1 year. 

By the Qregorian Calendar every year whose number is divisible by 4 Is a 
leap year, and contains 866 days, the other years containing 865 days, ex- 
cept that the centesimal years are leap years only when the number of the 
year is divisible by 400. 

The comparative values of mean solar and sidereal time are shown by the 
following relations according to Bessel : 

866.84223 mean solar days = 866.21282 sidereal days, whence 
1 mean solar day = 1.00273T91 sidereal days; 
1 sidereal day = 00726957 mean solar day; 
24 hours mean solar time = 24* 8f 56•..^^5 sidereal time; 
24 hours sidereal time = 2S>' 56n 4«.091 mean solar time, 

whence 1 mean solar day is S» 56^.01 longer than a sidereal day, reckoned in 
mean solar time. 

BOARD AND TIMBER REBASirBfi. 

Board measure. 

In board measure boards are assumed to be one inch in thickness. To 
obuiii the number of feet board measure (B. M.) of a board or stick of 
square timber, multiply together the length.in feet, the breadth in feet, and 
tb** thickness in inches. 

To compute tbe measure or surface in square feet.—When 
r11 diiiiensiuus are in feet, nmltiplv the length by the breadth, and the pro- 
duct will give the surface lequired. 

When either of the dimensions are in inches, multiply as above and divide 
the product by 12. 

When all dimensions are in inches, multiply as before and divide prodoci 
by 144. 

Timber Measure. 

To compute tbe volume of round timber.— When all dimen- 
sions ai-e in feet, multiply the length by one quarter of the product of the 
mean girth and diameter, and the product will Ktve the measurement in 
cubic feet. Wlien length is given in feet and girth and diameter in Indies, 
divide the product by 144 ; when all the dimensions are in inches, divide bv 
172R. 

To compute tbe volume of square timber.— When all dimen- 
sions ara in feet, multiply together the length, breadth, and depth; the 
£roduct will t>e tbe volume in cubic feet. When one dimension ia given in 
lehee, divide by 12; when two dimensions ai« in Inches, divide by 144; when 
all tJiree dimensioDS are in iuchea, divide by 1728. 



WEIGHTS AND MlsiASCRES. 



21 



€ont«nto In Feet of Jotirta, BeaMillng, and Timber. 

Length in Feet. 



12 14 16 18 



32 84 









Feet 


Board Measure. 










«X 4 


8 


9 


11 


18 


18 


15 


16 


17 


19 


90 


8X 6 


12 


14 


16 


18 


20 


22 


91 


86 


28 


80 


2X 8 


16 


19 


21 


24 


27 


29 


82 


85 


87 


40 


8X 10 


90 


28 


27 


80 


88 


87 


40 


48 


47 


50 


2X IS 


24 


28 


S2 


36 


40 


44 


48 


52 


ts 


60 


2X 14 


28 


88 


87 


42 


47 


51 


56 


61 


65 


70 


8x 8 


24 


88 


82 


86 


40 


44 


48 


52 


56 


60 


8X 10 


ao 


85 


40 


46 


60 


65 


80 


65 


70 


75 


3X 1« 


88 


42 


48 


54 


60 


66 


72 


78 


84 


90 


SX14 


42 


49 


66 


68 


70 


77 


84 


91 


96 


105 


4X 4 


16 


10 


21 


24 


97 


99 


8-i 


85 


87 


40 


4X 6 


24 


28 


82 


86 


40 


44 


48 


52 


56 


60 


4X 8 


82 


87 


48 


48 


58 


50 


64 


69 


75 


80 


4X10 


40 


47 


58 


60 


67 


78 


80 


87 


93 


100 


4X W 


48 


66 


64 


72 


80 


88 


96 


104 


112 


120 


4X14 


56 


65 


75 


84 


93 


108 


112 


121 


181 


140 


«X 6 


86 


42 


48 


64 


60 


66 


72 


78 


84 


90 


«X 8 


48 


66 


64 


7S 


80 


88 


96 


104 


118 


120 


cxio 


60 


70 


80 


90 


100 


110 


190 


180 


140 


150 


6X12 


72 


81 


96 


108 


120 


182 


144 


166 


168 


180 


6X14 


84 


96 


112 


126 


140 


164 


188 


182 


196 


210 


8X 8 


64 


75 


86 


06 


107 


117 


128 


139 


149 


160 


8x 10 


80 


98 


107 


120 


188 


147 


160 


178 


187 


200 


8X12 


96 


lis 


128 


144 


160 


176 


192 


208 


224 


240 


8X14 


112 


181 


149 


168 


187 


205 


224 


248 


261 


280 


roxio 


100 


117 


188 


160 


167 


ISS 


200 


917 


988 


250 


10 X 12 


120 


140 


160 


180 


200 


2-iO 


240 


960 


980 


300 


10 X 14 


140 


J68 


187 


210 


288 


257 


280 


803 


827 


850 


U X 12 


144 


168 


192 


216 


240 


264 


288 


312 


886 


360 


12 X 14 


168 


188 


sm 


252 


280 


806 


886 


864 


892 


420 


14 X 14 


196 


229 


261 


' 294 


827 


859 


802 


425 


457 


490 



FRBNCn OB nSTBIO aiBASCTRSS. 

Tbe metric unit of lenfrth is the metre s 39.87 inches. 
The metric unit of weight is the gram = 15.482 grains. 
The following prefixes are used for subdivisions and multiples ; Millf = x^n, 
Centi = 1^9. Dec) = ^, Deca = 10, Hecto = 100, Kilo = 1000, Myria = 10,000. 

VBEHCM AND BBITI8H (AND AIHBRICAlf) 
BQ1JITAI.BNT MBASVBSflU 

Keaenres of lienstb. 

FBiorcH. British and U. S. 

1 metre = 89.87 inches, or 8.28068 feet, or 1.09861 yards. 

.8048 metre ts i foot. 

1 centimetre s .8887 inch. 
2154 centimetres s l inch. 

1 millmetre = .08037 inch, or 1/25 inch, nearly. 
86. 4 millimetres s l inch. 

1 kilometre s 1008.61 yards, or 0.62197 mlla 



22 ARITHMETIC. 

Measnres of SarOtcA. 

Frknch. Britisb and U. 8. 

1 <x«..a.^ ...<>»»> i 10.761 square feet, 

1 square metre = -^ ,.196 square yartlB. 

.886 square metre = 1 square yard. 

.09si9 square metre = 1 square foot. 

1 square centimetre = Abb square inch. 
6.452 square centimetres = I square incb. 

1 square millimetre = .00155 square inch. 
646.^ square millimetres = 1 square inch. 
1 centiare = 1 sq. metre = 10 764 square feet. 

1 are = 1 Hq. decametre = 1076.41 *' 

1 hectare = 100 ares = 107641 " '* = 9.4711 acres. 

1 sq. kilometre = .886109 sq. miles s 247.U *' 

1 sq. myriametre = 88.6109 " '< 

or Volume, 

Frbnob. British and U. S. 

.7645 cubic metre = 1 cubic yard. 

.02832 cubic metre = 1 cubic foot. 

.cubic decimetre = ^'iSSaSSaSi^S."- 

28.83 cubic decimetres = 1 cubic foot. 
1 cubic centimetre = .061 cubic inch. 

16.387 cubic centimetres = 1 cubic inch. 
1 cubic centimetre = 1 miUilitre = .061 cubic inch. 
1 centilitre = = .610 " 

1 decilitre = = 6.108 " " 

1 litre = 1 cubic decimetre = 61.033 " " = 1.05671 quarts, U. a 

1 hectolitre or decisiere = 8.6814 cubic feet = 2.8875 burtiels, " 

1 stere, kiloUtre, or cubic metre — 1.806 cubic yards = 28.37 bushels, " 

or Capacity, 

Frbvoh. British and U. S. 

{61.023 cubic inches, 
i^'g^lonfrm-ericiu.). 
2.202 pounds of water at 62^ F. 
88.317 litres = 1 cubic foot. 

4.543 litres = 1 gallon (British). 

8.785 litres = 1 gallon (American). 

or l¥elsht. 

French. British and U. 8. 

1 gramme = 15.482 grains. 

.0648 gramme = 1 grain. 

28.35 gramme = 1 ounce avoirdupois. 

1 kilogramme = 2.2046 pounds. 

.4536 kilogramme = 1 pound. 

1 tonne or metric ton = ( '^^ J.^° ^' «^ ^^^^ 
1000 kilogrammes = ] 2^6 ^?inds. 

1.0 1 « metric tons = i 1 ton of SSUn tv^nnHa 

1016 kilogrammes ^ ^ 1 ton or -»40 pounds. 

Mr. O. H. Titmann, in Bulletin No. 9 of the U. 8. Coast end Geodetic Sur- 
vey, discusses the work of various authorities who have compared the yard 
and the metre, and by referring all tJie observations to a common standard 
has Hucceeded in reconciling the discrepancies within very narrow limits. 
The following are his results for the number of inches in a metre according 
to the comparisons of the authorities named: 

1817. Hassler 89.86991 inches. 

1818. Kater 39.86990 " 

1835. Bailv 39.86978 " 

1866. Clarke 39.86JW0 " 

1885. Conistock 39.86984 " 

The mean of these is 80.86988 ** 



METEIC WEIGHTS AND MEASURES. 23 

ntSTBIO CONVBR8ION TABLES. 

The followfne tables, with the subjoined memoranda, were published in 
1890 by the United States Coast ana Qeodetic Survey, office of standard 
weights and measures, T. C. Hendenhall, Superintendent. 

Tables for CoBvertlns IT. 8. mrelffbtai and IIEeaaiire»-i 
Customary to netrle. 

LINEAR. 





Inches to Hllli- 
metres. 


Feet to Mecree. 


Tarda to Metres. 


Miles to Kilo- 
metres. 


i = 

2 = 

3 = 

4 = 

5 = 

6 = 

7 = 

8 = 

9 = 


86.4001 
60.8001 
76.8002 
101.6008 
127.0008 

168.4008 
177.8004 
803.2004 
288.6006 


0.804801 
0.609601 
0.914402 
1.219202 
1.524006 

1.828801 
8.188604 
8.488406 
8.743206 


0.914402 
1.828804 
2.748206 
8.657607 
4.672009 

5.486411 
6.400813 
7.815215 
8.829616 


1.60935 
821860 
4.82804 
6.43789 
8.04674 

9.65606 
11.86548 
12.87478 
14.48412 



SQUARE. 





Square Inches to 
Square Centi- 
metres. 


Square Feet to 
Square Deci- 
metres. 


Square Yards to 
Square Metres. 


Acres to 
Hectares. 


1 = 

2 = 
8 = 

4 = 

5 = 

6 = 

8 = 

9 = 


6.453 
12.906 
19.850 
25.807 
82.268 

88.710 
45.161 
51.618 
68.066 


9.290 
18.581 
27.871 
37.161 
46.458 

56.748 
65.032 
74.823 
83.618 


0.836 
1.672 
2.608 
8.844 
4.181 

6.017 
6.858 
6.680 
7.626 


0.4047 
0.8094 
1.2141 
1.6187 
2.Q23I 

8.4881 
2.8328 
3 2375 
8.6422 



CfUBIO. 





Cubic Inches to 
Cubic Centi- 
metres. 


Cubic Feet to 
Cubic Metres. 


Cubic Yards to 
Cubic Metres. 


Bushels to 
Hectolitres. 


1 = 

2 = 

3 = 

4 = 

5 = 

6 = 

8^ 
9 = 


16.387 
88.774 
48.161 
65.549 
81.996 - 

98.383 
114.710 
181.097 
147.484 


0.02832 
0.a'V668 
0.08405 
0.1 1327 
0.14158 

0.16990 
0.198i» 
0.226M 
0.85486 


0.765 
1.529 
2.294 
3.058 
S.tfiS 

4.587 
5.3.'S2 
6.116 
6.881 


0.35212 
0.70485 
1.06727 
1.40969 
1.76211 

2.11454 
2.46696 
2.81938 
8.17181 



24 



ARITHMETIC. 
CAPACITY. 





Fluid Dracbms 










to Millilitres or 


Fluid Ounces to 


Quarts to Litres. 


QalloDs to Litres. 




Cubic Centi- 


Millilitres. 








metres. 








1 = 


3.70 


29.57 


946.36 


8.78544 


2 = 


7.39 


59.15 


1.89272 


7 57088 


8 = 


11.09 


88.72 


2.83908 


11.86682 


4 = 


14.79 


11880 


8.78544 


15.14176 


6 = 


18.48 


147.87 


4.78180 


18.927a) 


6 = 


22.18 


177.44 


5.67816 


82.71264 


7 = 


25.88 


207.02 


6.62462 


26.49808 


8 = 


29.07 


286.59 


7.57088 


80.28352 


9 = 


88.28 


266.16 


8.61724 


84.06896 



WEIGHT. 





Grains to MUli- 
gramroes. 


ATolrdupois 
Ounces to 
Gfluiiniee. 


AToirdupois 

Pounds to Kilo- 

gramtues. 


Troy Ounces to 
Grainmes. 


1 = 

2 = 
8 = 

4 = 

5 = 

6 = 

7 = 

8 = 

9 = 


64.7989 
129.5978 
194.8968 
259.1957 
828.9946 

888.7986 
453.5924 
518.3914 
683.1903 


28.8495 
66.6991 
85.0486 
118.8981 
141.7476 

170.0972 
198.4467 
226.7962 
255.1457 


0.45859 
0.90719 
1.86078 
1.8HS7 
8.80796 

S.72156 
8.17516 
3.62874 
4.08288 


81.10348 
62.20696 
98.81044 
121.41892 
166.51740 

186.62089 
217.72487 
248.82785 
279.98188 



1 chain zz 20.1169 metres. 

1 square mile = 259 hectares. 
1 fathom = 1.829 metres^ 

1 nautical mile = 1853J27 metres. 
1 foot = 0.804801 metre. 

1 avoir, pound r= 458.6924277 gram. 
16432.35689 grains = 1 kilogramme. 



Tableii for Convertlns IT, fl. Welerhta and jlleasares 
metric to Onstoinary. 



LINEAR, 





Metres to 
Inches. 


Metres to 
Feet. 


Metres to 
Yards. 


Kilometres to 
Miles. 


1 = 
8 = 
6 = 
6 = 

8 = 

9 = 


89.8700 
78.7400 
118.1100 
157.4800 
196.8500 

286.2200 
275.5900 
314.9000 
854..3300 


3.28088 
6.56167 
9.84250 
13.12333 
16.40417 

19.68500 
82.96.^83 
26.24667 
89.62750 


1.093611 
2.187222 
3.-.>80833 
4.374444 
5.468a56 

6.561667 
7.655278 
8.748889 
9.842500 


0.62187 
1 .24274 
1 86411 
2.48548 
8.10685 

8.72822 
4.34959 
4.97096 
6.69283 



METRIC COKrEBSlON TABLES. 
SQUARE. 



25 



Square Oenti- 

metresto 
Sqiuire Inches. 



Square Metres 
to square Feet 



Square Metres 
to Square Tarda. 



Hectares to 
Acres. 



0.1550 
0.8100 
0.4650 
0.6200 
0.7750 

0.9800 
1.0680 
1.2400 
1.8860 



10.764 
21.688 
82.292 
48.065 
63.819 

64.688 
75.847 
86.111 
06.874 



1.196 
2.892 
8.588 
4.784 
6.960 

7.176 
8.873 
9.668 
10.764 



2.471 
4.942 
7.418 
9.884 
12.855 

14.826 
17.297 
19.768 
28.880 



CUBIC. 





Cubic Oeiitl- 

metres to Cubic 

Inches. 


Cubic Deci- 

metres to Cubic 

Inches. 


Cubic Metres to 
Cubic Feet. 


Cubic Metres to 
Cubic Yards. 


1 = 


0.0610 


61.028 


86.814 


1.808 


2 = 


0.12% 


122.047 


70.629 


2.616 


8 = 


0.1881 


188.070 


106.948 


8.924 


4 = 


0.M41 


244.098 


141.256 


6.282 


5 = 


0.8061 


806.117 


176.572 


6.540 


6 = 


0.8661 


866.140 


211.887 


7.848 


7 = 


0.4m 


427.168 


»I7.801 


9.156 


8 = 


0.4888 


488.187 


282.616 


10.464 


9 = 


0.6488 


649.810 


817.880 


11.771 



CAPACITY. 



MtlUlitres or 
Cubic Oenti 
litres to Fluid 
1 Drachms. 


Oentilitrw 
to Fluid 
Ounces. 


Litres to 
Quarts. 


Dekalitres 

to 

QalloDB. 


Hektolitres 

to 

Bushels. 


1 = 0.87 
2= 0.54 
3 = 0.81 
4= 1.08 
5= 1.85 

6=r 1.02 
7= , 1.89 
H= 2.16 
9= 8.4S 


0.888 
0.676 
1.014 
1.852 
1.691 

2.020 
2.868 
8.706 
8.048 


1.0667 
2.2181 
8.I70O 
4.2207 
5.2834 

6.3401 
7.3968 
8.4534 
9.5101 


2.6417 
5.2834 
7.9261 
10.6668 
18.2086 

15.8602 
18.4919 
21.1886 
23.7758 


2.8875 
5.6760 
8.6125 
11.3500 
14.1875 

17.0250 
19.8625 
22.7000 
25.5376 



36 



ARITHMETIC. 
WEIGHT. 





MillifframmeB 
to Grains. 


Kilogrammes 
to Grains. 


HectoRrrammes 
(100 grammes) 
to Ounces Av. 


Ktlograrnmes 

to Pounds 

Avoirdupois. 


1 = 

8 = 
8 = 

4 = 
6 = 

6 = 

8 = 

9 = 


0.01648 
0.08086 
0.046.« 
0.06178 
0.07716 

0.09850 
0.10808 
0.18846 
0.18880 


15488.86 
80864.71 
46897.07 
61789.48 
77161.78 

92594.14 
108086.49 
188458.86 
188891.81 


8.6274 
7.0648 
10.6888 
14.1006 
17.6870 

21.1644 
24.6918 
28.2192 
81.7466 


2.20462 
4.40984 
6.61886 
8.81849 
11.08811 

18.28778 
15.48885 
17.68607 
10.84150 



WEIGHT— (Continued). 



1 = 

2 = 
8 = 

4 = 

5 = 

6 = 

7 = 

8 = 
= 



Quintals to 
Pounds Ay. 



280.46 
440.98 
661.88 
881.84 
1108.80 

1828.76 
1548.88 
1768.68 
1984.14 



Milliers or Tonnes to 
Pounds Av. 



22046 
4400.2 
6613 8 
8818.4 
11083.0 

18287.6 
15482.2 
17686.8 
19641.4 



Grammes to Ounces, 
Troy. 



0.08215 
0.06480 
0.09645 
0.18860 
0.16075 

0.19890 
0.8S506 
0.25781 
0.8 



The only authorized material statidard of customary length is the 
Troughton scale belonging to this office, whose length at 59*.62 Fahr. con- 
forms to the British standard. The yard in use in the United States is there- 
fore equal to the British yard. 

The only authorized material standard of customaiy weight is the Troy 
pound of the mint. It is of brass of unknown density, and therefore not 
suitable for a standard of mass. It was derived from the British standard 
Troy pound of 1758 by direct comparison. Tlie British Avoirdupois pound 
was aiHO derived from the latter, and contains 7000 grains Troy. 

The grain Troy is therefore the same as the grain Avoirdupois, and tiie 
pound Avoirdupois in use in the Uuited States is equal to the British pound 
Avoirdupois. 

The metric system was legalized in the United States in 1866. 

By the concurrent action of the principal governments of the world an 
International Bureau of Weights and Measures has been established near 
Paris. 

The International Standard Metre is derived from the Mdtre des Archives, 
and its length is defined by the distance between two lines at 0** OenClgrade, 
on a platinum-iridium bar deposited at the lotematlonal Bureau. 

The International Standard Kilogramme is a mass of platinum-lrtdlum 
deposited at the same place, and its weight in vacuo is the same as that of 
tlie Kilogramme des Archives. 

Copies of these international standards are deposited in the office of 
Btauaard weights and meaHures of the U. S. Coast and Geodetic Survey. 

The litre is equal to a cubic decimetre of water, and it is measured by the 

auantity of distlUed water which, at its maximum density, will counterpoise 
iie standard kilogramme in a vacuum; the volume of such a quantity of 
water being, as nearly as has been ascertained, equal to a cubic decimetre. 



WEIGHTS AND MBA8UBES — COMPOUND UNITS. 37 



COMPOUND UNITS. 

leasures of Pressure and IVeli^lit. 



1 lb. per square inch. 



1 atmosphere (14.7 lbs. per sq. Id.). = 



I inch of water at es? F. 



1 inch of water at 88* F. 



1 foot of water at O^^" F. 



1 inch of mercury at eaf F. 



144 lbs. per square foot. 

sj.0856 108. of mercury at 82* F. 

2.0416 («• F. 

2.809 ft. of water at 6*j* F. 
27.71 ins. *♦ " '» 6-2* F. 
2116.3 lbs. per square foot. 

83.947 ft. of water at 6-2* F. 
80 ins. of mercury at 62* F. 

29.922 Ins. of mercury at 8-2« F. 
.760 millimetres of mercury at 3;!* F. 
.0861 lb. per square inch. 
6.196 lbs. " " foot. 
.0786 in. of mercury at 62* F. 
. . 5.2021 lbs. per square foot. 
-} .086126 lbs. per *' inch. 
.488 lb. per square inch. 
62.356 lbs. " '» foot. 
.88:i in. of mercury at 02* F. 
.49 lb. per square inch. 
msOlbs. " '• foot. 
1.132 ft. of water at 62* F. 
18.58 ina 62* F. 



HFeUrlit of One Cubic Foot of Pure Water. 

At 82* F. (frwalnff-pofnt) 02.418 lbs. 

'' 89.1* F. (maximum deusity) 6*2.425 " 

- 62* F. (Standard temperature) 62.8.^5 " 

" 21 <• F. (boiling-point, under 1 atmosphere) 69.76 " 

American gallon = 281 cubic Ins. of water at 62* F. = 8.3856 lbs. 

British " = 277.274 " " '^ = 10 lbs. 

measures of Work, Poirer, and Duty. 

Work.— The sustained exertion of pressure through space. 

Unit of irork.— One foot-pound, i.e., a pressure of one pound exerted 
through a space of one fopt. 

Horse-poirer.— The rate of work. Unit of horse-power = 83,000 ft. - 
lbs. per minute, or 550 ft.-lbe. per second == 1,960,000 ft. -lbs. per hour. 

Heat imlt = heat required to raise 1 lb. of water 1* F. (from 89* to 40*). 

88000 

Hone-power expressed in heat units = -==^- = 42.416 heat units per min- 
ute = .707 heat unit per second = 2545 heat units per hour. 

1 Ih of tnt^ Mr H l» mr honi— i ^WO.OOO ft.-lb8. per lb. of fuel. 
I ID. or ruei per u. i . per nour= ^ g^g^ j^^^^^ ^^^^ „ 

1.000,000 ft.-lbs. per lb. of fuel s 1.98 lbs. of fuel per H. P. per hour. 
5280 22 
Yeloelty.— Feet per second = ^^^ ~ 15 ** ni>>«« per hour. 

Gross tons per 



lUe = ^3^ = ~ lbs. per yard (single rail.) 



Vrenek and Brtttsk Bqnlvalents of UTel^kt and Press- 
ure per Unit of Area. 

FanrcH. British. 

1 gramme per square millimetre - 1.422 lbs. per square inch. 

1 kUoirramme per square ** =1422.82 " " 

1 '* ** »• centimetre = 14.228 *» " •* " 

1.0835 kilogrammes per square centimetre (.-.it'- •< .« >. >« 

(1 atmosphere) \ ' 

0.070908 kilogramme per square centimetre = I lb. per square inch- 



98 



ABITUJIBTIC. 



WIBB AND SHBBT-lVBTAEi 


CtAVGBS GOMPARBD. 


Number of 
Gauge. 


III 


III 

^ CO 


hi 


" 1 


Erillsh Imperial 

Standard 

Wire Gauge. 

(Legid Standard 

in Great Britain 

since 

Marvh 1, 18Si.) 


Iflilt 


II 




incb. 


Inch. 


inch. 


iDCU. 


inch. 


milUm. 


inch. 




0000000 






.49 




.500 


12.7 


.5 


7/0 


000000 






.46 




.464 


11.78 


.409 


6,^0 


00000 






.43 




.432 


10.97 


.488 


5/'0 


0000 


.454 


.46 


.898 




.4 


10.16 


.406 


4/0 


000 


.425 


.40964 


.862 




.878 


9.46 


.875 


8/0 


00 


.88 


8648 


.881 




.848 


8.84 


.844 


2/0 





.84 


.82480 


.807 




.824 


6.38 


.818 





1 


.8 


.2898 


.288 


.227 


.8 


7.88 


.261 


1 


2 


.284 


.26768 


.268 


.219 


.276 


7.01 


.266 


8 


8 


.259 


.22942 


.244 


.812 


.252 


6.4 


.85 


8 


4 


.288 


.20181 


.236 


.207 


.283 


6.89 


.284 


4 


5 


.22 


.18194 


.207 


.204 


.212 


6.88 


.219 


6 


6 


.208 


.16202 


.192 


.201 


.192 


4.88 


.203 


6 


7 


.18 


.144-28 


.177 


.199 


.176 


4.47 


.188 


7 


8 


.165 


.12849 


.162 


.197 


.16 


4.06 


.172 


8 





.148 


.11443 


.148 


.194 


.144 


8.66 


.156 


9 


10 


.184 


.10189 


.136 


.191 


.128 


8.26 


.141 


10 


11 


.18 


.09074 


.12 


.188 


.116 


2.96 


.125 


11 


18 


.109 


.08081 


.106 


.185 


.104 


9.64 


.109 


12 


18 


.095 


.07196 


.092 


.1S8 


.092 


9.84 


.094 


18 


14 


.068 


.06408 


.08 


.180 


.06 


2.08 


.078 


11 


15 


.078 


.05707 


.072 


.178 


.072 


1,88 


.07 


15 


16 


.065 


.06068 


.063 


.175 


.064 


1.68 


.0625 


16 


17 


.058 


.(M526 


.054 


.172 


.056 


1.48 


.0568 


17 


18 


.049 


.0408 


.047 


.168 


.048 


1.22 


.06 


18 


10 


.042 


.08569 


.041 


.164 


.04 


1.01 


.0488 


19 


20 


.085 


.03196 


.085 


.161 


.086 


.91 


.0875 


20 


21 


.033 


.02846 


.082 


.157 


.082 


.81 


.0844 


81 


22 


.028 


.02535 


.028 


.156 


.028 


.71 


.0318 


88 


83 


.085 


.02257 


.025 


.158 


.034 


.61 


.J0981 


88 


24 


.022 


.OiiOl 


.0)28 


•^51 


.022 


.66 


.086 


91 


25 


.02 


.0179 


.03 


.148 


.02 


.61 


.0919 


85 


26 


.018 


.01594 


.018 


.146 


.018 


.46 


.0188 


86 


27 


.016 


.01419 


.017 


.143 


.0164 


.42 


.0178 


87 


98 


.014 


.01864 


.016 


.139 


.0148 


.88 


.0156 


88 


29 


.018 


.01126 


.015 


.184 


.0186 


.85 


.0141 


29 


80 


.012 


.01009 


.014 


.187 


.0124 


.81 


.0125 


80 


81 


.01 


.00698 


.0185 


.120 


.0116 


.29 


.0109 


81 


82 


.009 


.00795 


.018 


.115 


.0108 


.27 


.0101 


32 


88 


.008 


.00708 


.011 


.112 


.01 


.85 


.0094 


83 


84 


.oor 


.0068 


.01 


.110 


.0092 


.28 


.0086 


84 


85 


.005 


.00561 


.0095 


.106 


.0084 


.21 


.0078 


35 


86 


004 


.006 


.009 


.106 


.0078 


.19 


.007 


86 


87 




.00445 


.0085 


.108 


.0068 


.17 


.0066 


87 


88 




.00896 


.00*< 


.101 


.006 


.15 


.0068 


36 


89 




.00853 


.0075 


.099 


.0052 


.18 




89 


40 




.00814 


.007 


.097 


.0048 


.12 




40 


41 








.006 


.0044 


.11 




41 


42 








.092 


.004 


.10 




48 


43 








.088 


.0038 


.09 




48 


44 








.065 


.0032 


.06 




44 


46 








.081 


.0098 


.07 




45 


46 








.079 


.0034 


.08 




46 


47 








.077 


.008 


.05 




47 


48 








.076 


.0016 


.04 




48 


40 








.072 


.0012 


.08 




40 


50 








.069 


.001 


.086 




60 



WIRE QAUa£ TABLES. 



29 



BBISOlf , OI 



OmCIJIiAR Mill OAUOK, FOB EIiBC* 
TBIOAIi uriRBS. 



(Huge 
Num- 


Circular 
Mils. 


Diam- 
eter 


Gauge 
Num- 


Circular 
Mils. 


Diam- 
eter 


Gauge 
Num- 


Circular 
Mils. 


Diam- 
eter 


ber. 


Id Mils. 


oor. 


in Mils. 


ber. 


in Mils. 


3 


8,000 


54.78 


TO 


70,000 


264.58 


190 


190,000 


485.89 


5 


6,000 


70.72 


75 


75,000 


273.87 


300 


200,000 


447.32 


8 


8,000 


80.45 


80 


80,000 


28*^.85 


220 


220.000 


469.06 


12 


l-.»,000 


109. .% 


86 


85,000 


291.55 


240 


240,000 


489.90 


IS 


15,000 


ldsi.48 


90 


90.000 


300.00 


260 


2Q0.000 


509.91 


80 


ao,ooo 


141.48 


95 


95,000 


308.23 


280 


280,000 


589.16 


» 


25.000 


168.1? 


100 


100,000 


816.23 


800 


800,000 


547.73 


80 


90,000 


17S.81 


no 


110,000 


331.67 


8:20 


320.000 


665.69 


35 


35,000 


187.00 


120 


120,000 


U6.42 


840 


840,000 


588.10 


40 


40,000 


800.00 


180 


180,000 


300.56 


860 


860.000 


600.00 


45 


45,000 


212.14 


140 


140,000 


874.17 








BO 


60,000 


a!8.61 


150 


160,000 


887.30 








65 


65,000 


234.58 


160 


160,000 


400.00 








60 


60.000 


244.96 


170 


170,000 


412.32 








65 


66,000 


254.96 


180 


180,000 


4-44.27 









TDTIST DRII^Ii AND 8TSBI« WIRE GAVGR. 

(Morse Twist Drill and Machine Co.) 



Xo. 


«».l 


No. 


8ise. 


No. 


Sise. 


No. 


Sise. 


jNo. 


Size. 


No. 


Size. 




ipcb. 1 




inch 




inch. 




locb. 




incli. 




Inch. 




.2«0 ' 


11 


.1910 


21 


.1590 


31 


.1200 


41 


.0960 


61 


.0670 




.2<10 < 


12 


.1890 


22 


.1670 


88 


.1160 


42 


.0985 


52 


.0635 




.2130 , 


18 


.1860 


23 


.1540 


38 


.1180 


43 


.0890 


53 


.0595 




.2000 


14 


.1820 


24 


.1580 


Zi 


.1110 


44 


.0860,1 54 


.0550 




.8055 1 


15 


.1800 1 


25 


.1495 


35 


.1100 


45 


.08301 55 


.0620 




.2010 16 


.1770 1 


26 


.1470 


86 


.1065 


40 


.0810 I 56 


.0465 




.2010 1 17 
.190J I 18 


.1730 


27 


.1440 


37 


.1040 


47 


.0786 57 


.0430 




.1605 1 


^8 


.1405 


38 


.1015 


48 


.0700 58 


.0420 




.1900 19 


.1660 


29 


.1860 


89 


.0995 


49 


.0780 59 


.0410 


10 


■'•"II* 


.1610 j 


80 


.1285 


40 


.0U80 


50 


.OruOi 00 


.0400 



ST1JR89 8TEBI« UTIRK GAUGE. 

(F*or Nos. 1 to 50 see table on page 28.) 



Ko. 


Size. 


No. 


Sise. ! 


No. 


Size. 


"No.I 


Size. 


'No.| Size.'; No.' Size. 




inch. 




inch. ' 


inch. 


,1 


inch. 


, inch , i inch. 


Z 


.413 


P 


.888 . 


F 


.257 


'■ 51 


.066 


, 61 1 038 1 71 .086 


\ 


.401 


O 


.816 1 

.802 1 


E 


.280 


t 62 


.068 


1 62 1 .037 li 72 .024 


X 


.897 


N 


D 


.846 


|l 63 


.058 


63 I .086 i| 73 ; .0« 


w 


.386 


M 


.895 


C 


.248 


|i 54 


.065 


64 I .035 \- 74 


.082 


V 


.877 


I> 


.290 1 


B 


.288 


1 ^ 


.060 


65 ; .088 li 75 


.020 


u 


.868 


K 


.881 1 


A 


.284 


.1 66 1 


.045 


, 66 1 .062 1 76 


.018 


T 


.358 


J 


.277 1 


1 


See 


i 57 


.042 


67 ' .081 77 


.016 


8 


.848 


1 


.872 1 


to 


-{page 


1 56 1 


.041 


1 68 1 .030 78 


.015 


K 


.888 


H 


.866 


SO 


a 


11 59 1 


.040 


69 1 .089 11 79 


.014 


<^ 


.382 


G 


.261 1 






II 001 


.089 


1 70 : .027 1' 80 
t 1 it 


.018 



The Stubs' Steel Wire Gauge is used in measuring drawn steel wire or 
drill rods of Stubs' make, and Is also used by many makers of American 
drill rods. 



30 ARITHMETIC. 

THB BDI80N •» €IBCVI«AR Bill. HimB GA1JGB. 

(For table of copper wires by thi8*Kttuge« Klviiif? weights, electrical resist- 
aoces, etc.. fwe Copper Wire.) 

Mr. O. J. Field {Stevens Iivdicator^ July, 1887) thus describes the origin of 
the Edison gauge: 

The Edison company experienced inconvenience and loss by not having a 
wide enough range nor suffleienr. number of sizes in the existing gauges. 
This was felt more particularly in the central-station woik in making: 
electrical determinations for the street system. Tliey were compelled to 
make use of two of the existing gauges at least, thereby introducing a 
complication that was liable to lead to mistakes by the contractors and 
linemen. 

In the incandescent system an even distribution throughout the entire 
system and a uniform pressure at the point of delivery are obtained by cal- 
culating for a given maximum percentage of loss from the potential ss 
delivered from the dynamo. In carrying this out, on account of lack of 
. regular sizes, it was often necessary to use larger sizes than the occnslon 
demanded, and even to assume new sizes for large underground conductors. 
It was also found that nearly all manufacturers cased their calculation for 
the conductivity of their wire on a varic^ty of units, and that not one used 
the latest unit as adopted by the BiitiKh Association and determined from 
Dr. Matthiessen's experiments ; and as this was the unit employed in the 
manufacture of the Edison lamps, there was a further reason for construct- 
ing a new gauge. The engineering department of the Edison company, 
knowing tlie requirements, have designed a gnuge that has the widest 
range obtainable and a large numbei* of sizes which increase in a regular 
and uniform manner. Tlie baMiti of the graduation is the sectional area, and 
the number of the wire corresponds. A wire of 100,000 circular mils area is 
No. 100 ; a wire of one half the size will be No. 60 ; twice the size No. 300. 

In the older gauges, an the number increased the size decrcai-^i. With 
this gauge, however, the number increases with the wire, and the number 
multiplied by 10U0 will K'ive the circular mils. 

Tlie weight per mil-foot, 0.0000030*2705 pounds, agrees with a specific 
gravity of 8.889, which is the latest figure given for copper. The ampere 
capacity which is given was deduced from experiments made in the com- 
pany's laboratory', and is based on a rise of temperature of 60® F. in the wire. 

In 1898 Mr. Field writes, concerning gauges In use bj' electricnl engineers: 

The B. and 8. gauge seems to be in general use for the smaller sizes, up 
to 100,000 c. m.. and in some cases a little larger. From between one and 
two hundred thousand circular mils upwards, the Edison gauge or ita 
equivalent is practically in use, and there is a general tendency to designate 
all sizes above this In circular mils, specifying a wire as 200,000, 400,000, 500,- 
000, or 1.000.000 c. m. 

In the electrical business there Is a large use of copper wire and rod and 
other materials of these large sizes, and in ordering them, speaking of them, 
specifying, and in every other use, the general method is to stniplj' Hpecify 
the cii^ular milage. I think it is going to be the only system in the ruture 
for the designation of wires, and the attaining of it means practically the 
adoption of the Rdison gauge or the method and basis of this gauge as the 
correct one for wire sizes. 

THB V. S. STANDARD GAITGE FOR SHKBT AND 
PLATK IRON AND STEKL, 1893. 

Th^'re Is in this country no uniform or standard gauge, and the same 
numbers in different gauges represent different thicknesses of sheets or 
plates. This has given rise to much misunderstanding and friction between 
employers and workmen and mistakes and fraud l>etween dealers and con- 
sumers. 

An Act of Congress in 1893 established the Standard Gauge for sheet iron 
and steel which is given on the next page. It is based on the fact that a 
cubic foot of iron weighs 480 pounds. 

A sheet of iron 1 foot square and 1 inch thick weighs 40 pounds, or 640 
ounces, and 1 ounce in weight should be 1/640 inch tliick. The scale has 
been arranged so that each descriptive number represents a certain number 
of ounces in weight and an equal nunibtfr of 640ths of an inch in thickness. 

The law enacts that on and after July 1, 1H98, the new gauge shall be used 
in det«miiniug duties and taxes levied on sheet and plate iron and steel; and 
tlu^t in its applicatiou a variation of "iy^ |)er cent either way may be allowed. 



GAUGlS FOK SHEET AND PLATK IRON AND STEEL. 31 



S. STANDARD GA176B FOR 8HBKT AND PI«ATB 
IRON AND STBKI., 1893. 



II 


Aiiproxiniate 

Thick neti-s in 

Fractions of 

an Inch. 


Approximate 
Thickness in 

Decimal 

ParU of an 

Inch. 


Approximate 
Thickness 

in 
Millimeters. 


Weight per 1 
Square Foot 

in Ounces 
Avoirdupois. 


Weightper 

Square Foot 

in Pounds 

Avoirdupois. 


^1 

III 




Weigiitper 
Square Meter 

m Pounds 
Avoirdupois. 


>3iJO0O0 


1-a 


0.5 


12 7 


820 


20. 


9.072 


97.65 


216.28 


ooixiao 


15-82 


0.46875 


11.00626 


800 


18.75 


S.505 


91.55 


201.92 


(W)00 


7-16 


0.4375 


11.1126 


280 


17.60 


7.938 


86 44 


188.37 


oooo 


18-32 


0.40625 


10.81875 


260 


16.25 


7.871 


79.38 


174.91 


000 


8-8 


0.875 


0.525 


240 


15. 


6.804 


73.24 


161.46 


OQ 


11-82 


0.84875 


8.78125 


220 


18.76 


6 287 


67.13 


148 00 


o" 


6-16 


0.8125 


7.9875 


200 


12.60 


5.67 


01.08 


184.55 


1 


9-82 


0.28125 


7.14875 


180 


11.25 


5.108 


54.9.S 


121.09 


1 


17-64 


0.266G25 


6.746875 


170 


10.625 


4.819 


51.88 


114.87 


8 


1-4 


0.25 


6.36 


160 


10. 


4.536 


48.82 


107.64 


4 


15-64 


0.234375 


5.953125 


150 


9.375 


4.252 


46.77 


100.91 


6 


7-82 


0.21875 


5.65625 


140 


8.75 


3.960 


42.72 


94.18 


6 


I!{-«4 


0.20»125 


5 159376 


180 


8.125 


3.6»> 


39.67 


87.45 




8-16 


0.1875 


4.7625 


120 


7.5 


3.402 


36.62 


80.72 


8 


11-64 


0.171875 


4.865025 


110 


6.875 


3.118 


38.57 


74.00 


9 


5-32 


0.156-^5 


8.95875 


100 


6.25 


2.835 


30.52 


67.27 


10 


9-64 


0.140626 


3.571875 


90 


5.625 


2 552 


27.46 


60.55 


11 


1-8 


0.125 


8.175 


80 


5. 


2.268 


24.41 


58.82 


\l 


7-64 


0.108875 


2.778125 


70 


4.375 


1.984 


21.86 


47.09 


IS 


8-32 


0.09875 


2.38125 


60 


8.75 


1.701 


18.81 


40.86 


14 


5-64 


0.07B12S 


1.084375 


50 


3.125 


1.417 


15.26 


38.64 


15 


9-128 


0.0708125 


1.7869375 


45 


2.8125 


1.276 


13.73 


80.27 


16 


1-16 


0.0625 


1.5875 


40 


2.5 


1.134 


12.21 


26.91 


17 


9-160 


0.06625 


1.42875 


86 


2.25 


1.021 


10.09 


24.22 


- 18 


1-20 


0.06 - 


1.27 - 


32 


2. 


0.9072 


9.765 


21.58 


19 


7-160 


0.04877S 


1.11125 


28 


1.75 


0.7988 8.544 


18.84 


90 


8-8U 


0.0375 


0.9525 


24 


1.50 


0.6804 7.324 


16.15 


%\ 


n-8i0 


0.084375 


0.878125 


22 


1 875 


0.6287 


6.713 


14.80 


« 


1-82 


0.08125 


0.793750 


20 


1.25 


0.567 


6.ia3 


13 46 


23 


8-^20 


0.028125 


0.714375 


18 


1.125 


0.5103 


5.493 


12.11 


t4 


1-^ 


O.0S5 


685 


16 


1. 


4536 


4.882 


10.76 


25 


7-320 


0.0-il875 


0.555625 


14 


0.875 


0.8960 


4.272 


9.42 


S6 


8-lGO 


0.01876 


0.47025 


12 


0.75 


0.3^02 


3.662 


8.07 


27 


11-M) 


0.0171875 


0.4865625 


11 


0.6875 


0.8119 


3.857 


7.40 


» 


1-64 


0.015625 


0.896875 


10 


0.625 


0.2835 


8.052 


6.73 


29 


9-640 


0.0140825 


0.8571875 


9 


0.5625 


0.2551 


2.746 


6.05 


30 


1«) 


0.0:25 


0.8175 


8 


0.5 


0.22G8 


2441 


5 as 


81 


7-6lrt 


0.0109375 


0.2778126 


7 


4375 


0.19H4 


2.l:i6 


4.71 


%Z 


IS~1-J80 


O.OIOI.VWS 


0.25796875 


6^ 


0. 40625 


0.1848 


l.fl«3 


4.37 


83 


8-820 


0.009875 


0.288125 


6 


0.375 


0.1701 


1.831 


4.04 


84 


11-1280 


00650875 


0.218S8125 


5^ 


0..S4875 


1559 1.6:8 


8 70 


^ 


5-640 


0.0078125 


0.1984375 


5 


0.3125 


0.1417 1.526 


3 86 


86 


»-13H0 


00708125 


17850375 


J^ 


o-esi-i.'. 


0.1276 1.373 


8.03 


87 


17-2BO0 


0.006640625 


0.168671875 


o.2t'>r,e25 


0.1205 1 2«7 


2.87 


« 


1-160 


0.00625 


0.15876 


4 


0.25 


0.1 1:M 1 221 

1 


2.69 



32 



MATHEMATICS. 



The Decimal Gauffe*— The legalization of the standard sheet-metal 
gaufce of 189S and its adoption by some manufacturers of sheet iron have 
only added to the existing confusion of grauges- A ioint committee of the 
American Society of Mechanical Enjdneers and the American Railway 
Master Mechanics' Association in 18!)5ajrreed to recommend the use of the 
decimal gauge, that is, a gauge whose number for each thickness Is the 
number of thousandths of bn inch In that thickness, and also to recommend 
** llie abandonment and disuse of the various other gauges now in use, as 
tending to confusion and error/* A notched gauge of oval form, as shown 
in the cut below, has come into general use as a standard form of the dec- 
imal gauge, but for accurate measurement Ites indications should be checked 
by the use of a micrometer gauge reading to thousandths of an inch. 
UTelffbt of Sheet Iron and SteeK Thlckne»» bj Decimal 
Gauge* 





09 

B 


1 


Weightper 
Square Foot 




OB 

c 


1 


Weik!ht_per 
Square Foot 


, 


£ 


? 


in Pounds. 


. 


o 


o 


in Pounds. 


1 


it 


B 

i 






1 

08 
O 




1 






a-: 




1? 


ty 




a«-i 

CO 


a 


g& 


|35 


"§ 


g^ 


g 


P 


"^^5 


ft 


-< 


< 


►"• 


CC 


ft 


< 


< 


GO 


0.002 


1/500 


0.05 


0.08 


0.082 


o.oocT 


1/16 - 


1.82 


8.40 


8.448 


0.00* 


1/250 


0.10 


0.16 


0.1G3 


0.065 


18/200 


1.65 


8.60 


2 6.52 


0.006 


3/r,oo 


0.15 


0.24 


0.245 


0.070 


7/100 


1.78 


8.80 


2.H56 


0.008 


1/125 


0.20 


0.82 


0.326 


0.075 


3/40 


1.90 


8.00 


3.060 


0.010 


1/100 


0.25 


0.40 


0.408 


0.080 


2/:i5 


8.08 


8.80 


8.261 


0.012 


3/250 


0.30 


0.48 


0.490 


0.085 


17/200 


2.16 


8.40 


3.4(W 


O.OU 


7/500 


0.86 


0.66 


0.571 


0.090 


9/100 


8.28 


8.60 


3.672 


0.016 


1/64 4- 


41 


0.64 


0G53 


0.095 


19/200 


8 41 


8.80 


3.876 


0.018 


9/500 


0.46 


0.78 


0.784 


0.100 


1/10 


2.64 


4.00 


4.080 


0.020 


1/50 


61 


0.80 


0.816 


0.110 


11/100 


8.79 


4.40 


4.488 


022 


11/500 


0.56 


0.88 


898 


0.125 


1/8 


8.18 


6.00 


6 100 


0.025 


1/40 


0.64 


1.00 


1.020 


0.135 


27/200 


3.48 


6.40 


5.508 


0.028 


7/2fS0 


0.71 


1.12 


1.142 


0.1.50 


8/20 


8.81 


6.00 


6.120 


0.0:12 


1/82 -h 


0.81 


1.28 


1.306 


0.165 


3:^/200 


4.19 


6.60 


6 73i 


0.036 


9/250 


0.91 


1.44 


1.469 


0.180 


9/.0O 


4.67 


7.20 


7.344 


0.040 


1/26 


1.02 


1.60 


1.632 


0.200 


l/.'S 


6.08 


8.00 


8. ICO 


0.045 


9,-200 


1.14 


1.80 


1.836 


220 


11. ro 


5.59 


8.80 


8.970 


oav) 


1/20 


1.27 


200 


2.040 


0.240 


6/-,'5 


6.10 


9 60 


9.79i 


O.OM 


11/200 


1.40 


2.20 


2.244 


0.250 


1/4 


0.86 


10.00 


10.200 




ALGEBRA. 88 



ALGEBBA. 

Addition.— Add a and b. Ans. a-f &• Adda»6,and~e. An8.a4-b-e. 

Adii iu and — 8a. Ans. — a. Add Sob, — 8a6, — o, — 8c. Ads. ^ab^4e, 

Bubirmetlon.— Subtract a from b. Ana. b - a. Subtract — a from — 6. 
Ap #. — ^ 4- a. 

Subtract 6 + c from a. An8.a — b-o. Subtract 8a*6— 9o from 4a*6 + c. 
Ana. a*& + lOe. Bhlk: Caiauge the signs of the subtrahend and proceed as 
in additioD. 

lIaltlPlleaUoii.-Multipl7 a by 6. Ana. ab. HulUpIy ob bya + b. 
Ana. a*b 4- ab^, 

MulUply o -h 6 by a + b. Ans. (a + b)(a + b) = o« + Sob + &*. 

Muldply — a by ~ b. Ans. ab. MulUply - a by b. Ans. - ab. Like 
signs jjpve plus, unlike siffiis minus. 

rowers of iiniiibers.— The product of two or more powers of any 
number is the number with an exponent equal to the sum of the powers: 
a* X a* B a*; a*b* xab = o«b»; - 7ab x 2ac = — 14 o'bc 

To multiply a polynomial by a monomial, multiply each term of the poly- 
nomial by the monomial and add the partial products: (Oa — 8b) x 8o c= ISiic 

— 9bc. 

To multiply two polynomials, multiply each term of one factor by each 
term of tlie other and add the partial products: (5a - 6b) x (8a - 4b) = 
15a*-88ab + a4b«. 

The square of the sum of two numbers = sum of their squares + twice 
their product. 

The square of the diiference of two numbers = the sum of their squares 

- twice their product. 

The product of the sum and difference of two numbera s the difference 
of their squares: 

(a + b)« =^a'» + idb + b*; (a - b)« =a« - 2ab + b«; 
(a+b) x{a--b) = a*'-bK 

The square of half the sums of two quantities is equal to their product plus 
the sqoara of half their difference: (^^)' = ob + (^-^)'' 



The square of the sum of two quantities is equal to four times their prod- 
octii, plus the square of their difference: (a -\- b)* = 4ab + (a - b)* 

The sum of the squares of two quantities equals twice their product, plus 
the square of their difference: a* -V b* = 2ab + (a — b)*. 

The square of a trinomial = the square of each term 4- twice the product 
of each term by each of the terms that follow it: (a + b + c)* = a* + b* -h 
e*4-2ab-{-2ae+9bc; (a - b -c)« = o«-f b«H-c« -8ab -8ac+«bc. 

The square of (any number -^H) » square of the number -f the number 
-r- li; = the number x (the number 4* 1) + M'y 

The product of any number + H hy any other number 4- ^ = product of 
th« numbers 4- half their sum + 3|. (a + H) X b 4- V^) =s ab 4- ^(a + b)+ ^. 
4Hx(H<=4X64-«(4+6) + ^-M4-r+M=3M. ., ,^ 

S^iuuro, enbe, 4th poiwer^ ete.^ of a binomial a + b. 

(o + b)« = a« +8ab +b"; (a-^b)* = a» -f 8o«b 4 8ab« 4-b«; 
(a4-b)« sza* 4-4aSb + 6a*b< +4ab* 4-b«. 

In each case the number of terms is one greater than the exponent or 
the power to which the binomial is raised. 

i. In the first term the exponent of a is the same aa the exponent of the 
power CO which the binomial is raised, and it decreases by 1 in each succeed- 
mfcterm. 

1. b appears in the second term with the exponent 1, and its exponent 
increases by 1 in each succeeding term. 

4. The coefficient of the first term is 1. 

5. The coefficient of the second term is the exponent of the power to 
which the binomial is raised. 

6. The coefficient of each succeeding term is found from the next pre- 
eedlog term by multtplying its coefficient by the exponent of a, and divid- 
|ax the product by a number greater by 1 than the exponent of b. (See 
Binomial Theorem, below.) 



84 iXGEBai. 

PArentliesMi*— When a narenthesis is preceded by a plus sign It may be 
remoTed without cbaniriDer the value of the expression: a + 6 + (u + o) = 
Sa + ^- When a parentaeeiB i8 preceded by a minus siffn it may be removed 
If we change the signs of all the terms within the parenthesis: 1 — (a — 6 
.— e) = 1— a + 6 + c. When a parenthesis is within a parenthesis remove 

the imer one first: o-r6-]o-(d-e)[l =a- r6-|c-d + c[] 

A multiplioation sign, X, has the effect of a parenthesis. In that the oper- 
ation indicated by it must be performed before the operations of addition 
or subtraction. o + 6xa-f6 = a+a!) + 6; while (a -f 6) x (a -f 6) = 
a« -f 2a6 4- ft", and (a -f 6) X o + 6 = a« -f a6 -f 5. 

IHtIbIoii.— The quotient is positive when the dividend and divisor 
have Ulce signs, and negative when they have unlike signs: abc -t-b — ae; 
abe H — b= —ae. 

To divide a monomial by a monomial, write the dividend over the divisor 
with a line between them. If the expressions have common factors, remove 
the common factors: 

,. . a*bx ax , a* a* 1 _t 

a«te + a*y=-^^5j^ =y; -,=a; - = _=«- 

To divide a polynomial by a monomial, divide each term of the polynomial 
by the monomial: (8a6 - 18ac) -♦- 4a = 86 — 3c. 

To divide a polynomial by a polynomial, arrange both dividend and divi- 
sor in the order of the ascending or descending powers of some common 
letter, and keep this arranirement throughout the operation. 

Divide the first term of the dividend by the flrst term of the divisor, and 
write the result as the first term of the quotient. 

Multiply all the terms of the divisor by the flrst term of the quotient and 
subtract the product from <;he dividend. If there be a remainder, consider 
It as a new dividend and proceed as before: (a* — 6*) -•- (a + 6). 
a»-6«|a-|-6. 
o* -f gfc I g - 6 . 

- o6 - fe«. 
-ab- fe «. 

The difference of two equal odd powers of any two numbers Is divisible 
by their difference and also by their sum: 

(a« - 6«/ -H (a - 6> = a« + oft 4- 6* ; (a* - b«) ■«- (o + 6) = o« - a5 -f 6«. 

The diffAenoe of two equal even powers of two numbers Is divisible by 
their difference and also by their sum: (a* — &*)-•- (a — 6) = a -f b. 

The sum of two equal even powers of two numben* is not divisible by 
either the difference or the stun of the numbers; but when the exponent 
of each of the two equal powers is composed of an odd and an even factor, 
the sum of the givenpower is divisible by the sum of the powers expressed 
by the even factor. Thus a?* -4- ^ is not divisible by a; -f tf or by a; — y, but is i 
Hvislbie by a?» -f- !(•. j -r m j m^ 

Simple eqaaUoiis*— An equation is a statement of equality between i 
twoexpressions;a8,a4-6= c-f d. I 

A simple equation, or equation of the flrst degi-ee. Is one which contains 
only the flrst power of the unknown quantity. If equal cltanges be iiiailel 
(by addition, subtraction, multiplication, or division; in both hides of aa| 
equation, the results will be equal. 

Anv term may be changed from one side of an equation to another, pro>{ 
Tided its sign be changed: o -f 6 = c 4- d; a = c 4- d - 6. To solve a 
equation having one unknown quAntitv, trannpose all the terms involvin 
the unknown quantity to one side of the equation, and all the other term 
to the other side; combine like terms, and divide both sides by the ooefflcien 
of the unknown quantity. 

Solve &p - 29 = 86 - &r. &t 4- «* = » 4- ^: 11« = 56; « = 5, ans. 

Simple algebraic problems containing one nnlcnoxAn quantity are solve 
by making x = the unknown qiianllry, and Kiating the conditions of ^^ 

Sroblem in the form of an algebraic **quation, and then solving the er 
on. What two numbers are those whose sum is 48 and difference 14 7 
X = the smaller number, a; 4- 14 the greater, x -\- x -\- \A = 4&. 8fl?=:84, i 
= 17; * + 14 = 81, ans. 

Find a number whose treble exceeds 50 as much as Its double falls shoi| 
of 40. Let X = the number. &c - 50 = 40 - 2ar; 5a; = M): a; = 18, ans. Pr 
iag, 64-60 = 40-80. 



ALGEBRAr 35 

LvmtlaBs coiitetiiliiir tiwo nBknoira qaaiitlll«i*~If one 

. , Jon contains two unknown quantities, x and y, an indefinite number of 

pairs of values of x and y maj be found tliat will aatiafy the equation, but if 
a second equation be gireu only one ]Miir of values can be found that will 
satisf J both equations. Simultaneous equations, or those that may be satis- 
fied by the same values of the unknown quantities, are solved by combining 
the equations so as to obtain a single equation containing only one unknown 
quantit V. This prooess is called eilralnation. 

JSUmtnation by addition or «u6/}-act<on. —Multiplv the equation by 
puch numbers ss will make the coeiBcients of one of the unknown quanti« 
ties equal in the resulting equation. Add or subtract the resulting equar 
Uona according as they have unlike or like signs. 

c^i— j ar + «y = 7. Multiply by 2: 4* + 6y = 14 

*^*^* 1 4« - 5y = 8. Subtract: 4ac - 5y = 8 lly := 11; y s 1. 

Substituting value of v in first equation, ftv + 8 » 7; » = 9. 

ElimincUion 6y 9uMitutiati,^From one of the equations obtain the 
value of one of the unknown quantities in terms of the other. Substltu- 
tnte for this unknown quantity Its value In the other equation and reduce 
the resulting equations. 

J. . iar + 8y=r8. a). From (1) we find a? = -i^ . 

Sobstitute this value hi (8): 8(^^^) + 7y = 7; =» 94 - «y + 14y = 1< 

whence y = ~ 9. Substitute this value in (1): e» — 8 = 8; x = 7. 

Elimination by oomparison.'—Vrom each equation obtain the value of 
one of the unknown quantities in terms of the other. Form an equatioo 
(rum these equal values, and reduce this equation. 



Solve 



2x-'9y=U, (1). From (1) we find xs^^-^t^. 
dx-4y = 7. (2). From(«)weflndaj = ^i^. 



EquaUng these values of a;, ^^ "^ ^'^ = '^\*^ ; l«y= - 1»; y=:-l. 

Substitute this value of y in (1): 2x + 9 = n;z = l. 

If three simultaneous equations are given containing three unknown 
quantities, one of the unknown quantities must be eliminated between two 
pairs of the equations; then a second between the two resulting equations. 

4|«m4nttle eqitatloiis*— A quadratic equation contains the square 
of ihe unknown quantity, but no higher power. A pure quadraiw contains 
the square only; an affected quadratic both the square and the first power. 

To joJve a pure quadratic^ collect the unknown quantities on one side, 
and the known quantities on the other; divide by the coefficient Of the un- 
known quantity and extract the square root of each side of the resulting 
eqastion. 

Solve &r< -15_=0. Sx* = 15; a;> = 6; a; = ^5 

A root like ^57 which is indfcated, but which can be found only approzi- 
Bately, Is called a turd. 

Solve 8x* + 18 = 0. 8aji = - 16; ar« =s - 6; a: = 4^^. 

The square root of — 6 cannot be found even approximately, for the square 
«f any number positive or negative is positive; therefore a root which is in- 
dicated, but cannot be found even approximately, is called imaginary. 

To tolve an </ffected mutdratie.^\. Convert the equation into the form 
S'Z* ± iabx = c, multipnring or dividing the equation if necessary, so as 
Id make the coefficient of x* a square number. 

S. Complete the square of the first member of the equation, so as to con- 
fm it to the form of a*dB* ± 2abx -f b*, which is the square of the binomial 
tJc ± b, as follows: add to each side of the equation the square of the quo- 
tient obtained by dividing the second terra by twice the square root of the 
8m term. 

1 Extract the square root of each side of the resulting equation. 

Solve Sas* — 42 = 89. To make the coefficient of x* a square number, 
femUiply by 8: ftr« - 12r = 96; ISiT -1- (2 X ar) = :i; 2« = 4. 

Complete the square: ite* - 12sb + 4 = 100. Extract the root: 8a; - 2 = ± 



I 



86 ALQBBft^ 

10, whenoe « n 4 or -• 9 v. The niiare rod of 160 It eMber 4< 10 or * 10, 
stnoe the square of >- 10 aa well aa + 10* s lOO. 

Problema involving quadratic equations bave apparently two aolutiOBS, ai 
a quadratic bas two roots. Bometlmes both will be true solutlont, but Reu- 
erally one oolv wiU be a solution and the other be inooualatent with ihe 
conditions of the problem. 

The sum of the squares of two consecutive positive numbers is 481. Find 
the numbers. 

I«tap = oneiiumber.« + l the other. a;«4- (x + 1)* = 481. 9tv* + ftr + l 
«481. 

«• + X 3 S40. Oompletinic the square, «• + 9 + 0.iK m M0.S5. EztractinK 
the root we obtain x 4- 0.5 s ± 16.5; « ss 15 or — 10. 

The positive root gpives for the numbers 15 and 16. The Begative root - 
16 is inconsistent with the conditions of the problem. 

Quadratic equations containinir two unknown quantities require different 
methods for their solution, according to the form of the equations. For 
these methods reference must be mside to works on algebra. 

Theory of exponeiit0.>-r o when n is « positive intecer is one of m 

equal factors of a. y a^ means a is to be raised to the mth power and the 
itih root extracted. 

(r "a)"" means that the nth root of a is to be taken and the result 
raised to the mth power. 

Va^ := Kya) s a «. When the exponent tea fraction^e numera- 
tor indicates a power, and the denominator a root a^ = r <** = a«; al = 

To extract the root of a quantity- raised to an indicated power, divide 
the exponent by the index of the required root; as, 

Vo^ = a'i'* Va« = o* = a*. 

Subtracting 1 from the exponent of a is equivalent to dividing by a : 

,o«-» =a> = a: a» -» = a« = - = 1; a«-> = a -» = -; o -» -« = a-«=-i 

A number with a negative exponent denotes the reciprocal of the number 
with the corresponding po:titive exponent. 

A factor under the radical sign whofle root can be taken may. by having 
the root taken, be removed from under the radical sign: 

^cflh B |/a* )( |/b » a 4/6? 

A factor outside the radical sign may be raised to the corresponding 
power and placed under it: 

Binomial Theorem.— To obtain any power, as the nth, of an ex« 

I esMon of the form x-^-a 

etc. 
The following laws hold for any term In the expansion of (a •{• ar)". 
The exponent of x is less by one than the number of terms. 
The exponent of a is n minus the exponent of x. 

The lost factor of the numerator is greater by one than the exponent of a. 
The laMi factor of the denominator is the same as the exponent of x. 
In the rth term the exponent of a; will be r - 1. 
The exponent of n will be n — (r — 1). or n - »' + 1. 
The last factor of the numerator will be n - »H- 2. 
The last factor of the denominator will be =s r — 1. 

Hence the rth term =-^ -—x ——■ ---— — o« " »^ + 1 «^-» 

1 . s t a . . . . ^i — 1) 



OaOVKTBlCAl. PROBIiBXa, 



87 



OSOXBTBXOAXi FBOBLEMS. 




f 


^ 


ft 


} 


■■; 


■•) 




"iC 




" 




>t 






Fia. 


5. 




c 






O 


1 
1 

1 




* 


1 


.. ..|. 


n« 


^ 


B 



1« To bl»«ct • strAlslit line, 
or an aro of a circle (Fig. i).— 
From the ends A-, B^ an centres, de- 
pcribe arci interaectinK at C and A 
and draw a line through C and D 
which will bisect the line at i? or the 
ftfcat^. 

S* To dntiv a perpendicular 
to a ctraUdlt Une, or a radial 
line to a arcnlar arc*— Same as 
in Problem 1. CZ> is perpendicular to 
the line A B^ and also radial to the arc. 

8. To draiw a perpendicular 
to a straight line front a slT«n 
point Intliat line (Fig. 2).-With 
any radius, from the given point A in 
the line B C, cut the line at B and C. 
With a longer radius describe arcs 
from B and C, cutting each <^her at 
i>, and draw the perpendicular D A. 

Am From the end ^ofatflTen 
line ^ l> to erect a perpendlc" 
nlar A M (Fig. 8).-^Prom any centre 
>', above A 2>, describe a circle patising 
through the given point JL and cut- 
ting the given line at D. Draw D F 
and produce it to cut the circle at E^ 
and draw the perpendicular A E. 

Second Method (Fig. 4).— From the 
given point A set off a distance A E 
equal to three parts, bv any scale : 
and oil the centres A and E^ M'lth radii 
of four and Ave parts respectlvfly, 
describe arcs intersecting at C Draw 
the perpendicular 4 C, 

NoTB.— This method Is most useful 
on very large scales, where straight 
edges are inapplicable. Any multiples 
of the numb«*rs 9. 4, 5 may be taxen 
with the same effect as 6, 8, 10, or 9, 
12, 15. 



6« Todra^ra 
to a 



To dra^r a perpendlcnli 

•tralfflit fine Arom ai 

It Wltfiont It (Fig. 5.)-Fn 



lar 



Solnt wltnont It (Fig. 5.)- From 
le point 4, with a sufficient radius 
cut the given line at F and (?, and 
from these points describe arcs cut- 
ting at E, Draw the perpendicular 

6. To dra^r a etralsbt line 
parallel to a slTcn line, at a 
fflTcn dUtance apart (Fig. O).— 
From the centres A^ /«, in the given 
line, with the given distance as radius, 
describe arcs C. D, and draw the par- 
allel lines C J> touching the arc9. 



GEOMBTRIOAL PROBLBUS. 





Fio. 8. 




To dlTlde a stnlcht Una 
» a namber of equal narto 

. 7).-To divide the line A B into, 



Into I 

say, Ave parte, draw the line ^ C at 
an angle from A; set off live equal 
parts: draw B 6 and draw parallels to 
it from the other points of division in 
A C. These parallels divide A B aa 
required. 

KoTB.— Bya similar process a line 
may be divided into a number of un- 
equal parts; setting off divisions os 
A C, proportional by a scale to the re- 
quired divisions, and drawing parallrl 
cutting A B, Tlie triangles All^ A^ 
^88, etc., are similar Manglea, 



•tralslit line to 
Kle eqnal to 



8* Upon 
draw an anjrle eqnal to a 
tfiwen anffle (Fig. 6).— Let A be the 
given angle and F G the line. From 
the point A with any radius describe 
the arc D E. From F witli the same 
radius describe J H. Set off the arc 
/tfequaltoDf.anddrawir'JEr. The 
angle F Is equal to ^, as required. 





9« To dra^r anarl^* of 60* 
and 80<* (Fig. »).— T'rom F, %vith 
any radius F /, describe an arq^ J H ; 
and fruiii 7, with the same radiAs, cut 
the arc ax H and draw FH to form 
the required angle 7 F JET. Draw the 
perpendicular HK to the base line to 
form the angle ot 20*FHK, 



10* To dra^r an angle or45« 

(Fig. 10).— Set off the distance FJi 
draw the perpendicular 1 H equal to 
I Ft and join HF to form the angle at 
F. The angle at His also 45«. 



11. To bleeet an angle (Fig. 
11).— Let AC Bhe the angle; with O 
as a centre draw an arc cutting the 
sides at A^ B. From A and B as 
centres, describe arcs cutting each 
other at D. Draw C D, dividing the 
angle Into two equal parts. 



Fio. 12. 



19* Tliroagli t^ro glwen 
points to deacrlbe an are or 
a circle wltli a glTen radlna 

(Fig. 12).— From the points A and B 
an centres, with the given radius, de- 
Bcri))e arcs cutting at C \ and from 
Cwith the same radius desoribe an 
arc A B. 



OBOMBTfilCAL PB0BLEH6. 



39 




Fio. 18, 




Fig. 14. 




Fio. 15. 



18. To flnil the eeiitre of a 
eirele or of an are of a circle 

(Fiff. laX'-Select three points, A^ B, 
C% In tbe oiteumferenoe, well apart; 
with the same radius describe arcs 
from these three points, cutting each 
other, and draw the two lines, D E^ 
F-&^ through their intersections. The 
point O, where they cut, is the centre 
of the circle or arc. 

To describe a circle paesliiir 
Uuronslft three kItch points* 
~Ijet A^ B,CY>e the given points, and 
proceed asin last problem to find the 
centre O, from which the circle may 
be described. 

14. To describe an arc of 
a circle passing thronsb 
three slTcn points ivhen 
the centre Is not aTallable 
(Fig.14) — From the extreme points 
A, B, as centres, describe arcs A H, 
B O. Through the third point O 
draw AE, B F, cutting the arcs. 
DlTide A F and B E into any num- 
ber of equal parts, and set off a 
series of equal i>arts of the same 
length on the upper portions of the 
arcs beyond the points E F. Draw 
straight lines, B L, B M, etc., to 
the divisions in A F, and AI,AK, 
etc., to the divisions in E Q. The 
succeissive intersections N^ O, etc., 
of these li es are points in the 
circle required between the given 

Soints A and C. which mav be 
rawn in ; similarly tbe reraunlng 
Sart of tbe curve B C may be 
escribed. (See also Problem 64.) 

15* To dra^r a tangent to 
a circle Aroni a slTcn point 
In the elrcomference (Fig. 15). 
—Through the fdven point A^ draw the 
radial line A C, and a perpendicular 
to it, F &, which is the tangent re- 
quired. 




16* To dra^r tansrents to a 
circle ttont a point without 

It (ITig. 16).— From A, with tht* radius 
A C, desci-ibe an arc B C D^ and from 
C, with a radius eaual to the diameter 
of the circle, cut the arc at B D, Join 
BC, CD, cutting the circle at EFy 
and draw A E^ A F, the tangents. 

Note.- When a tangent Is already 
drawn: the exact point of contact may 
be found by drawing a perpendicular 
to it from the centre. 



17. Betireen tw^o Inclined lines to draur a series ofelT^ 
eles tonchlnc these lines and tonchlns each other (Fig. IT). 
-Bbtect the iiicUnation of the given lines A /?, CD, by the line N O. From 
a point F In this line draw the perpendicular P ^ to tbe line A B^ and 




GEOMETRICAL PBOBLEMS. 



on P describe the circle B P, touching 
the lines and cutting the centre line 
&t E. From JSdraw JCFperpendicular 
to the centre line, cutting ^ i^ at ^, 
and from ir* describe an arc £ 6. cut- 
ting ABaXG. Draw O H parallel to 
B P„ giving J7, the centre of the next 
circle, to be described with the radius 
H E, and so on for the next circle IN. 
Inversely, the largest circle may be 
described nrst, and the smaller ones 
in succession. This problem is of fre- 
quent use in scroll-work. 

18* Bet'lveen two Inclined 
llnea to drm^r a drenlar •««*- 
ment tansent to the lines and 
paaalnfT throoffli a point F 
on the line F C whleh bisects 
the ancle of the lines (Fig. 18). 
—Through jPdraw D ^ at right angles 
to FC; bisect the angles A and A as 
in Problem 11, by lines cutting at C, 
and from C with radius CFdraw the 
arcHFO required. 

19. To dra^r a clircniar are 
that will be tangent to t^ro 
flTlven lines ^ Ji and V J> In- 
clined to one another, one 
tangential point M helnc 
alTcn (Fig. 19).>-Draw the centre 
hne O F. From Ednvr EFat right 
to angles A B ; then F is the centre 
of the circle required. 

20* To describe a circular 
arc Joining tw^o circles, and 
tonchlns one of them' at a 
SlTcn point (Fig. 90).— To johi the 
circles AB^FQ.hy 9Ji arc touching 
one of them at F, draw the radius B F, 
and prodiiceit both ways. Setoff jPfi 
equal to the radius A C ot the oUier 
circle; join CH and bisect it with the 
perpendicular LI, cutting EF At I. 
On the centre I, with radius IF, de- 
scribe the arc F il as required. 

21 • To draw a circle wrfth a 

STcn radios M that will be 
nsent to two fclwen dreles 

A and B (Fig. 21).— From centre 
of circle A with radius equal R plus 
radius of ^, and from centre of B with 
radius equal to /Z + radius of B, draw 
two arcs cutting each other in C, which 
will be the centre of the circle re- 
quired. 

ft2. To construct an eanl- 
lateral trlanale« the sides 
belns tflTcn (Fig. «<).— On the ends 
of one side, A^ B, witli ABas radius, 
describe arcs cutting at O, and draw 
A O, CB. 



OBOXBIBIOAL FB0BLBM8. 



41 




28. To convCntet m, trlansle 
of nneaiua atde* (Fl«r. 88).~Oa 
either end of the hase A D, with the 
Bide B ag radius, describe an arc; 
and with the side C as radius, on the 
other end of the base as a centre, cut 
the arc at E, Join AE,DE, 



Fio. 23. 



" ';< 




Hi fi 




S4« To iKoiistniet H aqnare 
on a ^Ten atmifflit line A B 

(FIs:. 84).— With AB B» radius and A 
and B as centres, draw arcs i4 Dand B 
C, intersection; at E. Bisect EBAiF. 
'With fas centre and RFbm radius, 
cut the>arc8 A D and B C In D and C. 
Join A C\ C i>, and i>^ to form the 
square. 



%S» Vo eonstraet a reet- 
ancle witlijrtTen Imum J7 :p 

maA beiclftt jTlf (Fig. 85).— On the 
base E Fartkw the perpendiculars EH, 
F equal to the height, and Join O H. 



aboi 



6. Vo describe a elrele 



_ Dai a triankle (Fifr. 86).- 
Bisect two sides AB, A C of the tri- 
angle at E Jr\ and from these points 
d raw perpendiculars cutting at k. On 
the centre K^ with the radius KA, 
draw the circle ABC, 



97. To tneeribe a elrele In 
a triangle (Fig. 87).~Bi8ect two of 
the angles A^ C, of the triangle by lines 
cutting At JD ; from D draw a per- 
pendicular De! to any side, and with 
D Eas radius describe a circle. 

When the triangle is equilateral, 
draw a perpendicular from one of the 
angles to the opposite side, and from 
the side set off one third of the per> 
pendicular. 

S8. To desert be a clrele 
about a aiinare. and to In- 
■crtbe a square In a elrele (Fig. 
£8).— To describe the circle, draw the 
diagonals ^ B, C D of the square, cut- 
tine at E. On the centre E, with the 
radius A JST, describe the circle. 



FlO.88. 



To Inseribe tMte sqni 

Draw the two diameters, AB, OD, at 
right angles, and join tlie points A, B, 
C D,Xo form the square. 

Note.— In the same way a circle may 
be described abvut A rectangle. • 



42 



GEOMETRICAL PROBLEMS. 



A Q C 







At 



29. To Inseiibe a drde tn • 
tqnare (Fi^. 29).— To inscribe the 

slrcle, draw the diagonals A B^ C D 
jf the square, cutting at E; draw the 
perpendicular E F to one side, and 
with the radius B F describe the 
circle. 



30* To dosoHbe a B^iiare 
aboot a cirelo (Fig. 80).— Draw two 
diameters A B, CD at right angles. 
ViXth the radius of the circle and J, B^ 
C and D as centres, draw the four 
half circles which cross one anotfaar 
in the comers of the square. 



81. To Inscribe apontacon 
tn a drele (Fig. 81).— Draw cflam- 
eters AC^B DtX right angles, cutting 
at o. Bisect Ao tX E, and from £, 
with radius £ ^, out ^ C at ^ ; from 
B, with radius B F, cut the circumfer- 
ence at Oy H, and with the same radius 
step round the circle to /and K ; join 
the points so found to fonn Uie penta- 
gon. 



Fio.84. 



Bflm To constraet a penta^ 
iron on a jglwen line A B (Fig. 
JW).— From ^ erect a perpendicular 
B G half the length of A B; join A C 
and prolong it to />, maicing C 2> = BC. 
Then B D in the radius of the circle 
circumscribing the pentagon. From 
A and ^as centres, with BDbb radius, 
draw arcs cutting each other lu O, 
which is the centre of the circle. 

88. To ronatmet a bozairon 
upon a slwen atralsbt nno 

(Fig. S8).~From A and BTthe ends of 
the given line, with radius A B, de- 
scribe arcs cutting at g ; from g, with 
the radius g A^ describe a circle; with 
the same radius set off the arcs A O, 
O F, and BD.DE, Join the points so 
found to form the hexagon. The aide 
of a hexagon = radius of its circum- 
scribed circle. 

34» To Inscribe a bexag^on 
In a circle (Ffg. 84).— Draw a diam- 
eter A CB. From A and B as centres, 
with the radius of the circle A C, cut 
the circumference at D, E^ F^ O, and 
draw A D,DE, etc., to form the hexa- 
gon. The radius of the circle is equal 
to the Ride of the hexagon; therefore 
the points D, Ej etc.. may also be 
found by stepping the radius six 
times round the circle. The angle 
between the diameter and the sidea of 
a hexagon and also the exterior angfle 
between a side and an adjacent side 
prolonged Is 60 degrees; therefore a 
hexagon may conveniently be drawn 
by the use of a 60-degree triangle. 



GEOMETRIGAL FBOBLEMS. 



43 




F10.8& 




85. To describe a hexagon 
aboot a elrele (FIr. 85).— Draw a 
diameter A D B, and with the radius 
A A on the centre A^ cut the circum- 
ference at C ; join A 0, and bisect it 
with the radius D E ; through E draw 
FQ^ parallel to ^ C, cutting the diam* 
eter at F, and with the radius D F de- 
scribe the circumscribing circle ^J7. 
Within this circle describe a hexagon 
by the preceding problem. A more 
convenient method is by use of a 00- 
degree triangle. Four of the sides 
make angles of 60 degrees with the 
diameter, and the other two are par- 
allel to the diameter. 

86* To deaerlbe an oeCasoB 
on a KlT«n atralfflit line (l«lg. 
86).— Produce the given line A B botl 

V} 

BV, 

equal to^B. Draw C X> and H O par- 
aflel toAE, and equal UiAB\ from 
the centres O^ A with the radius A B, 
cut the perpendiculars at ff, F, and 
draw EFto complete the ocUgon. 

87. To convert a Minare 
Into an octagon (Fig. 87).— Draw 
the diagonals of the square cutting at 
e ; from the corners A^ B, C7, D, with 
^ « as radius, describe arcs cuttlne 
the sides at gn, fk, Am, and ol, and 
join the points so found to form the 
octagon. Adjacent sides of an octa- 
gon make an angle of 186 degrees. 



88. To Inscribe an octagon 
In a circle (Fig. 88).— Draw two 
diameter8» A C, B D at right angles; 
bisect the arcs A B^ B C, etc., at e/, 
etc., and join Ae^eB, etc., to form 
the octagon. 



89* To describe an octagon 
about a circle (Fig. 99).— Describe 
a square about the given circle A B ; 
draw perpendiculars h k, etc., to the 
diagonals, touching the circle to form 
the octagon. 



40. To describe a polygon of any number of sides upon 
ft Klwen strali^t line (Fig. 40).— Produce the given line A B, and on A, 



44 



OBOHKtRtOAL t>fiOBLfiKS. 







with the radius A S^ di»ffcribe a semi- 
circle; divide the seml-oircumference 
into as roftAy equal parts as therts are 
to be sides in the polj-gon^say, in this 
example, Ave Sides. Draw lineH from 
A through the divisional points D, b. 
and c, omitting one point a ; and on 
the centres B, Z>, u ith tlie radius A B, 
cut ^5 at J? and ^ cat ii^. DrawDf, 
^ ^, ^B to complete the polygon. 
41,. Vo ln(i«rtbe a eirele 



HVlUlln a poljrg^oil (Figs. 41, 45!).— 
When thepoTygon hai« an even number 
of sides (Fig. 41), bisect two opposite 



sides at ^ and ^; draw ^ B. and bisect 
it at C by a diagonal D E, and with 
the radius C^ describe the circle. 

When the number Of sides is odd 
(Fig. 43), bisedt two of the sides at A 
and B. and dr4w lines A E,BD to the 
opposite Angles, intersecting at C; 
from a with the radius C A, describe 
the circle. 



42. IPo deseiilM m drele 

'Wlthoot a polyKton (Figs. 41, 4*.'). 
—Find tiie Centre (J as before, and with 
the radius C Z> describe the circle. 



48. To inijerlbe a polygon 
of anj nomber of ftlde* iv^lUf 
In a circle (Fig. 48).— Draw the 
diameter A B and through the centre 
^draw the perpendicular EC^ cutting 
the circle at S*. Divide E F into four 
equal parts, shd set off three parts 
equal to those from F to C. Divide 
the diameter A B into as many equal 
parts as the polygon is to have siden ; 
and from C draw CA through the 
second point of division, Cutting the 
circle at D. Then ^ Z> is equal to one 
side of the polygon, and by stepping 
round the dircumfet'ence wiih the 
length A D the polygon may be com' 
pleted. 



TABLE OF POLYGONAL ANGLES. 



Number 


Angle 


Nttmbel» 


AnRle 


Number 


Angle 


of Bides. 


at Centtv. 


, Of Sides. 


at Centre. 


of Sides. 


at Centre. 


No. 


Degrees. 


No. 


Degrees. 


No. 


Degrees. 




lao 


9 


40 


16 


94 




90 


10 


86 


16 


^ 




72 


11 


88A 


17 




60 


18 


80 


18 


80 




S» 


13 
14 


^ 


IS 


10 
18 



OlSOMBtRlCAL PROBLEMS. 



45 



In this table the Mgle at th« centre is round hf dividing 860 deRroes, the 
number oC deKreee in a cfreir, by the number of sides fn Uie polytcon; and 
by seititir off roand the centre or the circle a succession of angles by means 
o( the protractor* equal to Vbin aiu^e in the table due to a fflven number of 
sHtee, the radii so drawn will divide the circumference into Uie sanae number 



of parts. 





Fio. 45. 




44. To deaerlbe mn «lllMe 
^rlieii tbe lenstli antt breadtli 
are clvett (Pig. 44). —^ B, transverse 
axis; C />, conjugate axis; F0t foci. 
The sum of the disianoes from C to 
F and Ot also the sum of ihe distances 
from F and O to any other poiut in 
the curve* is equal to the transverse 
axis. From the centre C, with A Em 
radius* cut the axis .^ If at Fand G, 
the foci; fix a couple of pins into the 
axis at F and O, and loop on a thread 
or cord upon them equail In length to 
the axis ^ £, so as when stretched to 
reach to the extremity C of the cou- 
jugate axis, as shown in dot-lining. 
Place a pencil inside ibe cord as at /f , 
and guiding the pencil in this way, 
keeping the Oord equally In tenBion* 
carry the pencil found the pins F^ <?, 
and so describe the ellipse. 

NoTS.— This method is employed in 
setting oif elliptical garden-plots, 
walks, etc. 

2d Method (Fig. 46). —Along the 
straight edge of a slip of stiff paper 
mark off a distance a c equal to A C, 
half the transverse axis; and from the 
same point a distance ab equal to 
C A half the conjugate axis. Place 
the slip so as to bring the point b on 
the line A B ot the transverse axis, 
and the point c on the line D E ; and 
set off on the drawing the position of 
the point a. Shifting the slip so that 
the point b travels on the transverse 
axis, and the point c on the conjugate 
axis, any number of points in the 
curve may be found, through which 
the curve may be traced. 

8d Mtthnd (Fig. 40).— The action of 
the preceding method may be em- 
bodied so as to afford the means of 
describing a large curve continuously 
by means of a bar m le, with steel 
points m, I, Jb, riveted into brass slides 
adjusted to the length of the semi- 
axis and fixed with set-screws. A 
rectangular cross E O, with guiding- 
slots is placed, coinciding with the 
two axes of the ellipse A C and B H. 
By sliding the points k^ I in the slots, 
and carrying round the point m. the 
curve may be continuously described. 
A pen or pencil may be fixed at m. 

Ath Method (Fig. 47).-BiseOt the 
transverse axis at C.and through 
draw the perpendicular D E, making 
O D and £ each equal to half the 
conjugate axis* From i> or E, with 
the radius A C, cut the transverse 
axis at F^ F\ for the foci. Divide 
A C into a number Of parts at the 



46 



GEOMETRICAL PROBLEMS. 



points 1, 2. 8, etc. With the radius A I on F and F' as oentres, describe 
Arcs, ana with the radius B / on the same centres cut these arcs as shown. 

Repeat the operation for the other 
diTisions of the transverse axis. The 
series of intersections thus made are 
points in the curve, through which the 
curve may be traced. 

6tk Method (FiR. 48).— On the two 
axes A B^ D Eaa diameters, on centre 
C, describe circles; from a number of 
points a, 5, etc., in the circumference 
AFB, draw radii cuttinf? the inner 
circle at a% b\ etc. From a, 6, etc., 
draw perpendiculars to AB; and from 
a\ b\ etc., draw parallels to ^ B, cut- 
ting the respective perpendiculars at 
n, o. etc. The Intersections are points 
in the curve, through which the curve 
may be traced. 

6th Method (Fig. 49). — When the 
transverse and conjugate diameters 
are given, A B, C 2>, draw the tansrent 
fJi^^ parallel to A B. Produce CD, 
and on the centre O with the radius 
of half A B, describe a semicircle 
H DK; from the centre O 4raw any 
number of straight lines to Uie points 
E, r, etc., in the line BF^ cutting the 
circumference at Z, m, n, etc.; from 
the centre O of the ellipse draw 
siraiKht lines to the points 17, r, etc.; 
and f iom the points I, m, 1^ etc., draw 
parallels to (? C, cutting ihe lines O J?, 
O r, etc., at L, Af, JV, etc. These are 
points in the circumference of the 
elllpKe, and the curve may be traced 
through them. Points in the other 
half of the ellipse are formed by ex- 
tending the intersecting lines as indi- 
cated in the figure. 

45, To deacribe an ellipse 
approxliiiatel]r by meaD* of 
clrcDlar area,— ^'»«^— With arcs 
of tuu radii (Kig. 50).— Find the dUTer- 
euce of the semi-axes, and set it oif 
from the centre O to a and c on O ii 
and O C ; draw a c. and set off half 
a c to d ; draw d i parallel to a c: set 
off O ^ equal to O a\ join e t, and draw 
tlie parallels em, dm. From m, with 
radius in C, describe an arc through 
C ; and from t describe an arc through 
B; from d and <^ describe arcs through 
A and B. The four arcs form the 
ellipse approximately. 

Nora.— rhis method does not applv 
satisfactorily when the conjugate axis 
is less than two thirds of the trans- 
verse axis. 

2d Method (by Carl G. Earth, 
Fig. 51).- In Fig. 51 a b is the major 
and c a the minor axis of the ^lipee 
to be approximated. Lay off 6 e equal 
to the semi-minor axis c O, and use a e 
as radius for the arc at each extremity 
of the minor axis. Bisect e o at / and 
lay off 6 (7 equal to e/, and use gbaa 
radius for the arc at each extremity 
Fio. 61. of the major axis. 




Pro 49. 




OBOHETBICAL FBOBLEHS. 



« 




Ths mettiod Ib not oonsldered sppUcable for cases in which the minor 
ujs is lees than two thirds of the major. . , 

Zd Method : With arcs of three radii 
(Fife. 5i0.— On the transverse axis A B 
draw the rectangle BGon the.beifrht 
OC; to the diagonal A C draw the 
perpendicular G H Di set off O Jt 
equal to O C, and deecribe a semi- 
circle on A K, and produce O C to L; 
set off O Jf equal to C L, and from X> 
describe an arc with radius D M ; from 
Aj with radius O L, cut ^ B at Ni from 
H, with radius HN, cut arc ad at a. 
Thus the five centres 2>. a» 6, H, H' 
are found, from which the arcs are 
described to form the ellipse. 

This process works well for nearly 
all proportions of ellipses. It is used 
in striking out vaults and stone bridges. 
Uh Method (by F. R. Honey, Figs. 58 and 54).— Three radii are emoloyed. 
With the shortest radius describe the two arcs which pass through the ver- 
tices of the major axis, with the longest the two arcs which pass through 
the vertices of the minor axis, and with the third radius the four arcs which 
counect the former. ^ . 

A. simple method of determinhig the radii of curvature is illustrated in 

Fig. 68. Draw the straight 
lines a/ and a c» forming any 
angle at a. With a as a centre, 
and with radii a b and a e, re- 
spectively, equal to the semi- 
minor and semi-major axes, 
draw the arcs 6 e and ed. Join 
ed, and through 6 and c re- 
spectively draw 6 g and e f 
parallel to e d, interseoting a e 
at a, and af at/; a/ is the 
radius of curvature at the ver- 
tex of the minor axis; and a g 
the radius of curvature at the 
vertex of the major axis. 
' - -- '— — ighthofftd. Join c^ and draw efe and 

longest radius {^R\al for the shortest 

„ - ean, or one half the sum of the semi-axes, 

for the third radius (=!>>, aud employ these radii for the eight-centred oval 
as follows: 

Let a b and c d (Fig. 54) 
be the major and minor 
axes. Lay off as equal 
to r, and a/ equal to p; 
also lay off cy equal to S, 
and c h equal to p. With 
9 as a centre and y A as a 
radius, draw the arc h k\ 
with the centre e and 
radius / draw the are / Jb, 
J. intersectingAib at le. Draw 
1^ the line gfc and produce it, 
makine g I equal to R, 
Draw k e and produce it, 
making h m equal to p. 
With the centre g and 
radius ge{=B) draw the 
arc c I : with the centre k 
and radius kl (=p) draw 
the arc { m, and with the 
centre e and radius em 
<= r) draw the arc m a. 
Fio. 54. The remainder of the 

work is symmetrical with 
respect to the axes. 




Tenex oi uie major axis. 

lAy oBdh (Fig. 58) equal to one eighth 
b Iparallel to e h. Take a k for the long< 
radius (= r). and the arithmetical mean, c 




48 



GEOMSTBICAL PBOBLEHS. 




Fig. 65. 



r 46. Tlfte Parabola*— A parabola 

(Z> ^ C, Fi«. 65} Is a curve nich that 
every point In the curve is equollj 
distant from the directrix KL axid the 
focus F. The focus lies in the axis 
A B drawn from the vertex or head of 
the curve ^4, so as to divide the figure 
into two equal i>arts. The vertex A 
is equidistant from the directrix and 
the focus, or A e^AF. Any line 
parallel to the axis is a diameter. A 
straight line, na EO or DC, dra^in 
across the figure at right angles to the 
axis is a double ordinate, and either 
half of it is an ordinate. The ordinate 
to the axis EFO, drawn through the 
focus, is called the parameter of the 
axia A segment of the axis, reckoned 
from the vertex, is an abscissa of the 
axis, and it is an abscissa of the ordi- 
nate drawn from the base of the ab- 
scissa. Thus, ^ B is an abscissa of 
... - ^ , . the ordinate B a 

AbscissflB of a parabola are as the squares of their ordinatea. 
To doaertbe a parabola wben an abrolsM and Ita or^- 
nate are fftven (Pig. 56).-Bi8ect the given ordinate B Cat a, draw A a, 
and then o 6 perpendicular to it, meeting the axis at b. Bet oil A e^ A F, 
each equal to B 6; and draw Ke L perpendicular to the axis. Then K L is 
the directrix and F is the focus. Through F and any number of points, o, o. 
etc., in the axis, draw double ordinates, non, etc., and from the centre F, 
with the radii Fe,o e, etc., out the respective ordinates at £L <?, n, t^ etc. 
The curve may be traced through these points as shown. --» » t •» "^ 



2d Method : By means of a square 
and a cord (Fig. 56).— Place a straight- 
edge to the directrix E N, and apply 
to it a square LEO. Fasten to the 
end O one end of a thread or cord 
equal in length to the edge E O, and 
attach the other end to the focus F ; 
slide the square along the straight- 
edge, holding the cord taut against the 
edge of the square by a pencil D^ by 
which the curve is described. 




Fia. 56. 




If a B a if cii 
Fia. 67. 



3d Method : When the height and 
the base are given (Fig. 67).— Let A B 
be the given axis, and C D &. double 
ordinate or base: to desciibe a para- 
bola of which the vertex is at A. 
Through A draw J^F parallel to CI>, 
and through C and D draw C E and 
Di^ parallel to the axis. Divide B O 
and BD into any number of equal 
parts, say five, at a, 5, etc., and divide 
C E and D F into the same number of 
parts. Through the points a, 6. c, d in 
the base CD on each side of the axis 
draw perpendiculars, and through 
a.&,c, din CE&nd D.F draw lines to 
the vertex A, cutting the perpendicu- 
lars at c. /, g, h. These are points in 
the parabola, and the curve CAD may 
be traced as shown, pasving throqgh 
then;. 



GEOHETRIGAL PROBLEMS. 



49 




Flo. 68. 




47* The 07perlN»la (Fi^. 58) .—A hyperbola Is a plane curve, such 
that the differentM of the distances from anv point of It to two fixed points 

is equal to a spi ven distance. The fixed 
points are called the foci. 

To conatmot a li]rperlN»la* 
—Let W and Jf" be the foci, and F' F 
the distance between them. Take a 
ruler longer than the distance J^' F, 
and fasten one of its extremities at the 
focus F'. At the other extremity, ff, 
attach a thread of such a length that 
the length of the ruler shall exceed 
the length of the thread by a given 
distance A B. Attach the other ex- 
tremity of the thread at the focus F. 

Press a pencil, P, against the ruler, 
and keep the thread constantly tense, 
while the ruler is turned around F' as 
a centre. The point of the pencil will 
describe one branch of the curve, 

2(i MeViod: By points (Fig. aO).— 
From the focus S" lay off a distance 
F' J^ equal to the transverse axis, or 
distance between the two branches of 
the curve, and take any other distance, 
as F'Hy greater than F'N. 

With J^' as a centre and F'H as a 
radius describe the arc of a circle. 
Then with Jr* as a centre and NU •»& 
radius describe an arc intersecting 
the arc before described at p and q. 
These will be jpohits of the hyperbola, torF^q-Fq is equal to the trans- 
verse axis AB, 

If, with F as a centre and F' H^ as a radius, an arc be described, and a 
second arc be described with i«^' as a centre and NHasA radius, two points 
in the other branch of the curve will be determined. Hence, by changing 
the oentma, each pair of radii will determine two points in each branch. 

Xlie BqntlAieiml Hyperbola.— The transverse axis of a hyperbola 
is the distance, on a line joining the foci, between the two branches of the 
curve. The conjugate axis is a line perpendicular to the transverse axis, 
drawn from its centre, and of such a length that the diagonal of the rect- 
angle of Uie transverse and conjugate axes is equal to the distance between 
the fod. The diagonals of this rectangle, indefinitely prolonged, are the 
cuympfoles of the hyperbola, lines which the curve continually approaches, 
but touches only at an infinite distance. If these asymptotes are perpen- 
dicular to each other, the hyperbola is called a rectanffular or equilateral 
hpperboia. It is a property of this hyperbola that if the asymptotes are 
taken as axes of a rectangular system of coordinates (see Analytical Geom- 
etry), the product of the abscissa and ordinate of any point in the curve is 
equal to the product of the abscissa and ordinate of any other point ; or, if 
p is the ordinate of any point and v its abscissa, and Pi and v, are the ordi- 
nate and abscissa of any other point, pv=pi v, ; or pv = a constant. 

48. The Cyelold 
(Fig. (iO).-lf a circle A d 
be roiled along a strarght 
line ^6, any point of the 
circumference as A will 
describe a curve, which Is 
called a cycloid. The circle 
is called the generating 
circle, and A the generat- 
ing point. 

To dranv a eyelold. 
— Divide the circumference 
of fbe generating circle Into an even number of equal parts, as A 1, 12, etc., 
and set off these distances on the base. Through the points 1, 8, 8, etc., on 
tbe circle draw horizontal lines, and on them set off distances la = Al, 
ibszA^^ = A^ etc. The points A^ a, 6, c, etc., wiU be points in the cycloid, 
throog^ which draw the curve. 




50 



GSOMETRICAL PROBLEMS. 




49. The Bpleyelold (Fig. 61) is 
irenerated by a point D in one circle 
D C roliinff upon the circumference of 
anotlier circle A C B, instead of on a 
flat surface or line; the former beini; 
tiie generatinK circle, and the latter 
the fundamental circle. The generat- 
ing circle is shown in four positions, in 
which the generating point Is succes- 
sively marked A ly, iy\ D"'. A W B 
is the epipydotd. 



50. The 0ypoeyelold(Fig. 02) 

Is generated bv a point in the gener- 
ating circle rolling on the inside of the 
fundamental circle. 

When the generatinflp circle = radius 
of the other circle, uie hypocydoid 
becomes a straight line. 



51* The Trmetrtx or 
Sehlele's mntl-fMetlon enrre 

(Fig. 08).^/; is the radius of the shaft, 

C, 1, 2, etc., the axis. From O set off 

on 17 a small distance, oa\ with radius 

B and centre a cut the axis at 1, join 

a 1, and set off a like small distance 

a h\ from h with radius B cut axis at 

S, Join 6 S, and so on, thus findinsr 

points o, a, &, c, d, etc., through which 

.^ ^ the curve is to be drawn. 

Fio. 68. 

62* The 8plral.-*The spiral is a curve described by a point which 

moves along a straight line accortltng to any given law, the line at the same 

time having a uniform angular motion. Tlie line is called the radius vector. 

If the radius vector increases directly 
as the ineasuring angle, the spires, 
or parts described in each revolution, 
thus gradually increasing their dis- 
tance from each other, the curve is 
known as the spiral of Archimedes 
(Fig. 64). 

This curve is commonly used for 
cams. To describe It draw the radius 
vector in several different directions 
around the centre, with equal angles 
between them; set off the distances 1, 2, 3, 4, etc., corresponding to the scale 
upon which the curve is drawn, as shown in Fig. 04. 

In the common spiral (Fig. 61) the pitch is uniform; that is. the spires are 
equidistant. Such a spiral is made by rolling up a belt of uniform thickness. 
/ 

To eonstrnct a spiral ^rltlt 
four centres (Fig. 66).— Qiven the 
pitch of the spiral, construct a square 
about the centre, with the sum of the 
four sides equal to the pitch. Prolong 
the sides in on^ direction as shown; 
the comers are the centres for each 
arc of the external angles, formlD£f a 
quadrant of a spire. 

Fig. 66. 




37 12 8 4 56 
s,^4 .«» -^ 



Fio. 64. 




GEOVETBIGAL PBOBLEKS. 



51 




53. To find tike dfmineter of a etrele Into urbleb a eertaln 
aamlMr of lines will fit on Its Inside iPig. 66).— For instanoe, 
what is ihe diameter of a circle Isto which twelve ^-inch rlnss will fit, as 
per aketch f AMume that we have found the diameter of the required 

ci rcle, and have drawn the riuss Ineride 
of It. Join the centres of the rinjirg 
bv straight lines, as shown : we then 
obtain a regular polygon with 18 
sides, each side being equal to the di- 
ameter of a Riven ring, we have now 
to And the diameter of a circle cir- 
cumscribed about this polygon, and 
add the diameter of one ring to it; the 
sum will be the diameter of the drole 
into whicli tlie rings will fit. Through 
the centres A andZ) of two adjacent 
rings draw the radii CA and CD; 
since the polygon has twelve sides the 
angle ACD = W> and ACB=l6\ 
One half of the side ^ D is equal to 
A B. We now give the following f^ro- 
portlon : The sme of the angle ACB 
isto^Baslisto the required ra- 
dius. From this we get the following 
Hjjd : IXIvide A B by the sine of the angle ACB ; the quotient will be the 
radius of the circumscribed circle : add to the corresponding diameter the 
diameter of one ring : the sum will be the required diameter FG. 

64. To describe an are of a elrcle nrlileli Is too lar^e to 
be drasm by a beam compass, by means of points In tbe 
are, radlns belns slTen.— Suppose the radius is )M) feet and it is 
desired to obtain five points in an arc whose half chord is 4 feet. Draw a 
line equal to the half chord, full size, or on a smaller scale if more con- 
venient, and erect a perpendicular at one end, thus making rectangular 
axes of coordinates. Erect perpendiculars at points 1, 8, 8, and 4 feet from 
tlM first perpendicular. Find values of y in the formula of the circle. 
j« + ^ = IP by substituting for x the values 0, 1, 8, 8, and 4, etc.. an d forJB* 
the^uare of ^e radius, or 400. The values will be y = ^K» -x*= «'400, 
<^a99, t".^, «'89]. 4^884; = 80, 19.975, 19.90, 19.774, 19.596. 
Subtract the smallest, 

or 19.S06, leaving 0.404, 0.879, 0.804. 0.178, feet, 

liiy off these distances on the Ave perpendiculars, as ordlnates from the 
hjjf chord, and the positions of five points on the arc will be found. 

Through these the curve may be 
drawn. fSee also Problem 14.) 

55* Tbe Catenary Is the curve 
assumed by a perfectly flexible cord 
when its ends are fastened at two 
points, the weight of a unit length 
being constant. 
The equation of the catenary Is 




e Is the 



y= ?(«*' + « "It *n which 

base of the Naperlan system of log- 
arithms. 
To plot the catenary,— Let o 

(Fig. 67) be the origin of coordinates. 
Assigning to a any value as 8, the 
equation becomes 



t = l(^ + e~^y 



M" 



Vl) = 8. 



52 



OBOHSTBIGAL PROBLEMS. 



Then put a; = 1; .*. y 



Put a; = 2; 






:|(i.896 + o.rir)a 



«.17. 



(1.948 4- 0.618) = 3.69. 



Put a; ^ 8, 4, ft, <itc., etc., aud And the correspondinft Taluee Ot y. For 
each value Of y we obtain two syiumetrical points, as for example p and p^ 
In this way, by making a successiyely equal to 2, 3, 4, 5, 6^ 7, and 8, the 
curves of Fig. 87 were plotted. 

In each case the distance from the origin to the lowest point of the curve 
is equal to a ; for ptittlng x=o, tlie general equation reduces to y = a. 

For values of a = 6, 7, and 8 the catenary closely approaches the parabola. 

For derivation of the equation of the catenary see Bowser^s Analytic 

Mechanics. For com po risen of the catenary with the parabola, see article 

by F. a. Hohev. Amer. Machinist, Feb. 1, 1W4. 

56* T]i« InTOlnte is a name given to the curve which is formed by 

the end of a string which is unwound 
from a cylinder and kept taut ; con- 
sequently the string as it is unwound 
will always lie in Ihe direction of a 
tangent to the cylinder. To describe 
the involute olf any given circle. Fig. 
68, take any point A on Its circum- 
ference, draw a diameter AB^ and 
from B draw B b perpendicular to AB. 
Make Bb equal in length to half the 
circumference of the circle. Divide 
Bb and the semi-circumference intt) 
the same number of equal part^ 
say six. Fi-om each point of division 
1, 2, 3, etc., on the circumference draw 
lines to the centre C of the circle. 
Then draw 1 a perpendicular to C 1 ; 
2 ns perpendicular to 02; and 00 on. 




Fta. 68. 



Make 1 a equal to b b, ; 2«r* equal 
to & 6) ; 8 as equal to 6 6| ; ana so on. 



Join the^polnts^, ai\ a^, a^, etc., by a curve; this curve will be the 
required involute. 
67« netliod of plotting angles ^irliliont nslnga prAtrae* 

tof.— The radius of a circle whose circumrerence is 360 is 57. S^ (more ac- 
curately 57.296). Striking a semicircle with a radius 57.3 by any scale, 
spacers set to 10 by the same scale will divide the arc into 18 spaces of 10^ 
each, and intermediates can be measured Indirectly at the rate of 1 by ecale 
for each iVor interpolated by eye according to the degree of accurtury 
required. The following table shows the chords to the above-mentioned 
radius, for every 10 degrees from 0^ up to 110"*. By means of one of these. 



Ani 



igle. Chord. 

!• 0.999 

lO* 9.988 

2(r» 19.899 

30«> 29.658 

40» 89.192 

60» 48.429 



Angle. Chord. 

60» 67.296 

70» 66.73? 

80" 78.668 

90** 81.029 

lWy> 87.782 

110» 93.809 



a 10° point is Oxed upon the paper next less than the required angle, azid 
the remainder is laid off at the rate of 1 by scale for each degree. 



GEOMETBICAL FBOt»0SITIONS. 5B 



QEOMETBICAL PROPOSITIONS. 

la a rightanKled trlaagl« the square on the hjpothcnuse is equal to the 
■uiD of the squares on tm Other two sidett. 

Ir a trianifle is equilateral, it is equiang^ular, and vice ver$a. 

If a straight line from the vertex of an isoMCeles trlauitle bisecti th« iMse, 
ft biveotA me verifeal angle and is perpendicular to the base. 

If one side of a triangle Ia produced, the exterior angle is equal to the earn 
of the two interior ahaopposlte angles. 

If two ti-laiigles are mutually equiangular, they are similar and theft* 
corresponding sides are proportional. 

If the sides of a polygon are produced la the same order, the ftum of the 
exterioir angles equals four rignt angles. 

In a quadrilateral, tbe sum of the Interior angles equals four right angles. 

In a parallelogram, th* opposite sides are equal t the oppoilte augles 
are equal; it Is oisected by Its diagonal; and its diagonals bisect e».ch 
other. 

If three points are not in the same straight line, a drole may ba paired 
throuith tbetn* 

If two arcs are intercepted on the same circle, they are proportional to 
the c or res ponding angles at the centre. 

If two arcs are simtuur, they are proportional to their radii. 

The areas of two oirdes are proportional to the square* of their radii. 

If a radius is perpendioular to a chord, it bisects the chord and It biseoti 
the are subtended by the chord. 

A straight line tangent to a circle meets it in only one point, and It It 
perpendicular to the radius drawn to that point. 

If frotn a point without a ch«le tangents are drawn to touch the circle, 
there are but two; they are equal, and they make equal angles wilh th« 
chord Joining the tangent points* 

ir two lines are parallel chords or * tangent and parallel chord, they 
intercept equal arcs of a circle. 

If an angle at the circumference of a circle, between two chords, H sub- 
tended 1^ the same arc as an angle at the centre, between two radii, the 
sfigla at tbe oinmmference is equal to half the angle at the centra^ 

Ir a triangle is inscribed in a semiolrcli*, it is rignt-anglrd. 

If an angle is formed by a tangent and chord, it is measured by obe half 
ef the aro intercepted by the chord; that is, it is equal to hair the angle at 
Ihe centre subtended by the chord. 

If two abords intersect each other in a drole, the rectangle of the seg- 
ments of the one equals the rectangle of the segments of the other. 

And if one chord is a diameter and the other perpendicular to ft-, the 
rectangle of the segments of the diameter Is equal to the square on half the 
other chord, and the half chord is a mean proportiotial betweeu the ^g- 
menta of the diameter. 



M XBKBUBATIOV. 

MENSUBATION. 

PI«ANE S17RFACE8, 

anadrllateral.— A four-sided figure. 

Paralleloffram.— A quadrllatemi with opposite sides parallel. 

Fanetiei.— Square : four sides e<iual, all angles right angles. Bectangle: 
opposite .sides equal, ali augles right angles. Rhouibus: four sides equal, 
opposite angles equal, angles not right angles. Rhomboid: opposite sides 
equai, opposite angles equal, augles not right angles. 

Trapeslom.— A quadrilateral with unequal sides. 

Trapexold.-A quadrilateral with only one pair of opposite sidea 
parallel. 

IMaffonal of a sqaare = 4/8 x i*ide« = 1.4148 x side. 



IHair, of a rectangle = 4/sum of squares of two adjacent sides. 

Area of any parallelogram = base x altitude. 

Area of rliombas or rEombold = product of two adjacent sides 
X sine of angle included between Lhem. 

Area of a trapezium = half the product of the diagonal by the sum 
of the perpendieulars let fall on it from opposite angles. 

Area of a trapezoid = product of naif the sum of the two parallel 
sides by the |>erpendicuiar distance between thefn. 

To find tlie area of any quadrilateral flcrure.— Divide the 
quadrilateral into two triangles; the sum of the areas of ih« triangles is the 



Or, multiply half the product of the two diagonals by the sine of the angle 
at their intersection. 
To find tlie area of a quadrilateral Inscribed In a circle* 

^From half the sum of the four sides subtract each side severally; multi- 
plythe four remainders together; the square root of the product is the area. 

Trlanfrle.— A three-sided plane figure. 

Fiat-ietiM.— Right-angled, having one right angle; obtuse-angled, having 
one obtuse anf^e ; isoweles, having two equal angles and two equal sides; 
equilateral, having three equal sides and equal angles. 

The sum of the three angles of every triangle = 180**. 

The two acute angles of a right-angled triangle are complements of each 
other. 

Hypothenuse of a right-angled triangle, the side opposite the right angla 

= |/sum of the squares of the other two sides. 
To find tbe area of a triangle s 

RuLK 1. Multiply the base by half the altitude. 

RuLB 8. Multiply half the product of two sides by the sine of the Included 
angle. 

RiTLK 8. From half the sum of the three sides subtract each side severally; 
multiply together the half sum and the three remainders, and extract the 
square root of the product. 

The area of an equilateral triangle is equal to one fourth the square of one 

of Its sides multiplied by the square root of 8, = - — ^ a being the side; or 

a« X .438018. 

Hypothenuse and o ne side of right- angled triangle given, to find other side, 
Requii'ed side = Vhyp« — given 8ide«. 

If the two sides are equal, side = hyp -•- 1.4148; or hyp X .7071. 

Area of a triangle given, to find base: Base = twice area -1- perpendicular 
height 

Area of a triangle given, to find height: Height = twice area •*- base. 

Two sides and base given, to fin^ perpendicular height (in a triangle in 
which both of the angles at the base are acute). 

RuLB.— As the base Is to the sum of the sides, so Is the difference of the 
sides to the diflTerence of the divisions of the base made by drawing the per- 
pendicular. Half this difference being added to or subtracted from naif 
the base will give the two divisions thereof. As each side and its opposite 



PLANE SUBFAGES. 



55 



dirisfoB of (he base constitutes a right-ang led triangle, t he perpendicular Is 
aso»lalned by the rule perpendicular = Vhyp* — base*. 

Polygon. — A plane figure haying three or more sides. Regular or 
irregular, according as the sides or angles are equal or unequal. Polj'gons 
are named from the number of their sides and angles. 

To find the area of an Irresnlar polyson.— Draw diagonals 
dlTiding the polygon into triangles, and find the suiu of the areas of these 
triangles. 

To find the area of a regular polygon s 

RuLX.— Multiply the length of a side by the perpendicular distance to the 
centre; multiply the product bv the number of sides, and divide it by 2. 
Or, multiple hau the perimeter by the perpendicular let fall from the centre 
on one of the sidea 

The perpendicular from the centre Is equal to half of one of the sides of 
the polygon multiplied by the cotangent of the angle subtended by the half 
side. 

The angle at the centre = SOO" divided by the number of sides. 







TABLE OF REGULAR POLYGONS. 












Radius of Cir- 
















cumscribed 


If 


4i 








1 


II 


Circle. 


^ 


% 


1 

6 




H 


< 




Triangle 


.4380127 


2. 


.6778 


.2887 


1.788 


120» 


60» 




Square 


1. 


1.414 


.7071 


.5 


1.4142 


90 


90 




Pentagon 


1.7204774 


1.288 


.8506 


.6888 


1.1756 


72 


106 




Hexagon 


8 5SM0782 


1.156 


1. 


.866 


1. 


60 


120 




Heptagon 


8.0389124 


l.U 


1.1524 


1.0388 


.8677 


51 2G' 


128 4-7 




Octagon 


4.8284271 


1.068 


1.3066 


1.2071 


.7658 


45 


185 




Nonagon 


6.1818212 


1.004 


1.4619 


1.8787 


.684 


40 


140 


10 


Decagon 


7.0942068 


1.061 


1.618 


1.5.^88 


.618 


36 


144 


11 


Undecagon 


9.3066890 


1.042 


1.7747 


1.70« 


.5634 


82 43' 


147 3-11 


» 


Dodecagon 


11.1961524 


1.087 


1.9319 


1.866 


.5176 


80 


150 



To And the area of a resnlar polygon, nrlieii the length 
of a side only to ciTen s 

KnjB. — Multiply the uquare of the side by the multiplier opposite to the 
Djinie of the polygon in the table. 

To And the area of an Ir- 
regular Acnre (Fig. 69).— Draw or- 
diiiates across its breadth at equal 
distances apart, the first and the last 
onlinate each being one half space 
from the endt of the figure. Find the 
average breadth by adding together 
the lengths of these lines included be- 
tw<*en the boundaries of the figure, 
and divide by the number of the lines 
added; multiply this mean breadth by 
tbti length. The greater the number 
of lilies the nearer the approximation. 

In a figure of very irregular outline, as an Indif^itor-diagram from a high- 
speed steam-engine, mean lines may be substituted for the actual lines of the 
figure, being so traced as to intersect the undulations, so that the total area 
of the spaces cut off may be compensated by that of the extra spaces in- 
closed. 




86 HE2)8URATIOK. 

fbd Method : Tm Trapbsoidal Rulb. — Divide the figure Into any euffl. 
dent nunibtr of equal parte: add half the gum of the two end ordinaten to 
the sum of all the other ordinates: divide by ihe number of spaoen ((hat is. 
one less than the number of ordinates) to obtain the mean ordinate, and 
multiply this by the length to obtain the area. 

8d Method : Simpsom^b Rulb.— Divide the length of the figure into any 
even number of equal parts, at the common distance D apart, and draw or. 
dinates through uie points of division to touch the boundary lines. Add 
together the first and last ordinates and call the sum^; add together the 
eveu ordinates and call the sum i'; add together the odd ordinates, except 
the first and last, and call the sum 0. Then, 

area of the figure = ^-^^^ + ^<^ x D. 

4th Method : Durand^s Rulb.— Add together 4/10 the sum of the first and 
last ordinates, 1 1/10 the sum of the second and the next to the last (or the 
penultimates), and the sum of all the intermediate ordinates. Multlplv the 
wum thus gained by the common distance between the ordinates to obtain 
the area, or divide this sum by the nmnber of spaces to obtain the mean or- 
dinate. 

Prof. Durand describes the method of obtaining his rule in Engineering 
Newt, Jan. 18. 1804. He (dalms that it is more accurate than Simpson's rule, 
and practically as simple as the trapezoidal rule. He thus describes Its ap- 
plication for approximate integration of diffenential equations. Any defi- 
nite integral may be represented graphically by an area. Thus, let 



Qz=yudx 



be an integral In which u Is some function of x, either known or admitting of 
computation or measurement. Any curve plotted with z as abscissa and u 
as ordinate will then represent the variation of u with x, and the area o«- 
tween such curve and the axis X will represent the integral in question, no 
matter how simple or complex may be the real nature of ibe function «. 

Substituting in the rule as above given the word ''volume" for "area ' 
and the word ** section ** for ** ordinate," it becomes applicable to the deter- 
mination of volumes from eqnidbtant sections as well as of areas from 
equidistant ordinates. 



nates + sum of the other ordinates) 1/10 of (sum of penultimates— sum of 
first and last) and multiplying by the common distance between the ordl- 

^th Method —Draw the figure on croes-sccilon poper. Count tlie number 
of squares that are entirely Included within the boundary; then esUmate 
the fractional parts of squares that are cut by the boundary, add together 
these fractions, and add the sum to the number of whole squares. The 
result is the area in units of the dimensions of the squares. The finer the 
ruling oi' tlie cross-section paper the more accurate the result. 

Mh Method.-Vrui a planimeter. 

7th Mt:thf)d.—'Witli a chemical balance, Rensitive to one milligram, draw 
thf flKure on paper of uniform thickness and cut It out carefully; weigh the 
piece cut out, and compare its weight with the weight per square Inch of the 
oaper as tested by weighing a piece of rectangular shape. 



THB CIBCLB. 



67 



THJB €Ilft€I<S* 

Circiiinferenoe s diameter x 8.1418, nearly; more McuraMy, 8.141RM66S50. 
Approximations, j = 8.148; ^ = 8.1415089. 
The ratio of circum. to diam. is represented by the symbol «- (called Pi), 



Multiples of V. 
lv= 8.14150965850 



«rs 6.88818580718 
a«-= 0.48477796077 
4« = 18.66687061486 
&r = 15.70706886:96 
6« = 18 84056598154 
7« = 21.00114857518 
8r = 25.18874128878 



' )c3 = l. 5707908 
' X 3 = 2.8561045 
X 4 = 8.1415027 
X 5 = 8.0800006 
X 6 = 4.7123890 
X 7 = 5.4077871 
X 8 = 6.8881858 
X = 7.068je85 
Batio of diam. to circumference = reciprocal of r = 0.8183009. 



Multiples of <. 
[* = .7853062 



Btfciprocal of |v = 1 .27824. 
Multiples of -. 



1 



.81881 

r 

= .68668 

= .05108 
= 1.27884 
= 1.50156 



•=1. 



- = 2.28817 

V 

5. = 2.54618 
ir 

- = 2.86470 

w 

-=8.18810 

n 

- = 8.81072 



^» = 1.5'n)796 
■ir= 1.647197 
> = 0.523600 



M= 0.261700 



^=z 0.0G87866 
— = 114.6015 



««= 0.86060 



-=- = 0.101321 






1 772458 
0.564180 



Logirs 0.49714067 
Lofi; ^» = T.895090 



^ = area. 



Diam. in Ins. = 18.5105 4^area in sq. ft. 

Area in sq. ft. = (.llam. In inches)^ x .0)54542. 

D = diameter, B = radius, G = circumference, 

O = irD; = 2irB; = 1^; = 2 V'l^^; = 8.545 VJ ; 
il=i>>x .7854; =^; =4i2« x .7854; =»/?•; =J«i>«; = ^; =.07058C»; = ^- 



1) = ^ 



= 0.81881 C; :=2V^; = 1.18888 V?; 
= 0.160156C; = V |-; = 0.564180 VI. 



Areas of drelss are to each other as the squares of their diameters. 
To And tlie len^^ of an are of a circle t 

anus 1. As 860 is to the number of degrees in ihe arc, so is the circum- 
ference of the circle to the length of the arc. 

BiTLa 2. Multiply the diameter of the circle by the number of degrees in 
the aic, and this product by 0.0067866. 



58 [mbnsubatiok. 

Relation* of Are, Cliord, CltoWl of Half the Are^ 
ITereed Sine, ete« 

Let B = radius, D = diameter, Ai-c = length of arc, 

Cd = chord of the arc, ch = chord of half the arc, 

F= versed sine, D—V = diam. minus ver. sin., 

Mi-cd, ,, v^aTTF* X ior« , „ , 
Arc = — (very nearly). = — i5c; a» + 8Sr« ^ ^'*' nearly. 



Chord of the arc = 2 Vcfc«-r«; = VDa-(2)-ar)«; =8cA-8^rc 

= 8ViJ»-(«-F)«; =2V(D-.r)x F. 
Chord of half the arc. cfc=:5^Ca» + 4K«; =«^^TF; ^ ^^^ + 0^, 



Diameter 



cfc« 



(^ca)«+r» 



Versed sine = ~ ; = g(0 " ♦'^- C«*; 



(or i(i) + Vz)a-Cd«), if F Is greater than radius. 



= / 



4 . 



Half the chord of the arc is a mean proportional between the versed sine 
and diameter minus versed sine: 



\cd = VK X (Z)- F). 



Ijenctli of a Circular Arc*— Hnjrjglieiis'e Approximation. 

Let C represent the length of the choid octhe arc ana c the length of the 
chord of half the arc; the length of the arc 

^ = ""8"- 
Professor Williamson shows that when the arc subtend 9 an angle of 80«, the 
radius being 100.000 feet (nearly 19 miles), the error by this formula is about 
two inches, or 1/600000 part or the radius. When the length of the arc la 
equal to the radius, i.e., when it subtends an angle of ST".:!, the error is Iei« 
than 1/7080 part of the radius. Therefore, if the radius Is 100,000 feet, the 

100000 
error is less than ^^ = 18 feet. The error increases rapidly with the 

increase of the angle subtended. 

In the measurement of an arc M'hich Is described with a short radius the 
eiTor is so small tl^at it may be neglected. Describing an arc with a radius 
of 12 inches subtending an angle of 80°, the error is 1/50000 of an Inch. For 
570.8 the error is less than 0''.U015. 

In order to measure an arc when It subtends a large angle, bisect it and 
measure each half as before— in this case making B = length of the chord of 
half the arc, and b = length of the chord of one fourth the arc ; then 
_ 166-25 

■^= — r~- 

Relation ot tlie Circle to Its Bqnai, Inscribed^ and €tr* 
enmscrlbed Squares* 

Diameter of circle x .88623 » _ , . - , ^..^^ 

. arcumference of circle x .28300 f - 8»"e of equal square. 
Circumference of circle x l.l28i = perimeter of equal square. 



THE ELLIPSfl. 59 

Diameter of circle x .70711 

Clrcumfereuoe of circle x .22508 > s side of inscribed square. 

Area of circle k .90061 •«- diameter ) 

Area of circle x 1.S78-2 = area of circumscribed square. 

Area of circle x .68663 = area of ioscrlbed square. 

Side of square x 1.414S = diam. of cii*cumscrA)ed circle. 

'* X 4.4428 =circum. * 

" •• X 1.1284 = diam. of equal circle. 

" 8.6*49 =circum. 



Perimeter of square x 0.8 

Square liiebes x LiiTS^ s circular inches. 

Sectors and Smgnkentm^—To find the ca-ea of a sector of a eirOe, 

RuLB 1. KultiplT the arc of the sector by half its radius. 

RuLK 8. As 860 18 to the number of degrees in the arc, so is the area of 
the circle to the area of the sector. 

RuLB 8. Multiply the number of degrees in the arc by the square of the 
radius and by .000787. 

To find the area of a segment of a eirde: Find the area of the sector 
which has the same arc, and also the area of the triangle formed by the 
diord of the segment and the radii of the sector. 

Then take the sum of these areas, if the segment is greater than a semi- 
circle, but take their difference if it is less. 

Another Method: Area of segment = -^ (<^*^ " ^^ ^) '^ which A Is the 

central angle, R the radius, and arc tlie length of arc to radius 1. 

To find the area of a segment of a circle when Its chord and height or 
rersed sine only are given. First find radius, as follows : 

., 1 fsquare of half the chord , , , . ."I 

8. Find the angle subtended by the arc, as follows: — ^. = sine 

of half the angle. Take the corresponding angle from a table of sines, and 
double it to get the angle of the arc. 
8. Find area of the sector of which the segment is a part ; 

M 1 , degrees of arc 
area of sector = area of circle x — = — 5^- . 

C Subtract area of triangle under the segment: 

Area of triangle = ^^^ x (radius - height of segment). 

Tlie remainder is the area of the segment. 

When the chord, arc, and diameter are given, to find the area. From the 
length of the arc subtract the length of the chord. Multiply the remainder 
by the radhis or one-half diameter; to the product add the chord multiplied 
by the height, and divide the sum by 2. 

Atiother rule: Multiply tlie chord oy the height and this product by .6884 
plus one tefith of the square of the height divided bv the radius. 

To find the chord: From the diameter subtract the height; multiply the 
remainder by four times the height and extract the square root. 

When the chords of the arc and of half the arc and the versed sine are 
given: To the chord of the arc add four thirds of the chord of half the arc; 
multiply the sum by the versed sine and the product bv .40426 (approximate). 

ClrcnlAr Wttn§[.—To find the area of a ring included betioeen the cir- 
cumferencea of two concentric circles: Take the difference between the areas 
of the two circles; or, subtract the sqnare of the less radius from the square 
of the greater, and multiply their difference by 8.14150. 

The area of the greater circle is equal to wR*; 
and the area of the smaller, • irr*. 

Their difference, or the area of the ring. Is tr(R* - r»). 

Tl^e BIltf«««— Area of an ellipse = product of Its semi-axes x 8.141S9 

¥= product of its a xes x .7 85898. 

TheEIUpae.'-CiTcamterence (approximate) = 8,1416 V^-^J — , D and d 

being the two axes. 

Trmutwine gives the following as more accurate: When the longer axis J) 
i< not more than five times the length of the shorter axis, d, 



60 HJBirSU&AnON, 

aroumferenoe =5 8.1416 Y — ^^ ti.H ' 

When P is mow tliim Sd, th^ divisor 8.8 ia to be replaced by the following : 
For5/d = 8 7 8 9 10 12 U 16 IS 80 80 40 60 
Divliior - 9 ».« » 3 9,35 9.4 9.5 9.6 9.C8 0.T5 9 8 0.93 9,98 10 

An accurate formula is Cm w(a + b){l + Y "^ 76 "^ 250 "^ 16384"^ . . .), in 
which A = ^-^.— /w«ni*ei*r» Taichenbuch, 1896. 

Carl G. Barth {Mo/ehiMry, Sept., 1900) give* as a very cIom approzliqation 
to thl» (grmula 

<?^^<<* + <>) e4^io^r 

^rea of a aegttkent of an eUip§e the baae of which ia parcel to em of 
the loea of the ellipia. Divide the height of th« Begment by the ox te nt 
whjch it 19 part, and And the area of a circular segment. In a table of circa- 
Ur HegmenU, of which the height is equal to the quotient; multiply the area 
thuH found by the product of the two axes of the ellipee. 

Cydoldf— A curve generated by the rolling of a circle on a plane. 
Length.of a cycloldal curve = 4 X diameter of the generating Qlrcle. 
Length of the base = circumference of the generating clrele. 
Ar^ft of a pycloid 3= S x area of generating circle. 

Helix (9er«W).-A Up© generated by the progressive rotation of a 
poiut around an axis and equidistant from its centre, ^, ^ ,. *». 

Length of a. helix.— To the gquare of the circumference described by the 
seneratUig-poInt add the square of the distance advanced in one re^rolutlon, 
and take the square root of their sum multiplied by the number of revobi- 
tlous of the generating point. Or, 

y(c« + h*)n = length, n being number of revolutions, 

Splrml««— Unes generated by the progressive rotation of a point around 
a fixed axis, with a constantly increasing distance from the axia. 

A vlane spiral Is when the point rotates in one plane. 

A conical spiral is when the point rotates around an axis at a progressing 
distance from ita centre, and advancing in the direction of the axis, as around 

* l^mgth of a plane spiral J»nP,— When the distance between the polls is 

RuLB.— Add together the greater and less diameters; divide their sum by 
2- multiply the quotient by 8.1416, and again by the number of revolutions. 
Or take the mean of the length of the greater and less circumferences and 
multiply ft by the number pf revolutions. Or, 

length ^ vn ^\^\ d and d' being the Inner and outer dlameteiv. 

Length of a conical spiral Wn«.-Add together the greater wd less diam- 
eters; divide their sum by 8 and multiply the quotient by 8.14J6, To the 
souare of the product of this circumference and the number of revolutions 
of the spiral add the square of t4ie height of ite axis and take the square 
root of Uie sum. 

Or, lengtli 1 



80LI9 BOPTBti, 

Xbe Priam,— TV) iind the surface of aright pri9m : Multiply the perim- 
eter of the base by the altitude for the convex surface. To thi« add the 
4feas of the two ends when the entire surface is required. 

Volume of a prism « area of ita base x ita altitude. 

The pywimld.-Convex surface of a regular pyramid = Pfrtmeter of 
its base X half the slant height. To tliis add area of the base if the whole 
surface is required. 

Volume of a pyramid = area of base X one third of the altitude. 



SOLID BODIES. 61 

To find the aurface vf afnutuni of a regular pyramid : Multiply half the 
slant height by the sum of the perimeters of the two bases for the couvez 
surface. To this add the areas of the two bases when the entire surface is 
reqalnHL 

To find the volume of a frustum of a pyramid : Add together the areas of 
tlie two bases and a mean proportional between them, and multiply the 
sum by one third of the altitude. (Mean proportional between two numbers 
= square root of their product.) 

wedipe*— A wedge is a solid bounded by Ave planes, yIz,: a rectangular 
base, two trapezoids, or two rectangles, meeting in an edge, and two tri> 
angular ends. The altitude is the perpendicular drawn from any point in 
the edge to the plane of the base. 

Tomtd the volume of a wedge : Add the length of the edge to twice the 
leogtli of the beae, and multiply the sum by one sixth of the product of the 
height of the wedge and Uie breadth of the base. 

Wt^eetmngutmr prlsmold*— A rectangular prismoid is a solid bounded 
by six plaues, of which the two bases are rectangles, baying their corre« 
sponding sides parallel, and the fotu* upright sides of the solidii are trape- 
zoids. 

To find the volume of a rectangular prigmoid: Add together the aresa of 
the two bases and four times the area of a pamllel section equally dlstaut 
from the bases, and multiply the sum by one sixth of the altitude. 

Cylinder*— Ck>n vex surface of a cylinder = perimeter of bane x altitude. 
To this add the areas of the two ends when the entire surface is required. 
Volume of a cylinder = area of base X altitude. 

CToae.— Convex surface of a oone =? oircumf erenoe of base x half the slant 
aide. To this add the area of the base when the entire surface is required. 

Volume of a oone s area of base x g altitude. 

3b find the aurfaoe of a frustum of a cone : Multiply half the side by the 
snm of the circumferences of the two bases for the convex surface; to this 
add the areas of the two bases when the entire surface is requirsd. 

To find the volume ofafnutum of a cone : Add together the areas of the 
tiro bases and a mean proportional between them, and multiply the sum 
by one third of the altitude. 

Spli«re.—7\> And the turf ace of a sphere : Multiply the diameter by the 
circumference of a great circle; or, multiply the square of the diameter by 
8.141seL 

Surface of sphere = 4 x area of its great circle. 

** •• *' =s convex surface of its circumscribing cylinder. 

Surfaces of spheres are to each other as the squares of their diameters. 

To find thm volume of a apJiere : Multiply the surface by one third of tbs 
radius; or, multiply the cube of the diameter by l/Ov; that is, by O.fi^. 

Value of |ir to 10 decimal places = .588^087766. 

The Tolume of a sphere = S/8 the volume of its dronmscribing cylinder. 

Volumes of spheres are to each other as the cubes of their diameters. 

8pherlesU tritLnjgle.^To find the areaqfa splieiioal triangle : Com- 
pute the surface of the quadrantal triangle, or one eighth of the surface of 
the sphere. From the sum of the three angles subtract two right angles; 
divide the remainder by SX), and multiply the quotient by the area of the 
quadrantal triangle. 

Spherleal poljgonm— To find the area of a spherical polygon: Com- 
pQie the snrface of the quadrantal tiiangle. From the sum of all the angles 
subtract the product of two right angles by the number of sides less two; 
divide the remainder by 90 and multiply the quotient by the area of the 
quadrantal triangle. 

The prlmnold.— The prismoid is a solid having parallel end areas, and 
mav be composed of any combination of prisms, cylinders, wedges, pyra- 
mids, or cones or frustums of the same, whose bases and apices lie in the 
end areas. 

Inasmuch as cylinders and cones are but special forms of prisms and 
pyramid^ and warped surface solids mav be divided into elementary forms 
of them, and since frustums may also be subdivided into the elementary 
forms, it Is sufflcient to sav that all prismoid s may be decomposed into 
prisms, wedges, and pyramids. If a formula can be found which is equally 
applicable to all of these forms, then it will apply to any combination of 
them. Such a formula is called 



62 MENSURATION. 

The Prlsmoldml Foratnta. 

Let A = area of the base of a prism, wedge, or pjramld; 
^j, ^t, Am = the two end and the middle areas of a pruunoid, or of any of 
its elementary solids; 
h = altitude of the prismoid or elementary solid; 
V= its volume; 

For a prism A^ Am and A^ are equal, =A\ V=-^x^A^ hA. 

For a wedge with paraUel ends, A^ = 0. ^m = |ii, ; r= |u, + 2Ao = —- 

For a cone or pyramid, At = 0, Am = 7^1 ; V s= ^(4, + Ai) = -^. 

The prismoldal formula is a rigid formula for all prismoids. The only 
approximation involved in its use is in the assumption that the given solid 
may be generated by a right line moving over tne boundaries of the end 



The area of the middle section Is never the mean of the two end areas if 
the prismoid contains any pyramids or coues among Its elementary forma 
When the three sections are similar in form the dimeiinoru of the middle 
area are always the means of the corresponding end dimensions. This fact 
often enables the dimensions, and hence the area of the middle section, to 
be computed from the end areas. 

Polyedronsa—A polyedron is a solid bounded by plane polygons. A 
regular polyedron Is one whose sides are all equal regular polygons. 

To find the aurface of a regular po/^edrou.— Multiply the area of one of 
the faces by the number of faces ; or, multiply the square of one of the 
edges by the surface of a similar solid whose edge is unity. 

A Tabub of the Rboular Poltbdbons whoss Edoks ark Unity. 

Names. No. of Faces. Surface. Volume. 

Tetraedron 4 1.7330608 0.1178513 

Hexaedron 6 6.O00O00O 1.0000000 

Octaedron 8 8.4041016 0.4714045 

Dodecaedron 18 I«).6I57288 7.6681189 

Icosaedron 90 8.060^540 2.1816950 

To flnd tb« Tola me of a reralar polyedron.— Multiply the 

surface by one third of the perpendicular let fall f 1 oin tliM centre on one of 
the faces ; or, multiply the cube of one of the edges by the solidity of a 
similar polyedron whose edge is unity. 

Solid of reTolatlon.— The volume of any solid of revolution Is 
equal to the product of the area of its generating surface by the length of 
the path of the centre of gravity of that surface. 

The convex surface of any soUd of revolution is equal to the product of 
the perimeter of Its generating surface by the length of path of its centre 
of gravity. 

Cylindrical rlnff.— Let d = outer diameter ; d' = inner diameter ; 

s (ri - d') = thickness = t ; 7 » f* = sectional area ; -Ad-^-d') = mean diam> 

s 4 « 

eter = 3f ; wt = circumference of section \ vM=^ mean circumference of 
ring; surface = irt x nM\ = iw«(d* - (!'«);= 9.86965 < If; = 8.46741 (cl» -d'«); 

volume = i » «« If it; = 8.4674U« M. 

4 

Splierleal momt*— Surface of a aphericaf gone or segment of a sphere 
= its altitude x the circumference of a great circle of the sphere. A great 
circle Is one whose plane passes through the centre of the sphere. 

Volume of a tone of a sphere.^To the sum of the squares of the radii 
of the ends add one third of the square of the height ; multiply the sum 
by the helirht and by 1.5706. 

Spherical •enrment.— Volume of a tpherical segment tcith one 6ase.~ 



SOLID BODIES. 63 

MaltlpW half the helRht ot the segment by the area of the base, and the 
cube or the heleht by .62S6 and add the two products. Or, from three times 
the diameter of the sphere subtracl twice the height of the segment; multi- 
ply the difference br the square of the height and by .5286. Or, to three 
times the square of the radius of the base of tne segment add the square of 
its height, and multiply the sum by the height and oy .5286. 

^berold or •lUpsold.— When the revolution of the spheroid Is about 
the trausverse diameter it is prolate^ and when about the conjugate it is 
oblate. 

Convex swf/aee of a §effment of a mAero^ —Square the diameters of the 
spheroid, and take the square root of half their sum ; then, as the diameter 
from which the segment is cut is to this root so is the height of the 
segment to the proportionate height of the segment to the mean diameter. 
Multiply the product of the other diameter and 8.1410 by the proportionate 
height. 

Convex mirfaee of a fnutwn or zone of a tpheroid.— Proceed as by 
prevloDS rule for the surface of a segment, and obtain the proportionate 
height of the frustum. Multiply the product of the diameter parallel to the 
base of the frustum and 8.1416 by the proportionate height of the frustum. 

Folttme of a mheroid is equal to theproduct of the square of the revolving 
axis by the fixed axis and by .6886. The volume of a spheroid is two thirds 
of that of the circumscribing cylinder. 

Volume of a tegment of a mkeroid.'-l. When the base is parallel to the 
revolviog axis, multipW the difference between three times the fixed axis 
and twice the height of the segment, by the square of the height and by 
jex. Multiply the product by the square of the revolving axis, and divide 
by the square of the fixed axis. 

t. When the base is perpendicular to the revolving axis, multiply the 
difference between three times the revolving axis and twice the height of 
the segment by the square of the height and by .6286. Multiply tlie 
product by the length of the fixed axis, and divide by the length of the 
revolving axis. 

Volume of ike middle fniBtum of a spheroid.—l. When the ends Are 
circular, or parallel to the revolving axis : To twice tlie square of the 
middle diameter add the square of the diameter of one end ; multiply the 
sum by the length of the frustum and by .2618. 

SL when the ends are elliptical, or perpendicular to the revolving axis : 
To twice the product of the transverse and conjugate diameters of the 
middle section add the product of the transverse and conjugate diameters 
of one end ; multiply the sum by the length of the frustum and by .2618. 

SpiDdles*— Figures generated bv the revolution of a plane area, when 
the curve is revolved about a chord perpendicular to its axis, or about its 
double ordinate. They are designated by the name of the arc or curve 
from which they are generated, as Circular, Elliptic, Paiabolic, etc., etc. 

Convex aurface of a circular apindle^ zone^ or tegmeiit of it —Rule: Mul- 
tiply the length by the radius of the revolving arc; multiply this arc by the 
central distance, or distance between the centre of the spindle and centre 
of the revolving arc ; subtract this product from the former, double the 
remainder, and multiply it bv 8.1416. 

Foliime o/ a c/nnttor «ptndle.— Multiply the central distance by half the 
area of the revolving segment; subtract the product from one third of the 
cube of half the length, and multiply the remainder by 12.5664. 

Viflume offi^utum or tone or a circular spindle.— From the square of 
half the length of the whole spindle take one third of the square of Irnlf the 
length of, the frustum, and multiply the remainder by the said half length 
of the frustnm ; multiply the central distance by the revolving area which 
f^enerates the frustum ; subtract this product from the former, and multi- 
ply the remainder by 6.888S. 

Volume of a aegment of a ciixular aptndle.— Subtract the length of the 
segment from the half length of the spindle : double the remainder and 
ascertain the volume of a middle frustum of this length ; subtract the 
result from the volume of the whole spindle and halve the remainder. 

Volume of a eydoidal mindle = five eighths of the volume of the circum- 
scribing cylinder.— Multiply theproduct of the square of twice the diameter 
of the generating circle and 8.927 by its circumference, and divide this pro- 
duct by & 

Parabolic conoid.— Fbliime of a parabolic conoid (generated by the 
revolution of a parabola on its axis).- Multiply the area of the base by half 
the height. 



64 MENSURATION. 

Or multiplxthe square of the diameter of the base by the height and bj 
•8827. 

Volume of a frustum of a parabolic eonotd.—Multipbr half the sum of 
the areas of the two ends by the heig^ht. 

Volume of a parabolic spindle (generated bv the reToluUon of a parabola 
on Its base).— Multiply the square of the middle diameter by the lengrth 
and by .4188. 

The volume of a parabolic spindle is to that of a cylinder of the same 
heif ht and diameter as 8 to 15. 

volume of the middle frustum of a parabolic spindle.— Add together 
8 times the square of the maximum diameter, 8 times the square of the end 
diameter, and 4 times the product of the diameters. Multiply the sum by 
the leoRth of the frustum and by .05286. 

This rule is applicable for calculating the content of casks of parabolic 
form. 

Caaks.— To find the volume of a cask of any /oitn.— Add together 88 
times the square of the bung diameter, 25 times the square of the head 
diameter, and 26 times the product of the diameters. Multiply the sum by 
tlie length, and divide by 81,778 for the content in Imperial gallons, or by 
«6j470 for U.S. gallons. 

This rule was framed by Dr. Hutton, on the supposition that the middle 
third of the length of the casir was a frustum or a parabolic spindle, and 
each outer third was a frustum of a cone. 

To find the ullage of a cask, the quantity of liquor in it when it is not full. 
1. For 8k lying cask : Divide the number of wet or dry inches by the bung 
diameter in Inches. If the Quotient is less than .5, deduct from it one 
fourth part of what it wants of .5. If it exceeds .5, add to it one fourth part 
of the excess above .5. Multiply the remainder or the sum by the whole 
content of the caslr. The product is the quantity of liquor in the cask, in 
gallons, when the dividend is wet Inches; or the empty space, if dry inches. 

2. For a standing cask : Divide the number of wet or dry inches by the 
length of the cask. If the quotient exceeds .6, add to it one tenth of its 
excess above .5; if less than .5, subtract from it one tenth of what it wants 
of .5. Multiply the sum or the remainder by the whole content of the cask. 
The product is the quantity of liquor in the cask, when the dividend is wet 
inches; or the empty space, if dry inches. 

Volume of cask (approximnte) U. 8. gallons = square of mean dlam. 
X lengtli in inches X .0084. Mean dlam. = half tlie sum of the bung and 
head dlams. 

Tolmne of mn Irre/gnlar solid.— Suppose it divided into parts, 
resembling prisms or other bodies measurable by preceding roles. Find 
the content of each part; the sum of the contents is the cubic contents of 
the solid. 

The content of a small part is found nearly by multiplying half the sum 
of the areas of each end by the perpendicular distance between them. 

The contents of small frr^ular solids mny sometimes be foimd by im- 
mersing them under water in a prismatic or cylindrical vessel, and observ* 
ing the amount by which the level of the water descends when the solid is 
w ithdrawn. The sectional area of the vessel being multiplied by the descent 
of the level gives the cubic contents. 

Or, weigh the solid in air and in water; the difference Is (he weight of 
water it displaces. Divide the weight In pounds by 62.4 to obtain volume in 
ciil>ic feet, or multiply it by 27.7 to obtain the volume In cubic inches. 

When the solid is very large and a great degree of accuracy is not 
requisite, measure its length, breadth, and depth in several oifferent 
places, and take the mean of the measui«ment for each dimension, and 
multiply the three means together. 

When the surface of (he solid is very extensive it Is better to divide it 
into triangles, to And the at-ea of each triangle, and to multiply it by the 
mean depth of the triangle for the contents of each triangular portion; ihs 
contents of the triangular sections are to be added together. 

The mean depth of a triangular section is obtained by measuring Uie 
depth at each angle, adding together the. three measurements, and taking 
one third of the sum. 



PLA.NB TBtaOlsrolCBTBT* 6S 



VLAITE TBIGK>HOMET&Y« 



Trtconom«trlcal FuneUons. 

Erery tiiAngto has six parts— three angles and thrM sides. When any 
hn-e of these parts are given, provided one of them is a side, the other 
>-irts may be determined, uy the solution of a triangle is meant the detet^> 
iiinaiion of the unknown parts of a triangle when certain parts are given. 

The complement of an angle or arc is what remains after subtracting the 
in^le or arc from Wf*., 

In general, if we represent any arc by A, its complement is 90^ — ^. 
Hf Dce the complement of an arc that exceeds 90** is negative. 

Since the two acute angles of a right-angled triangle are together equal to 
I right angle, each of them Is the complement of the other. 

The supplement of an angle or arc is what remains after subtracting the 
ingle or arc from I9ff*. If ^ is an arc its supplement Is ISO** — A, The sup- 
^rnent of an arc that exceeds 180° is negative. 

Tke Bum of the three angle$ of a triangle u eguaZ to 180». Either angle is 
\be supplement of the other two. In a nght-angled triangle, the right angle 
being equal to 90*, each of the acute angles is the complement of the other. 

In ail right-angled triangles having the same acute angle, the sides have 
lo each other the same ratio. These ratios have received special names, as 
billows: 

If ^ is one of the acute angles, a the opposite side, b the adjacent side, 
md c the hypothenuse. 

The sliie of the angle A is the quotient of the apposite tide divided by the 
a 
kfpothenum. Sin. A^ -^ 

Tlie ta.Bsemt of the angle A is the quotient of the opposite side divided by 
a 
tke adjacent tide. Tang. A = ^ 

The — eant of the angle A is the quotient of the hypothenuse difftded by 
o 
Ike adj€teeni side. Sec. A = -y 

The eoslBe^ cotmnipeiit, and eoseeant of an angle are respeo- 
tirely the sine, tangent, and secant of the complement of that angle. The 
terms sine, cosine, etc., are called trigonometrical functions. 

In adrcle whose radius is unity, tbe sine of an arc^ or of the angle at tfie 
centre metuured by that are^ is the perpendicular let fall from one extreme 
ii\*}f the arc upon the diameter passing through the other extremity. 

The tfluscent of an arc is ths line lokieh touches the circle at one extreme 
i(y o/ the arc. and is limited by the diameter {produced) passing through 
&> other extremity. 

The eeesiht o/ an arc is that pari of the produced diameter which is 
intfrcepted bet'reen the centre and the tangent. 

The wemed sine of an arc is that part of the diameter intercepted 
btttceen the extremity of the arc and the foot of the sine. 

Tn A circle whoee radius is not unitv, the trigonometric functions of an arc 
viil be equal to the lines here defined, divided by the radius of ihe circle. 

If / C ^ (Fig. 70) Is an angle in the first quadrant, and Fsa. radius, 

Tbescoeof th.««le=^. 00. = ^ = ^^ 

T««. = iSd.- Secant = ^^. Cot. = ^ ; 
CL „ , CfA 
^^**^^' = Bad.* VerslD. = g^. B 

If radius Is 1, then Rad. In the denominator is 
C'lnitLed, and sine as FQ^ ete. 

The erne of an arc a half the chord of tioiee ihe 
srv. 

Tbe sine of the supplement of the arc is the same fj/ 
SR that of the arc itself . Bine of arc B i^ i^' = J^ (? = 
riaare^*^ Fio. 70. 




68 



PLAKB TBIGOKOHETBY. 



The tangent of the supplement is equal to the tangent of the arc, but 
a contrary sign. IVing. BDF= B M. 

The secant of th6 supplement is equal to the secant of the arc, but with 
contrary sign. Sec. BDF= CM. 

Mens of the nmetloniB in tbe four qamdrante,— If 
dWide a circle into four quadrants by a vertical and a hurisontal dif- 
fer, the upper right-hand quadrant is called the first, the upper left the f 
ond, the lower left the thira, and the lower right the fourth. The signs ( 
the functions in the four quadrants are as follows: j 

F«r«fquadL 5eoo»dquad. STb^rdquad. Jrburf ft qusd 
Sine and cosecant, - + + — — i 

Cosine and secant, 4- — — 4- ! 

Tangent and cotangent, + -* + — 

The values of the functions are as follows for the angles specified: 



Angle 

Sine 

Ckwine 

Tangent 

Cotangent.. 

Secant 

Cosecant... 

Versed sine 



80 
j_ 

2 
2 



45 

_L 

1 
1 

V2 

V5-1 



^2 



90 180 

8 

_ 1 
8 

-8 
8 

8 



185 
_1_ 

1_ 

^2 
-1 
-1 

V2 

£?+-' 

V8 



150 

1 

. 8 

8 

J_ 
V8 

• V8 

8 

8 



1801870 
-1 

-1 




e.. J 



TRIGONOlOLBTIUCAIi FORJIiUIi.SU 

The following relations are deduced from the properties of similar U 
angles (Kaditis = 1): 

sin A 

cos .^ : sin ul n 1 1 tan ji, whence tan A = r : 

cos .<1* 



sin A t cos ^ a 1 1 cot A, ** cotan A = 

cos ul : 1 a 1 : sec .<1, " sec ^ = 

sin ^ 1 1 a 1 : cosec A, " oosec A = 

tanul:! alxcot^ " tan^ = 



cos.<i 

8ln A 

_J 

cos A 

J[ . 

sin A' 

_1 

cot.<l' 



The sum of the square of the sine of an arc and the square of its cosi^ 
equals unity. Sln« A 4- cos* A = 1. 
AlBo, 1 4- tan« A = sec* A: 1 + cot« A = cosec* A, 

Fimctlons of tlte anna and dllTerenee of tivo nnsloa i 

• Let the two angles be denoted by A and i?, their sum A-i- B = C, a^ 
their difference A - P hj D, 

sin (^ 4- 1;) = sin ^ cos £ + cos ^ sin 2^; 



TBIGOKOMBTBICAL FOBMULiB. 



o(M (A -{- S) = CM A cot B - Bin A tkn B; . . • 

9ln {A ^ B) = iAn A oob B — ooa ^ sin B; . . • 

COS {A — B) = cos A cw B i- sin A aia B. . . . 
From tbflte four formiilaB by addition and subtraction we obtain 

Bin U + ^) + sin U - B) = 2 sin ^ cos B; . . . 

BinU + iO-8in U~B) = 8co8^8la B; . . . 

O08 M + B) + 000 (il - B) = 2 ooe >. cofl J^; . . . 

coBU-Bj^CM(A + B) = 9tinAaiBB.. . . 



(«) 
(8) 

(4) 

(5) 
(«) 
(7) 
(8) 

If we put wl + B = C. and ^ - B = B, then A s mc + D) and B » ^C - 
B), and we hare 

sinC+BinBsS8inH<^+B)co«Vi(0- B); . ... (9) 
rinC-ilnB=:2coeH<C+I>)8lnH(C- B): . . . . (10) 
co8C4-cosB=r2co8H(C+B)co8H(C-B); . . . . (11) 
oosB-oos(7=28in H(C+B)BinH(C- B) (12) 

Equation (9) may be enunciated thus: The sum of the sines of any two 
amrles ifl equal to. twice the sine of half the sum of the ang^les multiplied by 
the cosine of lialf th«>ir difference. These f ormulss enable us to transform 
a imm or difference into a product. . . 

Th«* sum of the sines of two angrles is to their difference as the tangent of 
Lalf the sum of yifise angles is to the tangent of half their differenoe. 

sin ^ -I- sin B 2 sin H(^ + B) cos H(wi - B) _ tnn H(^ 4-B) ^ 
Bin ^ - sin B "^ 2 cos y^A + B) sin H(4 - B) ~ lati ^A - BY ^ ' 
The sum of the cosines of two angles is to their differencaas the ootangenl 
of half the sum of those angles Is to the tangent of half their difference. 
CM^+oosB 2oosH<i4 + B)cos^M-B) cot ^ (^ + B) 
ooeB-cOS-4 * «8in H(^ + B) sin H(^ - B) "^ tan M(^ - B)' ^ ' 
The moe of the sum of two angles is to the sine of theirdifreneooe as the 
sum of the tangents of those angles is to the difference of the tangents. 

sin (^ + B) _ tan ui + tan B 



sin (A - B) 

^iLl4Jl^=tanu4 + tanB; 
oos^oosB ^ 



■^^^j*--^ = tan^-tanB; 
oos^cosB 

®?lil±:^ = 1 - tan ^ tan B; 
cos^cosB 

???Ltd— -^ = 1 + tan 4 tan B; 
cos^oosB 

FaisetloBS of thrice an anffle i 

«in2^ s 2 sin ^ cos ^; 

.^ 2tan^ 
^«^ = l^tan«a - 
Fanettons of Half an an^le i 



tan ^ >- tan B* * 

tan (^ + B) = 
tan (^ - B) = 
cot (^ + B) = 



(15) 



cot (il - B) = 



tan ^ 4- tan B . 
1 ~ tan ^ tan B' 

tan i4 - tan B ^ 
1 + tan ii tan B* 
co t >4 cot B - 1 . 

cot B 4- cot ^ * 
cot ^ cot f? 4- 1 

cot B — cot A ' 



cos ZA =■ cos* A - sin* A\ 

* « . «"t« A - 1 
cot ZA = 



2co(^ 



HA 



lux HA 






— cos^ 



cosH^ 



../^ 



4- cos A 



- cos^ . 
cos^* 



cot Hi4 = ± , 



/I -f cow A 
' 1 - cos -4* 



68 PLAKB TJUOOKOKET&Y. 

flolutlOB oir Plan* Rlcl&t-«ii«l«d TrUmcIes. 

liBt A and S be the two acute angles and C tbe rig^t an«:le, and a, 6, and 
c the sides opposite these angles, respectiFely, then we have 

1. sin ^ = oosB «= - ; 8. tan^ = cot B = ^; 

S. COS ^ = sin B XI -; 4. cot ^ = tan B = -. 
c a 

1. In any plane right-aogled trlMiffle the sine of either of the acute angles 
is equal to the quotient ofthe opposite leg divided by the hypothenuse. 

2. Tlie cosine of eiiher of the acute angles is equal to the quotient of the 
adjacent leg divided by the hypothenuse. 

3. The tangent of either of the acute angles Is equal to the quotient of the 
opposite leg divided by the adjacent leg. 

4. The cotangent or either of the acute angles is eqnal to the quotient of 
the adjacent leg divided by the opposite leff. 

5. The square of the hypothenuse equals the sum of the squares of the 
other two sides. 

SolatlojOL of Obliqne-SLiiffled Trlancles, 

The following propositions are proved In works on plane trigonometry. In 
any plane triangle— 

Theorem, 1. The sines of the angles are proportional to the opposite aides. 

Tlworem 9. The sum of any two siden is to their difff rence as tbe tangent 
of half tbe sum of the opposite angles is to the tangent of half tbeir differ- 



Theorem 3. If from any angle of a triangle a perpendicular be drawn to 
the opposite side or base, the whole base will be to the sum of the oUier two 
sides as the difference of those two sides is to the difference of the aegments 
of the base. 

Cask I. Given two angles and a side, to And the third angle and tbe other 
two sides. 1. The third angle = 1B0<> — sum of the two angles. S. The sides 
may be fonnd by the following proportion : 

The sine of the an^le opposite the given side is to the sine of the angle op- 
posite the required side as the pri^en s^de is to the required side. 

Cabs II. Given two sides and an angle opposite one of them, to find the 
third side and the remaining angles. 

The side opposite the given angle is to the side opposite the required angle 
as the sine of the given angle is to the nine of the required angle. 

The third angle is found by subtracting the sum of the other two from 180*, 
and the third Ride is found as in Case I. 

Case m. Given two sides and the included angle, to And the third aide and 
the remaining angles. 

The sum of the required angles is found by subtracting the given angle j 
from 180**. The difference of the required angle:* is then found by Theorem 
II. Half the difference added to half the sum gives the greater angle, audi 
lialf ihe difference subtracted from half the sum gives the less angle. The 
third Bide 1» then found by Theorom I. 

Another method : 

Given the sides c, 6, and the included angle A^ to find the remaining side a 
and the remaining angles B and C. 

From either of the unknown angles, as B, draw a perpendlcuUr ^ e to tht 
opposite side. 

Then 

Ae = cQo» A^ Be = c^XnA^ eC=h - Ae, Be-*-eC= tan C. 

Or, in other words, solve Be.Ae and B e CaA right-angled triangles. 
Ca8B IV. Given the three sides, to And the angles. 



Let fall a perpendicular upon the longest side from the opposite angk| 
lividiiig the given triangle into two right-angled triangles. The two sea 
meiits of the base may be fonnd by Theorem III. There will then be givef 



the hypothenuse and one side of a right-angled triangle to flud the angles. 
For areas of triangles, sec Mensuration. 



A^^ltlQAJU a5Q¥OTllT. S9 



AlTALYTICAXi (JBOMBTBY. 

Anmljrtleml geometry is that branch of Mathematics which hap for 
its ob^i ih« deiermioatiou of the fonns and magultudea of geometrical 
masmi'iideA by means of analysis. 
OrdlDstes and abselMAS*— In analytical geometry two intersecting 
lilies YY\ XX' are used as coordinate axea, 
XX' being the axis of abscissas or axis of X 
and YY' Uie axis of ordlnatea or axia of Y. 
A. the intersection, is ealled tba origli\ of oo* 
Qrdinates. The distance of any point P from 
the axis of Y measured parallel to the axis of 
X is called the abscissa o( the point, as AD or 
(7P, Fig. 71. Its distance from the axis of X. 
measured parallel to the axis of F, is called 
the or din ate, as AC or PD. The abscissa and 
ordinate taken together are called the coor- 
dinates of the point P. The angle of intersec- 
tion is usually takeo as a right angle, in which 
case ihe axes of X and Fare called rectangu- 
lar oooi'd/na^es. 

The abscissa of a point is designated by the letter x and the ordinate by y. 

The equations of a point are the equations which express the distances of 

(he point from the axis. Thun a; = », 9 = 6 ara the equations of the point P. 

Eqn«ttoii0 referred to rectADKnlar cotfrdtnates.— The equa- 

UoQ of a lint) expresiMM the rehition which exists between the qoOpdinatea of 

every poini of ine line. 

Equation of a straight line, y = ax±h^\n which a is the tangent of the 
angle the line makes with the axis of X, and b the distance above A in which 
the line cuts the axis of F. 

Every equalioii of the flrat degree between two variables is the equation of 
» straight line, as ^y + Bx -j- C = 0, which can be reduced to the form y = 
ax ±b. 
Equation of the distance between two points: 




coordinates of the t^ 
; through a given poi 

j^ - y' = a(a; - x'). 



In which x'y', x"y" are the coordinates of the two points- 
Equation of a hne passing through a given point : 



in which rr'w' are the coordinates of the given point, a, the tangent of the 
angle the line makes with the axis of x, being undetermined, since any num- 
ber of lines may be drawn through a given point. 
Equation of a line passing through two given points : 

Equation of a lUie parallel to a given line and through a given point; 

y ^ y' cz aix — a?'). 
Equation of an angle K included between two given lines: 

'^ 1 + a'a 

in which a and a' are the tangents of the angles the lines make with the 
axis of abscissas. 
H. the lines are at right angles to each other tang T =: oo , and 

1 4- ^'o> = 0, r 
Equation of an intersection of two lines, whose equations are 
y = ax -\- by and y = a'x + fc', 

b- h' ^ ah' - a'b 

a; = ;, and y = -. 

a-^ a' a - a' 



70 AKALTTIOAL GEOMETBT. 

EquAtion of a perpendicular from a given point to a given line: 

Equation oC the length of the perpendicular F: 

The elrele.— BquaUon of a drole, the origin of ooOrdlnatee being at the 
centre, and radius s B : 

If the origin is at the left extremity of the diameter, on the axis of X : . 

If the origin is at any point, and the coOrdlDates of the centre are x'y^ : 

(X - ar')* + (y - y')* = i2». 

Equation of a tangent to a circle, the coordinates of the point of tangency 
being x' V' and the origin at the oentre, 

The ellipse,— Equation of an ellipse, referred to rectangular coOrdi' 
•nates with axis at the centre: 

^ V + B^x^ = A^B*, 

in which A is half the transverse axis and B half the conjugate axis. 

Equation of the ellipse when the origin is at the vertex or the transverse 
axis: 

The eccentricity of an ellipse is the distance from the centre to either 
focus, divided by the semi-transverse axis, or 

_ VA*^B* 
'- A • 

The parameter of an ellipse is the double ordinate passing through the 
focus. It is a third proportional to the transverse axis and its conjugate, or 

SB* 
iiA:2B ::2B : parameter; or parameter = -j-. 

Any ordinate of a circle circumscribing an ellipse is to the corresponding 
ordinate of the eiiipse as the semi-transverse axis to the aemi-conjugate. 
Any ordinate nf a circle inscribed in an ellipse is to the corregponding ordi- 
nate of the ellipse as the semi-conjugate axlH to the seuii-transversa 

Equation of the tangent to an ellipse, origin of axes at the centre : 

A*w" -f B^xx" = A^B*, 

yf'xf' being the coordinates of the point of tangencv. 

Equation of the normal, passing through the point of tangency, and per- 
pendicular to the tangent: 

The normal bisects the angle of the two lines drawn fron} the point of 
tangency to the foci. 

The lines drawn from the foci make equal angles with the tangent. 

Tlie parabola. -Equation of the parabola referred to. rectangular 
coordinates, the origin being at the vertex of its axis, y* = Sfpa;, in which 2p 
is the parameter or double ordinate through the focus. 



ANALYTICAL OEOMETBY. 71 

The parameter is a thfrd proportional to any abeciflsaand lUoorrespoiidiiig 
ordinate, or 

Equation of the tangent: 

tr''x" beinff coordinates of the point of tangenoy. 
Equation of the normal: 

The sub normal, or projection of the normal on the axis, Is constant, and 
equal to half the parameter. 

The tangent at any point makes equal angles with the axis and with the 
]in«* drawn from the point of tangenoy to the focus. 

Tlie liyperbola.— Equation of the hyperbola referred to rectangular 
coOrdioates, origin at the centre: 

in which A is the fleml-transverse axis and B the semi-conjugate axis. 
Equation when the origin is at the right Tert«sx of the transverse axis: 

1/*^^^{2Ax + X*). 

ConJncACe and eqallateral ltyperbola» .— If on the conjugate 
axis, as a transverse, and a focal distance equal to VA* + B*, we construct 
the two branches of a hyperbola, the two hyperbolas thus constructed are 
called coidugate hyperbolas. If the transverse and conjugate axes are 

' , the iiyperboiaiB are called equilateral, in which case ifi—afl= — A* 

^ is the transverae axis, and jt* - y* = - i?* when B is the trans- 



equal, 
when 



The parameter of the transverse axis is a third proportional to the trans- 
verse axis and its conjugate. 

iA:ZB::fiB : parameter. 

The tangent to a hyperbola bisects the angle of the two lines drawn from 
the point of tangenoy to the foci. 

Tne msymptotes of a byperbola are the diagonals of the rectangle 
described on the axes. Indefinitely produced in both directions. 

In an equilateral hyperbola the asvmptotes make equal angles with the 
transveme axis, and are at right angles to each other. 

The asymptotes continually approach the hyperbola, and become tangent 
to it at an infinite rtlstanoe from the centre. 

Conle seetlon8.>-Every equation of the second degree between two 
vsriables will represent either a circle, an ellipse, a parabola or a hyperbola. 
These curves are those which are obtained by intersecting the surface of a 
cone by planes, and for this reason they are called conic sections. 

liC^Cwrftltiiile curve.— A logMrithmic curve is one in which one of the 
coftrdiiiates of any point is the logarithm of the other. 

The coordinate axis to « hich the lines denoting the logarithms are parallel 
is called the axU of logarithmt, and the other the axis of numberg. If y is 
the axis of logarithms and x the axis of numbers, the equation of the curve 
isv = \ogx. 

If the base of a system of .logarithms is a, we have aV = x, in which y Is the 
knrarithm of x. 

Each system of logarithms will give a different logarithmic curve. If y = 
0. X = 1. Hence every lomrithmic curve will intersect the axis of numbers 
St a distance from the origin equal to 1. 



72 DIFFERENTIAL CALCULUS. 



DIFFEBENTIAL CALCULUS. 

The differential of a variable quantity is the difference between any two 
of its consecutive values; hence it is indefinitely small. It is expressed bjr 
writing d before the quantity, as dx, which is read differential of x. 

The term ^ is called the differential coefficient of y regarded as a f uno- 

tion of X. 
The differential of a function is equal to its differential coefficient inul< 

tiplied by the differential of the independent variable; thus, -^dx = dy. 

The limit of a variable quantity Is that value to which it contiiinallj 
approaches, so as at last to differ from it by less than any assignable quan- 
tity. 

The differential coefficient is the limit of the ratio of the increment of the 
independent variable to the increment of the function. 

The differential of a constant quantity is equal to 0. 

The differential of a product of a constant by a variable Is equal to the 
constant multiplied by the differential of the variable. 

If u =: AVf du = Adv. 

In any curve whose equation is y—f{x\ the differential coefficient 

:r = tfl^n <z; hence, the rate of increase of the function, or the ascension of 

cut 

ihe curve at any point, Is equal to the tangent of the angle which the tAOgent 

line makes witn the axis of abscissas. 

All the operations of the Differential Calculus comprise but two objects: 

1. To find the rate of change In a function when it passes from one 8tat« 
of value to another, consecutive with It. 

S. To find the actual change in the function : The rate of change te the 
differential coefficient, and the actual chaufre the differential. 

Dlfferenttals of algebraic fnnctlons,— The differential of the 
sum or difference of any number of functions, dependent on the same 
variable, is equal to the sum or difference of their differentials taken sepa- 
rately : 

If u — y-\-z — v}^ du — dy-\-dz— dw. 

The differential of a product of two functions dependent on the same 
variable is equal to the sum of the products of each by the differential of 
the other : 

,. . , , , diuv) dn , dv 
dCuv) = vdu + udv. — - ==ii^ + —' 

The differential of the product of any number of functions Is equal to the 
sum of the products which arise bv multiplying the differential of each 
function by the product of all the others: 

d{ut*) 3 tedu + nsdt -4- utd$. 
The differential of a fraction equals the denominator Into the differential 
of the numerator minus the numerator into the differential of the denom- 
inator, divided by the square of the denominator : , 



dt 



„/u\ vdu — udv 



If the denominator Is constant, dv = 0, and dt = —^ = ~<. 

udv 
If the numerator is constant, du = 0, and dt = r- 

The differential of the square root of a quantity is equal to the differen* 
tial of the quantity divided by twice the square root of the quantity: 

If V = it^, or v=i'u, dv = — -; = -u^^du* 
8 i^u 8 



DIFFERENTIAL CALCULUS. 73 

The differential of any power of a f onotion is equal to the exponent multi- 
plied by the function raixed to a power less one, multiplied by the differoR- 
lial of the function, d(M") = nt** - irfu. 

Ponnnlaa for OUTerenttrnttiig nlKebrale Aincttoiui, 

1. d (o) = 0. 



2. d (ax) =3 adx, 

4. d (x — y) = dx — dy, 

5. d (jcy) = xdy + ycte» 



e jj /?\ _ ytto-ardy 



\vf y* 

7. d («"•)= m*"*- 'da?. 
dx 



9. d 



2 Va: 
t"*)r=-ra?''*"W 



To find the differential of the form « = (a+ to")*": 

Multiply the exponent of the pareuthesis into tlie exponent of the varia- 
ble within the parenthesis, into the coefficient of the variable, into the bi- 
nomial raised to a power leas 1, into the variable witbin the pareuthesis 
raised to a power less I, into the differential of the variable. 

dw = d{a + te*)** = mnb{a + 6x*)* " * ar » " *da?. 

To find the rate of change for a given valtie of the variable : 
Find the differential coefficient, and substitute the value of the variable in 
the second member of the equation. 

ExAifPT^.— If X fs the side of a cube and u its volume, tt = «», ~ = 8a:*. 

ax 
Henoe the rate of change in the volume is three times the square of the. 
edge. If the edge la denoted by 1, the rate of change is 3. 

Application. The ooeffloient of expansion by heat of the volume of a body 
is three times the linear coefficient of expansion. Thus if the side of a cube 
exnands .001 inch, its volume expands .008 cubic inch. 1.001* = 1.003003001. 

A pajrttal diflRDFentlal eoeflletent is the differential coefficient of 
a function of two or more variables under the supposition that only one of 
them has changed its value. 

A partial differential is the differential of a function of two or more vari- 
ables under the supposition that only one of them has changed its value. 

The total differential of a function of any number of variables is eqiuil to 
the sum of the partial differentials. 

If tt =/(ary), the partial differentials are 7-dar, -fdy. 

Ifu = a:« + y*-»,dttr=^*daj + ^dy + ?^; =8a;dar4-3i/«dy-d«. 
ax ay dz 

Intesralfl.— An Integral is a functional expression derived flrom a 
differential. Integration Is the operation of finding the primitive function 
from the differential function. It is Indicated by the sign /, which is read 
'*the integral of." Thus/SLrdx = x* ; read, the integral of 2xdx equals x*. 

To inteirrate an expression of the form nu^ ~ ^d» or x^dx. add 1 to the 
exponent of the variable, and divide by the new exponent and by the differ- 
ential of the variable: fSx^dx = x*, (Applicable in all cases except when 



For /x dx see formula 2 page 78.) 



The integral of the product of a constant by the differential of a vari- 
able is equal to the constant multiplied by the integral of the differential; 



fas^dx = a/l^dx = a~-r- a-m + » 



The integral of the algebraic sum of any number of cdfferentials is equal to 
the algebraic sum of their integrals: 

du = ikuMx - bydy - z^dz; fdu = ^ax* - sJ'* - ^• 

fince the differential of a constant is 0, a constant connected with a vari- 
able by the sign -for - disappears in the differentiation; thus dia + a?*) = 
ds^ ss ms^ ' 'd«. Hence in integrating a differential expression we must 



74 DIFFEBENTIAL CALCULUS. 

annex to the intofcnd obtained a oonstant represented by O to compensate 
for tbe term whion may have been lost in differentiation. Thus if we hare 
dt s adx'y Jdy = afdx. Integrating, 

y = ax±C. 

The oonstant C, which is added to the first intefrral, must have such a 
YiUue as to render thefunctiooal equation true for every possible value that 
may be attributed to the variable. Hence, after having found the flrst 
integral equation and added the constant C, if we then make the variable 
equal to zero, the value which the function assumes will be the true value 
of a 

An indefinite Integral is the first integral obtained before the value of the 
constant Cis determined. 

A particular Integral is the integral after the value of C has been found. 

A definite integral is the integral corresponding to a given value of the 
variable. 

Intecimtton iMtween limtta.— Having found the indefinite Inte- 
gral and the particular integral, the next step Is to find the definite integral, 
and then the definite integral between given limite of the variable. 

The integral of a function, taken between two limits, indicated by given 
values of X, is equal to the difference of the definite integrals oorreepond- 
ing to those limits. The expression 



i dy = a j dx 



Is read: Integral of the differential of y, taken between the limits x^ and tee- 
the least limit, or the limit corresponding to the subtractlve integral, being 
placed below. 

Integrate du = fisesda; between the limits x = 1 and a; = 8, u being equal to 
81 when x s 0. /du s/to'dx s &r* + C; C= 81 when x = 0, then 



I 



»X-8 

du = 8(8)> + 81, ndnus 8(1>* + SI s 7& 
x»l 



Int^gmtlon of partlenlar forms. 

To integrate a differential of thefoi-m du = {a-\- bx^y^x* ' 'dx. 

1. If there is a constant factor, place it without the sign of the Integral, 
and omit the power of the variable without the parenthesis and the differ, 
ential; 

2. Augment the exponent of the parentheslsi by 1, and then divide this 
quantity, with the exponent so increased, bv tlie exponent of the paren< 

Into the exponent of the variable within the parenthesis, into the co> 
'" variabi '"" 



quantity, 
thesis, in 



efficient of the variable. Whence 



> 



{m + l)nb 



The differential of an arc Is the hypothenuse of a right-angle triangle of 
which the base Is dx and the perpendicular dy. 

Ifsisanarc, d£= VcU* + dy^ z=fVdx* + dy*\ 

anadrature of a plane flu^nre. 

The differential of the area of a plane surface is equal to the ordinate into 
the differential of the abscissa. 

dt = ydx. 

To apply the principle enunciated in the last equation, In finding the area 
of any particular plane surface : 

Find the value of y in terms of x, from the equation of the bounding line; 
substitute this value In the differential equation, and then integrate between 
the required limits of x. 

Area oftlte parabola ^^Fiud the area of any portion of the com- 
mon parabola whose equation Is 

y* = ftpx; whence y = j^2px. 



DIFFERENTIAL CALCULITS. 75 

SubsUtuUiiK this value of y in the differential equation d$ = ydx gives 
/ <^= / f^«ete= i/^J x^dx= — ^— al + C; 

or. •=-^ =3*F+C. 

If we eetimate the area from the principal vertex, x :=: 0. y = 0, and (7=0; 

2 
and denoting the particular Integral by y, «' =» £ 'V* 

Thai is, the area of any portion of the parabola, estimated from the ver- 
tex, is eanal to % of the rectangle of the abscissa and ordinate of the extreme 
point. 'The curve is therefore quadrable. 

Aaadrmtnre of ■nrteeMi of reTolatlon.— The differential of a 
surface of revolution is equal to tlie circumference of a circle perpendicular 
to the axis into the differential of the arc of the meridian curve. 



d$ = ftwy^dx* •{- dy*; 

in wliicli y is the radius of a circle of the boundfaig surface in a plane oer- 
pendlcular to the axis of revolution, and x is the abscissa, or distance of the 
plane from the orif in of coordinate axes. 

Therefore, to fina the volume of anv surface of revolution: 

Find the value of y and dy from the equation of the meridlao curve in 
terms of x and dx, then substitute these values in the differential equattoui 
and integrate between the proper limits of x. 

Bv application of this rule we may And: 

The curved surface of a cylinder equals the product of the circumference 
of the base Into the altitude. 

The convex surface of a cone equals the product of the circumference of 
the base Into half the slant height. 

The surface of a sphere is equal to the area of four great dreles, or equal 
tn the curved surface of the circumscrlbinjc cylinder. 

Cakttf «re of Tolvns^s of reTolatlon.— A volume of revolution 
ii a volume fceneraied by the revolution of a plane figure about a fixed line 
called the axis. 

If we denote the volume by F, dV = «y" dx. 

The area of a circle deecrlbed by any ordinate y is «y*; hence the differ* 
ential of a volume of rovolution Is equal to the area of a ciixsle perpendicular 
to the axis into the differential of the axis. 

The differential of a volume generated by the revolution of a plane figure 
about the axis of Y Is 9X*dy. 

Tn find the value of Ffor any fj^ven volume of revolution : 

Find thfl value of y* In terms of x from the equation of the meridian 
curve, substitute this value In the differential equation, and then integrate 



between the required limits of x. 
By application of this rule we may find : 
The volume of a cylinder Is equal to the 1 



i cylinder Is equal to the area of the base multiplied by the 
sltitnde. * -• 

The volume of a cone is equal to the area of the base into one third the 
siatiide. 

The volume of a prolate spheroid and of an oblate spheroid (formed by 
the revolution of an ellipse around its transverse and Its conjugate axis re- 
spectively) are each equal to two thirds of the clreumscriblng cylinder. 

If the axes are equal, the spheroid becomes a sphere and its volume => 

^IP sc i> = 2 *^« ^ being radius and D diameter. 
9 o 

The volume of a paraboloid is equal to half the cylinder having the same 
hsse and altitude. 

The volume of a pyramid equals the area of the base multiplied by one 
third the alUtnda 

floeonid^ tliirdf ete*^ dlil^reiitlals.— The differential coefficient 
bdnir a function of tne independent variable, it may be differentiated, and 
«e thus obtain the second dCfferantial coefficient: 

d(^) s ^. Dividtaig by dx, we have for the second differential ooefll- 
\fgx-^ ax 



76 DlFFBRfiKTlAL CALCULUS. 

dent -z-^, which is read: second differential of u divided by the square oi 

the differential of t (or dx squared). 

The third differentiHi coefficient ^ is read: third differential of %i divided 

by dx cubed. 
The differentials of the different orders are obtedned by multiplyio^ the 

differential coeffloients by the corresponding powers of rte; thus^-g <to*= 

third differential of u. 

St^rn of tlte first diiTereiltliil coettelent.— If we hate a curre 
wboHe equation is ^ = /s, referred to rectangular coordinates^ the curve 

will recede from the &xis of X when ^ is positive, and approach Uie 

axis when it Is negatiTe^ when the curve lies within the first ant^le of the 
'eo<(rdinate aares. For all angles and every relation of v and x the cur%*e 
will recede from the axis of X when the ordinate and fli*8t differential co- 
efflcient have the some sign, and approach it when they have different 
signs. If the tangent of the curve becomes parallel to the axis of X at any 

p6ilit x^ — 0. If the tangent becomes perpendicular to the astis of X nt any 

point ^ = 00. 

Slj^ ortlk« second dlflnsreAtlal coellleleiit.— tte second dif- 
fsrenual coefficient has the same sign as the ordinate when the curve Is 
convex toward the axis of abscissa and a contrary sign when it is concave. 

MaclauHn's Theorem.— For developing into a series any function 
M a single variable &a n-A^Bx + <}afl-\- Dx^ + Ex*^ etc.. In which A, £, 
C, etc., are independent of v: 

In applying the formnia, omit the expressioBs « s 0^ althougli tfa* ooeffi- 
oiente are always found under tiiis hypothesis. 
^Examples : 

+ -(j!!^<J!ir«)«--»^. + etc. 



« -f « a o« ' «» a* ^ ' * ' <ju + 1 ' 

Taylor's Theorem.— For develop! ng'into a series any function of the 

sum or difference of two independent variables, as a' te /(<r ± y): 

in which u is what u' becomes when y "= 0» x. ^ what -.- beoomes when 

y ts 0. etc. 

niaixima and mintmn.— To find the maximum or minimum value 
of a function of a single variable: 

1. Find the first differential coeffiolent of the f unctloa, place it equal to 0, 
and determine the roots of the equation. 

8. Find the second differential coefficient, and substitute each real root, 



In succession, for the variable m the second member of tlie equation. £acb 
root which gives a negative result will correspond to a maximum value of 
the function, and eaon which gives a positive result will correspond to a 
minimum value. 

JSXAIIPI.B.— To find the value of x which will render the ftttntlon y s 
maximum or minimum in the equation of the circle, y* 4- «• sx J2«; 

^= - * making - - = Ogives a? = 0. 



DIFITEBBKTIAL CALCULUS. 7T 

The second differential coefficient is: ^ = - ^'"t*^ . 

jf-. « 

When X = 0, y = R; hence ~ ~ "B* ^^^h being negatlTO, y is a maxi- 
mum for B poeitiva 

In applying the rule to practical examples we first find an expression for 
(he function which Is to be made a maximum or minimum. 

S. If in such expressloD a constant quantity is found as a faotor, it mav 
be omiOed in the operation; for the praduct will be a maximum or a mini- 
mum when the variable factor is a maximum or a minimum. 

S. An J Talue of Uie independent variable which rendera a function a maz- 
imura or a minimum will render any power or root of that function a 
maximum or minimum; hence we may square both members of an equa- 
tion to free it of radicals before differentiaung. 



Bf thane ruk* we may find: 
Tliei 



) maximum rectangle which can be inscribed in a triangle Is one whose 
altitude is luilf the altitude of the triangle. 

The altitude of the maximum cylinder which can be inscribed in a cone is 
one third the altitude of the cone. 

The surface of a cylindrical vessel of a given volume, open at the top, is a 
minimum when the altitude equals half the diameter. 

Tlie altitude of a cylinder inscribed in a sphere when its convex surface is 
a maximum is r ^2. r = radios. 

The altitude of ajcyllnder inscribed in a sphere when the volume is a 
maximum is 2r -«- VS. 

(For nuucima and minima without the calculus see Appendix, p. 1070.) 

JMfl'erentlal of an exi^onentlal Ametlon. 

If « = o* 0) 

thendttcd^^ =a^fcda;, (9> 

hi which Jb is a constant dependent on a. 

The relation between a and I; is a^ = e; whence a s e*, ..... <8) 

in which e = 2.718S818 . . . the base of the Naperian system of logarithms. 
Iiosmritliiiis.— The logarithms in the Naperian system are denoted by 
2, Nap. log or hyperbolic log, byp. log, or log^; and in the common system 
always by log. 

k =3 Nap. log a, log a = I; log e (4) 

The common logarithm of e, =r k)g 2.718S8I8 . . . = .4842M5 .... is called 
the modulus of the common system, and is denoted by M. Hence, if we have 
the Naperian logarithm of a number we can And the common logarithm of 
tlie same numwr by multiplying by the modulus. Beciprocally, Nap. 
ioK ^ com. log X 2.S0858S1. 

U in equation (4) we make a = 10, we have 

1 8 fc log e, or - s log e = If. 

That Is, the modulns of the common system is equal to 1, divided by the 
Naperian logarithm of the common base. 
From equation (IQ we have 

« a* 
If we make a s 10; the base of the common system, « s log ti, and 
jy. * J di* 1 dw _. 

d(IOg tt)ad«S Kr-CB — xjtf. 

That is, the differential of a common logarithm of a quantity is equal to the 
differential of the quantity divided by the quantity, into the modulus. 
If we make a =s e, the base of the Naperian system, x becomes the Nape* 



78 DIFFERENTIAL CALCULUS. 

rlan logarltnm of u, and k becomeB 1 (see equation (8)); bonce If » 1, and 

iKNap.lo«t*)r=ifx=^; = ^. 
a* u 

That is, the differential of a Naperlan logarithm of a quantity is equal to the 
differential of the quantity divided by the quantity; and In the Naperlan 
system tlie modulus is 1. 

Since fc is the Naperlan logarithm of a, du s a^ I a cte. That Is, the 
differential of a fuuctlon of the form a^ is equal to th« fuoction, into the 
Naperlan logarithm of the base a, into the differential of the exponent. 

If we have a .differential in a f raetioual form. In which the numerator is 
the differential of the denominator, the integral is the Naperlan logarithm 
of the denominator. Integrals of fractional differentials of other forms are 
given lielow: 

IMllbreiitla] forma iv^bleli Itave knoiv^n InteKral*} esc- 
ponenUal ftanettoiui* d = Nap. log.) 

1. I aP^ladx = ar^-\-C; 

dxx~ * = lx-\-0\ 






dx 

dx 



= Kx + j^x*±a*) + C; 



^x* ± lUix 

Clrenlar Ainetloiia.-Let z denote an arc in the first quadrant, y fts 
Bine, X its cosioe, v its versed sine, and i Its tangent; and the following nota- 
tion be employed to designate an arc by any one of its functions, vis., 

sin "^ If denotes an arc of which y is the sine 

^jQg-ljp M « « u «. a; is the cosine, 

tan-*« " •• " ** " Ms the tangent 



OIFFEBENTIAL CALCULUK 



79 



*md "arc whom sine is y,** etc.).— we hare the foUowing differential fonns 
vhidi hare known integrals (r = radius): 



/ cos sdx = sin c -f C^f 
/ - sin s dx = cos m+C; 



r^i 



dx 






= coe-*af C; 



= yer-sin -*« + (?; 



—==z =co» • 



«+C; 






Kdz = yer-sin Z'{-C\ 



dz 

C06*S 

rd t> 



= tan c + C; 



4/*;;;+^ = ^«''-"*» "*»+<?; 






/: 

—: rr:^. = COS * 



■»^ + C; 



l/'^tai* - u* 






u. =*--*i + <^. 



The eyelold*— If a circle be ro11<*<1 along a straight line, any point of 
the circumference, as P, will dt^sciibe a curve which \» called a cycloid. The 
circle is called the generating circle. ao<l Pthe geuerating point. 

The transcendental equation of the cycloid i« 



X = ver-sln- > y — J^try - p\ 



tad the differential equation is dx - 



^^-y - y*. 



The area of the cycloid is equal to three times the area of the generating 
tircle. 

The Mjrfaee descrllMrd by the arc of n cycloid when revolved about its base 
fe eqtial to 04 thii'dM of the enierHtiii? circle 

The volume of the M'>lid K^^nnrnted hv revolving a cycloid about its base is 
Moal to fli-e eiphUifi of fh- ciri-iiniscribinir cylinder. 

InCearrsil eftlcolna«~In the integral calculus we have to return from 
tiie differencial ii> the fiiiH-tlon from which it waM derived A number of 
liffervntial «>xpressions are triven abovf». «>aoh of which has a known in- 
vgral corresponding 10 it, and which being different.ated, will produce the 
iziveii differential 

In all cUumes of functions any differential expression may be integrated 
when it ia retiuceil to one of ilin known forms; and the operations of the 
loreirral calculiiH cotisisr nminlv In making such irai informations of given 
(iifferentiaJ expressions as Mhalf nxlnce them to equivalent ones whose in- 
ieirra>ls are known. 

For inctliods of making these transformations reference must be made to 
tie text^booka on differential and Iniegral calculus. 



80 



HATHEMATICAL TABLES. 
BBCIPBO€AI<9 OF NUMBERS. 



No. 


Recipro- 
cal. 


No. 


Reclpro- 


No. 
127 


Reclpro- 


No. 


Recipro- ' 


No. 

253 


Recipro- 
cal. 




1.00000000 


64 


.01662500 


.00787402 


190 


.00626816 


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.60000000 


6 


.0158^61 


$ 


.00781280 


1 


.00628560 


4 


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.3S8833W 





.01516161 


9 


.00^75194 


2 


.00620888 


6 


.008021.51 




,26000000 


7 


.01492687 


180 


.0076928: 


3 


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6 


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.90000000 


8 


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1 


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4 


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7 


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.16666«ST 


9 


.01449275 


8 


.(10757576 


5 


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8 


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.14«85rN 


70 


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4 


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7 


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200 


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5 


.01838 533 


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6 


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7 


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140 


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.00084981 


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210 


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6 


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150 


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6 


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14 


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280 


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.01080928 


160 


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8 


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6 


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1 


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6 


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8 


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1 


.00432900 


4 


.00*40131 




.02325681 


6 


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9 


.00591716 


2 


.00431034 


6 


.003.H8aH; 




.023?^727 


7 


.00934579 


170 


.00588235 


3 


.00429184 


6 


.oo3rr.<a 




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8 


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1 


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4 


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9 


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8 


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.02127660 


no 


.00909091 


8 


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6 


.00423729 


9 


.0a88144l 




.02083;Wi 


11 


.00900901 


4 


.00674713 


7 


.00421941 


300 


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.02040816 


12 


.00892g57 


6 


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8 


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1 


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18 


.0088495U 





.00.568182 


9 


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a 


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.01900784 


14 


.00877193 


7 


.00564972 


340 


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3 


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.01»2:i077 


16 


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6 


.0a56l798 


1 


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4 


.0082894: 




.01886792 


16 


.00862009 


9 


.00558059 


2 


.004l:«»8 


5 


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.01851852 


17 


.00854701 


180 


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8 


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6 


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18 


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4 


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7 


.0031-::^^ 




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19 


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2 


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8 


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.01784886 


120 


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8 


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9 


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7 


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810 


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2 


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5 


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8 


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11 


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.01686607 


8 


.00.Si:W(W 





.0053ro:i4 


9 


.00401000 


12 


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4 


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•J 


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250 


.00400000 


18 


. 0031 94 W 




.01012903 


5; .oasooooo 

01 .00793651 


8 


.0«).581914 


1 


.00398406 


14| .0a81Kl7I 




.01587802 


1 9 


.00529100 


21 .00.J96S25 


l&l .0081740< 



BECIPBOOAIA 07 KUMBXBS. 



81 



No 


Rc«ipro* 


No. 


Raclpro- 


No. 
446 


Reclpro- 


No. 


Redpro- 


No. 


R^r<. 


816 


.00316456 


381 


.00862467 


.00224215 


611 


.00195695 


576 


.00173611 


17 


.0031M67 


2 


.00281780 


7 


.00223714 


12 


.00195812 


7 


.00i;8810 


18 


.00314465 


3 


.00;»1097 


6 


.00223214 


18 


.00104932 


8 


.00178010 


1» 


.00818480 


4 


.00860417 


9 


.00222717 


14 


.00194552 


9 


.00172712 


3390 


OOSl^iQOO 


6 


.00259740 


460 


.00228222 


15 


.00194175 


580 


.00172414 


1 


.00311S86 


6 


.002«K)fi7 


1 


.00221729 


16 


.00193798 


1 


.00172117 


a 


.0O3tO55O 


7 


.0025K898 


2 


.00221289 


17 


.00193424 


2 


.00171821 


8 


.0090958: 


8 


.002577821 


8 


.00220751 


18 


.00193060 


8 


.00171527 


4 


.00908642 


9 


.00257009! 


4 


.00220264 


19 


.00192678 


4 


.0017123:) 


6 


.00807692 


890 


.00256410 


6 


.00219780 


KO 


.00192808 


5 


.00170940 


e 


.00906748 


1 


.00255754 


6 


.00219298 


1 


.00191939 


6 


.00170648 


7 


.00805810 


9 


.00255102 


7 


.00218818 


2 


.00191571 


7 


.00170358 


8 


.00804878 


8 


.00354453 


8 


.00218341 


8 


.00191905 


8 


.OO170OG8 


9 


.00808951 


4 


.00253807 


9 


.00217865 


4 


.00190840 


9 


.00169779 


sao 


.00303030 


6 


.00253ia5 


460 


.00217,391 


6 


.00190476 


590 


.00169491 


1 


.0090-2115 


6 


.0O,>52525 


1 


.00216920 


6 


.00190114 


1 


.00169-205 


8 


.00301306 


7 


.00251889 


2 


.00216450 


7 


.00189753 


2 


.00168019 


8 


.00800300 


8 


.00251250 


8 


.00215983 


8 


.00189394 


3 


.00168634 


4 


.00299101 


9 


.00250627 


4 


.00216617 


9 


.00189036 


4 


.001083.50 


6 


.oojfl8:.o: 


400 


.0025000n 


6 


.00215054 


580 


.00188679 


5 


.001680«i7 


6 


.00297619 


1 


.00249377 


6 


.00214592 


1 


.00188324 


6 


.00167785 


7 


.00296736 


2 


.0024S75C 


7 


.0O21413;J 


2 


.00187970 


7 


.00167504 


8 


.00295858 


3 


.00248139 


8 


.00218675 


8 


.00187617 


8 


.00107-224 


9 


.0029408.-, 


4 


.00247525 





.00213220 


4 


.00187266 





.00166945 


SIO 


.00294118 


» 


.00246914 


470 


.00212766 


5 


.00186916 


600 


.00166667 


1 


.ooaBefts 


6 


.00246305 


1 


.00212:^14 


6 


.00186567 


1 


.00166389 


9 


.OO-.-OiSOP 


7 


.00245700 


2 


.00211864 


7 


.00186220 


2 


.00166118 


8 


.00291 M5 


8 


.00245098 


8 


.00211416 


8 


.00185874 


8 


.00165837 


4 


.0029060* 


9 


.002444911 


4 


.00210970 


9 


.00186528 


4 


.00165563 


5 


.00SiS98&5 


410 


.00248902 


5 


.00210526 


640 


.00185185 


5 


.00165280 


6 


.00289017 


11 


.00243309 


6 


.00210064 


1 


.C0181<43 


6 


.00165016 


7 


.(0288184 


12 


.00242718 


7 


.00209644 


2 


.00184502 


7 


.00164746 


A 


.00287!»6 


13 


.00242131 


8 


.00209205 


8 


.00184162 


8 


.00164474 


9 


.00286.V» 


14 


.00241546 





.00208768 


4 


.00183823 


9 


.00164204 


850 


.00285714 


15 


.00210961 


480 


.00208:^ 


5 


.00183486 


610 


.00163934 


1 


.00284000 


16 


.00240:«5 


1 


.002a790(' 


6 


.00188150 


11 


.00168666 


9 


.00*284091 


17 


.00239808 


2 


.00207469 


7 


.0018-2815 


12 


.00163399 


8 


.0ftA)8288 


18 


.00J89284 


8 


.002070:11) 


8 


.00182482 


13 


.00163132 


4 


.0O-.'8248C 


10 


.00238603 


4 


.00206612 


9 


.00182149 


14 


.00162866 


b 


.00281680 


420 


.0028809*) 


5 


.00-206180 


C50 


.00181818 


15 


.00162602 


6 


.Qoamm 


1 


.00237530 


6 


.00205761 


1 


.00181488 


16 


.001623.38 


7 


.002801 U 


2 


.00286967 


7 


.002058.^9 


2 


.00181159 


17 


.00162075 


8 


.00279:530 


3 


.002.16407 


8 


.00204918 


8 


.00180832 


18 


.00161812 


9 


.002785.M 


4 


.0023.5849 





.00-2(M4D9 


4 


.00180505 


19 


.00161551 


860 


.00277778 


6 


.0023.5294 


490 


.0020408-2 


5 


.00180180 


6-20 


.00161-290 


I 


.0027700F 


6 


.00*^742 


1 


.ooeo36r.G 


6 


.00179856 


1 


.ooioiasi 


2 


.00276243 


7 


.0023419? 


2 


.002032.52 


7 


.00179533 


2 


.00160772 


8 


.00276482 


8 


.0O2:i'J64r) 


8 


.00202840 


8 


.00170211 


3 


.00160514 


4 


.0027472.T 


9 


.0023dl0i) 


4 


.00202429 





.00178891 


8 


.00160256 


5 


.002789r3 


430 


.00232^^8 


6 


.00202020 


560 


.00178571 


5 


.00160000 


6 


.00/78224 


1 


.00282019 


6 


.00201613 


»1 


.00178253 


6 


.001.59744 


7 


.002W480 


9 


.00231481 


7 


.00-20120'; 


2 


.00177936 




.00159490 


8 


.00271:39 


8 


,00230947 


8 


.00200803 


8 


.00177620 


8 


.001 59. '36 


9 


.owioiw 


4 


.002:^0115 


9 


.00200401 


4 


.00177305 


9 


.00158982 


870 


.00270270 


6 


.00229885 


600 


.00200000 


6 


.00176091 


630 


.001.58730 


3 


.00961L542 


6 


.00-.'2»3.-8 


1 


.001996(^1 


6 


.00170678 


1 


.00158479 


9 


.0U2et«l7 


7 


.C0228a3:5 


2 


.00199-203 




.0017C367 


2 


.001.58-228 


8 


.00268096 


8 


.002CH.S10 


3 


.0(M 98807 


8 


.00176056 


8 


.00157978 


4 


.0Oi6788() 


9 


.00227790 


4 


.00198413 


9 
570 


.0017.5747 


4 


.001577-29 


5 


.00260607 


440 


.00227273 


5 


.00198020 


.00175439 


5 


.00157480 


6 


.00265957 


1 


.00226757 


6 


.001976-28 


1 


.0017.5131 


6 


.00157233 


7 


00265652 


a 


.00228244 


7 


.0019?239' 


2 


.001748-25 


r- 


.00156986 


8 


.(xmaao 


3 


.00225784 


8 


.001968oO; 


3 


.00174520 


8 


.00156740 





.00260852 


4 


.0022.'i225 


9 


.01U1J6464 


4 


.00174216 


9 


.00156491 


880 


.00288158 


s 


.00294719 


510 


.001960781 


6 


.00173913 


640 


.00166250 



82 



MATHEMATICAL TABLES. 



No. 


Htjtfpro- 
cal. 


No. 


Reclpro 
cul. 


No 

l_ 

771 


Recipro- 


No. 


Recl->rD' 
eal. 


No. 
901 


Redpro- 


on 


.00156006 


T06 


00141643 


.00129702 


886 


.00119617 


.00110968 


2 


.00l5r,T6:j 


7 


.00141443 


S 


.001296*1 


7 


.00119474 


2 


.00110865 


8 


.00 .'VVS-.M 


8 


.00141243 


8 


.00129366 


8 


.00119&32 


3 


.001 10742 


4 


.00j55;?r9 


9 


.00141044 


1 4 


.00129199 


9 


.00119189 


4 


.00110619 


5 


.00:. 55089 


710 


.00140815 


6 


.00129082 


840 


.00119048 


6 


.001 10497 


6 


.001.54799 


11 


.00140647 


6 


.0012aS66 


1 


.00118006 


6 


.00110375 


7 


.00I.',45.'»9 


12 


.00110149 


7 


.00128700 


2 


.00118765 


7 


.00110254 


8 


.00r»432I 


13 


.00140252 


8 


.00128535 


8 


.00118024 


8 


.00110192 


9 


.0«)l540b3 


14 


.00140056 


9 


.00128370 


4 


.0011»188 


9 


.00110011 


6.V) 


.00ir)88lii 


15 


.00189880 


780 


.O0128'a05; 


5 


.00118843 


910 


.00109890 


1 


.00I.586I0 


16 


.00139665 


1 


.00128041 


6 


.00118203 


]l 


.00109760 


2 


.001.1*^74 


17 


.00189170 


8 


.00127877, 


7 


.00118064 


12 


.00109649 


8 


.00 53140 


IH 


.00139276 


3 


.00127714 


8 


.00117924 


13 


.00109539 


4 


.00I.53905 


19 


.00139082 


4 


.00127551' 


9 


.00117786 


14 


.00109409 


5 


.00I5«7,J 


720 


.00138889 


6 


.00127888: 


850 


.00117647 


15 


.00109890 


6 


.00152489 


1 


.00i:i8696 


6 


.00127226 


1 


.00117509 


16 


.00109170 


7 


.oovrijn: 


2 


.00138501 


7 


.00127065, 


2 


.00117871 


17 


.00109051 


8 


.00151975 


3 


.00138313 


8 


.00126904 


8 


0011?283 


18 


.00108982 


9 


.00151745 


4 


.00138121 


9 


.00126743 


4 


.00117096 


19 


.00108814 


6G0 


.00151515 


5 


.00187981 


790 


.00121/582, 


ft 


.OOI169ft9 


920 


.00108696 


1 


.00151*6 


6 


.00137741 


1 


.00126422 


6 


.00116822 


1 


.00106578 


2 


.00151057 


7 


.00187552 


2 


.001»263l 


7 


.00116686 


8 


.00108460 


3 


.00150830 


8 


.0013736:^ 


3 


.00126108 


8 


.00116650 


8 


.0010834^ 


4 


.0015060^ 


9 


.00187174 


4 


.00125945! 


9 


.00116414 


4 


.001082S5 


5 


.OOloOSItt 


730 


.00180986 


6 


.001257861 


860 


.001^279 


fi 


.00108106 


6 


.00130150 


1 


.00136799 


6 


.00125628, 


1 


.00116144 


6 


.00107991 


7 


.001499-25 


2 


.00i:J6612 


1 ' 


.00125170 


2 


.00116009 


7 


.00107875 


8 


.00149701 


3 


.00130426 


8 


.00125313 


8 


.00115875 


8 


.00107759 


9 


.00119177 


4 


.00136240 


1 


.00125156 


4 


.00116741 


9 


.0010764.H 


670 


.00149;.'54 


5 


.00136054 


• 800 


.00125000 


6 


.00115607 


090 


.00107527 


1 


.OOHOaSl 


6 


.00135870 


: 1 


.00124844' 


6 


.00115473 


1 


.0010741 t 


2 


.00148809 


^ 


.00135685 


! 2 


.001246881 


7 


.00115840 


2 


.00107296 


8 


.00148588 


8 


.00135501 


' a 


.0012458:1, 


8 


.00116207 


a 


.00107181 


4 


.00148368 


9 


.00185;J18 


4 


.00124878 


9 


.00115075 


4 


.00107066 


5 


.00148148 


740 


.00185135 


B 


.00124224 


870 


.00114942 


5 


.00100962 


e 


.001 79',»9 


1 


.00184953 


1 ^ 


.00124009 


1 


.00114811 


6 


.00106888 


7 


.00147710 


2 


.00134771 


7 


.00123916 


2 


.00114679 


7 


.00106724 


8 


.00147498 


3 


.00134589 


8 


.00123762 


8 


.00114547 


8 


00I0601O 


g 


.CO 14727 5 


4 


.001341(»9 
.00134228 


1 8 


.00128609 


4 


.00114416 


9 


.00106496 


68U 


.00147059 


6 


1 810 


.001284571 


5 


.00114286 


940 


.00106888 


1 


.0014r>813 


6 


.00134018 


11 


.00123305' 


6 


.00114155 


1 


.00106270 


2 


.001406 JH 


ly 


.00l33v%9 


1 ^'-^ 


.00123153, 


7 


.00114025 


8 


.00106157 


8 


.00I4G413 


8 


.0013:1690 


' 13 


.00128001! 


8 


.0011.3^95 


8 


.00106044 


4 


.0<M 46199 


9 


.00138511 


1 54 


.00128850 


9 


.00118766 


4 


.oaio.5a32 


5 


.00145985 


750 


.0013*138 


1 15 


.00122699 


880 


.00113636 


ft 


.00105880 


G 


.00 1 4:.: 78 


1 


.00133156 


; 16 


.00122549 


1 


00118507 


e 


.00105708 


7 


OOH-Vj^JO 


2 


.00132979 


1 17 


.001223901 


2 


.00113879 


7 


.0niaVM»7 


8 


.OU145J4U 


3 


.0013J802 


( '« 


.00122249 
.00122100 


3 


.00113250 


8 


.00105485 


9 


.00145137 


4 


.00182026 


! 19 


4 


.00118122 


9 


.00105874 


690 


.0OI419-,>7, 


5 


.00:32450 


, 820 


.00121951 


ft 


.00112994 


950 


.00106268 


1 


001 1471 S 


6 


.00132275 


1 


.00121808 


6 


.00112867 


1 


.00105153 


j» 


.00144501), 


7 


.00182100 


1 2 


.(X)121654| 


r- 


.00112740 


8 


.00106048 


S 


.00144:101)1 


8 


00I3I926 


« 


.00121507 


8 


.00112613 


8 


.00104982 


4 


.00I4401« 


9 


.001317.52 


1 4 


.00121:159' 


9 


.00112486 


4 


.00104822 


5 


,0014888.-. 


700 


.00181579 


1 6 


.001212l:» 


890 


.00112360 


ft 


.00104712 


6 


.OOII36<S' 


J 


00181106 


6 


.00121065 


1 


.0011228:1 


6 


60104602 


7 


.00M817-.'| 


2 


.00 1 81 •.'84 


7 


. 00 1 ,'0919 


2 


.00118108 


7 


.00104493 


8 


OONJ:^;^ 


3 


.OOI8:0<>2 
.001. 10890 ' 


8 


.0012077.81 


8 


.00111982 


8 


.00104884 


» 


OOI4*)6l 


4 


9 


.0012U627I 


4 


.00111867 


9 


.00104275 


700 


.00 14. '.157! 


5 


.0011071 HI 


830 


.001-J04K2. 


ft 


.00111782 


960 


.00104167 


1 (nii r-.'Gr,?i 


6 


.00 .805181 


1 


.001203371 


6 


.00111607 


1 


.00104058 


2 (K)l l-.U.'iit' 


7 .00j:w:i78: 


2 


. 0012019.'! 


7 


.001114&8 


2 


.00106960 


s .o^mtu:' 


8 .0();i0w'08| 


8 


.001200481 


8 


.00111859 


8 


.00108842 


4 00 H'^ir.l 


9 .00i:^008«M 


4 


.00119904 


9 


.00111235 


4 


.00108764 


•M iw..ii>lii 


::o .001'J9F70,' 


ft .001197601' 


900 


.00111111 


6 


.00108087 



BECIPBOCALS OF ITUHBEBS. 



83 



Ho. 


Recipro- 


No. 


Recipro- 


No. 


BecfDro- 


No. 


Eecjyro- 


No. 


Recipro- 


986 


.0O1035» 


1061 


.00096993-2 


1096 


.000918409 


1161 


.000861826 


1826 


.000616661 


r- 


.00108413 


2 


.000968992 


7 


.000911577 


8 


.000860585 


7 


.009614996 


8 


.00103906 




.000088054 


8 


.000910747 


8 


.000859845 


8 


.000614388 





.00108199 




.000967118 


9 


.000909918 


4 


.000859106 


9 


.000618670 


970 


.00103093 




.000966184 


1100 


.000909091 


6 


.000868869 


1880 


.000818006 


1 


.0O10S987 




.000965251 


1 


.000908265, 


6 


.000857688 


1 


.000812846 


S 


.00l0e2881 




.000964320 


8 


.000907441 


7 


.000856898 


8 


.000811686 


s 


.00102775 




.000968391 


8 


.000906618 


6 


.000856164 


8 


.000611080 


4 


.00109069 




.000962464 


4 


.000905797 





.000855432 


4 


.000610878 


5 


.00102564 


1040 .0009615881 


5 


.000904977 


1170 


.000854701 


5 


.000809717 


6 


.00108459 




.000960615 


6 


.000904159, 


1 


.000858971 


6 


.000809061 


7 


.0010SS54 




.000959693 


7 


.000903342 


2 


.00085&!42 


7 


.000806407 


8 


00102860 




.000956774 


8 


.0009025-,'7 


8 


.000852515 


6 


.000807754 


9 


.00109145 




.000957854 


9 


.000901713 


4 


.000851789 


9 


.000807102 


960 


.00109041 




.000956988 


1110 


.000900901 


5 


.000851064 


1840 


.000806452 


J 


.00101987 




.000956028 


11 


000900090 


6 


.000850840 


1 


.000805802 


2 


.00101833 




.000956110 


12 


.000899281 


7 


.000849618 


8 


.000806153 


S 


.00I01?29 




.000954198 


13 


.000898473 


8 


.000848896 


8 


.000604506 


4 


.00101696 




.000958289 


14 


.0008976C6 


9 


.000848176 


4 


.000808856 


5 


.00101583 


1060 


.000952381 


15 


.0008968611 


1180 


.000847457 


6 


.000803218 


6 


.00101420 




.000951475 


16 


.000896057 


1 


.000816740 


6 


.000802568 


7 


.00101817 




.000950570 


17 


.000805255 


8 


.000846024 


7 


.000801925 


e 


.00101215 




.000949668 


18 


.000894454 


8 


.000845308 


8 


.000801288 




.00101112 




.00094871^7 


19 


.000893655 


4 


.000844595 


9 


.000800640 


99i] 


.00101010 




.000947867 


1120 


.000892867 


6 


.000843882 


1260 


.000800000 




.00100906 




.000046070 


1 


.000892061 


6 


.000843170 


1 


.000799860 




00100606 




.000946074 


2 


.000801266 


7 


.000842400 


8 


.000798;'28 




.00100706 




.000945180 


8 


.000890472 


8 


.000841751 


8 


.000798085 




.00100601 




.000944287 


4 


.000889680 


9 


.000841043 


4 


.000797446 




.00100602 


1()6C 


.000948396 


6 


.000888889 


1190 


.00084033€ 


5 


00071*6818 




.00100102 




.000942507 


6 


.000888099 


1 


.000889631 


6 


.000796178 




.00100801 




.000941.620 


7 


.000687311 


8 


.OOOi'38926 


7 


.000795645 




.00100200 




.000940734 


8 


.000886525 


8 


.000838222 


8 


.000794913 




00100100 




.000939650 


9 


.000885740 


4 


.000837521 


9 


.OOO". 94281 


lOOC 


.00100000 




.000938967 


1180 


.000884956 


5 


.000836820 


1260 


.000793051 




.000990001 




000938086 


1 


.00C8&I173 


6 


.000836120 


1 


.000798021 




.000988001 




.000937207 


2 


.000683892 


7 


.00083:>422 


2 


.000792893 




.000997009 




.0009^6330 


8 


.000882612 


8 


.O0OSM724 


8 


.000791766 








.000935454 
.000934579 


4 


.000881834 


9 


.000884028 


4 


.000791139 




000095025 


107( 


6 


.000881067 


1200 


000838833 


6 


.000790514 




!000094O86 




.000938707 


e 


0008Maj82 


1 


.00U832689 


6 


.000780889 




.000993049 




.000932836 


7 


.000879508 


2 


.000881947 


7 


.000789266 




.0009fti063 




.000931966 


8 


.000878735 


8 


.000831255 


8 


.00078H643 




.000991060 




.000931099 


9 


.0006779e8 


4 


.00088a>65 


9 


.000788022 


1011 


.000990090 




.000980233 


1140 


.000877193 


5 


.000829875 


1270 


.000787402 




.000969120 




.000929868 


1 


.000876424 


6 


.0008V9187 


1 


.000786782 




.000968142 




.000988605 


a 


.000875057 


7 


.00082K500 


8 


.000786163 




.000087167 




.000927644 


s 


.000874891 


8 


.000827815 


8 


.OOir, 85546 




000966199 




.000926784 


4 


.000874126 


9 


.000827180 


4 


.000784929 




.000985822 


loec 


.000925926 


6 


.000873362 


1210 


.000826446 


5 


.000784314 




.0cn0S4262 




.000925069 


6 


.000872600 


11 


.000825764 


6 


.000783699 




.000083-^84 




.000984..M4 


7 


.000871840 


12 


.000825082 


7 


.000783085 




.000962318 




.000923361 


8 


.000871080 


18 


.000824402 


8 


.000782473 




.000961351 




.000922509 


9 


.000870322 


14 


.00a^«8728 


9 


.000781861 


ICtiO 


.000980892 




.000921659 


1150 


.000869565 


15 


.000823045 


1280 


.000781250 




.000970488 




.000920610 


1 


.000868810 


16 


.000822;HC8 


1 


.0007J-43640 




0000:6474 




.000919963 


2 


.000868056 


17 


.00082169.M 


2 


.000780031 




.000977517 




.000919118 


8 


.000867303 


18 


.000821018 


8 


.000779423 




.000076562 




000918274 


4 


.000866551 


19 


.000820344 


4 


.000778816 




.000975610 


1000 .000017431 


5 


.00086r.801 


1280 


.000819672 


5 


.000778210 




.000974659 


1 .0U0916590 


6 


.00l)«6r»0.V2 


1 


.000819001 


6 


.000777-605 




.000078710 


2 .000915761 


7 


.000861304 


2 


.000818381 


7 


.000777001 


s 


.000978783 


8 .000914913 


8 


.001J86a'i.'>8 


3 


.000817661 


8 


001)770397 


9 


.000071817 


4I.0U09HO77 


9 


.oc-osoaais 


4 


.000816993 


9 


.0m)77.^795 


KM 


.O00OSW74 


5 


.000913242 


1160 


.OOOHOOOTO 


5 


.000816326 


1290 


.000775194 



84 



MATHEMATICAL TABLES. 



No. 


ReciDro- 


No. 


Recinro- 


No. 


Recipro- 


'no. 


Recioro- 


No. 


Recioro- 


1801 


.000774503 


1856 


.000737468 


1421 .000708780 


1486 


.000672048 


1651 


.0006447* 


S 


.000778994 


7 


000736920, 


2.000708235 


7 


.000678405 


8 


.000644a3< 


8 


.000778395 


8 


.000786:177 


8 000702741 


8 


.000679048 


8 


.00064891,' 


4 


.000772797 


9 


.0U07358:i5 


4 .000702247 


01.000671502 


4 


.000&485O 


K 


.0007WJ01 


1880 


.00a7S5'294i 


6 .OU0701754 


14901.000671141 


5 


.000643081 


6 


.000771605 


1 


.0007^4754 


6 .000701262 


1 1.000670601 


6 


.000645KJ7; 


7 


.OOOTHOIO 


2 


.000794214, 


7 .000700771 


2;. 000670241 


7 


.00064226J 


8 


.000770416 


8 


.0007380761 


8 .010700280 


81.000660702 


8 


.00061 IS It 


9 


.000760&J3 


4 


.0007^138 


9 .000090790 


4 .000660344 





.00064143' 


1800 


.000:69-^1 


5 


. 00073 aoit 


1430 .000690;j01 


51.000668806 


1500 


.00064 10:>( 


1 


.00076t(639 


6 


.0007^«064 


1 .000698812 


6'. 000668440 


1 


.00064061,'! 


2.000768W9 


7 


.000781529 


2 .000008324 


7.000668008 


S 


.0006402QE 


8 .0007tf7459 


8 


.000730094 


8 .000697837 


8 .000667557 


8 


.000630793 


4 .0007C6871 


9 


.000730480 


4 .000U97.i60 


1 9.000667111 


4 


.OOOOSKISO 


6 .00076C883, 


1870 


.0007<0927 


6.000606664! 


1500.000666667 


5 


.000638978 


6;. 000755697 


1 


.0007203951 


6 .0006JW379| 


1 11.000666228 


« 


.000638570 


7.. 0007651 111 


2 


.000728863 


7 .000695894' 


2'. 000665779 


7 


.000688162 


8 


.000761526 


8 


.0007^»2l 


8.000605410; 


8 .000665836 


8 


.000687755 





.000768942 


4 


.000727802 


.000604027, 


4 .0006648041 


9.000637819 


1810 


.00076S:«9 


5 


.0007Vr278 


1440 .000604444' 


5 .000664452 


I570;.00063C943 


11 


.000762776 


6 


.000726744 


11.000693062; 


6.000664011 


1 .0000866:^7 


18 


.000762195 


7 


.000726216 


2.000693481' 


7 .000668570 


2' .0006381 .%> 


18 


.000761616 


8 


.000726689 


8 .0006980011 


8 .000668130 


8.0006857-2S 


14 


.000761035 


9 


.00072516-^ 


4 .00060-25211 


0<. 000602891 


4 .000685324 


15 


.000760466 


1890 


.000724638 


6 .000602041 


15101.000662252 


5 .000634021 


16 


.000750878 


1 


.000724113 


61.0006015631 


111.000661813 


6 .000684518 


17 


.000750301 


2 


.000723589 


7 .0006010851 


I2I .000661876 


71.000634115 


18 


.000768725 


3 


.00072:»06 


8<. 000600608 


18.000660930 


8 .000633714 


10 


.000758150 


4 


.000722543 


.000600131 


14 .000660502 


. 000688:112 


1820 


.000757576 


6 


.000722022 


1450;.00068D655 


15.000600066 


1580 .000682911 


1 


000767002 


6 


.000721501 


1 .000680180 


16;.OH0660631 


11.000682511 


2 


.000766430 


7 


. 0007520980 


2'. 000688705! 


17i.000(i50lP6 


«!. 000632111 


8 


.000755858 


8 


.000720401 


8.000688231 


18!. 000658761 


8!. 000631712 


4 


.000765-J87 


9 


.000719912 


41.000687758 


191.000658328 


4'. 000631318 


5 


.000754717 


1300 


.000719424 


6'. 000687285 


1520 .0006578951 


5. 000030915 


6 


.000754148 


1 


.000718907 


6.000686813 


1 


.000657462 


6.000630517 


7 


.000753679 


2 


000718391 


7 .000086341 


2 


.0006.57030 


7 .000680I'20 


8 


.0007^8013 


8 


.000717875 


8 .000685871 


8 


.000656598 


8.000629723 


»!. 000752445 


4 


.000717300 


.000685401 


4 


.000656168, 


.0006203-27 


isao 


.000751880 


6 


.000716846 


1460 .0006849321 


5;. 000665738 


1600 .0006289.^1 


1 


.000751315 





. 00071 63:« 


1 .000684463, 


6.000655308 


1 .000628.>36 


2 


.000r507.')0 


7 


.000715820 


2 .00008:1994! 




2 .00062X141 


8 


.000750187 


8 


.0007153081 


8 .0006885271 


81.000654450; 


8 .000627746 


4 


.0007496.25 


9 


.000714790i 


4 .0006830001 


0|. 000654022' 


4 


.000627353 


5 


.000749064 


1400 


.000714286 


5. 00068-2594' 


1580'. 000653595 


6 


.000626958 


6 


.000748503 


1 


.0007187761 


6' .000682128 


It. 000658168 


6 


.000626566 




.000747943 


2 


.0007132671 


7. 000681 663 


2.000652742 


7 


.000626174 


8 


.000747384 


8 


.000712758 


8 .000681199 


8;.0006523I6 


8 


.000625788 


9 


.000746826 


4 


.0007122511 


0'.000(«0735 


4l.CH)0651890 





.000625391 


1340 


.OOOJ 46269 


6 


.000711741' 


1470 .000680272 


JSI.00065146C 


1600 


.000625000 


1 


.000746712 


6 


.000711238 


1 .000079810 


61.0006.51042 


2 


.0006-24219 


2 


.000745156 


7 


.000710732 


2 .000079318 


71.000650618 


4 


.000623441 


8 


.00074400.; 


8 


.0007102271 


8 .000678S87 


8 .000650195 


6 


.00062J665 


4 


.000744018 


9 


.0007097-28] 


4 .000678426 


91.000649778 


8 


.000621H90 


5 


.000743494 


1410 


.000709220 


5 .000077966 


15401.000649351 


1610 


.000621118 


6 


00074294-2 


11 


.000708717' 


6 .00O(r775O7 


1:. 000648929 


8 


.000620847 


7 


.000742390 


12 


.0007082151 


71. 000677048 


2 


.000648508 


4 


.000619578 


8 


.000741840 


13 


.000707714' 


8 .000676590 


3 


.000(*^OH8 


6 


.0006ISS1S 





.000741290 


14 


000707214, 


9 .000076182 


4 


.000047608 


8 


.00061804/ 


1850 


.000740741 


16 


.000706714 


1480 000675676 


5 


.000647249 


1620 


,00061728* 


1 


.000740192 


16 


.000700215 


1 .00^575219 


6 


.000646a30 


2 


.0006165^ 
.0006157^ 


2 


000739645 


17 


.000705716 


2 .000«r4-64 


7'. 0006464 12 


4 


3 


.000789098 


18 


.0(»705219 


8 .000674309 


8. 000045995 


6 


.000615006 


4 


.000788552 


19 


.000704722 


4 .0no«}7TJ8.M 


1 9| .000645.578 


8 


.0006142M 
.000618407 


6 


.000788007. 


1420 


.000704225 


5'.0006'^3»01 


15501.000645161 


1680 



BBGIPROCALS OF NUMBERS. 



86 



No. 



Redpro- 
coil. 



ISttJ .•00612745 
4' .000611996 
6' 0006ll«47 
R 000610300 

16i0{ .000609:96 
S. 000600018 
.000608272 



8 
16S0 



i|.oooeon»3 

.000606796 
^ .000606061 
ti. 0006058-27 
4i.0006015l» 
V .000609865 
8; .000808186 

16601.000602410 
2i. 000601685 
4.00n600W9 
6.000600MO 
8.0006995SO 

26^)1 .OOQSflWOf 
». 000698066 
4 .000697371 
e: .000596658 
8.00059504T 

16<» .000595i38 
ft. 000594530 
4).000988«M 
6i.O0OGO31id0 
8 .00(fi9d417| 

1680 .000601716 
8 .000601017; 
4.000680819 
6.0005806'S 
8.000688928 

1700.000688386 
8.00058iS44 
4.00(680854 



No. 



1706 
8 

1710 
12 
14 
18 
18 

IT* 
2 
4 
6 
8 

1780 
9 
4 
6 
8 

1740 
2 
4 
6 
8 

1790 
8 
4 
6 
8 

1760 
8 
4 
6 
8 

1770 
2 
4 
6 
8 



Recipro- 
cal. 



.000686166 
.0005854801 
.000584795 
.000584112 
.0005834801 
.000582750 
.0005820721 
.000581395 
.000580720 
.00(»80046 
.000579374 
.0005787t)4 
.0005780:^5 
.000577367, 
.000676701 
.0005760^7 
.0005759r4 
.000574713 
.000574avJ 
.000573394, 
.00ai727a7 

0005720Hi 
.000571429 
.000570776! 
.0005701»5 
.000569476 
.000568828 
.000568182 
.000667587 

000566898 
.000566251 

000565611 

000564334; 
,000563698 
.00056806;i 
.0005624301 



No. 



Reclpro- 



tecipi 



4' 
6 

8| 
1700 

4 
6 
8 
1800. 

2; 

4. 

6 

8 

1810 
12i 
14 
161 
18) 
1820 

2 

^! 
«; 

8l 
1830 
2 
4 
6 
8 

1840. 
2 
4 

1850| 
2 



No. 



000561798 1854 
000561167 I 6 

.0005605381 8 

.000659910. I860 

.000550284 

.000558659; 

.000558aS5 

.000557413 

.000556798 

.000556174 

.000555556 

.000554989 

.0005543:24; 

.OOa'^710 11880 

.000563097 

.000552486 

.000551876 

.000651268 

.00a'j50661 

.000550055: 

.000549451 

.000548848 

.000548240 

.000547645 
000547040! 

. 0005464 ]8| 

.000545851 

.000545259 

.000544002 

.000544069, 

.000543478] 

.00054-JH88I 

.000542299, _ 

.000541711! 1920 

.000541125 1 a 

.000540540 4 

.00058095711 6 



1870 
2 

4 
6 
8 



2 

4 

6 

8 

1890 

2 

4 

6 

8 

1900 

2 

4 

6 

8 

1910 

12 

14 

16 

18 



Recfpro- 



'^: 



No. 



.000539874 1928 

.000538793 1980 

.000588213!! 2 

.0005876841 

.000587057, 

.000586480; 

.000535905; 

.000535332 

.0006847591 

.000534188 

.000538618 

0005S8049 
.000582481) 
.000531915 
.000581350 

aKl530785' 
.000530222 
.000529661 
.000520100 
.0005:28541 
.000527983 
.000527426 
.000526870, 
.000526816 
.000525762 
.000525210 
.000624050, 
.000524109 
.000528560 
.000523012 
.000622466 
.000521990; 
.000521376 
.000520833, 

.ooa')202gi 

000519750, 



4 
6 
8 

1940 
9 
4 
6 
8 

1950 
2 
4 
6 
8 

1660 
2 
4 
6 
8 

1970 

4 

8 

8 
1980 
2 
4 
6 



1990 
2 
4 
6 

8 



000519211 1 2000 



Reel 



ar 



,000518678 
000518188 
,000517599 
.000517003 
.000516528 
.000515990 
.000515464 
.000514988 
.000514408 
.000513974 
.000513847 
.000512880 
.000512295 
.000511770 
.00tt51l247 
000510785 
.00U5102(»4 
.00U5O9684 
.000509166 
.000508647 
.000508180 
.0005071)14 
.000507099 
.000506585 
.000506078 
.000506661 
.000605051 
.000604541 
.000504082 



.000503018 
.000502^18 
.000502006 
.000501504 
.000501009 
.000500601 
000500000 



Use of reciprocals.— Reciprocals niay be conveniently used to facili- 
tate computations in longr division. Instead of dividing as usual, multiply 
ihe divideud by the reciprocal of the divisor. The method is especially 
useful whcMi many different dividends are required to be divided by the 
same divisor. In this caso find the reciprocal of the divisor, and make a 
small fable of iis multiples up to 9 times, and use tliis as a multiplication- 
table instead of actually performing the multiplication in each case. 

Example. —9671 and several other numbers are to be divided by 1688. The 
reciprocal of 1688 is .000610600. 
XolUp 



lolUtiles of the 
redprocAl: 
I. .0006106 
SL .0019910 
8. .0018816 

4. .O0S4420 

5. .0080525 

6. .0096680 

7. .0042786 

8. .0046640 

9. .0064045 
10. .0061060 



The table of multiples Is made by continuous addition 
of 6105. The tenth Ime Is written to check the accuracy 
of the addition, but it is not afterwards used. 
Operation: 

Dividend 9871 

Take from table 1 0006105 

7 0.0427:« 

8 00.48S40 

9 005.4945 



Quotient 6.02G9455 

Oorteet quotient by direct diviKion 6 .026251 5 

Tbe result will generally be correct to as many figures as tliere are signifi- 
cant figures in the reciprocal, less one, and the error of the next figure will In 
general not exceed one. In the above example the reciprocal has six aig^* 
niflcaat figures, 610600, and the result is correct to five places of figures. 



86 



HATHEHATICAL TABLES. 



MII7ARE8, CUBBS, S<|UARE ROOTS AflD CtJBB 
ROOTS OF MUniBKRS FROM .1 TO 1600. 



No. 


Square. 


Cube. 


8q. 
Boot 


Cube 
Root. 


No. 
8.1 


Square. 


Cube. 


• 

8q. 
Boot. 


Cube 
Boot. 


.1 


.01 


.001 


.8162 


.4642 


0.61 


29.791 


1.761 


1.4.58 


.15 


0225 


.0084 


.3878 


.5313 


.2 


10.24 


32.768 


1.789 


1.474 


.2 


.04 


.006 


.4472 


.5848 


.8 


10.89 


85.987 


1.817 


1.489 


.26 


.0625 


.0156 


.500 


.6300 


.4 


11.56 


89.804 


1.844 


1.604 


.3 


.09 


027 


.6477 


.6694 


.5 


12.25 


42 875 


1.871 


1.518 


.85 


.1225 


.0429 


.5916 


.7047 


.6 


12.96 


46.666 


1.897 


1.5SS 


.4 


.16 


.064 


.6825 


.7868 


.7 


18.69 


60.663 


1.924 


1.547 


.45 


.2025 


.0911 


.6708 


.7668 


.8 


14.44 


64.878 


1.949 


1.500 


.5 


.25 


.125 


.7071 


.7987 


.9 


16.81 


59.819 


1.975 


1.674 


.55 


.3025 


.1664 


.7416 


.8193 


4. 


16. 


64. 


8. 


1.5874 


.6 


.86 


.216 


.7746 


.8434 


.1 


16.81 


68.921 


8 085 


1.601 


.66 


.4225 


.2746 


.8062 


.8662 


.2 


17.61 


74.088 


2.049 


1.618 


.< 


.49 


.843 


.a367 


.8879 


.8 


18.49 


79.607 


8.074 


1.G26 


.75 


.56% 


.4219 


.8660 


.9086 


.4 


19.36 


86.184 


8.096 


1.680 


.8 


.64 


.512 


.8944 


.9283 


.5 


20.25 


91.186 


8.181 


1.651 


.85 


.7225 


.6141 


.9219 


.9478 


.6 


21.16 


97.836 


8.145 


1.668 


.0 


.81 


.729 


.M«7 


.9655 


.7 


22.09 


108.828 


2.168 


1,675 


.96 


.9025 


.8574 


.9747 


.9830 


.8 


23.04 


110.602 


2.101 


1.687 


1. 


1. 


1. 


1. 


1. 


.9 


24.01 


117.649 


2.814 


1.006 


1.05 


1.1025 


1.158 


1.025 


1.016 


6. 


25. 


126. 


8.8861 


1.7100 


1.1 


1.21 


1.881 


1.049 


1.032 


.1 


26.01 


182.661 


8.868 


1.721. 


1.15 


1.8225 


1.521 


1.072 


1.0-18 


.2 


27.04 


140.606 


8.280 


1.788 


1.2 


1.44 


i.-na 


1.095 


1.063 


.8 


28.09 


148.877 


2.802 


1.744 


1.25 


1.56^5 


1.953 


1.118 


1.077 


.4 


29.16 


167.464 


8.884 


1.754 


1.8 


1.69 


2.197 


1.140 


1.091 


.5 


30.25 


166.875 


8.846 


1.765 


1.85 


1.8225 


2.460 


1,162 


1.105 


.6 


81.36 


175.616 


8.366 


1.776 


1.4 


1.96 


2.744 


1.183 


1.119 


.7 


82.49 


185.193 


8.387 


1.786 


1.45 


2.1025 


8.049 


1.204 


1.182 


.8 


83.64 


195.118 


2.406 


1.797 


1.5 


2.25 


8.375 


1.2247 


1.1447 


.9 


34.81 


205.379 


2.429 


1.807 


1.55 


2.4025 


8.721 


1.245 


1.157 


6. 


86. 


216. 


2.4495 


1.8171 


1.6 


2. .56 


4.096 


1.265 


1.170 


.1 


87.21 


226.981 


2.470 


1.827 


l.Oo 


2.7225 


4.492 


1.285 


1.182 


.2 


88.44 


2:i8.328 


2.490 


1.837 


l.T 


2.89 


4.918 


1.804 


1.193 


.8 


39.69 


250.047 


2.510 


1.847 


1.75 


3.0625 


5.359 


1.323 


1.205 


.4 


40.96 


268.144 


2.5.30 


1.867 


1.8 


8.24 


5.832 


1.842 


1.216 


.5 


42.25 


274.625 


2.660 


1.866 


1.85 


3.4225 


6.832 


1.360 


1.228 


.6 


48.56 


287.496 


2.569 


1.876 


1.9 


3.61 


6.&19 


1.878 


1.239 


y 


44.89 


800.763 


2.688 


1.885 


1.05 


8.8025 


7.415 


1.396 


1.249 


.'8 


46.24 


314 482 


2.608 


1.895 


S. 


4. 


8. 


1.4142 


1.2599 


.9 


47.61 


328.509 


2 627 


1.004 


.1 


4.41 


9.261 


t449 


1.281 


7. 


49. 


848. 


2.6458 


1.9129 


.2 


4.84 


10 648 


1.483 


1.801 


.1 


50.41 


857.911 


2.665 


1.928 


.8 


5.29 


12 167 


1.617 


1.3J0 


.2 


51.84 


378.248 


2.688 


1.981 


.4 


5.76 


18.8','4 


1.549 


1.839 


.3 


53.29 


389.017 


2.708 


1.940 


.5 


6.25 


15.625 


1.581 


1.357 


.4 


54.76 


405.224 


2.780 


1.949 


.0 


6.76 


17.576 


1.612 


1.375 


.5 


56.25 


421.876 


2.789 


1.967 


.7 


7.29 


19.083 


1.643 


1.392 


.6 


57.76 


438.976 


8.757 


1.966 


.8 


7.84 


21.952 


1.673 


1.400 


.7 


59.29 


456.533 


8.775 


1.975 


.9 


8.41 


24.889 


1.703 


1.426 


.8 


60.84 


474.652 


2.793 


1.983 


3. 


9. 


27. 


1.7321 


1.4422 


.9 


62.41 


498.080 


2 811 


1.998 



SQUABES, CUBES, SQUASB AND GCBE BOOTS. 87 



No. 


Square. 


Cube. 


Sq. 


Cube 
Boot 


No. 


Square. 


Cube. 


Bq. 
Root. 


Cube 
Root. 


8. 


G4. 


618. 


2.8884 


8. 


45 


20SS 


91185 


6.7088 


8.5569 


.1 


65.81 


581.441 


8.846 


8 008 


46 


^ 


97886 


6.7883 


3.5830 


.« 


S:IS 


551.868 


8.864 


8.017 


47 


103888 


6.8557 


8.6068 


.8 


571.787 


8.881 


8 085 


48 


8804 


110598 


0.9888 


3.6348 


.4 


70.56 


6O0.7O4 


8.808 


8.038 


49 


8401 


117649 


7. 


8.6598 


.5 


72.85 


614.185 


8.015 


8.041 


50 


8500 


185000 


7.0711 


8.6840 


.6 ' 73^ 


686.056 


8.938 


8.040 


51 


8601 


138651 


7.1414 


3.7084 


.7 


75.0d 


666.508 


8.950 


8.057 


58 


8704 


140606 


7 8111 


3.7885 


.8 


77.44 


681.478 


8.966 


8.065 


53 


2809 


148877 


7.2801 


3.7B63. 


.9 


7!9.«1 


701.969 


2.968 


8.078 


54 


8916 


167464 


7.3485 


3.7798 


t. 


81. 


780. 


8. 


8.0601 


55 


8085 


166875 


7.4168 


3.8030 


.1 


».81 


758.571 


8.017 


8.088 


56 


8186 


175616 


7.4883 


8.8859 


i 


81.64 


778.688 


8.088 


8.095 


57 


3849 


185193 


7.5498 


8.S485 


.8 


86.40 


804.857 


8.050 


8.108 


58 


8364 


ia51I8 


7.6158 


8.8700 


.4 


88.96 


880.564 


8.066 


8.110 


69 


3481 


905879 


7.6811 


8.8930 


.5 


S0.85 


857.875 


8.089 


8.118 


60 


3600 


816000 


7.7460 


3.9149 


.6 


flS.16 


881.786 


3.096 


8.126 


61 


8781 


286981 


7.8108 


8.9865 


.7 


M.Od 


918.678 


3.114 


8.188 


08 


8844 


288888 


7.8740 


8.9579 


.8 


96.M 


•41.108 


8.130 


8.140 


63 


8969 


260047 


7.9378 


3.9791 


.9 


98.01 


Sy70.890 


8.146 


8.147 


64 


4096 


868144 


8. 


4. 


10 


100 


1000 


3.1683 


8.1544 


66 


4885 


274685 


8.0623 


4.0307 


11 


121 


1381 


3.3166 


8.2240 


66 


4856 


287496 


8.1340 


4.0418 


w 


144 


1783 


3.4641 


8.2894 


67 


4489 


800768 


8.1854 


4.0615 


13 


160 


8197 


3.6056 


8.3513 


68 


4684 


814438 


8.2462 


4.0817 


14 


196 


8744 


3.7417 


8.4101 


69 


4761 


S-J8509 


8.3066 


4.1016 


15 


»5 


8875 


3.8780 


8.4668 


70 


4900 


348000 


8.8666 


4.1818 


16 


956 


4096 


4. 


8.5198 


71 


5041 


3.57911 


8.4861 


4.1408 


17 


889 


4918 


4.1831 


8.5713 


72 


5184 


373848 


8.4858 


4.1608 


18 


884 


5888 


4.3486 


8.6907 


73 


5389 


389017 


8.5440 


4.1798 


19 


861 


60B9 


4.3589 


8.6684 


74 


5476 


405284 


8.6083 


4.1983 


90 


400 


8000 


4.4731 


8.7144 


75 


5635 


481875 


8.6603 


4 2172 


21 


441 


0961 


4.5686 


8.7689 


76 


6776 


438976 


8 7178 


4.8358 


Si 


484 


10048 


4.6904 


8.8080 


77 


6039 


456533 


8.7760 


4.8543 


SS 


m 


18167 


4.7968 


8.8439 


78 


6064 


474558 


8.8318 


4.2787 


24 


576 


13894 


4.8990 


8.8845 


70 


6841 


498039 


8.8888 


4.8908 


O 


095 


15685 


5. 


8.9840 


80 


6400 


512000 


8.9443 


4.3089 


a 


676 


17576 


5.0990 


8.9685 


81 


6561 


531441 


9. 


4.3867 


97 7» 


19688 


5.1968 


3. 


88 


6724 


551368 


9 05.54 


4.3(45 


88 784 


8108S 


5.8915 


3 0866 


88 


0880 


571787 


9.1104 


4.3681 


29 


841 


94889 


5.3858 


3.0788 


84 


7056 


59-^704 


9.1658 


4.3795 


10 


900 


87000 


5.4778 


3.1072 


85 


7285 


614125 


0.8195 


4.3968 


81 


961 


89791 


5.5678 


3.1414 


86 


7896 


636056 


9.8736 


4.4140 


82 


1QS4 


88768 


5.6560 


3.1748 


87 


7.'>69 


65&'n8 


9 3-.T(5 


^.4^J10 


83 


1089 


38987 


5.7446 


3.8075 


88 


7744 


0^1478 


9.3H08 


4.4480 


SI 


1156 


89804 


5.8810 


8.8886 


80 


7981 


704969 


0.4M0 


4.4647 


& 


1285 


48875 


5.9161 


3.2711 


90 


8100 


739000 


9.4868 


4.4814 


88 1996 


46660 


6. 


8.8010 


91 


8881 


7Jmn 


9.5394 


4.4979 


37 1809 


50668 


6.0888 


3.8388 


98 


8464 


778688 


.5917 


4.5144 


88 


1444 


64879 


6.1644 


8.8690 


93 


8649 


804357 


9 6437 


4.5307 


39 


1581 


69319 


6.8450 


3.8918 


94 


8836 


830584 


9.6954 


4.5468 


40 


1600 


64000 


6.8848 


3.4900 


95 


9025 




9 7466 


4.5680 


41 


1081 


68»»1 


6.4081 


3.4488 


96 


9216 


884736 


9.7980 


4.5780 


48 


1764 


74088 


6.4807 


3.4700 


97 


9409 


918678 


9.84K0 


4.5947 


4S 


1840 


79607 


6.5574 


3.5034 


9H 


9604 


941198 


9.8995 


4.6104 


44 


1986 


851M 


6.6888 


3.5808 


99 


9801 


970899 


9.9499 


4.6861 



88 



HATHEUATIOAL TABLES. 



No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 
Root. 


Cube 
Root. 


100 


10000 


1000000 


10. 


4.6416 


155 


24025 


8728875 


12.4499 


58717 


101 


ioa)i 


IU.30801 


10.0199 
10.0905 


4.6570 


156 


243:i6 


8796416 


12.4900 


5.3832 


10;! 


10404 


1061208 


4.6728 


157 


24649 


3869893 


12.5800 
12.660i 


5.3947 


108 


10609 


1092727 


10.1489 


4.6875 


158 


24964 


8944313 


6.4061 


104 


10816 


1124864 


10.1980 


4.7027 


159 


25281 


401967V 


12.6005 


6.4175 


106 


1102S 


1157626 


10.2470 


4.7177 


160 


25600 


4096000 


18.M91 


6.4288 


106 


112 6 


1191016 


10.2956 


4.7326 


161 


25921 


4178261 


12.6686 


6.4401 


107 


11449 


1225043 


10.3441 


4.7476 


162 


26244 


4251528 


12.7879 


5 4514 


108 


11664 


1259712 


10.3923 


4.7622 


163 


20569 


4380747 


12.7671 


5.4«26 


109 


11881 


1295029 


10.4403 


4.7769 


164 


26896 


4410044 


12.6062 


5.4737 


110 


12100 


1831000 


10.4881 


4.7914 


166 


27225 


4492185 


12.8452 


5.4848 


111 


ia8;Jl 


1367631 


10.6857 


4.8050 


166 


27556 


4574296 


12.8841 


6.4959 


112 


12514 


1404928 


10.5880 


4.8203 


167 


27889 


4657463 


12.9228 


5.5069 


118 


12709 


1442897 


10.6301 


4.8S46 


108 


28v»24 


4741682 


12.9615 


5.5178 


114 


12996 


1481544 


10.6771 


4.8488 


169 


28561 


4826800 


18.0000 


5.5288 


115 


132« 


1520676 


10.7288 


4.6629 


170 


28900 


4918000 


18.0884 


5.53fR' 


116 


13456 


1660896 


10.7708 


4.8770 


171 


29241 


6000211 


18.0767 


5.5505 


117 


18689 


1601613 


10.8167 


4.8910 


172 


29584 


6088448 


13.1149 


5.5613 


118 


18924 


1613032 


10.8628 


4.9049 


173 


29929 


5177717 


18.1529 


5.5721 


110 


14161 


1685159 


10.9087 


4.9187 


174 


80276 


5268024 


13.1909 


5.5886 


120 


14400 


1728000 


10.9545 


4.9324 


175 


80625 


5859375 


18.2288 


5.5934 


121 


14641 


1771561 


11.0000 


4.9461 


176 


80976 


6451778 


13.2665 


5.6041 


li^ 


14884 


1815848 


11.0454 


4.9597 


177 


31329 


5545238 


13.8041 


6.6147 


123 


15129 


1800867 


11.090) 


4.9732 


178 


31684 


66:«758 


18.8417 


6.6253 


m 


15876 


1906624 


11.1355 


4.9866 


179 


&2011 


5735889 


13.8791 


5.6357 


125 


15025 


1953125 


11.1808 


5.0000 


180 


32400 


6833000 


13.4164 


5.6468 


190 


15876 


2000876 


11.2250 


5.0138 


181 


32761 


5929741 


13.4.')36 


6.6567 


1« 


16129 


•.•018883 


11.2694 


5 0265 


182 


38124 


602^566 


13.4907 


66671 


128 


16884 


2097152 


11.8187 


5.0897 


183 


88489 


6128487 


13.5277 


6.6774 


120 


16641 


2146660 


11.3578 


5.0528 


184 


83856 


6229504 


13.5647 


6.6«77 


180 


16900 


2197000 


11.4018 


6.0658 


186 


342-25 


6831685 


13.6015 


6.6060 


ISI 


17161 


2248091 


11.4455 


5.0788 


186 


84596 


6434856 


13.&S82 


6.7088 


laz 


17424 


229'J968 


11.4891 


5.0916 


187 


34960 


6639208 


13.6748 


6.7165 


188 


17689 


2352637 


n. 5-^26 


*.1045 


188 


35344 


6644672 


18.7113 


5.r287 


184 


17956 


2406104 


11.5758 


5.1172 


189 


85?21 


6751269 


18.7477 


5.7388 


185 


18226 


2460375 


11.6190 


6.1299 


190 


86100 


6850000 


13.7840 


6.7469 


136 


18496 


2515456 


11.6619 


5.1486 


101 


86481 


6967871 


13.8203 


5.7590 


187 


18769 


2571353 


11.7047 


5.1551 


192 


36864 


7077888 


13.8564 


5.7690 


188 


190J4 


•J628072 


11.7473 


5.1676 


198 


37249 


7189a'57 


13.8924 


6 7790 


139 


19321 


'J68otil9 


11.7893 


5.1801 


194 


37636 


7301384 


13.9284 


5.7890 


140 


19600 


2744000 


11.a332 


5.1926 


195 


38025 


7414875 


18.9642 


6.7989 


141 


19881 


2803221 


11.8743 


5.2048 


196 


38116 


76295^16 


14.0000 


5.8068 


14*2 


20164 


2863288 


11.9164 


5.2171 


197 


JJ8809 


7615378 


14.0357 


5.8188 


143 


W449 


29i4207 


11.9583 


5.2293 


196 


39204 


7762392 


14.0712 


5.8286 


144 vH)r3U 


2985984 


12.0000 


5.2425 


199 


89601 


7880599 


14.1067 


5.8383 


145 


21085 


3048625 


12.0416 


5.2536 


200 


40000 


6000000 


14.1421 


B.84S0 


146 


21316 


8112136 


12.08.W 


5.2656 


201 


40401 


8120601 


14.1774 


5.8578 


147 


21609 


3170523 


12.1244 


5.2776 


202 


40804 


8242406 


14.2127 


6.867B 


148 


21M)4 


3241792 


12.16."« 


5.2896 


203 


41209 


ft365427 


14.2478 


5.8771 


149 


22201 


3307949 


12.2066 


5.8015 


204 


41016 


8489064 


14.2829 


6.8808 


ISO 


22500 


3375000 


12.2474 


5.8133 


205 


42025 


6615125 


14.3178 


6.6964 


151 


2-2801 


3442051 


12.2H82| 5.1251 


206 


42436 


6741816 


14.3527 


69069 


152 


23104 


3511808 


12.328.**, 5,8368 


2t>7 


42849 


8860743 


14.8875 


6.9166 


168 


23409 


3681577 


12.36'.<3i 5.3485 


208 


48264 


8998913 


14.4288 


6.98fiO 


154 


23716 


3652264 


12.40971 5.8601 


209 


48681 


9129339 


14.4566 


6.9846 



SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 89 



Ko. 

«10 


Square. 


Cube. 


&. 


Cube 
Root. 


No. 


Square. 


Cube. 


Sq. 


Cube 
Root. 


44100 


9961000 


14.4914 


5.9489 


865 


70886 


18609086 


16.8788 


64888 


SU ' 44»ei i 


9893981 


14.6866 


5.9588 


866 


707A6 


18881096 


16.8095 


6.4818 


itt 


44014 


06S8138 


14.5608 


5.9687 


807 


71889 


19084168 


16.*401 


6.4398 


»a 


46809 


9668597 


14.5945 


5.9?81 


868 


71884 


19848838 


16.8707 


6.4478 


814 


45700 


9800844 


14.0887 


5.9814 


800 


?2861 


19466100 


16.4018 


6.4568 


SIS 


40989 


9988870 


14.6089 


59907 


«70 


78900 


10688000 


16 4817 


6.4688 


SIO 40666 


loonooo 


14.6960 


6.0000 


871 


73441 


100086U 


16.4681 


6.4718 


tn 470B0 


10«18818 


14.7809 


6 009-^ 


878 


73984 


80188648 


16.4984 


6.4798 


«1H 


47924 


1086098;] 


14.7648 


6.0185 


873 


74589 


20^6417 


16,5287 


6.4878 


S19 


47061 


10B00450 


14.7986 


6027T 


874 


76076 


80570834 


16.5589 


6.4961 


2» 


48400 


10648000 


14.8884 


6.0866 


875 


75685 


80706876 


16.6881 


6.6080 


sa 


48841 


10793801 


I4.866t 


6.0459 


876 


76176 


81084576 


16.6182 


6.5108 


ae ' 4ne4 


10941048 


14.8997 


60560 


877 


70'i'89 


81858088 


16.6438 


6.5187 


ea 


497SI9 


11089567 


14.03:i8 


6.0641 


878 


77884 


21484052 


16.6733 


6.5866 


SM 


60176 


118891i4 


14.9666 


6.0788 


879 


77841 


81717680 


16.7088 


6.6348 


tta 


50085 


11890885 


15.0000 


6.0888 


880 


78400 


81068000 


16.7882 


6.5481 


Stf 


51076 


11548176 


15.0888 


6.0918 


881 


78961 


82188041 


16.7681 


6.r>499 


an 


515S9 


11697088 


15.0666 


6.100;> 


.J88 


79584 


224«5768 


16.7989 


6.5677 


«e 


51084 


11868868 


15.0997 


6.1091 


888 


80039 


28066187 


16.8886 


6.6664 


XO SM41 


18008989 


15.1887 


6.1180 


884 


80666 


28006304 


16.8683 


6.5781 


tso 


SSOOO 


18167000 


15.1696 


6.186P 


886 


81886 


88140186 


16.8819 


6.6808 


»1 


53861 


I88»Sd91 


15.1987 


6.1856 


286 


81796 


8:^08656 


16.9115 


6.5886 


i8i 


53884 


19487168 


15.8815 


6.1446 


887 


S8869 


23680008 


16.9411 


6.5968 


8» 


51889 


18649887 


15.8648 


6.15^ 


888 


88944 


23887878 


16.9706 


6.60B9 


ai 


54796 


1«1^9904 


15.8971 


6Am 


880 


88581 


84137560 


17.0000 


6.6116 


S95 


56889 


18977875 


15.8897 


6.1710 


890 


84100 


84880000 


17.0894 


6.6191 


8» 


50606 


1S144866 


15.8688 


6.179? 


8tfl 


84681 


24642171 


17.0587 


6.6967 


»7 


56169 


18818058 


15 3948 


6.1885 


898 


85864 


24807088 


17.0880 


6.6848 


»8 


56644 


14461878 


15.4178 


6.197a 


808 


85849 


2516:3757 


17.1179 


6.6419 


SW 


571«1 


18651919 


15.4596 


6.8058 


894 


86436 


25418184 


17.1464 


6.6494 


MO 


67600 


18884000 


15.4919 


6.8145 


895 


87025 


2667^875 


17.1766 


6.6600 


«1 


56081 


18097681 


15.6848 


6.8831 


81i6 


87016 


25934336 


17.2047 


6.6644 


S41 


58564 


14178488 


15.6668 


6.8817 


897 


88209 


26198073 


17.8:337 


6.6710 


M3 


50040 


14848907 


15.5885 


6.»4OT 


298 


88804 


264G3592 


17.2687 


6.6794 


m 


58086 


14586784 


15.6803 


6.8488 


899 


89101 


26730899 


17.8916 


6.6869 


i» 


60085 


14706185 


15.6685 


6.8573 


800 


90000 


-27000000 


17.8806 


6.6948 


M 


00016 


14886086 


15.6814 


6.8658 


;301 


90601 


27270901 


17.3494 


6.7018 


247 


61009 


15009488 


15.7168 


6.8743 


;«B 


91804 


2764360S 


17.3781 


6.709B 


:;44 


C1604 


16858998 


15.7480 


6.8888 


:i03 


91800 


27818127 


17.4069 


6.7166 


»19 


eeooi 


15488M9 


15.7797 


6.8912 


304 


08416 


28094464 


17 4866 


6.7840 


SO 


02500 


15696000 


15.8114 


6.8996 


305 


98085 


28372685 


17.4648 


6.7318 


»1 : 64001 


15818«1 


16.8480 


6.8060 


306 


93636 


28658616 


17.4929 


6.7387 


« 


6S604 


16008008 


15.8745 


6 81(^4 


307 


04849 


28934443 


17.5214 


6.7460 


ss 


04009 


16194 <77 


16.9060 


6.a»47 


306 


94864 


29218112 


17.6499 


0.7538 


»i 


54516 


16887064 


15.9074 


6.8880 


309 


95481 


29503629 


17.5784 


6.7606 


» 


68005 


16681875 


15.9687 


6.8413 


310 


96100 


29791000 


17.6068 


6.7670 


a6 


66080 


16777816 


16.0000 


6.3496 


311 


96781 


800602:)1 


17.6:352 


6 7768 


»7 


66049 


16974508 


16.0812 


6.3579 


313 


97314 


30371328 


17.6635 


6.7884 


2» 


66664 


17178613 


16.06-24 


6.8661 


313 


97960 


30664^97 


17.6918 


6.7897 


2S0 1 67061 


17878979 


16.0985 


6.3743 


314 


98596 


30959144 


17.7800 


6.7969 


SGO 


67000 


17576000 


16.1845 


6 8385 


315 


99225 


81255875 


17.7482 


6.8041 


»1 


OKlil 


17779581 


16.1555 


6.3007 


316 


i«)S5C 


31554496 


17.7764 


6.8118 


»» 


68644 


17964788 


16.1864 


6.89«8 


■in 100489 


8185.5013 


17.8045 


6.8186 


»3 


eoi60 


18191447 


16.8178 


6.4070 


318 1101134 


32157432 


17.83-,»G 


6.8256 


asi 


60606 


18889744 


16.9461 


6.4151 


319 1101761 


32461759 


17.8606 


6.8:388 



90 



VATREUATICAL TABLES. 



No. 


Square. 


Cube. 


8q. 
Root. 


Cube 
Root. 


No. 


Square. 


Cube. 


Root. 


Cube 
Root. 


8?0 


102400 


82768000 


17.8886 


6.8800 


•375 


140625 


58784875 


19.8610 


7.2118 


321 


108041 


88076161 


17.9165 


6.8470 


876 


141376 


53167876 


19.8007 


7.2177 


92i 


103C84 


33386248 


17.9444 


6.8641 


877 


142129 


58582688 


19.4166 


7.2240 


S^ 


104329 


38696267 


17.9722 


6.8612 


378 


142884 


54010158 


19.4422 


7.2804 


8^ 


104976 


34012224 


18.0000 


6.8683 


879 


148641 


64489980 


19.4670 


7.2868 


8:25 


105625 


84888125 


18.0278 


6.8768 


880 


144400 


54872000 


10.4986 


7.8482 


8S6 


106276 


84645076 


18.0556 


6.8824 


381 


146161 


65806341 


10.5102 


7.2405 


937 


106929 


84965788 


18.0631 


6.8804 


882 


146924 


55748008 


10.5448 


7.2558 


328 


107584 


85287552 


18.1108 


6.8064 


888 


146680 


66181887 


10.5704 


7.2622 


829 


108241 


85611289 


18.1884 


6.0084 


384 


147456 


50683104 


10.5050 


7.8685 


830 


108900 


35987000 


18.1659 


6.0104 


886 


148225 


67066025 


10.6814 


7.2748 


881 


109561 


86264691 


18.1034 


6.0174 


886 


148006 


57512456 


10.6460 


7.2811 


832 


110224 


86594368 


18.2209 


6.0244 


387 


140760 


57060608 


10.6«8 


7.2874 


883 


110689 


80926087 


18.2488 


6.0318 


388 


160644 


68411078 


10.6977 


7.2936 


884 


111556 


37259:04 


18.2757 


6.0382 


889 


151821 


58868860 


19.7281 


7.2009 


885 


112225 


87595875 


18.3080 


6.0451 


890 


162100 


50810000 


19.7484 


7 8061 


830 


112806 


37983056 


18.3303 


6.9521 


891 


162881 


60776471 


10.7787 


7.8184 


887 


118569 
114214 


38272753 


18.8576 


6.9589 


892 


158604 


60836288 


10.7990 


7. .31 88 


888 


38614472 


18.8848 


6.9658 


898 


154440 


60608467 


19.8848 


7.8248 


880 


114921 


88958219 


18.4120 


6.97JJ7 


894 


155286 


61168064 


19.8404 


7.3310 


840 


115600 


89304000 


18.4891 


6.9795 


305 


156025 


61620875 


19.8746 


7.8872 


841 


116281 


89661821 


18.4602 


6.9864 


306 


156816 


68009136 


19.8907 


7.8434 


842 


116964 


40001688 


18.4932 


6 9982 


307 


1.57600 


62670778 


10.0240 


7..3496 


843 


117649 


40358607 


18.5203 


7.0000 


396 


158404 


68044792 


10.9400 


7.8568 


344 


118836 


40707684 


18.5472 


7.0068 


809 


150001 


68581100 


10.9760 


7.8619 


845 


11902S 


41063625 


18.5742 


7.0186 


400 


160000 


64000000 


20.0000 


7.8681 


846 


119716 


41421786 


18.6011 


7.0208 


401 


160801 


64481201 


20 02S(^ 


7.8742 


847 


120409 


41781923 


18.6279 


7.oen 


402 


161604 


64064806 


20.0499 


7. .3803 


848 


121104 


42144192 


18.6548 


7.0338 


408 


162400 


65450S27 


20 0749 


7.8864 


849 


121801 


42506549 


18.6815 


7.0406 


404 


168216 


65989264 


20.0908 


7.8925 


860 


122500 


42875000 


187083 


7.0478 


406 


164085 


66430125 


20.1846 


7.8086 


851 


123201 


43248561 


18.7850 


7.0540 


406 


164886 


6692.3416 


80.1404 


7.4047 


852 


123904 


43614208 


18.7617 


7.0607 


407 


166640 


67419148 


20.1748 


7.4108 


853 


124609 


43986977 


18.7883 


7.0674 


408 


166464 


67917812 


20.1990 


7.4169 


854 


125316 


44361864 


18.8149 


7.0740 


409 


167881 


68417929 


2Q.8287 


7.4229 


855 


126025 


44788876 


18.8414 


7.0607 


410 


108100 


66921000 


20.8486 


7.4290 


8-)6 


126786 


45118016 


18.8680 


7.0873 


411 


168021 


69426681 


80.8731 


7.4850 


857 


127449 


45499293 


18.8944 


7.0940 


412 


160744 


69934528 


80.2976 


7.4410 


858 


128164 


45H82712 


18 9209 


7.1006 


413 


170560 


70444997 


80.8284 


7.4470 


859 


128881 


46268279 


18.9473 


7.1072 


414 


171896 


70967944 


80.8470 


7.468t) 


360 


129600 


46666000 


18.9737 


7.11.38 


416 


172225 




80.8715 


7.4690 


861 


130321 


47045881 


19.0000 


7.1204 


416 


178056 


71991296 


20.8961 


7.4650 


362 


181044 


47437«28 


19.0263 


7.1269 


417 


173889 


72511713 


80.4206 


7.4710 


863 


131769 


47832147 


10.0626 


7.1385 


418 


17472* 


73084688 


80.4450 


7.4rro 


864 


13-^496 


48228544 


19.0788 


7.1400 


419 


175661 


78560059 


20.4605 


7.4829 


365 


183225 


48627125 


19.1060 


7.1460 


420 


176400 


74068000 


80.4080 


7.4889 


866 


133956 


49027896 


19.1311 


7.1,581 


421 


177241 


74618461 


20.6188 


7.4948 


867 


134689 


49430863 


19.1572 


7.1596 


422 


178084 


75151448 


80.5426 


7.5007 


868 


1354-.'4 


49836032 


19.1833 


7.1661 


423 


178929 


7.5686967 


20.6670 


7.5067 


869 


130161 


50243409 


19.2094 


7.1726 


424 


179776 


76225024 


80.6018 


7.5126 


870 


130900 


60658000 


19.2354 


7.1T91 


425 


180625 


7676.5625 


20.61.55 


7.5185 


871 


187041 


51064811 


19.2614 


7. law 


426 


181476 


77308776 


20.6896 


7.5244 


872 


138384 


5I4r884sS 


19.2873 


7.1920 


427 


182329 


778,5448:1 


20.6640 


7.5302 


:-73 


189129 


51895117 


19.3132 


7.1984 


428 


188184 


78402752 


20 6888 


7.5.^1 


874 


139876 


62313624 


19.3391 


7.2018 


429 


184041 


78953589 


80.7188 


7.6420 



SQUASES, 0UBS8^ SQUABS AJXD CUBE SOOTS. 91 



No. 



Square. 



18KV1 
180ftM 
187480 
188896 



«4 



441 
4«i 
4a 
441 

415 

446 
447 
448 



490 

4SI aOMOl 

4M ! 804804 

493 

494 806110 



101844 
1W781 

196000 
194481 
190804 
196849 
197180 



198910 
199809 

ooonM 

801001 



430 
496 

497 

<99 



461 
403 



407 
408 



470 
471 
473 
478 
474 

475 
470 
477 
478 
479 



481 
482 



404 



807020 
807988 
806840 
8D0;04 

210061 

911000 
81481 
813444 

814860 
SlSfiVO 

810889 
817180 



219084 

819901 



821841 

2887B4 



tMOTO 



226976 



828484 
289441 



280400 
831861 



284896 



Cube. 



TKOrooo 

ooooeooi 

80021968 
81182787 
61746004 



6S40845S 

81087673 
84001510 

86184000 
85766181 



86988807 
87888881 

88181185 
S8716986 
89814688 
80015898 
90918U9 

91199000 

91788891 
9:2845108 
9-2998677 
88570064 

94196875 
94818B16 



06071918 
90709979 



onaoooo 

97938181 
98011128 



99887811 

100544ee5 
101194090 
101647068 
108508888 
108101709 

108898000 
104487111 
109154048 
109ae8817 
10049M24 

107171875 
107800170 
1065SM88 
109816808 
108908289 

110608000 
111984641 
111980108 



118879904 



Sq. 
Boot. 



90.7804 
20.7805 
:M.7840 
80.8087 
«).8887 

80.8007 
80.8806 
90.9015 
80.9tt4 
80.9088 

90.0768 
81.0000 
81.0888 
81.0476 
81.0718 

81.0960 
81.1187 
81.1484 
21.1660 
81.1696 

91 .8182 

81.8868 

81.8606 

81.81 

81.8073 

81.8807 
81.3948 
81.8776 
81. « 
81.4818 

31.4476 
81.4709 
81.4948 
81.5174 
81.5407 

81.5089 

81.9870 

81.6108 

81.6 

81.0604 

81.6795 
81.7025 
81.7856 
81.7486 
31.7715 

21.7945 
81.8174 
21.8406 
21.8688 
21.8661 

81.9080 
21.9617 
21.0645 
81.9778 
«2.0000 



Cube 
Boot. 



7.5478 
7.5587 
7.9595 
7.5664 
7.5718 

7.5770 
7.58S» 
7.9886 
7.5044 
7.0001 

7.6090 
7.6117 
7.6174 
7.6832 
7.0889 

7.6846 
7.6408 
7.6460 
7.6917 
7.6974 

7.0681 
7.0688 
7.6744 
7.0800 
7.0837 

7.6914 
7.0970 
7.7026 
7.7082 
7.7188 

7.7194 
7.7850 
7.7806 
7.7863 

7.7418 

7.7478 
7.7530 
7.7984 
7.7689 
7.7695 

7.7780 
7.7806 
8.7800 
7.7915 
7.7970 

7.8089 
7.8079 
7.8184 
7.8188 
7.8846 

7.8897 
7.8852 
7.8406 
7.8460 
7.8914 



No. 



485 
460 
487 
488 
480 

490 
491 
408 
498 
494 

499 
496 
497 
498 
499 

600 
901 
608 
508 
504 

905 
506 
507 
908 
509 

910 
911 
913 
518 
614 

915 

616 
517 
518 
519 

530 
681 
528 
988 
584 

596 
586 
587 
688 
939 

980 
981 
982 
988 
984 

986 
986 
587 
588 
589 



Square. 



886196 
337160 
838144 
889131 



Cube. 



114064139 
114791396 
119601308 
116214273 
116880160 



940100 117649000 
341081 118370771 
342064 '119009488 
843049 !n9628l67 
944066 180598784 



349005 
346016 
947000 



121287875 
132038936 
133708473 



M8004 : 123909993 
949001 134391499 



390000 

351001 
392004 
858009 



135000000 

139791901 
126506006 
187268527 



394016 138034064 



399085 
256086 
267040 

858064 

£>ya8i 

360100 
261131 
262144 
363169 
2(M196 



366896 
867289 
368324 
369361 

270400 
371441 
372484 
373939 
374976 

275685 
276676 
377739 
278784 
279841 

280900 
281961 



384069 
389196 



387396 



289444 
300521 



138787889 
129»43IC 
130823813 
131096512 
131»72:tt9 

133661000 
133432831 
184217728 
1:^5005697 
135796744 

! 186600875 
,187388096 
138188418 
1188991882 
I 189798859 

' 140608000 
141420761 
142286648 
143055667 
|1438778-,^ 

^44708138 
146531576 
146363183 
147197952 
148085868 

'148877000 
149731291 
150568768 
151419487 
152278304 



1153130875 
153990656 
154854153 
.155720873 28.1948 
'15C590619 .23.3164 



Sq. 
Root. 



22.0837 
32.0454 
22.0681 
32.0907 
22.1183 

33.1869 
22.1586 
22.1811 
83.3086 
32.3261 

33.3480 
22.2711 
22.2985 
22.3159 
22 8868 

33.8607 
32.1»30 
22.4054 
32.4277 
22.4499 

32.4732 
22.4944 
22.5167 
32.5889 
3-^.9610 

32.5832 
22.6053 
23.6274 
2 i. 6195 
32.6716 



32.7156 
2i.7876 
22.7590 
22.7816 

32 8035 
22.8254 

22.8473 
22.8692 
23.8910 

23.9129 
23 931' 
22.9665 
22.9783 
33.0000 

28.0317 
23.0434 
23.0651 
28.0868 
23.1064 



38.180 

28.1517 

23.1733 



Cube 
Root 



7.8568 
7.8683 
7.8676 
7 8780 
7.8re4 

7.8887 
7.8891 
7.6944 
7.8996 
7.9051 

7.9105 
7.9158 
7.9211 
7.9264 
7.9317 

7.9870 
7.9438 

7.9476 
7.9528 
7.9581 

7.9684 
7.9686 
7.9789 
7.9791 
7.9848 

7.9896 
7.9948 
8.0000 
8.0053 
8.0104 

8.0166 
8.0208 
8.0260 
8.0311 
8.U3G3 

8.0415 
8.0166 
8.0517 
8.0:>69 
8.0620 

8.0671 
8.0728 
H.0774 
8.0835 
8.0676 

8.0987 
8.0978 
8.1028 
8.1079 
8.1130 

8.1180 
8.1231 
8.1281 
8.1&32 
8.1382 



92 



MATHEMATICAL TABLC8. 



Square. 



291600 



896764 
994849 
995986 

897025 
298116 



800804 
801401 

803500 
808601 
804704 
805809 
806916 

806085 
809186 
810249 
811864 
312481 

818600 
314721 
815844 
810909 
818096 

819225 
320856 
321489 



383761 

324900 
820041 
327184 
328329 
829476 



331776 
332929 
334084 
335241 

886400 
837561 
338724 



311056 

342225 
843396 
344569 
345744 
346921 

348100 
349281 
350464 
&51649 



Cube. 



157464000 
158840421 
159220088 
160103007 
160969184 

161878625 
162771886 
168667823 
1&;666592 
165469149 

166876000 
167284151 
168196606 
169112877 
170031464 

170958875 
171879616 
172806698 
173741112 
174676879 

175616000 
176558481 
177504328 
178453547 
179406144 

180862125 
181321496 
1S2284268 
183250432 
184220000 

185193000 
186169411 
187149248 
188182517 
189119224 

190109875 
191102976 
192100033 
193100552 
194104539 

195112000 
196122941 
197137368 
1981 55-^7 
199176704 



Sq. 
Boot. 



28.2879 
28.2594 
23.2809 
23.3024 
23.3238 

23.3452 
28.3666 
28.3880 
28.4094 
28.4307 

23.4521 
23.4734 
28.4947 
23.5160 
23.5372 

28.5584 
23.5797 
2:^.6008 
23.6220 
28.6432 

23.6648 
23.6854 
28.7066 
23.7276 
28.7487 

23.7697 
28.7908 
23.8118 
23.8328 
23.8537 

28.8747 
23.8956 
2:3.9165 
28.9374 
23.9583 

23.9792 
24.0000 
24.0208 
24.0416 
24.0(fiM 

24.0882 
24.1039 
24.1247 
24.1454 
24.1661 



Cube 
Boot. 



200201625 24.1868 

201280056 24.2074 

202262008 24.2281 

203297472 24.2487 

204386469 24.2693 



205379000 
206425071 

207474688 
208527«S7 
209.'>S4584 



24.2899 
24 3105 
24.3311 

24.3516 
24.3721 



8.1488 
8.1488 
8.1533 
8.1583 
8.1083 

8.1688 
8.1738 
8.1788 
8.1838 
8.1882 

8.1982 
8.1962 
8.2081 
8.2081 
8.2180 

8.2180 
8.2229 
8.2278 
8.2327 
8.2877 

8.2426 
8.2475 
8.2524 
8.2.'i73 
8.2621 

8.2670 
8.2719 
8.2768 
8.2816 
8.2865 

8.2918 
8.2962 
8.3010 
8.3059 
8.3107 

8.8155 
8.3203 
8.8251 
8.3300 
8.8348 

8.8396 
8.3443 
8.3491 
8.3539 
8.3587 

8.3634 
8.3682 
8.3730 
8.3777 
8.3825 

8.3872 
8.3919 
8.3<i67 
8.4014 
8.4061 



No. 



Square. 



680 



854085 
855816 
866409 
857604 



860000 
861801 
862404 



864816 

866025 
867286 
868449 
869664 
370881 

878100 
878881 
874544 
875769 
376996 



879456 



381924 
883161 



384400 
385641 



388129 



390625 
391876 
393129 
394384 
805641 



398161 



400689 
401956 



404496 
405769 
407044 
408821 

409600 
410881 
412164 
413449 
414736 

416025 
417816 
418609 
419904 
421201 



810644875 
211706786 
212776178 
818847192 
214981799 

816000000 
817081801 
818167808 
219856287 
880348864 

881445185 
228545016 
228648548 
284755718 
225866589 



228009181 
229220088 
280346397 
231475544 



238744896 
234885118 
286029088 
237176659 



Cube. 



239483061 
240641848 
241804867 
242970624 

244140685 
24.5314376 
246491883 
5M7678152 
248858189 

350047000 
251239591 
252435968 
253636137 
2&48401O4 

256047875 
257259456 

258474863 
25969407S 
260917119 

262144000 
263374721 
264609288 
265847707 
267069984 

268886185 
269586130 
270840028 
272097792 
27.^3.^449 



Boot. 



84.3886 
84.4181 
94.4886 
84.4540 
84.4745 

84.4949 
84.5158 
84.5857 
84.5861 
84.5764 

a4.6M7 
84.6171 
84.6874 
84.0677 
81.6779 

94 6888 
91.7184 
94.7866 
84,7688 
84.7790 

84.7908 

84.8198 
94.8895 
84.8596 
84.8797 

94.8096 
84.9199 
84.9899 
84.9600 
94.9800 

85.0000 
25.0200 
26.0400 
85.0599 
25.0799 

85.0998 
85.1197 
25.1896 
25.1605 
25.1794 

85.1999 
85.2190 
25.2889 
85.2587 
25.2784 

25.2969 
35.8160 
85.8877 
25.3674 
26.3772 

25.8969 
25.4165 
35.4868 
35.4568 
25.4755 



SQUARES, CUBES, SQUABB AND CUBE BOOTS. 93 



No. Sqnare. 



C50 



6U 

657 
6&S 
«5» 

660 
661 
662 
66S 
664 



667 

6e» 

609 

670 
671 
67^ 
674 
674 

675 
676 
677 
67d 
679 

680 
6S1 

eisi 

688 

6M 

6m 

6« 
6M7 
0« 



601 

oaee 



606 
697 
606 
690 

TOO 

^9i 
70S 
704 



4.3801 
425101 
4:96400 
4:^716 

4S9QaS 

4ao«» 

431649 
4.t:S64 
43*281 

48S600 

4.M0Oi 
436;B44 
480969 

440606 

44iSS6 
4469G6 

444809 
446i»4 
447561 

448000 

45QM1 
451564 



974699000 
275804451 
-.{77167808 
978445077 
3i797MU64 

981011875 
;t8a»(Nl6 



284890819 
986191179 

S87496000 
^i8880478l 
9901I7S98 
991484ai47 
9SEi754944 

994079686 
295408896 
996740968 
998077689 
299418809 



454276 

456076 
458890 



461011 

469400 
468761 
465194 
466489 

467866 



470396 
471969 

474791 

476100 
477481 
478864 
480949 
481636 



484416 
486809 



487204 

488601 



491401 
492801 
494909 



Cube. 



802111711 
806464448 
801821917 
806182094 

80n»l6876 
808915776 
810988783 
811665799 
418046889 

814488000 

816891941 
817914668 
818611987 
8J0018904 

831419195 



894949708 
8SS660879 
:«706976O 

388609000 

829989871 
881878888 
882812567 
884955884 

885708876 
887158586 
888608878 
840068899 
841689099 

848000000 

844479101 
845048408 
847498997 
848918064 



8q. 
Root. 



25.4961 
25.5147 
85.6848 
25.5580 
95.5784 

95.5980 
95.6125 
95.6820 
25.6615 
25.6710 

95.6905 
25.7099 
95TS94 
25.7488 
25.7689 

25.7876 
25.8070 
25.8968 
25.8467 
25.8650 

25.8844 
25.0087 
25.9980 
25.9492 
25.0615 

95.9808 
96.0000 
26.0192 
26.0884 
26.0576 

96.0768 
26.0060 
26.1151 
96.1848 
26.1584 

96.1795 
26.1916 
98.9107 
26.9986 
26.2488 

26.9679 
96.9860 
26.8060 
26.8249 
26.8489 

96.8089 
26.8816 
26.4006 
28.4197 
26.4886 

96.4576 
96.4764 
26.4968 
20.6141 
26.5880 



Cube 
Root. 



8.6694 
8.6666 
6.6713 
8.6757 
8.6801 

8.6645 
8.6890 
8.6084 
8.6978 
8.7022 

8.7060 
8.7110 
8.7154 
8.7198 
8.7941 

8.7985 
8.7829 
8.7873 
8.7416 
8.7460 

8.7608 
8.7547 
8.7590 
8.7681 
8.76; 

8.7721 
8.7764 
8.7807 
8.7850 
8.7898 

8.7987 
8.7980 
8.8023 
8.8066 
8.8109 

8.8152 
8.8194 

8.8287 
8.8280 
8.8823 

8.8! 

8.8408 

8.8451 

8.8493 

8.8586 

8.8678 
8.8021 
8.8663 
8.870C 
8.8748 

8.8790 
8.8883 
8.8875 
8.8917 
8.8959 



No. 



Square. 



Cube. 



497025 850402695 

498486 851895816 

499849 868808918 

501261 354894912 

502681 856100629 



604100 
505521 
506044 



509796 

511225 
512656 
514089 
515591 
516961 

518400 
519841 
521281 
522729 
524176 



527076 
62bo29 
529984 
531441 

589900 
5:34361 
5358-^ 

5:«'289 
538756 

510225 
541696 
.513169 
514611 
546121 

547600 

510801 

552049 
558536 



566095 
556516 
558009 
559501 
561001 



568500 
564001 
566504 
567009 
568516 



857911000 
359125(81 
860944128 
862467007 
868994344 

866686875 
867061696 
868601818 
870146288 
871694950 

878948000 
871805361 
876367048 
877983067 
879503424 

881078195 
882G57176 
38l240.')88 
3a')828359 
387120489 

889017000 
890617891 



89383S837 
395446901 



307065375 



40031.55.53 
401947272 
403568119 

405294000 
406869021 
408518488 
4101?2407 
411830784 



Sq. 
Root. 



26.6518 
26.5707 
26.5895 
26.6088 
26.6271 

26.6458 
26.6646 
26.6683 
26.7091 
26.7208 

96.7896 
26.7689 
26.7769 
26.7965 
26.8142 

26.8888 
26.8514 
26.8701 
28.8887 
26.9079 

26.9858 
26.9444 
26.9629 
26.9815 
27.0000 

27 0185 
27.0870 
27.0656 
27.0740 
27.0924 

27.1109 
27.1998 
27.1477 
27.1662 
27.1846 

97.2029 
27.2213 
27.289' 
27.2580 
27.2764 



Cube 
Root. 



418498625 27.2947 9.0654 

4I5I60936 27.3130 9.0694 

4IC832723 '£7.3313 9.0785 

418508992 e7.8496 9.0775 

420189749 27.8679 9.0816 



421875000 
42a')64751 
425259008 

426957777 
428661064 



570025 430366875 

571536 432081216 

573040 1433798093 

574.564 1435519512 

.576081 1417245479 



27.8861 
27 4044 
27.4226 
27.4408 
27.4591 

27.4778 
27.4955 
27.5136 
27.5318; 9.1178 
27.5500! 9.1218 



94 



UATHEHATIGAL TABLES. 



No. 


Square. 


760 


577600 


781 


5791-il 


7te{ 


580644 


763 


582169 


764 


58d6»6 


766 


585325 


766 


580756 


767 


588289 


768 


5898..>4 


76U 


591361 


770 


592900 


771 


594141 


772 


595984 


773 


697539 


774 


599076 


775 


600625 


TTC 


602176 


777 


6a3r.J9 


778 


605:.'84 




779 600841 

780 008400 

781 609961 
61]5'J4 
613089 
614656 



783 

784 

786 
786 
78^ 
78S 
789 

790 
791 
792 
7l>8 



616225 
617796 



6J0944 
628521 

624100 
C25681 
G572C4 
G28M9 



794 680436 



795 
796 

797 
798 
799 

800 
801 
8U2 
803 
804 

806 
806 

807 
808 
800 

SIO 
811 
812 
818 
814 



682025 
633616 
635209 

036804 
638401 

610000 
641U01 
G43^>04 
G14N09 
646416 

648025 
649636 
651249 
652864 
654481 

656100 
657721 
659;jM 
660969 
6«.»596 



Cube 
Root. 



9.1258 
9.1298 
9.1888 
9.1378 
9.1418 

9.1458 
9.1498 
9.1587 
9.1677 
9.1617 

9.1657 
9.1696 
9.1786 
9.1775 
9.1815 



465484376 27.8388 9.1865 

4672885T6 27.8668 9.1894 

469097433 27.8747 9.1938 

470910952 27.89271 9.1973 

472729139:27.9106 9.2012 

474552000 27 9285' 9.2052 
47687954 1]27. 9464 V.2091 
47821 1 70H 27.9643 9.2180 
4800l8li8T;27.98-.'l, 9.2170 
48l89O3a4'28.UO00 9.2209 



27 7489 
27.7869 
27.7849 
27.8029 
27.8209 



4a3736625' 
4a'i58T()5« 
487444408: 
489303ST2 
491169009 

4980390001 
494918071 
49679:^088! 
498677257 
5005661841 



28.0179 
28.0357 
28.0583 
28.0713 
28.0891 

28.10C9 
28.1247, 
2<4.1425' 
28.16031 
28.1780; 



9.2248 
9.2287 
9 2326 
9.2365 
9.;M04 

9.2443 
9.2482 
9,2521 
9.2560 
9.2599 



502469875 28.1957! 9.2638 
504a5a336 28.21851 9.2677 
50C86l5r<j:28.2312 9.2716 
508169592 28.2489; 9.2754 
510082399|28.2666 9.2798 

51200000o'28.284S 9.2832 
618922401 128.8010' 9.2870 
615849608 28.81961 9.2909 
517781627128.3373, 9.2918 
5197l8464;28.3&49l 9.2986 



621660125 28 
523606616 28 
525557948 28 
627514112,28 
629475129 28 

531441000 28 
53341173128 
535887328 28. 
537867797 28 
5:)93.53144 ^ 



87251 
890l! 
4077 
4253 
4420 

4605 
,4781 
49.')6 
51*2 
5307 



9.3025 
9.3003 
9.3102 
9.3140 
9.8179 

9.8217 
9.8255 
9.3294 
9.3.S:i2 
{) aS70 



No. Square. 



815 664225 

816! 665856 

817' 667489 

8181 669184 

819| 670761 

8S0l 672400 

821 1 674041 

8221 675684 

828 677829 

824 678976 

680025 
826 682276 
827 



829 



831 



837 



685584 
687241 



690561 
692224 



695556 
697S25 



700569 
702244 
708921 

706600 
707281 
708964 
710649 



Cube. 



64184^376 
543388496 
&4688851S 
647843482 
649853S59 

551868000 
658387061 
656412246 
557441767 
559476S24 

661515626 
668659976 
665609283 
667663562 
66972S789 

671787000 
573866191 
575980368 
678009587 
68008S704 

582182875 
684277066 
586876253 
688480472 
590689719 

592704000 
694823321 
6969476S8 
599077107 



12336 601211684 



714025 
715716 
717409 
719104 
720801 

722500 
724201 
72.')904 
727009 
729316 

731025 
782736 
734449 
736164 
787881 

739600 
741321 
743044 
744769 
746496 

748285 
749956 
751689 
753424 
755161 



Sq. 
Boot. 



28.566: 
28.5882 
28.6007 
28.6182 

28.6856 
28.6531 
28.6705 
28.6880 
28.7064 

28.7288 
28.7402 
28.7576 
28.7750 
28.7984 

28.8097 9.8978 

28.8871 9.40l« 

28.8444 9 4058 

28.8617 9.4091 

28.8791 9.41S9 

88.8964 
28.9187 
28.9810 
28.9482 
28.9666 



29.0000 
29.0172 
29.0845 
29.0617 



89.0689 
29.0861 
29.1033 



608851125 
605495r86 
607645428 
609800192 29.1204 
611960049:29.1876 

614125000*89.1548 
616295051 189.1719 
618470208,29.1890 
620650477 29.2062 
628885864|29.228S 

625026375 29.8404 
6272-J20I6.29 2575 
629422:03 29.2746 
631628712 29.8916 
638889779 29.8067 

686056000 29.8858 
63827T881 ]29 8428 
640508928:29. a598 
6427^5647129.8769 
644972544 89.8989 

647214685 29.4109 
649461896 29.4279 
651714863 89.4449 
653972a32|29.4618 
656234909129.4788 



SQUARES, CUBES, BQUAHE AND CUBE ROOTS. 95 



No. 



Square. 



87D 

871 
B» 
873 
674 

875 
876 
877 
878 
St9 

880 
881 
882 
8tt 
864 



887 



97i6000 
758641 
760884 
76ei.» 
763876 

7666» 
767378 
7691M 
770884 
77:»4l 

774400 
776161 
TTTftW 



781456 

TR895 
n)4996 
796760 
788541 
i9a»l 



708100 
^83881 
80S; 795664 
88S. 797440 



800 
891 



801035 
806 802816 
807, 804600 
808' 806404 

806;»1 



900 

901 

mm 



903 815400 



904 

flOO 
90S 
907 
908 

900 

010 

oil 
ot« 

918 
014 

015 
916 
01 

919 

3» 

0:i 

m 

088 
084 



810000 
811801 
818004 



8IK16 

810K5 
820886 
8SW49 
824464 
&MH81 

828100 



831744 



835896 



837'2!» 
839066 



84i73i4 
844561 



Cabe. 



658B0000O 

660770811 
668064848 



667687684 

6600S1875 
67tfci8l876 
674SJM1S8 
67B8S6150 
679161489 



Sq. 
BooC. 



89.4968 
W.6127 
29.6;!96 
29.6466 
29.5685 

99.5804 
20 5978 
VO.6142 
29.6811 
29.6479 



681472000 29.6648 



688797841 



688465887 
690807104 

0961541S5 
605506456 
607864103 
7002SiO7^ 
702805800 



29.6816 
29.6085 
29.7158 
80.7821 

90.7489 

29, 

29.7825 

29.7998 

29.8161 



704969000 20.ai20 
7t)7347971 129.8496 
700732288 29.8664 
712WlQ67'.».R88t 
714510084 29.8098 



716017875 
710828186 
7'il734«73 
724160702 
79657«»0 



29.9166 
29.9338 
89.9500 
29.9606 
89.9888 



73900000080 0000 



781432701 
733870806 
786814827 
788703264 



80.0167 
8U.0388 
80.0600 
80.0066 



741817085 80.0882 
748677416 80.0906 
746148648 80.1164 
748618812 80.1880 
751080429 80.1496 



758571000 
756068031 



761048497 
768551944 

760060875 
768576896 
771096818 
773690882 
7761515S0 



846400 778688000 

848041 781829961 

860084 783777448 

851089 , 786380167 

K8T7B ' 



30.1062 
30.1888 
30.1998 
80.2159 
80.8324 

80.8400 
80.8655 
80.2820 
80.8886 
80.8150 

80.3315 
30.3480 
30.8645 
80.8809 
30.8974 



Cube 
Root. 



0.6464 
0.5601 
0.5537 
0.5574 
9.6610 

9.5647 
0.60&8 
9.5719 
9.6756 
9.5792 

0.5828 
9.5365 
9.5901 
0.5937 
0.5078 

0.6010 
9.6046 
9.6082 
9.6118 
0.6154 

9.6190 
9.6226 
9.6262 
9.6896 
9.6334 

9.6406 
9.6442 
9.6477 
9 6513 

9.6549 
9.0686 
9.6620 
9.6056 
9.6692 

6727 
9.6763 
9.6790 
9.6334 
9.6870 

9.6905 
9.6941 
9.6976 
9.7012 
9.7047 

9.7082 
9.7118 
9.7153 
9.7188 
9.7824 

9.7S6fl 
9.7294 
9.7329 
9.7864 
9.7400 



No. 




857476 
859329 
861184 
868041 



864000 
866761 



980 
981 
982 

933 870489 

934 8^2366 



0391 



874326 
876096 
877969 
870844 
881781 



040 883600 

041! 885481 

942 887854 

943 889249 

944 891186 

945' 893085 

946| 804916 

947 806809 

948 898704 

949 900001 

050 908500 

951 904401 

962 906804 

953 008209 

954 910116 

955 912025 

956 913986 
957; 915849 

958 917764 

959 919681 



960 
961 
962 
968 
964 

905 
966 
067 
9G8 
969 

970 
971 
972 
973 
974 

975 
976 
97' 
078 
979 



981600 



985444 
927860 



981885 
933156 
935089 
937024 
038961 

940900 
942841 
R44784 
946729 
948676 

950635 
952576 
954r)2H 
956484 
958441 



791463185 30.4138 
794022776:30.4302 
796597983 80.4467 
799178752'80.4631 
801766080:80.4795 

80485700o'80.4959 
806954491180.5123 
800657568 30.5287 
812166237130.5450 
814780504 80.6614 

817400875I30.5778 
8«)02S8S6<30.6941 
828666058'80.6105 
825293672|80.6268 
887906019 30.6431 

880584000I80 6504 
888287621 ;80.675^ 
835806888|30.6920 
888661807 80.7083 
841282884 80.7246 

843908625 30.7409 
84659a'>36 80.7571 
8492781^8 30.7734 
851971S92.30.7896 
854670849 30.8068 



857375000 
860085351 
862801406 
805528177 
868250604 



80.8281 
30.8388 
30.8545 
30.8707 
30.8869 



870988875 30.0031 
873728816 30:9192 
876467493 30.9354 
879217912 80.9516 
88197407% 30.9677 



S84736000 
887503681 
8902771-^8 
89805(3347 
885841344 

898682125 
901428696 
904281063 
907OT9V82 
909853209 

912673000 
915498611 
918880048 
921167317 
924010424 

926859375 
929714176 
932574aS8 
93.'V44ia52 

g3a3].'n'89 



80.9839 
31.0000 
31.0161 
31.0322 
31.0483 

31.0644 
31.0805 
31.0966 
31.1127 

31.1288 

31.1448 
81.1609 
81.1769 
31.1929 
:n.2090 

81.2250 
31 .2410 
31.2570 
31.2730 
31.2890 



96 



HAtHEMATIOAL TABLES. 



Square. 



960400 
964801 
904894 
966889 



970S95 
9792196 
974169 
976144 
978181 

960100 
982061 
964064 
986049 
988086 

990085 
998016 
994009 
996004 
998001 

1000000 
100900] 
1004004 
1006009 
1006016 

1010085 
1018036 
1014040 
10HH)64 
1018081 

10-20100 
1088181 
1084144 
1086169 
1038196 

1090885 
1088856 
1034880 
10368:14 
1038861 

1040400 
1048441 
10444S4 
1046589 
1048576 

1050635 
105867C 
1054789 
1056784 
1058841 

1060900 
1068961 
1065084 
1067089 
10691 5C 



Cube. 



941198000 
944070141 
946966168 
94986S1U87 
958768904 

056671685 
958686856 
961504808 
964480878 
967861600 

970890000 
978S4*i871 
976191488 
979146657 
988107784 

966074875 
988047986 
991086078 
994011098 
99700S099 



1000000000 
1008006001 
1006018008 
1009087087 81.6708 
1018048064 81.6660 



8q. 
Root. 



81.8050 
81.8800 
81.8869 
81.8588 
81. 

81.8847 
81.4006 
81.4166 
81.4885 
81.4484 

81.4643 
81.4808 
81.4960 
81.5119 
81.5878 

81.5486 
81.6{»5 
31.5758 
81.5911 
81.6070 



Cube. 
Root. 



9.9880 
9.9868 
9.9896 
9.9480 
9.9464 

9.9497 
9.0681 
9.9565 
9.9596 
9.9688 

9.9666 
9.9699 
9.9788 
9.9766 
9.9600 

9.9888 
9.9866 
9.9900 
9 9988 
9.9967 



81.688610.0000 
81.6386 10.0088 



81.6544 



1016075125 
1018106816 
1081147848 
1084198518 
1087843789 



10.006: 
10.0100 
10.0188 



81.7017 10.0166 
81. '5175 10.0800 
81.7888110 0288 
81. 74901 10. 0866 
81.7648,10.0899 



1080801000 81.7605 10.0888 
10338643;^ 31 7968 10.0865 
10S6433788I8I .81 19, 10.0898 
1039509197 31 .88771 10.0481 
1048590744 81.8484 10.0465 



1046678375 
1048778096 
1051871918 
1054977888 
1058089859 

1061808000 
1064338801 
1067468648 
1070)9916: 
1078741884 

1076890685 
1080045570 
1083806688 
1086373952 
10e9647<J80 

1092787000 
1095918791 
1099104768 
1108:)08037 
1105507804 



81.8601 
81.8748 
81.8904 
81.9061 
81.9818 

81.9874 
81.9531 
31.9687 
31.9844 
38.0000 



10.0498 
10.0581 
10.0568 
10.0696 
10.0689 

10.0668 
10.0605 
10.0788 
10.0761 
10.0704 



32.0156 10.0626 
88.0318 10.0659 
82.0468 10.0898 
32.0684 10.0985 
88.078010.0957 

88.0986 10.0090 
.33.1098 10.1088 
32. 12481 10. 1055 
.32.1403 10.1088 
32.1559110. 1121 



No. 



Square. 



1065 
1080 
1087 
1088 
1039 

1040 
1041 
1048 
1043 
1044 

1045 
1040 
1047 
1048 
1049 

1050 
1051 
1058 
1053 
1054 

1065 
1050 
1067 
1058 
1059 

1060 
1061 
1068 
1063 
1064 

1065 
1066 
106: 
1068 
1060 

1070 
1071 
1078 
1078 
1074 

1075 
1076 
1077 
1078 
1079 

1060 
1081 
1068 
1083 
1084 

1065 
1066 
1087 
1088 
1089 



Cube. 



8q. 
Root. 



1071835 1106717876 82.1714 
1073896 1 11 1934656i82. 1870 
1075809 1115167653188.3085 
1077444i 11 18386878 88.8180 
1079081 U81688819 88.8885 

108160o'l 194804000 88.9400 
1088681 1 1186111981 88.8645 
1065764 1181866088 88.8800 



109784911134086507 
1080936 1187603184 



1098086 
1094116 
1096809 
1098804 
1100401 

1108600 
1104601 
1106704 
1108800 
1110916 

1118085 
1110186 
1117849 
1119364 
1181481 

1188000 
1185781 
1187644 
1189969 
1188096 



88.8955 
82.8110 



1141166125 82.8866 
1144445886 82.8419 
1147730888,88 8574 
11510885981.38.8788 
1154880649 88.8688 

1157689000 88.4037 
1160986651 88.4191 
1164858606.88.4845 
1167576877 82.4500 
1170905464 83.4654 



1174841875 
1177688616 
1180988198 
1184887118 
1187648879 



82.4606 
82.4968 
88.6116 
88.6860 
88.5488 



1191016000 82.8676 
119438998138.5780 
119777C388 33.5868 
1801157047 33.6066 
1804660144 88.6190 



1184885 1807940686 
1136356' 181 1855496 
118648911814767763 
1140684' 1818186438 
1148761 1881611509 

11449001 1325049000 
1147041; 1886480911 
1149184 1831985848 
11618891236876017 
1158476<1888883«I4 



88.6497 
88 6660 
88 
38 

82.n09 

as.'nwi 

38.7414 
83.7567 
38.7719 



1156685!l348896875 38.7878 
1157776 lS4,'i76697« 88.8084 
1150989 12498485.33 38.8177 
1168084 1858726658 88.8389 
1164841 1 !856816089 38.6481 



116640011859718000 
116866111268814441 
1170784' 1866728868 
1178889,1270888787 
1176056 1878760704 

1177885 1277S99195 
1179896 1280684056 
1181569 1884.365503 
11637441 J 88791 8478 
1185981,1891467960 



82.8684 
88.8766 
33.8986 
33.9090 
88.9848 

82.9898 
88.9545 
38.960; 
38.9848 
33.0000 



Cube 
Root. 



10.115& 
10.1186 
10.1818 
10.1851 
10.1868 

10.1816 
10.1848 
10.1881 
10.1418 
10.1446 

10.14TB 
10.1510 
10.1548 
10.1576 
10.1607 

10.1640 
1O.1078 
10.1704 
10.1786 
10.1760 

10.1801 
10.1688 
10.1865 
10.1897 
lO.lflECO 

10 1901 
10.1998 
10.8086 
10 8067 
10.8068 

10 8191 
10.8158 
10 8186 
10.8817 
10.«M9 

10.9881 
10.8813 
10.8845 
10.8876 
10.8406 

10.8440 
10.847^ 
10.8608 
10.8586 
10.8067 

10.8599 
10.8680 
10.8668 
10.8098 
10.8786 

10.9757 
10.9788 
10.8880 
10.9861 
10.9688 



SQUARES, CUBES, SQUARE AKD CUBE ROOTS. 97 



No. 



Square. 



1090 
S091 
lOBJ 
109S 
10M 

10B6 
1096 
1097 
1096 
1009 

1100 
1 101 

1103 
1 104 

1105 
1106 
1107 
1108 
1109 

1110 

nil 

1112 
1113 
1114 

1115 
1110 
1117 
1118 
1119 

11» 

mi 
im 

1128 
ItM 

1125 

11« 
1127 
IIJB 
1120 

iiao 

II31 
1182 
11S8 
1184 

im 

1188 
1187 
1138 
1180 

1)40 
1141 
!I43 
1148 
1144 



1188100 
1190881 
119aM8l 
11M649 
11988M 

1199085 
1201216 
1208409 
1206004 
1207801 



1210000 188100000088 



1212901 
1214404 
1218009 
1218810 

12210S> 



1225449 

1227604 
l;89681 

1282100 
12S4821 
1288544 

1288709 
1240096 

1248225 
1245456 
1247089 
1249924 
12»161 



1267876 
1270129 
1272884 
1274641 

1270900 
U79161 
1281424 
1288689 
1285066 



1290496 
1292709 
I29S044 
1207821 

1209600 
1801881 
1804164 
1806449 

18087861 



Cube. 



1206029000 

1296606971 
1808170688 
l.^i06751857 
1809888664 

1812982875 
1816582786 
1820189678 
1828758198 
1827878899 



88.0151 
83.0803 
8S.0454 
83.0606 
0767 

88.0906 
88.1069 
83.1910 
83.1861 
88.1618 



18<f468880l 
1888278906 
1841919797 
1845572664 



1352899016 
185657'J048 
I86025m2 
1868088099 

1807681000 
1871880681 
1875086926 
1878749697 
1882409644 

188619687S 
1889928696 
1808668618 
1897415082 
I40I 166150 



1254400 1404988000 



1296641 

1286884 

1261189 i41flM7867 

1288876 



Sq. 

Root. 



1662 
88.1818 
88.1964 
88.2114 
88. 

88.9415 
88.2566 
88.8716 
83.2666 
88.8017 

83.8167 
83.8817 
83.8467 
88.8617 
88.8766 

88.8916 
88.4066 
83.4n5 
88.4865 
88.4515 



88.4664 

1408694561 88.4618 

1412407646 83.4968 

83.5119 

1420084624 38.5261 

1428826125 88.5410 
1427626876'33.6660 
1481485868 83.5706 
1485249159 83.5667 
1489009689^38.6006 

1442697000 88.0165 



1446781091 
1450571966 
1454419687 
1458274104 

1462185875 

146600846688 

I460W8858 

1478760072 

1477646619 



33.6803 
33.6452 
33.6601 
83.0749 

38.6686 
7046 
33.7174 
88.7342 
83.7491 



146154400088 



1485446^21 
1489355268 
1498271907 



149719098488 



7688 
83.7787 

7986 
83.8068 

8881 



Cube 
Root. 



10.9014 
10.2946 
10.2977 
10.8009 
10.8040 

10.8071 
10.8108 
10.8184 
10.8165 
10.8197 

10.88S8 
10.8269 
10.8290 
10.8822 
10.8853 

10.8864 
10.3415 
10.8447 
10.8478 
10.8509 

10.8540 
10.8571 
10.8602 
10.3688 
10.8664 

10 8606 
10.8726 
10.8767 
10.8788 
10.8619 

10.8660 
10.8681 
10.8912 
10.8948 
10.8978 

10.4004 
10.4086 
10.4066 
10.4007 
10.4127 

10.4158 
10.4189 
10.4219 
10.4260 
10.4261 

10.4811 
10.4842 
10.4378 
10.4404 
10.4434 

10.4464 
10.4405 
10.4525 
10.4656 
10.4566 



No. 



1146 
1146 
1147 
1148 
1149 

1150 
1151 
1152 
1158 
1154 

1166 
1166 
1157 
1158 
1159 

1160 
1161 
1162 
1168 
1164 

1165 
1166 
1167 
1168 
1160 

1170 
1171 
11T2 
1178 
1174 

1175 
1176 
1177 
1178 
1179 

1180 
1181 
1182 
1188 
1184 

1185 
1186 
1167 
1188 
llb9 

1190 
1191 
1192 
1193 
1194 

1195 
1196 
1197 
1198 
1199 



Square. 



1311085 
1818816 
1315009 
1317904 
1320901 

1829500 
1824801 
1327104 
1389409 
1881716 

1884025 
1386886 
1338649 
1840864 
1343281 

1345600 
1347921 
1350244 
1SS2069 
1854896 

1357285 
1859656 
1361889 
1364224 
1366561 

1366900 
1371^1 
1373564 
1375929 
1378276 

1360825 
1382976 
1385829 
1867684 
1890041 

1302400 
1394761 
1397124 
1399489 
1401656 

1404225 
1406586 
1406869 
1411344 
1413721 

1416100 
1418481 
1420664 
1423249 
1425636 



1430416 
1432809 
1435204 
1187601 



Cube. 



1501128825 
1505060136 
1509003523 
1512958708 
1516910949 



33.8878 



83.8674 
83.8821 



1520676000 
158484596133 



9116 10, 



1532606677 
1536800264 

1540796875 
1544804416 
1548816803 
1558836812 
1556862679 

1560696000 
1564936S81 
1568968528 
1578087747 
1577006044 

1681167125 



1689324463 
1593418632 
1597509600 

1601618000 
1605723211 
1609640448 
1618964717 
1618096024 

1622234875 
16S8379776 
168058-^233 
1684601762 



33. 

9264 
33.9411 
33.9559 
33.9706 

33.9W} 
34.0000 
34.0147 
34.0294 
31.0441 

34.0688 
34.0735 
34.0881 
34.1098 
84.1174 

34.1821 

84.1467 

34.1614 

34.1 

34.1906 



1648082000 
1647212741 
1051400566 
1655505487 
1659797504 



1664006625 34 



1672446203 
16766T6672 
1680914269 

1685159000 
1689410871 



1697986057 
1702209884 

170M89875 
1710777536 
1715072873 
1719874892 
1723683590 



Sq. 
Root. 



'60 10 



34.9053 
34.2109 
84.2345 
34.2491 
34.2837 

84.2788 
34.2029 
34.8074 
34.3220 
34.3866 

34.3511 
34.3657 
34.8804 
34.394S 
34.4098 



34.4384 
34.4529 
34.4674 
34.4819 

34.4964 
34.5109 
34.5254 
34.539S 
34.5543 

34.6688 
34.683:2 
34.6977 
34.6121 
34.6266 



Cube 
Root. 



10.4617 
10.4647 
10.4678 
10.4706 
10.4789 



.4769 

10.4799 
10.4880 
10.4860 
10.4890 

10.4921 
10.4961 
10.4961 
1U.501I 
10.504:2 

10.6072 
10.5102 
10.5182 
10.5162 
10.5198 

10.5228 
10.5258 
10.5288 
5818 
10.5848 



10.5878 
10.5408 
10.5438 
10.5468 
10.5498 

10.5528 
10.5558 
10.5688 
10.5618 
10.5642 

10.5678 
10.5708 
10.5782 
10 5768 
10.5791 

10.5821 
10.5651 
10.5681 
10.5010 
10.5910 

10.5970 
10.6000 
10.6(»9 
10.6050 
10.6068 

10.6118 
10.6148 
10 6177 
10.6207 
10.6236 



HATHBMAtlCAL lAfl££8. 



Square. 



1900 1440000 

l:i01 144:M01 

3S0-J 1444804 

1908' 1447209 

1904 1449616 

1452035 
1454436 
1466849 
1469«64 
1461681 

1464100 
14665»!1 
1468944 
1471S69 
1478796 

1476885 
1478656 
1481089 
1488594 
1465961 

1488400 
1490641 
I498S84 
1495729 
1496176 

1500685 
1506076 
15055'J9 
1507964 
1510441 

151)f900 
1515861 
15178M 



1&22756 

II 

1597696 

153U169 

158^644 

1585191 

1587600 
1540061 
1549564 
1545049 
1547686 

1560095 
1552516 
1665009 

1557504 
1560001 

1569500 
1565001 
1567504 
1570009 
lOT'^lO 



Cube. 



Sq. 
Boot. 



1798000000 84.6410 
1789898601 34.6554 
1786654406 84.6699 
17409994<i7 84.6843 
1745387664 84.6967 

1749690195 34.7181 
1754049816,34.7975 
175641674384.7419 
176:nW9l2ld4.7563 
1767179389 34.7707 



1771561000 
1775056031 
1780860198 
1784770597 
1789188344 

1798618875 
1796045690 
1809485313 
1806982*^39 



84.7851 
31.7994 
34.8138 
34.8981 
34.8425 

34.8569 
34.8712 
34.8865 
34.80D9 



1811886459 34.914 



1815848000 
I8906l6b61 
18:M798048 
l829-.rt-6:H>7 
1H83767424 



1838'2fi5625 35.0000 



34.9285 
34.0428 
84.9571 
:W.J»7I4 
84.9857 



1842771176 
1847284083 
1851804352 
1850381069 

1860667000 
1865409391 
1869959168 
187451683; 
1879060904 

1888659875 



1892819053 
1897418272 
1909014919 

1906694000 

1911940521 
1015864488 
1920495907 
1925184764 

1999781125 
1984484986 
193809(»28 
1943764992 
1948441949 



:^.0148 
:i'j.0286 
35.0428 
35.0571 



Cube 
Root. 



10.6966 
10.6295 
10.6385 
10.6354 
10.6884 

10.6418 
10.6443 
10.6479 
10.6501 
10.6580 

10.6660 
10.6590 
10.6619 
10.6648 
10.6678 

10.6707 
10.6786 
10.6765 
10.0795 
10.6894 

10.68iS8 
10.6882 
10 6911 
10.6940 
10.6970 

10.61 

10.7028 

10.7057 

10.7086 

10.7115 



35.0714 10.7144 
35.0656 10.7173 
35.0909 10.7202 
35.1141 10.7281 
35.1963 10.7960 

85.149610.7980 
35.15681 10.7818 
85.1710 10.7347 
35.1852 10.7876 



85.1994 

85.9186 

35.2278 
35.9490 
35.9562 
35.2704 

85.9846 
35.2087 
35.3129 
35.3270 
36.8412 



10.7406 

10.7434 
10.7463 
10.7491 
10.7520 
10.7549 

10.7578 
10.7*507 
10.7635 
10.7664 
10 7693 



1958125000 35.8553 10.77^ 
1967816251 85.8695|10.r7.'j0 
1962515008i;«.3K36| 10.7779 
1967221277,:» 3077.10 7808 
1971935004!3.5 41imi0 7sa7 



No, 



1955 
1956 
1257 
1958 
1959 

1260 
1261 
1962 
1968 
1964 

1966 
1966 
1967 
1268 
1969 

1970 
1971 
1272 
1273 
1974 

1975 
1276 
1977 
1978 
1979 

1980 
1981 
1262 
1268 
1284 

1986 
1286 
1287 
1988 
1269 

1990 
1291 
1299 
1203 
1994 

1295 
1296 
1207 
lJi9H 
121^9 

1800 

1801 
1302 
1303 
1304 

1805 
i:)06 
1307 
1808 
1309 



Square. 



1575095 
1577586 
1580049 
15825M 
1565061 

1587000 
\1590121 
1592644 
1595169 
159769G 

1600995 
1602756 
1605269 
16078;M 
1610361 

1619900 
1615441 
1617964 
1690520 
1693076 

1695695 
162817C 
1630799 
1038284 
1635841 

1688400 
1640961 
1643524 
1646069 



1 
1976666375 85. 
1961885216 85. 
1966121593 36. 
1990865.519 85. 
1996616979 85. 

9000876030 86. 
9005142581 35. 
2009916798 35. 
2014698447 35. 
9019487744,85. 



9094964695 85.6668 
9099069096 86.5809 



1661925 
1653796 
1656369 
1658944 
1661521 

1664100 
1666681 
1669264 
1671849 
167418G 



Cube. 



Sq. 
Root. 



.4960 
.4401 
4549 
.4688 
4894 

4965 
6106 
6946 
5887 
5S96 



2088901163 
9036790839 
9043548109 

2048888000 

9058925511 
2056075648 
206293341 
9067706894 

2079671875 
2077652576 
2089440933 
2087886952 
2099940689 

9007159000 
9102071041 
2106997768 
2111982187 



1648656 9116874304 



9191894196 

9126781666 
2131746903 
2136719879 
2141700560 

2146689000 
2151685171 
2156689068 
2161700757 
216G?i0164 



167702.*) 91717473:5 

16796lC.,2l7678-.i836 
168;i209| 21 HI 820073 
1684804 218687.5502 
1667401 2191933890 



1690000 2197000000 
1692001 2202073901 
1695204 2207155608 
1697800 1 22 12945127 
1700416.2217342464 

170802512229447695 
1705636 9297660616 
1708249 22.32681443 
1710864 2287810112 
171348119242946629 



85.5049 
&5.6000 
36.6930 

35.6871 
85 6511 
35.6651 
35.6:91 
35.6081 

86.7071 
85.7911 
35,7351 
35.7491 
86.7631 

88.7771 
35.7911 
35.8050 
35.8190 
35.8899 

35.8460 
35.8606 
35.8748 
36.8887 
36.9096 

35 .0166 
35.9.^') 
35.9444 
35.9583 
35.9722 

35.9861 
36 0000 
36.0139 
36.0278 
36.0416 

86.0655 
36.0604 
36.0689 
36.0971 
36.1109 

36.1948 
36.1866 
36.1596 
36.1663 
36.1801 



Cube 
Root. 



10.7865 
10.7694 
lO.TOiS 
10.7951 
10.7880 

10.8006 
10.8087 
10.8065 
10.8094 
10.8129 

10.81.M 
10.8179 
10 K906 
10 8236 
10.8265 

10.8998 
]0.Ki92 
10.K»50 
10 8.^78 
10.6407 

10.8485 
10.6463 
10.6409 
10.8&40 
10.6548 

10.8577 
10.8806 
10.6688 
10.8661 
10.6690 

10.8718 
10.8746 
10.8774 
10.8809 
10.8631 

10.8859 
10.8887 
10.61115 
10.8043 
10.6971 

10.8990 
10.9097 
10.9065 
10.0083 
10.0111 

10.0189 
10.9167 
10.9195 
10.99S3 
10.9951 

10.9979 
10.9807 
10.988& 
10.986S 
10.9891 



SQUARES, CUBES, SQUARE AND CUBE ROOTS. 



No. 



1S10 
ISlI 
]3iS 
131$ 
1314 



SqiiAre. 



Cube. 



1716100 m»mof» 

171 tmi StiSZTH'^l 
17il344 2458408388 
17S890O 2868371297 
17MBM£i8874ni4 



36.1939 
36.«r77 
3tf.£8l6 
36.3858 
36.8491 



1315 17^90385 28nn3067& 36.8689 

13161 17818S6 8879188496 86.2767 

1317 1734489 2»4.%88013 86.8905 

1318i 1787184 888998943836.3048 

1319, 1739761 2891744729 86 3180 

ISaol 1749400 22999660OO 36.8818 

1381 > 1745041 8806199161 86.3456 

1882, 1747684 8310438848 86.3698 

lS8di 1790389 8415666067 86.3^11 

IXUi 1733)9:6 8380940884 86.8888 



13S5 1733685 
1386; 1758876 

i3s; 

IttSl 

1389 1760841 



8831478976 
8336738783 
«)48089568 
8317334889 



1760M9 8336738783 
1768664 «)48089568 



Boot. 



86.4005 

86.4143 
86.4880 
86.4417 
36.4566 



1330i 17698008358637000 36.4698 
1331 mi561hi%7947691 36.4889 
191' 1774884 8368866868 36.4966 
1333 1776889 8868603067 36.5106 
13MJ 1779666 J873987704 86.5840 



:31S| 

1386; 
1337 
1318 
1339 



8379870673 
1784896 2884681056 



1787669 
179W44 
1798881 



2889979758 
3895346478 
8400»1219 86. 



1348 

1^ 
1344 



:M0610400036. 
1798881 2411494881 

8416898088 86 
2428300607 



1340 1796000 
ISII 

1800964 

1808649 



1806830 2427715664 



1345 1800085 

1346 1811716 
1347! 1814409 
1348! 1817104 
l»l»: 1810801 



ISO] 

mi 



1353 
I»4 



188B00 
18i5-J0] 



13S8 1S87904 



1880009 

1838816 



1355, __ 

166 1838786 

last, : ~ 

13S6 1844164 

1S9 1846881 2609911879 



8615486000 
Ml 1862881 2681008881 



86.6377 
86.6513 
36.5650 
36.5787 



2433188085 
2488660786 
2444006028 
2449456198 
2454911549 

2460675000 
2465846561 
2471880806 
8476813077 
24fre809864 



2487813875 36 
2498886016 86 



1841449 9498846886 
2904874712 



1849600 

1862881 

1886044 
I 18B7760 9688189147 
I 1860496^87716544 



6060 
86.6197 

6888 
86.6460 
86.6606 

86.6748 
86.6879 
86.7015 
86.7151 
86.7887 



86.7560 



7881 
7967 



.810811 
11 
8875 
8611 
8646 



86.8788 
86.8917 
86.9053 
36.9188 
86.9884 



Cube 
Boot. 



10.9418 
10.9446 
10.9474 
10.9608 
10.9580 

10.0657 
10.9565 
10.9613 
10.9640 
10.9668 

10.9696 
10.9784 
10.9758 
10.9779 
10.9807 

10.9684 
10.9868 
10.9890 
10.9017 
10.9945 

10.9978 
11.0000 
11.0088 
11.0065 
11.0088 

11.0110 
11.0138 
11.0165 
11 0193 
ll.(»20 

11.0247 
11.0875 
11.0808 
11.0830 
11.0857 

11.0884 
11.0418 
11.0439 
11.0466 
11.0494 



86.748811.0681 
.0648 
.057^ 
.0603 
.0680 



.0657 
.0684 
.0712 
.0789 
0766 



11.0798 
11.0680 
11.0847 
11.0875 
11.0902 



No. 



1365 
1866 
1887 
1868 
1869 

1870 
1871 
1878 
1873 
1374 

1375 
1876 
1377 
1378 
1379 

1380 
1381 
1382 
1388 
1884 

1885 
1886 
1887 
1888 
1889 

1390 
1891 
1398 
1803 
1894 

1895 
i:i96 
1391 
1896 
1899 

1400 
1401 
1408 
1403 
1404 

1405 
1406 
1407 
1408 
1400 

1410 
1411 
1413 
1418 
1414 

1415 
1416 
1417 
1418 
1419 



Square. 



1868885 
1865056 
1868689 
1871484 
1874161 

1876000 
1879641 
1888384 
1885120 
1887876 



8543808125 
8548895896 
8554497868 
8560108088 
8565786409 

8671868000 
8576987811 
8588630848 
8688888117 
8593941684 



1890085 2699600875 



1698876 
1896180 
1896884 
1901641 

1904400 
1907161 
1909984 
191-2689 
1915456 

1918885 
1980996 
1988769 
1986544 
1989881 

1988100 
1984881 
1937664 
1940449 
1948836 

1946085 
1948816 
1951609 
1954404 
1960^1 

1960000 
196-2801 
1066604 
1968409 
1971816 

1974085 
1976886 
1979649 
1988464 
1985881 

1988100 
1900981 
1993744 
1996569 
1999896 

2008285 

20t)r889 
20107^4 
8013561 



8605885876 
8610969688 
8616668158 
8688868989 



2688079000 
8683780341 
8639514968 
8645848887 
8650991104 87 



865674168687 
8668500466 



Cube. 



86.9469 
86.9594 
86.9780 
86.9865 
37.0000 

87.0185 
37.0870 
87.0405 
87.0540 
87.0675 



87.081011.1199 
87 0945 11.1*226 



87.1080 
87.1814 
37.1849 



867404307:1 



2685619000 
8691419471 



2708045457 
8708870964 

8n4704876 
878054n86 
8786397773 
873-2266798 
8788184199 



8744000000 
■2749681801 
8755776808 
2761677827 87.4566 
8767587864 87.4700 



8q. 
Boot. 



Cube 
Boot 



11.0989 
11.0966 
11.0988 
11.1010 
11.1087 

11.1064 
11.1091 
11.1118 
11.1145 
11.1178 



87.1484 
37.1618 
87.1768 
87.1887 
8081 



.8166 
37.8290 
37.8424 
87.2559 
37.8090 



87.8827 
87.2961 
87.8095 
87.3889 
87.8863 

87.8497 

87.3681 
37.8765 
87.; 
37.4088 



87.4166 
87.4899 
87.44-18 



8773505185 
8779481416 
2785866143 
2791309318 
8797860989 

8808881000 
8H09189581 
2815166528 
2821151997 
■2887145944 



28891 .^9■296 
2845178713 
2851206632 
2857843059 



87.4838 
37.4967 
87.5100 
87.5238 
87.5866 



11.1858 
11.1880 
11.1307 

11.1834 

11.1861 
11.1387 
11.1414 
11.1441 

11.1468 
11.1495 
11.1528 
11.1548 
11.1575 

11.1608 
11.1689 
11.1665 
11.1688 
11.1709 

11.1736 
11.1768 
11.1789 
11.1816 
11.1842 

11.1809 
11.1896 
11.1928 
11.1949 
11.1975 

11.8008 
11.8088 
11.8055 
11.8088 
11.8106 



37.5500,11.2185 
37.563311.8161 



•S7.5766 
87.5899 
87.1 

87.6165 
87.6898 
87.6481 
87.6r>63 



11.2188 
11.8214 
11.8240 

11.2887 
11 2293 
11.2820 
11.2846 
11.2373 



100 



ITATHEMATICAL TABLES, 



No. 



1490 
1491 
14^ 
148S 
1444 

1485 
14M 
1427 
1428 
1429 

14S0 
14.S1 
1482 
1433 
1484 

1485 
148(5 
1487 
1488 
1489 

1440 
1441 
1443 
1443 
1444 

1446 
1446 
J447 
1448 
1449 

14.% 
1451 
145^ 
1453 
1434 

1466 
1456 
145: 
Ur^H 
1459 

1460 
1461 
146-2 
1403 
1464 

1465 
]4«fl 
1467 
1408 
1469 

1470 
1471 
1472 
1473 
1474 



Square 



2016400 
2019-^41 
20-J2084 
9024929 
2087770 

9080025 
9088476 
90868S9 
9089184 
9042041 

9044900 
9047761 

9058480 
9066856 



9069096 
9U64969 
8067844 

9079000 
207W81 
S079W4 
9082240 
9085186 



2090916 
1^098809 
9096704 
9099601 

9102500 
9105401 
9108304 
2111209 
2114116 



Cube. 



8q. 
Boot. 



S9«Ib»8000'87.6899 
2869841461 37.6962 
2875408448 37.7094 
2881478967 87.7227 
2887558024|87.7859 

2808840635'87.7492 
28997867T6;87.7644 
2905841488 87.7767 
39119647B2I87.7889 
2918076689 87.8021 

2994207tX)0 87.8158 
298084509187.8986 
..>986493568 87.8418 
2042649787 87.8660 
2948814504187.8089 

2954987879 87.8814 
2961l6g656'87.8946 
2907860458,87.9078 
2978559672187.9210 
9979767519,87.9342 



Cube 
Root. 



11.9899 
11.9485 

11.2459 
11.9478 
11.9605 

11.8581 
11.8557 
11.9588 
11.9010 
11.2686 

11.9009 
11.9689 
11.9715 
11.9741 
11.9767 

11.9798 
11.2fi«) 
11.9846 
11.9879 
11.9896 



9986984000 87.9478 11 .9994 
2992209121 87.9006 11.9950 
299844J888 87.9787 11 .9977 
8004685307,87.9888 11.3008 
8010986884 88.0000,11.9099 

8017190196 88.018211.8066 
8O234045d6}88.O268 11 .8061 
8029741628 88.0895 11.8107 
80:)6027892 38.0626 11.8188 
8042821849 88.0057111.8159 



8048695000 
3a-}4g:)6851 
306] 257408 
8067686677 
8073924064 



2117086 8060271875 

21 I993G 3086626818 
2122849,3092990998 
212570l';?0ft9863912 
212808113105745579 



88.078911.8186 
88.0920 11.8211 
88.1051 11.8237 
:». 1182' 11. 8268 
88.1814|ll.Se89 

88.1445111. 8816 
88. 1576' 11. 8841 
38.1707111.8867 
38. 1888M 1.8898 
88. 1969111. 8419 



213100013119186000 88.9099 11.8446 
21315. M 13118535181 38.9880 11 8471 
2137444 3124948128 38.9361 11.8496 
2140;i09l8181.359847 88.2492 11 .a')28 
2143296 3187785344 88.2693 11.8548 



9146325 
2149156 
21.'i2089 
2155<)24 
2157961 

9160900 
9163841 
2106784 
2169729 
9172676 



8144219625 88.977>8 
815066:696 3S.2884 
31571H563'38.8014 
810357523813^.8145 
31700447T)9 38.8276 



8176628000 

318801011 
818{)500048 
8190010817 
8202.524124 



88.8400 
3S.8680 
88.3607 
38.3797 
38.3997 



11.&574 
11.3600 
11.8626 
11.8(»2 
11.8677 

11.8708 
11.8729 
11.8756 
11.8780 
11.8806 



No, 



Square. 



1476 
1476 
1477 
1478 
1479 

1480 
1481 
1489 
1488 
1484 

1486 
1480 
1487 
1488 
1489 

1490 
1491 
1499 
1498 
1494 

1496 
1496 
1497 
1498 
1499 

1600 
1601 
1502 
1503 
1604 



9175685 
9178676 
9181599 
9184484 
9187441 

9190400 
9108861 
9190891 
91099B9 



8909046876 
8916578170 
3822118888 
8228687868 



8841799000 
8948807041 
8254959108 
8961646687 
8908147904 88 



9906190 
8911109 
9214144 



8974760195 
8961S70968 



8994040972 



9317191 8801298109 



9890100 



9890004 
2930049 



9988010 
2941009 
9244004 
9947001 

9960000 
6968001 
9950004 
9859009 
9969016 



1607 
1508 
1509 



1510 
1511 
1512 
1513 
1514 

1515 
1510 
1517 
1518 
1619 

1690 
1591 
1522 
1598 
1524 

1595 
1526 
1527 
1528 
1529 



1505 9906025 

1606 

2271049 
2274004 
2277081 



Cube. 



86.4067 
88.4187 
88.4818 
88.4448 
88.4678 



88.4706 

88!4908 
88 5007 



8807940000 
8814018771 
8821987488 
8827970167 
8084001784 88, 



8841809876 
8818071980 
8864790478 
8861517992 
8868954499 

8875000000 
8881764601 
8888518006 
8895290627 
840-.K>79064 



Sq. 
Root. 



88.5857 
88.6467 
86.6010 
88.6740 
88.6870 



88.0006 
88.0185 
88.0904 
88.0804 



88.00ia 

88.0789 
88.0911 
88.7040 
88.7109 

88.7996 
88.7497 
88.7660 
88.7086 
88.7814 



8408808025 88.7948 11 4598 



Cube 



11.8882 
11.8868 
11.8888 
11.8909 
11.8985 

ll.MGO 
11.8966 
11.4012 
11.4087 
11.4063 

1.4089 
11.4114 
11.4140 

n.4i» 

11.4191 

11.4910 
11.4942 
11.4268 
11.4203 
11.4819 

11.4844 
11.4370 
11.4895 
11.4421 
11.4446 

11.447! 
11.4497 
n .45<f2 
1I.4.'>4R 
11.4673 



3415602816 
8422470843 
3429288512 
3486116999 

8449951000 
8449795831 
3466649728 
3468512697 



9880100 
2288121 
2980144 
9289109 
9292196 8470884744 



8298950 
2301280 
8304324 
8807861 

8810400 
8818441 
2310484 
2819520 



3477206875 88.9980 
3484150096 88.9868 
8491055418 88.9487 
8497968882 88.9615 
3504881859 88.9744 



2«i1729 
2834784 
2887841 



88.8079,11.4624 
88.6901 11. 4n49 
38.83.W1 1.4615 
88.846811.4700 

88.8667 11.4725 
88 8716 1I.4T.M 
88 8844'll.4r70 
88.8973 11. 4Kn 
88.9102 11. 4tfi2a 



3611808000 
8618748701 
3525688048 
3582042667 
8589605824 

8540678196 
8553560670 
8560550188 
3667549969 
3574558869 



88.9678 
89.0000 
89.0198 
89.0260 
89.0884 

80.0618 
80.0040 
89.0768 
89.0600 
89.1094 



11.4a'(2 
11.4877 
11.4909 
11.4997 
11.4958 

11.4078 
11 6O08 
11.5028 
11 (X)54 
11.5079 

11.5101 
11.6129 
11.6154 
11.6179 
11.5W4 



SQUABBS, CUBES, SQUARE AND CUBE BOOTS. 101 



Xo 



Square. 



1530 
15M 



2SIO0OO 

SM7(»4 
SKOOtO 



3581577000 
»a8604«)l 
85Q5(H07l{8, 

360S74130A 



1S«I 2tS62&5 

1536 2859296 

1587 

158s 238&U4 

lS39t 8868581 

IMO 8871600 

IMl ^f7468l 

U42 S3T7761 

1518 3880849 
1544 



1346 

15t7 



$616S0S375 89.1791 
8688878656 88.1918 
8880961158 88.8046 
868^<06d8?e88.S178 
8645153819 88.8801 



869SQ64000 89. 
369088^)481 
8866512086 89 
3678660007 
8680797184 



8408500 
8405601 
8406704 

8411809 
8414916 

8418085 
8421188 
8484^19 
848r364 
8480481 

8483600 



3695119836 
37(l;>894888 
3709476698 
3716678148 

8?18875000 
8731087151 
87.18306606 
8745538877 
3758779464 



\^ 888^)85 86^7968685 
"•• *«»I16**-"'**"-^ 

8803809 

8896301 
1549i 8880101 



1330 

1551. 

1553! 

ISolj 

1S55 
1556 
1357 
1558 
156S^{ 

!360l 
1561 
156sS 
15G3 
1564 



84861^1 



8448969 
8446006 



Cube. 



Bq. 
Boot. 



39.1158 
88.1880 
89.1408 
89.1635 
39.1663 



39.8556 



8810 
8938 

8065 
3198 
8319 
8446 
8578 



39.8700 
39.3837 
89.8834 
89.4081 
39.4806 



8760098875 39.4336 
8797887610:394468 
877455569830.4588 
3781833112I89.4715 
8789119879 89.4648 



8796416000 
3803781481 



8489844 3d 1 1086388 



381836054' 
8883604144 



Cube 
Root. 



11.5830 
11.6865 
11.6880 
11.5305 
11.5830 

11.5365 
11.5380 
11.5406 
11.5430 
11.6455 

11.5480 
11.5605 
11.5680 
11.5565 
11.5580 

11.5605 
11.5680 
11.9666 
11.5680 
11.5706 

11.5790 
11.5764 
11.5779 
11.5804 
11.5829 

11.6664 
ll.58t9 
11.5908 
11.5988 
11.5958 



80 496811.5978 
89 eoa-) 11.6003 
39.588111.6037 
89. 634d 11.6053 
89.5474 U. 6077 



No. 



1565 
1566 
1567 
1568 
1569 

1579 
1571 
157% 
1573 
1574 

1575 
1576 
1577 
1578 
1579 

1580 
1581 
1583 
1588 
1584 

1586 
1586 
1587 
1588 
1589 

1590 
1591 
1593 
1593 
1594 

1505 
1596 
1597 
1598 
1599 



Square. 



Cube. 



8449335 3833087135 
8453856;3840889496 
8455469!3847751363 
8468634 386513348*^ 
8461761 j386250«)09 

8464000 3890693000 
8468041387739-^411 
a4711B4,3()84701848 
8474339 3893119517 
8477476,38095478^ 

8480635 3906984875 
8483770 3914430076 
8466939,3981887038 
8490084, 4930a')355<J 
84933413936837539 

9406400 8044313000 
84095613651805941 
2603734 3969309668 
8605889 896683»S87 
8609066 3Or4344704 

051283513981876635 
85158963989418056 
8518569 3996069008 
3681744 4004539473 
3584931 4013099469 



8588100 
8581281 
3584464 
85B7649 



4019679000 
4037368071 
4034866688 
404347485; 



8540836 4050094684 



2544035 
3547316 
3560409 
3553604 



4057719875 
4065356736 
4073003173 
4060659193 



3556801,4088334790 



1600 3360000'4Q96000000 40.0000'll.6961 



Sq. 
Root. 



39.5601 
89.5737 
39.5854 
;i9.508O 
39.6106 

89.6383 
89.6358 
89.6485 
30.6611 
80.6737 

89.6868 
89.6980 
89.7115 
.H9.7340 
89.7366 

30.7403 
89.7618 
89.7744 
369 
89.7995 



89.8181 11.6604 
39.894611.6619 
30.887311.6648 
80.8407ill.8668 
30,863311.6693 

89.874811.6717 
89.8873 11.6741 
39.8999 11.6765 
89.9134 11.6790 
39.984911.6614 

89.9375'll.7839 
89.0900 11.6868 
9635|11.6668 
89.075011.6913 
89. 9875,11. 6936 





Sai^ABBS ANB CVBBS OF BBCIMAM. 


5Eo. 


Square. 


Cube. 


No. 


Square. 


Cube. 


No. 


Square. 


Cube. 


.1 


.01 


.001 


.01 


.0001 


.000 001 


.001 


.00 00 01 


.000 000 001 


.2 


.04 


.006 


.03 


.0004 


.000 008 


.008 


.00 00 04 


.ooaooo 003 


8 


.09 


.037 


.03 


.0009 


.000 037 


.003 


.00 00 09 


.000 000 037 


4 


.16 


.064 


.04 


.0016 


.000 064 


.004 


.00 00 16 


.000 000 064 


.5 


.25 


.135 


.05 


.0035 


.000 135 


.005 


•00 00 35 


.000 000 125 


.6 


.36 


.816 


.06 


.0036 


.000 216 


.006 


.00 00 36 


.000 000 316 


.7 


.49 


.843 


.07 


.0040 


.000 34:^ 


.007 


.00 00 49 


.000 000 843 


.« 


.64 


.618 


.08 


.0064 


.000 513 


.008 


.00 00 64 


.000 000 51 -i 


.0 


.81 


.730 


.00 


.0061 


.000 739 


.009 


.00 00 81 


.000 000 739 


1.C 


1.00 


1.000 


.10 


.0100 


.001 000 


.010 


.00 01 00 


.000 001 000 


It 


1.44 


1.788 


.13 


.0144 


.001 738 


.013 


.00 01 44 


.000 001 728 



Nolo that the f^qiiare has twice as many decimal places, and the cube three 
times as many decimal places, as the root. 



102 



MATHEMATICAL TABLES. 



FIFTH ROOTS ANB FIFTH POWERS. 

(Abridged from Trautwinb.) 



g^ 




o ^ 




^o 




o^' 




H 




ll 


Power. 


68 


Power. 


6 


Power. 


dS 


Power. 


c 


Power. 




»« 




5?« 




&» 




J5« 




.10 


.000010 


3.7 


098.440 


0.8 


90392 


21.8 


4083507 


40 


102400000 


.15 


.000076 


3.8 


792.852 


9.9 


05099 


22.0 


61&3632 


41 


11585«-?01 


.90 


.000820 


8.9 


002.242 


lO.U 


100000 


22.2; 6392188 


42 


130691-^32 


.85 


.000977 


4.0 


1024.00 


10.2 


110408 


22.4| 5639403 


48 


147006443 


.80 


.00;i430 


4.1 


1158.56 


10.4 


121665 


22.6, 6895793 


44 


164916224 


.86 


.005252 


4.2 


i;«6.9i 


10.6 


133828 
140933 ' 


22.8 6161827 


45 


184588125 


.40 


.010240 


4.3 


1470.08 


10.8 


23. o; 6436343 


46 


805062976 


.45 


.018453 


4.4 


1649.16 


11. 


161051 


28.2, 6721093 


47 


289345007 


.CO 


.031250 


4 6 


1845.28 


11.2 


176284 


23.41 7015834 


48 


254803008 


.66 


.0508aJ 


4.6 


2050.63 


n.4 


192541 


28.6, 7820825 


40 


288475240 


.60 


.077760 


4.7 


2293.46 


11.6 


2101184 


23.8 7686832 


60 


318500000 


.06 


.116029 


4.8 


2548.04 


11.8 


228776 


24.0 706J624 


61 


345025851 


.70 


168070 


4.9 


2824.75 


12.0 


248832 


24.2 8299976 


58 


38020ltW3 


.75 


.2:J7805 


60 


8126.00 


12.2 


270271 


24.4 864^666 


53 


418105498 


.80 


.827680 


6.1 


8450.26 


12.4 


293163 


24.61 900H978 


64 


4501C5024 


.86 


.443705 


5.2 


8802 04 


12.6 


817580 


24.8, 9881200 


66 


503284375 


.90 


.600490 


6.3 


4181.95 


12.8 


843597 


85 9765626 


66 


550731776 


.06 


.778781 


6.4 


4591.65 


13.0 


871298 


25.2' 10162550 


67 


6016iKa'J7 


1.00 


1.00000 


6.6 


6032.84 


13.2 


400746 


25.4 10572278 


68 


666356708 


1.05 


1.27628 


6.6 


5.5<J7.32 


13.4 


482040 


25.6 10995118 


50 


714924899 


1.10 


1.61061 


5.7 


6016.92 


13 6 


465259 


85.8 11431377 


60 


777eCKW00 


1.16 


2.01135 


6.8 


«6«3.57 


13.8 


600490 


26.0 11881376 


61 


84459G301 


i.ao 


2.48832 


6.9 


7149.24 


14.0 


687824 


26.2 12345487 


62 


9161. '^»38 


1.S5 


3.05176 


6.0 


7776.00 


34.2 


677358 


26.4! 12828^86 


68 


992436543 


1.80 


8.71298 


6.1 


8445.96 


14.4 


619174 


26.6 133170f>5 


64 


1W3741824 


1.85 


4.46408 


6.2 


9161.33 


14 6 


663388 


26.8' 1882f)281 


65 


11602900V5 


1.40 


5.87824 


6.8 


9924.87 


14.8 


710082 


27.0, 14348907 


66 


1252882r76 


1.45 


6.40978 


64 


10737 


15.0 


759375 


27.2 14888280 


67 


1850185107 


l.SO 


7.59875 


6.5 


11603 


16 2 


811368 


27. 4 1 15413762 


68 


14589:«568 


1.56 


8.94661 


6.6 


125J8 


15.4 


866171 


27.6 16015681 


60 


1564U31349 


1.60 


10.4858 


6.7 


18501 


15.6 


923H96 


27.81 16604430 


70 


16W)7a)m>0 


1 66 


12.2898 


6.8 


14539 


15.8 


984658 


28.0 17210308 


71 


1804289351 


1 70 


14.1986 


6.9 


15C40 


16.0 


1048576 


28.2 17t'a3K0« 


72 


1934917032 


1.75 


16.4i:n 


7 


1C807 


16.2 


1115771 


28.41 1W;5309 


78 


2073071693 


1.80 


18.8957 


7.1 


18042 


16.4 


1186367 


28.6' 1«1850:6 


74 


22l90U6fr.'4 


1.85 


21.6?00 


7.2 


19849 


16.6 


1260408 


28.8 198I8&57 


75 


23': 8046875 


l.«0 


24.7610 


7.8 


20781 


16 8 


13:«'v>78 


29.0 20511149 


76 


95:^5516376 


1.95 


28.1951 


7.4 


22190 


17.0 


1419857 


29.8 212-J8253 


77 


2706784157 


:i.00 


32.WH)0 


7.5 


28780 


17.2 


1505386 


20. 4i 21965275 


78 


2887174368 


2.(H> 


36.2051 


7.6 


25356 


17.4 


15041)47 


a9.6, 2272x'628 


70 


3077066399 


2.10 


40 8410 


7.V 


27068 


17.6 


1688742 


29 8' yS.^00728 


80 


3276800000 


2.15 


45.9101 


7.8 


28872 


17.8 


1786H90 


30.0; 24dO0(K)O 


81 


3486781401 


2 v'O 


61.5:^ 


7.9 


aorri 


18.0 


1880568 


30. 5I 20H93634 


82 


3707308482 


2.25 


67.6650 


8.0 


32768 


18.2 


1996908 


81.0, 28629151 


83 


3989040648 


2.30 


64.3634 


8.1 


84868 


18.4 


2109061 


31.61 81013642 


84 


41H21 19424 


2.. 35 


71.6708 


82 


37074 


18.6 


2226203 


32.0, 835M432 


86 


4437053125 


2.40 


79.6-J62 


8.8 


89390 


18.8 


2348498 


82. 5I 3625«()«2 


66 
1 87 


4704270176 


2.45 


88.2785 


8.4 


41821 


19.0 


2476099 


33. 0| 39185303 


4984209207 


2.50 


97.6562 


8.5 


44371 


19.2 


2609193 


33.5' 42191410 


88 


5277819168 


2.55 


107.820 


8.6 


47048 


19.4 


2747M9 


34 0, 45435424 


80 


6684060449 


2.60 


I 118.814 


8.7 


49842 


19.6 


2892547 


81.5 48875980 


00 


5904900000 


2.70 


143.489 


8.8 


527:8 


19.8 


3043168 


35.0 52:j21875 


01 


6840821451 


2.80 


172.104 


8.9 


65841 


20.0 


8200000 


35.5 66382167 


02 


6500616288 


8.(» 


' 206.111 


9.0 


61K)49 


20.2 


3863232 


86.0' 60466176 


08 


6966888603 


3.00 


243 000 


9.1 


62403 


20.4 


3533a59 


86 5; W:a3487 


94 


73300)0824 


8.10 


286.292 


9.2 


65908 


20.6 


3709677 


37.0 60343957 


96 


TnJ7«00875 


8.20 


335.544 


9.3 


69569 


20.8 


8893289 


37.5, 74157716 


96 


8158786076 


3.80 


391.354 


9.4 


73390 


21.0 


4084101 


38. 0: 79235168 


97 


8587340257 


8.40 


454.854 


9.5 


7r378 


21.2 


4282822 


88.5, 84587005 
89.01 00224199 


98 


9030807068 


3.60 


625.219 


9.6 


81587 


21.4 


4488166 


99 


9509900400 


8.60 


604.662 


9.7 


85878 


21.6 


4701'^50 


39.5, 96158012 







OIBCUKFEBENCES AND ABEAS OF CIBOLES. 103 



OUr€tinFKKJfiM€l£ll ANJil AREAS OF CIBOIiBS. 


Ptom. 


ClrcQin. 


Area. 


Dlam. 


Circum. 


Area. 


Diaro. 


Circum. 


Area. 


1 


8.1416 


0.7854 


65 


204.20 


8818.31 


129 


406.87 


13069.61 


t 


e.ssae 


8.1416 


66 


207.84 


3421.19 


ISO 


406.41 


18278.83 


8 


0.4:M8 


7.0666 


67 


210.49 


8525.66 


131 


411.56 


18478.28 


4 


19.5664 


12.5664 


68 


218.68 


8631.68 


182 


414.69 


13684 78 


5 


15.7060 


19.635 


69 


216.77 


8739.28 


133 


417.88 


13802.91 


6 


18.850 


88 274 


70 


219.91 


8848.45 


134 


420.97 


14102.61 


7 


21.901 


88.486 


71 


223.00 


8859.19 


185 


424.12 


14318.88 


8 


«6.138 


60.266 


72 


226.19 


4071.50 


136 


427.26 


14526.72 


9 


88.274 


68.617 


78 


229.34 


4185.89 


187 


480.40 


14741.14 


10 


81.416 


78.510 


74 


232.48 


4300 84 


138 


433.54 


14957.18 


11 


84.556 


95.083 


75 


235.62 


4117.86 


139 


486.68 


15174.66 


S3 


87.699 


118.10 


76 


288.76 


4536.46 


140 


439.82 


15383.80 


28 


40.641 


132.73 


77 


811.90 


4656.68 


141 


442.96 


15614.50 


11 


48.962 


153.94 


78 


245.04 


4778.36 


142 


446.11 


15886.7? 


15 


47. la* 


176.71 


79 


248.19 


4901.67 


143 


449.25 


16060.CI 


16 


50.265 


201.06 


80 


251.88 


6026.55 


144 


452.89 


16286.02 


17 


53.407 


8^.96 


81 


254.47 


5158.00 


115 


455.58 


16513.00 


18 


56.549 


854.47 


82 


257.61 


5281.02 


146 


458.67 


16741.55 


19 


50.080 


283.53 


88 


860.75 


5410.61 


147 


461.81 


16971.67 


SO 


03.8»S 


814.16 


84 


863.89 


5541.77 


148 


464.96 


17203.86 


21 


65.973 


816.36 


85 


267.04 


5674 50 


149 


468.10 


17486.W 


a 


60.115 


880.13 


86 


270 18 


6808.80 


loO 


471.84 


17671.46 


23 


T5f.857 


415.48 


87 


273.88 


5914.68 


151 


474.88 


17907 86 


24 


75.896 


452.39 


88 


276.46 


&0tSi.l2 


152 


477.62 


18145.84 


25 


78.540 


490.87 


89 


279.60 


6^1.14 


153 


480.66 


18885.89 


» 


81.661 


530.98 


00 


282.74 


6381.78 


154 


483.81 


18626.50 


27 


64.828 


572.56 


91 


285.86 


6.V)8.88 


155 


486.95 


18869.19 


28 


67.965 


615.75 


92 


289.03 


6647.61 


156 


490.09 


19113.45 


» 


01.106 


660.62 


98 


292.17 


6792.91 


157 


498.23 


19359.26 


SO 


04 816 


706.86 


94 


205.31 


6939.78 


158 


496.37 


19606.68 


81 


97.889 


754.77 


95 


296.46 


7088.22 


159 


499.51 


19856.65 


» 


100.58 


804.25 


96 


801.69 


7236.23 


100 


502.66 


20106.19 


S3 


103.67 


855.30 


97 


801.78 


7389.61 


161 


505.80 


20858.81 


81 


106.81 


907.92 


96 


807.88 


7542.96 


162 


608.94 


20611.99 


S 


109.96 


962.11 


99 


311.02 


7697.69 


168 


512.06 


80867.24 


86 


118.10 


1017.88 


100 


814.16 


7853.96 


164 


515.22 


21124.07 


87 


116.84 


1075.21 


101 


817.30 


8011.85 


165 


.M8.86 


21882.46 


as 


119.86 


1184.11 


102 


3^.44 


8171.28 


166 


521.60 


21642.48 


89 


128.58 


1191.59 


103 


323.58 


8382.29 


167 


524.65 


21908 97 


40 


125.60 


1256.64 


104 


326.73 


8494.87 


168 


527.79 


22167 06 


41 


128.81 


1320.25 


105 


829 87 


8659.01 


169 


580.98 


22481.76 


42 


131.95 


1885.44 


106 


333.01 


8824.78 


170 


534.07 


28606.01 


43 


135.00 


1452.20 


107 


336.15 


8992.0S 


171 


537.21 


22965.88 


44 


188.28 


1520.53 


106 


389.29 


9160.88 


172 


510.85 


2:«i85.22 


45 


141.87 


1590.43 


109 


342.48 


9831.32 


173 


543.50 


28506.18 


45 


14451 


1661.90 


110 


345.58 


9503.32 


174 


546.64 


28778.71 


47 


147.65 


1734.94 


111 


348.78 


9676.89 


175 


549 78 


24052.82 


48 


150.80 


1809.56 


112 


351.86 


9852.08 


176 


552.92 


24828.49 


49 


158 94 


1885.74 


118 


355.00 


10028.75 


177 


566.06 


24605.74 


(0 


157.08 


1968.50 


114 


858.14 


10207.08 


178 


559.20 


84884.56 


51 


160.23 


2012.82 


115 


861.28 


10886 89 


179 


562.85 


25164.94 


52 


168.86 


2128.72 


116 


864.42 


10568.32 


160 


665.49 


25446 90 


&3 


166.50 


2806.18 


117 


867.57 


10751.82 


181 


568.68 


25780.43 


54 


109.05 


2290.22 


118 


370.71 


10985.88 


182 


571.77 


26015.58 


96 


1W.79 


2875.88 


119 


873.85 


11122.02 


183 


574.91 


26302.20 


56 


175.93 


2463.01 


180 


876.99 


11809.78 


181 


578.05 


26.')90.44 


57 


179.07 


2551.76 


121 


380.18 


11409.01 


185 


581.19 


26880.85 


58 


182.21 


2612.06 


122 


383.27 


11689.87 


186 


584.34 


27171.63 


59 


185.35 


8788.97 


188 


886.42 


11882.29 


187 


587.48 


27464.50 


60 


188.60 


8827.48 


124 


389 56 


13078.28 


188 


590.62 


2n59.ll 


61 


191.64 


2932.47 


125 


392.70 


12-J71.85 


189 


593.76 


28055.81 


ti 


194.78 


8019.07 


126 


395.81 


12468.98 


100 


596.90 


28:«»2 87 


63 


107.18 


8117.25 


127 


896.96 


iaK57.69 


191 


GOO.Ol 


28652.11 


64 


HOI .00 


8216.99 


128 


402.12 


12867.96 


192 


603.19 


28952.92 



104 



MATHBMATICAI. TABLES. 





Clrcum. 


Area. 


Dlam. Clrcum. 


Area. 


Diain. Clrcum. 

1 


Area. 


103 


606.88 


89865.80 


860 


816.81 


53092.92 


827 


1027.30 


89981.84 


194 


600.47 


29559.25 


261 


819.96 


53502.11 


828 


1030.44 


&I496 28 


m 


612.61 


29864.77 


262 


823.10 


53912.87 


829 


1033.68 


85013.28 


196 


615.75 


80171.86 


268 


820.24 


54325.21 


880 


1036 78 


a'il)2d.86 


197 


618.89 


80480.62 


264 


829.83 


64739.11 


331 


1039.87 


86049.01 


196 


622 04 


80790.75 


265 


aS2.52 


65154.59 


832 


1043.01 


86609.73 


199 


625.18 


81102.65 


266 


8:«.66 


55571.63 


838 


1046.16 


871098.02 


soo 


628.32 


81415.93 
81730.87 


267 


838.81 


55990.26 


8»l 


1049.29 


87615.^ 


201 


681.46 


268 


841.95 


66410.44 


885 


1052.48 


88141.31 


90-i 


684.60 


82047.89 


269 


845.09 


66832.20 


886 


1055.58 


88668.81 


903 


687.74 


82365.47 


870 


848.88 


67255.5JJ 


837 


1058.72 


89106.88 


204 


640.88 


82685.18 


271 


851.37 


67680.48 


9SS 


1061 86 


89727.03 


205 


644.06 


88006.86 


272 


854.51 


68106.90 


889 


1065.00 


9025x«4.74 


200 


647.17 


88329.16 


273 


857.65 


58584.94 


840 


1068.14 


90rJl8 U3 


207 


650.31 


88663.53 


874 


660.80 


58964.56 


841 


1071.28 


91326.88 


206 


653.46 


88979.47 


276 


863.94 


60395.74 


842 


1074.42 


91863.81 


209 


656.59 


84306.96 


876 


867.08 


69^8.49 


843 


1077.57 


98401 81 


210 


659.78 


346:i6.06 


877 


870.22 


60262.82 


344 


1080.71 


92940.88 


811 


662.88 


84966.71 


878 


873.36 


60698.71 


845 


1063.86 


93482.02 


2ri 


666.08 


85298.94 


279 


876.50 


61186.18 


846 


1066.99 


94021.73 


SIS 


669.16 


86688.78 


880 


879.66 


61575.82 


847 


1090.18 


94669.01 


S14 


672.80 


85968.09 


281 


882.79 


62015.88 


848 


1093.87 


95114.86 


215 


675.44 


86805.08 


882 


885.93 


62458.00 


849 


1096.42 


95662.28 


916 


678.58 


36648.64 


888 


889.07 


62901.75 


860 


1099.56 


90811.88 


817 


681.78 


86983.61 


284 


892.21 


68347.07 


851 


1108.70 


96761.84 


8)8 


084.87 


37325.26 


286 


895.85 


68798.97 


868 


1106.84 


97B18.97 


819 


668.01 


87668.48 


286 


898.50 


64242.48 


868 


1106.98 


97B67.68 


880 


G91.15 


88013.87 


287 


901.64 


64692.46 


854 


1118.12 


98422.96 


221 


694.29 


88359.68 


888 


904.78 


65144.07 


355 


1115.27 


98079.80 


2i» 


697.48 


88707.66 


889 


907 92 


65597.81 


856 


1118.41 


99538.22 


»d 


700.58 


89067.07 


290 


911.06 


66051.99 


a^7 


1121.55 


100098.81 


8si4 


703.72 


89408.14 


291 


914.20 


66508.30 


858 


1124.69 


100659.77 


iB5 


706.86 


89760.78 


292 


917.35 


669e6.l9 


859 


1127.88 


10182:2.90 


226 


710.00 


40115.00 


293 


920.49 


67425.65 


800 


1130.97 


101787.60 


227 


718.14 


40470.78 


294 


923.68 


67886.68 


861 


1134.11 


108853.87 


828 


716.28 


40828.14 


895 


926.77 


68349.28 


362 


1187.26 


102921.72 


829 


719.42 


41187.07 


206 


929.91 


68813.45 


868 


1140.40 


108491.13 


2S0 


722.67 


41547.56 


297 


9.33.06 


69279.19 


364 


1148.54 


104062.12 


281 


725 71 


41909.68 


298 


936.19 


69746.50 


365 


1146.68 


104634.67 


fi»2 


7:28.85 


42278.27 


299 


939.34 


70215 88 


SC6 


1149.82 


105808.80 


233 


781.99 


42688.48 


800 


942.48 


70685.63 


867 


1158.96 


105784.49 


284 


735.18 


43005.26 


801 


945.62 


71157.86 


368 


1156.11 


106861 76 


835 


788.27 


48373.61 


802 


948.76 


71681.45 


369 


1159.25 


106940.60 


836 


741.42 


48743.54 


808 


951.90 


72106.62 


870 


1168.89 


107521.01 


887 


744.56 


44115.08 


804 


955.04 


72588.86 


371 


1105.58 


108102.99 


888 


747.70 


44488.09 


805 


958.19 


78061.66 


372 


1168.67 


108686.51 


239 


750.84 


44862.73 


806 


961.83 


78541.64 


87« 


1171.81 


109871.66 


240 


753 98 


452:)8.98 


807 


964.47 


74022.99 


874 


1174 98 


109858.35 


241 


757,12 


45616.71 


806 


967.61 


74506.01 


875 


1178.10 


110446. 68 


212 


760.27 


45996.06 


809 


9T0.75 


74990.60 


376 


1181.24 


111036.45 


243 


763.41 


46876.98 


810 


973.89 


75476.76 


877 


1184.88 


111627.86 


244 


766.55 


46759.47 


811 


977.04 


75964.60 


878 


1187.52 


118220.83 


215 


769.69 


47143.52 


812 


980.18 


76468.80 


879 


1190.66 


112815.88 


846 


772.88 


47529.16 


818 


963.32 


76944.67 


380 


1198.81 


118411.49 


247 


775.97 


47916.86 


814 


986.46 


77487.12 


381 


1196.95 


ll40m).18 


248 


779.11 


48305.18 


815 


989.60 


77981.13 


382 


1200.08 


114608.44 


249 


782.26 


48695.47 


816 


992.74 


784-26.?2 


888 


1208.28 


115909.87 


850 


785.40 


49087 89 


817 


995.88 


78923.88 


384 


1206.87 


115811.67 


251 


788.54 


494H0.87 


318 


999.03 


79422.60 


C% 


12C9.51 


116415.64 


2.M 


791.68 


49875.92 


819 


1002.17 


79922.90 


386 


1218.65 


117021.18 


253 


794.82 


50272.56 


880 1005.31 


80424.77 


387 


1815.80 


117628.80 


254 


;97.96 


50670.75 


821 1008.45 


60928.21 


888 


1218.94 


118886.98 


855 


801.11 


61070.52 


822 1011.59 


81483 22 


380 


1223.08 


118847.24 


856 


804.25 


51471.55 


823 1014 73 


81939.80 


800 


1226.82 


119159.0(> 


257 


807.89 


51874.76 


824 


1017.88 


82447.96 


391 


1228.36 


180078.46 


256 


810.58 


52279.21 


325 


1021.02 


82957.68 


392 


1281.80 


180687.4:tt 


259 


813.67 


52085.29 


326 


1024. IG 


83468.98 


898 


1234.65 


181808.96 



CIRCUHFERBNCES AI^D AREAS OF CIRCLES. J 05 



Oum. 


Circum 


Area. 


Dlam. 


Clreum. 


Area. 


Dlam. 


Clrcam. 


Atml 


394 


1^87.79 


121922.07 


461 


1448.27 


166918.60 


628 


1668.76 


SI 8956.44 


3» 


1340.03 


122541.75 


462 


1451 .42 


167638.53 


529 


1661.90 


219786.61 


3« 


1-444.07 


123168.00 


468 


1454.56 


168366.02 


680 


1665.04 


220618.84 


•W 


1-^47.21 


123786 82 


464 


1457.70 


169098.08 


531 


1G68.19 


2214.51.65 


3* 


ri5U.35 


1-^4110.21 


465 


1460.84 


169822.72 


58-4 


1671.33 


a42286.68 


3)9 


1^^33 50 


125036 17 


466 


1463.98 


170553.9-4 


633 


1674.47 


228122.98 


«oa 


1456.64 


125663.71 


467 


1467.1« 


171286.70 


584 


1677.61 


228961.00 


101 


1239.7« 


128292.81 


408 


1470.27 


1720-41.05 


535 


1680.75 


224800.59 


•XH 


ISOi^i 


1:W623.48 


469 


1473.41 


172756.97 


636 


1683.89 


225641.75 


4^ 


1:366.06 


127S55.73 


470 


1476.65 


173494.45 


637 


1687.04 


226484.48 


^iM 


1269.20 


128189.55 


471 


1479.69 


174-438.61 


538 


1690.18 


227828.79 


K» 


l3S7i.85 


128884.93 


472 


1482.88 


174974.14 


539 


1C93.32 


228174.66 


*« 


1;!73.49 


129461.69 


473 


1485.97 


1757l«.ai 


640 


1696.46 


229022.10 


■lor 


I3r78.6» 


180100.42 


474 


1489.11 


176460 12 


Ml 


1699.60 


229871.12 


•lOS 


iai.77 


130740.52 


475 


1498.26 


177205.46 


542 


1702.74 


280721.71 


4'.i9 


1C&I.91 


181382.19 


476 


1496.40 


177952.37 


548 


17«»5.8b 


2815:8.66 


410 


i:08S.o& 


183»26.43 


477 


1498.54 


178r00.8ti 


544 


1709.08 


282427.59 


411 


1:291.19 


18i670.24 


478 


1501.68 


179450.01 


546 


1712.17 


2332;j2.89 


412 


1294.84 


133316.63 


479 


1504.8* 


180202.64 


646 


1715.81 


284139.76 


413 


1297. 4« 


133964.68 


480 


1607.96 


180955.74 


547 


1718.45 


284098.20 


4U 


\300.64 


181614.10 


481 


1511.11 


181710.60 


548 


1721.59 


285858.21 


415 


1303.76 


1855»6.20 


482 


1514.25 


]8:?466.84 


549 


17-44.73 


286719.79 


419 


1306.90 


185917.86 


483 


1617.89 


188-2d4.75 


650 


17-47.88 


287682.94 


417 


1310.04 


180672.10 


484 


155».53 


183984.23 


651 


1731.0-4 


288447.67 


4ie 


1313.10 


187227.91 


485 


1523.67 


184746.28 


652 


1734.16 


239818.9G 


419 


131633 


187885.29 


486 


1526.81 


185507.90 


553 


1737.80 


240181.88 


4iO 


1319.47 


188544.24 


487 


1529.96 


186278.10 


554 


1740.44 


241051.26 


Ul 


13;8.6i 


139»l.76 


488 


1633.10 


187087.86 


556 


1743.58 


241922.27 


4tS 


1135. 73 


139866.85 


480 


]5;j6.24 


187805.19 


656 


1746.73 


242794.85 


4S 


I3«.t« 


140530.51 


490 


1539.38 


183574.10 


557 


1749.87 


243668.09 


4:il 


133)2 04 


141195.74 


491 


1642.52 


189344.57 


556 


1758.01 


2445t4.n 


445 


1335.18 


141862 M 


492 


1645.66 


190116.62 


559 


1756.15 


245422 00 


4:90 


13«.8i 


142580.92 


498 


1548.81 


190890.24 


600 


1759.29 


246300.86 


ur 


1341.46 


143200 86 


491 


1651.05 


191666.48 


561 


1762.43 


247181.80 


4S 


1344 60 


143872.88 


495 


1555 09i 194442.181 


562 


1765.58 


248063.80 


4t» 


1347.74 


144545.40 


496 


1558.281 19:^420.51 1 


668 


17tfs.72 


248046.87 


IM 


1350.88 


145220.1-4 


497 


1661.37 


194000.41 


664 


1771.86 


240632.01 


431 


13:V4.03 


145896.35 


498 


1564.51 


194781.89 


566 


1775.00 


250718 73 


43« 


1837. (7 


146574 15 


499 


1567.65 


195564.93 


666 


1778.14 


251607.01 


in 


1360.31 


147258.52 


600 


1570.80 


196849.54 


667 


1781.28 


862496.87 


431 


1363.40 


14798146 


601 


1578.94 


197185.72 


566 


1784.42 


253386.30 


4« 


1366.69 


14^16.97 


502 


1677.08 


197948.48 


669 


1787.57 


254281 .29 


4» 


1369.73 


149301.05 


508 


1580.22 


198712.80 


670 


1790.71 


255175.66 


437 


137«.88 


149986.70 


604 


1688.86 


199508.70 


671 


1793.85 


256072.00 


m 


1376.0:: 


150673.93 


605 


1586 50 


200296.17 


672 


1706.99 


250969.71 


4» 


1379.16 


151362.7^ 


506 


1680.65 


201090.20 


678 


1800.13 


267808.99 


4I# 


138^.30 


152053.08 


607 


1594.79 


201885.81 


674 


1803.27 


25H7t>9.85 


411 


1385.44 


l&-i743.U2 


608 


1595.93 


202682.99 


r75 


1806.42 


259672.27 


4«e 


1386.68 


163438 53 


600 


1599.07 


203481.74 


576 


1809.56 


360576.26 


4U 


1391.78 


]54l3i.60 


610 


1602.21 


204282.06 


577 


1812.70 


261481 83 


4i4 


1394.87 


154880.25 


511 


1G05 85 206083.95 


678 


1815 84 


«6238H.06 


443 


1398 01 


155548.47 


512 


1608.501 805887.42 


679 


1818.98 


268497.67 


444 


1401.15 


15e2«.26 


613 


1611.641 20G692.45 


680 


1842 12 


264207.9^ 


447 


1404.29 


160M9.62 


614 


1614.78 207499.05 


581 


1843.27 


265119.70 


444 


1407.43 


157632 55 


616 


1017.92 208307.23 


bSi 


1828.41 


266088.21 


449 


1410.68 


158337 06 


616 


1621.06 209116 97 


583 


1831.55 


266948.20 


4»9 


14I37< 


159043.13 


617 


16*44.20 2099:88.29 


584 


18:)4.69 


267864.76 


451 


1416 86 


160790.77 


618 


1627.84; 210741.18 


586 


1837.83 


268782.89 


45i 


t4:«>.00 


160489.90 


510 


16:».49i 211555 63 


586 


1840.07 


269702.69 


45S 


1423.14 


161170.77 


420 


1638.63 212J71.66 


6S7 


1844.11 


270628.66 


454 


14:36.28 


161883.13 


521 


1636.77: 213189.26 


588 


1847.26 


271546.70 


tfft 


1429 42 


16«97 06 


522 


1689.91 214008.43 


589 


18.50.40 


272471.12 


«9t 


1432.67 


168312.55 


628 


1643.05, 2148-49.17 


600 


1853.54 


273897.10 


457 


1435.71 


161029.02 


524 


1646.10 21 .7651.49 


591 


1856.68 


974:^24.66 


4SB 


1438.85 


16474S.26 


6« 


1619.84 216475.37 


5U2 


1859.82 


275258.78 


4J9 


1441.99 


16546S.47 


526 


1652.48, 217300.8:4 


693 


1862.96 


276184.48 


4W 


1445.18J 


106199.26 


697 


1656 621 218127.85 


594 


1866.11 


277116.75 



106 



MATHEMATICAL TABLES. 



Biam. 


Clrcam. 


Area. 


Dlam- 


Clrcam. 


Area. 


Dlam* Clrcum. 


Area. 


&»6 


1869.86 


878050.58 


663 


8088.88 


845836.69 


731 i 8896.60 


419686.15 


596 


18^^.89 


278985.99 


664 


9086.08 


846278.91 


788 1 2890.65 


480835.19 


597 


1875.53 


279988.97 


665 


2089.16 


347888.70 


733 1 2808.79 


421985.79 


5y8 


1878.67 


280861.58 


666 


2098.80 


348368.07 


781 


2305 93 


428187.97 


599 


1881.81 


881801 65 


667 


2095.44 


849415.00 


735 


2309.07 


424891 .78 


600 


1884.96 


283743.84 


668 


2098.58 


350463.51 


736 


2812.81 


425447.04 


001 


1888.10 


883686.60 


669 


8101.78 


851518.59 


737 


2315.85 


486003.94 


608 


1891.24 


884631.44 


670 


8104.87 


858565.24 


738 


23i8.50 


427768.40 


60S 


1894.38 


885577.84 


671 


2108.01 


853618.45 


739 


2881.64 


428982.43 


604 


1897.68 


286585.88 


678 


2111.15 


854673 24 


740 


2384.78 


480084.03 


605 


1900.66 


887475.36 


673 


2114.29 


355789.60 


741 


2827.98 


431247.81 


606 


1908.81 


888486.48 


674 


2117.43 


856787.54 


742 


8381.06 


432411.95 


607 


1906.96 


889879.17 


675 


2120.58 


857847.04 


748 


2834.80 


438578,87 


606 


1910.09 


290688.43 


676 


2123.72 


358908.11 


744 


2837.34 


484746.16 


600 


1918.23 


291.889.26 


677 


2186.86 


859970.75 


746 


2840.49 


435915.68 


«10 


1916.87 


898846.66 


678 


2130.00 


861034.97 


746 


2343.68 


487086.61 


611 


1919 51 


293205.63 


679 


2188.14 


362100.75 


747 


£846.77 


488259.84 


61SS 


1928.65 


294166. r< 


680 


8186.28 


868168.11 


748 


2349.91 


439488.41 


618 


1985.80 


295188.88 


681 


2139.42 


864237.04 


749 


2353.05 


440009 16 


614 


1928.94 


296091.97 


688 


2142.57 


365807.54 


760 


2856.19 


441786.47 


615 


1988.08 


897057.82 


688 


2146.71 


886379.60 


751 


2859.34 


448965.35 


616 


1935.22 


298084.05 


684 


2148.85 


867458.84 


762 


2362.48 


444145. ») 


617 


1938.36 


898998.44 


685 


2151.99 


368528.45 


753 


2365.62 


445887.553 


618 


1941.50 


899968.41 


686 


2155.13 


869605.88 


754 


2868.76 


446511.48 


619 


1914.65 


300933.95 


687 


2158.27 


870683.59 


755 


2871.90 


447696.f.9 


680 


1947.^9 


301907.05 


688 


2161.42 


371768.51 


756 


8875.04 


448888.38 


63il 


1950.93 


808881.73 


689 


2164.56 


878845.00 


757 


2378.19 


450071 .63 


622 


1964.07 


803857.98 


690 


2167.70 


878988.07 


758 


2881.83 


451861.51 


688 


1957.21 


304885.80 


691 


2170.84 


875012.70 


759 


8384.47 


462452.96 


684 


1960.35 


805815.20 


692 


2173.98 


376098.91 


760 


8387.61 


458645 98 


685 


1963.60 


806796.16 


693 


2177.12 


877186.68 


761 


2390.75 


464840.57 


686 


1966.64 


3Ui7'i^.69 


694 


2180.87 


878876.08 


762 


2893.89 


456036.73 


627 


1969 78 


808768; 79 


695 


2188.41 


879366.95 


763 


S897.04 


457284.46 


688 


1972.98 


309748.47 


696 


2186.55 


380459.44 


764 


M400.18 


468483.77 


689 


1976.06 


810785 71 


697 


2189.69 


381558.50 


766 


2408.88 


469684 64 


680 


1979.80 


311784.53 


698 


2192.83 


888649.13 


766 


8406.46 


460837.08 


631 


1988.35 


818714.92 


699 


2195.97 


883746.88 


767 


2409.60 


468041.10 


638 


1985.49 


818706.88 


700 


2199.11 


384845.10 


768 


2412.74 


463846.69 


688 


1988.63 


314^00.40 


701 


2208.26 


885945.44 


769 


2416.88 


464458 84 


834 


1991.77 


815695.50 


702 


2805.40 


887047.36 


770 


2419.08 


46566B2.57 


635 


1994.91 


816692.17 


703 


2208.54 


388150.84 


771 


2422.17 


466872.87 


636 


1908.05 


817«90.48 


704 


2211.68 


889256.90 


778 


2485.81 


468084 74 


687 


2001.19 


818690.23 


705 


2814.82 


890368.52 


778 


2488.45 


469208.18 


638 


8004.34 


319691.61 


706 


2817.96 


891470.72 


774 


2431.59 


470518.19 


689 


2007.48 


380694.56 


707 


2281.11 


892580.49 


776 


2434.78 


471729.77 


640 


8010.68 


381690.09 


708 


2284.25 


893691 .82 


776 


2487.88 


47^2947.0^! 


641 


8018.76 


388705.18 


709 


2887.89 


894804.73 


777 


2441.02 


474167.05 


64:? 


8016.90 


383718.85 


710 


2830 53 


89r>919.81 


778 


2444. 1C 


475888.94 


643 


2080.04 


38478?2.09 


711 


2833.67 


397085.26 


779 


JW47.80 


476611.81 


644 


2083.19 


825788.89 


712 


2836.81 


398158.89 


780 


2450.44 


47T836.24 


645 


8086.33 


888745.87 


718 


2239.96 


899872.08 


7bl 


2458.58 


470U62.85 


646 


8089.47 


887759.88 


714 


2248.10 


400392.84 


782 


£466.73 


480280.83 


647 


208261 


888774.74 


715 


2246.24 


401515.18 


7«3 


2459.87 


481618.97 


648 


2035.75 


889791.83 


716 


2849.38 


408689.08 


784 


2463.01 


482749.69 


649 


8038.89 


3:30810.49 


717 


2858.58 


403764.50 


785 


2466.15 


488081.98 


660 


9012.04 


331830.78 


718 


2855.66 


4(M891.60 


786 


2469.29 


485815.84 


661 


2045.18 


338858.58 


719 


2858.81 


406080.82 


787 


2472.48 


486451.28 


668 


8048.88 


388875.90 


720 


8861 95 


407150.41 


788 


2475.58 


487688.88 


653 


2051.46 


8JH900.86 


721 


2865.00 


408282.17 


789 


8478.72 


488986.85 


654 


8054.60 


a3.5U87.36 


782 


2868.23 


409415.50 


700 


2481.86 


490166.99 


655 


2057.74 


336955.45 


783 


2871.87 


410550.40 


791 


2485.00 


491408.71 


656 


8060.88 


337985.10 


721 


2874.51 


411686.87 


792 


2488.14 


492651 .99 


657 


8064.03 


839016.33 


725 


2-.'77.65 


412824.91 


798 


2491.28 


403896.85 


658 


8067.17 


840049.13 


786 


2880.80 


418964.62 


794 


2494.42 


495148.28 


659 


8070.31 


341083.50 


787 


88f>5.94 


415105.71 


796 


2497.57 


496391.27 


660 


2073.45 


348119.44 


728 


2887.08 


416848.46 


796 


2500.71 


497640 84 


661 


8076.59 


3431.')6.93 


789 


8890.88 


417392.79 


797 


2508.85 


498891.98 


668 


8079.73 


344196.03 


780 


8898.36 


418538.68 


796 


2606.99 


600144.69 



CIECUMFEREKCES ANt> AREAS OF CIRCLES. 107 



DlanLJClrciun. 


Area. 


Diain.|Clrcum. 


Area. 


Dlnm. 


CIrcum. 


Area. 


799 


2510. M 


50l89ri.97 


867 


2723.76 


590375.16 


936 


2987.89 


686614.71 


800 


2518.27 


502654.82 


868 


2?26.n0 


591787.8:3 


936 


2940.63 


688084.19 


801 


2516.42 


50:«I2.25 


869 


2730.04 


598102 06 


937 


2948.67 


689555.24 


8(tt 


2519.56 


505171.24 


870 


2783.19 


694167.87 


938 


2946 81 


691027. 86 


m 


252;!. 70 


506431.80 


871 ] 2736.83 


595835.25 


939 


2949.96 


692502.05 


8M 


•i5i5.84 


507693.94 


872 1 2739.47 


597204.20 


940 


2953.10 


693977.82 


805 


2528 98 


5U8057.61 


873 2742.61 


598574.72 


941 


2956.24 


695455.15 


9» 


2532.121 510222.92 


874 2745.75 


599946.81 


942 


2959.38 


696934.06 


W7 


2585.27 


511489.77 


875 


2748.89 


601320 47 


948 


2962.52 


698414.58 


«M 


:»38.41 


512758. 19 


876 


2752.04 


602695.70 


944 


2965.66 


699896.58 


H09 • 2311.55 


514028.18 


877 


2755.18 


604072.50 


945 


8968.81 


701380.19 


910 


2344.69 


515299.74 


878 2758.82 


(J05450.88 


946 


2971.95 


702865.88 


811 


2547.88 


51(3372.87 


879 


2761 .46 


606830.82 


947 


2975.09 


704352.14 


812 


2550.97 


517847.57 


TJ80 


2764.60 


606212.34 


948 


2978.28 


705840.47 


813 


2554.11 


51912^) 84 


881 


2767.74 


009595.42 


949 


2981.87 


707830.37 


814 


)»57.26 


520401.68 


882 


2770.88 


610960.08 


960 


2984.51 


706821.84 


813 


2.MS0.40 


521681.10 


883 


2774.03 


612366.31 


951 


2987.65 


710314.88 


81« 


2563.54 


522962 08 


884 


2777.17 


613754.11 


952 


2990.80 


711809.50 


817 


2566.68 


524244.68 


883 


2780.81 


615143.48 


958 


2990.94 


718305.66 


818 


2569.82 


525528.76 


886 


2783.45 


616534.42 


954 


2997.08 


714803.43 


819 


2572 S6 


526814 46 


887 


2786.59 


617926.93 


955 


3000.22 


716302.78 


htO 


2376.11 


528101.78 


888 


2789.78 


619321.01 


956 


3008.36 


717806.66 


tiil 


2579.25 


529390.56 


889 


2792.88 


620716 66 


957 


8006.50 


719306 12 


9U 


2582.39 


530680.97 


800 


2796.02 


622113.89 


968 


8009 65 


720610.16 


8i3 


^85.53 


531972.96 


891 


2799.16 


623512.68 


959 


3012.79 


722315.77 


im 


2588.67 


533266.50 


892 


28(12.30 


624913.04 


900 


8015 93 


728822.96 


8» 


2591.81 


534561.62 


893 


2805.44 


626314.98 


961 


3019.07 


725331.70 


8:96 


2594.96 


535858.32 


894 


2808.58 


027718 49 


962 


8022.21 


726842.02 


847 


2598.10 


537156.58 


896 


2811.78 


629123.56 


968 


3025.85 


728853.91 


838 


2601 .24 


53S456 41 


896 


2814.87 


630530.21 


964 


3028.50 


729867.37 


(fi» 


2604.38 


539757.82 


897 


2818.01 


6:31938.48 


965 


8031.64 


731882.40 


sso 


260r.52 


541060.79 


808 


2821.16 


63:3:348.22 


966 


8034.78 


782890.01 


Ml 


2610.66 


512365.34 


899 


2824.29 


634759. 5S 


967 


3037.92 


734417.18 


«3-* 


2613.81 


54SG71.46 


000 


2827.43 


636172.51 


968 


8041.06 


rd6936.93 


884 


2616.95 


544979.15 


901 


2«».58 


6:37587.01 


969 


3044.20 


737458.24 


834 


2620.09 


546288.40 


902 


2833.72 


639003.09 


970 


8047.34 


738981.13 


8» 


2623.23 


517599.23 


903 


2836.86 


640420.7:3 


971 


3050.49 


740505.59 


836 


2626.37 


518911.68 


904 


2840.00 


641839.93 


972 


3058.63 


742031.62 


837 


2629.51 


530225.61 


905 


2843. 14 


643260.78 


978 


8056.77 


743569.22 


8« 


2682.65 


551541.15 


906 


2846.28 


644683.09 


974 


8059.91 


745068.39 


K» 


2635.80 


552868.26 


907 


2849.42 


640107.01 


975 


3063 05 


746619.13 


840 


2688.94 


554176 94 


S08 


2852.57 


6473:32.51 


976 


3066.19 


748151.44 


841 


2642.08 


555497.20 


909 


2853.71 


648959.58 


m 


3069.84 


749685.82 


812 


2645.22 


656819.02 


910 


2858.85 


68a388.22 


978 


3072.46 


751220.78 


^) 


2648 36 


558142.42 


911 


2861.99 


651818.48 


979 


8075.62 


752757.80 


84t 


2651.50 


559467.39 


912 


2865.13 


653230.21 


980 


8078.76 


754296.40 


»15 


2654.65 


56U'/»8.92' 


913 


2868.27 


654683.56 


961 


3081 90 


755886.56 


816 


2857.79 


562122.08 


914 2871.42 


650118.48 


982 


8085.04 


757878.80 


847 


2660.94 


563451.71 


915 1 2874.56 


6.37554.98 


983 


3088.19 


758921.61 


818 ■ 9664.07 


564782.96 


916 , 2877.70 


658993.04 


984 


3091.33 


760466 48 


849 ^667. 21 


566115.78 


917 i 2880.81 


660432.68 


966 


3094.47 


762012.93 


8eO 


2670.35 


567450.17 


918 28K3.98 


661873.88 


986 


8097.61 


7t>35(;0.95 


851 


2678.50 


568786.14 


919 2887.12 


663316 66 


967 


3100.75 


765110 54 


ssa 


3676.64 


670128.67 


920 : 2890.27 


664761.01 


988 


3108.89 


766661.70 


858 


2679.78 


571462.77 


921 ; 2893.41 


666206.92 


9S9 


8107.04 


768214.44 


854 


2682 92 


572803.45 


922 . 2896.55 


667654.41 


990 


3110.18 


769768.74 


S5S 


2686.06 


574145.69 


923 


2899.69 


669103.47 


991 


3113.32 


771324.61 


856 


2689.20 


575480.61 


924 


:>90J.83 


670554.10 


992 


3116.46 


772882.06 


857 


2692.34 


576884.90 


926 


8905.97 


672006.30 


993 


811960 


774441,07 


858 


9696.49 


578181.85 


926 


2909.11 


6'i:3400.08 


994 


3122.74 


776001.66 


859 


2896.68 


579580.38 


927 1 2912. 2fi 


674915.42 


995 


3125.88 


777563.82 


800 


2701.77 


580880.48 


928 2915.40 


67(^72.33 


996 


3129.03 


779127.54 


861 


2704.91 


6822SS.15 


929 • 2918.54 


677h80.82 


997 


3132.17 


780(i92.84 


ett 


S708.06 


568585.89 


9S0 1 2921.68 


(579290.87 


998 


3135.81 


782259 71 


888 


9711.19 


684940.20 


931 1 2924. K2 


680r.V2.50 


999 


3138.45 


783R28.1fi 


864 


2714 84 


588*96.69 


932 2927.% 


6^2215.69 


1000 


8141.59 


785398.16 


880 


8717.48 


68^54.54 


988 29:11.11 


68:3680.46 








886 


2390 6^ 


980014.07 


984 2934.25 


685146.80 









108 



MATHEMATICAL TABLES. 



CIBCUMFERENCBS AND AHBAS OF CIACJLBS 

Advanolus by Elslitli*. 



Dlam. 


Ciroum. 


Area. 


Plain. 


Clrcum. 


Area. 


Dlam. 


Ciroum. 


Area. 


•1/M 


.04009 


.00019 


2 H 


7.4613 


4.4301 


H 


19.949 


99.465 


^m 


.00818 


.00077 


7/16 


7.6576 


4.6664 


H 


10.630 


80 680 


8/64 


.14796 


.00173 


H 


7.8540 


4.9067 


% 


90.098 


31.019 


1/10 


.19635 


.00807 


V 


8.0508 


5.1579 


V6 


90.490 


38.183 




.99459 


.00690 


8.9467 


5.4119 


^ 


90.613 


34.479 


^ 


.89270 


.01997 


11/16 


8.4480 


5.8797 


P 


91 900 


35.7«5 


b/oi 


.490B7 


.01017 


18/16 


8.6394 


5.9896 


% 


91.598 


87. 199 


8/10 


.58905 


.09761 


8.8357 


6.9196 


7. 


91.091 


88.485 


7/ai 


.68799 


.08758 


k 


9.0891 


6.4918 




99.884 


39.871 








15/16 


9.9984 


6.7771 


^ 


99.770 


41 982 


9%i 


.78540 


.04909 








3|l 


93.109 


49.718 


.88857 


.06;il3 


8. 


9.4948 


70680 


7* 


93.5«i 


44-179 


5/16 


.98175 


.07670 


1/16 


9.6911 


7.8069 


M 


98.956 


46 664 


lim 


1.0799 


.00981 


k 


9.8175 


7.6699 


% 


94.847 


47.173 




1.1781 


.11045 


8/16 


10.014 


7.9798 


yi 1 


94.740 


48.707 


izm 


1.9768 


.1900.J 


M 


10.910 


8.9968 


8. 


95.183 


50 96.-) 


7/16 


1.8744 


.160:33 


6/16 


10.407 


8.6179 


/i 


95.595 


61.849 


15/8;) 


1.47y6 


.17937 


H 


10.008 


8.9469 


M 


95.918 


58.456 








7/16 


10.7119 


9.9806 




96.3U 


50 088 


1754l 


1 5708 


.19635 


9% 


10.996 


9.6911 


1 1 


90.704 


56.745 


1.0000 


.99166 


11.199 


9.0678 


H \ 


97.090 


58.426 


»/16 


1.7«71 


.94850 


H 


11.888 


10.881 


k 


97.480 


00.189 


19^3 


1.8668 


.97688 


4' 


11.585 


10.680 


H 


97.889 


01.869 




1.9G85 


.80080 


11.781 


11.045 


9. 


98.974 


08.617 


31^ 


a 061? 


.38894 


13/16 


ii.»rr 


11.416 


M 


98.667 


0.5 807 


11/10 


9.1598 


.87199 


155^6 


19.174 


11.793 


^ 


99.060 


67.901 


98/84 


2.9580 


.40674 


19.870 


19.177 


fl 


90.459 


69.0199 








4. 


19.566 


19.666 


L , 


90.845 


70.889 


95/1) 


9.8569 


.44179 


■a* 


19.768 


19.909 


yi 


30.988 


7«.Teo 


2.4544 


.47937 


19.059 


13.864 


^ 


30.681 


74.069 


13/10 


9.5.595 


.51849 


3/10 


13.156 


13.779 


tZ 


31.098 


76.689 


97/89 


9.6507 


.55914 


5/16 


13.358 


14.186 


10. 


31.416 


78 640 


k 


9.7489 


.60139 


13.548 


14.607 


H 


31.809 


80.616 


90/B-2 


9.8471 


.64504 


7/16 


13.744 


15.083 


H 


89.901 


89 516 


15/16 


9 9459 


.69099 


18.941 


15.466 


% 


89.594 


84.541 


81/39 


3.0484 


.78708 


i2 


14.187 


15.9(H 


K 


39.987 


86.590 








f 


14.834 


16.849 


K 


88.879 


86.664 


1. 


3.1416 


.7854 


14.530 


16.800 


» 


88.779 


90.7li3 


1/16 


3.8879 


.8866 


11/16 


14.796 


17.987 


U 


34.165 


d9.886 


H 


8.5848 


.9940 


N 


14 998 


17.781 


11. 


84.558 


96.083 


V 


3.7806 


1.1075 


X 


15.110 


18.190 


H 


34.050 


97.905 


8.9S70 


l.«79 


15.815 


18.665 


u 


85.348 


99.409 


5/16 


4.1988 


1.8580 


15/16 


15 519 


19.147 


?i 


85.780 


101.69 


9^ 


4.8197 


1.4849 


6. 


15.708 


19.685 


a 


86.198 


108.87 


7/T6 


4.5160 


1.6980 


1/16 


15.004 


90.199 


H 


86.591 


106.14 


k 


4.7194 


1.7671 


k 


16.101 


90.699 


S 


80.914 


108.43 


9/16 


4 0087 


1.9175 


3/16 


16.997 


91.185 


H 


87.800 


110.75 


5.1051 


9.0780 


5/l6 


16.499 


91.648 


12. 


87.090 


118.10 


11/16 


5.8014 


9.9365 


16.690 


99.166 


H 


88.099 


115.47 


6.49« 


9.4053 


% 


16.886 


99.691 


b 


88.485 


117.86 


la/te 


5.6941 


9.6909 


7/16 


17.088 


98.991 


n 


88.877 


190.98 


H 


58905 


9.7619 


H 


17.979 


93.758 


» 


80.970 


199.73 


13/J6 


6.0668 


9.9488 


''a' 


17.475 


94.801 


H 


89.068 


195.19 








17.671 


94.850 


H 


40.066 


197.68 


S. 


6.9889 


3.1416 


11/16 


17.868 


95.406 


H 


40.448 


1W.19 


'a" 


6.4795 


8 8410 


H 


18.064 


95.907 


18. 


40.841 


13Bi.73 


6.6759 


8 5466 


13-16 


18.961 


96.685 


H 


41.988 


185.80 


8^6 


6 8799 


8.7583 


15^6 


18.4.'i7 


97.109 


M 


41.690 


187.89 


H 


7.0086 


3.97G1 


IB. 653 


97.088 


% 


49.019 


140.50 


6/16 


7.9649 


4.9000 


«. 


Ift-R-Ml 


98.974 


« 


49.419 


148.14 



CIRCUMFERBNOES AND AREAS OP CIRCLES. 109 



Dtein 


L 


Cfrcttui. 


Area. 


13^ 


42.804 


145.80 


Sa 


«.!97 


148 49 


% 1 


48.»W 


151.90 


14. 


4B.mA 


153.94 


H 


44 JW6 


156.70 


4 


44.768 


159.48 


i 


45.160 


16:i.a0 


i 


45.583 


165.13 


^ 


45.846 


167.99 


% 


46.8a8 


170.87 


H 


46.731 


173.78 


15 


47.1J4 


176.71 






47.617 


179.67 






47.900 


182.65 






4S.30ii 


185.66 






48.605 


183.69 


'h 




49.087 


191.75 


!|| 




49.480 


194.88 


i2 




49.873 


197.03 


i« 


50.985 


201.06 


^ 


£0.668 


204.9-<i 


1 


51.051 


a07.89 


i 


51.414 


210.60 


4 


61.816 


213.82 


'« 


52.229 


217.08 


^ 


59.692 


220.35 


5i 


58.014 


238.ftS 


17 


58.407 


226.98 


H 


68.800 


280.33 


: ; 


54.199 


233.71 


'^f 


54.585 


237.10 


*.l 


ti4.978 


240.63 


'^; 


56.371 


248.98 


■fc 


65.768 


247.45 


fi 


56.156 


250.95 


18 


56.549 


254.47 


u 




56.941 


258.02 


■i 


, 


57.:^ 


261.59 


■^ 


1 
( 


57.727 


265.18 


^ 


■ 


68.119 


268.80 


'^ 




66.518 


279.45 


14 




56.905 


276.19 




69.288 


279.81 


19 


69.600 


288.53 


^ 




60 063 


287.27 


•^ 




60.476 


291.04 


'^ 


1 


60.8W 


294.83 


■' 


1 


61.961 


208.65 


'1 


1 


61.654 


309.49 


«< 




62.046 


306.33 


Jl 




61.439 


310.24 


t% 


6S.889 


314.16 


h 




63.23!5 


318.10 


■i 




63.617 


329.06 


ji 




64.010 


328.05 


■ 1 




6I.40S 


330.06 


ii 




6I.7B5 


3:i4.10 


ife 




65 188 


838.16 


'I 




66.S8t 


349.25 


« 


66.978 


346.36 


4 


66.886 


890.60 


t 


66.759 


354.66 


^\ 


67.158 


358.84 


1 ; 


67.544 


868.a5 


'^1 


67.987 


807.28 


k 


I 


68.8» 


S71.M 



Diam. 



89. 



28. 



24. 



26. 



26. 



27. 



28. 



29. 



90. 



CirouiB. 



68.729 
69.116 
69.508 
69.900 
70.296 
70.686 
7l.0i9 
n.471 
71.864 
72.257 
79.649 
78.049 
78.435 
78.827 
74.280 
74.613 
75.006 
75.898 
75.791 
76.184 
76.576 
76.969 
77.862 
77.754 
78.147 
78.540 
78.933 
79.825 
79.718 
80.111 
80.508 
80.886 
81.289 
81.681 
82.074 
82.467 
82 860 
83.252 
88.645 
»1.088 
84.480 
84.883 
85.216 
85.608 
86.001 
86.304 
86.786 
87.179 
87.679 
87.966 
88.357 
88.750 
89.143 
89.535 
89.988 
90.391 
90.713 
91.106 
91.499 
91.89B) 
92.284 
92. 6?? 
9.^.070 
93.462 
98.8.55 
94.»t8 



Area. 



375.88 
380.13 
884.46 



397.61 
402.04 
406.49 
410.97 
415.48 
420.00 
424.66 
429.13 
4^3.74 
438.86 
443.01 
447.69 
452.89 
457.11 
461.86 
466.64 
471.44 
476.26 
481.11 
485.96 
490.87 
495.79 
500.74 
506.71 
510.71 
515.72 
520.77 
525.84 
530.93 
536.05 
541.10 
546.35 
551.55 
556.76 
562.00 
567.27 
.079.56 
577.87 
583.91 
588.57 
593.96 
599.37. 
604.81 
610.27 
615.75 
621.26 
626.80 
632.36 
637.94 
643. .55 
649.18 
654.84 
600.52 
066.23 
671.96 
677.71 
1083.49 
689.30 
696.13 
700.98 
706 86 



Diam 



81. 



92. 



SS. 



94. 



86. 



87. 



88. 



Circum* 



91.640 
95.038 
96.426 
95.819 
96.211 
96.604 
96.997 
97.889 
97.789 
98.175 
98.667 
98.960 
99.358 
99.746 
00.138 
00.531 
00.924 
.316 
101.709 
102.109 
02.494 
102.887 
08.280 
103.678 
04.065 
104.458 
104.851 
105.243 
105.636 
106.029 
106.421 
106.814 
107.207 
07.600 
107.999 
106.385 
08.778 
09.170 
109.563 
09.966 
10.848 
10.741 
11.134 
11.527 
11.919 
12.312 
12.705 
13.097 
13.490 
13.883 
14.275 
14.G68 
16.061 
15.454 
15.846 
16.239 
16.6;i2 
17.024 
17.417 
17.810 
18.202 
18.596 
18.988 
19.381 
19.773 
20.166 



110 



ItATHEMATICAL TABLES. 



Diam. 


Circum. 


Area. 


Diiuii. 


Circum. 


Area. 


DianL 


Circum. 


Area. 


ZSfi 


120.. '»9 


1156.6 


I<5?.^ 


146.477 


1707.4 


o4% 


172. 395 


2865. 


,i 


120.951 


1164.2 


■}4 


146.869 


1716.5 


55. 


172.788 


2875.8 


'1 


121.344 


1171.7 




147.262 


1725.7 


M 


178.180 


8886.6 


;l 


121.787 


1179.3 


47 "" 


147.655 


1734.9 


173.673 


2897.5 


% 


12:1.129 


1186.9 


l-H 


148.048 


1744.2 


9^ 


173.966 


8408.3 


Z9 


122.52-^ 


1194.6 


^l 


148 440 


1753.5 


l2 


174.358 


8410.:: 




1-42.915 


1202.3 


'?M 


148 883 


1762.7 


7B 


174.751 


2480.1 


]a 


m.308 


1210.0 


H 


149 226 


1772.1 


i/t 


175.144 


2441.1 


vk 


123.700 


1217.7 




149.618 


1781.4 


175.686 


2458. 


'ii 


1*44 093 


1225.4 


?1 


150.011 


1790.8 


56. 


175.929 


2463. 


1^ 


124.486 


12.33.2 




150.404 


1800.1 


1^ 


176.828 


2474.0 


ill 


1:^4. 87« 


1241.0 


i^'"^ 


150.796 


1809.6 


L 


178.715 


2485. 


''1 


125.271 


1248.8 


'-i 


151.189 


1819.0 


& 1 


177.107 


2196.1 


40. 


125. 004 


1256.6 


' i 


151.582 


1828.5 


£2 


irr.500 


2507. « 


H 


128.056 


1264.5 




151.975 


1837.9 


7\ 1 


177.893 


2518.3 


H 


126.419 


12^2.4 


! . 


162.367 


1847.5 


7* 


178.285 


8520.4 


i| 


126.842 


1280.3 




152.760 


1857.0 


yk 


178.678 


8540.6 


1 


127.2.i5 


1288.2 


' 1 


158.153 


1866.5 


57. 


179.071 


2551.8 


S 


127.627 


1296.2 




153 545 


1876.1 


^ 


179.468 


2563.0 


^ 


128. 0*^ 


1304.2 


151 


158.988 


16*5.7 


^ 


179.858 


2574.2 


^ 


128.413 


1312.2 


'h 


154. 3;)! 


1895.4 


98 


180.249 


2585.4 


41 


1^.805 


1320.3 


' i 


154.723 


1905.0 


Lc 


180.642 


2506.7 


H 


129.198 


1328.3 




155.116 


1914.7 


tN 


181.081 


2608.0 


^ 


129.591 


13:36 4 


Uj 


155.509 


1924.4 


s 


181.427 


2610.4 


^1 


129.983 


1344.5 


■'h 


1.^5.l>02 


1984 2 


181.820 


2630.7 


1 


180.370 


135;'.7 


'>.] 


1.'>(;.294 


1943.9 


58. 


182.212 


2042.1 


3 


130.769 


1360.8 




150 Gb7 


1958.7 


^ 


182.605 


2658.5 


^ 


131.161 


1309.0 


.'lO ■■ 


157.080 


1968.5 


i3 


182.998 


2604.9 


^ 


181.554 


1377.2 


1 ^ 


157.472 


1978.3 


84 


183.390 


2676.4 


4a 


131 947 


1385 4 


' ( 


157. MJ5 


1983.2 


79 


183.788 


2C87.8 


H 


182.. ^40 


1393.7 




158.256 


1993.1 


^ 


184.176 


2690.3 


1 


132.732 


HUJ.O 


1 ., 


168.650 


2003.0 


ft 


184.569 


2710.9 


% 


133. 12^ 


1410 3 


■• ^ 


159.048 


2012.9 


184.961 


2722.4 


1 


13:^.518 


1418.6 


■'■J 


159.436 


2022.8 


59. 


185. a54 


27JM.O 


^ 


13i.9l0 


1427.0 




159.829 


2032.8 


H 


185.747 


2745.6 


^ 


134.303 


1435.4 


.M ^ 


160 221 


2042.8 


186.189 


8757.2 


fl 


134.696 


1443 8 


1 -^ 


160.614 


2052 8 


a^ 


166.532 


2768.8 


4S 


m.oes 


1452.2 


1 , 


161.007 


2062.9 


zi 


186.925 


2:80.5 




135.481 


1460.7 


:-.^ 


161.399 


207:3.0 


z8 


187.317 


2792.2 


/4 


135.874 


1469.1 


1 .. 


161.792 


2083 1 


n 


187.710 


2H03.9 


b2 


136.267 


1477 6 


r. ^ 


162.185 


2093 2 


188.108 


2815.7 


S 


136.659 


1486.2 


" ] 


162 577 


2103 8 


60. 


188.496 


2827.4 


7ft 


187.052 


1494.7 




162.970 


2113.5 


i 


188.888 


2889 2 


^ 


137.445 


1503.3 




163.363 


2128.7 


189.281 


2851.0 


137.887 


1511.9 




163.756 


21:33.9 


189.674 


2862.0 


44. 


138.230 


1520 5 


■ J 


164.148 


2144.2 


h^ 


190.066 


2874 8 




138.023 


1529.2 




164.541 


2154.5 


78 


190.450 


2886.6 


L 


189.015 


1587.9 




164.934 


2164. H 


f4 


190.852 


2898.6 


9f i 


139.408 


1546.6 


■''4 


165.326 


2175.1 


yk 


191.244 


2010.5 


12 


139.801 


1655.3 


■' I 


165.719 


2185.4 


61. 


191 687 


2922.5 


ill 


140 194 


1664.0 




166.112 


2195.8 


^ 


192 030 


2934.5 


i 


140.586 


1572.8 


:»:t 


166.504 


2206.2 


192.42:1 


2946.5 


140.979 


1681.6 


) 


166.897 


2216.6 


&^ 


192.816 


2058.5 


46. 


141.372 


1690.4 


' 1 


167.290 


2227.0 


zi 


103.208 


2070.6 


^ 


141.764 


1699.8 


■'K 


167.8*3 


2237.5 


&g 


193.601 


2082.7 


'1 


142.157 


1608.2 


i^4 


168.076 


2248.0 


a 


193.998 


2004.8 


j| 


142.550 


1617.0 


4 


168.488 


2258.5 


104.386 


8006 9 


ll 


142.942 


16:i6.0 


^\ 


168.861 


2269.1 


68. 


194.770 


8010.1 


<l 


148.3:35 


1634.9 


"^-ri 


160.253 


2279.6 


^ 


195.171 


8U81.3 


2 


143.728 


1643.9 


r,i 


169.646 


2290.2 


M 


195.664 


8043.5 


% 


144.121 


1652.9 


1 . 


170. a39 


2800.8 


s2 


195.957 


8055.7 


46. 


144.513 


1661.9 


^■1 


170,431 


2311.5 


1^ 


196 850 


3068.0 


li 


144.906 


1670.9 


'•^H 


170.824 


2322.1 


78 


196.742 


808»i..^ 


' < , 


145.299 


1680.0 


'•i 


171.217 


2332.8 


81 


197.186 


8002 6 


' 


14.). 691 


1689.1 


iV " 


171.609 


2343.5 


78 


197.528 


3104.9 


1 ' ' 
H 


146.064 


1698.2 


H 


172.002 


2354.3 


68. 


197.980 


8117.8 



CIBOUMFERENCES AND AREAS OF CIRCLES. Ill 



Diam 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


Diam. 


Circum. 


Area. 


M^ 


196.313 


81296 


71H 


224.281 


4001.1 


250.149 


4979.5 


^ 


1W.706 


8142.0 


H 


284.624 


4015.2 


» 


250 542 


4995.2 


199.096 


3154.5 


II 


225.017 


4029.2 


% 


2S0.935 


5010.0 


L^ 


199.491 


8166.9 


R 


225.409 


4043.3 


80. 


251.327 


5026.5 


71 1 


199.864 


3179.4 


% 


225.802 


4067.4 




251.720 


5042.3 


^ 


aoo.«77 


8191.9 


72. 


!d26.195 


4071.5 


7' 


252.118 


5058.0 


^ t 


U00.660 


3201.4 


'% 


228.687 


4085.7 


9t 1 


252.506 


5073.8 


C4. 


201 .(XU 


8217.0 


i 


886.980 


4099.8 


■ii 


252.898 


5069. G 


4 


8)1.455 


3229.6 


227.378 


4114.0 




25».201 


5106.4 




aOl.847 


8242.2 


227.786 


4128.2 


\\i 


258.684 


5121.8 


' v 


202.240 


8254 8 


228.158 


4142.5 


M I 


254.076 


5187.1 


^ 


202.688 


3267.5 


228.551 


4156.8 


81. 


254 469 


5153.0 


1 14 


208.026 


3280.1 


228.944 


4171.1 


M 


254.862 


5168.0 


' ''^ 


208.418 


8298.8 


*% 


229.386 


4165.4 




255.254 


5184.9 


' i 


208.811 


8305.6 


U 


229.7S9 


4199.7 


78 


255.647 


5200.8 


6S. 


204.204 


3318.8 


'•.t 


230.122 


4214.1 


I4 


256.040 


5216.8 


^ 


204.506 


8831.1 


% 


280.514 


4226.5 K 


256.433 


5282.8 


^ 


204.969 


8343.9 


i.i 


230.907 


4242.9 


» 


256.826 


5248.9 


9l 1 


205.882 


3856.7 


f^^ 


231.300 


4257.4 


H 


257.218 


5264.9 


u , 


206.774 


3369.G 


i^t 


231.692 


4271.8 


82. 


257.611 


5281.0 


78 


-•06 167 


8382.4 


""h 


232.065 


4286.3 


A 


258.008 


5297.1 


» 


2U6.5a0 


3395.3 


74 


282.478 


4300.8 


A 


258.396 


5313.8 


H 


206.952 


8406.2 


H 


282.871 


4315.4 


'1 


258.789 


5829.4 


M. 


207.345 


3421.2 


'4 


233.268 


4829.9 


I 


259.181 


5345.6 


^ 


207.736 


3434.2 


H 


233 656 


4344 5 


'1 


259.574 


5361.6 


H 


206.131 


8447.2 


U. 


234.049 


4359.2 


H 


259.967 


5378.1 


H 


208.52S 


3460.2 


^S 


234.441 


4373.8 


% 


260.359 


5394.8 


R 


206.916 


3473.2 


^ 


234. 8:M 


4388.5 


88. 


260.752 


5410 


^ 


209.309 


8486.3 




285.227 


4403.1 


H 


261.145 


5426.0 


H 


209.701 


3499.4 


To'"* 


235.619 


4417.9 


H 


261.538 


6443.8 


?2 


210.094 


8512.5 


'k 


286.012 


4432.6 


y 


261.930 


5459.6 


t7. 


210.487 


3525 7 


'1 


236.406 


4447.4 


4 


262.823 


5476.0 


-6 

'4 


210.879 


8538.8 


H 


236.796 


4462.2 


rk 


268.716 


5492.4 


211 272 


8552.0 


I,. 


237.190 


4477.0 


S 


263.106 


5608.6 


^ 


211.665 


S)65.2 


'*l 


237»583 


4491.8 


% 


283.501 


5625.8 


4 


212.058 


3578.6 


ly 


237.976 


4506.7 


84. 


268.804 


5541.8 


<s 


212.450 


3591.7 


- ^ 


288.868 


4621.5 


H 


264.286 


6658.8 


'^ 


212.843 


8605.0 


70 


238.761 


4536.5 


» 


264. 6T9 


6574.8 


% 


213.236 


8618.8 


1 K 


239.154 


4551.4 


% 


265.072 


5591.4 


68 


218.628 


8631.7 


'l 


239.546 


4566.4 


% 


265.465 


5607.9 


4 


214.021 


8645.0 


''n 


239.989 


4581.3 


1 


266.857 


6624.6 


214.414 


8658.4 


'\,j. 


240.832 


4596.8 


R 


266.250 


5641.2 


i 


214.806 


3671.8 


'''h 


240.725 


4611.4 


H 


266.643 


5657.8 


i 


215.199 


8685.3 


'\ 


241.117 


4626.4 


80 


267.085 


5674.5 


i 


215.592 


3698.7 




241.510 


4641.6 


% 


267.428 


6U91.2 


^ 


215.961 


8712.2 


J« . 


241.903 


4656.6 




267.821 


5707.9 


^ 


216.377 


8725.7 




242.295 


4671.8 


'98 


268.213 


5724.7 


w 


216.770 


3789.8 


7* 


242.688 


4686.0 


2 


268.606 


5741.5 


^ 


217.163 


8752.8 


7I 1 


243.081 


4702.1 


9r 


2C8.9D9 


5758.8 


H 


217.555 


3766.4 


Z9 


243.478 


4717.3 


R 


269.392 


5775.1 


!^ 


217.948 


8780.0 


rk 


243.866 


4732.5 


% 


269.784 


5791.9 


1 


218.311 


3793.7 


» 


244.259 


4747.8 


86 


270.177 


5808.8 


'^ 


218.738 


3807.3 


^ 


244.652 


4763.1 




270.570 


5825.7 


;1 


219 126 


3821.0 


78. 


245. OU 


4778.4 


7^ 


270.962 


5842.6 


^ 


219.519 


8884.7 


H 


245 437 


4798.7 


9s 


271.355 


5859.6 


;o 


819.911 


3'<48.5 


h 


245.830 


4809.0 


79 


271.748 


6876.6 


4 


220.304 


8862.2 


H 


246.222 


4824.4 


78 


272.140 


5898.5 


4 


220.697 


3876.0 


» 


246.616 


4839 8 


8^ 


272.533 


5910.6 


i 


221.000 


8889.8 


n 


247.008 


4855.2 


tZ 


272.926 


5927.6 


i 


221.462 


3908.6 


s 


247.400 


4870.7 


87. 


273.319 


5944.7 


'3 


221.875 


3917.5 


% 


247.793 


4880. 2 




273.711 


6961.8 


3 


222.268 


8881.4 


79. 


248.186 


4901.7 


t2 


274.104 


5978.9 


« 


222.660 


8045.8 


H 


248.579 


4917.2 


9h 


274.497 


5996.0 


71. 


228.058 


8959.2 


^ 


248.971 


4932.7 


1^ 


274.889 


6018.2 


^ 


223.446 


8973.1 


K 


249 364 


4948.3 


tS 


275.282 


6030.4 


228.8I« 


8987.1 


« 


249.757 


4963.9 


94 


275.675 


6047.6 



112 



UATHBMAllCAL TA13LE8. 



DIam. 


Cfrcum. 


Area. 


Diam. 


Cli*cuni. 


Area. 


Hmm. 


Circuin. 


Arecu 


87% 


2r:tjo«7 


6064 9 


03. 


2H9.027 


0647 € 


301 986 


7!Ki7 1 


88 


27«.4*iO 


UW2 1 




289.419 


6665.7 


' i 


302 378 


755TG U 


v& 


37».85:i 


6099.4 


C 


289.818 


6683.8 


''B 


;302.771 


7>!«4 9 


lA 


277^6 


6118.7 


ftj 1 


290.206 


6701.0 


l2 


.S03 llU 


7813. S 


' 'M 


877. (W8 


6134.1 


L . 


290.597 


67-.'0.1 


h\ 


30:4.666 


7S«,' 8 


' "i 


278 081 


6151.4 


7\\ 


290.990 


6;-88.32 


'i 


8(« 949 


7».M.rt 


1 2 


278. 45M 


6168.8 


Bj 


291.888 


6756.4 


304 .M42 


7«ro.s 


< 4 


'J78.S16 


6186.2 


y| I 


291.775 


6774.7 


i'^4 


304 734 


73S$> H 


''Z 


279 209 


6203.7 


98. 


298.168 


6792.8 


'4 


806. 127 


74ll^* M 


«». 


279. 602 


6221.1 


H 


292.561 


68tl.2 




:^05.520 


74^8 


^ 


279.994 


623K.0 


H 


292.954 


6889.5 


i.t| 


305.918 


7447 I 


^ 


280.387 


6350.1 


U 


298.816 


6847.8 


4 


306.805 


74«;ti V 


a2 


280 780 


6273.7 


7% 


298.789 


6866.1 


''1 


.306.098 


74H.-i :■% 


ZS 


281.178 


6291.2 


fl 


294.182 


6884.5 




807 091 


75iM 5 


^ 


ySl.wC". 


6 08.8 


n 


294.. •>24 


6902 9 


307.488 


7fi'».S 7 


•>4 


281.958 


6826.4 


H 


294 917 


6921.3 


IIS 


307.876 


754-^ O 


'/i. 


282.351 


6344. 1 


94. 


295 310 


69898 


'\ 


808 289 


7.''.<W '2 





2H2.748 


6361.7 




295.702 


6058.2 




308.661 


7581.5 


^ 


28:) ]8tf 


63T9 4 


V 


296 095 


6976 7 


309 054 


76CIO.S 




288.529 


6397.1 


7\\ 


206.488 


6995.3 


.309.447 


765JO 1 


f^ 


28:1921 


6414.9 


12 


296.881 


7013 8 


309 840 


7WIW 5 


^ 


281.814 


6432.0 


Tt 


297.278 


7a3i.4 


. ? 


310.232 


7tt.%8 9 


2^.707 
£«6J00 


6450.4 


A^ 


297.666 


7051.0 


810.625 


767K 3 


^ 


64682 


7\\ 


298 059 


7069.6 


li;i 


811.018 


7GI>7 7 


285.492 


6486 


9o. 


298.451 


7088. '2 


II 


311.410 


7717-1 


91. 


2a%.88B 


6503 9 


L^ 


298.844 


7106.9 


311.803 


TT^ttJ t$ 


t^ 


286.278 


6521.8 


M 


299.237 


7125.6 


:412 198 


77R6 1 


C 


288.670 


6539.7 


s2 


299 629 


7144.8 


312.588 


7r7r,.6 


ii2 


287.063 


6557.6 


12 


300.022 


7io:).o 


812.981 


7r9r> ;; 


2 


5-87.456 


6575.5 


zB 


300.415 


7181.8 


313 874 


7814 8 


'^ 


287.848 


6593.5 


a 


300.807 


7200.6 


813 767 


7834 4 


'1 


288.241 


6611.5 


301.200 


7219.4 


1 00 


814.159 


7864.0 


8 


288.634 


6629.6 


90. 


801.598 


7238.2 









BECimALS OF A FOOT EQUIVALCNT TO INCHKS 
AND FRACTIONS OF AN INCH. 



Inches. 





% 


H 


% 


% 


% 


% 


H 








.01012 


.02083 


.03125 


.01166 


.06208 


.00250 


.mii»i 


1 


.0833 


.09;J7 


.1042 


.1146 


.1250 


.1.354 


.1459 


.15<i3 


2 


.1067 


.ITil 


.1875 


.1979 


.2063 


.2188 


.2292 


2896 


8 


.2500 


.2001 


.2708 


.2813 


.2917 


.3021 


.8125 


.8229 


4 


.83.^3 


.8137 


.3542 


..3646 


.:C50 


..-{854 


.3958 


.40(S8 


5 


.4167 


.4271 


.4:575 


.4479 


.4588 


.4688 


.4792 


.4896 


6 


.5000 


.5104 


.5208 


.5313 


.5417 


.5521 


.5025 


.57"^ 


7 


.5833 


.6937 


.6042 


.6146 


.62.V) 


.6a'>4 


.6159 


.6568 


8 


.6667 


.em 


.6875 


.6979 


.7083 


.7188 


.7292 


.7396 


9 


.7500 


.7604 


.7708 


.7818 


.7917 


.8021 


.8125 


.8.S9 


10 


.«m 


.84:^7 


.8512 


.8646 


.8750 


.88M 


.8958 


.9068 


11 


.9167 


.9271 


.9375 


.9479 


.9583 


.9688 


.9792 


.9696 



CIBCVJIFERBNCBS OF CIRCLES. 



113 






8 ! 






m 

V 



5^^ggg^.|fH2?^^««fg^^jr;?.-^fl^ 



je«<oxc«iAao<-< 
























^ ~»»asaS88S8a5?*8S51SS888EeeSg3E5S$!§ 



>-• «o^ • floli ^ "^ ** ^ w sTct ^2 •-1 n Vw 00 o» »- •-I ©* •^«6 i-ft r-*w-^ 5i^ ^T^'ob' 

1^ «««a!S9S883S8??^S5SSgS8SKl5f5£S$sSfe8 



fMi^ooeeoio 



^ "••S«S2s^«SS^§gi^gSSg3S§?fSP5o5£5S68 






«-<4teo^iOttc*Qoa»o^e«eo^tc«Dt>Qoa»0'«2»eo'^ 



8»S{S$^3^^^gS8»SiS 



114 



KATHEKATICAL TABLES. 



liENOTHS OF CIBCUIiAB ARCS. 
(Mk^tpreeu betns ffiveii* Hmdtlis ofCttrele = !•> 

Formula.— Length of arc = -- X radius X number of def^rees. 

RuLB.— Multiply the factor in table for any given number of degi^tsa by 
the radius. 

ExAMPLB.— Given a curve of a radius of 6S feet and an angle of 78<* 90'. 
What is the length of same in feet ? 

Factor from table for 780 1.8618548 

Factor from table for aK .0058178 

Factor 1.3671746 

1.8671746X55 = 75.19 feet. 



I>effree8. 


' Mlnntca. 




^0t71&33 


61 


) n^^&ofl 


121 


2.11IM}^ 


1 


.M08M8 






68 


LOAiimL 


128 


8.1?^WS7 


% 


.0006818 




*fXMSSt9 


68 


i.vmi^H 


123 


tM^:i^*i 


8 


.0008787 




iMHI82i 


61 


MiTom; 


124 


8 1«*#L^ 


4 


.6011636 




.gmtm 


65 


I 1^4040 


128 


i.1HLe«l6 


6 


.0014M4 




.IHTlif^ 


66 


1 iM«i7:^ 


126 


a.iswiuv 


6 


.0017463 




.1111780 


67 


1 IftKtVOft 


127 


1 SIOMffi 


7 


.0080968 




.1198183 


68 


1 lS4^flE» 


128 


a ijruosn 


8 


.0083871 




.lATmM 


69 


1.304Ejr« 


199 


^ se5H747 


• 


.ogoiao 


10 


J74tlS& 


70 


1.1217305 


130 


l.lie^^1^!llO 


10 




11 


.l9]fM3 


71 


i 'e?'jin;i,K 


131 


t iMA.tML3 


U 


.0031808 


Ifl 


.10M»& 


78 


1 ■. ■ ■ I 


138 


2.lci«c34a 


18 


.0084907 


IS 




78 


< mt-wvwvm 


133 


;E, 321979 


18 


omrmj 


14 


'f 443441 


74 


1.2915438 


134 


a.M«71l? 


14 


OMu7t4 


lA 


.»1T»M 


76 




136 


2 3^1945 


U 


iMHSAn 


16 


.wnawj 


76 


l!326l&08 


138 


t S73fl47Jt 


16 


,a(H£6ii 


17 


.IBSTWU 


n 


1.3439086 


137 


2 SUilOll 


17 


OOiMtl 


18 


.5141m:i 


78 


1.3618668 


138 


j!.1(W>£pH 


18 


jtoMsm 


19 


.TSTflt^fl 


79 


1.3788101 


139 


% imMTi 


19 


OQfiCkMf 


» 


."1 1 \H \t\ \'f 


80 


1.3868834 


140 


^ 4434(110 


80 


JKHUTI 


21 


.KHi,.-,'.'il 


81 


1.4137167 


141 


^mum 


81 


Mnm 


tt 


:iA ur^Ji 


88 


1.4311700 


148 


« I7SM75 


88 


.oooaat 


» 


.*iiHi'N 


88 


1.1488833 


143 


t.i9M8(M 


23 


.OOOBfiil 


U 


.IIW7lhP 


84 


1.4660706 


144 


8.613*741 


84 


.OQWlt 


» 




86 




146 


8.65107874 


86 


-ttWllTPf 


S6 


A^lT^'*y 


86 


: ..-...;__: 


146 


i.Mdiwg 


86 


mi^^\ 


S7 


+ ; 1 ;<.'it 


87 


i-M.S4!^ 


147 


B.flif'rfim-i 


87 


.SMt'imftti 


» 


l-A-Vi;. 


88 


1 J)3&llMr 


148 


2.^*;-!'^«t:! 


88 


.008144V 


S9 


'A^.-\ \\.. 


89 


1 &5XMi»l 


140 




89 


.0004364 


80 


'.■.■■L',.»M^v 


90 


1 SiTCI^WEI 


150 


8.6179990 


80 


.0007816 


31 


.■.JI-'.JL 


91 


l.Mt<249a 


161 


8.6354478 


81 


.0000175 


SS 


■,,,K-*1-,| 


98 


t QOTNirai 


188 


8.66C9006 


as 


.0003084 


88 


". ; 'rv .'^ ; 


93 


i.«ji3t.'Vfli 


153 


8.6708588 


88 




34 


I'l; 1 1 1'J 


94 


i^^voBfrn 


164 


2.6878070 


84 


.0098988 


85 


1. rl-.. ' 


96 


1 65IMM8 


166 


2.7068608 


36 


.0101811 


88 


1 . ; h ' 1 . ', 


96 


i.nauii 


166 


m >Ma».Aa 


86 


.0104780 


87 


u\ ..-.•'■^ 


97 


i.i6«m»4 


157 




S7 


.0107089 


38 


I"' "'.'.■ 


98 


I.JIMOT 


168 


^."ki'y^K 


88 


.0110638 


39 


l',f^|^*!.7^| 


99 


1 TSTS7»f) 


160 


a,776or3a 


80 


.0113446 


40 


\,-\*-\ .1- 


100 


1 utisim. 


160 


%7w^m. 


40 


.0116366 


41 


1 - ^' 


101 


1 rnsjusw 


161 


twmmi 


41 


.0U99U 


48 


iwwvwow 


102 


1 7Wȣ>.'* 


168 


%^ir{iifA% 


48 




43 


!7501916 


103 


1 7970BSH 


163 


2 lt4iJ4?«7 


48 


.0185082 


44 


.7679449 


104 


} HI5UT1 


164 


H.wyjioo 


44 


.0187991 


a 


.7853968 


106 


t ^:>-I,-p*7 


166 


2.»TV793a 


46 


.0130900 


40 


.80)!»516 


106 


L -..Vh.J>-' 


166 


s^-aiM 


46 


.0133809 


47 


oa/>oA4^ 


107 


k.tmiiiM-^ 


167 


1 SlMiKW 


47 


.0186717 


48 




106 


1.8^9566 


168 


2 KttlMI 


48 


.0139086 


49 




109 


1.9024069 


169 


12 MMOOi 


49 


.0148636 


fiO 




no 


1.9198622 


170 


z wfhJir*: 


60 


.0146444 


51 




111 


1.9373165 


171 


1 mkfx\M 


61 


.014838& 


68 




112 


1.9647688 


178 


N ip'LiMfi:! 


68 


.0161862 


63 


"f;. '1 ■|."i 


113 


1.9782281 


173 




63 


.0164171 


64 


•>ivr - 


114 


1.9896753 


174 


VUUDI*^ 


64 


.0167080 


66 


■y »i.::i 


116 


8.0071286 


176 


3.0643868 


66 


.0160989 


58 


.yi::sJiHHi 


116 


2.0245819 


176 


8.0717796 


86 


.0188897 


87 


wijii;: 


117 


2 0420352 


177 


8.0892328 


67 


.0166806 


68 


1 MlS.lJ](i 


118 


2.0694886 


178 


8.1066861 


68 


.0168715 


69 


1 flPn-Ttrt 


119 


2.0769418 


179 


3.1241394 


69 


.0171684 


60 


1 i.lTi'A-i. 


120 


2.0943951 


180 


S.1416087 


iO 


.0174633 



LSKGTHS OF CIROULAB AllCS. 



116 



ILKNGTHS OP €m€ri<A» ARCH. 
CBlmmeter = 1« Given tlte Cbord mud Heicl^t of tlie Arc.) 

Bulk worn. Uss or the Tabxa.— Divide the height by the chord. Find in the 
eolnmn of beifrhtii the number equal to this quotient. Take out the corre- 
spODdtns number from the column of lengths. Multiply this last number 
by the length of the given chord ; the product will be length of the arc. 

1/ the are is ffreaier than a 9emicircle, flrst And the diameter from the 
formiila, Dijun. = (square of half chord -*- rise) + rise; the formula is true 
whether the arc exceeds a semicircle or not. Then And the circumference. 
FYom the diameter subtract the given height of arc, the remainder will be 
height of the smaller arc of the circle; find its length according to the rule, 
and subtract It from the circumference. 



Hgts. 


Lffths. 


Hgts. 


Lgths. 


Hgts. 


Lgths. 


Hgts. 


Lgths. 


Hgts. 


Lgths. 


.001 


i.ooooe 


.15 


1.06806 


.288 


1.14480 


.386 


1.86288 


.414 


1.40788 


.005 


1.00007 


.168 


1.06061 


.24 


1.14714 


.328 


1.26588 


.416 


1.41145 


.01 


1.00027 


.154 


1.06200 


.842 


1.14951 


.38 


1.26892 


.418 


1.41608 


.015 


1.00061 


.156 


1.06368 


.844 


1.15189 


.832 


1.27196 


.42 


1.41861 


.02 


1.00107 


.158 


1.08580 


.846 


1.15428 


.384 


1.27502 


.422 


1.42221 


.OS 


1.001G7 


.16 


1.06608 


.248 


1.15670 


.386 


1.27810 


.424 


1.42583 


.01 


1.00040 


.162 


1.06858 


.25 


1.15912 


.838 


1.28118 


.426 


1.42945 


.036 


1.00327 


.164 


1.07025 


.252 


1.16166 


.84 


1.28428 


.428 


1.43300 


.01 


1.004S6 


.166 


1.07194 


.254 


1.16402 


.342 


1.28T39 


.43 


1.43678 


.045 


1.00689 


.168 


1.07365 


.256 


1.16660 


.344 


1.29059 


.432 


1.4^039 
1.44405 


.06 


1.00065 


.17 


1.07537 


.258 


1.16899 


.346 


1.80366 


.484 


.005 


1.O0606 


.178 


1.07711 


.86 


1.17150 


.848 


1.29681 


.436 


1.44778 


.06 


1.00057 


.174 


1.07988 


.262 


1.17408 


.85 


1.29907 


.488 


1.46142 


.065 


1.01128 


.176 


1.08066 


.864 


1.17657 


.352 


1.80315 


.44 


1.45613 


.07 


1.01802 


.178 


1.06246 


.266 


1.17912 


.354 


1.30634 


.448 


1.45883 


.0» 


1.01498 


.18 


1.06428 


.268 


1.18169 


.356 


1.80054 


.444 


1.46856 


.08 


1.01098 


.182 


1.08611 


.27 


1.18429 


.358 


1.81276 


.446 


1.46628 


.085 


1.01916 


.184 


1.08797 


.278 


1.18689 


.86 


1.81599 


.448 


1.47008 


.00 


1.02146 


.186 


1.08964 


.874 


1.18951 


.862 


1.81923 


.45 


1.47377 


.005 


1.02889 


.198 


1.O0174 


.876 


1.19214 


.364 


1.82849 


.458 


1.47753 


.10 


1.08646 


.19 


1.00365 


.278 


1.19479 


.366 


1.32677 


.454 


1.48131 


.102 


1.02752 


.192 


1.09557 


.88 


1.19746 


.368 


1.32905 


.466 


1.48609 


.101 


1.08860 


.194 


1.09752 


.282 


1.90014 


.37 


1.38234 


.458 


1.48689 


.106 


1.02970 


.196 


1.09949 


.284 


1.20284 


.378 


1.33564 


.46 


1.49269 


.108 


1.0:1062 


.198 


1.10147 


.286 


1.2a5.\5 


.374 


1.33896 


.462 


1.49651 


.11 


1.03196 


.20 


1.10847 


.288 


1.20827 


.376 


1.34229 


.464 


1.60088 


.m 


1.03312 


.202 


1.10548 


.29 


1.21102 


.378 


1.34568 


.466 


1.S0416 


.114 


1.03480 


.204 


1.10752 


.292 


1.21377 


.88 


1.34899 


.468 


1.60600 


.tio 


1.08561 


.206 


1.10968 


.294 


1.21664 


.382 


1.86237 


.47 


1.M185 


.118 


1.08672 


.208 


1.11165 


.296 


1.21938 


.384 


1.35575 


.478 


1.51571 


.18 


1.09797 


.21 


1.11374 


.298 


1.22213 


.386 


1.35914 


.474 


1.51958 


.128 


1.09928 


.212 


1.11584 


.30 


1.22495 


.388 


1.86254 


.476 


1.52346 


.194 


1.04061 


.214 


1.11796 


.802 


1.22778 


.89 


1.36596 


.478 


1.62736 


.126 


1.04181 


.216 


1.12011 


.804 


1.23068 


.392 


1.86939 


.48 


1.53126 


.128 


1 04818 


.218 


1.12226 


.806 


1.23849 


.394 


1.37288 


.488 


1.58518 


.13 


1.04447 


.22 


1.12444 


.308 


1.23686 


.896 


1.87628 


.484 


1.58910 


.IS 


1 04584 


.222 


1.12664 


.81 


1.28926 


.398 


1.37974 


.486 


1.54308 


.184 


1.04722 


.824 


1.12886 


.813 


1.24216 


.40 


1.38322 


.488 


1.54696 


.130 


1 04862 


.820 


1.18108 


.814 


1.24507 


.402 


1.88671 


.49 


1.55091 


.196 


1 06008 


.828 


1.13881 


.816 


1.24801 


.404 


1.39021 


.492 


1.55487 


.14 


1 05147 


.28 


1.18657 


.318 


1.25095 


.406 


1.39372 


.494 


1.65854 


.142 


1.'05298 


.2S8 


1.18786 


! .88 


1.25391 


.408 


1.89784 


.496 


1.66282 


144 


1 06441 


.284 


1.14015 


.822 


1.25689 


.41 


1.40077 


.496 


1.66681 


.145 


1.06691 


.286 


1.14247 


.824 


1.85988 


.412 


1.40438 


.50 


1.57080 


.148 


1.06748 






1 


1 1 





116 



MATHEMATICAL TABLB3. 



AREAS OF THB 8BOHBNT8 OF A €IB€I<S. 

(Illameter = 1 : Alse or Versed Sine In parts of nimmeter 
fcelugr Stveu.) 

RDI.K SOR UsB or tBB TABi*B,~Dlvide the rise or height of the segment by 
the diameter to obtain the Tersed sine. Multiply the area ih the table coi> 
responding to this yeraed sine by the square of the diameter. 

If the aegmtnt escceedt a «emictrdc its area is aroa of oirole*area of seg- 
ment whose rise is (diam. of circle— rise of giveo segm^it). 

Given chord and rifie, to And diameter. Diam. s (square of half chord •«- 
rise) ■+ ri«e. The half chord in a mean proportional between the two parts 
into which the ohord divides the diameter which is perpendicular to f u 



V«r»d 
Sine. 


Araiu 


VwMd 

Sine. 


Area. 


Vetwd 
Sine. 


Are.. 


Vened 

Sine. 


Ana. 


VmmiI 

Sine. 


An*. 


.001 


.00004 


.054 


.01646 


.107 


.04514 


.16 


.08111 


.218 


.ia2:i5 


.004 


.00018 


.056 


.01691 


.108 


.04576 


.161 


.08185 


.914 


.18817 


.Oft3 


.00038 


.056 


.01737 


.109 


.04638 


.168 


.08858 


.216 


.18399 


.004 


.00034 


.057 


.01783 


.11 


.01701 


.163 


.08338 


.216 


.18481 


.005 


.00047 


.058 


.01830 


.111 


.04763 


.164 


.08406 


.217 


.18S68 


.006 


.00068 


.069 


.01877 


.118 


.04836 


.165 


.08480 


.918 


.12646 


.007 


.00078 


.06 


.01934 


.113 


.04889 


.166 


.08654 


.919 


.187^ 


.008 


.00005 


.061 


.01973 


.114 


.04958 


.167 


.08689 


.89 


.19611 


.009 


.00118 


.068 


.08080 


.115 


.05016 


.168 


.08704 


.981 


.198»1 


.01 


.00183 


.068 


.08068 


.116 


.06080 


.169 


.08779 


.999 


.12977 


.011 


.00153 


.064 


.08117 


.117 


.05146 


.17 


.08854 


.993 


.18060 


.018 


.00175 


.066 


.08166 


.118 


.06800 


.171 


.08939 


.994 


.18144 


.018 


.00197 


.066 


.08815 


.119 


.05874 


.ITS 


.09004 


.986 


.18927 


.014 


.0088 


.067 


.08865 


.19 


.05838 


.178 


.09080 


.996 


.18811 


.015 


.00844 


.068 


.08315 


.131 


.05404 


.174 


.09155 


.997 


.18805 


.016 


.00868 


.069 


.08366 


.133 


.03469 


.175 


.09331 


.998 


.18478 


.017 


.00891 


.07 


.08417 


.183 


.05585 


.ITO 


.09307 


.999 


.135G8 


.018 


.00*8 


.071 


.0i468 


.134 


.05600 


.177 


.09:384 


.33 


.18646 


.019 


.00347 


078 


.08580 


.136 


.06666 


.m 


.09460 


.931 


.137^1 


.0» 


.00375 


.076 


.08571 


.136 


.05783 


.179 


.09587 


.289 


.18815 


.041 


.0040:3 


.074 


.08684 


.127 


.05799 


.18 


.09613 


.383 


.13900 


.Oti 


.00438 


.075 


.08676 


.138 


.05866 


.181 


.09690 


.384 


.18984 


.0^3 


.00468 


.076 


.03789 


.139 


.05938 


.188 


.09767 


.385 


.14069 


.094 


.00498 


.077 


.03788 


.18 


.06000 


.188 


.09^5 


.286 


.14154 


.005 


.00SS3 


.078 


.08886 


.181 


.06067 


.184 


.09938 


.987 


.14239 


.o-w 


.00655 


.079 


.03889 


.133 


.06186 


.186 


.10000 


.988 


.148,*4 


.087 


.00587 


.08 


.08943 


.188 


.06808 


.186 


.10077 


.989 


.14400 


.0« 


.00619 


.081 


.08998 


.184 


.06271 


.187 


.10155 


.94 


.144W 


.029 


.00658 


.088 


.08053 


.185 


.06339 


.188 


.10933 


.941 


.14580 


.06 


.00687 


.063 


.08108 


.186 


.06407 


.189 


.10313 


.949 


.14666 


.081 


.00781 


.084 


.08168 


.137 


.06476 


.19 


.10890 


.343 


.147^1 


Mi 


.00756 


.085 


.03819 


.138 


.06545 


.191 


.10469 


.944 


.14S17 


.OW 


.00791 


.086 


.03875 


.139 


.06614 


.193 


.10547 


.846 


.149M8 


.081 


.00887 


.087 


.08331 


.14 


.06683 


.103 


.10686 


.346 


.l.WOO 


.085 


.00864 


.088 


.08887 


.141 


.06758 


.m 


.10705 


.247 


.15095 


.096 


.00901 


.089 


.08444 


.143 


.06832 


.195 


.10784 


.248 


.15188 


.067 


.009:18 


.09 


.08.101 


.143 


.06892 


.196 


.10864 


.949 


.lo268 


.088 


.00976 


.091 


.a3559 


.144 


.06968 


.197 


.10948 


.95 


.16866 


.0?» 


.01015 


.098 


.08616 


.145 


.07083 


.198 


.11093 


.851 


.16441 


.04 


.01054 


.093 


.08674 


.146 


.07103 


.199 


.11108 


.969 


.15606 


.041 


.01098 


.094 


.08788 


.147 


.07174 


.9 


.11182 


.963 


.15615 


.048 


.01133 


.096 


.03791 


.148 


.07845 


.301 


.11362 


.964 


.1571)8 


.048 


.01173 


.096 


.03850 


.149 


.07316 


.308 


.11843 


.965 


.157W 


.044 


.01814 


.097 


.03909 


.15 


.07887 


.308 


.11493 


.956 


.15876 


.045 


.01355 


.096 


.08968 


.151 


.07469 


.9M 


.11504 


.857 


.15964 


.046 


.01897 


.099 


.04088 


.158 


.07531 


.305 


.11584 


.938 


.16051 


.047 


.01339 


.1 


.04087 


.158 


.0760} 


.906 


.11665 


.959 


.16189 


.048 


.01888 


.101 


.04148 


.164 


.07675 


.907 


.11746 


.96 


.lftM6 


.049 


.01435 


.108 


.04208 


.l.'iS 


.07747 


.808 


.11887 


.961 


.16314 


.06 


.01468 


A(Xi 


.04-JG9 


.156 


.07819 


.809 


.11908 


.263 


.16408 


.051 


.01518 


.104 


.04;^30 


.157 


.07898 


.21 


.11990 


.908 


.16490 


.052 


.01556 


.105 


.04391 


.158 


.07965 


.211 


.12071 


.264 


.165T8 


.068 


.01601 


.106 


.04458 1 


.169 


.08038 


.218 


.18153 


.265 


.16666 



AKBA8 OF THE SEGMBKT8 OF A CIRCLE. Ill7 



ttrmd 




Vencd 


t ^ 


Vened 








Vencd 




^.M. 


Am. 


StD«L 


Area. 


SilM. 


At»«. 


Sin*. 


Anm, 


Sin*. 


An*. 


JG6 


.iers6 


.813 


.»016 


.36 


.25456 


.407 


.30024 


.454 


.34676 


XT 


.16843 


.814 


.31108 


.861 


.85551 


.408 


.80122 


.455 


.84770 


Sffi 


.10K» 


.315 


.aUK)! 


.862 


.25647 


.409 


.8O.>20 


.156 


.84876 


J0» 


.17UM 


.316 


.91294 


.863 


.2574.3 


.41 


.30319 


.457 


.»1875 


J?7 


.17100 


.317 


.01.387 


.364 


.25839 


.411 


.30417 


.458 


.85076 


.n 


.17198 


.818 


.81480 


.9bC 


.25936 


418 


.80516 


.459 


.85175 


.872 


.i?«r 


.319 


.31578 


.366 




.418 


.80'-.14 


.46 


.85874 


//n 


.17376 


.88 


.31667 


.367 


.26128 


.414 


.80712 


.401 


.?5874 


.274 


.17465 


.881 


.31760 


.868 


.26:e5 


.415 


.80811 


.462 


.86474 


J873 


.I75S4 


.88:2 


.91868 


.860 


.26:ii»l 


.416 


.80910 


.463 


.35673 


.«7B 


.175*4 


.883 


.31M7 


.37 


.26418 


.417 


.31008 


.404 


.85673 


.2:7 


.17738 


.884 


.:9040 


.871 


.26514 


.418 


.81107 


.405 


.85778 


.27S 


.17828 


.885 


.8SJ84 


.878 


.96611 


.419 


.31205 


.406 


.35873 


jrrv 


.1791« 


.826 


.8^^ 


.73 


.96708 


.42 


.31.304 


.467 


.35972 


JJ9 


.18009 


.8*7 


.3S8S9 


.374 


.26805 


.421 


.31403 


.468 


.36072 


.•>1 


.180M 


.888 


.32416 


.375 


J»901 


.428 


.SLVW 


.409 


.86172 


.«J 


.18188 


.889 


.36509 


.376 


.26098 


.423 


.31600 


.47 


.36272 


JW3 


ASJ72 


.88 


.«3003 


.877 


.270a'i 


.424 


.81(599 


,471 


.86872 


.5M 


.18383 


.881 


.38697 


.378 


.27192 


.425 


.31798 


.472 


.36471 


.2S5 


.184.U 


.882 


.32799 


.879 


.a7J89 


.428 


.31597 


.473 


.36871 


.^ 


.18548 


.883 


.««» 


.88 


.87386 


.427 


.31996 


.474 


.86671 


Jf!7 


.18633 


.884 


.38980 


.881 


.27488 


.428 


.32095 


.475 


.36771 


J2» 


.I8r» 


.885 


.88074 


.382 


.27560 


.429 


.82194 


.476 


.86871 


.2» 


.18814 


.886 


.881G0 


.883 


.27878 


.43 


.88293 


.477 


.80971 


JS 


.18005 


.887 


.83308 


.384 


.27776 


.431 


.32303 


.478 


.87071 


^n 


.18896 


.888 


.88358 


.885 


.37872 


1 .488 


.32491 


.479 


.87171 


.292 


.19086 


.889 


.83468 


.386 


.87969 


.4.38 


.32590 


.48 


.37270 


;33 


.19177 


.84 


.83547 


.887 


.98067 


, .434 


.32689 


.481 


.87870 


^4 


.19868 


.841 


.88642 


.888 


.28164 


.436 


.82788 


.482 


.3747D 


.•35 


.19860 


.848 


.83787 


.889 


.28268 


.436 


.328^7 


.488 


.87570 


•296 


.lfM51 


.343 


.83838 


.89 


.88359 


1 .4.37 


.82987 


.484 


.37670 


.387 


.1954)2 


.844 


.83987 


.891 


.884.57 


.438 


.33086 


.485 


.3'iTrO 


JS» 


.19634 


.845 


.84088 


.892 


.28554 


, .439 


.83185 


.486 


.87870 


.2» 


.19785 


.346 


.34117 


.893 


.28652 


1 .44 


.38284 


.487 


.37970 


.3 


.10817 


.347 


.»4312 


.894 


J»750 


1 .441 


.83:^84 


.488 


.88070 


.»! 


.19908 


.848 


.84307 


.895 


.28848 


1 .448 


.33483 


.488 


.88170 


;m 


.90000 


.849 


.84408 


.896 


.28945 


.448 


.33582 


.49 


.88270 


.«3 


.90092 


.85 


.94486 


.897 


.29(M3 


.444 


.ai682 


.491 


.88370 


.304 


.90184 


.861 


.84598 


.898 


.29141 


.446 


.33781 


.498 


.88470 


^V5 


jaQS76 


.852 


.84680 


.899 


.292:» 


, .446 


..3;^880 


.498 


.88570 


.908 


.80368 


.853 


.14781 


.4 


.298:17 


.447 


.a3980 


.494 


..38670 


^ 


J80460 


.354 


.84880 


.401 


.2W35 


1 448 


.34079 


.495 


.38770 




.365 


.84976 


.402 


.89538 


.449 


.34179 


.496 


.88870 


.«g 


.20645 


.»i6 


.85071 


.408 


Jidm 


.45 


.84278 


.497 


.38970 


;)i 


.90738 


.867 


.85167 


.404 


.89729 


j .4.M 


.34378 


.498 


.39070 


.311 


jaoeso 


.858 


.95368 


.405 


.298-^ 


1 .4.18 


.34477 


.499 


.39170 


.Sid 


.809-^ 


.859 


.85360 


.406 


.29926 


.4.58 


.34.^77 


.5 


..39270 



For rules for finding the area of a tegment see Mensuration, page 99. 



118 MAtHEMATlCAL TABLES. 

SPHERES. 

(Some errors of 1 In the last figure only. Fi-om Tratjtwine.) 



TMam 


Sur. 


Solid- 


uiAm. 


faee. 


ity. 


1-^ 


.00807 


.00002 


1-ie 


.01287 


.00018 


8-«S 


.02761 


.00048 


^ 


.04909 


.00102 


.07670 


.00200 


8-16 


.11045 


.00845 


7-32 


.16088 


.00548 


^ 


.19685 


.00818 


.24851 


.01165 


5-16 


.80680 


.01598 


11-82 


.87128 


.02127 


isJ 


.44179 


.02761 


.51848 


.03511 


7-16 


.00182 


.04886 


15-88 


.69028 


05898 


^^ 


.78540 


!06545 


.99403 


.09319 


ii-?i 


1.2278 


.12783 


1.4849 


.17014 


IS-^ 


1.7671 


.22069 


2.0739 


.28084 


,5-?i 


2.4058 


.85077 


2.7611 


.43148 


1. 


8.1416 


.52360 


1-16 


8.5466 


.63804 


s-ll 


8.9761 


.74551 


4.4801 


.87681 


ja 


4.9068 


1 .02^i7 


5.4119 


1.1839 


T-?i 


5.9896 


1.3611 


6.4919 


1.5558 


>.\i 


7.0686 


1.7671 


7.6699 


1.9974 


itM 


8.2957 


2.2468 


8.9461 


2.5161 


li-^ 


9.6211 


2.8062 


10.321 


8.1177 


15-li 


11.044 


3.4514 


11.798 


3.8063 


t. 


12.566 


4.1868 


1-16 


18.304 


4.5939 


^11 


14.188 


5.0248 


15.083 


5.4809 


^^ 


15904 


6.9641 


16.800 


6.4751 


r-?i 


17.781 


7.0144 


18.666 


7.5829 


H 


19.635 


8.1813 


»-ltt 


20.629 


8.8108 


11-16 


21.648 


9.4706 


22.691 


10.164 


18-?^ 


28.758 


10.880 


24.890 


11.649 


15-?l 


25.967 


13.448 


27.109 


18.272 


8. 


88.274 


14.187 


1-16 


29.465 


15.080 


^11 


80.680 


15.979 


81.919 


16.957 



Diam. 



Sur- 
face. 



Solid. 

ity. 



17.974 
19.031 
20.129 
21.268 
22.449 
28.674 
24.942 
26.254 
27.611 
29.016 
80.466 
81.965 
33.510 
36.751 
40.196 
48.847 
47.718 
51.801 
56.116 
60.668 
65.450 
70.482 
75.767 
81.306 
87.118 
93.180 
99.541 
106.18 
118.10 
120.81 
127.88 
135.66 
148.79 
152.25 
161.03 
170.14 
179.59 
180.89 
199 53 
210.08 
220.89 
282.18 
348.73 
255.72 
268.08 
280.85 
294.01 
807.58 
321.56 
835.95 
350.77 
360.02 
381.70 
897.83 
414.41 
431 .44 
448.92 
466.87 
485.81 



Dlam. 



% 
10. 



11. 



12. 



18. 



14. 



15. 



16. 



18. 



19. 



20. 



21. 



22. 



Sur- 
face. 



306.86 
814.16 
822.06 
830.06 
838.16 
846.86 
354.66 
868.05 
871.54 
880.18 
888.83 
897.61 
406.49 
415.48 
424.50 
438.73 
448.01 
452.89 
471.44 
490 87 
510.71 
580.93 
551.55 
5TO.55 
593.05 
615.75 
637.96 
660.52 
683.49 
706 85 
730.68 
754.77 

rr9.82 

804.25 
829.57 
855.29 
881.42 
907.98 
934.88 
962.12 
989.80 
1017.9 
1046.4 
1075.2 
1104.5 
1134.1 
1164.2 
1194.6 
1225.4 
1256.7 
1288.8 
1320.3 
1852.7 
1385.5 
1418.6 
1462.2 
1486.2 
1520.5 
1655.8 



Solid- 
ity. 



501.21 
58:{.G0 
543 48 
503 m 
584.74 
606 13 
6:i8.04 
650.46 
673.42 
696.91 
7^^.95 
745.51 
770. M 
796.83 
823 58 
640.40 
8TC.79 
904 78 
962. r.2 

iafci.7 

1085.8 
1150.3 
1218.0 
1288.8 
13G1 .2 
1486.8 
1515.1 
1596.8 
1680.8 
1767.2 
1857.0 
1949.8 
2045.7 
|2144.7 
2:246.8 
.2862.1 
2460.6 
2572.4 
12687.6 
12806.2 
2928.2 
|3a>8.6 
13182.6 
3815.8 
,3451 .5 
3591.4 
3735.0 
13882.5 
4083.7 
4188.8 
4847.8 
14510.9 
4677.9 
4849 1 
5024.8 
5208.7 
5387.4 
5575.3 
5767.6 



SPHERES. 



119 



flPU lUftE»-<Cbnf IniMd.) 



Diam. 


Sur- 
face. 


Solid- 
ity. 


Diam. 


Sur- 
face. 


Solid- 
ity. 


Diam. 


Sur- 
face. 


Bolid- 

tty. 


*^ 


150C.4 


6064.1 


«) H 


5158.1 


84788 


70 H 


15G15 


183471 


1«S6.0 


6165.8 


41.'' 


5881.1 


86067 


71. ^ 


15687 


187408 


«. 


1661. 


6870.6 


H 


5410.7 


87488 


H 


16061 


191880 


4 


1696.8 


6S80.6 


42. ^ 


6541.9 


88792 


72. 


16886 


196488 


:2 


1785.0 


6796.8 


H 


5674.5 


40194 


H 


16518 


199582 


s 


1778.1 


7014.8 


43. 


5808.8 


41680 


78. 


16742 


908689 


24. 


1800.6 


7288.8 


H 


5944.7 


48099 


^ 


16978 


80790S 


t£ 


1847.5 


7466.7 


44. 


6088.1 


44608 


74." 


17804 


818175 


L 1 


1885.8 


7700.1 


K 


6S21.2 


46141 


K 


17487 


816605 


m 


1984.4 


7088.8 


45. 


6861.7 


47718 


75. " 


17678 


880894 


». 


1968.5 


8181.8 


H 


6608.9 


40681 


^ 


17906 


225841 


'4 


900S.9 


8429.2 


46. 


6647.6 


60965 


76." 


18146 


889848 


'Hi 


2042.8 


8688.0 


H 


6792.9 


58645 


K 


18886 


284414 


iS 


8083.O 


8980.9 


47. 


6939.9 


54362 


77. 


18626 


289041 


». 


8128.7 


9d02.8 


M 


7TJ88.3 


66115 


H 


18809 


848786 


i,4 


8164.7 


9470.8 


48. 


7288.8 


57006 


78." 


19114 


848475 


i^ 


2206.2 


9744.0 


K 


7«9.9 


69784 


K 


19860 


868884 


Ik 


8848.0 


looee 


49. 


7543.1 


61601 


79. 


19607 


858155 


27. 


2890.2 


10806 


H 


7697.7 


63506 


H 


19656 


868068 


M 


2838.8 


10595 


50. 


7854.0 


65450 


80. 


80106 


268068 


4 


2875.8 


10889 


K 


8011.8 


67488 


H 


80868 


278141 


H 


8419.2 


11189 


61. 


8171.2 


60456 


81. 


80618 


278868 


». 


8468.0 


11494 


» 


8882.3 


71519 


H 


80807 


288447 


iA 


2807.2 


11905 


68. 


8494.8 


78682 


88. 


81124 


888606 


!^ 1 


8651.8 


12181 


H 


8658.9 


75767 


H 


81388 


894010 


i|i£ 


2596.7 


12448 


68. 


8824.8 


77952 


88. 


81643 


899388 


». 


8642.1 


12770 


H 


8992.0 


80178 


^ 


81904 


804881 


H 


8887.8 


18108 


64. 


9160.8 


88448 


84. 


32167 


810340 


j4 


8784.0 


18442 


^ 


9881.2 


84760 


H 


88432 


815915 


^ 


2780.6 


18787 


66. 


9508.2 


87114 


86. " 


88698 


881566 


8D. 


8887.4 


14187 


K 


9676.8 


89511 


H 


88966 


887864 




2874.8 


14494 


56. ^ 


9852.0 


91958 


86. 


83285 


888089 


n 


8988.5 


14866 


H 


10029 


94488 


H 


88506 


888888 


S 


2970.6 


15284 


67. 


10907 


96967 


87. 


83779 


844798 


SI. 


8019.1 


15599 


H 


10887 


99541 


^ 


24058 


850771 




8068.0 


15079 


68. 


10568 


102161 


88. 


84888 


856819 


L , 


8117.8 


16866 


H 


10751 


104826 


H 


S4606 


862985 


£ 


8166.9 


16758 


59. 


10936 


107536 


89. 


24885 


869122 


«. 


8817.0 


17157 


H 


lirJ8 


110294 


H 


26165 


875878 


:^ 


ai«7.4 


17868 


60. 


11810 


118098 


90. 


25447 


381704 


"11 


8818.8 


17974 


^ 


11499 


115949 


^ 


25780 


888102 


^ 


8860.6 


18892 


61. 


11090 


118847 


91. 


26016 


804570 


as. 


3481.8 


18817 


H 


11882 


181794 


H 


26802 


401109 


H 


8478.8 


19248 


62. 


ijore 


124789 


92. " 


86590 


407781 


-, 


8586.7 


19685 


H 


12278 


127882 


^ 


86880 


414405 


r 


8578.5 


80129 


68. 


12469 


130925 


98. 


27172 


421161 


84. 


8681.7 


80580 


H 


18668 


134067 


^ 


27464 


427991 


g 


8685.8 


21087 


64. 


18868 


137259 


94. 


27759 


484894 


878B.8 


81501 


^ 


13070 


140601 


H 


28055 


441871 


85. 


8848.5 


82449 


65. 


13278 


143794 


95. 


28858 


448920 


M 


8050.2 


88485 


H 


18478 


147138 


H 


88652 


466047 


86. 


4071.5 




66. 


13685 


150538 


96. 


28958 


463848 


M 


4186.5 


85461 


H 


13893 


158980 


M 


29255 


470684 


ST. 


4800.9 


86528 


67. 


14106 


157480 


W. 


29559 


477874 


H 


4417.9 


27612 


H 


14814 


161032 


M 


89665 


485802 


38. 


4586.5 


88781 


68. 


14527 


164637 


98. 


8017^2 


492808 


H 


4656.7 


29880 


H 


14741 


1(W«95 


>li 


30481 


500888 


89. 


4778.4 


81059 


69. 


14957 


172007 


99. 


80791 


606047 


M 


4901.7 


SisSTO 


H 


15175 


1T5774 


^ 


81108 


516785 


40. 


50S6.5 


83610 


70. 


15394 


17^595 


100. " 


81416 


828598 



120 



MATHEMATICAL TABLES. 



contehts in oreio fbet anb ir. s. oali^ons of 

PIPKS AND Ciri.INDKR8 OF VAHIOU8 DIABKSTS1K8 
ANB ONK FOOT IN liENGTH. 





1 gallon = 231 oublo inches. 1 cubic foot 


= 7.4805 gaUoDS. 






For 1 Foot in 




For 1 Foot in 




For 1 Foot in 


o 


Length. 


a 


Lenf?th. 




•Length. 


5*^ 


Cubic Ft. 
also Area 


U.S. 
Oals., 

281 
Cu. In. 


Cubic Ft. 
also Area 


u.a 

Oats., 

231 
Cu, In. 


Oublo Ft. 
also Area 


U.S. 
Oals.. 

231 
Cu.Id. 


in Sq. Ft. 


in Sq. Ft. 


in Sq. Ft. 


5-la 


.OOOB 


.0025 


en 


.2486 


1.859 


19 


1.969 


14.73 


.0005 


.004 


7 


.2878 


1.909 


19^ 


2.074 


15.51 


T?f6 


.0006 


.0067 


7U 


.28117 


2.146 


20 


2.182 


16.a2 


.001 


,0078 


TJ^i 


.8068 


2.296 


20^ 


2.292 


17.15 


H 


.0014 


.0102 


7* 


.8276 


2.45 


21 


2.406 


17.99 


9-16 


.0017 


.0120 


8 


.8491 


2.611 


21^ 


2.881 


18.86 


H 


.0031 


.0150 


^ 


.8712 


2.777 


22 


2.640 


19.7.5 


1&« 


.0036 


.0108 


* 


.8941 


2.948 


22^ 


2.761 


80.66 


,&. 


.0061 


.0280 


* 


.4176 


3.126 


28 


2.885 


81.58 


.0036 


.0269 


9 


.4418 


8.805 


88H 


8.012 


22.58 


ll^l« 


.0042 


.0312 


m 


.4667 


8.491 


94 


8.149 


23.50 


.0048 


.0850 


^y} 


.4922 


8.682 


96 


8.409 


25.50 


1 


.0066 


.0406 


m 


.5186 


8.879 


26 


8.687 


27.68 


iM 


.0066 


.0686 


10 


.5454 


4.08 


27 


3.976 


29.74 


iS 


.0128 


.0918 


io« 


.5780 


4.286 


28 


4Ji76 


81.99 


19< 


.0167 


.1240 


i^ 


.6018 


4.496 


29 


4.687 


84.81 


8 


.0918 


.1682 


.6808 


4.715 


80 


4.909 


86.78 


3^ 


.0^6 


.2068 


11 


.66 


4.937 


81 


6.841 


89.21 


S^ 


.0641 


.2550 


l1^ 


.6908 


5.164 


88 


6.585 


4I.7S 


2% 


.0412 


.3065 


.7m 


6.896 


88 


6.940 


44.48 


8 


.0491 


.8672 


ii9i 


.7580 


6.683 


84 


6.805 


47.16 




.0576 


.4809 


18^ 


.78.V4 


5.8i1( 


86 


6.681 


49.96 


3yi 


.0668 


.4996 


12^ 


.8522 


6.375 


36 


7.069 


62.88 


89i 


.0767 


.6788 


18 


.9218 


6.806 


87 


7.467 


55.86 


4 


.0878 


.6528 


18« 


.904 


7.436 


88 


7.87« 


58.92 


^ 


.0986 


.7360 


14 


1.069 


7.997 


89 


8.296 


62.06 


4t 


.1104 


.8268 


14^ 


1147 


8.578 


40 


8.727 


65.28 


^ 


.1881 


.9206 


18 


1.227 


9.180 


41 


9.168 


68.68 


r* 


.1864 


1.020 


16^ 


1.810 


9.801 


42 


9.6i1 


71.97 


5M 


.1506 


1.125 


16 


1.896 


10.44 


48 


10.066 


75.44 


^ 


.1650 


1.234 


16^ 


1.486 


11.11 


44 


10.560 


?8.99 


.1808 


1.340 


17 


1.676 


11.79 


45 


11.045 


82.68 


r 


.1968 


1.469 


\7% 


1.670 


12.49 


46 


11.541 


86.88 


§1 


.2131 


1.594 


18^ 


1.768 


18.28 


47 


18.048 


90.18 


.2804 


1.7*4 


18« 


1.867 


18.06 


48 


12.666 


94.00 



To find the capacity of pipes greater than the largent given In the fable, 
look in the table for a pipe of one half the ^iven sixe, and multiply its capac- 
ity by 4; or one of one third its siee, and nuiliiply its capacity by 9, etc. 

To find the weight of*water in any of the given sizes multiply the capacity 
In cubic feet by 62^ or the gallons by M^, or, if a doner approximation m 
required, by the weight of a cubic foot of water at the actual temperature in 
the pipe. 

Oiven the dimensions of a cylinder in inches, to find its capacity in U. 8. 
gallons: Square the diameter, multiply by the length and by .OOCM. ltd z^ 
d« X .7864 X I 



diameter, I ^ length, gallons = - 



IMl 



= .0034dn. 



CAPACITY OF CYLINDRICAL VESSELS. 



121 



CTI«IND»I€AI. TBMBLS, TANKS, 0I9TBBN8, ST€« 

]HaBiet«r in Feet mod Incites, Area In Square Feet, and 
IT. 8. Gallons Capacity for One Foot In Depth* 



1 gallon = 3Sl cubic inches : 



1 ciibic foot 
7.4805 



= 0.18868 cubic feet. 



Diam. 


Area. 


Gals. 


DIam. ' 


Area. 


Gals. 


DIam. 


Area. 


Gals. 


Fi. In. 


Sq.ft. 


Ifoot 


Ft. In. 


Sq.ft. 


Ifoot 
depth. 
1^66 


Ft. In. 


Sq.ft. 


Ifoot 
depth. 
2120.9 


I 


.785 


5 8 


25.22 


19 




288.53 


1 1 


.g8;2 


6.89 


5 9 


25.97 


194.25 


19 




291.04 


2177.1 


1 i 


1.069 


8.00 


5 10 


26 78 


199.92 


19 




298.65 


22S4.0 


1 3 


1J887 


9.18 


5 11 


27.49 


205.67 


19 




806.86 


2291.7 


1* 4 


1.306 


10.44 




28.27 


211.51 


20 




314.16 


2360.1 


1 5 


1.576 


11.79 


6 3 


80.68 


229.50 


20 




3J2.06 


2409.2 


1 6 


1.767 


13.22 


6 


83.18 


248.28 


20 




330.06 


2469.1 


I 7 


1.969 


14.73 


6 9 


35.78 


267.60 


20 




88816 


2529.6 


1 8 


2.1&2 


16.32 




38.48 


287.88 


21 




:M6 36 


2591.0 


1 9 


2.405 


1799 


7 8 


41.28 


808.81 


21 




354.66 


26C3.0 


1 10 


2.640 


19.75 


7 6 


44.18 


8*).49 


21 




868.05 


2715.8 


1 U 


2.885 


21.68 


7 9 


47.17 


852.88 


21 




871.54 


-2779.3 


S 


3.14;i 


23.60 


8 


50.27 


876.01 


22 




.380.18 


2843.6 


i i 


8.409 


25.50 


8 8 


53.46 


399.88 


22 




888.82 


2908.6 


a 2 


8.687 


27.58 


8 6 


56.75 


424.48 


22 




397.61 


a)74.3 


2 3 


8.976 


29.74 


8 9 


60.18 


449,82 
47^89 


22 




406.49 


3040.8 


2 4 


4.276 


81.99 


9 


63 62 


28 




415.48 


3J08.0 


2 5 


4387 


8181 


9 8 


67.20 


602.70 


23 




424.56 


8175.9 


3 e 


4.909 


86.72 


9 6 


70.88 


680 24 


23 




438.74 


3244.6 


2 7 


5.:»ll 


39.21 


9 9 


74.66 


558.51 


23 




448 01 


8314.0 


2 8 


5l565 


41.78 


10 


78.54 


887.52 


24 




452.89 


8884.1 


2 9 


5.940 


44.43 


10 8 


83.62 


617.26 


24 




401.86 


3455.0 


2 10 


6.305 


47.16 


10 6 


86.59 


647.74 


24 




471.44 


8626.6 


2 11 


6.081 


49.98 


10 


90.76 


678.95 


24 




481.11 


8598.9 


3 


7.069 


52 88 


11 


95.08 


710.90 


25 




490.87 


8672.0 


3 1 


7.467 


65.86 


11 8 


99.40 


748.58 


25 




500.74 


3745.8 


S 2 


7.876 


68.09 


11 6 


108.87 


776.90 


25 




510.71 


8820 8 


8 3 


8.296 


62.06 


11 9 


108.43 


811.14 


25 




520.77 


3895.6 


3 4 


8.727 


65.28 


12 


113.10 


846.03 


26 




580.93 


8971.6 


3 5 


9.16S 


66.58 


12 8 


117.86 


881.65 


26 




641.10 


4048.4 


3 6 


9.621 


71.97 


12 6 


122.72 


918.00 


26 




651.55 


4125.9 


8 7 


10.066 


75.44 


12 9 


127.68 


955.09 


26 




662.00 


4204.1 


3 8 


10.950 


78.99 


13 


132.73 


992.91 


27 




572.56 


4V83.0 


8 9 


11.015 


68 62 


13 8 


187.89 


1031.5 


27 




583.21 


4362.7 


3 10 


11.541 


86^ 


13 6 


143.14 


1070.8 


27 




593.96 


4448.1 


8 11 


12.WS 


90.18 


13 9 


148.49 


1110.8 


27 




604.81 


4524.3 


4 


12.566 


94,00 


14 


153.94 


1161.5 


28 




615.75 


4606.2 


4 1 


13.005 


97.96 


14 8 


159.48 


1193.0 


28 




62G.P0 


4688.8 


4 2 


18.635 


102.00 


14 6 


165.13 


1235.3 


28 




687.94 


47i;> 1 


4 8 


14.186 


106.12 


14 9 


170.87 


1278.2 


28 




649.18 


4^56 2 


4 4 


14.748 


110.82 


15 


176.71 


1321.9 


20 




660.52 


4941.0 


4 5 


15.821 


114.61 


15 8 


182.65 


1366.4 


29 




671.96 


5C26 


4 6 


15.90 


118.07 


15 6 


188 69 


1411.6 


29 




G88.49 


5112.9 


4 7 


16.50 


123 42 


15 9 


194.8:3 


1457.4 


29 




69r).13 


5199.U 


4 8 


17.10 


127.95 


16 


201.06 


lf04.1 


80 




7C6.8e 


5:187.7 


4 9 


17.72 


132.56 


16 8 


207.89 


1561.4 


80 




718.09 


5376.2 


4 10 


18.85 


187.25 


16 6 


213 82 


1599.5 


30 




730.62 


5465 4 


4 11 


18.99 


142 02 


16 9 


2^.35 


1648.4 


80 




742.64 


56r.5.4 


5 


19.63 


146.^ 


17 


2^6.96 


1697.9 


81 




754.77 


5646.1 


5 1 


20.29 


151.82 


17 8 


231.71 


1748.2 


31 




766.99 


5737.5 


5 2 


20.97 


156.83 


17 6 


240.53 


1799.8 


31 




779.81 


5829.7 


5 3 


21.65 


161.93 


17 9 


247.45 


1851.1 


81 




791.73 


5922.6 


5 4 


22.34 


167.12 


18 


264 47 


1903.6 


82 




804 26 


6016.2 


5 5 


23.04 


172.88 


18 8 


261.59 


1956.8 


32 




816.86 


6110.6 


5 6 


28 76 


177.72 


18 6 


268 80 


2010.8 


32 




829.58 


0e05.7 


5 7 


24.48 


188.15 


16 9 


276. :2 


2065 5 1 


82 




842.39 


6301.5 



122 MATHEMATICAL TABLES. 

OAIiliONS AND G1TBIC FEBT. 

ITiitteil States Omllons In a stven Namber of €able Feet* 

1 cubic foot = 7.480610 U. S. gallons; 1 gaUon = 281 cu. in. = .18868056 cu. ft. 



Cubic Ft. 


Gallons. 


Cubic Ft. 


Gallons. 


Cubic Ft. 


Gallons. 


0.1 
0.2 
0.8 
0.4 
0.5 


0.75 
1.60 
2.84 
2.99 

8.74 


60 
60 
70 
80 
90 


874.0 
448.8 
528.6 
696.4 
678.8 


8,000 
9,000 
10,000 
80,000 
80,000 


59,844.2 
67,824.7 
74,805.8 
140,6l0f4 
884,415.6 


0.6 
0.7 
0.8 
0.9 

1 


4.49 
5.24 
5.96 
6.78 
7.48 


100 
800 
800 
400 
600 


748.0 
1,496.1 
8,244.2 
2,992.2 
8,740.8 


40,000 
50,000 
60,000 
70,000 
80,000 


899,220.8 
874,025.0 
448,881.1 
588,686.8 
698,441.6 


8 

8 

4 
6 
6 


14.96 
22.44 
29.92 
87.40 
44.88 


600 
700 
800 
900 
1,000 


4,488.8 
5,286.4 
5,984.4 
6,782.5 
7,480.5 


90,000 
100,000 
200,000 
800,000 
400,000 


678,346.7 

748,051.9 

1,496,108.8 

2,-,'4l,l55.7 

2,992,207.6 


7 
8 

10 
80 


52.86 
59.84 
67.82 
74.80 
149.6 


•8,000 
8,000 
4,000 
5,000 
6,000 


14,961.0 
22,441.6 
29,922.1 
87,402.6 
44,888.1 


500,000 
600,000 
700.000 
800,000 
900,000 


8.740.8S9.5 
4,488,811.4 
5,286,863 8 
5,984,415.2 
6,732,467.1 


80 
40 


2S4.4 
299.8 


7,000 


68.868.6 


1,000,000 


7,480,519.0 



Gable Feet In a stven Namber of Gallons. 



Gallons. 


Cubic Ft. 


Gallons. 


Cubic Ft. 


Gallons. 


Cubic Ft. 


1 
2 
3 

4 
5 

6 
7 
8 
9 
10 


.184 
JW7 
.401 
.585 
.668 

.808 

.986 

1.069 

1.808 

1.887 


1,000 
2,000 
8,000 
4,000 
5,000 

6,000 
7,000 
8,000 
9,000 
10,000 


133.681 
267.861 
401. OIJ 
534.722 
668.408 

802 088 

935.764 

1,06!).444 

1,20:5.125 

1,380.806 


1,000,000 
2,000,000 
8,00U,000 
4,000.000 
5,000,000 

6,000,000 
7,000,000 
8,000.000 
9,000,000 
10,000,000 


188,680.6 
267,861.1 
401,041.7 
684.722.2 
668,408.8 

802.088 8 

985.768.9 

1,060.444.4 

l,208,ltr,.0 

1,886,805.6 



NUMBER OF SQUARE FEET IN PLATES. 



123 



inmBBS OF Sai^ABB FBBT IN PI^ATBS 8 TO 92 
FBBT I«ONG, AND 1 INCH DTIDB. 

For other widths, multiply by the width in inches. 1 Bq. in. = .0060| sq. ft. 



h. and 
In. 


Ins. 
Long. 


Square 
Feet. 


Ft. and 
Ins. 
Long. 


Ins. 
Long. 


Square 


Ft. and 
Ins. 
Long. 


Ins. 
Long. 


Square 


S. 


88 


.25 


7.10 


94 


.6628 


1«.8 


152 


1.056 




87 


.2569 




95 


.6597 




158 


1.068 




38 


.2639 


8. 


96 


.6667 




154 


1.069 




99 


.2708 




97 


.6736 




155 


1.076 




40 


.2778 




96 


.6806 


18.0 


156 


1.088 




41 


.2817 




99 


.6876 




157 


1.09 




42 


.2917 




100 


.6944 




158 


1.097 




43 


.2986 




101 


.7014 




159 


1.104 




44 


.3056 




102 


.7088 




160 


1.114 




45 


.8125 




103 


.7158 




161 


1.118 


10 


40 


.8194 




104 


.7822 




162 


1.125 


11 


47 


.8264 




105 


.7292 




163 


1.132 


4. 


48 


.3388 




106 


.7361 




164 


1.189 




49 


.8406 




107 


.7481 




165 


1.146 




50 


.3472 


9.0 


108 


.75 




166 


1.168 




51 


.8542 




100 


.7569 




167 


1.159 




53 


.3611 




no 


.7689 


14.0 


168 


1.167 




53 


.8881 




111 


.7708 




169 


1.174 




54 


.876 




112 


.7778 




170 


1.181 




55 


.3819 




113 


.7847 




171 


1.188 




06 


.8889 




114 


.7917 




173 


1.194 




57 


.8958 




115 


.7986 




173 


1.201 


10 


58 


.4028 




116 


.8066 




174 


1.208 


11 


59 


.4097 




117 


.8125 




175 


1.215 


5. 


60 


.4167 




118 


.8194 




178 


1.822 




61 


.4286 




119 


.8264 




177 


1.229 




9i 


.4306 


10.0 


120 


.8388 




178 


1.286 




63 


.4375 




121 


.8408 




179 


1.248 




61 


.4444 




122 


.8472 


16.0 


180 


1.25 




65 


.4614 




123 


.8542 




181 


1.257 




66 


.4583 




124 


.8611 




182 


1.264 




67 


.4668 




125 


.8681 




188 


1.271 




68 


.4782 




126 


.875 




184 


1.278 




69 


.4792 




127 


.6819 




185 


1.285 


10 


70 


.4861 




128 


.6889 




186 


1.292 


11 


71 


.4931 




129 


.8958 




187 


1.299 


«. 


72 


.6 




180 


.9028 




188 


1.S06 




73 


.6069 




181 


.9097 




189 


1 .313 




74 


.5189 


11.0 


182 


9167 




190 


1.319 




75 


.1^08 




183 


.9286 




191 


1.336 




76 


.5278 




184 


.9306 


16.0 


192 


1.388 




77 


.6847 




185 


.9875 




198 


1.34 




78 


.6417 




186 


.9444 




194 


1.347 




T9 


.5486 




187 


.9514 




195 


1.854 




80 


.6656 




188 


.9588 




196 


1 861 




61 


.6625 




189 


.9658 




197 


1.868 


10 


82 


.5694 




140 


.9722 




196 


1.375 


11 


83 


.5764 




141 


.9792 




199 


1.882 


1. 


84 


.5834 




142 


.9861 




200 


1.889 




85 


.5905 




143 


.9981 




201 


1.396 




86 


.5972 


18.0 


144 


1.000 




202 


1.408 




87 


.6042 




145 


1.007 




203 


1.41 




88 


.6111 




146 


1.014 


17.0 


201 


1.417 




89 


.6181 




147 


1.021 


1 


205 


1.424 




90 


.625 




148 


1 028 




206 


1.431 




91 


.6819 




149 


1.085 




207 


1.438 




93 


.6389 




150 


1.042 




20S 


1.444 




93 


.6458 




151 


1.049 




209 


1.461 



124 MATHEMATICAL TABLES. 

i^tTARB FBBT IH PIiATB8-(C<mfMi(ed.) 



Ft. and 
Ins. 
Long. 


Ins. 
L«.ng. 


Square 

F66t. 


Ft. and 
Ins. 
Long, 


Ins. 
Long. 


%■" 


Ft. and 
Ins. 
Long. 


IllB. 

Long. 


Square 
Veet. 


17. C 


210 


1.458 


88.6 


269 


1.868 


87.4 


828 


8.878 


7 


211 


1.465 


6 


870 


1.875 


6 


829 


2.885 


8 


2i-.a 


1.472 


r 


871 


1.882 


6 


830 


2.292 


U 


U18 


1.479 


8 


272 


1.689 


7 


881 


8.J99 


10 


214 


1.486 


9 


278 


1 696 


8 


BSi 


8.80t. 


11 


tfin 


1.498 


10 


274 


1.903 


9 


B83 


8.818 


18.0 


216 


1.5 


11 


875 


1.91 


10 


834 


8.819 


1 


217 


1.607 


118.0 


878 


1.917 


11 


835 


2.826 


a 


218 


1.514 


1 


877 


1.924 


88.0 


836 


2..^3 


3 


219 


1.521 


8 


878 


1.931 


1 


887 


2.84 


4 


220 


1.528 


8 


279 


1 988 


8 


888 


8.347 


5 


221 


1.586 


4 


880 


1.944 


8 


889 


2.854 


6 


2*12 


1.542 


6 


281 


1.951 


4 


840 


2.861 


7 


223 


1.649 


6 


882 


1.958 


6 


841 


2.868 


8 


224 


1.556 


7 


283 


1.965 


6 


848 


2 875 


9 


2-J5 


1.663 


8 


884 


1.972 


7 


843 


2 882 





226 


1.669 


9 


885 


1.979 


8 


844 


2.380 


11 


2^ 


1.5T6 
1.588 


10 


280 


1.966 


9 


845 


8.896 


19.0 


228 


11 


887 


1.993 


10 


846 


2.«U3 


1 


239 


1.59 


84.0 


888 


2. 


11 


817 


8.41 


2 


230 


1.597 


1 


889 


2.007 


88.0 


848 


2.417 


3 


m 


1.604 


8 


890 


2.014 


1 


849 


2.424 


4 


232 


1.611 


8 


891 


2.0-31 


8 


frV) 


2.4n 


5 


233 


1.618 


4 


892 


2028 


8 


851 


8.438 


6 


234 


1.655 


5 


893 


2.a35 


4 


&52 


2.441 


7 


236 


1.682 


6 


294 


2.042 


6 


853 


2.451 


8 


m 


1.6:S9 


7 


895 


2.049 


6 


854 


2.458 


9 


287 


1.645 


8 


296 


2.066 


7 


8.55 


2.4es 


10 


238 


1.653 


9 


897 


20C8 


6 


856 


2.472 


11 


289 


1.659 


10 


898 


2.069 


9 


867 


2.4^9 


80.0 


240 


1.667 


11 


299 


2.076 


10 


853 


2.486 


1 


241 


1.6T4 


86.0 


800 


2.088 


11 


859 


2.41.3 


2 


242 


1.681 


1 


801 


209 


80.0 


860 


2.5 


3 


243 


1.688 


2 


802 


2.097 


1 


861 


2.607 


4 


214 


1.6W 


8 


8a3 


2.104 


8 


862 


2.614 


5 


245 


1.701 


4 


804 


2.111 


8 


863 


2.5ei 


6 


246 


1.708 


5 


8a5 


2.118 


4 


864 


2.688 


7 


247 


1.715 


6 


806 


2,125 


5 


865 


8.&S5 


8 


248 


1.722 


I 


807 


2.182 


6 


866 


2 642 


9 


249 


1.729 


803 


2.139 


7 


867 


2.649 


10 


2.'50 


1.736 


9 


809 


2.146 


8 


868 


2.556 


11 


251 


1.743 


10 


810 


2.153 


9 


869 


2.663 


21.0 


252 


1.75 


11 


811 


2.10 


10 


ro 


2.560 


1 


253 


1.757 


86.0 


812 


2.167 


11 


871 


2 578 


2 


254 


1.764 


1 


813 


2. 174 


81.0 


^ 


8.583 


3 


255 


1.771 


8 


814 


2.181 


1 


8.50 


4 


256 


1.778 


8 


815 


2.188 


8 


874 


8.597 


5 


257 


1.786 


4 


816 


2.194 


8 


875 


8.004 


6 


258 


1.792 


6 


817 


2.201 


4 


SI? 


8.611 


7 


259 


1.799 


6 


818 


2.208 


6 


2.618 


8 


260 


1.806 


I 


819 


2.215 


6 


878 


8.6i5 


9 


£61 


1.818 


8v'0 


2.222 


r 


879 


2.6.S» 


10 


262 


1.819 


9 


821 


2.229 


8 


8£0 


2.6:)9 


11 


203 


1.826 


10 


822 


2.236 


9 


881 


2 61G 


9d.o 


]»4 


1.883 


11 


823 


2.248 


10 


883 


8 c.'sa 


1 


265 


1.84 


87.0 


824 


2.26 


11 


883 


2 60 


2 


266 


1.847 


1 


825 


2.257 


88.0 


884 


2.667 


3 


267 


1.854 


2 


826 


2.264 


1 


885 


8 674 


4 


268 


1.861 


8 


827 


2.871 


8 


880 


8.681 



CAPACITY OP RBOTANGULAB TAKES. 



135 



CAPACITIKS OF KBCTAlfGITIiAK TAHK8 IIV 17. 8. 
GAIiIiONS, FOB BACH FOOT IK DBPTH. 

1 cubic foot = 7.4806 U. a gallons. 



Width 


Length of Tank. 


of 
Tank. 


feet. ft. In. 


feet. 

S 


ft. in. 
8 • 


feet. 

4 


ft. In. 
4 6 


feet. 
6 


ft. In. 
6 6 

88.80 
108.86 
188.43 
H4.00 
104.67 

185.14 
80.-1.71 
886.28 


feet. 
6 


ft. In. 
6 6 


feet. 
7 


ft. in. 
2 C 


ao.os 


37.40 
46.75 


44.88 
56.10 
67.89 


58.36 
65.46 
78.54 
01.04 


50.84 
74.80 
80.77 
104.78 
110.60 


67.38 
84.16 
100.00 


74.81 
03.51 
iia.2i 


80.T7 
112.81 
184.05 
157 00 
170.58 

801.07 
884.41 
846.86 
800.30 


07.25 
181.56 
145.87 
170,18 
194.40 

81S.80 
843.11 
867.43 
»81,74 
316.05 


104.78 
180.01 
157.00 


U G 






117.881 180.01 


18:187 


4 
4 6 








134.65 
151.48 


149.61 

168.31 
187.01 


800.45 
835.63 


5 












861 88 


i G 










** 




888.00 


6 
















314.18 


u 6 


















840.86 
366.54 



























Width 


Length of Tank. 


of 
Tiuk. 


ft. in. 
7 6 


feet. 
8 


ft. In. 
8 « 


feet. 
9 


ft. In. 
9 6 


feet. 
10 


ft. In. 
10 6 


feft. 
11 


ft. In. 
11 6 


feet. 
12 


fi in. 
2 


11«.« 
140J86 
168.31 
106 80 
'tiiAl 

258.47 
280..'^ 
80H.57 
336.68 
364.67 

802.78 
480.78 


110.08 
140.01 
170.53 
800 45 
830.37 

860.30 
890.82 

820.14 
850.00 
888.98 

418.91 
448.83 
478.75 


187.17 
1.58.06 
10a75 
888.54 
854.34 

236.13 
317.92 
340.71 
381.50 
418.30 

445 00 
476.88 
508.67 
540.46 


134.65 
168.81 
208.07 
835.03 
860.80 

302.06 
330.62 
370.88 
40804 
487.60 

47187 
604.03 
588.60 
572.85 
605.02 


14813 
177.66 
213.19 
848.73 
884.86 

319.79 
355..32 
390.85 
426.89 
461.92 

497.45 
538.98 
.508.51 
604.05 
089.58 

075.11 


140.61 
187.01 
884 41 
801.88 
890.82 

336.62 
874.08 
411.43 
448.a3 
480.83 

683.64 
66104 
698.44 
6:«.&4 
678.85 

710 65 
748.06 


187.09 
196.80 
885.03 
874.90 
814.18 

858.45 
898.?^ 
438.00 
471.27 
510.54 

540.81 
580.08 
688.86 
607 6:J 
706.00 

740.17 
785.45 
884.78 


104.57 
806.71 
240.80 
288.00 
329.14 

370,88 
411.43 
452.67 
498.71 
5^.85 

67fi09 
617 14 
058.88 
690.42 
740.66 

781.71 
888.86 
804.00 
005.14 


178.05 
815.00 
258.07 
801.09 
344 10 

387.11 
4.3018 
473.14 
516.16 
650.10 

008.18 
04.5.19 
088.80 
731.21 
774 23 

817.24 
800.86 
903.86 
946.87 
060.80 


170 53 
884.41 
809.30 
814.18 
350.00 

403.04 

448.83 
403.71 
538.50 

583.47 

628 86 
673.84 
718.18 
763 00 








807.88 










852 77 












897 66 














042.56 
















987.43 


















1082.3 


i; 


















1077.2 

























126 



HATHEMATIOAL TABLES. 



NUIIKBBB OF BARBEIjS (31 1-2 GAI«I«ON8) Ilf 
CISTERNS AND TANKS. 



1 Barrel = 81^ gallons = 



8 1.5X28 1 
1788 



= 4.21094 cubic fpefc. Reciprocal = .8S7477. 



Depth 

in 
Feet. 


Diameter in Feet 


6 


« 


7 


8 


9 


10 


11 


12 


IS 


14 


1 


4.663 


6.714 


9.189 


11.987 


15.108 


18.652 


22.560 


26.859 


81.522 


36.557 


6 


28.8 


83.6 


45.7 


59.7 


75.5 


98.8 


112.8 


184.3 


157.6 


183.8 


6 


88.0 


40.8 


54.8 


71.6 


90.6 


111.9 


185.4 


161.2 


189.1 


219 3 


7 


82.6 


47.0 


64.0 


88.6 


105.8 


180.6 


158.0 


188.0 


2j».7 


S55.9 


8 


87.8 


68.7 


78.1 


96.5 


120.9 


149.2 


180.6 


214.9 


2S2.2 


^»i.S 


9 


42.0 


60.4 


82.8 


107.4 


186.0 


167.9 


208.1 


241.7 


288.7 


329.0 


10 


46.6 


67.1 


91.4 


119.4 


151.1 


186.5 


228.7 


268.6 


815.2 


865.6 


11 


51.8 


78.9 


100.5 


181.8 


166.9 


205.2 


948.8 


295.4 


846.7 


402.1 


12 


66.0 


80.6 


109.7 


148.2 


181.8 


223.8 


270.8 


882.8 


878.8 


488.7 


18 


60.6 


87.8 


U8.8 


155.2 


196.4 


242.6 


298.4 


849.2 


409.8 


47B.2 


14 


65.8 


94.0 


127.9 


ler.i 


211.5 


261.1 


816.0 


8T6.0 


441.8 


SI 1.8 


16 


69.0 


100.7 


187.1 


179.1 


226.6 


289.8 


888.5 


402.9 


472.8 


548.4 


16 


74.6 


107.4 


146.2 


191.0 


241.7 


296.4 


361.1 


429.7 


504.4 


584.9 


17 


79.8 


114.1 


156,4 


202.9 


256.8 


317.1 


883.7 


456.6 


685.9 


U21.5 


18 


88.9 


180.0 


164.5 


214.9 


271.9 


386.7 


406.2 


483.5 


667.4 


058.0 


10 


88.6 


127.6 


178.6 


226.8 


287.1 


354.4 


488.8 


510.8 


606.9 


604.6 


20 


988 


184.8 


182.8 


288.7 


802.2 


878.0 


451.4 


687.2 


680.4 


731.1 



Depth 

in 
Feet. 








Diameter in Feet 






16 


16 


17 


18 


19 


20 


21 


82 


1 
6 
6 
7 
8 


41.966 

209.8 

251.8 

885.7 


47.748 

288.7 

286.5 

834.2 

882.0 


58.903 

269.5 

828.4 

377.8 

431.2 


60.481 
302.2 
862 6 
423.0 

483.4 


67.8S2 

836.7 

404.0 

471.8 

538.7 


74.606 

873.0 

447.6 

622.2 

596.8 


82.258 

411.8 
498.5 
575.8 
658.0 


90.«rs 

451.4 
541.6 
631.9 
722. S 


9 

10 
11 
12 
18 


8rr.7 

419.7 
461.6 
608.6 
545.6 


429.7 
477.5 
526.2 
578.0 
620.7 


4851 
539.0 
592.9 
046.8 
700.7 


543.9 
604.8 
664.7 
725.2 
785.6 


606.0 
678.3 
740.7 
806.0 
875.8 


671.5 
746.1 
830.7 
805.3 
969.9 


740.8 
822.5 
904.8 
987.0 
1069.8 


812.5 
902.7 
993.0 
1083.3 
1178.5 


14 
15 
16 

17 
18 


687.5 
629.5 
671.6 
713.4 
756.4 


668.5 
716.2 
764.0 
811.7 
859.5 


754.6 
808.5 
662.4 
916.4 
970.3 


846.0 
906.5 
966.9 
1027.8 
1087.8 


942.6 
1010.0 
1077.8 
1144.6 
1212.0 


1044.5 
1119.1 
1198.7 
1268.3 
1842.9 


1151.6 
1283.8 
1316.0 
1398.8 
1480.6 


1268.8 
1854.1 
1444.4 
1534.5 
1624.9 


19 
20 


797.4 
889.8 


907.2 
955.0 


1024.2 
1078.1 


1148.2 
1206.6 


1279.3 
1346.6 


1417.5 
1492.1 


1562.8 
1645.1 


1715.9 
1805.5 



LOOAEITHMS. 



127 



HITKBER OP BABRBIiS (81 1-S OAIitiONS) IN 
GISTJEBN8 AND TANKS.— Continued. 



Depth 


Diameter in Feet. 


in 
FeeL 


<s 


24 


25 


28 


27 


28 


29 


80 


1 
5 

7 
8 


96.668 
498.8 
5»0 
090.7 
789.3 


107.488 
537.2 
344.6 
752.0 
859.5 


116.571 
582.9 
699.4 
8160 
988.6 


126.068 
680.4 
756.5 
682.6 

1006.7 


185.966 
679.8 
615.8 
951.6 

1087.7 


146.226 
781.1 
677.4 
1028.6 
1169.6 


157.658 

784.8 

941.1 
1098.0 
1264 9 


167.688 
889.8 
1007.2 
1175.0 
1342.9 


9 

10 
11 
1« 
13 


888.0 
966.7 
1065.8 
1184.0 
»82.7 


966.9 
1074.8 
1181.8 
1289.2 
1896.6 


1049.1 
1165.7 
1282.8 
1896.8 
1515.4 


1184.7 
1260.6 
1886.9 
1518.0 
1689.1 


1228.7 
1859.7 
1495.6 
1681.6 
1787.6 


1816.0 
1462.2 
1606.5 
1754.7 
1900.9 


1411.7 
1568.6 
1725.4 
1882.8 
2089.2 


1510.6 
1678.6 
1846.5 
2014.4 
2182.3 


14 

15 
]« 
17 

IS 


1881.8 
1480.0 
1578.7 
1677.8 
1776.0 


1904 
1611.5 
1718.9 
1826.8 
1988.8 


1682.0 
1748.6 
1665.1 
1961.7 
2096.3 


1765.2 
1891.2 
2017.8 
2148.4 
2369.5 


1908.6 
2069.5 
2176.5 
S311.5 
2447.4 


2047.2 
2196.4 
2889.6 
2485.8 
3633.0 


2106.0 
2852.9 
2509.7 
9666.6 
2828.4 


28S0.1 
2517.9 
2686.8 
2858.7 
8001.5 


19 
SO 


1874.7 
1978.8 


9041.2 
2148.6 


2214.6 
2821.4 


2895.6 
2521.7 


2588.4 
2719.4 


2778.8 
2924.5 


2960.8 
8187.2 


8189.4 
8857.8 



liOGARITSMS. 

If#KAritlims (abbreviation log).— The log of a number is the exponent 
of Che power to which it is neo o o na nr to raise a fixed number to produce the 
invcn Domber. The fixed number is called the base. Thus if the base is 10. 
the loff of 1000 is 8, for 10> = 1000. There are two systems of logs in general 
OK, the commofiy in which the base is 10, and the Naperian, or hyperbolic, 
ia which the base is 2.718881838 .... The Kaperian base is commonly de* 
aax^ by e» aa in the equation e* = «« in which y is the Nap. log of x. 

la any sjsteni of logs, the log of 1 is 0; the log of the base, taken in that 
fTHem. is 1. In any system the base of which is greater tlum 1, the logs of 
aQ cambers greater than 1 are positive and the logs of all numbers less than 
I lie negative. 

The modulus of any system is equal to the redproeal of the Naperian log 
of the base of that system. The modulus of the naperian system is 1, that 
^ the common system is .4842945. 

The* lo^ of a number in anv system equals the modulus of that system x 
ib^ !9aperian log of the number. 

The^|>er6o<«; or .kaperian log of any number equals the common log 

ISvefy log consists of two parts, an entire part called the charcuiteriatic, or 
m^x, and the decimal part, or mantiasa. The mantissa only is given in the 
oroal tables of common logs, with the decimal point omitted. The charac- 
ttrimie is found by a simple rule, vis., it is one less than the number of 
flcores to the left of the decimal point in the number whose log is to 1>e 
found. Thus the characteristic of numbers from 1 to 9.99 + Is 0, from 10 to 
»J9+is 1, from 100 to 990 + is 2, from .1 to .99 + is - 1, from .01 to .009 -i- 
it - 2, etc Thus 

log of IHWO is 8.80108; log of .2 is - 1.80108; 

.P u jioo .. 2.80108; " " .02 ** - 2.80108; 

•• •• 20 " 1.80108; •* " .008 •• - 8.80108; 

•• •• 2 " 0.80103; " " .0002 " - 4.80108. 



128 MATHEMATICAL TABLES. 

The minus lign ii frequently written a\)OTe the characteristic thus: 
log .002 s= 7.80108. The characteristic only is negatiTd, the decimal |»art, or 
mantiasaf being always positive. 

When a log consists ora negative index and a positive mantissa, it Is usual 
to write the negative sign over the Index, or else to add 10 to the index, and 
to indicate the subtraction of 10 from the resultinsr logarithm. 

Thus log .2 = T.aolftJ, and this may be written 9.30103 - 10. 

In tables of logarithmic sines, etc., the - 10 is generally omitted, as l>eing : 
lin«lMn«tood. 

Rules for use of the table of liOgarlthms.- To flna «lie ; 
log of any ivliole nnmber.— For 1 to lOU inclusive Uie log is given 
oouiplete in the small table on page 139. 

For 100 to 999 inclusive the decimal part of the log Is given opposite the i 
given number In the column headed in the table (including the two fi^ires 
to the left, making six figures). Prefix the cliaracteristic, or index. 2. 

For 1000 to 0999 inclusive : The last four figures of tho log are found i 
opposite the first three figures of the given number and in the veriical : 
column headed with the fourth figure of the given number ; prefix the two 
figures under column 0, and the index, which is 3. 

For numbers over 10,000 having five or more digits : Find the decimal part 
of the log for the first four digits as above, multiply the difference figure 
in the last column by the remaming digit or digits, and divide by 10 If there 
be only one digit more, by 100 If there be two more, and so on ; add the 
quotient to the log of the first four digits and prefix the index, which, is 4 
Ix there are five digits, 6 if there are six digits, and so on. The table of pro- 
portional parts may be used, as shown below. 

To find the loff of a deeinal firaetlon or of m ivliole 
number and a deelaial*— First find the log of the quantity as if there 
were no decimal point, then prefix the index according to rule ; the Index is 
one less than the number of figures to the left of the dedmsl point. 

Be<}uired log of 3.141598. 

log of 8.141 =0.497068. Piff. a 188 

From proportional parts B = 690 

*• .. 09 = 1248 

•• - •» 003 « 041 



log 8.141593 0.4971498 

To find the number correspondins to a slven los*— l^^nd 

A) the table tbe log nearest to the decimal part of the given log andtake the 
first four digits of the required number from the column N and tfie top or ^ 
foot of the column containing the log which Is the next less than the given 
log. To find the 5th and 6th digits subtract the log in the table from the 

given log. multiply the diffei-enoe by 100, and divide by the figure in tlie 
iff. column opposite the log ; annex the quotient to the four digits already 
found, and place the decimal point aocoralug to the nile ; tbe number oC 
figures to the left of the decimal point is one greater than the Index. 

Find number corresponding to the log 0.407150 

Next lowest log in table oorresponds to 8141 497068 

DIff. a 89 

Tabular diff. = 188; 82 -•- 188 = .59 + 

The Index being 0, the number Is therefore 8.14159 +. 

To multiply i^svo numbers by tbe use of loararltl&ms. 

AAd together the logs of the two numbers, and find the number whose log i 
1b the sum. i 

To dlTlde ty^o numbers.— Subtract the log of the divisor from ; 
the log of the dividend, and find the number whose log Is the differetioe. 

To raise a number to any ctven poirer.— Multiply the log o( ; 
the number by tbe exponent of the power, and find the number whose log U 
the product. ' 

To find any root of a Klven number.—Diyide the los of the ' 
Dumber by the index of the r(X)t. The quotient is the log of the root. 

To find tbe reelproeal of a anmber.— Subtract the decimal ^ 
part of the log of the number from 0, add 1 to the Index and change the sigq ' 
of the index. The result is the log of the reciprocal. 



L0GARITHH6. 



12» 



Beqiiired the reciprocal of 8.141603. 

Lotrof8.141S03,a8foundaboTe 0.4971406 

Subtract decimal part from eives 0.SO28b0i 

Add 1 to the Index, and changing sign of the Index gives.. T.6038QQ3 
irbiefa Is the loff of 0.81W1 . 

Vo MmA Uia Iburth ttmn of a vropoitlon by lofEaxitluns* 
—Add the loearithms of tJie aecond and third terms, and from their sum 
subtract the logarithm of tLe first term. 

When one logarithm is to be subtraoted from another, it may be more 
convenient to convert the subtraction Into an addition, which may be done 
by first subtracting tUfjgiven logarithm from 10. adding the difference to the 
Of her losrarithm, and afterwards rejecting the 10. 

Tht» dlflerenoe between a given logarithm and 10 is called its arWunetioal 
complement, or cologarithm. 

To subtract one logarithm from another Is the same aa to add its comple- 
ment and then reject 10 from the result. For a — 6 s 10 — 6 -p «* - 10. 

To work a proportion, then, by logarithms, add the complement of the 
logarithm of the first term to the logarithms of the second and third terms. 
The characteriHtic must afterwards be diminished by 10. 

Example In locarltlmia nrltlt a nesatlve Index. -Solve by 

logaritiims (^) ^t which means divide 596 by 1011 and raise the quotient 

to the S.45 power. 

log 596 s 9.790986 
log 1011= 8.004751 



Jog of quotient = - 1.716986 
Jffitftiply by 2.45 



- 2.561175 
-2J 64940 
-^ l.4» 2470 
-1.80 477576 = .20178, Ani. 



In multiplying - 1.7 by 5. we i , 
f- ;< oarriea = — 9. In adding -U4-B-\-Z-^l carried from previous oolumn, 
we say: 1 -f 8 + 8 =: 19, minu82 = 10. set down and carry 1; 14-4-2 = & 



*y- 5 ^ I =: *^' ' *9 cftiry ; 5 X — 1 =3 — 5 less 



liOOARTTBHS OF N^IMBKBS FROM 1 TO 100. 



N. 


Log. 


N. 


Log. |! N. 


Log. 


N. 


Log. 


N. 


Log. 


1 


o.oooooo 


21 


I.a32»9 , 


41 


1.619784 


61 


1.786880 


81 


1.906485 


2 


0.801080 


22 


1.842428 


42 


1.028949 


62 


1.792892 


82 


1.913814 


8 


0.477121 


28 


1.861796 


48 


1.638468 


68 


1.799341 


88 


1.919078 


4 


0.602060 


24 


1.880211 


44 


1.648458 


64 


1.806180 


84 


1.921279 


5 


0.098970 


25 


1.897940 


46 


1.668218 


65 


1.812918 


85 


1.929410 


6 


0.778151 


26 


1.414978 


46 


1.662758 


66 


1.819544 


86 


1.934466 


7 


0.845096 


27 


1.481864 


47 


1.679006 


67 


1.826075 


87 


1.939519 


8 


O.9O009O 


28 


1.447166 


48 


1.681241 


68 


1.882509 


88 


1.944488 





0.964M3 


29 


1.462896 


40 


1.600196 


69 


1.838849 


89 


1.949390 


10 


l.OOOOOO 


80 


1.477121 


60 


1.666970 


70 


1.846096 


90 


1.954248 


11 


1.041888 


81 


1.491362 


51 


1.707670 


71 


1.851958 


91 


1.959041 


12 


1.079181 


82 


1.505160 


52 


1.716008 


72 


1.857382 


99 


1.963788 


18 


1.118948 


83 


1.618514 


58 


1.7S4276 


78 


1.868828 


98 


1.96&188 


14 


1.146198 


84 


1.581479 


54 


1.789894 


71 


1.869282 


94 


1.978198 


15 


1.176001 


85 


1.544066 


55 


1.740868 


75 


1.875061 


95 


1.977794 


16 


1.20«120 


86 


1.656806 


66 


1.748188 


78 


1.880614 


96 


1.982271 


17 


1.280449 


87 


1.668902 


57 


1.756875 


77 


1.88W91 


97 


1.086778 


18 


1.256278 


88 


1.679784 


56 


1.763498 


78 


1.892095 


98 


1.991i226 


19 


1.278754 


89 


1.601066 


50 


1.770852 


79 


1.8i)7097 


99 


1.095686 


90 


1.801090 


40 


1.6O906O 1 


60 


1.778151 , 


1 ^ 


1.908000 


100 


2.000000 



L30 



IiOGARITHMS OF XUHBEBS. 



Ko. 100 L. 000.] 



[No. 109 L. (HO. 



N. 





1 


2 


S 


4 


6 


6 


7 


8 


9 


Difl. 






0434 
4751 
9026 


0868 
5181 

QdAI 


ISOl 
5609 


1784 
6088 


2166 
6466 


2596 
6894 


8029 
7821 


8461 
7748 


8891 
8174 




lOU 

1 
2 


000000 

8600 


438 
436 


VMOl ««.« 


0800 
4521 
870O 


0724 
4940 
9116 


1147 1 1570 
5860 5779 
9582 I 9947 


1998 
6197 


2415 
6616 


404 

4s» 


8 
4 


012837 
7083 


8259 
7461 


^ 


4100 
8284 


0861 
4486 
85n 


0775 
4896 
8878 


416 
41S 
408 


6 
6 

7 


021189 
5306 
9384 


1608 
5715 
9789 


2016 
6125 


2428 
6588 


2841 
6942 


8252 
7850 


3864 1 4075 
7757 1 8164 


0195 
4227 
8228 


0600 
4628 
8620 


1004 
5029 
9017 


1406 
5480 
9414 


1812 
5880 
9611 


2216 
6280 


2610 
6629 


3021 
7088 


404 

4U0 


6 
9 


osatsM 

7436 
04 


8826 
7825 


0907 


0602 


0998 


897 



Pbopobtional Pabts. 



Diff. 


1 


2 


8 


4 


6 


6 


7 


6 


9 


484 


48.4 


86.8 


180.2 


173.6 


217.0 


260.4 


303.8 


847.2 


890.6 


48:^ 


48.8 


88.6 


129.9 


178.2 


216.5 


259.8 


803.1 


846.4 


889.7 


482 


48.2 


86.4 


129.6 


172.8 


216.0 


259.2 


802.4 


845.6 


888.8 


481 


48.1 


86.2 


129.8 


m.4 


215.6 


258.6 


801.7 


344.8 


887.9 


480 


48.0 


86.0 


129.0 


172.0 


215.0 


256.0 


801.0 


844.0 


887.0 


429 


42.9 


85.8 


128.7 


171.6 


214.6 


257.4 


800.8 


343.2 


386.1 


428 


42.S 


85.6 


128.4 


171.2 


214.0 


256.8 


299.6 


812.4 


365.2 


427 


42.7 


85.4 


128.1 


170.8 


213.6 


256.2 


298.9 


841.6 


884.8 


426 


42.6 


85.2 


127.8 


170.4 


213.0 


255.6 


29S.2 


840.8 


883.4 


425 


42.5 


65.0 


127.6 


170.0 


212.6 


255.0 


297.5 


840.0 


882.5 


424 


42.4 


84.8 


127.2 


109.6 


212.0 


854.4 


896.8 


880.2 


381.6 


423 


42.8 


84.6 


126.9 


169.2 


211.6 


258.8 


296.1 


888.4 


880.7 


422 


42.2 


84.4 


126.6 


168.8 


211.0 


258.2 


295.4 


887.6 


879.8 


421 


42.1 


84.2 


126.8 


168.4 


210.5 


252.0 


294.7 


886.8 


878.9 


420 


42.0 


84.0 


126.0 


168.0 


210.0 


252.0 


894.0 


886.0 


878.U 


419 


41.9 


83.8 


125.7 


167.0 


209.5 


251.4 


293.8 


835.2 


877.1 


418 


41.8 


83.6 


125.4 


167.2 


209.0 


250.8 


292.6 


884.4 


876.2 


417 


41.7 


83.4 


125.1 


166.8 


206.6 


260.2 


291.9 


888.6 


875.3 


416 


41.6 


83.2 


124.8 


166.4 


208.0 


219.6 


291.2 


882.8 


874.4 


416 


41.5 


88.0 


124.5 


166.0 


207.6 


M9.0 


290.5 


882.0 


878.5 


414 


41.4 


82.8 


124.2 


165.6 


207.0 


»I8.4 


289.8 


831.2 


»ra.6 


418 


41.8 


82.6 


123.9 


165.2 


206.5 


847.8 


289.1 


880.4 


871.7 


412 


41.2 


82.4 


123.6 


164.8 


206.0 


247.2 


288.4 


829.6 


870.8 


411 


41.1 


82.2 


128.8 


164.4 


205.5 


246.6 


287.7 


328.8 


809.9 


410 


41.0 


82.0 


128.0 


164.0 


205.0 


246.0 


287.0 


828.0 


869.0 


409 


40.9 


81.8 


122.7 


168.6 


2(M.5 


245.4 


286.3 


827.2 


868.1 


408 


408 


81.6 


122.4 


163.2 


204.0 


244.8 


285.6 


826,4 


867.2 


407 


40.7 


81.4 


122.1 


162.8 


208.5 


244.2 


284.9 


825.6 


866.8 


406 


40.6 


81.2 


121.8 


162.4 


208.O 


2436 


284.2 


824.8 


866.4 


406 


40.5 


81.0 


121.5 


162.0 


202.6 


248.0 


283.5 


824.0 


864.5 


404 


40.4 


80.8 


121.2 


161.6 


202.0 


242.4 


282.8 


828.2 


868.6 


403 


40.8 


80.6 


120.9 


161.2 


201.5 


241.8 


282.1 


822.4 


362.7 


402 


40.2 


80.4 


120.6 


160.8 


301.0 


211 2 


281.4 


821.6 


861.8 


401 


40.1 


80.2 


120.3 


160.4 


200.5 


210.6 


280.7 


820.6 


860.9 


400 


40.0 


800 


120.0 


160.0 


200.0 


240.0 


280.0 


820.0 


880.0 


399 


89.9 


79.8 


119.7 


159.6 


199.5 


239.4 


279.3 


819.2 


869.1 


398 


89.8 


79.6 


119.4 


159.2 


199.0 


238.8 


278.6 


318.4 


858.2 


397 


89.7 


79.4 


119.1 


l.'i8.8 


198.5 


2:w.2 


277.9 


317.6 


867.3 


896 


39.6 


79.2 


118.8 


158.4 


198.0 


287.0 


277.2 


816.8 


856.4 


896 


1 39.5 


79,0 


118.5 


158.0 


197,5 


237.0 


1 278.5 


316 


885.6 



LOOARITHHS OF XTJXBEB8. 



131 

LNo. 119 L. (i7^ I 



No UOLuOtL] 



N. 



041908 
5323 
S&18 



069078 
0905 



000808 
4468 
8180 



071888 
5647 



1787 
67J4 
9006 



8468 
7286 



1075 
4832 
8567 



2S60 
6012 



218S 
6105 
9993 



8846 
7066 



1452 
5a06 



8617 
6276 



6495 



0680 
4280 
8046 



5580 



6610 



6885 



0766 
461S 



5958 
9668 



70M 



8862 
7275 



1158 
4996 
8805 



6826 



8718 
7868 



87B6 
7664 



1588 
587B 
9185 



2958 



6699 



0407 
4085 
7781 



414B 
8058 



1924 
5760 
9668 



7071 



0776 
4451 
8094 



8 9 Difl. 



4540 
8442 



6142 
9942 



3709 
7448 



1145 
4816 
8457 



4932 I 
88:>0 1 



2691 I 886 



6524 



0820 
4088 
7815 



1514 870 
5182 866 
8619 



^mopoamovAi, Pabt& 



Difl. 



805 
9M 
808 
9B8 
391 
880 
389 
388 
887 
886 
885 

381 



381 
380 

378 
377 
876 
375 

874 
373 
832 
371 
370 
369 
368 
167 
366 

364 
363 
302 
391 
360 
360 

35T 
8B6 



89.5 
89.4 
80.8 
80.2 
80.1 
80.0 
86.9 
38.8 
88.7 
88.6 
88.5 

38.4 
88.3 
88.2 
38.1 
88.0 
87.9 
87.8 
87.7 
87.6 
87.5 

87.4 
87.3 
87.2 
87.1 
87.0 
86.9 
86.8 
86.7 
36.6 
86.5 

86.4 
86.3 
86.2 
86.1 
36.0 
85.9 
85.8 
85.7 
35.0 



79.0 
78.8 
78.6 
78.4 
78.2 
78.0 
77.8 
TT.6 
77.4 
77.2 
77.0 

78.8 
76.6 
78.4 
78.3 
78.0 
75.8 
75.6 
75.4 
75.8 
75.0 

74.8 
74.6 
74.4 
74.2 
74.0 
78.8 
78.6 
78.4 
78.2 
73.0 

73.8 
78.6 
79.4 
78.3 
78.0 
71.8 
71.6 
71.4 
71.3 



118.5 
118.2 
117.9 
117.6 
117.3 
117.0 
116.7 
116.4 
116.1 
115.8 
115.5 

115.2 
114.9 
114.6 
114.3 
114.0 
118.7 
118.4 
118.1 
112.8 
112.5 

112.2 
111.9 
111.6 
111.8 
111.0 
110.7 
110.4 
110.1 
109.8 
109.5 

109.2 
108.9 
106.6 
108.8 
106.0 

iar.7 

107.4 

107.1 
106.8 



158.0 
157.6 
157.2 
166.8 
156.4 
1S6.0 
155.6 
155.2 
154.8 
164.4 
154.0 

158.6 
158.2 
152.8 
152.4 
152.0 
151.6 
151.2 
150.8 
150.4 
150.0 

149.6 
149.2 
148.8 
148.4 
148.0 
147,6 
147.2 
146.8 
146.4 
146.0 

145.6 
145.2 
144.8 
144.4 
144.0 
148.6 
148.2 
1« 8 
142.4 I 



197.5 
197.0 
196.5 
106.0 
195.5 
195.0 
194.5 
194.0 
198.5 
198.0 
192.5 

192.0 
191.5 
191.0 
190.5 
190.0 
189.5 
180.0 
188.5 
188.0 
187.5 

187.0 
186.6 
186.0 
185.5 
185.0 
184.5 
184.0 
188.5 
188.0 
182.5 

182.0 
181.5 
181.0 
180.5 
180.0 
179.5 
179.0 
-78.5 
178.0 



287.0 
286.4 
285.8 
286.2 
234.6 
234.0 
288.4 
282.8 
282.2 
281.6 
231.0 

230.4 
229.8 
229.2 
228.6 
228.0 
227.4 
226.8 
226.2 
825.6 
225.0 

224.4 
228.8 
223.2 
222.6 
222.0 
221.4 
220.8 
220.2 
219.6 
210.0 

218.4 
217.8 
217.2 
216.6 
216.0 
215.4 
214.8 
S14.2 
218.0 



276.5 
27V.8 
275.1 
274.4 
278.7 
278.0 
272.3 

2n.6 

270.9 
270.2 
269.5 

268.8 
268.1 
267.4 
286.7 
266.0 
265.8 
264.6 
268.9 
268.2 
262.5 

261.8 
261.1 
260.4 
259.7 
259.0 
258.8 
267.6 
2S6.0 
256.2 
255.7 

254.8 
254.1 
253.4 
262.7 
252.0 
251.3 
250.6 
349. 9 
249.2 



816.0 


866.5 


816.2 


854.6 


814.4 


858.7 


813.6 


8G8.8 


812.8 


851.9 


812.0 


851.0 


811.2 


850.1 


810.4 


349.2 


809.6 


848.8 


808.8 


347.4 


808.O 


846.6 


80r.2 


845.6 


806.4 


844.7 


805.6 


348.8 


804.8 


842.9 


804.0 


342.0 


808.2 


841.1 


802.4 


840.2 


801.6 


889.8 


800.8 


338.4 


800.0 


887.5 


299.2 


336.6 


206.4 


885.7 


297.6 


834.8 


206.8 


838.9 


296.0 


838.0 


205.2 


882.1 


294.4 


881.2 


203.6 


s;:o.3 


292.8 


829.4 


202.0 


U28.5 


291.2 


827.6 


200.4 


326.7 


289.6 


825.8 


288.8 


324.9 


288.0 


324.0 


287.2 


;23.1 


amy 


322.2 


2H5.0 


C21.3 


284.8 


820.4 



2Si 



UMASITHMS OF KUHBRBS. 



No. 


190 L. 079.) 














[No. 184 L. 1«L 


N. 





1 


2 


8 


1 


6 


« 


7 


8 


9 


Diff. 








990i 
















190 


07^181 1 WHO 


0966 1 0086 ! 


0067 
4576 
8186 


1847 
4984 

8490 


1707 

6291 
8815 


2007 

5W7 
9198 


M26 
00O4 
9552 


800 


1 
2 
8 


082785 
6900 
9905 


8144 
6716 


8508 
7071 


8861 
7498 


4219 ' 
7781 


887 
865 


0238 0611 
8772 1 4122 

T*>"»7 TftfU 


0003 

4471 


1315 
4^ 


1067 
5169 
86M 


2018 
5518 
8990 


2370 
5866 
9385 


2721 
6215 
9681 


8071 
6668 


862 


4 

5 


(mm 

0910 


849 




0026 
Si68 
6871 


S46 
843 


6 
7 

8 


180 

2 


lOOfin 0715 1059 , 1403 
38M 4146 4487 4S» 
TOIO 7M9 7988 8227 


1747 
5109 


2091 
5510 


2484 

6851 


2777 
6191 
9579 


8119 
6581 
9016 




0858 
8609 

6940 


838 

S35 

883 


110590 

8943 
?^71 


0926 

4277 
7003 


1263 

4611 
7931 


1509 

4944 
8265 


1984 

5278 
8595 


2270 

6611 
8926 


2605 

5948 
flG»6 


2940 

6876 
9586 


3273 

6006 
9915 


0246 
8525 
0781 


830 
825 


13a574 
3852 
7105 

13 


0003 1231 
4178 4504 
7429 ffffsi 


1560 
4830 
8076 


1888 
5156 
8899 


8S16 
fr481 
87« 


9644 

5806 
9015 


2871 
6181 
9868 


8198 
(M56 
9090 






0012 


8S3 



Pboportionai. Farts. 



DIff. 


1 


8 


8 


4 


6 


6 


r 


8 


9 


a-is 


35.5 


71.0 


106.5 


142.0 


177.6 


218 


848.6 


884.0 


819.5 


354 


35.4 


70.8 


106.2 


141.6 


177.0 


212.4 


m.% 


888.8 


818.6 


fm 


35.8 


70.6 


106.9 


141.2 


176.6 


211.8 


847.1 


888.4 


817.7 


852 


35.2 


70.4 


105.0 


140.8 


178.0 


211.8 


846.4 


281.6 


816.8 


351 


35.1 


70.2 


106.8 


140.4 


175.5 


210.6 


845.7 


880.8 


816.9 


350 


a5.o 


70.0 


106.0 


140.0 


175.0 


210.0 


845.0 


280.0 


816.0 


819 


34.9 


60.8 


104.7 


189.6 


174.5 


209.4 


844.8 


879.8 


814.1 


348 


84.8 


69.6 


104.4 


189.2 


174.0 


208.8 


1M8.6 


278.4 


818.8 


847 


34,7 


69.4 


104.1 


138.8 


178.6 


206.8 


842.9 


877.6 


818.8 


846 


84.6 


69.8 


108.8 


138.4 


178.0 


807.6 


842.8 


8716.8 


SU.4 


345 


84.5 


69.0 


103.6 


188.0 


172.6 


807.0 


841.5 


876.0 


810.5 


344 


34.4 


68.8 


108.2 


137.6 


172.0 


806.4 


840.8 


87K.8 


800.6 


34.) 


84.3 


68.6 


102.9 


187.8 


m.5 


806.8 


840.1 


874.4 


808.7 


312 


84.2 


08.4 


102.6 


136.8 


171.0 


805.2 


239.4 


873.6 


807.8 


SMI 


84.1 


68.2 


102.8 


136.4 


170.5 


204.6 


838.7 


872.8 


306.9 


340 


34.0 


GK.O 


102.0 


136.0 


170.0 


801.0 


238.0 


872.0 


806.0 


330 


33.9 


67.8 


101.7 


136.6 


160.5 


808.4 


237.8 


8n.8 


aooj 


338 


33.8 


67.6 


101.4 


185.2 


100.0 


802.8 


836.6 


270.4 


804.8 


337 


88.7 


07.4 


101.1 


184.8 


168.5 


808.2 


835.9 


869.6 


806.8 


836 


33.6 


67.2 


100.8 


184.4 


108.0 


801.6 


835.8 


868.8 


808.4 


835 


33.5 


67.0 


100.5 


134.0 


167.5 


201.0 


881.5 


868.0 


801.5 


331 


:33.4 


66.8 


100.2 


138.6 


167.0 


800.4 


838.8 


867.8 


800.6 


838 


.3.3.3 


66.6 


99.9 


188. 2 


166.6 


199.8 


888.1 


866.4 


290.7 


XiJ 


3.3.2 


664 


99.6 


132.8 


166.0 


199.8 


832.4 


865.6 


298.6 


3:J1 


33.1 


66 2 


99.3 


132.4 


166.6 


198.6 


831.7 


864.8 


897.0 


830 


33.0 


66.0 


99.0 


132.0 


165.0 


198.0 


881.0 


864.0 


897.0 


3d9 


32.9 


65.3 


98.7 


131.6 


164.5 


197.4 


830.8 


868.8 


ami 


328 


32.8 


t>5.6 


98.4 


181.2 


164.0 


196.8 


8sH).6 


868.4 


896.8 


ftJ7 


38.7 


65.4 


98.1 


130.8 


168.5 


196.2 


828.9 


861.6 


894.8 


826 


32.0 


65.2 


97.8 


130.4 


168.0 


195.6 


828.2 


860.8- 


888.4 


325 


32.5 


65.0 


97,5 


180.0 


168.5 


195.0 


827.5 


860.0 


898.5 


821 


32.4 


&4.8 


97.2 


120.6 


162.0 


194.4 


226.8 


850.8 


291.6 


323 


32 3 


&4.6 


96.9 


129.2 


161.5 


198.8 


226.1 


868.4 


890.7 


822 


32.2 


64.4 


96.6 


128. h 


161.0 1 


103.2 


825.4 


SS7.6 


880.8 



LOGAUITHMS OF KITHBBB8* 



189 



No. 185 L. laO.] 



[No. 149 L. 17B. 



N. 



130884 



8.^ 
9879 



143015 

8148 
flei9 



06» 
8856 

TOW 



0977 
4177 
7854 



OlM 
33S7 

6488 
9BS7 



6886 



1C1868 
43fS3 
7817 



17Q90S 
8186 



6640 
8664 



1667 
4650 
7618 



0665 

8478 



0606 
8639 

8748 



S900 

9948 
8966 



1967 
4917 
7906 



0648 
8769 



1298 
4496 

Ten 



8961 
7066 



0148 
8906 
0946 



8906 



1141 
4060 



1619 
4614 
7987 



1196 
4263 

7867 



8610 
6649 
9567 



2564 

6541 
6497 



1434 
4351 



1939 
6188 
680B 



2900 
6451 
8618 



1450 
4574 

7870 



1768 



7966 



0756 
8816 
6658 
9668 



2868 

5838 
8792 



1726 
4641 



1068 
4120 
7154 



0168 
8161 
6184 
9066 



2019 
4932 



2660 
5760 
8984 



2000 

6066 
9049 



9076 
6196 

8894 



1870 
4424 
7457 



0469 
8460 
6480 
9880 



2311 



2389 
6507 

8008 



1676 
4728 
7r69 



0769 
8766 
6796 
9674 



9608 
6612 



8819 
6406 
9664 



2702 
5618 

8911 



1962 
6082 

6061 



1068 
4066 
7098 
99G8 



2895 



PROPORTioNAii Parts. 



DUE. 



821 
890 
319 
818 
817 
816 
815 
814 
818 
812 

811 
810 
809 
806 
807 
806 
806 
804 
806 
802 

801 
800 
299 
896 
897 
896 
296 
994 



891 
890 



287 



82.1 
82.0 
81.9 
81.8 
81.7 
81.6 
81.5 
81.4 
81.8 
81.8 

81.1 
81.0 
80.9 
80.8 
80.7 
80.6 
80.5 
80.4 
80.8 
80.2 

80.1 
80.0 
89.9 
89.8 
29.7 
89.6 
S9.5 
29.4 
89.8 
29.8 

S9.1 
89.0 
£8.9 
88.B 
28.7 
88.6 



64.2 
64.0 
68.8 
68.6 
68.4 
68.2 
68.0 
68.8 
02.6 
62.4 

68.9 
68.0 
61.8 
61.6 
61.4 
61.2 
61.0 
60.8 
60.6< 
60.4 

60.2 
60.0 
69.8 
69.6 
60.4 
69.2 
69.0 
68.8 
56.6 
68.4 

58.9 
68.0 
67.8 
OT.6 
W.4 
M,2 



8 


4 


6 


6 


7 


8 


96.8 


126.4 


160.5 


192.6 


294.7 


256.8 


96.0 


188.0 


160.0 


192.0 


294.0 


856.0 


95.7 


127.6 


159.6 


191.4 


228.8 


265.2 


95.4 


187.2 


159.0 


190.8 


222.6 


261.4 


95.1 


126.8 


156.5 


190.2 


221.9 


258.6 


94.8 


126.4 


158.0 


189.6 


221.2 


262.6 


94.5 


126.0 


157.5 


189.0 


220.5 


2S2.0 


94.2 


125.6 


157.0 


188.4 


219.8 


251.8 


98.9 


126.2 


156.6 


187.8 


219.1 


260.4 


96.0 


m.8 


166.0 


187.2 


218.4 


249.6 


96.8 


1»4.4 


166.6 


186.6 


217.7 


248.8 


96.0 


124.0 


155.0 


186.0 


217.0 


248.0 


9S.7 


128.6 


164.5 


185.4 


216.8 


247.2 


0B.4 


128.2 


154.0 


184.8 


216.6 


240.4 


96.1 


122.8 


153.6 


184.2 


214.9 


2466 


91.8 


128.4 


153.0 


188.6 


214.2 


244.8 


91.6 


182.0 


162.5 


188.0 


218.6 


244.0 


01.2 


121.6 


152.0 


183.4 


212.8 


243.2 


90.9 


121.2 


151.6 


181.8 


212.1 


242.4 


90.6 


120.8 


151.0 


181.2 


211.4 


241.6 


90.8 


120.4 


160.6 


180.6 


210.7 


240.8 


00.0 


120.0 


160.0 


180.0 


210.0 


240.0 


89.7 


119.6 


149.6 


179.4 


209.8 


289.2 


80.4 


119.2 


149.0 


178.8 


206.6 


288.4 


80.1 


118.8 


148.6 


178.2 


907.9 


287.6 


88.8 


118.4 


148.0 


177.6 


207.2 


288.8 


88.6 


118.0 


147.6 


177.0 


206.6 


286.0 


88.2 


117.6 


147.0 


176.4 


206.8 


286.2 


87.9 


117.2 


146.5 


175.8 


206.1 


2^.4 


87.6 


116.8 


146.0 


176.2 


204.4 


283.6 


87.8 


116.4 


146.5 


174.6 


208.7 


282.8 


87.0 


116.0 


145.0 


174.0 


208.0 


282.0 


86.7 


115.6 


144.6 


178.4 


202.8 


281.2 


86.4 


115.2 


144.0 


172.8 


201.6 


280.4 


86.1 


114.8 


148.5 


172.2 


200.9 


229.6 


86.6 


114.4 


148.0 


171.6 


200.2 


228.8 



.84 



lOOABITHMS OF KUlfBEBS. 



No.160Ll176.j 














CNo. 169L.880. 


N. 





1 


f 


» 


4 


i 


• 


7 


8 


e 


Diff. 


150 

1 


178091 
8877 


6881 
9264 


6670 
9652 


6959 
0889 


7248 


7536 


7S25 


8113 


8401 


8689 


S80 


0126 
2986 
5a25 
8647 


0413 
8270 
6108 
89S8 


0099 
8555 
0391 
9209 


0966 
8839 
6674 
9490 


1272 
4128 
6056 
9771 


1668 
4407 
7239 


887 
885 
888 


2 
8 

4 


181844 
4691 
7SS1 


2129 

4975 
7808 


2415 
5259 
8064 


2700 
5542 
8366 


0051 
2846 
5028 
8882 


8S1 
871) 
878 
876 


S 
6 
7 

8 


190632 
8125 
6000 
8657 


0612 
3403 
6176 
8982 


0802 
3681 
6458 
8206 


1171 
8059 
6729 
9481 


1451 
4237 
7006 
9755 


1780 
4514 
7281 


2010 
4702 
7556 


2289 
5060 
7882 


2567 
5846 
8107 


0029 
2701 

5175 
8173 

0R53 
8518 
6166 
8798 


0806 
8088 

5746 
8441 


0577 
8805 

6016 

8no 


0850 
8577 

6286 
8979 


1124 
8848 

6566 
9247 


874 
878 

971 
880 



160 

1 
2 


201897 

4120 
6820 
9515 


1670 

4891 
7096 
9788 


1943 

4663 
7365 


2210 

4934 
76M 


2188 

5204 
7904 


0061 
2720 
5378 
8010 


0619 
2986 
5638 
8273 


0586 
3252 
5002 
8536 


1121 
87^ 
6430 
9060 


1388 
4049 
6694 
9823 


1654 
4314 
0057 
9586 


1921 
4579 
7221 
9846 


867 
866 
264 
868 


8 

4 
5 


212188 
4844 
7484 


2454 
5100 
7747 


8 
7 
8 
9 


220106 
2716 
5809 
7887 

28 


0870 
2976 
5668 
8144 


0631 
8236 
5826 
8400 


0802 
3496 
6084 
8657 


1153 
8755 
6342 
8013 


1414 
4015 
6600 
9170 


1676 
4274 
6858 
9426 


1986 
4.^38 
7115 
9682 


2196 
4792 
7872 
9988 


2456 
5051 
7680 


861 
258 
8S8 


0193 


256 



PBOPORTIONAL PARTS. 



BIfl. 


1 


8 


8 


4 


5 


6 


7 


8 


9 


286 


28.5 


57.0 


85.5 


114.0 


142.5 


171.0 


199.5 


888.0 


856.5 


284 


28.4 


56.8 


85.2 


113.6 


142.0 


170.4 


198.8 


227.2 


255.6 


288 


28.8 


56.0 


84.9 


118.2 


141.5 


169.8 


198.1 


226.4 


854.7 


282 


28.2 


56.4 


84.6 


112.8 


141.0 


169.2 


197.4 


226.0 


868.8 


281 


28.1 


66.2 


84.3 


112 4 


140.5 


168.6 


196.7 


284:8 


8S8.9 


280 


28.0 


66.0 


840 


112.0 


140.0 


168.0 


100.0 


224.0 


858.0 


279 


27.9 


55.8 


88.7 


111.6 


189.5 


lff7.4 


195.8 


228.8 


861.1 


278 


27.8 


55.6 


83.4 


111.2 


139.0 


166.8 


194.6 


222.4 


860.8 


277 


27.7 


56.4 


83.1 


110.8 


188.5 


166.2 


198.9 


281.6 


848.8 


276 


2r.6 


55.2 


82.8 


110.4 


188.0- 


165.6 


198.2 


280.8 


848.4 


275 


27.5 


56.0 


88.5 


110.0 


187.5 


166.0 


192.5 


820.0 


847.5 


274 


27.4 


64.8 


82.2 


109.6 


187.0 


164.4 


191.8 


210.8 


846.6 


278 


27.8 


54.0 


81.9 


109.2 


136.5 


168.8 


191.1 


218.4 


845.7 


272 


27.2 


64.4 


81.6 


108.8 


186.0 


168.2 


190.4 


217.6 


844.8 


271 


27.1 


54.2 


81.8 


108.4 


135.5 


102.6 


180.7 


216.8 


848.0 


270 


27.0 


54.0 


81.0 


108.0 


135.0 


168 


189.0 


216.0 


848.0 


269 


26.9 


58.8 


80.7 


107.6 


184.5 


161.4 


188.3 


815.8 


848.1 


288 


26.8 


58.6 


80.4 


107.2 


134.0 


160.8 


187.6 


814.4 


841.2 


867 


26.7 


58.4 


80.1 


106.8 


183.5 


160.2 


186.9 


818.6 


840.8 


866 


26.6 


53.2 


70.8 


106.4 


183.0 


159.6 


186.2 


818.8 


880.4 


865 


26.5 


53.0 


79.5 


106.0 


182.6 


159.0 


185.6 


818.0 


888.5 


264 


26.4 


52.8 


79.2 


105.6 


1S2.0 


158.4 


184.8 


811.8 


887.6 


863 


26.3 


52.0 


78.9 


105.2 


131.5 


157.8 


164.1 


810.4 


886.7 


262 


28.2 


62.4 


78.6 


104.8 


181.0 


157.2 


183.4 


809.6 


885.8 


861 


26.1 


52.2 


78.3 


104.4 


130.5 


156.6 


188.7 


808.8 


884.9 


860 


86.0 


68.0 


78.0 


104.0 


180.0 


156.0 


182.0 


808.0 


884.0 


259 


25.9 


51.8 


T7.7 


103.6 


129.5 


155.4 


181.8 


807.8 


8SS.1 


258 


25.8 


51.6 


77.4 


103.2 


129.0 


154.8 


180.6 


806.4 


888.8 


257 


25.7 


51.4 


TT.l 


102.8 


128.5 


154.2 


179.9 


806.0 


881.8 


256 


25.6 


51.2 


70.13 


102.4 


128.0 


158.6 


179.2 


804.8 


880.4 


255 


25.5 


51.0 


76.5 


102.0 


lSt7.5 


153.0 


178.6 


804.0 


889.5 



LOGABnmfS OF KITMBSBa 



135 



]sa.l70I^2a(Kl 



[No. lao L. 27a 



N. 


O 


1 


f 


s 


4 


ft 


6 


7 


8 


9 


Diff. 


170 
1 
9 
3 


Sa0449 
S906 
5528 


0704 

aaso 

6781 

0007 


0000 
8604 
e088 
8548 


1215 
8757 
6886 
8799 


1470 
4011 
6637 
9019 


1724 
4264 
6799 
9299 


1979 
4617 
7041 
9660 


2234 
4770 
7202 
9800 


2488 
6028 
7544 


2742 
6276 
7795 


256 
268 
252 




0050 
2541 
6019 
7482 
9982 


0800 
2790 
5266 
7?28 


250 
249 
248 
246 


4 
5 
G 

7 


840549 07D9 
.3088 3S85 
6518 6750 
7»;8 8219 


1<M8 
8584 

6006 
8464 


1297 
8782 
62S2 
8709 


1546 
406O 
6499 
8054 


1795 
4277 
6745 
9198 


2044 
4526 

6991 

9m 


2293 
4772 
7237 
9687 


0176 
2610 
6061 

7489 
9833 


245 
248 
242 

241 
389 


8 
9 

180 

1 


2S0i20 
£858 

5S78 
7B79 


0864 
8096 

S514 
7Vf8 


0906 
8888 

6756 
8158 


1151 
8580 

6996 

8898 


1895 
8882 

6287 
8837 


1088 
4064 

6477 
8877 


1881 
4806 

6718 
9116 


2125 
4648 

6968 
9856 


2868 
4790 

7198 
9694 


2 
3 
4 

6 


260071 
8451 
4818 
7172 
9518. 


081O 
2688 
5064 
7406 
9740 


0548 
2825 
6290 
7M1 
9980 


a?87 
816S 

7875 


1025 
3899 
6761 
8110 


1268 
8686 
6996 
8844 


1601 
8878 
6232 
8578 


1789 
4109 
G467 
8812 


1976 
4846 
6702 
9046 


2214 
4562 
6937 
9279 


288 
237 
236 
284 


0318 
2588 
4850 
7151 


0446 

27ro 

6081 
7880 


0679 
8001 
6811 
7809 


0912 
8233 
5642 
7888 


1144 
8164 
6T72 
8067 


1877 
3606 
6002 
8296 


1609 
892r 
6232 
8625 


288 
232 
280 
229 


7 

8 
9 


271842 
4158 
6402 


2074 
4889 
6692 


2306 
4620 
61ttl 



PsopomxoNAi* Pabtb. 



Diff, 



2B5 
254 
258 
252 
261 
250 
249 
248 
2f7 
246 



944 
248 
242 
HI 
210 



296 
235 

234 



281 
280 



227 
2» 



25.6 
25.4 
25.8 
25.2 
25.1 
25 
24.9 
24.8 
24.7 
24.6 
24.5 

24.4 
24.8 
24.2 
24.1 
24.0 
23.9 
28.8 
28.7 
28.6 
28.6 

28.4 
28.8 
28.2 
28.1 
28.0 
22.9 
22.8 
22.7 
22.6 



62. 

»>.8 
«^ *5 
6<). I 
6(» :3 
6(1 (i 
41^^ 

4&.4 
49.2 
49.0 

4B.4 

48, S 

47,8 
47.0 
47.4 
47 2 

17 '* 

46.8 
46.6 
46.4 
46.9 
46.0 
45.8 
4S.0 

^i I 
4S.9 I 



76.6 
76.2 
75.9 
75.6 
75.3 
75.0 
74.7 
74.4 
74.1 
73.8 
TO.6 

78.2 
72.9 
72.6 
72.8 
72.0 
71.7 
71.4 
71.1 
70.8 
70.6 

70.2 
69.9 
69.6 
69.8 
69.0 
68.7 
68.4 
68.1 
67.8 



lOS.O 
101.6 
101.2 
100.8 
100.4 
100.0 
99.6 
99.2 
96.8 
96.4 
96.0 

97.6 
97.2 
96.8 
96.4 
96.0 
96.6 
96.2 
94.8 
94.4 
94.0 

93.6 
98.2 
02.8 
92.4 
92.0 
91.6 
91.2 
90.8 
90.4 



127.5 
127.0 
1S8.5 
128.0 
125.5 
125.0 
124.6 
124.0 
123.6 
123.0 
122.6 

122.0 
121.6 
121.0 
120.6 
120.0 
119.6 
119.0 
118.5 
118.0 
117.6 

117.0 
110.6 
116.0 
116.5 
116.0 
114.6 
114.0 
113.5 
113.0 



153.0 
152.4 
151.8 
151.2 
160.6 
150.0 
149.4 
148.8 
148.2 
147.6 
147.0 

146.4 
145.8 
145.2 
144.6 
144.0 
148.4 
142.8 
142.2 
141.6 
141.0 

140.4 
139.8 
139.2 
138.6 
188.0 
187.4 
136.8 
136.2 
135.6 



175.6 
177.8 
ITT.l 
176.4 
175.7 
175.0 
174.8 
173.6 
172.9 
172.2 
171.6 

170.8 
170.1 
109.4 
168.7 
168.0 
167.8 
166.6 
165.9 
165.2 
164.6 

168.8 
168.1 
162.4 
161.7 
161.0 
160.3 
169.6 
158.9 
1582 



204.0 

203.2 
202.4 
201.6 
200.8 
200.0 
199.2 
198.4 
197.6 
196.8 
196.0 

195.2 
194.4 
196.6 
192.8 
192.0 
191.2 
190.4 
180.0 
188.8 
188.0 

187.2 
186.4 
185.6 
181.8 
184.0 
188.2 
1SJ.4 
181.6 
180.8 



L30 



LOGAUTTHMS OF KUMBXSa 



No. 1ML.9T8.] 














[Ko. 214 L. aas. 


H. 





1 


i 


S 


4 


h 


6 


7 


8 


9 


Diff. 




278754 


8982 


0811 


9489 


9667 














190 


9896 


0128 

fBm 

4666 
6905 
9148 


0861 
2622 
4882 

7180 
9866 


0678 
2649 
5107 
7854 
9689 


0806 

6882 

7578 
9612 


886 
827 
826 
825 
828 


1 

8 
4 


261088 
8801 
6067 
7800 


1281 
8627 
5788 
8096 


1486 
8798 
6007 
8849 


1715 
8079 
8288 
8478 


1942 
4205 
6466 
8606 


2169 
4481 
6081 
8920 


5 

? 

8 
9 


290085 

4460 
6685 
88S8 


OOBff 
2478 
4687 
6684 
9071 


0480 
26U9 
4907 
7104 
9260 


ma 
asm 

5127 
7888 
9607 


0985 
8141 
6347 
7548 

was 


1147 
8868 

6667 
7761 
9943 


1869 
8584 
57«7 
7979 


IGOl 
8804 
6007 
8108 


1818 
4025 

8416 


2084 
4246 
6446 
8085 


829 
821 
820 
810 


0181 

2881 
4491 

8^ 


08» 

0654 
8901 


QC95 

2764 
4921 
7008 
9804 


0818 

2080 
5186 
7)»« 
9417 


tl8 

217 
816 
815 
818 


200 

1 
2 
8 

4 


801080 
8198 
6851 
7498 
9680 


1847 
8412 
6068 

7710 
9848 


1484 
86tt8 
6781 
7984 


1681 
8644 
5006 
8187 


1808 
40B9 
8211 
8861 


2114 
4275 
6425 
8664 


00B6 
2177 
4280 
6800 
8481 


0268 
2868 
4499 
0909 
8080 


<VI81 
2600 
4710 
6809 
8608 


0693 
2812 
4020 
7018 
9106 


0906 
9028 
5180 
7227 
0814 


1118 
aBi84 
5840 
7488 
9688 


1880 
8445 
6551 

7648 
9780 


1548 
8666 
6760 
7854 
9986 


818 

811 
810 
800 

806 


5 

e 

7 
8 


811754 
8887 
6070 
8068 


^ 
^ 


9 
210 
1 
2 
8 


820146 

2219 
4282 
6886 
8380 


0854 

2426 

4488 
8541 
8588 


0602 

8688 

4004 
6745 
8787 


0TG9 
28m 
4899 
6050 
8901 


0977 

8046 

5105 
7155 
9194 


1184 

8868 

5310 

7X59 

! 9308 


1801 

8458 
5516 
7568 
9601 


1568 

8665 
5721 
7767 
9805 


1805 

3871 
5926 
7972 

O0G8 
2094 


2012 

4077 
6181 
8176 


207 

206 
206 
204 


^ 


903 


4 


880414 


0617 


0819 


1022 


1225 


1 1427 


1630 


1832 


802 



Pbopobtional Parts. 



Dur. 


1 


8 


8 


4 


5 


6 


7 


8 


9 


226 


88.5 


45.0 


67.5 


90.0 


112.5 


135.0 


157.5 


180.0 


908.6 


224 


22.4 


44.8 


67.8 


69.0 


112.0 


134.4 


166.8 


179.8 


201.6 


228 


22.8 


44.6 


66.9 


69.2 


111.5 


1338 


356.1 


178.4 


200.7 


222 


22.2 


44.4 


66.6 


66.8 


111.0 


133.8 


155.4 


177.6 


199.6 


221 


82.1 


44.2 


66.8 


68.4 


110.5 


132.6 


154.7 


176,8 


198.0 


220 


82.0 


44.0 


66.0 


88.0 


110.0 


182.0 


154.0 


176.0 


198.0 


219 


21.9 


48.8 


65.7 


87.6 


109.5 


131.4 


153.3 


175.8 


197.1 


218 


81.8 


48.6 


65.4 


87.2 


109.0 


130.8 


152.6 


174.4 


196.8 


217 


81.7 


48.4 


65.1 


66.8 


106.6 


180.2 


151.9 


178.6 


195.8 


216 


21.6 


48.8 


64.8 


B6.4 


108.0 


129.6 


151.2 


172.8 


194.4 


215 


21.6 


48.0 


64.5 


86.0 


107.5 


129.0 


160.5 


178.0 


198.5 


214 


81.4 


48.8 


64.8 


85.6 


107.0 


128.4 


148.8 


171.8 


188.6 


218 


81.8 


48.0 


68.9 


85.8 


106.5 


127.8 


149.1 


170.4 


191.7 


218 


81.9 


48.4 


68.6 


84.8 


106.0 


127.2 


148.4 


169.6 


190.8 


211 


81.1 


48.8 


68.8 


84.4 


106.5 


126.6 


147.7 


168.8 


189.9 


210 


21.0 


42.0 


68.0 


84.0 


105.0 


126.0 


147.0 


168.0 


189.0 


209 


80.9 


41.8 


68.7 


88.6 


104.6 


125.4 


146.8 


197.8 


188.1 


206 


90.8 


41.6 


68.4 


88.8 


IW.O 


1248 


145.6 


166 4 


187.8 


207 


80.7 


41.4 


68.1 


82.8 


103.5 


124.8 


144.9 


165.0 


186.8 


206 


80.6 


41.8 


61.8 


82.4 


108.0 


123.6 


144.8 


164.8 


185.4 


206 


80.5 


41.0 


CI .6 


82.0 


1025 


128 


143.5 


164.0 




204 


80.4 


40.8 


61 8 


81.0 


102.0 


122.4 


142.8 


168.8 


188.6 


208 


80.8 


40.0 


60.9 


81.2 


101.5 


121.8 


148.1 


108.4 


iSi 


m 


80.2 


40.4 


60.0 


'».8 


101.0 


121.2 


141.4 


161.6 



liOOABTTHHS OF KUMBBRS. 



No. S15Lu 338:3 



[NaS89L.88 



K. 



215 


7 
8 

9 

290 
1 

3 

4 
5 
6 
7 
8 
9 



4404 

640O 
8490 



840444 



4808 
68eS8 
8305 



2188 
4108 



3817S8 
8618 
5488 
7866 
9816 



871008 
2018 
4748 
6B77 



06«e 

9000 
^4580 
6&40 
8500 



044S 
S87& 
4801 
e817 
81)85 



loir 

8800 

6675 
7542 
04O1 



1868 



8680 



884S 
48M 

6860 



0841 

S817 
4786 
6744 
8604 



0668 



e408 

8816 



OS16 

sno6 



7729 
9687 



1487 



5115 
6048 

87B1 



8044 

8067 

Toeo 

9054 



1080 

8014 
4981 
09S0 



oeeo 

8761 
4686 
6669 
8606 



0104 



4176 
6049 
7915 
9772 



1682 
8464 



7124 
8948 



8246 
6867 
72fX> 
9868 



1287 

8212 
6173 
7186 
9068 I 



1088 
2964 

4876 
6790 
8096 



0606 



8101 
9958 



1806 
8647 
5481 
7806 
9124 



8447 
6466 
7469 
9451 



1486 

8409 
6874 
7880 
9878 



1216 
8147 
6008 

6981 
8686 



0788 

2671 
4661 
6488 
8»7 



8649 
6668 
7669 
9660 



1682 

8006 
6670 
76S5 
9472 



1410 



6260 
7172 
9076 



0972 

2859 
4739 
6610 
8478 



8860 
6869 
7868 
9849 



1880 
8809 



5766 
7J20 
9666 



1008 



6462 
7868 
9206 



1161 

8048 
4986 
6796 
8669 



0148 
1991 
8881 
6664 
7488 
9806 



2175 
4015 
6846 
7870 
9487 



0618 
2800 
4198 
6Q8Q 
iSS 
9068 



4061 

6069 
8006 



0047 
2088 

8990 

6968 
7915 
9860 



1796 
8724 
6648 
7554 
9466 



1860 



5118 
6963 
8646 



FsopoRTzoNAL Parts. 



0098 
2644 

4888 
6812 
8084 
9640 



8900 
8867 



0846 
8886 

4196 
6167 
8110 



0064 
1989 
8916 
6884 
7744 



1589 

8424 
5801 
7169 
9060 



2788 
4565 

6894 
8816 



0080 



DIff. 


1 


» 


8 


4 


5 


S 


7 


S 


9 


aoB 

201 
200 
199 

196 
197 
198 
106 

194 


80.8 


40.^ 


60.6 


80.8 


101.0 


121.2 


141.4 


101.6 


181. 


"20.1 


40.8 


60.8 


80.4 


100.5 


120.6 


140.7 


100.8 


180. 


20 O 


40.0 


60.0 


§2-2 


100.0 


120.0 


140.0 


160.0 


180 


19.9 
19.8 
19.7 
19.6 
19.5 
19.4 


89.8 


59.7 


79.6 


99.5 


119.4 


189.8 


150.2 


179 


m.t 


59.4 
59.1 


79.2 

78.8 


99.0 
98.5 


118.8 
118.2 


188.6 
187.9 


158.4 
167.6 


178. 
177, 


ao.d 


58.8 


78.4 


98.0 


117.6 


187.2 


166.8 


176. 


ao.o 


68.5 


78.0 


97.5 


117.0 


180.6 


166.0 


175. 


88.8 


56.2 


77.0 


97.0 


116.4 


185.8 


155.2 


174. 


198 
198 
191 
190 
189 
18B 
187 
186 


19.8 
19.8 
19.1 
19.0 
18.9 
i8.S 

16.6 


80.0 


B7.9 


r7.2 


96.6 


116.8 


186.1 


164.4 


178 


S.4 


67.6 


78.8 


96.0 


115.2 


184.4 


158.6 


m 


^.S 


67.8 


76.4 


96.6 


114.0 


138.7 


152.8 


171 


as.o 


57.0 


76.0 


96.0 


114.0 


188.0 


162.0 


171 


^.8 


66.7 


76.6 


94.5 


118.4 


188.8 


161.2 


170 


97.8 


56.4 


75.2 


94.0 


118.8 


181.6 


160.4 


169, < 


S7 41 


^l 


74.8 


96.5 


112.2 


180.9 


149.6 


108 i 


87.d 


66.8 


74.4 


08.0 


111.6 


180.2 


148.8 


167.- 


18.8 

18.4 

J8.a 

18. a 
18.1 
18.0 
17.^ 


^T.O 


86.6 


74.0 


92.5 


111.0 


129.5 


148.0 


106 1 


186 


ailB 


66.2 


78.0 


98.0 


110.4 


128.8 


147.2 


106 { 


181 


88.0 


54.9 


78.2 


91.5 


109.8 


128.1 


146.4 


104 ' 


188 


86 ^ 


54.6 


72.8 


91.0 


109.2 


127.4 


145.6 


108 1 


188 


^ - 


64.8 


72.4 


90.5 


108.6 


126.7 


144.8 


102 1 


181 
180 
179 


54.0 


72.0 


90.0 


108.0 


120.0 


144.0 


102 < 


88.7 


71.0 


89.6 


107.4 


125.8 


148.2 


101.: 









138 



IX>OARITHyS OF XrUMBERS. 



No. 940 L. 880.1 














[No. 868 L. 431. 


K. 





1 


t 


S 


4 


ft 


6 


7 


8 


8 


DifL 


240 


S8oeii 

8017 
8815 
6606 
7390 
9166 


0898 
2197 
8995 
5785 
7568 
9843 


0578 
8877 
4174 
5964 
7746 
96ii0 


0754 
8557 
4358 
6148 

7U24 
9696 


0984 
8787 
4588 

6881 
8101 
9675 


1116 
8917 
4718 
6499 
8279 


1886 
8097 
4801 
6677 
8456 


1476 
8877 
6070 
6866 
8634 


1666 

8466 
6849 
7084 
8811 


1887 
8686 
648S 
7818 

8969 


in 

180 

ira 

178 
17« 


0061 
1817 
8675 
6826 
7071 

8808 


0288 
1993 
8751 
6601 
7845 

8881 


0105 
8169 
8086 
5676 
7419 

8154 


0688 
fBMR 
4101 
6850 
7608 

9388 


0759 
8581 
4277 
6025 
7786 
9601 


17T 


2S0 


890985 
S697 
4458 
6199 

T5M0 
9674 


1112 
8878 
4627 
6874 

8114 
9847 


1288 
8048 
4802 
6548 

8887 


1464 
8284 
4977 
6788 

8461 


1641 
8400 
5162 
6806 

8684 


176 
17« 
175 
174 
173 


0020 
1745 
8464 
6178 
6881 
8679 


0198 
1917 
8686 
5346 
7051 
8749 


0866 
8080 
.880r 
5517 
7881 
8918 


0638 
8861 
88:8 
6688 
7891 
9087 


0711 
8488 
4140 
6868 
7561 
9857 


0888 

8606 
4380 
6080 
7781 
9426 


1066 
8777 
4498 
6199 
7901 
9605 


1888 
8918 
4668 
6870 
8070 
9764 


173 
179 
171 
171 
170 
169 




401401 
8181 
4884 

6640 
8240 
9938 


1578 
3292 
6006 
6710 
8410 


0108 
1788 
8467 

5140 
680r 
8467 


0271 
1956 
8635 

5307 

69ra 

8638 


0440 
8124 
8808 

5474 
7139 
8798 


0009 
8898 
8870 

6641 
7806 
8964 


0777 
8461 
4187 
5808 


0946 
2629 
4806 

6074 
7688 
9806 


1114 
2796 
4479 

6141 
7804 
9460 


1283 
8964 
4638 

6808 
7970 
9625 


1451 
8138 

4906 

6474 
8185 
9791 


109 


S60 


411620 
8800 

4978 
6641 
6801 
qaka 


168 
167 

167 
16G 
165 




0121 
1768 
MIO 
6045 
6674 
8297 
9014 


'0286 
1938 
8574 
6806 
6886 
8459 


0451 
8097 
8737 
6371 
6999 
8621 


0616 
1^1 
8901 
6634 
7161 
8788 


0?81 
8486 
4065 
6697 
7884 
8914 


0945 
8590 
4228 
5860 
7486 
9106 


1110 
27TW 
4392 
6023 
7848 
9268 


1275 
8918 
4566 
6186 
7811 
9429 


1488 
8068 

4718 
6349 
7978 
9691 


165 
164 
104 
163 
168 
168 


8 
9 


421604 
8246 
4882 
6611 
8135 
9762 

48 


0075 


0286 


0898 


1 0560 


0720 


0881 


1048 


1808 


161 



PBopoimoNAL Parts. 



Diff. 


1 


8 


8 


4 


5 


6 


7 


8 


9 


178 


17.8 


85.6 


63.4 


71.2 


89.0 


106.8 


184.6 


142.4 


160.2 


177 


17.7 


85.4 


53.1 


70.8 


88.5 


106.8 


123.9 


141.6 


160.8 


176 


17.6 


85.8 


52.8 


70.4 


88.0 


106.6 


128.2 


140.8 


158.4 


175 


17.5 


85.0 


52.5 


70.0 


87.5 


106.0 


122.5 


140.0 


157.5 


174 


17.4 


84.8 


62.8 


69.6 


87.0 


104.4 


121.8 


189.8 


156.6 


178 


17.8 


ai.o 


61.9 


69.8 


86.5 


106.8 


121.1 


188.4 


155.7 


172 


17.8 


84.4 


61.6 


68.8 


86.0 


103.8 


120.4 


187.6 


154.8 


171 


17.1 


84.8 


51.3 


68.4 


85.5 


102.6 


119.7 


186.8 


158.9 


170 


17.0 


84.0 


51.0 


68.0 


85.0 


102.0 


119.0 


186.0 


168.0 


169 


16.9 


88.8 


60.7 


67.6 


81.5 


101.4 


118.8 


185.8 


1S8.1 


168 


16.8 


88.6 


60.4 


67.2 


84.0 


100.8 


117.6 


184.4 


151.2 


167 


16.7 


88.4 


60.1 


66.8 


83.5 


100.8 


116.9 


183.6 


160.3 


166 


16.6 


38.8 


49.8 


66.4 


83.0 


99.6 


116.8 


188.8 


149.4 


165 


16.5 


88.0 


49.5 


66.0 


82.5 


99.0 


115.5 


188.0 


148.5 


164 


16.4 


82.8 


49.8 


65.6 


82.0 


98.4 


114.8 


181.8 


147.6 


168 


16.3 


88.6 


48.9 


65.2 


81.5 


97.8 


114.1 


180.4 


146.7 


162 


16.8 


884 


48.5 


51.8 


81.0 


97.8 


113.4 


189.6 


145.8 


161 


16.1 


82.2 


48.3 


(M.4 


80.5 


96.6 


112.7 


188.8 


144.9 



LOOAinTHHS 09 KUMBBSa 



I No. 



S70 I^ 431.] 




1846 
8450 
5048 
0640 
6896 
9606 



1881 



4518 
6071 

7088 
9170 



0711 
aS47 
8777 



9645 



1348 

2847 
4840 



7S12 
8790 



0008 
178S 
8196 
4658 
6107 



2007 
8610 
6fi07 
6790 
8884 
9964 



1588 
8106 
4660 
6886 

7778 
0894 



0666 
8400 
8980 
5454 
6073 
8487 
9996 



1499 

8997 
4490 
69T7 
7400 
8988 



0410 
1878 
8341 
4799 



2167 
8770 
5807 
69«' 

8548 



0128 
1696 



7988 
9478 



1018 



4068 
6606 
7185 



0146 
1649 

8146 
4689 
6126 

7808 
9065 



0557 
8085 



8487 
4944 
6397 



8030 
5686 

7116 
8701 



0879 
1858 
8419 
4961 
6587 

8068 

9688 



1178 
8706 
4885 
6738 
7«76 
8780 



0896 
1799 

8896 
4788 
6874 
7756 



0704 
2171 



5090 
6&t8 



2488 

4000 



7875 



0487 
8009 
8976 
5187 

8M8 

9787 



8659 
4887 
5910 
7428 
8940 



0447 
1948 

8445 
4986 
6488 
7904 
9660 



0651 
8818 
877D 
5835 
6687 



[Na 299 L. 47 
I Dif 



8649 
4^9 
5644 
7488 
9017 



0694 
8166 
8788 
6898 
6818 



9941 



1479 
8018 
4M0 
6068 
7570 
9091 



0597 
8096 

8504 
5065 

6671 



0587 



8464 
8085 
6881 



8809 
4409 
6004 
7608 
9176 



0758 



5449 
7003 



0096 
1633 
8165 
4698 
6814 
7781 
9848 



0748 
2^ 

8744 
6884 

6719 
8800 
9675 



1145 
2610 
4071 
5586 
6970 



Pbofobtional Parts. 



8 


4 


5 


6 


7 


8 


48.8 


64.4 


80.6 


96.6 


112.7 


128.8 


48.0 


640 


80.0 


96.0 


112.0 


128.0 


47.7 


68.6 


79.6 


95.4 


111.8 


127.2 


47.4 


68.2 


79.0 


94.8 


110.6 


128.4 


47.1 


62.8 


78.5 


94.2 


109.9 


125.6 


46.8 


02.4 


78.0 


98.6 


109.2 


124.8 


46.5 


68.0 


77.6 


98.0 


108.5 


124.0 


46.2 


61.6 


77.0 


08.4 


107.8 


123.2 


45.9 


61.2 


76.5 


91.8 


107.1 


1.22.4 


45.6 


60.8 


76.0 


91.2 


106.4 


121.6 


46.8 


60.4 


75.6 


90.6 


105.7 


180 8 


46.0 


60.0 


7B.0 


90.0 


106.0 


120.0 


44.7 


50.6 


74.5 


80.4 


104.3 


119.8 


44.4 


50.2 


74.0 


88.8 


103.6 


118.4 


44.1 


58.8 


73.5 


88.2 


102.9 


117.6 


48.8 


68.4 


78.0 


87.6 


102.2 


116.8 


48.5 


58.0 


72.6 


87.0 


101.5 


116.0 


48.8 


57.6 


72.0 


86.4 


100.8 


115.2 


42.9 


67.8 


71.6 


86.8 


100.1 


114.4 


42.6 


56.8 


71.0 


SR2 


994 


118.6 


42.8 


66.4 


70.5 


84.0 


96 t 


112.8 


42.0 


56.0 


70.0 


84.0 


96.0 


112.0 



140 








LOGARITHVS 


OF KUMBEM. 








No, 80OL. 177.1 


[Ka339U63l. 1 


N. 





1 


% 


« 


4 


ft 


• 


7 


S 


f 


DOT. 


800 


477121 


7266 


7411 


7565 


rroo 


7844 


7980 


8188 


8278 8422 


145 


1 




8566 


8711 


8655 


8909 


9148 


9287 


0481 


0675 


0710 9888 


144 

144 


480007 


0151 


0804 


0488 


0609 


0795 


0608 


1012 


1156 1 1900 




1448 1586 


1729 


18718 


2016 


2150 


9808 


d445 


2688 


9781 


148 




9674 


8016 


8150 


8800 


8445 


8587 


8780 


8872 


4015 


4167 


148 




4800 


4442 


4585 


47^ 


4869 


5011 


6158 


6295 


5487 


6679 


148 




5721 


6868 


6005 


6147 


6880 


6480 


6572 


6714 


6856 


6907 


142 




7188 


7280 


7421 


7568 


7704 


7B45 


7986 


8127 


8260 


8410 


141 




8651 


8082 


8888 


8874 


9114 


9055 


9806 


0537 


9077 


9618 


141 




9066 


























0009 
1808 


0080 
1642 


0880 

17W 


0620 
1088 


0661 
2008 


0801 
2201 


0041 
0841 


1061 
2481 


1900 

2621 


140 
140 


810 


401868 




8760 


2000 


8040 


8170 


8810 


8468 


8507 


8787 


8876 


4015 


180 




41S5 


4294 


4488 


4572 


4711 


4850 


4989 


5128 


6207 


5406 


ISO 




5544 


5^8 


5882 


60G0 


6000 


6288 


6876 


6516 


6058 


6701 


189 




6960 


7068 


7906 


7844 


7488 


7621 


7759 


7807 


8085 


817^ 


188 




8811 


8448 


8586 


87^a4 


8868 


8080 


9187 


9070 


9412 


0550 


laj 




9687 


9824 


9062 






















0009 
1470 


0280 
1607 


VM 


0611 


0648 
2017 


0785 
9154 


0002 
8291 


187 
187 




601080 


1106 


1888 




84S7 


2564 


2700 


2887 


2978 


8100 


8246 


8862 


8518 


8666 


186 




8791 


8087 


4068 


4100 


4885 


4471 


4607 


4748 


4878 


5014 


ISO 


820 


6150 


6286 


5421 


6657 


6698 


6828 


5064 


6099 


6284 


6870 


186 




6506 


6640 


6776 


6011 


7046 


7181 


7816 


7451 


7586 


7721 


186 




7866 


7901 


8126 


8SQ0 


8895 


8580 


8664 


8790 


f984 


9068 


185 




9908 


98S7 


9471 


9606 


9740 


9874 














0009 
1849 


0148 
1488 


0877 
1616 


0411 
1760 


in 




610545 


0670 


0818 


0047 


1081 


1215 




1668 


2017 


2151 


2284 


2418 


2551 


2084 


2818 


2851 


8084 




8818 


8851 


8484 


8617 


8750 


8888 


4016 


4140 


4282 


4415 


183 




4548 


4681 


4818 


4946 


6079 


WU 


5344 


5476 


6600 


0741 


183 




5874 


6006 


6180 


6271 


6108 


6586 


6CC8 


ot-oo 


6088 


7064 


182 




7196 


7826 


7460 


7592 


7724 


7855 


7987 


8110 


8261 


8888 


182 


880 


8514 


8646 


8777 


8909 


9010 


om 


0S03 


9434 


9666 


9097 


181 




9886 


9960 




















0090 
1400 


0221 
1580 


1661 


0484 

1702 


0615 

1U28 


0745 

2058 


Of 76 
21P8 


1007 
2314 


181 
131 




621188 


1289 




2444 


»75 


2705 


2886 


2966 


8096 


3836 


8356 


8486 


8616 


180 




8746 


8878 


4006 


4186 


4266 


4396 


4526 


4656 


47W 


4015 


180 




5015 


6174 


5304 


6484 


6563 


5698 




5951 


60R1 


0210 


129 




6880 


0460 


6596 


0787 


6856 


6965 


7114 


?^48 


7872 


7601 


190 




7C80 


7780 


7888 


8016 


8145 


1 8274 


8402 


85.S1 


8660 


8768 


199 


8 


8917 


9045 


0174 


9802 


9480 


1 9560 


9687 


0H15 


9043 






0070 
1851 


128 

128 





580!ii00 


QS26 


10456 


0584 


0710 


1 0640 


0968 


1096 1 1223 


Pbopobtxonai. Parts. 


Dlif. 


1 


». 


s 


4 


a 


4 


T 


8 


9 


189 


18.9 


27.8 


41.7 


55,6 


69.5 


83.4 


97.8 


111.8 
110.4 


125.1 


188 


18.8 


«r.6 


41.4 


55.2 


60.0 


82.8 


96.6 


104.2 


187 


18.7 


27.4 


41.1 


54,8 


68.5 


82.2 


05.9 


109.6 


128.3 


186 


18.6 


97.9 


40.8 


64.4 


68.0 


81.6 


95.2 


108.0 


122.4 


185 


18.5 


27.0 


40.6 


64.0 


67.5 


81.0 


04.5 


108.0 


12? .5 


184 


18.4 


26.8 


40.2 


68.0 


67.0 


80.4 


93.8 


107.0 


120.6 


188 


18.8 


26.6 


809 


58.2 


66.5 


79.8 


88.1 


106.4 


110.7 


188 


18.2 


26.4 


80.6 


52.8 


66.0 


79.2 


92.4 


106.6 


118.8 


181 


18.1 


S6.8 


^9.3 


62.4 


65.5 


78,6 


91.7 


104.8 


117.9 


180 


18.0 


26.0 


89.0 


62.0 


65.0 


78.0 


91.0 


104.0 


m.o 


188 


Ui 


96.8 


88.7 


61.6 


64.5 


77.4 


90.8 


106.8 


116.1 


188 


25.6 


88.4 


61.2 


64.0 


76.8 


89.0 


102.4 


116.9 


m 


W7 


96.4 


88.1 


50.8 


63.5 


76.2 


88.9 


101.6 


114.8 



XfOOAKlTHMS OV KTTHBSB& 




9 


8 


4 


5 


6 


7 


S5.6 


88.4 


51.8 


64.0 


76.8 


89.8 


».4. 


88.1 


60.8 


63.6 


76.8 


88.9 


W.S 


87.8 


60.4 


68.0 


76.6 


88.8 


S.o 


87.5 


60.0 


62.6 


75.0 


87.6 


M.8 


87.8 


49.0 


62.0 


74.4 


86.8 


M.e 


86.9 


«.8 


61.5 


73.8 


86.1 


M.4 


86.8 


48.8 


61.0 


78.2 


86.4 


MS 


86.8 


48.4 


60.5 


72.6 


84 7 


94.0 


86.0 


48.0 


60.0 


72.0 


81.0 


0.8 


86.7 


47.6 


69.5 


71.4 


83.3 



142 



lOaXRITRMS OP KtTKBBBS. 



No. 88a um,] 


[No. 414 L. 617. [ 


N. 





1 


a 


8 


4 


i 





7 


8 





Diff. 


380 
1 


579784 


9806 




















0012 
1158 


0126 

1267 


0041 
1881 


; 0865 

' 1496 


0460 1 0688 


0607 

1836 


0611 
19R0 


114 


fi800S25 


1030 


1606 


1723 


2 


2003 


2177 


2291 


2404 


2518 


2681 


2746 


2K58 


2972 


8085 




8 


8199 


3312 


3426 


8580 


8652 


8766 


8879 


8992 


4105 


4218 




4 


4331 


4444 


4557 


4670 


4783 


4806 


nooo 


6122 


5235 


5^48 


lie 


5 


5461 


5574 


5686 


5799 


5912 


6024 


6187 


6250 


6362 


6475 




6 


6587 


6700 


6K12 


6025 


7037 


7149 


7988 


7874 


7486 


75r.9 




7 


7711 


7823 


7935 


8047 


8160 


8272 


6884 


8496 


8G06 


87ao 


113 


8 


8838 


8044 


9056 


9167 


9279 


9891 


9508 


9015 


9726 


0838 







99SO 




















0061 


0178 


0284 


0806 


0507 


0610 


0780 


084S 


0953 






880 


601065 


1178 


1237 


1399 


1510 


1621 


1782 


1848 


1955 


2066 




1 


2177 


2288 


2G;>9 


2510 


262] 


2782 


2848 


2954 


8064 


8175 


111 


2 


8386 


8307 


3508 


8618 


8729 


8810 


8950 


4061 


4171 


4282 




8 


4308 


4508 


4C14 


4724 


4884 


4945 


50B6 


5165 


5276 


5386 




4 


5196 


5606 


5717 


5827 


6087 


6047 


6157 


6867 


6877 


6487 




R 


0597 


6T07 


0817 


6027 


7087 


'|-?46 


7256 


79Ba 


7476 


7586 


110 





7596 


7805 


7914 


8024 


8184 


S248 


8858 


6462 


8572 


8081 




7 


mi 


8900 


0(X)9 


9110 


0228 


9387 


0446 


9556 


9065 


9774 




8 


9888 


9092 




















0101 


0210 


C819 


0428 


0587 


0646 


0755 


0864 


109 




GOOOn 


106i 


U91 


1290 


1406 


1617 


1Q» 


1734 


1848 


1951 




400 


2060 


2109 


2277 


2386 


2404 


2008 


2711 


2819 


2828 


8096 






S144 


8253 


3361 


ftl09 


3577 


3C86 


8794 


8902 


4010 


4118 


108 




4296 


4384 


4442 


4550 


4656 


4706 


4874 


4962 


6069 


5197 






5305 


5113 


5521 


5G28 


5736 


6844 


5951 


6059 


6166 


6274 






6381 


6489 


6596 


6704 


6811 


6019 


7026 


7138 


7»41 


7848 






7465 


7568 


7869 


rrrr 


7884 


7991 


8098 


8206 


8312 


8419 


107 




8526 


8638 


8740 


8ft47 


8064 


9061 


9167 


9274 


0381 


0488 




9504 


9n)l 


9808 


9914 


















0021 
1086 


0128 
1192 


^ 


0641 
1405 


0417 
1511 


0S54 

1617 






G10660 


07G7 


573 


0979 






1788 


1829 


1936 


204S 


2148 


2254 


2800 


M66 


8572 


8078 


106 


410 


2784 


2890 


2996 


8102 


8S07 


8818 


8419 


3525 


8680 


87S6 






8842 


8047 


4053 


4150 


4264 


4870 


4475 


4581 


4686 


4792 






4807 


5003 


5106 


5218 


5819 


5424 


5529 


5634 


5740 


6845 






5050 


6055 


6160 


6265 


6370 


6476 


6581 


6686 


6790 


6885 


105 




7000 


7106 


7210 


7315 


7420 


7525 


7629 


7734 


7839 


7948 




FlK>POBTZOKAl« PaB!I& 


DU 


1 


3 


8 


4 


6 


6 




7 


8 


. 1 


118 


11.8 


23.6 


85.4 


47.2 


50.0 


70.8 


82.6 


04.4 


106.2 


117 


11.7 


23.4 


85.1 


46.8 


58.5 


70.2 


81.0 


08.6 


105.8 


116 


11.6 


23.2 


84.8 


46.4 


68.0 


60.6 


81.2 


08.6 


104.4 


115 


11.5 


28.0 


84.6 


46.0 


67.6 


68.0 


80.5 


08.0 


108.6 


lU 


Vi 


22.8 


84.2 


45.6 


W.O 


68.4 


79.8 


01.8 


102.6 


118 


11.8 


22.6 


88.0 


45.2 


56.6 


67.8 


79.1 


80.4 


101.7 


llii 


ii.a 


22.4 


88.6 


44.8 


66.0 


67.2 


78.4 


80.6 


100.6 


111 


U.l 


82.9 


88.8 


44.4 


66.6 


66.6 


77.7 


88.8 


98.0 


110 


11.0 


22.0 


83.0 


44.0 


55.0 


66.0 


77.0 


88.0 


80.0 


100 


10.9 


21.8 


82.7 


48.6 


54.6 


65.4 


70.8 


87.2 


86.1 


108 


10.8 


21.6 


82.4 


48.2 


54.0 


64.8 


76.6 


86.4 


07.2 


107 


10.7 


21.4 


82.1 


42.8 


58.5 


64.2 


74.0 


85.6 


06.8 


106 


10.6 


21.2 


81.8 


42.4 


68.0 


68.6 


74.8 


84.8 


86.4 


US 


10.6 


21.0 


.81.6 


42.0 


68.5 


68.0 


78.5 


84.0 


04.6 


104 1 10.4 1 


20.8 


81.2 


41.6 


52.0 


62.4 


72.8 1 88.2 1 


88.6 



r^OaAKITHMS OF KUHBEBS. 







PnrtpoRTIONAL PaSTS. 




8 


8 


4 


5 





7 


81 .0 


ai.6 


42.0 


62.6 


63.0 


73. 


1 S0.8 


81.3 


41.8 


53.0 


63.4 


721 


\ 20.6 


80.9 


41.3 


61.5 


61.8 


72 


1 SO. 4 


80.6 


40.8 


61.0 


61.3 


71 ' 


1 20.3 


80.3 


40.4 


60.5 


60.8 


70' 


1 8O.0 


800 


40.0 


60.0 


60.0 


70 1 


10.8 


20.7 


80.6 


49.5 


69.4 


69.i 



LOGARITHMS OP NUMBIBBS 



0OL.6ae.] 



a¥a4gOL.60S. 



37B6 
4786 
6675 

oeid 

7546 
8479 
9410 



1965 

3190 
8118 
40S1 
4958 
6870 
6785 
7696 
8609 
9610 



0126 

1883 
S285 

8137 
4087 
4985 
5831 
6736 
7618 
8609 
9606 



0285 
1170 
S053 



2935 
8815 
4698 
6569 
6444 
7817 
8188 



2947 
8869 
4880 
5769 
6705 
7640 
8579 
0608 



0481 
1858 



4126 
6045 
5962 
6876 
7789 
870O 
9610 



0617 
1422 



8227 
4127 
6025 
6021 
6815 
770r 
8698 
9486 



0878 
1268 
214'^ 
802:) 
8908 
4781 
5667 
6581 
7404 
8275 



8041 



4924 



6799 
7788 
8666 
9596 



0624 
1461 

2875 
8297 
4218 
5187 
6068 
6968 
7881 
8791 
9700 



0607 

1518 
2410 
3817 
4217 
5114 
6010 
6904 
7796 
8687 
9575 



0462 

1847 
2^80 
3111 
8991 
4866 
5744 
6618 
7491 
8862 



8185 
4078 
5018 
5966 
6892 
7826 
6760 
9689 



0617 
1548 

2467 
8890 
4810 
6228 
6146 
7069 
7973 
8882 
9791 



0098 

1608 
2606 
8407 
4807 
6204 
6100 
6994 
7886 
8776 
9664 



0660 
1485 
2818 
8199 
4078 
4966 
6833 
6706 
7678 
8449 



4172 
5112 
6060 
6986 
7920 



9782 



0710 
1686 

2660 
8482 
4402 
5820 
6286 
7161 
8068 
8978 



0789 

1698 
2696 
8497 
4896 
6294 
6189 
7068 
7975 
8865 
9758 



1624 
2406 
3287 
4166 
6044 
6019 
6793 
7666 
8685 



8824 
4266 
6206 
6148 
7079 
8018 
8945 
9875 



0802 
1738 



8674 
4494 
&112 
6828 

8154 
9064 
9978 



0879 

1784 
2680 
8687 
4486 
5888 



6279 
7172 
8064 
8958 
9841 



0728 
1612 
2494 
8875 
4254 
6181 
6007 
6880 
7753 



8418 
4800 
5299 
6237 
7178 
8106 
9068 
9967 



0695 
Itfil 

2744 
8666 
4686 
5603 
6419 
7388 
8315 
9155 



0068 
0970 

1874 
2777 
8677 
4576 
6473 
6866 
7261 
8158 
9012 
9980 



0816 
17C0 
2688 
8468 
4842 
6219 
6094 
6968 
7889 
8709 



3612 
4464 

6898 
6881 
7266 
8199 
9181 



0060 
0988 
1918 

2886 
8768 
4677 
5506 
6611 
7424 



9246 



0154 
1060 

1964 
2667 
8767 
4666 
6608 
0466 
7861 
8242 
9181 



0019 

0905 
1789 
2671 
8661 
4480 
6807 
0182 
7065 
7026 
8790 



8607 
4M8 
5487 
6424 
7860 
8298 
0224 



0158 
1080 
2005 

2929 
8860 
4769 
5687 
6602 
7516 
8427 
9337 



0245 
1161 

2065 
2967 
8867 
4756 
5662 
6547 
7440 
8881 
0220 



0107 

0098 
1877 
2799 
8688 
4617 
6894 
6209 
7142 
6014 
8883 



DUE. 



n 



90 



89 



87 



rR0P0RT10NAX« PaRTS. 



1 


2 


8 


4 


5 


« 


7 


8 


9 


9.8 


19.6 


29.4 


89.2 


40.0 


08.8 


68.6 


78.4 


88.9 


9.7 


19.4 


29.1 


88.8 


48.5 


68.2 


67.9 


77.6 


87.8 


9.6 


19.2 


28.8 


88.4 


48.0 


67.6 


67.2 


76.8 


864 


9.5 


19.0 


28.5 


88.0 


47.5 


67.0 


66.5 


76.0 


85.5 


9.4 


18.8 


28.2 


87.6 


47.0 


66.4 


65.8 


76.3 


64.6 


9.3 


18.6 


27.9 


87.2 


46.5 


66.8 


65.1 


74.4 


88.7 


9.3 


18.4 


27.6 


36.8 


46.0 


65.2 


61.4 


78.6 


82.8 


9.1 


18.2 


27.8 


30.4 


45.5 


64.6 


63.7 


73.6 


61.9 


9.0 


18.0 


37.0 


36.0 


45.0 


64.0 


63.0 


73.0 


81.0 


8.9 


17.8 


26.7 


85.6 


41.5 


58.4 


63.8 


71.2 


89.1 


8.8 


17.6 


26.4 


85.2 


44.0 


02.8 


61.6 


9D.4 


%i 


8.7 


17.4 


96.1 


84.-8 


48.B 


7S8.2 


60.9 


60.« 


8.6 


17.2 


26.8 


84.4 


48.0 


51.0 


60.2 


68.8 


77.4 



liOOARITHMS OF ]IUMBBB8» 



1 





^fiooi^ 6^ai 




</ • 


/ ■ 


f 


8 


4 


6 


• 


7 


i 


f ^ oesa j 9024. 


9144 


9S81 


9817 1 


9404 


9491 


9^ 


96 
















OOll 
0677 


0098 


0184 
1060 


0871 


0668 
1838 


0444 
1809 


06 
18 


$ ; 7O07O4 • irsito 


0968 


1186 


n / IGOB j ICHVi 
^ 2431 f 2517 


1741 


1887 


1018 


1999 


8086 


8178 


SS 


S608 


2689 


2775 


2861 


3947 


8038 


81 


^ 


aS91 3377 


3463 


8540 


8636 


srai 


3807 


3898 


8S 


ft 


2[di ^isao 


4822 


4408 


4494 


4579 


4665 


4751 


4C 




0008 0004 


6179 


tti65 


5850 ' 


6430 


5532 


560r 


6( 




0BeM 0CM9 


8035 


6120 


62G6 


6891 


6876 


(M68 


66 




^B 08O3 


88oo 


0974 


7069 


7144 


7229 


7816 


74 


^^0 




7O0B 


7740 


78B6 


7911 


7996 


8061 


8166 


88 




0401 


flQOO 


8C91 


8876 


8761 


8846 


8081 


9016 


91 




9070 




9440 


9SS4 


9609 


9694 


9779 


9868 


90 




710117 
0008 


QSQS 


CS87 


0871 


0156 


0540 


0635 


0710 


07 


1048 


1132 


1217 


1301 


1386 


1470 


1554 


]( 


1807 
8650 


1B02 


1976 


2060 


2144 


2329 


8813 


3897 


S^ 


2^734 


2818 


2902 


29H6 


8070 


8154 


8338 


88 


3101 
4S30 

B167 


«57S 


3650 


3742 


8826 


8910 


8994 


4078 


41 


^4414 


4497 


4581 


4665 


4749 


4883 


4916 


6( 


SS&l 


0885 


M18 


6602 


6586 


5669 


6763 


•5fc 


ssfiao 


6008 


eoerr 


6170 


8864 


6887 


6421 


6504 


6686 


M 


OOSl 


'•004 


7088 


7171 


7854 


7888 


7421 


71 


? ' 7071 


77&4 


7887 


7900 


8008 


8086 


8169 


6888 


« 


asBo 


8668 


8751 


8884 


8917 


9000 


9068 


91 


8 

4 


osai 


04X4 


0497 


9680 


9668 


9746 


9838 


9911 


9ti 


5 


O068 

tftft 


CfiS'lS^ 


0325 


Oi07 


0490 


0578 


0656 


0788 


oe 


loeB IIM 


1288 


1816 


1398 


1481 


1568 


16 


C 


4^S lOTs 


9(}R8 


2140 


2322 


8305 


2887 


*M 


* 


2^ \ 27ie 2798 


2881 


2968 


8046 


8137 


8209 


8S 


8 
9 


ISI 1 SM8 ^ 


8020 


8702 


8784 


8866 


8948 


4030 


41 




^ ^ 1 4asB 


4440 


4588 


4604 


4686 


4767 


4849 


40 


sao 


3358 


6840 


6488 


5608 


6686 


6667 


57 




6073 
6800 


6166 
8978 


6B8U 
7068 


6880 
7184 


0401 
7316 


6488 
7297 


66 
78 


77<M 


7786 


7866 


7948 


8089 


8110 


81 


t \ R-A I ^S »16 


8597 


8678 


8759 


8841 


8923 


90 


SI ^r4\«S 


g8l?7 


M08 


9489 


9570 


9651 


9788 


98 


•3 
1 


' '^* COW 


0186 


0817 
lOSM 
1880 


0208 
1106 
1911 


0878 
1186 
1991 


0459 
1366 
30?i 


0540 
1347 
3158 


06 


» 1 168© \ 186» 


0944 
1750 


14 

23 


siol aa»4 \ »«;4 


2B65 


8685 


2715 


8796 


2878 


2956 


80 


8868 


8488 


8518 


8598 


3679 


8759 


88 


4160 


4240 


4330 


4400 


4480 


4560 


46 


4960 


6010 


5130 


5300 


5379 


5359 


54 


\ *.\ ^ 


oo \ e«7fl 


5759 


5888 


5918 


5096 


6078 


6157 


62 


\ 


C 


Pbopoktional Pabts. 




V\ M • 


3 


4 


5 


6 


7 


\ « \ UtI 17.4 


36.1 


84.8 


48.5 


53.2 




60.9 


\ » \ ¥.6 17.2 
\ ^ 1 1 5 17.0 


25.8 


84.4 


43.0 


51. C 




60.2 


26.5 


84.0 


42.5 


51. C 




59.5 




» 


1 8.4 1 16.8 


85.3 


88.6 


42.0 


50.4 


1 58.8 



146 



LOGABITHMS OF S^UMBEBS. 



Na M6 L. 786.1 



N. I 



545 i 786907 I 

6 7103 , 

7 7987 I 

8 8781 I 

9 9572 



660 
1 
2 
8 
4 
6 
6 
7 
8 
9 

660 

1 
2 

3 
4 

6 
6 
7 
8 
9 

5ro 

1 

2 
8 
4 
5 

6 

7 
8 
9 

580 

1 
2 
3 
4 



6476 
7272 
8067 
8860 
9651 



1152 
1939 
27S85 
3510 
4293 
6075 
5855 
6634 
7412 

8188 
8903 
9736 



750506 
1279 
2018 
2816 
3688 
4348 
5112 

8875 
6686 
7306 
8155 
8912 
9668 



760422 
1170 
1928 
2679 

8438 
417G 
4923 
5660 
6413 



0442 
1230 
2018 
2801 
3588 

4:m 

5158 
5933 

ona 

7489 



9614 



0686 
1356 
2125 



5189 

6951 
6712 
74« 



9743 



6556 
7352 
8146 
8039 
9781 



0621 
1800 
2096 



8667 
4449 
5281 
6011 
6790 
7567 

8843 
9118 
9691 



0496 
1251 
2003 
2751 

3508 
4251 
4998 
5743 
6487 



0063 
1433 



2970 
8736 
4501 
6266 

6027 
6788 
7548 
8300 
9063 
9819 



0573 
1326 
2078 
2829 

8678 
4326 

5072 
5818 
6562 



6685 
7481 



9018 
9810 



0600 
1888 
2175 
2961 
3745 
4528 
5309 
6060 
6868 
7645 

8421 
9195 



6715 'I 6796 

— 7590 

8384 

917r 

9968 



r511 
8305 
9097 
9889 



0740 
1510 
2279 
8017 
8813 
4678 
5841 

6103 
6864 
7624 
8382 
9139 
98&1 



0649 
1402 
2153 
2901 

8653 
4400 
5147 
5892 



0678 
1467 
2254 
3080 
3823 
4606 
5387 
6167 
6045 
7722 

8496 
9272 



0045 
0617 
1587 
2356 
3128 
S880 
4654 
5417 

6180 
6910 
7700 
8158 
9214 
99TO 



ora4 

1477 
2228 
2978 

8727 
4475 
5221 
5966 
6710 



0757 
1546 



3118 
3902 



5465 



7028 
7800 



9350 



LNo. 6841x767. 



6874 
7670 
8463 
9256 



0047 

0636 
1624 
2411 
8196 
8960 
4762 
6543 
6323 
7101 
7878 

8663 
9427 



8 



Diff. 



T749 



0126 

0016 
1703 
2489 
3275 
4058 
4810 
6621 
6401 
7170 
7965 

8781 
9504 



0000 
0971 
1741 
2509 
8277 
4042 
4807 
6570 

6332 

709S 

7851 

8533 i 8609 

i 9366 



0O15 I 0121 
0790 I 0875 



0128 
0894 
1664 



3966 
4730 
5494 

6256 
7016 
7775 



1562 
2308 
3053 

3808 



1627 
2878 
3128 

8877 



4560 
5296 I 5370 
6041 6116 
6785 6860 



0277 
1048 
1818 
2686 



4119 



5646 

6408 
7168 
7027 



9441 



0196 
0060 
1708 
2453 



8062 
4000 
5445 
6190 



70ai i 7113 

7829 I 7908 

8622 8701 

0414 9493 



0806 

0094 
1782 
2568 



4136 
4919 
5699 
6479 
TfSbG 



8806 
0582 



0854 
1125 
1805 
2663 
3480 
4106 
4060 
6722 

6484 
7S44 
8003 
8761 
9517 



0272 
1085 
17?« 
2580 
8278 

4087 
4774 
5620 



7007 



0864 

Km 

1860 
2647 
3431 
4215 
4007 
6777 
6566 
7884 
8110 



9650 



0431 
1202 

lo;^ 

2740 
3506 
4872 
5086 

5798 

6660 
7880 
8079 
8886 

0598 



0847 
1101 
1858 
8604 
8368 



4101 
4848 
6604 
6388 
7068 



TO 



77 



76 



73 



Proportional Vartb, 



Diff. 


1 


2 


8 


4 


5 


6 


7 


8 





83 


8.3 


10. 6 


21.9 


83.2 


41.5 


49.8 


68.1 


66.4 


74.7 


82 


8.2 


16. 4 


34.6 


32.8 


41.0 


49.2 


57.4 


65.6 


ra.8 


81 


8.1 


1G.2 


24.3 


32.4 


40.5 


48.6 


56.7 


64.8 


72.9 


80 


8.0 


IG.O 


24.0 


32.0 


400 


48.0 


56.0 


64.0 


78.0 


79 


7 


15.8 


23.7 


31.6 


39 5 


47.4 


55.3 


68.2 


71.1 


78 


7 8 


15.6 


28.4 


31.2 


39.0 


46.8 


54.6 


02.4 


70.2 


77 


7 7 


15.4 


2;m 


30.8 


38.5 


46.2 


58.9 


61.6 


60.3 


76 


7 6 


15.2 


2iJ.8 


30.4 


38.0 


45.0 


63.2 


60.8 


68.4 


75 


7.5 


15.0 


22.5 


30.0 


37.5 


46.0 


52.5 


60.0 


67.5 


74 


7.4 


14.8 


22.2 


29.6 


37.0 


44.4 


61.8 


50.8 


06.6 



rOGABITHHS OF KItHBEnS. 



L 


'*b, 58S r-. 


7B7.1 
















^/ • 


J ' 


8 


8 


4 


5 





7 




r^ 1 TBTlsa 1 TSaO I 7304 7879 


7458 


! 7827 


lioF 


7675 


7 


/ S ^»8 7»72 ' 8046 8120 


8194 


8268 


8342 


8410 


£ 


/ ^ 8SS8 


8ns • 8786 : 8860 


8934 


1 9008 


9082 


9156 


G 


L 


^ ♦ 03TT 


1 9451 1 9625 ; 9599 


9678 


9746 


9820 


9894 


fi 


^ ; 770115 


i 0189 : 0963 


0336 


0410 


0484 


0657 


0631 





/^; OS5S 


1 O006 O990 


1078 


1146 


1220 


1298 


1867 


1 


( X MtiftT 
% / Si3SSS 


< 1661 1734 


1806 


1881 


1966 


2028 


2102 


S 


1 :£!«» ^$408 


8542 


2615 


2688 


276^; 


2835 


2 


t ^ . S05& 


31SJB aaoi 


ai»74 


3348 


8121 


8194 


3567 


3 


r ^ / 378a 


3860 


3933 


4006 


4079 


4152 


4225 


4298 


4 


^ 4^17 


4500 


4668 


4736 


4809 


4882 


4956 


6028 


5 


^ &»46 


5319 


5392 


5466 


5638 


5610 


6683 


5756 


5 


"5- S974 


604T 


6120 


6193 


6866 


6888 


6411 


6488 


6 


^ 6701 


67T4 


6S46 


6919 


6992 


TOW 


7137 


7209 


rj 


9 74:27 


7499 


7572 


7644 


7n7 


7789 


7862 


7984 


8 


^^O^ I 81S1 


8204 


»396 


8368 


8441 


8513 


8585 


Bliss 


8 


1 i 8874 


8947 


9019 


9091 


9163 


9236 


9308 Vfm 


9 


9669 


9741 


9613 


9686 


9957 






nfk9& f.iiM 


a 














\MCv 


wji'i 


3 "SSeir 

A , ICKTT 

g 1 l-TFiS 


08B9 


0461 


0633 


0605 


0677 


0749 


OHai 





1109 


1181 


1253 


1^4 


1396 


1468 


IMO 


1 


iwrr 


1809 


1971 


2042 


2114 


2186 


mB 


2. 


6 

7 


53473 


2544 


2616 


2088 


2759 


2831 


2905i 


2j?rj 


9 


31^9 


aSBGO 


3338 


»I08 


8175 


3546 


8618 


S6K9 


» 


8 
11 


3»04 

46 rr 


39^75 


4046 


4118 


4189 


4261 


4832 ^m 


4 


4689 


4760 


4831 


4902 


4974 


5045 5J16 


6 


€Srio ' 5S30 
2 i 0751 

8 74oa 

4 1 81C8 

5 887S 


S401 
6112 


5472 
6183 


5M3 
6254 


5615 
6325 


5680 
6390 


5757 
6467 


6rt2S 


5 


68BSd 


6893 


69W 


7035 


7100 


7m 


'i-Ml^ 


7 


7531 


7602 


vd'i-a 


7744 


7815 


7885 


TUTiii 


B 


8:^239 


8310 


8381 


8151 


8522 


8593 


mm 


V( 


8946 
9651 


9016 
9722 


9087 
9792 


9157 
9863 


9238 
9933 


9299 D3IJJ* 1 


fl" 


G 


©oei 


0004 i^rTA : 


~0 
01 


7«1S85 


0356 
1059 


0426 
1129 


M96 
1199 


0567 
1269 


0637 
1810 


0707 
1410 


orrs 

14W 


S 1 1G91 


1761 


1831 


1901 


1971 


2011 


2111 


2m 


^ 


Gao 230^ 
^ 1 I 3093 

> a 4488 

1 6 ; 5880 


«S.t«>3 


S532 


2602 


2672 


2742 


2812 


3fiH2 ' 


^ 


3 n v^ 


3281 


3301 


3871 


8141 


3511 3.Vii j 


3( 


a^-o 


3930 


4000 


4070 


4189 


4209 4^*7\> 


4,^ 


, 4 ,',8 4627 


4697 


4767 


4836 


4906 4'.m ' 


U 


B^^4 saw 


6393 


5463 


5532 


5602 T^\"2 


:^i 


5,* id 6019 


6088 


6158 


6227 


6297 fl*J6 (^J 


I^VVt 6718 


6782 


6853 


6921 


6990 1 71)00 1 r 


?^^rr 7406 


7475 


7545 


7614 


7683 rrs3 i 7$ 


t t^rJQ 80O8 


8167 


8236 


8805 


8374 »m K 




5 wSi 


1 tTr-JO 8789 


8868 


8027 


8996 


9066 6134 1 m 



Proportional Parts. 




d 


8 


4 


5 


6 


7 


[ i&.o 


22.5 


30.0 


87.5 


45.0 


52.5 


14-8 


22.2 


29.6 


37.0 


41.4 


51.8 


\ 14.6 


21.9 


29.2 


86.6 


43.8 


51.1 


1 14.4 


21.6 


28.8 


86.0 


43.2 


50.4 


14.2 


21.8 


28.4 


85.5 


42.6 


49.7 


14.0 


21.0 


28.0 


35.0 


42.0 


49.0 

4K.^ 


18.8 


20.7 


27.6 


84.5 


41.4 



118 



XfOOARITHMS OF KUHBBR8. 



Ka 680 L. TNi] 



[N<fc674L.an. 



N. 





1 


8 


• 


4 


6 


6 


7 


8 


9 


Dlff. 


680 


790841 


9409 


9478 


9647 


9616 


9686 


9754 


9883 


9608 


9961 




1 

8 
4 

6 
6 
7 
8 
9 

640 
1 
8 
8 
4 
5 


800QB9 

0717 
14M 
8060 

8774 
8457 
4189 
4881 
6601 

806180 
6858 
7686 

iSi 

9660 


1478 
8168 
8818 
8088 

4808 
4889 
6669 

6848 
6086 
7608 
8879 
8068 
9687 


0167 
0664 

1541 
8806 
2910 
8604 
4876 
4967 
6687 

6816 
6904 
7670 
8846 
9081 
9694 


0886 
0088 
1609 
8885 
2979 
8668 
4844 
6085 
6706 

6884 
7061 
7788 
8414 
9088 
9708 


0606 
0908 
1678 
8868 
8M7 
8780 
4418 
5008 
6T78 

6461 
7189 
7806 
8181 
9156 
9629 


0878 
1061 
1747 
8488 
8116 
8798 
4480 
5161 
6841 

6619 
7197 
7878 
8549 
0888 
9896 


0448 
1189 
1815 
8600 
8184 
8867 
4548 
6880 
6906 

6687 
7864 
7941 
8616 
0890 
9964 


0511 
1196 
1884 
8658 
32r)2 
8865 
4616 
6897 
6978 

6656 

7838 

8006 
8684 
9868 


0680 
1866 
1968 
8687 
8881 
4006 
4686 
5865 
6044 

67S8 
7400 
8076 
8751 
9485 


064Q 
1836 
8021 
8706 
3389 
4071 
4758 
5488 
6118 

0790 
7487 
8148 
8818 
9488 


06 


0081 

oroo 

1874 
8044 
8718 

8881 
4018 
4714 
5378 
6048 
6705 
7807 
8088 
86H8 
9^16 


0006 

vno 

1441 
8111 
8780 

3448 
4114 
4780 
5446 
6109 
6771 
7488 
8094 
8784 
9418 


0166 
0687 
1606 
8178 
8617 

8514 
4181 
4847 
65U 
017S 
6888 
7499 
8160 
8880 
9478 




6 
7 

§ 

650 
1 
8 
8 

t 

6 
7 

8 
9 


810B88 
WH 
1675 
SMS 

8916 

i 

61138 

8886 
9644 


0800 
0971 
1648 
8818 

8080 
8648 
48M 
4880 
6644 
6806 
6070 

S£ 

8961 
9610 


0867 
1089 
1709 
8879 

8047 
8n4 
4881 
6046 
6711 

9017 

9m 


0484 
1106 
17?8 
8445 

8114 
8781 
4447 
6118 
67r7 
6440 
7108 
7764 
8184 
9068 

9741 


0601 
1178 
1848 
8618 

8181 
8848 
46H 

5179 
6818 

6606 
7169 
7880 
84SO 
9149 

9607 


0669 
1840 
1910 
8679 

8847 
8914 
4581 
6846 
5910 
6578 
7885 
7886 
8666 
9815 

9878 


0686 
1807 
1977 
8646 

8814 
8081 
4647 
5818 
6078 
6630 
7801 
7968 
8688 
9881 

9089 


07 
66 


660 


0004 
0661 
1817 

19:8 

8826 
8279 
8930 
4681 
6831 
6680 

6688 
7175 
78U1 
8407 
91U 


O07O 
07S7 
18W 
8087 
8691 
8344 
8986 
4646 
6896 
0945 

6698 
7840 
7886 
8681 
9176 


0186 
0798 
1448 
8106 
8786 
8400 
4061 
4711 
5861 
8010 

i 




1 
8 
8 
4 
5 
6 
7 
8 


670 

1 
2 
8 

4 


880801 
0868 
1614 
8168 
8888 
8474 
4186 
4776 
6486 

6090 

o»a 

790d 
8016 
8660 


0967 
0684 
1579 

^ 

8689 
4191 
4841 
6491 

6140 

6787 
7484 
8080 


8968 
8606 
4866 
4906 

6804 
6888 
7409 
8144 
8389 


0899 
1065 
1710 
8964 
3018 
8670 
4881 
4971 
6681 

6869 
6917 

SS 

8868 


0164 
1180 
1775 
8480 
8083 
8785 
4886 
6086 

6884 

6961 

8918 


0630 
1186 
1841 
8496 
3148 
8800 
4461 
6101 
6751 

6809 

7046 
7808 
8888 

aw8 


0606 
1861 
1906 
8660 
8813 
8865 
4616 
6166 
6815 

6464 
7111 
7757 
8408 
9046 


65 



Pbopobtxohal Pabtb. 



Dlff. 


1 


8 


S 


4 


5 


8 


7 


8 


8 


68 




18.6 


80.4 


87.8 


34.0 


40.8 


47.6 


544 


61.8 


67 




18.4 


80.1 


86.8 


88.5 


40.8 


46.0 


686 


60.8 


66 




13.8 


19.8 


86.4 


83.0 


89.6 


40.8 


688 


09.4 


65 




18.0 


19.5 


86.0 


88.5 


80.0 


45.6 


680 


06.5 


64 




U.6 


19.8 


85.8 


88.0 


38.4 


44.8 


619 


97.6 



rOGAKITHHS OF Km 




150 



tOGARlTHMd Cr KlTMBBRS^ 



No. 730 L. 857.] 

N. 



[No. 7«4 L. J 



1 

2 

8 
4 

5 
6 
7 
8 
9 

730 
1 
2 
8 
4 
5 
6 
7 
« 


-;40 

1 

2 
8 
4 

5 
6 

7 
8 
9 

730 
1 
2 
8 
4 
5 
6 



760 
1 
2 
3 
4 



7985 
8587 
9188 
9789 



0987 
1581 
2181 
2728 

8828 
8917 
4511 
5101 



6878 
7467 
8066 
8614 

9282 

9818 



870401 
0989 
1578 
2156 
2789 
8321 
8902 
44&3 

6001 
5610 
G218 
6795 
^71 
7917 
8522 
9096 



880212 

0614 
1385 
1955 
2525 
8008 



7898 
7905 
8607 
9198 
9799 



0996 
1594 
2191 
2787 



8977 
4570 
6168 
6755 
6346 
6987 
7526 
8115 
8708 

9290 
9877 



0462 
1017 
1031 
2215 
2797 
8379 
3960 
4510 

5119 
5098 
6276 
6853 
7129 

8oai 

8579 
9153 
9796 



0299 

0871 
1412 
2012 
2581 
8150 



7458 
8066 
8657 
9258 
9860 



0458 
1066 
1654 
2251 
8847 

8442 
4086 
4680 
5222 

5814 
6106 



7585 
8174 
8762 

0840 
9935 



0521 
1100 
1690 
2273 
2855 
8137 
4018 
4508 

sirr 

5736 
6338 
6910 

7187 
8062 
8637 
9211 
9784 



0356 

0928 
1499 
2069 
26:« 
82C7 



7518 
8116 
8718 
9318 
9918 



0518 
1110 
1714 
2310 
2006 

8601 
4096 
4680 
5282 
5874 
6465 
7055 
7644 
8238 
8821 

9406 
9991 



0579 
1164 
1748 
2331 
2913 
819e 
4076 
4656 

5285 
5818 
6391 
6968 
7514 
8119 
8694 
9288 
0841 



0113 

0966 
1556 
2126 
2695 
8264 



7574 
ei76 
8778 
9879 
9978 



0578 
1176 
1778 
2870 



8661 
4166 
4748 
6341 



7114 



8870 
0466 



0063 
0688 
1233 
1806 
<S89 

29ra 

8658 
4134 
4n4 

5298 
5871 
&I49 
7026 
7602 
8177 



0171 

1012 
1013 
2183 
2752 



7684 



9480 



0068 
0637 
1289 
1888 
2130 
8025 

8620 
4214 
4808 
5400 



7178 
7762 
8360 



0111 
0696 
1281 
1865 
2448 
8080 
8611 
4192 
I 4772 

5851 
5929 
6507 
7083 
7659 
8234 
8809 
9383 
9966 



1009 
1670 



2809 
8377 



7604 
8997 



0499 



0096 
0607 
1296 
1898 
»189 
8066 

8680 
4274 
4867 
5450 
6061 
G648 
7282 
7821 
8409 
8007 

0684 



0170 
0755 
1880 
1928 
2506 
8088 
8660 
4250 
4830 

5400 

5987 
6561 
7111 
7n7 



9440 



0013 
0585 

1150 
1727 

2m 

2866 
8134 



7786 

8857 



0550 



0158 
0767 
1866 
1062 
8549 
8114 

8780 



4986 
6519 
6110 
0701 
7201 
7880 
8168 
0066 

0642 



0226 
0818 
1306 
1961 
2604 
8146 
87»7 
4308 
4888 

5466 
6015 
6622 
7199 

rr74 

8319 
8924 
0497 



0070 
0642 

1218 
1784 
S354 
2923 
8401 



7815 
&117 
0018 
0619 



0218 
0817 
1416 
8018 
8606 
8204 

8799 
4882 
4086 
6578 
6160 
0760 

nso 

7080 
8627 
0114 

0701 



0287 
0678 
1456 
8040 
8628 
8201 
8786 
4366 
4945 

5501 
6102 
6680 
7256 



8407 
8981 
9655 



0127 
0609 

1271 
1841 
2411 
2980 
8648 



7875 
8477 
0078 
0070 



0278 
0677 
1476 
8078 



4468 
5(MS 



esir 

7400 
7996 



01 iB 
0760 



0646 
0080 
1515 
8008 



8844 
4434 
6006 



6100 
6737 
7814 
7860 
8464 
0080 
0612 



0185 
0756 

1828 
1806 
8468 
8087 
S606 



Diff. 



00 



W 



pROPORTioNAi. Parts. 



DIff. 


1 


2 


8 


4 


5 


6 


7 


8 


59 


6.9 


11.8 


17.7 


23.6 


29.5 


36.4 


41.8 


47.2 


58 


5.8 


11.6 


17.4 


23.2 


29.0 


S4.8 


40.6 


46.4 


57 


5.7 


11.4 


17.1 


22.8 


28.5 


31.2 


89.9 


45.6 


56 


5.0 


11.2 


16.8 


22.4 


28.0 


88.6 


39.8 


44.8 



68.1 
68.2 
51.8 
60.4 



XiOOARITHMS OP K UMBERS. 



151 



Nq.7Q5L.8B8.] 



[No. 800 L. 006. 



785 
6 

7 
8 

770 
1 
2 
S 
4 
5 
6 

7 
8 
9 

1 
2 
8 
4 
5 
6 
7 
8 


730 

1 
2 
3 

4 

6 
G 
7 
8 
9 

800 
1 
2 
8 
4 
£ 
6 
7 
8 
9 



888681 



4*^ 

can 



6491 
7064 
7017 
8179 
8741 
9002 
0602 



8B0«n 
0080 
1587 

2006 
2651 

an? 

8702 
4810 
4870 
5488 
5875 
6586 
7077 

7627 
8170 
8725 
9273 
96S1 



900867 

0018 
1468 
2008 

2647 

8000 
8688 

4174 
4716 
5256 
R96 
6885 
6874 
7411 
7949 



8718 

4852 
5418 
5068 

6647 
7111 
7674 
8286 
8797 
0858 
9018 



0177 
1086 
1508 

2150 
2707 



S817 
42^71 
4905 
5478 
60d0 
6581 
7182 

7682 
8291 
8780 
0828 
0875 



04S2 
0068 
1518 
20^ 
2601 

8144 
8087 
4220 
4770 
5310 
5860 
6880 
6927 
7465 
8008 



8775 
4842 
4000 
5474 
6080 

6604 
7167 

Tjao 



9414 
9974 



0533 

1001 
1649 

2206 

2762 
8318 
8878 
4427 
4080 
5633 
6065 
6686 
7187 

7787 



9680 



0476 
1032 
1567 
2112 
2065 



8199 
8741 
4288 
4824 
5364 
5904 
6448 
6981 
7519 
8066 



4800 
4066 
6531 
6006 



6660 
7223 

7?86 
8348 
8000 
9470 



0080 
0660 
1147 
1705 



2818 
3878 



4482 
5036 
5588 
6140 
6602 
7242 

7702 
8311 
8600 
9437 
9085 



058? 
1077 



2166 
2710 



8795 
4387 
4878 
5418 
6056 
6497 
7085 
7578 
8110 



4466 
5022 

5687 
6152 

6716 
7260 

7642 
6404 
6065 
9526 



0086 
0645 
1208 
1760 

2317 
2678 
3429 
3064 
4536 
5091 
5644 
6105 
6747 
7897 

7847 
8396 
6044 
9492 



0039 
0586 
1131 
1676 
2221 
2764 



8807 
8649 
4391 
4032 
M72 
6012 
6551 
7060 
7626 
6168 



8045 
4512 
5076 
5644 
6200 

6778 
7386 

7606 
6460 
0021 



0141 
0700 
1250 
1616 

2373 
2029 
3464 
4030 
4598 
6146 
5609 
6251 
6802 
7352 

7002 
&151 
6900 
0547 



0094 
0640 
1186 
1731 
2275 
2818 

8861 
8004 
4445 
4966 
5526 
6066 
6604 
7143 
7680 
8217 



4002 



5186 
5700 



7302 
7965 
8516 
0077 
0636 



0197 
0756 
1314 
1872 

2428 

2865 
8540 
4094 
4618 
5201 
57M 
6306 
6857 
7407 

7957 
6506 
9(»4 
9002 



0149 
0G95 
1240 
1785 
2329 
2878 

8416 
8956 
4499 
5010 
5580 
6119 
6656 
7196 
7784 
8270 



4059 
4625 
5102 
5757 



7449 
8011 
8573 
9134 
9604 



0612 
1370 
1026 

2484 
8040 
8505 
4160 
4704 
6257 
5809 
6361 
6912 
7402 

6012 
8661 
9109 
9(356 



0749 
1295 
1610 
2384 

2927 

8470 
4012 
4558 

5094 
5634 
6178 
6n2 
7250 
7787 



8 9 Diff. 



4115 
4682 
6246 
5613 
6378 

0912 

7605 
8067 
8629 
9190 
9750 



1426 
1068 

2540 
8006 
8651 
4206 
4750 
5312 
6864 
6416 
6967 
7517 

8067 
6615 
9164 
9711 



06O1 
1349 
1894 
2436 
2961 

8524 

4066 
4607 
5146 
5688 
6227 
6766 
7304 
7811 
6878 



4172 

4789 
5305 
5870 
6434 

6906 
7561 
8123 
8665 
0246 
9606 



0365 
0024 
1462 



2595 
8151 
87106 
4261 
4614 
53C7 
5920 
6171 
7022 
7572 

8122 
8670 
0218 
9766 



0812 
0659 
1401 
1946 
2492 
8086 

8578 
4120 
4661 



5742 
6281 
6830 
7358 
7895 
8431 



56 



53 



54 



PSOPOBTXONAL PARTS. 



DUt 


1 


2 


8 


4 


5 


6 


7 


8 


9 


57 


5.7 


11.4 


17.1 


22.8 


28.5 


S1.2 


39.9 


45.6 


51.3 


56 


5.6 


11.2 


16.8 


22.4 


28.0 


33.6 


89.2 


44.8 


50.4 


56 


5.5 


11.0 


16.5 


22.0 


27.5 


33.0 


86.5 


44.0 


49.5 


54 


5.4 


10.8 


16.2 


21.6 


27.0 


82.4 


87.8 


48.2 


48. G 



152 



LOGARITHMS OF KUMBEBB. 



llo.8tOLi.Ma] 



[No.864L.9BL 



N. 



810 908486 

1 I 9091 

2 9666 



910001 
0004 
1168 
1090 

S7S8 



8814 
4848 
4878 
6400 
6027 
6464 

eoeo 

7S06 
8000 
8866 

9078 
9001 



900198 
0646 
1166 
1086 
S906 
2796 
8944 
8709 

4S79 
4796 
6818 
6888 
6848 
6887 
7870 



8908 

9410 
9080 



080440 
0019 
1466 



8688 
90r4 
9010 



0144 
0678 
1211 
1748 
2275 
8806 
8837 

S867 
4896 
4996 
5458 
5080 
0507 
7088 
7566 
8068 
8007 

0180 



0176 
0007 
1218 
1788 
2268 
2r?7 
8296 
8814 

4881 
4848 
6364 
5879 
6804 
6006 
7498 
7986 
8447 
8959 

9470 
9081 



0401 
1000 
1500 



869B 
9128 
9668 



0107 
0781 
1264 
1797 



8800 

8020 
4440 
4077 
5506 
0068 
6560 
7066 
7811 
6186 
8060 

9188 
0706 



02&8 
0740 
1270 
1790 
2810 
2829 
8348 



4888 

4800 
6116 
6081 
6446 
6059 
7478 
7086 
8t08 
0010 

0621 



0088 

0549 

1051 
1560 



8046 
9181 
9716 



0261 

0784 
1817 
1860 
2881 
2918 
8418 



8073 
4502 



5668 

0065 
6612 
7188 
7068 
8186 
8712 

9236 
9758 



0801 
1822 
1842 



8017 

4484 
4051 
5407 
5062 
0407 
7011 
7684 
8087 
8540 
0061 

9679 



0068 

0502 
1102 
1610 



8090 
9885 



0804 
0638 
1871 
1008 
2485 
2960 
8496 



4656 

6068 
5611 
0186 
0604 
7100 
TTIO 
8240 
8784 

0287 
0810 



0668 
1374 
1804 
2414 



8451 
8000 

4486 
5008 

6518 
0004 
6648 
7062 
7576 
8068 
6001 
9112 



0184 
0018 
1158 
1661 



0868 
0691 
1424 
1066 
2466 
8U19 
8549 

4070 
4008 
5186 
5004 

0191 

ffri7 

7248 
7766 
8208 
8810 

0840 
0602 



9848 

8677 



8800 
0896 
9000 



0884 
0006 
1420 
1940 
2406 
2065 
8606 



5054 

5670 
6065 
0000 
7114 
7627 
8140 
6052 
9108 



9074 



0166 
0094 
1204 
1712 



0411 
0914 
1477 
2009 
2541 
8072 
3602 

4182 
4000 
0180 
671d 
6243 
6770 
7295 
7820 
8845 
8869 

0892 
9914 



0006 
1580 
20O8 
2591 
8125 
8655 

4184 
4713 



8014 
9449 
9084 



• ]>ifl. 



8987 



0618 
1051 
1564 
2116 
2647 
8178 
8708 

4287 
4766 



5241 5294 
5709 5882 



(822 
7848 
7878 
8397 
602t 

9444 
9907 



6875 
7400 
7925 
8450 
8078 

9490 



0486 
0956 
14;^ 
1908 
2518 
8087 
8555 
4079 

4589 
5106 
5621 
6187 
0051 
7106 
7078 
8191 
8706 
0215 

9785 



0746 
3254 
1763 



0460 
1010 
1580 
2060 
2670 
8069 
8607 
4124 

4641 

5167 
5678 
0166 
0702 
7^216 
7780 
B242 
6754 



97?6 



0019 
0641 
1062 
1562 
2102 
2622 
8140 



4176 

4098 
6209 

5785 
6240 
0764 
7266 
7781 
6206 
6605 
0817 

9627 



0067 

oon 

1104 
1087 
2109 
2700 
3281 
8701 

4200 
4819 
5847 
5875 
6401 
6027 
7458 
7078 
8608 
9026 



0287 
0796 
1305 
1814 



0847 
1366 
1865 



wn 

0608 
1114 
1634 
2104 
2074 
8192 
8710 
4226 

4744 
&961 
5778 
6291 

6606 
7819 
76» 
6845 
6667 
0906 

0679 



53 



6S 



0889 
0696 
1407 
1916 



61 



pROPORTioNAi. Parts. 



Dili. 


1 


2 


8 


4 


6 


6 


7 


8 


9 


58 
62 
51 
60 


6.8 
5.2 
5.1 
5.0 


10.6 
10.4 
10.2 
10.0 


15.9 
15.6 
15.3 
15.0 


21.2 
20.8 
20.4 
20.0 


30.5 
2fi.O 
2f).5 
25.0 


81.8 
81.2 
3().6 
30.0 


87.1 
86.4 
85.7 
85.0 


42.4 
41.6 
40.8 
40.0 


47.7 
40.6 
45.9 
45.0 



rt>. 








roc 


^A1 


an 


rHHS 


OF 1 


^tJMB 


BES. 






i^ 


WBi:.. 881.1 1 


/^ 


; . / . . 


8 


4 


i 


• 


9 


• 


t ^^ ' 082000 / aWT 1 9068 


8118 


8160 


8a» 


8821 


8881 


8871 


/ $ 


««rtf / as04 asTs 


WW 


a«7 


8797 


8778 


8aw 


887! 


9081 / 9oai ao8s 


8188 


8188 


8884 


8886 


8886 


888f 


^ 


8497 / >tftfW SBSO 


8680 


8600 


8740 


8701 


8841 


880i 


«^ 




4094 


4145 


4106 


4846 


4896 


4847 


489" 


4406 / 'tfCMO 


4599 


46B0 


4700 


4761 


4801 


4868 


490i 


1 


So08 / GO(V4 


G104 


5154 


5806 


5865 


5806 


5356 


540( 


2 


-">r i osfis 


6606 


5658 


5709 


6759 


5809 


5660 


591( 


8 


'^Ul 


0Ofll 


OUl 


6168 


6818 


6868 


6813 


6368 


6412 


4 


i^^I4 


Gse^ 


0614 


0665 


6715 


6765 


6815 


6665 


691( 


i 5 


7010 


7oa0 


T118 


7167 


7817 


7967 


7817 


7867 


74n 


1 e 


75 18 


7B4S8 


7816 


7868 


T718 


7780 


7819 


7860 


791t 


I I 


hjhO 


«)O0O 


8119 


6160 


8819 


8869 


8390 


8870 


842L 


^^U0 


8fy70 


8620 


8670 


8780 


8770 


^ 


8870 


808( 


I " 


;*.r^JO 


00^70 


O10O 


9170 


9880 


9870 


8860 


9411 


1 »70 


9510 


osae 


9619 


9680 


9719 


9760 


9819 


9669 


991f 


\ 1 

7 
8 
g 


M0018 
0616 


o^^« 


0118 


0168 


0818 


(B67 


0817 


0367 


041^ 


a- -0 


0616 


0606 


0716 


0766 


0815 


0665 


091£ 


1014 
1511 
9006 
9604 
8000 
8499 
8089 


il 1 


1114 


1163 


1818 


1868 


1813 


1368 


14U 


li I 


1611 


1660 


1710 


1760 


1809 


1859 


190( 


^ 


9107 


2157 


8907 


8866 


8806 


8855 


840e 


le i 


9008 


8658 


8708 


8758 


2801 


8851 


8901 


» ^^ 


8099 


8146 


8106 


8M7 


8807 


8846 


880( 


2=, 8 


8096 


8648 


8608 


8748 


8791 


8841 


889C 


2 rti 


4066 


4187 


4186 


4886 


4886 


4886 


4884 


1 
« 
8 

4 
5 
6 

7 
6 

\ 9 


4488 
4078 
IV4fi9 


41 s 


C074 


4681 
5194 


4680 
5178 


4739 
5883 


4779 
6872 


4888 
6S81 


4873 
687C 


& 8 


6667 


6616 


5665 


5715 


5704 


5818 


586S 


»» S ? 

A^fifi 1 O 1 


6069 


6108 


6157 


6807 


6856 


6305 


6354 


6661 


6600 


6649 


6096 


6747 


6796 


684S 


•KM24 1 *: i*^s J oQai 

8418 ^t» »11 


7090 


7180 


7180 


7888 


7387 


7886 


7581 


7680 


T679 


7788 


7777 


7886 


8070 


8119 


8168 


8817 


8866 


8816 


8560 


8608 


8C57 


8706 


8755 


8804 


9048 


9097 


9146 


9196 


9844 


9898 


\J 


S2S2 5i^2 


9488 
9975 


9686 


9685 


9684 


9688 


9781 


9780 


1 J 


0B7S 


--=^ 


0084 
0611 


0078 
0660 


0121 
0608 


0170 
0657 


0219 
0706 


0887 


f 


ctfstfWIA 


0«68 


0754 


1338 I 1 g 
'. 8f7«0 \ 1 « 


CM8 


0997 


1046 


1005 


1148 


1198 


1840 




4 
5 
6 
7 


1435 


1488 


1538 


1580 


1689 


1677 


1726 




IMO 


1960 


8017 


9066 


8114 


8166 


88H 




9105 


8458 


8508 


asso 


8599 


8647 


8696 




«»9 


8088 


8986 


8084 


8088 


8181 


8180 




8878 


8481 


3470 


8518 




8615 


8663 




8866 


8905 


8058 


4001 


4049 


4066 


4146 


v 


Pboportional Parts. 




\ 


T>%ff.\ 1 \ « 


8 


4 


5 


6 


7 


r 




) 


^ ^\ \ ^M 12? 


15.8 


80.4 


85.5 


30.6 


85.7 






\ «> \ 5.0 10.0 


15.0 


80.0 


25.0 


30.0 


35.0 






\ « *? S2 


14.7 : 19.6 1 


84.5 


20.4 


34.8 








^ 48 I 4.8 1 9.6 14.4 I 19.8 I 


84.0 


28.8 


33.6 





154 



LOOAtlltttMS OF irtJHBEBS. 



No 900 L. 864.1 



[No.M4L.9ni 



N. 



064948 
47!ffi 

6307 
S688 
6168 
6649 
7128 
TB07 
8086 
8564 

9041 
9518 
9996 



960471 
0916 
1421 
1896 
2869 
28IS 
8816 

87B8 
4260 
4781 
5203 
6672 
6142 
6611 
7080 
7548 
8016 

8488 
8950 
9416 



970847 
0812 
1278 
1740 



8128 
8590 
4051 
4512 
4972 



4291 
4778 
5255 
6786 
6216 
6607 
7176 
7655 
8184 
8612 

9089 
9666 



0043 
0618 
0994 
1460 
1948 
2417 
2890 



4807 
4778 
5240 
6719 
6189 
6658 
7127 
7505 
8068 

8580 
8096 
9468 



0608 



1786 
2249 
2712 

8174 
8686 
4097 
4568 
5018 



4821 
5808 

6784 

6745 
7224 
Tr08 
8181 
8650 

9187 
9614 



0090 
0666 
1041 
1516 
1990 
24&I 
2987 
8410 

8882 

4854 
4825 
6296 
5766 
6286 
0706 
7178 
7642 
8109 

8576 
9M8 
9609 
9975 



0440 
0904 
1369 
1832 
2395 
2758 



4148 
4604 
5064 



4887 
4860 
6861 



6818 
8788 
7272 

rrai 

8220 
8707 

0185 
0661 



0188 
0618 
1080 
1568 
2038 
2511 
2065 
8457 



4401 
4873 
6818 
5818 
6283 
6752 
71220 
7688 
8156 

8628 

0090 
9656 



0021 
0486 
0961 
1415 
1879 
2342 
2804 

8266 
3728 
4189 
4660 
6110 



4485 
4918 
6899 

5880 
6861 
6840 
7820 
7799 
8277 
8755 



9709 



0185 
0661 
1136 
1611 
2085 



3082 
86(M 

8077 
4448 
4010 
5890 
5860 
0320 
6790 
7867 
7785 
8308 

8070 
0136 
0602 



0068 
0538 
0097 
1461 
1025 
2388 
2851 

8818 

8774 
4285 
4606 
5156 



4484 
4066 
5447 
5028 
6400 
6888 
7368 
7847 
8325 



8S80 
0757 



0238 
0700 
1184 
1668 
2188 
2606 
8070 
8552 



4024 
4405 
4066 
5487 
6007 
6876 
6845 
7814 
7782 
8240 

8716 
0188 
0640 



0114 
0579 
1014 
1508 
1071 
3484 
2807 



4281 
4742 
6308 



6014 
5485 
6076 
6457 
6086 



7416 
7804 
8878 



9804 



0756 
1281 
1706 
2180 
2668 
3126 
8500 

40n 
4542 

5018 
5484 
6064 
6428 
6803 
7861 
7880 
8306 

8768 
0220 
0696 



0161 
0626 
1090 
1554 

2018 
2481 
2948 

8405 
3866 
4827 

4788 



4680 
5068 
5548 
6024 
6605 
6964 
7464 
7942 
8421 



9675 
9668 



0804 
1279 
1768 
2227 
2701 
8174 
8646 

4118 
4500 
6061 
6631 
6001 
6470 
6060 
7408 
7875 
8848 

8810 
0276 
0742 



0307 
0673 
1187 
1601 
2064 
2527 
2060 

8451 
8018 
4874 
4884 
5204 



4828 
5110 
6682 
607S 



7082 
7512 
7000 
8468 
8046 



0676 
0661 
1826 
1801 
2275 
2748 
8221 
8608 

4165 
4687 
5108 
6578 
6048 
6517 
6066 
7454 
7828 



8866 

0828 
0780 



0264 
0710 
1188 
1647 
2110 
2678 
8065 



8407 
8050 
4420 
4880 
5840 



4677 
5158 
5640 
6120 
6601 
7080 
7550 
8088 
8516 
8004 

0471 
0947 



0428 
0600 
1874 
1848 
8822 
2796 
8268 
8741 

4212 
4684 
5155 
6625 
6096 
6564 
7088 
7801 
7969 
8486 

8906 
9860 



0800 

0765 
1899 
1698 
2167 
2619 
8082 

8548 
4005 
4466 
4086 
5886 



DifL 



47 



Peoportiomal Pabti. 











4 


5 








1 


DIff. 


1 


3 


8 


6 


7 


8 





47 
46 


4.7 
4.6 


9.4 
9.2 


14.1 
13.8 


18.8 
18.4 


23.5 
23.0 


28.2 

27.6 


82.0 
S3. 2 


87.6 
86.8 


42.8 
41.4 













X.0GAKITHM8 OF KUXBBBS. 








15fi 


NaMfil^gVBLl INa.fl»L.fi86.| 


H. 





a 


8 


1 
69 


1 


4 


6 


• 


7 


8 


9 


Difl. 


M5 


97MK 


5478 


6804 


ro 


6010 


6062 


5707 


6768 


5799 


6846 






fi801 £987 


0068 


6089 


0076 


0121 


0107 


6212 


6258 


0804 






tfBfio eao6 


em 


0488 


0688 


05T9 


6026 


6871 


6n7 


0763 






6M8 


0854 


6000 


0946 


0992 


7087 


7088 


7189 


7175 


7280 






7806 


7812 


7358 


7408 


7449 


7495 


7M1 


7680 


7082 


7878 




960 


7794 


7700 


7815 


7861 


7906 


7952 


7996 


8048 


8089 


8135 






8181 aseas 


8273 


8817 


8308 


8409 


8454 


8600 


8540 


8501 






86S7 , 8688 


8728 


8774 


8810 


6665 


8911 


8050 


9002 


9047 






9003 0138 


0184 


9290 


9275 


9321 


9360 


9412 


W57 


9503 






964S 96M 


0639 


9685 


9730 


97ro 


9821 


9807 


9012 


9958 




980008 0040 


0094 


0140 


0185 


0231 


0276 


0S22 


0907 


0412 




0468 0508 


0649 


0694 


0640 


0065 


0780 


0776 


0821 


0867 






0012 0067 


lOOB 


1048 


1096 


1189 


1184 


1229 


1275 


1320 






1366 1411 


1460 


1501 


1&47 


15» 


1637 


1663 


1728 


ITTS 






1810 1804 


1909 


1954 


8000 


2045 


2090 


8186 


8181 


2220 




MO 


ssn 


2810 


2888 


8107 


2458 


8497 


2548 


2588 


8688 


8678 






2728 


2700 


2814 


2859 


2904 


2949 


2994 


8010 


8086 


8130 






8175 


8880 


3965 


8310 


8850 


8401 


8440 


8491 


8530 


8561 






3086 


8071 


3716 


8762 


3807 




8897 


3942 


8967 


4032 






4077 


4122 


4167 


4213 


4257 


4302 


4847 


4398 


4437 


4482 






4587 4578 


4617 


4662 


4707 


4758 


4797 


4848 


4887 


4932 


46 




4877 


6082 


6067 


5112 


5157 


5202 


6847 


5298 


5337 


5382 






5486 


5471 


5616 


5601 


5000 


5651 


5090 


5741 


57«0 


5830 






5875 


50B0 


6965 


6010 


6055 


0100 


6144 


6189 


6284 


6279 




0?0 


€884 


6800 


6413 


6458 


0506 


0548 


C596 


6687 


6688 






6772 


0817 


6861 


0800 


0951 


0990 


704O 


7085 


7130 


7175 






7210 


7204 


r-J09 


7S58 


7398 


7443 


7488 


7582 


7577 


7622 






7666 


7711 


7756 


7800 


7845 


7H90 


7934 


7979 


6024 


60C8 




3 8113 


8157 


8308 


8^7 


8291 


63S0 


8881 


8425 


ai70 


K514 




4 8r.59 


8604 


8CI8 


8698 


8787 


8:32 


86S6 


8871 


8016 


8(MK) 






0006 


0040 


9094 


9188 


9188 


9227 


9272 


0310 


0361 


9105 






OlSO 


OI94 


9689 


9G88 


9G28 


Hfffli 


9717 


9761 


9600 


9860 






9H06 


0030 


9988 




















0028 
0478 


0072 


0117 


0161 
0605 


0200 


0850 


0204 
0TS8 




8 1 OWIMQ 


0988 


0428 


0510 


0561 


omo 


0604 






ores 


0837 


0871 


0910 


0900 


1004 


1049 


1098 


1137 


1182 




960 


1286 


1270 


1315 


1859 


1408 


1448 


1498 


1680 


1580 


1625 






1609 


1713 


1758 


1602 


IWO 


1890 


1936 


1979 


2023 


2067 






2111 


2150 


2900 


8^44 


2288 


2838 


2377 


8121 


2465 


2509 






2554 


2506 


9548 


2660 


2730 


2774 


2819 


2863 


2907 


2U51 






2006 


3080 


8068 


8127 


8172 


8316 


8260 


3904 


mis 


3302 






8486 


8480 


8684 


8568 


8013 


8657 


8701 


»r45 


3789 


3833 






8877 


8021 


8065 


4000 


4053 


4097 


4141 


4185 


4229 


4273 






4317 


4301 


4405 


4449 


4493 


4537 


4581 


4025 


4669 


4713 


44 


8 


4757 


4801 


4845 


4689 


4983 


4977 


6021 


5065 


5108 


5152 




9 


5106 


0fifi40 


6284 


6328 


5378 


5410 


5460 


5604 


5547 


5501 




VaapoKTiovAjj Pakts. 


DUf. 


1 


2 


a 


4 


6 





7 


8 


9 


46 


4.0 


9.8 


13.8 


18.4 


23.0 


27.6 


322 


3C.8 


41.4 


45 


4.5 


9.0 


18.5 


18.0 


22. 5 


27.0 


81.5 


3G.0 


40.6 


44 


4.4 


8.8 


13.2 


17.0 


22.0 


2HA 


30. H 


35.2 


39.6 


4S 


4.8 


8.e 


1S.9 


17.2 


21.5 


25.8 


30.1 


34.4 


88.7 



156 

No. 900 L. 906.] 



KATUEHATIOAL TABLES. 



[No. 999 L. 09k 



N. 





1 


a 


8 


4 


6 


6 


7 


8 


9 


Dift. 


900 


99S6SS 


6079 


6788 


5767 


6611 


6854 


6898 


SM2 


6066 


6000 




1 


9074 


6117 


6161 


0905 


6949 


6896 


6887 


6880 


6484 


6468 


44 


2 


6619 


6665 


6699 


6648 


6687 


0m 


6774 


6618 


6868 


6906 






6949 


6098 


TOB7 


7080 


7184 


7168 


7812 


7206 


7899 


7848 






7886 


7480 


7474 


7517 


7B61 


7006 


7648 


7698 


7796 


77T9 






7888 


7867 


7910 


7054 


7908 


8041 


8086 


8189 


8178 


8il6 






8269 


8808 


8847 


8890 


8484 


8477 


8081 


8564 


8608 


8668 






8605 


8780 


8788 


8826 


8860 


8918 


8956 


9000 


90«,t 


9067 




8 


9181 


9174 


9218 


9261 


9805 


9848 


9302 


9485 


9470 


9028 




9 


9065 


9609 


9668 


9000 


9789 


9788 


0686 


9870 


9918 


9967 


48 







BTPEBBOI.IC 


I.OOABITIIM8. 






No. 


Log. 


No. 
1.45 


Log. 


No. 


Log. 


No. 


Lo.. 


No. 


Log. 


1.01 


.0009 


.8?16 


1.80 


.6866 


2.88 


.8468 


2.77 


1.0188 


1.08 


.0198 


1.46 


.8784 


1.00 


.6419 


2.84 


.8002 


2.78 


1.0825 


1.08 


.0296 


1.47 


.8858 


1.91 


.6471 


2.85 


.8544 


2.79 


1.0260 


1.04 


.0308 


1.48 


.8020 


1.92 


.6628 


2.86 


.8567 


2.80 


1.0206 


1.05 


.0488 


1.49 


.8988 


1.98 


.6676 


2.87 


.8689 


2.81 


1.06S2 


1.06 


.0688 


1.50 


.4066 


1.94 


.6687 


2.88 


.86n 


2.S2 


1.0867 


1.07 


.0677 


1.51 


.4121 


1.96 


.6678 


2.89 


.8718 


2.88 


1.0408 


1.08 


.0770 


1.58 


.4187 


1.96 


.6720 


2.40 


.8706 1 


2.84 


1.0438 


1.09 


.0668 


1.58 


.4858 


1.97 


.6780 


2.41 


.8796 1 


2.86 


1.0478 


1.10 


.0058 


1.54 


.4318 


1.98 


.6831 


2.42 


.8888 


2.86 


1.0606 


1.11 


.1044 


1.56 


.4888 


1.99 


.6881 


2.48 


.8879 ! 


2.87 


1.0548 


1.18 


.1138 


1.56 


.4447 


8.00 


.6031 


2.44 


.8080 1 


2.88 


1.0578 


1.18 


.1822 


1.57 


.4511 


2.01 


.6081 


2.45 


.8961 . 


2.89 


1.0613 


1.14 


.1310 


1.56 


.4574 


2.02 


.7031 


2.46 


.0008 , 


2.90 


1.0647 


1.15 


.1396 


1.59 


.4687 


8.08 


.7080 


2.47 


.0042 


2.91 


1.068u» 


1.16 


.1484 


1.60 


.4700 


8.04 


.7120 


2.48 


.0083 ; 


2.92 


1.0716 


1.17 


.1570 


1.61 


.4768 


8.05 


.7178 


2.49 


.9123 1 


2.03 


1.0750 


1.18 


.1666 


1.68 


.4824 


2.06 


.7T227 


2.60 


.9168 < 


2.94 


1.0784 


1.19 


.1740 


1.63 


.4886 


2.07 


.7275 


2.61 


.9208 ; 


2.95 


1.0618 


1.20 


.1828 


1.64 


.4947 


2.08 


.7824 


2.52 


.9248 ' 


2.96 


1.0852 


1.21 


.1006 


1.65 


.6008 


2.09 


.7872 


2.58 


.9282 


2.97 


1.0886 


1.22 


.1088 


1.66 


.6068 


2.10 


.7419 


2.54 


.9382 


2.98 


1.0910 


1.33 


.2070 


1.67 


.5128 


8.11 


.7467 


2.55 


.9361 


2.99 


1.0963 


1.24 


.2161 


1.68 


.5188 


8.18 


.7614 


2.56 


.9400 


8.00 


1.09H6 


1.85 


.2281 


1.69 


.6247 


8.18 


.7561 


8.57 


.9489 


3.01 


1.1019 


1.26 


.8811 


1.70 


.5806 


8.14 


.7608 


2.58 


.9478 


8.09 


1.1053 


1.27 


.2300 


1.71 


.5865 


8.16 


.7655 


2.59 


.9617 


3.08 


1.1086 


1.28 


.2469 


1.78 


.5428 


8.13 


.7701 


2.60 


.9556 


8.04 


1.1119 


l.-,>9 


.2546 


1.73 


.5481 


2.17 


.7747 


2.61 


.9594 


8.05 


1.1151 


1.80 


.2624 


1.74 


.5589 


2.18 


.7793 


2.62 


.9682 


3.06 


1.1184 


1.81 


.2700 


1.75 


.6606 


8.10 


.7830 


2.63 


.9670 


8.07 


1.1817 


1.82 


.2776 


1.76 


.5653 


2.20 


.7b85 


2.64 


.9708 


8.08 


1.1249 


1.88 


.2852 


1.77 


.5710 


2.21 


.7930 


2.65 


.9746 


8.00 


1.1288 


1.34 


.2027 


1.78 


.6766 


2.22 


.7975 


2.66 


.9783 


8.10 


1.1814 


1.85 


.3001 


1.79 


.5822 


2.28 


.8020 


2.67 


.9821 


8.11 


1.1846 


1.86 


.8075 


1.80 


.5878 


8.24 


.8065 


2.68 


.9658 


8.12 


1.18:8 


1.87 


.8148 


1.81 


.5033 


2.25 


.8109 


2.69 


.9895 


8.18 


1.1410 


1.88 


.3221 


1.82 


.5088 


2.26 


.8154 


2.70 


.9933 


8.14 


1.1442 


1.80 


.8208 


1.88 


.6043 


2.27 


.8198 


2.71 


.9969 


8.15 


1.1474 


1.40 


.8865 


1.84 


.6098 


2.28 


.8242 


2.72 


1.0006 


8 16 


1.1600 


1.41 


.34.S6 


1.85 


.6162 


2.29 


.82fi6 


2.73 


1.0043 


8.17 


1.1587 


1.48 


.8607 


1.86 


.<;206 


2.30 


.8829 


2.74 


1.0080 


8.18 


1.1009 


1.43 


.8677 


1.87 


.6v'50 


2.81 


.8372 


2.75 


1.0116 


8.19 


1.1600 


1.44 


.8646 


1.88 


.6818 


2.32 


.841G 


2.76 


1.0162 


8.20 


1.1632 



HYPERBOLIC LOGARITHMS. 




Log. 


No. 


Log. 


No. 


Log. 


l.%88 


4.58 


1.5107 


6.19 


1.6467 


1.8598 


4.54 


1.61» 


5.80 


1.6487 


1.8M4 


4.56 


1.5151 


5.81 


1.6606 


1.3610 


4M 


1.5178 


5.28 


1.6585 


l.!»35 


4.57 


1.5195 


5.83 


1.6614 


1.8661 


4.58 


1.5817 


6.24 


1.6563 


1.3686 


4.59 


1.5839 


6.85 


1.6588 


1.87W 


4.60 


1.5861 


5.96 


1.6601 


1.8737 


4.61 


1.5288 


6.87 


1.6620 


1.8768 


4.68 


1.5804 


5.88 


1.6630 


1.8788 


4.63 


1.58-J6 


5.89 


1.6658 


1.8818 


4.64 


1.5847 


5.80 


1.6677 


1.8888 


4.65 


1.5869 


6.81 


1.6606 


1.8868 


4.66 


1.5390 


5.82 


1.6716 


1.8888 


4.67 


1.5418 


5.83 


1.6734 


1.3913 


4.68 


1.5433 


5.34 


1.6758 


1.8988 


4.60 


1.5451 


5.35 


1.6771 


1.39(3 


4.70 


1.5476 


5.86 


1.6790 


1.3987 


4.71 


1.5497 


5.87 


1.6808 


1.401-^ 


4.78 


1.5518 


5.88 


1.6887 


1.4086 


4.73 


1.5589 


5.89 


1.6845 


1.4061 


4.74 


1.5560 


5.40 


1.6864 


1.4086 


4.75 


1.5581 


5.41 


1.6888 


1.4110 


4.76 


1.5808 


5.42 


1.6901 


1.4184 


4.77 


1.5683 


5.48 


1.6919 


1.4150 


4.78 


1.5644 


5.44 


1.69.« 


1.4188 


4.79 


1.5665 


5.45 


1.6956 


1.4207 


4.80 


1.5686 


5.46 


1.6974 


1.4^1 


4.81 


1.5707 


5.47 


1.6993 


1.42.Vf 


4.88 


,1.5788 


5.48 


1.7011 


1.4279 


4.83 


1.5748 


5.49 


1.7089 


1.4803 


4.84 


1.5769 


5.60 


1 .7047 


1.4887 


4.85 


1.5790 


5.51 


1.7066 


1.4351 


4.86 


1.6810 


5.52 


1.7084 


1.4875 


4.87 


1.5831 


5.54 


1.7108 


1.4398 


4.88 


1.5851 


5.54 


1.7180 


1.4488 


4.89 


1.5872 


5.55 


1.7188 


1.4446 


4.90 


1.5892 


5.66 


1.7156 


1.4469 


4.91 


1.5913 


5.57 


1.7174 


1.4193 


4.98 


1.6933 


5.58 


1.7192 


1.4516 


4.93 


1.595:^ 


5.59 


1 .7210 


1-4540 


4.94 


1.6974 


5.60 


1.7288 


1.4563 


4.95 


1.5994 


?i.61 


1.7246 


1-4586 


4.96 


1.6014 


5.68 


1.7263 


1.4609 


4.97 


1.6084 


5.68 


1.7281 


1.4683 


4.96 


1.6054 


5.64 


1.7299 


1.4656 


4.99 


1.6074 


5.65 


1.7317 


1.4679 


5.00 


1.6094 


5.66 


1.7334 


1.4708 


6.01 


1.6114 


5.67 


1.7852 


1.4785 


5.02 


1.6134 


5.68 


1.7370 


1.4748 


5.08 


1.6154 


5.69 


1.7887 


1.4770 


5.04 


1.6174 


5.70 


1.7405 


1.4793 


5.05 


1 .6194 


5.71 


1.7422 


1.4816 


5.06 


1.6214 


5.72 


1.7440 


1.4889 


5.07 


1.6283 


5.73 


1.7457 


1.4861 


5.08 


1.6253 


5.74 


1.7475 


1.4884 


5.09 


1.6278 


5.75 


1.7492 


1,4907 


5.10 


1.6292 


5.76 


1.7509 


1.4929 


5.11 


1.6312 


5.77 


1.7.527 


1.4951 


5.18 


1.6332 


5.78 


1.7544 


1.4974 


5.18 


1 .6351 


5.79 


1 .7561 


1.4996 


6.14 


1.6371 


5.80 


1.7579 


1.5019 


5.15 


1.6390 


5.81 


1.7596 


1.5041 


5.16 


1.6409 


5.82 


1.7613 


1.5068 


5.17 


1.6429 


5.83 


1.7630 


1.5085 


5.18 


1 .6448 


5.84 


1.7647 



158 



HATHEMATICAL TABLB8. 



No. 


Lev. 


Na 


LoK. 


No. 


1 


No. 


Log. 


No. 


Log. 


6.61 


1.8783 


7.16 


1 
1.9671 


7.79 


s.osw 


8.66 


8.1687 


9.04 


8.8966 


6.52 


1.8749 


7.16 


1.9685 


7.80 


2.0641 


8.68 


8.1610 


0.06 


8.2066 


6.5S 


1.8764 


7.17 


1.9699 


7.81 


2.0654 


8.70 


8.1688 


O.OB 


8.8006 


6.54 


1.8779 


7.18 


1.9718 


7.82 


8.0567 


8.78 


2.1656 


10.00 


8.8096 


6.56 


1.8795 


7.19 


1.9727 


7.88 


2.0580 


8.74 


2.1679 


10.85 


8.8279 


6.56 


1.8810 


7.20 


1.9741 


7.84 


2.0592 


8.76 


8.1708 


10.60 


8.8513 


6.67 


1.8825 


7.21 


1.9754 


7.86 


2.0605 


8.78 


8.1786 


10.75 


8.8740 


6.68 


1.8840 


7.22 


1.0769 


7.66 


8.0618 


8.80 


8.1748 


11.00 


8.80T0 


6.59 


1.8856 


7.28 


1.9782 


7.87 


8.0631 


8.82 


2.1770 


11.25 


8.4801 


6.60 


1.8871 


7.84 


1.9796 


7.88 


2.0648 


8.84 


8.1798 


11.60 


2.44.W 


6.61 


1.8886 


7.26 


1.9810 


7.80 


8.0656 


8.86 


8.1816 


11.75 


8.4636 


6.6.2 


1.8901 


7.26 


1.9824 


7.90 


8.0669 


8.86 


2.1888 


18.00 


8.4819 


6.68 


1.8916 


7.27 


1.9638 


J.91 
102 


8.0681 


8.90 


8.1861 


18.85 


8.6062 


6.64 


1.8031 


7.28 


1.9651 


2.0694 


8.98 


8.1888 


18.60 


8.5262 


6.65 


1.8916 


7.29 


1.9865 


7.98 


2.0707 


8.94 


8.1906 


18.76 


8.6455 


6.66 


1.8961 


7.80 


1.9879 


7,94 


2.0719 


8.96 


8.1928 


18.00 


8.5649 


6.67 


1.8976 


7.81 


1.9892 




2.0782 


8.98 


8.1950 


18.25 


8.5840 


6.68 


1.8991 


7.82 


1.9906 


7.96 


8.0744 


0.00 


8.1072 


18.60 


2.6027 


6.69 


1.9006 


7.88 


1.9920 


7.97 


8.0757 


0.02 


2.1004 


18.75 


8.ti811 


6.70 


1.9021 


7.84 


1.9938 


7.{k 


2.0769 


0.04 


2.2017 


14.00 


8.6301 


6.71 


1.9086 


7.35 


1.9947 


7.90 


2.0782 


0.06 


8.2039 


14.25 


2.6567 


6.7-^ 


1.9051 


7.86 


1.9961 


8.00 


2.0794 


0.06 


8.8061 


14.60 


8.6740 


6.78 


1.9066 


7.87 


1.9974 


8.01. 


2.0607 


9 10 


8.2088 


14.76 


8.6013 


6.74 


1.9081 


7.38 


1.9988 


8.02 


2.0819 


9.12 


8.8105 


16.00 


8.7061 


6.75 


1.9095 


7.89 


2.0001 


8.03 


2.0832 


9.14 


8.2187 


15.50 


8 7408 


6.76 


1.9110 


7.40 


2.0015 


8.04 


2.0644 


9.W 


2.2148 


16.00 


8.7726 


6.77 


1.9125 


7.41 


2.0028 


8.06 


2.0857 


9.18 


2.2170 


16.60 


2.8084 


6.78 


1.9140 


7.42 


2.0041 


8.06 


2.0869 


9.80 


2.8198 


17 00 


2.8332 


6.79 


1.9155 


7.43 


2.0056 


8.07 


8.0S82 


9.22 


8.2214 


17.60 


2.662) 


6.80 


1.9169 


7.44 


2.0069 


8.08 


2.0894 


9.84 


2.2835 


18.00 


2.8904 


6.81 


1.9184 


7.45 


2.0088 


8.09 


2.0906 


9.26 


8.2857 


18.50 


2.917H 


6.K2 


1.9199 


7.46 


2.0096 


8.10 


2.0919 


9.28 


2.8U70 


10.00 


2.9144 


6.83 


1.9218 


7.47 


2.0108 


8.11 


2.0981 


9.80 


2.2800 


10.50 


2.970:1 


6.84 


1.9228 


7.48 


2.0122 


8.12 


2.0943 


9.82 


2.2322 


80.00 


2.9057 


6.85 


1.9242 


7.49 


2.0186 


8.18 


2.0956 


9.34 


2.2348 


21 


8.0445 


G.86 


1.9257 


7.60 


2.0149 


8.14 


2.0968 


9.36 


2.2364 


82 


8.0010 


6.87 


1.9272 


7.51 


2.0162 


8.16 


2.0980 


9.88 


2.2886 


83 


8.1855 


6.88 


1.9286 


7.52 


2.0176 


8.16 


2.0992 


9.40 


2.2407 


84 


8.r78l 


6.89 


1.9301 


7.53 


2.0189 


8.17 


2.1005 


0.42 


2.2428 


26 


8.8Ib0 


6.90 


1.9315 


7.54 


2.0202 


8.18 


2.1017 


0.44 


2.2450 


86 


8.2661 


6.91 


1.9330 


7.55 


2.0215 


8.19 


2.1029 


9.46 


2.2471 


87 


8.2958 


6.9-^ 


1 .9344 


7.56 


2.0220 


8.20 


2.1041 


9.48 


2.2402 


28 


8.8^22 


6.93 


1.9369 


7.57 


2.0248 


8.22 


2.1066 


9.60 


2.2518 


80 


8.86::) 


6.94 


1.9878 


7.68 


2.0256 


8.24 


2.1090 


9.62 


8.2584 


80 


8.4012 


6.95 


1.9387 


7.69 


2.0268 


8.26 


2.1114 


9.64 


2.-J665 


81 


8.4340 


6.96 


1.9402 


7.60 


2.0281 


8.28 


2.1138 


9.66 


2.2576 


88 


3.4667 


6.97 


1.9416 


7.61 


2.0295 


8.30 


2.1163 


9.58 


2.2697 


88 


8.4965 


6.98 


1.9480 


7.62 


2.0306 


8.82 


2.1187 


9.60 


2.2618 


84 


8.6263 


6.99 


1.9445 


7.63 


2.0321 


8.34 


2.1211 


9.62 


2.2638 


85 


8.5553 


7.00 


1 .9459 


7.64 


2.0384 


8.36 


2.1235 


9.64 


2.2659 


86 


8.5835 


7.01 


1.9478 


7.65 


2.0347 


8.88 


2.1268 


9.66 


2.2680 


87 


8.6109 


7.08 


1.9488 


7.66 


2.0360 


8.40 


2.1282 


0.68 


2.2701 


88 


8.0876 


7.a3 


1.9302 


7.67 


2.a373 


8.42 


8.1306 


0.70 


8.2721 


80 


8.6638 


7.W 


1.0516 


7.68 


2.0386 


8.44 


8.1330 


0.72 


2.8742 


40 


3.6889 


7.05 


1.9530 


7.69 


2.0899 


8.46 


2.1353 


0.74 


2.2762 


41 


8.7136 


7.06 


1.9544 


7.70 


2.0412 


8.48 


8.l87r 


0.76 


8.8r83 


48 


8.7377 


7.07 


1.9559 


7.71 


2.0425 


8.50 


8.1401 


0.78 


8.2808 


48 


8.7612 


7.08 


1.9573 


7.72 


2.0488 


8.52 


2.1424 


0.80 


8.8824 


44 


8.7H42 


7.09 


1.9587 


7.73 


2.0451 


8.54 


2.1448 


0.88 


8.8844 


45 


8.8067 


7.10 


1.9601 


7.74 


2.0464 


8..V) 


2.1471 


0.81 


8.8865 


46 


8.8286 


7.11 


1.9615 


7.75 


2.0477 


8.58 


2.1494 


9.86 


2.2886 


47 • 


8.8501 


7.12 


1.9629 


7.76 


2.0490 


8.60 


2.1518 


9.88 


2.2905 


48 


8.8712 


7.18 


1.964.3 


7.77 


2.0ri03 


8.62 


2.1.'S41 


9.90 


2.2925 


40 


8.8018 


7.14 


1.9657 


7.78 


2.0516 


8.64 


2.1564 


9.92 


2.2946 


60 


3.9180 



KATURAL TBIGOKOMETRICAL FUNCTIONS, 



159 



HATITRAIi TRIGONOniBTBICAIi FrNCTIONS. 



• 


M. 


SIlM. 


C«-V«n. 


Cons. 


T-». 


Cetan. 1 Bacant. 


Vpr. 8lB. Coniu*. 


90 




*""" 







1.0000 


Infinite 


.00000 


[nflnite! 1.0000 


.000001.0000 






.09000 







15 


.00488 


.90664 


ifi».18 


.00486 


229.18 ' l.OOOO 


.00001 .99999 




45 




80 


.00873 


,99127 


114.59 


.00873 


114.59 1.0000 


.00004 .99996; 


80 




45 


.01809 


.96691 


76.897 


.01309 


76.390 1.0001 


.00009 .99991 


15 







.01745 


.98255 


97.299 


.01745 


57.290 1.0001 


.00015 .99985 89 







15 


.OseiSl 


.97819 


45.840 


.02182 


45.829 . 1.0002 


.00024 .99976 


45 




80 


.(y»\s 


.97»& 


88.202 


.02618 


38.188 1.0008 


.000:i4 


.99966 


80 




45 


.oao-Ji 


.9U946 


82.746 


.03055 


32.730 1 1.0006 


,00047 


.9995.-), 


15 







.03490 


.96310 


28.054 


.03492 


28.636 ; 1.0006 


,00061 


.99939 88 







15 


.08^» 


.96074 


25.471 


.06929 


25.452 1 1.0008 


.00077 


.99923 




45 




80 


.01308 


.95685 


22.926 


.04366 


22.904 ; 1.0009 


.00095 


.99905 




80 




45 


.047W 


.93202 


20.843 


.04803 


20.819 1.0011 


.00115 .99885 




15 







.OTisa* 


.94766 


19.107 


.05241 


19.081 1.0014 


.00137 .99863 
.00161 .99889 


87 







15 


.05609 


.948:M 


17.639 


.056« 


17.611 1.0016 
16.350 1.0019 




45 




80 


.05106 


.9S695 


16.880 


.06116 


.00187, .99813 




30 




15 


.00510 


.98160 


15.290 


.06551 


15.257 


1.0021 


.00214 .09786 




15 







.oooro 


.98024 


14.3% 


.06998 


14.801 


1.0024 


.00244 .99756 


86 







15 


.omi 


.9^569 


13.494 


.07431 


IS.W 


1.0028 


.00275 .99725 




45 




80 


.07810 


.92154 


12.745 


.07870 


12.706 


1.0061 


.00308 


.99692 




80 




45 


.0*fcJI 


.91719 


12.076 


.08809 


12.035 


1.0084 


.00843 


.99656 




15 







.06716 


.912S4 


11.174 


.08749 


11.430 


1.0038 


.00881 


.99619 


86 







15 


.00150 


.90850 


10.929 


.09189 


10.838 


1.0042 


.00420 


.99580 




45 




80 


.09585 


.90415 


10.433 


.09629 


10.885 


1.0046 


.004601 .99540 




80 




45 


.10019 


.89961 


9.9BI2 


.10069 


9.9310 


1.0061 


.00503 .90497 




15 







.10453 


.89547 


9.5668 


.10510 


9.5144 


1.0055 


.00548. .99452 


84 





15 


.ia«7 


.80118 


9.1855 


.10952 


9.1309 1.0060 


.0(1504, .994U6 




45 


80 


.tl%i0 


.88680 


8.8387 


.11393 


8.7769 1.0065 


.00648 


.99357 




SO 


45 


.11754 


.88^46 


8.5079 


.11836 


8.4490; 1.0070 


.00693 


.99807 




15 


7 


.UlS? 


.87818 


8.2055 


.12278 


8.1443: 1.0075 


.00745 


.99255 


88 





15 


.l^SfrJO 


.87380 


7.9210 


.12722 


7.8606; 1.0081 


,00800 


.99200 




45 


80 


.11038 


.86947 


7.6613 


.13165 


7.5958 1.0086 


.00856 


.99144 




80 


45 


.13IS5 


.86315 


7.4158 


.18609 


7.8479 1.0092 


.00013 .99086 




15 


8 


.13017 


.8608) 


7.1853 


.14054 


7,1154 1.0098 


.00078 


.99027 


82 







15 


.14319 


.85631 


6.9690 


.14499 


6.8969| 1.0105 


.01035 


.98965 




45 




80 


.14781 


.85^19 


6.7655 


.14945 


6.69121 1.0111 


.01098 


.98902 




80 




45 


.15812 


.84788 


6.5786 


.15391 


6.4971 


1.0118 


.01164 


.98836 




15 







.15618 


.84357 


6.3924 


.15883 


6.3138 


1.0125 


.01281 


.98769 


81 







15 


.16074 


.839« 


6.2211 


.16286 


6.1402 


1.0132 


.01300 


.98700 




45 




80 


.16505 


.83495 


6.0589 


.16734 


6.9758 


1.0139 


.01371 


.98629 




80 




46 


.10085 


.88065 


5.90411 


.17183 


5.8197 


1.0147 


.014441 .98556 




15 


10 





.17865 


.82635 


5.7568 


.17633 


5.6713 


1.0154 


.015191 .98481 


80 







15 


.17794 


.82206 


6.6196 


.13083 


6.5301 


1.0162 


.01596 


.98404 




45 




80 


.18»4 


.81776 


5.4874 


.18534 


5.3955 


1.0170 


.01675 


.98326 




30 




45 


.18668 


.81848 


6.8612 


.18966 


5.2672 


1.0179 


.01755 


.98245 




15 


u 





.19081 


.80919 


5.2408 


.19488 


5.1446 


1.0187 


.01887 


.98168 


79 







15 


.19809 


.80491 


6.1258 


.19891 


6.0273 


1.0196 


.01021 


.9H079 




45 




80 


.19937 


.80063 


6.0158 


.20345 


4.9152 


1.0205 


.02008 


.97992 




.'JO 




45 


.:HB6I 


.79686 


4.9106 


.20800 


4.8077 


1.0214 


.02095 


.97905 




15 


IS 





.:a0791 


.79209 


4.8097 


.21236 


4.7046 


1.0223 


.02185 


.97815 


78 







15 


.81218 


.7878;J 


4.7180 


.21712 


4.6057 


1.0283 


.02877 


.977^-M 




45 




80 


.21644 


.78356 


4.0202 


.22169 


4.5107 


1.0243 


.02370 


.97630 




30 




45 


.2AV70 


.77900 


4.5311 


.22628 


4.4194 


1.0853 


.02466 


.97584 




15 


IS 





.turn 


.77506 


4.4454 


.23087 


4.8815 


1.0263 


.02563 


.974.37 


77 







15 


.229^ 


.77080 


4.3680 


.28547 


4.2468 


1.0273 


.02662 


.97838 




45 




80 


.£»45 


.76668 


4.2837 


.24008< 4.1651 


1.0284 


.02768 


.97287 




80 




45 


.tirm 


.78231 


4.2072 


•24470 


4.0667 


1.0295 


.02866 


.97184 




15 


U; 


.8419si 


.75808 


4.1336 


.24933 


4.0108 


1.0306 


.02970 


.97090 


76 







15 


.24619 


.75380 


4.0025 


.25397 


8.9:)7S 


1.0817 


.08077 


.96928 




45 




80 


.25088 


.749QJ 


8.9989 


.25862 


8.8667 


1.0829 


.06185 


.96815 




30 




45 


.2546Q 


.7454C 


3.9277 


.26328 8.7989 


1.0341 


.03296 


.95705 




16 


tk ' o 


.8S68il 


.741 IC 


8.8637 


.26795 8.782C 


1.0358 


.08407 


.96598 


76 

o 



M. 








1 


G»lM 


V«. Sis 


Smut. 


CoteB. 1 lanff. 


COMC 


Co-Vwt. 


Sloa. 



Vrom 76° to 90" vead flrom bottom of table npivards. 



160 



ItATHSltATICAL TABLIS. 



• 


M. 


Sloe. 


Co-Vm 


Cotee. 


TE«r 


C«tan. 


Secant. 


Vw. »n. 


Coda*. 






16 





.9669 


.74118 


8.8687 


.26705 


8.7880 


1.0358 


.08407 


.96503 


74 







16 


.90808 


.78607 


8.8018 


.27868 


8.6660 


1.0866 


.06681 


.90470 




45 




ao 


.967iM 


.7W.16 


8.7420 


.277Si 


8.6060 


1.0877 


.O06S7 


.00868 




SL 




46 


.87144 


.78856 


8.6840 


.f»m 


8.6457 


1.0800 


.06754 


.00240 




15 


16 





.97664 


.7^4486 


8.6280 


.28674 


3.4874 


1.0403 


.08874 


.06186 


74 







15 


.27068 


.79017 


8.6736 


.20147 


8.4308 


1.0416 


.08006 


.06006 




49 




SO 


.88402 


.71508 


8.5209 


.20621 


8.37Q0 


1.0429 


•04118 


.96688 




80 




45 


.itsm 


.71180 


8.4609 


.80006 


8.8»6 


1.0448 


.04243 


.05757 




15 


17 





.2W87 


.70763 


8.4208 


•80573 


8.2709 


1.0457 


.04370 


.06680 


7t 







15 


.2U6d4 


.70846 


8.8722 


.81051 


8.2205 


1.0471 


.04406 


.05602 




45 




ao 


.80070 


.69029 


8.8255 


.81530 


8.1716 


1.0485 


.04628 


.05878 




80 




•16 


.80486 


.69514 


8.2801 


.82010 


8.1240 


1.0600 


.04760 


.06840 




15 


18 





.aOMn2 


.69098 


8.2361 


.32402 


8.0777 


1.0615 


.04804 


.05100 


78 







15 


.81316 


.68684 


3.1982 


.82975 


8.0386 


1.0630 


.06080 


.04970 




45 




ao 


.81730 


.68270 


3.1515 


.83450 


8.9887 


1.0645 


.06168 


.04688 




80 




45 


.38144 


.07856 


8.1110 


.83045 


8.9159 


1.0560 


.05807 


.94608 




15 


lf» 





.82667 


.67448 


8.0715 


.34438 


8.0O42 


1.0676 


.05448 


.94558 


71 







15 


.iom 


.67081 


8.0331 


.84021 


8.8686 


1.0592 


.06601 


.04400 




45 




ao 


.83881 


.66619 


8.0957 


.35412 


8.8289 


1.060S 


.05786 


.04264 




30 




45 


.88792 


.U6208 


8.0593 


.85004 


8.786fi; 


1.0625 


.068R2 


.04118 




15 


80 





.8430-^ 


.66708 


2.9236 


■86897 


8.7475 


1.0642 


.06091 


.03800 


70 







15 


.84618 


.66888 


8.8802 


•36808 


8.7106 


1.0669 


.06181 


.03819 




45 




80 


.86081 


.64979 


2.8554 


-87888 


8.6746 


1.0676 


.06388 


.08667 




30 




45 


.86429 


.64671 


8.8225 


.87887 


8.6305 


1.0694 


.06486 


.93614 




15 


81 





.85887 


.64168 


8.7904 


.38886 


8.6051 


1.0711 


.06642 


.03868 


•9 







15 


.36244 


.68756 


2.7591 


.88888 


8.6715 


1.0729 


.06700 


.OSMl 




45 




ao 


.36650 


.08360 


2.7285 


.39891 


8.5886 


1.0748 


.06068 


.08042 




80 




45 


.37056 


.62944 


2.6986 


.89696 


8.5065 


1.0766 


.07110 


.92881 




15 


82 





.87461 


.62589 


2.6695 


.40408 


8.47C] 


1.0785 


.0ri68 


.02718 


68 







15 


.8786ft 


.00135 


8.6410 


.40911 


84448 


1.0804 


.07446 


.08564 




46 




ao 


.88268 


.61782 


8.6181 


.41421 


8.4142 


1.0624 


.07612 


.02888 




80 




46 


.8867: 


.61829 


2.5869 


.41988 


88847 


1.0644 


.07780 


.08220 




15 


88 





.89078 


.60027 


8.6593 


.42447 


88559 


1.0664 


.07050 


.08060 


67 







15 


.89474 


.60526 


8.5383 


•42968 


8.8276 


1.0684 


.08121 


.91879 




46 




80 


.89875 


60185 


8.6078 


.43481 


88008 


1.0904 


.08804 


•91706 




30 




45 


.40275 


.59725 


8.4829 


.44001 


28727 


1.0995 


.06469 


•91581 




15 


84 





.40674 


.69326 


8.4586 


.44528 


82460 


1.0046 


.06645 


.91856 


66 







15 


.41072 


.68028 


8.4348 


.46047 


8.2109 


1.0968 


.08834 


.91176 




45 




80 


.41469 


.6853] 


2.4114 


.45678 


2.1943 


1.0989 


•00004 


.90906 




80 




46 


41866 


.68184 


8.3886 


.46101 


8.1692 


1.1011 


.00186 


.00614 




15 


85 





.42262 


.67788 


8.8662 


.46631 


8.1445 


1.1034 


.09369 


.90681 


66 





• 15 


.42667 


.67848 


8.8443 


•47)68 


2.1203 


1.1066 


.09564 


.00446 




45 


. 80 


.48061 


.66949 


8.8228 


.47607 


80965 


1.1079 


.09741 


.00260 




80 




46 


.48445 


.66565 


8.8018 


.48234 


8.0782 


1.1102 


.09930 


.90070 




15 


8G 





.43837 


.5616:) 


2.2812 


.48778 


8.0603 


1.1120 


.10121 


.80579 


64 







15 


.44-229 


.66771 


8.2610 


.40814 


8.0278 


1.11.W 


.10813 


.89087 




45 




30 


.44620 


.55380 


8.2412 


.40658 


8.0U67 


1.117^ 


.10607 


.69498 




SG 




45 


.45010 


.54990 


8.2217 


•60404 


1.9640 


1.1198 


.10702 


.80298 




16 


87 





.4.'>399 


.54601 


8.2027 


.50052 


1.9026 


1.1823 


.10890 


.89101 


68 







15 


,45787 


.64218 


8.1840 


.61608 


1.94ri 


1.1848 


.11098 


.88002 




45 




30 


.46175 


.58825 


2.IC67 


.69067 


1.9210 


1.1971 


.11299 


.88701 




80 




45 


.46561 


.53439 


8.1477 


.52612 


1.9007 


1.1800 


.11501 


.88499 




15 


8S 





.46947 


.58058 


8.1800 


.58171 


1.8807 


1.1326 


.11705 


.88896 


68 







15 


.47332 


.5206R 


2.1127 


.53782 


1.8611 


l.l85-.i 


.11911 


.88089 




45 




80 


.47716 


.52284 


2.0957 


.64205 


1.8418 


1.1379 


.12118 


.87882 




80 




45 


.48009 


.51901 


8.0790 


.54862 


1.8228 


1.1400 


.12327 


.87678 




15 


80 





.48481 


.51519 


8.0627 


.55431 


1.8040 


1.1433 


.12538 


.87468 


61 


6 




1ft 


.48862 


.6113« 


2.0466 


.56003 


1.7866 


1.1461 


.12750 


.87850 




45 




ao 


.49242 


.60758 


8.0808 


.66577 


1.7675 


1.1490 


.18964 


.87036 




80 




45 


.49622 


.60876 


8.01.^8 


.67165 


1.7496 


1.1518 


.13180 


.86820 




16 


JIO 





.50000 


.50000 


3.0000 


.57785 


1.7820 


1.1547 


.18897 


.86608 


JM 


_0 






C««n«. 


V«r. Sio. 


i^Mt. 


CoUn 


Twr. 


Come. 


G>.Vm. 


8««. 


o 


M. 



From 60" to 75"* read trom. bottom of table npirards* 



NATURAL TRIGONOMETRICAL FUNCTIONS. 



161 



e 


M. 




SlM. 


Oh\tr». 


COMC 


T»g. 


1 1 
Cotaa. ; BmuI. Vrr. Sin. 


Coaisc 






SO 


.60000 


.50000 


8.0000 


.57785 


1.7890' 1.1647 


.13897 


.86603 


00 







15 


.60877 


.49628 


1.9860 


.58818 


1.7147; 1.1576 


.18616 


.86884 




45 




ao 


.60754 


.40^6 


1.9703 


.68904 


1.89771 1.1606 


.18837 


.86168 




80 




45 


.511S9 


.48671 


1.9668 


.69494 


1.8606 1.10S6 


.14060 


.86M1 




15 


SI 





.61504 




1.9416 


.60066 


1.6643 


1.1666 


.14388 


.86n7 


59 







15 


.61877 


!481S8 


1.9276 


.60681 


1.6479 


1.1807 


.14509 


.86491 




45 




80 


.&9«0 


.47750 


1.9189 


.61280 


1.6819 


1.1788 


.14786 


.85264 




80 




45 


.5^621 


.47879 


1.9004 


.61888 


1.6160 


1.1780 


.14966 


.86035 




15 


ss 





.5»08 


.47008 


1.8871 


.69487 


1.6008 


1.1792 


.15195 


.84805 


68 







15 


.53861 


.46689 


1.8740 


.68096 


1.6849 


1.1824 


.15437 


.84578 




♦45 




SO 


.58780 


.46S70 


1.8618 


.68707 


1.5697 


1.1867 


.15661 


.81889 




80 




45 


.54007 


4600S 


1.8485 


.61882 


1.6547 


1.1800 


.16696 


.84104 




16 


ss 





.54464 


!45586 


1.8361 


.64941 


1.S399 


1 1924 


.16188 


.88867 


67 







15 


.54820 


.45171 


1.8288 


.66503 


1.5253 


1.1958 


.16371 


.88629 




45 




SO 


.55194 


.44806 


1.8118 


.66186 


1.6106 


1.1992 


.16611 


.88889 




80 




45 


.66657 


.44448 


1.7909 


.66818 


1.4966 


1.9027 


.16858 


.88147 




15 


S4 





.66919 


.44061 


1.7883 


.67461 


1.4886 


1.9068 


.17096 


.82904 


66 







15 


.5«a» 


.48») 


1.7768 


.68087 


1.4687 


1.9098 


.17341 


.83659 




45 




80 


.'66511 


.48859 


1.7665 


.68728 


1.4660 


1.3184 


.17587 


.82418 




80 




45 


.57000 


.48000 


1.7544 


.69S7V 


1.4415 


1.2171 


.17835 


.82165 




15 


S6 





.n8&8 


.4»48 


1.7434 


.70031 


1.4281 


1.8»6 


.180S) 


.81916 


66 







15 


.67715 


.42886 


1.7337 


.70673 


1.4150 


1.8245 


.18888 


.81664 




45 




80 


.68070 


.41980 


t.TWO 


.71320 


1.4019 


1.3288 


.18588 


.81413 




80 




45 


.68435 


.41575 


X.71J6 


.71990 


1.8891 


1.8322 


.18848 


.61157 




15 


S6 





.68779 


.41231 


1.7013 


.72664 


1.8764 


1.2361 


.19098 


.80903 


64 







15 


.60181 


.40669 


1.6912 


.78323 


1.3638 


1.2400 


.19356 


.80644 




45 




SO 


.60482 


.40518 


1.6812 


.73996 


1.8514 


1.2440 


.19614 


.80366 




80 




4^ 


.60682 


.40168 


1.6713 


.74673 


1.8892 


1.2480 


.19876 


.80126 




16 


S7 





.60181 


.39619 


1.6616 


.Teaw 


1.8270 


1.2521 


.90136 


.79664 


68 







15 


.60629 


.89471 


1.6021 


.78042 


1.8161 


1.2568 


.20400 


,79600 




45 




ao 


.00876 


.39m 


1.6427 


.76783 


1.8082 


1.2605 


.90665 


.79886 




80 




45 


.^ifta 


.38778 


1.6884 


.77428 


1.2916 


1.2647 


.20981 


.79069 




15 


S8 





.61566 


.884:^ 


1.6248 


.78129 


1.2799 


1.2690 


.21199 


.78801 


69 







15 


.61909 


.86091 


1.6153 


.78884 


1.2685 


1.2734 


.21468 


.78532 




45 




80 


.88251 


.87749 


1.6064 


.79543 


1.2572 


1.2778 


.21^739 


.78261 




8?^ 




45 


.6i59d 


.87403 


1.697G 


.802J58 


1.2460 


1.2822 


.22012 


.77988 




15 


S9 





.8si9a? 


.37068 


1.689D 


.80978 


1.2349 


1.2868 


«>Qgg5 


.77715 


61 







15 


.63271 


.38729 


1.6805 


.8170:^ 


1.2239 


1.2913 


122561 


.77489 




45 




80 


.64608 


.8639-J 


1.672i 


.82431 


1.2181 


1.2960 


.92688 


.77168 




80 




45 


.6:3944 


.360o6 


1.6639 


.83169 


1.9024 


1.8007 


.38116 


.76884 




15 


40 





.64279 


.85721 


1.8557 


.KIRBIO 


1.1918 


1.3054 


.88396 


.76601 


60 







15 


.64619 


.853S8 


1.6477 


.84656 


1.1812 


1.3102 


.88677 


.76328 




45 




30 


.64945 


.85056 


1.5398 


.85406 


1.1708 


1.3161 


.28969 


.76041 




80 




45 


.69876 


.34724 


1.5820 


.86165 


1.1606 


1.8200 


.24244 


.75756 




15 


41 





.65606 


.84)94 


1.6242 


.86929 


1.1504 


1.8250 


.24529 


.76471 


49 







15 


.6582)6 


.34065 


1.5166 


.87008 


1.1408 


1.3301 


.34816 


.75184 




45 




80 


.66«i 


.88738 


1 6092 


.88472 


1.1806 


1.8358 


.25104 


.74896 




80 




45 


.66588 


.38412 


1.6018 


.89258 


1.1201 


1.8404 


.35394 


.74606 




16 


42 





.66918 


.33087 


1.4945 


.90040 


1.1106 


1.84.W 


.25686 


.74314 


48 







15 


.67«7 


.82763 


1.4878 


.90884 


1.1009 


1.3509 


.25978 


.74022 




45 




80 


.07359 


.82441 


1.4808 


.9168:J 


1.0013 


i.sm 


.96272 


.78728 




80 




45 


.67880 


.88120 


1.4732 


.92489 


1.0818 


1.3618 


.96568 


.78482 




15 


4S 





.09900 


.81800 


1.4668 


93251 


1.07*4 


1.8673 


.26865 


.78135 


47 







15 


.68518 


.31482 


1.4.595 


.94071 


1.0630 


1.3?-J» 


.27163 


.?«37 




46 




80 


.68835 


.81165 


1.4597 


.94896 


1.0588 


1.8786 


.27468 


.72587 




80 




45 


.69151 


.80849 


1 .4461 


.95729 


1.0446 


1.8848 


.27764 


.72236 




15 


44 





.69466 


.30684 


1.48te 


.96569 


1.08M 


1.8908 


.28066 


.71984 


46 







16 


.69779 


.80221 


1.4831 


.97416 


1.026r) 


1.8961 


.28370 


.71630 




45 




80 


.70091 


.8990G 


1.4267 


.98270 


1.0176 


1.4020 


.28675 


.71825 




30 




45 


.70401 


.29509 


1.4204 


.99131 


1.U0S8 


1.4081 


.28981 


.71019 




15 


jW 





.wni 


.29289 


1.4142 


1.000(j' 1.0000 


1.414;.' 


.29289 


.70711 


45 

o 









Coda*. 


Ver.Sia. 


Soomt. 


CoUn. 


Tmjf. 


CoMC 


Co-Vm, 


Sine. 


M. 



From 45° to 60« read ft*om bottom oftable upwardii... 



162 



MATHEMATICAL TABLES, 
I^OGARlTBiniC 8INBS, ETC. 



Iii.Neg. Inflnite. 
8.94186 11.76814 
8.64888 11.45718 
8.71880 11.88120 
8.84858 11.16648 



8.94080 

0.01983 

9. 

9.14856 

9.19488 

9.88967 
9.28060 
9.81788 
9.85809 
9.38868 

9.41300 
9.440S4 
9.46694 
9.48996 
9.61:;»4 

9.58406 
9.55488 
9.5786H 
9 A91H8 
9.60981 

9.64184 
9.66?m 
9.67161 
9.68567 

9.69897 
9.71184 
9.78481 
9.73611 
9.74766 

9.75859 

9.77946 
9.7Rltt4 
9.79887 

9.80807 
9.81694 
9.8;2661 
9.88878 
9.84177 

9.84949 



Cosine. 



11.06970 
10.98077 
10.91411 
10.85644 
10.8056^ 

10.76088 
10.71940 
10.68812 
10.64791 
10.61638 



Yonln. 



Tugent. 



In.Ne«:. 
G.18^1 
6.78474 
7.18687 
7.88667 

7.68089 
7.78H63 
7.87888 



8.09088 

8.18162 
8.26418 
8.83950 
8.40675 
8.47288 



10.58700 8.68848 

10.559ri6 8.58814 

10.68406 8.64048 

10.510 8 8.08969 

10.48736 8.736;!5 



10.40695 
10.44567 
10.4264:j 
10.40812 
10. 



8.78037 
8.82280 
8.86223 
8.90034 
8.93U79 



10 87405 8.97170 

lo.a^sie 9.00581 

10.84295! 9.06740 
10..S2889 9.06888 
10.81448 9.09883 



10.80106 

10.88816 

10.27679 

10.26; 

10.85844 



9.18708 
9.16483 
9.18171 
9.90771 
9.93290 



10.94141 9.96781 

10.28078 9.88099 

10.88064 9.80398 

10.81066 9.38681 

10.20113 9.84802 



10.19193 
10.18806 
10.17449 
10.16688 



10.15838 9.44818 



10.15052 



9.36918 
9.38968 
9.40969 

9.42918 



9.46671 



In. Nee. 
8.94192 
8.54308 
8.71940 
8.84464 

8.94195 
9.02168 
9.08914 
9.14780 
9.19971 

9.94638 
9 88865 
9.32747 
9.86886 
9.89677 

9.48805 
9.467.'i0 
0.48584 
9.51178 
9.&3697 

9.56107 
9.5&118 
9.60641 
9.627»5 
9.64858 

9.6C867 
9.r881R 
9.707r 
9.7256^ 
9.74375 

9.76144 
9.77877 
9.79579 
9.81952 
9.88899 

9.84583 
9.86126 
9.8771! 
9.89881 
9.908S7 

9.98381 
9.9.H916 
9.95444 
9.9f)96() 
9.96484 

10.00000 



InfiDite. 
11.75808 
11 45692 
11.88060 
11.10586 



Secant. Corora. Cotan. Tangent Yersln. 



Cotan. 



11 

10.97838 

10.91086 

10.85280 

10.80029 



10 

10.71135 

I0.672.'i8 

10.68664 

10 



10.57195 
I0.54;.»r>0 
10.5146G 
10.4t«22 
10.46:303 

10.43^93 

10.41.582 

10.89! 

10.87215 

10.85142 

lO.SSl&i 
10.31182 
10.8928:) 
10.274.3:3 
10.25625 

10.83866 
10.82123 
10.20481 
10.18748 
10.17101 

10.15477 
10.i;«74 
10.12289 
10.10713 
10.00163 

10.07619 
10.0C0S4 
10.04•^^6 
10.03aS4 
10.01516 

lO.OOOOO 



Coven. 



10.00000 
9.99285 
9.9K457 
\).97665 
9.96860 

9.90040 
9.95205 
9.94366 
9.93492 
9.98618 

9.91717 
9.90805 
9.89877 
0.88983 
9.87971 

9.8G998 
9.a'J996 
9.84061 
9.83947 
9.82891 

9.81881 
9.80729 
9.79615 
9.78481 
9.77885 

9.7614f 
9.74945 
9.73720 
9.72471 
9.71197 

9.69897 
9.68571 
9.67217 
9.65836 
9.64485 

962964 
9.61.512 
9.6(XI0H 
9.. 58471 
9.66900 

9.5.5208 
9 5:3648 
9.51966 
9.60848 
9.48479 

9 46671 



10.00000 
10.00007 
10.00086 
10.00060 
10.09106 

10.00166 
10.00839 
10.00895 
10.00426 
10.00588 

10.00665 
10.00805 
10.00960 
10.01186 
10.01310 

10.01506 
10.01716 
10.01940 
10.08179 
10.0S438 

10.02701 
10.08065 
10.03288 
10.03.597 
10.03987 

10.04879 
10.04684 
I0.0&018 
10.05407 
10.05818 

10.06947 
10 06693 
10.07158 
10.07641 
10.06148 

10.08664 
10.09804 
IO.01I765 
10.10347 
10.10960 

10.11575 
10.12829 
10.12898 
10.1.3587 
10.14807 

10.1.5062 



Coeeo. 



Dee. 



10.00000 90 
9.09998 80 
9.1*99741 88 
9.99940' 87 
9.90894. 86 

9.908341 a*) 

0.99761! 84 

9.99675f 83 

9.9057r)| H8 

9.90468 81 

9.998351 80 
9.99105 79 
0.99040) 7B 
9.988?2, 77 
0.9h690 76 



9.9B494 
9.982841 
0.98060i 
9.9:«2I 
9.97667! 

0.97899' 
0.97T)I5I 
0.06717 
0.06408 
0.06078 

0.067881 
0.05866 
0.949W, 
9.94593 
9.94182< 



70 



65 
64 
6S 
09 
61 



9.937581 CO 

9.93807! f>9 

9.92848 58 

9.92359 .57 

9.91857. £6 

9 91836' 55 
9.90796 .^4 
9.90-^:35 ?A 
9.WI6.VJ b'i 
9.89050 51 

9.88486' 50 

9 87778 49 

9.87107, 48 

9.86418 47 

9.85693 46 

9.84040 4!S 



From 45<> to 90» read IVoiti boUom of table apwards. 



SPECIFIC GRAVITY. 



168 



MATEBIAIiS. 



THK CHEniCAIj BliElHFNTS. 

Xha Common Elements (43). 



^1 


Name. 


il 


ll 


Name. 


S if 


it 


Name. 


II 


t^ 




<^ 


ga 




<^ 


g^ 




<^ 


Al 


Aluminum 


87.1 


P 


Fluorine 


10. 


Pd 


Palladium 


106. 


Sb 


Antimony 


l:».4 


Au 


Gold 


107.8 


P 


Phosphorus 


SI. 


As 


Arsenic 


75.1 


H 


Hydrogen 


1.01 


Pt 


PlaUnum 


ie«.o 


Ba 


Barium 


187.4 


I 


Iodine 


1S6.8 


K 


Potassium 


30.1 


Bi 


Bi«muth 


S09.1 


Ir 


Iridium 


193.1 


Si 


Silicon 


88.4 


B 


Boron 


10.9 


Fe 


Iron 


56. 


Ag 


SUver 


107.0 


Br 


Broujine 


79.0 


Pb 


Lead 


S206.9 


Ni 


Sodium 


88. 


Cd 


Cadmium 


111.9 


U 


Lithium 


7.08 


Sr 


Strontium 


87.6 


Ca 


Calcium 


40.1 


Mg 


Magnesium 


24.3 


S 


Sulphur 


98.1 


C 


Carbon 


Vi. 


Mn 


Manganese 


55. 


Sn 


Tin 


lie. 


CI 


Chlorine 


85.4 


^ 


Mei-cury 


200. 


Tl 


Titanium 


48.1 


C» 


Chromium 


6S.I 


Nickel 


68.7 


W 


Tungsten 


184.8 


Co 


Cobalt 


59. 


N 


Nitrogen 


14. 


Va 


Vanadium 


61.4 


Cu 


Copper 


63.6 


O 


Oxygen 


10. 


Zn 


Ziuc 


66.4 



The atomic weights of many of the elements vary In the decimal place aa 
in'ven by different authorities. The above are the most recent valuee re- 
fctrtsd to O = 16 and U s 1.008. When H is taken as 1, O = 16.879, and Uie 
other fljrures are diminished proportionately. (See Jour. Ant. Chem, Soc.% 
liarch, 1806.) 



The Rare Element* (S7)« 



Berjiliuiii, Be. 
Cseidum, Ob. 
C»»rium, Ce. 
Didymium, D. 
Erbium, E. 
(;alUum, Oa. 
(iermaoium, Ge. 



Qluclnum, G. 
Indium, In. 
Lanthanum, La. 
Molybdenum, Mo. 
Niobium, Nb. 
Osmium, Os. 
Bhodium, R. 



Rubidium, Rb. 
Ruthenium, Ru. 
Samarium, Sm. 
Scandium, Sc. 
Selenium, Se. 
Tantalum, Ta. 
Tellurium, Te. 



Tliallium, Tl. 
Thorium, Th. 
Uranium, U. 
Ytterbium, Yr. 
Yttrium, Y. 
Zirconium, Zr. 



SPECIFIC 6RATITT. 



The specific gravity of a substance is its weight as compared with the 
vHsfat of an equal bulk of pure water. 
To And. the ■peclile sraTtty; of a ■nbstanee. 

W s weight of body in air; w = weight of body submerged in water. 

w 

Specific gravity = ^_^ . 

If the substance be lighter than the water, sink it by means of a heavier 
Hibstaace, and deduct the weight of the heavier substance. 

Specific-gravity determinations ai*e usually referred to the standard of the 
wdght of water at 68« F., 68.365 lbs. per cubic foot. Some experiineuters 
have used 60<* F. as the standard, and others dZ^ and 89. 1<* F. There is no 
pcneral agreement. 

Given sp. gr. referred to water at 89.1° F., to reduce it to the standard of 
ei* F. muTuply it by l.OOlld. 

Given sp. gr. referred to water at 62*» F.. to find weight per cubic foot mul- 
tiply by 02.K6. Given weight per cubic foot, to find sp. gr. multiply by 
0J)16087. Given sp. gr., to find weight per cubic inch multiply by .036065. 



164 



MATERIALS. 



fVelgbt and Specific GraTlty of fflleUil*. 



Aluminum 

Antimony 

Bismuth 

Brass: Copper + Zinc ^, 

TO 80 \ 

to 40 

50 60 J 

n^^^m^ i Copper, 85 to 80 1 

BrouieJTii*^' 6to20f 

Cfulmium 

CfUcium 

Chromium 

Cobalt 

Gold, pure 

Copper 

Iridium.....* .. 

Iron, Caat 

** Wrought. 

Lead. . ....... 

Manganese 

Magnesium 



i 9i* 

i 60< 

212< 



Mercury -j 60« 



Nickel 

Platinum.., 
Potassium. 

Silver 

Sodium — 

Steel 

Tin 

Titanium... 
Tungsten.., 
Zinc 



Specific Gravity. 
Range accord- 
ing to 
several 
Authorities. 



2.56 to 2.71 
6.M to 6.86 
9.74 to 9.90 



7.8 to 8.6 



8.52 to S.96 

8.6 to 8.7 
1.58 
6.0 
8.5 to 8.6 
19.245 to 19.861 
8.69 to 8.98 
to 28. 
to 7.48 
to 7.9 
to 11.44 
to 8. 
to 1.75 
to 18.62 
13..58 
to 13.38 
8.-.iT9 to 8.93 
80.88 to 82.07 

0.865 
10.474 to 10.511 

0.97 
7.69* to 7.932^ 
7.291 to 7.409 

6.8 
17. to 17.6 
6.86 to 7.20 



28.88 
C.85 
7.4 

11.07 
7. 
1.69 

18.60 

18.87 



Specific Grav- 
ity. Approx. 
Mean Value, 

used in 

Calculation of 

Weight. 


Weight 

per 
Cubic 


Weight 

Cubic 


Foot, 
lbs. 


Inch, 
lbs. 


2.67 


166.5 


.0963 


6.76 


421.6 


.2139 


9.82 


612.4 


.3M4 




[8.60 


686.8 


.3108 




8.40 


628.8 


.3031 




8.86 


681.8 


.301? 




L8.80 


611.4 


.8950 


8.868 


658. 


.8195 


8.65 


539. 


.8121 


19.858 


1200.9 


.6049 


8.858 


558. 


.31% 




1896. 


.8076 


7.218 


4.V). 


.!200t 


7.70 


480. 


.2779 


11.88 


709.7 


.4106 


8. 


499. 


.2aS7 


1.75 


109. 


.0641 


13.62 


849.8 


.4915 


13.58 


846.8 


.4900 


18.38 


834.4 


.4838 


8.8 


548.7 


.3175 


81.5 


1347.0 


.7758 


10.506 


655.1 


.3791 


7.864 


489.6 


.2834 


7.350 


458.8 


.»JC52 




7.00 


486.5 


.2536 



* Hard and burned. 

t Vet'y pure and soft. The sp. gr. decreases as the carbon is increased. 

In the ni-st column of figures the lowest are usually those ot cast metals, 
which are more or less porous; the highest are of metals finely rollecl or 
drawn Into wire. 

Specific GraTlty ot I^lquld* at eo*" F. 



Acid, Muriatic 1.200 

»• Nitric 1.217 

** Sulphuric 1.849 

Alcohol, pure 794 

** 96 per cent 816 

** 50 '• " 934 

Ammonia, 87.9 per cent 891 

Bromine 2.97 

Carbon disulphide 1 .26 

Ether, Sulphuric .72 

Oll.Llnseed 94 

ComprcMlon of tbe followlns Fluid* under a Fressnre of 
15 Iba. per Square Inch. 

Water 0000466:^ I Kther 00006158 

Alcohol 0000216 (Mercury 00000865 



Oil,01ive 93 

** Palm 97 

*' Pftroleum 78 to .88 

** Rape 92 

•* Turpentine 87 

*' Whale 98 

Tar 1. 

Vinegar 1.08 

Warer 1. 

*' sea 1.026tol.03 



SPECIFIC GRAVITY. 



165 



The Hydrometer. 

The hydrometer is an instrument for dcterminiDg the density of liquids. 
It is usually made of glass, and consfKts of three parts: (1) the upper part, 
a gradoftted stem or fine tube of uniforni diameter; (2) a bulb, or enlarge- 
m»*nt of the tube, containing air ; and (3) a small bulb at the bottom, con- 
raining shot or mercury which causes the instrument to float in a vertical 
position. The graduations are figures representinrr either specific gravities, 
or the numbers of an arbitrary scale, as in Baum6's, Ti^addell's, Beck's, 
and other hydrometers. 

There is a tendency to discard all hydrometers with arbitrary scales and 
u> use only those which read in terms of the specific gravity directly. 

Bmune^e Hydrometer and Specific Gravities Compared. 

Liquids' 
Lighter 

than 
Water, 
sp.gr. 



li 


Uquida 


Uquids 


Heavier 
than 


Lighter 
than 


>c 


Water, 


Water, 


ar 


sp. KT. 


sp.gr. 





1.000 
1.007 
1.018 
l.OSSO 




T 




? 




8 




4 


1.027 
1.0S4 
1.041 




•» 




6 




~ 


1.048 
1.056 
1.06S 




8 




9 




10 


1.070 


1.000 


11 


1.07« 


.993 


13 


1.085 


.986 


n 


l.OM 


.980 


14 


1.101 


.973 


:5 


1.100 


.967 


1*5 


1.118 


.960 


17 


1.126 


.9M 


IS 


1.131 


.W8 



II 


Liquids 
Heavier 

than 
Water, 
Rp. gr. 


19 


1.148 


20 


1.1 W 


ii 


1.160 


22 


1.169 


5J8 


1.178 


94 


1.188 


25 


1.197 


26 


1.206 


27 


1.216 


28 


1.226 


29 


1.8:^6 


30 


1.246 


81 


1.256 


32 


1.267 


33 


1.277 


34 


1.288 


&5 


1.299 


86 


1.310 


87 


1.822 



Liquids 


93 . 


Liquids 


Lighter 
tlan 




Heavier 


than 


Water, 


Is 


Water, 


»P. JfT. 


ep.RT. 


.912 


88 


1.333 


.936 


89 


1.345 


.930 


40 


1.357 


.924 


41 


1.369 


.918 


42 


1.382 


.918 


44 


1.407 


.907 


46 


1.484 


.901 


48 


1.462 


.896 


50 


1.490 


.890 


62 


1.520 


.885 


64 


1.551 


.880 


56 


1.583 


.874 


58 


1.617 


.869 


60 


1.652 


.864 


65 


1.747 


.859 


70 


1.854 


.854 


75 


1.974 


.849 


76 


2.000 


.844 







839 
881 
810 
826 
8M 
811 
802 
794 
785 

768 
760 
7M 
745 



SpeclAc GraTlty and UTelclit or UTood. 









W«lKht 








Welffhl 




Spcdflc Gravity. 


C?5c 




Sp«clfic Gnrlty. 


Cubic 








*r' 








is?- 


AMer 


Avge. 
0.56 to 0.80 768 


42 


Hornbeam. . . 


Avge. 
.76 !76 


47 


Apple 


.73 to .79 


.76 


47 


Juniper 

Larch 


.56 


.56 


85 


-U 


.6010 .84 


.72 


45 


.56 


.56 


35 


Bamboo.. .. 


.31 to .40 


.85 


88 


Lignum vitie 


.65 to 1.33 


1.00 


62 


fcrwh 


.62 to .85 


.78 


46 


Linden 


.604 




87 


B:rch 


.58 to .74 


.65 


41 


Locust 


.738 




40 


b-x, 


.91 to 1.83 


1.12 


70 


Mahogany... 


.56 to 1.06 


.81 


SI 


C<*lar 


.49 to .75 


.62 


80 


Maple 

Mulberry.... 


.57 to .79 


.68 


42 


'.berry 


.61 to .72 


.66 


41 


.56 to .90 


.73 


46 


' *:««tnut . . 


.46 to .66 


.56 


85 


Oak, Live... 


.96 to 1.28 


1.11 


69 


'I'tt 


.24 


.24 


15 


" White.. 


.69 to .86 


.< < 


48 


'Tpress.... 


.41 to .66 


.M 


88 


" Red... 


.73 to .75 


.74 


46 


I^-^wood . . . 


.76 


.7C 


47 


Pine, White.. 


.35 to .55 


.45 


88 


E'^oy 


1.13 to 1.83 


i.ai 


76 


*• Yellow. 


.46 to .76 


.61 


38 


am.' 


.65 to .78 


.61 


88 


Poplar 


.38 to .58 


.48 


30 


» 


.48 to .70 


.59 


87 


Spruce, 


.40 to .50 


.4.') 


28 


'»t.ai 


.8410 1.00 


.92 


67 


Sycamoi-e.... 


.59 to .62 


.60 


87 


HacJnnatock 


.50 

.86 to .41 


.59 
.38 


87 
34 


Teak 


.66 to .98 
.50 to .67 


.82 

.58 


51 


a-mlock .. 


Walnut 


86 


r-ckory 


.69 to .94 


.77 


48 


Willow 


.49 to .59 


.54 


84 


H'llr .... 


.76 


.76 


47 











-^ 



166 



MATERIALS. 



urelffbt and Specific GraTlty of Stones, Brick, 
Cement, etc. 



Asphalium 

Brick, Soft 

•* Common 

" Hard 

" PreHsed 

•♦ Fire 

Brickwork In mortar 

•* " cement 

Cement, Rosendale, loose 
*' Portland, ** 

Clay 

Concrete , 

Earth, Ktose 

rammed 

Emery , 

GIosA 

♦' flint , 

Qneiss I 

GraniteJ 

Gravel , 

Ghrpsum 

Hornblende 

Lime, quick, In bulk 

Limestone 

Magnesia, Carbonate 

Marble 

Masonry, dry rubble 

•• dressed 

Mortar 

Pitch 

Plaster of Paris 

Quartz 

Sand 

Sandstone 

Slate 

Stone, various 

Trap 

Tile 

Soapstone 



Pound K Iter 
Cubic Foot. 



87 
100 
113 
]2r> 
185 
HO to 
100 
IIS 

60 

78 

l^to 
1:20 to 

72 to 

00 to 
260 
IWto 
180 10 



150 



160 
140 
80 
110 

178 
196 



100 to 170 



100 to 
i;*)to 
2()(Uo 
Wto 
ITO to 
150 
l«50tO 
140 to 
140 to 

floto 

72 

74 to 
165 

90 to 
140 to 
170 to 
1»5 to 
170 to 

no to 

166 to 



190 
150 
tSJO 
56 
•JOO 

180 
liX) 
180 
100 

80 

110 
150 
180 
WO 
iiOO 
120 
173 



Spec'iftc 
Gravity. 



1.80 
1.6 
1.79 
2.0 
2.16 

2.34 to 2.4 
1.6 
1.70 
.96 
1.35 

1 .92 to 2.4 

1.93 to 2.24 
1.16 to 1.28 
1.44 to 1.76 
4. 

3.5 to 2.75 
2.88 to 3.14 

2.56 to 2.7S 

1.6 tol.9^ 
2.aSto2.4 
8.2 to8..^^ 

.8 to .S8 
2.?2 to 3.a 
2.4 

2.56to2.R8 
3.24 to 2.56 
3.34 to 2.88 
1.44 to 1.6 
1.15 

1.18tol.28 
2.64 

1.44 to 1.76 
2.34 to 2.4 
2.72toS.68 
2.10 to 3.4 
2.7^3 to 3.4 
1.76 to 1.98 
3.65 to 2.8 



SpeelAo Gravity and IVelsht of Gases at Atmosp]iert« 
Pressure and 32° Fo 

(For other temperatures and pressures see pp. 459, 479.) 



Air 

Oxyjfen 

H^drop:en 

Nitrogen 

Carbonic oxide, CO 

Carbonic acid, C()« 

Marsh gns. methane, Cn4 
Ethylene, C^ H4 



Density, 
Air = 1. 



1.0(KTO 
1.1051 
0.0693 
0.9714 
0.9074 
l.r.3'.K) 
0..5.560 
0.9SI7 



Q PR mm OS 
per Uire. 



1.0931 

1 . 1390 

0.0S9S7 

1.35(51 

1.351 

1.9:7 

o.;i9 

1.373 



libs, per 
Cu. FU 



0.0S0738 

0.08931 

0.00561 

0.07812 

0.07810 

0.12.J43 

0.04488 

0.07949 



Cubic Ft. 
per Lb. 



12.387 
ll.30y 
1*^.23 
13.752 
12.801 
8.103 
83.301 
12.&80 



PROPERTIES OF THK tJBBFUL 305TAtS. 167 

PROPBRTIB8 OF THB USBFITIi MBTAIiS. 

Almlnuiii, AI.— Atomic wel^rht 27.1. Sprciflo Kravity 2.0 to 2.7. 
Th«* Ufrhtt^t of all the usef al metals except mafirneiiiiiin. A soft, ductile, 
mall«'abl« metal, of a white color, approachiDft Bilver, but wiUi a bluii^ cast. 
Very ooa-corroeiTe. Tenacity about one third that of wroueht-Iron. For- 
meriy a ntre metal, but since 1H90 its production and une hnve greatlv in- 
creased on account nf the discovery of cheap proces8«'B for reduciufr It from 
the ore. Melts at about 1160<* F. For further description see Aluminum, 
vnAfr Strength of Materials^. 

Antlmonir (Stibium), 8b.- At. wt. 120.4. gp. gr. 6.7 to 6.8. A brittle 
nwtal of a bluish-white color ana highly crystalline or laminated structure. 
Iff Its at e4a° F. Heated in the open air It bums with a blulsli-whlte flame. 
Its chief use is for the manufacture of certain allovs, as type-metal (anti- 
mony ], lead 4), britannia (antimony I, tin 9), ana various anti-fricrion 
metHlft (ffiee Alloys). Cubical expansion by heat from 2B? to 212? F., 0.0070. 
8fw>^fk: heat .OAO. 

Bfnnatli, BI.— At. wt. 206.1. Bismuth Is of a peculiar Ii)?bt reddish 
color, highly cryBtalline. and so brittle that it can readily be pulverized It 
n>elti$ at &10* F.. and boils at about 2^i00^ F. Sp. gr. 9.8S3 at 54<> P., and 
I0.eS5 jnst above the meltincr-point. Specific heat about .0301 at ordinanr 
temperatures. Coelllcient of cubical expansion from 8S* to 2I'-»®, 0040. Con- 
du<-tiv1tv for heat about l/HQ and for electricity only about I/K) of that of 
»lr(>r. Ita tensile strenf^th iH about G400 lljs. per square ir)ch. Bismuth ex* 
piipds in cooling, and Tribe has shown that this expansion does not take 
iilAce until after eolidification. Bismuth is the most diamagneiic element 
kiiowD, a sphere of ic Ixding repelled by a mnernet. 

CmdmilEiil, €d.— At. wt. lis. Sp. gr. 8.6 to 8.7. A bluish- white metal, 
lusirtfus, with a fibrous fracture. Melts below 500° F. an<l volatilizes at 
aUiut mo^ F. It is used as an ingredient in some fusible alloys with lead, 
tin. and • ismutb. Cubical expansion from S^*' to 2Vi'* F., 0.0004. 

Copper, On*— At. wt. (HVi. Sp. gr. 8.81 to 8.95. Fuses at about 1930o 
F. i>ii$tinguished from all oiher metals by its reddish color. Very ductile 
and malleable, and its tenacity is next to iron. Tensile strength 20,000 to 
aO.liQO lbs. per square inch. Heat couductiviiy 73.0^ of that of silver, and su- 
perior to that or other metals. Electric conauctivity equal to that of gold 
and silver. Exfiansion by heat from 92? to 21 2® F., 0.0051 of its volume. 
Sf»^*iflc heat .093. (See Copper under Rtrenfcth of Materials: also Alloys.) 

Gold (Aivum). Aa«— At. wt. 197.2. Sp. gr., when pure and pres.Hed in a 
ilif, 19.81. Melts at about 1915** F. The most malleable and ductile of all 
metals. One ounce Troy may be beaten so as to cover itK) sq. ft. of surface. 
The average thickness of golc leaf is 1/2S-J000 of an inch, or 100 «q. ft. per 
ounce. One grain may be drawn into a wire 500 ft. in length. The ductil* 
itT i«* destroyed by the presence of 1/2000 jwirt of lead, bismuth, or an imoiiy. 
<to:d la hardened by the addition of silver or of copper. In U S. j^old com 
tiiere are 90 parts fsold and 10 partes of ailo}', which is chiefly copper with a 
li.tle silver. By jewelers the flueness of gold is expressed in carats, pure 
gii'd ^}n*ing 24 carats, tliree foiirth.s fine 18 carats, etc. 

Irldlam. — Iridium Is one of the rarer metuls. It has a white lustre, re- 
A*(nbling that of steel: its hardness Is about equal to that of the ruby; in 
th^ cold it is quite brittle, but at a white heat it is somewhat inalleal>le. It 
i« one of the Heaviest of nietals, having a speciHC gravity ot 2-J.3a. It is ex- 
tremely infusible a:id almost absolutely inoxidizable. 

For uses of iridium, methods of manufacturing it, etc., see paper by W. D- 
Dudley on the "Iridium Industiy." Trans. A. I. M. E. 18«l. 

Iron (Ferrum), Fe.— At. wif 56. Sp. gr.: Cast, 6.rt to 7 48; Wrought. 
7.4 to 7.9. Pure Iron is extremely infusible, itK melting point being above 
3000^ F., but its fusibility increases with the addition of carbon, cas^t iron fns' 
io? about 2900° F. Conductivity for heat 11.9, and for electricity 12 to 14.8, 
Eihrer being lOO. Expansion in bulk bv heat: cast iron .0<V5:j, and wrought iron 
0035. from 32** to 212^ F. Specific heat: cast iron .1208. wrouKht iron .11:38, 
stwl .1165. Cast iron exposed to continued ht-at becomes permanently ex- 
panded IH to 3 per cent of Its length. Qrate-bars should therefore be 
allowed about 4 per cent play. (For other properties see Iron and Steel 
ander Strength of xM aterlals.) , ^ 

Leftd (Plumbum), Fb.— At. wt. 206.**. Sp. gr 11.07 to 11.44 by dUTerent 
auihorities. Meltw at about (i2r)0 F., softens and becomes pasty at about 
8:t* F. If broken bv a sudden blow when just below the melting point it is 
[|i.lte brittle and the fraciui-e appears crystalline. l«eftd is very malleable 



168 MATEKIALS. 

and ductile, but its tenadtv la such Ihnt it can bo drawn into wire wiih prpat 
difficulty. Tensile strenj^th, 1600 to -^MitO lbs. per sqimr« inch. Its elast ii-ii y 15 
very lo*v, and the mntnl flows under very slight Htralii, Lead diasolv^* to 
some extent In pure water, but water oontalninpr carbonaten or sulphatet 
forms over it a film of insoluble salt whicli prevents f uriher action. 1 

BlasneBlum, Mg.-At. wt. 'J4. Sp. pr. 1.69 to 1.76. Silver-white, | 
brilliuni, malleable, und ductile. It Ih one of the lightest of metals, wei^hiDf^ 1 
only about two thirds as much as aluminum. In the form of fliinfi^s, wire. 
or thin ribbons it Is hig^hlv combustible, burning? with a light of daulinj; 
brilliancy, useful fur signal-liKhU and for flash-lights for photographers. It 
is nearly non -corrosive, a ihin film of carbonate of magnesia forming^ on ex- i 
posure to damp air, which protects It from further corrosion. It may be 
alloyed with alumlrmm, 5 per cent Mg added to Al giving about as much in- 
crense of strength and hardness as 10 percent cf copper. Cubical expaDsioo 
by heat 0.0083, from 32«» to ilZ^ F. Meits at VXXy F. Si>eciflc heat Ji5. 

jnEanffaneBe. Mil.— At. wl 55. 6p. gr. 7 to 8. The pure metal is not 
used in me aits, out alloys of manganese and iron, called spiegeleisen when 
containing below ib per cent of manganese, and ferro-mangMuese wb«n o<»n- 
taiiiing from 25 to 90 per cent, are used in the manuf ctureof steel. Metallic 
manganem. when alloyed with iron, oxidizes rapidly in the air, and Its func- 
tion in cteel manufacture is to remove the oxvgen from the bath of sun-l 
whether it exists as oxide of iron or as ocoluded gas. 

Stercnry (Hydrargyrum), Hg.— At. wt. 199.8. A silver-whit© metal 
liquid ai leniperatures above— 39'* !•'., and boils at 660^ F. Unchangeable as 
gold, silver, and platinum in the atmosphere at ordinary temperatures, but 
oxidizes to the red oxide when near its boiling-point. 8p. gr.: when liquid 
13.58 to 18.59, when frozen 14.4 to 14.5. Easily tarnished by sulphur fume**, 
also by dust, from which it may be freed by straining through a cloth. No 
metal except iron or platinum should be allowed to touch mercury. The 
smallest portions of tin, lead, zinc, and even copper to a less extents cause it 
to tarnish and lose its perfect liquidity. Coefficient of cubical ex|>all^ii>n 
from 88«» to 5fia« F. .0182; per deg. .000101. 

Nickel, Nl. -At. wt. 58.8. Sp. gr. 8.37 to 8.08. A silvery-white metal 
with a strong lustre, not tarnishing on exposure to Uie air. Ductile, lianl, 
and as tenacious as iron. It Is attracted to the maguet and may be mode 
magnetic like iron. Nickel is very difficult of fusion, melting at alntut 
aoOU* F. Chiefly used In alloys with copper, as german-sUver, nickel siiver, 
etc., and recently in the manufacture of steel to lncr(>ase its hardness and 
strength, also for nickel-plating. Cubical expansion from SH? to 212^ F., 
0.0088. Specific heat .109. 

Platinum, Pt.— At. wt. 195. A whitish steel-gray metal, malleable. 
Tery ductile, and as unalterable by ordinary agencies as gold. When fiu«ed 
ancf refined it is as soft as copper. 8p. gr. i!1.15. It is fusible only by the 
oxyhydrogen blowpipe or in strong electric currents. When combined with 
iridium it forms an alloy of great hardness, which has been used for gun- 
vents and for standard weights and measures. The most important us«*s of 
platinum ip the arts ai'e for vessels for chemical laboratories and manufac- 
tories, ana for the connecting wires in incandesoent ehnztric lamps. Cubical 
expansion from m* to *^12* F., 0.00S7, less than that of any other metal ex- 
cept the rare metals, and almost the same as glass. 

Silver (Argentum), Ag. -At. wt. 107.7. Sp. gr. 10.1 to 11.1, according to 
condition and purity. It is the whitest of the metals, very malleable and 
ductile, and in hardness intermediate between gold and copper. AleltR at 
about 1750* F. Specific heat .050. Cubical expansion from 3a<» to 81S« F., 
0.0058. As a conductor of electricity it is equal to copper. As a conductor 
of heat it is superior- to ail other metals. 

Tin (Stanniun) Sn.— At. wt. 118. Sp. gr. 7.298. White, lustrous, soft, 
malleable, of little stivngtii, tenacity about 3600 lbs. per square inch. F'ui^es 
at 442® F. Not sensibly volatile when melted at onimary heats. Heat con- 
ductivity 14.5, electric conductivity ia.4; silver being 100 In each caF»e. 
Expansion of volume by heat .0069 from 9^ to 2VZ? F. Specific heat .055. Ite 
chief uses are for coating of sheet-iron (called tin plate) and for making 
alloys with copper and other metals. 

asinc, Zn.-At. wt. tt5. Sp. gr. 7.14. Melts at 780« F. Volatiltsee and 
burns in the air when melted, with bluish-white fumes of zinc oxide. It is 
ductile and malleable, but to a much less extent than copper, and its tenacity, 
about 50UU to 6000 ll>s. per s(}uare inch, is about one tenth that of wrougiit 
iron. It is practically noii corrosive in the atmosphere, a thin film of car- 
bonate of zinc forming uik>ii it. Cubical expansion between 82* and 212** F., 



MEASUEES AND WEIGHTS OF VARIOUS MATERIALS, 169 



0.0088. Specific heat .096. Electric conductivity 29, heat conductivity 86, 
ulver beinj? 100. Its principal uses are for coating iron surfaces, called 
"^ galvanizing," and for making brass and other alloys. 



MftlleablUtj. 

Gold 

SUver 

Aluminum 

Copper. 

Tin 

J^^ead 

Zinc 

Platinum 

Iron 



Table Sboirliic the Order of 



Bnetlllty. 


Tenacity. 


InmslbUUy. 


PlatiDum 


Iron 


Platinum 


Silver 


Copper 


Iron 


Iron 


Aluminum 


H^ofr 


ssrr 


Platinum 


Silver 


Silver 


Aluminum 


Zinc 


Aluminum 


Zinc 


Gold 


Zinc 


Tin 


Tin 


Lead 


Lead 


Lead 


Tin 



rOWMJJlAM AND TABIiK FOB CAI^CVLATING 

irBieHT OP ROD», BAR8, PI4 ATBS, TUBES, AND 
8PSBBBS OF DIFFJBRENT JHATBRlAIiS. 

Notation : b = breadth, i ss thickness, b s side of square, d s external 
iiameter, d] = internal diameter, all in inches. 

Sectional areas : of square bars s= a*; of flat bars cz bt; ot round rods s 
:S54d«; of tubes = .7854(d' - d,«) = Z.UlfUdt - t*). 

Volume of 1 foot in length : of square bars = ISfs^; of flat bars = 126^; of 
round bans = 9AUiitP; or luueb = 9.45J48(ct'' - di«) s 8T.699(dt - /»>, In en. in. 

Weight per foot length = volume x weitcht per cubic inch of the materials 
Weight of a sphere = diam.* x .&286 x weight per cubic inch. 



Cast iron 

"^rijusht Iron 

Steel 

Copper A Bronze I 
• j'Dpper and tin)f 
p-.^> 85 Chopper.. 

L«d 

Aiaminmn 

"jLaas 

Pine Wood, dry . . . 



7.S18460, 

7,7 

7.854 



8.855 

8.898 

11.88 
2.67 
2.62 
0.481 



I 



480. 
180.6 

55S, 



528. 

709.6 
IG6.5 
168. 4 

ao.o 



1^ 



87.6 

40. 

40.B 

4C. 



50. 
13.0 



2.5 



s^i 



^ 



833fi3 



43.6 3.G8.35' 



4.03*« 
1.16b« 



18.61.18«« 



0.21«3 



lit 



8^6/ 

3.8336/ 
3.6836/ 

A.m>t 

\AGbt 
1.186* 
0.2lbt 



III 






.2604,16-16 
.2TT9 1, 
.28331.02 

.3195|1.15 



2.464d» 
2.6l8d» 
2.670d« 

S.Ollda 



.3020'I.09 2.864d« 

.410cll.48 8.870d* 
.(WC3 0.347i0.90«d9 
.0W.')0.34 O.SOtrf' 
.01741 1-16 0.164d3 



S8 



.136Sd» 
.146fid« 
.H84d«' 

.1673d» 

.1586d« 

.2150d«- 
.06O4d>- 
.0495d» 
.0091d« 



WHebt p»»r cylindrical in., 1 in. lone, = coefficient of d« in nintii col. -•- 12. 

For tabes use the cotrfflcieut of d^ in ninth column, ns for rod». and 
riTiitiply it into (d* — dx*); or take four times this coefficient and multiply it 
krofdt — f*). 

For bollo-vr spheres use the coefflcient of d> in the last column and 
&u.Upiy it into (d« — d,»). 

JHEASVBBS AND IFEIGIIT'9 OF VABIOUS 
MATEBIALS (APPBOXIRKATE). 

Brlelnvor]C«— Brickwork is estimated by the thousand, and for various 
tLickneeaes of wall runs as follows: 

K^-in. wall, or 1 brick In thickness^ 14 bricks per superficial Si^^'i. 
ISS m 21 ** 

SIM " '* " 2M '* " " 85 " *• " »• 

An ordhiary brick measures about 8>^ x 4 X 2 inches, which is equal to 66 
ci;Uc inches, or 26.3 bricks to a cubic foot. The average weight Is i^i Ibe. 



170 



MATERIALS. 



Foel.^A bushel of bltuminona coal wei^^hs 76 f>ound8 and contains 8688 
cubic inches = 1.564 cubic feet. 20.47 bushels = J ^ross ton. 

A bueliel of coke weighs 40 lbs. (85 to 42 lbs.). 

One acre of bituminous coal contains lOOU tons of 8240 lbs. per foot of 
thickness of coal worked. 15 to i^ per cent luust be deducted for waste In 
minine. 

41 to 45 cubic feet bituminuus coal when broken down = 1 ton, "HW Ibe. 

d4 to 41 " '* anthrncite, prepared for market = 1 ton, :iii40 11^. 

123 *' •* of charcoal =s 1 ton, «f40 lbs. 

70.9 '* *' -coke as 1 ton, aa40 ll**. 

1 cubic foot of anthracite coal (see also page Qi5) = 55 to 6t> lbs. 

1 bituminous" a60to561b6. 

1 •* ** Cumberland coal =531bs. 

1 " " Cannel coal... = 60.3 lb& 

1 ♦* " charcoal (hardwood) = 18.6 lb«L 

1 (pine) =181bs. 

A busbel of cliarcoal.— In 1881 the American Charcoal Iron Work- 
ers* AHSOciailou adupte<l for use in its official publications for tJm standard 
bushel of charcoal 2<4K cubic inches, or '20 pounds. A ton of charco>al is to 
be taken at 2000 pounds. This figure of SM) pounds to the bushel was taken 
as a fair averago of different bushels used throughout the country, and it 
has since been established by law in some States. 

Ores, Bartbs, etc, 

13 cubic feet of ordinary gold or silver ore, in mine = 1 ton = SOOO lbs. 

20 •• " " broken quartz = 1 ton = aoOO Ib& 

18 feet of gravel in bank = 1 ton. 

27 cubic feet of gravel when dry = i ion. 

25 ** '* *' sand = 1 ton. 

18 " *• '* earth In bank = 1 ton. 

27 ** •* *' " when dry = 1 ntn. 

17 " ** *' clay = 1 ton. 

Cement.— English Portland, sp. gr. 1.25 to 1.51, per bbl .... 400 to 430 lbs, 

Rosendale, U. S., a struck bushel 02 to 7ull>». 

Mme.— A stiiick bushel 72 to 75 lbs 

Grain*— A struck bushel of wheat = QO lbs.; of corn = 56 lbs.; of oaUi = 
30 lbs. 

Salt.^A struck bushel of salt, coarse, Syracuse, N. Y. = 5G lbs. ; Turk's 
Island = 76 to 80 lbs. 

irelfftat of Barth FllUn^. 
(From llowe^s ** Retaining Walls.") 

Average weight in 
lbs. ];>er cubic foot. 

Earth, common loam, loose 72 to HO 

** " '* shaken ftij to 93 

** ** " rammed moderately 90 to 100 

Gravel 90 to 106 

Sand 90toUt6 

Soft flowing mud 104 to 1«) 

Sand, perfectly wet 118tol29 

COnmiBRCIAIi SIZB8 OF IRON BAAS. 

Flats. 



Width. 



Thickness. 



^to n 

J^ to 15/16 
y^tol 

kto 1^ 
V6toi^ 

3/10 to 1>J 



Width. 



Thickness. 




Width. 



4 



Tbickneas. 




WEIGHTS OF WROUGHT IRON BARS. 



171 



Koondfl : M to 19^ inches, advaDciog: by IGtbs, and 19^ to 5 inches by 
Squares z 5/16 to 1^ inches, advancing by lethst, and 1^ to 8 inches by 

Hair ronnd«: 7/16, ^, %, 11/16, 94» 3. If6^ ^H. 1H» % 2 inches. 

Hexagons : 9^ to lU iuchfs, advancing by 8ths. 

Ovals I H X J4. % X 5/16, ^ X ^. % X 7/16 inch. 

Hair ovals: Ji X J6, % X 5/3:.>, 94 X 3/16, % X 7/88, IJi X «, 1« X 5i 
l?lx H >nth. 

Bonnd-edse flats: 1^ X K. l^ X ^ 1% X ^ inch. 

Bands: Vie to lU inches, advancing by 8tbs, 7 to 16 B. W. gauge. 

IH to 5 inchtfs, advancing by 4ths, 7 to 16 gauge up to S inches, 4 to 14 
9&o£e, S^ to 5 inches. 

1¥1BI6HT8 OF SaUARE AND ROUNH BARS OF 
WROWHT IRON IN POUNHS P£R lilNBAIi FOOT. 



Iron weighing 480 lbs. per cubic foot. 


For steel add S 


per cent. 


c . . 

15^ 




-oh 


hi 






III 




- i>5 




U3»s 


tic§2a 




•^ c^ 


tlCp 43P 


t)£3 » C 




ftlJ 


l&5i 




13-53 


-00 


|5 = 


i=?o3 


po5 









11/16 


24.06 


18.91 


t 


96.80 


75.64 


1/16 


.013 


.010 


H 


25.21 


19.80 


98.55 


77.40 


H 


.OW 


.041 


13/16 


26.37 


20.71 


100.8 


79.19 


^l« 


.117 


.092 


% 


27.55 


21.64 


V 


103.1 


81.00 


H 


.208 


.164 


15/16 


28.76 


22.59 


105.5 


82. H3 


.VI6 


.S* 


.256 


8 


30.00 


23.56 


11/16 


107.8 


84.69 


H 


.469 


.368 


1/16 


31.26 


24.55 


94 


110.2 


86.56 


T/l« 


.638 


.501 


u 


82.55 


25.57 


13/16 


112.6 


88.45 


H 


.833 


.654 


3/16 


33.87 


26.60 


% 


115.1 


90.36 


9/16 


1.065 


.828 


k 


85.21 


27.65 


15/16 


117.5 


92.29 


^ 


i.ao-i 


1.023 


5/16 


36.58 


28.73 


6 


120.0 


94.25 


11/16 


1.576 


1.237 


k 


87.97 


29.82 




125.1 


98.23 


H 


1.875 


1.473 


7/16 


39.39 


30.91 


i2 


130.2 


102.3 


IM6 


2.201 


1.728 


% 


40.83 


3.».07 


s/ 


135.5 


100.4 


?^ 


2.^-a 


2.001 


9/lG 


42.30 


33.23 


1^ 


140.8 


110.6 


TVI6 


2.930 


2.:W1 


k 


43.80 


34.40 


76 


H6.3 


114.9 


1 


8.333 


2.618 


11/16 


45.33 


a5.60 


% 


151.9 


119.3 


1/16 


3.763 


2.9rx'S 


^ 


46.88 


36.82 


% 


157 6 


12:3.7 


H 


4.219 


3.313 


13/16 


48.45 


38.05 


7 


163.8 


128.3 


3/16 


4.701 


3.692 


% 


60.05 


3S.31 




169.2 


132.9 


^4 


5.308 


4.091 


15/16 


61.68 


40.59 


14 


175.2 


137.6 


5/16 


6.742 


4.510 


4 


53.33 


41.89 


79 


181.8 


142.4 


H 


6.302 


4.950 


1/16 


55.01 


43.21 


l2 


1S7.5 


147.3 


7716 


6.S88 


5.410 


H 


66.72 


44.55 


7H 


19:i.8 


152.2 


^ 


7.500 


5.890 


3/16 


58.45 


46.91 


&i 


200.2 


157.2 


S?16 


8.138 


6.392 


ii 


60.21 


47.29 


yk 


200.7 


162.4 


^ 


8.802 


6.913 


6/16 


61.99 


48.69 


8 


213.3 


167.6 


11/16 


9.492 


7.455 


% 


63.80 


50.11 


vc 


226.9 


178.2 


H 


10.21 


8.018 


7/16 


65.64 


61.55 


\L 


240.8 


189.2 


1-1/ 16 


10.95 


8.601 


u 


67.50 


5:1.01 


a^ 


255.2 


200.4 


Ti^ 


11.72 


9.204 


V 


69.39 


54.50 





270.0 


212.1 


15716 


12.51 


9.828 


71. ;» 


60.00 


M 


285.2 


224.0 


9 


13.33 


10.47 


T 


73.21 


57.52 


i2 


300.8 


236.3 


" 1/16 


14.18 


11.14 


75.21 


59.07 


% 


816.9 


248.9 


H 


15.05 


11 .82 


13/16 


77.20 


00.63 


10' 


3:«.3 


261.8 


3/16 


15.95 


12.53 


% 


79.22 


62.22 


^ 


350.2 


275.1 


H 


16.88 


13.25 


15/16 


81.20 


63.82 


l2 


31)7.5 


288.6 


5/16 


17. S3 


14.00 


& 


83.3:3 


65.45 


74 


385.2 


302.5 


H 


18.80 


14.77 


1/16 


85.43 


07.10 


11 


403.3 


316.8 


V16 


19.80 


15.55 


3% 


87.55 


68.76 


\/i 


421 .9 


331.3 


^ 


20.83 


16.86 


89.70 


70.45 


i2 


440.8 


346.2 


V16 


21.89 


17.19 


k 


91.88 


72.16 


9« 


460.2 


361.4 


H 


88.97 


18.04 


5/16 


94.08 


73.89 


12 


430. 


377. 



172 



MATERIALS. 






Sal 

MO U 

fits. 
Si I 

2 S 

fa ^ 
fa 

H 

n 

M 



§;|iggSS35§SS^^SSSS5g.o5^?;S{:?eesss?!r:f5gS^' 



-^e*eo^rf:«t-QDao — "vec^towc^oDCs 






♦-iwooTTiotefXOfco — wec^^-oiccoi-occiOj-i 



^ i{:SS5S58SSSn:2si«V:gsa;5t:?5518?82S5fSSg$5' 



I 



r-«90^i0<Ot-l-000»O-^i?»0i?rfiCift«t-0006O^ 



s?c;^S^«S&5 



" ©i eo TT •« irj CD t>^ x os o (b «' c* c<? ^ lo »r5 «" i>I oo o» o c -- 



SSMSS*?^^! 









^ 



^!?8r:§$;:S8fc!$SSSSSs2SS«8S,i8gS88Ss8S| 



*^c<oeo'^<vino«Dt>acxiOkOO^c)9*eoTP^io«o«0C^acaoo» 



SSc 



" T-1 1-i e* « e*^' »o rt to 5D t- db od »" o" o »^ f-^ <?*" eo eo •^ ift »« »" w i> od oc ci c 



■^ r-I (Tl <Ji 00 "** ^ tcj lO «C d t^ 00 OD Oi ci d d " CJ ©» 00 CO 3J "^ «ft 



■'r.i**cjcoeo^"^io»rjddi-t'-odoooiddd»-'«--o»eooo'^-^ir}tr:d« 



^5* * * i-i »-^ 01 7» » oo' rr ^ ^ d d d c* t* c^ 00 d d d d d •- 1-^ M o* CO 00 -^ "^ rt I 



' T-i t-J o* e» oj CO d ^ ^ d d d d d I' t>^ t- X X d d d d d -^ — e* c* or d ' 



in 



i^r-ir-idoto(dd-<^-^^dddddd(.>t^xdo6dddddoi-^«^i 



;ii?R2??Sg5DS^r:8?!?S8SS3Efj2§^2S«$25 

* ^ ^ ,J d ot o< d eo d '^ <«' -"i^ d d d d d d d t<^ t<^ i<^ d oc oe oa o» a» d 



' ^ ^ r-! f-i 7« ot 01 d d d d d '^' '^ -^ -^ d d d d d d d i-^ I* t> c^ oc d 



' »-i T-i »-l »- ^ d o» d d o< d d d d d f •^ ^ ■^ d d d d d d « d d 






to «o to o « 



•O O O CO O (0 



I^5-''5*.I^»«I*'5-^5 j:*5-'J«:i*5«;-«d^i 



WEIGHTS OF FLAT WROUOHT IRON. 



17S 





l> 


S8S8S8S8S8S8SSS88S8S8888 




5' 












§gj2S3S§282Bf2Sgg«S8f2SS8SSa8 




-«-'^»;:«SS28?JS8aS8fc^^5SgSg 




i 






So 


&:S8^S8'«8S8idSS8$oS8&8SSio8S3lD88 




^ 






1: 




as 

i 


^ 






i 






i 


^oied««^:o;o;;«;5j22a2«gg5gg55ggg5. 




^' 


SSi^8S2^83Sr^83Sf38S8S8S|8S8 




i 






1 


S85SS88V:«5?Si?8S28S85;SS88S 




i 


8SS8^SSr.SSS^$i;;?SS388^9SaS8 




2) 






11 


C3 (O iD V V to *0 

^ ^ ii t^ cfc ;. ^ -^^S^^^i^?^^ 



I 



88{28 



! Sails 

I sa 

I 

hS 
55 






174 



MATERIALS. 



WEIGHT OF IRON AND STEBIi SHEETS. 

ITelKliUi per Sqoare Foot. 

(For weights by Decimal Gauge, see page 32.) 



Thickness by BinniDgham Gauge. 


Thickness by American (Brown and 
Sharpens) Gauge. 




Thick- 








Thick- 






No. of 


nessin 


Iron. 


Steel 


No. of 
Gauge. 


nes8in 


Iron. 


SteeL 


Gauge. 


Inches. 






Inches. 






0000 


.454 


18.16 


18.58 


0000 


.46 


18.40 


18.77 


000 


.425 


17.00 


17.84 


000 


.4096 


16.88 


16.71 


00 


.88 


15.80 


15.50 


00 


.8648 


14.59 


14.88 





.84 


18.60 


18.87 





.8249 


13.00 


18.86 


1 


.8 


12.00 


18.84 


1 


.2898 


11.57 


11.80 


s 


.284 


11.36 


n.59 


8 


.2576 


10.80 


10.. M 


8 


.259 


10.86 


10.57 


8 


.2294 


9.18 


9.S8 


4 


.288 


9.52 


9.71 


4 


.2048 


8.17 


6.34 


6 


.28 


8.80 


8.98 


6 


.1819 


7.28 


7.48 


6 


.803 


8.12 


8.28 


6 


.1620 


6.48 


6.61 


7 


.18 


7.20 


7.34 


7 


.1443 


5.77 


5.89 


8 


.165 


0.60 


6.73 


8 


.1285 


6.14 


5.24 


9 


.148 


5.92 


6.04 


9 


.1144 


4.58 


4.07 


10 


.184 


5.86 


5.47 


10 


.1010 


4.08 


4.16 


11 


.18 


4.80 


4.00 


11 


.0007 


8.68 


8.70 


18 


.109 


4.86 


4.45 


18 


.0808 


8.23 


8.80 


18 


.095 


SHO 


8.88 


18 


.0780 


2.88 


2.94 


14 


.088 


8.32 


8.30 


14 


.0641 


856 


8.62 


15 


.072 


2.88 


2.04 


15 


.0571 


8.S8 


8.33 


16 


.065 


2.60 


265 


16 


.0508 


8.08 


8.07 


17 


.or>8 


282 


2.87 


17 


.0453 


1.81 


1.85 


18 


.049 


1.96 


2.00 


18 


.0^03 


1.61 


1 64 


19 


.042 


1 68 


1.71 


19 


.0359 


1.44 


1.46 


90 


.085 


1.40 


1.48 


20 


.0820 


1.28 


1.31 


21 


.032 


1.28 


1.31 


81 


.02*) 


1.14 


1.16 


S3 


.028 


1.12 


1.14 


22 


.0258 


1.01 


1.08 


28 


.025 


1.00 


1.02 


23 


.02-J6 


.004 


.922 


S4 


.028 


.88 


.808 


24 


.0201 


.804 


.820 


85 


.08 


.80 


.816 


25 


.0170 


.716 


.730 


86 


.018 


.78- 


.734 


26 


.0159 


.686 


.649 


87 


.016 


.64 


.ess 


27 


.0142 


.568 


.579 


88 


.014 


.56 


.571 


28 


.0126 


.504 


.514 


89 


.018 


.52 


.580 


29 


.0113 


.458 


.461 


80 


.018 


.48 


.490 


80 


.0100 


.400 


.408 


81 


.01 


.40 


.408 


31 


.0089 


.856 


.863 


88 


.009 


.86 


.367 


82 


.0080 


.3:20 


.826 


88 


.008 


.82 


.3-^6 


33 


.0071 


.284 


.290 


84 


.007 .28 


.286 


84 


.0063 


.358 


.257 


86 


.006 .80 


.204 


85 .0056 


.2^ 


.288 






Iron. SWel. 






Specfflc graTity . 
Wefglit per cubic 




ij- 


.7 7.854 
480.6 






foot!::.'! 


::.:!; 48o 






»• 


». tt 


Inch... 




.2778 


.2888 





As there are many gau?e8 in use diffi^rlng from eaich other, and even the 
thicknesses of a certain KptKjifled ^auge. aHthe Birmingham, are not assumed 
the same by all mannractiirers. orders for sheets and wires sliould always 
state the weight per sauare foot, or the thickness in thousandths of an inch. 



WEIGHT OF PLATE IRON.' 



175 





- 


8S&8S&8S&8S&8S&8S&ot.c9o».»ot-e9ot-«50fc.Mo 






S3i°S8S3SoSSi?S8S^S(SS!^o».oooo««QOOe«.oooo».<» 




;se 


8SSSf2'oSS7^^i;88s8g{?'«3S!SV;o«>t. .0 00 (MOOD t-c>oo»o 
S3SS^9§^S18SSSS'g?i:«-|255SteS882S|{^g|§s§^ 




2 






sr 


828S8S8S8S8S8S8S8S88888ooooooooooo 




«D 


ge828fessS§82??t:8SS8?:^SSSS8fegJao^o««oo«o»« 




;i^ 


8St:SS$SSgSSS8S^SS^8&S8^SSS&Soo««»«».QDo 


1 




a 


:« 


S&S8&S8^S81»S8&88&»S8§8&8Sg8SS^8S&8S^o 
^'s:Si^S^8sSS^S?;;9^^^8SS8S8^S5S^SS8SSSS8 


i 


2 






;i! 


838i^8^S:^8'6{S^83Sr383S8S8S8S888S888S8 




2 






3? 


8SiDSI3^8SSDSS^8S&8S^8^38lDS8&S8^S8lD88 




«9 


SS23S8SSSDS?Sr2S83^SS^ie8^Si'38;SS^838:28»8 




:« 


8?S8fe8SSSr2t;SS?§S8lS$g85:8S8&SS5:8S&S??i:8 
io«rf3tD«t^t^t-<»o6o»o»ooo---2»2S2{SSS5^S28SS3iSS^ 




o 
1 


gr:SSSSeSt^??SeS5;SSSS«S$S8St-2?:S8?fSSfe$S 

99 99 oi «0 90 09 09 09 -T ^ ^ -^ to lO »6 i6 to fO ^ V>tAt^ I' QD^ciai a ^O ^^^99 


1' 


^1 


?5S2:SS5;SSS8saaSSSSia«8SS88S^?5gl58S22S28 



176 MATERIALS. 

WEIGHTS OF STEEIi BLOCKS. 

Soft steel. 1 cubic inch = 0. SSI lb. 1 cubic foot = 4S0. 75 lbs. 



Sizes. 


Lengths. 




























1" 


6" 
82 


18" 

164 


18" 
245 


84" 
827 


30" 
409 


86" 
491 


42" 


48" 


64" 
736 


60" 
816 


6©" 


12" x4" 


18.63 


573 


664 


900 


11 X 6 


i8.';5 


118 


225 


388 


450 


563 


675 


788 


900 


1018 


1125 


ia*B 


X 5 


15. 6i 


94 


188 


281 


375 


469 


562 


656 


750 


843 


9:^ 


1031 


x4 


1«.60 


75 


160 


225 


800 


376 


450 


525 


600 


675 


750 


825 


10 X 7 


19.88 


120 


889 


358 


477 


596 


715 


835 


955 


iar4 


1193 


1319 


x6 


17.01 


102 


204 


807 


400 


511 


618 


716 


818 


920 


1022 


1125 


X 5 


14.*^ 


65 


170 


256 


841 


426 


511 


596 


682 


767 


852 


037 


X 4 


11.36 


68 


186 


205 


2r3 


341 


409 


477 


516 


614 


682 


780 


X 8 


8.53 


51 


102 


153 


204 


256 


806 


858 


409 


460 


511 


509 


9x7 


17.89 


107 


215 


322 


430 


537 


644 


751 


859 


966 


1078 


1181 


x6 


15.84 


92 


184 


276 


868 


460 


552 


614 


736 


838 


920 


1018 


X 5 


18.78 


77 


153 


280 


307 


888 


460 


587 


614 


690 


767 


844 


x4 


10.23 


61 


123 


184 


245 


307 


368 


429 


490 


552 


613 


674 


8x8 


18.18 


109 


218 


327 


436 


545 


656 


784 


878 


962 


1001 


1200 


X 7 


15.9 


95 


191 


286 


382 


477 


578 


668 


763 


a^9 


954 


1040 


x6 


18.68 


82 


164 


245 


327 


409 


491 


578 


654 


T36 


818 


900 


x6 


11.86 


68 


186 


205 


278 


841 


409 


477 


546 


614 


683 


750 


X 4 


9.09 


55 


109 


164 


218 


278 


327 


382 


436 


491 


545 


600 


f x7 


13.02 


88 


167 


251 


334 


418 


801 


565 


668 


752 


886 


919 


x6 


11.93 


72 


143 


215 


286 


358 


430 


501 


573 


644 


716 


788 


X 6 


9.94 


60 


119 


179 


2:)8 


296 


358 


417 


477 


536 


596 


656 


X 4 


7.95 


48 


96 


148 


191 


239 


286 


384 


382 


429 


477 


5:25 


X a 


5.96 


86 


72 


107 


148 


179 


214 


250 


286 


822 


858 


893 


(^x6^ 


12. 


72 


144 


216 


388 


360 


432 


504 


576 


648 


730 


798 


x4 


7.3R 


44 


89 


183 


177 


221 


266 


310 


854 


889 


443 


487 


6x6 


10.2-2 


61 


123 


184 


245 


307 


868 


429 


490 


651 


613 


674 


x6 


8.5-» 


51 


102 


153 


204 


255 


807 


858 


409 


460 


511 


569 


x4 


6.83 


41 


82 


128 


164 


304 


245 


286 


327 


368 


409 


450 


x8 


5.11 


81 


61 


92 


128 


158 


184 


214 


»I5 


276 


307 


837 


fiHx5^ 


8.59 


S2 


106 


155 


206 


258 


800 


861 


412 


464 


515 


567 


X 4 


6.25 


87 


75 


112 


150 


1H8 


235 


263 


300 


837 


375 


41 S 


6x5 


7.10 


43 


85 


128 


170 


213 


356 


298 


341 


383 


436 


469 


x4 


6.68 


84 


68 


102 


136 


170 


205 


239 


278 


307 


841 


875 


4Hx4H 


5.75 


35 


69 


104 


138 


178 


207 


242 


276 


311 


345 


380 


x4 


5.11 


31 


61 


92 


133 


153 


184 


215 


246 


278 


307 


338 


4x4 


4.54 


27 


65 


82 


109 i 186 


164 


191 


318 


246 


272 


300 


x8Hi 


3.97 


24 


48 


72 


96 1 110 


143 


167 


181 


216 


388 


262 


x3 


3.40 


20 


41 


61 


82 102 


122 


143 


IKi 


181 


304 


224 


8>ix8H 


3.48 


21 


42 


68 


84 1 104 1 125 1 146 


167 


188 


209 


2:i0 


x 8 


2.98 


18 


36 


54 


72 ^ 89 107 135 14''. 


161 


179 


197 


8x8 


2.56 


15 


81 


46 


61 \ 


77 


92 


108 


138 


m 


154 


169 



L 



SIZES AND WEIGHTS OF STRUCTURAL SHAPKS. 



SIXB8 AND WBTOHTS OF 8TRITCT17BAI4 8HAPB8. 

MlnlBuniB, JVaxlmnm, and Intermediate ireiffhts and 





DlmenBlona of Carne^^o Steel I-Beama* 




Sec- 
tion 
laJex 


Depth 

of 
Beam- 


Weijfht 
l^'t. 


^fdT 


Weh 
Thick- 
ness. 


Sec- 
tion 
[ndex 


Depth 

of 
Beam- 


WeiKht 


FlanKP 
Width. 


Web 
Thick 
ness. 




ins. 


IbH. 


ios. 


ins. 




ins. 


lbs. 


ins. 


ins. 


Bl 


24 


100 


7.85 


0.76 


610 





17 25 


8.58 


0.48 




»• 


05 


7.10 


0.69 


*» 


• • 


14.75 


3.45 


0.35 


*» 


•* 


00 


7.13 


0.63 


»' 


♦• 


12.25 


8.33 


23 


'• 


•• 


85 


7.07 


0.57 


Bei 


5 


14.75 


8.20 


0.50 


" 


•• 


80 


7.00 


0.60 




»• 


12.26 


8.15 


0.30 


m 


90 


75 


6.40 


0.65 


*♦ 


• ( 


0.75 


8.00 


0.21 






70 


C.8:J 


0.58 


B.J3 


4 


10.5 


2 S8 


0.41 


♦• 


•* 


65 


6.« 


0.50 


'♦ 


*• 


0.5 


2.81 


0.:M 


Bao 


18 


70 


6.26 


0.7J 


'» 


(» 


8 5 


2.73 


26 


** 


»* 


65 


6.18 


0.C4 


" 


'* 


7.5 


2.G0 


0.19 


•• 


•* 


60 


6.10 


0.56 


B77 


3 


7 5 


2.52 


0.36 


»• 


*• 


55 


6.00 


46 






C 5 


8.42 


0.26 


B7 


15 


53 


5.75 


CO 


'» 


»• 


6.5 


2 3^) 


17 


•• 


•• 


50 


5.65 


0.56 


Bi 


20 


lOU 


728 


0.88 


•• 


»* 


45 


5.55 


0.46 




" 


05 


7 21 


0.81 


** 


*t 


42 


5.50 


041 


♦* 


t« 


00 


7.14 


0.74 


B9 


IS 


35 


.5.00 


044 


•* 


tt 


85 


7,06 


0.68 


*• 


*» 


31.5 


5.00 


085 


*» 


** 


80 


7.00 


0.60 


Bll 


10 


40 


5.10 


0.75 


B4 


15 


100 


6.77 


1.18 


»• 


•' 


35 


4.05 


0.60 


»' 


•* 


05 


6.68 


1.00 


»^ 


»» 


80 


4.81 


0.46 


♦' 


" 


00 


6.58 


0.09 


•* 


»• 


liO 


4.66 


0.31 


'1 


'• 


86 


6.48 


0.80 


B13 


9 


35 


4.77 


0.73 




" 


80 


6.40 


0.81 


•♦ 


k« 


30 


4.61 


0.57 


B5 


1.5 


75 


6.20 


88 


»• 


•* 


25 


4.45 


0.41 


»* 


" 


70 


6.19 


0.78 


•• 


*t 


21 


4.33 


0.29 


" 


*' 


65 


6.10 


0.69 


BIS 


8 


23.5 


4.87 


0.54 


»• 


'* 


60 


6.00 


0.50 




** 


23 


4.18 


45 


B8 


1-2 


65 


5.61 


0.82 


•* 


•ft 


20.5 


4.00 


0.86 


" 




50 


5.40 


0.70 


" 


•» 


18 


4.00 


27 


" 


*' 


45 


5.37 


0.58 


Brr 


7 


SO 


3.87 


4G 


'* 


" 


40 


5.25 


0.46 


•» 


ft* 

«ft 


17.5 

15 


3.76 
3.66 


0..S5 
0.25 


Sections 
•' spei'ial " 


B2, B4. ] 

beams, 


B5. and ] 
11 le otbe 


B8 are 
rs are 












*'8I0 


Lndard. 









Sectional area = weight in lbs. per ft. ■♦-3.4, or x 0.2041. 
Weigbt ill lbs. per foot = sectional urea x 3.4. 

Xaxinaiam and minimum HVelgrbta and IMnienalona of 
Carnegie Steel Heck Bcama. 



Section 


Depth 
of 


WHpht per 
Foot, lbs. 


Flange Width. 


Web 
Tliickness. 


Increase* of 
Web and 


Index. 


B«*atn, 
iuches 








Flange per 














lb. Increa.««j 






Min. 


Max. 


Min. 


Max. 


Min. 


Max. 

.63 


of Weight. 


BlOO 


10 


27.33 


a'>.70 


5.25 


5.50 


.38 


.020 


HlOl 


9 


26.00 


30.00 


4.0i 


6.07 


.44 


57 


.im 


BlftJ 


8 


20.15 


24.48 


6.00 


5.10 


31 


.47 


.0:^7 


B'.03 


7 


18.11 


28.46 


4 87 


5 10 


31 


.54 


.042 


BIOS 


6 


15.30 


18.36 


4.y8 


4.53 


.28 


.43 


.040 



178 



MATERIALS. 



nUnlmiiiii, Mmxtmniii, and Intermeillate UTelsbt* and 
Dimensions of Carnegie Standard Cbannela* 





|ll 




III 

8.83 






i 


^^1 
pi 






01 


15 


55 


0.82 


C5 


8 


16.25 


2.44 


0.40 


•• 


• 4 


50 


3.72 


0.72 




»» 


18.75 


2.35 


0.31 


•• 


•« 


45 


3.62 


0.62 


»» 


• « 


11.25 


2.26 


0.22 


" 


it 


40 


8.52 


0.52 


C6 


7 


19.75 


2.61 


0.C3 


** 


'* 


35 


8.43 


0.43 




*' 


17.25 


241 


0..^8 


•' 


** 


38 


8.40 


0.40 


»* 


** 


14.75 


2.80 


0.42 


i^ 


12 


40 


8.42 


0.76 


<« 


*' 


12.25 


2-20 


O.Si 


*• 


'• 


35 


8.80 


0.64 


»« 


»* 


9.75 


2.09 


0.21 


" 


'* 


30 


8.17 


0.51 


C7 


6 


15.50 


2.28 


0.50 


i> 


*' 


A 


8.05 


0.89 




*» 


13 


2.16 


0.44 


" 


'* 


flO.5 


2.M 


0.28 


•' 


•' 


10.50 


2.04 


0..32 


C:^ 


10 


35 


3. IS 


0.82 


** 


«* 


8 


1.92 


o.ao 


tt 


tk 


30 


8.04 


0.68 


C8 


5 


11.50 


2.04 


0.48 


'* 


*' 


25 


2.89 


0.53 






9 


1.89 


0.33 


•' 


** 


20 


2.74 


0.88 


** 


" 


6.50 


1.75 


0.10 


»t 


»* 


15 


2.60 


0.24 


C9 


4 


7.25 


1.73 


0.33 


C4 


9 


25 


2.82 


0.62 




♦' 


6.25 


1.65 


O.^K 


'* 


*» 


20 


2.65 


0.45 


** 


** 


5.25 


1.58 


0.18 


" 


" 


15 


2.49 


0.29 


072 


8 


6 


1.60 


o.ae 


*• 


•• 


13 25 


2.43 


083 


t( 




5 


i.ao 


o.*-» 


ta 


8 


21.25 
18.75 


2.62 
2.53 


0.58 
0.49 


M 




4 


1.41 


0.17 



H^elshts and Dimensions of Carnegie Steel Z-Bars. 







Size. 






• 


Size. 






11 




15.6 


jl 

Z6 


u 

so 






c t1 

.2^ 


3 ^ 


i 




1 


P 


Zl 


6 


H 


8 5/16 


5 1/16 


36.0 


" 


7/10 


3 9/16 


6 1/16 


18.3 


" 


13/16 


3 H 


B >6 


28.3 


*' 


^ 


3 % 


6 H 


21.0 


Z7 


H 


3 1/16 




8.2 


Z2 


9/lC 


8 ^ 


6 


22.7 


*' 


5/16 


3 3/16 


4 1/10 


10.3 


•• 


% 


3 9/16 


6 1/16 


25.4 


** 


% 


4 H 


12.4 


'* 


U/IC 


3 % 


6 H 


28.0 


ZH 


7/16 


8 1/16 




13.8 


Z3 


Va 


3 Yi 


6 


29.8 


'* 


H 


3 8/16 


4 1/16 


15.8 


" 


13/16 


3 9/lU 


6 1/16 


32.0 


»« 


9/16 
% 


4 ^ 


17 9 


'* 


% 


8 % 


6 ^ 


84.6 


Z9 


8 1/16 




18.9 


Z4 


5/16 


3 M 


5 


11.6 


" 


11/16 


U% 


ra" 


S0.9 


'* 


% 


3 5/16 


5 1/16 


18.0 


** 


5/l6 


22.9 


'* 


7/.6 


l'^ 


5 14 


10 4 


ZIO 


2 11/16 




6.7 


Z5 


5 


17.8 


'* 


'-J H 


3 1/16 


8.4 


*' 


X 


3 5/10 


5 1/16 


20.2 


Zll 


H 


2 11/18 




9.7 


** 


3 ^ 
3 \i 


5 >6 


22.0 


" 


7/16 


a H 


8 1/16 


11.4 


Z6 


11/10 


5 


23.7 


Z12 


H 


2 11/16 


8 


ia.5 














D/IO 


2 « 


3 1/16 


14.2 



SIZES AND WEIGHTS OF STRUCTURAL SHAPES. 179 











Pen 


Goyd Steel 


Anffles. 












EVEN Lisas. 




Approximate Wei«:iit in Pounds per Foot Tor Various 




Thicknesses in Inches. 


Size in 




Inches. 


j& Win 


!i 


"ilS 


% 






% 


|]M0 % 


"/■IS 


% 


"^2 


1 
1.00 


8> s 
















3H.2 


ae.fi'ag.o 


42.4 


45.8 


49.8 


52.8 


Cxi 










I4.KI7.3 


]» T'-.'-J.O 


'M 4 


'^.f-^^.? 


81.0 


38.4 


85.9 




5 *i 










1^.8^14^ 


;e.jj 


m.2 


aoj 


rj-O'ja.s 


25.6 


27.4 


29.4 




4 xA 








S.3 


9.«11.8rJ.H 


11.,^ 


IE? t* 


17^ m.G 










^^^^ 








T,l 


fiSi\ 1kfi\\\.l 


t'^.l 


1.1.7 














8 >% 






4.0 


fl.l 


7,y 


HX 9.\ 


10. 1 


n.& 














^ V ia^ 






4.B 


ft.S 


B.e 


7 7 


s.d 


















^-m 




SJ 


4 1 


R.O 


s.e 


eiij 


T.H 


















2^4 xiZ; 




3.;i3.6 


4.5 


&.4 






















2 ^a* 




a.S,3.3 


4.0 


4.e 






















lH^l^ 




S.l ii.S 


3.S 


4.1 






















l^xlij 


l.S 1.8|3.4 


3.8 


S.5 






















1^-T>4 


l.rt;; i.5:iJ.O 


























1 X 1 


."••i 'V'* 


























UNEVEN LEGS. 




Approximate Weight in Pounds per Foot for Various 
Thicknesses in Inches. 


Size in 


Inches. 


.!*» 


.1875 


1 


%1S 


^. 


^iil! 


^ 


.MS.^ 


^i 


11/16 
.6876 


M 


'.V.i§ 


^» 


Xt 


1 
1.00 


H x8 














$3.0 


^h 


'^^^.7 


81.7 


3JI.R 


30.6 


39. r, 


42.5 


45.6 


: x3H 

6H*4 
6 x4 














iro 


so.o 


Jl.O 


23.0 


24.8 


28.7 


28.6 


30.5 


32.5 










12.9 


15.0 


17.0 


lt>.i> 


i^.2 


23.4 


25.6 


27.8 


29.8 


31.9 












Vi.2 


14.3 


ia.3 


3K.I 


iu.l 


22.0 


28.8 


a&.ts 


27.4 


29.4 




6 xSU 










11.6 


13.6 


l.K'. 


IT 1 


]:.o 


20.8 


22.6 


24.5 


26.5 


28.6 




54-^ 










11.0 


la.s 


] + .rl 


](i.'J 


]-.9 














5 x4 










11.0 


12.ti 


n.G 


10. li 


j:.9 


19.6 


21.8 










I If" 








8.7 


10.3 


I«.0 


13.6 


15.2 


16.8 


18.4 


20.0 
















8.2 


Q.7 


11.2 


12.8 


14.2 


15.7 


17.2 


18.7 










44x3 
4 «3H 
4 X.) 








T.7 


9.1 


10.5 


11.9 


13.3 


14.7 


16.0 


17.4 
















7.7 


9.1 


10.5 


11.9 


13.3 


14.7 


16.0 


17.4 
















7.1 


8.5 


9.8 


11.1 


12.4 


13.8 














2^x3 

34«2H 

3^x2 

3 xe^ 








6.6 


7.8 


9.1 


10.3 


11.6 


12.9 


















«•? 


6.1 
5.5 


7.8 
6.6 


8.3 


9.4 






■ 
















*•? 


5.5 


6.C 


7.7 


8.7 


















8 xi 






^•i 


5.0 


5.9 


6.9 


7.9 


















S4x2 
»4 X IH 




2.7 


^'n 


4.5 


6.4 


6.2 


7.0 




















2.8 


K 


3.7 


4.4 






















2 xlS 




2.1 


'^l 


3.6 


4.3 






















3 xl^ 




l.» 


2.« 


8.8 


8.9 






















ANGLE-COVERS. 


SizA in 
Izicheft. 


8/16 


M 


5/16 


% 


7/16 


H 


9/16 


H 


3 x3 




4.8 


5.9 


7.1 


8.2 


9.3 


10.4 


11.5 




8.0 


4.4 

4.0 


5.5 
5.0 


6.6 
6.0 


7.7 
7.0 


8.8 
8.1 






«^y 2W 


2.6 


8.5 


4.4 


5 3 










i x2 


2.4 


8.3 


4.0 


4.8 











180 



MATERIALS. 



SQUARE-ROOT ANGLES. 





Approximate Weight in Pounds 
l)er Foot for Various Tbiclcnesi»es 




Approximate Weight to 
Poiiuda per Foot for 
Various Thiclcuesses 






Size in 


iu luclies. 


Size in 


in Inches. 


luches 




Inches. 






1 


'^^ 


If* 


7/16 


% 


9/16 % 


^ 


3/16 


H '5/16| H 




.4375 


,5625 .6:^5 




.1«5 


.18Y5 


.25 


.3125' 375 


4 x4 






8 


11.4 


18.0 


14.616.2 


2 k2 






38 


4.1 


4.9 


8V6x3»^ 




7.1 


8.5 


9.9 


11.4 






\n:\n 






2.9 


8.0 


4.4 


8 x3 


4.9 


6.1 


7.2 


8.3 


9.4 








1.80 


2.4 


8.0 




3^ x2<^ 


4.5 


5.6 


6.7 


7.8 


8.0 






1)4x14 




l.M 


2.04 


2.56 




2j^ xiJ^ 


4.1 


6.1 


C.l 


7.1 


8.:^ 






1x1 


0.82 


1.16 


1.53 






2V4 >< -M 


8.6 


4.5 


5.4 














1 







Peneoyd Tees. 



Section 
Number. 



Size 
iu Indies. 



Weight 
per Foot. 



Section 
Number. 



Size 
in lucliea. 



Weight 
per tooU 



EVEN TEES. 



440T 
44 IT 
385T 
886T 

8srr 

35)OT 
83iT 
225T 
226T 
227T 
Ji22T 
223T 
220T 

iirr 

115T 
112T 

HOT 




2>,4x2^ 
2 x2 

134 « l^ 

1^4 «n4 

1 xl 



UNEVEN TEES. 



6 IT 
C."iT 
631' 
54T 
42T 



6x4 
Cx5W 
5x3^ 
5x4 



10 9 
13.7 
7.0 
9.0 
11.0 
6.5 
7.7 
5.0 
6.8 
6.6 
4.0 
4.0 
3,5 
2.4 
2.0 
1.5 
1.0 



17.4 
:39.o 
17.0 
15.3 
6.5 



48T 
44T 
4.'>T 
88T 
89T 
»i»T 
31 T 
82T 
33T 
34T 
35T 
3CT 
28T 
SOT 
2oT 
26T 
27T 
24T 
20T 
22T 
21T 
23T 
17T 
18T 
15T 
12T 



UNEVEN TEES. 



0.0 
10.2 
18.5 
7.0 
8.5 
4.0 
5.0 
6.0 

r.o 

8.0 
8.31 
9.5 
6 8 
7. -J 
8.3 
5.7 
6.0 

s.a 

2 
2 O 
2.5 
8.0 
1.9 
3.5 
1.4 





Peneoyd Mlaeellaneous Shapes 




Snction 
Number. 


Section. 


Size in Inches. 


Weight per Foot 
in Pounds. 


21 7M 

210>I 

aeoM 


Hoavy rails. 
Floor-bars. 


6 

3 1/10x4x3 l/If)xiito Jl^ 

•46xOx2j^xUlo% 


50.0 
7.1 to 14.8 
9.8 to 14.7 



SIZES AND WEIGHTS OP ROOFING MATERIALS. 181 

SIZES AND WBTGHTB OF ROOFING MATKRIAIiS. 

Corr abated Iron* (The Ciaciniiati Corrugating Co.) 

8CHKDCLB OF WEIGHTS. 



I 



x«. as 

No. 26 
No. 91 
No. 2i 
No. JO 
No. ^^ 
No. 16 



Tlilckiiess in 

deciiiial parts 

of All inch. 

Flat. 



.015685 

.01875 

.03.1 

.0SI25 

.0673 

.05 



Weight per 

100 sq. ft. 

Flat, Painted. 



75 " 

100 " 
1« 
150 

aoo •* 

850 •• 



Weight per 

100 sq. ft. 

Corrugated 

and Painted. 



70 IbR. 

84 *' 

111 •* 

i:« *• 

1«5 " 

S«0 *» 

876 " 



Weight per 

100 sq. ft. 

Corrugated 

and 
Ga1vatil»>d. 



06 lbs. 

09 •* 

1«7 " 

154 '♦ 

IKS •• 

«.% " 

«01 *• 



Weight in os. 

|)«r sq. ft. 

Flat, Uuiran* 

Uttfd. 



26M 



The abOT« table is on the basin of sheets rolled according to the U. S. 
Sundard Sheet-metal Gauge of ISM (see page 81). It is also ou the basis of 
^X%*n. corrugations. 

To estimate the weight per 100 sq. ft. on the roof when lapped one corru- 
gation at aides and 4 in. at ends, add approximately 12H< to the weighto per 
100 sq. ft., respectively, given above. 

0>migatfons S4 in. wide by ^ or ^ in. deep are recognisced generally aa 
the staiMlard siae for both rooflng and siding; sheets aie manufactui'ed 
usually in lengths 6, 7, 8, 9, and 10 ft., and iiave a width of 2CU or !36 in. out- 
»ide width— ten corrugations,— and will cover 2 ft. when lapped one corruga- 
tion at sides. 

Ordinary oomigated sheets should have a lap of IK or 2 corrugations s^lde- 
lai> for rooflng in order to secure water-tight side seams; if the roof Is 
rather steep 1}% comigations will answer. 

Some manufacturers make a special high-edge corrugation on sides of 
<>heeis ( rhe Cincinnati Corrugating Co.), and thereby are enabled to secure 
a water-proof side-lap with one corrugation only, thus saving from 6^ to 1^ 
of material to cover a given area. 

The usual width of flat" sheets used for making the abov>e corrugated 
matMial is 88^ inches. 

No. 28 gauge corrugated iron is generally used for applying to wooden 
buildiags; but for applying to iron framework No. S4 gauge or heavier 
should be adopted. 

Few manufacturers are prepared to corrugate heavier than No. 20 gauge, 
but noma have facilities for corrugating as heavy as No. 18 gauge. 

Ten feet is the limit in length of corrugated slieets. 

Galvanizisg sheet iron adds about 2>i oz. to its weight per square foot. 

Corroffated Arebea. 

For corrugated curved sheets for floor and ceiling construction in flre- 
ITiwt buildings, No. 16, 18, or 80 gauge iron is commonly used, and sbi^ets 
inav be curved from 4 to 10 in. rise— the higher the rise the stronger ib« 
arrh. 

By a series of tests It has been demonstrated that corrugated arches give 
tite moKt satisfactorr results with a base length not exceeding 6 ft., aiid 5 
fi. («r even lens is prerenible where great strength is required. 

These corrugated ai-ches are usually made with 2>^ X H ">• corrugations, 
snd in same width of slieet as above mentioned. 

Terra-€otta« 

P^vrous terra-cotta roofing 3" thick weighs 16 lbs. per square foot and 2" 
thick. 13 lbs. per square foot. 
Ceiling maae of the same material 2'' thick weighs 11 lbs. per square foot. 

TUes. 

Flaf files 6^" X t(H*' X %'* weigh from 1480 to 1850 lbs. per square of 
roof (100 square feet), the lap belnc: one-half the length of the tile. 
r«/M with grooves atul fillets weij^h from 740 to 92^} lbs. per square of roof. 
PttnrtiUs 14Mt" X lOMi" laid 10" to the wcatlier weigh BoO lbs, per square. 



182 



MATERIALS. 



Tin PIate«yri nned Sheet Sloel. 

ftie Usual sizes for rooflit^ tin are 14" X 90" ami 20" x 28". "Without 
allowance for lap or waste, tin rooflnK weighs from 50 to 62 lbs. per square. 

Tin on the roof weighs from fl*i to 76 lbs. per square. 

RooflnK plates or teme plates (8teeJ plates coaled with an alloy of tin 
and lead) are made only in IC and IX thicknesses (S7and 20 Birinliij^ham 
gauee). **Coke*' and *' charcoal'* tin plates, old names used when iron 
made with coke and charcoal was used for the tinned nlHle, are still used in 
the trade, although sleel plates have been substituted for iron: a coke plate 
now commonly meaning one made of Bessemer steel, and a charcoal plate 
one of open-hearth steel. The thickness of the tin coating on the plates 
varies with different ** brands.'* 

For valuable information on Tin Roofing, eee circulars of Merchant & Co.. 
Philndelpliia. 

The thickness and weight of tin plates were formerly designated in tho 
•trade, botb in tiie United States and England, by letters, such as I.C\, I>-C'., 
I.X., D.X., etc. A new system was iniriHiuced in tlie United Statrs in iwu«, 
knuivn as the "American base-box system.'* The base-box is a puck:tKe 
•containing 3*^000 square inches of plaie. The aciuai boxfsutied in llie trade 
•contaiu 60, 120, or 240 sheets, according to liio size. The ituiuber o: squuro 
inc les in any given box divided by 8^,000 is known as liie *' box ratio.*' Tltis 
ratio mnltiplied by the weigiit or price of the base-box givfS the weight i»r 
price of the given box. Thus tlie ratio of a box of 120 sheets 14 x 20 in. is 
83,600 -«- 82,000 = 1.0.->. and tlie price at $;).00 base is $3.00 X 1.05 = $3.15. I he 
following tables are fumisiied by the Anif rican Tin Piute Co.. Chicnj^o, ill. 
Comparison of Ganfireft and UTeletatii of Ttn Plates. 
(Butted oil U. a. oianduid Sheei-ineial UuuKe.) 



AMERICAN BASE-BOX. 
(.i2,000 sq. iu.) 
Weight. Qau{?e. 

^51os Ko. 38.00 



00 
(«3 
70 
75 
80 
H5 
90 
i« 
100 

no 

130 

no 

160 

180 

200 

220 

240 

2(30 ' 

280 

140 

IHt) 

220 

2i0 

280 



8U.TJ 

" 35. C4 

»' 81.9;i 

" 34 20 

" .33.4S 

•• 32.70 

" 32.04 

*• 81.32 

" 80.80 

** 30.0rt 

** 28. C| 

** 27.1»;i 

'* 20.18 

" 25. 5i 

*' 24. Ho 

•• 21. Os 

*' 23.30 

" 22.C4 

" 21.92 

, " 27.92 

" 2.J.5.' 

" 21.0.^ 

" 2.1. 3;i 

•» 21.92 

Amcracan PackaKcn Tin Plato. 



ENGLISH BASE-BOX. 
(31,^60 bq. hi ) 
Gauge. Weight. 

No.»<.00 54.4nits. 

'• 87.00 67.84 " 

*• 86.00 G1.24 ** 

" a*). 00 08. (V) " 

'• 34.00 74.8.-) *» 

" 8:1.24 WVOO " 

•• 82..'V0 85.00 '* 

'* 31.77 90.00 " 

*' 81.04 1«5 tW ** 

" 80.a5 100.00 •• I.C.L. 

" 30.00 108.00 •• I.e. 

" 28.74 120.00 *' 1 X U 

•• 28.00 i:«.00 ♦* 1 X. 

" 26. 4« 157. «H) *• l.-.:X. 

" 25.40 ]7S.0«» •* 1..3X. 

*• 24. C8 199. tX) " 1. IX. 

" 23. HI 2J0.00 " I r^X. 

•' 23.14 241.00 '* I.OX. 

'* 22..•^7 202.00 '* 1.7X. 

*' 21.00 28:1.00 " 1. i<X. 

'♦ 27. Sn 130 a) •' DC. 

'* 25. as 1<»0.00 * D.X. 

'* 24.24 211.00 *• D 2X. 

'• 23.12 212.00 " 1>. "iX. 

" 22 00 273.00 *' D. 4X. 



l4engLh. 



Inches 
Wide^ 

to 16?>^ S(]uare. 

17 " 2."j% Square. 

86 '* 30 8<iiiHre. 

9 " 10% .\11 lengths. 

n " 1111 To 18 in long, incl. 

11 " 1194 I8I1J and longer. 

12 •* 12J^To 17 in. loiiK, incl. 



Sheet > 
I)er Hoxj] 

240 
120 
GO 
240 
240 
120 
240 



In hfs I 
Wi;le. 



Length. 



12941 I7V4 and lonK«T. 

jl3 " 13»4 To 16 in. lonjr. InH. 
fll3 to Vi^4 10>4and longer. 
Dl4 '• 14514 To 1. "3 in. long, incl 
IJ14 •* n->4 ir)»4 rtnd longer. 

2:>?4 All li*njrtlis. 
I.'O '* 80 Alllenfirthit, 



Small aizen of light basH weights will be paclced in double lL>ozes. 



_ Sli»*»-ts 
'^H-r Box 

120 
240 
IJO 
2 40 
120 
390 
60 



SIZES AND WEIGHTS OF ROOFING MATERIALS^ 183 



Slate. 

Number and superficial area of slate required for ooe square of roof. 
(1 square = 100 square feet.) 



Dliiieiiflioiis 

in 

Inches. 


Number 

per 
Square. 


Superficial 
Area in 
Sq. Ft. 


Dimensions 

in 

Inches. 


Number 

per 
Square. 


Superflrial 
Area in 
Sq. Ft. 


6x18 
7x12 


588 
4S7 
400 
855 
874 
827 
291 
261 
277 
846 
221 
213 
192 


867 


12x18 
10x20 
11x20 
12x20 
14x20 
16x20 
12x22 
14x28 
12x24 
14x24 
16x24 
14x26 
16x26 


160 
169 
154 
141 
121 
187 
l;J6 
106 
114 
98 
86 
80 
78 


840 
285 


8x18 






9x12 
7x14 
8x14 
9x14 


254" " 


281 


10x14 






8x18 
9x16 


846 


228 


10x16 
9x18 
10x18 


240" 


225 









An slate is usually laid, the number of square feet of roof covered by one 
sJate can be obtained from the following formula : 
width xden g g.- 8 inches) ^ ^^^ ^^^ ^^ ^^^ ^^ ^, ^, ^,^^ 

Weight of slate of various lengths and thicknesses required for one square 
of roof : 



Length 

in 
Inches. 


Weight In Pounds per Square for the Thickness. 


«" 


3-16" 


J4" 


%" 


Ji" 


H" 


«" 


1" 


13 


483 


734 


967 


1450 


1936 


2419 


2902 


8872 


14 


460 


688 


920 


1379 


1842 


2801 


2760 


8683 


16 


445 


667 


890 


ia36 


1784 


2229 


2670 


3.^67 


18 


434 


650 


869 


l:«W 


1740 


2174 


2607 


8480 


80 


425 


637 


a*)! 


1276 


1704 


2129 


25.'i8 


3108 


28 


418 


626 


836 


1254 


1675 


2098 


2508 


88.'50 


94 


412 


617 


825 


12:38 


1653 


2(m 


2478 


8306 


26 


407 


610 


815 


1223 


1631 


2039 


2445 


3263 



Tiie weigiita given above are based on the number of slate required for one 
square of roof, taking the weight of a cubic foot of slate at 175 pounds. 

Pine Sblnsles. 

Number and weight of pine shingles required to cover one square of 
roof: 



Xumberof 
laches 



Exposed { 
Weather 



to per 



4 



Number of 

Shingles 

er Square 

of Roof. 



900 
800 
790 
665 
600 



Weight in 
Pounds of 
Shingle on 
One-square 

of Roofs. 



216 
192 
178 
167 
144 



Remarks. 



The number of shingles per square is 
for common gable-roofs. For liip- 
roofs add five percent, to these figures. 

The weights per square are based on 
the number per square. 



J 84 



MATEUIALS. 



Skylislkt GlaM. 

The wel{i:ht8 of various ftizes and thicknesses of fluted or rough p>a.e«^s:ias8 
required for one squara of roof. 



DlmenKioiis in 
Inches. 


Thickness in 
Inches. 


Area 
In Square Feet. 


WeiKht in Lbs. j>er 
Square of TUnyt. 


]Sx48 
15x00 
20x100 
04x156 


8-16 

i 


8.907 

6.246 

18.8H0 

101. T68 


350 

35C 
600 
TOO 



In the above taUle no allowance is made for lap. 
If oixlinary wludow-glass is used, siimle thick f^\a»» (about 1-16") will 'weSarh 
about 82 Um'. per square, and double thick glass (about %") will weigh about 
164 IbK. pel- pquare, no aUotrance being macie for lap. A box of ordinary 
M'indo\v-gla«<s contains at* nearly 50 vquare feet as the size of the panes wlU 
admit of. Panes of any size ai*e made to order by the manufacturers, but a 
great variety of sizes are usually kept in stock, ranging from 6x8 inches to 
S(5 X 00 inches. 

APPROXiniATE WBIGHT8 OF VARIOCS ROOF* 
COTKRINGS. 

For preliminary estimates t)ie weights of various roof coverings mcyilOl 
taken as tabulated below (a square or roof = 10 ft. square = 100 sq. ft.); 

Name. ^fiSKe^^ir 

Cost-iron plates (^'' thick) IfiOO 

Copper 80-185 

Felt and asphalt , 100 

Felt and gravel 800-1000 

Iron, corrugated 100-879 

Iron, galvanized, flat 100- 860 

Lath and plaster OOO-lOOO 

Sheathing, pine, V Uiick yellow, northern.. 800 

" ** •' *' southern.. 400 

Spruce, 1" thick 90O 

Sneaihing, chestnut or maple, V thick 400 

aKh, hickory, or oak, 1" thick.... 600 

Sheet iron 0-10" thick) 800 

" ** ** andhiths 500 

Shingles, pine 200 

Slates (J^4'^ thick)..-. SOO 

SkylightH (glass 8-16" to Ji" thick) 265. 700 

Sheet lead 600- 800 

Thatch 060 

Tin 70-125 

Tiles, flat l.'iOO-aOOO 

*• (grooves and fillets) TOO-lOOO 

" pan 1000 

" withmortar. 2000-yOOO 

ZIdc lUO- )iOO 

Approximate Ijoad* per Square Foot for Roofh ot t 
nnder 75 Feet, Includlnar H^elfl^tat of TruMU 

(Carnegie Steel Co.) 

Roof covered with corrngated sheets, unboarded 8 lbs. 

Roof covered with corruicated sheets, ou boards 11 " 

Roof covered with slate, on laths 18 ** 

Same, on boardK, U4 in. thick 16 " 

Roof covered with shingles, on laths 10 ** 

Add to above if plastered below rafters 10 ** 

Snow, light, weighs |>er cubic foot 6 to 12 ** 

For spans over 75 feet add 4 lbs. to the above loads per square foot. 

It is customary to add 80 lbs. per square foot to the aboTo fore 
wind when separate caloulatlous are not made. 



WEIGHT OF CAST-IRON PIPES OR COLUMNS. 185 



'WSI6HT OF €A8T-IRON PIPBS OR COIiVniiB. 

In I^bs. per I^Ineal Foot. 

Cast iron = 450 Iba. per cubic foot. 



; Thick. 
Bore. of 
Metal. 



Ins. ' Ins. 
3 



3H 
4 

5 

6 

t 

8 

9 

10 



Weight 
per Foot. 



Lbs. 
12.4 
17.2 
82.3 
14.3 
10.0 
S5.3 
16.1 
SW.l 
38.4 
17.9 
^.5 
31 5 
19.8 
87.0 
34.4 
31.6 
39.4 
37.0 
23.5 
81.8 
40.7 
35.3 
84.4 
43.7 
37.1 
36.8 
46 8 
39.0 
39.8 
49.9 
30.8 
41.7 
52.9 
44.3 
56.0 
68 1 
46.6 
59.1 
71.8 
49.1 
62.1 
73.5 
51.5 
65.3 



Bore r ^'of • ^'«'>»'t 



Ins. 

10 

ICW 

11 

n^ 

13 

12H 

18 

14 

15 

16 

17 

18 

19 

80 

31 

82 



Lbs. 
79.3 
54 
68 3 
82.8 
56.5 
71.3 
86.5 
58.9 
74.4 
90.3 
61.8 
77.5 
98.9 
68.8 
80.5 
97.6 
668 
88.6 
101.3 
71.3 
89.7 
108.6 
95.9 
116.0 
186.4 
108.0 
138.8 
145.0 
108.3 
180.7 
153.6 
114.8 
138.1 
168.1 
180.4 
145.4 
170.7 
136.6 
158.8 
179.8 
138.7 
160.1 
187.9 
i:«.8 



Bore, 



Ins. 
83 
S4 
25 
86 
87 



SO 
81 
88 
88 
34 
85 
86 



IDICK. 

of 
Meuil. 


Weight 
per Ifool, 


Ins. 


Lbs. 


A^ 


167.5 


yk 


196.5 


a \ 


174.9 


A 1 


305.1 


1 


23.^.6 


^ 


183.8 


818.7 


1 


845.4 


g 


189.6 


228.8 


1 


855.3 


§ 


197.0 
380.9 


1 


265.1 


^ 


204.3 


239.4 


1 


8T4.9 


^ 


211.7 
248.1 


1 


284.7 


^ 


219.1 


256.6 


1 


294.5 


% 


265.8 


1 


3W.8 


'^ 


8437 


873.8 


1 


814.3 


'^ 


a54.8 
882 4 


1 


824.0 




3C5.8 


^ 


391.0 


1 


3:«.8 


'^ 


376.9 
299.6 


1 


343.7 


'^ 


388.0 


3l)S.l 


1 


8.')3.4 




3H9.0 
310 6 


1 


363 I 


1« 


410.0 



The weight of the two flanges may be reciconed = weight of one foot. 



186 



MATERIALS. 



WBIGHT8 OF OJLST-IBON PIPE TO I«JLT 12 FEBT 
I«ENGTH. 

Welsbts are Gross yBVeigiktUf Including Hub. 

(Calculated by F. H. Lewis.) 



Thickness. 



i-bes. ^!j;;;- 



7-16 
15-JJ;f 

9-16 
19-3;^ 

.% 



I 



tiials. 



.875 

.4875 

.4d«7 

.5 

.nSl25 

.56^.5 

.5JW75 

.0-25 

.0875 

.76 

.8123 

.875 

.9375 

1. 

1.125 

1.25 

i.87r> 



Inside Diameter. 



209 
228 
217 
260 
286 
806 



304 
881 
858 
886 
414 
44-i 
470 
498 



8" 10" 12" 14" 16" 18" 20" 



400 
485 

470 
505 
541 
577 
018 
649 



581 
624 
668 
712 
756 
801 
845 

a35 

1026 



092 
744 
795 
846 
899 
951 
1U08 
3110 
1216 
1824 
1432 



804 
863 
922 
983 
1043 
1103 
1163 
1285 
1408 
1531 
16.'^ 
178:1 
1909 



1050 
1118 
1186 
1254 
1322 
1460 
lf>98 
17:i8 
1879 
iXtil 
2163 



1177 
1258 
1829 
1405 
1481 
16:tf) 
17l«9 
1945 
2101 
2259 
!»18 
2738 
806t 



1640 

1HI0 
1980 
21,52 
2»24 
2-4118 
2672 
3024 
3:iHa 



3389 I 3; 39 



Thickness. 


22" 






Inside Diameter. 






Inches. 


Equiv. 
Decimals. 


24" 


27" 


30" 


33" 


30" 


42" 


48" 


60" 


% 


.625 


1799 


















11-16 


.6875 


1985 


2160 


2422 














94 


.76 


2171 


2362 


2648 


21)84 


3221 


a507 








13-16 


.8li» 


2359 


2565 


2875 


81S« 


34WJ 


3«<-6 


4426 






H 


.875 


2:>47 


2709 


3ia3 


81i7 


3771 


4105 


4773 


6442 




I.C16 


.9375 


2737 


2975 


8832 


8690 


4048 


4406 


5122 


5839 






1. 


2927 


3180 


3562 


3942 


4325 


4708 


54?^ 


6236 




iWt 


1 125 


a3io 


3598 


4027 


4456 


4886 


5.<10 


6176 


7034 






1.25 


86y8 


4016 


4492 


4970 


5447 


.')924 


(5880 


7h33 


9r4£ 


1^2 


1.875 




4439 


4964 


5491 


(KH5 


6540 


75H1 


h640 


10740 


Jl? 


1.5 
1 625 
1.75 
1.875 
2. 






5439 


6012 
6539 


6584 
7159 
1737 


7lf)8 
77K2 
8405 


8:«3 

9022 
9742 
1()46H 
lllil7 


9447 

nweo 

11076 
11898 
12725 


117^ 


19? 






12744 


2 








13750 










14:62 














15776 




2.25 
















14885 


17821 


01/ 


2 fS 
2.75 


















19880 


hi 


















»966 



CAST-IRON PIPE FITTINGS. 



187 



CAST-IRON PIPB FITTINGS. 

Approximate l¥elsht. 

(Addyftton Pipe nnd 8t«?el Co., Cincinnati, Ohio.) 



SMze III 
In«.-h«*. 



tin Li)8. 



I ROSSES. 



•> 


41 


3 


110 


3x2 


9J 




1-30 


4x3 


114 


4x4 


90 




£00 


6x4 


160 


6x8 


160 


8 


325 


8xfS 


280 


Hx4 


265 


Sx3 


2-25 


10 


575 


10x8 


4 IS 


10x6 


450 


10x4 


31K) 


10x8 


;i50 


1-2 


740 


«xTO 


6.<)0 


lix8 


6-^J 


1-2x6 


540 


1-2 X 4 


525 


12x3 


495 


14x10 


750 


lix8 


U:» 


14x6 


570 


16 


1100 


16x14 


1070 


iexi2 


1000 


16xlo 


1010 


16x8" 


825 


lGx6 


700 


lGx4 


650 


18 


1560 


ao 


1790 


•20x12 


1370 


'JOxlO 


1-225 


20x8 


1000 


a)x6 


1000 


a0x4 


1000 


24 


2400 


•24x20 


20!» 


£4 X 6 


1^0 


ajxao 


2635 


30x12 


2250 


:«x8 


1995 



TEFS. 



2 

3 

1x2 

4 

4x3 

4x2 

6 

6x4 

6x3 

6x2 

8 

8x6 



76 
100 

90 

87 
IJW 
145 
145 

75 
300 
270 



8ize iu |VV>iKht 
Inches. I III Lbs 



TEES. 



8x4 
8x3 
Id 

10x8 
10x6 
10x4 
10x3 
1-2 

r-*xio 

1-2x8 

12x6 

12x4 

14 X 12 

14x10 

14 xS 

14x6 

lix4 

14x3 

16 

16x14 

10x12 

16x10 

16x8 

16x6 

16x4 

18 

20 

•20x16 

20x12 

20x 10 

20x8 

20x0 

20x4 

20x10 

'24 

24x12 

24x8 

24x6 

30 

30x24 

30x20 

80x12 

30x10 

80x6 

36 

36x30 

36x12 



2^i0 
2'iO 



370 

850 

810 

600 

555 

515 

550 

625 

6.-10 

650 

575 

545 

525 

490 

790 

850 

850 

H50 

765 

680 

665 

1235 

1475 

1115 

1025 

1090 

900 

875 

845 

1465 

2000 

1425 

1875 

1450 

30*25 

2640 

2:i00 

2035 

2050 

1825 

5140 

4200 

4050 



45'» BRANCH 
PIPES. 



3 

4 

6 
6x6x4 

8 

8x6 
24 
24x24x20 
SO 
86 



90 

125 

205 

145 

8-)0 

3:^ 

2765 

2:45 

4170 

10800 



Size iu iWeiKliL 
Inches. I in Lbs. 



SLEEVES. 



2 
3 
4 

6 
8 
10 
12 
14 
16 
18 
20 
24 
80 
86 



10 
25 
45 
65 
80 
140 
190 
208 
850 
375 
500 
710 
965 
1200 



90« ELBOWS. 



2 


14 


3 


34 


4 


55 


6 


120 


8 


ISO 


10 


360 


12 


870 


14 


450 


16 


660 


18 


8S0 


20 


900 


24 


1400 


30 


3000 



^ or 45° BENDS. 
3 



4 
6 
8 
10 
12 
16 
18 
20 
24 
80 



70 
95 
150 
i!00 
290 
510 
580 
780 
1425 

atoo 



1/16 or 22^0 
BENDS. 



8 
10 
12 
16 
24 
30 



15-5 
205 
260 
4.50 
1-280 
-2000 



REDUCERS. 



3x2 
4x3 
4x2 

6x4 
6x3 
8x6 

Hx4 



25 
42 
40 
95 
70 
126 
116 



Sizeiu iWeiKht 
Inches. I in Lbs. 



REDUCERS. 



8x3 
10x8 
10x0 
10x4 
12x10 
12x8 
12x6 
12x4 
14 X 12 
14 X 10 
14x8 
14x6 
16 X 12 
16x10 
20x16 
20x14 
80x12 
20x8 
24x20 
80X-24 
30x18 
86x30 



116 
212 
170 
160 
820 
250 
250 
260 
475 
440 
800 

475 
435 
690 
675 
540 
400 
990 
1305 
13S5 
1730 



ANGLE REDUC- 
ERS FOR UA S. 
6x4 I 95 
6x 8 J^ 70 

S PIP ES. 
4 I 105 



PLUGS. 



2 


3 


8 


10 


4 


10 


6 


15 


8 


80 


10 


46 


12 


66 


14 


90 


16 


100 


18 


180 


20 


150 


24 


185 


30 


370 





CAPS. 


3 




20 


4 




25 


6 




60 


8 




75 


10 




100 


12 




120 



DRIP BOXES. 



4 
6 
8 
10 
20 



295 
330 
375 
8:5 
14-20 



188 



MATERIALS. 



WEIGHTS OF CAST-IRON WATER- AND GAS-PIPB. 

(Addyston Pipe and Steel Co., Cincinnati, Ohio.) 

I Standard Qos-pipe. 





Standard Water-pipe. | 


Per Foot. 


Thick- 
ness. 


Per 

LeiiRth. 


2 


7 . 


5/16 


63 


3 


15 


9i 


180 


3 


ir 


1^ 


m 


4 


2i 


^ 


364 





83 


\2 


396 


8 


42 


^ 


504 


8 


45 


H 


540 


10 


CO 


9/10 


7iK) 


12! 


75 


9/16 


000 


14 


117 


?4 


1400 


10 


];» 


^ 


1500 


18 


167 


U 


i?000 


20 


iiOO 


iS/lC 


2400 


Hi 


^50 


1 


aooo 


90 


350 


1% 


4','00 


80 


475 


5700 


42 


600 


]K^ 


7:J00 


48 


775 


IH 


9300 


(SO 


1330 


3 


15960 


72 


lais 


^H 


JKOJO 



9 
8 

4 

6 

8 

10 
13 
14 
16 
18 
30 
24 
30 
36 
43 
48 
6U 





Thick- 


Per 


1 Per Foot. 


ness. 


LeiiKth. 


6 


H 


4S 


12^ 


5/16 


150 


17 


H 


2W 


80 


7/16 


860 


40 


7/16 


4b0 


50 


7/16 


600 


70 


H 


640 


84 


0/16 


1000 


100 


9/10 


13U0 


134 


11/16 


1600 


150 


11/16 


1J<00 


184 




3300 


250 


9% 


3000 


sno 


rl 


4300 


417 


15/16 


5000 


543 


]^ 


6500 


900 


1^6 


lOHX) 


1.150 


IH 


15000 



THICKNESS OF CAST-IRON WATER-PIPES. 

P. H. Baermann, In a paper read before the Enfrineers^ Club of Phila- 
delphia in 1H83, {;ave tweniy fliffereiit forniulag Tor determininK the thick- 
nesH of east-iron pipes under presHure. The formulas are of three classes: 

1. Depend injc upon the diameter only. 

2. Those dependiog upon tiie diameter and head, and which add a ccn* 
stant. 

3. Those dependinfc upon the diameter and head, contain an addlUre .)r 
Bubtractive term dependinf? upon the diiimeter. and add a constant. 

The more modern fornmlos are of tue thiixl class, and are as follows: 

t = .OOOOHAd -4- .Old + .36 She<ld, No. 1. 

t = .00i)0G/id -f Oisati + .296 Warren Foundry, No. 2. 

t = .OOOOSH/td 4- .0133d -|- .812 Francis, No. 8. 

t= .0000l8/id 4- .0l5d 4- .33 Dupuit, No. 4. 

<=: .00004ftd + .l 4/d-f .15 Box, No. 5. 

t = .00U135/id + .4- .OOlld Whitman, No. 6. 

t = .0<X)06(/i 4- 230)d 4- .mi - .0038d Fanning, No. 7. 

t = .000l5/»d + .35 - .0053d ....Meggs, No. 8. 

In which t = thickness in inches, h = bead in feet, d =3 diameter in inches. 
Rankine, *' Civil Engineering," p. 721, says: ** Cast-iron pipes should be 
maile of a soft and louKh qualitv of Iron. Ureat attention should l>e paid 
to iiiouldingthem correctly, BO that the thick nf^ss may be exact Ij' uniform all 
roil lid. Each pipe should l)e tested for Jr-bub^les and flav h by ringing !e 
with a hammer, and for aren^th by ex)X>8mg 't to "ou »e th- iiiteiidv'd 
greatest working presbure.' The rule for comp..ting the .hickness of a p:;.© 

to resist a given working pressure is < =s ^, where r Is the radius In Inch;t.i, 

p the pressure in pounds per square inch, and / the tenacity of the iron iwr 
square inch. When/ = IHtXX), and a factor of safety of 5 is ujmkI, the abuvs 
expressed in terms of d and h becomes 

' = "3600"== 16lii>8= •^^*^^^^^^^*'*- 

"There are limitations, however, arising from difficulties In casting, aud 
by the strain produced by shocks, wtiicli couse tlie thickness to be uiado 
greater than tliat given by the above formula.** 



THICKNESS OP CAST-IRON PIPE. 



189 



TIftfekneas of Metal and Weight per I^en^tli for IMIDDreiit 
Sizes of Cast-iron Pipes under Various Heads of UTater, 

(WarreD Foundry and Machine Co.) 





M 


100 


laO 


soo 


250 


soo 




Ft. Head. 


Ft. Hea<l. 


Ft. Mvad. 


Ft. J 


tad, 


Ft. Head. 


Ft. Head. 








8I». 


11 






en? 


P 

ll 


il 


11' 




1-3 




II 


1^ 




go 


§o 


So 


''a 


s», 


"i 


g^ 


^l 


go 


"1 


S 


.944 


144 


.858 


III) 


.fl«a 


isa 


.B71 


157 


.880 


161 


.890 


166 


4 


.861 


107 


.878 


2<U 


,26^ 


ni 


.3JI7 


318 


.409 


228 


.421 


285 


& 


.S78 


254 


.803 


»i.> 


.40t+ 


27Ji 


.4x':J 


286 


.488 


298 


.453 


.309 


6 


.803 


315 


.411 


t'A} 


.4l» 


S«i 


.-147 


;»] 


.466 


877 


.483 


393 


8 


.422 


445 


.450 


i-r> 


.474 


WIS 


41^ 


!529 


.622 


657 


.546 


684 


10 


.459 


600 


.480 


eit 


..MH* CRi 


.hV.i 


723 


.679 


766 


.609 


808 


It 


.491 


768 


.627 


e-*; 


..^^3 


*tH-i 


.SJri) 


1M4 


.685 


1004 


.671 


1064 


U 


.BcM 


95S> 


.666 


10;n 


.mW 


un 


.fCHI 


noi 


.692 


1272 


.734 


1352 


16 


.567 


1152 


.601 


12:.:^ 


.65:2 


lae^ 


.7(^1 


1168 


.748 


1568 


.796 


1673 


18 


.560{ 


1370 


.648 


IS^Nh 


.01»7 


Tf-ftii 


.7:. 3 


1761 


.805 


1894 


.ft'iO 


2026 


iO 


.tt23{ 


1603 


.682 


17^:^ 


.74iJ 


1EW4 


.NH.^ 


.086 


.862 


2248 


.92-.> 


2412 


24 


687 


81-20 


.759 


28iLi 


.R1I 


S^'jfiO 


.tMi:i 


J ^11 


.975 


8045 


1.047 


3379 


to 


.785 


80-JO 


.876 


8]]^'t; 


.H65 


8735 


!Jir.:. 


J')95 


1.145 


4458 


1.235 


4822 


S6 


.8^ 


4O70 


.990 


4fiM 


I m>H 


B*M>6 


l,i.^5 


:n8 


1.814 


6188 


1.422 


6656 


4i 


.980 


&265 


1.106 


6flr.^ 


] Si;! 


(5057il.S5K 


:m 


1.484 


8070 


I.CIO 


8804 


48 


1.078 


6616 


1.222 


75:j; 


1.366 


SI.*Jl 


1.510 


mio 


1.654 


10269 


1.798 


11105 



AH pipe cast Tertlcally In dry sand; the 8 to 12 inch in lengths of 12 feet, 
&li larger sises in lengths of 12 feet 4 inches. 



talto Pressures and EqulTalent Beads of Water for Cast* 
Iron Pipe of miTerent Sizes and Tlilcknesses. 

(Calculated by F. H. Lewis, from Fanning's Formula.) 





SlixnttlHpc 


\ ..It* 

■ it 

r 

i 1* 


4" 


«*' 


g*# 


10" 


IS" 


14" 


,«.. 


18" 


20" 


^1 

II 

Sf 


s . 

II 


II 
11 

St c 




II 

1 


5j 

n 

*S9 


^:4 
II 


ll 

3Jit 

m 


li 

SJ 
Hi 


c 



aifl 

bit 
.... 


If 
IS 

13* 
iMi 




lit 

130 


Mil 


ll 

t£ 

41 

m 

Ul 

Lta 

HI 

ivi 

Sll^ 


ll 

l.-ra 
:£1<1 


74 

111 

ISi 


|l 

lis 
i:<i 
ifcji 

saft 

€81 

BSD 



190 



HATERIALS. 



Safe PresMureii, etc., 


for € 


iifit-lroii 


Plp«.-<Cbn(toi<«t.) 






hijf:n 1.1 E iiyiv. 




312" 


ti" 


87" 


30' 


»»" 


30" 


42" 48" 


or* 


Whkk^ 












































1- 


5j 
II 

3 


pi 


ll 




I 5 


ti 


5 , 

is 










HI ' 


~iW 


It 


fill 




h 


E- 




£": 


''^ ^ a 






Ih 


c; 


— c 


3" 




s 






4(t 


9i 


»> 


1H» 


la 


«l 






^ 


















ll-t0 
















" 




3-^ 


IHI 


\9X 


is 


113 


M 


(in 


u 


Fkl 






















IS- 1ft 


HO 


IM 


w 


irj7 


fkU 


]»i 


3> 


Wl 






















7H 


101 


403 


iW 


1BA 


rtfl 


la* 


&4 


t^ 


« 


in 


3S 


7, 














l&-t« 


in 


-•TV 


lift 


n^\ K,1 


ii« 


^ 


litf 


.y, 


1,^ 


44 


10 














t 


Mi 


Hi? 


1^4 


.i>i^ niy 


sety 


Ki 


m 


fiy 


IMJ 


Tn 


l!l 


3* 


«^ 


m 


«i 






1 t-H 


i*i 


410 


ir,l 


j;] i;tj 


ail 


111 


-IhI 


Mti L^-il 




m 


W 


134 


ti 








t 1-4 


m 


&1Q 


IW 


IM l«9 


ami 


IJI 




I'l i^ 


107 


M 


lit 


117 


« 


113 


r ■ 


m 


I ^^ 






«S7 


&A5 HK 


w> 


in 


IMI 


III, :hs 


13^ 


» 


]«9 


137 


SI 


lit7 


4» 


tti 


i 1 « 








...H JPfl 


w* 


aM 


i-ii ITS 


4]il 


411? 


3ft 


»U 


HI 


ft 


M 


M 


117 


1 34 














:£H 


.??• 




4;^ 183 

&CT7, 5Sii: 


fit 14A 


31ft 


HA 

m 


^3 


ft 


IS? 








.... 






4" 


1*7 


17-« 






, ..d 


,+.* 




fiik 


.11 












lot 


iV 


m 


»7 


1«* 


«1 


■ 















t*^. 


... 




.,» 




.... 


tVi 


Ml 


m 


4#t 


t«l 




i J^ 


*.t* 




.... 








kl.1 


«... 




,. + - 


TF-. 


-.#. 


...4 


*-.. 


m 


4i» 


t9» 


iMB 


t 1-1 






r... 


.... 






»... 


i... 




+ ..« 




-+-► 


.... 




ait 


IM 


ISI 


m 


Kl-i 


—* 












*... 


.,.. 




.... 




***■ 




.... 






1*T* 
314 




^*" 





1 










I""" 




»» , 






*,.. 



Note.— The absolute safe static pressure which may be 

put upon pipe Is Riven by the formula P s= ^ X ■=-, in 

which formula P is the pressui-e per square inch; T. the 
thickness of the shell; 5, the ultimate strentrth per square 
inch of the metal in tension; and D, the inside diameter of 
the pipe. In the tables S is taken as 18000 pounds per 
square inch, with a worlcinf? strain of one flfth this amount 
or 3600 pound*: per square inch. The formula for the 

T200r 
absolute safe static pressure then is: P = . 

It is, however, usual to allow for •* water-ram " by In- 
creasinjc tho thickness enoueh to provide for 100 pounds 
atiditional static pressure, and, to insure sufficient metal for 
good casting,' and for wear and tear, a further increase 

equal to .333 (l - ^^). 

The expression for the thickness then becomes: 

and for safe working pressure 

P=^(r-.«»(t-5^))-m 

The additional section provided as above represents an 
increased value under static pressure for the different sizes 
of pi|^ as follows (see table in margin). So that to test 
the pipes ud to one fifth of the ultimate strength of the 
material, the pressures in the marginal table should be 
added to the pressure- values given in the table above. 



r = 



Size 




of 


Lbs. 


Pipe. 




4" 


6T6 


6 


476 


8 


346 


10 


3IG 


1:3 


276 


14 


848 


16 


',>28 


18 


)a09 


SO 


396 


Si 


185 


24 


176 


27 


3 as 


80 


356 


88 


149 


80 


14.^ 


43 


J 34 


48 


128 


60 


116 



SHEET-IRON HYDRAULIC PIPE, 



191 



SHSBT-IBON HTJIIRA17I«I€ PIPE* 

(Pel too Water-Wheel Co.) 
Weight per foot, with safe head for varioas sizes of double-riveted pipe. 



z 

u 

is 


)i 


III 


ife Head 
in Feet the 
Pipe will 
stand. 




la 




ill 


ife Head 
in Feet the 
Pipe will 
stand. 


III 




< 


p ^ 


S 


^ 




< 


3i 


15 


in. 


sq.in. 


B.W.G. 


feet. 


lbs. 


in. 


so. in. 


B.G.W. 


feet. 


lbs. 


3 


7 


18 


400 


2 


18 


854 


16 


165 


16^1 


4 


12 


18 


850 


m, 


38 


254 


14 


258 


ai? 


4 


12 


16 


685 


8 


38 


2.M 


12 


8S5 


27]^ 


5 


a> 


18 


885 


^ 


38 


254 


11 


484 


80 


5 


20 


16 


500 


38 


254 


10 


505 


34 


5 


20 


14 


675 


6 


80 


814 


36 


148 


18 


6 


88 


38 


296 




SO 


814 


14 


227 


28^ 


6 


28 


16 


4S7 


80 


814 


18 


846 


80^ 


6 


28 


14 


748 


7^ 


20 


814 


11 


880 


^ 


T 


» 


18 


251 


5^ 


20 


814 


30 


456 


!• 


W 


16 


419 


69j. 


22 


880 


36 


1.% 


80 


7 


38 


14 


640 


8U 


28 


3KG 


14 


206 


244^ 


8 


SO 


16 


867 


71^ 


28 


880 


18 


816 


82^ 


8 


50 


14 


500 


91^ 


28 


880 


11 


847 


3591 


8 


50 


12 


854 


18 


22 


380 


30 


415 


40 


9 


& 


10 


887 


8^ 


24 


458 


14 


188 


'^ 


9 


63 


14 


499 


1041 


24 


4.V2 


38 


890 


9 


GS 


12 


781 


14H 


84 


458 


13 


318 


SfT 


10 


78 


16 


895 


0^4 


24 


452 


30 


8:9 


43^ 


10 


78 


14 


450 


iis^ 


24 


452 


8 


466 


53 


10 


T8 


12 


687 


1591 


86 


580 


34 


175 


20^ 


10 


78 


11 


754 


17^ 


26 


580 


32 


267 


8H^ 


10 


78 


10 


90U 


JOW 


86 


m) 


33 


294 


42 


11 


95 


16 


869 


9^ 


26 


530 


10 


858 


47 


n 


05 


14 


418 


13 


26 


&30 


8 


432 


57W 


n 


96 


12 


686 


1^ 


88 


616 


14 


168 


31^ 


11 


95 


11 


687 


28 


615 


12 


247 


41^ 


11 


95 


10 


880 


81 


28 


615 


11 


278 


45 


li 


IW 


16 


246 


11^ 


28 


615 


30 


387 


50>4 
61^ 


12 


113 


14 


377 


14 


28 


615 


8 


400 


li 


118 


12 


674 


38W 


80 


706 


38 


231 


44 


li 


113 


11 


680 


199a 


SO 


706 


33 


254 


48 


It 


113 


10 


758 


289l 


80 


706 


30 


304 


54 


n 


\9i 


16 


288 


18 


80 


706 


8 


875 


66 


n 


138 


14 


348 


15 


80 


706 


7 




74 


u 


m 


12 


6H0 


20 


86 


1017 


31 




58 


IS 


138 


11 


688 


28 


86 


1017 


10 




6? 


u 


138 


10 


696 


84^ 


86 


1017 


8 




78 


14 


153 


16 


211 


13 


86 


1017 


7 




bS 


14 


158 


14 


884 


16 


40 


1256 


30 




71 


14 


153 


12 


494 


^ 


40 


1850 


8 




86 


11 


153 


11 


643 


40 


1256 


7 




97 


14 


163 


10 


648 


26 


40 


3256 


6 




108 


15 


176 


16 


197 


!?« 


40 


18r,6 


4 




126 


15 


178 


14 


308 


42 


1885 


10 




74>< 


15 


176 


12 


460 


23 


48 


1885 


8 




91 


15 


176 


11 


507 


24^ 


48 


13W> 


7 




108 


15 


178 


10 


006 


28 


42 


1 885 


6 




114 


I'j 


201 


16 


385 


14^ 


48 


laa-i 


4 




133 


15 


m 


14 


283 


17^ 


42 


138.') 


M 




187 


IS 


801 


12 


4S8 


84^4 


48 


188.'> 


3 




145 


15 


801 


n 


474 


.26^ 


42 


18.S5 


5-16 




177 


IS 


901 


10 


667 


m 


48 


1385 


H 




216 



193 



HATRRIALI. 



8TANBARB PIPE FLAIfOBS. 

Adopted Aufcust. 1804, at a conference of committees of the American 
Society of Mechanical Engineers, and the Master Steam and Hot Water Fjr- 
ters' Association, with represent atives of leadinfc manufacturers and it^rs 
of pipe.— Trnns. A. S. M. E.. xxi. •^. (The standard dimensions Riven have 
not yet, 1901, Iwi-n adopted by some nrianufncturers on account of Uieir un 
willingness to make n chan^^e In their patterns.) 

The list Is divided into two grroups; for medium and hljrh pres.«nreR, lh€ 
first ranfirinir up to 76 lbs. per square incli. and the second up to 'JOO lt». 



s. 

2 

4 

« 

7 

8 

9 
10 
12 
14 
15 
10 
18 
JiO 
2-i 
24 
20 
28 
30 

m 

42 

48 



I 






r-f I 



.409 
.429 
.44S 

.4G0 
.480 
.498 
.525 
.568 
.60 
.639 
.678 
.713 
.79 
.864 
.904 
.946 
1.02 
1 09 
1.18 
1.25 
1.30 
l.:« 
1.18 
1.71 
1.8? 
2.17 




2^^ 

I2H 



31 tl 3l«li24 1 

•1V4 -m 34 - ■ 

m ir>^ 36 

47„42 429432 

r.J^ \^}i 49^ :i6 1 
h^ r^4^ 56 44 I 



the 



2041) ^^ :i6 
20iH) 14 :i8 
1920 ^4 44^ 
210»H4.')1 

z^'-^UJ^'}^ _^M[;: 

NoTKS.— Siz«^s up to 24 inches are desipjiied for 200 Ihs. or less. 

Sizes from 24 to 48 inches are dividetl into two scales, one for 200 lbs, 
other for les.«». 

The sizes of bolts ^ven are for high pressure. For medium pressures the 
diameters are ^ in. less for pipe.s 2 10 \H) \n. diameter Inclusive, and^ in. 
less for larger hizes. except 4H-jii. pipe, for ^liich the size of bolt is 1^ in. 

When two lines of figures occur under one hemling, the single ctilumns are 
for both medium and high pressures. Beginning with 24 inches, the left-hand 
columns are for medium and the right-hand lines are for hlgli pressures. 

The sudden increase in diameters at 10 inches is due to the (X)sslble inser- 
tion of wroiight-iron pipe, making with a nearly constant width of g^asket a 
greater diameter desirnble. 

When wrought-iron pipe is used, if thinner flnnges than those given are 
surtlcient. It is prop* 've*! that bosses be use«l to bring the nuts up to the 
standard lengths. This avoids the iir<" of a reinforcement around tne pipe. 

Figures in the S*!, 4th, 5th, and last columns refer only to pipe for high 
pressure. 

In drilling valve flanges a vertical line parallel to the spindles should be 
midway between two lioles on the upper side of tlie flanges. 



OAST-IROK PIPE AND PIPB PLAKGES, 



193 



TCBISNSlOIfS OF PIPB FLANGES AND CA8T-IRON 
PIPK8« 

(J. £. Codman, Engineers' Club of Philadelphia, 1889.) 






2 
8 
4 
5 
6 
8 
10 
12 
14 
16 
18 
20 
22 
24 
28 
28 
30 
8^ 
81 
86 
88 
40 
42 
44 
46 
48 






PI 



15 






4 
4 
6 
6 
8 
8 
10 
12 
14 
16 
16 
18 
20 
22 
24 
24 
26 
28 
80 
82 
82 
84 
84 
86 
88 
40 



n 



H-ie 



ftae 

1 

1 1-16 
11,6 

11-16 

1 Mtf 

1 11-16 
ll8-16 

2 

2 1-16 



Thickoess 
ol Pipe. 



Frac. Dec. 



t-16 
7-16 
15-82 

^\fi 
19-82 
21-82 
11-16 



27-32 

?ii6 

81-82 
1 
1 1-16 

15-82 
1 8-16 

11,6 
11182 



.878 

.896 

.420 

.448 

.466 

.511 

.657 

.603 

.649 

.695 

.741 

.787 

.838 

.879 

.925 

.971 

1.017 

1.063 

1.109 

1.155 

1.201 

1.247 

1.298 

l.J 

1.885 

1.481 



|1& 



6.96 

11.16 

15.84 

21.00 

26.64 

39.86 

54.00 

70 .56 

89.04 

109.44 

131 .76 

156.00 

182 16 

210.24 

240.24 

272.16 

806.00 

841.76 

879.44 

419.04 

460.56 

604.00 

549.36 

696.64 

645.84 

696.96 






4.41 
5.96 
7.66 
9.68 
11 82 
10.91 
23.00 
80.18 
88.34 
47.70 
58.28 
70 00 
88.05 
97.42 
113.18 
180.35 
149.00 
169.17 
190.90 
214.^26 
289.27 
266.00 
294 49 
324 78 
3.^6.94 
391.00 



D = Diameter of pipe. All dimensions In inches. 
FoBMULX.— Thickness of flange = 0.033D + 0.56. 
Thiciiness of pipe == 0.023Z> -f 827. 
Weight of pipe per foot = 0.24I>2 -f BD. 
Weight of flange = .001 Z)» 4- 0.1 D« + D -f 2. 
Diameter of flange = 1 . 1252) + 4 . 25> 
Diameter of bolt-circle = 1 .092D + 2.666. 
Diameter of bolt s O.OllD + 0.78. 
Number of bolts = 0.78D -f 266. 

PIPB FLANGES FOR CIIGH 8TEAJH-PBE88CBE. 

(Oliapman Valve Mfg. Co.) 



Size of 


Diameter 


Number of 


Diameter 


Diameter of 


Length of 
Pipe-Thread. 


Pipe. 


of Flange. 


Bolts. 


of Bolts. 


Bolt Circle. 


Inclies. 


Inches. 




Inches. 


Inches. 


Inches. 


r 


V 



6 




g 


6^ 


lit 


m 


9 


7 




,' 


7Jc 


1 7-16 




10 


8 




i 


7?2 


1 9-16 


m 


10^ 


8 




i 


^^ 


1 11>]6 




rT 


9 




\\ 


9Vi 


1 18-16 




18 


10 




^ 


10^ 


i% 




14 


12 






11^ 


1 15-16 




15 


12 




4t 


IS 


2 




16 


18 




2 


14 


2 


10 


mi 


15 




^ 


15V4 


^H 


IS 


20 


18 




^ 


i 


2^ 


14 


88 


18 


1~ 


^n 


U 


28W 


18 


1 


296 



194 



HATERIALS. 



■qooijod 
tpvdjqx 



t 
h 

il 

P 

^=- 

an 



'pvantx 

iJMJO 






i«fc?2SS2^ri^^** ****** ******** 









■adidjo 
•uoiivo 

_:8 a 



iiiiii|yillili|i§§is§igsiissii 



I if.iuoa 



h|iJ$'4SS2 






- *- — »" et a«» oe 









^ ^ ^ « 91 e* 



§Hpil§ggS^giii!^ygi§i^i§g|g§ 



i-iMoe^c»Q»7*ioe» 



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qijfiuri 

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9pi«ui 

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adMjo 



Ui§i§lliliii§l.s§lisri§§§ga§§SS§ 



s^ 

Cb C» I- lO '«• 00 91 M Tt ^ <-• »^ o 



i5iBS|gmsp;sgss§§$ssp.gssss 



Ifc-TCi*»'9'eo*»©»i-*-i^.-0 



ii 

5 0. 



-jojiuno ! 
•JKJIwa I 



i?.isii5Siigii5iaigiig=ii5iigsg 



'*^»e«cieo^ietni.-c»09*^ot- 



3!:::s5:S$?i;SSS$$^SSSSc^ 



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OPIMUI 

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rroftuox 






|iiil=2SHiSpipgiiigggSggggg 



g ^ «*> o oc c5 « e c» CO « 35 (_, «?S c tK «o « w » f- Fi P- 

^ * ' 1-^ •-^ ^ •-• c« 91 m '<9^ 'V to to to {<J Qc o» o •^ 9f ^ tn «o 00 ^ or ^ 



|3S3t3aw„s^„^„^,x 



io<ot»aDa»Oi-io>««^to 



WROUGHT-IRON PIPE. 



195 



F6r diflcuaslon of the Brigg^ Standard of Wroufcht-lron Pipe Dimensions, 
see Report of the Committee of the A. B. M. E. in " Standard Pipe and Pipe 
Tbreads,'' 1886. Trans., Vol. VIII, p. 29. The diameter of tlte bottom of 

the thread is derived from the formula D — (0.05Z> + 1.9) x — , in which 

X> = outside diameter of the tubes, and n the number of threads to the 
inch. The diameter of the top of tlie thread is derived frf)m the formula 

0.8— X S + c2, or 1.6 f- d, ia which d is the diameter at the bottom of the 

H n 

thread at the end of the pipe. 

Morris, Tatker & Ou.'s sixes for the diameters at the bottom and top of 
the threiid at the end of the pipe are as follows: 



Diam. 


DUm. 


Diam. 


Diam. 


Diam. 


Diam. 


Diam. 


Diam. 


Diam. 


of Pipe, 


atBot- 


at Top 


of Pipe, 


at Bot- 


at Top 


of Pipe, 


at Bot- 


at Top 


Nom- 


torn of 


of 


Nom- 


tom of 


of 


Nom- 


tom of 


of 


ioal 


Thread. 


Thread. 


inal. 


Thread. 


Thread. 


inal. 


Thread. 


Thread. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


in. 


H 


.384 


.893 


''^^ 


2.0-iO 


2.820 


8 


8.334 


8.534 




.463 


.sa 


8 


8.841 


8.441 


9 


9.887 


9.587 


C 


.668 


.658 


^H 


8.T38 


3.938 


10 


10.445 


10.645 


^ 


.701 


.815 


4 


4.284 


4.434 


n 


11.439 


11.689 


94 


.911 


1.085 


4^ 


4.731 


4.931 


18 


18.488 


18.688 


1 


1.J44 


1.888 


6 


5.890 


5.490 


18 


13.675 


13.875 


iii 


1 488 


i.e«7 


6 


6.346 


6 546 


14 


14.669 


14.869 


IH 


1.7-27 


1.866 


7 


7.310 


7.640 


15 


15.668 


15.b68 


a 


2.£i3 


s.aw 















Having; the taper, length of full- threaded portion, and the sizes at bottom 
SfKi top of thread at the end of the pipe, as ffiveii in the table, uips and dies 
can ite made to secure these points correctly, the length of the imperfect 
threaded portions on the pipe, and the length the tap in run into the fittings 
bt>jrond the point at which the size is as given, or, in otlier words, beyond 
the f lid of the pipe, having no effect upon the standard. The angle or the 
thread is 00*. aD<i it is slightly rounded off at top and bottom, so that, instead 
of lU depth being U.866 its pitch, as is tlie case with a full V-t bread, it is 
4/5 the pitcti^ or equal to 0.8 -h n, n being the number of threads per inch. 

Ta|»er of oooksal tube ends, 1 in 88 to axis of tube = 94 inch to the foot 
total taper. 



196 



HATfiRtALS. 



WBOVGBV-ntON WBIiABO TVBBS, BXTBA SniOBrO. 

StandiiWl IMMenslonii. 





Actual Out- 


Thickness, 
Extra 


ThtoknesB, 


Actual Inside 


Actual Inside 


Nominal 


side 


Ppuble 
Extra 
Strong. 


Diameter, 


Diameter, 


Diameter. 


Diameter. 


Btrong. 


Extra 
Strong. 


Double Extra 
Strong. 


Inches. 


Inches. 


Inches. 


Inches. 


Inches. 


Ihclies. 


'■i 1 


0.405 

0.54 

0.075 


0.100 
O.ltt 

o.m 




0.206 
0.204 
0.421 




: 






s 


^ 




;1 


084 


0.149 


0.208 


0.549 


0.944 


^ 


1.05 


0.167 


0.814 


0.780 


0.499 


1* 


1.815 


0.18^ 


0.804 


0.961 


0.687 


iM 


l.tfO 


194 


0.888 


1.279 


0.8M 


iH 


1.9 


0.20.1 


0.400 


1.404 


1.068 


5 


8.875 


221 


0.442 


1.988 


1.401 


2^ 


2.875 


0.280 


0.500 


S.815 


1.736 


8 


8.5 


404 


0.608 


9.892 


9.9M 


8^ 


4.0 


0.821 


0.049 


8.858 


9 710 


4 


4.5 


0.841 


0.682 


8.818 


8.186 



STANDARD SIZltS, BT€«, OF IiAP^-WBIitlfiD CAAB- 
COAIi-IBOlf BOIIiBtt-TrBBS« 

(Morris, tasker & Co.. Inc., Philadelphia, Pa.) 



1 1.S..VH ' 
,14.4H:> 
16.4.Vi, 

17.4101 
I1H.4M) 
l»..l«0 



v\ 


^i 


ai 




11 


rf 


r 


|S 


.11 

In. 


In. 


tn. 


.OM 


IMS 


.lus 


.ow 


3.39U 


1977 


.»& 


4.116 


*.7H 


ss 


4.B0I 


5.491 


u.m 


«.«83 


.09f) 


«.472 


7.0W 


.109 


7.I69 


7.8B4 


.109 


7.»:a 


«.S39 


IU9 


8.740 


9.42.'^ 


120 


9AiA 


10.810 


.130 


10.242 


10.99« 


l;M 


11.027 


11.781 


AM 


11.724 


l2.iiM 


.m 


13.205 


14.137 


M% 


U.77X 


l&.7i« 


.m 


17.M13 


l8.RriO 


A^ 


20.954 


21.991 


.i«r. 


24.(J'JC 


2.'5.133 


Am 


27.143 


2«.5.'74 


.'9X\ 


30.141 


31.410 


.«0 


XKAl^ 


34.,'yW 


.«« 


36.2A0 


37 699 


.2** 


».:m.'> 


40.8.1 


.2IK 


42.424 


43.982 


.i>» 


4.^497 


47.124 


.•,»7l 


48.603 


/XI.206 


.•»! 


.'il.623 


M.4(»7 


.W-! 


M.714 


iAM9 


..■WO 


hl.WXt 


.'>9.09« 


330 


B0.821 


02 8.r2 


.a»o 


IWKt: 


0:).974 









1 = 8 

^4 


n^ 


«5| 


i 


Internal 
Aruu 


External 
Are*. 


14 




^'"5 








""TtT" 

4.«7« 


I.I4« 


flq. In. 


"iSi 


nq. In uq.ft, 
^ .785 0065 


IkM 

.90 


.888 


.0061 


1.227 0085 


laoi 


3.0M 


s-aio 


1.18 


1.348 


.0094 


1.707 .0128 


t.916 


8.547 


878S 


1.40 


1.911 


.0133 


«.405 


.0167 


2.448 


2.188 


«.3i6 


1.65 


<87S 


0179 


8.142 


.0218 


2.110 


1.910 


?:?« 


l.tl 


8.833 


.02:u 


8.970 


0276 


1.854 


1.698 


8.16 


4 090 


.0284 


4.909 


.0341 


1.674 


1.628 


-.601 


2.T5 


6.035 


0350 


5 940 


0412 


1.608 


1.889 


1.449 


S.04 


0.079 


.0422 


7.069 


0491 


1.878 


1.278 


1.8S8 


333 


. 7.110 


.0494 


8.290 


.0576 


1.269 


1. 176 


1.228 


:t.96 


8.347 


avto 


9.0n 


0608 


1.172 


1.091 


1.132 


4.28 


9.070 


0672 


11.045 


0767 


1.088 


1.019 


1.064 


4.60 


10 939 


OTCO 


12.500 


0872 


1.024 


.966 


.990 


5.47 


I4.0rt6 


.0977 


16.904 


1104 


.903 


.849 


.876 


6.17 


17.379 


.1207 


19.035 


.1.361 


.812 


764 


.788 


7..VI 


26.2.^ 


.1750 


28.- 74 .1962 


.674 


.087 


.660 


lO.lti 


.'U.942 


2427 


38. 485 1 .2873 


.578 


.546 


.660 


ll.OM 


40.2IH 


.I'209 


50.206, 3491 


.498 


ATI 


.488 


13.65 


58.830 


.4072 


63.617' .4411 


.442 


.494 


.488 


16.7* 


72.292 


.50V0 


78..'>40| 5454 


.9M 


.388 


.390 


«I.Ot> 


H7.583 


.0082 


95.093, 66m 


.362 


.847 


.855 


25.0a 


104.829 


.7200 


113.0981 .7854 


831 


.318 


.825 


a-.'M 


12:5.190 


.8655 


132.7.8' .9217 


.305 


.294 


.800 


38.06 


143.224 


.9940 


153 938 l.fl«9( 


.288 


.273 


.878 


36.0a 


104.721 


1.1439 


176.716 1.2272 


.864 


.255 


860 


40.60 


187.071 


1.30:« 


1 «)». 062, 1.390; 


.247 


.230 


.243 


4A.av 


212.000 


1.4727 


1 226.981 1 1.576; 


.232 


.225 


.829 


49.»i» 


2-18.825 


1.6.543 


251.470 1.7671 


.219 


.212 


.816 


M.t« 


28.'i.905 


1.8400 


! 283.629 1.969( 


.208 


.201 


.305 


ee.48 


294.- 75 


2.0443 


314. 159 2.1817 


.197 


.191 


.194 


66.77 


324.291 


2.2520 


; 346.301 


S.4063 


.188 


.182 


.186 


7a.4« 



In CMtimatintr the effective Ktram-heatiuff or boiler sarface of tubes, the •urface in 
contort with air or gaMe» of combustion (wiielher internal or external to the tubes) U to 
be taken 

For heatinir liqnidrt by steam. supcrheatin(r Ht«>ani. or trunsferrlncr heat from one 
liquid ur ifOM to another, the nieun surface of ibe tubes Is to be taken. 



BITBTKI> IBOK PIPB. 



197 



To find the square feet of surfaM, S, tn a tob« of a gfrai length, L, in feet, 
and diameter, d, In fitches, multiply the length in feet oy the diameter in 

iBelieeaBdliy.Ml& Or, 8 » !^^^^ = .S6l8dL. For (he diameters in the 

table below, multiply Ihe length in feet by the flgnres given o|»pOsltQ the 
diameter. 



Inches, 
Diameierj 



Square Feet 
p«r Foot 
Xeiigth. 



Inches, 
Diameter. 



Square Fleet 
TCrFoot 
Length. 



.6890 
.6645 
.7199 
.7864 
.8506 
.9163 
.9617 



Inches, 
Diameter. 



6 
6 

7 
8 


1? 

19 



Square Feet 
per Foot 
iiength. 



1.8090 
1.6708 

i.sa^e 

t.0044 

1.6180 
S.87SB 
8.141d 



BIVETED IRON PIPB. 

CAbendroth & Boot Mf^. Co.) 

Sheets pom^bed and rolled, ready for riveting, are paclred in eonvenient 
form for Hliipiiient. The following table shows the iron and rivets required 
fitr punched and formed sheets. 



Wninbrr Square reel of Iron 
rvqnlred to inalce ItO Llneai 
r«cC PsmdMd and Porniod 
She«ta when put tog«th6r. 



Width of 
La»ltt 
Incfaea 



i 

4 
5 
6 
7 
8 

10 

n 

It 

18 



Sgiu 
Fe< 



90 

116 
160 

Its 

£06 

m 

289 
814 
848 
809 




1,600 
1.700 
1.800 
1.900 
2.000 
2,900 
S,800 
2.400 
2.500 
2,600 
2,700 



Kamber Square Peel of Iron 
requirM to make 100 Lineal 
Feet l^nohed and Funned 
aheeta when put together. 



iMam- 
eteria 
Inches. 



14 
15 
16 
18 
20 
22 
24 
26 



Width of 
I.Apln 
Inches. 



Sqtian 
Feet. 



807 
426 

4»J 
506 
502 
617 
670 
W5 

rro 

886 




WKIGRT OF ONB $<|VARK FOOT OF SStBEf-IlfcON 
FOR RIVSTBR PIPB. 

Thlcknesa Jij the Blrmlncliaiii TTlre-Gause. 



!to. of 
Qauge. 


Thick- 
ness ia 
Decimals 
0f an 
Inch. 


Weight 


Qalrati- 
ised. 


Ko. of 
Gauge. 


Thick- 
ness hi 
Deoimala 
of an 
Inch. 


Black. 


Weight 

in iGs., 

Galvani. 

izc-d. 


S 

SO 


.018 
.012 
.088 
.085 


.80 
1.00 
1.25 
1.56 


.01 
1.16 
1.40 
1.67 


18 
16 
14 
U 


.049 
.065 
.083 
.109 


1.H8 
2.M) 
8.12 
i.\37 


2.1& 

2r 

8.M 

4.78 



198 



HATSBIALS. 



SPIRAIi RITBTBD PIPB. 

(Abendroth &. Boot Mfg. Co.) 



Thickness. 


Diam- 
eter, 
Inches. 


Approximate Weight 

in lbs. per Foot in 

Length. 


Approximate Burst- 
ing Pressure in lbs. 
per Square Inch. 


B. w. a. 

No. 


Inches. 


26 
84 

16 
14 
18 


.018 
.(ha 
.038 
.085 
.049 
.065 
.068 
.109 


8to 6 
8tol8 
8 to 14 
8to24 
8 to 21 
6to84 
8to24 
9to84 


Ib8.rs 

'* =Hofdiam.inin8. 

" = .5 " *• 

" = .6 " 

»* = .8 " " 

" =1.1 " 

" =1.4 *• 


860O" -H " 
4800" H- " 
6400 " -4- " " 
8000 " ^ " 



The above are black pipes. Galvanized weighs 10 to 80 )( heavier. 
Double Galvanised Spiral Blveted Flanged Pressure Pipe, tested to 150 lbs. 
hydraulic pressure. 



Inside diameters, inches.... 

Thickness, B. W. G 

Nominal wt. per foot, lbs.. . 



81 9 
,8|,8 



13] 14115 
16 14 14 
16'80l28 



80 84 



40 50 



BinBNSIONS OF SPIRAL PIPB FITTINGS. 



Inside 
Diameter. 



ins. 

8 

4 

5 

6 

7 

8 



10 

11 

18 

18 

14 

15 

16 

18 

80 

88 

24 



Outside 
Diameter 
Flanges. 



Number 
Bolt-holes. 



4 

8 

8 

8 

8 

8 

8 

8 

It 

13 

18 

18 

18 

18 

16 

16 

16 

16 



Diameter 
Bolt-holes.i 



ins. 



11/16 
11/16 



,1/16 



Diameter 
Circles on 
which Bolt- 
holes are 
Drilled. 



ins. 



Sixes of 
Bolts. 



SBAAILBSS BRASS TUBB. IRON-PIPB SIZBS. 

(For actual dimensions see tables of Wrought-iron Pipe.) 



Nominal 
sue. 


Weight 
per Foot. 


Nom. 
Size. 


Weight 
per Foot. 


Nom. 
Sise. 


Weight 
per Foot. 


Nom. 
Siae. 


Weight 
per Foot. 


Ins. 

\ 


lbs. 
.85 
.48 
.68 
.00 


ins. 


lbs. 
1.85 
1.70 
8.60 
8. 


ins. 
8 

9^ 


lbs. 
4.0 
5.75 
8.80 

10.90 


ins. 
4 

6 


lbs. 

18.70 

18.90 

15.75 

18.81 



BRASS tubing; coiled pipes. 



199 



SSAIHLBSS DRADTH BRASS TUBING. 

(Uandolph & Clowes, Waterbury, Uonn.) 
Outside diameter 8/16 to TH inches. ThtcknesB of walls 8 to 85 Stubs* 
Gauge, length 18 feet The following are the standard siaes: 



Lenirth 



14 

13 
13 

n 

12 

n 
u 

12 
IS 
13 
12 



or Ol*l 
GaUfs?, 



20 
19 
19 
IB 
18 
17 
17 
IT 
17 
Ifi 
10 

la 



IMam- F^S*" ^^ Old 
eter. *^^^'" tlniige. 



t« 


14 


t« 


14 


13 


IS 


IS 


13 


w 


IS 


n 


lii 


113 


IS 


J a 


12 


]« 


IS 


12 


12 


IS 


n 


13 J 


11 



OyUlde 
Dlum 
eter. 



Feet. 



n 

J? 

1« 

IS 

!S 

10 to IS 
10 to IS 
10 to IS 
10 to 12 
10 to IS 
Id to IBi 



SlubbB' 
or Old 



It 
It 
11 
11 
11 
]1 
11 

n 

11 

n 
11 



BBNT ANB COILBB PIPBS. 

(Natioual Pipe ijending Co., New Haren. Conn.) 
COILS AND BENDS OF IRON AND STEEL PIPE. 



Siae of pipe Inches 

Least outside diameter of 
ooa Inches 

Siae of pipe Inches 

Least outside diameter of 
C(^l Inches 



H 



18 



84 



8» 


4 


<H 


5 


6 


7 


8 


9 


10 


40 


48 


S3 


58 


68 


ao 


98 


lOS 


ISO 



18 
156 



Lengths continuous welded up to Ji-iu. pipe or coupled as desired. 
OOnJB AND BENDS OF DRAWN BRASS AND COPPER TUBING. 



Size of tube, outside diameter Inches 

Least outside diameter of coil Inches 

Siae of tube, outside diameter Inches 

Least outside diameter of coil In<^>e6 



.^ 



H 



J« 



,J« 



.?* 



16 



,^ 



Lengths continuous braxed, soldered, or coupled as desired. 

90« BENDS. EXTRA-HEAVY WROUGHT-IRON PIPE. 



Diameter of pipe Inches 

Radius Inches 

Centre to end Inches 



4 


*H 


5 


6 


7 


8 


9 


10 


72 


84 


86 


80 


36 


48 


48 


60 


S6 


s^ 


31 


86 


43 


60 


37 


70 



30 



The radii Ri^cu are for the centre of the pipe. *' Centre to end ** means 
the perpendicular distance from the centre of one end of the bent pipe to a 
plane passing across the other end. Standard Iron pipes of sizes 4 to 8 In. 
are bent to radii 8 in. larger than the radii in the above table; siaes 9 to 18 in. 
to radii 18 in. larger. 

Hr elded Solid Brawn-sCeel Tabes, imported by P. S. Justice A 
Co., Philadelphia, are made in sizes from W to 4^ in. external diameter, 
Tarying by ^ths, and with thickness of walls from 1/16 to 11/16 in. The 
msTimnm length is 16 feet. 



200 



MATERIALS. 



WBIG97 OV PPASS, COPPBB, AND ZINC TITBING. 

Per Foot. 

Thickness bjr Brown & Sliarpe's Gauge. 







Copper, 


Brass, Np. 17. 


Erase, No. 90. 


LigbtDlnfc-rod Tutie, 
No. 28. 


iDCta. 


Lbs. 


Inch. 


Lbs. 


Inch. 


Lbfl. 


.\ 


.107 
.157 


A 


.083 
.089 


^1. 


.165 
.178 


% 


.186 


^U 


.068 


% 


.186 
.ill 


7^6 


.884 


.106 


11-16 


^1. 


.906 
.818 


7?f. 


.l'.>6 
.158 


H 


.880 






.883 
.877 


^, 


.189 
.208 


Zino, No. 30. 


' « 


.4frj 




.890 




1 


.542 
.675 


i 


.2.53 
.284 




^4 


L^ 


.161 


1; 


.740 


1 


.878 





.185 


'i 


.915 


1^ 


.500 


i 


.2*1 


I'S 


.980 


.580 


.2W 


8 


1.90 






1 


811 


•iH 


1.506 






IS 


.380 


8 


2.188 






.452 



LEAP PIP^ lOr liBNGTHS OF 10 FEET. 



Iq. 


8-8 Thick. 


5-18 Thick. 


M Thick. 


3 16 Thick. 




lb. oz, 

17 
20 
28 
25 

81 


lb. oz. 

It 
16 
18 
21 


lb. oz. 

11 
IS 

15 

16 
18 
80 


lb. oz, 

8 

9 

9 6 

» s . 



I.EAD HIFASTE^PIPB. 

l}i in., 8 lbs. per foot. I 8Hi In., 4 lbs. per foot. 

8 " 8 and 4 lbs. per foot. 4 "5, 6. and 8 lbs. 

9 *' m and 5 lbs. per foot. f 4^ ** 6 and 8 lbs. 

5 in. 8, 10, and IsTlbs. 

LEAI> AND TIN TUBING. 

H inch. H inch. 

SHEET I4EAII. 

Weight per scuare foot, 9U. 8, 8^ 4. i^, 5, 6, 8, 9, 10 lbs. and i]p«ttWi& 
Oihei' weights rolled to order. 

BIiO0K*TIN PIPE. 

in., 15, and 18 os. per fool. 



( la , 4U, 6H> and 8 oz. per foot. 

i •* 6, 7j<. and 10 oz. ** 

I ♦* 8 and 10 oz. 

I '* 10 and 12 oz. ** 



H4 " 11.4 UHd lUlbq. ** 
1^ " aandawfbi. ♦♦ 
2 '' 8^and81ba. *« 



LBAO PIPB. 



201 





I^BAD AlfB TIN-LINKO liEAB PIPK. 








(TiMhani & Bros., New York.) 






1 


1 


Weight per 
Foot and Rod, 


a . 


1 


1 


Foot and Bad. 


II 


§ 


s 




|2 


3 




^7 


«in. 


£ 


7 lbs. per rod 




1 in. 


E 


1^ lbs. per foot 


10 




D 


10 oz. per foot 


6 


14 


D 


2 " " 


11 


4i 


C 
B 


12 " 
1 lb. " 


8 
12 


!! 



B 


11;: :: 


14 
17 


to 


A 


m " ** 


16 


«4 


A 


f* 4. 4. 


21 


*• 


AA 


]L^ *i ** 


19 


II 


AA 


m ** 


24 


** 


AAA 


194 *' ** 


27 


** 


AAA 


?• *4 


30 


7-16 in. 




IS OB. " 




l^In. 


E 


2 •» 


10 


*' 




1 lb. " ^ 




D 


2K *' 


12 


Hm. 


E 


lbs. per rod 


7 


it 





3^ ;; ;; 


14 




D 


94 lb. per foot 


9 


II 


B 




10 


*• 


C 


1 •* 


n 


1: 


A 


4^ ** ** 


19 


** 


B 


IW ** " 


13 


II 


AA 


5 5 «« ** 


85 


** 




1)4 ** ** 




14 


AAA 


89a ** ** 




•* 


A 


1« •* 


16 


l^in. 


£ 


8 " 


18 


• 4 


AA 


U " 


19 


.4 


D 


?^" " 


14 


•* 




«^ .. - 


^ 


»» 


C 


17 


*> 


AAA 


8 " 


85 


44 


B 


5 " 


19 


Hfn. 


E 


IS " per rod 


8 


»• 


A 


6« " 


83 


»» 


D 


1 *; per foot 


9 


II 


AA 


s" " " 


87 


** 


C 


5« " 


13 


41 


AAA 


Q " " 




** 


B 


2 ** »* 


16 


194 In. 


C 


4 " 


iz 


•» 


A 


M^ II II 


80 


"^7* 


B 


5 ;; ;; 


17 


t* 


AA 


294 ** " 


22 


'♦ 


A 




81 


** 


AAA 


3Vb ** ** 


25 


«* 


AA 


qlZ '• *' 


87 


94 in. 


m 


1 " perfect 


8 


2 In. 


C 


4^ " " 


16 


•• 


fi 


1J4 ** '* 


10 


'♦ 


B 


6 *» 


18 


*4 


§ 


]9i ** *' 


18 
16 


it 


A 
AA 


7 " 
9 " 


28 
87 


«* 


A 


8^1. II 


80 


41 


AAA 


1194 " 




M 


AA 


8U *• " 


88 










M 


AAA 


SO 











WBIGttT OF LEAD PIPK WH1C0 SH017LB BE USED 
FOB A GIVEN BEAB OF WATBB. 

(Tatham & Bros., New York.) 



H«ador 

Number 

of Feet 

FaU. 


fretfure 

per 
sq. inch. 




Calibre and Weight per Foot. 




Letter. 


9^ Inch. 


^ Inch. 9i Inch. 


94 inch. 


Itnch. 


lJ4in. 


30ft. 

soft. 
:5 ft. 

100 ft. 
190 ft. 
JOOffc. 


151be. 
85 lbs. 
88 lbs. 
50 lbs. 
75 lbs. 
100 Iba. 


8 

B 

A 
AA 
AAA 


10 OB. 

IS oz. 
1 lb. 

m lbs. 

1)2 lbs. 
194 lbs. 


94 lb. 1 lb. 
1 lb. IHIbs. 
l^lbs. 8 lbs. 
194 lbs. 2Ulb8. 
8 lbs.' 294 lbs. 
3 lbs. 3^ lbs. 


m lbs. 
194 lbs. 
2^ , lbs. 
8 lbs. 
3U lbs. 
4«lbs. 


8 bs. 

4 lbs. 
494 lbs. 
6 lbs. 


4^rs: 

6 H>8. 
694 lbs. 



To lliid ili6 ttilokiiefMi of l«a4 pipe required ivlieiE tbe 
]ie«4 or ir*ter Is tfTren. (Chadwlck Lead Works). 

Rtlb.— Multiply the head in feet by sisee of pipe wanted, expressed deci- 
maQy, and diride by 750; the quotient will give thickness required, in one- 
hunaredtbBOf an inch. , ^^ 

IXilltfUt.--«iKa(|tUrM tblclcneffl of half -inch pipe for a head of 85 feet. 

SB X 0.50 -i-7fiOs 0.16 inch. 



202 



MATERIALS. 



II 

I 



A< 

M i 

^ 2 I 

S '^ ^ 

fi I i 

w: id Z 

9"! 

S: § 
S fe 

9 

Ed 

s 






^<5 



IS 






iJ »-i r-l * OC 






§ 



^sSisiBii^i^sss^iiiiii s s 



j«^„^ 



i«3ZS!§.g§iS§iSI§iiiiii I 



J«rH«^ 



S^t-SSsS 






II 

8 









I 

6 



riS8SS!S&S«^SZSS3=SSSS8SSS9 



t7 

el 

'I 



I 



^1 



.oeoioeookoo^Q* 



9SSS93r:S9SSSSSS 






liiJ^Psiii^siigiiiiii^ 



o 3 






BOLT COPPEB— SHEET AND BAB BBASS. 



203 



IHTBIGST OF R0171f B BOIiT COPPSB* 

Per Foot. 



IllCl>€fr 



Pounds. 



.4125 
.7M 
1.18 
1.70 
S.81 



Inches. 



Pounds. 



8.0S 
8.88 
4.73 

6.81 



Incbes. 



Pounds. 



7.M 
9.87 
10.64 
12.10 



WBIGST 


OF 89IHBT 


ANB BAB BBAS0« 




Thickness, 


Sheets 


Square 


Bound 


Thickness, 


Sheets 


Square 


Bound 


aide or 


per 


Barsl 


Barsl 


Side or 


per 


Barsl 


Barsl 


Diam. 


sq.ft. 


ft. long. 


ft. long. 


Diam. 


sq.ft. 


ft long. 


ft. long. 


Inches. 








Inchps. 








1-16 


2.7S 


.014 


.011 


1 1-16 


46.82 


4.10 


8.82 


H 


5.45 


.066 


.045 


11,. 


40.06 


4.50 


8.61 


S-i6 


8.17 


.12S 


.100 


61.77 


5.12 


4.02 


e^l. 


10 90 


.227 


.178 


1 5-16 


54.50 


5.67 


4.45 


18.62 


.855 


.8« 


67.22 


6.86 


4.91 


7?f6 


16.85 


.510 


.401 


il... 


60.05 


6.86 


5.80 


19.07 


.096 


.645 


68.67 


7.60 


5.89 


,^6 


21.80 


.907 


.712 


It,. 


66.40 


8.16 


6.41 


24.52 


1.15 


.902 


68.12 


8.86 


6.95 


ll56 


27.25 


1.42 


l.ll 


1^ 


70.85 


9.60 


7.58 


29.97 


1.72 


1.85 


l7l-16 


78.57 


10.84 


8.18 


1^6 


82.70 


3.04 


1.60 


lis-ia 


76.80 


11.12 


8.78 


85.42 


8.40 


1.88 


79.02 


11.98 


0.86 


1536 


88.15 


3.78 


3.18 


^% 


81.75 


12.76 


10.01 


40.87 


8.19 


2.90 


1 15>16 


84.47 


18.68 


10.70 


1 


48.00 


8.68 


2.85 


2 


87,20 


14.52 


11.40 



COHPOSmOlf OF VABIOTT8 GBADB8 OF BOLIiBB 
BBASS, BTC. 



Trade Name. 


Copper 


Zinc. 


Tin. 


Lead. 


Nickel. 


fVifnmnn liiirh bnuR 


61.5 
60 

60 
60 


88.5 
40 

40 
40 








Teliow in4*Ml 








OartrMge brass. ................... 








Low brass ........r...... ......rt 








Clock brasB 


"iii" 


.«'S« 




Drill rod. 




Spring brass. 




18 per cent German slWer 




18 



The above table was furnished bj the superintendent of a mill in Connec- 
ticut in 1894. He sajrs: While each mill has Its own proportions for various 
mixtures, depending upon the purposes for which the product is intended, 
the figures gvven are about the average standard. Thua« between cartridge 
brass with 3^ per cent zinc and common high brass with 88U per cent 
sine, there are any number of different mixtures known general^ as ** high 
brass/* or speeiflcally as "spinning brass,'' ** drawing brass/' etc., wherein 
the amount of cine Is dependent upon the amount of scrap used in the mix- 
ture, the degree of workmg to which the metal is to be subjected, etc. 



204 



ICATBRIALS. 



ARISBlOAlf BTANSABll 8tSE8 #9 BROP^HOT. 















^i 






^S 






if 




Diameter. 


^1 




Diameter. 


^1 




DIam. 
«ter. 






4^ 


No. 8 




^5 






igS 


Fine Dust. 


a-100" 


10784 


Trap Shot 
9-100" 


471 


No. a.... 


15-100" 


86 


Dust 


4-100 


4>(» 


" 8 


89f 


" 1.. . 


W-100 


71 


No. IB 


5-100 


ai26 


" 7 


Trap Mtot 


838 


" B... 


17-100 


59 


" 11. ... 


5-100 

Trap Shot 
7-100" 


1H6 


•• CllO-lOiK' 


^ 


•* BB. 


ie-100 


50 


" 10 


I'ttO 


" 6 11-100 


" BBB 


19-100 


4i 


** 10 


848 


«• 6: 12-100 


J68 


'• T. . 


20-100 


96 


" 


Trap Shot 


688 


" 4ia-ioo 


18-2 


" TT.. 


21-100 


31 


" 8 


MS 


« a 14^00 


1M 


" P.. 


t:MOO 


27 








1 




" FF.. 


2S-10Q 


94 





COntPRESSBD Br€K-«ROT. 






Diameter. 


No-ofBallB 
to the lb. 




Diameter. 


No. of Balls 
to the lb. 


Ko. 8 ■ .-. 

" 2 


25-100" 
87-100 
:8a-l00 
i82-100 


284 

m 

Its 

140 


No 00 

" 000 

Balls 


84 100" 
C6-100 
88-100 
44.100 


•s 


•* 1 


ift 


* a.::...:. 


50 









8CRIBW-11BBEA1M) 8ELLBRS 9U IT. S. STAl|iRAlil». 

It 1804 a comtiilttee of the FrttnkliQ Imititute recofnmendrd the adca>tion 
6f ^e system of screvv-threads and bolts which was Revised bv Mr. w<niiani 
Sellers, of Pbiladolphia^ This same system was subseQuentW adopted af 
the Btatdard by both the Army and Navy Detartmenta or the united States, 
a-Dd by the Master Mechanics' and Master Car Builders* Asseciatlona, so 
that it teay how be regarded, and ia fact Is called, the United States Stan* 
dard. 

Tlie rule ghren by Mr. Setlerti for proportioning the thread is as follows : 
Divide the pitch, or, w<hat is the same thiagr* the side of the thread, into 
.eight e^ual parU; take off one part from the top and All in one part in thA 
bottom of the thread; then the flat top and bottom will equal one eighth of 
the pitch, the wearing stirface will be three qiiartei-s of the pitch, and the 
diameter ol screw at bottom of -the thivad will be eKprattfd br Mm /^r 
nmla 

1 299 
diameter of bolt — ; 



For a 



no. threads per inch 
V thread 'wl«h«agle df 00* the formula is 

diameter of bolt — • 



no. of threads per inah 
The afeigle of dhe thread in the Sellers system is 60*. In the Wbitirorth or 
English system it is 56°, and the point and root of tlie tbpi»ad ara rounded. 
Sere w-Threaaa, United States StandaHl, 



^16 



Ci« 



lT-16 



20 
18 
16 
14 
18 
12 
11 
11 



18-16 

15-16 

1 

1 1-16 

1^ 



10 
10 


8 
7 
7 



.s 

Q 



7 
6 
6 
6 

6 



s 

3 



15-16 



I 



2 18-16 
8 

85-18 



V. S. OR SELLERS aYSTEM OP SCREW-THREADS. 300 



9m 


mw«TIwmUU, Wliltwovtb (BiwUtll) fiUliAlirtf 


f 


1 


1 


1 


iC 

n 
11 

10 

1 


1 


1 


1 


1 


1 




1" 




18 


8 
If 

t 





5 
4 


8 
8Vi 


f 


S?1« 


13 


1?-16 


9 


li, 


5 


aU 


3^ 







BOLTS AMD THREADS. 




UaaT GAU6KS FOH IRON P09 9€1^|SW TUUMA^f^. 

In AdepliVg (lie 9e)l«f9, or FrankUn Iu«titato, or Uniua States Standard, 
w it ip variOfMlT ClMJl<9a, a (liffiKnilty ai'oee f i ona the fact that it is l))e babk 
«f iisp mpwx^WW^ U> miftl^e iron oyer-ai^, and aq th#ro ar« no 0¥9r-ai8a 



206 



MATERIALS. 



screws in the Selleni ssrstem, If Iron is too lar^e it Is necenary to cut it away 
with the dies. So great is this diflBculty, that the practice of making tafx 
and dies over-sixe lias become very general. If the Sellers system is adopted 
it is essential that iron should be obtained of the correct size, or very nearly 
so. Of course no high degree of precision is possible in rolling iron, and 
when exact sizes were demanded, the question arose how much allowable 
Tariatlonjthere should be from the true size. It was proposed to make limit- 
gauges for inspecting iron with two openings, one larger and the other 
smaller than the standard size, and then specify that the iron should enter 
the large end and not enter the small one. The following table of dimen- 
sions for the limit-gauges was recommended by the Master Car-BuUders* 
Association and adopted by letter ballot in 1888. 





Size Of 


Size of 






Size of 


Size of 




Size of 


Large 


Small 


Differ- 


Size of 


Large 


Small 


Differ- 


Iron. 


End of 


End of 


ence. 


Iron. 


End of 


End of 


ence. 




Qauge. 


Gauge. 






Gauge. 


Gauge. 




^in. 


0.8550 


0.3450 


0.010 


^in. 


0.6880 


0.6170 


0.016 


6l|6 


0.8180 


0.8070 


0.011 


» 


0.7586 


0.7415 


0.017 


7?f. 


0.8810 


0.8690 


0.012 


jI 


0.8840 


0.8660 


018 


0.4440 


0.4810 


9.013 


1 


1.0005 


0.0905 


0.019 


^. 


0.60T0 


o.4ino 


0.014 


v.i 


1.1850 


1.1160 


o.oao 


0.6700 


0.5660 


0.015 


m 


1.2605 


1.2895 


Q.Oil 



Caliper gauges with the above dimensions, and standard reference gauges 
for testing them, are made by The Pratt & Whitney Co. 

THE nAXUHUJH VARIATION IN 8IZR OF ROUGH 
IRON FOR 17. S. STANDARD ROLTS. 

Am. MacK, May IS, 1892. 

Bv the adoption of the Sellers or U. H. Standard thread tapnand dies keep 
their Hize much longer in use when flatted in accordance with this system 
than when made .sharp "V,'* though it has been found advisable in practice 
in most cases to mnke the taps of somewhat larger outside diameter than 
the nominal sfz**, thuK carryuig the threads further towards the V-shape 
and giving corresponding clearance to the tops of the threads when iu the 
nuts or tapped holes. 

Makers of taps and dies often have calls for taps and dies, U. S. Standard. 
" for rough Iron." 

An examination of rough Iron will show that much of it is rolled out of 
round to an amount exceeding the limit of variation in size allowed. 

In view of this it may be desirable to know what the extreme variation in 
iron may be, consistent with the maintenance of U. S. Standard threads, i.e.« 
threads which are standard when measured upon the angles, the only plac«> 
where it seems advisable to have them fit closely. Mr. Chas. A. Bauer. Uie 

general manager of the Warder. Biishnell A Glessner Co., at Sprinsfleld, 
hio, in 1884 adopted a plan which may be stated as follows: AH bolts, 
whether cut from rough or finished stock, are standard size at the bottom 
and at the sides or angles of the threads, the variation for flt of the nut and 
allowance for wear of taps being made in the machine taps. Nuts are 
punched with holes of such size as to give 85 per cent of a full thread, expe- 
rience showing that the metal of wrought nuts will then crowd into the 
threads of the taps sufficiently to give practically a full thread, while if 
punched smaller some of the metal will be cut out bv the tap at the bottom 
of the threads, which is of course undesirable. Machine taps are made 
enough larger than the nominal to bring the tops of the threads up sharp, 
plus the amount allowed for flt and wear of taps. This allows the iron to 
oe enough above the nominal diameter to bring the threads up full (sharp) 
at top, while if it is small the only effect is to give a flat at top of threads ; 
neither condition affecting the actual size of the thread at the point at which 
it is intended to bear. Limit gauges are furnished to the mills, by which tho 
iron is rolled, the maximum size being shown in the third column of the 
table. The minimum diameter is not given, the tendency in rolling bein|^ 
nearly always to exceed the nominal diameter. 

In making the taps the threa<1ed portion is turned to the size given in the 
eighth column of the table, which gives 6 to 7 thousandths of an Inch allow- 
Uice for fit and wear of tap. Just above the threaded portion of the tap a 



SIZES OP 8CRBW-THBBAD8 FOR BOLTS AND TAPS. 207 

place is tamed to the size giveii In the ninth column, these sizes befne the 
same as Uioee of the regular U. S. Standard bolt, at the bottom or the 
thread, phis the amount allowed for fit and wear of tap ; or, in other words, 
d' = U. S. Standard d + (IX - D). Gauses like the one in the cut. Fig. 
7i^ are fumlslied for ihis sizing:. In flnlBbing the threads of the tap a tool 




Fig. 7a. 
is oaed which has a removable cutter finished aocurately togauge bj grind- 
ing, this tool being correct U. 8. Standard as to angle, and fULt at the point. 
It is fed <n and the threads chased uutil the flat point just touches the por- 
tion of the tap which has been turned to size a'. Caj*e having been taken 
with the form of the tool, with its grinding on the top face (a fixture being 
{HDvided for this to insure its being ground properly), and also with the set- 
ting of the tool properly in the lathe, the result is that the threads of the tap 
are correctly sized without further attention. 

It is evklent that one of the points of advantage of the Sellers system Is 
sacrificed, i.e., instead of the taps being flatted at the top of the threads 
they are sharp, and are consequently not so durable as they otherwise would 
be ; but practically this disadvantage is not found to be serious, and is far 
overbalanced by the greater ease of getting iron within the prescribed 
Umiu ; while any rough bolt when reduced in size at the top of the threads, 
by filing or otherwise, will fit a hole tapped with the U. S. Standard hand 
taps, thus affording proof that the two kinds of bolts or screws made for the 
two different kind^ of work are practically interchangeable. ])y this system 
i" iron can be .000" smaller or .0106" larger than the nominal diameter, or, 
in other words. It may have a total variation of .01A8", while iy* iron can be 
.0106" smaller or .oaolK' larger than nominal— a total variation of .0414"— 
and within these limits it is found practicable to procure the iron. 

«TAlfl»AIt]> SIZES OF SCRBIV-THBKADS FOB BOIiTS 
AND TAPS. 

(Chas. a. Bapkr.) 



1 


8 


8 


4 


5 





7 


8 


9 


10 ' 


A 


n 


D 


d 


h 


/ 


Iiches. 


D' 


d' 


H 






Inches. 


Inches 


Inches. 


Inches. 


Im*he». 


Inches. 


Inches. 


^. 


xo 


.2606 


.1855 


.0J»79 


.oo«--» 


.006 


.2668 


.1915 


.2024 


18 


.3845 


.2408 


.0421 


.0070 


.006 


.3805 


.2468 


.2589 


H 


16 


.«85 


.«9iW 


0174 


.0078 


.006 


.8945 


.2996 


.3189 


7-\6 


14 


.4.'M0 


.844? 


.0511 


.on«o 


.006 


.4590 


.8507 


.3670 


H 


13 


.51(50 


.4000 


.05W 


.0096 


.OOrt 


.6220 


.4060 


.4286 


»^« 


ite 


..•MW, 


.4518 


.Ol«Jl 


.0104 


.007 


.5875 


.4018 


.4802 


H 


11 


.0447 


.rxm 


.OrtRQ 


.0114 


.007 


.6517 


.5189 


.6846 


H 


10 


.7717 


.O-JO! 


.0:58 


.0125 


.007 


.7787 


.6871 


.6499 


9 


.8091 


.;807 


.084-i 


0180 


.007 


.9061 


.7877 


.7680 


r 


8 


1 0271 


.8:i7« 


.0047 


.0!5« 


.007 


1.0341 


.8446 


.8731 


i¥. 


7 


1.1M9 


WW* 


.lOKS 


.0179 


.007 


1.1689 


.M64 


.9789 


iS 


7 


1.2809 


1.0644 


.1088 




0179 


.007 


1.2879 


1.0714 


1.1089 


A - nominal diameter of bolt. 




3165 




D = aetual diameter of bolt. 


D-A+ ^^ . 




d = dUmeter of bolt at bottom of 


d^A^'"^. 




thread, 
n = number of threads per Inch. 


. .7577 i) - <f 
^- n - 2 
^ .125 




/= flat of bottom of thread. 




h = depth of thread. 


^=ir- 






ld' = 
lolei 


3 dlame 
in nut be 


tersofti 
>f ore tap 


tp. 

ping. 




H 


= !>'- 


^- 


y-.85a 


UL) 



208 



MATERIALS. 



STANDARD SKT-SCBBWS AND GAP«B€RBW8« 

Americau, Hattford, and Worcester Machine-Screw Companies. 
(Compiled by W. 8. Diz.) 



Diameter of ScrAW. . . . 

Threads per Inch 

Sisse of Tap Drill* 



(A) 
No. 48 



(B) 

S4 

No. 80 



(C) 



No. 5 



(D) 
5-16 
18 

ir-w 



(E) 

!^ 

31-64 



(F) 

7-16 

14 

% 



(Q) 



Diameter of Sci-ew.. 
Thi-eac)8 per Inch. .. 
Size of Tap Drill*... 



(H) 

9-16 

12 

31-64 



(I) 
17-8i 



(J) 
21-32 



(K) 
49-61 



(L) 
1 
8 
% 



(M) 

'I* 

68-64 



(N) 

ij4 



Set Screws. 



of Hem I of Head 







iC\ H 


JJ5 


(Dt 5-ia 


Ji 


It) M 


.53 


(F> 7™tt 


,f?i 


i,Qj ^ 


.71 


H;9^C 


M 


d> % 


.89 




3.(« 
I 'i4 


(L} r 


1.4> 


(M) 1^ 


!.&> 


IN) lis 


1-77 




Round and Filister Head 
Cap-screws. 



Flat Head Cap-screws. 



Button -head Cap- 
screws. 



DIsm. of 
Head. 



(A) 
(B) 
(C) 
(0) 
it) 
(F) 

(K) 



3-16 



16 
9-16 

^ 

13-16 

1^ 



Lengths 

(under 

Head). 



Diam. of 
Head. 




K 



13-t6 



Lengths 

(including 

Head). 



biam. of 
Head. 




7-82 (.221 
5-16 
7-16 
9-16 
% 

13^16 

15-16 

1 



Lengths 
(under 
Head). 




* For cast iron. For numbers of twist-drills see p. 29. 

Threads are U. S. Standard. Cap screws are threaded H length up to and 
Including T'diam. x 4" long, an»l V6 It'ugth above. Lengths increatce by ^" 
each regular size between tlie limits given. Lengths of headH, except flat 
and button, equal diam. of screws. 

The angle of the cone of the flat head screw is 76*, the sides making angleg 
of 5^ with the top. 



8TAKDARD MAOHIHE BGBEW8. 
STANDARS 9EACHINE 8CBEW8. 



209 



No. 


Threads Der 
Inch. 


Diam. of 
Body. 


Diam. 
of Flat 
Head. 


Diam. of 
Round 
Head. 


Dfam. of 
Flllster 
Head. 


LeDgtbs. 


From 


To 


2 


66 


.0842 


.1681 


.1644 


:i^ 


8-16 


H 


8 


46 


.0078 


.1804 


.1786 


»-l6 


ll 


4 


82,86,40 


.1105 


.2158 


.2028 


.1747 


8-16 


1 


6 


» 8ft 40 


.1288 


.9421 


.2270 


.1966 


8-16 


% 


9 


30,82 


.1388 


.2684 


.2512 


.2175 


8-16 


1 


7 


80 8;) 


.ISOO 


.2047 


.trtA 


.2892 




^H 


8 


80,88 


.1631 


.8210 


.8086 


.2610 


o 


iS 


9 


S4.80.82 


.1:68 


.8474 


.8^288 


.2805 


h 


vl 


10 


24. d0,ft( 


.1804 


.87«7 


.8480 


.8085 


h, 


'1 


12 


20,24 


.2136 


.4288 


.3928 


.8449 


1 


111 


14 


20,24 


.2421 


.4790 


.4864 


.8888 


'1 


2 


IB 


16. 18, 90 


.2684 


.5816 


.4866 


.4800 


'1 




16 


16,18 


.2047 


.584-2 


.5248 


.4710 


4 


214 


20 


16,18 


.8.il0 


.8808 


.5690 


.5900 




294 


« 


16,12 


.^H 


.0894 


.8106 


.8567 


J 


8 


24 


14. 16 


.87^ 


.7430 


.6522 


.0005 


1^ 


8 


» 


14,16 


.4000 


.74^ 


.6988 


.6425 


' % 


8 


« 


14.16 


.4J63 


.7946 


.7854 


.6920 


12 


8 


ao 


14, 16 


.45:30 


.81^3 


.7T70 


.7240 


1 


3 



Lenirths varj bj lOths from 8>16 to ^, by Sths from H'to l^, by 4ih8 from 
iHtoS. 

8ISfe£8 ANli WJ610BTS OF SOlJAlftB AND 

0E3LAGONAli Num. 

rnltea llMites Stanclard fltsea. Gkainfereji and tHmmed. 

Panebed to salt IT, 8. Standard Yap*. 



Square. 




1.04 
1.48 
1.72 
2.27 
2.94 
8.33 
4.35 
5.26 
8.83 
fl.ll 
18.64 



Hexagon. 



1" 
^1 



7615 
5200 
800O 
2000 
1430 

iioo 

740 

450 

309 

216 

148 

HI 

85 

68 

56 

40 

87 

29 

21 

16 

11 



h 



.0181 

.0192 

.0838 

.aM) 

.070 

Ml 

.135 

.222 

.324 

.468 

.676 

.901 

1.18 

1.47 

1.79 

250 

270 

3.45 

4.76 

6.67 



9.09 
11.76 



210 



MATERIALS. 



Q 



M 



o 



o 
o 

O 



"' U : : ; : ■ ■ ib--i;!tl®i:^S?g«ii^¥?|^t7^r?ll 


1" 






2^ S ; : :i5lsS^esi^^S5;:^iS3itiS?^Sii=.g|^5i 


^ |i : : :g|S^^iiig^iS§iiisii£i 




?SzM^i 


^ j|:HSSgSgS§l§iSISii^SSI 


MM 




1 . ']-t:i:X}0'7'fi'tSi9t7f*^aQ'^^7'QOOOOOOOOOOQ5SO 




^' jl :i^gr?!g§35iE5SS=;|jlilrIiiSSi?il*^l§ 




1 .C'TliJCi^^CiT. qirt?*K^«C4''CT>T-tP']tC:'i*^O^CCOOO 




1 ^oci"MJinii-oi^Mf»x>cJOOi»«sFn^r^3*roi<;oir:oooo 




-i llfeg^^s5ss^t:?j:^?s^i;i;5?-^§^^siigi 




tp . o O QJ -* t- O H3 s CI 5* .r. x- ^- .-r: rp r.. t - . ■. ? - - - _ =: O C 












^ 1 ,'-r^o^ic.^7*ag!r'j'u-T*i"ClOinOiCC}*iCnrtC*^^ = 














S5SI 


- 


-:- 


- 


- 


:- 


I'll— <- — , — ^ — ^ ^ ^^ . 


: i ; 
























^ f « » ® -^ S> Ofc <J 1^^ C^ t - -T -^ ii; 1.T t* _ ■ • ; ^ ; ■ 


; : : 












«"§ = J" :? J* 3! ^ J? LR 


J ; : 3 


^ 


- 1 


31 







a j^j: 



TRACK BOLTfl. 

'WlCb fTnlCed State* Standard 0exacoii Nuts. 



Rails used. 



45to851b0... 



aoto 



40 lbs... I 



aotoaoibs. 



Bolts. 



x3 



Nuts. 



No. in Keg, 
900 lbs. 



2S0 
240 
264 
260 
260 
268 

375 
410 
435 
466 

715 
760 
800 
8;iO 



Kegs per BiUe. 



6.3 

6. 

6.7 

6.5 

6.4 

5.1 

4. 

8.7 
8.8 
8.1 

2. 
2. 
8. 
2. 



KIVETS — ^TUBKBUCKLES. 



211 



€01fE-HEAI> BOILER RIVETS, TITEIGHT PER 100. 

(Hoopes Sc Townsend.) 



Scant. 


1/S 


9/16 
lbs. 


5/8 


11/16 


H 


18/16 


% 


1 


1^* 


IM* 


I^enRth. 


lbs. 


n>8. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


lbs. 


fi'mch 


8.7B 


18.7 


16.80 
















,^ 


M 


0.86 
10.00 


14.4 
15.8 


17.28 
18.85 


81.70 


86.66 












^H 


•• 


10.70 


16.0 


19.88 


83.10 


88.00 












i?l 


•• 


11.40 


16.8 


90.81 


84.50 


29.45 


87.0 


46 


60 






11) 


4» 


18.10 


17.6 


21 .34 


25.90 


80.90 


88.6 


48 


68 


95 




IV. 


t< 


IS.W 


18.4 


28.87 


87.80 


82.36 


40.2 


50 


66 


08 


188 


}% 


*• 


18.50 


19.8 


88.40 


88.70 


33.80 


41.9 


62 


67 


lot 


187 


iH 


M 


14.90 


20.0 


84.48 


30.10 


85.85 


48.6 


64 


69 


104 


141 


i4 


44 


14.90 


80.8 


85.46 


31.50 


86.70 


46.8 


56 


71 


107 


145 


8 


(k 


15.60 


81.6 


86.49 


82.90 


38.15 


47.0 


68 


74 


no 


149 




•• 


16.80 


88.4 


87.58 


34.80 


80.60 


48.7 


60 




114 


158 


2^ 


(t 


17.00 


83.8 


28.56 


35.70 


41.05 


50.3 


62 


80 


118 


157 


2t2 


♦• 


17.70 


84.0 


89.58 


37.10 


48.60 


61.9 


64 


83 


121 


161 


2C4 


•• 


18.40 


84.8 


80.61 


38.60 


48.96 


63.5 


66 


86 


184 


165 


wZ 


»» 


19.10 


85.6 


31.64 


39.90 


46.40 


66.1 


68 


89 


187 


169 


S9i 


• » 


19.80 


86.4 


88.67 


41.80 


46.85 


66.8 


70 


92 


180 


178 


2^ 


** 


ao.so 


87.8 


83.70 


48.70 


48.90 


68.4 


72 


06 


188 


177 


8^ 


*♦ 


81.90 


88.0 


84.78 


44.10 


49.75 


60.0 


74 


98 


137 


181 




• * 


28.60 


89.7 


86.79 


46.90 


58.66 


63.8 


78 


103 


144 


189 


3^ 1 


• « 


84.00 


81 5 


88.8.') 


49.70 


65.55 


66.6 


88 


108 


151 


197 


3^ 


•• 


85.40 


83.8 


40.91 


52.50 


58.45 


69.8 


86 


118 


158 


205 


4 


•4 


86.80 


85.8 


48.97 


55.30 


61.85 


780 


90 


118 


165 


218 




4« 


88.90 


86.9 


45.08 


68.10 


64.86 


76.3 


04 


124 


178 


221 


4^ 


M 


89.60 


88.6 


47.09 


60.90 


67.15 


79.5 


98 


130 


179 


229 


^9% 


•• 


81.00 


40.8 


49.15 


63.70 


70.05 


88.8 


102 


136 


186 


237 


5 


«• 


38.40 


48.0 


51.81 


66.50 


78.95 


86.0 


106 


142 


193 


845 


5^ 


44 


ai.8o 


48.7 


53.27 


69.20 


7^.85 


89.3 


110 


148 


200 


254 


5^ 


4« 


35.90 


45.4 


55.33 


72.00 


78.75 


92.5 


114 


154 


206 


268 


fi^ 


• • 


86.60 


47.1 


57.39 


74.80 


81.05 


96.7 


118 


160 


812 


272 


6 


*• 


8M.0O 


48.8 


59.45 


77.60 


84.56 


99.0 


122 


166 


818 


281 


«H 


** 


40.80 


58.0 


63.67 


88.30 


00.86 


105.6 


180 


177 


831 


297 


*t 


48.60 


65.8 


C7.69 


88.90 


96.15 


112.0 


138 


188 


845 


814 


Heads 


6.50 


8.40 


11.60 


18.20 


18.00 


28.0 


29.0 


88.0 


56.0 


77.5 



* These two sizes are calculated for exact diameter. 
RiTets with button heads weigh approximately the same as cone-head 
riTets. 

TfTRNRrCKIiEl. 

(Cleveland Citj Forge and Iron Co.) 
Standard sizes made with right and left threads. D : 



outside diameter 




j-B -^ — A—^-B-^ 

Fio. 78. 



of ncrew. A = lengrth in clear between heads =s 6 ins. for all sizes, 
length of tapped heads = ly^D nearly. L = H ins. + 3D nearly. 



B = 



318 



iJiTWIALS, 



SIXE9 OF WXSf9VM9, 



Diameter in 
kiclies. 



SIse of Hole, in 
Incliefl. 



5-15 

i^ii 
?f.i6 

18.-16 
81-fti 



ThiokneM, 
Btrmloffbam 
Wire-gauge. 



Ko. 16 
** 16 
•• 14 
" 11 
•' 11 
•* 11 



6 
6 

7 
6 



Bolt In 
iucheg. 



No. in 100 lbs. 



89,800 
18,000 
7,600 

iaoo 

i,180 

S.SSO 

1,680 

1,140 

680 

476 

860 

860 





TBAC^ BFIl^BS* 




BaB0 u«ed. 


Spike*. 


number Iq Keg, 
800 IbB. 


between Centres. 


45 to 65 
40 •'58 
8ft'* 40 
84 "85 
84 *' 80 
18 *' 84 
16 "80 
14 " 16 
8 " 12 
8 *' 10 




880 
400 
490 
550 

880 
1850 
1850 
1550 
8:200 


80 

81! 

ai 
111 
i;i 
II 
II 
" 
6 



8TBBIBT BAII^WAT SPfKBS. 



Spikes. 


Number in Keg, 800 lbs. 


Kegs per MUe. Ties »li«. 
between Oentrek. 


4Mixt-16 


400 
6tQ 
800 


80 
10 
18 



BOAT 0FIBBS. 



Length. 


H 


5-16. 


9^ 


H 


4 inch. 


2875 
2050 
18:i» 










1280 

1176 
900 
880 


940 
800 
650 
600 

475 




6 '* 

7 *• 


450 

875 


8 '* 




885 


9 ** 




800 


10 " 






S?5 











8PIKX8; CUT ITAILS. 



m 



VmOVQU-T SPIKES. 

Bfumber of Nail* in Kee of 15# Pounds* 



sua. 


Min. 


5-16 In. 


«lii. 


7-16 In. 


Hin. 


a Ib<.i«A9 


2290 
1890 
1090 
1484 
1880 
1292 
1161 










?* - ::;::: 


1206 
118S 
1064 
030 
868 
663 
686 
678 















t"^ " :;•::: 









748 

070 

482 
465 
424 
891 






6 •• 






7 - ...... 

8 - 

9 • 

It • — .. 


446 
884 
800 
270 
849 
886 


ao6 

866 
840 
822 


u ■• 






806 


18 • 








180 













DTIKB 8PIKK8. 



Sixe. 


Approx. Size 
ofWire Nails. 


Ap. No. 
in 1 lb. 

60 
85 
26 
26 
16 
12 


Size. 


Approx. Si»e 
ofWireNaiiB. 


inilb? 


Wd Spike 

16d " 

20d " 

80d •• 

40d •• 

SOd •* * 


8 In. No. 7 

?« :: :: I 
'^ :: :: i 

5« " •* 2 


aod Spike 

fii^W 

8 " *• !!.."..' 

9 ••" 


6 in. No. J 

fi :: r. j 

8 " •* #0 

9 " " #0 


16 

? 

6 



I^VNCTH AND NtTlHRBm OF CUT KAILS TO THB 
POUND. 



Sin. 


J3 

1 


6 
o 


-; 

i 


£ 


1 


1 


1 

i 


tic 

a 

I 


1 
1 


o 


1 


U 








800 
600 
876 
224 

180 








2 . . 




















§.:::;:::. 


fiOO 

480 

886 

200 

168 

124 

88 

70 

68 

44 

84 

23 

18 

14 

10 

8 


95 
74 
68 
53 
46 
42 
88 
88 
20 


64 
48 
86 
80 
24 
20 
16 


1100 

720 

623 

410 

268 

188 

146 

180 

102 

76 

62 

64 


1000 
760 
868 










8d 


"898 








4d. 




M 




130 
96 

68 




6d 






224 


126 
OS 
75 
65 
65 
40 
87 




7d. 








8d 






128 
110 
91 
71 
64 
40 
83 
27 




9d 








lOd 








?8 


lad 








16d 








Wl 


aod 








SOd, 










12v2 


40d 














9vl 


50d- 












8' 


eod. 



















6 























214 



XATBBIAia. 



1^ 

M 

S 



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^ i 

K ca 



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0) 



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•BOJiids wiM 


::*•:: 


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•auian 


8S8 i : : : 
s55 : : : : 


*!i:tiz!!i:r 


'ooovqox 


: : : : ! iS^SS^S* i : : : i : : 


•aianms 


!::•::: gi^SH® : : i : i : i 


*XauooH PdqJVfl 


SI5 :S :§?§ i i i i i i : i : : : 


•aapins 


: :; :5 :i^ i : i : : i : i i : : : 


1^. 


i 


i : ; : i ;|£|J2835S«S«SS2:S 


1 


: : h i JSISSaSfe^f^SSSUS 


•«pwa auijooij ::::::: :i2|S8SeS9 : i : : i 


•xoa P«»q4tiH 
puv qiooius 


: ;| "2 :2||§gSgS?JS^8 i ' 


•iai4«a 


1500 
1000 
875 
775 
500 
890 
850 


•oaij 


"II ; ■§ ; H i n ! h ^ n : 


•aumsiujj paq.raa 1 : :g is -SSggSfSfeSSi: : : : : 


•«»a»j I i i i i i i iggaSSSSSS : . 1 : 


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1 


l«qS 


: i i •§ i i 

: : . .5 : : 
j :3S8 i^S 


^gj^sdsg^ISS' 



APPROXIMATE NUMBER OP WIRE NAILS PER POUND. 215 



? 



;!!: 



3f 



s; 



;t? 












35 

lO to t« 000-4 



<Ofc«aoo»*«»ioao 



t^oooj^egjoajjjQgj 



«<*rSiSSgj88jg;;8 



•SS3i:;8S8Sg5Sg 



S;:S:5SS»5So92|g?S 



2SSSSJa88S5?2C8S:S 



;:;SS?33SS;:S2o8S8Ss^g 



'Ml 

Mis 






^ i— 63 a 
•1" 3 --fiS 



SS:S«88iSS8888§gg|§ ; ; ; 



8agS8!gS8t588§5|g§§gg : 



SSSi59S8<:SSS|gg§|§g|8 



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«o^lO«^■aoo»o*«o>eo^lO<o^•aDO»os 



$s;8i 



216 



HATBBIALSi 



SIZB, WBIGBT, liBWOT H, A ND STBBNGTK OP IBOH 
WIBB* 

(Treaton Iron Co.) 





DIanl. 


No. by 


in Deci- 


Wire 


mals of 


aftuge. 


One 




Inch. 


00000 


.480 


0000 


.400 


000 


.860 


00 


.880 





.805 


1 


.285 


8 


.965 


8 


.915 


4 


.295 


5 


.905 


6 


.190 


7 


.175 


8 


.160 


9 


.145 


10 


.180 


11 


.1175 


IS 


.106 


18 


.0995 


14 


.080 


15 


.OTO 


16 


.081 


17 


.05« 


18 


.045 


19 


.040 


90 


.035 


91 


.031 


89 


.098 


28 


.095 


94 


.09% 


95 


.090 


90 


.018 


97 


.017 


98 


.016 


29 


.015 


80 


.014 


81 


.018 


39 


.019 


88 


.011 


SI 


.010 


85 


.0093 


80 


.009 


87 


.0UH6 


88 


.OOH 


89 


.OOiA 


4i 


.wr 



Area of 
SecUon In 


Feet to 

the 
Pound. 


Decimals of 
One Inch. 


.15904 


1.863 


.12566 


2.358 


.10179 


8.911 


.08568 


8.465 


.07306 


4.057 


.06879 


4.645 


.05515 


6.874 


.04714 


6.986 


.lWtf/6 


7.464 


.03301 


8.976 


.02885 


10.458 


.02105 


12.322 


.02011 


14.736 


.01651 


17.950 


.01327 


29.383 


.01084 


87.840 


.00^ 


84.219 


.00679 


44 099 


.00.-i08 


68.016 


.00385 


70.984 


.00999 


101.488 


.00216 


187.174 


.00159 


186.885 


.0019.560 


285.084 


.0009621 


808.079 


.0007547 


392. na 


.0006167 


481.234 


.0001909 


003.863 


.0008976 


745.710 


.0003149 


943.806 


.0002545 


1164.680 


.0002270 


1805.670 


.0002011 


1476.869 


.0001767 


1676 969 


.0001589 


1925.321 


.0001327 


2282.658 


.0001131 


96d0.607 


.0000950 


8119.092 


.00007854 


8778.584 


.00007088 


4182.508 


.00006362 


4657.798 


.00005675 


5222.035 


.00005027 


5896.147 


.00004418 


6794.201 


.00008848 


7698.958 



Weight of 
One Mile 
in pounds. 



Tennile Strpngth (Ap. 

proximal*') of Charcoal 

Iron Wire In Pounds. 




TESTS OF TELEGRAPH WIRE. 



21? 



GALTAN IZBD IRON ITIRE FOR TEI.EGRAPB AND 
TEIiEPHONE LINES, 

(Trenton Iron Ck).) 
Wnovr PBB Milb-Obv.— This term !« to be undei-stood as distingulBhlng 
the mistanee of material only, and means the weight of such material re- 
quired per mile to giTe the resistance of one ohm. To ascei'tain the mileage 
resistance of any wire, divide the '* weight per mile-ohm '' by the weight of 
the wire per mito. Thus in a grade of Extra Best Best, of wtiicb the weiglit 
per mlte-obm is 8000, the mllea^ resistance of No. 6 (weight per mile fiSiS 
lbs.) wotild be about 9^ ohms: and No. 14 steel wire. 6A0O lbs. weight per 
mile-ohm (ftS lbs. weight per mile), would show about 69 ohms. 

Sixes of Wire need In Telecrapli and Teleplione Lines* 

No. 4. Has not been much used until recently; to now used on important 
lines where the multiplex systems are applied. 

NOb 6. Little used in the United States. 

No. C Used for Important circuits between oltles. 

Jfo. 8. Medium slxe for circuits of 400 miles or less. 

No. 9. For similar locations to No. 8, but on somewhat shorter circuits ; 
until lately was the size most largely used in this country. 

Nos. 10. II. For shorter circuits, railway telegraphs, private lines, police 
and fire^lann lines, etc. 

No. IS. For telephone Hues, police and fire-alarm lines, etc. 

No9. 18, 14. For telephone lines and short private lines: steel wire is used 
D>o^ generally in these sixes. 

The coating of telegraph wire with sine as a protection against oxidation 
is now generally admitted to be the most efflcacious method. 

The grades of line wire are generally known to the trade as " Extra Best 
Best " (E. B. B.), *' Best Best " (B. B.), and *♦ Steel." 

** Extra Best Best *' is made of the very best iron, as nearly pure as any 
commerelal iron, soft, tough, uniform, and of very high oonauctivity, its 
««*ii^t per mile-ohm being about 6000 lbs. 

The ** Bemt Best** is of Iron, showing in mechanical tests almost as good 
results as the E. B. B., but not quite as soft, and being somewhat lower in 
conductivity; weight per mile-ohm about 6700 lbs. 

The Trenton *' Steel ^* wire is well suited for telephone or short telegraph 
lines, and the weight per mile-ohm is about 6SO0 lbs. 

The following are (approximatelv) the weights per mile of various sixes of 



[appro ^ 

gaivanixed telegraigh wire, drawn by Trenton Iron Co.'s gauge: 



No. 4, l. 



7, 



18. 14. 



Lbs. 7;», 610, &25, 450, 875, 8l0, ^, 200, 160, 1S5, 95. 

TESTS OF TEI.EORAPH DTIRE. 

Ilie following data are taken from a table given by Mr. Prescott relating 
to tests of E. B. B. galvanized wire furnished the Western Union Telegrwh 
Co.: ^ ^ 



Size 
of 


Dlam. 

Parts of 
One 
Inch. 


Weight. 


Length. 

Feet 

I)er 

pound. 


ReslsUnce. 
Temp. 75. 8« Fahr. 


Ratio of 

Breaking 

Weight to 

Weight 

per mile. 


Wire. 


Grains, 
per foot. 


Pounds 
per mile. 


Feet 
per ohm. 


per mile. 


10 
11 
12 
14 


.888 
.2W 
.90S 
.180 
.165 
.148 
.134 
.ttO 
.100 
.063 


1048.3 
891.8 
758.9 
696.7 
501.4 
408.4 
880.7 
966.8 
818.8 
126.9 


886.6 
678.0 
57i.S 
449.9 
878.1 
804.8 
249.4 
800.0 
166.0 
95.7 


600 
7.86 
9.20 
11.70 
14.00 
17.4 
21.2 
28.4 
82.0 
55.2 


958 
727 

618 
578 
409 
828 
269 
216 
179 
104 


6.51 
7.26 
8.54 
10.86 
12.92 
16.10 
19.60 
24.42 
29.60 
51.00 


8.05 
8.40 
8.07 
8.88 
8.87 
2.97 
3.43 
8.06 



Joisrrs IN Tklkoraph Wires. — ^The fewer the joints in a Hue the better. 
All Joints should be carefully made and well soldered over, for a bad joint 
nay cause aa much resistance to the electric current as several miles of 
»ire. 



218 



KATEBIALS. 



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Dm Bl^SIONS, WBIOHT, RESISTANCE OF COPPEB WIRE. 219 




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MATBBIALS. 



S8g9*4M*9«««ei 



t*«e«o«4w<i:«io«(««ee»< 



asjsasisiiSiSssseesKsiistssKs 






9k 



SI 

«<!a 

4 

ii 

■H 

m 

O 

H 

H 








.St;83SS.8S3.38S38i93S8Ei!3Sii§il!iiliilisii|i 

PiiilliililiP2SSSISSS«S8S8ssa2s---''*-'^---» 



iiiliiiliiilSiliilis8S§§eS3§gisSSisis.Rssfle8 



4P«o«Mn««r«»>»ies 



bSs8! 



asssissgsgsiiiiiiiiiiililsisiiiiiiiiiiiiiil 



III 



alJz 



Bl 



iilliiiiiiiiiiiiiiiiiiiiii^^^^ 



n. 



iiiiiiiiiiiiiiii^ii^^i^^§^^^^^^^^^^^ 



ES 



;;aES9i§;ss§iii§ 



ligi'asssi'issiiirsiiiliiiiisiiiii 



iiii 



||8o-««^-5«»t^acoio;j2.23SSS«2«C«8S|g«fc«88a«ft«IIJI6RR5 



HARD-DRAWN COPP^ft WIRE} INSULATED WIRE. 221 



lAVnt COFPBS TBIiBettAFli WtBK* 

(J. A4 Roebllng^B Bona Co.) 
FumMied in hAlf-mne coils, either bare or insulated. 



UK. B. ft 6. 
Gauge* 



9 
10 
11 
18 
18 
14 
15 
16 



Reslsta&cein 

Ohms 

per Mile. 



4.90 
0.40 
9.90 
8.TO 
10.00 
18.10 
17.40 
2^.10 



Breaklnir 
Sti-enKth. 



can 

4«) 



213 

IS! 



Weight 
per Mile. 



ttoo 

160 
131 
104 
88 

41 



Attt>lK»±imftte 

BtfeeofB B.B. 
Iron Wire 
eqiial to 
Copper. 



'I 

n(3 

If 
JO I 



lit haudiin^ tlii^ wire tlie rreatest care should be obseKed to avbid Iclnlcs, 
i««fndB. ecratches, or cats. Joints should be made only with Mclutire Cuu- 
nectors. 

On account of ila cendoeHvitj being about fire times thai of Ex. B. B. 
Iron Wire, and its breaking Strength oter three times its weight per niilef 
copper maybe used of which the section is smaller and the weight less ihnn 
an equivaMst if6n wire, allowing a gi^aier number of wires lo be strung on 

Beaides this advAntage, the rediiotlon of section materially decreases the 
electroetatlc capacity^ while its non-magiietlc character lessens the self-in- 
duction of the line, both of which features tend to increase thepo.«sible 
fpeed of signalllitg in telegraphing, and to give greater clearness ofenuuci- 
aiioii over telephone lines, especially Ihoee Of great lengili. 

I1f917I<ATBII COPPER WlRB^ WfiATHEBPttOOJP 

iivscri<ATlON. 





Doable Braid. 


Triple Braid. 














Approziroate 
weights, 


Kum- 










henu 
B & B. 


Outside 


Weights, 


Outside 


Weights. 


Pounds. 


Diame- 


Pounds. 


Diattie- 


Pounds. 




Gau^. 


ters in 
88d8 




ters in 
ftMs 








1000 




1000 










Inch. 


Feet. 


Mile. 


Inch. 


Feet. 


Mile. 


Reel. 


Coil. 


0000 


20 


716 


8781 


24 


776 


4098 


2000 


250 


000 


18 


675 


8096 


22 


080 


8820 


2000 


2.')0 


00 


17 


46S 


a4.w 


18 


490 


2587 


500 


2.'iO 





10 


875 


1980 


17 


400 


2112 


600 


250 


1 


15 


885 


1505 


10 


800 


1616 


600 


250 


2 


14 


845 


1204 


15 


268 


1415 


600 


250 


t 


13 


190 


1003 


14 


210 


1109 


600 


250 


4 


11 


15a 


808 


12 


104 


860 


250 


125 


5 


10 


m 


684 


11 


115 


700 


200 


130 


6 


9 


98 


618 


10 


112 


591 


275 


140 


8 


8 


66 


349 


9 


78 


412 


200 


100 


!0 


7 


45 


2:J8 


8 


55 


290 


200 


100 


12 


6 


80 


158 


7 


85 


186 


.... 


25 


24 


5 


20 


106 





20 


U7 




25 


Id 


4 


14 


74 


5 


20 


100 


.... 


25 


»s 


8 


10 


&l 


4 


10 


85 


.... 


S5 



222 



HATBRIAL8. 



Power Cable*. I<e«d Ineaeetf, JTute or VwLpvr iBsnlated* 

(John A. BoebliDg's Bons Co.) 



N08,. 


circular 
Mils. 


Outside 
Diain. 
Inches. 


Weights, 
1000 feet. 
Pounds. 


Nos.. 
B.&S.O. 


Circular 
Mils. 


Outside 
Dlam. 
Inches. 


Weights, 
1000 feet. 
Founds. 




1000000 
900000 
800000 
750000 
700000 
660000 
600000 
650000 
600000 
460000 
400000 
850000 


1 18/16 
1 88/82 
1 21/32 

1 19/82 
1 9/16 
1 17/82 

•y 

1 11/32 
1 5/16 


6685 
6228 
6778 
6543 
5316 
6088 
4857 
4630 
4278 
8928 
3619 
8410 




800000 

260000 
811600 
168100 
138835 
105625 
88581 
66564 
62441 
41616 
iJ6844 


11/.. 

1 3/38 
1 1/16 

15/16 

i/82 
lf/16 


8000 




"0606* 
CM 
00 

1 
8 
8 
4 
6 


8782 
8633 




8300 




.8021 




1772 




1683 
1482 




1360 
1851 




1046 











Stranded "Weatlter-proof Feed "Wire* 



Circular 


Outside 

Diam. 

Inches. 


Weighto. 
Pounds. 


CO 

ll 
Si 

m 


Circular 
Mils. 


Outside 

Diam. 

Inches. 


Weights. 
Pounds. 


5 


Mils. 


1000 
feet. 


Mile. 


1000 
feet. 


Mile. 


III 


1000000 
900000 
800000 
750000 
700000 
660000 
600000 


ll8/32 
1 11/32 
1 5/16 
1 9/82 

1 7/32 


3350 
3215 
2«80 
2718 
8545 
2378 
2210 


18744 
16975 
15206 

13438 
12.556 
11608 


800 
800 
860 
8.^) 
900 
900 
1000 


650000 
500000 

450000 
400000 
3.50000 
300000 
250000 


1 3/16 

\%^ 

1 1/16 

1 
15/16 
29/38 


2043 
1875 
1708 
1590 
1868 
1185 
1012 


10787 
9900 
8998 
8078 
7170 
6257 
5843 


1800 
1320 
1400 
1450 
1500 
1600 
1600 



The table Is calculated for concentric strands. Rope-laid strands are 
larger. 



8TSBL WIBE CABLES. 



OAltTANIZED STBBIi-lTIBE imftAHD. 

If or Smokestack Gnys, Slffiial Strand, etc* 

(J. A. Roebling's Sons Co.) 
This strand is composed of 7 wires, twisted together into a single strand. 



i 




llf 


J ♦ 




lit 

1 


in. 


^1 

^8 
^- 

lbs. 


Hi 


in. 


lbs. 


lbs. 


in. 


lbs. 


lbs. 


llM. 


15% 


5t 


&8-J0 


9/82 


18 


2,600 


5/8» 


tx 


700 


48 


7.500 


17/64 


15 


2,i»0 


9/64 


525 


'4' 


ST 

m 


«,000 
4.700 


7%l 


im 


1,750 
1.300 


8^ 


f^ 


875 
820 


57l« 


21 


8,aoo 


a/16 


1,000 









For special purposes these strands can be made of 50 to 100 per cent 
f^reater tensile strength. When used to run over sheaves or pulleys the use 
of soft-iron stock is advisable. 

FI«EXIBI.E STBBI«-WimE CABI«E8 FOR TBSSBI^S. 

(Trenton Iron Co., 1886.) 

With numerous disadvantages, the system of working ships* anchors with 
tiiafn cables is still in vogue. A heavy chain cable contributes to the hold- 
ing-power of the anchor, and the facility of increasing that resistance by 
1)a\tng out the cable is prized as an advantage. The requisite holding- 
power is obtained, however, by the combined action of a comparatively 
iij;ht anchor and a correspondingly great mass of chain of little service in 
proportion to its weight or to the weight of the anchor. If the weiffbt and 
size of tb« anchor were increased so as to give the greatest holding-power 
required, and it were attached by means of a light wire cable, Ww combined 
wftght of the cable and anchor would be much less than the total weight of 
the chain and anchor, and the facility of haiidline would be much greater. 
English shipbuilders have taken the initiative in this direction, and many of 
the large«t and most serviceable vessels afloat are fitted with steel -wire 
cables. They have given complete satisfaction. 

The Trenton Iron Co.'s cables are made of crucible cast-steel wire, nnd 
guaranteed to fulfil Lloyd's requirements. Thev are composed of 7*J wires 
subdivided into six strands of twelve wires each. In order to obtain great 
flexibility, hempen centres are introduced in the strands as well as in the 
completed cable. 

FI.BXIBI«B 8TBBI.-WIBE RAWSERS. 

These hawsers are extensively used. They are made wlrh six strands of 
twelve wires each, hemp centres being inserted in the individual strands as 
wdl as In the completea rope. The material employed is crucible cast steel, 
galvanised, ond gitarantetra to fulfil Lloyd's requirements. They are only 
one ihini the weight of hempen hawsers; and are sufTlcientl v pliable to work 
ronnd any bitts to which hempen rope of eqnivalent strength can be applied. 

13-tnih lArred RnsKian hemp hawser weighs nbout 89 IhH. per fathom. 

10-inch white manila hawser weighs about 20 lbs. per fathom. 

1^-inch stud chain weighs about 68 lbs. per fathom. 

4-Hic^ galwvuixed ateel hawaer loeighs tibont 12 Ibt. per faViom. 

Each of the above named has about the same tensile strength. 



iu 



irAtBBIALS. 



SPBOinCATIOWS WOU OAI^TAHimBD IROV «7IIiB. 

Issued bjr tlie Vrtttsh Postal Telegrapli Autborltles. 



WeiRht per Mile. 



Allowed. 



Diameter. 



Allowed. 






Teste for Strength and 
Ductility. 









h5 



I 



ix 



lbs. 

dOO 
iOO 
490 
400 
200 



lbs. 

T87 
571 
494 
877 
190 



lbs. 

010 
47T 
424 
218 



mils. 

242 
SiOe 
181 
171 
121 



mils. 

237 
204 

176 

166 
118 



mils. 

247 
814 
186 

176 
125 



lbs. 

2480 
1880 
1890 
1240 



lbs. 

2550 
1910 
1425 
1270 
638 



lbs. 

20S0 
1960 
1460 
1800 
655 



ohms. 

6.75 
9.00 
12.00 
18.60 
27.00 



5400 
5100 
6400 
5400 
5400 



STRENGTH OF PIAN4>-1¥IRE. 

The average strength of English piano- wire is given as follows by Web> 
ster, Ho^sfals & Lean: 



Numbers 


Equivalents 
in Fractions 


Ultimate 


Numbers 


Equivalents 
In Fractions 


Ultimate. 


in Music- 


Tensile 


in Music. 


Tensile 


wire 


of Inches In 


Strength in 


wire 


of Inches in 


Strength in 


Gauge. 


Diameters. 


Pounds. 


Qauge. 


Diameters. 


Pounds. 


12 


.0«0 


286 


18 


.041 


805 


18 


.081 


260 


19 


.048 


«» 


14 


.088 


286 


20 


.045 


600 


15 


.065 


805 


21 


.047 


640 


16 


.087 


840 


28 


.052 


050 


17 


.089 


860 









These strengths range from 800.000 to 340,000 lbs. per so. In. The compo- 
sition of this wire is as follows: Carbon, 0.570; silicon, 0.090; sulphur, 0.011; 
phosphorus, 0.018; manganese, 0.425. 

<<PI«017GH"-STBEI. WIRE. 

The term "plough," given in England to steel wire of high quality, was 
derived from the fact that such wire is used for the coustruciion of ropes 
ttsed for ploughing purposes. It is to be hoped that the term will not be 
ised in this oouutry, as it tends to confusion of terms. Plough-steel is 
tnown here in some steel- works as the quality of plate steel used for the 
mould-boards of ploughs, for which a very ordinary grade is good enough. 

Experiments by Dr. Percy on the English plough-steel (so^Mdled) gave the 
following resultji: Specific gravity, 7.614 ; carbon, 0.828 per cent; manga- 
nese, 0.587 per cent; silicon, 0.143 per cent; sulphur, 0.009 per cent; phos- 
phorus, nil; copper, 0.030 per cent. No traces of chromium, titaaiam, o7 
tungsten were found. The breaking strains of the wire were as follows: 

Diameter, inch 098 .188 .159 .191 

Pounds per sq. inch 814.960 237,600 224,000 201,600 

The elongation was only from 0.75 to 1.1 per cent. 



SPECIFICATIONS FOR HARtMDRAWN COPPER WIRE. 225 



WIBBS OP DIFFHBBNT nBTALS AND AI«I.OT8. 

(J. Buckoall Smith's TreaUae on Wiro.) 

RnuM Wire to eommonly compoMd of an alloy of 1 8/4 to S parts of 
•opper to 1 part of sine. The tensile strength rangos from SO to 40 tons per 
square inch, increaslnjT with the perceatace of zfno fn the alloj. 

OevBUtn or NlcMl Ml-ver^ an atfojr of copper, sine, and nickel, is 
practically brass whitened bj the addition of nickel. It has been drawn into 
wire as line as .008" diam. 

yiatiimm wire may be drawn info the finest sizes. On account of its 
hl^ii price Its use Is praetieaUy conflned to pedal scientlflj Instruments and 
eleetrical i^plianaes in which reslstanoee to high temperature, oxygen* and 
acids are easentiaL It expands leas than other metals when heated, which 
property permits its being sealed In ghiss without fear of cracking. It is 
iharefore used in Incandescent electrio lamps. 

FIsospMoi^broiuM IITlre contains from 8 to 6 per cent of tin and 
from l/SU to 1/8 per cent of phosphorus. The presence of pbosptioms is 
d^rrfmental to electric conductivitv. 

** Delta«!BieCiil " wire Is made from an alloy of 
Its strength fsnges from 46 to 08 tons per 84|nare inch. 



comwr. Iron, and sine. 

_ 84|nare inch. It Is vied for some 

kinds of wirs rope, also for wire gauie, It Is Dot shbJeot to deposits of Yer- 



\ great touglmess, even when Its tensile strength is over 00 
elneh. 

B been drawn as line as 1 1 ,400 yards to 



avity .868. T^'nslle strength only 



Sigris. It : 

tons pM- square Inch. 

Al«SBiimBBi tirtre* — Speciflo 
alwai 10 tons per square Inch. It li ^ 
tbe ouooe. or .OCajrralAs per yard. 

Al— irawm mrowtrntf W copper, lO ataunlnum, has ' Igb strength and 
ddciility I is inozkUzablev sonorous. Its eleotric coddcKStiTit;^ Is IS .0 per oent 

Hlteon Bronze, patented In 1883 by L. Weller of Paris, is mads n^ 
foUuws : Flnosilicate or potash, pounded glass, chloride of sodium and cal- 
dom, earbooate of soda and lime, are heated in a plumbago Crucible, and 
after the i^eaction takes place the contents are thrown Into the molten 
bronze to be treated. Billcon-bronce wire has a conductivity of from 40 to 
98 per cent of that of copper wire and four times more than that of iron, 
vhile Its tensfle strength is nearir that of steel, or 80 to 55 tons per sqtiarO 
inch of SMtion. The conductivity decreases as the tensile strengih in- 
creajics. Wire whose conductivity equals 95 per cent of that of pure copper 
fives a tensile strength of 88 tons per square inoh, Imt when its conductivity 
u 31 per cent of ptire copper, its strength is 60 tons per square inoh. It is 
bein^ largely used for tel4*graf^ wires. It has great resfittAnce to oicldatlon. 

Ordinary UrAurn and Annealed Copper Wire has a strength 
of from U to 80 tons per square inch, 

nnSCIFICATIOllS VOB HABl»«miAWN CO^Pttm 



Tbe British Post Office authorities require that hard-drawn copper wire 
supplied to them shall be of the lengths, sises, weights, strengths, and Con- 
doctivities as set forth in the annexed table. 



Weight p^ Statute 


Approkimate Squlf a- 
lent Diameter. 


1 

1 


is 
P 


fill 


In 

lit 


11 
11 


1 


1 


1 


1 


1 


lbs. 
100 
ISO 
»0 
460 




lbs. 
410 


158 


mils. 
78 


mils. 
80 


lbs. 
890 

490 
C'^ 
1800 


80 
85 
SO 
10 


ohmS« 
0.10 
0.06 
4.58 
8.87 


lbs. 
so 

60 



226 



MATERIALS. 



WIRE ROPBS. 

List adopted by manufacturer in 189 J. See pamphlets of John A. 
Boebling's Sooa Co., Trenton Iron Co., and other makers. 

Pliable filoUtlns Rope* 

With 6 strands of 19 wires each. 

IRON. 



1 



1 
9 
S 
4 
ft 

7 

10 

T 
\L 



I 
I 



P 



4 

m 

P 



id 



g.OO 

9.05 

1.58 
l.VD 

0.4tt 



^1 

OQS 



74 
85 
64 

44 

89 
88 
27 
80 

16 

11.50 
8.64 
5.18 
4.27 
8.48 
8.00 
2.50 



§4 
I sal 



11 
It 

18 
12 

20 

P 



OA8T STKBL. 



1 

8 
8 

4 
S 

7 
8 
9 
10 

lo^ 

i 

lOa 
10% 




8.00 
6.80 
6,^5 
4.10 
8.65 
8.00 
2.50 
2.00 
1.58 
1.20 
0.88 
0.60 
0.48 
0.39 
0.29 
0.23 



155 
125 
106 
86 
77 
63 
52 
42 
83 
25 
16 
12 
9 
7 



sn 






Cable-Traction Ropes* 

According to English practice, cable tiHcti ^n ropeK, of about 8>{ in. in 
circumference, are commonly constructed with six strands of seven or fif- 
teen wires, the lays in the strands varying from, say. 8 in. to 8H in., and the 
lays in the ropes from, say, 7^ in. lo 9 in. In the United States, however, 
strands of nineteen wires are generally preferred as being more flexible; 
but, on the other hand, the smailler external wires wear out more rapidly. 
The Marlcet street Street Railway Company, Snn Francisco, has used ropes 
1^ in. in diameter, composed of six strands of nineteen steel wires, weighing 
2^ lbs. per foot, the longest continuous length being 24,125 ft^ The Chicago 
City Railroad Company has employed cables of i Vntical construction, the 
lonxesl length being 27,700 ft. On the New York and T. .olslyn Bridge cable- 
railway steel ropes of 11,500 ft. long, containing 114 wires, have been used. 



WIEE BOPES. 



227 



Truismlsaloii anA Standlns Rope. 

With 6 strandB of 7 wires each. 

XBON. 



a 

s 



11 

12 
IS 
14 
15 

le 

17 
18 
19 
SO 

» 

SS 
M 



IK 

Hi 



8.87 
«.77 
2.88 
1.83 
1.60 
1.18 
0.92 
0.70 
0.57 
0.41 
0.81 
O.SS 
0.21 
0.16 
0.125 






88 
80 
25 
20 
16 
12.8 
8.8 
7.6 
6.8 
4.1 
2.88 
2.18 
1.66 
1.88 
1.03 



4 
8 



Ml. 

Ijlf 

ill 



as ^ w 



CAST 8TBKL. 



11 


iH 


^ 


8.87 


62 


18 


18 


BH 


13 


i^^i 


2.77 


52 


10 


12 


8^ 


18 


]< 


4 


2.28 


44 





11 


75(, 


14 


iii 


3^ f 


1.88 


86 


7H 


10 


^ 


1ft 


1 


312 


1.50 


80 


6 


9 


S 


16 
17 


L 


2^ 


1.12 
0.92 


22 

17 


n 


8 

7 


5^ 
4^ 


18 


sS 


0.70 


14 


8 


6 


4^ 


19 
90 


1,. 


2 


0.57 
0.41 


11 

8 




^ 


?* 


21 


^.r. 


u 


0.31 


6 


]L 1 


4* 


in 


28 


1^ 


0.28 


4H 


1^ 


1 


28 
M 


t,. 


{^ 


0.21 
0.16 


4 
8 


1 


8 


96 


0-82 


» 


0.12 


2 


iS 



Plon8:h-8teel Rope. 

Wire ropes of Tery high tensile strenRth, which are ordinarily called 
"Plougfa-steel Ropes/* are made of a high grade of crucible steel, which, 
when put in the form of wire, will bear a strain of from 100 to 150 tons per 
noare indi. 

w^here it Is necessary to use very long or very heavy ropes, a reduction of 
the dead weight of ropes becomes a matter of serious consideration. 

It is advisable to reduce all bends to a minimum, and to use somewhat 
larger drums or sheaves than are suitable for an ordinary crucible rope hav- 
ing a strength of 60 to 80 tons per square inch. Before using Plough-steel 
Hopes it is oest to have advice on the subject of adaptability. 



MATERIALS. 



WItli fltrands of 19 wires each. 



Trade 


Diameter In 


Weight per 
foot ia 
pounds. 


Breaking 
Strain in 


Proper Work- 


Min. Size of 
Drum or 


Number. 


inches. 


tons of 

aoooibB. 


iDg Load. 


Sheave in 
feet. 




2H 


8.00 


240 


46 







¥ 


6.ao 


189 


37 


8 






5.25 


167 


31 


7^ 




I'lg 


4.10 


128 


25 


6 




] ^ 


8.66 


110 


22 


^ 


9H 


lis 


3.00 


90 


18 




T^H 


2.60 


75 


15 


6 




IH 


2.00 


60 


12 


41^ 




1 


1.58 


47 


9 


4i± 






l.<» 


87 


7 


2B^ 




. 1^ 


0.88 


27 


5 


8l4 


low 


tH 


0.60 


18 


8^ 


3 


10^ 


0-16 


0.44 


18 


^ 


2H 


m 


H 


0.30 


10 


r 


2 



With 7 Wires to the Strand, 



15 


1 


1.80 


45 


9 


^H 


16 


L 


1.12 


83 


•H 


5 


17 


0.«2 


25 


6 


4 


18 


0.70 


21 


4 


3^ 


19 


^16 


0.57 


16 




8 


20 


0.41 


12 


212 


n 


21 


^-16 


0.81 


9 


ifft 


23 


0.23 


5 


1V6 


2 


23 


% 


0.21 


4 


1 


iH 



OalTanlzed Iron Wire Rope* 

For Ships' Rigging and Quys for Derricks. 
CHAROOAL ROPE. 



Circum- 
ference 
in inches. 



6« 



4 
2« 



Weight 
per Fath- 
om in 
pounds. 



22 
21 
19 

14^ 



ar. of 

new 

Manila 

Rope of 

equal 

Strength. 

11 

101^ 

10 



n 



Break- 

ifig 
Strain 
In tons 
of 2000 
pounds 



48 

40 
35 
33 
80 
26 
23 
20 
16 
H 
IS 
10 



circum- 
ference 
f n inches 



Weight 

per 
Fathom 



CIr. of 

new 
Manila 



i_ I Hope of 



Brwtk- 

iiig 
Strain 
in tons 
of 2000 
pounds 



WIBE HOPES. 



229 



Galyatoed CMt-iitMl Taebt 



Tjir^ 'per Path. 



Cir. of 

new 

Vanina 

Bopeof 

equal 

Strei^h. 



Break-; 

Strain 
in tons 
of 9000 
pounds 



18 
11 



n 



06 

49 
8t 
£7 
8S 
18 



Girram- 

farence 
kiinoiies 



Weight 
per 

Fathom 

in 
pounds. 



Cir. of 

new 

Manilla 

Kopeof 

equal 

Strength. 



Break- 
ing 
Strain 
in toiis 
of 2000 
pounds 





Steel Hawsers. 

Por Mooring, Sea, and Lake Towing. 




CircnmfBT- 
ence. 


BTCMklflff 

Strengtii. 


Size of 
Manilla Haw- 
ser of eqoal 
StrengOi. 


GlreimfQr- 
eaee. 


Breaking 
Strength. 


Size of 
Manilla Haw- 
ser of eqiial 
Strength. 


Inches. 


Tons. 
15 
18 


Inches. 


Inches. 

- f" 


Tons. 
29 
85 


Inrbes. 
9 
10 



Steel Flat Ropes* 

(J. A. Boebling's Sons Co.) 
Steel-wire Flat Ropes are composed of a number of strands, aitematelf 
twisted to the right and left, laid alongside of each other, and sewed together 
with soft iron wires. These ropes are used at times in place of round ropes 
Id the shafts of mines. They wind upon themselves on a narrow wiudinf;- 
dnua, which takes up less room than one necewary for a round rope. Tiie 
softriroa sewing-wires wear out sooner than the steel strands, and then it 
bfgoroes necessary to sew the rope with new iron wires. 



Width and 
Thickness 


^/cSi^sr 


Strength in 
pounds. 


Width and 
Thickness 


Weight per 
foot in 


Strength hi 
pounds. 


in Inches. 


pounds. 


in inches. 


pounds. 




Kx> 


1.10 


85,700 


Ux8 


2.86 


71.400 




i*^ 


1.88 


55.800 


'2k3^ 


2.97 


89.000 




i^^9 


2.00 


80,000 


2x4 


8.80 


99,000 




Z^m 


S.50 


75,000 


i^*H 


4.00 


120.000 




2«4 


8.86 


85,800 


:|x6 


4.27 


128,000 




i^4H 


8.12 


88,600 


2x5^ 


4.82 


144.600 




^ x6 


8.40 


100,000 


^k6 


6.10 


153,000 


^ 


«k5« 


8.90 


110,000 


Hx7 


5.90 


177,000 



For safe working load allow from one fifth to one seventh of the breaking 
stress. 

** liani: Lay »' Rope. 

la wire rope, as ordinarily made, the component strands are laid up Into 
rope in a direction opposite to that in which the wires are laid into strands; 
tiMt is, if the wires in the strands are laid from right to left, the strands are 
laid into rope from left to right. In the ** Lang Lay," sometimes known as 
■* UDiTersai Lay,** the wires are laid into strands and the strands into rope 
in the same db-ection; that is, if the wire iB laid in the strands from right to 
left, the strands are also laid into rope from right to left. Its nse has been 
fonnd desirable under certain conditions and for certain purposes, mostly 
for bauUge plants, inclined planes, and street railway cables, although ft 
has also been used tor Tertlcia hoists in mines, etc. Its advantages are that 



230 



MATERIALS. 



OALTANIKED STBEIt CABLB8. 

For Suspension Bridges. (Roebling's.) 



200 

180 



I 

I 



13 

11.8 

10 



2 

1% 



155 

no 

100 



8.G4 

6.5 

6.8 



a 
a 



95 
75 
65 



t 



t 



5.6 
485 
3.7 



COniPABATITE 8TBBNGTH8 OF FLEXIBIiE GAI.- 
VANIZBD STEEIi-HriBB HADTSBRS, 

'With. Olialn Cable, Tarred Russian Hemp, and DTlUte 
Manila Ropes. 



Pftt<?iit Flpxiblfl 






Tarred Riib- 


White 


Sttiel-wjna llawmerN 


Chatn Cable 




sian Hemp 


Blaiiilla 


mil Ct^h}^ 






Rope. 


Ropes. 




e 
^ 


3c 






E 


g 


1 




1 


3 




i 


1 


ts 


a 


c 


IX"-^ 




« 


5 


c 




cO 


a 




t 


o" 


1 

g 

s 




II 


a. c ^ 




! 


5 


1 




fa 

2 


c 




& 

I 


1 

be 
c 


J 


i 

^4 


6 




It 


1 


3 . 
n 


1 


1 


1 

n 


1 


1 


!S 


1 








2" 


iH 


! 


'Hi 


7H 


>4 


H 


4HJ 





3 


2m ' 


8 


^ 


ii 


IH 


i 


» 










4 


^ 


3k! 


m 


8 


S^ 


f-H 


104 


J-tO; ir 


A» 


7J4 


5 


6^ 


5 


4 


8 


r* 


a 


!K 


T 


13 


1 






i 


8 


7 


6 


4H 


7H 


m 


^ 





1^^ 


JQ-JO 


31 


V 


0^ 


10 


9 


^ 


6 


ic^ 


m 


3^ 1^ 1 


15 










13 


11H 


6M 


7 


18? 


■i^ 


«j 


IS 
IS 


]S« 


11-16, 


Sfl' 


tSI 


17 8-10 


s« 


16 
10 


14 
6^ 


7 


1^ 


16^ 
18 


^ 


B 


ShJ 


J»14 


1S-1C 


s-% 


t i^K 


10 


23 


20 


18^ 


2SIC 





art 




liWtfl 


<ti 


1&8-10 


23 7-10 


11 


28 


24H 


9 


14^ 


25^ 


4^ 


ra 


3-1 


'^1 


1 


r.( 


IS 


27 


12 


33 


29 


10 


18 


1^ 


4^ 


ts 


Sfl 


o; 


IV4 


ft^,-'« 


34^ 

rr.u 


13 


89 


34 


11 


22 


f. 


■ii4 'i* J 


sn 


t iT-^IS* 


fr;r!7L2 


m 


Wl 


50 


1^ 


29V4 


51 


f.H 


liH 


Tl 1 


33 


m 


nn.g 


17 


C7 


GO 


85H 


62 


c 


.^1 


ftB 


M ; 


IS 


Uki, u 


771.^ 


'0 


m 


72 


15^ 


42 


^^ 


«H 


iJ7 \m 1 


:;:> 


I I?uffi 


siu b:2 


94U 

in? 1-10 


21 


106 


89 








7 


11 


116 


43 


3 l-1fl 


a^i 7. a 


23 


123 


106 








m 


tr 


ISO 


45 


2 3-16 


i*^ ^vQ 


134^ 


24 


134 


115 








ft 


« .» 1 


4S ' 


a 5-;c 


yWKjJ 


25 


146 


125 









Note.— This is an old table, and its authority is uncertain. The figures In 
the fourth column are probably much too small for durability. 



WIRB R0P2S* 231 

It is somevhat more flexible than rope of the same diameter and composed 
of the same number of wires laid up in the ordinary manner; and (especi- 
ally) that owinfc to the fact that the whies are laid more axially in the rope, 
longer surfaces of the wire are exposed to wear, and the endurance of the 
rope is thereby increased. (Trenton Iron Co.) 

Note* on the Vme of Wire Rope. 
(J. A. Boeblincf's Sons Co.) 

SeTsral kinds of wire rope are manufactured. Tlie moftt pliable variety 
eontains nineteen wires in the strand, and is geuerally used for liolstiug and 
running rope. The ropes with twelve wires and seven wires In the strand 
are stiffer, and are better adapted for standing rope, gu^'s, nod riguing. Or- 
dem should state the use of the rope, and sdvice will be given. Itopes are 
made up to three Inches in diameter, upon application. 

For safe working load, allow one fifth to one seventh of the ultimate 
strength, according to 8pee<l. so as lo get good wear from the rope. When 
substituting wire rope for hemp rope, it Is good economy to allow for tho 
former the same weight per foot which experience has approved for tho 
latter. 

Wire rope is a«( pliable as new liemp rope of the same strength: the for- 
mer will tnerefore run over the same-sized sheaves and pulleys as the latter. 
Bat the greater the diameter of tlie sheaves, pulleys, or drums, the longer 
wire rope irill last. The minimum size of drum is given in the table. 

Experience has demonstrated that the wear incivases with the t:peed. It 
Is, tiierefore, better to increase the lond tlian the s|»ee<l. 

Wire rope is manufactured either with a wii-e or a hemp centre. The lat- 
ter is more pliable than the former, and will wear better where there is 
short bending. Orders should specify what kind of centre is wanted. 

Wire rope musi^ not be coiled or uncoiled like henip rope. 

Milken mounted on a reel, the latter should be mounted on a Kplndle or flat 
turn-table to pay off the rope. WJien forwarded in a small coll. witliout rrel. 
roll it over the ground like a wheel, and run off (he rope in that way. All 
uniwisting or kinking must be avoided. 

To preserve wire rope, apply raw linseed-oil with a piece of eheepskln, 
wool inside; or mix the oil with equal parts of Spanish brown or lamp-black. 

To preserve wire rope under water or under jrround, take mineral or vege- 
table tar, and add one bushel of fresh-slacked lime to one barral of tar, 
which wQl neutralize the acid. Boil it well, and saturate the rope wi^ the 
hot tar. To give the mixture body, add some Kawdust. 

The grooves of cast-iron pulleys and sheaves should be flileil with well- 
sMHoned blocks of hard wood, set on end, to be renewed when worn out. 
TTiis end-wood will save wear and increase adhesion. The smaller pulleys 
or rtdlers which support the ropes on inclined planes should be contttrucied 
on the santie plan. When large sheaves run with very great velocity, the 
^roovea should be lined with leather, set on end, or with India rubber. This 
IS done in the cose of sheaves used in the transmiaeioH of power between 
distant points by means of rope, which frequently runs at the rate of 4000 
feet per minute. 

Steel ropes are takin? the place of iron ropes, where It Is a special object 
to combine lightness with strength. 

But in substituting a steel rope for an iron running rope, the object in view 
should be to gain an increased wear from the rope rather ihau to reduce the 
size. 

Locked "Wire Rope. 

Fig 74 ohows what Is known as the Patent Locked Wire Rope, made by 
the Trenton Iron Co. It is chiimed to wear two to three times as long as an 




Fio. 74. 



ordinary wif« rone of equal diameter and of like material. Sizes made aro 
irum ^to 1^ incmes diameter. 



232 



HATEBIAL8. 



OBANB GHAINS, 

(PencQyd Iran Works.) 



*' D. B. a/* Special Craoe. 



Crane. 



^16 
?1C 

H 
18-16 

16<16 
1 
1 1-16 



II 

r 



25-92 

Sl-SS 
15-83 
111*88 

i2S~sa 

127-83 

181-88 

8S-88 

2 7-82 

815-82 

81<M8 

8^8-82 

8 27-82 

8 6^88 

8 7-88 

815-88 



3 

8 81-SS2 



1 

17-10 

8 

SM 

8-^10 

J« 

8 

9 

10 7-10 
118-10 

18 7-10 
16 

18 4^10 

19 7-10 
21 7-10 



1938 



89568 
89264 
87576 
41888 
46800 
00518 
66748 
60868 
66588 



I 



o 



8864 
6796 
8878 
11593 
15456 
19880 



28980 
84776 
40579 
44968 
61744 
60186 



79158 
88776 
98400 
101024 
111496 
180736 
138059 



¥ 



1988 
8790 
8864 
5188 
6440 
7948 
9660 
11598 
18594 
14989 
17848 
19718 
88176 
85060 
87995 
80800 
88674 
87165 
40245 
44858 



1680 
8500 
8640 
6040 
6790 
8400 
10660 
18600 
15180 
17640 
9044O 
88080 
86880 
80840 
84160 
88080 
42000 
45080 
50680 
64880 
60480 



8860 
6040 
7880 
10080 
18440 
160OO 
90780 



40880 
47040 
68760 
60480 
68880 
76160 
84000 
91840 
101860 
109760 
120960 



1180 
1680 
8477 
8880 

4480 
6600 
0907 
8400 
10080 
11760 
18627 
18680 
17980 
80160 
887T8 
96887 
48000 
8061S 
89787 
86687 
4O380 



The dUtanoe from centre of one link to centre of next ie equal to the In- 
side length of link, but in practice 1/82 inch is allowed for weld. This is ap- 
proximate, and where exactness is required, cliain should be made so. 

Fob CBAiif SBBAVB(i.»The diameter, if possible, should be not lass than 
twenty times the diameter of chain used. 

BxAMFLB.— For 1-inoh chain use 20-inch sheaTes. 

DTBIGHTS OF LOGS, liVRIBBR, ETC. 
ITelfflit of Green liOffs to Scale 1,000 Feet, Board lIlMunire* 

Yellow pine (Southern) 8,000 to 10.000 lbs. 

Norway pine (Michigan) 7,000 to 8,000 " 

whitepine(Mich.gan,]°««'i,-;>™5-:;;:;:::::;:;:::::: f^^ iz " 

White pine (Pennsylvania}, bark off 5,000 to 6,000 " 

Hemlock (Pennsylvania), bark off 6.000 to 7,000 •• 

Four acres of water are required to store 1,000,000 feet of logs. 
Wel^rbt of I9OOO Feet of Lnmber, Board Rleaaare. 

Yellow or Norway, pine Dry, 8.000 lbs. Green, 5,000 lbs. 

White pine •• 2,800" " 4,000 " 

l¥elfflit of 1 Cord or Seasoned Wood, 188 €ttMe Feet per 

Cord* 

Hickory or sugar maple 4,500 Ibe. 

Whiteoak 8,860 *• 

Beech, red oak or black oak , a»860 ^* 

Poplar, chestnut or elm 8,860 " 

Pine (white or Norway) 8,000 •* 

Hemlock bark, dry S;K)0 " 



8IZ£S OP FIBE-BBICK. 



233 




\ OF FIRB-BBICK. 

9-ineh straight 9x4Hx3U Incliei. 

Soap 9x8Hx2U ** 

Jamb \ Checker 9x8 kST " 

2-inch 9x4Uxa »» 

fxiUxSU / SpUt 9x4UxlJ4 " 

»««<«i« / jj;^^^ 9x42xau •' 

No. 1 key 9 x 2^ thick x4H to 4 Inches 

wide. 
«.^. \ 118 bricks to circle 12 feet inside diam. 

^V__A No.2kej 9x2^ thick x 4H to S« 

f" ik'Ue't'4^ iDches wide. 

b *«»« y^^;^ 03 ijricks to drcfe 6 ft. inside dIam. 

No.Skey 9 x 2^ thick x 4H to 8 

inches wide. 

8S bricks to circle 8 ft. inside diftni. 

Weds. \ ^^'ijj^^de ^'^^ ^^^ « <« to 2J4 

/'•w «tf- imtTTTT^ 25 bricks to circle 1 W ft. inside diani. 

* *^'^^ 'Mr No. 1 wedge (or bullhead). 9x4H wide x 2^ to 2 in. 
thick, tapering lengthwise. 
A ■ V 96 bricks to circle 5 ft. inside diam. 

/\ AKk \ No.2wedge 9x4Wx2U to 1^ in. thick. 

/ y -:: \ aO bricks to circle 2^ ft. inside dlam. 

/ A*iH*{tH:iH/ No. larch........ ........ 9x4^x2V< to 8 in. thick, 

V/ / tapering breadthwise. 
^ -^ 78 bricks to circle 4 ft. inside dIam. 
No.2areh 9x4^x2XtoiW. 

C\ 42 bricks to circl« 2 ft. inside diam. 
No.lBk«w\ No. 1 skew 9to7x4Hto2^. 
V \ Bevel on one end. 
> ^ No.2Bkew 9xaUx4Uto2Ji. 
^ViiiH*S}f/ Equal berel on both edges. 
^ No. 8skew 9x2Hx4HtolH. 

Taper on one edge. 

/sr — , ,„, \ 24inchcircle ^to!S^x4V<xaH. 

< \ go«8Uw\ Edjpes curred, 9 bricks Tine a 2*.ineh circle. 

\ r 7 a«-incbcircle W to ew x 4^ x J%. 

\ /f X tJtx tiiCK£i 18 bricks line a Sd-inch circle. 

\i n i»-«»i 4ft4ochdrcle 8^ to 7J4 x 4^ ^ 2^. 

^« » 17 bricks Une a 48-inch circle. 

ISU-lnch straight inix2Hx6. 

13>2-inch key No. 1 IJ^ x 2^ x 6 to 5 inch. 

. No s Skew — \ *> bricks turn a 12-ft. circle. 

' ^ IS^lncb key No. 2 13^x2^x0 to 49^ inch. 

/0x9UMtAU\uV ^2 briclcs turn a 6-ft. circle. 

^ ^^^*V Bridge wall, No. 1 13x6^x6. 

Bridge wall, No. 2 18x6>i x 3. 

38to.OI«4e MlUtUe 18.20, or 24x6x8. 

sV "^ Stock-hole tiles 18, 20, or 24x9x4, 

\ IS-inchblock 18x9x6. 

,^ •« \ Flatback 9xQxt^, 

' ^ ^ Flatbackarch 9x6x8«to2H. 

22-inch radius, 66 brlcJcs to circle. 

Locomotive tile 82x10x8. 

84x10x8. 
CopoU"***^ Wx 8x3. 

40x10x8. 
Tiles, slabs, and blocks, yarioita sises 12 to 30 inches 
long, 8 to 80 Inches wide. 2 to 6 inches thick. 
Cupola brick, 4 and 6 inches high, 4 and 6 inches radial width, to line shells 
S3 to 66 in diameter. 

A 94ach straight brick weighs 7 lbs. and contains 100 cubic inches. (=120 
lbs. pfw cubic foot. Specific gravity 1.98.) 

One cubic foot of wall requires 17 9>lnch bricks, one cubic yard requires 
400. Where keys, wedges, and other *' shapes " are used, add 10 per cent la 
ettloMttiig the nuoibar required. 




234 



MATERIALS. 



One ton of fire-clay should be sufficient to \aj 8000 ordinary bricks. To 
secure the best results, fire-bricks should be laid in the same clay from which 
they are manufactured. It should be used as a thin paste, and not as moi^ 
tar. The thinner the joint the better the furnace walL In ordering bricks 
the service for which they are required should be stated. 



NURIBKR OF FIRE-BRICK REI^ITIRED FOB 
TARIOU8 CIRCIiES. 





KEY BRICKS 




ARCH BRICKS. 


WEDGE BRICKS. 


H 


























^ 


eo 


o» 


^ 


i 


ei 


-J 




•a 


o« 


- 




^ 




o 


o 


d 


o 


o 


o 


o 






c 


o 








\B, 


» 


» 


^ 


H 


SS 


SQ 


d» 


H 


sz: 


^ 


d» 


H 


ft. in. 




























1 6 


25 
17 
9 








85 
80 
84 


















2 


18 
25 






42 

81 






42 
49 










8 6 


18 




60 






60 


8 




88 






88 


21 


86 




67 


48 


20 




68 


3 6 




82 


10 




42 


10 


54 




64 


86 


40 




76 


4 




S5 


21 




46 




72 




72 


24 


69 




83 


4 6 




19 


82 




5! 




72 


8 


80 


12 


79 




91 


6 




18 


42 




56 




72 


15 


87 




98 




96 


5 6 




6 


58 




69 




72 


88 


95 




96 


8 


106 


6 






68 




68 




72 


SO 


102 




98 


15 


118 


6 






58 


9 


67 




72 


88 


110 




98 


28 


121 


7 






52 


19 


71 




72 


45 


117 




98 


80 


128 


7 6 






47 


29 


76 




78 


58 


126 




98 


88 


186 


8 






42 


38 


80 




72 


60 


138 




98 


46 


144 


8 6 






87 


47 


84 




72 


68 


140 




98 


58 


161 


9 






81 


57 


88 




72 


75 


147 




9H 


61 


150 


9 6 






26 


66 


92 




72 


88 


155 




98 


68 


166 


10 






21 


76 


97 




72 


90 


182 




96 


76 


174 


10 






16 


85 


101 




72 


98 


170 




98 


88 


181 


11 






11 


94 


106 




78 


106 


177 




98 


91 


189 


11 6 






5 


104 


109 




72 


118 


186 




98 


98 


196 


12 








118 
118 


118 
117 




72 


121 


198 




96 


106 


20i 


12 6 



































For larf^r circles than 12 feet use 113 No. 1 Key, and as many 9-inch brick 
as may be needed in addition. 



ANAIiYSES OF MT. 8ATAOE FIBE-CIiAT. 

(1) (2) (t) (4) 

1871 1877. 187a 1885. 

Institute of New^lreev flSrvev^ Dr. Otto ' 

Technology, pj^^. if.^^k. PenS^lTanla. ^uth. 

60.467 66.80 Silica 44.396 66.16 

86.904 80.06 Alumina 83.558 88.895 

1.15 Titanic acid 1.680 

1.604 1.12 Peroxide iron 1.080 0.60 

0.133 ... . Lime trace 0.17 

0.018 Magnesia 0.108 0.115 

trace .80 PotOHh (alkalies) 0.247 

12.744 10.50 Water and inors> matter. 14.675 9.68 ; 

100.760 100.450 J00.498 100.000 



MAGKE8IA BktOItd. <635 

KAONK8IA BRICKS. 

** Foreign Abstracts " of the Institution of Ci^U Engineers, 1893, gives a 
(taper by C Btschof on the production of magnesia bricks. The material 
most in favor at present is the magueslte of Styria, which, altliough less 
pure eonsidervd as a source of magnesia than the Greek, has the property 
of fritting at a high temperature without melting. The composition of the 
two subaitances, iu the natural and burnt stales, is as follows: 

Hagnesite. Styrian. Greek. 

Carbonate of magnesia 90.0to96.0:t 94.46)( 

" lime 0.6 to 2.0 4.49 

" h-on 8.0to 6.0 FeOO.OS 

Silica 1.0 0.6a 

Manganous oxide 0.6 Water 0.54 

Burnt Magnesite. 

Magnesia 77.6 88.46-95.86 

Lime 7.8 0.88—10.92 

Alumina and ferric oxide 18.0 0.66— 8.64 

Silica 1J8 0.78—7.98 

At a red heat magnesium carbonat-e is decomposed into carbonic acid and 
caustic magnesia, which resembles lime in becoming hydrated and recar- 
bonated when exnosed to the air, and possess e s a certain plasticity, so that 
it can be moulded when subjected to a heary pressure. By long-continued 
or stronger heating the material becomes dead-burnt, giving a form of mag- 
necua of high deonity, sp. gr. 8.8, as compared with 8.0 in the plastic form, 
which is unalterable in the air but devoid of plasticity. A mixture of two 
voiuiiies of dead-burnt with one of plastic magnesia can be moulded into 
bricks which contract but little in firing. Other binding materials that have 
been used are: clay up to 10 or )6 per cent; gas-tar, perfectly freed from 
water, soda, silica, vinegar as a solution of magnesium acetate which is 
readily decomposed by heat, and carbolates of alkalies or lime. Among 
magnetduin compounds a weak solution of magnesium chloride may also be 
used. For setting the bricks lightly burnt, caustic magnesia, with a small 
proportion of silica to render it less refractory, is recommended. The 
strength of the bricks may be inci^eased by adding iron, either as oxide or 
vilicate. If a porous product is required, sawdust or starch may be added 
to the mixture. When dead-burnt magnesia is used alone, soda is said to be 
the best binding material. 

See also papers by A. £. Hunt, Trans. A. I. M. E., xvl, 7^, and oy T. Egles- 
ton. Tran>«. A. I. M. E., xlv. 458. 

Asbestos.— J. T. Donald, Eng. and M. Jour., June 27, 1891. 

AHALTBIB. 

Canadian. 

Italian. Broughton. Templeton. 

Silica 4O.809( 40.67){ 4O.S0j( 

Magnesia 48.87 41.60 42.06 

Ferrous oxide 87 2.81 1.97 

Alumina 2.27 .90 2.10 

Water 18.72 18.65 13.46 

100.68 99.88 100.10 

Chemical analysis throws light upon an Important point in connection 
with asbestos, i.e., the cause of the harshness of the fibre of some varieties. 
A««be«toe is piinclpally a hjdrouM silicate of magnesia, i.e.. silicate of mag- 
nesia combined with water. When harsh fibre is anaijsed it is found to 
eontain less water than the soft fibre. In fibre of very fine quality from 
BUck Tjake analjrsis showed 14.88j( of water, while a hareh-fibred sample 
gave only ll.TOjt. If soft fibre be heated to a temperature that will drive off 
a portion of the combined water, there results a substance so brittle that it 
may be crumbled between thumb and Anrer. There is evidently some con- 
nection between the oonsistenoy of the fibre and the amount of water in its 
eomposition. 



236 8TBBN6TH OF MATEBIALS. 



8!FB£NGTH OF MATESEIAIiS. 

stress an4 Straim.— There Is mocb oonlusion amouff writers oo 
strength ot msteriala as to the defloiUon of these terms. An ezterual force 
applied to a body, so as to pull it apart, is resisted by an interual force, or 
resists Doe, and the action of these forces catises a displacement of the mole- 
cules, or deformation. By some writers the external force is called a stress, 
and the internal force a strain; others call the external force a strain, and 
the internal force a stress: this confasiou of terms is not of importance, as 
the words stress and strain are quite commonly used synonymously, but the 
use of the word strain to mean molecular displacement, deformation, or dis- 
tortion, as is the custom of some, Is a corruption of the languacre. See JSH- 
gineeiing Newt, June 28, 1892. Definitions by leading authorities are given 
Delow. 

Stres$,—A stress is a force which acts in the Interior of a body, and re- 
sists the external forces which tend to change its shape. A deformation is 
the amount of change of shape of a body caused by the stress. The word 
strain is often used as synonymous with stress and sometimes It Is also used 
to designate the deformation. iMei'riman.) 

The force by which the molecules of a body resist a strain at any point is 
called the stress at that point. 

The summation of the displacements of the motocnles of a body for a 
given point is called the distortion or strain at the point considered. (Burr). 

Btreesee are the forces which are applied to bodies to bring into action 
their elastic and cohesive properties, 'niese forces causs allerattons of the 
fonns of the bodies upon which they set. Strain is a name given to the 
kind of alteration produced by tho stresses. Tiie distinction betwe«ni Btn*f>« 
and strain Is not always obeerred, one being used for the other. (Wood.) 

Stresses are of different kinds, vis. : termie^ eomprettive, froiwvrras, tor- 
s/i/Mo/, and ahectring stresses. 

A tensiU »tre»a^ or pull, is a force tending to ekmgats a pises. A eorn- 
presBive stresa^ or pnsh, Is a force tending to shmten it. A Irantwne wtrem 
tends to bend it. A torsional atresa tends to twist it. A ahearinQ ttreaa 
tends to foroe one part of it to slide o\'er the adjacent part. 

Tensile, compressive, and shearing stresses are called simple stresses. 
Transverse stress is compounded of tensile and compressive stresses, and 
torsional of tensile and shearing stresses. 

To these five varieties of stresses might be added feaWno stress, which Is 
either tensile or shearing, but in which the resistance of different portions 
of the msterfsl are brought into play in detail, or one after the other, in- 
stead of siroultaneous]3% as in the simple stresses. 

KITects of fltressss.^The following genera) laws for casss of simple 
tension or compression have been established by experiment. (Merritnan): 

1. ^hen a small stress is applied to a body, a small deformation is pro- 
duced, and on the removal of the stress the body springs back to its original 
form. For small stresses* then, materials may be regarded as perfectly 
elastic. 

2. Under small stresses the deformations are approximately proportional 
to the foix;es or stresses which produce them, and also approximately pix>- 
poriionai to tlie length of the bar or t>ody. 

3. When the stress is great enough a deformation is produced which is 
partly permanent, that is, the body does not spring back entirely to its 
original form on removal of the stress. This permanent part is teimed a 
set. In such cases the deformations are not proi>ortional to the stress. 

4. When the stress is greater still the deformation rapidly increases and 
the body finally ruptures. 

5. A sudden stress, or shook, Is more hs jurkHis than a steady stress or than 
s stress gradually applied. 

Blastie Limit.— The elastic limit is defined as that point at which the 
deformations cease to be proportional to the stresses, or, the point at which 
the rate of stretch (or other deformation) begins to increase. It is als«> 
defined as the point at which the first permanent set becomes visible. The 
last definition is not considered as good as the first, as It is found that with 
some materials a set occure with any load, no matter how small, and that 
with others a set which might be called permanent vanlsheB with lapse of 
time, and as it is impossible to get the point of first set without removtB^r 



8TBB88 AKD STRAUr. 237 

Uie whole load atUr each increMe of load, wblch hi tnqa&uVfy li ieo n f —iwit. 
Thr eJastie limit, defined, however, as ihe point at which the mxtoatAoom he* 
gin to incr eaae at » higher ratio than the applied streaaea, uauaily comaponda 
Tery nearly with thepolnt of first measurable permaneDt aefc. 

Tl«ld*potat«— llie term yield-point has recently been introduoed into 
the literature of the strength of materials. It ia deOaed as that point a4 
which the rate of stretoh soddenlf Inereaaea rapk^y. Th9 differeoos be* 
tween the elastic limit, strictly defined aa the point at which tlw rate of 
stretch begins to increase, and the vield-point, at which Ihe rate hiereassa 
suddealy, may in some cases be considerable. This difference, however, will 
not be (Uflcorersd hi short test*pieoes unless the readings of ehMigatioBS an 

made by mn cxeeedlBgfy Am instrument, as a mlcroiiieter readhig to t==jl 

of aa iaelu In nainf a coarser Instmmeni, such as oaUpen reading to 1/lOt 
of aa inch, the eiaaOe Usdt and the rleld'pofnt will appear to be smiixltane* 
ousL UaCortmiataly for nreeWon of language, the term yield-point was not 
introduced until long after the term elastic limit had been almost unirer- 
laUy adcMted to ainify the saine physical fact which is now defined by the 
Iran xMd-poiat, tfaaft ia, not the point at which the first change In rate^ 
obserTable <jDly by a microsoope, occurs, but that later polDt (pioto or less 
IndeOnlK* as to ila precias positioa) at which the Increase Is rreat enough to 
be seen by the naked eye. A most oonvenlettt method of deterrainitig the 
poteS ai whieh a soddea Increase of rate of stretch occurs in atntrt speci- 
mstm, when a tesUng-machlne in which the puIHng Is done by screws is 
uaed, is to note the weight on the beam at the tnstaat that the beam ** drops.** 
DttriaiK tbe eartter portioa of the lest, as the extension is steadily increased 

2r tka uniform but slow rotation of the screws, the potee is mored steadi]|- 
... . . ... ... 

thet 



aJoo^ the beam to keep it in equipoise; suddenly a point fs reached at which 
the beam drops, and will not nae until the elongation has been eonslderaUy 



r the further rotation of the scnsws, the advaochig of the poise 

jMng suspended. This point oorreq>onds practical^ to the point 

St wliieh the rate or eloogatlon suddenly Increases, and to the peint at 
which an appreciable permanent set is first found. It is also the point which 
haa hltJierto betn. caltod in practioe and in text-books the elastic limit, and 
it wUl probably continue to be so called, although tbe use of the newer term 
"yieW^oiat *^ for it, and the restriction of the term elastic limit to mean 
the earuer polat at which the rate of stret^ begins to Increase, aa determin* 



B aaif by mieromelrio measnreaoents^ Is more precise and sdentiflc. 
IB taUea of strength of ssatsrlals hereafter given, the term elastte ttmit is 
used in ha easteoary raeaatag; the point at which the rate of stress has be* 
gun to increase, as observable by ordinary instruments or by the drop of 
the beam. With thia definlthm It Is practically synonymous with yleld- 



Jit ior ■•Aiditt^ of Blaatleity*— HkfB fa a term expi Bss- 

lag tbe reladoa between the amount of extenoion or c otiipress fon of a mat» 
rial aad the load prodocfaiff that eztensioii or compression. 

It ma^^ be defined aa the load per unit of sectlOD divided by tlie estcnsion 
per unk of Isngth; or the lecftprocal of the f mclion espresnng the efonga* 
tkm per Inch of length, divided by the poands per square inch of section 
prodacfaiff that elongation. 

Let P be the applfed load, h the seethmal area of the piece, I the length of 
the part exteadad» A the amoaat of the extensioD, ana IT the ooefilcient of 
elasticity. Then 

p 

jr B theloadooaniiltof sectfon; 

J m the eleogation of a unit of length. ' 
- P A PJ 



The coefilcl e pt of eUutlcity is sometimes defined as the figure expressina 
the load which would be necessary to elongate a piece of one iquare inch 
seetioa lo double its original length, provided Che piece would not break, and 



the ratio of extension to the force producing it remaiaed ooiwtant. This 
definitioa follows from the formula above given, thus: If Jrasoue squace 
Iseh, I and A each ss one inch, then B= P. 
WSUn the efawtie limit, when the deformations are proportional to thf 



238 ST&ENGTH OF MATERIALS. 

stresses, the ooefflcient of elasticity Is constant, but beyond tLe eliistic limit 
it decreases rapidly. 

In cast iron there is generally do apparent limit of elasticity, the deforms^ 
Uons increasing at a faster rate than the stresses, and a permanent set being 
produced by small loads. The coefficient of elasticity therefore Is not con- 
stant during any portion of a test, but grows smaller as the load increaises. 
The same is true in the case of timber. In wrought iron and steel, however, 
there is a well-defined elastic limit, and the coefficient of elasticity within 
that limit Is nearly constant. 

ResUlenee, or WorlL of Resistance of a Material.— Within 
the elastic limit, the resistance increasing uniformly from zero stress to the 
stress at the elastic limit, the work done by a load applied gniduaily is equal 
to one half the product of the final stress by the extension or other deforma- 
tion. Beyond the elastic limit, the extensions increasing more rapidly than 
the loads, and the strain diagram approximating a parabolic form, the work 
is approximately equal to two thirds the product of the maximum stress by 
the extennion. 

The amount of work required to break a bar, measured usually in inch- 
pounds, is called its resilience; the work required to strain it to the elastic 
limit is called its elastic resilience. 

Under a load applied suddenly the momentary elastic distortion is equal 
to twice that caused by tlie same load applied gradually. 

When a solid material is exposed to percussive streis, as when a weight 
falls upon a beam transversely, the work of resistance is measured by the 
product of the weight into the total fall. 

BleTatlon of Ultimate Beaiatanee an4 Blaatio I<lmit«— It 
was first observed by Pruf. R. H. Thurston, and Commander L. ▲. Beards- 
lee, U. 8. N., independenthr, in 1878, that if wrought iron be subjected to a 
stress beyond its elastic limit, but not beyond its ultimate resistance, and 
then allowed to " rest ^* for a definite interval of time, a considerable in- 
crease of elastic limit and ultimate resistance may be experienced. In other 
words, the application of stress and subsequent ** rest " mcreases the resist- 
ance of wrought iron. 

This *' rest " may be an entire release from stress or a simple holding the 
test-piece at a (riven intensity of stress. 

Commander Beardslee prepared twelve specimens and subjected them to 
an Intensity of stress equal to the ultimate resistance of the material, with- 
out breaking the specimens. Thene were then allowed to rest, entirely free 
from stresii, from S4 to SO hours, after which period they were again stressed 
until broken. The gain in ultimate resistance by the rest was found to Taiy 
from 4.4 to 17 per cent. 

This elevation of elastic and ultimate resistance appears to be peculiar to 
iron and steel: it lias not been found in other metals. 

Relation of tl&e Blastie lilmit to Kndaranee under Re* 
peated Stresses (condensed from Engineering^ August 7, 10B1).~ 
When engineers first began to test materials. It was soon recognised that 
if a specimen was loaded beyond a certain point it did not recover its origi- 
nal dimensions on removing the load, but took a permanent set: this point 
was called the elastic limit. Since below this pointa bar appeared to recover 
completely its original form and dimensions on removing the load, it ap* 
peared obvious that it had not been injured by the load, and hence the work- 
ing load might be deduced from the elastic limit by u^g a small factor of 
safety. 

Experience showed, however, that in many cases a bar would not carry 
safely a stress anywhere near the elastic limit of the material as determined 
hr these experiments, and the whole theoi y of any connection between the 
elastic limit of a bar and its working load became almost discredited, and 
engineers employed the ultimate strength only in deducing the safe working 
load to which their structures might be subjected. Still, as experience accu- 
mulated it was observed that a higher factor of safety was required for a live 
load than for a dead one. 

In 1871 W5hler published the results of a number of experiments on bars 
of iron and steel subjected to live loads. In these experiments the stresses 
were put on and removed from the specimens without impact, but it was, 
nevertneless, found that the breaking stress of the materials was in every 
case much below the statical breaking load. Thus, a bar of Krupp*s axfe 
steel having a tenacity of 49 tons per square inch broke with a stress of 88.6 
tons per square inch, when the load was completely removed and replaced 
without impact 170,0iD0 timea These experiments were made on a large 



BTBE8S AND STBAIK. 889 

BDmber of different bnnds of iron and stael, and the remiltg were ooneor- 
dant Id flhowiiiff that a bar would break with an alternating stren of only, 
■aj, one third the statical breaking strenflrtii of the material, if the repetitions 
of itreea were sufficiently numerous. At the same time, howerer, it ap- 
peared from the generaltrend of the experiments that a bar would stand an 
iDdellBite number of alternations of stress, prorided the stress was kept 
below the limit. 

Prof. Bauflchineer defines the elastic limit as the point at which stress 
eeaaea to be sensibly proportional to strain, the latter being measured with 

a mirror apparatus reading to ggg^tb of a millimetre, or about ^jq^qq In. 

This limit Is always below the yield-point, and may on occasion be sero. On 
loading a bar above the yield-point, this point rises with the stress, and the 
rise continues for weeks, months, and possibly for years if the bar Is left at 
rest under Its load. On the other band, when a bar Is loaded beyond its true 
elastic limit, but below its yield-point, this limit rises, but reaches a maxi- 
mum as the yield-point, is approached, and then falls rapidly, reaching eren 
to zero. On leaving the bar at rest under a stress exceeding that of its 
primitive breaking-down point the elastic limit begins to rise again, and 
may. If left a sufficient time, rise to a point much exceeding Its previous 
value. 

This property of the elastic limit of changing with the histoiy of a bar has 
done more to discredit it than anything else, nevertheless it now seems as If 
ic« owlDff to this very property, were once more to take its former place in 
the estimation of engineers, and this time with fixity of tenure. It had long 
been known that th<9 limit of elasticity might be raised, as we have said, to 
almost any point within the breaking load of a bar. Thus, in some experi- 
ments 1^ Professor Styfle, the elastic limit of a puddled-steel bar was raised 
ie,QOO lbs. by subjecting the bar to a load exceeding its primitive elastic 
Bmlt. 

A oar has two limits of elasticity, one for tension and one for compression. 
Baiischlnger loaded a number of oars in tension until stress ceased to be 
ienslbly proportional to strain. The load was then removed and the bar 
tested in compression until the elastic limit In this direction had been ex- 
ceeded. This process raises the elastic limit in compression, as would be 
found on testing the bar in compression a second time. In place of this, 
however. It was now again tested in tension, when It was found that the 
artificial raising of the limit in compression had lowered that in tension be- 
low its previous value. By repeating the process of alternately testing in 
tension and compression, the two limits took up points at equal distances 
from the line of^no load, both In tension and compression. These limits 
Baoschinger calls natural elastic limits of the bar, which for wrought Iron 
correspond to a stress of about 8^ tons per square inch, but this is practically 
the limiting load to which a bar of the same material can be strained alter- 
mutely In tension and compression, without breaking when the loading is 
repeated sufficiently often, as determined by wahler's method. 

As received from the rolls the elastic limit of the bar in tension Is above 
the natural elastic limit of the bar as defined by Bauschinger, having been 
artificially raised by the deformations to which it has been subjected in the 
process of manufacture. Hence, when subjected to alternating stresses, 
the limit In tension is immediate^ lowered, while that in compression Is 
raised nntU they both correspond to equal loads. Hence, in Wohler's ex- 
periments, in wnich the bars broke at loads nominally below the elastic 
omits of the material, there is every reason for concluding that the loads 
vere really greater than true elastic limits of the material. This is con- 
firmed by tests on the connecting-rods of engines, which of course work 
under alternating stresses of equal Intensity. Careful experiments on old 
rods sliow that the elastic limit in compression is the same as that in ten- 
sion, and thAt both are far below the tension elastic limit of the material as 
raoeived from the rolls. 

The common opinion that straining a metal beyond Its elastic limit injures 
it appears to be untrue. It is not the mere straining of a metal beyond one 
elastic limit that Injures It, but the straining, many times repeated, beyond 
its two etastic limits. Sir Benjamin Baker has shown that In bending a shell 
piste for a boiler the metal is of necessity strained beyond its elastic limit, 
■othat stresses of as much as 7 tons to 15 tons per square inch may obtain 
is It as it comes from the rolls, and unless the plate is annealed, these 
itiesses will still exii^ after it has been built Into the boiler. In such a case, 
hofrever, when exposed t9 %^9 9<|dlt|ooal stress due to the pressure inskio 



240 STBBNaTH OF MATERIALS. 

Ihe boJ]«r, the ovsntratned poitkmft of the plate will relieve Chemselvee by 
9tretchinff and tekinc e permanent eet, 80 that probably after a Tear'ii work- 
ing verj Hctle dlflerenoe couid be detected in (he RtreMee In a plate built in- 
to the boiler ac it eame from tlie bending rolU, and in one mrhioh had b««fli 
aonealed. before rlTetloK into plaoe, and the flrit, la spite of Ite having been 
strained beyond iU eleetio llnuU, and not aubeequentiy annealed, would be 
as strong as the other. 

Bestotanee ot Het«l« to Repeated ffbocke* 

More than twelve years were spent by WOhler at the instance of thePrue- 
■laQ Oovemment In experimenting upon the resistance of iron and steel to 
repeated stressea The reeulte of his experimeDts are expressed in what is 
known as WOhlei^s law, which le given in the following words In Duboto'e 
translation of Weyrauch: 

" Rupture may be caused not only by a steady loed which e x ceeds ib# 
carrir ing strength, but also hf repeated applications of stresses, none of 
whi<$h are equal to the carrying strength. The differences of these stresses 
are measures of the disturbance of continuity, in so far as by their increase 
the minimum stress which Is still neoesaary for rupture dinuniabes," 

▲ practical illustration of the meaning of the first portion of this law may 
tie Ki^B thus: If 60,000 pounds once applied will just break a bar ot iron or 
steel, a stress very much less than wjw pounds will break it if repeated 
suiUciently often. 

This is fully oonflrmed by the experiments of Fairbaim and Spaogenbery. 
as well as tnose of WOhler; and, as is remarked by Weyrauch, it may be 
considered as a long-known result of common experience. It parUally ac^ 
counts for what Mr. Holley has called the *' hitrineically ridiculous factor of 
safety of six.*' 

Another ** long-known result of experience ^ Is the fact that rupture may 
be caused by a succession of thocka or impacUt none of which alone wottkl 
be sufficient to oauae it. Iron axles, the piston-rods of steam hammers, and 

gther pieces of metal subject to continuously repeated shocks. inTariably 
raak after a oertain leogtb of service. They have a "Uf^ ** which is lim- 
ited. 

Several years ago Fairbahm wrote: ** We know that In some cases wrouirbt 
iron subjected to oontbiuous vibration assumes a cnrstalline structure, *nd 
that the cohesive powers are much deteriorated, but we are ignorant of the 
causes of this change.^' We are still ignorant, not only of the causes of this 
change, but of the conditions under which It takes place. Who knows 
whether wrought iron subjected to very slight continuous vibration will co- 
dure forever? or whether to insure final rupture each of the continuous small 
shocks must amount at least to a certain percentage of single heavy shock 
(both measured in f oot-poundK), which would cause rupture with one applies^ 
tlon ? WOhler found in testing iron by repeated stresses (not impacts) tliat 
in one case 400,000 applications of a stress of 600 centners to the square inch 
caused rupture, while a similar bar remained sound after 48,000,000 applica- 
tions of a stress of 800 centners to the sauare inch (1 centner =* llO Ji Aw.)' 

Wbo knows whether or not a similar law holda true in regard to re|»eaiad 
shocks 7 Suppose that a bar of iron would break under a shigle impact of 
lOOO foot-pounds, how many times would It be likely to bear the repetition 
of 100 foot pounds, or would it be safe to allow it to remain for fifty y^Lrs 
subjected to a continual succession of blows of even 10 foot-pounds each r 

Mr. WUllam Metcalf published in ibe Metallurgical Review, Pea 1877, the 
results of some tests of the life of steel of different percentages of carbon 
under impact. Some small steel pitmaos were made, the specifications for 
which required that the unloaded machine should run 4^ hours at the rate 
of rJOO revolutions per minute before breaking. 

The steel wss all of uniform quality, except as to carbon. Here are the 
results: The 

.ao C. ran 1 h. 21 m. Heated and bent before breaking. 

.49 C. ** Ih. 88 m., »* " *• " ** 

.43 0. " 4 b. 67 m. Broke without heating. 

•46 a " <h. 90 m. Broke at wekl where Imperfect. 

•fiOC. '* 5h.40m. 

.84 0. "18h. 

•87 0. Broke in weld near the end. 

M 0. Ban 4.66 m., and the machine broke down. 

Some other experlmenta by Mr. Metcalf confirmed hia <;0D0luBion, vis,. 



6TR£S6 AND STRAIN. 241 

that liigb-carbon steel was better Adapted to reeist repeated shocks and vi- 
brations than low-carboD steel. 

Tliese results, however, would scarcely be suiflcient to induce any eu- 
^neer to use .84 carbon steel In a car-axle or a bridge-rod. Further experi- 
ments are needed to confirm or overthrow them. 

(See description of proposed apparatus for such an investigation in the 
aiitbor*s paper in Trans. A. I. M. £., vol. viii , p. 76, from which the above 
extract is taken.) 

Mresfles Prodneed 1»f Suddenly AppUed Forces and 
Sltocks* 

(Mansfield Herriman, B. B. dt Eng. Jour., Dec. 1889.) 
Let P be tiie weif?ht which la dropped from a height h upon the end of a 
bar, and let y be the maximum elongation which is produced. The work 
performed bjr the falling weight, then, is 

and this must equal the internal work of the resisting molecular stresses. 
The stresA in the bar, which is at first 0, increases up to a certain limit Q, 
which is greater than P; and if the elastic limit be not exceeded the elonga- 
tion increases uniformly with the stress, so that the internal work is equal 
to the mean stress 1/2Q multiplied by the total elongation y, or 

Whence, neglecting the wori^ that may be dissipated in beat, 

if e be the elongation due to the statio load P, within the elastic limit 
y= %e; whence 



Q.p(iV»+«')' 



(t) 

which gjkves the momentaiy maximum stress. Substltnting this value of Q, 
there rnnlta 

»=«(i+i^i+2^) (») 

whiefa is tbe value of the momentary maximmn elongation. 

A. shoek resolts when the force P, before its action on the bar. Is moving 
with velocity, as is the case when a weight P falls from a heignt h. The 
above formulas show that this height h may be small if e is a stnall quan- 
tity, and yet very great stresses and deformations be produced. For in- 
stanoe, let A s 4€, then Q=s4P and tf 3 4e ; ahm let A » iSe, then Q = 8P 



steady load of 6000 Ikm. tbis will be compressed about 0.018 in., supposfncr 
that no lateral flexure occurs; but if a weight of 5000 lbs. drops upon its end 
from the amaU height of 0.046 in. there will be produced the stress of 90,000 



A soddeoly applied force Is one which acts with the uniform taCensity P 
upon the end of the bar, but which has no velocity before acting upon it. 
This corresponds to the case of A s in the above formulas, and gives Q = 
2P and y s= 2e for the maximum stress and maximum deformation. Prob- 
ably the action of a rapidly-moving train upon a bridge pivduces stresses 
of this character. 

Inereaatms tbe Tenalle Streng^tli of Iron Bars by Tirlat- 
iBg: tlaeiit«— Ernest L. Ransoms of San Francisco has obtaiued an English 
Patenu No. 16'<S^1 of 1888, for an *' improvement in strengthening and testing 
vrought metal and steel rods or bars, consisting in twisting the same in a 
cold state. . . . Any defect In the lamination of the metal which would 
otherwiaebe concealed is revealed by twisting, and imperfections are shown 
at once. The treatment may be applied to bolts, suspension-rods or bars 
subjected to tensile strength of any description." 

Besnlta of tests of this process were reported by lieutenant F. P. Qilmore, 
U. 8. v., in a paper read before the Technical Society of the Pacific Ooast, 
published in the Transactions of the Society for the month of December, 
lilSA. The experiments Include trials with thirty-nine bars, twenty-nine of 
which were variously twisted, from three-eighths of one turn to six turns per 
loot. Tbe test-pieces were cut from one and the same bar, and accurately 



243 



STRENGTH OF MATERIALS. 



measured and numbered. From each lot two pieces without twist were 
•«8ted for tensile streuKth and duclilit v. One group of each set was twisted 
until the pieces broke, as a guide for the amount of twist to be giyen those 
to be tested for tensile strain. 

The following is the result of one set of Lieut. Qilmore's tests, on iron 
bars 8 in. long, .719 in. diameter. 



No. of 
Bars. 


Conditions. 


Twists 

in 
Turns. 


Twists 
per ft. 


Tensile 
Strength. 


Tensile 
per sq. in. 


Gain per 
ceoL 




Nottwisfed. 
Twisted cold. 

«« 4» 



8 





SS,000 
28,900 
85,800 
86,800 
86,400 


54,180 
60,080 
68,600 
64,750 
65,000 


9 
17 
19 
20 



Tests that corroborated these results were made by the University of 
California in 1889 and by the Low Moor Iron Works, England, in 1890. 

TENSIIiE STRENGTH. 

The following data are usually obtained in testing by tension In a testing- 
machine a sample of a material of construction : 

The load and the amount of extension at the elastic limit 

The maximum load applied before rupture. 

The elongation of the piece, measured between gauge-marks placed a 
stated distance apart before the test; and the reduction of area at the 
point of fracture. 

The load at the elastic limit and the maximum load are recorded in pounds 
per aquare inch of the original area. The elongation is recorded as a per- 
centage of the stated length between the gauge-marks, and the reduction 
area as a percentage of the original area. The coefficient of elasticltv is cal- 
culated from the ratio the extension within the elastic limit per inch of 
lengfth bears to the load per square inch producing that extension. 

On account of thedifflculty of making accurate measurements of the frac- 
tured area of a test-piece, and of the fact that elongation is more valuable 
than reduction of area as a measure of ductility and of redlience or work 
of resistance before rupture, modem experimenters are abandoning the 
custom of reporting reduction of area. Tne " strength per square inch of 
fractured section " formerly frequently used in reporting tests is now almost 
entirely abandoned. The data now calculated from the results of a tensile 
test for commercial purposes are: 1. Tensile strength in pounds per square 
inch of original area. 8. Elongation per cent of a stated iengu between 
gauge-marks, usually 8 inches. 8. Elastic limit in pounds per square inch 
of original area. 

The short or grooved test specimen gives with most metals, especially 
with wrought iron and steel, an apparent tensile strength much higher 
than the real strength. This form of test-piece is now almost entirely aban- 
doned. 

The following results of the tests of six specimens from the same 1^" steel 
bar illustrate the apparent elevation of elastic limit and the changes in 
other properties due to change in length of stems which were turned down 
in each specimen to .796'' diameter. (Jas. £. Howard, Eng. Congress 1898^ 
Section O.) 



Description of Stem. 



Elastic Limit, 
Lbs. per Bq. In. 



Tensile Strength, 
Lbs. per Sq. In. 



Contraction of 
Area, per cent. 



1.00" long 

.50 »» 

.25 " 

Semicircular groove, 

.4" radius 

Semicircular groove, 

W' radius 

V-shaped groove 



64,900 
65,880 
68,000 

75,000 

86.000, about 
90,000, about 



94,400 
97,800 
108,480 

116,880 

184,960 
117,000 



49.0 
48.4 
80.6 

81.6 

S8.0 
Indeterminate. 



TEK81LB BtRENGTH. 



243 



Teito plate made by the author in 1879 of straight and grooved tert-pleoes 
of boilerplate steel out from the same gare the following results : 
5 straight pieces, 56,606 to 59,012 lbs. T. 8. Arer. 57,566 lbs. 

4 grooved " 64,841 lo 67.400 65,450 " 

Excesi of the short or grooved specimen, 21 per cent, or 12,114 lbs. 

Heaaurement of Elonciitioii.— In order to be able to compare 
records of elongation, it is necessary not only to have a uniform length of 
sectioo between gauge-marks (say 8 inches), but to adopt a uniform method 
of measuring the elongation to compensate for the duferenee between the 
apparent elongation when the piece breaks near one of the sauge-marks, 
and when it breaks midway between them. The following method is rec- 
ommended (Trans. A. 8. M . E., voL zi., p. 622): 

Kark on the specimen divisions of 1/2 inch each. After fracture measure 
from the point of fracture the length of 8 of the marked spaces on each 
fractured portion (or 7 -f on one side and 8 + on the other if the fracture is 
not at one of the marks). The sum of these messurements, less 8 inches, is 
the elongation of 8 inohes of the original length. If the fracture is so 
near one end of the specimen that 7+ spaces are not left on the shorter 
portion, then take the measurement of as many spaces (with the fractional 
part next to the fracture) as are left, and for the spaoes lacking add the 
measoreDoent of as many corresponding spaces of the longer portion as are 
necessary to make the 7 -f> spaoes. 

^ apes of SpacimeMa i6r Tenalle Taata.— The shapes shown 
:. iS were reoommended by the author in 1883 when he was connected 



No. 1. Square or flat bar, aa 
roUed. 




No. 2. Round bar, as ndled. 



No. 8. Standard shape for 
flats or squares. Fillets Vi 
inch radius. 



No. 4. Standard shape for 
rounds. Fillets |^ in. radius. 

No. 5. Government shape for 
marine boiler-plates oi: iron. 
Not recommended for other 
tests, as results are generally 
in« 



Fio. 75. 
with the Ptttabargfa Testing Laboratory. They are now in most general 
USB, the earlier forms, with 6 inches or less in length between shoulders, 
bring almost enUrety abandoned. 

^nemmUonm Baanlred In maklms Tenalle Teata*— The 
tMtine-mac^iine itself snould be tested, to detormine whether its weighing 
spparatus is accurate, and whether it is so made and adjusted that in the 
test of a properly maae specimen the line of strain of the testing-machine 
h ahaolutely in line with flie axis of the specimen. 

The specimen should be so shaped that It will not give an hioorrect record 
of strength. 

It should be of uniform minimum section for not less than five inches of 
its length. 

Regard must be had to the time occupied in making tests of certain mate- 
rials. Wrought iron and soft steel can be made to show a higher than their 
•etoal an*****"*' strengdi by keeping them under strain for a great length 

of lime, 
la testing soft alloys, copper, tin, sine, and the like, which flow under con- 

asat snm their highesf apparent strength is obtained by testing them 

raaidly. In reoording teste of such materials the length of time occupied in 

ihs test ahonld be stated. 



244 STRENGTH OF MATERIALS. 

For Terv accurate measurements of eloDgatioo, corresponding to Incre- 
ments of load during tlte tests, the electric contact micrometer, described 
in Trans. A. 8. M. E., vol. vl., p. 479, will be found coDvenient. When read- 
ings of elongation are then taken during the test, a strain diagram may be 
plotted from the reading, which is useful in comparing the qualities of dif- 
ferent specimens. Such strain diagrams are made automatically by the new 
Olsen testing-machine, described in Jiimr. Frank. Trut 1891. 

The coefficient of elasticity shookl be deduced from measaremoiit ob- 
served between fixed increments of load per nni( section, say between 9000 
and 12,000 pounds per square inch or between 1000 and 11,000 poonds instead 
of between and 10^000 pounds. 

COMPHSSSIVB mrBENGTtt. 

What la meant by the term ** compressive strength " has not yet been 
settled by the authmities, and there exists more confusion in regard to this 
term than in regard to any other used by writers on strength of mnteriiUa. 
The reason of this may be easily explained. The elfect of a compresif^ 
stress upon a material varies with the nature of the material, and with the 
shape and sise of the madmen tested. While tho effect of a tensffe stress la 
to produce rupture or separation of particles in the direction of the line of 
strain, the effect of a compressire strsis on a piece of material may be either 
to cause it to fly into splinters, to separate into two cr more wedge-shaped 
plecesaadfly apart, to bulge, buckle, or bend, or to flatten out and utterly re- 
sist rupture or separation of particles. A piece of speculum metal nnder 
compressive stress will exhibit no change of appearance until rupture takes 
place, and then it will fly to pieces as suddenly as If blown apart by gua- 
powder. A pieco of cast Iron or of stone will generally split into wedifo- 
shaped fragments. A piece of wrought Iron will buckle or bend. A piece of 
wood or zinc may bulge, but its action will depend upon its shape and sise. 
A piece of lead will flatten out and resist compression till the last degree; 
that is, the more it is compressed the greater becomes its resistance. 

Air and other gaseous bodies are compressible to any extent as long as 
they retain the gaseous condition. Water not confined in a vessel is com^ 
pressed by its own weight to the thickness of a mere film, while when con- 
fined in a vessel it is almost incompressible. 

It is probable, although it has not been determined experimentally, that 
solid bodies when confined are at least as incompressible as water. Wh«*n 
they are not confined, the effect of a compressive stress is not only to 
shorten them, but also to increaso their lateral dimenskwis or ba^ them. 
Lateral strains are therefore Induced by compressive stresses. 

The weight per square Inch of original section required to produce any 
given amount or percentage of shortening of any material is not a constant 
quantifty, but varies with Doth the length and the sectional area, with the 
stiape of this secttonal area, and with the relation of the area to tlie len^h. 
The '^ compressive strength*' of a material, if this term be sopposed to mean 
the weight in pounds per square inch necessary to cause rupture, may vary 
with every size and shape of specimen experimented upon. Still more diffi- 
cult would it be to state what is the " compressive strength " of a material 
which does not rupture at all, but flattens out. Supposa we are tssiliivc n 
cylinder of a soft metal Ifke lead, two inches In length and one inch in diam- 
eter, a certain weight will shorten It one per cent, aiK>Cher wedlght ten per 
cent, another fifty ber cent, but no weight that we caa place upon ii will 
rupture it, for it win flatten out to a thin sheet. What, then, is its ooroprea* 
sive strength f Again, a similar cylinder of soft wrought iron would prob> 
ably compress a few per cent, bulging evenly all arouna ; it would ihea c?odi- 
mence to bend, but at first the bend would be imperceptible to the eje and 
too small to be measured. Soon this bend would be great enough to be 
noticed, and finally the piece might be bent nearly double, or otherwise dis- 
torted. What is the '' compressive strength'' of this piece of Iron ? In it 
the weight per square inch which compresjses the piece one per cent or five 
per cent, that which causes the first bending (impossible to oe discovered^, 
or that which causes a perceptible bend? 

As sliowing the conrusion concerning the definitions of eompreasive 
strength, the following statements from different authorities on the strength 
of wrought iron are of Interest. 

Wood*s Resistance of Materials states, " comparatively few ezperimenta 
have been made to determine how much wrought iron will sustain at the 
point of crushing. Rodgkinson gives 65,000, Rondulet 7O»6O0^ Weisbach *8«000 



COMPBESilTE STREl^GTH. 245 

BunWim 80^000 koiOtOOO. It Ift generally aasumed that wrought Iron wiU rerist 
about two thirds as much crushing as to tenskm, but the experiments fail 
to give a twy deOnite ratio." 

Mr. Whippto, ia his treatise on bridge-buildiug, sUtes that a bar of good 
vrooght iron wiU sustain a tensile strain of about (X),000 pounds per square 
inch, and a cooapreesive strain, in pieces of a length not exceeding twice the 
leasK diameter, of about 90,000 pounds. 

The following values, said to be deduced from the experiments of Maior 
Wade, Hodgkinson. and Oapb. Meigs, are given by Hasweil : 

American wroqght Iron 197,720 Iba 

" *• (mean) 86.600 " 

™«™" i 40,000 " 

Stooej states that the strength of short pillars of any given material, all 
having the same diameter, does not vary much, provided the length of the 
pi<^ is not less than oue and does not exceed four or five diameters, and 
tliai the weight which win jost crush a short prism whoee base equals one 
»qaan9 mch, and whose height is not less than 1 to IW and does not exceed 
4 or 6 diameters, is ealled the crushing strength of the material. It would 
be n'ell if experimenters would all agree upon some such definition of the 
term ^* crushing strength," and insist that all experiments which are made 
for the purpose of testing the relative values of different materials in com* 
prt'^sion be made on specimens of exactly the same shape and size. An 



srbitrary siie and shape should be assumed and agreed upon for this pui^ 
pORBw The slae mentioned by 8tone;r is definite as regards area of section, 
vis^ ooa square inch, but is Indefiiute as regards length, vis., from one to 



five dianteters. In some metals a specimen five diameters long would bend, 
and give a much lower apparent strength than a specimen having a length of 
one disifliet«r. The words '* will jost crush " are fUso indeflttit4» for ductile 
materialB, in which the resistance increases without limit if the piece tested 
does not bend. In such cases the weight which causes a certain percentage 
Gt compression, as five, ten, or fifty per cent, should be assumed as the 
crushing strength. 

For fuuire experiments on crushing strength three things are desirable : 
First, an arbitrary standard shape and sixe of test specimen for comparison 
of ail materials. Secondly, a standard limit of compression for ductile 
materials,' which shall be considered equivalent to fracture in brittle mate- 
risls. Thirdly, an accurate knowledge of the relation of the crushing 
scrength of a specimen of standard shape and size to the crushing strength 
of specimens of all other shapes and sizes. The latter can only be 
secured by a very extensive and accurate series of experiments upon all 
kinds of materials, and on specimens of a great number of different shapes 



The anchor proposes, as a standard shape and sise, for a compressive test 
ipecimcm for all metiuB, a cylinder one inch in length, and one half square 
inch in sectional area, or Ql798 inch diameter: and for the limit of compree* 
fliCiQ equivalent to flracture, ten per cent of the original length. The term 
"oompreasive strength,*' or *' compressive strength of standard specimen," 
vould then mean the weight per square inch required to fracture by com- 
pf^asive stress a cylinder one inch k)ng and 0.798 inch diameter, or to 
reduce Ita length to 0.9 inch if fracture does not take place before that reduc- 
tioD in lengthis reached. If sncb a standard, or any standard sise whatever, 
lisd been used hy the earlier authorities on the strength of materials, we 
never wovdd have had such diserepancies in their statements in regard to 
the oompresaive strength of wrought iron as those given above. 

Thereaaotts why this particular sise is recommended are : that the sectional 
trea. one^half square inch. Is as large as can be taken in the ordinary test- 
iog-rnachlnea of 100,000 pounds capadty, to include all the ordinary metals 
of oonstructkm, cast and wrought iron, and the softer steels: and that the 
length, one Inch, Is convenient for caksulatlon of percentage of oompresFion. 
If (lie length were made two Inches, many materials would bend in testing, 
•sd give incorrect resoRs. Even in cast iron Hodgkinson f onnd as the mean 
«r several experiments on various gradee, tested in spectmens 94 inch in 
beifht, a compressive streneth per square inch of 94,730 pounds, while the 
aeaa of fbe same number of spectmens of the same irons tested in pieces 1^ 
iBcfaes in heig:fat was only 88,800 pounds. The best sine and shape of standard 
^Mwimen- should, however, be settled upon only after oonauitation and 
I several authorities. 



!i46 



S1)H£KGT& O^ ^ATERtAtid. 



The Oommittee on Standard Tests of the American Society of Mechanical 
Engineers say (vol. xi., p. 6S4) : 

'• Although compression tests hare heretofore been made on diminutive 
sample pieces, it is highly desirable ihat tests be also made on long pieces 
from 10 to ISO diameters in length, corresponding more nearly willi actual 
practice, in order that elastic strain and cliange of shape may be determined 
by usinff proper measuring apparatus. 

The elastic limit, modulus ur coefBcient of elasticity, maximum and ulti- 
mate resistances, should be determineti, as well as the Increase of section at 
various points, vis., at bearing surfaces and at crippling point. 

The use of long cumpressi on-test pieces is i^ecomnienaed, because the in- 
vestigation of short cubes or cylinders has led to no direct application of 
tlie constants obtained by their use in computation of actual structures, 
which have always been and are now designed according to empirical for 
mul89 obtained from a few tests of long columns." 

COIiVllINS, PIIiliARS, OR STBITTS. 

HodfslLiiisoii's Formula for Columns. 

P — crushing weight in pounds; d — exterior diameter in inches; d, = in- 
terior diameter in inches; L =■ length in feet. 



Kind of Column. 



Both ends rounded, the 
length of the column 
exceeding 15 times 
its diameter. 



Solid cylindrical col- ) 

umns of cast iron ) 

Hollow cylindrical col- } 

umns of cast iron ) 

Solid cylindrical col- ) 

umns of wrought iron. ( 
Solid square pillar of ) 

Dan tzic oak (dry) ) 

Solid square pillar of \ 

red deal (dry) ) 



P = 88,880 



,d»-»« 



i»-» 



P = 29,120 



d»-'«-d^« 



IS't 






Both ends flat, the 
length of the column 
exceeding 80 times 
its diameter. 

ds«ftB 
p= 98.920-^ 




P = 24,540— 
P= 17.5101, 



The above formulss apply ouly in cases in which the length is so great that 
the column breaks by bending and not by simple crushing. If the column 
be shorter than that given in the table, and more than four or five times its 
diameter, the strength is found by the following formula : 



Tr = 



PCjr 
P-i-J^CiT 



L 



In which P s the value given by the preceding formuIcB, K =s the transverse 
^«ectlon of the column in square inches, C =: the ultimate compressive resis- 
tance of the material, and W = the crushing strength of the column. 

Hodgkinson's experiments were made upon comparatively short columns, 
the greatest length of cast-iron columns being GO^ inches, of wrought iix>a 
90^ inches. 

'Die following are some of his conclusions: 

1. In all long pillars of the same dimensions, when the force is I4>plied in 
the direction of the axis, the strength of one which has flat ends is about 
three times as great as one with roun led ends. 

2. The strength of a pillar with "^ne *nd rounded and the other flat is an 
arithmetical mean between the two given in the preceding case of the same 
dimensions. 

8. The strength of a pillar having both ends firmly fixed is Uie same as 
one of half the length with both ends rounded. 

4. The strenirth of a pillar is not increased more than one seventh by en- 
larging it at the middle. 



XOMENT OF INERTIA AND RADIUS OF GYRATIOK. 247 

Ck»rAoii'fl rormnUD deduced from HodfrklnaoD^s expert roenls are mora 
fimirralJj used than UodfirkinflOD^s own. They are: 

Colnmns with both ends fixed or flat, P = — ^—7,; 

C6lumD8 with one end flat, the other end round, P = — ^tt 

I + IA.^ 



Columns with both ends round, or hinged, P =s — ^ p; 



l + ia-^ 



8 = area of crofw-section in inches; 
P = ultimMte retdfitanoe of odumn, in pounds; 
/ = cniBhlng strength of the material In lbs. per square inch; 
, . ., - ^111^ m Moment of inertia 

r = least radius of xyration, in inches, r* = rz—rt ; 

^- • area of section ' 

I = length of column in Inches; 
a = a corfncient depending upon the material; 
/and a are usually taken as constants: tliey are really empirical ▼ariables, 
dependent upon the dimensions and character of the column as well as upon 
th«* material. (Bnrr.) 

For solid wroufrht-iron columns, values commonly taken are: / a 86,000 to 
40,000: a ^ 1/%,000 to 1/40.000. 
For solid cast-iron columns, / = 80,000, a = 1/0400. 

For hollow casMron columns, flxed ends, p = V~7i* ' ^ length and 

d = diameter in the rame unit, and p s strength in lbs. per square inch. 

The coefflt'ient of f/d* is given various values, as 1/400, 1/500. 1/UUO, and 
y^uo. by fliirei-ent writers. The use of Gordon's formula, with any coef- 
ficients derived from Hodt^kinnon's experiments, for cast-Iron cplumns is to 
be deprecated. See Strength of Cast-iron Columns, pp. 250, S51. 

Sir Benjamin Baker gives. 

For mild steel, / = 07,000 ib.i., a = 1/22,400. 
For strong? steel,/ = 114,000 lbs , a = 1/14,400 

Prof. Burr considers these only loose approximations for the ultimate 
resintanceK. See his formulas on p. 980. 

For dry timber Runkine givt-s/ s= 7800 lbs., a = l/SOOO. 

HOHBFfT OP INBRTIA AHB BAB1U8 OF GTBATION, 

Tlie momeiit of liierlla of a section is the sum of the product^i of 
each elemental^ area of the section into the square of its distance from an 
aeaiiimed axis of rotation, as the neutral axis. 

The mdliis of synitloit of the section equals the square root of the 
onotieDt of the moment of inertia divided by the area of the section. If 
B = Miaa of gyratloii. Is moment of inertia and A =■ area. 

Hie moments of inertia of various sections are as follows: 
d = diameter, or outside diameter; d, = inside diameter; b s breadth; 
k = depth; b\' ^\- inside breadth and diameter; 

Solid rectanKle I = \/\2bh*\ Hollow rectangle 7 = 1/{S(bA> - b./ii*); 

Solid square I=^/\^b*l Hollow square /= 1/12(M - 5,*); 

Solid cylinder /= l/64«d<; Hollow cylinder J = l/64»(d* - d,*). 

Homentfl of Inertia and Radlnw of Gyration for Tarlons 
Sections, and their IJse In the Formulas lOr Strength of 
Girders and Columns*— The strength of sections to resist strains, 
eiciier e^ KtrUers or as columns, depends not only on the area but alno on the 
form of the section, and the property of the section which fonns the basis 
of the constants used in the formulas for strength of fdrders and columns 
to express the effect of the form, is Its moment of inertia about Its neutral 
axis. The modulus 0( resistance of any section to transverse bending is its 



248 ' STRENaXH OF MATERIALS. 

moment of Inertia divided by the distance from the neutral axis to tte 
fibres farthest removed from that nxis; or 

_ .. , , Moment of inertia ^_ Z 

Section modulus = Distance of extreme fibre from axis' ^ ' y- 

Koment of resistance s section modulus X unit stress on extreme fibre. 

BEoment of ImerUa of Compound Shapes, (Pencoyd Iron 
"Worlia)— The moment of inertia of any section about any axis is eaiial to the 
/ about a parallel axis passing through its centre of gravity 4- (the area of 
the section X the square of the distance between the axe^). 

By this rule, the moments of Inertia or radii of gyration of any single sec- 
tions being known, corresponding values may be obtained for any combhiar 
tion of these sections. . ^ 

Omdius of Gyration of Compound Shapes .--In the case of a 
pair of any shape without a web the value of R can always be found with- 
out considering the moment of inertia. 

The radius of gyration for any section around an aads paraUel to another 
axis passing through Its centre of gravity is found as follows: 

Letr = radius of gyration around axis through centre of gravity; R = 
radius of gyration around a nother axis parallel to above; d = distance be- 
tween axes: R = Vd* -h i*. 

When r is small, R may be taken as equal to d without material eiror. 

Oraphteal Hethod for FlBdink Badtas of Gyration.— Ben J. 
F. La Kue, Eng. Neicsy Feb. 2, 1898, gives a short graphical method for 
finding the radius of gyration of hollow, cylindrical* and rectangular col- 
umns, as follows: 

For cylindrical colunms: 

Lav off to a scale of 4 (or 40) a right-angled triangle, in which the base 
equals the outer diameter, and the altitude equals the inner diamet^sr of the 
column, or vice versa. The hypothenuse, measured to a scale of unl^ (or 
10). will be the radius of ^ration sought. 

This depends upon the formula 

/Mom. of Inertia ^D* + d^ 



o = ^- 



Area 4 

In which A = area and D = diameter of outer circle, a = area and d a« dia- 
meter of inner circle, and O = radius of gyration. i^D* + d« is the expres- 
sion for the hypothenuse of a right-angled triangle. In which D and d are the 
base and altitude. 

The sectional area of a hollow round column is .7854(D« — cP). By con- 
structing a right-angled triangle in which 2> equal s the hypothenme and d 
equals the altitude, the base will equsl VD» - d\ Calling the Tolue of this 
expression for the bsse B^ the area wUl equal .78MB*. 

value of 6 for square columns: 

Lay off as before, but using a scale of 10, a right-angled triangle of whicL 
tlie base equals Z> or the side of the outer square, and the altitude equals d, 
the side of the inner square. With a scale of 8 measure the hypothenuse, 
which will be, approximately, the radius of gyration. 

This process for square columns gives an excess of slightly more than 4%. 
By deducting 4% from the result, a close approximation will be obtained. 

A very close result is also obtained by measuring the hypothenuse with 
the same scale by which the base and altitude were laid off, and multiplyiug 
by the decimal 0.29; more exactly, the decimal is 0.128867. 

The foi-mula is 

/5 _ . /Mom. of inertia _ 1 , ^ oosuvr , • 

^ y aSS^ 7^^^" + ^'» = 0.288«7 ^TJS + d^ 

This may also be applied to any rectangular column by using the lesser 
diameters of an unsupported column, and the greater diimieters If the col< 
umn is supported in the direction of its least dimenKlons. 

BliKlHBNTS OF ITSVAL SECTIONS. ' 

Moments refer to horufontal axis through centre of gravity. This table is 
Intended for convenient application where extreme accuracy la not impor- 
tant. Some of the terms are only approximate; those marked * are corrects 
Values for radius of gyration in flanged beams apply to standard minimum 
•ections onl> . A = area of section ; b ^ k^readth; A = depth; D = diameter. 



ELEVBKT8 OF CSDAL 8ECTION8. 



249 





ShAp* of aeeCkm. 


Moment 
of Inertia. 


Section 
Modulus. 


Sqniareof 

Radius of 
aeration. 


Least 
Radius of 
Gyration. 









SoUdBect- 
angle. 


6A' • 
12 


bh** 
6 


(Lea«t Bide)9* 


Lea«t8ide» 




12 


8.46 


u-i-. 






H 


T 


HoUow Beot- 
angle. 




6;i»-5,74,i* 


12 


7t4-Ai 




Hh 


4.89 


« 


ft-- 







Solid Olrole. 


16 


AD* 
8 


16 


5* 
4 


^ 


Hollow Circle. 
A, area of 
large section; 
a, area of 
small section. 


Aiy-ad* 


8X> 


16 


D-f d 


w 


Itt 


6.W 


■A^ 


Solid Triangle. 


bh* 
30 


bh* 

Ah 
7.2 


The least of 
of the two: 

'*' .>r ^' 
18 '''^24 


The least of 
the two: 
h b 


P-6-A 


4.24 ^^ 4.9 


- 


Q 


Bven Angle. 


Ah* 
1C.2 


6» 
25 


6 
6 


L 


-fr-l 






p 


Uneren Angle. 


Ah* 
9.6 


^7i 
6.5 


Oib)* 
Wi*+b*) 


Jib 




2.e(/t 4- 6) 


^ 


£f en Cross. 


Ah* 
19 


Ah 
9.6 


h* 
22.5 


4.74 


M 


SfenTee. 


Ah* 
11.1 


Ah 

a 


22.5 


b 
4.74 


c 




I Beam. 


0.66 


AJi 
82 


b* 
21 


b 

4.68 


am 


ChaimeL 


Ah* 
7.34 


Ah 
3.67 


6» 
12.5 


6 
8.54 


e 


m 


Deck Beam. 


e.o 


4 


b* 
86.6 


b 
6 



Distaace of base from centre of gravity, solid triangle, r; even angle, -^